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Vibration Control of Vehicle Suspension Systems This book covers complex issues for a vehicle suspension model, including non-linearities and uncertainties in a suspension model, network-induced time delays, and sampled-data model from a theoretical point of view. It includes control design methods such as neural network supervisory, sliding mode variable structure, optimal control, internal-model principle, feedback linearization control, input-to-state stabilization, and so on. Every control method is applied to the simulation for comparison and verification. Features: • Includes theoretical derivation, proof, and simulation verification combined with suspension models • Provides the vibration control strategies for sampled-data suspension models • Focuses on the suspensions with time-delays instead of delay-free • Covers all the models related to quarter-, half-, and full-vehicle suspensions • Details rigorous mathematical derivation process for each theorem supported by MATLAB®-based simulation This book is aimed at researchers and graduate students in automotive engineering, vehicle vibration, mechatronics, control systems, applied mechanics, and vehicle dynamics.

Vibration Control of Vehicle Suspension Systems

Haiping Du, Panshuo Li and Donghong Ning

MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software.

Contents About the Authors........................................................................................................................... viii Preface...............................................................................................................................................ix Acknowledgements.............................................................................................................................x Chapter 1 Introduction...................................................................................................................1 1.1 Introduction........................................................................................................1 1.2 Vehicle Suspension System................................................................................ 2 1.2.1 Suspension Components........................................................................ 2 1.2.2 Modelling..............................................................................................3 1.2.3 Evaluation..............................................................................................6 1.2.4 Conclusion............................................................................................. 8 References.....................................................................................................................8 Chapter 2 Active Suspension Control............................................................................................9 2.1 Introduction........................................................................................................9 2.2 Robust Control.................................................................................................. 10 2.2.1 Static Output Feedback Control with GA........................................... 10 2.2.2 Non-Fragile Control............................................................................ 17 2.2.3 Multi-Objective Control with Uncertainties....................................... 22 2.3 Control with Time Delay.................................................................................. 47 2.3.1 Input Delay.......................................................................................... 47 2.3.2 Time Delay Using T-S Fuzzy Approach............................................. 61 2.4 Preview Control................................................................................................ 85 2.4.1 System Modelling................................................................................ 85 2.4.2 Augmented System with Wheelbase Preview Information................. 88 2.4.3 Multi-Objective Disturbance Attenuation........................................... 91 2.4.4 Simulation Results...............................................................................94 2.4.5 Conclusions.........................................................................................97 2.5 Parameter-Dependent/LPV Control................................................................. 98 2.5.1 System Modelling................................................................................ 98 2.5.2 Velocity-Dependent Controller Design............................................. 102 2.5.3 Linear Parameter-Varying Controller Design................................... 105 2.5.4 Simulation Results............................................................................. 107 2.5.5 Conclusions....................................................................................... 110 2.6 Motion Mode Control..................................................................................... 110 2.6.1 System Description........................................................................... 110 2.6.2 Subsystem Establishment.................................................................. 111 2.6.3 Extended State Observer Design....................................................... 113 2.6.4 PD and Fuzzy-PD Controller Design................................................ 113 2.6.5 Actuator Force................................................................................... 114 2.6.6 Simulation Results............................................................................. 115 2.6.7 Conclusions....................................................................................... 120

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vi

Contents

Chapter 3 Semi-active Suspension Control................................................................................ 121 3.1 Introduction.................................................................................................... 121 3.2 Varying Damping........................................................................................... 122 3.2.1 Magnetorheological Damper............................................................. 122 3.2.2 Electrorheological Damper............................................................... 140 3.3 Varying Inerter .............................................................................................. 158 3.3.1 System Description........................................................................... 158 3.3.2 Inerter Effects on Suspension............................................................ 160 3.3.3 Adaptive Inerter Control of Suspension System............................... 162 3.3.4 Simulation Results and Discussions.................................................. 164 3.3.5 Conclusions....................................................................................... 167 3.4 Varying Equivalent Stiffness and Inerter....................................................... 167 3.4.1 The Semi-active Device with Variable Electrical Networks............. 167 3.4.2 The Electromagnetic VESI Device with a Variable Electrical Network............................................................................................. 171 3.4.3 The VESI Suspension for Vehicles................................................... 175 3.4.4 Simulation Analysis.......................................................................... 181 3.5 Conclusions..................................................................................................... 184 References ................................................................................................................ 184 Chapter 4 Integrated Suspension Control.................................................................................. 187 4.1 Introduction.................................................................................................... 187 4.2 Integrated with Lateral Dynamics.................................................................. 187 4.2.1 Observer-Based Multi-Objective Integrated Control for Vehicle Lateral Stability and Active Suspension Design.................. 187 4.2.2 Fuzzy Control for Nonlinear Uncertain Electrohydraulic Active Suspensions with Input Constraint........................................208 4.3 Integrated with Seat Suspension and Driver Model and Cabin Model..........224 4.3.1 Integrated Vehicle and Seat Model...................................................224 4.3.2 Controller Design.............................................................................. 227 4.3.3 Numerical Simulation ...................................................................... 237 4.3.4 Conclusion.........................................................................................244 References.................................................................................................................246 Chapter 5 Interconnected Suspension Control...........................................................................248 5.1 Introduction....................................................................................................248 5.2 Motion Mode Control Strategy .....................................................................248 5.2.1 Switched Control...............................................................................248 5.3 Electromagnetically Interconnected Control................................................. 267 5.3.1 Introduction of EIS............................................................................ 267 5.3.2 EIS System Design and Analysis...................................................... 268 5.3.3 System Verification .......................................................................... 273 5.3.4 Vibration Control of Half-Car EIS System....................................... 276 5.3.5 Conclusions....................................................................................... 283 References ................................................................................................................284

Contents

vii

Chapter 6 Suspension Control for In-Wheel Motor Driven Electric Vehicle............................ 286 6.1 Introduction.................................................................................................... 286 6.2 In-Wheel Motor System Modelling................................................................ 286 6.3 Parameter Optimization of Suspension and DVA.......................................... 290 6.4 Output Feedback Controller Design............................................................... 291 6.5 Simulation Results..........................................................................................300 6.5.1 Parameter Optimization Results.......................................................300 6.5.2 Proposed Control Methods Validation..............................................302 6.6 Conclusions.....................................................................................................307 References.................................................................................................................308 Index............................................................................................................................................... 311

About the Authors Haiping Du received a PhD from Shanghai Jiao Tong University, Shanghai, China, in 2002. He is a Senior Professor at the School of Electrical, Computer and Telecommunications Engineering, University of Wollongong, Australia. He was a research fellow with the University of Technology, Sydney, from 2005 to 2009, and was a postdoctoral research associate with Imperial College London from 2004 to 2005 and the University of Hong Kong from 2002 to 2003. He is a subject editor of various journals. His research interests include vibration control, vehicle dynamics and control systems, robust control theory and engineering applications, electric vehicles, robotics and automation, and smart materials and structures. He is a recipient of the Australian Endeavour Research Fellowship (2012). Panshuo Li r eceived her B.S. and M.S. in Mechanical Engineering from Donghua University and Shanghai Jiao Tong University, Shanghai, China, in 2009 and 2012, respectively. She obtained a PhD in Mechanical Engineering from University of Hong Kong, Hong Kong, in 2016. She was a research associate with the Department of Mechanical Engineering, University of Hong Kong, Hong Kong and the School of Electrical, Mechanical and Mechatronic Systems, University of Technology Sydney, Australia. She is currently a Full Professor with the School of Automation, Guangdong University of Technology, Guangzhou, China, and in the editorial board of Proc. IMechE Part I: Journal of Systems and Control Engineering. Her current research interests include switched systems, time-varying systems, and intelligent vehicle control. Donghong Ning received a B.E. in Agricultural Mechanization and Automation from the College of Mechanical and Electronic Engineering, North West Agriculture and Forestry University, Yangling, China, in 2012 and a PhD in 2018 from the University of Wollongong, Australia. He was an associated research fellow at the University of Wollongong. He is currently a Professor at the College of Engineering, Ocean University of China, Qingdao, China. His research interests include active and semi-active vibration control, multiple degrees of freedom vibration control, interconnected suspension, electromagnetic suspension, and marine motion compensation.

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Preface This book is targeted as a reference for scientists and engineers in vehicle suspension design and control. Suspension plays a significant role in both traditional and intelligent vehicles. The authors of this book are all researchers dedicated to developing high-performance vehicle suspension and exploring innovative suspension technologies. Especially, Professor Haiping Du has worked in this area for over 20 years. In this book, the authors have summarized their outcomes into several topics, including most of the enthusiastically discussed technologies by researchers and those potentially boosting the future suspension industry. This book briefly introduces the suspension model and evaluation, discusses the active and semi-active suspension control methods, and introduces three emerging suspension technologies, including integrated suspension, interconnected suspension and suspension with in-wheel motors. Some of the advanced technologies in this book have been adopted by industry, and some only have academic outcomes, such as papers and patents. Therefore, the authors believe this book would inspire readers to explore new knowledge of vehicle suspension. MATLAB® is a registered trademark of The MathWorks, Inc. For product information, please contact: The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 USA Tel: 508-647-7000 Fax: 508-647-7001 E-mail: [email protected] Web: www.mathworks.com

ix

Acknowledgements Vehicles play a significant role in our lives nowadays, and there is no doubt that vehicle suspension is one of the most important parts, no matter if they are conventional fuel vehicles or new energy vehicles. This book summarizes our research outcomes about vehicle suspension in the last 20 years, and we are still working in this field to explore innovative suspension design and control methods. We would like to express our great gratitude to our dear colleagues and friends, James Lam, Nong Zhang, Weihua Li, Dongpu Cao, Xu Wang, Hongyi Li, and Shuaishuai Sun, who have made their valuable suggestions to our relevant research and played important roles in making this book a reality. Also, the PhD students, Yulin Liao, Xiangjun Xia, Pengxu Li, Jiawei Luo, and Zhixiang Wen, have helped to archive and edit chapters. Their time and effort are really appreciated.

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1

Introduction

1.1 INTRODUCTION With the emergence of various new technologies, vehicles have become sophisticated technological machines during the past few decades. Breakthroughs have been made in different aspects to improve vehicle performance such as power supply, fuel economy, ride comfort, and safety. The suspension system, as one of the most important components of vehicles, plays a vital role in improving handling safety and ride comfort. In general, vehicle suspension systems connect the vehicle body to the wheels and support the vehicle’s static weight. A qualified suspension system should be able to isolate the body from road disturbances (ride quality) and provide good road holding and handling (ride safety). Suspension systems can be roughly classified into passive, semi-active, and active suspensions. The traditional passive suspension is mainly composed of some forms of springs and dampers, which are widely used due to their simple structure and low cost. However, the performance of passive suspensions is limited since there always exists a significant trade-off in the performance of ride safety and ride comfort. With the development of electromagnetic technology and the increased requirements of suspension performance from customers, electromagnetic semi-active and active suspensions have been developed rapidly. Both semi-active and active suspensions introduce electronic signal control systems, and the difference is that semi-active suspension only changes the dissipation resistance of the semi-active device, whereas active suspension will input control force to the system through the actuator. In semi-active suspensions, controllable dissipative components such as magnetorheological (MR) and electrorheological (ER) dampers are usually used as semiactive actuators. The dissipative force provided by dampers can be controlled by adjusting the electromagnetic field. In active suspension, electronically controlled actuators are used to actively apply an adjustable force according to the control signal. Therefore, more flexible control schemes can be designed for active suspension with better performance and higher power consumption. In recent years, the research on semi-active suspension mainly focuses on the control method of MR/ER dampers. For active suspension, due to the existence of electronically driven actuators, the suspension performance depends on the appropriate control force; hence, many control methods have been developed in the design of vehicle active suspension systems, such as linear control strategies, nonlinear control strategies, robust control strategies, and some intelligence control strategies. With advanced control strategies, suspension performances, including ride quality, road holding, and handling performance, are improved significantly. This book discusses the vibration control of vehicle suspension systems and different control methods of active, semi-active, and integrated suspensions. In the following sections of this chapter, some preliminaries on the suspension design are provided. The dynamics models of quarter-car, half-car, and full-car suspensions are given and the evaluation criteria of suspension performance which is used in the following studies of this book are presented. Chapter 2 investigates the control problem of active suspension systems. The external disturbance, parameter uncertainty, actuator dynamics, static control, non-fragile control, and multi-objective control methods are considered to design the appropriate controllers. Moreover, suspension systems that have a time delay, wheelbase preview information, and time-varying parameter are also studied, and the corresponding controllers are designed to ensure the suspension performance. Chapter 3 presents the control problem of semi-active suspension systems. The MR and ER dampers are introduced and modelled, and some control strategies are developed to design the controllers including direct control, indirect control, T-S fuzzy control, etc. The suspension parameter DOI: 10.1201/9781003265665-11

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Vibration Control of Vehicle Suspension Systems

uncertainties, control input constraints, and measurement noises are also considered. Then, for the semi-active suspension equipped with the inerter, H 2 controller and adaptive control law are designed to improve the suspension performance. In addition, a semi-active variable equivalent stiffness and inertance device are developed by an electrical network and an electromagnetic device. Chapter 4 studies the control problem of the integrated suspension system, including vehicle vertical, lateral, and seat suspension dynamics with the driver model and cabin model. For the integrated system consisting of lateral and vertical dynamics, an observer-based gain-scheduling integrated control strategy is proposed. For the integrated system consisting of vertical and roll dynamics, fuzzy control strategy and H ∞ performance index are used to design the controller. A motion-mode-based switch control strategy is designed for the integrated system including vertical, pitch, and roll dynamics. Moreover, an integrated seat and suspension have been developed, and a static output feedback controller is designed for it. Chapter 5 investigates the control problem of interconnected suspension. A switched control method of vehicle suspension is proposed based on motion-mode detection, which is suitable for the control of interconnected suspensions. Then, a novel controllable electrically interconnected suspension for improving vehicle ride comfort is studied and experimentally validated. Chapter 6 studies the active suspension control problem of electric vehicles driven by in-wheel motors. Considering the actuator faults and time delay, output feedback H ∞ controller is designed. The dynamic damping in-wheel motor-driven system, in which the in-wheel motor is designed as a dynamic vibration absorber, is developed to improve the ride quality and isolate the force transmitted to motor bearings. And the parameters of vehicle suspension and dynamic vibration absorber are optimized based on the particle swarm optimization to achieve better suspension performance.

1.2  VEHICLE SUSPENSION SYSTEM 1.2.1  Suspension Components Suspension systems, which connect the chassis and vehicle body, are the core components of modern vehicles. Suspension systems play an important role in attenuating vibration transmitted from uneven roads to drivers and cargoes [1]. In addition, suspension systems also have a non-negligible effect on vehicle handling stability, including anti-roll, anti-pitch, dynamic tyre load, and suspension deflection [2–4]. Suspension systems can be classified into three types: passive suspension, semi-active suspension, and active suspension. The passive suspension usually consists of a spring, a damper, and a stop block. Passive suspensions have been widely used in vehicles for their simple structures, high reliability, and low cost. However, passive suspensions also have limited performance because the stiffness and damping parameters cannot be adjusted after being fixed. The active suspension consists typically of a spring, an active actuator, sensors, and controllers. In addition, active suspensions may need transmission devices sometimes to transform vertical motion to rotary motion. There are many types of active actuators: electromagnetic actuators, hydraulic actuators, electrohydraulic actuators, and servo-valve hydraulic actuators [5–7]. Active suspensions can output ideal active force to meet multi-objective performance requirements. However, the energy consumption and cost of active suspensions are very high, which obstructs their actual application. Therefore, although active suspension can provide excellent performance, they are currently only applied to a few high-end vehicles. For semi-active suspensions, they have similar structures to active suspensions. The major difference between active suspensions and semi-active suspensions is that the output force of semi-active suspensions is limited. There are several types of semi-active suspensions: magnetorheological suspension, electrorheological suspensions, and electromagnetic suspensions. The advantages of semi-active suspensions are less energy consumption and cost compared to active suspensions. In addition, semiactive suspensions also have comparable performance compared to traditional passive suspensions.

3

Introduction

1.2.2 Modelling Vehicle dynamic modelling is an important step in the design of vehicle suspension systems. Vehicle suspension dynamic models can be classified into quarter-car, half-car, and full-car models based on the requirement of suspension and controller design. In this section, a two-degree-of-freedom (2-DOF) quarter-car suspension model, a 4-DOF half-car suspension model, and a 7-DOF full-car suspension model are introduced first. 1.2.2.1  Quarter-Car Suspension Model When an individual suspension design problem is considered, a quarter-car suspension model is suitable for dynamic modelling and controller design. Figure 1.1 illustrates a 2-DOF quarter-car active or semi-active suspension model, where ms and mu are the sprung and unsprung mass; ks and kt are the suspension and tyre stiffness, respectively; cs is the damping of the suspension system; zs , zu, and zr are the displacements of sprung mass, tyre, and road roughness, respectively; F is the controllable force controlled by an active or a semi-active controller. The dynamic equation of the quarter-car active or semi-active suspension model can be obtained according to Newton’s second law:  ms   zs = − ks ( zs − zu ) − cs ( zs − zu ) + F (1.1)  zu = ks ( zs − zu ) + cs ( zs − zu ) − F − kt ( zu − zr )  mu 

Since traditional passive suspension usually consists of a spring, and a damper, there is no external force input. Therefore, the term F in equation (1.1) should be removed. The dynamic equation of the traditional passive suspension model can be obtained according to equation (1.1). Remark 1.1 For active and semi-active suspension, an actuator is adopted to replace traditional dampers. However, suspension systems still exhibit damping characteristics because of friction and the actuator’s transmission mechanisms. Therefore, a damping coefficient cs is considered in the active or semi-active quarter-car suspension model. 1.2.2.2  Half-Car Suspension Model As we can see from Figure 1.1 that only the vertical motion of vehicle systems can be studied by the quarter-car suspension model. However, there are vertical, roll, and pitch motions for real vehicles when

FIGURE 1.1  Quarter-car active suspension model.

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Vibration Control of Vehicle Suspension Systems

FIGURE 1.2  Half-car suspension model.

vehicle systems are exposed to terrible vibration environments. To get closer to the real vibration state of vehicles, a half-car suspension model and a full-car suspension model are established in this section. To simplify the analysis, a left-right symmetrical half-car model is introduced in this subsection. Figure 1.2 shows a 4-DOF half-car active suspension model, where ms , I s , muf , and mur are the sprung mass, roll moment of inertia of the sprung mass, left and right tyre mass, respectively; ks and kt are the suspension and tyre stiffness; cs is the damping coefficient; zs and ∅ are the sprung mass centre displacement and the roll angle around the rotary centre; zls and zrs are the left and right displacements of suspension mounting locations; zlu and zru are the displacements of left and right tyres; zlr and zrr are the left and right road displacements; l is the distance of the left and right suspensions to the mass centre of the sprung mass; Fl and Fr are the left and right output forces of semi-active or active actuators. Similarly, the dynamic equation of the half-car active suspension model can be obtained

zs = − ks ( zls − zlu ) − cs ( zls − zlu ) − ks ( zrs − zru ) − cs ( zrs − zru ) + Fl + Fr ms    I s∅  = − ( − ks ( zls − zlu ) − −cs ( zls − zlu ) + Fl ) l + ( − ks ( zrs − zru ) − cs ( zrs − zru ) + Fr ) l  (1.2)  mu  zlu = k s ( zls − zlu ) + cs ( zls − zlu ) − Fl − kt ( zlu − zlr )  mu   zru = k s ( zrs − zru ) + cs ( zrs − zru ) − Fr − kt ( zru − zrr )

Similar to quarter-car suspensions, the half-car suspension model also consists of springs, actuators, and damping parts. The left and right output forces need to be removed to obtain the dynamic equation of passive half-car suspension models. 1.2.2.3  Full-Car Suspension Model To obtain vehicles’ bounce, roll, and pitch motions and the vertical bounce of all four wheels, a 7-DOF full-car suspension model is established in this subsection. Figure 1.3 shows the full-car suspension model, where ms , I x, and I y are the sprung mass, roll moment of inertia, and the pitch moment of inertia of the sprung mass; m fl , m fr , mrl , and mrr are the mass of left front tyre, right front tyre, left rear tyre, and right rear tyre; k fls , k frs, krls, and krrs are the stiffness of left front spring, right

5

Introduction

FIGURE 1.3  Full-car suspension model.

front spring, left rear spring, and right rear spring, respectively; k flt, k frt , krlt , and krrt are the stiffness of left front tyre, right front tyre, left rear tyre, and right rear tyre; c fl , c fr , crl, and crr are the damping of left front suspension, right front suspension, left rear suspension, and right rear suspension; Ffl , Ffr , Frl , and Frr are the output of left front actuator, right front actuator, left rear actuator, and right rear actuator; zs , ϕ , and ∅ are the sprung mass centre displacement, roll angle around the rotary centre of the sprung mass, and the pitch angle around the rotary centre of the sprung mass, respectively; z fls, z frs , zrls , and zrrs are the displacements of left front sprung mass, right front suspension mass, left rear suspension mass, and right rear suspension mass; z flu , z fru , zrlu, and zrru are the displacements of left front tyre, right front tyre, left rear tyre, and right rear tyre; z flr, z frr , zrlr , and zrrr are the displacements of left front road roughness, right front road roughness, left rear road roughness, and right rear roughness; a and b are the displacement of the front and rear suspensions to the mass centre of the sprung mass, respectively; l is the distance of the left and right suspensions to the mass centre of the sprung mass. The suspension displacements can be obtained by sprung mass displacement, roll angle, and pitch angle:

      

z fls = zs + lϕ + b∅ z frs = zs − lϕ + b∅ zrls = zs + lϕ − a∅

(1.3)

zrrs = zs − lϕ − a∅

Therefore, the dynamic equation of the 7-DOF full-car model can be obtained by Newton second’s law:

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Vibration Control of Vehicle Suspension Systems

zs = Frls + Ffls + Frrs + Ffrs ms    I xϕ = −l ( Frrs + Ffrs ) + l ( Frls + Ffls )     I yφ = − a ( Frrs + Frls ) + b ( Ffls + Ffrs )  z flu = − Ffls + Fflt (1.4) m fl   z fru = − Ffrs + Ffrt m fr   zrlu = − Frls + Frlt mrl   mrr  zrru = − Frrs + Frrt 

Frls , Ffls, Frrs, Ffrs , Fflt , Ffrt, Frlt , and Frrt can be obtained by:

 F = − k z − z − c z − z + F ijs ( ijs iju ) ij ( ijs iju ) ij  ijs (1.5)   Fijt = − kijt ( ziju − zijr ) 

where i = f , r ; j = l ,  r .

1.2.3 Evaluation 1.2.3.1  Road Excitation When vehicles drive on uneven roads, the road surface introduces most of the vibration to vehicles. Since suspension systems mainly transmit vertical motion, the road surface models need to be considered in this section. There are two typical road excitations: bump road excitation and random road excitation. In real road conditions, there are many speed bumps. Therefore, bump excitations are usually used to evaluate the transient performance of suspension systems. A typical bump road excitation is given as:

 A L 2πv     1 − cos  t , 0≤t ≤  L    2 v zr ( t ) =  (1.6) L  0, t>  v 

where A is the bump height; L is the bump length, and v is the forward velocity of vehicles. In general, the change in the height of the road surface relative to the datum plane along the length of the road trend is regarded as the road roughness function. When measuring road unevenness, a level gauge or a special pavement gauge is usually used to obtain the unevenness of the pavement profile. The measured data can be processed to obtain the parameters of the power spectral density (PSD) or variance of the road roughness. The road roughness, as the vehicle vibration input, mainly uses the road PSD to describe its statistical characteristics [8]. The road PSD of pavement displacement can be expressed as:

 n Gq ( n ) = Gq ( n0 )    n0 

− w0

( nq ≤ n ≤ nm ) (1.7)

where n is the spatial frequency;  n0 is the reference spatial frequency, usually taken as 0.01 m−1; nq and nm are the upper and lower cut-off frequencies of spatial frequencies; Gq ( n0 ) is the road roughness coefficient; w0 is the frequency index.

7

Introduction

When vehicles are driving on the road at a certain speed and a spatial frequency, the equivalent time-frequency f = un. In the meantime, when the frequency tends to zero at the low-frequency domain, the road PSD will tend to infinity. Then, the road PSD can be expressed as: Gq ( f ) = Gq ( n0 )( n0 )

2

u (1.8) f + fd2 2

where f is the time-frequency, fd is the lower cut-off frequency. The random road excitation model can be obtained as: zr (t ) = −2πnquzr + 2π Gq ( n0 ) uw (1.9)

where w is the standard Gaussian white noise. To study the full-car ride comfort and handling stability, the four-wheel road excitation model needs to be established. To simplify the analysis, the tracks of the front and rear wheels are assumed to coincide. In addition, the rear-wheel road excitation is different from the front-wheel road excitation with time delay. In addition, there is coherence between the left-wheel road excitation and the right-wheel road excitation. Therefore, the road excitation of the four wheels of the car can be expressed as:

           

z flr (t ) = −2πnquz flr + 2π Gq ( n0 ) uw    2u  2u z frr (t ) =  2πnqu +  z flr − 2π Gq ( n0 ) uw − z frr  Ld  Ld  2u  2u zrlr (t ) =  2πnqu +  z flr − 2πn0 Gq ( n0 ) uw − zrlr  Td  Td

(1.10)

 2π 2u   2π 2u  +  z frr − zrrr (t ) =  −2πnqu +  z flr + 2πn0 Gq ( n0 ) uw +  zrrr  Ld Td   Ld  Td

where Ld = 2 * l ; Td = a + b; a and b are the displacement of the front and rear suspensions to the mass centre of the sprung mass, respectively; l is the distance of the left and right suspensions to the mass centre of the sprung mass. 1.2.3.2  Performance Indexes As mentioned above, vehicle suspension systems play an important role in isolating uneven vibration and ensuring vehicle handling stability. There are three main performance indexes for modern suspension system design: ride comfort, suspension deflection, and road-holding ability. • Ride comfort: it is acknowledged that ride comfort is an important performance for suspension design, which is normally evaluated by the sprung acceleration in the vertical, roll, and pitch directions. • Maximum suspension deflection: since the physical constraint of the suspension structure, the maximum suspension deflection needs to be considered to prevent the suspension from hitting the stop block. • Road-holding stability: since suspension systems also output forces to the tyre, the roadholding performance of tyres also needs to be considered in the suspension design process. The dynamic tyre load should not exceed the static tyre load to ensure firm uninterrupted contact of wheels to the road.

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Vibration Control of Vehicle Suspension Systems

1.2.4  Conclusion In this section, the basic concept of suspension systems and the main contents of this book are introduced first. Then, basic dynamic models of quarter-car, half-car, and full-car suspension are established. The components of passive suspensions, semi-active suspensions, and active suspensions are introduced. Finally, the evaluation indexes and road excitation models are briefly introduced in this section.

REFERENCES

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2

Active Suspension Control

2.1 INTRODUCTION Active suspensions have drawn much attention in recent years because of their excellent performance in improving ride comfort, reducing suspension deflections, and keeping tyre contact. This chapter focuses on various active suspension control methods such as robust control, preview control, LPV control, and motion mode control. Robust control is a typical modern control algorithm, which has high robustness to system disturbance. Robust control is an approach to controller design that explicitly deals with uncertainty. In this chapter, several typical robust control problems are studied. First, this chapter presents an approach to design multi-objective static output feedback H 2 /  H ∞ /  GH 2 controller for vehicle suspension by using linear matrix inequalities (LMIs) and genetic algorithms. Then, a new approach to design the non-fragile H ∞ controller for active suspension is presented. The designed controller not only can achieve the optimal performance for active suspensions but also can preserve the closedloop stability in spite of the controller gain variations. To deal with changes in vehicle inertial properties and the existence of actuator time delay, a parameter-dependent controller design approach is proposed in this chapter. The problem of nonfragile H ∞ control for uncertain active suspension systems with time delay is investigated through a fuzzy control approach in the finite frequency domain. A Takagi-Sugeno (T-S) fuzzy model is constructed to describe the non-linear suspension dynamics based on a typical quarter-automobile active suspension system model. And the input delays and gain perturbations of the actuator are considered to approximate the real physical device situations in the control systems. Preview information of the road profile could further improve the suspension performance since the information of road profile can be obtained by the controller in advance. A multi-objective control method with a wheelbase preview for active vehicle suspension is proposed in this chapter. The disturbances at the front wheel are obtained as preview information for the rear wheel. H ∞ norm and generalized H 2 norm are used to improve ride quality and ensure that hard constraints are satisfied. Static output feedback is utilized in designing controllers, the solution is derived by iterative linear matrix inequality (ILMI) and cone complementarity linearization (CCL) algorithms. Simulation results confirm that multi-objective control with wheelbase preview achieves a significant improvement in control effect. In previous works, the controllers are designed under the unrealistic assumption that the velocity of the vehicle is constant. However, the velocity of vehicle is time-varying or uncertain in practical system. For the system with uncertainties, based on the parameter-dependent idea, the polynomial parameter-dependent Lyapunov function can be obtained from the affine parameter-dependence, which could reduce the conservatism of designed controller. In this section, a more practical vehicle system is considered which has the uncertain velocity, and assuming that the forward velocity of the vehicle resides in a prescribed interval and can be measured online, a systematic procedure is formulated to design a velocity-dependent multi-objective preview control solution for vehicle suspension. The preview information is incorporated to a half-car model through Padé approximation of the delay. Then two controllers are designed for the velocity uncertainty and time-varying situations based on the homogeneous polynomial parameter-dependent approach and linear parametervarying approach, respectively. Considering a more practical real vehicle suspension system, the vibration attenuation problem of a non-linear full-car suspension system is studied in this chapter to stabilize the vehicle attitude and provide good ride quality. For this purpose, the dynamic equations of full-car suspension are DOI: 10.1201/9781003265665-29

10

Vibration Control of Vehicle Suspension Systems

extracted as three subsystems which correspond to heave motion, pitch motion, and roll motion, respectively. For each subsystem, corresponding motion-based active disturbance rejection control (ADRC) controllers are designed to attenuate the vibration of the sprung mass. For each motionbased controller design, the couple dynamics and the external disturbances combined with system uncertainties are viewed as the total disturbance, which is estimated through the extended state observer (ESO), and a proportional-derivative (PD) controller and a fuzzy-PD controller are designed with the disturbance rejection strategy. Finally, four actuator inputs are computed online according to the three motion-based controllers obtained. Simulations are carried out for different road conditions to illustrate the advantages of the proposed control method.

2.2  ROBUST CONTROL 2.2.1  Static Output Feedback Control with GA 2.2.1.1  Problem Formulation This section will mainly concern the multi-objective static output feedback controller designs for active suspension systems. A quarter-car model consists of one-fourth of the body mass, suspension components, and one wheel as shown in Figure 2.1. The dynamic equations of the quarter-car model with active suspension are given as:

ms  zs = − ks ( zs − zu ) − cs ( zs − zu ) + u  (2.1)

mu  zu = ks ( zs − zu ) + cs ( zs − zu ) − u − kt ( zu − zr ) (2.2)

where ms and mu are the sprung and unsprung mass, respectively; ks and cs are the stiffness and damping of the suspension system; kt is the tyre stiffness; zs , zu, and zr are the displacements of sprung mass, unsprung mass and road profile; u is the active output force of the active suspension.

FIGURE 2.1  Quarter-car model with active suspension.

11

Active Suspension Control ˙

T

x =  zs − zu zu − zr zs zu  is selected as the state variable; w = zr is defined as the system disturbance caused by the road roughness. The state-space model can be given as: x = Ax + B1w + B2u (2.3)

 0  0 where A =   − k s / ms  ks / mu 

0 0 0 kt / mu

1 0 −cs / ms cs / mu

−1 1 cs / ms −cs / mu

  0   ; B1 =  −1   0   0  

  0   0  ; B2 =  1 / ms    −1 / mu   

  .   

As introduced before, there are three main design performance requirements for a vehicle suspension: ride comfort, road holding capacity, and suspension deflection limitation. The sprung acceleration  zs , suspension deflection zs − zu , and tyre deflection zu − zr are selected as the performance indices of the above performance requirements. In accordance with the aforementioned requirements, a multi-objective H 2/H ∞ / GH 2 control problem to deal with the three different objectives for vehicle suspensions. To satisfy performance requirements, the controlled outputs are selected as z1 =  zs , z2 = zu − zr , z3 = zs − zu . Hence, the vehicle suspension control system can be described by equation of the form:        where y is the measured output; C1 = 

x = Ax + B1w + B2u z1 = C1 x + D12u z 2 = C2 x (2.4) z 3 = C3 x y = Cx

− ks / ms 0 −cs / ms cs / ms  ; D12 = 1 / ms ;  1 0 0 0  C2 =  0 1 0 0 ; C3 =  1 0 0 0 ; C =  .  0 0 1 0  For H ∞ controller design, the multi-objective static output feedback control problem can be summarized as: find the control gain matrix K such that the closed-loop system with control input u = Ky is stable and the system Tz1w H is minimized subject to Tz2w H∞ < γ ∞ and Tz3w GH < α , 2 2 where Tzi w denotes the closed-loop transfer function from w to zi for i = 1,2,3; γ ∞ > 0 and α > 0 are performance indices; and the performances, Tz1w H2 , Tz2w H∞ , Tz3w GH are defined in the following 2 contents. 2.2.1.2 Multi-Objective Controller Design with GA In this section, different performances are expressed by matrix inequalities, and the controller gain matrix K is obtained by combining the solution of LMIs and the randomized search of GAs The H 2 norm of Tz1w is defined by [1]: +∞

Tz1w

:= tr H2

1 π Tz1w ( jw ) Tz1w ( jw ) dw (2.5) 2

−∞

which corresponds to the asymptotic variance of the output z1 when the system is driven by the noise w. The static output feedback H 2 problem for system (2.4) is presented by finding matrices P2 > 0, R > 0 and control matrix while realizing the control objective Tz1w H2 < γ 2 for γ 2 > 0 , such that the following inequalities hold

12

Vibration Control of Vehicle Suspension Systems

 ( A + B KC ) P + P ( A + B KC )T 2 2 2 2   B1T   R   P2 (C1 + D12 KC )T 

B1   < 0 − I 

(C1 + D12 KC ) P2 

 > 0  

PQ

(2.6)

(2.7)

trace  ( R ) < γ 2 (2.8)

The H ∞ norm gives the system input–output gain when both the input and the output are measured in the finite energy. The H ∞ norm of Tz2 w can be calculated from:

Tz2w

H∞

:= supwδ Tz2w ( jw )  (2.9)

The static output feedback H ∞ problem for system (2.4) is presented as to find matrix P∞ > 0 and control gain matrix K , while realizing the control objective Tz2w ∞ < γ ∞ for γ ∞ > 0, such that the following inequality holds  ( A + B KC ) P + P ( A + B KC )T B P2C2T  2 ∞ ∞ 2 1    B1T  < 0 (2.10) −γ ∞ I 0   0 −γ ∞ I   C2 P  If the input is quantified by its energy and the peak amplitude of the output is kept to a certain level, this leads to the so-called generalized H 2 ( GH 2 ) control problem or energy-to-peak control problem. The GH 2 norm of system Tz3w is defined by [2] +∞

Tz3w

GH 2

:= τ max

1 π Tz3w ( jw ) Tz3w ( jw ) dw (2.11) 2

−∞

The static output feedback GH 2 problem for the suspension system is commonly presented as finding matrix PG2 > 0 and control gain matrix K , while realizing the control objective Tz3w GH < α for 2 α > 0, such that the following inequalities hold

 P ( A + B KC )T + ( A + B KC ) P 2 2 G2  G2  B1T   PG 2   C3 PG 2

PG 2C3T −α I

B1   < 0 (2.12) −1 

  < 0 (2.13) 

The multi-objective static output feedback H 2 / H ∞ / GH 2 problem presented to find matrices P2 > 0,  R > 0,  P∞ > 0,  PG 2 > 0 and control gain matrix K such that Tz1w H2 is minimized subject to Tz2 w H∞ < γ ∞ and Tz3 wGH2 < α . This requires that equations (2.6)–(2.8), (2.10), (2.12), and (2.13) are satisfied simultaneously. Normally, mixed state feedback control problem (where C should be identity matrix) is to set P2 = P∞ = PG 2 = P > 0 and define Q = KP, and to find P and Q to satisfy these equations. It is convex optimization problem and can be solved by MATLAB LMI toolbox in spite of its conservatism. However, when considering the static output feedback problem, these equations

13

Active Suspension Control

are bilinear matrix inequalities (BMIs) and cannot be solved by numerically tractable methods. Therefore, genetic algorithm is presented to find the solutions based on its stochastic search capacity. Similar to the approach presented in [3] to design a static output feedback controller based on GA, the multi-objective H 2 / H ∞ / GH 2 static output feedback controller design problem is resolved by a binary-coded GA approach through the following minimization problem:

minK ∈K Tz1w

H2

s.t. Tz2 w

H∞

< γ ∞  and  Tz2 w

GH 2

< α (2.14)

where K := { K : Tzi w ( s; K ) is stable, i = 1,2,3}. The GA-based scheme can be found in [4] and references therein, and they are omitted here for brevity: The evolution process will repeat for N g generations or being ended when the search process converges with a given accuracy. The best chromosome is decoded into real values to produce again the control gain matrix K . In this approach, we do not require that P2 = P∞ = PG2 = P > 0,  and hence, the conservatism of the mixed control problem is reduced. Remark 2.1 When the disturbance is zero, i.e., w = 0, the closed-loop system is expressed as x = ( A + B2 KC ) x . From Lyapunov stability theory, we know that the closed-loop system matrix A + B2 KC is stable T if and only if K satisfies the matrix inequality ( A + B2 KC ) P + P ( A + B2 KC ) < 0 for some P > 0. Now, Pcan be P2, P∞, or PG2 . So we can use the proposed design procedure to find K and then to guarantee the closed-loop system stability no matter what the initial conditions of the system are.

Remark 2.2 The efficiency of the proposed approach will be evaluated by simulations in the next section. Although it is only applied to a quarter-car model in this section, the approach can be applied to more complicated suspension models. Certainly, the computational complexity and time will be different.

Remark 2.3 The existence of the static output feedback control gain K for a given system can be checked by the theorem presented in [5]. The proposed approach only tries to find the possibly existing controller gain. However, it does not guarantee finding the solutions all the time without any constraint conditions. To increase the opportunity to find feasible solutions, two methods can be used. If the approach does not find a candidate solution that can stabilize the closed-loop system, a value that is related to the feasibility solution of LMIs can be used as a fitness value for this candidate so that it can be evolved to find the feasible solution at least. The second method is to try different parameter setting for GA especially the search range of the controller gain to find a possible solution. Nevertheless, as will be shown in the next section, the feasible solution is easily found for the given example without resorting to these two methods.

14

Vibration Control of Vehicle Suspension Systems

Remark 2.4 In practice, there are always parameter uncertainties in the system due to the modelling problem and components ageing, etc. When these uncertainties exist, the system equation can be expressed as x = ( A + ∆A ) x + ( B1 + ∆B1 ) w + ( B2 + ∆B2 ) u, where ∆A, ∆B1, and ∆B2 are real-valued unknown matrices representing parameter uncertainties and are assumed to be the form of ∆A = H A FE A , ∆B1 = H1FE1, ∆B2 = H 2 FE2, where H A, H1, H 2, E A , E1, and E2 are known real constant matrices. F is unknown matrix satisfying F T F ≤ I . Then, the aforementioned three parameters can be expressed by LMIs with some derivations. and the presented controller design approach can still be applied to find the appropriate static output feedback controller to stabilize the system with required performance. 2.2.1.3  Simulation Results In this section, the proposed approach will be applied to design the static output feedback controllers based on the quarter-car model described in this section. The parameters of the quarter-car model are given as: ms = 504.5 kg, mu = 62 kg, ks = 13,100 N/m, cs = 400 Ns/m, kt = 252,000 N/m. For comparison purposes, a full state feedback controller is designed by setting P2 = P∞ = PG2 = P > 0 , defining Q = KP, and finding P and Q to satisfy LMIs. And for brevity, we denote the active suspension realized by this state feedback controller as Active Suspension I. Then, to show the effectiveness of the presented approach, we take the case that assumes only the suspension deflection zs − zu and the velocity of sprung mass zs are measurements available as an example, and we use the approach presented in this section to design the static output feedback controller through LMIs and GA. For brevity, we denote the active suspension realized by this static output feedback controller as Active Suspension Ⅱ. The parameters used in the genetic algorithm are selected as: N p = 80, N g = 100, pc = 0.7, pm = 0.02, and the performance indices are set as γ ∞ = 1, α = 0.3. Note that the parameter selections for γ ∞ and α are made by trial and error in terms of the three performances realized by the state feedback control. Hence, the static output feedback control will also use these parameters to make fair comparison. The parameters for GA are selected according to previous experiences and some simulation results. To show the effects of GA parameter on the design results, several examples are given in Figure 2.2, which show, for example, the different settings on population size and search range make different effects on the evolution results. It can be seen that GA is really a random algorithm. In general, it needs to run many times to get a fair result (20 times in this study). Parameter setting on population size does not affect the results too much. Similar observations are found in the parameter settings on crossover probability and mutation probability. The search range will affect the result largely. However, from the practical point of view, the controller gain cannot be given arbitrarily. Also, for fair comparison, the search range, which is given as  −10 4 ,10 4 , is set refer  ring to the state feedback control case. The generation size can be selected as a large number, but it will spend more time for the evolution process. Observing the simulation experiments, the evolution results will converge to small values after 50 generations for most cases, so the generation size is selected as 100 which can satisfy the requirement. Even though the different parameter settings may affect the evolution results, it can be seen from Figure 2.2 that the proposed approach is able to find the desired result very efficiently with the given parameter setting. This validates the efficiency of the proposed approach. Since the ride comfort performance is frequency sensitive and the human body is much sensitive to vertical vibrations in the frequency range 4–8 Hz according to ISO 2631, we mainly evaluate the ride comfort performance in the frequency domain. The frequency responses for the above-designed active suspensions from disturbance to sprung mass acceleration are depicted in Figure 2.3. For comparison, the frequency response for the passive suspension is plotted in the same figures as well, it can be clearly seen from Figure 2.3 that the designed active suspensions achieve

15

Active Suspension Control Range=[-103, 103]

28

Range=[-105, 105]

28

(b)

Range=[-106, 106]

28

Range=[-103, 103]

28

Range=[-105, 105]

28

26

24

24

24

24

24

24

22

22

22

22

22

22

20

20

20

Fitness

26

Fitness

26

Fitness

26

Fitness

26

20

20

20

18

18

18

18

18

18

16

16

16

16

16

16

14

0

50 Generation

100

14

0

50 Generation

100

14

0

50 Generation

100

14

0

50 Generation

100

14

0

50 Generation

Range=[-106, 106]

28

26

Fitness

Fitness

(a)

100

14

0

50 Generation

100

FIGURE 2.2  (a) Evolution process with different population size setting; (b) evolution process with different parameter search range setting.

FIGURE 2.3  Frequency response of the sprung mass acceleration.

significant improvement on ride comfort for the active suspension systems. The sprung mass accelerations of active suspensions are smaller than the uncontrolled suspension, especially in the range of sprung mass resonance. Note that the frequency response of body acceleration is invariant at the second resonance frequency (10.15 Hz) due to no tyre damping being considered in the present model [6]. In spite of the simplicity of active suspension Ⅱ, active suspension Ⅱ even realizes a better ride comfort performance than active suspension Ⅰ. The frequency responses from disturbance to suspension deflection, tyre deflection for both active suspensions and passive suspension are plotted in Figure 2.4. It can be seen that active suspensions improve the suspension deflection and tyre deflection performances as well compared with passive suspension. In order to show the time-domain performance of the new designed active suspensions (tyre deflection and suspension deflection are normally regarded as time-domain performances), the time responses of sprung mass acceleration, tyre deflection, and suspension deflection for a bump road input are plotted in Figure 2.5, which shows the sprung mass acceleration, tyre deflection, and suspension deflection, respectively, for the active suspensions and the passive suspension; and the active force for the active suspensions.

16

Vibration Control of Vehicle Suspension Systems

FIGURE 2.4  Frequency response of (a) sprung deflection; (b) tyre deflection.

FIGURE 2.5  Bump response: passive suspension (dot-dash line), active suspension | (dot line), active suspension || (solid line).

The corresponding displacement for the bump road input is given by:

 A  2πV    t , 1 − cos   L    2  zr ( t ) =   0,  

0≤t ≤ L t> V

L V

(2.15)

17

Active Suspension Control

where A and L are the height and the length of the bump, and A = 0.1 m, L = 5  m and the vehicle forward is V = 45 km/h. It can be seen from Figure 2.5 that the active suspensions experience smaller body acceleration, tyre deflection and suspension deflection, respectively, than those of passive suspension for the same bump disturbance input. It proves that, in spite of its simplicity, the static output feedback controller realizes the active suspension performances very well. 2.2.1.4 Conclusions This section presents a multi-objective static output feedback controller design approach with application to vehicle suspensions. The multi-objective control problem is expressed as minimizing the ride comfort performance (H 2 norm) subjected to the tyre deflection (H ∞ norm) and suspension deflection (GH 2 norm) being constrained to given limitations. Due to the difficulties in resolving such a multi-objective control problem, a genetic algorithm is used to search for the final result together with the feasible solution of LMIs. The designed static output feedback controller is more applicable in engineering because it just uses the measurable variables for the suspension system. Numerical simulation validates that the vehicle suspension performances are improved with such a controller in spite of its simple structure.

2.2.2 Non-Fragile Control 2.2.2.1  Problem Formulation In this section, the quarter-car model introduced in Section 2.2.1 is also adopted, which can be seen in Figure 2.1. To compensate for the effect induced by the variations of the designed controller, a priori norm-bounded gain variation is considered in the design process. Therefore, considering a controller gain variation ∆K , we formulate the following state-feedback non-fragile H ∞ control problem for vehicle suspensions. The system is given by:

x = Ax + Bu + Bw w (2.16)

z = Cx + Du (2.17)

 − ks / ms 0 −cs / ms cs / ms    where x ,  w,  A,  B,  Bw are defined in equation (2.3), and C =  α 0 0 0 ;   0 β 0 0 T    1  D =     0 0   ; α and β are positive weighting scalars for the suspension deflection and tyre  ms  deflection, respectively. The state-feedback gain matrix K to be designed is:

u = ( K + ∆K ) x (2.18)

where K is the state-feedback gain matrix to be designed, and ∆K is an priori norm-bounded gain variation of the form [7]:

∆K = LFE (2.19)

where L and E being known constant matrices of appropriate dimensions, and

F

≤ 1 (2.20)

18

Vibration Control of Vehicle Suspension Systems

such that the resulting closed-loop system is asymptotically stable and the H ∞ norm of the closedloop transfer function matrix from w to z is bounded by a constant γ > 0 for any uncertainties F satisfying equation (2.20). 2.2.2.2 Non-Fragile H ∞ Controller To derive the main results, we need the following lemma. Lemma 2.1 Given real matrices Y ,  M , and N of appropriate dimensions, then: Y + M∆N + N T ∆ T M T < 0

For all ∆ satisfying ∆ ≤ 1 if and only if there exists a constant ε > 0 such that Y + ε MM T + ε −1 N T N < 0

System (2.17) with the static state-feedback control law u = ( K + ∆K ) x becomes

x = Ax + B ( K + ∆K ) x + Bw w (2.21)

z = Cx + D ( K + ∆K ) x (2.22) In order to establish that system (3.22) is asymptotically stable with a disturbance attenuation γ > 0, it is required that the associated Hamiltonian H ( x , w, t ) = V ( x , t ) + z T z − γ 2 wT w < 0 (2.23)

where V ( x , t ) is a Lyapunov function given by: V ( x , t ) = x T Px (2.24)

for some P > 0. We take the derivative of V ( x , t ) along the state trajectory of system (2.22): V ( x , t ) = x T

{[ A + B( K + ∆K )] P + P [ A + B( K + ∆K )]} x + 2w B Px (2.25) T

T

T w

then we obtain that H ( x , w, t ) = x T

{[ A + B( K + ∆K )] P + P [ A + B( K + ∆K )]} x T

+2 wT BwT Px + x T [C + D ( K + ∆K )] [C + D ( K + ∆K )] x − γ 2 wT w T

T   [ A + B ( K + ∆K )] P  = ϕ T  + P A + B ( K + ∆K ) + C + D ( K + ∆K ) T C + D ( K + ∆K ) [ ] [ ][ ]  T Bw P  

 (2.26)  PBw  ϕ  −γ 2 I  

19

Active Suspension Control T

where ϕ =  x T   wT  . Therefore, the requirement that H ( x , w, t ) < 0 in equation (2.23) for all ϕ ≠ 0 is implied by T T    [ A + B ( K + ∆K )] P + P [ A + B ( K + ∆K )] + [C + D ( K + ∆K )] [C + D ( K + ∆K )] PBw  < 0  −γ 2 I  BwT P   (2.27)

Using the Schur complement, it is equivalent to T   [ A + B ( K + ∆K )] P + P [ A + B ( K + ∆K )] PBw  * −γ 2 I  0  * and it is further equivalent to

 ( A + BK )T P + P ( A + BK )   *   *  PBL  F  E + 0     DL 

0

PBw −γ 2 I 0

   0 is implied by  ( A + BK )T P + P ( A + BK ) + ε PBLLT BT P + ε −1E T E PB w  2  * −γ I  *  *

(C + DK )T + ε PBLLT D T  0

− I + ε DLL D T

T

 (2.30)  

Using the Schur complement, it is further equivalent to

       

( A + BK )T P + *

PBw

(C + DK )T

ε PBL

* * * *

−γ 2 I * * *

0 −I * *

0 ε DL −ε I *

   0  (2.31) 0   0 −ε I  ET

Pre- and post-multiplying equation (2.31) by respectively diag(X ,  I ,  I ,  I ,  I ) and its transpose, we can obtain

20

Vibration Control of Vehicle Suspension Systems

       

XAT + AX + Y T BT + BY

Bw

XC T + Y T D T

* * * *

−γ 2 I 0 * −I * * * *

XE T    0 0  (2.32) ε DL 0  −ε I 0  −ε I  *

ε BL

where X := P −1 and Y = KX. This implies that equation (2.32) is satisfied, and hence H ( x ,  w, t ) < 0 for all ϕ ≠ 0. Therefore, we can summarize the above development on the non-fragile H ∞ controller design procedure for a suspension system in the following theorem. Theorem 2.1 Consider suspension system (2.4), a non-fragile H∞ state-feedback control gain can be found such that the closed-loop system is asymptotically stable with disturbance attenuation γ for all admissible controller gain uncertainties described by equations (2.19) and (2.20), if there exist matrices X > 0, Y and scalar ε > 0 , satisfying LMI (2.32). Moreover, a desired control gain is given by K = YX −1. 2.2.2.3  Simulation Results Now we apply the proposed approach to design a non-fragile suspension controller based on the quarter-car model described in the above section. The parameters of the quarter-car model are as follows (Table 2.1). For later comparison, a state-feedback H ∞ controller without fragility consideration is designed. In other words, controller gain variations are not taken into account. This controller is referred to as a fragile controller here. The obtained controller gain is

K =  −404.7

−7824

−4659.9

29.9 

with  K  = 9115.6, and the achieved H ∞ -norm of the closed-loop transfer function matrix from w  to z is γ 0 = 7.8325. The closed-loop frequency responses from disturbance to sprung mass acceleration, suspension deflection, and tyre deflection are depicted in Figure 2.6. It can be seen from these figures that the H ∞ controller gives a closed-loop that satisfies the three different performance criteria. Both the sprung mass acceleration and the suspension deflection are better than the uncontrolled suspension, especially in the range of sprung mass resonance, apart from a slight deterioration of the tyre deflection between the sprung mass resonance and unsprung mass resonance. To test the

TABLE 2.1 Parameters of the Quarter-Car Model Parameters ms mu ks cs kt

Values 504.5 kg 62 kg 13,100 N/m 400 Ns/m 252,000 N/m

21

Active Suspension Control

FIGURE 2.6  Transfer function from road disturbance to (a and b) sprung mass acceleration; (c and d) suspension deflection (e and f) tyre deflection.

fragility of the controller, we construct the minimum norm destabilizing perturbation on the controller gain by computing the real stability radius. Such a controller perturbation is given by

∆K =  6.9148

−6.8961

1.2068

−443.6770 

with ∆K = 443.7861, which is 4.87% of K . Then, we consider the controller gain variations defined by L = ρ × [1, 1, 1,1], E = diag (1,1,1,1), and ρ = 250 in equation (2.19). Using the proposed method in the above section, we set α = 16 and

22

Vibration Control of Vehicle Suspension Systems

β = 32 in matrix C, γ = 12 in equation (2.32), and we obtain the following non-fragile controller constructed based on K = YX −1 where X > 0 and Y satisfy equation (2.32):

K = 10 4 × [ 0.2275 1.8020  −0.4338 0.1181]

with K = 1.8711 × 10 4 The achieved H ∞ -norm of the closed-loop transfer function matrix from w to z is γ 0 = 7.8470 , which has no significant difference from the performance using the fragile controller. The closed-loop frequency responses from disturbance to sprung mass acceleration, suspension deflection, and tyre deflection are depicted in Figure 2.7. It can be seen from these figures that the performance of the sprung mass acceleration, suspension deflection, and tyre deflection are better than the uncontrolled suspension system. To test the fragility of the controller, we also construct the worst perturbation to the controller gain. The minimum norm destabilizing controller perturbation is given by

∆K = 10 3 × [ 0.0081  −0.0080 0.0208  −1.6406 ]

with ∆K = 1.6403 × 10 3. Thus, the required destabilizing norm is much larger than previous one and is nearly 8.77% of K, and hence it has better non-fragility characteristics. To further validate the results, we compare the two controllers with a road bump disturbance input. We generate 10 sets of randomly generated perturbations to the two controllers, respectively. The time-domain simulation results for the system with fragile controller and non-fragile are shown in Figure 2.7. It is observed that the fragile controller will give a destabilizing effect under certain perturbations, while the non-fragile controller maintains its performance throughout. The frequency responses from disturbance to sprung mass acceleration, suspension deflection, and tyre deflection for the non-fragile controller are shown in Figure 2.8. Notice that the perturbed controllers not only maintained closed-loop stability, their performances are also acceptable. Different non-fragile controllers can be obtained by varying ρ in L. In Table 2.2, a number of values of ρ are used to obtain different non-fragile controllers with different corresponding H ∞ performance with α = 10 and β = 20 in matrix C, and γ = 12 for all instances except γ = 120 for ρ = 2000 and γ = 1200 for ρ = 5000 in equation (2.32). It can be seen from these results that when ρ increases, the obtained destabilizing norm ∆K increases with the increase of K . From these results, we can conclude that when gain variations are taken into consideration in the design, the approach presented in this section gives controllers that are more non-fragile under the same level of performance. 2.2.2.4 Conclusions In this section, the non-fragile H ∞ controller design method for active suspensions is addressed. Based on the solvability of an LMI, we present an approach to design a non-fragile static statefeedback H ∞ controller using a quarter-car model. The extension of the proposed approach to more complex vehicle models does not present any conceptual difficulty. The performance of the nonfragile controller is validated using numerical simulation. It is also demonstrated that the deterioration of the performance a controller designed without fragility consideration may be unacceptable in practice.

2.2.3 Multi-Objective Control with Uncertainties 2.2.3.1  Problem Formulation In this section, a 2-DOF quarter-car active suspension model is shown in Figure 2.9 [8], where zs ( t ) and zu ( t ) denote the vertical displacements of the sprung mass and unsprung mass, respectively; zr ( t ) is the rough road excitation; ms ( t ) and mu ( t ) stand for the varying sprung mass and

23

Active Suspension Control (b)

Sprung mass acceleration (m/s2)

4

Open-loop Closed-loop

3

3

Sprung Mass Acceleration (m/s2)

(a)

2 1 0 –1

Open-loop Closed-loop

2

1

0

–1

–2 –3 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

–2 0.0

2.0

0.2

0.4

0.6

Time (s)

(c)

0.06

Open-loop Closed-loop

(d) Suspension Deflection (m)

Suspension Deflection (m)

0.04

0.02

0.00

–0.02

–0.04

–0.06

1.2

1.4

1.6

1.8

2.0

Open-loop Closed-loop

0.06 0.04 0.02 0.00 –0.02 –0.04

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.0

0.2

0.4

0.6

Time (s)

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Frequency (Hz)

0.008

Open-loop Closed-loop

(f)

0.000

–0.004

Open-loop Closed-loop

0.006 0.004

Tire Deflection (m)

0.004

Tire Deflection (m)

1.0

–0.06

0.0

(e)

0.8

Frequency (Hz)

0.002 0.000 –0.002 –0.004 –0.006

–0.008

0.0

–0.008

0.2

0.4

0.6

0.8

1.0

Time (s)

1.2

1.4

1.6

1.8

2.0

–0.010 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Frequency (Hz)

FIGURE 2.7  (a and b) Sprung mass acceleration; (c and d) suspension deflection; (e and f) tyre deflection.

the unsprung mass, respectively; ks and cs are the stiffness and damping coefficient of the suspension system, respectively; kt and ct represent the compressibility and damping coefficient of the tyre, respectively; u ( t ) stands for actuator input of the active suspension system; d ( t ) is the actuator delay, which satisfies 0 ≤ d ( t ) ≤ τ . Assuming no wheel lift-off and no slippage, therefore, according to the Newton’s second law, the equations of vertical motion can be obtained as follows:

24

Vibration Control of Vehicle Suspension Systems

FIGURE 2.8  Transfer function from road disturbance to sprung mass acceleration.

25

Active Suspension Control

TABLE 2.2 Summary of Destabilizing Gains for Different Non-fragile Controllers ρ

γ0

50

7.8114

100

7.8128

∆K

K

∆K

10 [ 2.6922 − 4.4310 − 3.9263 0.1083]

[−4.3893 4.3767 1.4919 −520.7143]

520.7534

10 3 [−1.0462 −5.1546 −5.1782 0.2195]

[1.9992 −1.9907 2.7397 −643.2013]

643.2134

3

200

7.8375

10 [−0.1291 5.0124 −4.9985 0.6693]

10 [0.1119 −0.1110 0.0092 −1.1137]

1124.9

500

7.8510

10 4 [0.0757 5.0124 −4.9985 0.6693]

10 3 [−0.0373 0.0362 0.0991 −3.2944]

3296.3

1000

7.8560

10 [−0.1224 3.7916 −0.1291 0.0763]

10 [−0.0563 0.0523 0.6822 −8.9923]

9018.4

2000

7.8543

10 [−0.0196 0.0180 0.1157 −1.2958]

13,013

5000

8.4498

10 [−0.3453 6.179610 −0.2617 0.1029] [−0.4247 7.6703 −0.3574 0.1423]

4 4

17,961

3

5 5

5

3

3

10 [−0.0330 0.0302 0.1684 −1.7876]

FIGURE 2.9  2-DOF “quarter-car” active suspension model.

 ms ( t )  zs ( t ) = − ks [ zs ( t ) − zu ( t )] − cs [ z s ( t ) − z u ( t )] + u ( t − d ( t )) ,   zu ( t ) = ks [ zs ( t ) − zu ( t )] + cs [ z s ( t ) − z u ( t )] − kt [ zu ( t ) − zr ( t )] (2.33) mu ( t )    −ct [ z u ( t ) − z r ( t )] − u ( t − d ( t )) .

To derive the corresponding state-space form of the active suspension system, the following set of state variables are chosen:

x1 ( t ) = zs ( t ) − zu ( t ) , x3 ( t ) = z s ( t ) ,

x 2 ( t ) = zu ( t ) − zr ( t ) , (2.34) x 4 ( t ) = z u ( t ) ,

where x1 ( t ) is the suspension travel, x 2 ( t ) is the tyre deflection, x3 ( t ) is the absolute vertical velocity of sprung mass, x 4 ( t ) is the absolute vertical velocity of unsprung mass.

26

Vibration Control of Vehicle Suspension Systems

Therefore, the dynamic equations in (2.33) can be transformed as the following state-space form: x ( t ) = A ( t ) x ( t ) + B1 ( t )ω ( t ) + B2 ( t ) u ( t − d ( t )) , (2.35)

where

 0  0  ks  A (t ) =  − ms ( t )  ks   mu ( t )   0  0  1  B2 ( t ) =  ms ( t )  1   − mu ( t ) 

0 0 0 −

kt mu ( t )

1 0 c − s ms ( t ) cs mu ( t )

−1 1 cs ms ( t ) c +c − s t mu ( t )

   0   1 −   0  , B1 ( t ) =    ct    mu ( t )  

    T  ,ω ( t ) = z r ( t ) , x ( t ) = [ x1 ( t )  x 2 ( t )  x3 ( t )  x 4 ( t )] .    

   ,    (2.36)

The main performance indexes have been introduced in Chapter 1. The three performance indexes can be evaluated by the following signals, that is,

1. Ride comfort: Ride comfort can be characterized by the vehicle body acceleration, especially in the frequency range of 4–8 Hz. Hence, in order to improve ride comfort, it is of significance to minimize the sprung mass acceleration  zs ( t ) within the concerned frequency band, namely min    zs ( t ) . (2.37) 2. Maximum suspension travel: In order to avoid damaging vehicle components and deteriorating handling stability, the suspension travel should be complied with the constraints of mechanical structure, namely zs ( t ) − zu ( t ) ≤ zmax , (2.38) where zmax is the maximum suspension travel limit.

3. Road holding ability: In order to ensure a good road holding ability and a firm uninterrupted contact of wheels to road, the dynamic tyre load is required to be smaller than its static load, namely,

kt ( zu ( t ) − zr ( t )) < ( ms ( t ) − mu ( t )) g, (2.39)

where g is the value of the acceleration of gravity. Therefore, the performance output z1 ( t ) and the constrained output z2 ( t ) are defined as follows:

27

Active Suspension Control

z1 ( t ) =  zs ( t ) ,

 z2 ( t ) =  

  z21 ( t )   = z22 ( t )     

  zmax  (2.40) , k t ( zu ( t ) − zr ( t ) )  ( ms (t ) + mu (t )) g 

( z s ( t ) − zu ( t ) )

where z21 ( t ) denotes the relative suspension travel, z22 ( t ) stands for the relative dynamic tyre load. Given that the suspension travel can be measured with a displacement transducer, the vertical velocities of the sprung mass and the unsprung mass can be measured with a velocity transducer. However, the tyre deflection cannot be measured online by commercial sensors [9]. Consequently, the following measurable output is chosen: y ( t ) =  z s ( t ) − zu ( t ) 

z s ( t )

T

z u ( t )  . (2.41) 

Hence, the vehicle active suspension system can be described by the state-space equation as follows: x ( t ) = A ( t ) x ( t ) + B1 ( t )ω ( t ) + B2 ( t ) u ( t − d ( t )) z1 ( t ) = C1 ( t ) x ( t ) + D1 ( t ) u ( t − d ( t ))

z 2 ( t ) = C2 ( t ) x ( t )

(2.42)

y ( t ) = Cx ( t ) x ( t ) = φ ( t ) , t ∈[ −τ ,0 ] where φ ( t ) is an initial continuous function, A ( t ), B1 ( t ) and B2 ( t ) are defined in (2.36), and  ks C1 =  − ms ( t ) 

 1  zmax C2 =    0 

0

cs ms ( t )

 cs  1   , D1 =   ms ( t )   ms ( t )  

0

0

kt ( ms (t ) + mu (t )) g

0

 0    0  

(2.43)

    As vehicle load is always uncertain due to the variation in the payload and the number of passengers, the sprung mass and unsprung mass are regarded as varying parameter in a given range, i.e. msmin ≤ ms ( t ) ≤ msmax and mumin ≤ mu ( t ) ≤ mumax . Next, the following equalities can be obtained:  1 C= 0   0

max max

0 0 0

0 1 0

0 0 1

1 1 1 1 = = mˆ s , min = = ms V ms ( t ) msmin ms ( t ) msmax 1 mu ( t )

=

1 mumin

= mˆ u , min

1 mu ( t )

=

1 mumax

= ms V

(2.44)

28

Vibration Control of Vehicle Suspension Systems

According to sector non-linear method [10], the 1/ms ( t ) and 1 /mu ( t ) can be represented as 1

˘

ms ( t ) 1

= M1 (ζ 1 ( t )) mˆ s + M 2 (ζ 1 ( t )) m s (2.45)

˘

mu ( t )

= N1 (ζ 2 ( t )) mˆ u + N 2 (ζ 2 ( t )) m u

where ζ 1 ( t ) = 1 /ms ( t ) and ζ 2 ( t ) = 1 /mu ( t ) stand for premise variables, and M1 (ζ 1 ( t )) + M 2 (ζ 1 ( t )) = 1 (2.46) N1 (ζ 2 ( t )) + N 2 (ζ 2 ( t )) = 1

Given that the uncertain parameters of ms ( t ) and mu ( t ) are bounded by their maximum values and minimum values, the membership functions M1 (ζ 1 ( t )), M 2 (ζ 1 ( t )), N1 (ζ 2 ( t )) and N 2 (ζ 2 ( t )) can be denoted as follows [11]: M1 (ζ 1 ( t )) =

˘

1 / ms ( t ) − m s ˘

mˆ s − m s

N1 (ζ 2 ( t )) =

˘

1 / mu ( t ) − m u ˘

mˆ u − m u

, M 2 (ζ 1 ( t )) = , N 2 (ζ 2 ( t )) =

mˆ s − 1 / ms ( t ) ˘

mˆ s − m s

(2.47)

mˆ u − 1 / mu ( t ) ˘

mˆ u − m u

The membership functions M1 (ζ 1 ( t )) and M 2 (ζ 1 ( t )) stand for “Heavy” and “Light” and the membership functions N1 (ζ 2 ( t )) and N 2 (ζ 2 ( t )) represent “Heavy” and “Light”, respectively. Which is shown in Figure 2.10. Hence, the uncertain and non-linear active suspension system can be denoted by the following T-S fuzzy model. Model Rule 1:

IF ζ 1 ( t ) is Heavy and ζ 2 ( t ) is Heavy, THEN x ( t ) = A1 χ ( t ) + B11ω ( t ) + B21u ( t − d ( t )) z1 ( t ) = C11 χ ( t ) + D11u ( t − d ( t )) z2 ( t ) = C21 χ ( t )

FIGURE 2.10  Membership functions, (a) M1 (ζ 1 ( t )) and M 2 (ζ 1 ( t )) , (b) N1 (ζ 2 ( t )) and N 2 (ζ 2 ( t )).

29

Active Suspension Control

Here, matrices A1, B11, B21, C11, D11, and C21 can be obtained by replacing 1/ms ( t ) and 1/mu ( t ) with mˆ s and mˆ u in the matrices of A ( t ), B1 ( t ), B2 ( t ), C1 ( t ), D1 ( t ), and C2 ( t ), respectively. Model Rule 2:

IF ζ 1 ( t ) is Heavy and ζ 2 ( t ) is Light, THEN x ( t ) = A2 χ ( t ) + B12ω ( t ) + B22u ( t − d ( t )) z1 ( t ) = C12 χ ( t ) + D12u ( t − d ( t ))

.

z2 ( t ) = C22 χ ( t )

Here, matrices A2, B12 , B22, C12, D12, and C22 can be obtained by replacing 1/ms ( t ) and 1/mu ( t ) with mˆ s and mˆ u in the matrices of A ( t ), B1 ( t ), B2 ( t ), C1 ( t ), D1 ( t ), and C2 ( t ), respectively. Model Rule 3:

IF ζ 1 ( t ) is Light and ζ 2 ( t ) is Heavy, THEN x ( t ) = A3 χ ( t ) + B13ω ( t ) + B23u ( t − d ( t )) z1 ( t ) = C13 χ ( t ) + D13u ( t − d ( t ))

.

z2 ( t ) = C23 χ ( t )

Here, matrices A3, B13 , B23, C13, D13, and C23 can be obtained by replacing 1/ms ( t ) and 1 / mu ( t ) with mˆ s and mˆ u in the matrices of A ( t ), B1 ( t ), B2 ( t ), C1 ( t ), D1 ( t ), and C2 ( t ), respectively. Model Rule 4:

IF ζ 1 ( t ) is Light and ζ 2 ( t ) is Light, THEN x ( t ) = A4 χ ( t ) + B14ω ( t ) + B24 u ( t − d ( t )) z1 ( t ) = C14 χ ( t ) + D14 u ( t − d ( t ))

.

z2 ( t ) = C24 χ ( t )

Here, matrices A4 , B14 , B24, C14, D14, and C24 can be obtained by replacing 1 / ms ( t ) and 1 / mu ( t ) with mˆ s and mˆ u in the matrices of A ( t ), B1 ( t ), B2 ( t ), C1 ( t ), D1 ( t ), and C2 ( t ), respectively. Based on a standard fuzzy inference approach, the overall fuzzy model can be shown as follows: 4

x ( t ) =

∑h (ζ (t ))  A x (t ) + B ω (t ) + B u (t − d (t )) i

i

1i

2i

i =1

= Ah x ( t ) + B1hω ( t ) + B2 hu ( t − d ( t )) 4

z1 ( t ) =

∑h (ζ (t )) C x (t ) + D u (t − d (t )) i

1i

1i

i =1

= C1h x ( t ) + D1hu ( t − d ( t )) 4

z2 ( t ) =

∑h (ζ (t ))C i

i =1

2i

x ( t ) = C2 h x ( t )

(2.48)

30

Vibration Control of Vehicle Suspension Systems 4

where ∂h =

∑h (ζ (t )) ∂ , ∂= A, B , B ,C , D ,C and i

1

i

2

1

1

2

i =1

h1 (ζ ( t )) = M1 (ζ 1 ( t )) × N1 (ζ 2 ( t )) h2 (ζ ( t )) = M1 (ζ 1 ( t )) × N 2 (ζ 2 ( t ))

(2.49)

h3 (ζ ( t )) = M 2 (ζ 1 ( t )) × N1 (ζ 2 ( t ))

h4 (ζ ( t )) = M 2 (ζ 1 ( t )) × N 2 (ζ 2 ( t )) It is noteworthy that the fuzzy weighting function hi (ζ ( t )) satisfies the following conditions, hi (ζ ( t )) ≥ 0,

4

∑h (ζ (t )) = 1 (2.50) i

i =1

The PDC method is adapted to design a fuzzy H ∞ SOF controller, and the following fuzzy control rules are presented. IF ζ 1 ( t ) is Heavy and ζ 2 ( t ) is Heavy,

Control Rule 1:

THEN u ( t ) = K1Cx ( t ) IF ζ 1 ( t ) is Heavy and ζ 2 ( t ) is Light,

Control Rule 2:

THEN u ( t ) = K 2Cx ( t ) IF ζ 1 ( t ) is Light and ζ 2 ( t ) is Heavy,

Control Rule 3:

THEN u ( t ) = K 3Cx ( t ) IF ζ 1 ( t ) is Light and ζ 2 ( t ) is Light,

Control Rule 4:

THEN u ( t ) = K 4Cx ( t )

Similarly, the overall fuzzy controller with actuator delay can be obtained as follows, 4

u (t ) =

∑h (ζ (t − d (t ))) K Cx (t ) = K j

j

fsof

Cx ( t ) , (2.51)

i =1

4

where K j is the local control gain and K fsof = tions are used:

∑h (ζ (t − d (t ))) K . For brevity, the following notaj

j

i =1

(

)

hi  hi (ζ ( t )) , h j  h j ζ ( t − d ( t )) . (2.52)

Based on (2.48) and (2.51), the closed-loop system of the uncertain and non-linear active suspension system with actuator delay can be written as follows: x ( t ) = Ax ( t ) + Ad ( x − d ( t )) + B1ω ( t ) ,

z1 ( t ) = C1 x ( t ) + C1d u ( t − d ( t )) , z 2 ( t ) = C2 x ( t ) ,

(2.53)

31

Active Suspension Control

where A = Ah ,

Ad = B2 h K fsof C , B1 = B1h ,

C1 = C1h , C1d = D1h K fsof C , C2 = C2 h , (2.54)

0 ≤ d (t ) ≤ τ . In a word, the purpose is to find a fuzzy finite frequency H ∞ SOF controller for the uncertain and non-linear active suspension system with actuator delay and output constraints, such that the following conditions are satisfied for all ω ∈L2 [ 0, 1) under the zero initial condition:

1. The closed-loop system in (2.53) is asymptotically stable; 2. For the closed-loop system in (2.53), the H ∞ performance satisfies the following condition: sup  Tω z1 ( jϖ ) ∞ < γ . (2.55)

ϖ 1 ≤ϖ ≤ϖ 2

3. For the closed-loop system in (2.53), the generalized H 2 performance satisfies the following conditions:

[ z2 (t )]q

≤ 1 q = 1,2. (2.56)

2.2.3.2  Design of Finite Frequency Fuzzy Controller Lemma 2.2 [8] The closed-loop active suspension system in (2.53) is asymptotically stable for any time delay d ( t ) satisfying 0 < d ( t ) ≤ τ , if there exist symmetric positive definite matrices P1h , R1h and S1h such that  A   I

T

 A  I

( Φ ⊗ P1h + ψ 0 ⊗ τ S1h ) 

−1  Ad   R1h − τ S1h + 0   *

τ −1S1h −1

− R1h − τ S1h

  < 0. (2.57)  

where  1 ψ0 =   0

 0 0  ,  Φ =   0   1

1  . (2.58) 0 

Lemma 2.3 [12] Considering the closed-loop active suspension system (2.53) and a given symmetric matrix P, the following two statements are equivalent if there exist symmetric positive definite matrices Qh , Z h and symmetric matrices Ph and X h.

1. The finite-frequency inequality

σ ( Tω z1 ( jϖ ) , Π ) < 0, ϖ 1 < ϖ < ϖ 2 . (2.59)

32

Vibration Control of Vehicle Suspension Systems

2. There exist symmetric matrices Ph and Qh satisfying Qh > 0 and  A   I

 C + 1  0

  

B1 0

C1d 0

 X h − τ −1 Z h  + τ −1 Z h  0 

T

 A  I

  

0 I

( Φ ⊗ Ph + ψ ⊗ Qh + ψ 0 ⊗ τ Zh )  T

 C  1  0

0 I

C1d 0

B1 0

  

   (2.60)

0   0  < 0,  0 

τ −1 Z h − X h − τ −1 Z h 0

where Ψ 0 and Φ is defined in (2.58) and  −1 Ψ=  − jϖ c

 I  jϖ c , Π =  −ϖ 1ϖ 2   0 

 0 (ϖ 1 + ϖ 2 ) . (2.61) , ϖc = 2 −γ I  2 

It is easy to know that (2.57) can be rewritten into the form as follows:

ς sT Ξsς s , (2.62)

where

  ςs =    

A I I 0

  Φ ⊗ P1h + Ψ 0 ⊗ τ S1h    , Ξs =  *     * 

0

0 −1

R1h − τ S1h

0

*

− R1h − τ −1S1h

   . (2.63)  

Similarly, the LMI in (2.60) can be expressed as below:

ς T Ξς , (2.64)

where

 A   I  ς =  C1  0  I   0    Ξ=  

Ad 0 C1d 0 0 I

B1 0 0 I 0 0

    ,    

(2.65)

Φ ⊗ Ph + Ψ ⊗ Qh + Ψ 0 ⊗ τ Z h * *

0 Π *

0 0 X h − τ −1Z h

0 0 τ −1Z h

*

*

τ −1Z h

− X h − τ −1Z h

   .  

33

Active Suspension Control

The LMIs in (2.62) and (2.64) can guarantee that the closed-loop system (2.53) satisfies the asymptotical stability and the H ∞ performance described in (2.55). Partly inspired by the approach in [13,14], the following theorem is obtained.

Theorem 2.2 Considering the closed-loop active suspension system in (2.53): Given positive scalar τ , γ , ρ and the fuzzy SOF controller in form of (2.51). For any time delay d ( t ) satisfying 0 < d ( t ) ≤ τ , the closed-loop system in (2.53) is asymptotically stable and satisfies the conditions described in (2.55) and (2.56), if there exist symmetric positive definite matrices P1h , R1h , S1h, Qh , Z h , and symmetric matrices Ph and X h, and general matrices Eh and Fh satisfying

ξ ST Ξsξ s + Ω S + ΩTs < 0, (2.66)

ξ T Ξξ + Ω + ΩT < 0, (2.67)  −I   * 

ρ {C2 h }q   < 0, q = 1,2, (2.68)  − P1h 

where Ξs is defined in (2.63), Ξ is defined in (2.65), and

 Ah  ξs =  I  I  0 

0 0 0 I

Ωs =  0 

EhT

Ω= 0 

EhT

B2 h 0 0 0

 Ah    I   , ξ =  C1h  0     I   0  T

− FhT   0

0 0 0 0 0 I

T

B2 h 0 D1h 0 0 0

    ,    

−I  , 

K fsof C

− FhT   0

0

B1h 0 0 I 0 0

K fsof  C

−I . 

0

Proof. Supposing the conditions in (2.66) and (2.67) are satisfied for some symmetric positive definite matrices P1h , R1h , S1h, Qh , Z h , and symmetric matrices Ph and X h, and general matrices Eh and Fh Note that

ξ S T1s ≡ ζ s ,  0 

where

ξ T1 ≡ ζ ,  0   I  T1s =  0  0

K fsof C

     = , T 1   K fsof C     0 I

− I  T1s ≡ 0, 

K fsof C 0

I 0 0 0

(2.69) − I  T1 ≡ 0,  0 I 0 K fsof C

0 0 I 0

  .   

34

Vibration Control of Vehicle Suspension Systems

After multiplying the left- and right-hand sides of the inequality in (2.66) by T1Ts and T1s , respectively, (2.62) can be obtained. Similarly, (2.64) can be obtained by applying T1 to the inequality in (2.67). It is easy to know that (2.62) can be rewritten into the form as (2.57) and (2.64) can be rewritten into the form as (2.60). According to Lemma 2.1 and Lemma 2.2, the closed-loop system in is asymptotically stable and satisfies the condition described in (2.55). In other words, (2.66) and (2.67) are sufficient conditions for that the closed-loop system in (2.53) satisfies the asymptotical stability and the condition described in (2.55). Next, the GH 2 performance of (2.56) is established. It should be noted that the following Lyapunov functional V(x(t)) was chosen to obtain Lemma 2.1 in [8]. V ( x ( t )) = x ( t ) P1h x ( t ) + T

0

t

∫x

T

t −τ

t

∫∫

( s ) R1h x ( s ) ds +   x T ( s ) S1h x ( s ) dsdβ , (2.70) −τ t + β

where P1h , R1h , and S1h are symmetric positive definite matrices. According to Lemma 2.1, the following inequality is true: V ( x ( t )) < 0 < ηω T ( t )ω ( t ) ∀η > 0. (2.71)

Integrating both sides of the inequality in (2.71) from 0  to   t , the following inequality can be obtained: t

V ( x ( t )) − V ( 0 ) ≤ η ω T ( t )ω ( t ) dt ≤ η  ω  22 = ηω max (2.72)

0

It is easy to know that x T ( t ) P1h x ( t ) ≤ V ( x ( t )) ≤ V ( 0 ) + ηω max = ρ . Then, a transformation is defined as x ( t ) = P11/h 2 x ( t ), which shows that x T ( t ) x ( t ) ≤ ρ . Hence, the following condition can be obtained 1 1 2   − − t max { z2 ( t )}q ≤ ρ ⋅ λmax  x T ( t ) P1h 2 {C2 h }q {C2 h }q P1h 2 x ( t ) , (2.73) t ≥0  

where λmax (⋅) denotes the maximum eigenvalue. Then, the desired condition in (2.56) is guaranteed if the following inequality holds:

ρ P1−h1/ 2 {C2 h }q {C2 h }q P1−h1/ 2 < I , q = 1,2, (2.74) T

which is equivalent to (2.68) according to Schur complement. Hence, the proof is completed. After taking the uncertainties of the active suspension system into account, the following theorem is further introduced to propose a new result for parameterizing a fuzzy SOF controller. Theorem 2.3 Considering the closed-loop active suspension system in (2.53): Given positive scalar τ , γ , ρ and the fuzzy SOF controller in form of (2.51). For any time delay d ( t ) satisfying 0 < d ( t ) ≤ τ ,

35

Active Suspension Control

the closed-loop system in (2.53) is asymptotically stable and satisfies the desired conditions described in (2.55) and (2.56), if there exist symmetric positive definite matrices P1 j, R1 j , S1 j , Q j, Z j, and symmetric matrices Pj and X j , and general matrices K j, L j and Fj satisfying following inequalities ( i, j = 1;2;3;4 ).

ξ sijTΞsijξ sij + Γ sij + Γ Tsij < 0, (2.75)

ξijT Ξijξij + Γ ij + Γ Tij < 0, (2.76)  −I   * 

ρ {C2i }q   > 0, q = 1,2. (2.77)  − P1 j 

Moreover, the fuzzy SOF control gain K fsof can be obtained as follows:

K fsof

 = 

 h j Fj   j =1 4

−1

  

 hj L j   j =1 4

where  Φ ⊗ P1 j + Ψ 0 ⊗ τ S1 j  Ξsij =  *   *    Ξij =    

0

    

0 −1

−1

R1 j − τ S1 j

τ S1 j

*

− R1 j − τ −1S1 j

Φ ⊗ Pj + Ψ ⊗ Q j + Ψ 0 ⊗ τ Z j

0

0

0

* *|

Π *

0 X j − τ −1Z j

0 τ −1Z j

*

*

τ −1Z j

− X j − τ −1Z j

 Ai    I   , ξ =  C1i  0     I   0 

 Ai  ξs =  I  I  0 

0 0 0 I

Γ sij =  0

K

− I  ×  0

Γ ij =  0

K

0

B2i 0 0 0

T

T

L jC

− I  ×  0

0 0 0 0 0 I

B1i 0 0 I 0 0

B2i 0 D1i 0 0 0

      

        

− Fj   L jC

0

− Fj  

Proof. According to the inequalities in (2.66) and (2.67), it is easy to obtain that FhT Fh < 0, T namely, Fh is invertible. After substituting Lh = Fh K fsof and K = Eh Fh−1 for the conditions in (2.66) and (2.67) and the following definitions are applied:

(

)

36

Vibration Control of Vehicle Suspension Systems

Ah =

∑h A , i

i

B1h =

i =1

D1h =

R1h =

i =1

Zh =

∑ i =1

Lh =

2i

,

j 1j

,

i

∑h S i =1 4

Ph =

hj Z j ,

∑h P , j

j

i =1

4

hj X j ,

∑h C 4

S1h =

h j R1 j ,

i =1

4

2i

i =1

4

h jQ j ,

i =1

Xh =

∑ i =1

4

Qh =

i

4

C2 h =

hi D1i ,

4

h j P1 j ,

∑h B , i =1

i =1

4

P1h =

B2 h =

i 1i

4

hiC1i ,

i =1

∑h B , i =1

4

C1h =

4

4

4

4

Fh =

hj L j ,

i =1

∑h F , j

j

i =1

The inequalities in (2.75)–(2.77) can be obtained. Hence, the proof is completed. Given the existence of matrix K , the conditions in Theorem 2.3 are not LMIs. However, if the matrix K is known as a priori and fixed matrix, the conditions in Theorem 2.3 become LMIs with respect to the remaining unknown matrices. Inspired by [13–15], Theorem 2.3 can be solved to find a fuzzy SOF controller with K fixed as an initial fuzzy SF controller. Moreover, Theorem 2.3 can be used to update and improve the fuzzy SF controller gain with measurement output matrix C replaced by identify matrix I. Theorem 2.4 Considering the closed-loop active suspension system in (2.53): Given matrices Ψ0 defined in (2.58), Φ and Ψ defined in (2.61), and positive scalars τ , γ and ρ . For any time delay d ( t ) satisfying 0 < d ( t ) ≤ τ , the closed-loop system in (2.53) is asymptotically stable and satisfies the desired conditions described in (2.55) and (2.56), if there exist a positive scalar α h , symmetric positive definite matrices P1h , R1h , S1h, Qh , Z h , and symmetric matrices Ph and X h, and general matrices Jh and Vh satisfying the following inequalities. Moreover, the state feedback control gain K sf can be obtained by K sf = Vh Jh−1.  τ S1h − α [ Jh ] s   *   * 

P1h − α h Jh T + α h Ah Jh

α h B2 hVh

R1h − τ −1S1h + α h [ Ah Jh ]s

α h B2 hVh + τ −1S1h

*

− R1h − τ −1S1h

−Qh + τ Z h < 0, (2.79)

        

   < 0, (2.78)   

−Qh + τ Z h

Ph + jϖ cQh − Jh T

0

0

0

Ph − jϖ cQh − Jh

−ϖ 1ϖ 2Qh + [ Ah Jh ]s + X h − τ −1 Z h

B2 hVh + τ −1 Z h

B1h

JhT C1Th

*

*

− X h − τ −1 Z h

0

VhT D1Th

* *

* *

* *

−γ 2 I *

0 −I

     < 0     (2.80)

37

Active Suspension Control

 P  1h  * 

T ρ JhT {C2 h }q   < 0, q = 1,2. (2.81)  − ρI 

Proof. Similar proof of this theorem can be found in [8]. To deal with the polytypic uncertainties of active suspension systems, the following theorem is proposed to calculate a fuzzy SF controller. Theorem 2.5 Considering the closed-loop active suspension system in (2.53): Given matrices Ψ0 defined in (2.58), Φ and Ψ defined in (2.61), and positive scalars τ , γ and ρ . For any time delay d ( t ) satisfying 0 < d ( t ) ≤ τ , the closed-loop system in (2.53) is asymptotically stable and satisfies the desired conditions described in (2.55) and (2.56), if there exist positive scalars α j, symmetric positive definite matrices P1 j, R1 j , S1 j , Q j, Z j, and symmetric matrices Pj and X j , and general matrices hj and V j satisfying the following inequalities ( i, j = 1;2;3;4 ).  τ S − α J  1j j  j s   *   * 

P1 j − α j J Tj + α j A j J j

α j B2iV j

R1 j − τ −1S1 j + α j  Ai J j  s

α j B2iV j + τ −1S1 j

*

− R1 j − τ −1S1 j

−Q j + τ Z j < 0 (2.83)

        

   < 0 (2.82)   

−Q j + τ Z j

Pj + jϖ cQ j − J Tj

0

0

0

Pj − jϖ cQ j − J j

−ϖ 1ϖ 2Q j +  Ai J j  s + X j − τ −1 Z j

B2iV j + τ −1 Z j

B1i

J Tj C1Ti

*

*

− X j − τ −1 Z j

0

V jT D1Ti

* *

* *

* *

−γ 2 I *

0 −I

 P  1j  * 

     < 0     (2.84)

T ρ J Tj {C2i }q   < 0, q = 1,2 (2.85)  − ρI 

Moreover, the fuzzy SF control gain can be obtained as:

K fsf

 = 

 h jV j     j =1 4

 hj J j   j =1 4

−1

( j = 1,2,3,4 ) .

Proof. Based on Theorem 2.4 and the inner property of the polytypic uncertain systems, the following definitions is applied to Theorem 2.4:

38

Vibration Control of Vehicle Suspension Systems

Ah =

∑h A , i

B1h =

i

i =1

∑ ∑ ∑ i =1

∑ i =1

∑h C

2i

,

j 1j

,

i

4

h j R1 j ,

S1h =

∑h S i =1 4

hj Z j ,

Ph =

∑h P , j

j

i =1

4

Jh =

hj X j ,

2i

i =1

i =1

4

Xh =

C2 h =

4

Zh =

h jQ j ,

i

4

hi D1i ,

i =1

4

∑h B , i =1

4

R1h =

h j P1 j ,

i =1

Qh =

B2 h =

i =1

4

P1h =

i 1i

4

D1h =

hiC1i ,

i =1

∑h B , i =1

4

C1h =

4

4

4

4

hj J j ,

Vh =

i =1

∑h V , j

j

i =1

The inequalities in (2.78)–(2.81) can be rewritten as follows:  τ S − α J  1j  j s     hi h j *  i =1 j =1  *  4

4

∑∑

P1 j − α j J Tj + α j Ai J j

α j B2iV j

R1 j − τ −1S1 j + α j  A j J j  s

α j B2iV j + τ −1S1 j

*

− R1 j − τ −1S1 j

   < 0, (2.86)   

4

∑h −Q + τ Z  < 0, (2.87)

j

j

j

j =1

 −Qh + τ Z h  Ph + jϖ cQh − Jh T 0 0 0   T T −1 −1 J h C1h  4 4  Ph − jϖ cQh − Jh −ϖ 1ϖ 2Qh + [ Ah J h ]s + X h − τ Z h B2 hVh + τ Z h B1h   − 1   hi h j * * 0 VhT D1Th  < 0 − Xh − τ Zh  i =1 j =1  *  * * −γ 2 I 0   * * * −I  *  (2.88)

∑∑

 P 1j   hi h j   * i =1 j =1  4

T ρ J Tj {C2i }q   < 0, q = 1,2. (2.89)  − ρI 

4

∑∑

4

4

Given that hi ≥ 0, h j ≥ 0,

hi = 1 and

i =1

∑h = 1, the inequalities (2.82)–(2.85) can be j

i =1

obtained. Hence, the proof is completed. There exist a few studies about fuzzy SF control of uncertain active suspension systems [16–19]. However, constant matrices are utilized to construct the Lyapunov function in the above fuzzy SF controllers. In this study, the corresponding matrices are replaced by 4

P1h =

4

∑h P , R = ∑h R j 1j

i =1

1h

j

i =1

1j

4

 V1h =

∑h V

j 1j

i =1

in the design of the fuzzy controllers, which is

39

Active Suspension Control

also dependent on the varying parameters ms (t ) and mu(t ). As a consequence, the conservative of the fuzzy controller is reduced compared with previous investigations. Moreover, in light of the constant Lyapunov function, a traditional fuzzy finite frequency SF controller can be obtained by the following Corollary for the active suspension system with time delay. Which is also utilized for comparative purpose of highlighting the performance of the proposed fuzzy controller. Corollary 2.1 Considering the closed-loop active suspension system in (2.53): Given matrices Ψ0 defined in (2.58), Φ and Ψ defined in (2.61), and positive scalars τ , γ and ρ . For any time delay d ( t ) satisfying 0 < d ( t ) ≤ τ , the closed-loop system in (2.53) is asymptotically stable and satisfies the desired conditions described in (2.55) and (2.56), if there exist positive scalar a, symmetric positive definite matrices P1, R1, S1, Q, Z , and symmetric matrices P and X, and general matrices J and V j satisfying the following inequalities ( i, j = 1;2;3;4 ).

 τ S1 − α [ J ] s   *   * 

P1 − α J T + α A j J

α B2iV j

R1 − τ −1S1 + α [ Ai J ]s

α B2iV j + τ −1S1

*

− R1 − τ −1S1

   < 0, (2.90)   

−Q + τ Z < 0, (2.91)

 −Q + τ Z   P − jϖ cQ − J   *  *   * 

P + jϖ cQ − J T

0

0

0

−ϖ 1ϖ 2Q +  A j  s + X − τ −1 Z

B2iV j + τ −1 Z

B1i

J T C1Ti

*

− X − τ −1 Z

0

V jT D1Ti

* *

* *

−γ 2 I *

0 −I

 P  1  * 

     < 0, (2.92)    

T ρ J T {C2i }q   < 0, q = 1,2. (2.93)  − ρI 

Moreover, the fuzzy SF control gain can be obtained as: 4

K c − fsf =

∑h V J j

j

−1

( j = 1,2,3,4 ) .

j =1

Proof. The proof of Corollary 2.1 can be easily completed by choosing constant matrices P1, R1, S1, Q, Z , P, X, J and following the similar manner of Theorem 2.4. After obtaining an initial fuzzy SF controller from Theorem 2.4, the following optimization problem can be solved to derive a desired fuzzy SOF controller with minimum H ∞ performance:

min

γ2

s.t.

( 2.75) − ( 2.77 ) , P1 j > 0, R1 j > 0, S1 j > 0, Q j > 0, Z j > 0

for

P1 j , R1 j , S1 j , Q j , Z j , Pj , X j , K , L j , Fj , i, j = 1, 2,,4,

with

K is priori and fixed; i.e: K = K initial − fsf

(2.94)

40

Vibration Control of Vehicle Suspension Systems

 = 

The fuzzy SOF controller is given as K fsof

 h j Fj   j =1 4

−1

  

 hj L j   j =1 4

Remark 2.5. The solution of optimization problem (2.94) depends on the choice of the initial fuzzy SF controller obtained by solving Theorem 2.5. However, if the optimization problem (2.94) cannot be solved, an algorithm is required to update and improve the initial fuzzy SF controller. Note that the initial fuzzy SF controller Kfsf should approach the form of a fuzzy SOF controller, namely,  K fsf − K fsof C  which should be as small as possible. Inspired by [13, 14] and recalling Remark 2.1 the following optimization problem can be presented to update the initial fuzzy SF controller. min

γ2

s.t.

( 2.75) − ( 2.77 ) , 

 −η I

for

  < 0, P1 j > 0, R1 j > 0, S1 j > 0, Q j > 0, ⊥ − I   LC (2.95) P1 j , R1 j , S1 j , Q j , Z j , Pj , X j , K , L j , Fj , i, j = 1, 2,,4,

with

K is priori and fixed; i.e: K = K initial − fsf

with

C replaced by identity matrix I .

*

where C ⊥ denotes an orthonormal basis of the null space of C, namely, CC ⊥ = 0 and

(C )

⊥ T

C = I . The updating fuzzy SF controller is given as K f − sof

 = 

 h j Fj   j =1 4

−1

  

 hj L j  .  j =1 4

Then, Algorithm 2.1 can be obtained update and improve the initial fuzzy SF controller. Moreover, Algorithm 2.2 is proposed for the fuzzy SOF controller design. Algorithm 2.1. Algorithm for initial fuzzy SF controller design Step 1: Find an initial fuzzy SF controller gain by solving Theorem 2.5, i.e. K initial − fsf = K fsf . Step 2: Solve the optimization problem in (2.95) to obtain the updated fuzzy SF controller gain as K update − fsf

 = 

 h j Fj   j =1 4

−1

  

4

∑h L  . j

j

j =1

Step 3: Output is defined as the initial fuzzy SF controller gain for solving the optimization problem (2.94). Algorithm 2.2. Algorithm for fuzzy SOF controller design Step 1: Find an initial fuzzy SF controller gain by solving Theorem 2.4, i.e. K initial − fsf = K fsf . Step 2: Solve the optimization problem in (2.95) to obtain the updated fuzzy SF controller gain as K update − fsf

 = 

 h j Fj   j =1 4

−1

  

4

∑h L  . j

j =1

j

41

Active Suspension Control i Step 3: With K initial − fsf as the initial fuzzy SF controller, solve the optimization problem in

(2.94) to obtain the fuzzy SOF controller as K fsof

 = 

 h j Fj   j =1 4

−1

  

 hj L j  .  j =1 4

Step 4: If a desired fuzzy SOF controller can be obtained, exit; Else i set i = i + 1,  K initial − fsf = K update − fsf and go back to Step 2. 2.2.3.3  Simulation Results In this section, some numerical simulations are presented to evaluate and verify the performance of the proposed fuzzy finite frequency SOF controller. The overall workflow of the proposed fuzzy SOF controller is shown in Figure 2.11. ms is defined to vary in the interval  256 kg,384 kg  and mu is defined to vary in the interval 35kg,45kg . The parameters of the quarter-car active suspension model are listed in Table 2.3 and the values are borrowed from [20]. For given scalars ϖ 1 = 4 Hz , ϖ 2 = 8Hz, τ = 5ms and ρ = 1, the fuzzy SF controller can be obtained by solving Theorem 2.5, and the fuzzy SOF controller can be derived from Algorithm 2.2. Recalling Remark 2.2, a traditional fuzzy finite frequency SF controller can be obtained by solving Corollary 2.1. To highlight the effectiveness of the fuzzy SOF controller, the traditional fuzzy finite frequency SF controller,

T-S fuzzy model Vehicle mass

T-S fuzzy approach with four fuzzy rules

Actuator delay

IF..., THEN...

Fuzzy SOF controller

Initial fuzzy SF controller

Measured ouputs Delay

Actuator (Motor)

Algorithm 2

Controller Design

FIGURE 2.11  Overall workflow of proposed fuzzy SOF controller.

TABLE 2.3 Parameters of Quarter-Car Model ms 320 kg

mu

ks

kt

cs

ct

umax

zmax

40 kg

18 kN/m

200 kN/m

1 kNs/m

10 Ns/m

2500 N

0.1 m

42

Vibration Control of Vehicle Suspension Systems

TABLE 2.4 Controller Gains Fuzzy SOF controller

K fsof = 10 4 × [1.2738 − 0.2684 − 0.0709 ]

Fuzzy SF controller

K fsof = 10 4 × [1.2759 − 0.2647 − 0.2649 − 0.0723]

Compared fuzzy SF controller

K fsos = 10 4 × 1.5025 0.5455 − 0.1015 − 0.0534 

FIGURE 2.12  Frequency response of sprung mass acceleration: (a) no delay, (b) d ( t ) = 5seconds.

regarded as compared SF controller, is utilized for comparative purpose. All corresponding control gains are listed in Table 2.4. The frequency responses of the sprung mass acceleration are plotted in Figure 2.12. In all results, the fuzzy SOF controller is denoted by the solid line, the fuzzy SF controller is represented by the dashed line, the dot-dashed line stands for the compared SF controller, and the dotted line stands for the passive suspension. Figure 2.12 shows that the fuzzy SOF controller yields smaller gain values than that with the compared SF controller and the passive suspension in the frequency band of 4 − 8Hz, even the time delay occurs in the closed-loop system. It reveals that the fuzzy SOF controller is more effective than the compared SF controller. Moreover, it can be clearly seen that the performance of the fuzzy SOF controller is almost the same as the fuzzy SF controller, which implies

43

Active Suspension Control

Algorithm 2.2 can obtain a less conservative solution. Hence, the results of frequency response verify the effectiveness of the fuzzy SOF controller. Now, a bumpy road profile is utilized to evaluate and verify the performance of the proposed controller as follows:

 A  2π v    t 1 − cos   l    2  Zr ( t ) =   0  

if 0 ≤ t ≤ l if t > v

l v

(2.96)

where A and l are the height and the length of the bump, respectively. The parameters can be set as A = 0.06m, l = 5m and the vehicle forward velocity is set as v = 35km/h. The bump response results are shown in Figure 2.13. Figure 2.13a and b shows that the fuzzy SOF controller can acquire a better ride comfort than that with the compared SF controller, despite the existence of time delay. Figure 2.13c–h illustrate that relative suspension travel, relative dynamic load and relative actuator force are all less than one. In other words, the hard constraints of suspension travel, tyre deflection and actuator saturation are simultaneously satisfied, despite the existence of time delay. Moreover, Figure 2.13a–h show that the fuzzy SOF controller and the fuzzy SF controller have similar performances, which indicates that Algorithm 2.2 is less conservative. Hence, the effectiveness and robustness of the designed fuzzy SOF controller are validated. Apart from the bump response test, a non-stationary random road profile is examined as follows [21]:

zr ( t ) + ( v0 + at ) 2π nc zr ( t ) = 2π n0 Gq ( n0 )( v0 + at )w ( t ) , (2.97)

where v0 is the initial speed, a is the vehicle acceleration, w ( t ) is a white noise in the timedomain, n0 = 0.1m −1 is the road spatial cut-off frequency, n0 = 0.1 m −1 is the standard spatial cut-off frequency and Gq ( n0 ) is the coefficient of road roughness. V0 = 20 m/s and a = 2m/s2 are selected in this simulation. According to ISO-2631, four different road roughness coefficients are defined as Gq ( n0 ) = 16 × 10 −6 m 3 (B grade-good), Gq ( n0 ) = 64 × 10 −6 m 3 (C grade-average), Gq ( n0 ) = 256 × 10 −6 m 3 (D grade-poor) and Gq ( n0 ) = 1024 × 10 −6 m 3 (E grade-very poor). For brevity, the C grade-average road roughness coeffificient is selected to plot the simulation results. Figure 2.14 plots the random response results. As expected, Figure 2.14a and b illustrate that the fuzzy SOF controller can yield a less value of sprung mass acceleration than the compared SF controller and pass suspension, despite the existence of time delay. Figure 2.14c–h show that the mechanical constraints (i.e. relative suspension travel, relative dynamic load, and relative actuator force) are all less than one. In other words, the three constraints’ requirements are simultaneously met, despite the existence of time delay. Besides, Figure 2.14c–h illustrate that the fuzzy SOF controller and the fuzzy SF controller pose similar performances, which also indicates that Algorithm 2.2 is effective to obtain a less conservative output feedback controller. Hence, the random response results cogently validate the effectiveness and robustness of the designed fuzzy SOF controller. Moreover, as the root mean square (RMS) values are strictly related to the ride comfort, it is often utilized to quantify the intensity of acceleration transmitted to the passenger location in random simulation [9]. Hence, the RMS values are employed to demonstrate the performance of the designed controller under non-stationary running. T

RMS x =

(1 / T ) χ T ( t ) x ( t ) dt . (2.98) 0

44

Vibration Control of Vehicle Suspension Systems

FIGURE 2.13  Bump response of: (a) sprung mass acceleration, d(t) = 0, (b) sprung mass acceleration, d(t) = 5 ms, (c) relative suspension travel, d(t) = 0, (d) relative suspension travel, d(t) = 5 ms, (e) relative dynamic load, d(t) = 0, (f) relative dynamic load, d(t) = 5 ms, (g) relative actuator force, d(t) = 0, (h) relative actuator force, d(t) = 5 ms.

45

Active Suspension Control Sprung mass acceleration (m/s/s)

0.08

Sprung mass acceleration (m/s/s)

0.08

(a)

0.04

0.04

0

0

(b)

Fuzzy SOF controller

-0.04

Fuzzy SOF controller

-0.04

Fuzzy SF controller

Fuzzy SF controller

Compared SF controller

Compared SF controller

Passive suspension

-0.08

0

2

1

3

-0.08

Passive suspension

0

Time (s) 8

(c)

0.005

4

0

0 Fuzzy SOF controller

-0.005

x 10-3

Fuzzy SOF controller

-4

Fuzzy SF controller

Fuzzy SF controller Compared SF controller

Passive suspension

0

1

-8 2

(d)

Compared SF controller

-0.01

3

Passive suspension

0

1

2

(e)

(f)

0.01

0.01

0

0 Fuzzy SOF controller

-0.01

Fuzzy SOF controller

-0.01

Fuzzy SF controller

Fuzzy SF controller

Compared SF controller

-0.02

Compared SF controller

Passive suspension

0

2

1

3

-0.02

Passive suspension

0

1

Relative actuator force

0.005

0.005

0

0

(h)

-0.005

Fuzzy SOF controller

Fuzzy SOF controller

Fuzzy SF controller

Fuzzy SF controller

Compared SF controller

-0.01 0

1

2 Time (s)

3

(g)

-0.005

2 Time (s)

Time (s) 0.01

3

Time (s)

Time (s) 0.02

3

Time (s)

2

1

Compared SF controller

3

-0.01 0

1

2

3

Time (s)

FIGURE 2.14  Random response of: (a) sprung mass acceleration, d(t) = 0, (b) sprung mass acceleration, d(t) = 5 ms, (c) relative suspension travel, d(t) = 0, (d) relative suspension travel, d(t) = 5 ms, (e) relative dynamic load, d(t) = 0, (f) relative dynamic load, d(t) = 5 ms, (g) relative actuator force, d(t) = 0, (H) relative actuator force, d(t) = 5 ms.

46

Vibration Control of Vehicle Suspension Systems

Tables 2.5 and 2.6 show the RMS values of the sprung mass acceleration, where the reduced rate is relative to the passive suspension. When no delay occurs, Table 2.5 shows that as compared with the passive suspension, the RMS values of sprung mass acceleration are reduced by 30.41% with compared SF controller, by 59.31% with fuzzy SF controller and by 56.91% with fuzzy SOF controller, respectively. When the time delay is encountered in the control loop, Table 2.6 shows that the RMS values of sprung mass acceleration are reduced by 28.12% with compared SF controller, by 46.63% with fuzzy SF controller, and by 45.44% with fuzzy SOF controller, respectively. It clearly reveals the negative impact of time delay on the active suspension system. As expected, it clearly indicates that the fuzzy SOF controller possesses a better disturbance attenuation performance than that with the compared SF controller. Moreover, it can be seen from Tables 2.5 and 2.6 that the disturbance attenuation performance of the fuzzy SOF controller is close to that of the fuzzy SF controller, which indicates that Algorithm 2.2 can derive an effective and robust fuzzy SOF controller. Hence, the above results verify the performance of the proposed fuzzy SOF controller. 2.2.3.4 Conclusions This section has dealt with the problem of fuzzy SOF control for non-linear active suspension systems with time delay and output constraints in finite frequency range. The T-S fuzzy model approach is applied to denote the non-linear active suspension system with taking sprung and unsprung mass variations, and the time delay into account. Based on a further generalization of the strict S-procedure and Lyapunov stability theory, a new fuzzy finite frequency SOF controller for the non-linear active suspension system with time delay and output constraints is proposed to improve ride comfort and simultaneously ensure suspension hard constraints. Simulation results validate the effectiveness and applicability of the proposed fuzzy SOF controller. In the future, more efforts will be devoted to full suspension systems or chassis-integrated systems. TABLE 2.5 RMS Values of Sprung Mass Acceleration with No Delay Roughness

Suspension Active Suspension

16 × 10−6 m3 64 × 10−6 m3 256 × 10−6 m3 1024 × 10−6 m3 Reduced rate

Passive Suspension 0.0109 0.0219 0.0437 0.0875 –

K c− fsf 7.6080 × 10−3 0.0152 0.0304 0.0609

K fsf 4.4441 × 10−3 8.8881 × 10−3 0.0178 0.0356

K fsof 4.7119 × 10−3 9.4238 × 10−3 0.0188 0.0377

−30.41%

−59.31%

−56.91%

TABLE 2.6 RMS Values of Sprung Mass Acceleration with Delay d ( t ) = 5 ms Roughness

Suspension Active Suspension

16 × 10−6 m3 64 × 10−6 m3 256 × 10−6 m3 1024 × 10−6 m3 Reduced rate

Passive Suspension 0.0109 0.0219 0.0437 0.0875 –

K c− fsf 7.8598 × 10−3 0.0157

K fsf 5.8302 × 10−3

0.0314 0.0629

0.0233 0.0466

0.0239 0.0477

−28.12%

−46.63%

−45.44%

0.0117

K fsof 5.9636 × 10−3

0.0119

47

Active Suspension Control

2.3  CONTROL WITH TIME DELAY 2.3.1 Input Delay The two-degree-of-freedom quarter-car model presented in Section 2.2.1 is considered for active controller design problems with time delay. The governing equation of motion for the sprung and unsprung mass can be seen in equations (3.1) and (3.2). By selecting the same state variables with Section 2.2.1, the state-space equation can also be obtained. In this chapter, three performance aspects of the quarter-car suspension system are taken into account, which has been introduced in Section 2.2. Ride comfort can be quantified by the sprung mass acceleration, therefore, the sprung mass acceleration is chosen as the first control output, where the sprung mass acceleration can be derived from equation (2.1). To design an active suspension to perform adequately in a wide range of shock and vibration environments, the H ∞ norm is chosen as the performance measure since H ∞ norm of a linear time-invariant (LTI) system is equal to the energy-to-energy gain and its value actually gives an upper bound on the root-mean-square (rms) gain. Hence, our goal is to minimize the H ∞ norm of the transfer function Tz1w from the disturbance w to the control output z1 (can be seen in equation (2.4)) to improve the ride comfort performance. The suspension travel space does not need to be minimal but its peak value needs to be constrained. Since an H ∞ norm of a mathematical function in the time-domain actually defines the peak value of the function, i.e., z∞ = supt ∈[0,∞ ) z T z (2.99)

it is able to optimize the H ∞ norm of the suspension deflection output under the energy-bounded road disturbance, that is, ∞

w2 =

∫w wdt < ∞ (2.100) T

0

i.e., w ∈ L2 [ 0, ∞ ), to realize the hard requirement for the suspension deflection. This is generalized H 2 ( GH 2 ) or energy-to-peak optimization problem [22]. The dynamic tyre is also a peak value optimization problem which can be dealt with the same way as the suspension deflection. Hence, we define the hard constrains on the suspension deflection and the tyre load as the second control output, i.e.,

 ( zs − zu ) / zmax z2 =   kt ( zu − zr ) / ( 9.8 ( ms + mu )) 

  (2.101)  

In summary, the vehicle suspension control system is described as

      

x = Ax + B1w + B2u z1 = C1 x + D12u (2.102) z 2 = C2 x y = Cx

where y ∈ R n is the measurement output, n(1 ≤ n ≤ 4 ) is the number of measurement variables, C1 ∈ R1× 4 , C2 ∈ R 2× 4 , D12 ∈ R1 , and  1 / zmax c c  1  k , C2 =  C1 =  − s  0  s   s  , D12 = ms ms  ms  ms  0

0 kt / 9.8(ms + mu )

0 0

0  . 0 

48

Vibration Control of Vehicle Suspension Systems

Matrix C ∈ R n × 4 is defined in terms of the available measurements. For state feedback controller design, C = I . When changes in vehicle inertial properties and actuator time-delays are considered in (2.39), the vehicle model becomes an uncertain model with input delay and this model can be expressed as a parameter-dependent model given by         

x = A (α ) x + B1 (α ) w + B2 (α ) u ( t − τ ) z1 = C1 (α ) x + D12 (α ) u ( t − τ ) z2 = C2 (α ) x y = Cx

(2.103)

x ( t ) = ∅ ( t ) , ∀t ∈[ −τ ,0 ]

where τ is the actuator time delay satisfying 0 < τ ≤ τ , where τ is the delay bound, ∅ ( t ) is the initial condition. Matrices A (α ) , B1 (α ) , B2 (α ), C1 (α ), D12 (α ), and C2 (α ) are functions of α which is the uncertain parameter vector. Assume matrices A (α ), B1 (α ) , B2 (α ), C1 (α ), D12 (α ), and C2 (α ) are constrained within the polytope ℵ given by     ℵ=     

  N  ( A, B1 , B2 , C1 , C2 , D12 )(α ) = α i ( A, B1 , B2 , C1 , C2 , D12 )i ,   (2.104) i =1  N  α i = 1, α i ≥ 0, i = 1,…, N  i =1 

( A, B1 , B2 , C1 , C2 , D12 )(α ) :

It is clear from equation (2.104) that the knowledge of the value of α i defines a precisely known system (2.103) inside the polytope ℵ described by the convex combination of its N vertices. Throughout this chapter, the vertices of the polytope ℵ are denoted as Ai, B1i , B2i, C1i , C2i , D12i, i = 1,…., N . In practice, based on the developed identification method, the inertial parameters could be measured or estimated online so that the value of α i can be found. In this chapter, the aim of the robust active suspension design is to find a parameter-dependent control law:  u = K (α ) Cx =  

1× n

where K i ∈ R

 α i K i  Cx ,  i =1 N

N

∑α = 1, i

α i ≥ 0, i = 1,…, N (2.105)

i =1

is the control gain matrix to be designed, such that the closed-loop system given by       

x = A (α ) x + B1 (α ) w + B2 (α ) K (α ) Cx ( t − τ ) z1 = C1 (α ) x + D12 (α ) K (α ) Cx ( t − τ ) z2 = C2 (α ) x

(2.106)

x = ∅ ( t ) , ∀t ∈[ −τ ,0 ]

has the following properties: (i) the closed-loop system (2.106) is asymptotically stable; (ii) the performance Tz1w ∞ < γ 1 is minimized subject to z2 ∞ < γ 2 w2 for all non-zero w ∈ L2 [ 0, ∞ ) and the

49

Active Suspension Control

prescribed constant γ 2 > 0 , where Tz1w denotes the closed-loop transfer function from the road disturbance w to the control output z1. The sufficient conditions for the robust asymptotically stable of closed-loop system (2.106) and performance requirements can be derived as follows. Define a parameter-dependent Lyapunov-Krasinski functional candidate as 0 t

V ( t ,α ) = x T P (α ) x +

∫∫

˙

x T (θ )Q (α ) x (θ ) dθ ds (2.107)

−τ t + s

where Pi ∈ R 4 × 4, Pi = PiT > 0, Qi ∈ R 4 × 4 , Qi = QiT > 0 are matrices to be determined. Then, the derivative of V ( t,α ) along the solution of system is given by ˙

V ( t,α ) = x T P (α ) x + x T P (α ) x + τ x T Q (α ) x −

t

˙

˙

∫ ()

()

x θ Q (α ) x θ dθ

t −τ

t

≤ x P (α ) x + x P (α ) x + τ x Q (α ) x − T

T

T

˙

˙

∫ ()

()

x θ Q (α ) x θ dθ (2.108)

t −τ

1 = τ

t

∫ ∅(t,θ ) dθ

t −τ

˙

˙

where ∅ ( t ,θ ) = x T P (α ) x + x T P (α ) x + τ x T Q (α ) x − x (θ ) Q (α ) x (θ ). By the Newton-Leibniz formula, we have t

x (t ) − x (t − τ ) =

T

˙

x (θ ) dθ (2.109)

t −τ

Then, for any matrices N

R (α ) =

N

α i Ri , S (α ) =

i =1

α i Si , T (α ) =

i =1

N

∑α T (2.110) i i

i =1

where Ri ∈ R 4 × 4 ,  Si ∈ R 4 × 4 , and Ti ∈ R 4 × 4, we have

2 τ

t

t −τ

 x T ( t ) R (α ) + x T ( t − τ ) S (α ) + x T T (α )  ˙    x ( t ) − x ( t − τ ) − τ x (θ ) dθ = 0

(2.111)

Moreover, according to equation (2.106), for any matrices U ∈ R 4 × 4 , V ∈ R 4 × 4, and W ∈ R 4 × 4 , we have

2 τ

t

t −τ

 x T ( t )U + x T ( t − τ )V + x T W 

 x (˙ t )− A (α ) x ( t ) − B (α ) w ( t ) − B (α ) K (α ) Cx ( t − τ )  dθ = 0 (2.112) 1 2  

50

Vibration Control of Vehicle Suspension Systems

According to equations (2.111) and (2.112) to equation (2.108) yields 1 V ( t , α ) ≤ τ

t

∫η

( t ,θ ) Π (α ) η ( t ,θ ) dθ (2.113)

T

t −τ

where η T ( t ,θ ) =  x T ( t )  x T ( t − τ )  x T   wT  and    II (α ) =     

II11

II12

II13

−τ R (α )

*

II 22

II 23

−τ S (α )

*

*

II 33

−τ T (α )

* *

* *

* *

−τ Q (α ) *

−UB1 (α )   −VB1 (α )  −WB1 (α )  (2.114)  0  0 

where           

II11 = R (α ) + R (α )T − UA (α ) − AT (α )U T II12 = − R (α ) + S T (α ) − UB2 (α ) K (α ) C − AT (α ) V T II13 = P (α ) + U + T T (α ) − AT (α ) W T II 22 = − S (α ) − S T (α ) − VB2 (α ) K (α ) C − C T K T (α ) B2T (α ) V T

(2.115)

II 23 = V − T T (α ) − C T K T (α ) B2T (α ) W T II 33 = τ Q (α ) + W + W T

and the asterisk symbol (*) represents a term that is induced by symmetry. For example, the last row of equation (2.114) is induced as  − B1T (α )U T   − B1T (α )V T   − B1T (α ) W T  0 0  . It is noted from equation (2.22) that if II (α ) < 0 , we have V ( t,α ) < 0, then system with w ( t ) = 0, parameter uncertainty, and time delay τ satisfying 0 < τ ≤ τ is robust asymptotically stable for all uncertain parameter α . Next, we will establish the H ∞ performance of the uncertain delay system under zero initial condition, that is, ∅ ( t ) = 0, ∀t ∈[ −τ ,0 ], and V ( t ,α )t = 0 = 0 . Consider the following index: ∞

J1 =

∫  z z − γ T 1 1

2 1

w T w  (2.116)

0

Then, for any non-zero w ∈ L2 [ 0, ∞ ), there holds, ∞

J1 ≤ [ z1T z1 − γ 12 w T w]dt + V ( t ,α )t = ∞ − V ( t ,α )t = 0

0

J1 ≤

 z1T z1 − γ 12 w T w  dt + V ( t ,α )t = ∞ − V ( t ,α )t = 0 =

0

∫  z z − γ T 1 1

2 1

w T w + V (t ,α )  (2.117)

0

After some algebraic manipulations, we obtain

z1T z1 − γ 12 w T w + V ( t ,α ) ≤

1 τ

t

∫ η (t,θ ) II (α )η (t,θ ) dθ (2.118) T

t −τ

51

Active Suspension Control

where    _  II (α ) =     

II13 −τ R (α ) −UB1 (α )   T II 22 + ( D12 (α ) K (α ) C ) ( D12 (α ) K (α ) C ) II 23 −τ S (α ) −VB1 (α )   * II 33 −τ T (α ) −WB1 (α )   * * −τ Q (α ) 0   * * * −γ 12 

II11 + C1T (α ) C1 (α ) II12 + C1T (α ) D12 (α ) K (α ) C * * * *

(2.119) Then, if II (α ) < 0, we have z1T z1 − γ 12 w T w + V ( t ,α ) < 0, therefore J1 < 0,  and hence,  z1  2 < γ 1  w  2 is satisfied for any non-zero w ∈ L2 [ 0, ∞ ).  Applying the Schur complement, II (α ) < 0 is equivalent to     II (α ) =     

II11

II12

II13

− R (α )

−UB1 (α )

C1T (α )

*

II 22

II 23

− S (α )

−VB1 (α )

( D12 (α ) K (α ) C )T

*

*

II 33

−T (α )

−WB1 (α )

0

*

−τ Q (α )

0

0

* *

−γ *

* * *

* * *

−1

* *

2 1

0 −I

     0, Qi > 0, Ri , Si, Ti , K i, i = 1,…, N , matrices U, V , W and scalars γ 1 > 0 , γ 2 > 0 such that matrix inequalities are satisfied simultaneously, then, closed-loop system is asymptotically stable with the performance Tz1W ∞ < γ 1 and z2 ∞ < γ 2 w 2 . When K i is unknown, matrix inequalities (2.124) and (2.125) are non-convex, and cannot be resolved using convex optimization algorithm. However, for the state feedback control case, i.e., C = I , two possible approaches may be used to design the controller. One possible method is to try to use some matrix inequalities to transform the non-convex optimization problem to the convex optimization problem [23]. Another possible method is to use the CCL method to solve the controller design problem as done in Ref. [24]. Nevertheless, for the present non-convex optimization problem, finding an appropriate matrix bounding inequality and designing a static output feedback controller are not straightforward using the two methods mentioned earlier. On the other hand, for this kind of problem, GA is found to be very effective [25]. Hence, in this chapter, when we assume τ is given, we will use GA to solve the problem of

minKi γ 1 subject to LMIs ( 2.124 ) , ( 2.125) , ( 2.126 ) , and ( 2.129 ) (2.130)

where K i  is initially randomly generated by GA and then evolved in terms of the objective presented in equation (2.130). For a known K i, matrix inequalities (2.124), (2.125), (2.126), and (2.129) are LMIs and can be efficiently resolved using MATLAB LMI toolbox. The flow diagram of the proposed GA/LMI algorithm for problem (2.130) is outlined in Figure 2.15. In Figure 2.15, parameter encoding is used to convert the feedback gain matrix K i, i = 1,…,  N into a row vector with binary-coded method. Population initialization is used to randomly generate an initial population of N p chromosomes for K i within a given search space. Much attention is paid to the objective evaluation step. In this step, the initial population produced in previous step is decoded into real value for every controller gain matrix K ij , j = 1, 2, …, N p; If problem (2.130) with K ij  is feasible, then determine the minimal γ 1 j by solving LMIs (2.124), (2.125), (2.126), and (2.129), and take every γ 1 j as the objective value corresponding to K ij and associate every K ij with a suitable fitness value according to rank-based fitness assignment approach, and then go to next step. If problem (2.130) with K ij is infeasible, the objective value corresponding to K ij will be assigned a large value in order to reduce its opportunity to be survived in the next generation. Since for a feasibility problem defined by LMI constraint of the form L ( x ) < R ( x ), where x is a feasible value of the vector of decision variables, a feasible solution can be found by using the auxiliary convex program, i.e., minimize β subject to L ( x ) < R ( x ) + β I , the solution to the LMI L ( x ) < R ( x ) is feasible if and only if the global minimum of β is negative, and the value of β can indicate the closeness of the decision variable to the solution. Therefore, a large value will be associated to the value of β so that more potential K ij can be evolved to find the solution. Tournament selection, uniform crossover, bit

54

Vibration Control of Vehicle Suspension Systems

FIGURE 2.15  Flow diagram of the proposed GA/LMI algorithm.

mutation, and elitist reinsertion are standard evolutionary operators, which can be found in most references about GA. When the evolution process repeats for N g generations, the algorithm stops, and the best chromosome is decoded into realvalues to produce again the control gain matrix K i. N

N

At last, the obtained parameter-dependent control gain is given as K (α ) = α i K i, α i = 1, α i ≥ 0, i = 1,…,  N . i =1 i =1 It is noted that GA is an evolutionary algorithm with built-in randomness. It cannot guarantee to find the optimal results every time. Hence, running the same algorithm for many times to find the possible optimal results is often necessary. In addition, if the presented GA/LMI algorithm cannot obtain a solution to a given problem after running several times, some parameters used in the

55

Active Suspension Control

TABLE 2.7 Parameter Values Used in a Quarter-Car Suspension Model Parameter Value

ms

ks

cs

kt

ct

mu

320 kg

18 kN/m

1 kNs/m

200 kN/m

0

40 kg

algorithm, such as the population size, the controller parameter search space, the time delay bound, and the uncertain parameter variation range, should be modified. However, there is no guarantee that a suitable solution can be located if the feasible solution set is small. Now, we will apply the proposed approach to design the parameter-dependent controller for a quarter-car suspension model shown in Figure 2.1. The parameters of the quarter-car suspension model selected for this study are listed in Table 2.7 and the maximum suspension deflection is defined as zmax = 0.08m. In this example, only vehicle sprung mass ms is assumed to be varying due to vehicle load variation and ms can fluctuate around its nominal value by 20%. The relationship between the uncertain parameter vector α = [α 1 ,α 2 ] and the estimation of sprung mass ms is given as

1 1 1 1 − − msmin ms ms msmax , α2 = α1 = (2.131) 1 1 1 1 − − msmin msmax msmin msmax

where msmax and msmin denote the maximum and the minimum sprung mass allowable, respectively. It can be seen from (2.131) that α 1 + α 2 = 1 and α 1 ≥ 0, α 2 ≥ 0 . Using the above-defined vector α , the parameter-dependent model can be defined. The actuator time delay bound τ = 50 ms is assumed. The basic GA parameters used in this chapter are given as the population size N g = 300. The controller parameter search space is defined as [−10 4  10 4] and the algorithm is independently run 50 times for each case. In order to validate the designed vehicle suspension performance in time domain, examinations of the response quantities will be done to evaluate the suspension characteristics taking into account the shock and vibration road profiles. In this section, two kinds of controller design problems will be studied. In this case, we assume that all the state variables for a quarter-car suspension model are measurement available, we can design a full state feedback parameter-dependent controller by solving the problem (2.130). The obtained controller gain are given as

K1 = 10 3 × [ −2.8529 2.8372 − 2.6676 − 0.0435], K 2 = 10 3 × [ −2.9608 2.8117 − 2.4160 − 0.0469 ].

The performance of active suspension with the designed controller is evaluated through the following simulations. Firstly, an isolated bump in an otherwise smooth road surface is used. The corresponding ground displacement for the wheel is given by

 a  2πv0     1 − cos  l t   ,  2 zr ( t ) =   0,  

0≤t ≤ l t> v0

l v0

(2.132)

56

Vibration Control of Vehicle Suspension Systems

where a and L are the height and the length of the bump. We choose a = 0.1m, l = 10 m, and the vehicle forward velocity as v0 = 45 km/h. When actuator time delay is zero, i.e., τ = 0 ms, the bump responses of the passive suspension and the active suspension are compared in Figure 2.16, where bump responses of the sprung mass acceleration, suspension deflection, tyre deflection, and active force are plotted. For clarity, only the nominal case where the sprung mass is used as its nominal value, 320 kg, and the two-vertex cases where the sprung mass is used as its maximum value, 384 kg, and its minimum value, 256 kg, respectively, are plotted. It can be seen from Figure 2.16 that all the responses of the sprung mass acceleration for active suspension are lower than those of the passive suspension no matter what value the sprung mass is. Compared to the passive suspension, the suspension deflection and the tyre deflection of active suspension are all guaranteed to be less than their hard limits as those in spite of the large bump energy. It can be seen from Figure 2.17 that the responses of the sprung mass acceleration, the suspension deflection, the tyre deflection, and the active force of active suspension are all similar to those shown in Figure 2.16 in spite of the presence of time delay. It is validated that the designed robust active suspension can achieve good suspension performance regardless of the sprung mass variation and the actuator time delay within given bounds. To clearly show the results, the maximum peak values of the bump responses for sprung mass acceleration, suspension deflection, tyre deflection, and active force are listed in Table 2.8. From Table 2.8, it can be seen that even when the actuator time delay presents, the maximum peak values realized by active suspension are similar to those without time delay cases. Secondly, when the road disturbance is considered as random vibration, it

FIGURE 2.16  Bump response for full state feedback control case with time delay τ = 0 ms. The legends shown in upper-left and lower-right plots are used for all plots in this figure.

57

Active Suspension Control

FIGURE 2.17  Bump response for full state feedback control case with time delay τ = 50 ms. The legends shown in upper-left and lower-left plots are used for all plots in this figure.

TABLE 2.8 Comparison of Maximum Peak Value for Bump Response (Full State Feedback Control Case) Active

τ = 0 ms

Passive ms (kg) x3max (m/s2) x1max (m) x2max (m) umax (N)

256 5.2255 0.0681 0.0070 –

320 4.8756 0.0799 0.0080 –

384 4.4959 0.0889 0.0087 –

256 2.5649 0.0481 0.0036 687.05

320 2.4700 0.0533 0.0042 706.18

τ = 50 ms 384 2.3771 0.0578 0.0048 708.42

256 2.6043 0.0396 0.0038 705.94

320 2.5978 0.0446 0.0046 742.23

384 2.5570 0.0502 0.0052 766.46

is consistent and typically specified as random process with a ground displacement power spectral density (PSD) of

− n1   Sg (ϕ 0 )  ϕ  ,  ϕ   0 Sg (ϕ ) =  − n2 ϕ  S ϕ ( )  g 0  ϕ  , 0 

if   ϕ ≤ ϕ 0 (2.133) if   ϕ > ϕ 0

58

Vibration Control of Vehicle Suspension Systems

where ϕ 0 = 1 / 2π is a reference spatial frequency, ϕ is a spatial frequency. The value Sg (ϕ 0 ) provides a measure for the roughness of the road. n1 and n2 are road roughness constants. The ISO has proposed road roughness classification using the PSD values as listed in Table 2.9. The random road profile can be generated according to Chapter 1. Taking into account the random nature of the excitation applied, the root mean square (rms) for the random responses of sprung mass acceleration, suspension deflection, and tyre deflection will be evaluated. For each case, 50 random tests are used to compute the expected rms values. To show the results clearly, we only plot the rms ratio between the active suspension and the passive suspension for sprung mass acceleration in Figure 2.18a, where the sprung mass is given as the nominal value and the two-vertex cases for one type of road roughness (C, average) and one selected vehicle forward velocity 72 km/h. It can be seen from Figure 2.18a that in spite of the variation of sprung mass, the sprung mass acceleration of active suspension is always less than the passive suspension (the response ratio is less than 1) within time delay τ ≤ 50 ms. It shows that the designed active suspension could achieve good ride comfort performance within the indicated delay bound in spite of the variation of sprung mass within allowable range. Figure 2.18b shows the PSD of sprung mass acceleration for the active suspension and the passive suspension. To show the curves clearly, only those cases that the sprung mass is given as the nominal value and the two-vertex cases are plotted, respectively. The road roughness (C, average), the vehicle forward velocity 72 km/h, and the actuator time delay τ = 10 ms are used. It is clearly seen from Figure 2.18b that the active suspension achieves a significant improvement on ride comfort regardless of the sprung mass variation and the actuator time delay. In practice, not all the state variables are measurements available for control. Therefore, a static output feedback parameter-dependent controller, which uses easily available measurements, needs

(

Degree of Roughness Sg ( ϕ 0 ) 10 −6 m3 Road class Range Geometric mean

B (Good) 8–32 16

) C (Average) 32–128 64

D (Poor) 128–512 256

FIGURE 2.18  (a) RMS ratio between active suspension and passive suspension for sprung mass acceleration versus actuator time delay (full state feedback control case); (b) PSD of sprung mass acceleration under a random road profile (C, average) at vehicle speed 72 km/h with time delay τ = 10 ms .

Active Suspension Control

59

to be designed. Since the suspension deflection x1 ( t ) can be measured using suitable displacement transducer, and the sprung mass acceleration can be straightforwardly measured using accelerometer, and in principle, the sprung mass velocity x3 ( t ) can be obtained by integrating the sprung mass acceleration signal accordingly, we prefer to using suspension deflection and sprung mass velocity as feedback signals to design such a static output feedback controller. Using the same algorithm as presented in the last section with an appropriate C matrix given by  1 0 0 0  C=   0 0 1 0  the static output feedback parameter-dependent controller gains are obtained as

K1 = 10 3 × [ −4.9709 − 2.8662 ], K 2 = 10 3 × [ −4.9356 − 2.3330 ]

The performance evaluation of the designed controller is done similarly to that for full state feedback control case. The bump responses are plotted in Figures 2.19 and 2.20, for time delay as 0 and 50 ms, respectively. The maximum peak response values are given in Table 2.10. For designed active suspension, it can be seen that the maximum peak values are similar for both τ = 0 and 50 ms cases. The RMS ratio between the active suspension and the passive suspension for sprung mass acceleration for one type of road roughness (C, average) and one selected vehicle forward velocity 72 km/h is plotted in Figure 2.21a, where the sprung mass is given as the nominal value and the two-vertex cases, respectively. Similarly, it can be seen that the sprung mass acceleration of active suspension

FIGURE 2.19  Bump response for static output feedback control case with time delay τ = 0 ms .

60

Vibration Control of Vehicle Suspension Systems

FIGURE 2.20  Bump response for static output feedback control case with time delay τ = 50 ms.

TABLE 2.10 Comparison of Maximum Peak Value for Bump Response (Static Output Feedback Control Case) Active Passive ms (kg) x3max (m/s2) x1max (m) x2max (m) umax (N)

256 5.2255 0.0681 0.0070 –

320 4.8756 0.0799 0.0080 –

384 4.4959 0.0889 0.0087 –

256 2.7887 0.0465 0.0040 650.87

τ = 0 ms 320 2.6073 0.0518 0.0045 694.34

384 2.4723 0.0561 0.0050 712.87

256 2.8675 0.038 0.0042 669.88

τ = 50 ms 320 2.7631 0.0438 0.0048 738.29

384 2.6776 0.0489 0.0055 780.7575

is less than the passive suspension within time delay τ ≤50 ms. Similarly, the PSD of sprung mass acceleration for the active suspension and the passive suspension is shown in Figure 2.21b, where the cases that the sprung mass is given as the nominal value and the two-vertex cases are plotted for clarity, and the road roughness (C, average), the vehicle forward velocity 72 km/h, and the actuator time delay τ =10 ms are used. It can be seen that a significant improvement on ride comfort is achieved by the active suspension regardless of the sprung mass variation and the actuator time delay. The simulation results verify that, using the same algorithm, the designed static output feedback controller can realize the similar performance to that of full state feedback controller when the sprung mass has uncertainty and the actuator time delay is present.

61

Active Suspension Control

FIGURE 2.21  (a) RMS ratio between active suspension and passive suspension for sprung mass acceleration versus actuator time delay (static output feedback control case); (b) PSD of sprung mass acceleration under a random road profile (C, average) at vehicle speed 72 km/h with time delay τ = 10 ms .

This section presents a parameter-dependent controller design approach for vehicle suspension with considerations on changes in vehicle inertial parameters and the existence of actuator time-delays. To reduce the conservativeness of the presented controller design conditions, parameter-dependent Lyapunov functional is used. Based on the identification technique recently developed for accurate estimations of inertial parameters, the presented parameter-dependent controller could be implemented in practice. Furthermore, the controller that only uses the easily available measurements, such as sprung mass velocity and suspension deflection, is designed in the same way. The designed controllers are applied to a quarter-car suspension model with a large change in sprung mass and large time delay in input. Numerical simulations have validated that the vehicle suspension performance is improved with the designed controller in spite of the sprung mass variation and actuator time delay.

2.3.2 Time Delay Using T-S Fuzzy Approach 2.3.2.1  System Modelling and Problem Description A typical quarter-automobile active suspension system model with two degrees of freedom is used to describe the vehicle dynamics, as shown in Figure 2.22 [26]. The notations of the model are listed in Table 2.11, and the dynamic equations of the system are obtained as

Define

 ms ( t )  zs ( t ) = − cs [ zs ( t ) − zu ( t )] − ks [ zs ( t ) − zu ( t )] + u ( t − τ )   zu ( t ) = cs [ zs ( t ) − zu ( t )] + ks [ zs ( t ) − zu ( t )] (2.134) mu ( t )    − cu [ zu ( t ) − zr ( t )] − ku [ zu ( t ) − zr ( t )] − u ( t − τ ) x1 ( t ) = zs ( t ) − zu ( t ) , x 2 ( t ) = zs ( t ) , x3 ( t ) = zu ( t ) − zr ( t ) , x 4 ( t ) = zu ( t )

z1 ( t ) =  zs ( t ) ,  z ( t ) − zu ( t ) k u ( zu ( t ) − zr ( t ) )  z2 ( t ) =  s zmax ( ms (t ) + mu (t )) g  

(2.135) T

62

Vibration Control of Vehicle Suspension Systems

FIGURE 2.22  Quarter-automobile active suspension system model.

TABLE 2.11 Notations of the Automotive Active Suspension System Model Parameters ms(t) mu(t) zs(t) zu(t) zr(t) cs τ ks ku u(t) cu

Physical Meaning (Units) Sprung mass (kg) Unsprung mass (kg) Displacements of sprung mass (m) Displacements of unsprung mass (m) Displacement inputs of road (m) Damping coefficients of suspension (N·s/m) Actuator delays (seconds) Stiffness of suspension (N/m) Stiffness of tyre (N/m) Actuator force (N) Damping coefficients of the pneumatic tyre (N·s/m)

where x1 ( t ) is the suspension deflection, x 2 ( t ) is the absolute vertical velocity of sprung mass, x3 ( t ) represents the tyre deflection, x 4 ( t ) represents the absolute vertical velocity of the unsprung mass. z max denotes the maximum suspension deflection and g represents the gravitational acceleration. T Then, by defining the state variable x ( t ) =  x1 ( t ) x 2 ( t ) x3 ( t ) x 4 ( t )  and ω ( t ) = zr ( t ),   the suspension system (2.134) can be rewritten as x ( t ) = A ( t ) x ( t ) + Bu ( t ) u ( t − τ ) + B ( t )ω ( t )

z1 ( t ) = C1 ( t ) x ( t ) + D1 ( t ) u ( t − τ ) z 2 ( t ) = C2 ( t ) x ( t ) x ( t ) = φ ( t ) , ∀t ∈[ −τ ,0 ]

(2.136)

63

Active Suspension Control

where φ ( t ) is an initial continuous function, and the time delay τ satisfies 0 ≤ τ ≤ τ . And the system matrices are given as  0   − ks ms ( t )  A (t ) =  0  ks   mu ( t ) 

1 c − s ms ( t ) 0 cs mu ( t )

−1 cs ms ( t ) 1 cs + cu − mu ( t )

0 0 0 k − u mu ( t )

  0   1     ms ( t ) , Bu ( t ) =  0   1     − mu ( t )  

 1  z max  − ks cs cs  1 − 0 , D1 = , C2 ( t ) =  C1 ( t ) =    ms ( t )  ms ( t ) ms ( t ) m s ( t )   0 

   0   0   −1 , B ( t ) =    cu   m u (t )   

0

0

0

ku ( ms (t ) + mu (t )) g

   ,   

 0  .  0  

The sprung mass and unsprung mass of a vehicle could be uncertain values on account of the changes in the cargo load or the passenger numbers and tyre wear, etc. Hence, the suspension system of the vehicle in (2.134) is regarded as an uncertain model, i.e., both the sprung mass ms ( t ) and the unsprung mass mu ( t ) are changing, and their values should be within a given range, i.e., ms ( t ) ∈[ msmin , msmax ] and mu ( t ) ∈[ mumin , mumax ]. Then we have 1 1 1 = , mˇ s = min ms ( t ) msmin ms ( t ) msmax 1 1 1 1 = = mˆ u = max , mˇ u = min mu ( t ) mumin mu ( t ) mumax mˆ s = max

1

=

(2.137)

According to the sector non-linear method [27], 1/ms (t ) and 1/mu ( t ) can be given as 1

= M1 (ξ1 ( t )) mˆ s + M2 (ξ1 ( t )) mˇ s ms ( t ) 1 = N1 (ξ 2 ( t )) mˆ u + N 2 (ξ 2 ( t )) mˇ u mu ( t )

(2.138)

where ξ1 ( t ) = 1/ms ( t ) and ξ 2 ( t ) = 1/mu ( t ) are the premise variables, and 1 = M1 (ξ1 ( t )) + M2 (ξ1 ( t ))

1 = N 1 (ξ 2 ( t )) + N 2 (ξ 2 ( t ))

(2.139)

To establish the fuzzy rules model, the membership functions can be described as 1

M1 (ξ1 ( t )) =

ms ( t )

mˆ s − mˇ s 1

N 1 (ξ 2 ( t )) =

− mˇ s

mu ( t )

− mu

, M2 (ξ1 ( t )) =

ˇ

mˆ u − m u ˇ

, N 2 (ξ 2 ( t )) =

mˆ s −

1

ms ( t )

mˆ s − mˇ s mˆ u −

1

mu ( t )

mˆ u − mˇ u

(2.140)

64

Vibration Control of Vehicle Suspension Systems

TABLE 2.12 Fuzzy Weighting Functions and Fuzzy Rules Rule Numbers

ξ1 ( t )

ξ2 ( t )

1

Heavy

Heavy

2

Heavy

Light

3

Light

Heavy

4

Light

Light

Fuzzy Weighting Functions k1 (ξ ( t )) = M1 (ξ1 ( t )) × N1 (ξ 2 ( t ))

k2 (ξ ( t )) = M1 (ξ1 ( t )) × N 2 (ξ 2 ( t ))

k3 (ξ ( t )) = M 2 (ξ1 ( t )) × N1 (ξ 2 ( t ))

k4 (ξ ( t )) = M 2 (ξ1 ( t )) × N 2 (ξ 2 ( t ))

The membership functions 1 (ξ1 ( t )) and 1 (ξ 2 ( t )) represent “Heavy”, and 2 (ξ1 ( t )),  2 (ξ 2 ( t )) represent “Light”. Then, the fuzzy weighting functions and fuzzy rules are listed in Table 2.12. Fuzzy weighting functions ki (ξ ( t )) conform to

4

∑k (ξ (t )) = 1, and k (ξ (t )) ≥ 0,(i = 1,…,4 ). For i

i

i =1

brevity, we define ki = ki (ξ ( t )) in the following analysis. Then, based on fuzzy blending, the whole fuzzy model can be described as follows 4

x ( t ) =

∑k [ A x (t ) + B u (t − τ ) + B ω (t )] i

i

ui

i

i =1 4

z1 ( t ) =

∑k [C x (t ) + D u (t − τ )] 1i

i

1i

(2.141)

i =1 4

z2 ( t ) =

∑k C i

2i

x (t )

i =1

x ( t ) = φ ( t ) , t ∈[ −τ ,0 ] where Ai , Bui , Bi , C1i , D1i , C2i can be obtained by replacing 1 / ms ( t ) and 1 / mu ( t ) with mˆ s (or m s ) and mˆ u (or m u) in matrices A ( t ) ,  B ( t ) ,  Bu ( t ) , C1 ( t ) ,  D1 ( t ) and C2 ( t ), respectively. Next, to simplify the writing, the system in (2.141) can be described as x ( t ) = Ak x ( t ) + Buk u ( t − τ ) + Bkω ( t ) z1 ( t ) = C1k x ( t ) + D1k u ( t − τ )

z 2 ( t ) = C2 k x ( t )

(2.142)

x ( t ) = φ ( t ) , t ∈[ −τ ,0 ] 4

where Ak =

ki Ai , Buk =

i =1

4

4

ki Bui , Bk =

i =1

ki Bi , C1k =

i =1

4

kiC1i , D1k =

i =1

4

ki D1i , C2 k =

i =1

4

∑k C . i

2i

i =1

Based on the system in (2.142) with the T-S fuzzy rules, a non-fragile fuzzy state feedback controller will be designed. Firstly, a non-fragile controller [28] without fuzzy rules is given as

u ( t − τ ) = ( K + ∆K ( t )) x ( t − τ ) (2.143)

65

Active Suspension Control

where ∆K ( t ) is the perturbation of controller gain and it is assumed to be a multiplication form between unknown time-varying continuous function F ( t ), real constant matrices H and E, and controller gain K as ∆K ( t ) = HF ( t ) EK (2.144)

where F ( t ) satisfies F T ( t ) F ( t ) ≤ I . Secondly, the fuzzy rules are described as Control Rule j: IF ξ1 ( t )  is Mθ (ξ1 ( t )) , and ξ 2 ( t )  is Nθ (ξ 2 ( t ))(θ = 1,2 ), THEN u j ( t − τ ) = ( K j + ∆K j ) x ( t − τ ) ,

j = 1,…,4 (2.145)

Further, considering the actuator delay, the overall non-fragile fuzzy controller can be rewritten as u ( t − τ ) = K k (1 + HF ( t ) E ) x ( t − τ ) (2.146)

4

where K k =

∑k K . j

j

j =1

Finally, based on (2.142) and (2.146), and considered with fuzzy rules, the overall closed-loop system with time delay and actuator uncertainties can be depicted as x ( t ) = Ak x ( t ) + Buk K k (1 + HF ( t ) E ) x ( t − τ ) + Bkω ( t )

z1 ( t ) = C1k x ( t ) + D1k K k (1 + HF ( t ) E ) x ( t − τ )

(2.147)

z 2 ( t ) = C2 k x ( t ) In designing a controller for active suspension systems, the first objective is to reduce body acceleration due to ground interference. Then, constraints on the performance of the suspension system, such as road stability and suspension deflection, need to be ensured. In addition, it is necessary to establish the transfer function from ground interference ω ( t ) to body output z1 ( t ) and optimize the parameter γ when the H ∞ control method is used. The transfer function for the closed-loop system in (2.147) from ω ( t ) to z1 ( t ) can be shown as

Gω z1 ( jϖ ) = (C1k + e −τ jϖ D1k K k )( jϖ I − Ak − e −τ jϖ Buk K k ) Bk (2.148) −1

And for the concerned frequency range, the frequency domain inequality (FDI in [29]) can be shown as

 σ ( Gω z1 ( jϖ ) , Π ) =   

( jϖ − A)

 Θ :=  C  0

I

−1

T

  B   Θ    

T

D  Π C   I   0

( jϖ − A) I D   I 

−1

 B  < 0, ∀ϖ ∈Ω  

(2.149)

 I  0 where A = Ak + e −τ jϖ Buk K k , B = Bk , C = C1k + e −τ jϖ D1k K k , D = 0, Π =  , 2  0 −γ I  Ω = {ϖ ∈ R | ϖ 1 ≤ ϖ ≤ ϖ 2 ,ϖ 1 ≤ ϖ 2 }. The H ∞ control problem for the aforementioned transfer function is to design a controller guaranteeing

66

Vibration Control of Vehicle Suspension Systems

sup  G ( jϖ )  ∞2 < γ 2 (2.150)

ϖ 1 0 such that  T   *  * 

µV − µI *

UT 0 − µI

   < 0.  

For the active suspension systems, a delay-dependent sufficient condition is established to satisfy the asymptotic stability performance in the finite frequency domain. Theorem 2.6  −1  jϖ c  1 0   0 1  Let matrices Φ =  ,Ψ =   ,Y =    and Π in (2.153) − − j ϖ ϖ ϖ 1 0 c 1 2      0 0  be given. A fuzzy controller is constructed in the form of (2.150) with ∆K j = 0. For a random

68

Vibration Control of Vehicle Suspension Systems

time delay 0 < τ ≤ τ , the closed-loop system in (2.151) is asymptotically stable and con2 2 forms to  G ( jϖ )  ∞ < γ for all non-zero ω ∈ L2 [ 0, ∞ ), if there exist symmetric matrices 4

4

P1k =

ki P1i > 0 , Q1k =

i =1 4

P2 k =

4

kiQ1i > 0, Z1k =

i =1 4

4

ki Z1i > 0 , Z 2 k =

i =1

4

ki Z 2i > 0, Q2 k =

i =1

∑k Q i

2i

> 0,

i =1

∑k P , and R = ∑k R  (i = 1,…,4) satisfying i 2i

k

i =1

i

i

i =1

FkT Ξ k Fk + Ω k < 0 (2.151)

FskT Ξsk Fsk + X skT ΠX sk + Ωsk < 0 (2.152) where  A Fk =  k  I

(

Buk K k 0

Bk 0

 , 

)

Ξ k = Φ ⊗ P1k + Y ⊗ τ 2 Z1k ,  Ξsk = ( Φ ⊗ P2 k + Ψ ⊗ Q2 k + Y ⊗ τ Z 2 k ) ,

 Ak   ,  Fsk =   I 

Buk K k 0

 Q1k − Z1k Ωk =  * 

 Rk − τ −1 Z 2 k  Z1k   ,  Ωsk =  τ −1 Z 2 k −Q1k − Z1k    0   C X sk =  1k  0

D1k K k 0

τ −1 Z 2 k − Rk − τ −1 Z 2 k 0

0   0 ,  0 

0  . I 

Proof. Firstly, the proof of asymptotic stability in (2.147) with ω ( t ) = 0 is given. A Lyapunov functional candidate is considered as V ( t ) = V1 ( t ) + V2 ( t ) + V3 ( t ) (2.153)

where

V1 ( t ) = xT ( t ) P1k x ( t ) , (2.154) t

V2 ( t ) =

∫x

T

(α ) Q1k x (α ) dα , (2.155)

t −τ 0

V3 ( t ) = τ

t

∫ ∫ x

T

(α ) Z1k x (α ) dα d β (2.156)

−τ t + β

where the matrices P1k = P1Tk > 0, Q1k = Q1Tk > 0 and Z1k = Z1Tk > 0 are to be defined. For taking the derivative of equations V1 ( t ), V2 ( t ) and V3 ( t ), we can obtain

V1 ( t ) = x T ( t ) P1k x ( t ) + x T ( t ) P1k x ( t ) (2.157)

69

Active Suspension Control

V2 ( t ) = x T ( t ) Q1k x ( t ) − x T ( t − τ ) Q1k x ( t − τ ) (2.158)

t

V3 ( t ) = τ 2 x T ( t ) Z1k x ( t ) − τ

∫ x

T

(α ) Z1k x (α ) dα (2.159)

t −τ

Based on Lemma 2.4, the second part in (2.159) can be scaled up to t

−τ

∫ x

T

(α ) Z1k x (α ) dα ≤ − [ x ( t ) − x ( t − τ )] Z1k [ x ( t ) − x ( t − τ )] (2.160) T

t −τ

Then, we have V ( t ) ≤ ζ T ( t ) Λ kζ ( t ) (2.161)

where

 x (t ) ζ (t ) =  x t −τ) ( 

 [ P1k Ak ] + Q1k + τ 2 AkT Z1k Ak − Z1k s Λk =   * 

 , 

P1k Buk K k + τ 2 AkT Z1k Buk K k + Z1k −Q1k + τ 2 ( Buk K k ) Z1k Buk K k − Z1k T

   

which is equivalent to inequality (2.151) after relevant transformation. If Λ k < 0 holds, the stability of the system in (2.147) is ensured. Secondly, the inequality (2.150) for H ∞ performance requirements is equivalent to the finite FDI (2.149), i.e., T

 δ   δ   Θ  < 0, ∀ϖ ∈Ω (2.162)   I   I 

where δ = ( jϖ I − Ak − e −τ jϖ Buk K k ) Bk and Θ can be found in (2.153). The inequality (2.162) can be rewritten as −1

T Θ1 < 0, ∀ϖ ∈Ω (2.163)

where  δ  −τ jw = e δ  I

  C1k   , Θ1 =  0  

D1k K k 0

0 I

T

  C1k  Π   0

D1k K k 0

0 I

 R   k + 0    0

0 − Rk 0

0 0 0

  .  

By defining a set as

{

}

W1 = ξ ∈C 2l + mξ = η ,η ∈C m ,η ≠ 0,ϖ ∈Ω (2.164)

70

Vibration Control of Vehicle Suspension Systems

Inequality (2.163) can be described as

ξ T Θ1ξ < 0, ∀ξ ∈W1 (2.165)

Besides, by defining Γ λ =  I l 

− λ I l  , λ = jϖ and  A Fsk =  k  I

Buk K k 0

Bk   , (2.166) 0 

{

we get Γ λ Fsk ξ = 0 for ∀ξ ∈W1. By defining a set as W2 = ξ ∈C 2l + m | ξ ≠ 0, Γ λ Fsk ξ = 0, λ ∈V from Lemma 2.6, which can be characterized by

{

W2 = ξ ∈C 2l + m | ξ ≠ 0, ξ T ξ ≥ 0, ∀ ∈ M

}

}

with M = { FskT ( Φ ⊗ P2 k + Ψ ⊗ Q2 k ) Fsk P2 k , Q2 k ∈ H n , Q2 k 0}

and W1 ⊂ W2 .

Next, we also define a set as

{

}

W = ξ ∈C 2l + mξ ≠ 0, ξ T Mξ ≥ 0 ∀M ∈ M , ξ T Nξ ≥ 0 (2.167)

and a matrix as

 −τ −1 Z 2 k   =  τ −1 Z 2 k   0

τ −1 Z 2 k −τ −1 Z 2 k 0

0   T 0  + Fsk (Y ⊗ τ Z 2 k ) Fsk (2.168)  0 

It is distinct that W ⊂ W2. Finally, the relationships between W and W1 should be shown. For ∀ξ ≠ 0 ∈W1 and any delay τ satisfying 0 < τ ≤ τ , we obtain

ξ T Nξ = ( Lη ) N ( Lη ) = ( −2τ −1 + 2τ −1cosτϖ + τϖ 2 ) (δη ) Z 2 k (δη ) T

T

τω τ 2 −τ2 2 T T =  −4τ −1sin 2 + τϖ 2 (δη ) Z 2 k (δη ) ≥ ϖ (δη ) Z 2 k (δη ) ≥ 0   2 τ2

(2.169)

Due to W1 ⊂ W2 , it follows that ξ ∈W . Therefore, we have W1 ⊂ W , which means W1 ⊂ W ⊂ W2 . Hence, ξ T Θ1ξ < 0, ∀ξ ∈W holds if ξ T Θ1ξ < 0 holds for ∀ξ ∈W1. M is a rank-one separable and admissible based on Lemma 2.6. From Lemma 2.7, ξ T Θ1ξ < 0, ∀ξ ∈W can be shown as Θ1 + M + τ N < 0 . By replacing τ Z 2 k with Z 2 k , Θ1 + M + N < 0 can be obtained, which is equivalent to inequality (2.152). The proof is accomplished.

71

Active Suspension Control

To satisfy the performance requirements in inequality (2.150), a sufficient condition is given in Theorem 2.6, which depends on the time delay’s upper bound. It cannot be ignored that a necessary and sufficient condition with specification (2.150) is equivalent to a restrictive inequality ξ T Θ1ξ < 0 with a constraint set ∀ξ ∈W1. However, it can not be converted to a convex unconstrained inequality directly. The constraint set is extended from W1 to W to solve this difficulty. Then, the second condition in Theorem 2.6 can be obtained, which is equivalent to ξ T Θ1ξ < 0, ∀ξ ∈W. Secondly, based on the inequalities (2.151) and (2.152) in Theorem 2.6, the projection theorem (Lemma 2.8) is used to associate the inequality (2.151) with (2.152) so that the delay-dependent system stability and the finite frequency domain conditions are satisfied simultaneously. Furthermore, the constraint factors of suspension deflection and road-holding stability are satisfied by transforming into matrix inequalities. Theorem 2.7  jϖ c  1 0  and Π in (2.153)  ,Y =  −ϖ 1ϖ 2  0 0    be given. A fuzzy controller is constructed in the form of (2.150) with ∆K j = 0. For a random time delay 0 < τ ≤ τ , the closed-loop system in (2.151) is asymptotically stable and conforms to  G ( jϖ )  ∞2 < γ 2 for all non-zero ω ∈ L2 [ 0, ∞ ), if there exist symmetric matrices  0 Let matrices Φ =   1

4

P1k =

 −1 1  ,Ψ =   0   − jϖ c

4

ki P1i > 0 , Q1k =

i =1

∑ i =1

4

P2 k =

∑k P , and R = ∑ i 2i

k

i =1

          

−Q2 k + τ Z 2 k P2 k − jϖ cQ2 k − O

4 i =1

4

ki Z 2i > 0, Q2 k =

i =1

∑k Q i

2i

> 0,

i =1

ki Ri (i = 1, 4) satisfying P1k − ε O + ε O T Ak

ε O T Buk K k

Q1k − Z1k + ε  AkT O  s *

ε O T Buk K k + Z1k

P2 k + jϖ cQ2 k − O T

4

ki Z1i > 0 , Z 2 k =

i =1

 τ 2 Z1k − ε [O ] s   *   * 

4

kiQ1i > 0, Z1k =

−ϖ 1ϖ 2Q2 k +  AkT O  s −1

+ Rk − τ Z 2 k

−Q1k − Z1k

   < 0 (2.170)   

0

0

0

O T Buk K k + τ −1Z 2 k

O T Bk

C1Tk

*

*

− Rk − τ −1Z 2 k

0

K Tj D1Tk

* *

* *

* *

−γ 2 I *

0 −I

 P1k    *   P1k    * 

      0 (2.172) γ2 I  2 ϑ 

C2Tk

K kT / umax    > 0 (2.173) γ2 I  2 ϑ 

72

Vibration Control of Vehicle Suspension Systems

Proof. Rewritting the inequality (2.151) as Γ QT1 I1T Ξ k I1Γ Q1 + Γ QTT Ω′k Γ Q1 < 0 (2.174)

1

where  I I1 =   0

 Ak 0  , Γ Q1 =  I 0   0

0 I

 0  , Ω =  0 ′k    0  

Buk K k 0 I

0 Q1k − Z1k Z1k

  .  

0 Z1k −Q1k − Z1k

And define matrices P1 =  I

I

0  , Γ P1 =  0

T

I  , Q1 =  − I

0

where P1Γ P1 = 0 and Q1Γ Q1 =  0

Ak

Buk K k  ,

0 . Then we have

Γ TPT I1T Ξ k I1Γ P1 + Γ TP1 Ω′k Γ P1 = −Q1k − Z1k < 0 (2.175)

1

Based on Lemma 2.8, if there is a general matrix ε O (ε is a scalar) which makes the following formula holding I1T Ξ k I1 + Ω′k + ε Q1T OP1  s < 0 (2.176)

it is equivalent to (2.170). Similarly, rewriting (2.152) as

Γ QT2 I 2T Ξsk I 2 Γ Q2 + Γ QT2 E2T ΠE2 Γ Q2 + Γ QT2 Ω′sk Γ Q2 < 0 (2.177)

where

 I I2 =   0

0 I

0 0

  0  , Γ Q2 =   0       Ω′sk =    

Ak I 0 0

Buk K k 0 I 0

    , E2 =  0   0  

Bk 0 0 I

0 0

0 Rk − τ −1 Z 2 k

0 τ −1 Z 2 k

0 0

τ −1 Z 2 k 0

− Rk − τ −1 Z 2 k 0

C1k 0

D1k K k 0

Ak

Buk K k

0 I

 , 

0   0  . 0   0 

Furthermore, three matrices of proper dimensions are defined as P2 =  0

I

0

0  , Γ P 2 =  I

where P2 Γ P 2 = 0 , Q2 Γ Q2 =  0

0

0

0

T

0  , Q2 =  − I

Bk  ,

0 . And the following inequality can be obtained

Γ TP2 I 2T Ξsk I 2 Γ P2 + Γ TP2 E2T ΠE2 Γ P2 + Γ TP2 Ω′sk Γ P2 = τ Z 2 k − Q2 k < 0 (2.178)

73

Active Suspension Control

Based on Lemma 2.8, if there is a general matrix O which makes the following LMI holding I 2T Ξsk I 2 + E2T ΠE2 + Ω′sk + Q2T OP2  s < 0 (2.179)

it becomes inequality (2.171) after using the Schur complement. Meanwhile, it also can guarantee τ Z 2 k − Q2 k < 0 . Finally, the dynamic deflection constraint and road-holding stability of the suspension system should be guaranteed. With the Lyapunov stability and H 2 performance satisfying (2.38) and (2.39), we can get z1T ( t ) z1 ( t ) − γ 2ω T ( t )ω ( t ) + V ( t ) < 0 (2.180)

which assures that V ( t ) < γ 2ω T ( t )ω ( t ). Then, by integrating from 0 to t > 0 on both sides of the inequality and omitting two positives of the Lyapunov function V ( t ), the following inequality can be obtained t

x T ( t ) P1k x ( t ) < γ 2 ω T ( s )ω ( s ) ds (2.181)

0

Assuming

 ϑ2  C2Tk C2 k <  2  P1k (2.182) γ 

t

x ( t ) P1k x ( t ) < γ T

2

∫ω

T

( s )ω ( s ) ds (2.183)

0

and we can obtain

 ϑ2  z ( t ) z2 ( t ) & = x ( t ) C C2 k x ( t ) <  2  x T ( t ) P1k x ( t ) < ϑ 2 ω T ( s )ω ( s ) ds γ  T 2

T

T 2k

∫ 0

(2.184)

x (t ) K K k x (t )  ϑ 2  T u (t ) u (t ) &= <  2  x ( t ) P1k x ( t ) < ϑ 2 ω T ( s )ω ( s ) ds umax 2 umax 2 γ  T

T

T k

∫ 0

By using the Schur complement, (2.184) can be transformed into (2.172) and (2.173). Then, the proof is accomplished. Thirdly, it is worth noting that in the aforementioned formula, the gain perturbations ∆K j are not considered. To improve the robustness of the suspension controller, ∆K j is taken into consideration, which is assumed to be bounded as ∆K j x ( t ) ≤ ∆umax . By using Lemma 2.9, the following theorem can be obtained. Theorem 2.8  −1  jϖ c  1 0   0 1  Let matrices Φ =  ,Ψ =   ,Y =    , Π in (2.149) and a −ϖ 1ϖ 2   − jϖ c  1 0   0 0   scalar ε be given. A fuzzy controller is constructed in the form of (2.146) with ∆K j = HF ( t ) EK j. For a random time delay 0 < τ ≤ τ , the closed-loop system in (2.147) is asymptotically stable

74

Vibration Control of Vehicle Suspension Systems 2 and conforms to G ( jϖ ) ∞ < γ for all non-zero ω ∈ L2 [ 0, ∞ ), if there exist symmetric matrices 2

4

P1k =

4

ki P1i > 0 ,Q1k =

i =1 4

P2 k =

4

kiQ1i > 0, Z1k =

i =1 4

4

ki Z1i > 0 , Z 2 k =

i =1

ki Z 2i > 0,Q2 k =

i =1

4

∑k Q i

2i

> 0,

i =1

∑k P , and R = ∑k R  (i = 1,…,4) satisfying i 2i

i =1

k

i

i

i =1

 T1k   *  * 

µV1k − µI *

U1Tk 0 − µI

   < 0 (2.185)  

 T2 k   *  * 

µV2 k − µI *

U 2Tk 0 − µI

   < 0 (2.186)  

 P1k    * 

  P1k   * 

   > 0 (2.187) γ2 I  2 ∂ 

C2Tk

K kT umax − ∆umax I

   > 0 (2.188)  

where  τ 2 Z1k − ε [O ] s   T1k = *   * 

    T2 k =     

−Q2 k + τ Z 2 k

P1k − ε O + ε O T Ak

ε O T Buk K k

Q1k − Z1k + ε  AkT O  s *

ε O T Buk K k + Z1k −Q1k − Z1k

P2 k + jϖ cQ2 k − O

0

  ,    0

0

P2 k − jϖ cQ2 k − O T −ϖ 1ϖ 2Q2 k +  AkT O  s + Rk − τ −1Z 2 k O T Buk K k + τ −1Z 2 k O T Bk C1Tk *

*

− Rk − τ −1Z 2 k

0

* *

* *

* *

−γ 2 I 0 −I *

 ε O T Buk H  V1k =  ε O T Buk H  0 

   , U 2 k =  0  

0

EK k  ,

K kT D1Tk

        

75

Active Suspension Control

   V2 k =    

0 O T Buk H 0 0 D1k H

    ,U 2 k =  0   

0

EK k

0

0  .

Proof. By replacing K k with K k + HF ( t ) EK k, (2.170) and (2.171) can be extended to T1k + V1k FU1k + [V1k FU1k ] < 0 T

T2 k + V2 k FU 2 k + [V2 k FU 2 k ] < 0 T

(2.189)

According to Lemma 2.9, if there exists a scalar µ > 0 so that the inequalities (2.185) and (2.186) are established, the inequality (2.189) can be fulfilled. Then, the proof is accomplished. Finally, due to the non-linear terms in inequalities (2.185) and (2.186), they can not be solved directly by using LMI optimization. Hence, some congruence transformations are performed and the following theorem is obtained. Theorem 2.9  −1  jϖ c  0 1   1 0  Let matrices Φ =  ,Ψ =   ,Y =    , ∏ in (2.149) −ϖ 1ϖ 2   − jϖ c  1 0   0 0   and a scalar ε be given. A fuzzy controller is constructed in the form of (2.146) with ∆K j = HF ( t ) EK j. For a random time delay 0 < τ ≤ τ , the closed-loop system in (2.147) is

asymptotically stable and conforms to G ( jϖ ) ∞ < γ 2 for all non-zero ω ∈ L2 [ 0, ∞ ), if there 2

exist symmetric matrices P1i > 0, Q1i > 0, Z1i > 0 , Z 2i > 0 , Q2i > 0 , P2i and Ri  ( i = 1,…,4 ) satisfying 4

 T1ij  ki k j  *  * j =1 

µV1ij −µI *

U1Tij

 T  2ij ki k j  *  * j =1 

µV2ij − µI *

U 2Tij

4

∑∑

i =1

4

4

∑∑

i =1

 P1 i  ki   * i =1  4

4

 P1i  ki k j   * j =1  4

∑∑

i =1

where

  0  < 0 (2.190) − µ I     0  < 0 (2.191) − µ I  

O T C2Ti    > 0 (2.192) ϑ2 I  2 γ  K Tj / ( umax − ∆umax )    > 0 (2.193) ϑ2 I  2 γ 

76

Vibration Control of Vehicle Suspension Systems

 τ 2 Z − ε O  1i  s   T1ij = *    *

    T2ij =     

P1i − ε O T + ε AiO

ε Bui K j

Q1i − Z1i + ε  AiO  s

ε Bui K j + Z1i

*

−Q1i − Z1i

−Q2i + τ Z 2i

P2i + jΦcQ2i − O T

P2i − jΦcQ2i − O

  ,   

0

0

0

−ϖ 1W2Q2i +  AiO  + Ri − τ −1 Z 2i

Bui K j + τ −1 Z 2i

Bi

O T C1Ti

*

*

− Ri − τ −1 Z 2i

0

K Tj D1Ti

* *

* *

* *

−γ 2 I *

0 −I

s

 ε Bui H  V1ij =  ε Bui H  0

 0  B H ui  V2ij =  0  0  D 1 iH 

(

    ,U 2ij =  0 

    ,U 2ij =  0    

0

0 EK j  , 

0

EK j

)

0  

(

)

Proof. Define J1 = diag O −1 , O −1 , O −1 , I , I , J2 = diag O −1 , O −1O,−1 , I , I , I ,   I and −1 J3 = diag O , I . Then, a congruence transformation to (2.185)–(2.188) is performed by multiplying the full rank matrix J1, J2, J3, and J3 on the right, and J1T , J2T , J3T and J3T on the left, respectively. By substituting

(

)

P1i = O − T P1iO −1 , Q1i = O − T Q1iO −1 , Z1i = O − T Z1iO −1 ,

P2i = O − T P2iO −1 , Q2i = O − T Q2iO −1 , Z 2i = O − T Z 2iO −1 , Ri = O − T RiO −1 , K j = K jO −1 , O = O −1 then, the conditions in Theorem 2.9 is obtained. If this theorem can be solved, the gain matrix of the proposed controller is calculated by K j = K jO −1. Due to the complex variables in (2.191), it cannot be handled by LMI techniques directly. However, it can be transformed into a real variable’s LMI of a larger dimension [33]. That means the S1 + jS2 < 0 which contains complex variables can be rewritten as . In this condition, the inequalities (2.190)–(2.193) can be solved by using LMI techniques.

2.3.2.3  Simulation Results In this section, the proposed non-fragile fuzzy H ∞ controller with time delay is implemented in the quarter-automobile numerical simulation model to verify the effectiveness of the control strategy in the finite frequency range. Meanwhile, different time-delays and actuator uncertainties are considered in different simulations. The parameters and values of the quarter-automobile active suspension system model are listed in Table 2.13. The value of sprung mass ms ( t ) is in the range [256 kg, 384 kg] and the value of unsprung mass mu ( t ) is in [35 kg, 45 kg]. Given the scalars ϖ 1 = 4 Hz,

        

77

Active Suspension Control

ϖ 2 = 8 Hz and ϑ = γ , set an appropriate time delay τ and uncertain parameter , the gain of nonfragile fuzzy H ∞ controller with time delay c∆K an be obtained by solving Theorem 2.9. To emphasize the effects of the proposed control method more clearly, the control methods in [34, 35] are adopted for comparative purposes. For the convenience of description, this group of the comparison is exemplified as Case 1. On the other hand, the time delay and fuzzy model are only considered in [35]. To highlight the effects of the finite frequency domain, we compared the proposed controller with the fuzzy entire frequency domain controller with the time delay in [35], which is listed as Case 2. The relevant parameters for Cases 1 and 2 are listed in Table 2.14. Table 2.15 lists the numerical simulation control gains of the proposed controller and the compared controller. For simplicity, the proposed controller and compared controller in Case 1 are denoted as K1 and K c1, respectively; the proposed controller and the compared controller in Case 2 are denoted as K 2 and K c2 , respectively. Then, the time delay and actuator uncertainties are considered separately to elucidate their influences on the performance of the active suspension systems. For the uncertainty ∆K = 0 (Case 3) and based on the conditions in Theorem 2.9, the different performance indexes γ min by choosing different parameters τ can be obtained in Table 2.16, which is used to guarantee the minimum H ∞ norm of closed-loop system. By giving a certain time delay τ = 5 ms and solving the conditions in Theorem 2.9, the different γ min can be obtained in Table 2.16, in which the values of H and F are set to 1 and different values of E (Case 4) are chosen. Furthermore, Table 2.16 shows that the system performance indexes γ min are larger with larger time delay and actuator uncertainties. However, the performance indexes γ min cannot highlight the effects of suspension acceleration in the concerned frequency domain. Next, further verifications will be performed. For the sake of brevity, the control gains in Table 2.16 are not listed. Finally, the varying sprung mass and the unsprung mass are considered separately to elucidate the influences on the active suspension systems. Considering that the practical varying masses are mostly non-continuous, so three sprung mass values (Case 5) and three unsprung mass values (Case 6) are considered to verify the effectiveness of the control strategy. The values of the corresponding parameters are listed in Table 2.17. The frequency responses are applied to display the interference attenuation performance in the concerned frequency domain. The vehicle body acceleration’s frequency responses for Cases 1–6 TABLE 2.13 The Parameters and Values of the Quarter-Automobile Suspension Model Parameters ks cs ∆umax

Values

Parameters

Values

Parameters

Values

18 kN/m 1.0 kNs/m 50 N

ku cu

200 kN/m 10 Ns/m [256, 384] kg

zmax umax

0.1 m 2500 N [35, 45] kg

ms ( t )

mu ( t )

TABLE 2.14 Time-Delays and Actuator Uncertainties in Cases 1 and 2 Cases

Masses (kg)

Case 1

ms  = 320 mu = 40 ms  = 320 mu = 40

Case 2

Delays (ms)

Uncertainties

γ min

τ  = 10−6

H = 0, F = 0, E = 0

0.8713

τ  = 5

H = 1, F = 1, E = 0.005

3.3207

78

Vibration Control of Vehicle Suspension Systems

TABLE 2.15 Control Gains in Cases 1 and 2 Cases

Designed Gains

Compared Gains

Case 1 Case 2

K1 = 104 × [−1.4387, −1.8615, −6.4414, −0.0936] K 2  = 104 × [−1.4387, −1.8615, −6.4414, −0.0936]

K c1 = 104 × [1.6026, −0.1181, −0.2097, −0.0333] K c2  = 104 × [0.0299, −1.6707, −0.2184, −0.0446]

TABLE 2.16 Time-Delays and Actuator Uncertainties in Cases 3 and 4 Cases

Masses (kg)

Case 3

ms  = 320 mu = 40

Case 4

ms  = 320 ms  = 40

Delays (ms)

τ  = 0 τ  = 5 τ  = 10 τ  = 15 τ  = 20 τ  = 5

Uncertainties H=0, F=0, E=0

H=1, F=1

E=0.001 E=0.005 E=0.010 E=0.015 E=0.020

γmin 0.8713 1.8809 2.8033 3.5267 4.1461 2.1634 3.3207 4.7337 6.0545 7.3228

TABLE 2.17 Varying Masses in Cases 5 and 6 Cases Case 5

Case 6

Masses (kg) ms  = 290 ms  = 320 ms  = 350 ms  = 320

Delays (ms)

Uncertainties

γmin

mu = 40

τ  = 5

H = 1, F = 1, E = 0.005

3.3207

mu = 37

τ  = 5

H = 1, F = 1, E = 0.005

3.3207

mu = 40 kg mu = 43 kg

are shown in Figure 2.23. As shown in Case 1, when the time delay is about τ = 0 ms, the values of responding acceleration of the active suspension systems with a controller K1 are less than that of the passive suspension significantly. Besides, it should be pointed out that the frequency responses of K1 are much smaller than K c1 due to the considerations of more Lyapunov functional parameters P1k , Q1k and Z1k in the derivation of K1. Meanwhile, as shown in Case 1, the proposed controller can not only handle the time delay problem but also ensures the control performance. This means that the much less conservative is realized in the proposed controller. Considering the time delay with τ = 5 ms both in the proposed controller K 2 and the compared controller K c2 , the results on Case 2 show that the K 2 has a more excellent damping effect than the K c2 which considers the entire frequency domain.

79

Active Suspension Control Case 1

20 10 0 –10 –20 –30 10–1

Passive Kc1 K1

100

Case 2

30

Magnitude (dB)

Magnitude (dB)

30

101

20 10 0 –10 –20 –30 10–1

102

Passive Kc2 K2

100

Frequency (Hz) Case 3

30

Magnitude (dB)

Magnitude (dB)

20

10 0

–20 –30 –40 10–1

Passive 0 ms 5 ms 10 ms 15 ms 20 ms

10 0 –10 –20 –30

100

101

–40 10–1

102

Passive 0.001 0.005 0.010 0.015 0.020

100

Frequency (Hz)

4

6 (Hz)

8

5 0 10–1

100

101

Frequency (Hz)

102

20 15

6

mu=37kg mu=40kg mu=43kg mu=37kg mu=40kg mu=43kg

(abs)

4 2

10

Magnitude (abs)

(abs)

Magnitude (abs)

15

102

Case 6

25

6

ms=290kg ms=320kg ms=350kg ms=290kg ms=320kg ms=350kg

101

Frequency (Hz)

Case 5

25 20

102

Case 4

30

20

–10

101

Frequency (Hz)

4 2

4

10

6 (Hz)

8

5 0 10–1

100

101

102

Frequency (Hz)

FIGURE 2.23  The frequency responses of vehicle body acceleration in Cases 1–6.

The results in Case 3 show that the ride comfort of the vehicles continues to decline along with the increase of time delay. Similarly, the performance of the active suspension systems is deteriorated by the actuator uncertainties as shown in Case 4. In short, the time delay and actuator uncertainties can lead to poor performances. Hence, the results of frequency responses show that the proposed controller is effective to simultaneously improve the ride comfort and maintain good stability and robustness. Case 5 and Case 6 plot the frequency responses of the vehicle body acceleration with different sprung mass and unsprung mass, respectively. It can be seen from Case 5 that when the varying sprung mass is not considered, the vehicle body acceleration value of the passive suspension systems has a significant change, and the proposed controller can effectively improve the ride comfort. However, Case 6 shows that the varying unsprung mass has a very small effect on the values of the vehicle body acceleration because the change in the unsprung mass affects mainly the driving safety of the vehicle.

80

Vibration Control of Vehicle Suspension Systems

During vehicle driving and handing, road disturbances can be divided into shocks and random vibrations generally. There are several kinds of relatively high-intensity and short-duration discrete events that can be regarded as shocks, such as obvious bumps or potholes on an otherwise flat road. Then, a raised isolated bump is used as a pavement incentive on a smooth road. This bump acts as a disturbance signal from the road is as  A  2π v    t , 1 − cos   l    2  zr ( t ) =   0,  

if 0 ≤ t ≤

l v

l if < t v

(2.194)

where v denotes the forward velocity of the vehicle, l and A represent the length and the height of the bump, respectively. The values of parameters can be set as v = 15 m/s, l = 3 m and A= 0.05 m, and the frequency is set as f  = 5 Hz which is at the 4–8 Hz range. (b)

Case 1

6

Kc1 K1 Passive

4 2 0 –2 –4

0

0.5

1

1.5

2

2.5

3

Sprung mass acceleration (m/s2)

Sprung mass acceleration (m/s2)

(a)

Case 2

6

Kc2 K2 Passive

4 2 0 –2 –4

0

0.5

1

Time (s)

(c) 0.2 0 –0.2 –0.4 –0.6

0

0.5

1

1.5

2

2.5

–0.2 –0.4

0

0.5

1

0.2 0 –0.2 –0.4 1

1.5

Time (s)

2

2.5

3

Case 2

0.6

(f) Kc1 K1 Passive

0.5

1.5

Time (s)

0.4

0

3

0

–0.6

3

Case 1

0.6

2.5

Kc2 K2 Passive

0.2

Time (s)

(e)

2

Case 2

0.4 Kc1 K1 Passive

Relative suspension travel

Relative suspension travel

(d)

Case 1

0.4

1.5

Time (s)

2

2.5

3

Kc2 K2 Passive

0.4 0.2 0 –0.2 –0.4 –0.6

0

0.5

1

1.5

2

2.5

3

Time (s)

FIGURE 2.24  Simulations under bump excitation: (a and b) sprung mass acceleration, (c and d) relative suspension travel, (e and f) relative dynamic load.

81

Active Suspension Control

Figure 2.24 plots the time-domain response results. Figure 2.24a shows that controller K1 can quickly reduce the sprung mass acceleration compared with controller K c1 and passive suspension systems. In Figure 2.24b, the sprung mass acceleration does not decrease so quickly due to the effects of the time delay and actuator uncertainties. Nevertheless, the proposed controller reduces greatly the sprung mass acceleration to the compared controller and passive suspension systems. Figure 2.24c and d indicate that the maximum relative dynamic travel of the active suspension systems is less than 1. Figure 2.24e and f indicate that the maximum relative dynamic load of the active suspension systems is also less than 1. They indicate that the performance constraints of the active suspension systems are met. In addition, to further elucidate the effects of the time delay and actuator uncertainties on the active suspension systems, Figure 2.25 and Figure 2.26 plot the time-domain comparison diagrams for Case 3 and Case 4, respectively. Figure 2.25 shows the responses of the sprung mass acceleration (a), relative suspension travel (c), and relative dynamic load (d) in Case 3. To show more clearly the performance of the comparison under different chosen time-delays τ , we truncate Figure 2.25a a very short response time of 0.2 seconds and show it in Figure 2.25b. It can be seen from Figure 2.25a and b that the values of the sprung mass acceleration are getting larger with the increase of the time delay, which means the ride comfort is getting worse. Moreover, it can be seen from Figure 2.25c and d that the relative suspension travel constraints and relative dynamic load constraints are less than the required value 1, which means the road holding stability is ensured by the proposed controller. Then, Figure 2.26 plots the responses of the sprung mass acceleration (a), relative suspension travel (c), and relative dynamic load (d) in Case 4. For showing different chosen uncertainties ∆K more clearly, we truncate Figure 2.26a a very short response time of 0.2 seconds and show it in Figure 2.26b. It can be found from Figure 2.26a and b that the actuator uncertainties have similar results with that of the time delay, i.e., the value of the sprung mass acceleration increase with the growth of the actuator uncertainties, which means that the ride comfort is decreased. This is consistent with the calculation results in Table 2.16. In short, both the time delay and actuator uncertainties have negative influence on the ride comfort of the vehicles. (b)

Case 3

6

0 ms 5 ms

4

10 ms 15 ms

20 ms Passive

2 0 –2 –4

0

1

0.5

1.5

2

2.5

Sprung mass acceleration (m/s2)

Sprung mass acceleration (m/s2)

(a)

Case 3

6

0 ms 5 ms 10 ms

4 2 0 –2 –4

0

Relative suspension travel

Time (s)

Case 3

0.4 0.2 0 –0.2 –0.4 –0.6

0 ms 5 ms 0

0.5

10 ms 15 ms 1

1.5

Time (s)

0.2

0.15

0.1

0.05

Time (s)

(c)

15 ms 20 ms Passive

20 ms Passive 2

2.5

Case 3

0.6

0 ms 5 ms

0.4

10 ms 15 ms

20 ms Passive

0.2 0 –0.2 –0.4 0

0.5

1

1.5

Time (s)

FIGURE 2.25  Responses of suspension system under bump excitation in Case 3.

2

2.5

82

Vibration Control of Vehicle Suspension Systems (b)

Case 4

6

0.001 0.005

4

0.010 0.015

0.020 Passive

2 0 –2 –4

0

1

0.5

1.5

2

2.5

Sprung mass acceleration (m/s2)

Sprung mass acceleration (m/s2)

(a)

Case 4

6

0.001 0.005 0.010

4 2 0 –2 –4

0

Relative suspension travel

Time (s)

Case 4

0.4 0.2 0

–0.2 –0.4 –0.6

0.001 0.005 0

0.5

0.010 0.015 1

1.5

Time (s)

0.2

0.15

0.1

0.05

Time (s)

(c)

0.015 0.020 Passive

0.020 Passive 2

2.5

Case 4

0.6

0.001 0.005

0.4

0.010 0.015

0.020 Passive

0.2 0 –0.2 –0.4 –0.6

0

0.5

1

1.5

2

2.5

Time (s)

FIGURE 2.26  Responses of suspension system under bump excitation in Case 4.

The road disturbances for a driving vehicle are considered as random vibrations, which are usually viewed as a stochastic process with ground displacement PSD. When the vehicle speed is constant, whether, in the time-domain or the space-domain, road disturbances can be regarded as a stationary process. However, when an automobile runs with a changeable speed in the time-domain, the road profile becomes a non-stationary process [36]. Therefore, in the light of the mimic filter white noise method based on the stable stochastic theory, we show the differential equation of road excitation in the time-domain as

zr ( t ) + ( v0 + at ) 2πnc zr ( t ) = 2πn0 Gq ( n0 )( v0 + at )ω ( t ) (2.195)

where n0= 0.1 m−1 denotes the standard spatial cut-off frequency, nc = 0.01 m−1 stands for the road spatial cut-off frequency, v0  = 20 m/s is the initial speed, a = 4 m/s2 is the vehicle longitudinal acceleration, Gq ( n0 ) represents the road roughness coefficient which is selected as 64 × 10 −6 m3 , and ω ( t ) denotes the road white noise in the time-domain. Then, for Cases 1–4, the time-domain responses of sprung mass acceleration under the above road excitation are shown in (a–d) of Figure 2.27, respectively. For simplicity, the figures for relative suspension travel and relative dynamic load meeting performance constraints are not shown. Figure 2.27a and b shows that the designed closed-loop system accomplishes a lower sprung mass acceleration while ensuring performance constraints. In addition, it is observed from Figure 2.27c and d that the actuator time delay and uncertainties deteriorate the performance of the active suspension system. The acceleration’s root mean square (RMS) values are often used to reflect a vehicle’s ride ­comfort as it can quantify the amount of acceleration that is passed from the road to the vehicle body [37]. Therefore, the RMS values of the non-stationary state are given to display the superior ­performance of the proposed controller more intuitively. RMS values of a signal x ( t ) can be gotten by

83

Active Suspension Control (b)

Case 1

0.1

K1

Kc1

Sprung mass acceleration (m/s2)

Sprung mass acceleration (m/s2)

(a)

Passive

0.05

0

–0.05

Case 2

0.1

K2

Kc2

0.05

0

–0.05

–0.1 0

0.5

1

1.5

2

2.5

3

–0.1 0

0.5

1

Time (s)

(c)

(d)

Case 3

0.1

0 ms 5 ms

10 ms 15 ms

1.5

2

2.5

3

2.5

3

Time (s)

Sprung mass acceleration (m/s2)

Sprung mass acceleration (m/s2)

Passive

20 ms Passive

0.05

0

–0.05

Case 4

0.1

0.001 0.005

0.010 0.015

0.020 Passive

0.05

0

–0.05

–0.1 0

0.5

1

1.5

2

2.5

3

Time (s)

–0.1 0

0.5

1

1.5

2

Time (s)

FIGURE 2.27  Acceleration responses under non-stationary excitation in Cases 1–4.

1 T RMS x =  x T ( t ) x ( t ) dt (2.196)  T 0

84

Vibration Control of Vehicle Suspension Systems

TABLE 2.18 RMS Values of Suspension Sprung Mass Acceleration in Case 1 Roughness

Passive

16 × 10−6 m3 64 × 10−6 m3 156 × 10−6 m3 1024 × 10−6 m3 Reduced rates

0.0123 0.0246 0.0492 0.0985 –

Active Designed K1 0.0021 0.0043 0.0086 0.0171 82.63%

Compared Kc1 0.0075 0.0150 0.0300 0.0600 39.02%

TABLE 2.19 RMS Values of Suspension Sprung Mass Acceleration in Case 2 Roughness 16 × 10−6 m3 64 × 10−6 m3 156 × 10−6 m3 1024 × 10−6 m3 Reduced rates

Passive 0.0123 0.0246 0.0492 0.0985 –

Active Compared Kc2 0.0106 0.0212 0.0424 0.0848 13.84%

Designed K2 0.0077 0.0155 0.0311 0.0621 37.03%

TABLE 2.20 RMS Values of Suspension Sprung Mass Acceleration in Case 3 Roughness

Passive

16 × 10−6 m3 64 × 10−6 m3 156 × 10−6 m3 1024 × 10−6 m3 Reduced rates

0.0123 0.0246 0.0492 0.0985 –

Active (H = 0, F = 0, E = 0)

τ = 0 ms 0.0034 0.0069 0.0138 0.0275 72.09%

τ = 5 ms 0.0066 0.0132 0.0265 0.0529 46.28%

τ = 10 ms τ = 15 ms 0.0087 0.0099 0.0173 0.0199 0.0345 0.0397 0.0695 0.0795 29.57% 19.31%

τ = 20 ms 0.0107 0.0214 0.0429 0.0855 13.01%

TABLE 2.21 RMS Values of Suspension Sprung Mass Acceleration in Case 4 Roughness

Passive

Active (H = 0, F = 0, E = 0)

16 × 10−6 m3 64 × 10−6 m3 156 × 10−6 m3 1024 × 10−6 m3 Reduced rates

0.0123 0.0246 0.0492 0.0985 –

HF ( t ) E = 0.001 HF ( t ) E = 0.005 HF ( t ) E = 0.010 HF ( t ) E = 0.015 HF ( t ) E = 0.020 0.0068 0.0077 0.0087 0.0091 0.0092 0.0136 0.0155 0.0174 0.0182 0.0184 0.0272 0.0311 0.0347 0.0364 0.0368 0.0543 0.0621 0.0695 0.0729 0.0737 44.76% 37.03% 29.36% 26.02% 25.20%

85

Active Suspension Control

new non-fragile fuzzy finite frequency control method has been proposed for the active suspension systems with time delay and actuator uncertainties to improve suspension performances. Numerical simulation examples on a quarter-automobile active suspension model are implemented and the simulation results display that the proposed controller can improve greatly ride comfort in the concerned frequency domain despite the existence of the time delay and actuator uncertainties.

2.4  PREVIEW CONTROL 2.4.1  System Modelling A half-car model with four degrees of freedom is used in this section, as shown in Figure 2.28, tyre damping is ignored in this model, because it is always much smaller than the suspension damping. Assume that the pitch angle is small, and the characteristics of the suspension elements are linear, the equations of motion can be given as Mzc = f f + fr , Jθ = af f − bfr ,

(2.197) m f  z1 = − k f 2 (η f − µ f ) − f f , mr  z2 = − kr 2 (ηr − µr ) − fr ,

where

( + bθ ) + b (η

)

f f = k f 1 (η f − zc − aθ ) + b f η f − zc − aθ + u f , fr = kr1 (ηr − zc

r

r

)

− zc + bθ + ur .

In the above equations and figure, a,  b denote the horizontal distances from the centre of mass to the front wheel and rear wheels, respectively. M and J are the sprung mass and its mass moment

FIGURE 2.28  Half-car model.

86

Vibration Control of Vehicle Suspension Systems

of inertia, m f and mr are the front and rear unsprung masses, respectively, µ f and µr are the control forces. k f 1, kr1 and b f , br denote the stiffnesses and damping coefficients of the passive suspension elements for the front and rear assembles. Similarly, k f 2 and kr 2 denote the front and rear tyre stiffnesses. The displacements at the front and rear wheels of the vehicle are given as z f = zc + aθ ,

z r = z c − bθ .

Choose the state vector as x =  x1

x2

x3

x4

x5

x6

x7

T

x8  , (2.198)

where x1 = zc + aθ , x 2 = zc − bθ , x3 = η f , x 4 = ηr ,

x5 = zc + aθ, x6 = zc − bθ, x 7 = η f , x8 = ηr .

And the control input and road input are introduced as follows  uf u=  ur

  wf , w =    wr

  µf =   µr

  , (2.199) 

a+b where wr is time delay of w f , that is, wr ( t ) = w f ( t − τ ), τ = , and v is the vehicle forward velocv ity (m/s). Then the dynamic equations can be presented in the following state-space form x ( t ) = Ax ( t ) + Bu ( t ) + Dw ( t ) , (2.200)

where

 04× 4 I4  A=    

    =          B=      

− a1k f 1

− a2 kr1

a1k f 1

a2 kr1

− a1b f

− a2 br

a1b f

a2 br

− a2 k f 2

− a3 kr1

a2 k f 1

a3 kr1

− a2 b f

− a3br

a2 b f

a2 br

k f1 mf

0

0

bf mf

0

0

kr 1 mr

0

0

br mr

0

0 4 ×1

0 4 ×1

a1

a2

a2

a3

− 0

1 mf

0 −

1 mr

k f1 + k f 2 mf

        , D =           

0 6 ×1 kf2 mf 0

kr 1 + kr 2 mr

bf mf

0 −

br mr

        

0 6 ×1    2 2 0  , a1 = 1 + a , a2 = 1 − ab , a3 = 1 + b . M J M J M J  kr 2  mr 

87

Active Suspension Control

According to the analysis of control objectives in the previous subsection, for this section, the control objectives can be briefly given as

1. Ride quality: minimize  zc and θ under disturbances. 2. Suspension deflection limit z f − η f ≤ z fmax ,

zr − ηr ≤ zrmax , (2.201)

where z fmax, zrmax are the maximum suspension deflections at the front and rear, respectively.

3. Road holding aM  bM  k f 2 (η f − µ f ) < 9.8  + m f  , kr 2 (ηr − µr ) < 9.8  + mr  . (2.202) a+b  a+b 

4. Actuator saturation: Considering the limited actuating power, the active control force should not exceed a certain limit u f ( t ) ≤ u fmax ,

ur ( t ) ≤ urmax , (2.203)

where u fmax , urmax are the maximum control inputs for the front and rear wheels, respectively. Based on the analysis above, the output variables can be defined as: T

  , z1 ( t ) =  c1  z c2θ    zf − ηf z2 ( t ) =  z fmax  

z r − ηr zrmax

k f 2 (η f − µ f )  bM  + mf  9.8  a+b 

kr 2 (ηr − µr ) aM + mr  9.8   a+b

T

uf u fmax

 ur  (2.204) urmax  .  

zc and θ. And z ( t ) can be where c1, c2 are the weighting coefficients to balance the performance of  rewritten as the following state-space form z1 ( t ) = C1 x ( t ) + D1u ( t ) + Dw1w ( t ) ,

z2 ( t ) = C2 x ( t ) + D2u ( t ) + Dw 2 w ( t ) ,

(2.205)

where  c1 C1 & =   0

 1  M D1 & =   a  J 

 k − f1 0   M  k f1 c2    −a J  1 M b − J

kr 1 M kr 1 b J

   , Dw1 = 0 2 × 2 ,   

k f1 M −a

kr 1 M kr 1 J

−b

bf M b −a f J

− kr 1 J

br M br b J

bf M b a r J

br M −b

br J

     

88

Vibration Control of Vehicle Suspension Systems

       C2 =         

1

0

z fmax

1

0

zrmax

1 z fmax

0 kf2  bM  9.8  + mf  a+b 

0

0

0

0

0

0 2 ×1

0 2 ×1

0 2 ×1

 0 4 ×1  1  D2 =  u fmax   0  

0 4 ×1 0 1 urmax

0

    , Dw 2    

     =      

01× 4 1

01× 4

zrmax

0

01× 4

kr 2 aM + mr  9.8  a+b  0 2 ×1 0 2 ×1 −

01× 4 02× 4

       ,        

0 2 ×1

kf2  bM  9.8  + mf  a+b 

0 kr 2 aM  9.8  + m f  a+b 

0

0 2 ×1

0 2 ×1

     .      

With system sampling period Ts, a discrete-time representation of the half-car model can be obtained as x ( k + 1) = Gx ( k ) + Hu ( k ) + Fw ( k )

z1 ( k ) = C1 x ( k ) + D1u ( k ) + D w1w ( k ) z 2 ( k ) = C 2 x ( k ) + D 2u ( k ) + D w 2 w ( k )

(2.206)

y( k ) = C y x ( k ) where G;  H;  F are the discrete-time equivalents of continuous-time system matrices A;  B;  D, respectively, obtained by step-invariant discretization.

2.4.2 Augmented System with Wheelbase Preview Information Let xα ( k ) denote the vector which represents all the preview information for the rear wheel,

    k = xa ( )    

in which N a ∈  0 is the integral part of τ / Ts .

w f ( k − Na ) w f ( k − N a + 1)  w f ( k − 1) wf (k )

       

89

Active Suspension Control

The unevenness of road surface w f ( t ) at the front wheel can be modelled as [38,39]: w f + α vw f = ξ (2.207)

where α is a constant related to type of road surface, ξ is a Gaussian white noise process with T  ξ ( t1 )ξ ξ ( t2 )  = 2α vσ 2δ ( t1 − t2 ) (2.208)

where σ is the standard deviation of road unevenness and δ ( t ) is the Dirac delta. In the current work, it is assumed that the vehicle velocity and the road profile are not varying with respect to time. However, in practice, these parameters are variables. For interested readers, there are some works about the road identification techniques, see [40–42]. The discrete-time representation of the disturbances at the front and rear wheels is thus given by w f ( k + 1) = GW w f ( k ) + φ ( k )

wr ( k ) = w f ( k − N a )

(2.209)

where GW = e − ∝ vTs and φ ( k ) can be given as a stochastic integral [43,44]: ( k +1)Ts

φ(k ) =

(

kTs

e

− α v [( k +1)Ts − s ]

dW ( s ) (2.210)

)

where W ( s ) is the Wiener process with  dW 2 = 2α vσ 2   dt , and φ ( k ) has the following statistical properties:  ( k +1)Ts  2 − α v k +1 T − s e [( ) s ] ds  = σ 2 1 − e −2α vTs (2.211)  (φ ( k )) = 0  φ ( k ) = 2α vσ     kTs Then the state-space equation of the augmented vector xa can be easily obtained as

(

2

)

2

(

)

x a ( k + 1) = G a x a ( k ) + Faψ ( k ) (2.212)

where

    Ga =    

0 0  0 0 0

1 0  0 0 0

0 1 0 0 0

0 0

1 0 0

0 0  0 1 Gw

        , Fa =       

0  0

σ 2 (1 − e −2α vTs )

      

and G a ∈ ( N a +1)×( N a +1) , Fa ∈ ( N a +1)×1. The effective input ψ is a unit variance white noise sequence. Denote the preview sampling period, used for collecting preview information from x a , as Tp. yα ( k ) denotes the measurement output vector which represents the available preview information for the rear wheel with Tp as the preview sampling period. The preview sampling period is an integral multiple of the system sampling period, that is, Tp = Ts ,  = 1,2,…. Then yα ( k ) can be written as

yα ( k ) = Cα xα ( k ) , (2.213)

90

Vibration Control of Vehicle Suspension Systems

where  Q1  0 Ca =   0   0

  1  Qi =   0 

0 Q2

0

0 0

0 0

0

0 QN a + 1

  ,   

Na + 1 − i  is an integer,   i = 1,…, N a + 1. N +1− i  when  a  is not an integer,    when

The augmented state vector and the measured output can be obtained as  x(k ) xg ( k ) =   xa ( k ) 

  y( k )  , yg ( k ) =    ya ( k )  

 .  

Then the augmented system with preview information can be described as x g ( k + 1) = A g x g ( k ) + B gu ( k ) + D gψ ( k ) , z1 ( k ) = C g1x g ( k ) + D g1u ( k ) ,

(2.214)

z 2 ( k ) = C g 2 x g ( k ) + D g 2u ( k ) , y g ( k ) = C gy x g ( k ) ,

where

F =  Ff 

Fr

08 ×( N a −1) Ga

Ff

 ,  

   0 8 ×1  H Bg =  , Dg =    ,   0( N a +1)× 2   Fa 

 G Fr  , Ag =    0( N a +1)× 8 

Dw1 =  Dwf 1  Dw 2 =  Dwf 2 

Dwr1  , C g1 =  C1  

Dwr1

0 2 ×( N a +1)

Dwf 1  , 

Dwr 2  , C g 2 =  C 2  

Dwr 2

0 6 ×( N a +1)

Dwf 2  ,  

Dgi = Di , i = 1,2.

According to the different numbers of previewed data of the road disturbance used in the controller design, the following different conditions are defined, and C gy can be specified as

1. Active control without preview:  Cy C gy =   0( N a +1)× 8 

08 ×( N a +1) 0( N a +1)×( N a +1)

   

91

Active Suspension Control

2. Active control with full-preview: Tp = Ts  Cy C gy =   0( N a +1)× 8 

08 ×( N a +1)   I N a +1  

3. Active control with partial-preview: Tp = Ts ,  = 1,2,…  Cy C gy =   0( N a +1)× 8 

08 ×( N a +1)   Ca  

2.4.3 Multi-Objective Disturbance Attenuation The H ∞ and generalized H 2 norms of a transfer function operator T from w to z are defined, respectively, as  Z 2 w ∈l2  w 2

 T ∞ = sup

(2.215)  Z ∞  T GH2 = sup , w ∈l2  W 2

where ∞

 z 2 =

∑ | w(k ) | ,

| z ( k ) |2 ,  w 2 =

2

k =0

k =0

 z ∞ = sup z ( k ) . k ∈ 0

Denote the transfer functions from disturbance ψ to the outputs z1 and z2 as T1 and T2, respectively. Then, the control objectives of this subsection are given as follows: PF: Find a controller such that, with wheelbase preview, the augmented closed-loop system is internally stable, the H ∞ norm of T1 is minimized, and the generalized H 2 norm of T2 is less than a given γ Remark 2.6 The H ∞ norm and generalized H 2 norm should be used under energy-bounded disturbances, while a white-noise signal is an unbounded energy signal. In practice, since the excitation is studied over a finite-time span, the disturbance signal may be regarded as energy-bounded, hence it is suitable to use the H ∞ norm and the generalized H 2 norm as the indicators. According to [45], the following lemmas are for the H ∞ and the generalized H 2 performance. Lemma 2.10 The system (2.214) with u = 0 is asymptotically stable and T1 ∞ < γ 1 if and only if there exists matrix P1 > 0 such that

     

P1

0

A g P1

*

γ 1I * *

C g1P1

* *

P *

Dg   0  > 0 (2.216)  0  γ 1I  

92

Vibration Control of Vehicle Suspension Systems

Lemma 2.11 The system (2.214) with u = 0 is asymptotically stable and T2 exists matrix P2 > 0 such that  P2   *  * 

A g P2 P2 *

 γ 2I   *

GH 2

< γ 2 if and only if there

Dg   0  > 0 (2.217)  I 

C g 2 P2   > 0. (2.218) P2 

Then, a static output feedback controller is adopted: u ( k ) = Ky g ( k ) (2.219)

with this static output feedback controller, following proposition can be obtained directly by considering Lemma 2.10 and Lemma 2.11 together. Proposition 2.1 Closed-loop system (2.214) is asymptotically stable and satisfies T1 ∞ < γ 1, T2 there exist matrices P > 0 and K1 such that       

P

0

*

γ 1I

( Ag + BgKC y ) P (C g1 + Dg1KC y ))P

* *

* *

P *

 P   *  * 

( Ag + BgKC y ) P P *

 γ I  2  *

< γ 2, if

Dg    0  > 0 (2.220)  0 γ 1I  

Dg    > 0 (2.221) 0  I 

(C g 2 + Dg 2KC y ) P P

GH 2

  > 0. (2.222) 

The above matrix inequalities cannot be solved directly, because of the bilinear terms, such as B gKC y P . The following ILMI iterations [46,47] are employed to compute the output feedback controller. ILMIs Algorithm: 1. Set i = 1. Select the initial matrix K1 which is obtained from asymptotically stable condition that system (2.214) is asymptotically stable if and only if there exists a matrix W > 0 and K1 such that

 W   *

A g + B gK1C y   > 0 (2.223) W −1 

93

Active Suspension Control

2. For fixed K i, solve the following optimization problem for K i > 0 and γ 1i . OP: Minimize γ 1i subject to the following constraints       

0

*

γ 1i I

( Ag + BgK iC y ) Pi (C g1 + Dg1K iC y ) Pi

* *

* *

Pi *

( Ag + BgK iC y ) Pi

 P  i  *   *

Pi *

   0  > 0 (2.224)  0  γ 1i I   Dg

Dg    > 0 (2.225) 0  I 

(C g 2 + Dg 2K iC y ) Pi

 γ I  2  * 

Pi

Pi

  > 0 (2.226)  

3. For fixed Pi > 0, solve the following feasibility problem for K i. FP: Find K i subject to LMIs (2.224)–(2.226)

4. If γ i − γ i −1 < ε , where ε > 0 is a prescribed tolerance, then STOP. else set i = i + 1 and K i = K i −1, then go to Step 2. The asymptotic stability condition in (2.223) cannot be solved directly by using convex programming techniques, a CCL algorithm [48–50] is proposed here, and we introduce a new matrix Q > 0. Then inequality (2.223) holds if  W   *

A g + B gKC   > 0 (2.227) Q 

 W   *

I   ≥ 0 (2.228) Q  

WQ = I. (2.229)

According to the above LMIs, the CCL algorithm can be formulated in the following algorithm. CCL Algorithm: 1. Set i = 1. Choosing initial W0 > 0 ; Q 0 > 0 randomly. 2. Solve

mintr ( W i Q + WQi )

subject to  ( 2.227 ) and ( 2.228 ).

94

Vibration Control of Vehicle Suspension Systems

Set W i = W; Q i = Q.

3. Substitute the obtained matrix W into (2.223). If one of the inequalities (2.223) and tr ( W i Q + W i ) − 2n < ρ , n = dim ( A g ) (2.230)

is not satisfied with some sufficiently small ρ > 0, then set i = i + 1 and go to Step 2. Otherwise, EXIT.

2.4.4  Simulation Results In this section, the results of multi-objective control with wheelbase preview for a half-car model are studied. The performances of vehicle suspension with passive control, multi-objective control with full-preview, partial-preview and without preview are compared. It is assumed that the preview information is obtained from system responses under the front wheel disturbance. The vehicle parameters for a compact sedan [51] used in simulation are listed in Table 2.22, and choosing z fmax = zrmax = 0.08m, u fmax = urmax = 1000 N. The parameters of road roughness are chosen as α = 0.45m −1;σ 2 = 300 × 10 −6 m 2 . The vehicle is assumed to run linearly with a constant velocity of 20 m/s, and the system sampling period is Ts = 0.025 seconds. Since the natural frequencies of the heave mode and pitch mode of this system are less than 20 Hz, this sampling period is sufficient to capture the essential vibration characteristics. According to the above parameters, N a = 5. Combining with w f ( k ), there are at most six preview data for the rear wheel. Considering the preview sampling period Tp, when  > 6, the available preview information is the same after preview sampling, that is, C a is the same, so we choose  = 1, …,6, for comparing the performances of different conditions of control with preview. The condition  = 1 corresponds to active control with full-preview, the condition  = 6 corresponds to preview information only including w f ( k ). Let us define  = 7 as the condition that multi-objective control without preview in subsequent discussions. Since different initial conditions of K1 used in Algorithm ILMIs will result in different controllers, hundreds of runs with different initial conditions are tested. The minimum γ 1 and the corresponding K of each case are used in the comparison. Figure 2.29 shows the γ 1 of different cases. Table 2.23 lists the minimum value of each condition. It can be seen that γ 1 of control with preview ( = 1, …,6) are significantly less than that of control without preview (  = 7 ) . As for different conditions of control with preview, the more preview information, the smaller γ 1 we get, that is, γ 1 of  = 1 is the smallest one, γ 1 of  = 2 is smaller than that of  = 4, and γ 1 for all of  = 1, …,5, are smaller than that of  = 6.

TABLE 2.22 Model Parameters M 500 kg kf1 10,000 N/m bf 1000 Ns/m

J

mf

mr

910 kg m2 kr1 10,000 N/m br 1000 Ns/m

30 kg kf2 100,000 N/m a 1.25m

40 kg kr1 100,000 N/m b 1.45m

95

Active Suspension Control 1.6 1.4 1.2

1

1 0.8 0.6 0.4 0.2

1

2

3

4

5

6

7

/ FIGURE 2.29  γ 1 comparison.

TABLE 2.23 Minimal γ 1 Under Different Conditions l

1

2

3

4

5

6

7

γ1

0.2342

0.2752

0.2640

0.2778

0.3025

0.4346

0.7601

Figures 2.30 and 2.31 show the vehicle vertical (heave mode) and angular (pitch mode) accelerations for different conditions separately. Figure 2.30 compares the responses of passive control, multi-objective control without preview, multi-objective with full-preview (  = 1). As expected, both the vertical and angular accelerations of control with preview, especially the angular acceleration, are lower than that of control without preview. This indicates that active suspension with preview can provide better ride quality. Figure 2.31 shows the response of multi-objective control with different numbers of preview information, accelerations of control with full-preview (  = 1) and with partial-preview (  = 6 ) are chosen to compare. The difference between vertical acceleration is slight, but the angular acceleration of control with full-preview is obviously less than that of control with partial-preview. To assess the performance of the vibration reduction in the heave and pitch modes, we provide zs and θ of different conditions. Denote Tθ as the the root mean square (RMS) values of the signals  transfer function from ψ to θ, then the RMS value of θ and H 2 norm of Tθ can be given by

θ

rms

2

 1 =   2π  1 =   2π

π

−π

π

−π

 2 Tθ ( e jω ) Sψ (ω ) dω    2 Tθ ( e jω )  dω 

96 (a)

Vibration Control of Vehicle Suspension Systems (b)

Heave mode

2 1.5

magnitude(m/s2)

1 0.5 0 –0.5 –1

1 0.5 0 –0.5 –1

–1.5 –2

Pitch mode

2

0

0.5

1

1.5

2

2.5

Time (s)

3

3.5

4

4.5

5

–1.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (s)

FIGURE 2.30 Response comparison of passive control, active control with and without preview. (a) Vertical acceleration comparison; (b) angular acceleration comparison.

FIGURE 2.31  Response comparison of active control with different numbers of preview information. (a) Vertical acceleration comparison; (b) angular acceleration comparison.

where Sψ is the PSD of disturbance ψ . Since ψ is white noise input, Sψ (ω ) = 1, the RMS value equals the H 2 norm. In this way, the RMS value of  zs and θ based on the H 2 norm of their respective transfer function is used to compare. The RMS value of  zs , θ under four different conditions is listed in Table 2.24 It can be clearly seen that active control with partial-preview reduces the vertical acceleration of control without preview by up to 19%, control with full-preview further reduces the acceleration of control with partial-preview up to 10%. As for angular acceleration, control with partial-preview provides 25% improvement comparing with that of control without preview, while control with full-preview provides 46% more improvement than that of control with partial-preview. In general, compared with control without preview, both vertical and angular accelerations are attenuated by control with preview, and angular acceleration is attenuated more significantly than vertical acceleration. Figure 2.32 compares  T1 ( e jw )  of vehicle suspension with passive control, active control without preview and with full-preview (  = 1) for 0 < ω < π . The heave mode and pitch mode correspond to the first peak and second peak, respectively. The highest peak value equals  T1 ∞ in each situation. It can be seen that both heave mode and pitch mode peak values have been reduced by control without preview, especially the frequency response of heave mode is almost flat. That explains why the vertical acceleration is only reduced slightly by control with preview when compared to control without preview. The figure also shows under control with full-preview, the peak value of the pitch

97

Active Suspension Control

TABLE 2.24 RMS zs; θ Under Different Conditions Signal

Control Passive

RMS z RMS θ

Multi-objective

With Preview

( l = 6)

With Preview

( l = 1)

0.7572

0.1923

0.1564

0.1417

0.5502

0.2722

0.2050

0.1104

FIGURE 2.32  Frequency responses comparison.

mode has also been significantly attenuated, which illustrates that control with full-preview can provide better ride quality than the control without preview. Figure 2.33 shows the active suspension disturbance responses of Z 2 under control with full-preview, the dashed lines denote the responses of the front assemblies, the responses of rear assemblies are shown by the solid lines. It can be seen that the suspension deflection, tyre deflection and actuator saturation are all having absolute values strictly less than unity (that is, they all do not violate the physical constraints).

2.4.5  Conclusions Based on the wheelbase preview method, a multi-objective control method for vehicle suspension has been proposed. H ∞ norm is used as the indicator of ride quality, and generalized H 2 norm is utilized to constrain suspension deflection, tyre deflection and actuator saturation. Algorithms ILMIs and CCL are used to derive the control gain of static output feedback. The performances of vehicle suspension under different conditions are compared. The results illustrate that the proposed control method achieves better performance than that of control without preview, and further improvement can be obtained with more preview information. The application of active control is important for high-performance vehicles but at the expense of higher power usage. It is generally recognized that this has limited its use in common vehicle systems.

Actuator saturation ratio

Tire deflection ratio

Suspension deflection ratio

98

Vibration Control of Vehicle Suspension Systems

0.5 0 –0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

1 0 –1 1 0 –1

Time(s)

FIGURE 2.33  z 2 responses comparison. , ----responses of front assemblies; ——, responses of rear assemblies.

2.5  PARAMETER-DEPENDENT/LPV CONTROL 2.5.1  System Modelling A half-car model with four degrees of freedom is used in this section, as shown in Figure 2.28. The suspension system used in this part and the control objective are also same with the system (2.205) and the control objective in Section 2.4.1. For the system (2.205), the road inputs at front and rear wheel w f , wr can be given as

w f ( t ) = aw w f ( t ) + bwφ f ( t ) , w r ( t ) = aw wr ( t ) + bwφr ( t ) .

(2.231)

where aw = − λ v, bw = σ 2λ v , λ is the constant related to type of road surface, σ is the standard deviation of road unevenness, φ f and φr are unit variance Gaussian white noise processes. φr is the delayed signal of φ f , and their relationship can be presented in Laplace transfer function as

φr = e −τ s . (2.232) φf

To approximate e −τ s by a finite-order transfer function, the Padé approximation is applied as

e −τ s 

Pmn ( −τ s ) (2.233) Qmn ( −τ s )

99

Active Suspension Control

where m! Pmn ( −τ s ) = ( m + n )!

∑ ( j!( m − j ))! ( −τ s ) , m

j

j=0

n! ( m + n )!

Qmn ( −τ s ) =

m+n− j !

∑ ( j!( n − j )!) (τ s ) . n

m+n− j !

j

j=0

The all-pass transfer function is usually used to approximate the term e −τ s. Choose n = m, and we have Pmn ( −τ s ) ( −1) s m + ( −1) bm −1s m −1 +  + ( −1) b1s + b0 = (2.234) s m + bm −1s m −1 +  + b1s + b0 Qmn ( −τ s ) m −1

m

where

bk =

( 2m − k )! , k = 1,2,…, m − 1. k !( m − k )!(τ )m − k

with the state vector

ξ = [ξ1 , ξ 2 ,…, ξ m ] , (2.235) T

based on the definition above, (2.234) can be rewritten as the state-space form as

ξ& = Aξ ξ + Bξφ f ,

φr & = Cξ ξ + Dξφ f ,

(2.236)

where    Aξ =     

− bm −1 − bm − 2  − b1 − b0

1 0  0 0

0 1  0 0

Cξ =  1

0 0  1 0

 −2 ( −1)m −1 b m −1     0    , Bξ =     −1 − ( −1)m b1     1 − ( −1)m b0  

( (

m 0  , Dξ = ( −1) .

0

1

0

0

0

0 1 τ 0

0

0

0 1

Similarly, we have Tξ−1 Aξ Txi = AT , where

    Tξ =     

0

τ m −1

        

)

)

    ,    

100

Vibration Control of Vehicle Suspension Systems

so that ξ = Tξ ε which is equivalent to the new vector, then (2.236) can be transformed to the following form

ε ( t ) = AT ε ( t ) + BTφ f ( t ) ,

(2.237)

φr ( t ) = C T ε ( t ) + φ f ( t ) ,

where

AT & = vAT ,

     1  AT = a+b     

− ( m + 1) m −  − −

    1  BT & = vBT , BT = a+b     

( m + 2 )( m + 1) m 2

( 2m − 1)! ( m − 1)! ( 2m )! m!

−2 ( −1)

m −1

(

m ( m + 1)

% ( −1)

m−2

− ( −1)

m

)

0 

(1 − ( −1) ) ( 2mm!)! m

1

0

0

1

0

0

0

0

0    0     ,  1    0  

  ( m + 2 )( m + 1) m   2  ,     

CT & = Cξ , DT = Dξ .

T

To establish the augmented system, the state vector is chosen as x g =  x ,  w f ,  wr ,  ε  , and define q1 = v, q2 = v , the augmented system can be presented as x g ( t ) = Ag ( q ) x g ( t ) + Bgu ( t ) + Bwg ( q )φ f ( t ) ,

z1 ( t ) = Cg1 x g ( t ) + Dg1u ( t ) ,

(2.238)

z2 ( t ) = Cg 2 x g ( t ) + Dg 2u ( t ) where

 A  Ag ( q ) =  0  0   B  Bg ( q ) =  0 2 × m  0 m × m Cgi ( q ) =  Ci

D aw I 2 01× 2

08 × m ( 2) ( m )T

bwe2 e1 AT

  ,  

 08 × m     1  B q = ,  b ( )  wg w m   ( −1)     BT

Dwi

  

   ,   

0 m × m  , Dgi ( q ) = Di , i = 1,2.

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Active Suspension Control

More explicitly, we have

 A  Ag ( q ) =  0   0

D

08 × m

− q1λ I 2

q2σ 2λ e2( 2)e1( m )T

01× 2

q1 AT

 08 × m       , Bwg ( q ) =  q2σ 2λ        q1 BT

1

( −1)

m

  

   .   

Assuming that the velocity v can be measured in real time, and the value of qi is in interval  qi , qi  , then the parameter q varies in a polytope Θ which has vertices ρ1 , ρ2 , ρ3 , ρ4, that is,  4  q ∈Θ := Co { ρ1 , ρ2 , ρ3 , ρ4 } =  α i ρi ,α ∈∆  ,  i =1  (2.239) 4   ∆ := (α 1 , α 2 , α 3 , α 4 ) : α i ≥ 0, α i = 1 , i =1  

where

 q1 ρ1 =   q2

  q1  q1   , ρ2 =  , ρ3 =    q2   q2 

  q1  ,  ρ4 =  q   2

 , 

with x=

q2 − q2 q1 − q1 , y= . q1 − q1 q2 − q2

And the matrices in system (2.238) belong to a given convex bounded polyhedral domain, that is, Ω  ( Ag ( q ) ,  Bq ( q ) ,  Bwg ( q ) , Cg1 ( q ) ,  Dg1 ( q ) , Cg 2 ( q ) ,  Dg 2 ( q ) ) ∈Π,

with

 Π  Ω := 

4

∑α Ω , α ∈∆  , i

i

i =1

 Agi  Ωi   Cg1i   Cg 2i

Bgi Dg1i Dg 2i

Bwgi   0 . 0 

Considering a velocity-dependent controller with the following structure

u ( q ) = K ( q ) x ( t ), (2.240)

the closed-loop augmented system can be presented as x g ( t ) = Aˆ g ( q ) x g ( t ) + Bwg ( q )φ f ( t ) ,

z g1 ( t ) = Cˆ g1 ( q ) x g ( t ) , z g 2 ( t ) = Cˆ g 2 ( q ) x g ( t ) ,

(2.241)

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Vibration Control of Vehicle Suspension Systems

where Aˆ g ( q ) = Ag ( q ) + Bg ( q ) K ( q ) , Cˆ g1 ( q ) = Cg1 ( q ) + Dg1 ( q ) K ( q ) ,

Cˆ g 2 ( q ) = Cg 2 ( q ) + Dg 2 ( q ) K ( q ) . For the state-feedback controller (2.240), the state information can be obtained by designing an observer. In this section, the state of system is assumed to be available. Then the control problem in this section can be summarized as: For an active suspension system (2.238) with wheelbase preview under velocity dependency, design a velocity-dependent controller such that, the augmented closed-loop system is robust stable, the H ∞ norm of closed-loop system is minimized, and the generalized H 2 norm is less than a given constant γ 2 , where the H ∞ norm and GH 2 norm are defined as

T1

= sup

φ f ∈L2

z1 2 , φf 2

T2

GH 2

= sup

φ f ∈L2

z1 φf

,

2

where 1

z1

2

∞ 2 =  z12 ( t )  dt  ,   0

1

φf

2

∞ 2 =  φ 2f ( t )  dt  ,   0

z2

= max z2 ( t ) . t ≥0

2.5.2 Velocity-Dependent Controller Design In this part, a multi-objective velocity-dependent controller will be designed, that is, the controller gain matrix K ( q ) is determined. Meanwhile, the asymptotically stable and H ∞ /GH 2 performance of closed-loop system can be guaranteed. The following two lemmas are useful for the controller design, which are related to the H ∞ and GH 2 performance, respectively. Lemma 2.12 [52] The closed-loop system in (2.241) is robustly asymptotically stable over Ω with T1 there exists a matrix P1 > 0 satisfying  Aˆ T q P q + P Aˆ q  g ( ) 1( ) 1 g ( )  *   *

P1 ( q ) Bwg ( q ) −I *

< γ ∞ if

Cˆ g1 ( q )   < 0, (2.242) 0  2 −γ ∞ I 

Lemma 2.13 [53] The closed-loop system in (2.241) is robustly asymptotically stable over Ω with T2 there exists a matrix P2 > 0 satisfying

 Aˆ T q P q + P q Aˆ q  g ( ) 2( ) 2( ) g( )  *

GH 2

< γ 2 if

P2 Bwg ( q )  < 0, (2.243) −I 

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Active Suspension Control

 −γ 2 I 2   * 

Cˆ g 2 ( q )  < 0, (2.244) − P2 ( q )  

These two Lemmas should be taken into account together to ensure the system performance. Pre- and post-multiply (2.242), (2.243), and (2.244) by diag P −1 ( q ) , I , I , diag P −1 ( q ) , I , and diag P −1 ( q ) , P −1 ( q ) , respectively, and define

{

{

}

}

{

}

X ( q ) = P −1 ( q ) , W ( q ) = K ( q ) P −1 ( q ) the following inequality can be obtained

  sym { Ag ( q ) X ( q ) + Bg ( q ) W ( q )}  *   *

Bwg ( q )

(C X ( q ) + D

−I *

0 −γ ∞2 I

 sym A ( q ) X ( q ) + B ( q ) W ( q ) { g } g   *   −γ 22 I   * 

g1

W ( q ))

g1

T

   < 0, (2.245)  

Bwg ( q )   < 0, (2.246)  −I 

Cg 2 ( q ) X ( q ) + Dg1 ( q ) W ( q )   < 0, (2.247)  − X (q) 

Here the homogeneous polynomial parameter-dependent controller (HPPD) will be designed. Assuming the velocity of vehicle is time-invariant but the values are within a polytope, to reduce the conservativeness of designed controller, the homogeneous polynomial parameterdependent approach is used. Let X (α ) , W (α )  ∈  4 × 4 take homogeneous polynomial forms of degree d ∈  and depend polynomial on the uncertain parameter α i ,  i = 1,2,3,4. That is, X (α , d ) =

J (d )

∑α

k1 1

α 2k2 α 3k3 α 4k4 X K j ( d ) ,

k1 1

α 2k2 α 3k3 α 4k4 WK j ( d ) , (2.248)

j =1

W (α , d ) =

J (d )

∑α j =1

[ k1 , k2 , k3 , k4 ] = K j ( d ). The notations used here are borrowed from [54, 55], they are explained as follows. Define K ( d ) as the set of 4-tuples obtained as all possible combination of [ k1 , k2 , k3 , k4 ], such that k1 + k2 + k3 + k4 = d . K j ( d ) is the jth 4-tuples of K ( d ), which is lexically ordered, j = 1,…, J ( d ). ( 3 + d )! The number of elements in K ( d ) is given by J ( d ) = . These elements define the subd !3! scripts k1 , k2 , k3 , k4 of the constant matrices

X[ k1 , k2 , k3 , k4 ]  X K j ( d ) , W[ k1 , k2 , k3 , k4 ]  WK j ( d ) , (2.249)

104

Vibration Control of Vehicle Suspension Systems

which are used to construct a homogeneous polynomial-dependent Lyapunov matrices in (2.248). For d = 0, X 0α = X , W0α = W which corresponds to a single Lyapunov matrix P = X −1. When d = 1, which corresponds to an affine parameter-dependent Lyapunov matrix. Note also that, imposing X K j ( d ) > 0, for j = 1,2,…, J ( d ) is a simple way to guarantee X dα > 0 under 4

the condition that

∑α = 1,α ≥ 0. i

i

i =1

For each set K ( d ), define the set F ( d ) with elements Fi ( d ) given by subsets of i, i ∈{1,2,3,4}, associated to the 4-tuples K j ( d ) of which ki are non-zero. For each i, i = 1,2,3,4, define the 4-tuples K ij ( d ) as being equal to K j ( d ) but with ki > 0 replaced by ki − 1. Note that, K ij ( d ) is only defined in the cases that the corresponding ki is positive. Note also that, when applied to the elements of K ( d + 1), the 4-tuples K li ( d + 1) define subscripts k1k2 k3 k4 of matrices X k1k2 k3 k4 , Wk1k2 k3 k4 associated to a homogeneous polynomially parameter-dependent matrices of degree d . Finally, define a scalar constant coefficients β ij ( d + 1) = d !/ ( k1 ! k2 ! k3 ! k4 !), with [ k1 , k2 , k3 , k4 ] ∈ K ij ( d + 1). By using this method, the following Theorem can be obtained for designing the controller. Theorem 2.10 Consider closed-loop system with velocity-dependent controller whose gain is given by −1 K ( q ) = W ( q ) X ( q ) , if there exist matrices X K j ( d ) > 0, WK j ( d ) , K j ( d ) ∈ K ( d ) , j = 1,2,…, J ( d ), such that the following LMIs hold for all K l ( d + 1) ∈ K ( d + 1) , l = 1,2,…, J ( d + 1):

  Ψ   * i ∈Fl ( d +1)   *

βli ( d + 1) Bgwi

(C

− βli ( d + 1) I

0

*

− βli ( d + 1)γ ∞2 I

g1i

X K i ( d +1) + Dg1iWK i ( d +1) l

l

)

T

   < 0 (2.250)   

 Ψ   * i ∈Fl ( d +1) 

βli ( d + 1) Bgwi   < 0 (2.251)  − βli ( d + 1) I 

 − βli ( d + 1)γ 22 I   * i ∈Fl ( d +1)  

Cg 2i X K i ( d +1) + Dg 2iWK i ( d +1)  l l  < 0 (2.252)  − X K i ( d +1) l 

(

)

T

where Ψ = Agi X K i ( d +1) + BgiWK i ( d +1) + Agi X K i ( d +1) + BgiWK i ( d +1) , then the homogeneous polyl

l

l

l

nomially matrices given by (2.248) assure that (2.245)–(2.247) hold for all α ∈∆. Moreover, if LMIs of (2.250)–(2.252) are fulfilled for a given degree d , then the LMIs corresponding to any degree d > d are also satisfied and smaller γ 1 can be found. Proof. Denote the matrices in (2.245), (2.250), (2.246), (2.251), and (2.247), (2.252) by Γ1, Ξ1, Γ 2 , Ξ2 and Γ 3 , Ξ3, respectively. Since X K j ( d ) > 0, K j ( d ) ∈ K ( g ) , j = 1,2,…, J ( d ), then X dα > 0 for all α i. Now, note that Γ1 , Γ 2 , Γ 3 for ( Ag ( q ) , Bq ( q ) , Bwg ( q ) , Cg1 ( q ) , Dg1 ( q ) , Cg 2 ( q ) , Dg 2 ( q ) ) ∈Ω and X dα , Wdα given by (2.248) are homogeneous polynomial matrix equations of degree d + 1 that can be written as

105

Active Suspension Control J ( d +1)

Γ1 =

∑α

k1 1

l =1

  α 2k2 α 3k3 α 4k4  Ξ1  , i ∈Fl ( d +1) 

J ( d +1)

Γ2 =

∑α l =1

k1 1

  α 2k2 α 3k3 α 4k4  Ξ2  , i ∈Fl ( d +1) 

J ( d +1)

Γ3 =

∑ l =1

  α 1k1α 2k2 α 3k3 α 4k4  Ξ3  , i ∈Fl ( d +1) 

[ k1 , k2 , k3 , k4 ] = K l ( d + 1). Conditions (2.250)–(2.252) imposed for all l , l = 1,2,…, J ( d + 1) ensure that Γ1 < 0, Γ 2 < 0 and Γ 3 < 0 for all α ∈∆. Then suppose that the LMIs in (2.250)–(2.252) are fulfilled for a certain degree d , that is, there exist J d symmetric positive definite matrices X Ki ( d ), and matrix WKi ( d ), i = 1,2,…, J d such that X ( dα ) , W ( dα ) are homogeneous polynomial parameter-dependent matrices ensuring Γ1 < 0, Γ 2 < 0, Γ 3 < 0. Then, the terms of the polynomial matrices X ( d + 1,α ) = (α 1 + α 2 + α 3 + α 4 ) X ( d ,α ), W ( d + 1,α ) = (α 1 + α 2 + α 3 + α 4 )W ( d ,α ) satisfy the LMIs of Theorem 2.10 corresponding to the degree d + 1, which can be obtained by linear combination of the LMIs of Theorem 2.10 for d . The smallest γ ∞ obtained with d is also feasible for d + 1. Due to the availability of more variables, smaller γ ∞ can be obtained for d + 1. The proof is completed. As the degree d of the polynomial increases, the conditions become less conservative because of the additional free variables. Although the number of LMIs is also increasing, each LMI becomes easier to be fulfilled due to the extra degrees of freedom.

()

()

Here is an example to illustrate the notation above. Assume d = 1, then J (1) = 4, K (1) = 1,0,0,0 ],[ 0,1,0,0 ],[ 0,0,1,0 ],[ 0,0,0,1 and

X (α ,1) = α 1 X[1,0,0,0 ] + α 2 X[0,1,0,0 ] + α 3 X[0,0,1,0 ] + α 4 X[0,0,0,1] , W (α ,1) = α 1W[1,0,0,0] + α 2W[0,1,0,0] + α 3W[0,0,1,0] + α 4W[0,0,0,1]. Moreover, J (1 + 1) = 10, K (1 + 1) =  2,0,0,0 ], [ 0,2,0,0 ], [ 0,0,2,0 ], [ 0,0,0,2  ,

1,1,0,0 ], [1,0,1,0 ], [1,0,0,1 ], [ 0,1,1,0 ], [ 0,1,0,1 ], [ 0,0,1,1, then F (1 + 1) = {{1} , {2} , {3} , {4} , {1,2} , {1,3} , {1,4} , {2,3} , {2,4} , {3,4}} and K11 = [1,0,0,0 ],

K 22 = [ 0,1,0,0 ], K 33 = [ 0,0,1,0 ], K 44 = [ 0,0,0,1], K 51 = [ 0,1,0,0 ], K 52 = [1,0,0,0 ], K 61 = [ 0,0,1,0 ], K 63 = [1,0,0,0 ], K 71 = [ 0,0,0,1], K 74 = [1,0,0,0 ], K 82 = [ 0,0,1,0 ], K 83 = [ 0,1,0,0 ], K 92 = [ 0,0,0,1],

3   K 94 = [ 0,1,0,0 ], K10 = [ 0,0,0,1], K104 = [ 0,0,1,0 ].

2.5.3 Linear Parameter-Varying Controller Design In the above section, the velocity v is considered to be a static parameter resides within a polytope, while if v is treated as a time-varying parameter, then system (2.238) will become a polytopic LPV system. Motivated by [53,56], the following theorem can be obtained for designing the LPV controller.

106

Vibration Control of Vehicle Suspension Systems

Theorem 2.11 Consider a continuous-time polytopic LPV system (2.238), there exists an LPV controller guaranteeing the system to be quadratically stable and satisfies T1 ∞ < γ 1, T2 GH2 < γ 2 along all parameter trajectories in the polytope 4  Θ :=  α i ( t ) ρi : α i ( t ) ≥ 0,  i =1

4

∑α (t ) = 1 i

i =1

if there exist X > 0 and Wi > 0, i = 1,…,4 such that T   Ag ( i ) X + BgiWi + ( Agi X + BgiWi )  *  * 

Bwgi

(Cg1i X + Dg1iWi )T

−I *

0 −γ ∞2 I

T   Agi X + BgiWi + ( Agi X + BgiWi )  * 

 −γ 22 I  * 

   < 0, (2.253)  

 Bwgi  < 0, (2.254) − I 

Cg 2i X + Dg1iWi   < 0, (2.255) −X 

under these conditions, an LPV controller can be obtained as K (q) =

4

∑α (t ) K (2.256) i

i

i =1

where

K i = Wi X −1 (2.257) Proof. (2.257) is a vertex controller, considering that the LPV controller is obtained as interpolant of these controllers, it is the polytopic controller. This LPV controller makes the closed-loop system polytopic. By using the vertex property in [53], (2.253)–(2.255) guarantee that for all q ( t ) ∈Θ, (2.245)–(2.247) hold. Then, the proof is completed. It should be noted that a naive interpolation of LTI controllers may fail to ensure stability and performance demand over Θ. The above approach is only valid in that a single Lyapunov function V ( x g ) = x gT X −1 x g is used over the entire range. The vertex controllers can be computed off-line, while the LPV controller gain must be updated in real time according to α i ( t ) based on the velocity measurement. The LPV control strategy proposed above has been developed under a quadratic framework, and operated by using a single Lyapunov function V ( x g ) = x g X −1 x g . However, it is different from the control strategy which utilizes homogeneous polynomially Lyapunov function with d = 0 which lead to X (α ,0 ) = X , W (α ,0 ) = W , the LPV controller leads to X ( x g ) = X , W ( x g ) = α 1W1 + α 2W2 + α 3W3 + α 4W4. The LPV controller is a velocity-dependent controller, while with homogeneous polynomially approach when d = 0, the resulting controller is a fixed controller for the whole range.

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Active Suspension Control

2.5.4  Simulation Results In this section, simulation is used to illustrate the advantage of the proposed controllers. The vehicle parameters are listed in Table 2.25, and choosing z fmax = zrmax = 0.08 m, u fmax = urmax = 1500 N. The parameters of road roughness are chosen as λ = 0.45m −1 ,σ 2 = 300 × 10 −6 m 2. The weighting parameters c1 , c2 are set to 1 and 1.3, respectively. The range of vehicle forward velocity is assumed as [10, 40] m/s, then q1 varies in the interval [10, 40 ], q2 varies in the interval  10 , 40  correspondingly. Refer to treating velocity as timeinvariant parameter or time-varying parameter, different situations are simulated. Homogeneous polynomially parameter-dependent approach is used to design the controller when the system is subject to fixed velocity over a polytopic domain, and an LPV controller is designed for the system subject to time-varying velocity. 2.5.4.1  Comparison of Different HPPD Controllers Three controllers with different degrees are considered here. For brevity, denoting the controller designed with d = 0 as HPPD0, the controller designed with d = 1 as HPPD1 and the controller designed with d = 2 as HPPD2 thereafter. The off-line (numerical) computational time increases with increasing HPPD degree (1.5 seconds, 10.1 seconds and 84.5 seconds for HPPD0, HPPD1, HPPD2, respectively). However, the online computational time of these controllers is generally negligible in practical applications when compared with the response time of the suspension system. Table 2.26 lists the minimized γ ∞ values of different conditions with three controllers, it can be seen that with the degree increases, smaller minimized γ ∞ values are obtained. In other words, smaller γ ∞ can be achieved with homogeneous polynomially parameter-dependent approach than that of with single Lyapunov function or affine parameter-dependent approach. Figure 2.34a shows the T1 ∞ among the whole range with three controllers. From this figure, it can be seen that for all admissible parameter v, all conditions have obtained less H ∞ norm compared with the open-loop system, and the smallest H ∞ norm corresponds to HPPD2. In addition, with HPPD2, for different admissible v, the generalized H 2 norm of elements in T2 are presented in Figure 2.34b–d, which clearly show that all of them are within constraints, the suspension deflection ratio and tyre deflection ratio are greater than those of open-loop system, that means the improvement of ride quality is at the cost of larger suspension deflection and tyre deflection. To verify the effects of the proposed controller in terms of the uncertainty of vehicle velocity, both vertical and angular accelerations of the open-loop and closed-loop systems with three different controllers are checked. Figure 2.34e and f compare the RMS values of the weighted vertical and angular accelerations under different velocities. It can be seen that HPPD1 and HPPD2 TABLE 2.25 Model Parameters M 500 kg k f1 10,000 N/m

J 910 kgm kr1 10,000 N/m 2

mf

mr

a

b

30 kg kf2 100,000 N/m

40 kg kr 2 100,000 N/m

1.25 m bf 1000 Ns/m

1.45 m br 1000 Ns/m

TABLE 2.26 γ ∞ Under Different Conditions Controller

HPPD0

HPPD1

HPPD2

γ∞

4.4145

2.5116

2.1993

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Vibration Control of Vehicle Suspension Systems

FIGURE 2.34  (a) T1 ∞ comparison, (b) suspension deflection performance, (c) tyre deflection performance, (d) actuator saturation performance, (e) RMS value of weighted vertical acceleration, and (f) RMS value of weighted pitch acceleration.

attenuated the vertical acceleration over the whole range, for HPPD0, both the vertical and pitch accelerations are greater than those of the passive system in the region of high velocities. Moreover, for the acceleration under the same velocity, with the degree increases, both vertical and angular accelerations are getting small, HPPD2 achieves the best performance. 2.5.4.2  Comparison of the Performance Under Different Road Conditions In order to verify the merits of the HPPD controller, two other different road conditions are considered here. σ = 80 × 10 −6 m 2, σ = 4000 × 10 −6 m 2 are chosen here for the better and worse road

Active Suspension Control

109

conditions and HPPD2 is selected here for comparing because of its relative higher degree. Figure 2.35 shows the RMS value of the vertical and pitch accelerations under two road conditions. It can be seen from the figures that no matter under which condition, the performance with HPPD2 controller is much better than that of the passive system. It indicates that the HPPD2 controller can significantly improve the performance under most road conditions. 2.5.4.3  Comparison of HPPD0 and LPV Controller Both HPPD0 and LPV controllers are formulated in quadratic framework by using a single Lyapunov function. The performances of these two controllers are compared. Figure 2.34a compares the T1 ∞ of two controllers. From the figure, it can be seen that among the entire velocity range, both H ∞ of with two controllers are smaller than that of the open-loop system, and the system with LPV controller obtains a little smaller T1 ∞ than that with HPPD0. It illustrates that the LPV controller yields less conservative results than HPPD0 corresponds to the discussion in the previous section. The RMS values under different velocities of two controllers are compared in Figure 2.34e and f. It can be seen that the LPV controller achieves smaller RMS values than that of HPPD0. That indicates that the LPV controller yields less conservative results than HPPD0, which corresponds to the discussion in the previous section. Moreover, both HPPD0 and LPV have larger accelerations than those of passive system at the high velocity range, which results from the conservatism of employing the single Lyapunov function.

FIGURE 2.35  Comparison of RMS value under σ = 80 × 10 −6 , (a) weighted vertical acceleration, (b) weighted pitch acceleration; σ = 4000 × 10 −6, (c) weighted vertical acceleration, (d) weighted pitch acceleration.

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Vibration Control of Vehicle Suspension Systems

2.5.5  Conclusions In this section, by employing wheelbase preview strategy, multi-objective controllers have been proposed for a half-car model, which gain is dependent on vehicle forward velocity. Due to the consideration of the velocity variation in the preview control, the proposed controller is more practical to be realized. By utilizing homogenous polynomially parameter-dependent approach, a less conservative controller can be obtained for velocities belong to a given polytope. On the other hand, a LPV controller is obtained for the vehicle within time-varying velocity. Numerical simulations have been presented to illustrate the performance of the designed controllers. The results show that with the degree increases, the HPPD controller yields less conservative results, obtains better performance and the LPV controller can achieve good performance when the system subject to varying velocity.

2.6  MOTION MODE CONTROL 2.6.1  System Description In this subsection, a seven-degree-of-freedom non-linear full-car suspension model is introduced for the control method design, as shown in Figure 2.36. The dynamic equations of this model can be written as mui  z0 i = Fsi + Fdi + Fti + ui , i = 1,2,3,4, 4

ms  zs = −

∑( F + F ) + u , si

di

z

i =1

Iθθ = a ( − Fs1 − Fd 1 − Fs 3 − Fd 3 ) − b ( − Fs 2 − Fd 2 − Fs 4 − Fd 4 ) + uθ , (2.258) Iφφ =

1 1 t f ( − Fs1 − Fd 1 ) + tr ( − Fs 2 − Fd 2 ) 2 2

1 1 ( − Fs 3 − Fd 3 ) − ( − Fs 4 − Fd 4 ) + uφ 2 2

where uz , uθ , uφ are control forces for heave, pitch and roll modes, respectively, which can be represented with ui ,  i = 1,2,3,4, as uz = − ( u1 + u2 + u3 + u4 ) ,

uθ = − ( au1 + au3 − bu2 − bu4 ) ,

uφ = −0.5 ( t f u1 + tr u2 − t f u3 − tr u4 ) .

(2.259)

The displacements of the sprung mass at each corner can be given as

1 z1 = zs + asinθ + t f sinφ , 2 1 z2 = zs − bsinθ + tr sinφ , 2 (2.260) 1 z3 = zs + asinθ − t f sinφ , 2 1 z4 = zs − bsinθ − tr sinφ . 2

111

Active Suspension Control Z4

Z2

tr

FS4

Fd4

u4

FS2

u2

Fd2

b

Z04 mu4 W2

Z02 mu2

ZS mS , I IS

Fr4

Fr2

w2

a t

Z3

FS3

Fd3

u3

Z1

FS1

Fd1

u1 Z01

Z03

W3

mu3

mu1

Fr3

Fr1

W1

FIGURE 2.36  Full-car suspension model.

It should be noted that the proposed method is based on a model-free strategy. In the following control method development, an integrated and accurate system model is not needed because the system dynamics can be estimated by an ESO. Good performance could be achieved in spite of highly non-linear or uncertain in the system.

2.6.2 Subsystem Establishment To implement the ADRC scheme, we should lump various known and unknown quantities which may affect the system performance into total disturbance [57]. Due to the complicated and coupled dynamic characteristics, it is difficult to convert the integrate full-car system as a single control problem within an ADRC formulation. Hence the full-car dynamic equations are divided as three subsystems by considering the heave, pitch, and roll motions of the sprung mass. First, considering the heave motion and denote q1z = zs , q2z = zs , the subsystem of the sprung mass can be established as  q1z = q2z ,  Σ z :  q2z = Fz ( q z , w, t ) + bz uz , (2.261)  yz = q1z , 

4

where Fz ( q z , w, t ) = − 1 ( Fsi + Fdi ) + Fz∆ is the total disturbance. Fz∆ is the unknown disturbance, ms i =1 1 bz = , and yz is the measured output. ms

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Vibration Control of Vehicle Suspension Systems

Next, for the pitch motion and roll motion, choosing q1θ = θ , qθ2 = θ and q1φ = φ , qφ2 = φ , the subsystems for them can be described as

 q1θ = qθ2 ,   θ Σθ :  q2 = Fθ qθ , w, t + bθ uθ , (2.262)  yθ = q1θ , 

 q1φ = qφ2 ,   Σφ :  qφ2 = Fφ ( qφ , w, t ) + bφ uφ , (2.263)  yφ = q1φ , 

(

)

where

Fθ qθ , w, t =

1 ( − a ( F1s + F1d + F3s + F3d ) + b ( F2s + F2d + F4 s + F4 d )) + Fθ∆ , Iθ

Fφ ( qφ , w, t ) =

1 (t f ( F3s + F3d − F1s − F1d ) + tr ( F4 s + F4 d − F2s − F2d )) + Fφ∆ 2 Iφ

(

)

and Fθ∆, Fφ∆ are the unknown disturbances for the pitch motion and roll motion subsystems, 1 1 bθ = , bφ = . Iθ Iφ In the following, the controller will be designed for the subsystems based on the ADRC scheme. For each subsystem, a motion-based controller will be established. The controller uz for the heave motion will be designed first, and the controllers uθ and uφ can be formulated by the same method. The schematic of ADRC, which includes ESO, tracking differentiator and controller design, is shown in Figure 2.37. It is worth noting that the motion-based controller designed in this part is not the same with the well-known mode control in vibration attenuation, which always operates in frequency domain with all the modes are uncoupled. The motion-based controller proposed here operates in time domain which means the dynamic properties may couple with each other. Usually, for system with coupled dynamics, it is not easy to attenuate the vibration from separated motions in time domain, while the ADRC scheme can address this issue, since the inherent coupled dynamics can be treated as a part of total disturbance. y2nf

Tracking differentiator

v21

w e1

e2

v22

PD/Fuzzy-PD uoz control –

1/ b0z

uz

b0z

pz2 pz1

FIGURE 2.37  The schematic of ADRC.

pz3

Linear extended state observer

z

yz

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Active Suspension Control

2.6.3 Extended State Observer Design For the heave motion subsystem, Fz ( q z , w, t ) is the total disturbance, which should be observed by an ESO. Denoting q3z = Fz ( q z , w, t ) as an additional state variable, assume that q3z is differentiable with respect to time and let Fz ( t ) = Gz, where Gz is unknown, then the original system (2.265) can be described as  q1z = q2z ,   q2z = q3z + bz uz , (2.264)  z z  q3 = Gz ( q , w, t ) ,  z  yz = q1 And the ESO can be established in the following form

      

ez = p1z − yz , z p1z = p2z − β 01 ez , z p 2z = p3z − β 02 ez + b0 z uz ,

(2.265)

z p 3z = − β 03 ez

z z z where p1z , p2z , p3z are the components of the ESO, and β 01 , β 02 , β 03 , b0 z are constants. It should be noted that the internal dynamics of the suspension and the external disturbance can be estimated as p3z by the ESO, and p3z is used to compensate the total disturbance in the feedback. Then the controller can be given by

uz =

u0 z − p3z (2.266) b0 z

where b0 z ≈ bz and u0 z will be designed in the following subsection.

2.6.4 PD and Fuzzy-PD Controller Design In this subsection, PD and fuzzy-PD are adopted to design u 0z, which amount to a linear and nonlinear PD control. PD controller is easy to implement in practice while fuzzy-PD controller may achieve better performance under some conditions. Considering the classic PD method, u0 z can be expressed by the following control law

u0 z = k pze1 + kdz e2 (2.267)

z z where k pz and kdz are the proportional and derivative gains, respectively. e1 = yref − p1z , e2 = e1 = yref − p2z, z and yref is the reference signal. The selection of parameters for the extended observer and classic PD controller can be referred to [58]. The classic PD controller gain can be chosen as k pz = 2ξω c , kdz = ω c2 , where ω c , ξ are the desired natural frequency and damping ratio of the closed-loop system, respectively. The bandwidth of the observer can be chosen as ω o = 3 ~ 5ω c , and the parameters of the observer can be selected z z z as β 01 = 3ω o, β 02 = 3ω o2 , β 03 = ω o3. Notice that, the upper bound of estimation error will monotonously decrease with the increase of the observer bandwidth in the absence of an accurate system model [59].

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Vibration Control of Vehicle Suspension Systems

Then considering the fuzzy-PD controller design, it can be derived from the classic PD controller. In this part, the zero-order Sugeno method [60] is used to construct the fuzzy-PD controller. The fuzzy control rules are expressed as follows Rule i: IF k pze1 is M1i and kdz e2 is M 2i , THEN u0 z is M ui. where M1i , M 2i and M ui are the fuzzy sets, and could be characterized by membership functions which named as “negative large” (NL, nl), “negative medium” (NM, nm), “negative small” (NS, ns), “zero” (ZE, zp), “positive small” (PS, ps), “positive medium” (PM, pm) and “positive large” (PL, pl). Triangular membership function is used to formulate the fuzzy sets. The universe of discourse for M1i , M 2i and M ui are taken to be in  M1−i , M1+i ,  M 2−i , M 2+i  and  M ui− , M ui+ . Each label for M ui takes 2 − 1 1 2 M ui , ns = M ui− , ze = 0, ps = M ui+ , pm = M ui+ , pl = M ui+ . For clarity, the 3 3 3 3 fuzzy rules are listed in Table 2.27. Then considering the different control objectives in different conditions, two strategies are introduced here for the reference signal selection. Strategy I: Considering the design of active suspension, the main objective is to attenuate the vibration of sprung mass and stabilize the vehicle attitude [61,62], in this case, the reference sigz z nal is selecting as yref = yref = 0. That leads to e1 = − p1z , e2 = − p2z, and the controller parameter u 0z becomes

values as nl = M ui− , nm =

u0 z = − k pz p1z − kdz p2z . (2.268)

Strategy II: Considering the stochastic excitation for all wheels and keeping enough rattle space, the reference signal could be chosen as the following form, which is related to the effective value of the unsprung mass displacement under the centre of gravity [63,64]. z yref =

c0  a ( z02 + z04 ) + b ( z01 + z03 )   (2.269) 2  a+b

z It should be noted that in the strategy II, the reference signal yref and its derivative are both required. z z z And a tracking differentiator [57] is constructed here with v1 tracking yref and v2z tracking yref .

2.6.5 Actuator Force In order to calculate the real actuator inputs u1, u2 , u3 and u4 , a new constraint is introduced to ensure that the actuator forces do not result in a torsional moment at the CG of the sprung mass, and the torsional moment may cause mechanical damage to the structure and have a negative impact on ride TABLE 2.27 Fuzzy Rules M2 i M1 i NL NM NS ZE PS PM PL

NL nl nl nl nl nm ns ns

NM nl nl nl nm ns ns ns

NS nl nl nm ns ns ns ns

ZE ns ns ns ze ps ps ps

PS ps ps ps ps pm pl pl

PM ps ps ps pm pl pl pl

PL ps ps pm pl pl pl pl

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Active Suspension Control

safety, i.e. t f u1 − t f u3 = tr u2 − tr u4 . When this is combined with equation (2.259), then we can obtain the nominal actuator inputs u01, u02, u03 and u04 as

u01 = −

uφ ( a + b ) + t f uθ + bt f uz 2t f ( a + b )

u02 = −

uφ ( a + b ) − tr uθ + atr uz 2tr ( a + b )

u ( a + b ) − t f uθ − bt f uz u03 = − φ 2t f ( a + b ) u04 = −

(2.270)

uφ ( a + b ) + tr uθ − atr uz 2tr ( a + b )

It should be noted that the structural parameters a, b, t f and tr are necessary in the computations of the four actuator force c, and these structural parameters of a vehicle can be easily obtained. The actuator saturation is also taken into consideration, the saturated actuator inputs ui can be obtained as

 u lim ,  i ui = sat ( u0 i )  u0 i ,  lim  −ui ,

u0 i ≥ uilim −uilim < u0 i < uilim ,

i = 1,2,3,4.

u0 i ≤ −u

lim i

where uilim > 0 denotes the force limitation of the ith actuator. A design diagram of the whole active vehicle suspension system is shown in Figure 2.38.

2.6.6  Simulation Results In this section, simulations are carried out to verify the proposed control method in different road conditions. For simulation convenience, the tyre force, the suspension spring force and the

FIGURE 2.38  Sketch of the full-car active suspension design.

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Vibration Control of Vehicle Suspension Systems

suspension damping force are described as follows. A tyre is equivalent to a spring, and its force Fti can be calculated as

Fti = − kti ( z0 i − wi ) ,

i = 1,2,3,4

The components of passive suspension system always have non-linear properties and may change during the vehicle lifespan, such as the springs and the dampers. The spring forces can be given in the form

Fsi = k fi ( zi − z0 i ) + kni ( zi − z0 i ) , 3

i = 1,2,3,4

where k fi > 0 and kni > 0; the term of first degree represents the linear part of the spring force, and the term of second degree represents the non-linear spring force characteristics. The suspension damper is a device with obvious complex non-linear behaviour; it may show asymmetric multistage force-velocity dynamic characteristics as shown in Figure 2.39, which can be described as

    Fd =     

cd 1∆v, cd 2 ∆v, cd 3 ∆v, cd 4 ∆v, cds ∆v, cd 6 ∆v,

∆v < ∆v1 ∆v1 ≤ ∆v < ∆v2 ∆v2 ≤ ∆v < 0 0 ≤ ∆v < ∆v3 ∆v3 ≤ ∆v4 < 0 ∆v ≥ ∆v4

where ∆v is the relative velocity between the two terminals. The parameters of the suspension system used in the simulations are listed in Table 2.28. 2.6.6.1  Road Conditions Two types of road model, namely a stochastic road model and a deterministic road model, are used to formulate different road conditions. The stochastic road profile can be approximated as a stationary stochastic process with the spectral density

S (ω ) =

σ αv (2.271) π ω 2 + (α v )2

FIGURE 2.39  Non-linear property of the suspension damper.

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Active Suspension Control

TABLE 2.28 Model Parameters of a Full-Car Suspension System Parameter

Value

Parameter

Value

Parameter

ms

1460 kg

b

1.803 m

cd 4

2850 N s/m

mu1 , mu3 mu 2 , mu 4 tf

40 kg 35.5 kg 1.522 m

∆v1 ∆v2 ∆v3

−0.3 m/s −0.1 m/s 0.1 m/s

cd 5 cd 6

660 N s/m 660 N s/m 1500 N

tr

1.510 m

∆v4

0.3 m/s

1200 N/m 1290 N/m 1.011 m

cd1 cd 2 cd 3

270 N s/m 375 N s/m 690 N s/m

kn1 , kn3 kn 2 , kn 4 a

u1lim , u2lim u3lim , u4lim k f 1, k f 3 kf 2,kf 4 kt

Value

1500 N 19,860 N/m 17,400 N/m 175,500 N/m

where ω is the circular frequency (rad/s), α is a constant related to the type of road surface, σ is the standard deviation of the road unevenness and v is the forward velocity. Then the road surface elevation process w ( t ) can be given as

w ( t ) + α vw ( t ) = ξ (2.272)

where ξ is a Gaussian white-noise process, and the mathematical expectation is given as

T E ξ ( t1 )ξ ( t2 )  = 2α vσ 2δ ( t1 − t2 )

with

 1, δ (s) =   0,

s=0 s≠0

A type of deterministic road profile, i.e. an isolated bump in an otherwise smooth road, is also studied. The road profile is given as

 h 1 − cos (8π t ) ],  [ w (t ) =  2  0, 

1 < t < 1.25

(2.273)

otherwise

where h is the height of the bump. Based on these two types of road profile, the road condition is considered in simulation as follows. The vehicle runs on a stochastic road surface; the road input at the rear wheels is the delay of front wheels, likes w2 ( t ) = w1 [ t − ( a + b ) / v ] and w4 ( t ) = w3 [ t − ( a + b ) / v ]. The inputs at the right wheels and at the left wheels are related, and it is assumed that the road inputs on the right side and on the left side have a correlation coefficient Rw . The model parameters for simulating the road surface in this condition are listed in Table 2.29. 2.6.6.2  Performance Comparison and Discussion The surface of input-output gain is provided in Figure 2.40; the surfaces of u0θ and u0φ are similar to that of u0 z , and the corresponding input-output ranges of them are listed in Table 2.30. Here, we choose r = 60 in the track differentiator and fc = 5, s = 1 to compute the corresponding parameters in the ESO and the classic PD controller. In this condition, strategy II is used to design the

118

Vibration Control of Vehicle Suspension Systems

TABLE 2.29 Road Profile Parameters Parameter

Value

V σ2

20 m/s 256 × 10−6 m2

α Rw

0.45 m−1 0.2021

FIGURE 2.40  Input–output surface of u0 z .

TABLE 2.30 Road Profile Parameters Sugeno Controller

M1−

M1+

M 2−

M 2+

M u−

M u+

u0 z u0θ u0φ

−0.4 −0.2 −0.8

0.4 0.2 0.8

−3 −0.8 −2

3 0.8 2

−3 −1 −2.8

3 1 2.8

controllers for heave motion to ensure that there is sufficient suspension deflection space, and strategy I is used to design the controllers for pitch motion and roll motion in order to attenuate the vibrations. Comparisons of zs , u and f for passive and active suspensions with the proposed control method, and results are shown in Figure 2.41.

119

Active Suspension Control (a)

0.15

open–loop system closed–loop system with PD control closed–loop system with fuzzy–PD control

0.1

Zs(m/s)

0.05

0

–0.05

–0.1

(b)

0

5

10

15 t (s)

20

25

30

0.015 0.01

0.005 0 –0.005 –0.01 –0.015

open–loop system closed–loop system with PD control closed–loop system with fuzzy–PD control

0

5

10

15

20

25

30

t (s)

(c)

0.1

open–loop system closed–loop system with PD control closed–loop system with fuzzy–PD control

0.08 0.06 0.04

0.02 0 –0.02 –0.04 –0.06 –0.08 –0.1

0

5

10

15

20

25

30

t (s)

FIGURE 2.41  (a) Comparison of the vertical displacements zs, (b) comparison of the pitch angles θ , (c) comparison of the roll angles φ .

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Vibration Control of Vehicle Suspension Systems

2.6.7  Conclusions In this part, a new control strategy for non-linear full-car suspension system is proposed. By considering heave motion, pitch motion and roll motion, three subsystems are extracted from the vehicle suspension dynamics. For each subsystem, a motion-based controller is designed based on the active disturbance rejection control principle. An ESO is designed to estimate the total disturbance including coupling dynamics and uncertainty. Based on the observation disturbance compensation, a PD controller and a fuzzy-PD controller are designed to suppress the heave vibration, the pitch vibrations, and the roll vibrations of the sprung mass. By calculating three motion-based controller forces, four actuator inputs can be obtained online. From the simulation of two typical road conditions, the effectiveness of the control method is verified.

3

Semi-active Suspension Control

3.1 INTRODUCTION This chapter introduces three semi-active suspension control methods, i.e. varying damping, varying inerter, and varying equivalent stiffness and inerter. Two typical varying damping dampers, variable magnetorheological and varying ER dampers, are introduced first. Then, a novel control strategy is developed to achieve variable inerter. Finally, an equivalent variable stiffness and inertance device is developed based on electric network way. This chapter presents a study on the direct voltage control of a magnetorheological (MR) damper for application in vehicle suspensions. As MR damper dynamics is highly nonlinear, the direct control system design for an MR damper is difficult. Representing an MR damper by a Takagi-Sugeno (TS) fuzzy model enables the linear control theory to be directly applied to design the MR damper controller. In this chapter, first the MR damper dynamics is represented by a TS fuzzy model, and then an H ∞ controller that considers the suspension performance requirements and the constraint on the input voltage for the MR damper is designed. Furthermore, considering the case that not all the state variables are measurable in practice, the design of an H ∞ observer with immeasurable premise variables and the design of a robust controller are proposed, respectively. Numerical simulations are used to validate the effectiveness of the proposed approaches. In this chapter, a novel robust control approach is presented for a semi-active suspension with (ER) dampers considering suspension parameter uncertainties, control input constraint, and measurement noises. By representing the suspension with parameter uncertainties in a polytopic form, applying a norm-bounded approach to handle control input constraint, and constructing an appropriate observer to estimate state variables from noisy measurements, the design of this controller is realized by solving a finite number of linear matrix inequalities with optimized H∞ performance on ride comfort and estimation error. Numerical simulations on a quarter-car suspension with an ER damper are performed to validate the effectiveness of the proposed approach. The obtained results show that the designed controller can achieve good suspension performance despite the presence of measurement noises and the variations on sprung mass and ER damper time constant with constrained control input. Inerter is a mechanical two-terminal element and the applied force F is proportional to the relative acceleration between these two terminals, and F = b (  z1 −  z2 ), where b is the constant of proportionality called the inertance, which has units of kilograms. The inerter can broaden the class of mechanical realizations of complex impedances which could only be operated with springs and dampers in the past. In the past research studies, the fixed inerter was used in the passive suspension, and the better performance could be obtained compared with the suspension without inerters. However, these suspensions are not suitable for general applications due to the more complex suspension layout. Although the structure consists of one damper, one inerter, and several springs for suspension have been proposed, the realization conditions are not easily met. In this section, assuming the inertance could be adjusted in real-time, the adaptive inerter is considered for improving suspension performance by employing a force tracking strategy. Different from previous semiactive control with inerter, in which a variable damper is under control while the inerter is a fixed element or both the damper and inerter can be adjusted. In this section, the only variable element is the inerter, which can be adjusted in real-time to track the desired force. The simplest suspension DOI: 10.1201/9781003265665-3

121

122

Vibration Control of Vehicle Suspension Systems

layout is employed, that is, a quarter-car suspension model with an inerter installed in parallel with a spring and a damper. First, the effects of a fixed inerter on a given suspension system are analyzed which illustrates its limitation in improving the vehicle performance. Then, considering the ride quality, suspension deflection and tyre deflection performances independently, an H 2 controller is designed, where the desired control force is approximated by adaptively varying the inertance. The control performance of the proposed scheme is verified through simulations. In this chapter, a semi-active variable equivalent stiffness and inertance (VESI) device is developed by an electrical network and an electromagnetic device instead of a mechanical system. Due to that the inerter and spring can store energy, the dynamic control of their mechanical properties without the external power to compensate the stored energy will cause the system discontinuity, which will affect the control performance. Therefore, the variable damping (VD) device is applied to connect with spring or inerter in series to obtain variable equivalent stiffness or inertance properties. Then, according to the force-current analogy, an electrical “VESI” network is designed to simulate the mechanical VESI network. An electromagnetic device equips with the variable electrical system can achieve the VESI mechanical properties, which is validated with experiments. The imaginary part of the admittance of the proposed device can vary to the positive phase and also the negative phase by controlling two resistors. The application of VESI device in the vehicle is investigated and analysed. A semi-active control strategy is designed for vibration control and rollover prevention of vehicles with VESI suspensions. According to the simulation result, the proposed semi-active system can significantly improve the vibration reduction performance at both vertical and roll directions; also, it can decrease the roll angle when vehicles in the cornering manoeuvre. Besides, the semiactive VESI device with an electrical network has the advantages in installation and maintenance.

3.2  VARYING DAMPING 3.2.1 Magnetorheological Damper 3.2.1.1  MR Damper Model To date, many different models have been proposed to describe the nonlinear dynamics of MR dampers. Among all the proposed models, a phenomenological model proposed by Spencer et al. [1] is regarded as a representative model to portray the behaviour of the MR damper. This phenomenological model is based on a Bouc–Wen hysteresis model, which is numerically tractable and is capable of exhibiting a wide variety of hysteretic behaviours. The parameters for the model are determined from the experimental data with an appropriate optimization method. The phenomenological model of MR dampers is regarded as the “state-of-the-art” semi-physical model and has been extensively used in modelling MR dampers with applications in vehicle suspension control. The phenomenological model is governed by the following seven simultaneous equations: fd = c1 yd + k1 ( x d − x 0 ) , yd =

1

α zd + c0 xd + k0 ( x d − yd )  ,

( c0 + c1 ) 

zd = −γ d xd − yd zd zd

α = α a + α b v, c1 = c1a + c1b v, c0 = c0a + c0b v, v = −η ( v − u ) ,

n −1

− β ( x d − yd ) zd + Ad ( xd − yd ) , n

(3.1)

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Semi-active Suspension Control

TABLE 3.1 Parameters for an MR Damper Parameter

Value

Parameter

Value

c0a

784 Ns/m

αa

12, 441Nm −1

c0b

1083 Ns/ ( mV )

αb

38, 430 N ( mV )

k0

3610 N/m

γd

136,320 m

c1a

14, 649 Ns/m

βd

2, 059, 020 m −2 58

−1

−2

c1b

34, 622 Ns/ ( mV )

k1

840 Nm −1 0.0245 m

n

2

η

190 second −1

x0

where fd is the force generated by the MR damper; x d is the displacement of the damper; yd is an internal pseudo-displacement of the MR damper; v is the output of a first-order filter; and u is the command voltage sent to the current driver. In this model, k1 is the accumulator stiffness; c0 and c1 are the viscous damping coefficients observed at high and low velocities, respectively; k0 is the gain to control the stiffness at large velocities; x 0 is the initial displacement of spring k1 associated with the nominal damper force due to the accumulator; γ d , β d , and Ad are hysteresis parameters for the yielding element; and α is the evolutionary coefficient. A set of parameters that were obtained by Lai and Liao [2] to characterize one MR damper using experimental data and a constrained nonlinear optimization algorithm is listed in Table 3.1. 3.2.1.2  Quarter-Car Model with MR Damper The quarter-car model with an MR damper shown in Figure 3.1 is considered here in designing the control law. The governing equations of motion for the sprung and unsprung masses are

ms  z s = − k s ( z s − z u ) − fd , mu  z u = − k s ( z u − z s ) − k t ( z u − z r ) + fd ,

(3.2)

where zs and zu are the displacements of the sprung and unsprung masses, respectively; ms is the sprung mass, which represents the car chassis; mu is the unsprung mass, which represents the wheel assembly; ks is the stiffness of the passive suspension system; kt stands for compressibility of the pneumatic tyre, respectively; zr is the road displacement input; fd represents the controllable damping force generated by the MR damper. It is noted that, in the model, the passive damping force cs ( zs – zu ), where cs is the damping coefficient of the passive suspension system, is replaced by the MR damping force fd . To simplify the representation of the MR damping force [3], a simple Bouc– Wen model, which can accurately predict the force–displacement behaviour of an MR damper like the phenomenological model (3.1) (also called modified Bouc–Wen model) [1] and is well suited for numerical simulation [4], will be adopted here. Then, (3.2) is written as ms  zs = − ks ( zs − zu ) − c0 ( zs − zu ) − k0 ( zs − zu ) − α zd , mu  zu = − k S ( zu − zs ) − kt ( zu − zr ) + c0 ( zs − zu ) + k0 ( zs − zu ) + α zd ,

zd = −γ d zs − zu zd zd c0 = c0a + c0b v,

α = α a + α b v, v = −η ( v − u ) .

n −1

− β d ( zs − zu ) zd + Ad ( zs − zu ) , n

(3.3)

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Vibration Control of Vehicle Suspension Systems

ks

Bouc-Wen

MR damper

suspension

zs

ms

car

zu

mu

wheel

k0

c0

kt

tyre

zr

FIGURE 3.1  Quarter-car model with MR damper.

Note that in the simple Bouc–Wen model, there are no c1, k1, and yd . The term x d in (3.1) is equivalent to zs − zu in (3.3). Defining the state variables as follows: x1 = zs , x 4 = zu ,

x 2 = zs , x5 = zd ,

x 3 = zu , (3.4) x6 = v ,

and the state vector as x = [ x1   x 2   x3   x 4   x5   x6 ] , (3.3) can be written as T

x2 =

1  −c0 a ( x 2 − x 4 ) − ( k0 + ks )( x1 − x3 ) −α a x5 − f1 x6 ] , ms 

x 4 =

1  −c0 a ( x 4 − x 2 ) − ( k 0 + ks )( x3 − x1 ) − kt ( x3 − zr ) + α a x5 + f1 x6  , (3.5) mu 

x5 = Ad ( x 2 − x 4 ) + f2 , x6 = −η ( x6 − u ) , where the two nonlinear functions f1 and f2 are defined as

f1 = C0 b ( x 2 − x 4 ) + α b x5 , (3.6)

n −1 n − β d ( x 2 − x 4 ) x5 . (3.7) f2 = −γ d x 2 − x 4 x5 x5

Note that an absolute value of a quantity δ implies

δ = δ sign (δ ) , (3.8)

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Semi-active Suspension Control

where  −1,  sign (δ ) =  0,  1, 

if δ < 0 if δ = 0 (3.9) if δ > 0

Based on Table 3.1, where n = 2 is commonly chosen, (3.7) can be represented as f2 = − x5 (γ d x 2 − x 4 + β d ( x 2 − x 4 ) sign ( x5 )) x5 . (3.10)

By defining

f2 = − x5 (γ d x 2 − x 4 + β d ( x 2 − x 4 ) sign ( x5 )) , (3.11)

we can write a state-space equation for (3.5) as x ( t ) = A ( f ) x + B1w + B2u, (3.12)

where  0  k  − 0 + ks ms   0 A(f ) =   k0 + ks  mu  0   0   0  0   0 B1 =  kt   mu  0  0 

1 c0 a − ms 0 c0 a mu Ad 0

0 k0 + ks ms 0 k + k s + kt − 0 mu 0 0

0 c0 a ms 1 c − 0a mu − Ad 0

0 α − a ms 0 αa mu f2 0

0 f − 1 ms 0 f1 mu 0 −η

     ,      

  0    0       ,   B2 =  0  ,   w = zr  0       0    η   

3.2.1.3  TS Fuzzy Modelling of Quarter-Car with MR Damper Since the state variables x 2, x 4, and x5 are actually bounded in practice for a stable system, the nonlinear functions f1 and f2 in (3.6) and (3.11) are also limited in operation. Suppose the nonlinear function f1 is bounded by its minimum value f1min  and its maximum value f1max , then using the “sector nonlinearity” approach [5] it is not difficult to represent the nonlinear function f1 by

f1 = M1 (ξ1 ) f1max + M 2 (ξ1 ) f1min , (3.13)

where ξ1 = f1 is denoted as a premise variable, and M1 (ξ1 ) and M 2 (ξ1 ) are membership functions which provide a measure for the degree of similarity and are defined as

126

Vibration Control of Vehicle Suspension Systems

f1 − f1min , f1max − f1min (3.14) f1max − f1 M 2 (ξ1 ) = . f1max − f1min M1 (ξ1 ) =

Similarly, the nonlinear function f2 is bounded by its minimum value f2 min and its maximum value f2 max so that it can be represented by f2 = N1 (ξ 2 ) f2 max + N 2 (ξ 2 ) f2 min , (3.15)

where ξ 2   =   f2 is also denoted as a premise variable, and N1 (ξ 2 ) and N 2 (ξ 2 ) are membership functions which are defined as f2 − f2 min , f2 max − f2 min (3.16) f − f2 N 2 (ξ 2 ) = 2 max . f2 max − f2 min N1 ( ξ 2 ) =

To make the description more understandable, we name the aforementioned membership functions M1 (ξ1 ), M 2 (ξ1 ), N1 (ξ 2 ) and N 2 (ξ 2 ) as “Positive,” “Negative,” “Big,” and “Small,” respectively, in terms of their possible ranges, in which case the nonlinear quarter-car suspension model (3.12) can be represented by the following models: Model rule 1:

IF THEN

ξ1  is 'Positive' and ξ 2  is 'Big', x = A1x + B1w + B2u.

Model rule 2:

IF THEN

ξ1  is 'Positive' and ξ 2  is 'Small', x = A 2 x + B1w + B2u.

Model rule 3:

IF THEN

ξ1  is 'Negative' and ξ 2  is 'Big', x = A3 x + B1w + B2u.

Model rule 4:

IF THEN

ξ1  is 'Negative' and ξ 2  is 'Small', x = A 4 x + B1w + B2u.

where matrices Ai, i = 1,,4, are obtained by replacing f1 and f2 in matrix A ( f ) of (3.12) with fimin and fimax , respectively, according to the above-defined rules.

127

Semi-active Suspension Control

The TS fuzzy model that exactly represents the nonlinear quarter-car model with an MR damper (3.12) under the assumption of bounded state variables is obtained as 4

x =

∑ µ (ξ ) A x + B w + B u = A x + B w + B u, (3.17) i

i

1

2

h

1

2

i =1

4

where A h =

∑µ (ξ ) A , i

i

i =1

µ1 (ξ ) = M1 (ξ1 ) N1 (ξ 2 ) , µ2 (ξ ) = M1 (ξ1 ) N 2 (ξ 2 ) ,

µ3 (ξ ) = M 2 (ξ1 ) N1 (ξ 2 ) , µ4 (ξ ) = M 2 (ξ1 ) N 2 (ξ 2 ) ,

µi (ξ ) ≥ 0, i = 1 − 4 and

4

∑µ (ξ ) = 1. i

i =1

Note that this TS fuzzy model can exactly represent the quarter-car dynamics with a simple Bouc– Wen MR damper model. If a simple Bouc–Wen model may not predict the system exactly, the modelling uncertainty can be included in the TS fuzzy model as 4

x =

∑µ (ξ )( A + ∆A ) x + B w + B u, (3.18) i

i

i

1

2

i =1

where ∆A i are norm-bounded matrices which are used to describe the uncertainty caused by the parameter variations and system modelling errors. As there is no great difference between the simple Bouc–Wen model and the modified Bouc–Wen model in modelling, the dynamics of an MR damper, (3.18) will not be applied in this section. In the simulation, all the tests will be validated by using the modified Bouc–Wen model, and it is confirmed by the simulation that the obtained results have no significant differences compared to the case of using the simple Bouc–Wen model. In addition, the input voltage sent to the MR damper is practically limited, and hence the actuator saturation needs to be considered. In general, a control input with saturation limitation is defined as u = sat ( u ), where sat ( u ) is a saturation function defined as

where umin ≤ 0 < umax

 umin , if u < umin ,  if umin ≤ u ≤ umax , (3.19) sat ( u ) =  u,  umax , if u > umax ,  are the control input limits. Therefore, the system (3.17) is modified as x = A h x + B1w + B2u. (3.20)

3.2.1.4  TS Fuzzy Control of Quarter-Car with MR Damper The fuzzy controller design for the TS fuzzy model (3.20) is carried out based on the so-called parallel distributed compensation (PDC) scheme [5]. For the TS fuzzy model (3.20), we construct the fuzzy state feedback controller through the PDC as

128

Vibration Control of Vehicle Suspension Systems 4

u=

∑µ (ξ )K x = K x, (3.21) i

i

h

i =1

4

where K h =

∑ µ (ξ )K , K are the state feedback gain matrices to be designed. i

i

i

i =1

For vehicle suspension design, ride comfort, road holding ability, and suspension deflection are three main performance criteria. Ride comfort, which can be usually quantified by the sprung mass acceleration, is the main control objective that needs to be optimized in the controller design process. In addition, due to the disturbances caused by road bumpiness, a firm uninterrupted contact of wheels with the road (good road holding) is important for vehicle handling and is essentially related to driving safety. To ensure good road holding, it is required that the tyre deflection, zu − zr , should be small. The structural features of the vehicle also constrain the amount of suspension deflection, zs − zu , with a hard limit. Hitting the deflection limit not only results in rapid deterioration in the ride comfort, but at the same time increases the wear of the vehicle. Hence, it is also important to keep the suspension deflection, zs − zu , small enough to prevent excessive suspension bottoming. To satisfy the above-mentioned performance requirements, the controlled output z will be composed zs , zs − zu , and zu − zr , for the quarter-car suspension model. Therefore, the controlled output is of  defined as z =  λ1 zs   λ2 ( zs − zu )  λ3 ( zu − zr ) = T

4

∑µ (ξ )C x + Dw = C x + Dw, (3.22) i

i

h

i =1

4

where C h =

∑ µ (ξ )C  , i

i

i =1

 k0 + ks  − λ1 ms Ci =   λ2   0

− λ1 0 0

c0 a ms

k0 + ks ms − λ2 λ3

λ1

λ1

c0 a ms

0 0

− λ1 0 0

αa ms

− λ1 0 0

fi ms

  0    ,  D =  0    − λ3  

   , 

where λ1   >  0 , λ2   >  0, and λ3   >  0 are weighting parameters. Note that it is standard to use weighting functions to shape and compromise different performance objectives, and the weighting parameters λ1, λ2 , and λ3 can be properly chosen to provide the trade-off among ride comfort, suspension deflection, and road holding [6]. To design a suspension control system to perform adequately in a wide range of shock and vibration environments, the H ∞ norm is chosen as the performance measure. The L2 gain of the system (3.20) with (3.22), which is defined as

Tzw ∞

where z

2 2

= z T ( t ) z ( t ) dt and w 0

 z 2 , (3.23) w2 ≠ 0  w 2

= sup

2 2

= w T ( t ) w ( t ) dt and the supremum is taken over all nonzero 0

trajectories of the system (3.20) with x ( 0 ) = 0, is chosen as the performance measure. Our aim is to

129

Semi-active Suspension Control

design a fuzzy controller (3.21) such that the fuzzy system (3.20) with controller (3.21) is quadratically stable and the L2 gain (3.23) is minimized. To design such a controller, the following lemmas will be used. Lemma 3.1 [7] For the saturation constraint defined by (3.19), as long as u ≤ u−

ulim , we have ε

1+ ε 1− ε  u , (3.24) u ≤ 2 2

and hence, 2

T

u − 1 + ε u  u − 1 + ε u  ≤  1 − ε  u T u, (3.25)    1   1   2 

where 0 < ε < 1 is a given scalar. To apply Lemma 3.1, system (3.20) is further written as x = A h x + B1w +

where d = u −

1+ ε 1+ ε   u B 2u + B 2  u −  2 2 

1+ ε = A h x + B1w + B 2u + B 2 d , 2

1+ ε u. 2

(3.26)

Lemma 3.2 [8] For any matrices (or vectors) X and Y with appropriate dimensions, we have X T Y + Y T X ≤  X T X +  −1Y T Y

where  > 0 is any scalar.

Let us define a Lyapunov function for the system (3.20) as

V ( x ) = x T Px , (3.27) where P is a positive definite matrix. By differentiating (3.27), we obtain V ( x ) = x T Px + x T Px T

1+ ε =  Ah x + B1w + B2u + B2 d  Px (3.28) 2   1+ ε + x T P  Ah x + B1w + B2u + B2 d  . 2  

130

Vibration Control of Vehicle Suspension Systems

By Lemmas 3.1, 3.2, and the definition of (3.21), we have T   1+ ε  1+ ε  V ( x ) ≤ x T  AhT P + PAh +  B2 K h  P + PB2 K h  x  2  2  

+ w T B1T Px + x T PB1w +  d T d +  −1 x T PB2 B2T Px

(3.29)

≤ x T Θx + w T B1T Px + x T PB1w where T  1+ ε   1+ ε Θ =  AhT P + PAh +  B2 K h  P + PB2 K h   2 2 

2   1− ε  T +  K K +  −1PB2 B2T P    2  h h 

(3.30)

and  is any positive scalar. Adding z T z − γ 2 w T w to the two sides of (3.29) yields  Θ + ChT Ch PB1 + ChT D V ( x ) + z T z − γ 2 w T w ≤  x T w T   T  B1 P + D T Ch −γ 2 I + D T D 

    x  . (3.31)  w  

Consider  Θ + ChT Ch Π= T  B1 P + D T Ch − γ 2 I + D T D 

PB1 + ChT D   < 0, (3.32)  

then V ( x ) + z T z − γ 2 wT w ≤ 0, and the L2 gain defined in (3.23) is less than γ > 0 with the initial condition x ( 0 ) = 0 [9]. When the disturbance is zero, that is, w = 0 , it can be inferred from (3.31) that if Π < 0, then V ( x ) < 0, and the fuzzy system (3.20) with the controller (3.21) is quadratically stable. Pre- and post-multiplying (3.32) by diag P −1 ,  I and its transpose, respectively, and defin-

(

−1

)

ing Q = P , Yh   =   K hQ, the condition Π < 0 is equivalent to

    ∑=    

QAhT + AhQ +

1+ ε T T 1+ ε Yh B2 + B2Yh 2 2

2

 1− ε  T Y Y +  −1 B2 B2T + QChT ChQ +   2  h h B1T + D T ChQ

By Schur complement equivalence, ∑ < 0 is equivalent to

   T B1 + QCh D   < 0. (3.33)   −γ 2 I + D T D  

131

Semi-active Suspension Control

    ∑=    

QAhT + AhQ +

1+ ε T T Yh B2 + B2Yh  +  −1 B2 B2T 2 

YhT

*

 2  − −1   1 − ε 

* *

* * 4

By the definitions Ah =

B1

DT

0

−I *

0 −γ 2

2

4

     < 0. (3.34)    

4

∑µ (ξ ) A , C = ∑µ (ξ )C , Y = ∑µ (ξ )Y , and the fact that i

h

i

i

i =1

µi (ξ ) ≥ 0 and

QChT

h

i

i

i =1

i

i =1

4

∑µ (ξ (t )) = 1, Ψ < 0 is equivalent to i

i =1

        

QAiT + AiQ +

1+ ε T T Yi B2 + B2Yi  +  −1 B2 B2T YiT 2 

QCiT B1

*

 2  − −1   1 − ε 

* *

* *

2

DT

0

−I *

0 −γ 2

     < 0, i = 1,…,4. (3.35)    

On the other hand, from (3.21), when assuming that umax = −umin = ulim, the constraint u u ( t ) ≤ lim can be expressed as ε 4

∑µ (ξ ) K x ≤ uε

i

i

lim

. (3.36)

i =1

It is obvious that if K i x ≤

2   ulim u , then (3.36) holds. Let Ω ( K ) =  x |  x T K iT K i x ≤  lim  , the   ε ε  

equivalent condition for an ellipsoid Ω ( P, ρ ) = { x |  x T Px ≤ ρ } being a subset of Ω ( K ), that is, Ω ( P, ρ ) ⊂ Ω ( K ), is [10]: −1

2  P u K i   K iT ≤  lim  . (3.37)  ε   ρ

By Schur complement equivalence, inequality (3.37) can be written as

 2   ulim    ε     P  −1 T    Ki   ρ 

 P Ki    ρ −1

−1

 P  ρ 

    ≥ 0. (3.38)   

132

Vibration Control of Vehicle Suspension Systems

Using the definitions Q = P −1, and Yi = K iQ, inequality (3.38) is equivalent to     

 ulim    ε 

   ≥ 0. (3.39) ρ −1Q    

2

Yi

YiT

Note that the input voltage sent to the MR damper is generally defined as 0 ~ umax  V, which means umin = 0 in (3.19) is required, and, therefore, an asymmetric saturation should be considered. However, the condition of (3.39) is derived with an assumption of symmetric saturation. To apply this condition to an asymmetric saturation, one possible modification is to shift the saturation centre as the average of both saturation limits; that is, define u + umax u umin + umax umax ulim = min = max = = with umin = 0 umin = 0 [11] and define the 2 2ulim 2 2 new saturation function limits as umin = −ulim , umax = ulim. umin = −ulim , umax = ulim . In the section, the input voltage is considered to be 0 − 2V, hence ulim = ulim = 1V will be used to design the controller. Nevertheless, in this case, the effects of different control levels need to be carefully examined and simulation is important to understand how control limits affect the system behaviour [12]. In summary, for a given number γ > 0, the TS fuzzy system (3.20) with controller (3.21) is quadratically stable and the L2 gain defined by (3.23) is less than γ if there exist matrices Q > 0, Yi, i = 1 − 4, scalar  > 0 such that (3.35) and (3.39) are satisfied. Moreover, the fuzzy state feedback gains can be obtained as K i = YiQ −1, i = 1 − 4. It is noticed that (3.35) and (3.39) are linear matrix inequalities (LMIs) to γ 2, hence, to minimize the performance measure γ , the controller design problem can be modified as a minimization problem of

minγ 2  s.t. LMIs (3.35) and (3.39). (3.40) This minimization problem is a convex optimization problem and can be solved using standard software.

3.2.1.5  TS Fuzzy Observer Design In practice, not all the state variables, in particular, the evolutionary variable zd , are available for measurement, and therefore, the state variables and premise variables for constructing the TS fuzzy controller (3.21) will be immeasurable, and the controller has to be constructed using the estimated state variables and premise variables; that is, the controller becomes

∑µ (ξˆ ) K xˆ = K xˆ, (3.41) 4

u=

i

i

h

i =1

where ξˆ is the estimated premise vector and xˆ is the estimated state vector. Assume that the measurement available output is the sprung mass velocity zs . Based on this measurement, we need to construct an observer based on the model (3.20) to estimate the state vector and premise vector. The proposed observer is defined as

xˆ =

∑µ (ξˆ )  A xˆ + L ( y − yˆ ) + B u  , 4

i

i =1

yˆ = Cxˆ ,

i

i

2

(3.42)

133

Semi-active Suspension Control

where xˆ is the observer state vector, yˆ is the estimated output, Li is the observer gain matrix to be designed, y = Cx is the measured sprung mass velocity, and the C matrix is defined as . C = [ 0 1 0 0 0 0 ]. Considering (3.20) and (3.42) the error dynamics model is defined as e = x − xˆ

∑ () 4

=

µi ξˆ ( Ai − LiC ) e +

i =1

∑( µ (ξ ) − µ (ξˆ )) A x + B w 4

i

i

1

i

i =1

(3.43)

∑ µ (ξˆ )( A − L C )e + w + B w 4

=

i

i

1

i

i =1

∑µ (ξˆ )( A e + G w), 4

=

i

i

i

i =1

where e = x − xˆ is the estimation error, Ai = Ai − LiC , Gi = [ I   B1 ], w = T w =  w T   wT  .

∑( µ (ξ ) − µ (ξˆ )) A x, and 4

i

i =1

i

i

∑( µ (ξ ) − µ (ξˆ )) A x 4

Note that several methods were proposed to deal with the term of

i

i

i

i =1

in (3.43). One of those methods is referred to as the uncertainty method [13] such that

∑( µ (ξ ) − µ (ξˆ )) A = HFE , where H = [ A   A   A   A ], F = dia  µ (ξ ) − µ (ξˆ )…  µ (ξ ) − µ (ξˆ ) 4

i

i

i =1

1

i

2

3

4

1

1

4

4

()

T and E = [ I   I   I   I ] . Since −1 ≤ µi (ξ ) − µi ξˆ ≤ 1, i = 1,…,4, F T F ≤ I , and Lemma 3.2 can be applied.

Another method is referred to as the bounding method [14–17], which uses the Lipschitz condition, matrix norm, differential mean value theorem, or simple assumptions to obtain a bound on µ (ξ ) − µ ξˆ which then results in an inequality. However, it has been found that the aforemen i

i

()

tioned methods may have problems in obtaining feasible solutions to the observer design of system

∑( µ (ξ ) − µ (ξˆ )) A x as a 4

(3.43). Therefore, a simple method is applied to (3.43) by defining w =

i

i

i

i =1

()

disturbance. Note that w is bounded due to the boundedness of µi (ξ ) − µi ξˆ and the assumption of a bounded state x for a practically stable system. To reduce the effect of disturbance on the estimation error, an observer is designed to achieve a given H ∞ -norm level. By defining an objective output as

zo = e = x − xˆ = Coe, (3.44)

it is easy to conclude that if there exists a matrix P > 0 such that the following LMI is satisfied [9]  PA + AT P PGi CoT  i i    < 0, i = 1,…,4, (3.45)  * 0 −γ o2 I   −I  *  * then the system (3.43) is stable with H ∞ disturbance attenuation γ o > 0 . Using the definition Yi = PLi in (3.45) and solving the LMI (3.45), the observer gain matrix can be obtained as Li = P −1Yi .

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Vibration Control of Vehicle Suspension Systems

To overcome the immeasurable problem of the evolutionary variable zd , another alternative controller design approach, which regards zd and its related terms as external disturbances, is proposed here. We now only define the state variables for the quarter-car suspension as x1 = zs , x 2 = zs, x3 = zu , T x 4 = zu , and define the state vector as x = [ x1   x 2   x3   x 4 ] , then from (3.3), we have x2 =

1  −c0 a ( x 2 − x 4 ) − ( k0 + ks )( x1 − x3 ) − α a zd − α b zd v − c0 b ( x 2 − x 4 ) v  , ms 

x4 =

1  −c0 a ( x 4 − x 2 ) − ( k0 + ks )( x3 − x1 ) − kt ( x3 − zr ) + α a zd mu 

(3.46)

+α b zd v + c0 b ( x 2 − x 4 ) v ]. If we define f = c0 b ( x 2 − x 4 ), and define fmax and fmin as the maximum and minimum values for f , respectively, by following a similar procedure to that in Section 2.3, we can write another TS fuzzy model for the quarter-car suspension with MR damper as 2

x =

∑ µ (ξ )[ Ax + B w + B v ], (3.47) 1

i

2i

i =1

where ξ = f is a premise variable, u1 (ξ ) =     A=    

0 −

1 k0 + ks ms

0 k0 + ks mu

 0   − fmin ms  B22 =  0   fmin  mu 

c0 a ms

0 c0 a mu

0 k0 + ks ms 0 k + k s + kt − 0 mu

f − fmin f −f , u2 (ξ ) = max , fmax − fmin fmax − fmin

0 c0 a ms 1 c − 0a mu

        , B1 =         

0

0

0

0

0 kt mu

0 αa mu

αa ms

αb ms

0 αb mu

  0    − fmax  ms   , B21 =  0    fmax   mu   

    ,    

    zr      , w =  z d .   zd v      

It is noted that the premise variable in (3.47) is only related to the relative speed between the sprung mass and unsprung mass, which is available for measurement in practice. In addition, zd and zd v are regarded as disturbances in (3.47). As the evolutionary variable zd is bounded [18] and the applied voltage v is saturated in practice, thus zd and zd v are all bounded. Therefore, the TS fuzzy control2

ler design procedure in Section 3.1 can be applied to design a robust controller v =

∑µ (ξ ) K x i

i

i =1

for system (3.47) so that the performance requirements on ride comfort, road holding ability and suspension deflection are achieved under the disturbances of road irregularity and unknown evolutionary variable. More details on solving this controller can be referred to in Section 3.1 and are omitted for brevity. After v is obtained, using the relationship v = −η ( v − u ), the control voltage u can be obtained and will be sent to the MR damper.

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Semi-active Suspension Control

3.2.1.6 Numerical Application In this section, we apply the proposed approaches to design controllers for the quarter-car model with an MR damper as described in Section 3.2. The parameter values for the MR damper are listed in Table 3.1. To validate the effectiveness of the designed controllers, the modified Bouc–Wen model will be used to represent the real MR damper in the simulation. The quarter-car suspension model parameter values are listed in Table 3.2 [19]. Two controllers will be designed for the quarter-car suspension with an MR damper and validated in this section. The first controller is an observer-based controller as described in Section 3.2. Assuming that only the sprung mass velocity is available for measurement and all the state variables defined in (3.4) will be estimated, the controller gain matrices are designed by solving the minimization problem in (3.40) and the observer gain matrices are obtained by solving the LMI (3.45). For description brevity, we denote this controller as TS fuzzy controller I. The second controller is designed using the method presented in Section 3.3 and assuming the state variables related to the quarter-car suspension as defined in (3.4) can be measured. For description brevity, this controller is denoted as TS fuzzy controller II. Furthermore, to validate the performance of the designed controllers, the skyhook controller and the hybrid controller [20,21] are also adopted for comparison purpose. The skyhook controller defines the desired damping force as  K sky zs , zS ( zs − zu ) ≥ 0,  (3.48) fsky =  zS ( zs − zu ) < 0,  0, where K sky is the skyhook gain, and the hybrid controller defines the desired damping force as

fhybrid   = κ fsky   + (1 − κ ) fground , (3.49)

where

 K ground   zu ,  fground   =   0,

− zu ( zs − zu ) ≥ 0, − zu ( zs − zu ) < 0,

(3.50)

K ground is the groundhook gain and κ is a weighting coefficient. Note that both the skyhook control and the hybrid control are often regarded as benchmark control strategies and used to validate any new control strategies for vehicle suspension with an MR damper. For example, the comparison between these two control strategies and other control strategies, such as linear quadratic Gaussian (LQG), sliding mode control, and fuzzy logic control, for a quarter-car suspension with an MR damper was done in [19], where the skyhook and hybrid controllers showed somehow improved performance compared to the three control strategies. The experimental comparison of these control strategies on a full vehicle was done in [22]. As stated in the introduction, most of the existing control strategies for MR dampers aim at designing the desired damping force first and then use different methods, such as the inverse dynamics model method [19,23–25] and the clipped-optimal method

TABLE 3.2 Parameter Values of the Quarter-Car Suspension Model Parameter Unit Value

ms

mu

cs

ks

kt

kg 424.5

kg 60.4

Ns/m 1000

N/m 60,000

N/m 181,000

136

Vibration Control of Vehicle Suspension Systems

[26], to compute the command voltage sent to the MR damper. In the simulation, the skyhook and the hybrid controllers are used to generate the desired damping forces with K sky = 2840 Nsm −1, K ground = 3280 Nsm −1 , and κ = 0.8 [19], and the command voltage to the MR damper is computed according to the clipped-optimal approach [26] as

u = umax H ( fdes   − fmeas   ) fmeas    , (3.51)

where H [·]1 is the Heaviside function, fdes is the desired damping force, which can be fsky or fhybrid , and fmeas is the actual MR damper force measured by a force sensor. Note that the command voltage computed by (3.51) can only be 0, umax / 2, and umax. However, the command voltage computed by the TS fuzzy controller I and TS fuzzy controller II can vary continuously between 0 and umax. To evaluate the suspension performance with respect to ride comfort, working space of the suspension, and road holding, the evaluation of the vehicle suspension is based on the examination of response quantities within a prespecified frequency range on the sprung mass acceleration, the suspension deflection, and the dynamic tyre load under typical road profiles. Two types of road excitations, which are chosen to be similar to the real-world road profiles and often used in evaluating suspension performance in the time domain [23,27], are considered in the simulation. A bump input, which is normally used to describe the transient response characteristic, is adopted as the first road excitation. Considering the case of an isolated bump in an otherwise smooth road surface, the corresponding ground displacement is given by

 a  2πv0     1 − cos  l t   , 2  zr ( t ) =   0,  

0≤t ≤ l t> , v0

l , v0

(3.52)

where a and l are the height and the length of the bump. We choose a = 0.07m , l = 0.8m and the vehicle forward velocity as v0 = 0.856ms−1 [27]. By applying the bump road profile (3.52) to the wheel, the responses of the passive suspension (denoted as conventional passive), where a conventional damper with a damping coefficient of cs = 1000 Nsm −1 is used instead of MR damper, the passive MR suspension (denoted as MR Passive), which is composed of a quarter-car model and an MR damper without controller (input voltage is set as zero), and the MR suspensions with skyhook controller (denoted as skyhook), with hybrid controller (denoted as hybrid), with TS fuzzy controller I (denoted as TS Fuzzy I), and with TS fuzzy controller II (denoted as TS Fuzzy II) are compared in Table 3.3 in terms of the peak-to-peak (PTP) values for sprung mass acceleration, suspension deflection, and dynamic tyre load. The improvement ratios compared to the conventional passive suspension are also given in Table 3.3. It can be seen from Table 3.3 that the controlled MR suspension with TS fuzzy controller I achieves the lowest peak values for the sprung mass acceleration, suspension deflection, and dynamic tyre load among all the suspensions, and reduces the maximum PTP values by 45%, 56%, and 42%, respectively, compared with the conventional passive suspension. The controlled MR suspension with TS fuzzy controller II achieves similar performance compared to skyhook and hybrid. It is confirmed by the simulation results that improved bump responses for sprung mass acceleration, suspension deflection, and dynamic tyre load are achieved by the proposed control approaches. The bump responses of sprung mass acceleration, suspension deflection, dynamic tyre load, input voltage, and MR damper force are shown in Figures 3.2 and 3.3, where only the responses for the conventional passive, TS Fuzzy I, and TS Fuzzy II are shown in Figures 3.2 and 3.3 for clarity. It can be seen from Figure 3.2 that better responses are obtained for the controlled MR suspensions compared to the passive suspension. It is also observed from Figure 3.3 that the voltage signals obtained

137

Semi-active Suspension Control

TABLE 3.3 Peak-to-Peak Values and Improvement Ratios Under Bump Road Disturbance Parameter

(

 zs   m/s 2

)

% zs − zu  ( m ) % kt ( zs − zr ) ( kN ) %

Conventional Passive

TS Fuzzy I

TS Fuzzy II

8.3278

8.0432

4.8145

5.3045

4.5656

4.9997

– 0.0584

−3.42 0.0512

−42.19 0.0297

−36.30 0.0330

−45.18 0.0256

−39.96 0.0264

– 3.7956

−12.26 3.6806

−49.15 2.2693

−43.53 2.3830

−56.22 2.2161

−54.74 2.3139

−40.21

−37.22

−41.61

−39.04

MR Passive

−3.03

Skyhool

Hybrid

from TS Fuzzy I and TS Fuzzy II are continuously varied and are almost zero when the system responses reach the steady state. The voltage signals computed from skyhook and hybrid are discrete values and are not always zero even when the system responses reach the steady state due to the sensitivity of Heaviside function to small errors between the desired force and the measured force. To further validate the effectiveness of the designed controllers, a second type of road excitation, a random road profile, is applied [27]. The road displacement of the random road profile is generated by

zr ( t ) + ρVzr ( t ) = VWn , (3.53)

where Wn is white noise with the intensity 2σ 2 ρV , ρ is the road roughness parameter, σ 2 is the covariance of road irregularity, and V is the vehicle speed. In the simulation, ρ = 0.45m −1, σ 2 = 300 mm 2 , and V = 20 ms−1 are chosen to simulate the scenario in which the vehicle is moving on the paved road with a constant speed of 20 ms−1 [27]. In this case, the responses of the passive suspension, the passive MR suspension, and the MR suspensions with skyhook controller, with hybrid controller, with TS fuzzy controller I, and with TS fuzzy controller II are compared in Table 3.4 in terms of the root mean square (RMS) values for sprung mass acceleration, suspension deflection, and dynamic tyre load. The improvement ratios compared to the conventional passive suspension are also given in Table 3.4. It can be seen from Table 3.4 that the controlled MR suspensions have lower RMS values for the sprung mass acceleration, suspension deflection, and dynamic tyre load compared to the passive MR suspension. In particular, the controlled MR suspensions with both TS fuzzy controller I and TS fuzzy controller II can reduce the RMS values for sprung mass acceleration, suspension deflection, and dynamic tyre load by about 16%, 34%, and 19%, respectively, compared with the conventional passive suspension. These improvement ratios are also the best values among all the controlled MR suspensions. It is further confirmed by the simulation results that good random responses for sprung mass acceleration, suspension deflection and dynamic tyre load are also achieved by the proposed control approaches. The random responses of sprung mass acceleration, suspension deflection, dynamic tyre load, input voltage, and MR damper force are shown in Figures 3.4 and 3.5, where only the responses for the conventional passive, TS Fuzzy I and TS Fuzzy II are shown in Figures 3.4 and 3.5 for clarity. Similar conclusion can be obtained for this case where the designed controllers are effective in improving suspension performance. 3.2.1.7 Conclusions In this section, a TS fuzzy modelling approach is adopted to model a quarter-car model with an MR damper. The TS fuzzy model can exactly represent an MR damper described by a simple

138

Vibration Control of Vehicle Suspension Systems 5

Conventional Passive TS Fuzzy I TS Fuzzy II

Sprung Mass Acceleration (m/s/s)

4 3 2 1 0 –1 –2 –3 –4 –5

0

0.5

1

1.5

2

0.03

2.5 Time (s)

3

3.5

4.5

5

Conventional Passive TS Fuzzy I TS Fuzzy II

0.02 Suspension Deflection (m)

4

0.01 0 –0.01 –0.02 –0.03 –0.04

0

0.5

1

1.5

2

2.5 Time (s)

3

3.5

1500

4

4.5

5

Conventional Passive TS Fuzzy I TS Fuzzy II

1000

500

0

–500

–1000

–1500

0

0.5

1

1.5

2

2.5 Time (s)

FIGURE 3.2  Dynamic response under bump road disturbance.

3

3.5

4

4.5

5

Semi-active Suspension Control

FIGURE 3.3  Input voltage and output force of MR damper.

139

140

Vibration Control of Vehicle Suspension Systems

TABLE 3.4 RMS Values and Improvement Ratios Under Random Road Disturbance Parameter

(

 zs m/s 2 %

)

zs − zu  ( m ) % kt ( zs − zr ) ( kN ) %

Conventional Passive

TS Fuzzy I

TS Fuzzy II

0.7910

0.8452

0.7264

0.7699

0.6652

0.6656

– 0.0052

+6.84 0.0051

−8.18 0.0043

−2.66 0.0046

−15.90 0.0034

−15.85 0.0034

– 461.4896

−1.57 461.9377

−16.28 419.9871

−11.06 433.8660

−33.59 370.2937

−35.20 373.8291

+0.09

−8.99

−5.99

−19.76

−19.00

MR Passive

Skyhool

Hybrid

Bouc–Wen model. Based on the TS fuzzy model, an H∞ controller under a constraint on the input voltage can be designed to improve suspension performance. Furthermore, a fuzzy observer design and a robust control design are proposed so that the implementation of the control systems is achievable in practice. These approaches have the advantage of designing a controller directly targeting system performance without considering the semi-active condition – that is, judging the direction of force or velocity and switching control laws – so that a continuous control signal can be obtained. In addition, no force sensor or estimation is needed, and no inverse dynamics of the MR damper needs to be calculated to determine the required input voltage; therefore, cost-effectiveness and fast control response are guaranteed. Numerical simulations are used to demonstrate the effectiveness of the proposed approaches. Further study on finding an effective method to reduce the conservatism caused by the asymmetric actuator saturation and considering system modelling uncertainty in the controller design process will be investigated.

3.2.2 Electrorheological Damper 3.2.2.1  ER Damper Model A cylindrical type of ER damper, which is applicable to a middle-sized passenger vehicle, was designed and manufactured in [28]. The ER damper can produce additional damping force owing to the yield stress of the ER fluid if a certain level of the electric field is supplied to the ER damper and this damping force of the ER damper can be continuously tuned by controlling the intensity of the electric field. The damping force of the proposed ER damper is given as [28]

˙

F ( t ) = ke x p ( t ) + Ce x p ( t ) + Fs ( t ) (3.54)

where ke is the effective stiffness due to the gaspressure, ce is the effective damping due to the fluid viscosity, x p ( t ) and x p ( t ) are the excitation displacement and velocity, respectively, and Fs ( t ) is the field-dependent damping force which is tunable as a function of applied electric field. Taking the dynamic characteristic of the ER damper into account, the controllable damping force Fs ( t )  is expressed by

l τ Fs ( t ) = − Fs ( t ) + ( Ap − Ar ) 2 α E β ( t ) sgn ( x p ( t )) (3.55) h

where τ is the time constant of damping force. It was experimentally identified by 380 ms for the studied ER damper. Ap and Ar represent piston and piston rod areas, respectively. Sgn (·) is a sign function, l is the electrode length, h is the electrode gap, and E ( t ) is the electric field. The α and β are intrinsic values of the ER fluid to be experimentally determined. Since the dynamic motion

141

Semi-active Suspension Control

Sprung Mass Acceleration (m/s/s)

3

Conventional Passive TS Fuzzy I TS Fuzzy II

2

1

0

–1

–2

–3

0.5

0

1

1.5

2

0.02

2.5 Time (s)

3

3.5

4.5

5

Conventional Passive TS Fuzzy I TS Fuzzy II

0.015 Suspension Deflection (m)

4

0.01 0.005 0 –0.005 –0.01 –0.015

0

0.5

1

1.5

2

2000

2.5 Time (s)

3

3.5

4.5

5

Conventional Passive TS Fuzzy I TS Fuzzy II

4

1000 500 0 –500 –1000 –1500

0

0.5

1

1.5

2

2.5 Time (s)

FIGURE 3.4  Dynamic response under random road disturbance.

3

3.5

4

4.5

5

142

Vibration Control of Vehicle Suspension Systems

FIGURE 3.5  Input voltage and output force of MR damper.

Semi-active Suspension Control

143

of ER fluid between the inner and outer cylinder of the ER damper can be regarded as flow mode, the intrinsic values α and β of the employed ER fluids are experimentally determined by using the flow mode type electroviscometer. In this study, the field-dependent yield stresses of the ER fluid which was experimentally obtained by 565.2 E1.55 ( t ) Pa [28], where the unit of E ( t ) is kV/mm, are used. 3.2.2.2  Quarter-Car Model with ER Damper In this section, a quarter-car suspension model as shown in Figure 3.6 is used for the controller design and performance evaluation. The equations of motion for the sprung and unsprung masses are expressed as ms  zs ( t ) + cs [ zs ( t ) − zu ( t )] + ks [ zs ( t ) − zu ( t )] = Fs ( t )

mu  zu ( t ) + cs [ zu ( t ) − zs ( t )] + ks [ zu ( t ) − zs ( t )] + kt [ zu ( t ) − zr ( t )] (3.56) + ct [ zu ( t ) − zr ( t )] = Fs ( t )

where ms is the sprung mass, which represents the car chassis; mu is the unsprung mass, which represents the wheel assembly; cs and ks are damping and stiffness of the passive suspension, respectively; kt and ct stand for compressibility and damping of the pneumatic tyre, respectively; zs ( t ) and zu ( t ) are the displacements of the sprung and unsprung masses, respectively; zr ( t ) is the road displacement input; Fs ( t ) represents the control force, which is provided by the ER damper placed between sprung mass and unsprung mass of vehicle suspension. By defining the state variables as follows:

FIGURE 3.6  Quarter-car suspension model.

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Vibration Control of Vehicle Suspension Systems

x1 ( t ) = zs ( t ) − zu ( t ) ,

suspension delfection,

x 2 ( t ) = zu ( t ) − zr ( t ) ,

tyre deflection,

x3 ( t ) = zs ( t ) ,

sprung mass velocity,

x 4 ( t ) = zu ( t ) ,

unsprung mass velocity,

x5 ( t ) = Fs ( t ) ,

damping force,

the state-space equation of the quarter-car suspension model is written as x ( t ) = Ax ( t ) + B1w ( t ) + B2u ( t ) (3.57)

T

where x ( t ) =  x1 ( t ) x 2 ( t ) x3 ( t ) x 4 ( t ) x5 ( t )  is the state vector, w ( t ) = zr ( t ) is the road dis  turbance, u ( t ) is the bounded control input to the ER damper. In a real application, the control input

to the ER damper can be bounded as u ( t ) = sat ( u ( t )), where sat(u(t)) is a saturation function of control input u(t) and is defined as

 −ulim if   u ( t ) < −ulim ,  sat ( u ( t )) =  u ( t ) if − ulim ≤ u ( t ) ≤ ulim , (3.58)  u if   u ( t ) > ulim ,  lim where ulim is a control input limit. The control input u ( t ) applied to the ER damper has the following semi-active condition imposed  u (t )  u (t ) =   0

if   u ( t ) ( zu ( t ) − zs ( t )) > 0,

if   u ( t ) ( zu ( t ) − zs ( t )) ≤ 0,

The control input is used to determine the input electric field applied to the ER damper by [28] 1

 β h E ( t ) = u ( t )  . (3.59) 2α l ( Ap − Ar )  

The matrices in (3.57) are defined as

 0  0   − ks  ms A=  ks  mu   0  

kt mu

1 0 c − s ms cs mu

−1 1 cs ms c − s mu

0

0

0

0 0 0 −

0 0 1 ms 1 − mu 1 − τ

    0   −1    , B1 =  0   0     0   

 0    0      0    , B2 =  0       1    τ 

145

Semi-active Suspension Control

It is noted that the vehicle sprung mass ms is often varied due to the change of loading conditions in practice, and the time constant τ  of ER damper can be slightly altered by some conditions such as operating temperature. Taking the varying parameters ms and τ into account, the suspension model is becoming a parameter-dependent model as

x ( t ) = Aθ x ( t ) + B1w ( t ) + B2u ( t ) (3.60)

where Aθ is a function of θ which is a varying parameter vector. Since sprung mass ms and ER damper time constant τ can only vary in a bounded space in practice, Aθ can be constrained to a polytope  given by

  =  Aθ : Aθ = 

4

4

∑ ∑ρ = 1, ρ ≥ 0,i = 1,…,4. , (3.61) ρi Ai ,

i =1

i

i

i =1

where ρ does not necessarily represent the actual varying parameter vector θ but there exists a linear relationship between θ and ρ that can be easily determined from the physical model whenever θ affects affinely the linear system. For example, we can define the varying parameters as

θ1 =

1 1 , θ 2 = , (3.62) ms τ

and define the four vertices of the polytope  as

 θ1min θ 2 min   θ1max θ 2 min   θ1min θ 2 max   θ1max θ 2 max  (3.63)        

where θ imin and θ imax , i  =  1, 2, denote the minimum and maximum values, respectively. Accordingly, the convex coordinates ρi , i  = 1, 2, 3, 4, are defined as

ρ1 = ζξ ρ2 = (1 − ζ ) ξ ρ3 = ζ (1 − ξ ) ρ4 = (1 − ζ ) (1 − ξ ) (3.64)

where ζ = (θ1max − θ1 ) / (θ1max − θ1min ), ξ = (θ 2 − θ 2 min ) / (θ 2 max − θ 2 min ). It is clear that the knowledge of the value of ρ defines a precisely known system inside the polytope  described by the convex combination of its four vertices. Therefore, the system (4) with parameter uncertainties on ms and τ can be represented by (3.60) in a polytopic form. Matrices Ai ,  i   = 1, ... , 4, are vertices of Aθ , and they can be obtained from matrix A in (3.57) by replacing 1 ms−1 and 1 τ−1 with their vertex values defined in (3.63), respectively. For example, A1 is obtained by replacing 1 ms−1 with θ1min (= 1 / msmax , where msmax denotes the maximum value of ms ) and 1 τ−1 with θ 2 min (=  1 / τ max , where τ max denotes the maximum value of τ ) in matrix A. In a real application, the state measurements cannot be perfect. Thus, the measured state variables should be corrupted by measurement noises as

y ( t ) = Cx ( t ) + n ( t ) (3.65)

where y ( t ) is the measured output, n ( t ) denotes the measurement noise, and C is a constant matrix (if all the state variables are measured, C is an identity matrix). To estimate the state variables from noisy measurements, we construct an observer as

xˆ ( t ) = Aθ xˆ ( t ) + B2u ( t ) + L ( y ( t ) − yˆ ( t )) , (3.66)

yˆ (t ) = Cxˆ (t ), (3.67)

146

Vibration Control of Vehicle Suspension Systems

where xˆ ( t ) is the observer state vector, L is the observer gain matrix to be designed, yˆ ( t ) is observer output. By defining the estimation error as e ( t ) = x ( t ) − xˆ ( t ) (3.68)

we obtain

e ( t ) = x ( t ) − xˆ ( t ) = Aθ x ( t ) + B1w ( t ) − Aθ xˆ ( t ) − L ( y ( t ) − yˆ ( t )) (3.69)

= ( Aθ − LC ) e ( t ) + B1w ( t ) − Ln ( t ) . Since ride comfort is an important performance for a vehicle suspension design, and ride comfort usually can be quantified by the sprung mass acceleration, the sprung mass acceleration is chosen as the control output as

za ( t ) =  zs ( t ) = Ca x ( t ) = Ca ( xˆ ( t ) + e ( t )) (3.70)

 ks c 0 − s where Ca =  − m m s s 

cs ms

1 ms

 . 

Remark 3.1 Other suspension performances like road holding ability and suspension deflection limitation can be easily included in the proposed controller design process which will be presented in the next section by using the constrained control idea [6,29,30]. However, for simplicity and comparison purposes, this section only focuses on ride comfort performance without loss of generality. To make the estimation error as small as possible, we also define another control output as ze ( t ) = Cee ( t ) (3.71)

where Ce is constant matrix which can be used to make the compromise between za ( t ) and ze ( t ) in the control objective. Then, the control output is defined as

 za ( t ) z ( t ) =   ze ( t )

  Ca =   0

Ca Ce

  xˆ ( t )    e ( t )

  = C z x ( t ) (3.72) 

 Ca Ca  T where x ( t ) =  xˆ T ( t ) eT ( t )  is the augmented system state vector, and C z =  .  0 Ce  As defined by (3.72), the control output z ( t ) is composed of two parts, za ( t ) and ze ( t ), and it

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Semi-active Suspension Control

will be optimized in the controller design process. The introduction of matrix Ce increases the freedom in optimising the control output z ( t ). If we choose bigger Ce, then the optimisation will focus on ze ( t ) part so that smaller ze ( t ) can be obtained. On the contrary, if we choose smaller Ce, then the optimisation will focus on za ( t ) part so that smaller za ( t ) can be obtained. Therefore, Ce can be chosen by a trial and error process so that a desired compromise between za ( t ) and ze ( t ) in the control output can be realized. Generally, Ce can be chosen as an identity matrix for simplicity In the section, the aim of the robust controller design is to find a state feedback control law based on the estimated state variables as u ( t ) = Kxˆ ( t ) (3.73)

where K is the control gain matrix to be designed, such that the closed-loop system is quadratically stable and Tzw  ∞ < γ is realized for all nonzero w ( t ) ∈ L2 [ 0, ∞ ) and the prescribed constant γ > 0, where Tzw denotes the closed-loop transfer function from the road disturbance w ( t ) to the control output z ( t ) and

Tzw  ∞

z ( t ) 2 (3.74) w( t ) 2 ≠ 0  w ( t )  2

= sup ∞

where  z ( t ) 22 = z T ( t ) z ( t ) dt and  w ( t ) 22 = w T ( t ) w ( t ) dt . 0

0

3.2.2.3  Controller Design To use the norm-bounded approach [7,31] to handle the control input constraint defined in (3.58), (3.66) is written as xˆ ( t ) = Aθ xˆ ( t ) + B2u ( t ) + L ( y ( t ) − yˆ ( t ))

= Aθ xˆ ( t ) + B2

(3.75) 1+ ε 1+ ε   u ( t ) + B2  u ( t ) − u ( t ) + LCe ( t ) + Ln ( t ) .   2 2

We now define a Lyapunov function as

V ( t ) = x T ( t ) Xx ( t )

 P 0  T T where X   = X T > 0. By defining X =  , where P = P > 0, Q = Q > 0, taking the time 0 Q   derivative of V ( t ) along equations (3.22) and (3.16) yields

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Vibration Control of Vehicle Suspension Systems

V ( t ) = xˆ T ( t ) Pxˆ ( t ) + xˆ T ( t ) Pxˆ ( t ) + eT ( t ) Qe ( t ) + eT ( t ) Qe ( t ) T

1+ ε 1+ ε                =  Aθ xˆ ( t ) + B2 u ( t ) + B2  u ( t ) − u ( t ) + LCe ( t ) + Ln ( t )  Pxˆ ( t )   2 2   1+ ε 1+ ε     + xˆ T ( t ) P  Aθ xˆ ( t ) + B2 u ( t ) + B2  u ( t ) − u ( t ) + LCe ( t ) + Ln ( t )    2 2   + ( Aθ − LC ) e ( t ) + B1w ( t ) − Ln ( t )  Qe ( t ) + eT ( t ) Q ( Aθ − LC ) e ( t ) + B1w ( t ) − Ln ( t )  T

T

1+ ε =  Aθ xˆ ( t ) + B2 u ( t ) + LCe ( t ) + Ln ( t )  Pxˆ ( t ) 2   1+ ε + xˆ T ( t ) P  Aθ xˆ ( t ) + B2 u ( t ) + LCe ( t ) + Ln ( t )  2   T

1+ ε 1+ ε     +  u (t ) − u ( t ) B2T Pxˆ ( t ) + xˆ T ( t ) PB2  u ( t ) − u ( t )     2 2 + ( Aθ − LC ) e ( t ) + B1w ( t ) − Ln ( t )  Qe ( t ) T

+ eT ( t ) Q ( Aθ − LC ) e ( t ) + B1w ( t ) − Ln ( t )  . Lemma 3.3 For any matrices (or vectors) X and Y with appropriate dimensions, we have X TY + Y T X ≤  X T X +  −1Y TY

where  > 0 is any scalar.

By using Lemma 3.3, we obtain T

1+ ε V ( t ) ≤  Aθ xˆ ( t ) + B2 u ( t ) + LCe ( t ) + Ln ( t )  Pxˆ ( t ) 2   1+ ε + xˆ T ( t ) P  Aθ xˆ ( t ) + B2 u ( t ) + LCe ( t ) + Ln ( t )  2  

+ ( Aθ − LC ) e ( t ) + B1w ( t ) − Ln ( t )  Qe ( t ) T

+ e T ( t ) Q ( Aθ − LC ) e ( t ) + B1w ( t ) − Ln ( t )  T

1+ ε 1+ ε     +   u (t ) − u ( t )  u ( t ) − u ( t ) +  −1 xˆ T ( t ) PB2 B2T Pxˆ ( t ) .     2 2

Using the fact [7] that

149

Semi-active Suspension Control

1+ ε 1− ε u (t ) ≤ u ( t ) (3.76) 2 2

u (t ) −

and hence, 2

T

u ( t ) − 1 + ε u ( t )  u ( t ) − 1 + ε u ( t )  ≤  1 − ε  u T ( t ) u ( t ) (3.77)     2   2 2

for the saturation nonlinearity defined in (3.58) as long as u ( t ) ≤ ulim / ε , where 0 < ε < 1, we then have, T

1+ ε V ( t ) ≤  Aθ xˆ ( t ) + B2 u ( t ) + LCe ( t ) + Ln ( t )  Pxˆ ( t ) 2   1+ ε + xˆ T ( t ) P  Aθ xˆ ( t ) + B2 u ( t ) + LCe ( t ) + Ln ( t )  2   + ( Aθ − LC ) e ( t ) + B1w ( t ) − Ln ( t )  Qe ( t ) T

+ eT ( t ) Q ( Aθ − LC ) e ( t ) + B1w ( t ) − Ln ( t )  2

 1− ε  T +  u ( t ) u ( t ) +  −1 xˆ T ( t ) PB2 B2T Pxˆ ( t )  2 

T  1+ ε 1+ ε   AθT P + PAθ +  B2 K  P + PB2 K   2 2  T = xˆ ( t )  2  1− ε  T  +  K K +  −1PB2 B2T P   2   

(3.78)   ˆ  x (t )   

+ eT ( t ) ( Aθ − LC ) Q + Q ( Aθ − LC ) e ( t ) − xˆ T ( t ) PLCe ( t ) T

+ eT ( t ) C T LT Pxˆ ( t ) + wT ( t ) B1T Qe ( t ) + eT ( t ) QB1w ( t ) + xˆ T ( t ) PLn ( t ) + n T ( t ) LT Pxˆ ( t ) − eT ( t ) QLn ( t ) − n T ( t ) LT Qe ( t ) = x T ( t ) Φx ( t ) + x T ( t ) Ξw ( t ) + w T ( t ) ΞT x ( t ) T

where w =  w T ( t ) n T ( t )  ,

    Φ=    

T

1+ ε  1+ ε  A P + PAθ +  B2 K  P + PB2 K   2 2 T

2

 1− ε  T +  K K +  −1PB2 B2T P  2  C T LT P

   PLC     T ( Aθ − LC ) Q + Q ( Aθ − LC ) 

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Vibration Control of Vehicle Suspension Systems

and  0 PL  Ξ=   QB1 −QL 

By adding z T ( t ) z ( t ) − γ 2 w T ( t ) w ( t ) to two sides of (3.78) yields V ( t ) + z T ( t ) z ( t ) − γ 2 w T ( t ) w ( t ) ≤ x T ( t ) Φx ( t ) + x T ( t ) Ξw ( t ) + w T ( t ) ΞT x ( t ) + x T ( t ) C zT C z x ( t ) − γ 2 w T ( t ) w ( t )

= x T ( t ) Θx ( t ) , T

where x ( t ) =  x T ( t ) w T ( t )  , and  

T  1+ ε   AθT P + PAθ +  B2 K P   2   2  1+ ε  1− ε  T K + K K   + PB2  2  2   −1 T T Θ =  + PB2 B2 P + Ca Ca   *    *   * 

PLC + CaT Ca

( Aθ − LC )T Q + Q ( Aθ − LC ) +CeT Ce + CaT Ca

0

QB1

*

−γ 2

*

0

     PL       −QL    0  −γ 2  

If Θ < 0, then V ( t ) + z T ( t ) z ( t ) − γ 2 w T ( t ) w ( t ) < 0 . Thus, the closed-loop system is stable when the disturbance w ( t ) = 0 and the H ∞ performance defined in (3.74) is satisfied when x ( 0 ) = 0. By the Schur complement, Θ < 0 is equivalent to

T  1+ ε  1+ ε  AθT P + PAθ +  B2 K  P + PB2 K   2 2   2   1− ε  T T T −1  K K +  PB2 B2 P + Ca Ca  +   2    *   T

 0 PL   0 PL  + γ −2     0,  β > 0 are weighting coefficients for suspension deflection and tyre deflection, respectively. These coefficients are used to trade off two different performance objectives. Parameters of the suspension chosen for analysis are listed in Table 3.6, and the value of inertance is assumed in [0, 500] kg.

3.3.2 Inerter Effects on Suspension The effects of fixed inerter on a given suspension system are investigated in this section, which shows that it is difficult to choose a fixed inerter to satisfy the performance of vehicle at the sprung mass natural frequency without significant deterioration at the unsprung mass natural frequency. Figure 3.12a–c show the frequency responses relative to different fixed inertances.

161

Semi-active Suspension Control

TABLE 3.6 Parameters of Quarter-Car Suspension System ms 320 kg

ms

ks

Cs

kt

40 kg

18 kN/m

1000 Ns/m

200 kN/m

(a)

(d)

10

2

10

Passive system Passive system with b=100 Passive system with b=300 Passive system with b=500

2

Singular value

Singular value

104

103

100

10–2 500 400 300 b 200 100 0

101

10–1

10–0

10–1

102

10 Frequency (Hz) 0

101

10–2 –1 10

100

101

102

101

102

101

102

Frequency (Hz)

(b)

(e)

100

100 10–1 10

Singular value

Singular value

–2

10–4

10 600

–6

10–2 10–3 10–4

400 b

200

10–5 0

10

–1

100 Frequency

101

Passive system Passive system with b=100 Passive system with b=300 Passive system with b=500

102 10–6 –1 10

100 Frequency (Hz)

(c)

(f)

101

100

100

10–1

Singular value

Singular value

101

10–2 10 600

–3

400 b

Passive system Passive system with b=100 Passive system with b=300 Passive system with b=500

10–1

10–2

200 0

10–1

100 Frequency

101

10

2

10–3 –1 10

100 Frequency (Hz)

FIGURE 3.12  Frequency response of: (a) vertical acceleration, (b) suspension deflection (c) tyre deflection; comparison of frequency responses of: (d) vertical acceleration, (e) suspension deflection, and (f) tyre deflection.

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Vibration Control of Vehicle Suspension Systems

Figure 3.12d–f show the comparisons of the frequency response of a passive suspension with no inerter and with three different fixed inerters. It can be seen that adding a fixed inerter to a given suspension system leads to a reduction in the natural frequencies and a change in the resonant peak amplitudes, similar results have been reported for adding parallel inerter to reduce the natural frequency in [33]. As the inerter changes the system inertia, the system dynamic characteristics have been modified. More specifically, the natural frequencies of both sprung and unsprung mass are reduced. The reduction in the natural frequency of the unsprung mass is more apparent and the larger inertance will result in greater reduction of the natural frequency. For vertical acceleration and tyre deflection, the first resonance peak decreases with the presence of inerter and the peak can be further removed by increasing inertance. However, the magnitude of the second resonance peak will increase significantly with increasing inertance. For suspension deflection, the change in resonance peak is not so prominent as those for vertical acceleration and tyre deflection. In other words, the reduction of the first resonance peak and the amplification of the second resonance peak are not obvious with the increasing inertance. In addition, for suspension deflection, increasing inertance will result in better highfrequency performance. The comparisons of the dynamics characteristics of passive suspensions with no inerter and three different fixed inerters are shown in Table 3.7, where f1, f2 are the natural frequencies of the sprung and unsprung mass. At the natural frequency of sprung mass, the table shows that the fixed inerter suspension system shows improvements in terms of reducing vertical acceleration, suspension deflection and tyre deflection compared to the passive system with no inerter. However, at the natural frequency of the unsprung mass, the magnitude of all three of these measures is much larger compared with the passive suspension system without inerter. Excessive amplification of the unsprung mass natural frequency is unacceptable, particularly for vertical acceleration and tyre deflection. These results show that although the addition of fixed inerter to a given suspension system could improve performance at the sprung mass natural frequency, it will also significantly exacerbate vibrations at the unsprung mass natural frequency and should be avoided.

3.3.3 Adaptive Inerter Control of Suspension System Inspired by the design of variable damper semi-active suspension control, an active H 2 controller is designed for the suspension system first, and then an adaptive inerter is used to apply force to the suspension system by tracking ideal control force.

TABLE 3.7 Comparison of Dynamic Characteristics for Passive System with Different Fixed Inerters Performance Natural frequency Vertical acceleration Suspension deflection Tyre deflection

Dynamic Characteristics f1 (Hz) f2 (Hz) Peak value (f1) Peak value (f2) Peak value ×10−2 (f1) Peak value ×10−2 (f2) Peak value ×10−2 (f1) Peak value ×10−2 (f2)

W/O Inerter 1.11 10.45 23.39 7.87 44.21 4.68 4.77 4.76

b = 100 1.02 6.45 18.63 55.58 43.23 15.07 3.79 18.43

b = 300

b = 500

0.85 4.85 13.40 237.70 38.32 33.91 2.78 58.75

0.75 4.40 10.42 419.30 36.51 46.18 2.18 103.10

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Semi-active Suspension Control

3.3.3.1  Formulation of H 2 Controller In general, ride quality, suspension deflection and road holding are the main performance indexes that should be considered in suspension design. Ride quality is related to the vertical acceleration of the sprung mass. It keeps the transfer function from road disturbance w to acceleration of the sprung mass  zs small. Suspension deflection requires a hard constraint on the suspension deflection zs − zu to avoid structural damage and deterioration of ride comfort when the deflection limit is reached. Road holding requires minimizing the transfer function from the disturbance w to the tyre deflection zu − zr to ensure firm contact of the wheels to road. An H 2 controller will be designed to meet the above requirements. Weighting factors α and β are used to shape and compromise the performance objectives. State feedback control law u = Kx is considered, where K is the control gain matrix which is designed to make the closed-loop system asymptotically stable and the H 2 norm (defined as T2 ) of the transfer function from the perturbation w to the output z of the closedloop system is bounded by a constant γ > 0. The following lemma gives a characterization on the H 2 performance [34]. Lemma 3.4 The suspension closed-loop system with state feedback controller u = Kx, where K = YP −1 is asymptotically stable with T2 ≤ γ if there exist matrices P > 0, W > 0 and Y satisfying tr (W ) < γ 2 ,

 PAT + AP + Y T BT + BY   *  −W  * 

Bw   < 0, (3.95) − I 

CP + DY   < 0. −P 

3.3.3.2 Adaptive Control Law for Inerter The desired active control force Fdes is obtained according to the active controller designed above. Figure 3.13a shows the adaptive inertance scheme. When Fdes and the relative suspension acceleration  zs −  zu are in the same direction, such an Fdes cannot be provided by the inerter since an inerter cannot actively supply energy. In this case, the value of the inertance b is set to be zero. When the active control force Fdes and the relative acceleration  zs −  zu are in the different directions, the inerter F can provide the desired force. In this case, the semi-active inertance is chosen as b   =   − des .  zs −  zu Considering the constraints 0 ≤ b < bmax , the inertance could be obtained as

 0,   − Fdes ,  b=  zs −  zu   bmax , 

if − Fdes (  zs −  zu ) ≤ 0, F if  0 < − des <   bmax , (3.96)  zs −  zu F if   − des ≥   bmax ,  zs −  zu

and the actual inerter force is calculated by F = − b (  zs −  zu ). The block diagram for the semi-active H 2 control of the vehicle suspension with adaptive inerter is shown in Figure 3.13b. The desired inerter force is calculated from the designed active controller, then it is approximately realized by the inerter with a time-varying inertance based on (3.96).

164

Vibration Control of Vehicle Suspension Systems z¨ s – z¨ u b=–

Fdes

z¨ s – z¨ u b ≤ bmax

b=0 Inerter Fdes b=–

b=0

b

Fdes

Inerter force

x¨ s – x¨ u

Inertance

z¨ s – z¨ u b ≤ bmax

Plant

Desired force

H2 controller

Fdes

(a)

(b)

FIGURE 3.13  (a) Adaptive law for inertance, (b) schematic diagram for vehicle suspension control with adaptive inerter.

TABLE 3.8 Weighting Constants and Corresponding Closed-Loop System Poles Case I II III

α

β

5

10

150

10

5

200

Poles −0.72 ± 63.23i

−1.29 ± 1.15i

−0.43 ± 0.19i

−13.70 ± 66.71i

−9.62 ± 57.55i

−11.15 ± 8.79i

3.3.4  Simulation Results and Discussions In this section, the proposed semi-active control method with adaptive inerter is investigated in simulation. Three cases are considered according to different performance priorities, Case I is with heavily weighted ride quality, Case II and Case III are with heavily weighted suspension deflection and tyre deflection, respectively. Three sets of α and β are adopted for controller gain calculations and the corresponding closed-loop system poles are listed in Table 3.8. The controller gain matrices corresponding to the above conditions are given as K1 = 10 4 × [1.8795, −0.1091, −0.0037, −0.0928 ],

K 2 = 10 4 × [ −4.8636, −7.5439, −0.6980,0.0080 ], K 3 = 10 4 × [1.9898,3.2976,0.0604,0.0363].

Figure 3.14a–c shows the open- and closed-loop frequency responses of the corresponding controllers from the disturbance zr to the vertical acceleration, suspension deflection and tyre deflection. In Case I, it can be seen that the designed controller can significantly reduce the vertical acceleration over a wide range of frequencies except at the natural frequency of the unsprung mass. This phenomenon is well known and can be found in previous works, see [34,35]. In Case II, the suspension deflection is improved at both the sprung and unsprung mass natural frequencies and also gives some improvements at high frequencies, but the performance is poor at low frequencies. In Case III, the tyre deflection is improved at both resonance frequencies. Table 3.9 compares the approximated H 2 norms of the vertical acceleration, suspension deflection and tyre deflection for suspensions with fixed and adaptive inerters. The H 2 norm is approximated by a trapezoidal rule on a given data set from 0.1 to 100 Hz. It can be seen that for vertical acceleration and tyre deflection, the H 2 norm of suspensions with adaptive inerter are significantly

Semi-active Suspension Control

165

FIGURE 3.14  Comparison of frequency responses from disturbance to: (a) vertical acceleration in Case I, (b) suspension deflection in Case II, (c) tyre deflection in Case III; comparison of scaled RMS value of: (d) vertical acceleration in Case I, (e) suspension deflection in Case II, (f) tyre deflection in Case III. (d–f) Show the scaled RMS values comparisons of vertical acceleration, suspension deflection and tyre deflection with and without adaptive inerter for the disturbance zr = 0.1sin ( 2πft ). Three adaptive inertance ranges are considered as b ∈[ 0, 100 ], b ∈[ 0, 300 ] and b ∈[ 0, 500 ] kg.

smaller than that of suspensions with fixed inerter. For suspension deflection, the difference of H 2 norms under different conditions is not obvious. Since the vibration at natural frequency is the focus in the vibration attenuation, the comparison of natural frequencies and their corresponding RMS values are presented in Tables 3.10–3.12. It can be seen from Figure 3.14d–f and Tables 3.10–3.12, the variations in the natural frequencies are not so obvious as those in the passive system with fixed

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Vibration Control of Vehicle Suspension Systems

TABLE 3.9 Comparison of the Approximated H 2 Norm Approximated H 2 norm Passive b = 100 b ∈[ 0, 100 ) b = 300 b ∈[ 0, 300 ) b = 500 b ∈[ 0, 500 )

VA

SD

TD

98.3245 89 44

0.8833 0.47 0.46

0.6824 0.29 0.26

189 65

0.45 0.50

0.49 0.34

340 81

0.49 0.48

0.84 0.38

VA, vertical acceleration; SD, suspension deflection; TD, tyre deflection.

TABLE 3.10 Natural Frequencies and RMS Accelerations Comparisons in Case I Passive System f1 (Hz) f2 (Hz) Peak value (f1) Peak value (f2)

1.11 10.45 1.65 0.54

b ∈ [ 0, 100 ] 1.01 8.10 1.24 0.92

b ∈ [ 0, 300 ]

b ∈ [ 0, 500 ]

0.93 7.06 0.96 1.75

0.87 6.14 0.82 2.34

TABLE 3.11 Natural Frequencies and RMS Suspension Deflection Comparisons in Case II f1 (Hz) f2 (Hz) Peak value ×10−2 (f1) Peak value ×10−2 (f2)

Passive System

b ∈ [ 0, 100 ]

b ∈ [ 0, 300 ]

b ∈ [ 0, 500 ]

1.11 10.45 3.09 0.34

1.01 6.58 3.31 0.32

0.93 5.72 4.34 0.43

0.93 5.72 4.67 0.54

TABLE 3.12 Natural Frequencies and RMS Tyre Deflection Comparisons in Case III f1 (Hz) f2 (Hz) Peak value ×10−2 (f1) Peak value ×10−2 (f2)

Passive System

b ∈ [ 0, 100 ]

b ∈ [ 0, 300 ]

b ∈ [ 0, 500 ]

1.11 10.45 0.34 0.33

1.00 7.56 0.27 0.84

0.93 5.72 0.21 1.05

0.87 5.33 0.18 1.13

Semi-active Suspension Control

167

3.3.5  Conclusions In this section, the adjustable inerter vehicle suspension is investigated. Different from the complex layout of designing a passive suspension with inerter, a suspension system consists of a spring, a damper and an inerter is considered. The effect of fixed inerter on the passive suspension system is investigated first, and the results show that the fixed inerter can attenuate the vibration at the natural frequency of the sprung mass, but the vibration at unsprung mass natural frequency is significantly amplified, particularly for ride quality and tyre deflection. Then for the practical applications, a force tracking control strategy with adjustable inerter is proposed to address this issue. In the control framework, the inerter tracks the desired control force of a suitably designed state feedback H 2 controller. Simulation results show that for both ride quality and tyre deflection, the designed control scheme reduces vibration at the sprung mass frequency at the cost of a small deterioration at the unsprung mass frequency; for suspension deflection, the performance is improved at high frequencies while the performance at low frequencies is also maintained.

3.4  VARYING EQUIVALENT STIFFNESS AND INERTER 3.4.1 The Semi-active Device with Variable Electrical Networks The semi-active device has been applied in numerous applications to improve system performance. However, the device implemented by a mechanical network is complicated and uneconomical when

168

Vibration Control of Vehicle Suspension Systems

FIGURE 3.15  Desired inerter force in: (a) Case I, (b) Case II, and (c) Case III; comparison of time responses in: (d) Case I, (e) Case II, and (f) Case III.

multiple mechanical properties are required to be controllable. This subsection will introduce the method to design the semi-active device with a variable electrical network, which can achieve similar mechanical properties with fewer efforts. 3.4.1.1  Variable Mechanical Networks The mechanical network with dampers, springs, and inerters can be characterized by admittance:

Yv ( w ) = 0 ( w ) + ρ0 ( w ) j (3.97)

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Semi-active Suspension Control

where j is the imaginary axis, and ω is the angular frequency. The real part 0 represents the mechanical conductance, and the imaginary part ρ0 is the mechanical susceptance containing the stiffness and inertance information. The equivalent damping ce , equivalent inertance be, and equivalent stiffness ke of a mechanical network can be obtained from its admittance, which are: ce = real (Yv ) (3.98)

be =

imag (Yv ) (3.99) ω

ke = − imag (Yv )ω (3.100)

where real( ) and imag( ) are the functions to obtain the real and imaginary parts of the admittance, respectively. When the inertance or stiffness in the mechanical network is dynamically varied without any external energy input to compensate for the stored energy in the spring or inerter, the system discontinuity will be caused easily. Therefore, the directly varying stiffness or inertance in the real-time controlled semi-active system will introduce disadvantages. In contrast, the VD device is easy to implement, which only requires efforts to change the damping property but no energy input to the system dynamics. The combinations of VD devices with spring or inerter have been proven to be effective for obtaining variable equivalent stiffness or inertance [36]. Table 3.13 shows four basic VMNs, where c is a VD device; b and c are the passive inerter and spring. The VMNs contain an inerter or a spring connecting with a VD device in serial or in parallel, respectively. Case 1 is a VMN that can vary the equivalent inertance by controlling the VD device. Assuming that b = 40 kg, and the damping c can vary from 0 to 2000 Ns/m, Figure 3.16a shows its admittance in the phasor diagram with different frequencies and damping. When damping c increases, the mechanical susceptance will increase. However, the mechanical conductance will increase first; then it will decrease when the phasor angle larger than 45°. As shown in Figure 3.16b, the equivalent inertance is decreasing with the increase of frequency, and the maximum equivalent inertance is determined by b. TABLE 3.13 Basic VMNs VMNs 1

2

3

4

Admittance b 2 cω 2 1 bc 2 = 2 + 2 ωj 2 2 1 1 c +b ω c + b 2ω 2 + Yb Yc Y2 = Yb + Yc = c + jω b Y = 1

Y = 3

1 1 1 + Yk Yc

=

k 2c kc 2ω 2 1 + 2 2 2 k +c ω k + c 2ω 2 jω 2

Y4 = Yk + Yc = c + k jω

170

Mechanical susceptance (Ns/m)

2000

(b)

1 Hz 2 Hz 3 Hz 10 Hz 100 Hz

1500

Inerter like

1000

c (0 to 2000 Ns/m) Damper like

500

0

0

500

1000

1500

40

c (Ns/m)

30

2000 1000 500 100

20 10 0

1

10 Frequency (Hz)

100

1

10 Frequency (Hz)

100

90 Phase (°)

(a)

Equivalent inertance (kg)

Vibration Control of Vehicle Suspension Systems

60 30 0

2000

Mechanical susceptance (Ns/m)

0

1 Hz 2 Hz 3 Hz 10 Hz 100 Hz

Damper like

–200

–400

(d)

10000 c (Ns/m) 2000 1000 500 100

5000

0 0

1

10 Frequency (Hz)

100

1

10 Frequency (Hz)

100

–600 c (0 to 2000 Ns/m) –800

–1000

Phase (°)

(c)

Equivalent stiffness (N/m)

Mechanical conductance (Ns/m)

Spring like

–30 –60 –90

0

200

400

600

800

Mechanical conductance (Ns/m)

1000

FIGURE 3.16  (a) Admittance of Case 1; (b) equivalent inertance of Case 1; (c) admittance of Case 3; (d) equivalent stiffness of Case 3.

Similar to Case 1, the variable equivalent stiffness can be achieved in Case 3. It is assumed that k = 10,000 N/m and the damping c can vary from 0 to 2000 Ns/m. Figure 3.16c shows the admittance in the phasor diagram, and Figure 3.16d shows the equivalent stiffness. The result illustrates that the equivalent stiffness is increasing with the increase of frequency and damping c. In Cases 2 and 4, elements are connected in parallel. The change of VD device has no influence on the imaginary part of the admittance; hence, it will not vary the equivalent stiffness or inertance. Therefore, Case 1 and Case 3 are used in this chapter to achieve variable inertance and stiffness. 3.4.1.2  Semi-active Device with an Electrical Network Researchers have found the similarity of theories between electrical and mechanical systems and proposed the force-current analogy theory. It provides the correspondences of mechanical systems and electrical systems [37]. For a two-terminal mechanical and electrical element, it can be characterized with impedance and admittance, which are reciprocal of each other. Table 3.14 shows that the mechanical damping c, stiffness k, and inertance b are corresponding with the electrical resistance R, inductance L, and capacitance C, respectively. The implementation of an electrical network is significantly more accessible than the mechanical structure. Also, the electrical system has advantages in installation and maintenance. However, with electrical elements barely cannot achieve the mechanical properties. The electromagnetic device can transfer the mechanical energy into electrical energy, and the current in its circuit is corresponding to the force output. Therefore, according to the force-current analogy, a variable electrical network can be designed by imitating the VMNs; then the electromagnetic device can be applied to achieve the desired mechanical properties.

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Semi-active Suspension Control

TABLE 3.14 Impedance and Admittance of Mechanical and Electrical Elements Mechanical (Electrical) Element c (R) k (L)

b (C)

Impedance 1 Z c =    ( Z c = R ) c Zk =

jω   ( Z L = jω L ) k

Zb =

1  1     ZC = jω b  jω C 

Admittance 1  Yc = c     YR =   R Yk =

1  k     YL = jω L  jω 

Yb = jω b  (YC = jω C )

3.4.2 The Electromagnetic VESI Device with a Variable Electrical Network The inertance control can improve the vibration reduction, especially at the resonance frequency [38]. Besides, for vehicles, the stiffness is essential in the anti-roll performance during cornering manoeuvres. Therefore, a device with the capability of controlling inertance and stiffness is expected to improve vehicle performances in both vibration control and rollover prevention. This subsection will present a semi-active VESI device with a controllable electrical network, which is inspired by the basic VMN structure, and then it is analyzed according to experiments. 3.4.2.1 The VESI Electromagnetic Device The parallel connection of mechanical networks means their properties are accumulated; ideally, Cases 1 and 3 of the VMN can generate a new VESI network by connecting them in parallel. Figure 3.17a shows the VESI network with electrical and mechanical elements, where the capacitor C, inductor L and variable resistors R1, 2 are corresponding to the inerter b, spring k, and VD devices c1, 2, respectively. The electromagnetic device is applied as the intermedia of the electrical system and mechanical system, which can transform mechanical energy into electrical energy. Figure 3.17b shows the physical configuration of the “VESI” electrical network with an electromagnetic device, where z1 and z2 are the two mechanical terminals (the rotor and stator of the electromagnetic device); e is the induced voltage by the device; Li is the inner inductance; Ri is the resistance of the electromagnetic device and circuit. The electrical VESI network is attached to the electromagnetic system. R1 and R2 are two variable resistors; Re1 and Re2 are the system inherent resistance coming from the circuit and coils of the inductor. Figure 3.17c shows the equivalent mechanical network of the electromagnetic VESI device, where T is the torque of the device, and z1 and z2 are the rotary angle of its two terminals. The equivalent mechanical properties of the electrical system are [39]:

b = kem 2C (3.101)

k=

2 kem (3.102) L

ki =

2 kem   (3.103) Li

ci =

2 kem (3.104) Ri

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Vibration Control of Vehicle Suspension Systems

FIGURE 3.17  (a) VESI network; (b) the VESI device with electromagnetic device; (c) the equivalent mechanical network of the electromagnetic VESI device; (d) admittance of the electromagnetic VESI device; (e) circuit of the electromagnetic VESI device; (f) variable resistor structure.

c1 =

2 kem (3.105) ( Re1 + R1 )

c2 =

2 kem (3.106) ( Re 2 + R2 )

where kem is the motor constant of the electromagnetic device. The admittance of the VESI device can be simplified to three components as shown in Figure 3.17d, where Ym is the admittance of the motor; Yi is the VI part with a capacitor; and Ys is the VS part with a inductor. The three admittances are:

ki ci jω Ym = (3.107) ki + ci jω

Yi =

k c2 jω Ys = (3.109) k + c2 jω

bjω c1 (3.108) bjω + c1

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Semi-active Suspension Control

Hence, the total admittance of the VESI device is:

Yvsvi =

(Yi + Ys )Ym (3.110) (Yi + Ys ) + Ym

Besides, the dynamic model of the VESI device can be built. The circuit of the electromagnetic VESI device is shown in Figure 3.17e. The induced voltage e is:

e = − kem ( z2 − z1 ) (3.111)

The mesh currents I 0 and I1 are assigned to two meshes. Applying Kirchhoff’s voltage laws (KVL), the system model is:

(

)

(

)

−e + I 0 Ri + I0 Li + ( I 0 − I1 )( Re 2 + R2 ) + I0 − I1 L = 0 (3.112)

I1 ( Re1 + R1 ) + U1 − ( I 0 − I1 )( Re 2 + R2 ) + I0 − I1 L = 0 (3.113)

CU1 = I1 (3.114)

The output torque T of the electromagnetic device is:

T = kem I 0 (3.115)

Therefore, the electromagnetic torque of the device can be evaluated with the current I 0. 3.4.2.2  Experimental Validation For building the electromagnetic VESI device prototype, the real-time controllable resistor needs to be designed first. Figure 3.17f shows the structure of the variable resistor [40], where r1~ n are n constant resistors; S0~ n are n + 1 switches. The constant resistors r1~ n are set according to their resistance from small to large. Hence, by controlling switches, the equivalent resistance of the variable resistor can vary from 0 to rn. In this test, three constant resistors, 3 Ω, 10 Ω, and 100 Ω, are applied. In theory, this variable resistor has a variation range from 0 to 100 Ω. CotoMos CS128 is applied as the switches, which is controllable with 5 V control signal. The whole experimental system is shown in Figure 3.18, where the VESI device prototype includes a DC motor (Maxon 353301) as the electromagnetic device, two variable resistors, a customized inductor and a non-polarized capacitor (modified by two polarized capacitors: Cornell Dubilier DCMX323U040AC2B). The servomotor can generate reciprocating rotation to drive the DC motor. The controller (NI CompactRio 9074) can control the variation of resistors and record the current and voltage of the circuit, as well as the rotation angle of the DC motor through its encoder. When the servomotor rotates in a sinusoidal wave with 1.5 Hz and 150°, and the two controllable resistors are varied from 0 to 100 Ω, respectively, the experimental result is shown in Figure 3.18b and c. According to test data and the specifications of the motor and the applied electrical elements, the parameters in the device model (3.111) to (3.114) are all identified as shown in Table 3.15. The device model is built based on the assumption that all the elements are with linear properties. However, the customized inductor and capacitor are nonlinear at low frequency, which has an

174

Vibration Control of Vehicle Suspension Systems

(a)

Variable resistors

Inductor

(b)

0 exp 3 exp 10 exp 100 exp 0 sim 3 sim 10 sim 100 sim

R2 (Ohm)

2.0 1.5

Capacitor

1.0

Servomotor

DC motor

0.5

Current (A)

Controller

0.0 –0.5 –1.0 –1.5 –2.0 –150

(c)

(d)

–50

0 50 100 Rotary angle (°)

150

200

0.012

1.0

0.010 Mechanical susceptance (Ns/m)

R1 (Ohm)

0.8

0 exp 3 exp 10 exp 100 exp 0 sim 3 sim 10 sim 100 sim

0.6 0.4 Current (A)

–100

0.014

0.2 0.0 –0.2 –0.4 –0.6

R1 (100 to 0 Ohm)

0.008 0.006

VI (R2 =100 Ohm)

0.004

1 Hz 2 Hz 3 Hz 10 Hz

0.002 0.000 –0.002

VS (R1 =100 Ohm)

–0.004

R2 (100 to 0 Ohm)

–0.006 –0.008

–0.8

1 Hz 2 Hz 3 Hz 10 Hz

–0.010 –50

0 50 100 Rotary angle (°)

150

200

(f)

0.0010

R1

0.0008 0.0006

0 Ohm 3 Ohm 10 Ohm 100 Ohm

0.0004 0.0002 0.0000 90 80 70 60 50 40 30 20 10 0 –10 –20

1

Frequency(Hz)

10 phase(°)

phase(°)

Equivalent inertance (kgm2)

(e)

0.010 0.015 0.020 0.005 Mechanical conductance (Ns/m) Equivalent Stiffness (Nm/rad)

–150 –100

1

Frequency(Hz)

0.30 0.25 0.20 0.15 0.10 0.05 0.00

0.025

R2 0 Ohm 3 Ohm 10 Ohm 100 Ohm

10 0 –10 –20 –30 –40 –50 –60 –70 –80

1

Frequency(Hz)

10

1

Frequency(Hz)

10

10

FIGURE 3.18  (a) Experimental system; (b) variable equivalent stiffness (R1 = 100 Ω); (c) variable equivalent inertance (R2 = 100 Ω; (d) phasor diagram of the VESI device; (e) equivalent inertance of the VESI device (R2 = 100 Ω); (f) equivalent stiffness of the VESI device (R2 = 100 Ω).

TABLE 3.15 Identified System Parameters Li

0.644 mH

Re1

0.2 Ω

Ri kem C

1.6 Ω 0.245 Nm/A 0.016 F

Re2 L

1.2 Ω 0.21 H

influence on the system dynamics to cause the mismatch of experimental and simulation results with low magnitude resistances. Despite the effect of nonlinearity, the proposed model can successfully describe the VESI properties, and the experimental and simulation results are reasonably matched. The electromagnetic torque output is proportional to the current. Therefore, the results imply that, by decreasing R1, the equivalent inertance will increase. Similarly, the equivalent stiffness increases

175

Semi-active Suspension Control

with the decrease of R2. The equivalent inertance and stiffness are controllable with the two variable resistors, respectively. This section aims to achieve the variable mechanical properties with an electrical network, which can be characterized by the proposed linear model. 3.4.2.3  Admittance Analysis of VESI Device The experimental data shows the VESI capability of the device. With the identified system parameters, the device in the frequency domain can be further analysed. According to equations (3.107)–(3.110), the admittance of the device in terms of the variation of frequency and resistance is obtained. Figure 3.18d shows the admittance in the phasor diagram, where the imaginary part of the admittance can vary to the positive phase and also the negative phase by controlling two resistors. Figure 3.18e and f shows the equivalent inertance and stiffness, respectively. The results are consistent with the experiments, where the variable equivalent inertance and stiffness can be achieved by the control of R1 and R2, respectively.

3.4.3 The VESI Suspension for Vehicles The rotary VESI device is applied in the vehicle to improve vibration control and rollover prevention performance in this subsection. 3.4.3.1  VESI Suspension Figure 3.19a shows an electromagnetic suspension design, where the ball screw is applied to transform the rotary and linear motions as well as the torque and force from the motor to the suspension. 2π The transformation ratio rg is , where ld is the lead of the ball screw. Therefore, the total force ld output of electromagnetic suspension is: Fe = −cs ( zds ) − bs (  zds ) + um (3.116)

where the equivalent inertance of the motor rotor is bs = rg 2 I m, and I m = 1.34 × 10 −4  kgm 2 is the rotor inertia of the motor; the electromagnetic force is um = rgT , and T is the controllable electromagnetic torque of the motor; zds is the deflection of the two suspension terminals; cs is the equivalent damping caused by the friction of the ball screw and motor. 3.4.3.2  Vehicle Model The VESI suspension will be applied in a half-car model to investigate its effectiveness in vibration control and rollover prevention. First, the bicycle model is applied to obtain the lateral vehicle dynamics [41], as shown in Figure 3.19b. The lateral motion is:

 y=−

k f + kr k l + kr lr k y − Vxψ − f f ψ + f σ (3.117) msVx msVx ms

where ms is sprung mass; k f and kr are the front and rear cornering stiffness, respectively; y is the lateral motion; Vx is the longitudinal velocity; ψ is the yaw rate; σ is the steering angle; l f and lr are distances of front and rear wheels to the centre of gravity (c.g.) of the vehicle, respectively. The yaw dynamics model is:

ψ = −

k f l f − kr lr k l 2 + kr lr 2 k l y − f f ψ + f f σ (3.118) I zzVx I zzVx I zz

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Vibration Control of Vehicle Suspension Systems

FIGURE 3.19  (a) Electromagnetic suspension; (b) lateral vehicle dynamics; (c) the control circuit for one suspension; (d) semi-active suspension control.

where I zz is the moment of inertia in yaw. The half-car vertical vibration model has been introduced in Chapter 1, and this section will not introduce this model in detail. However, the roll dynamics of the sprung mass is different from the passive one because the bicycle model is introduced in this section. The roll dynamics of the sprung mass is:

(I

xx

)

l + ms hR 2 θs = ms a y hR cosθ s + ms ghR sin θ s + s ( − Fsr + Fsl ) (3.119) 2

where I xx is the roll moment of inertia about the c.g. and g is the gravity acceleration. The dynamic suspension forces can be obtained by using the VESI device:

Fsr = − ks ( zsr − zur ) − cs ( zsr − zur ) − bs (  zsr −  zur ) + ur (3.120)

Fsl = − ks ( zsl − zul ) − cs ( zsl − zul ) − bs (  zsl −  zul ) + ul (3.121)

ls l sinθ s and zsl = zs + s sinθ s ; ur and ul are the electromagnetic forces at right and 2 2 left sides, respectively. ks and kt are the stiffness of suspension and tyre, respectively.

where zsr = zs −

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Semi-active Suspension Control

With the aforementioned model, the investigation about the performance of VESI suspension in vibration control and rollover prevention can be implemented. 3.4.3.3  Vibration Control The VESI suspensions for vibration control and rollover prevention are at two different modes. Thus, separate controllers are required to be designed. H ∞ controller, which has been proven to be effective for suspension, is applied in this paper for controlling the road vibration. According to the system model, the state variables of the system are selected as T  X =  zsr − zur zsr zur − zrr zur zsl − zul zsl zul − zrl zul  . Hence, the half-car suspension model can be built as:  = AX + B1U + B2 d (3.122) X

 0  −a k 1 s   0  ks   mu A= 0  − a 2 ks   0   0 

where

  0 B1 =    0 

a1 a2

0 0

u2 = bs (  zsl −  zul ) + ul ,

1 − mu 0

1 − a1cs 0 cs mu 0 − a2cs 0

0 0 0 k − t mu 0 0 0

−1 a1cs 1 c − s mu 0 a2cs 0

0

0

0

0 − a2 ks 0

0 − a2c2 0

0 0 0

0 a2cs 0

0

0

0

0

0 − a1ks 0 ks mu

1 − a1cs 0 cs mu

0 0 0 k − t mu

−1 a1cs 1 c − s mu

       ,      

T

0 0

 zlr  d =  zrr   Fa

a2 a1   ,  

0

0

0

1 − mu

 0  0 B2 =   0  

   ,    0 0 l − s 2I s

 u1 U=  u2

 , 

u1 = bs (  zsr −  zur ) + ur ,

−1 0

0 0

0 0

0

0

0

0 0 ls 2I s

0 −1 0

T

0  0  , 0  

1 1 l2 l2 + s , a2 = − s , I s = I xx + ms hR 2, Fa = ms a y hR cosθ s + ms ghR sin θ s . sin θ s ≈ θ s is applied ms 4 I s ms 4 I s to simplify the model. In the suspension control, the ride comfort, suspension deflection and road-holding ability are generally taken into account [42]. Therefore, the control outputs are defined as:

a1 =

Z = CX + D1U + D 2 d (3.123)

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Vibration Control of Vehicle Suspension Systems

 zs where Z =   

θs

 ks  − m s   k s ls   2I s  1  z  fmax C = α 0    0     0 

zsr − zur z fmax cs ms cs ls 2I s

zsl − zul z fmax 0 0

kt ( zur − zrr ) Ff

cs m cl − ss 2I s

ks ms kl − ss 2I s

cs ms cl − ss 2I s

kt ( zul − zrl )   , Ff  0 0

cs ms csls 2I s

0

0

0

0

0

0

0

0

0

0

1 z fmax

0

0

0

0

kt Ff

0

0

0

0

0

0

0

0

0

0

kt Ff

0

          ,        

 1 1    ms   ms α = diag (α 1 ,α 2 ,α 3 ,α 4 ,α 5 ,α 6 ) is a weighting matrix, D1 = α  ls ls   ,  −   2I s 2I s   0 4 ×1 0 4 ×1     01× 2 0    1  D 2 = α  01× 2 ; 0 is a zero-matrix with specified dimensions; z fmax is the maximum suspen Is     0 4 × 2 0 4 ×1  sion deflection hard limits; Ff = ( ms + 2mu ) g /2 is the static tyre load. The H ∞ state feedback control law is given by U = KX, where K is the feedback gain matrix to be designed. Therefore, K can be obtained by solving

 P ( A + B1K ) + * PB2 *    * −γ 2 I *  < 0 (3.124)   D2 − I   C + D1K

where P = P T > 0, γ is the desired level of disturbance attenuation. Pre- and post-multiplying (3.124) by diag P −1 , I, I and its transposition, respectively, and defining Q = P −1, W = KQ, we have

(

)

 AQ + B1W + * B2 *    −γ 2 I *  < 0 (3.125)  *  CQ + D W D2 − I  1  

which can be solved with the LMI toolbox in MATLAB and K = WQ −1.

179

Semi-active Suspension Control

Therefore, the desired electromagnetic forces of the right and left side suspensions are:

ur = u1 − bs (  zsr −  zur ) (3.126)

ul = u2 − bs (  zsl −  zul ) (3.127)

3.4.3.4  Rollover Prevention When the vehicle is under the cornering manoeuvre, the lateral acceleration a y has the risk to cause the vehicle rollover. A sliding mode controller is applied in the VESI suspension for rollover prevention. The system roll dynamics is written as:

(I

xx

)

+ ms hR 2 θs = ms ghR sin θ s + Tϑ + dϑ (3.128)

l where dϑ = ms a y hR cosθ s is the system disturbance with the assumption dϑ ≤ D; Tϑ = s ( − Fsr + Fsl ) 2 is taken as the total control input. The tracking error of the controller is: e = θ sd − θ s (3.129)

where θ sd is the desired roll angle, which is assumed as zero. Therefore, the sliding surface is defined as: s = ce + e (3.130)

where c > 0. The derivative of the sliding surface is:

s = −

1 ( ms ghR sinθ s + Tϑ + dϑ ) + ce (3.131) I xx + ms hR 2

Considering the Lyapunov functions as:

L=

1 2 s (3.132) 2

So, its derivative is:

  1 L = ss = s  − ( ms ghR sinθ s + Tϑ + dϑ ) + ce  I xx + ms hR 2 

Therefore, the sliding mode controller can be designed as:

(

)

Tϑ = − ms ghR sin θ s + I xx + ms hR 2 ce + ηsgn ( s ) (3.133)

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Vibration Control of Vehicle Suspension Systems

where η > D. Then, we have

L = −

s (ηsgn ( s ) + dϑ ) = − η s + sdϑ 2 ≤ 0 (3.134) I xx + ms hR 2 I xx + ms hR

To avoid the chattering phenomenon in (3.133) caused by sgn(s), the saturated function sat(s) is applied to replace it:  1  sat ( s ) =  ks  −1 

s>∆ 1 s ≤ ∆ ,   k = (3.135) ∆ s < −∆

where ∆ is the “boundary layer”. The input torque Tϑ is composed of the suspension forces on the right and left sides, which are assumed to be equal with the magnitude. Therefore, we have

Fsr = −

Fsl =

Tϑ (3.136) ls

Tϑ (3.137) ls

Hence, the desired electromagnetic forces are:

ur = −

Tϑ + ks ( zsr − zur ) + cs ( zsr − zur ) + bs (  zsr −  zur ) (3.138) ls

ul =

Tϑ + ks ( zsl − zul ) + cs ( zsl − zul ) + bs (  zsl −  zul ) (3.139) ls

3.4.3.5  Semi-Active VESI Suspension Control No matter whether the controller is working in the vibration control or rollover prevention mode, it can only give the desired active electromagnetic forces of the VESI suspension. A semi-active strategy is required to control the proposed system according to the desired forces and system states. Figure 3.19c shows the control circuit for one suspension, where U 0 is the voltage of the control circuit, I L is the current through the inductor, and U1 is the voltage of the capacitor. According u to the desired electromagnetic forces, the desired current I d is obtained as i , where i = r  or  l . rg kem Therefore, the semi-active control is to make I 0 tracking I d by varying the two resistors R1 and R2. I 0 is composed of the inductor and capacitor currents; thus, the two branch currents need to be controlled. The upper and lower bounds of the resistance-related admittances in the two branches are obtained:

Y1max =

1 (3.140) R1min + Re1

Y1min =

1 (3.141) R1max + Re1

Y2 max =

1 (3.142) R2 min + Re 2

181

Semi-active Suspension Control

Y2 min =

1 (3.143) R2 max + Re 2

In the first step, R1 can be determined. R2 is assumed to be the maximum value; thus, the minimum current in the inductor branch is:

I Lmin = (U 0 − U L )Y2 min (3.144)

where U L = IL L is the voltage of the inductor. Thus, the desired admittance of the resistance in the capacitor branch is:

Y1d = ( I d − I Lmin ) / (U 0 − U1 ) (3.145)

The admittance should be within Y1min and Y1max , thus, we have:

Y1 = min ( max (Y1min ,Y1d ) ,Y1max ) (3.146)

Therefore, R1 can be obtained:

R1 =

1 − Re1 (3.147) Y1

In the second step, R2 can be determined similarly. The desired admittance of the resistance in the inductor branch is:

Y2 d = ( I d − (U 0 − U1 )Y1 ) / (U 0 − U L ) (3.148)

Also, the resistance admittance should be within the upper and lower bounds.

Y2 = min ( max (Y2 min ,Y2 d ) ,Y2 max ) (3.149)

Thus, R2 can be obtained:

R2 =

1 − Re 2 (3.150) Y2

Due to that the anti-roll controller only works in the cornering state, a trigger needs to be designed to activate the controller. When the steering angle σ > σ 0 , the suspension will switch from vibration control mode to rollover prevention mode. σ 0 is a threshold to enable the sliding mode controller. When σ ≤ σ 0 and the state keeps longer than a period of time t0 , the suspension control will change back to vibration control. The whole semi-active suspension control strategy is shown in Figure 3.19d, where the desired electromagnetic forces can be obtained by the active controller, and then the semi-active controller will figure out proper resistances to track the desired forces.

3.4.4  Simulation Analysis The developed VESI suspension and controllers are validated with numerical simulation. 3.4.4.1  Simulation Setup A passive suspension is also simulated for comparison, in which a passive damper c p is applied to replace the electromagnetic suspension. Table 3.16 shows the suspension parameters in the simulation.

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Vibration Control of Vehicle Suspension Systems

In the vibration controller, α = diag(1, 0.1, 0.05, 0.05, 0.05, 0.05) and γ = 7 are applied. The control 26,572 −1781 −1889 −127 2705 1068 1372 107  ler gain is obtained as K =  . 1068 1372 107 26,572 −1781 −1889 −127   2705 For the rollover prevention controller, c = 20, η = 2000 and ∆ = 0.01 are applied. The trigger condition is defined as σ 0 = 0.015 and t0 = 1. 3.4.4.2  Vibration Control The random road profile, as shown in Figure 3.20a, is applied to test the suspension performance in the vertical and roll vibration control. Figure 3.20b shows the vertical acceleration of the vehicle body, and Figure 3.20c shows the PSD of accelerations. The proposed VESI system can effectively control the vertical vibration, especially TABLE 3.16 Suspension Parameters ms

800 kg

ls

1.5 m

mu kf kr lf lr I zz ld

ks cs kt I xx hR

2240 kg*m2 0.008 m

g cp

30,000 N/m 300 Ns/m 200,000 N/m 390 kg*m2 0.4 m 9.81 m/s2 3000 Ns/m

FIGURE 3.20  (a) Random road profiles; (b) vertical acceleration at time domain; (c) vertical acceleration at frequency domain; (d) roll acceleration at time domain; (e) roll acceleration at frequency domain; (f) force tracking performance.

Semi-active Suspension Control

183

at low frequencies. The RMS acceleration of the passive and semi-active ones are 0.9085 m/s2 and 0.5904 m/s2, respectively. The proposed system achieves a reduction of 35.01% in the vertical vibration. The roll vibration control result is shown in Figure 3.20d and e, where the RMS acceleration of the passive and semi-active suspensions are 1.0265 and 0.7535 rad/s2, respectively. The proposed system has improved the performance of roll vibration control by 26.6%.

FIGURE 3.21  (a) Steering angle input of the tyre; (b) roll angle of sprung mass; (c) control modes (0: vibration control, rollover prevention).

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Vibration Control of Vehicle Suspension Systems

Figure 3.20f shows the force-tracking performance of the right suspension. The semi-active device output depends on the system states and control signals. Therefore, it is impossible for the semi-active device to track a random force precisely. The result indicates that the proposed control method can enable the device to track most parts of the desired active force. Besides, at the points where the semi-active device can only generate a force opposite to the desired one, the output forces are at low magnitudes. The result implies the effectiveness of the proposed semi-active control strategy. 3.4.4.3  Rollover Prevention The steering angle σ as shown in Figure 3.21a is applied for the vehicle bicycle model and the longitudinal velocity Vx, is assumed as 40 km/h. Figure 3.21b shows the roll angle of the sprung mass, where the PTP roll angle has decreased by 38.68%. The proposed VESI system can decrease the roll angle of the sprung mass during vehicle cornering, which will increase the safety of the vehicle. Besides, with the proposed strategy, the steering is successfully detected as shown in Figure 3.21c.

3.5 CONCLUSIONS In this section, an electromagnetic VESI device with an electrical network has been designed and tested; the application of the device for vehicle vibration control and rollover prevention has been investigated. According to the force-current analogy, a “VESI” electrical network has been designed to simulate the VESI mechanical network. The electrical system has been equipped with an electromagnetic device in order to achieve the VESI properties. The VESI device has been built and validated with experiments. Then, the application of the VESI suspension in the vehicle has been investigated and analysed. Two controllers have been proposed for the suspension to control the vibration and prevent the rollover. A semi-active strategy has been designed to control the halfcar system with VESI suspensions. The simulation result shows the proposed semi-active system can significantly improve the vibration control performance as well as the rollover prevention in the cornering manoeuvre.

REFERENCES

1. Spencer, B.F., et al., Phenomenological model for magnetorheological dampers. Journal of Engineering Mechanic, 1997. 123(3): 230–238. 2. Lai, C.Y. and W.H. Liao, Vibration control of a suspension system via a magnetorheological fluid damper. Journal of Vibration and Control, 2002. 8(4): 527–547. 3. Felix-Herran, L., et al., Control of a semi-active suspension with a magnetorheological damper modeled via Takagi-Sugeno. in IEEE 18th Mediterranean Conference on Control & Automation (MED), 2010: 1265–1270. 4. Wang, D.H. and W.H. Liao, Magnetorheological fluid dampers: A review of parametric modelling. Smart Materials and Structures, 2011. 20(023001): 1–34. 5. Kazuo Tanaka, H.O.W., Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach. 2002, New York: Wiley. 6. Chen, H. and K. Guo, Constrained H∞ control of active suspensions: An LMI approach. IEEE Transactions on Control Systems Technology, 2005. 13(3): 412–421. 7. Kim, J.H. and F. Jabbari, Actuator saturation and control design for buildings under seismic excitation. Journal of Engineering Mechanics, 2002. 128(4): 403–412. 8. Zhou, K. and P.P. Khargonekar, An algebraic Riccati equation approach to H∞ optimization. Systems & Control Letters, 1988. 11: 85–91. 9. Boyd S, et al., Linear Matrix Inequalities in System and Control Theory. 1994, Philadelphia: Society for Industrial and Applied Mathematics. 10. Cao, Y.Y. and Z. Lin, Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation. IEEE Transactions on Fuzzy Systems, 2003. 11(1): 57–67. 11. Boada, J. Satellite control with saturating inputs. Mathematics, 2010.

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40. Liu, P., et al., Torque response characteristics of a controllable electromagnetic damper for seat suspension vibration control. Mechanical Systems and Signal Processing, 2019. 133: 106238. 41. Rajamani, R., Vehicle Dynamics and Control. 2011, Berlin: Springer Science & Business Media. 42. Du, H. and N. Zhang, Constrained H∞ control of active suspension for a half-car model with a time delay in control. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 2008. 222(5): 665–684.

4

Integrated Suspension Control

4.1 INTRODUCTION This chapter introduces integrated suspension control methods with lateral dynamics and vertical dynamics. First, an observer-based robust gain-scheduling integrated control strategy for the vehicle lateral and active suspension system control is investigated. The uncertainty of tyre cornering stiffness, the actuator saturation and the hard constrains of suspension design are taken into account. The longitudinal velocity is considered time-varying and a polytope with trapezoidal structure is used to describe the velocity-dependent parameters. An observer is designed to solve the problem where states are hard to be measured. The simulation results illustrate the effectiveness of the robust saturated gain-scheduling H ∞ / GH 2 controller under different road profiles and manoeuvres. This chapter also presents a Takagi-Sugeno (T-S) model-based fuzzy control design approach for electrohydraulic active vehicle suspensions. The T-S fuzzy model is applied to represent the nonlinear uncertain, such as actuator, sprung mass variation, and constraints on the control input. Then, the fuzzy state feedback controller is designed for the obtained T-S fuzzy model with optimized H ∞ performance for ride comfort by using the parallel distributed compensation (PDC) scheme. Numerical simulations on a full-car suspension model are performed to validate the effectiveness of the proposed approach. An integrated vehicle seat suspension for a quarter-car with a driver model and a cabin model is proposed to improve suspension performance on driver ride comfort. An integrated suspension model that includes a quarter-car suspension, a seat suspension, and a 4-degree-of-freedom (DOF) driver body model is presented first. This integrated model provides a method to evaluate ride comfort performance in terms of driver head acceleration responses under typical road disturbances and to develop an integrated control of the seat and car suspensions. Based on the integrated model, a H ∞ state feedback controller is designed to minimize the driver head acceleration under road disturbances. Since state variables of a driver body model are not measurements available in practice, a static output feedback controller, which only uses measurable state variables, is designed. Further discussion on robust multi-objective controller design, which considers driver body parameter uncertainties, suspension stroke limitation, and road-holding properties, is also provided. Finally, simulations are conducted to evaluate the effectiveness of the proposed control strategy. Simulation results show that the integrated seat and suspension control can effectively improve suspension ride comfort performance compared with the passive seat suspension, active seat suspension control, and active car suspension control.

4.2  INTEGRATED WITH LATERAL DYNAMICS 4.2.1 Observer-Based Multi-Objective Integrated Control for Vehicle Lateral Stability and Active Suspension Design 4.2.1.1  System Modelling This session focuses on the integrated control of vehicle lateral and vertical dynamics based on the coordination of DYC and ASS. Therefore, four wheels vehicle model and bicycle model are used to describe the vehicle lateral dynamic, and a half-car suspension model is utilized to illustrate the vehicle pitch dynamic in this work. As the four wheels vehicle model shown in Figure 4.1a, the δ denotes the front tyre steering angle; β , r are the vehicle sideslip angle and the yaw rate of vehicle, respectively; Vx, Vy are the vehicle DOI: 10.1201/9781003265665-4187

188

Vibration Control of Vehicle Suspension Systems

FIGURE 4.1  (a) Four wheels vehicle model and (b) bicycle model.

longitudinal and lateral velocity, respectively; I z denotes the vehicle yaw inertia; l f , lr stand for the d­ istance from the vehicle; Fxi denotes the longitudinal tyre force of the ith tyre, where i = 1, 2, 3, 4; ∆M z represents the external yaw moment that used to control vehicle lateral stability, and can be produced by the driving/braking force differences between the wheels as follows:

∆ M z = Fx1 ( − cos δ ⋅ ls + sin δ ⋅ ls ) + Fx 2 ( cos δ ⋅ ls + sin δ ⋅ ls ) (4.1) − Fx 3 ⋅ ls + Fx 4 ⋅ ls

To simplify the calculation process, the four wheels vehicle model can be replaced by a bicycle model as shown in Figure 4.1b. Then, the vehicle lateral dynamic characteristics can be established as following equations:

(

)

mVx β + r = Fyf + Fyr , I z r = l f Fyf − lr Fyr + ∆M z

(4.2)

where m is the total mass of vehicle, Fyf ,  Fyr are the front and rear vehicle lateral tyre forces, respectively. Suppose the vehicle tyre slip angle is small enough and the tyres work in a linear region, the vehicle lateral tyres force can be rewritten as: Fyf = C yf α yf ,  Fyr = C yrα yr ,

αf =δ −

lfr lr − β ,α r = r − β Vx Vx

(4.3)

As shown in Figure 4.2, a half-car suspension model is adopted to describe the vehicle vertical and pitch motion, where ms is the vehicle sprung mass; musf ,   musr denotes the vehicle front and rear unsprung masses, respectively; Z s and θ are the vertical displacement and the pitch angle of the

189

Integrated Suspension Control

FIGURE 4.2  Half-car suspension model.

sprung mass at CG, respectively; Z sf and Z sr represent the displacement of front and rear sides of sprung mass, respectively; Zusf , Zusr are the displacement of front and rear unsprung masses, respectively; Zcf , Zcr denote the road displacement at front and rear wheels, respectively; Fuf , Fur are the front and rear active suspension forces, respectively; K sf , K sr and K tf , K tr are the spring coefficients of the front and rear suspension and tyres, respectively. Then, the vehicle vertical dynamics equations can be written as:

(

)

ms Z s = K sf ( Zusf − Z sf ) + Csf Z usf − Z sf + Fuf ¨

(

)

+ K sr ( Zusr − Z sr ) + Csr Z usr − Z sr + Fur ,

(

¨

(

)

I p θ = lr K sr ( Zusr − Z sr ) + Csr Z usr − Z sr + Fur

(

(

)

)

)

− l f K sf ( Zusf − Z sf ) + Csf Z usf − Z sf + Fuf ,

(4.4)

musf Z usf = K tf ( Zcf − Zusf ) − ( K sf ( Zusf − Z sf ) ¨

(

)

+Csf Z usf − Z sf + Fuf ), ¨

musr Z usr = K tr ( Zcr − Zusr ) − ( K sr ( Zusr − Z sr )

(

)

+Csr Z usr − Z sr + Fur ). ¨

¨

¨

where Z s and θ represent the car-body vertical acceleration and pitch acceleration, respectively; Z usf , ¨ Z usr denote the vertical acceleration of front and rear unsprung masses; Z sf , Z sr are the vertical velocity of front and rear sides of sprung mass, respectively. The displacements at the front and rear wheels of the vehicle can be written as Z sf = Z s − l f θ , Z sr = Z s + lrθ . (4.5)

Synthesize the aforementioned analysis, define the state vector, control input and disturbance as

x ( t ) =  β 

r

Z sf

Z sr

Z usf

Z usr

Zusf − Z sf

Zusr − Z sr

Zcf − Zusf

T

Zcr − Zusr  . 

190

Vibration Control of Vehicle Suspension Systems

u ( t ) =  ∆M z 

Fuf

T

Fur  , w ( t ) =  δ  

T

Z cr  . 

Z cf

Combining the lateral and vertical dynamics equations (4.2) and (4.4), the integrated vehicle dynamics system can be described as x ( t ) = Ax ( t ) + B1u ( t ) + B2 w ( t )

(4.6)

where

 A1 A=  08 × 2  −Csf a1   −Csf a2  C sf   musf  A2 =  0   −1  0   0  0 

02×8 A2

 C + C yr  − yf  mVx   , A1 =    − C yf l f − C yr lr  Iz 

C l −C l    −  1 + yf f 2 yr r      mVx , 2 2 C yf l f + C yr lr  −  I zVx 

−Csr a2

Csf a1

Csr a2

K sf a1

K sr a2

0

0

−Csr a3

Csf a2

Csr a3

K sf a2

K sr a3

0

0

K sf musf

0

K tf musf

0

0 Csr musr 0 −1 0 0

  0   B1 =  0    0 

Csf musf 0

0 −

1 0 −1 0

Csr musr 0 1 0 −1

0 0 0 0 0

K sr musr 0 0 0 0

0 0 0 0 0

K tr musr 0 0 0 0

              

T

1 Iz

0

0

0

a1

a2

0

a2

a3

 C yf  mVx B2 =   0  0 

C yf l f Iz 0 0

0

0

01× 4

1 musf

0

01× 4

1 musf

01× 4

0

     ,    

T

01× 6

0

01× 6 01× 6

1 0

 0   , 0   1 

with

a1 =

1 l 2f 1 l f lr 1 lr2 , a3 = + , a2 = − + . ms I p ms I p ms I p

The tyre cornering stiffness is an unknown parameter due to the change of driving condition, it can be considered as an uncertain parameter of the vehicle dynamics system and apply a normbounding approach to deal with.

191

Integrated Suspension Control

FIGURE 4.3  Structure of the polytope.

Thus, the tyre cornering stiffness in (4.3) can be expressed as C yf = C0 yf + ∆C yf = C0 yf + η f ∆C fm , C yr = C0 yr + ∆C yr = C0 yr + ηr ∆Crm , where η f , ηr ∈[ −1,1], C0 yf , C0 yr and ∆C fm , ∆Crm are the nominal values and maximum variations of front and rear tyre cornering stiffness, respectively. Considering the vehicle longitudinal velocity Vx is time-varying and satisfies Vx ∈Vx ,Vx  , define 1 1 two velocity-dependent auxiliary time-varying parameters as ρ = [ ρ1 , ρ2 ] =  , 2 . Then, the V  x Vx  1 1  1 1  values of ρ1 , ρ2 are bounded in  ,  and  2 , 2 , respectively.  Vx Vx   Vx Vx  The possible trajectory of the auxiliary parameter ρ is shown as the red curve MP in Figure 4.3, to reduce the conservative, a trapezoidal polytope MPRS with a small area is adopted to replace the rectangular polytope, the coordinates of four vertices can be given as

 5V 2 + 2V V + V 2 1   1 1   1 3V 2 + 2VxVx − Vx2   1 1  M : , 2, S: x 2 x x 2x , 2, P: , 2, R: , x   Vx Vx  4Vx2Vx2 Vx   Vx Vx   Vx  4 Vx Vx + VxVx

(

)

All the possible variable values of ρ can be described with these four new vertices and the coefficients α i , i = 1,2,3,4.α i can be given as

 1  1   V − ρ1   V 2 − ρ2  x x α1 ( ρ ) = ,α 2 ( ρ ) =  1 1  1 1  − −  V V   V 2 V 2  x x x x  1  V x α3 (ρ) =  1  V x

 1  1  ρ2 − V 2   ρ1 − V  x x ,  1 1  1 1 − −  V 2 V 2   V V  x x x x  1    1  1 − ρ1   ρ2 − 2   ρ1 − V   V − ρ2  Vx   x x ,α 4 ( ρ ) = .  1 1  1 1 1 1 1 − ( 2 − 2  V − V   V − V  Vx  Vx Vx ) x x x x

(4.7)

192

Vibration Control of Vehicle Suspension Systems

Based on the above analysis, the state-space representation of vehicle dynamics system can be rewritten as follows:

(

)

(

)

x ( t ) = A0 ( ρ ) + ∆ A ( ρ ) x ( t ) + B1u ( t ) + B02 ( ρ ) + ∆ B2 ( ρ ) w ( t ) (4.8)

where  A01 ( ρ ) A0 ( ρ ) =   08 × 2 

 C0 yf + C0 yr ρ1  − m  A01 ( ρ ) =  C0 yf l f − C0 yr lr  − Iz 

 ∆C yf + ∆C yr ρ1  − m  ∆C yf l f − ∆C yr lr ∆ A( ρ ) =   − Iz   0 8 ×1 

∆C yf l f − ∆C yr lr ρ2 m ∆C yf l 2f + ∆C yr lr2 − ρ1 Iz 0 8 ×1

 ∆Crm ρ1  − m  ∆Crmlr ∆ Ar ( ρ ) =   Iz   0 8 ×1 

C0 yf l f Iz 0 0

 ∆C yf ρ1  m  ∆B2 ( ρ ) =  ∆C yf l f  Iz   0 8 ×1

  ,   

C0 yf l f − C0 yr lr ρ2 m C l 2 + C0 yr lr2 ρ1 − 0 yf f Iz

−1 −

 ∆C fm ρ1  − m  ∆C fm l f ∆ Af ( ρ ) =   − Iz   0 8 ×1 

 C0 yf ρ1  m B02 ( ρ ) =   0  0 

02×8  , A2 

 01× 8    = η f ∆ A f ( ρ ) + ηr ∆ Ar ( ρ ) , 01× 8   08 × 8 

∆C fm l f ρ2 m ∆C fm l 2f − ρ1 Iz 0 8 ×1

 01× 8   , 01× 8   08 × 8 

∆Crmlr ρ2 m ∆Crmlr2 ρ1 − Iz 0 8 ×1

 01× 8   , 01× 8   08 × 8 

T

0

0

0

0

0

0

0

0 0

0 0

0 0

0 0

0 0

0 0

1 0

0

0

0

0

0 8 ×1

0 8 ×1

    = η f ∆ B2 f ( ρ ) ,   

 0   , 0   1 

193

Integrated Suspension Control

 ∆C fm ρ1  m  ∆B2 f ( ρ ) =  ∆C fm l f  Iz   0 8 ×1

0

0

0

0

0 8 ×1

0 8 ×1

   .   

Using the norm-bounding approach, ∆ A ( ρ ) and ∆B2 ( ρ ) can be rewritten as ∆ A ( ρ ) = H1F1E1 ( ρ ) , ∆ B2 ( ρ ) = H 2 F2 E2 ( ρ )

where

 1 H1 =  0  08 ×1

0 1

1 0

0 1

0 8 ×1

0 8 ×1

0 8 ×1

 ∆C fm ρ1  − m   ∆C fm l f  − Iz E1 ( ρ ) =   ∆Crm ρ1  − m   ∆Crm lr  Iz   µf F2 =   0

 ηf   , F =  0  1  0    0

µf

0

0

ηf 0 0

0

0

ηr 0

0 ηr

 01× 8    01× 8   1 , H =  0  2  01× 8   08 ×1   01× 8  

∆C fm l f ρ2 m ∆C fm l 2f − ρ1 Iz ∆Crmlr ρ2 m ∆Crmlr2 − ρ1 Iz

0

0

 ∆C fm ρ1   m  , E2 ( ρ ) =   ∆C fm l f   Iz 

0 1 0 8 ×1

   ,  

 ,  

 0  .  0  

0 0

A0 ( ρ ) , E1 ( ρ ) , B02 ( ρ ) and E2 ( ρ ) can be described with polytope vertices and coefficients as A0 ( ρ ) = B02 ( ρ ) =

4

4

∑α A , E ( ρ ) = ∑α E , 0i

i

1

1i

i

i =1

i =1

4

4

∑α B , E ( ρ ) = ∑α E i

2i

2

i =1

i

(4.9)

2i

i =1

4.2.1.2  Problem Statement In practice, the vehicle yaw rate can be directly measured by the on-board sensor. However, the vehicle lateral velocity, the suspension deflection and tyre deflection are not easy to be measured directly. Hence, a measured output y ( t ) including vehicle yaw rate r, the vertical velocity of front and rear sprung and unsprung masses Z sf , Z sr , Z usf , Z usr is chosen as

y ( t ) = Cx ( t )

194

Vibration Control of Vehicle Suspension Systems

where

   C=   

0 0 0 0 0

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

   .   

Then, the state observer is given as ˙

xˆ ( t ) = A0 ( ρ ) xˆ ( t ) + L ( ρ ) ( y − yˆ ) + B1u ( t ) ,

yˆ ( t ) = Cxˆ ( t )

(4.10)

where xˆ ( t ) is the observer state vector, L ( ρ ) is the observer gain matrix, yˆ ( t ) is the observed output. Based on the observed state vector, a parameter-dependent controller can be designed with state feedback as u ( t ) = K ( ρ ) xˆ ( t )

(4.11)

where K ( ρ ) is the desired controller gain. Then, the observed state xˆ ( t ) and estimation error e ( t ) can be expressed as

(

˙

)

xˆ ( t ) = A0 ( ρ ) + B1K ( ρ ) xˆ ( t )( t ) + L ( ρ ) Ce ( t ) , ˙

e ( t ) = x ( t ) − xˆ ( t )( t ) = A0 ( ρ ) ( x ( t ) − xˆ ( t )) + ∆ A ( ρ ) x ( t )

(

) = ( A ( ρ ) − L ( ρ ) C ) e ( t ) + ∆ A ( ρ ) ( xˆ ( t ) + e ( t )) + ( B ( ρ ) + ∆ B ( ρ )) w ( t ) = ∆ A ( ρ ) xˆ ( t ) + ( A ( ρ ) − L ( ρ ) C + ∆ A ( ρ )) e ( t ) + ( B ( ρ ) + ∆ B ( ρ )) w ( t ) .

+ B02 ( ρ ) + ∆ B2 ( ρ ) w ( t ) − L ( ρ ) C ( x ( t ) − xˆ ( t ))

(4.12)

0

02

2

0

02

2

Combining (4.10), (4.11), (4.12) and defining ξ ( t ) = [ xˆ (t )T   e ( t )T  , the state-space representation of vehicle dynamics system can be further expressed as T

(

)

(

)

ξ ( t ) = A0 ( ρ ) + ∆ A ( ρ ) ξ ( t ) + B02 ( ρ ) + ∆ B2 ( ρ ) , w ( t ) = Acl ( ρ ) ξ ( t ) + Bcl ( ρ ) w ( t ) ,

where

 A0 ( ρ ) + B1K ( ρ ) A0 ( ρ ) =   0 

 , A0 ( ρ ) − L ( ρ ) C   L ( ρ )C

(4.13)

195

Integrated Suspension Control

 0 ∆ A( ρ ) =   H1F1E1 ( ρ )

 0 , H1F1E1 ( ρ )  

  0 0 . , ∆ B2 ( ρ ) =  B02 ( ρ ) =  H 2 F2 E2 ( ρ )   B02 ( ρ )  To improve driving stability, safety and ride comfort, different control objectives are considered in the following. To obtain a good performance in vehicle lateral dynamics control, the vehicle sideslip angle β should be minimized and the yaw rate r should follow its reference value rd , which can be given as 1 [1]: rd = δ and K u is the stability factor determined by the mass of vehicle. (l f + lr ) KuVx

Furthermore, the limit of the external yaw moment should be considered ∆ M z ≤ ∆ M zmax

where ∆M zmax is the maximum value of the external yaw moment. In the active suspension design, the impact of road interference should be attenuated by the suspension control force under multiple constrains to improve ride safety and comfort. The control object and hard constrains can be given as [2] Ride comfort: A well-designed vehicle suspension means the forces transmitted from wheel to ¨ ¨ vehicle body should be reduced, that is, Z s and θ should be minimized under road disturbance. Suspension space limit: Suspension space is limited by its physical structure, it should be considered to avoid the exceeding physical limit when the suspension is acting, that is,

Z sf − Zusf ≤ Z fmax , Z sr − Zusr ≤ Zrmax

where Z fmax , Zrmax are the maximum suspension stroke space of the front and rear sides of vehicle. Road holding: For safety, the tyre should be in close contact with the road at all times. In other words, the dynamic tyre load should not exceed its static value, that is,

 l m   lm  ktf ( Zusf − Zcf ) < g  f s + musf  , ktr ( Zusr − Zcr ) < g  r s + musr   l f + lr   l f + lr 

where g is the acceleration of gravity, it is usually recorded as 9.8 m/s2. Actuator saturation: Considering the physical limit of the actuator, the active suspension force should not exceed the designed value, that is,

Fuf ≤ Fufmax , Fur ≤ Furmax

where Fufmax , Furmax are the given maximum active suspension control force on the front and rear sides of vehicle, respectively. Based on the above-mentioned control objectives in lateral dynamics stabilization and suspension design, the output vectors can be defined as

T

Z1 ( t ) =  c11β  Z 2 ( t ) =  c21 Z s  ¨

c12 ( r − rd )  , 

T  , Z ( t ) = Z sf − Zusf , 3 c22 θ  Z sfmax ¨

196

Vibration Control of Vehicle Suspension Systems

Z sr − Zusr , Z5 ( t ) = Z srmax

Z4 (t ) =

Z6 ( t ) =

 l m  9.8  f s + musf   l f + lr 

,

F K tr ( Zusr − Zcr ) , Z 7 ( t ) = uf ,  lm  Fufmax 9.8  r s + musr   l f + lr 

Z8 ( t ) =

K tf ( Zusf − Zcf )

∆ Mz Fur , Z9 ( t ) = ∆ M zmax Furmax

where c11, c12 and c21, c22 are the weighting coefficients that use to balance vehicle sideslip angle β , the difference between the yaw rate and its reference value r − rd and vertical acceleration of sprung ¨ ¨ mass Z s , pitch acceleration of sprung mass θ . Then the closed-loop system can be obtained as

ξ ( t ) = Acl ( ρ ) ξ ( t ) + Bcl ( ρ ) w ( t ) , Z1 ( t ) = Cc1ξ ( t ) + Dw1 ( ρ ) w ( t ) ,

(4.14)

Z 2 ( t ) = Cc 2ξ ( t ) , Z3 ( t ) = Cc 3ξ ( t ) , Z 4 ( t ) = Cc 4ξ ( t ) , Z5 ( t ) = Cc 5ξ ( t ) , Z6 ( t ) = Cc 6ξ ( t ) , Z 7 ( t ) = Cc 7ξ ( t ) , Z8 ( t ) = Cc8ξ ( t ) , Z9 ( t ) = Cc 9ξ ( t ) , where Cc1 =  C1

 c11 C1  , C1 =   0

 c11 Dw1 ( ρ ) =   0

0 c12

 1    0

0 1

0 0

 0  1  − ρ1   l f + lr ) K u ( 

0

0    0  , Cc 2 =  C2 + D2 K ( ρ )  

0 c12

 c21 D2 =   0

 Csf  0 0 − ms 0   l f Csf c22   0 0 Ip 

 c21 C2 =   0

Cc 3 =  C3

0 c22

  0     0 

Csr ms lC − r sr Ip −

0

0 0

1 ms l − f Ip

Csf ms lC − f sf Ip

 C3  , C3 =  01× 6 

0 0

1 ms lr Ip Csr ms lr Csr Ip

1 Z sfmax

0 0

0 0

0 0

0 0

0  , 0 

C2  , 

  ,    K sf ms l K − f sf Ip

K sr ms lr K sr Ip

 01× 3  , 

 0 0    0 0  

197

Integrated Suspension Control

 C4  , C4 =  01× 7 

Cc 4 =  C4

Cc 5 =  C5

  01× 8  , C5 =    

C5

Cc 6 =  C6

C6

 l m  9.8  f s + musf  l l +  f r 

  01× 9  , C6 =    

  ,  

K tr  lm  9.8  r s + musr  l l +  f r 

Cc 7 =  D7 K ( ρ ) 

 0  , D7 =  0  

Cc8 =  D8 K ( ρ ) 

 0  , D8 =  0  

 0 , 

1 Fufmax 1

0

Furmax

 1 0  , D9 =  ∆ M  zmax 

Cc 9 =  D9 K ( ρ ) 

 01×1  ,  

K tf

 01× 2  , 

1 Z srmax

0

 ,   0 . 

In this work, the observer-based gain-scheduling controller is designed in the presence of the uncertainties of cornering stiffness, time-varying longitudinal velocity, actuator saturation and suspension hard constrain to achieve good performances in vehicle lateral and vertical dynamics. The H∞ norm and GH 2 norm are introduced to measure the output variables Ti

= sup w ∈l2

zi w

2

, Tj

GH 2

2

= sup w ∈l2

zi w

∞ 2

where the subscript ∞

1 2

1 2

1

i = 1,2, j = 3,…,9, zi 2 = ( z ( t ) zi ( t ) dt ) , z j ∞ = sup( z ( t ) z j ( t )) , w 2 = ( wT ( t ) w ( t ) dt ) 2 . 0

T i

t ≥0

T j

0

The H∞ and GH 2 norm of the closed-loop system are adopted to measure the output energy and peak amplitude of the output signal in the case of the worst input condition, respectively. Therefore, ¨ ¨ under the designed controller, β , r − rd and Z s , θ are simultaneously minimized, and the control inputs ∆M z , Fsf and Fsr are limited to a reasonable level, the changes of suspension strokes Z sf − Zusf , Z sr − Zusr and tyre loads K tf ( Zusf − Zcf ), K tr ( Zusr − Zcr ) can be bounded within a designed range. Then, the control objectives of this paper can be summarized as: design an observer-based control that guarantee (1) the closed-loop system (4.13) is asymptotically stable, (2) Ti ∞ ≤ γ 1 ,  i = 1,2,γ 1 is minimized, (3) Tj GH2 ≤ γ 2 , j = 3,…,9,γ 2 is a given positive scalar. 4.2.1.3  Observer-Based Gain-Scheduling Controller Synthesis In this section, an observer-based multi-objective gain-scheduling controller considering the uncertain parameters and multiple pre-set limits is designed to improve ride comfort and driving safety. To handle the parameter uncertainty and design the observer and controller, the following lemma is introduced.

198

Vibration Control of Vehicle Suspension Systems

Lemma 4.1 [3] Given matrices Y , D and E of appropriate dimensions, where Y is symmetric, for all matrices F that satisfy F T F ≤ I , then Y + DFE + E T F T D T < 0.

if and only if there is a constant ε > 0 such that Y + ε DD T + ε −1E T E < 0.

Based on the above lemma, a general condition concerning the H∞ and GH 2 performance of system (4.14) under observer-based control is derived in the following theorem. Theorem 4.1 Consider a vehicle dynamics system (4.14), given positive constants γ 1, γ 2 , if there exist positive scalars ε1, ε 2, ε 3 , a positive definite matrix P = diag {P1 , P2 }, such that the following conditions hold

 Γ1 ( ρ )   *   *  *   *   *  * 

 Γ1 ( ρ )   *   *  *   *   *  * 

P1 L ( ρ ) C

0

C1T

Γ2 ( ρ )

P2 B02 ( ρ )

E1T ( ρ )

C1T

0

*

−I

P1 L ( ρ ) C

* * * *

T w1

D

−γ * * *

* * * * 0

(ρ) 2 1

0 E1T ( ρ )

0 −ε1 * *

0 0 −ε 2 *

0 0 0 −ε 3

0

0

C2T

0

*

−I

0

* * * *

* * * *

−γ * * *

 Γ1 ( ρ )   *   *  *   *  * 

P1 L ( ρ ) C

2 1

0

E1T ( ρ )

E

E1T ( ρ )

0

0

0 −ε1 * *

0 0 −ε 2 *

P2 B02 ( ρ )

*

−I

0

0

* * *

* * *

−ε1 * *

0 −ε 2 *

0 E

T 2

(ρ)

0 0 0 −ε 3

0

Γ2 ( ρ )

0

(ρ)

0

E1T ( ρ )

P2 B02 ( ρ )

0 T 2

0

C2T + K T ( ρ ) D2T

Γ2 ( ρ )

0

0

E1T ( ρ )

0 E

T 2

(ρ)

0 0 −ε 3

      < 0, (4.15)             < 0, (4.16)      

     < 0,      

(4.17)

199

Integrated Suspension Control

 −γ 22   *  * 

 −γ 22   *  * 

Ci − P1 *

   < 0,  

Ci 0 − P2

DjK ( ρ )

0

− P1 *

0 − P2

(4.18)

   < 0,  

(4.19)

where the subscript i = 3,4,5,6, j = 7,8,9, and Γ1 ( ρ ) = P1 A0 ( ρ ) + A0T ( ρ ) P1 + P1 B1K ( ρ ) + K T ( ρ ) B1T P1 ,

Γ 2 ( ρ ) = P2 A0 ( ρ ) + A0T ( ρ ) P2 − P2 L ( ρ ) C − C T LT ( ρ ) P2 + ε1P2 H1 H1T P2 + ε 2 P2 H1 H1T P2 + ε 3 P2 H 2 H 2T P2 . Then the closed-loop system (4.14) is asymptotically stable and has z1 2 < γ 12 w 2 , z2

2

< γ 12 w 2 , zi

< γ 22 w 2 , z j

< γ 22 w 2 .

Proof. Construct a Lyapunov function as V ( t ) = ξ T ( t ) Pξ ( t ) = xˆ T ( t ) P1 xˆ ( t ) + eT ( t ) P2e ( t )

For brevity, the dependent of ρ is omitted in the derivation of this theorem, to establish the H∞ and GH 2 performance indicators T1 < γ 1 and T2 < γ 2 for closed-loop systems, one has ∞

(

)

V ( t ) + γ 1−2 z1T ( t ) z1 ( t ) − w T ( t ) w ( t ) = xˆ T ( t ) P1 A0 + A0T P1 + P1 B1K + K T B1T P1 + γ 1−2C1T C1 xˆ ( t )

( (t )( P A

)

(

)

+ xˆ T ( t ) P1 LC + E1T F1T H1T P2 + γ 1−2C1T C1 e ( t ) + eT ( t ) C T LT P1 + P2 H1F1E1 + γ 1−2C1T C1 xˆ ( t ) + eT

2

0

)

+ A0T P2 − P2 LC − C T LT P2 + E1T F1T H1T P2 + P2 H1F1E1 + γ 1−2C1T C1 e ( t )

(

)

+ xˆ T ( t )γ 1−2C1T Dw1w ( t ) + wT ( t )γ 1−2 DwT1C1 xˆ ( t ) + wT ( t ) − I + γ 1−2 DwT1 Dw1 w ( t )

( (t )( B

)

+ eT ( t ) P2 B02 + γ 1−2 DwT1C1 + P2 H 2 F2 E2 w ( t ) + wT

)

P + γ 1−2 DwT1C1 + E2T F2T H 2T P2 e ( t )

T 02 2

According to Lemma 4.1, one obtains V ( t ) + γ 1−2 z1T ( t ) z1 ( t ) − wT ( t ) w ( t )

≤ xˆ T ( t ) ( P1 A0 + A0T P1 + P1B1K + K T B1T P1 + γ 1−2C1T C1

(

)

+ ε1−1E1T E1 ) xˆ ( t ) + xˆ T ( t ) γ 1−2C1T C1 + P1LC e ( t ) + eT ( t ) (γ 1−2C1T C1 + C T LT P1 ) xˆ ( t ) + eT ( t ) ( P2 A0 + A0T P2 − P2 LC − C T LT P2 + γ 1−2C1T C1 + ε1P2 H1H1T P2 + ε 2 P2 H1H1T P2T + ε 2−1E1T E1 + ε 3 P2 H 2 H 2T P2T )e ( t )

(4.20)

200

Vibration Control of Vehicle Suspension Systems

+ xˆ T ( t ) γ 1−2C1T Dw1w ( t ) + w T ( t ) γ 1−2 DwT1C1xˆ ( t ) + eT ( t ) ( P2 B02

(

)

T + γ 1−2C1T Dw1 ) w ( t ) + w T ( t ) B02 P2 + γ 1−2 DwT1C1 e ( t ) + w T ( t ) (− I

+ γ 1−2 DwT1Dw1 + ε 3−1E2T E2 ) w ( t )

 xˆ ( t )  =  e (t )  w t () 

    

T

 ∆  1  *    *

P1 LC ∆2

0 P2 B02

*

− I + ε 3−1E2T E2 T

 xˆ ( t )  =  e (t )  w t () 

where

 C1T    −2 T  + γ 1  C1   T  Dw1 

  C1T    C1T  T   Dw1

    

T

 ˆ x (t )    e (t )    w ( t )

    

   (4.21)  

  xˆ ( t )    Ω1  e ( t )   w t ()  

∆1 = P1 A0 + A0T P1 + P1 B1K + K T B1T P1 + ε1−1E1T E1 , ∆ 2

= P2 A0 + A0T P2 − P2 LC − C T LT P2 + ε1P2 H1 H1T P2 + ε 2 P2 H1 H1T P2 + ε 2−1E1T E1 + ε 3 P2 H 2 H 2T P2 . Then, similar to the derivation above, one has

 xˆ ( t )  2 − T T V ( t ) + γ 1 z2 ( t ) z2 ( t ) − w ( t ) w ( t ) ≤  e ( t )  w t ()   C2T + K T D2T  + γ 1−2  C2T  0 

  C T + K T D2T  2 C2T   0 

    

T

    

  ∆1   *  * 

T

P1 LC ∆2

0 P2 B02

*

− I + ε 3−1E2T E2 T

    

  xˆ ( t )   xˆ ( t )   xˆ ( t )        e (t )  =  e (t )  Ω2  e (t )  w t   w (t )   w (t )  ()      

(4.22)   .  

Then applying Schur complement to (5.15) and (5.16), one obtains

 ∆1   *   *  *   ∆1   *   *  * 

C1T

0

P1 LC ∆2

T 1

P2 B02 −1 3

C

*

− I + ε E E2

DwT1

*

*

−γ 12

P1 LC

0

∆2

P2 B02

*

− I + ε 3−1E2T E2

*

*

T 2

    < 0,   

C2T + K T D2T   C2T   < 0. 0  2  −γ 1 

(4.23)

(4.24)

201

Integrated Suspension Control

With the Schur complement, (4.23), (4.24) implies Ω1 < 0 and Ω 2 < 0. Then the following condition is derived

V ( t ) + γ 1−2 z1T ( t ) z1 ( t ) − w T ( t ) w ( t ) < 0,

V ( t ) + γ 1−2 z2T ( t ) z2 ( t ) − wT ( t ) w ( t ) < 0. Integrating it with zero initial condition V ( 0 ) = O , it is obtained that  z1  2 < γ 1  w  2,  z2  2 < γ 1  w  2, and the H∞ performance is satisfied. When the disturbance is zero, that is, w = 0, it is obtained that obtain V ( t ) < 0 , which means that the closed-loop system is asymptotically stable. To establish the GH 2 performance indicators  Ti  GH2 < γ 2 ,  Tj  GH2 < γ 2 for closed-loop systems, similar to the derivation processes of H∞ performance, one has V ( t ) − w T ( t ) w ( t )

 xˆ ( t )  ≤  e (t )  w t () 

    

T

 Γ11   *  * 

P1 LC Γ 22

0 P2 B02

*

− I + ε 3−1E2T E2

 xˆ ( t )  =  e (t )  w t () 

T

  xˆ ( t )    Ω3  e ( t )   w t ()  

  xˆ ( t )    e (t )   w (t )  

    

(4.25)

  .  

Applying Schur complement to inequalities (5.17)–(5.19), one obtains

 Γ11   *  *   P1 −  0  P1 −  0

P1 LC Γ 22 *

−1 3

(4.26)

CiT Ci   < 0, CiT Ci 

(4.27)

− I + ε E E2

 CiT Ci 0  −2 +γ 2  P2   CiT Ci  0 P2

   < 0,  

0 P2 B02 T 2

 K T D Tj D j K  −2 +γ 2  0  

0   < 0. 0 

(4.28)

With the Schur complement, (4.26) implies Ω3 < 0 . Then the following condition is derived V ( t ) − w(t )T w ( t ) < 0.

Combining (4.27)–(4.29), one obtains ∞

γ z ( t ) zi ( t ) < V ( t ) < wT ( t ) w ( t ) dt , −2 T 2 i

0

γ 2−2 z Tj ( t ) z j ( t ) < V ( t ) < wT ( t ) w ( t ) dt. 0

(4.29)

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Vibration Control of Vehicle Suspension Systems

Take the maximum when t ≥ 0 and for all nonzero w ∈ L2 [ 0,∞ ), it is obtained that  zi  ∞ < γ 2  w  2 and  z j  ∞ < γ 2  w  2 , and the GH 2 performance is satisfied. The proof is completed. In Theorem 4.1, the H∞ and GH 2 performance conditions of the closed-loop system (4.14) are derived. Then, based on Theorem 1, an observer-based controller can be constructed as in Theorem 4.2. Theorem 4.2 Consider a vehicle dynamics system (4.14), given positive constants γ 1, γ 2 , if there exist a family of matrices Qk , Rk , positive scalars ε1, ε 2, ε 3 , a positive definite matrix W = diag {W1 , W2 }, and a nonsingular matrix W3, satisfying, for 1 ≤ k ≤ 4,

 Θ1, k   *   *   *  *   *  *   Θ1, k   *   *   *  *   *  * 

Rk C

0

W1C1T

W1E1,T k

0

0

Θ 2, k

B02, k

W2C1T

0

W2 E1,T k

0

*

−I

T w1, k

D

0

0

E2,T k

* * * *

* * * *

−γ 12 * * *

0 −ε1 * *

0 0 −ε 2 *

0 0 0 −ε 3

      < 0, (4.30)      

Rk C

0

W1C2T + QkT D2T

W1E1,T k

0

0

Θ 2, k

B02, k

W2C2T

0

W2 E1,T k

0

*

−I

0

0

0

E2,T k

* * * *

* * * *

−γ 12 * * *

0 −ε1 * *

0 0 −ε 2 *

0 0 0 −ε 3

 Θ1, k   *   *  *   *  * 

0

Rk C

W1E1,T k

∏ 2, k

B02, k

0

* * * *

−I * * *

0 −ε1 * *

 −γ 22   *  * 

CiW1 −W1 *

0

0 T 1, k

W2 E

0 0 −ε 2 * CiW2 0 −W2

   < 0,  

0 E2,T k 0 0 −ε 3

     < 0,     

      < 0,      

(4.31)

(4.32)

(4.33)

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Integrated Suspension Control

 −γ 22   *  * 

D jQ ( ρ )

0

−W1 *

0 −W2

   < 0  

(4.34)

where the subscript i = 3,4,5,6, j = 7,8,9. and

Θ1, k = A0, k W1 + W1 A0,T k + B1Qk + QkT B1T ,

Θ 2, k = A0, k W2 + W2 A0,T k − Rk C − C T RkT + ε1 H1 H1T + ε 2 H1 H1T + ε 3 H 2 H 2T . Then the closed-loop system (4.14) is asymptotically stable and has  z1  2< γ 12  w  2,  z2  2< γ 12  w  2,  zi  ∞ < γ 22  w  2,  z j  ∞ < γ 22  w  2. Moreover, assume that CW2 = W3C , the singular value decomposition of C is C = U [C0 ,0 ]V , where C0 > 0,and U ,V = [V1 ,V2 ] are unitary matrix, the singular value decomposition of W2 is W2 = [V1 ,V2 ] diag {W21 , W22 }[V1 ,V2 ]T . Then, the controller and observer gain can be expressed as K ( ρ ) =

4

∑α Q W k

k

−1 1

and L ( ρ ) = R ( ρ )UC0W C U −1 21

−1 0

4

−1

k =1

=

∑α RUC W i

0

i

−1 21

i =1

Q ( ρ ) = α 1 ( ρ ) Q1 + α 2 ( ρ ) Q2 + α 3 ( ρ ) Q3 + α 4 ( ρ ) Q4 ,

C0−1U −1, with

R ( ρ ) = α 1 ( ρ ) R1 + α 2 ( ρ ) R2 + α 3 ( ρ ) R3 + α 4 ( ρ ) R4 .

(4.35)

where α k ( ρ ) is given in (4.7). Proof. Applying Schur complement to (4.30) and (4.31), one obtains  Λ1, k ( ρ )   *   *  *  

 Λ1, k ( ρ )   *   *  *  

Rk C

0

W1C1T

Λ 2, k ( ρ )

B02, k ( ρ )

W2C1T

*

− I + ε 3−1E2,T k ( ρ ) E2, k ( ρ )

DwT1, k ( ρ )

*

*

−γ 12

0

Rk C Λ 2, k ( ρ )

B02 ( ρ )

*

− I + ε E2,T k ( ρ ) E2, k ( ρ )

*

*

−1 3

    < 0, (4.36)    

W1C2T + QkT ( ρ ) D2T    W2C2T  < 0. (4.37)  0  2 −γ 1  

where

Λ1, k = A0, k W1 + W1 A0,T k + B1Qk + QkT B1T + ε1−1W1E1,T k E1, k W1 ,

Λ 2, k = A0, k W2 + W2 A0,T k − Rk C − C T Rk + ε1 H1 H1T

+ε 2 H1 H1T + ε 2−1W2 E1,T k E1, k W2 + ε 3 H 2 H 2T . Then, construct Q ( ρ ) R ( ρ ) as in (4.35), based on the vertex property of the polytopic LPV system, one has

204

Vibration Control of Vehicle Suspension Systems

 Λ1, k  4  * αk   * k =1   *

 Λ1 ( ρ )   * =  *  *    Λ1, k  4  * αk   * k =1   *

 Λ1 ( ρ )   * =  *  *  

Rk C

0

W1C1T

Λ 2, k

B02, k

W2C1T

*

− I + ε 3−1E2,T k E2, k

DwT1, k

*

*

−γ 12

R ( ρ )C Λ2 ( ρ )

0

W1C1T

B02 ( ρ )

W2C1T

*

− I + ε 3−1E2T ( ρ ) E2 ( ρ )

DwT1 ( ρ )

*

*

−γ 12

Rk C

0

Λ 2, k ( ρ )

B02, k

*

− I + ε 3−1E2,T k E2, k

*

*

R ( ρ )C Λ2 ( ρ )

W1C + Q T 2

B02 ( ρ ) −1 3

−I + ε E

T 2

*

(4.38)     < 0,    

W1C2T + QkT D2T    W2C2T  0   2 −γ 1 

0

*

      

T

W2C

( ρ ) E2 ( ρ )

(ρ)D

T 2

0 −γ 12

*

T 2

(4.39)     ulim

where ulim is the control input limit. The matrices are

 Aˆ A (t ) =   Aˆ a 

  ˆ  B1 =  B1  0 Aa ( t )   Bˆ 2Ca

  0  B2 =   Ba 

 . 

It is noted that the system matrix A ( t ) is a nonlinear and time-varying matrix due to the nonlinear time-varying behaviour of the actuator dynamics and the variation of the sprung mass. 4.2.2.2  T-S Fuzzy Modelling The full-car electrohydraulic suspension model (4.50) includes actuator nonlinearities, and the controller design is required to consider both the parameter uncertainty and the control input saturation, which leads to a challenging control problem. To design a controller for the model through the fuzzy approach, the T-S fuzzy modelling technique will be applied, and the idea of “sector nonlinearity” [9] is employed to construct an exact T-S fuzzy model for the nonlinear uncertain suspension system (4.50). Suppose that the actuator force Fi ( t ) (where i denotes fl , fr, rl, and rr, respectively) is bounded in practice by its minimum value Fimin and its maximum value Fimax; the nonlinear function fi ( t ) is then bounded by its minimum value fmin and its maximum value fmax . Thus, using the idea of “sector nonlinearity” [9], fi ( t ) can be represented by fi ( t ) = M1i (ξi ( t )) fmax + M 2i (ξi ( t )) fmin

(4.52)

where ξi ( t ) = fi ( t ) is a premise variable, M1i (ξi ( t )) and M 2i (ξi ( t )) are membership functions, with M1i (ξi ( t )) =

fi ( t ) − fmin f − fi ( t ) M 2i (ξi ( t )) = max . fmax − fmin fmax − fmin

(4.53)

Similarly, the uncertain sprung mass ms  ( t ) is bounded by its minimum value msmin and its maximum value msmax , and can thus be represented by 1

ms ( t )

= N1 (ξ m ( t )) mmax + N 2 (ξ m ( t )) mmin

(4.54)

where ξ m ( t ) = 1 / ms ( t ) is also a premise variable, mmax = 1 / msmin, mmin = 1 / msmax , and N1 (ξ m ( t )) and N 2 (ξ m ( t )) are membership functions that are defined as 1 1 − mmin mmax − ms ( t ) ms ( t ) ,  N 2 (ξ m ( t )) = N1 ( ξ m ( t ) ) = . (4.55) mmax − mmin mmax − mmin

To brevity, we name the aforementioned membership functions M1i (ξi ( t )), M 2i (ξi ( t )), N1 (ξ m ( t )), and N 2 (ξ m ( t )), shown in Figure 4.7a, as “big,” “small,” ‘light,” and “heavy,” respectively. The nonlinear uncertain suspension model (4.50) can then be represented by a T-S fuzzy model composed of 32  25 fuzzy rules, as listed in Table 4.5, where B, S, L, and H represent ‘big,” ‘small,” ‘light,” and

( )

214

Vibration Control of Vehicle Suspension Systems

FIGURE 4.7  (a) Membership functions and (b) membership functions for sprung mass obtained from TP model transformation.

TABLE 4.5 List of Fuzzy Rules Rule No.

Premise Variables

ξ fl S B S B S B S B S B S B S B S B

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

ξ fr S S B B S D B B S S B B S S B B

ξrl S S S S B B B B S S S S B B B B

ξrr S S S S S S S S B B B B B B B B

Rule No.

ξm L L L L L L L L L L L L L L L L

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

Premise Variables

ξ fl S B S B S B S B S B S B S B S B

ξ fr S S B B S D B B S S B B S S B B

ξrl S S S S B B B B S S S S B B B B

ξrr S S S S S S S S B B B B B B B B

ξm H H H H H H H H H H H H H H H H

‘heavy,” respectively. To describe the T-S fuzzy model more clearly, several examples of the fuzzy IF-THEN rules corresponding to Table 4.5 are explained as follows. Model Rule 1: IF ξi ( t ) (i denotes fl , fr, rl, and rr, respectively) are small and ξ m ( t ) is light, Then x ( t ) = A1 ( t ) x ( t ) + B1w ( t ) + B2u ( t ) where matrix A1 is obtained from matrix A ( t ) in (4.50) by replacing fi with fmin and 1 / ms with mmax .  Model Rule 16: IF ξi ( t ) (i denotes fl , fr, rl, and rr, respectively) are big and ξ m ( t ) is light,

215

Integrated Suspension Control

Then x ( t ) = A16 ( t ) x ( t ) + B1w ( t ) + B2u ( t ) where matrix A16 is obtained from matrix A ( t ) in (4.50) by replacing fi with fmax and 1 / ms with mmax . Model Rule 17: IF ξi ( t ) (i denotes fl , fr, rl, and rr, respectively) are small and ξ m ( t ) is heavy, Then x ( t ) = A17 ( t ) x ( t ) + B1w ( t ) + B2u ( t ) where matrix A17 is obtained from matrix A ( t ) in (4.50) by replacing fi with fmin and 1 / ms with mmin .  Model Rule 32: IF ξi ( t ) (i denotes fl , fr, rl, and rr, respectively) are big and ξ m ( t ) is heavy, Then x ( t ) = A32 ( t ) x ( t ) + B1w ( t ) + B2u ( t ) where matrix A32 is obtained from matrix A ( t ) in (4.50) by replacing fi with fmax and 1 / ms with mmin . Thus, the T-S fuzzy model that represents exactly the nonlinear uncertain suspension model (4.50) under the assumption of bounds on actuator forces Fi ( t ) ∈[ Fmin , Fmax ] and sprung mass ms ( t ) ∈[ msmin , msmax ] is obtained as 32

x ( t ) =

∑h (ξ (t )) A x (t ) + B w (t ) + B u (t ) (4.56) i

1

i

2

i =1

where

h1 (ξ ( t )) = M 2 fl (ξ fl ( t )) M 2 fr (ξ fr ( t )) M 2rl (ξrl ( t )) × M 2rr (ξrr ( t )) N1 (ξ m ( t )) ,

h2 (ξ ( t )) = M1 fl (ξ fl ( t )) M 2 fr (ξ fr ( t )) M 2rl (ξrl ( t )) × M 2rr (ξrr ( t )) N1 (ξ m ( t )) ,

h32 (ξ ( t )) = M1 fl (ξ fl ( t )) M1 fr (ξ fr ( t )) M1rl (ξrl ( t )) × M1rr (ξrr ( t )) N 2 (ξ m ( t )) ,

hi (ξ ( t )) ≥ 0, i = 1,2,…,32, and

32

∑h (ξ (t )) = 1 i

i =1

In practice, the actuator force Fi ( t ), the spool valve position x vi ( t ), and the sprung mass ms ( t ) can be measured; thus, the T-S fuzzy model (4.56) can be realized. The T-S fuzzy model (4.56) can be obtained via the “sector nonlinearity” approach based on the analysis of the nonlinear function fi ( t ) and the variation of sprung mass ms , the bounds of which can be estimated in a real operating situation. The construction of a T-S fuzzy model from a given nonlinear dynamic model can also utilize the idea of “local approximation” or a combination of “sector nonlinearity” and “local approximation” [9]. In general, these are analytic transformation techniques that are used for models described analytically. Since analytic techniques need problem-dependent human intuition and cannot be easily solved in some cases, recently, a higher order-singular-value-decomposition (HOSVD)-based tensor product (TP) model transformation approach was proposed to automatically and numerically transform a general dynamic system model into a TP model form, including polytopic and T-S model forms [10]. The TP model representation has shown various advantages for LMI-based controller design [11,12], and relaxed LMI conditions can be further obtained for closed-loop fuzzy systems with TP structure [13]. There

216

Vibration Control of Vehicle Suspension Systems

is also a MATLAB Toolbox for TP model transformation (available for download together with documentation and examples at http://tptool.sztaki.hu/tpde). For our question, in fact, using the convex normalized (CNO) type of TP model transformation can obtain the same membership functions as those described before when fi  ( t ) and 1 / ms  ( t ) are used as time-varying parameter variables. However, if ms  ( t ) is used as a time-varying parameter variable instead of 1 / ms  ( t ), different membership functions can be generated with the TP model transformation, as shown in Figure 4.7b. It can be seen that the membership functions are nonlinear, and they are different from those shown in Figure 4.7a. However, this new type of membership function does not affect the design process of the LMI-based controller or the design results obtained with the membership functions defined in (4.55). Generally, Fi ( t ) and x vi ( t ) can be seen as the time-varying parameter variables to obtain the T-S model using the TP model transformation. However, the tradeoff between approximation accuracy and complexity should be considered in the TP model transformation. For the studied problem, when using the derived membership functions (4.53) and (4.55), only 32 fuzzy rules need to be applied. Therefore, in this paper, the derived membership functions (4.53) and (4.55) are used. And, despite the authors’ effort, no other types of membership functions are found to yield better performance than the derived membership functions (4.53) and (4.55), although there may exist methods to automatically generate affine decompositions. 4.2.2.3  Fuzzy Controller Design To avoid the problem associated with having a large number of inequalities involved in the controller design, the norm-bounded approach [14,15] is used to handle the saturation nonlinearity defined in (4.51). Hence, (4.56) will be written as 32

x ( t ) =

∑h (ξ (t )) A x (t ) + B w (t ) + B u (t ) i

1

i

2

i =1 32

=

∑h (ξ (t )) A x (t ) + B w (t ) + B 1 +2 ε u (t ) + B  u (t ) − 1 +2 ε u (t ) i

1

i

2

2

(4.57)

i =1

= Ah x ( t ) + B1w ( t ) + B2

1+ ε u ( t ) + B2 v ( t ) 2

32

where Ah =

∑h (ξ (t )) A and v (t ) = u (t ) − 1 +2 ε u (t ), 0 < ε 0 is any scalar. Theorem 4.3

For a given number γ   >  0, 0  < ε <  1, the T-S fuzzy system (4.57) with controller (4.60) is quadratically stable and the L2 gain defined by (4.62) is less than γ if there exist matrices Q   >  0, Yi ,  i   =  1, 2,..., 32, and scalar  > 0, such that, (4.63) and (4.64) hold.

218

Vibration Control of Vehicle Suspension Systems

 1+ ε T T T T −1  QAi + AiQ + Yi B2 + B2Yi  +  B2 B2 2   *    *  *  2  u   lim  I   ε   YiT 

YiT

QCiT 2

 2  − −1  I  1 − ε 

0

* *

−I *

    0  < 0 (4.63)   0  2 −γ I   B1

   ≥ 0 (4.64) −1 ρ Q   Yi

Moreover, the fuzzy state feedback gains can be obtained as  K i = YiQ −1  ,  i = 1, 2,..., 32 . Proof: Let us define a Lyapunov function for the system (4.57) as V   ( x ( t ) )   =  x T   ( t ) P  x ( t )

(4.65)

where P is a positive definite matrix. By differentiating (4.65), we obtain 1+ ε V ( x ( t )) = x T ( t ) Px ( t ) + x T ( t ) Px ( t ) =  Ah x ( t ) + B1w ( t ) + B2 u ( t ) + B2 v ( t )  2   1+ ε Px ( t ) + x ( t ) P  Ah x ( t ) + B1w ( t ) + B2 u ( t ) + B2 v ( t )  . 2  

T

(4.66)

T

By Lemma 4.2, Lemma 4.3, and definition (4.60), we have T   1+ ε  1+ ε  V ( x ( t )) ≤ x T ( t )  AhT P + PAh +  B2 K h  P + PB2 K h  x ( t ) + wT ( t ) B1T Px ( t ) + x T ( t ) PB1w ( t )   2 2  

+  v T ( t ) v ( t ) +  −1 x T ( t ) PB2 B2T Px ( t ) ≤ x T ( t ) Θx ( t ) + w T ( t ) B1T Px ( t ) + x T ( t ) PB1w ( t ) (4.67) where

T 2   1+ ε  1+ ε   1− ε  T Θ =  AhT P + PAh +  B2 K h  P + PB2 Kh +   K h K h +  −1PB2 B2T P  (4.68)      2 2 2  

and  is any positive scalar. Adding z T ( t ) z ( t ) − γ 2 w T ( t ) w ( t ) on both sides of (4.67) yields  Θ + ChT Ch V ( x ( t )) + z T ( t ) z ( t ) − γ 2 wT ( t ) w ( t ) ≤  x T ( t ) wT ( t )    B1T P 

PB1   x ( t )  −γ 2 I   w ( t ) 

  . 

(4.69)

Let us consider

 Θ + ChT Ch ∏=  B1T P 

PB1   < 0, (4.70) −γ 2 I  

219

Integrated Suspension Control

then, V ( x ( t )) + z T ( t ) z ( t ) − γ 2 wT ( t ) w ( t ) and the L2 gain defined in (4.62) is less than γ   >  0 with the initial condition x ( 0 ) =  0  [16]. When the disturbance is zero, i.e., w ( t ) = 0, it can be inferred from (4.69) that if Π   <  0, then V ( x ( t )) < 0, and the fuzzy system (4.57) with the controller (4.60) is quadratically stable. Pre- and post-multiplying (4.70) by diag P −1    I and its transpose, respectively, and defining Q = P −1   and Yh =   K hQ, the condition Π   <  0 is equivalent to

(

)

 1+ ε T T 1+ ε T Yh B2 + B2Yh  QAh + AhQ + 2 2  2 ∑=  1− ε  T −1 T T  +   Yh Yh +  B2 B2 + QCh ChQ 2   B1T 

    < 0. (4.71)   −γ 2 I   B1

 1+ ε T T T Yh B2 + B2Yh  +  −1 B2 B2T YhT QChT  QAh + AhQ + 2  2   2  * 0 − −1  I Ψ=   1− ε    * * −I  * * *  By the Schur complement, Σ   <  0 is equivalent to (Ψ). 32

By the definitions Ah =

hi (ξ ( t )) Ai, Ch =

i =1

hi (ξ ( t )) ≥ 0 and

32

hi (ξ ( t )) Ci, Yh =

i =1

    0  < 0.  0   −γ 2 I   B1

(4.72)

32

∑h (ξ (t ))Y , and the fact that i

i

i =1

32

∑h (ξ (t )) = 1, Ψ   v0

. (4.79)

where a and l are the height and the length of the bump. We choose a = 0.1 m, l = 10 m, and the vehicle forward velocity as v0 = 45 km/h. For the nominal sprung mass under the road disturbance (4.79), Figure 4.9a shows the bump responses of the sprung mass heave acceleration, pitch acceleration, and roll acceleration for the passive suspension, the active suspension with H ∞ controller (4.77), and the active suspension with the electrohydraulic actuators and the designed fuzzy controller. It is again observed that the proposed fuzzy control strategy achieves suspension performance very

222

Vibration Control of Vehicle Suspension Systems

FIGURE 4.8  (a) Time-domain responses under a test road profile, (b) acceleration responses under a bump road profile. (Solid line is for active suspension with fuzzy controller. Dotted line is for active suspension with H∞ controller. Dot-dashed line is for passive suspension.)

Integrated Suspension Control

223

FIGURE 4.9  (a–c) Heave, pitch, and roll acceleration responses for different sprung mass, (d) RMS ratios for sprung mass heave acceleration, pitch acceleration, and roll acceleration versus sprung mass.

similar to the active suspension with an optimal H ∞ controller. The active suspension performance is significantly improved compared to the passive suspension. To illustrate the effect of sprung mass variation, Figure 4.9a shows the bump responses of the sprung mass heave acceleration for the passive suspension (passive) and the active suspension with electrohydraulic actuator and fuzzy controller (active) when the sprung mass is 1120 and 1680 kg. It is observed that, despite the change in sprung mass, the designed fuzzy controller achieves significantly better performance on heave acceleration, where a lower peak and shorter settling time are obtained. Figure 4.9b and c show the bump responses of the sprung mass pitch acceleration and roll acceleration, respectively. It can be seen from Figure 4.9b that the sprung mass affects the pitch acceleration significantly. However, the active suspension can keep the pitch acceleration low, regardless of the sprung mass variation. In Figure 4.9c, the active suspension achieves lower roll acceleration compared to the passive suspension, although the sprung mass variation does not affect roll acceleration due to the symmetric distribution of the sprung mass about the vehicle’s roll axis. Figure 4.9a–c indicates that the improvement in ride comfort can be maintained by the designed active suspension for large changes in load conditions. When the road disturbance is considered as random vibration, it is typically specified as a stationary random process that can be represented by

zr ( t ) = 2πq0 G0V ω ( t ) , (4.80)

224

Vibration Control of Vehicle Suspension Systems

where G0 stands for the road roughness coefficient, q0 is the reference spatial frequency, V is the vehicle forward velocity, and ω ( t ) is zero-mean white noise with identity power spectral density. For a given road roughness G0 =  512  ×  10 −6  m 3 and a given vehicle forward velocity V   =  20 m/s, the RMS values for sprung mass heave acceleration, pitch acceleration, and roll acceleration are calculated as the sprung mass changes from 1120 to 1680 kg. The RMS ratios between the active suspension with fuzzy controller and the passive suspension are plotted against sprung mass in Figure 4.9d. It can be seen that the designed fuzzy controller maintains a ratio below 1, regardless of the large variations in the sprung mass. When the road roughness and vehicle forward velocity are given different values, very similar results are obtained. For brevity, these results are not shown. Figure 4.9d further validates the claim that the proposed fuzzy control strategy can realize good ride comfort performance for electrohydraulic suspension even when the sprung mass is varied significantly. 4.2.2.5 Conclusion In this paper, we have presented a fuzzy state feedback control strategy for electrohydraulic active suspensions to deal with the nonlinear actuator dynamics, sprung mass variation, and control input constraint problems. First, using the idea of “sector nonlinearity,” the nonlinear uncertain electrohydraulic actuator was represented by a T-S fuzzy model in defined regions. Thus, by means of the PDC scheme, a fuzzy state feedback controller was designed for the obtained T-S fuzzy model to optimize the H∞ performance of ride comfort. At the same time, the actuator input voltage constraint was incorporated into the controller design process. The sufficient conditions for designing such a controller were expressed by LMIs. Simulations were used to validate the effectiveness of the designed controller.

4.3 INTEGRATED WITH SEAT SUSPENSION AND DRIVER MODEL AND CABIN MODEL 4.3.1 Integrated Vehicle and Seat Model The integrated vehicle model includes a quarter-car suspension model, a seat suspension model, and a 4-DOF driver body model, as shown in Figure 4.10, where ms is the sprung mass, which represents the car chassis; mu is the unsprung mass, which represents the wheel assembly; m f is the seat frame mass; mc is the seat cushion mass; and the driver body is composed of four mass segments, i.e., thighs m1, lower torso m2, high torso m3, and head m4, where the arms and legs are combined with the upper torso and thighs, respectively. zu, zs , z f , zc , and z1− 4 are the displacements of the corresponding masses, respectively, and zr is the road displacement input. cs and ks are the damping and stiffness of the car suspension, respectively. kt and ct . kt and ct stand for the compressibility and damping of the pneumatic tyre, respectively. cs , css , c1− 4 , ks , kss  and k1− 4 are defined in Table 4.8. us and u f represent the active control forces that are applied to the car suspension and the seat suspension, respectively. In practice, electrohydraulic actuators or linear permanent-magnet motors could be applied to generate the required forces us and u f . The dynamic vertical motion of equations for the quarter-car suspension, seat suspension, and driver body is given by

(

¨

˙

)

˙

(

˙

˙

)

mu zu = − kt ( zu − zr ) − ct zu − zr + ks ( zs − zu ) + cs zs − zu + us

(

)

(

(4.81)

)

ms zs = − ks ( zs − zu ) − cs zs − zu + kss ( z f − zs ) + css z f − zs − us + u f (4.82)

m f z f = − kss ( z f − zs ) − css z f − zs + kc ( zc − z f ) + cc zc − z f − u f (4.83)

¨

¨

˙

˙

(

˙

˙

)

˙

˙

(

˙

˙

)

225

Integrated Suspension Control

FIGURE 4.10  Integrated seat and suspension model

TABLE 4.8 Parameters of the Seat-Driver Suspension Model

css

Damping of seat suspension

kss

Stiffness of seat suspension

cc c1 c2 c3 c4

Damping of seat cushion Damping of buttocks and thighs Damping of lumber spine Damping of thoracic spine Damping of cervical spine

kc k1 k2 k3 k4

Stiffness of seat cushion Stiffness of buttocks and thighs Stiffness of lumber spine Stiffness of thoracic spine Stiffness of cervical spine

( ) ( ) m z = − k ( z − z ) − c ( z − z ) + k ( z − z ) + c ( z − z ) (4.85)

mc zc = − kc ( zc − z f ) − cc zc − z f + k1 ( z1 − zc ) + c1 z1 − zc (4.84) ¨

˙

¨

1 1

1

1

c

c

˙

˙

˙

1

c

2

2

1

2

˙

˙

˙

˙

2

1

226

Vibration Control of Vehicle Suspension Systems ¨

( )

(

(

(

˙

˙

˙

˙

˙

˙

˙

˙

)

m2 z2 = − k2 ( z2 − z1 ) − c2 z2 − z1 + k3 ( z3 − z2 ) + c3 z3 − z2 (4.86)

m3 z3 = − k3 ( z3 − z2 ) − c3 z3 − z2 + k4 ( z4 − z3 ) + c4 z4 − z3 (4.87)

m4 z4 = − k4 ( z4 − z3 ) − c4 z4 − z3 (4.88)

¨

)

(

¨

˙

˙

)

)

Note that the quarter-car suspension model (4.81) and (4.82), with kss = 0, css = 0, and u f = 0, has been used by many previous studies in researching the active or semi-active control of vehicle suspensions. The seat suspension model (4.83) and (4.84) or the seat suspension with driver body model (4.83)–(4.88), with ks = 0, cs = 0, and zs = zr , has been applied in studying active or semi-active seat suspension control. An integrated model (4.81)–(4.83) or (4.81)–(4.84), with us = 0 and u f = 0, has been used in studying the seat or suspension optimization problem [18,19]. Currently, no integrated model (4.81)–(4.88) has been found in the literature to study active seat and suspension control together. The system state variables are defined as: x1 = zu − zr, x 2 = zu , x3 = zs − zu , x 4 = zs , x5 = z f − zs , ˙ ˙ ˙ ˙ ˙ x6 = z f , x 7 = zc − z f , x8 = zc , x9 = z1 − zc, x10 = z1 , x11 = z2 − z1, x12 = z2 , x13 = z3 − z2, x14 = z3 , ˙

x15 = z4 − z3, and x16 = z4 . Then, state vector x =  x1

T

x16  , the control input vec˙ T tor u = u f us  , and the road disturbance w = zr can be defined. The dynamic equations (4.81)– (4.88) can be rearranged as state-space form: x2

x = Ax + Bw w + Bu (4.89)

where matrices A, Bw , and B can be obtained from (4.81)–(4.88). In practice, all the actuators are limited by their physical capabilities, and actuator saturation needs to be considered for the active suspension control [20] and car suspension [21]. The equation (4.89) can be modified by taking actuator saturation into account: x = Ax + Bw w + Bu (4.90)

where u = sat ( u ), and sat ( u ) is a saturation function of control input u, defined as:  −ulim ,                             if   u < −ulim  (4.91) sat ( u ) =  u,                      if   −ulim ≤ u ≤ ulim  ulim ,                                     if   u > ulim 

where ulim is the control input limit. The following lemma will be used and introduced to solve the actuator saturation problem. Lemma 4.4 [22] u For the saturation constraint that was defined by (4.91), as long as u ≤  lim  , we have  ε 

227

Integrated Suspension Control

u−

1+ ε 1− ε u≤ 2 2

u (4.92)

and T

  1− ε 2 T    1+ ε 1+ ε u u (4.93) u   u− u ≤  u−  2  2 2    

where 0 < ε < 1 is a given scalar. Equation (4.90) is further written to apply Lemma 4.4 in the next section. x = Ax + Bw w + B

1+ ε 1+ ε  1+ ε  u + B u − u  = Ax + Bw w + B u + Bv (4.94)  2 2  2

where v = u − (1 + ε / 2 ) u . The following lemma is also used to derive the main result. Lemma 4.5 For any matrices (or vectors) X and Y with appropriate dimensions, we have: X T Y + Y T X ≤  X T X +  −1Y T Y (4.95)

where  > 0 is any scalar.

4.3.2  Controller Design To improve the system performance, a state feedback controller is designed as u = Kx (4.96)

where K is the feedback gain matrix to be designed. Once K is known, u can be calculated using equation (4.96). For further understanding, Figure 4.11 shows a block diagram of the controller, of which inputs are the state variables x1 to x8, which are assumed to be measurable in practice as an example, and outputs are us and u f . For the car and seat suspension design, the performance on ride comfort is mainly described by the driver head acceleration [23,24] and, therefore, the driver head acceleration as: z = z4 = Cx (4.97)

where C is the last row of matrix A, which is defined as the control output. To achieve good ride comfort and make the controller adequately perform for a wide range of road disturbances, the L2 gain between the road disturbance input w and the control output z is defined as

Tzw

= sup w ≠ 0

z

2

w

2

(4.98)

228

Vibration Control of Vehicle Suspension Systems 8

(a)

1500

Active Seat Active Suspension Integrated Seat Integrated Suspension

1000

4 2

500 Force (N)

(b)

Passive Active Seat Active Suspension Integrated

6

0 –2

0 –500

–4 –6

–1000

–8 –10

0

0.2

0.4

0.6

(c)

0.8

1 1.2 Time (s)

1.4

1.6

–1500

2

8

0.4

0.6

0.8

1 1.2 Time (s)

1.4

2

1.6

1.8

2

Static Feedback - Seat Static Feedback - Suspension State Feedback - Seat State Feedback - Suspension

1000

4

Force (N)

500

0 –2 –4

0 –500

–6

–10

–1000

0

0.2

0.4

0.6

0.8

1 1.2 Time (s)

1.4

1.6

1.8

–1500

2

(e)

(f) 8

Suspension stroke (mm)

2 0 –2 –4 –6 –8

0.6

0.8

1 1.2 Time (s)

1.4

1.6

1.8

2

Passive Static Feedback Robust Static Feedback

20 0 –20 –40 –60 –80

0

0.2

0.4

0.6

0.8

(g) 20

1 1.2 Time (s)

1.4

1.6

1.8

2

–100 0

(h) Passive Static Feedback Robust Static Feedback

15

0.5

1 Time (s)

1.5

2

4000

Passive Static Feedback Robust Static Feedback

3000

10 5 0 –5

–10

2000 1000 0 –1000 –2000

–15 –20

0.4

40

4

–10

0.2

0

60

Passive Static Feedback Robust Static Feedback

0.2

1500

Passive Static Feedback State Feedback

–8

Seat stroke (mm)

0

(d)

1.8

0

0.2

0.4

0.6

0.8

1 1.2 Time (s)

1.4

1.6

1.8

2

–3000

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (s)

FIGURE 4.11  Bump responses on driver head acceleration and control forces for different control systems ∞

where z

2 2

= z ( t ) z ( t ) dt , and w 0

T

2 2

= wT ( t ) w ( t ) dt, is chosen as the performance measure. A 0

small value of ║Tzw║∞ generally means a small value of driver head acceleration under energylimited road disturbances. Therefore, the control objective is to design a controller (4.96) such that the closed-loop system, which is composed by substituting (4.96) into (4.90), is asymptotically stable and the performance measure (4.98) is minimized.

229

Integrated Suspension Control

4.3.2.1  Controller Design for a Nominal System To design such a controller, we now define a Lyapunov function for (4.90), which is assumed to be a nominal system without parameter uncertainties, as V ( x ) = x T Px (4.99)

where P is a positive definite matrix. By differentiating (4.99) and using (4.94), we obtain T

1+ ε 1+ ε     V ( x ) = x T Px + x T Px =  Ax + Bw w + B u + Bv  Px + x T ( t ) P  Ax + Bw w + B u + Bv  2 2     (4.100) By using Lemma 4.4, Lemma 4.5, and (4.96), we have V ( x ) T   1+ ε  1+ ε ≤ x T  AT P + PA +  B K  P + PB K x   2 2  

+ w T BwT Px + x T PBw w +  v T v +  −1 x T PBBT Px

T   1+ ε  1+ ε ≤ x  AT P + PA +  B K  P + PB K x   2 2  

(4.101)

T

2

 1−  T + w T BwT Px + x T PBw w +   u u +  −1 x T PBBT Px  2  = x Tθ x + wT BwT Px + x T PBw w T   1+ ε  K where θ =  AT P + PA +  B   2 

P + PB

2  1+ ε  1−  T K + K K +  −1PBBT P  and  is   2  2 

any positive scalar. Adding z T z − γ 2 w T w, γ > 0, which is a performance index, to the two sides of (4.101) yields

V ( x ) + z T z − γ 2 w T w ≤  x T

 θ + CTC wT    BwT P 

PBw   x  T  = x −γ 2 I   w   

 x  wT  Π    w  (4.102)

 θ + CTC where Π =   BwT P 

PBw  . −γ 2 I  

Based on (4.102), it is now deduced that, if Π < 0 , then V ( x ) + z T z − γ 2 wT w < 0, and Tzw ∞ < γ with the initial condition x ( 0 ) = 0 [25]. When the road disturbance is zero, i.e., w = 0, based on

230

Vibration Control of Vehicle Suspension Systems

(4.102), it can be inferred that, if Π  0, ε > 0, ρ > 0, and ulim , (4.90) with the controller (4.96) is quadratically stable, and Tzw ∞ < γ if there exist matrices Q > 0, Y and a scalar ∈> 0 such that LMIs (4.104) and (4.108) are feasible. Moreover, the feedback gain matrix is obtained as K = YQ −1. It is noticed that (4.104) and (4.108) are LMIs to γ 2; hence, to minimize the performance measure γ , the controller design problem can be modified as a minimization problem of          min γ2 s.t. LMIs (4.104) and (4.108)

(4.109)

This minimization problem is a convex optimization problem and can be solved by using some available software such as the MATLAB LMI toolbox. Because the solution to (4.109) will be dependent on the values of ε and ρ, it is a suboptimal solution for a given ulim. Choosing values for ε and ρ is a trail-and-error process. Typically, small values of ε and ρ may get a high-gain controller design. Note that the aforementioned state feedback controller assumes that all the state variables are measurement available. This case is not true, particularly when considering a high DOF human body model where most of the state variables, e.g., torso displacements and velocities, are not measurable or not suitable for measurement when a driver is driving. Therefore, a control strategy that uses only available measurements needs to be developed. An observer-based output feedback or dynamic output feedback [27] could be applied using the available measurements; however, it makes the design and implementation tasks expensive and hard, particularly when the model order (even after model reduction [28]) is higher. On the contrary, controllers that use static output feedback are less expensive to implement and are more reliable. Therefore, a static output feedback controller will further be considered for the integrated seat and suspension control. A static output feedback controller is a challenging issue from both the analytical and numerical points of view due to its nonconvex nature [29]. Although genetic algorithms can be applied to design a static output feedback controller [30], a computationally efficient numerical algorithm [31] will be applied here. The static output feedback controller is designed as

u = KCs x (4.110)

where Cs is used to define the available state variables. For example, if only in (4.89) is available for feedback, then Cs is defined as Cs = 1 [ 0 ]1×15  . By using (4.110) instead of (4.96) in (4.100), defining WCs = CsQ and Y = KW , and following a similar procedure as derived for the state feedback controller design, we can get the conditions as given in (4.111) and (4.112), which are similar to (4.104) and (4.108), respectively, for the static output feedback controller design. In addition, the static output feedback gain matrix is obtained as K = YW −1.  1+ ε  T T T T  +  −1 BBT  QA + AQ +  Cs Y B + BYCs  2   *    *  * 

2  u   lim    ε   T T  Cs Y

CsT Y T

QC T 2

 2  − −1  I  1 − ε 

0

* *

−I *

    0  < 0   0  2 −γ I   (4.111) Bw

   ≥ 0 (4.112) −1 ρ Q   YCs

232

Vibration Control of Vehicle Suspension Systems

It is observed that the static output feedback controller design is the feasibility problem of LMIs (4.87) and (4.112) with the equality constraint WCs = CsQ . The equality constraint WCs = CsQ can equivalently be converted to [46]. T tr (WCs − CsQ ) (WCs − CsQ )  = 0 (4.113)

By introducing the condition

(WCs − CsQ )T (WCs − CsQ ) ≤ µ I (4.114)

where µ > 0, the equation (4.114) can be rewritten based on the Schur complement:  − µI (WCs − CsQ )T  ≤ 0 (4.115)   WCs − CsQ  −I   −10 If the µ is a very small positive number, such as 10 , then we can design a static output feedback controller by solving the following minimization problem:

min γ 2 s.t. to LMIs (4.111), (4.112), and (4.115)

(4.116)

4.3.2.2  Robust Multi-Objective Controller Design In practice, the mass of the driver body may be varied when a driver’s physical condition is changed or a different driver who has a different weight is driving the vehicle. To make the controller have similar performance despite the changes in the driver’s mass, the variation to the driver’s mass will be considered. Referring to the driver model that was used in this chapter, it is shown that the driv4

er’s mass is composed of the masses of the thighs, lower torso, high torso, and head, i.e., m =

∑m . i

i =1

It is reasonable to assume that the mass variation ratio to each segment of the driver body is equal and the driver’s mass is, in fact, varied in a range of [ mmin , mmax ], where mmin and mmax are the possible minimum and maximum driver’s masses, respectively. Therefore, it is not difficult to represent the uncertain driver’s mass that appeared in the model as 1 1 1 = h1 + h2 (4.117) m mmin mmax

where

h1 =

1 / m − 1 / mmax 1 / mmax − 1 / m , h2 = (4.118) 1 / mmin − 1 / mmax 1 / mmin − 1 / mmax 2

It is shown that hi ≥ 0, i = 1,2, and , mmax = (1 + δ ) m = δ max m = δ max

hi = 1. If we define mmin = (1 − δ ) m = δ min m = δ min

i =1

4

∑m , where 0 < δ < 1, and δ i

4

∑m

i

i =1

max

= 1 + δ , the vehicle model (4.90)

i =1

with an uncertain driver’s mass can be defined as 2

x =

∑h A x + B w + Bu (4.119) i

i =1

i

w

233

Integrated Suspension Control

where matrices Ai, i = 1,2 are obtained by replacing m j , j = 1, 2, 3, 4 in matrix A with δ min m j and δ max m j, respectively. On the other hand, parameter uncertainties may happen to the damping coefficient and stiffness of each segment of the driver body, of which values are, in fact, hard to accurately measure in practice. To describe these uncertainties in the model, a norm-bounded method can be used. Let us assume that the stiffness and damping coefficient with uncertainties can be described as k = k0 (1 + d kδ k )  and c = c0 (1 + dcδ c ), respectively, where k0 and c0 are the nominal values, δ k and δ c are the uncertainties, with δ k ≤ 1 and δ c ≤ 1, and d k ( dc ) indicates the percentage of variation that is allowed for a given parameter around its nominal value. Then, taking a matrix T with uncertain k and c as an example, it can be expressed as  k T=  #

c   k0 (1 + d kδ k ) = #   # 

c0 (1 + dcδ c )   k0 = #   #

 d k k0   0

1  δk  0   0 

c0   1 + #   0

0   δ c 

  = T0 + HFE 

0 dcc0

 δk  d k k0  k  1 1  0  0  c0  where T0 =  0 , F= , E =   with , H =   dcc0  δ c  #   0  0  0 0   # F T F ≤ I , and # represents an arbitrary element in the matrix. Following a similar principle, (4.119) with parameter uncertainties on stiffness and damping coefficients can be defined as 2

x =

∑h ( A + ∆A )x + B w + Bu (4.120) i

i

i

w

i =1

where ∆Ai = H a FEi represents the uncertainty that was caused by the uncertain stiffness and damping coefficients on matrix Ai, H a and Ei are known constant matrices with appropriate dimensions, which can be defined in terms of the locations and variation ranges of the uncertain parameters that appeared in matrix Ai, and F is an unknown matrix function that is bounded by F T F ≤ I . For 2

description simplicity, Ah =

∑ i =1

2

2

hi Ai, ∆Ah =

2

hi ∆Ai =

i =1

hi H a FEi = H a FEh, where Eh =

i =1

∑h E , and i

i

i =1

 Ah = Ah + ∆Ah . Then equation (4.120) can be expressed as: x =  Ah x + Bw w + Bu (4.121)

Similarly, the control output (4.97) can also be rewritten as ¨

h x (4.122) z = z4 = C

2

h = Ch + ∆Ch, Ch = where C

hiCi, and ∆Ch =

i =1

2

hi ∆Ci =

i =1

2

∑h H FE = H FE . i

i =1

c

i

c

h

234

Vibration Control of Vehicle Suspension Systems

Note that the parameter uncertainties on the stiffness and damping coefficients of car and seat suspensions and sprung and unsprung masses can be dealt with in the same way, which, however, will not be further discussed here. For the uncertain system (4.121) and the control output (4.122), (4.111) is also applied and can be obtained as in (4.123), which is further expressed as (4.124). Equations (4.123) and (4.124) are shown as:  T 1+ ε T T T Cs Y B + BYCs  +  −1 BBT Ah +  AhQ +  Q 2   *    *  *   Q ( A + ∆A )T + ( A + ∆A )Q h h h h    + 1 + ε CsT Y T BT + BYCs  +  −1 BBT   2    *   *   * 

T s

C Y

h T QC

CsT Y T 2

 2  − −1  I  1 − ε 

0

* *

−I *

Q (Ch + ∆Ch )

T

T

2

 2  − −1  I  1 − ε 

0

* *

−I *

    0  < 0   0  2 −γ I  (4.123)  Bw

  Bw     0 is a performance index. Note that the weighting parameters α 1, α 2 , and α 3 can properly be chosen to provide the tradeoff among different requirements such as ride comfort and road holding [32]. In general, if a small suspension stroke is required, a big weighting value for α 1 or α 2 should be chosen, and if good road-holding performance is required, a big value for α 3 should be chosen. By using the Schur complement, the feasibility of the following inequality guarantees that CcT Cc < P :

 P   Cc

CcT I

  > 0 (4.132) 

237

Integrated Suspension Control t

At the same time, based on (4.99) and (4.102), it can be derived that x Px < γ T

2

∫w ( s ) w ( s ) ds T

0

if ∏ < 0 is guaranteed. Then, based on (4.131) and (4.132), it can easily be established that, for all t ≥ 0 ∞

t

z z = x C Cc x < x Px < γ T 2 2

T

T c

T

2

∫w ( s ) w ( s ) ds ≤ γ ∫w ( s ) w ( s ) ds (4.133) 2

T

0

0

is satisfied. Taking the supremum over t ≥ 0 yields z2

(

−1

multiplying and post-multiplying (4.132) by diag P   I −1

Q = P is defined, and equation (4.132) is equivalent to  Q QCcT  I  CcQ

T

< γ w 2 for all w ∈ L2 [ 0, ∞ ). Pre-

)

and its transpose, respectively.

  > 0 (4.134) 

Considering parameter uncertainties and the multi-objective control requirement, we now summarize the robust multi-objective controller design problem as follows. For given scalars, i.e., ρ > 0 and ε > 0, and matrices H a, Hc , Ei, i = 1,2, the uncertain system (4.121) with a controller (4.110) is quadratically stable, the L2 gain that is defined by (4.98) is less than γ , and z2 ∞ < γ w 2 if there exist matrices Q > 0 and Y and scalars, i.e.,  > 0 and 1 > 0 , such that the following minimization problem is feasible: min γ2 s.t. LMIs (4.112), (4.115), (4.127), and (4.134)

(4.135)

By solving the problem of (4.135), the controller gain matrix can be obtained as K = YW −1. Note that the performance requirement that was enforced on the control output z2 is subjected to the performance index γ and the energy of the road disturbance ║w║2. Even when γ is minimized, the constraints on the suspension stroke and the dynamic tyre load may be deteriorated in practice if the road disturbance is very strong. Nevertheless, when designing a controller, an appropriate weighting on the control output z2 can provide a good compromise among the ride comfort performance, suspension stroke limitation, and road-holding capability.

4.3.3 Numerical Simulation 4.3.3.1  Validation on a Quarter-Car Model Numerical simulations are conducted in this section to show the effectiveness of the proposed integrated seat and suspension control to improve the driver ride comfort. The parameters used in the simulations are listed in Table 4.9, where the quarter-car suspension parameters have been optimized in terms of driver body acceleration in [33], and the seat suspension and driver body model parameters are discussed in [34]. In the simulation, the actuator force limitation for the quarter-car suspension is considered 1500 N, and for the seat suspension, the actuator force limitation is 500 N. The scalars ε = 0.9 and ρ = 10 −3 are chosen for designing the controllers. To show the effectiveness and advance of the proposed control strategy, several different controllers will be designed and compared. First, we design a state feedback controller for the seat suspension model only, i.e., (4.83)–(4.88), with ks = 0 and

238

Vibration Control of Vehicle Suspension Systems

TABLE 4.9 Parameter Values of the Proposed Suspension Model Mass (kg)

Damping Coefficients (Ns/m)

mu ms

20 300

mf mc m1 m2 m3 m4

15 1 12.78 8.62 28.49 5.31

Spring Stiffness (N/m)

ct cs

0 2000

kt ks

180,000 10,000

css cc c1 c2 c3 c4

830 200 2064 4585 4750 400

kss kc k1 k2 k3 k4

31,000 18,000 90,000 162,800 183,000 310,000

cs = 0, by solving the minimization problem of (4.109) without considering the suspension stroke limitation and road-holding performance. The obtained controller gain matrix is given in

 −2.0237 K = 10 6   1.4845

−0.0083 −0.09073

−0.6569 0.9270

−0.0079 −0.33368

−1.0691 0.39880

−0.1164  (4.136) 0.0792 

This controller will use the state variables x5 ~ x16 of the model (4.89) as feedback signals in the simulation and is denoted as controller 1 for description simplicity. Then, we design another state feedback controller for the quarter-car suspension model only, i.e., (4.81) and (4.82), with kss = 0, css = 0, and u f = 0, by solving the minimization problem of (4.109) without considering the suspension stroke limitation and road-holding performance. The obtained controller gain matrix is given as

K = 10 3 [ 0.4456  −  1.8543 9.5208 1.1960 ] (4.137)

This controller will use the state variables x1 ~ x 4 of the model (4.89) as feedback signals in the simulation and is denoted as controller 2 for description simplicity. Then, we design a state feedback controller for the integrated seat and suspension model, i.e., (4.81)–(4.88), by solving the minimization problem of (4.109) without considering the suspension stroke limitation and road-holding performance. The obtained controller gain matrix is given as K = 10 6 [−0.0061  −  0.0000  −  0.0052  −  0.0006 0.0198  −  0.0035 0.2834  −  0.0021 0.2195  −  0.0280 0.9059

−  0.0119 1.1534 0.0101  −  26.103 0.0284;  0.0553  −  0.0001 0.0041  −  0.0096 0.1501  −  0.0015 0.1983

(4.138)

−  0.0000 0.1636 0.0002 0.0564 0.0021  −  0.0954 0.0162  −  3.4882  −  0.0085] This controller will use the state variables x1 ~ x16 of the model (4.89) as feedback signals in the simulation and is denoted as controller 3 for description simplicity. This controller will provide two control inputs to the seat suspension and car suspension, respectively. To validate the suspension performance in the time domain, two typical road disturbances, i.e., bump road disturbance and random road disturbance, will be considered in the simulation and applied to the vehicle wheel. Bump excitation: The road displacement under bump excitation is given by

239

Integrated Suspension Control

 a  2πv0     1 − cos  l t   ,  2 zr ( t ) =   0,  

0≤t ≤ t>

l v0

l v0

(4.139)

where a and l are the height and the length of the bump excitation, respectively. v0 is the vehicle forward speed. In this research, a = 0.1 m,  l = 2 m, and v0 = 30 km/h are selected in the simulation. The bump responses of the driver head acceleration for the integrated seat and suspension system with different controllers are compared in Figure 4.11a, where Passive means that no controller has been used, Active Seat means that controller 1 is used for seat suspension only, Active Suspension means that controller 2 is used for car suspension only, and integrated means that controller 3 is used for both seat suspension and car suspension. In Figure 4.11a, it is shown that the integrated control achieves the best performance among all the compared control strategies on ride comfort in terms of the peak value of driver head acceleration. Further comparison on the control forces is shown in Figure 4.11b, where the integrated control provides two control forces, which are denoted as Active Seat and Active Suspension to the seat suspension and the car suspension, respectively. As stated previously, the state feedback controller is not practically realizable, particularly when the human body model is included. We now design a static output feedback controller for the integrated seat and suspension model (4.81)–(4.88) by solving the minimization problem of (4.116) without considering the suspension stroke limitation and road-holding performance. By assuming that all the state variables for car suspension and seat suspension are available for measurement by using displacement and velocity sensors or using accelerometers with integration functions and all the state variables for the driver body model are not measurement available, the controller gain matrix is obtained as  −0.4665 0.0000  − 0.4759  − 0.0080  − 0.1965  − 0.1023 8.6420  − 0.1991 K = 10 5   (4.140)  8.2020 0.0171 1.4630  − 0.1564 9.4831 0.0284 6.1010 0.1435  This controller uses only the measurement available state variables x1 ~ x8 of the model (4.89) as feedback signals in the simulation and is denoted as controller 4 for description simplicity. To clearly show the performance of the designed static output feedback controller, the bump responses on driver head acceleration for the integrated seat and suspension system with no controller, state feedback controller, and static output feedback controller are compared in Figure 4.17c, where state feedback means that controller 3 is used, and Static Feedback means that controller 4 is used. In Figure 4.11c, it is shown that the static output feedback controller achieves similar performance to the state feedback controller in terms of the peak value on driver head acceleration despite its simple structure. The comparison on the control forces is shown in Figure 4.12d. In Figure 4.11d, it is shown that both the state feedback controller and the static output feedback controller provide two control forces to the system and that their forces to seat suspension and car suspension are quite similar. It is noticed that controller 4 achieves good ride comfort performance with limited information. However, for vehicle suspension, aside from the ride comfort that needs to be focused on, the car and seat suspension stroke limitation and road-holding performance also need to be considered. In addition, parameter uncertainties, which may often happen to the system in practice, should also be dealt with. Furthermore, the measurement of tyre deflection x1 and velocity x 2 may not be easily available in practice. Therefore, a robust controller that compromises the performance among ride comfort, car

240

Vibration Control of Vehicle Suspension Systems

and seat suspension stroke limitation, and road-holding capability and considers parameter uncertainties and measurement availability is finally designed by solving the problem of (4.135). The obtained controller gain matrix is given as (4.141), which uses the measurement available state variables x3 ~ x8 of the model (4.89) as feedback signals and is denoted as controller 5 for description simplicity.

 0.0661 0.0065  − 0.2115 0.0336  − 2.7173  − 0.0167    K = 10 5   (4.141)  −0.1255 0.0378 − 0.3292  − 0.0205 1.3831 0.0042 

To show the difference between controllers 4 and 5 on different performance aspects, the driver head acceleration, car suspension stroke, seat suspension stroke, and dynamic tyre load under bump road input are shown in Figure 4.11e–h, respectively. It is shown that controller 4, which is indicated as Static Feedback in the figures, achieves better ride comfort in terms of the peak value on driver head acceleration in Figure 4.11e compared with controller 5, which is indicated as Robust Static Feedback. However, it generates bigger suspension stroke and dynamic tyre load, as shown in Figure 4.11f and g, compared with controller 5. This condition may result in suspension end-stop collision and cause the wheels to lift off the ground. The dynamic tyre load of controller 5 is quite similar to the passive suspension in terms of the maximum peak value. Although controller 5 requires bigger seat suspension stroke than controller 4 and passive suspension, in Figure 4.11g, it is observed that the stroke is still within ±20 mm, which is acceptable for seat suspension [35]. Therefore, controller 5 achieves a good tradeoff among different performance requirements. This controller will further be tested on a full-car model in the next section. On the other hand, from an implementation point of view, note that, for a real vehicle, the aforementioned controller can be integrated into a suspension control module, which is designed as an embedded electronic control unit that controls one or more of the electrical systems in a car. This module will receive signals from sensors that were installed at wheels and seat frame and calculate the required control forces in terms of the designed controller gain matrix. The control forces will then be generated by the actuators and applied to the vehicle and seat. Note that the controller gain matrix is a constant matrix that does not need to be recalculated in a real-time implementation and can easily be stored in a microprocessor memory (random access memory or read-only memory). The calculation of the control forces is straightforward, without high computational power. This condition enables the implementation of the controller on a microcontroller board. 2) Random excitation: When the road disturbance is considered as vibration, it is typically specified as a random process with a ground displacement power spectral density of

− n1   Sg ( Ω 0 )  Ω  ,     if   Ω ≤ Ω 0  Ω   0 Sg ( Ω ) =  (4.142) − n2  Ω   Sg ( Ω 0 )  Ω  ,     if   Ω ≥ Ω 0 0 

where Ω 0 = (1 / 2π ) is a reference frequency, Ω is a frequency, and n1 and n2 are road roughness constants. The value Sg ( Ω 0 ) provides a measure for the roughness of the road. In particular, samples of the random road profile can be generated using the spectral representation method [36]. If the vehicle is assumed to travel with a constant horizontal speed v0 over a given road, the road irregularities can be simulated by the following series: Nf

zr ( t ) =

∑s sin ( nw t + ϕ ) (4.143) n

n =1

0

n

241

Integrated Suspension Control

(

)

˙ ˙ Vt ˙ PL = QL − Ctp PL − Ar x s − xu (4.144) 4βe

where PL is the pressure drop across the piston, Ar is the piston area of the hydraulic actuator, βe is the effective bulk modulus, Vt is the total actuator volume, Ctp is the coefficient of the total leakage due to pressure, and QL is the load flow. The parameter values are given as Ar = 3.35 × 10 −4 m2,  Vt   4βe  13 5  4 β   = 4.515×10 N/m , and Ctp =  V  . e t To validate the system performance, the bump road disturbances as shown in Figure 4.14a will be applied to the vehicle wheels. Figure 4.14a shows that the road disturbances, which are applied to the front and rear wheels, have the same peak amplitude, with a time delay of ( l f + lr ) / v0 . However, to excite the roll motion of the vehicle, the road disturbances to the left and right wheels are applied with different amplitude [17]. In the simulation, the designed controller 5 will be applied to calculate the desired control force in terms of the measured signals for each actuator, and then, the desired forces will be tracked and applied to the vehicle and seat suspension through electrohydraulic actuators. For simplicity, a proportionalintegral-derivative (PID) controller will be applied to each actuator as an inner control loop so that each actuator can track its desired force. More advanced strategies for controlling electrohydraulic actuators can be found, for example, in [37,38], which, however, will not be discussed in this chapter.

242

Vibration Control of Vehicle Suspension Systems 2.6

Suspension stroke (mm)

(a)

0.6 0.5 0.4 0.3 0.2 0.1 0 60

70

80 Speed (km/h)

90

2.4 2.2 2 1.8 1.6 1.4

100

0.6 0.5 0.4 0.3 0.2 60

70

80 Speed (km/h)

90

Suspension stroke (mm)

0.4 0.2 0 60

70

80 Speed (km/h)

90

100

90

100

70

80 Speed (km/h)

90

100

70

80 Speed (km/h)

90

100

4.5 4 3.5 3 2.5 60

Seat stroke (mm)

80 Speed (km/h)

5

0.8

1.2 1 0.8 0.6 70

80 Speed (km/h)

90

280 260 240 220 200 180 160 60

100 Active

Passive

2.5

10 Suspension stroke (mm)

70

90

Passive

0.6

2 1.5 1 0.5 0 60

70

80 Speed (km/h)

90

2 1.5

70

80 Speed (km/h)

90

5 60

80 Speed (km/h)

90

100

70

80 Speed (km/h)

90

100

500 450 400

Passive

Suspension stroke (mm)

1 0 –1 –2 0

70

550

Active

2

2

4 Time (s)

60 40 20 0 –20 –40

6

0

2

4 Time (s)

6

0

2

4 Time (s)

6

15 10 5 0

–5 –10 –15

6

350 60

100

3

–3

8 7

600

2.5

1 60

9

100

3 Seat stroke (mm)

60

100 80

1

100

110

100

1.2

Seat stroke (mm)

90

120

1.4

0.4 60

(d)

80 Speed (km/h)

130

1.4

(c)

70

140

Active

(b)

60 150 Dynamic tyre load (N)

Seat stroke (mm)

0.7

0

2

4 Time (s)

2000 1000 0

–1000 –2000 –3000

6 Active

Passive

Integrated Suspension Control

243

FIGURE 4.13  Full-car suspension model with a driver seat.

244

Vibration Control of Vehicle Suspension Systems

(a)

(b)

2 Passive Active

0.1 Front right Rear right Front left Rear left

Displacement(m)

0.08

1

0.06

0.04

0.02

0 –1

–2

–3 0

0

0.2

0.4

0.6

0.8

1 Time (s)

1.2

1.4

1.6

2

1.8

Suspension stroke (mm)

1

1.5 Time (s)

2

2.5

0 –20 –40

0

0.5

1

40

2 1.5 Time (s) Rear left

2.5

3

–20 0

0.5

1

0

2 1.5 Time (s)

2.5

6

–20 –40 –60 0

0

Passive Active

8

20

3

20

–40

40

Seat stroke (mm)

20

0.5

1

2 1.5 Time (s) Rear right

2.5

3

40 20 0 –20

4 2 0 –2

–4

–40

–6

–60 0

3 Passive

0.5

Active

1

2 1.5 Time (s)

2.5

3

–8

0

0.5

1

1.5 Time (s)

2

2.5

3

(f) Front right

0 –500 –1000 0

0.5

1

1.5 Time (s)

2

2.5

0

0

0

0.5

1

1.5 Time (s)

0.5

1

2

2.5

3

1.5 Time (s)

2

2.5

3

Rear right

2000

–500 –1000

–1000

3

500

Front left Front right Rear left Rear right Seat

1000

0

Rear right

1000

1500

1000

Force (N)

500

–1500

Front right

0.5

10

Front right

(e)

0

(d) Front left

40

Suspension stroke (mm)

Suspension stroke (mm)

Suspension stroke (mm)

(c)

–4

500

0

1000

–500

0

–1000

Passive

0 Active

0.5

1

1.5 Time (s)

2

2.5

3

–1000

0

0.5

1

1.5 Time (s)

2

2.5

3

FIGURE 4.14  (a) Road disturbance; (b) driver head acceleration; (c) car suspension strokes; (d) seat suspension strokes; (e) dynamic tyre loads; (f) actuator output forces.

4.3.4  Conclusion In this chapter, an integrated seat and suspension have been developed and used for an integrated controller design. Because some state variables are not measurement available in practice, a static output feedback controller design method has been presented. Considering the limited capability of actuators, the actuator saturation constraint is included in the controller design process. Numerical simulations are used to validate the performance of the designed controllers. The results show that the integrated seat and suspension control can provide the best ride comfort performance compared with the passive seat and suspension, active seat suspension control, and active car suspension control. The static output feedback control achieves compatible performance to the state feedback control with a realizable structure.

1

Suspension stroke (mm)

(a)

Integrated Suspension Control

0.8 0.6 0.4 0.2 0 60

65

70

75 80 85 Speed (km/h)

90

95

100

4 3 2 75 80 85 Speed (km/h)

90

95

70

75 80 85 Speed (km/h)

90

95

100

65

70

75 80 85 Speed (km/h)

90

95

100

800 700 600 60

100 Active

Passive

1

0 –0.5 –1 0

1

2

3 4 Time (s)

5

6

7

20 10 0 –10 –20 0

1

2

3 4 Time (s)

5

6

7

1

2

3 4 Time (s)

5

6

7

0.5 Seat stroke (mm)

65

900

Suspension stroke (mm)

70

65

0.5

0

–0.5

10

Seat stroke (mm)

6

(b)

12

60

5

1 60

16 14

0

(c)

1

2

3 4 Time (s)

5

6

1000 0 –1000 –2000

7

0

Passive

Active

2 Passive Active

1

0

–1

–2

–3

0

1

0.5

10 Front Left

5 0 –5 –10 0

1

2

2

1.5 Time (s)

3

Wheel acceleration (m/s2)

(d)

Wheel acceleration (m/s2)

–4

10

Front Right

5 0 –5 –10 0

1

10 Rear Left

5 0 –5 1

2 Time (s)

2

3

Time (s)

3

Wheel acceleration (m/s2)

Wheel acceleration (m/s2)

Time (s)

–10 0

3

2.5

20 Rear Right

10 0 –10 0

1

2

3

Time (s)

FIGURE 4.15  (a) RMS of random responses under E-grade road disturbance with different vehicle speed levels; (b) random responses under D-grade road disturbance with a vehicle speed of 100 km/h; (c) bump responses on driver head acceleration for a full-car suspension with parameter uncertainties and measurement noises; (d) wheel vertical accelerations with measurement noises.

246

Vibration Control of Vehicle Suspension Systems

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28. Ho, D.W.C. and Y. Niu, Robust fuzzy design for nonlinear uncertain stochastic systems via sliding-mode control. IEEE Transactions on Fuzzy Systems, 2007. 15(3): 350–358. 29. Syrmos, V.L., C.T. Abdallah, P. Dorato, and K. Grigoriadis, Static output feedback: A survey. Automatica, 1997. 33(2): 122–137. 30. Du, H. and N. Zhang, DesigningH∞/GH2static-output feedback controller for vehicle suspensions using linear matrix inequalities and genetic algorithms. Vehicle System Dynamics, 2008. 46(5): 385–412. 31. Du, H. and N. Zhang, Static output feedback control for electrohydraulic active suspensions via T-S fuzzy model approach. Journal of Dynamic Systems, Measurement, and Control, 2009. 131(5): 051004–11. 32. Hong, C. and G. Kong-Hui, Constrained H-infinity control of active suspensions: an LMI approach. IEEE Transactions on Control Systems Technology, 2005. 13(3): 412–421. 33. Kuznetsov, A., et al., Optimization of a quarter-car suspension model coupled with the driver biomechanical effects. Journal of Sound and Vibration, 2011. 330(12): 2937–2946. 34. Choi, S.-B. and Y.-M. Han, Vibration control of electrorheological seat suspension with human-body model using sliding mode control. Journal of Sound and Vibration, 2007. 303(1–2): 391–404. 35. Maciejewski, I., L. Meyer, and T. Krzyzynski, Modelling and multi-criteria optimisation of passive seat suspension vibro-isolating properties. Journal of Sound and Vibration, 2009. 324(3–5): 520–538. 36. Verros, G., S. Natsiavas, and C. Papadimitriou, Design optimization of quarter-car models with passive and semi-active suspensions under random road excitation. Journal of Vibration and Control, 2016. 11(5): 581–606. 37. Alleyne, A. and R. Liu , Systematic control of a class of nonlinear systems with application to electrohydraulic cylinder pressure control. IEEE Transactions on Control Systems Technology, 2000. 8(4): 623–634. 38. Thompson, A.G. and B.R. Davis, Technical note: Force control in electrohydraulic active suspensions revisited. Vehicle System Dynamics, 2001. 35(3): 217–222.

5

Interconnected Suspension Control

5.1 INTRODUCTION In this chapter, we proposed a switched control method of vehicle suspension based on motion-mode detection, which is suitable for the control of interconnected suspensions. Then, a novel controllable electrically interconnected suspension (EIS) for improving vehicle ride comfort is systematically studied and experimentally validated. This switched control strategy based on the motion-mode detection can be potentially implemented through interconnected suspensions such as hydraulically interconnected suspension [1–4] and EIS [5] by actively switching its interconnection configuration in terms of the dominant vehicle body motion-mode [6,7]. The design of the switched control law is developed focusing on three vehicle body motion-modes: bounce, pitch, and roll. At first, an H∞ optimal controller will be designed for each motion-mode with the use of a common quadratic Lyapunov function [8–10], which guarantees the stability of the switched system under arbitrary switching functions. Then, a motion-mode detection method based on the calculation of the motion-mode energy is introduced. And then, the possible implementation of the control system in practice is discussed. Finally, numerical simulations are used to validate the proposed study. Different from the mechanical or hydraulic energy in the traditional interconnected suspensions, the EIS system is interconnected with the electrical energy that is transformed from vibration energy by independent electromagnetic suspensions. We introduce an EIS system comprising a controllable electrical network (EN) and two independent electromagnetic suspensions to present the system design and analysis method. The electromagnetic suspension’s mechanical characteristics are related to the applied electrical elements [11–13]. Similarly, the EN of the EIS can determine the heave and roll dynamics. Experiments are implemented to validate the system model. The EIS system is then applied in a half-car model with one kind of EN topology. The frequency and time domain simulations show that the vehicle performance at heave and roll are both improved by controlling resistors in the EN. The proposed EIS system can determine the suspension characteristics (damping, stiffness, and inertance) in different DOFs with electrical elements, and the EN is easy to embed into the vehicle system. Besides, the system only requires energy for controlling the electrical elements, which is very low. The EIS system shows great potential in practical applications.

5.2  MOTION MODE CONTROL STRATEGY 5.2.1  Switched Control 5.2.1.1  Switched Suspension Model A full-car suspension model, as shown in Figure 5.1, is considered in this chapter. This is a 7-degreeof-freedom (DOF) model where the sprung mass is assumed to be a rigid body with freedoms of motion in the vertical, pitch, and roll directions and each unsprung mass has freedom of motion in the vertical direction. In Figure 5.1, zs is the vertical displacement at the centre of gravity, and θ and φ are the pitch and roll angles of the sprung mass. ms , muf , and mur denote the sprung and unsprung masses, respectively. Iθ and Iφ are pitch and roll moments of inertia. The front and rear displacements of the sprung mass on the left and right sides are denoted by zsfl , zsrl , zsfr , and zsrr . The front

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Interconnected Suspension Control

249

FIGURE 5.1  Full-car suspension model.

and rear displacements of the unsprung masses on the left and right sides are denoted by zufl , zurl , zufr , and zurr. The disturbances, which are caused by road irregularities, are denoted by w fl , wrl , w fr , and wrr . The front and rear suspension stiffness and the front and rear tyre stiffness are denoted by ksf , ksr , and ktf , ktr , respectively. The front and rear suspension damping coefficients are csf and csr . Four cylinders are placed between the sprung mass and the unsprung masses to generate pushing forces, denoted by Ffl , F _ rl, Ffr , and Frr. Note that the four actuators can be arbitrarily connected to each other through hydraulic fluid by switching the connection valve of circuits in the hydraulically interconnected suspension. Assuming that the pitch angle θ and the roll angle φ are small enough around their static equilibrium positions, the following linear approximations are applied:

zsfl ( t ) = zs ( t ) + l f θ ( t ) + t f φ ( t )

zsfr ( t ) = zs ( t ) + l f θ ( t ) − t f φ ( t )

zsrl ( t ) = zs ( t ) − lrθ ( t ) + trφ ( t )

zsrr ( t ) = zs ( t ) − lrθ ( t ) − trφ ( t ) (5.1)

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and a kinematic relationship between x s ( t ) and q ( t ) can be established as x s ( t ) = LT q ( t ) (5.2)

where

T

q ( t ) =  zs ( t ) θ ( t ) φ ( t )  ,  

T

x s ( t ) =  zsfl ( t ) zsfr ( t ) zsrl ( t ) zsrr ( t )  ,  

 1  l L= f  tf 

1 lf

1 −lr

−t f

tr

1  −lr  . − tr  

By applying Newton’s second law of motion and using the static equilibrium positions as the origins for the displacement of the centre of gravity, the pitch, and roll angles of the car body, and the displacements of the unsprung masses, the motion equations of the full-car suspension model can be formalized in terms of mass, damping, and stiffness matrices as ¨

M s q ( t ) = LBs ( xu ( t ) − xs ( t )) + LK s ( xu ( t ) − x s ( t )) − LF ( t )

¨

M u xu ( t ) = Bs ( x s ( t ) − xu ( t )) + K s ( x s ( t ) − xu ( t )) + K t ( w ( t ) − xu ( t )) + F ( t ) (5.3)

T

where  xu ( t ) =  zufl ( t ) zufr ( t ) zurl ( t ) zurr ( t )  ,  

T

w ( t ) =  w fl ( t ) w fr ( t ) wrl ( t ) wrr ( t )  ,  

T

F ( t ) =  Ffl ( t ) Ffr ( t ) Frl ( t ) Frr ( t )  , and the matrices are given as    muf 0 0 0  ms 0 0   0 m 0 0   uf  M s =  0 I θ 0 , M s =  mur 0 0 0  0 0 Iφ     0 0 mur  0  ksf   0 Ks =  0   0

0

0

0

ksf

0

0

0 0

ksr 0

0 ksr

  ktf     0 = , K  s  0     0

  csf 0 0 0     0 csf 0 0 B = ,  s  0 0 c 0 sr     0 0 0 csr

0

0

0

ktf

0

0

0 0

ktr 0

0 ktr

   ,  

   .  

Substituting equation (5.2) into equation (5.3), we obtain

¨

˙

M m zm ( t ) + Bm zm ( t ) + K m zm ( t ) = K mt w ( t ) + Lm F ( t ) (5.4)

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Interconnected Suspension Control T

Where zm ( t ) =  q T ( t ) xuT ( t )  , and    LBs LT − LBs  Ms 0  Mm =   , Bm =   − Bs LT Bs  0 M u    −L  Lm =   .  I 

  LK s LT − LK s  , K m =   − K s LT K s + K t  

  0    , K mt =   ,   K t  

The state-space form of equation (5.4) can be expressed as

x ( t ) = Ax ( t ) + B1w ( t ) + B2 F ( t ) (5.5)

T  0 I ˙   where x ( t ) =  zmT ( t ) zmT ( t )   , A =  −1 −1    − M m K m − M m Bm

  0  , B1 =  −1   M m

  0   , and B2 =  −1   M m Lm

  . 

For the 7-DOF full-car suspension model, it has seven motion-modes. As the three-vehicle body motion-modes: bounce, pitch, and roll are more relevant to the ride comfort performance, we mainly focus on these motion-modes to design the control system in this study. For the bounce motion-mode, which is a vertical motion, to achieve the best control performance on it and reduce the energy consumption required by the actuators, we expect that the control forces generated in four wheels are in the same direction with the same amplitude, that is, we only need to find one desired control force F ( t ) and apply this force to four wheels. This can be easily done through, for example, hydraulically interconnected suspension by connecting four wheels with one circuit. In this case, we define

Lm1 =  −4 0 0 1 1 1 1 

T

And obtain

  0 B21 =  −1   M m Lm1 

For the pitch motion-mode, which is a rotation around the pitch axis, to achieve the best control performance and reduce the energy consumption required by the actuators, we expect that the control forces generated in the two front wheels are opposite to the two rear wheels but with the same amplitude, that is, we only need to find one desired control force F ( t ) and apply this force to the two front wheels, and apply −F ( t ) to the two rear wheels. This can also be implemented by switching the connection of circuits in, for example, a hydraulically interconnected suspension. In this case, we define

Lm 2 =  0 − 2 ( l f + lr ) 0 1 1 − 1 − 1   

And obtain

  0 B22 =  −1   M m Lm 2 

T

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Vibration Control of Vehicle Suspension Systems

Similarly, for the roll motion-mode, which is a rotation around the pitch axis, to achieve the best control performance and reduce the energy consumption required by the actuators, we expect that the control forces generated in the two left wheels are opposite to the two right wheels but with the same amplitude, that is, we only need to find one desired control force F ( t ) and apply this force to the two left wheels, and apply −F ( t ) to the two right wheels. In this case, we define

Lm 3 =  0 0 − 2 ( t f + tr ) 1 − 1 1 − 1   

T

And obtain

  0 B23 =  −1   M m Lm 3 

It can be seen that for different motion-modes, the control system configuration can be set differently. To consider all of them together, a switched suspension system is constructed as

x ( t ) = Ax ( t ) + B1w ( t ) + B2σ (t )u ( t ) (5.6)

where u ( t )   =  F ( t ) represents the control input and σ ( t ) is the switching rule defined by the mapping

σ ( t ) : R + →  ,  = {1,2,3},

that is, the linear system ( A,  B1 ,  B2 )i is active if σ ( t )   =  i with i  ∈   . It is assumed that the switching rule σ ( t ) is not known a priori but it is available based on the motion-mode detection in real time. We now need to design a switched control law so that the optimal control performance can be achieved by switching the control system configuration according to the dominant motion-mode, which will be discussed in the next section. Remark 1 It is noted that the main assumption for deriving the current vehicle suspension model is that the pitch angle θ  and the roll angle φ are small enough around their static equilibrium positions, and hence, the linear relationships as shown in equation (5.1) between the displacements of the sprung mass at the four suspension positions and the displacement, pitch angle, and roll angle of the vehicle body centre of gravity can be established. This assumption is commonly adopted by many research papers in building full-car suspension model, and it is more reasonable for semi-active or active suspension where the pitch or roll angle could be controlled to be smaller with appropriate control actions. Remark 2 In practice, the spring and damper of a vehicle suspension system may show nonlinear characteristics. In addition, when considering the dynamics of damper or actuator in the semi-active or active suspension, the nonlinearity may come from the damper, such as Magnetorheological (MR) damper, or the actuator, such as the electro-hydraulic actuator, such that the suspension model becomes a nonlinear system. To deal with the nonlinearities, different approaches have been proposed. For example, an adaptive backstepping control was proposed to deal with the spring nonlinearity and the piecewise linear behaviour of the damper [14], a Takagi–Sugeno (T–S) fuzzy approach was proposed to deal with the nonlinearity of the electro-hydraulic actuator used in vehicle active suspension [15], and a robust control strategy was proposed in [16] to deal with the nonlinear dynamics of the MR damper. As discussed that the currently proposed switched control strategy could be possibly implemented in practice by the hydraulically interconnected suspension. Therefore, the modelling and control of the nonlinear electro-hydraulic system will be a challenging task. To this end, it may consider to use the T–S fuzzy modelling technique [15,17] and the switched fuzzy control approach [18,19]. Thus, each sub-system of the switched system (5.6) will be

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represented by a T–S fuzzy system, and the overall representation of the switched system (5.6) in the form of a T–S fuzzy system can be given in terms of the following rule base:

Model Rule Rσj (t ) :

IF ξ1  is Mσ (t )1 , … , and ξ p  is Mσ (t ) p

THEN  x ( t ) = A j x ( t ) + B1 j w ( t ) + B2σ (t ) j u ( t ) , j = 1,2, … , Nσ (t )

where Rσj (t ) denotes the jth fuzzy inference rule in the σ th switched sub-system, Nσ (t ) is the number of inference rules in the σth switched sub-system, ξ1, ξ2, … , and ξ p are the premise variables, Mσ (t )1, … , Mσ (t ) p are the fuzzy sets in the σ th switched sub-system, A j, B1 j , and B2σ (t ) j are known constant matrices of appropriate dimensions for the σ th switched sub-system. 5.2.1.2  Switched Control System Design 5.2.1.2.1  Control Objective For a full-car suspension design, the performance on ride comfort will be mainly described by the sprung mass vertical acceleration, pitch acceleration, and roll acceleration, thus, the sprung mass accelerations will be defined as the control objectives as ¨

z11 ( t ) = z s ( t ) = C11 x ( t ) + D21u ( t ) (5.7)

z12 ( t ) = θ ( t ) = C12 x ( t ) + D22u ( t ) (5.8)

z13 ( t ) = φ ( t ) = C13 x ( t ) + D23u ( t ) (5.9)

¨

¨

where C1i , i  =  1, 2,3, are defined as the 8th, 9th, 10th rows of A matrix, respectively, and D2i , i   =  1, 2,3, are defined as the 8th, 9th, 10th rows of B2 matrix, respectively. Equations (5.7)–(5.9) are written together as

z1 ( t ) = C1σ (t ) x ( t ) + D2σ (t )u ( t ) , σ ( t ) = 1,2,3

On the other hand, for vehicle suspension systems, besides the ride comfort performance, the ­ suspension deflection limitation and the road holding ability should be considered as well. To keep the suspension deflection to be within its limitation, the suspension deflections zsi ( t ) –  zui ( t ) ,  i   =  fl , fr , rl , rr are required to be smaller. Similarly, to keep the wheel contact with ground, the wheel vertical relative displacements zui ( t ) –  wi ( t ) ,  i   =  fl , fr , rl , rr are required to be smaller so that a good road holding performance can be achieved. Thus, we will define the suspension deflections and wheel relative displacements as other control objectives, that is,   z2 ( t ) = diag  zsfl ( t ) − zufl ( t ) zsfr ( t ) − zufr ( t ) zsrl ( t ) − zurl ( t ) zsrr ( t ) − zurr ( t )  = C2 x ( t ) (5.10)

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Vibration Control of Vehicle Suspension Systems

And

z3 ( t ) = diag  zufl ( t ) − w fl ( t ) zufr ( t ) − w fr ( t ) zurl ( t ) − wrl ( t ) zurr ( t ) − wrr ( t )  = C3 x ( t ) + D31w ( t )

(5.11)

where C2, C3, and D31 can be defined appropriately in terms of the state vector x ( t ) and distur­ bance w ( t ). As these control objectives are conflicted with each other, they cannot be optimized at the same time. To compromise these control objectives and convert the multiple objective problem into a single objective problem in the controller design procedure, the final control objective vector is defined as  z1 ( t )  z ( t ) =  α z2 ( t )  β z (t ) 3 

   = Cσ (t ) x ( t ) + D1w ( t ) + Dσ (t )u ( t ) , σ ( t ) = 1,2,3 (5.12)  

where α and β are weighting parameters used to provide trade-off among z1 ( t ) to z2 ( t ) and z3 ( t ), and Cσ (t ) =  C1Tσ (t )  D1 =  0 

α C2T

0

Dσ (t ) =  D2Tσ (t ) 

T

β C3T  ,  T

T  , β C31 

0

T

0  . 

To achieve good suspension performance and make the controller perform adequately for a wide range of road disturbances, the L2 gain between the road disturbance input w ( t ) and the control objective vector z ( t ), which is defined as

Tzw ∞

 z 2 (5.13) w ≠ 0  w 2

= sup

where  z  = z ( t ) z ( t ) dt and  w  = wT ( t ) w ( t ) dt , is chosen as the performance measure. 2 2

2 2

T

0

Small value of Tzw disturbances.

0

generally means small value of control output under the energy limited road

5.2.1.2.2  Switched Controller Design To achieve the required objectives, a switching state feedback control law is designed as

u ( t ) = Kσ (t ) x ( t ) (5.14)

where Kσ (t ) is the switched control gains to be found. Remark 3 Following Remark 2, when the switched fuzzy system is considered, the controller could be designed using the parallel distributed compensation scheme and is defined as

Control Rule Rσj (t ) :

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Interconnected Suspension Control

IF ξ1  is Mσ (t )1 , … , and ξ p  is Mσ (t ) p ,

THEN  u ( t ) = Kσ (t ) j x ( t ) , j = 1,2, … , Nσ (t )

where Kσ (t ) j is the controller gain matrix under the jth rule for the σ th sub-system. Nevertheless, it should be noted that the controller synthesis problem for the switched fuzzy system is still an open problem, which needs further investigation. In addition, the T–S fuzzy modelling of the interconnected suspension system also needs to be further studied. These issues together with the experimental validation for the nonlinear suspension are all under planned, which however are out of the scope of this chapter and will not be further discussed. From equations (5.6), (5.12), and (5.14), the closed-loop switched system can be written as follows:

x ( t ) = Aσ (t ) x ( t ) + B1w ( t )

z ( t ) = Cσ (t ) x ( t ) + D1w ( t ) (5.15)

where Aσ (t ) = A + B2σ (t ) Kσ (t ) , Cσ (t ) = Cσ (t ) + Dσ (t ) Kσ (t ). The objective is then to determine the state feedback gains Kσ(t) such that the closed-loop switched system (5.15) is quadratically stable and the performance measure (5.13) is minimized. A sufficient condition ensuring Tzw ∞ < γ is given by the following matrix inequality [8,20]:

 A X + XAT B1 XCσT(t )  σ (t )   σ (t )  −I D1T  < 0,σ ( t ) = 1,2,3 (5.16) *   * * −γ 2 I   

where X is a symmetric positive definite matrix. It is further written as

T T   ( A + B2σ (t ) Kσ (t ) ) X + X ( A + B2σ (t ) Kσ (t ) ) B1 X (Cσ (t ) + D2σ (t ) Kσ (t ) )  * −I D1T   * * −γ 2 I 

   < 0 (5.17)   

By defining Yσ (t ) = Kσ (t ) X , equation (5.17) is equivalent to

 AX + XA + B2σ (t )Yσ (t ) + YσT(t ) B2σ (t ) B1 XCσT(t ) + YσT(t ) D2σ (t )   −I D1T *  −γ 2 I * * 

   < 0,σ ( t ) = 1,2,3 (5.18)  

We now state the controller design problem as: for given number γ > 0, the system (5.15) is quadratically stable and Tzw ∞ < γ if there exist matrices X > 0, and Yσ (t ) ,  σ ( t )   =  1,2,3 such that linear matrix inequalities (LMIs) (5.18) are feasible. Moreover, the feedback gains are obtained as Kσ (t ) = Yσ (t ) X −1 ,  σ ( t ) = 1,2,3. It is noticed that equation (5.18) is LMIs to γ 2, hence, to minimize the performance measure γ the controller design problem can be modified as a minimization problem of minγ 2

s.t. LMIs (5.18). (5.19)

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This minimization problem is a convex optimization problem and can be solved by using the standard software. 5.2.1.2.3  Motion-Mode Detection Vehicle body-wheel motion-modes refer to vehicle relative motions between body and wheels, and can be classified into several distinct modes in terms of the characteristic frequencies and modal shapes. For a two-axle four-wheel vehicle, there are seven body-wheel motion-modes. The three body-dominated motion-modes are usually referred as vehicle bounce, roll and pitch. Vehicle bodydominated motions mostly contribute to vehicle ride comfort. The research on vehicle dynamics is mostly dedicated to the study and control of these three motion-modes. The motion-mode energy method [21,22] will be introduced to detect the dominant motion mode. In this method, the vehicle dynamic energy for bounce, pitch, and roll motion-modes will be calculated and used to provide a switching signal for the above-designed control law. The dominant motion-mode is the motion-mode that has the highest level of energy relative to other motion-modes at one time instant. From the modal analysis of the characteristic matrix of an n-DOF system, we can form a modal matrix as Γ =  Ψ ΨΛ  (5.20)  

where Λ = diag  λ1λ2 , … , λn  is the eigenvalue matrix and Ψ =  ψ 1ψ 2 , … ,ψ n  is the eigenvector   matrix, where λi and ψ i , i   =  1, 2, . . . , n , are the ith eigenvalue and its corresponding eigenvector, respectively. The system state vector can then be represented as a superposition of the motion-modes through modal transformation as x ( t ) = Γp ( t ) (5.21)

where p ( t ) is the modal coordinate vector. As Γ is a known matrix determined by a given system, the modal coordinate vector p ( t ) can be obtained using the least-square method based on the measured state vector x ( t ). Once the modal coordinate p ( t ) is obtained, we can identify the interested motion-mode pi ( t ) and can transfer its modal coordinate into physical coordinate as  di ( t ) di ( t )  , i = 1,2, … , n, where di ( t ) = real (ψ i pi ( t )) , di ( t ) = real (ψ i λi pi ( t )), and di ( t ) can be further written as  d qi ( t ) dui ( t )  referring to zm ( t ) defined in equation (5.4). In fact,  di ( t )  di ( t )  defines a virtual sub-system in a physical coordinate frame corresponding to this motion-mode, where the energy in this sub-system does not exchange with others according to the orthogonal property of modal transformation. In each sub-system, its energy is continuously converted between potential energy and kinetic energy, and decays due to the damping effect. The kinetic energy eki ( t ) and potential energy epi ( t ) stored in the ith mode can be derived as eki ( t ) =

epi ( t ) =

1 M m di2 ( t ) (5.22) 2

T 1  L d qi ( t ) − dui ( t ) H 2  dui ( t ) − w i ( t ) 

  , i = 1,2, … , n (5.23)  

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Interconnected Suspension Control

 Ks 0  Where H =   and w i ( t ) is the projection of the road input in the same coordinate K t   0 frame with the vehicle motion-mode, of which derivation follows the same principle of modal transformation and is omitted here for brevity. The sum of the energy for each motion-mode is defined as

ei ( t ) = eki ( t ) + epi ( t ) (5.24)

After obtaining the energy for each motion-mode, the mode energy contribution ratio can be calculated as

ηi ( t ) =

ei ( t ) (5.25) E (t )

where E ( t ) represents the sum of the energy from all motion-modes, which is written as n

E (t ) =

∑e (t ) (5.26) i

i =1

Note that ηi ( t ) is a function of time and is calculated at every time instant. The energy contribution ratio of the ith motion-mode will be used for the determination of the priority of each control configuration. As we mainly consider bounce, pitch, and roll three motion-modes, we only need to calculate the energy for three body motion-modes and calculate their corresponding energy contribution ratio. Furthermore, to reduce the effect of discontinuity when switching between different configurations on ride comfort, a soft switching strategy will be applied. This strategy is implemented by linearly interpolating the switched controllers as

K = η1 ( t ) K1 + η2 ( t ) K 2 + η3 ( t ) K 3 (5.27)

where η1 ( t ) + η2 ( t ) + η3 ( t ) = 1, thus the stability of the system can be guaranteed [23] and the impulse effect on the transient response is reduced. The overall control system block diagram is shown in Figure 5.2.

FIGURE 5.2  Block diagram of the switched control system structure.

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Vibration Control of Vehicle Suspension Systems

FIGURE 5.3  Diagram of the reconfigurable hydraulically interconnected suspension.

5.2.1.2.4  System Implementation As mentioned before, a possible implementation of the proposed control strategy is for the interconnected suspension. A hydraulically interconnected suspension system was designed and assembled onto a Ford Territory sport utility vehicle in the Dynamics Laboratory of University of Technology, Sydney. This suspension system consists of four double-direction hydraulic actuators which are hydraulically interconnected and powered by an electric pump. The diagram of this system is shown in Figure 5.3. The interconnecting circuits are reconfigurable and can be controlled by a compact manifold fitted with a group of directional valves. The circuits’ pressure is controlled by a pressure control valve. When the system is in operating, it is able to re-configure its circuits in accordance with the required control mode, such as anti-roll control. Conceptual design and verification of this suspension system have been conducted in [24–26]. This suspension is supposed to be adopted to install the above-designed control system and test its performance in practice. In principle, once the energy level of the dominant vehicle motion-mode reaches the threshold value, the mode selection block will switch the circuits to the corresponding configuration and the pressure control block will provide the required control force in the circuits to achieve the designed control performance. 5.2.1.3  Simulation Results The effectiveness of the proposed suspension control system for improving ride comfort is evaluated by numerical simulations. The parameters for the full-car suspension model are listed in Table 5.1 [27]. When choosing α = 305 and β = 2  by a trial-and-error method, the controller gains are obtained by solving the minimization problem of equation (5.19) as

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TABLE 5.1 Parameter Values of the Full-Car Suspension Model Parameter

Value

Unit

Parameter

Value

Unit

ms Iθ Iφ muf mur Lf lr

1400 2100 460 40 40 0.96 1.44

kg kgm2 kgm2 kg kg m m

csf csr ksf ksr ktf, ktr tf tr

1000 1100 23,500 25,500 190,000 0.71 0.71

Nm/s Nm/s N/m N/m N/m m m

 1.0046 K1 = 10 5 ×   0.6836

0.0458 0.2624

0.0000 0.0000

−0.6214 0.0011

−0.6214 0.0011

−0.3556 0.0000

−0.3556  , 0.0000 

 −0.0342 K 2 = 10 5 ×   0.1906

1.2056 0.7903

0.0000 0.0000

−0.5162 0.0007

−0.5162 0.0007

0.4607 −0.0006

0.4607  , −0.0006 

 0.0000 K 3 = 10 4 ×   0.0000

0.0000 0.0000

7.5983 2.7902

−4.5759 −0.0002

4.5759 0.0002

−4.5759 0.0044

4.9030  . −0.0044 

It is noted that many control approaches have been proposed for linear suspension systems, such as H∞ control, linear-quadratic regulator/linear-quadratic-Gaussian control, and intelligent control including fuzzy control and neural network control. Particularly, H∞ control has been paid more attention in the last two decades due to its robustness and its flexibility in dealing with multiobjective control problem. However, most of these approaches were proposed for one suspension configuration and the controller designed for one configuration will be used to deal with multiple control objectives. To show the advances of the proposed control strategy in dealing with multiple control objectives, a comparison between the switched control and the H∞ control, which is effective and often used in the literature for suspension control [28–30], will be conducted in this section. The H∞ controller is designed by using the similar optimization procedure as defined in equation (5.19). However, in this case, four control forces as defined in equation (5.4) with the Lm matrix, and three main control objectives as defined by equations (5.7)–(5.9) as well as the control objectives of equations (5.10) and (5.11) will be included in the controller design procedure. As three accelerations (displacement, roll angle, and pitch angle) are simultaneously involved in the control objective vector when designing the H∞ controller, which is different from the switched control where only one acceleration is involved for each sub-system, the values of α   =  10  and β   =  1 are chosen by trial and error such that the H∞  control can achieve a compatible performance to the switched control for fair comparison. The designed H∞ controller gain matrix is obtained as K = 10 4 × [ K z   K z ]

where

  Kz =    

−1.0226 −1.0226 −1.3367

−0.5289 −0.5289 1.6503

−0.3877 0.3877 −2.1138

0.0092 0.9424 0.6358

0.9424 0.0092 −0.6311

−0.3190 0.5140 1.0988

0.5140 −0.3190 −0.2111

−1.3367

1.6503

2.1138

−0.6311

0.6358

−0.2111

1.0988

  ,   

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Vibration Control of Vehicle Suspension Systems

  K z =    

0.2793 0.2793 0.2036 0.2036

0.3230 0.3230 −0.3146 −0.3146

0.2542 −0.2542 −0.0723 0.0723

−0.0228 0.0255 −0.0227 0.0227

0.0255 −0.0228 0.0227 −0.0227

−0.0238 0.0240 −0.0208 0.0240

0.0240 −0.0238 0.0240 −0.0208

  .   

In the simulation, the typical bump road profile will be used to test the system performance. The ground displacement for an isolated bump in an otherwise smooth road surface is given by

 a l  2πv0   t  , 0  t    1 − cos   v l 2 0 (5.28) zr ( t ) =  l  t> 0, v0 

Interconnected Suspension Control

261

FIGURE 5.4  (a) Bump road input of the first case; (b) energy distribution ratio of passive suspension of the first case; (c) energy distribution ratio of active suspension of the first case; (d–f) bump response of sprung mass accelerations of the first case.

motion-mode is dominant. This is coincident with the fact that vehicle motions are actually coupled and cannot be explicitly separated, and hence, the control effort of pitch motion also takes effect in the bounce motion control. In addition, no big impulse disturbance is observed from Figure 5.4d and e when switching between two control modes by using the soft switching strategy.

262

FIGURE 5.5  Bump response of the first case.

Vibration Control of Vehicle Suspension Systems

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263

FIGURE 5.6  Force output under bump road input of the first case.

By comparing the switched control and the H∞ control from Figures 5.4–5.6, it is found that both control approaches achieve similar performance on sprung mass vertical acceleration, where the reductions 65.21% and 64.66% are calculated in terms of the maximum peak value for the switched control and the H∞ control, respectively. However, for the pitch acceleration, the switched control achieves more reduction (53.13%) than the H∞ control (44.50%). This is because the pitch motion is identified as dominant motion-mode at early beginning and more control effort is therefore applied by the switched control. This can be clearly observed from Figure 5.6 that, before the rear wheels touch the bump, the rear forces for the switched control already become non-zeros while they are still zeros for the H∞ control. We now consider the second case that the road disturbances to the left and right wheels are applied with different amplitudes to excite the roll motion of the vehicle. [31] In this case, the road input is shown in Figure 5.7a. The energy contribution ratio of each motion-mode for the passive suspension and the designed active suspension under this road input is shown in Figure 5.7b and c. It can be seen from these figures that the ratio for the roll motion-mode is not zero because the roll motion is excited when applying different bump height values to the left and right wheels. The bump responses of the sprung mass vertical acceleration, pitch acceleration, and roll accelera­tion for the passive suspension and the designed active suspensions are compared from Figure 5.7d–f, respectively. As the roll motion is excited under this road input, roll accelerations for both passive suspension and active suspensions are not zero. It can be seen that an improved

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Vibration Control of Vehicle Suspension Systems

FIGURE 5.7  (a) Bump road input of the second case; (b) energy distribution ratio of passive suspension of the second case; (c) energy distribution ratio of active suspension of the second case; (d–f) bump response of sprung mass accelerations of the second case.

performance on ride comfort in terms of the maximum peak values of vertical, pitch, and roll accelerations is obtained by the designed active suspensions when compared to the passive suspension. Similarly, the comparisons on suspension deflections and tyre dynamic loads are shown in Figure  5.8, from which we can see that the designed active suspensions generate less or similar suspension deflections and tyre dynamic loads to the passive suspension due to the consideration of these constraints in the controller design process. The generated control forces on four wheels are

Interconnected Suspension Control

FIGURE 5.8  Bump response of the second case.

265

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Vibration Control of Vehicle Suspension Systems

FIGURE 5.9  Force output of the second case.

shown in Figure 5.9. Due to the soft switching of the controllers among the detected three motionmodes, the generated control forces are different for all of the wheels. By comparing the switched control and the H∞ control from Figures 5.7–5.9, it is found that both control approaches also achieve similar performance on sprung mass vertical acceleration, where the reductions 63.37% and 64.66% are calculated in terms of the maximum peak value for the switched control and the H∞ control, respectively. However, for the pitch acceleration, the switched control achieves more reduction (50.67%) than the H∞ control (44.50%). In particular, for the roll acceleration, the switched control achieves much more reduction of 41.51% than the H∞ control, which is only about 6.68%. In addition, the peak control forces required by the switched control are less than those of the H∞ control as shown in Figure 5.9. From the comparison results, we can see that the H∞ control may achieve good performance on one control objective, but cannot guarantee to achieve good performance on multiple control objectives as it has to balance different control objectives. On the contrary, the switched control can switch the control objective in terms of the dominant motion-mode, and therefore can achieve good performance on all of the control objectives under the similar suspension deflection and tyre deflection constraints. This control strategy is more suitable for multi-objective control problem. 5.2.1.4 Conclusions In this subsection, a switched control strategy is applied to the vehicle suspension control. The switching of the controllers is based on the detected dominant motion-mode. Three-vehicle body motion-modes: bounce, pitch, and roll, are considered. The identification of motion-mode is achieved by calculating the energy of each motion-mode. This control strategy is applicable to the

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Interconnected Suspension Control

interconnected suspension, which is re-configurable to different motion-modes. Numerical simulations are used to validate the performance of the designed active suspension. The results show that the switched control strategy can effectively improve vehicle ride comfort performance. This motion-mode based switched control strategy can possibly lead to a reduced power consumption by focusing on the dominant vehicle dynamic issue and setting motion-mode as control target instead of motion. This, however, needs to be further studied and compared. In addition, experimental implementation and validation of the proposed control strategy through the hydraulically interconnected suspension will be conducted in the near future.

5.3  ELECTROMAGNETICALLY INTERCONNECTED CONTROL 5.3.1 Introduction of EIS This section introduces the motivations of this section from the research of interconnected suspension, electromagnetic suspension, and force-current analogy. Then, an interconnected suspension system with a controllable EN is proposed. 5.3.1.1 Motivations The traditional interconnected suspensions apply mechanical, hydraulic or pneumatic networks; they can help to specify heave, pitch, roll and warp dynamics independently, and hence, it has excellent performance. The HIS, which has two hydraulic cylinders, shows the advantage of the interconnected suspension for the anti-roll application [1]. The two hydraulic cylinders of independent suspensions are in the roll plane of a vehicle. With a designed hydraulic circuit, the roll stiffness of the system can increase significantly, which can obstruct the roll motion of the vehicle to improve the anti-roll function. The HIS can transfer the vibration energy into hydraulic energy, and the energy is interconnected with a hydraulic circuit. However, there are drawbacks with HIS. The seal and maintenance of the hydraulic fluid will undoubtedly increase the cost. Also, though the hydraulic line can be curved to fit in the vehicle, it will require long lines for heavy duty vehicles which have a large dimension. Physical disadvantages are inevitable for the conventional interconnected suspensions. The force-current analogy is one of the mechanical-electrical analogies, in which the ideas of through and across variables are introduced [11]. One benefit of the force-current method is that the EN analysis methods, the mesh and nodal analyses, can be applied in mechanical system designs. With the force-current analogy, the following correspondences can be set up: force ↔ current     velocity ↔ voltage mechanical ground ↔ electrical ground

spring ↔ inductor damper ↔ resistor

inerter  ( mass ) ↔ capacitor Electromagnetic suspension, which applies an electromagnetic device to generate damping force or harvest energy, has been widely studied for vehicles. The mechanical part of the electromagnetic suspension consists of a motor and a ball screw, and it also has a circuit in its electrical part.

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Vibration Control of Vehicle Suspension Systems

The  ball  screw can translate the linear motion of the suspension to the rotational motion of the motor. Hence, the vibration energy is transformed into electrical energy through the motor. The motor is an intermedia between the mechanical and electrical systems. We can use the electromagnetic suspension to validate the force-current analogy, as the usage of the resistor, inductor and capacitor in the electrical part of the electromagnetic suspension can generate the mechanical characteristics of the damper, spring, and inerter, respectively [32,33]. As the electrical device is easy to fit into vehicles, it is more feasible than the mechanical system. 5.3.1.2  Electrically Interconnected Suspension We propose a novel EIS where the electromagnetic suspensions are interconnected by the EN instead of mechanical, hydraulic, or pneumatic ones to overcome the drawbacks of existing interconnected suspensions and provide new advantages in controllability, flexibility, response speed, structure complexity, maintenance, and cost, et al. In this suspension, the vibration energy is transformed into electrical energy through independent electromagnetic suspensions. The EN can determine the characteristics of the system, and can be designed to fulfil the performance requirements based on the force-current analogy. Figure 5.10 shows an EIS system in roll plane where two independent electromagnetic suspensions El and Er are interconnected by an EN, and the EN can be controlled by a controller. This simplified EIS system has two DOFs, the heave zs and roll θ s , without considering wheels. It is noted that, multiple electromagnetic suspensions can be interconnected to form a more complicated system with more DOFs. zls and zrs are the left and right sprung mass displacements, respectively. Fl and Fr are two vertical force exerted on the sprung mass by the two electromagnetic suspensions. l is the distance of the left and right suspensions to the centre of sprung mass. Theoretically, the heave and roll characteristics of the EIS system can be determined by the EN and can be controlled by varying certain elements, which will be discussed in the following sections. The EIS and HIS have correspondences in terms of the energy flow. The electric current is equivalent to hydraulic volume flow rate, and the electric potential is equivalent to hydraulic pressure. Though the vibration energy is transformed to electric one instead of hydraulic, the two interconnected suspensions have many similarities.

5.3.2 EIS System Design and Analysis In this section, the electrical impedance method is used to model the EIS system; two impedance network topologies are proposed and analyzed.

FIGURE 5.10  Schematic of a semi-active half-car EIS system.

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Interconnected Suspension Control

5.3.2.1  A Two-DOF EIS System The characteristics of the two-DOF EIS system shown in Figure 5.10 are determined by the equivalent force and torque of the two independent electromagnetic suspensions exerted on the sprung mass. The total vertical force to the sprung mass is

F = Fl + Fr (5.29)

The torque about the mass centre is

T = − Fl l + Fr l (5.30)

Assuming that the roll angle θ s is small enough, the displacements of the two sides of sprung mass are given by

zls = zs − lθ s (5.31)

zrs = zs + lθ s

(5.32)

The ball screws in the electromagnetic suspensions can translate the suspensions’ linear motions to the rotary motions. Hence, the rotary angles of the left and right motors are

α l = rg zls (5.33)

α r = rg zrs (5.34)

2π , ld is the lead of the applied ball screw. ld The application of ball screws could introduce nonlinearities, such as the backlash has an influence on the reciprocating rotation of the motor and the friction could exert nonlinear force on the system. Hence, these nonlinearities will affect the performance of the EIS. To deal with these possible nonlinearities, one way is to regard them as external disturbances and design a robust controller to achieve satisfying performance under the effect of the disturbances; another way is to include the models of the nonlinearities into the system and design an appropriate nonlinear controller to deal with them. In this chapter, all of these possible nonlinearities are regarded as disturbances and H_∞ controller will be designed to deal with these nonlinearities. In addition, ball screws can amplify the torque from the moment of inertia of the motor rotor. This effect is also regarded as a disturbance force. where rg =

5.3.2.2  Two-Port Impedance Network Impedance, which is a complex quantity, can describe the network’s frequency characteristics. Admittance is the reciprocal of impedance. For single mechanical and electrical elements, their impedances are shown in Table 5.2 where c, k, and b represent mechanical damping, stiffness, and inertance, respectively; R, L, and C represent electrical resistance, inductance and capacitance, respectively; ω is the angular frequency; j is the imaginary unit. The EN is a complicated circuit composed of resistors, inductors, and capacitors, which can be taken as a two-port impedance network as shown in Figure 5.11a. The electromagnetic suspension is generally modelled as a voltage source, an inner resistance Ri and an inner inductor Li which are

270

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TABLE 5.2 Impedance and Admittance of Passive Elements Mechanical (Electrical) Element c (R) k (L) b (C)

Impedance 1 Z =  ( Z = R ) c jω Z=  ( Z = jω L ) k Z=

Admittance 1 Y = c    Y =   R Y = k   Y = 1  jω L  jω 

1  1  Y = jω b  (Y = jω C )   Z= jω b  jω C 

FIGURE 5.11  (a) Two-port network; (b) topology one; (c) topology two; (d) mesh analysis for EN Topology One.

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connected in series. zm is the impedance of the Ri and Li . el and er are the induced voltages by the left and right electromagnetic devices, respectively; il and ir are corresponding currents. The electromagnetic device-induced voltages are proportional to the rotary rate of the motors. Hence, the generated voltages are ˙

el = ke zls (5.35)

er = ke zrs

˙

(5.36)

where ke = rg ki, ki is the torque constant of motors. The forces generated by the two electromagnetic devices are

Fl = − keil

(5.37)

Fr = − keir

(5.38)

The two-port network is an EN with two separate ports for input and output. For the two-port network, EN, the system can be described as

el − zm il = z11il + z12ir

(5.39)

er − zm ir = z21il + z22ir

(5.40)

where z11 and z22 are driving-point impedances of the two-port network; z12 and z21 are transfer impedances. When the two-port network is symmetrical, z11 = z22; and when the EN is linear and has no dependent sources, z12 = z21. There is no dependent source in the EN, and we assume that the EN is symmetrically designed. Then, the physical property of the EN is describable with only two parameters, z11 and  z12 . We can assume that EN is a black box first. Therefore, according to (5.29) and (5.40), the equivalent force and torque of the EIS exerted on the sprung mass are ˙

2 ke 2 z s F=− z11 + zm + z12

T =−

(5.41)

˙

2 ke 2l 2 θ s z11 + zm − z12

(5.42)

Hence, by finding the driving-point impedance z11 and the transfer impedance z12 of the EN, the frequency domain characteristics of the F and T can be defined. 5.3.2.3  Example Analysis of EN Topologies As we can use two impedances to describe any no dependent source and symmetrical EN composed of resistors, inductors, and capacitors, there are many applicable designs. We will introduce two EN topologies as examples to show the design method. The two examples are classic EN in the electrical system study. The suspension has two-DOF of motions; hence, we assume that the EN

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is composed of two kinds of controllable branch circuits with impedances z1 and z2, respectively. Because it will be easy to design the controller if the number of control variables is matching with the number of output variables. Topology One is shown in Figure 5.11b, where the EN consists of two z1 and two z2. According to the knowledge of two-port impedance networks, the driving-point and transfer impedances can be evaluated by setting il = 0 and ir = 0, respectively; it means that one of the input ports is opencircuited. The impedances can be calculated by

z11 =

el |ir = 0 il

(5.43)

z12 =

el |il = 0 ir

(5.44)

Therefore, the driving-point impedance z11 and the transfer impedance z12 of this EN topology can be obtained as

z11 =

z1 + z2 2

(5.45)

z12 =

− z1 + z2 2

(5.46)

Substituting (5.45) and (5.46) into (5.41) and (5.42), we have ˙

2 ke 2 z s zm + z2

F1 = −

2 k 2l 2 θ s T1 = − e zm + z1

(5.47)

˙

(5.48)

It means that the force and torque of the two electromagnetic suspensions to the sprung mass are decoupled with the branch circuits z2 and z1. The heave and roll characteristics of the system can be independently controlled. Figure 5.11c shows another EN topology, Topology Two, which is composed of one z1 and two z2. The driving-point impedance z11 and transfer impedance z12 of the EN are derived as

z11 =

z1z2 + z2 2 2 z2 + z1

(5.49)

z12 =

z2 2 2 z2 + z1

(5.50)

Substituting (5.49) and (5.50) into (5.41) and (5.42) yields ˙

2k 2 z F2 = − e s zm + z2

(5.51)

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Interconnected Suspension Control ˙

2 ke 2l 2 θ s T2 = − zz zm + 1 2 2 z2 + z1

(5.52)

z z1z2 is the impedance of z2 and 1 in parallel. 2 2 z2 + z1 Hence, with the EN Topology Two, the vertical force of the two electromagnetic suspensions to the sprung mass is determined by the branch circuit z2, while the toque is controlled by z2 and z1 together. The force F and torque T are corresponding to the heave and roll characteristics of the suspension, respectively. Therefore, the impedance analyses, which describe the F and T in the frequency domain, can reveal the damping, stiffness, and inertance features of the suspension in a certain DOF. Theoretically, designing proper branch circuits with resistors, inductors, and capacitors, we can characterize the EIS system in the heave and roll DOFs. The two examples show that the different EN topologies can be implemented according to practical applications. Though the EN Topology Two requires one less branch circuit, the EN Topology One is more comfortable to be controlled as it can decouple the roll and heave characteristics of the suspension by the circuit.

where

5.3.3 System Verification In this section, the specific branch circuits of z1 and z2 are designed, then the system with EN Topology One is evaluated in the frequency domains and time domain. 5.3.3.1  Branch Circuits The branch circuits of zm , z1 and z2 are identified first. zm consists of a resistor and an inductor, which are connected in series. z1 is a variable and an inductor, which are also connected in series. z2 is a variable resistoe. Their impedances are Ri + Liω j, RL + Leω j and Re , respectively. The branch circuits z1 and z2 are designed based on the knowledge of vehicle suspensions. Commonly, variable damping device is applied for the heave vibration control of suspensions, thus, a variable resistor Re can be used in branch z2 to emulate a variable damper according to the force-current analogy [34]. For the suspension’s roll dynamics, the controllable stiffness is generally considered beneficial. The inductor is able to emulate a spring in mechanical network, however, it is hard to tune an inductor in real time as it is an energy storage element. Inspired by the mechanical variable equivalent stiffness device design, the branch z1 is designed as a variable resistor RL  connected with a inductor Le in series; by changing the value of RL , the equivalent inductance of the branch circuit is controllable. Similarly, z1 and z2 can be other kinds of configurations when different suspension characteristics are required, such as the capacitor can be applied to simulate a mechanical inerter. 5.3.3.2  Experimental Setup Generally, the MR fluid damper is effective to the low-frequency vibration control, which is due to that the device needs a response time to vary its damping. The performance of the seat suspension with high inertance and low inertance has an intersecting frequency f0 which can be found in the vibration transmissibility experiment. When the vibration frequency is lower than f0 , the high inertance is preferred. On the contrary, the low inertance can lead to a better performance. Therefore, we can apply the modified on-off controller when the vibration frequency is lower than f0 , and keep 0 A current when there is a higher frequency vibration. An experimental setup is applied to verify the EIS system. Two direct current (DC) motors (Maxon 353301) are rotated by two servo motors (Panasonic, 400 W) to simulate two electromagnetic suspensions’ heave and roll motions. Two inductors are customized for the EN. Variable resistors are manually tuned in the experiments; they can be easily changed to digitally controllable

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resistors for a real-time control system. An NI Compact Rio 9074 is the control unit which can control the motions of servo motors, record the actual rotary angles of motors, and measure the currents and voltages in the EN. According to equations (5.29)–(5.38), the equivalent force in heave and the torque in roll of the two-DOF EIS system can be obtained by the measured currents; the heave displacement and roll angle can be calculated by the measured rotary angles of motors. When the heave motion of the EIS system is simulated, the two servo motors are controlled to do in-phase rotations

α l = α r = 200° sin ( 4πt )

(5.53)

When the roll motion is simulated, the two servo motors are controlled to do reverse-phase rotations

α l = 200° sin ( 4πt )

(5.54)

α r = 200° sin ( 4πt + π )

(5.55)

The parameters of the two-DOF EIS system are shown in Table 5.3 where Ri0 is the nominated inner resistance of the applied DC motors. Considering the resistance of cables in the circuit and the inner resistance of inductors, the resistances in the system are amended as

Ri = Ri 0 + 0.2 Ω

(5.56)

Re = Re 0 + 0.2 Ω

(5.57)

RL = RL 0 + 0.9 Ω

(5.58)

where Re0 and RL 0 are the variable resistances in the branches z2 and z1, respectively. 5.3.3.3  Frequency Domain Performance With the practical branch circuits, the frequency domain verification of the EIS system is implemented first. According to (5.47), the system admittance in the heave of the two-DOF EIS system with EN Topology One can be written in the frequency domain as

Yh =

 2 ke 2 Lω2  R + Re + i  2 2 2  i Ri + Re + Li ω  ωj 

(5.59)

Hence, it is equivalent to that a damper and a spring are connected in parallel. The equivalent damping of the suspension in heave is the real part

2 ke 2 ( Ri + Re ), and the equivalent stiffness Ri + Re 2 + Li 2ω 2

TABLE 5.3 Parameters for the Two-DOF EIS System Ri0 1.5Ω Le 0.22 H 0.000644 H 0.016 m Li ld 0.245 Nm/A 0.8 m ki l

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is

2 ke 2 Liω 2 . The equivalent damping is shown in Figure 5.12a, which can vary from about Ri + Re 2 + Li 2ω 2

356 to 6384 Ns/m when Re0 is changed from 50 to 1 Ω. Figure 5.12b shows that the equivalent stiffness is small at low frequency, and the stiffness caused by the inner inductor Li is ignorable except when Re0 is 1 Ω. Similarly, based on (5.48), the system admittance in roll can be written in frequency domain

Yr =

 2 k e 2l 2 ( L + Le )ω 2  R + RL + i 2  2  i ωj Ri + RL + ( Li + Le ) ω  2

Hence, the equivalent damping of the suspension in roll is

(5.60)

2 ke 2l 2 ( Ri + RL ), and 2 Ri + RL + ( Li + Le ) ω 2 2

2 ke 2l 2 ( Li + Le )ω 2 . 2 Ri + RL 2 + ( Li + Le ) ω 2 In Figure 5.12c, the equivalent stiffness in roll is presented; it is increased with the decrease of RL 0 , and it is frequency dependent; the maximum roll stiffness can reach about 50,000 Nm/rad. The equivalent damping in roll is shown in Figure 5.12d; it is decreased with the increase of frequency.

the equivalent stiffness is

5.3.3.4  Time Domain Performance The mesh analysis is one of the general procedures for analyzing circuits, and it is applied to the EN Topology One. In Figure 5.11d, the circuit for the EN Topology One is presented. The mesh currents i1, i2 and i3 are assigned to the three meshes. Kirchhoff’s voltage laws (KVL) is applied to the meshes, then using Ohm’s law to express the voltages in terms of the mesh currents yields:

FIGURE 5.12  (a) Equivalent damping in heave; (b) equivalent stiffness in heave; (c) equivalent stiffness in roll; (d) equivalent damping in roll.

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−el + Ri i1 + Li i1 + RL ( i1 − i3 ) + Le ( i1 − i3 ) + Re ( i1 − i2 ) = 0

(5.61)

er + Li ( i2 − i3 ) + Ri ( i2 − i3 ) + Re ( i2 − i1 ) + Lei2 + RL i2 = 0

(5.62)

−er + Li ( i3 − i2 ) + Ri ( i3 − i2 ) + Rei3 + RL ( i3 − i1 ) + Le ( i3 − i1 ) = 0

(5.63)

i1 = il

(5.64)

i3 − i2 = ir

(5.65)

By rearranging (5.61)–(5.65), the dynamics of the EN can be obtained as:

 Li + Le − ( Li + Le ) L m1 =  Li Li   el   1 −1  E m1 =  ,E= .   er   1 1  where

˙

L m1 I m = R m1I m + E m1E  , 

 il  Im =  ,  ir 

(5.66)  −( Ri + RL ) Ri + RL  R m1 =  ,  −( Ri + Re ) −( Ri + Re ) 

The heave motion of the EIS system is tested first. In Figure 5.13a, by setting the RL 0 as 50 Ω and varying the values of Re0, the experimental and simulation results of the force-velocity graph of the two-DOF EIS system are presented, which indicates that the system heave damping is increased with the decrease of Re0 and the model (5.66) can accurately describe the system. The damping varies from 358 to 6382 Ns/m, which can match the result in frequency domain analysis. By setting Re0 as 5 Ω and varying the values of RL 0 , the experimental results are shown in Figure 5.13b. The results are almost overlapped with different values of RL 0 . The two tests indicate that the heave damping of the suspension can be only controlled by Re0. The roll motion test of the EIS system is implemented. By setting Re0 as 50 Ω and varying the values of RL 0 , the experimental and simulation results are shown in Figure 5.13c. The experimental and simulation results are well matched except when RL 0 = 1 Ω, which may be caused by the nonlinearity of the inductor at low frequency. However, both results show that the roll stiffness is increased with the decrease of RL 0 , as the slopes of the figures represent the equivalent stiffness. By setting RL 0 as 5 Ω and varying the values of Re0, the results in Figure 5.13d are almost overlapped, which indicates that Re0 has very little influence on the roll characteristics. In theory, Re0 and RL 0 can be thoroughly decoupled to control the heave and roll characteristics of the suspension, respectively. However, it is hard to build a totally symmetric circuit, in which, hence, the small differences in Figure 5.13b and d are inevitable. The two tests imply that the frequency and time domain analyses for the EN Topology One are correct; they can be applied for the controller design in the future. In the practical application, the inductor and capacitor may be nonlinear, which can be solved by building a nonlinear model or designing robust controllers to reduce the impact.

5.3.4 Vibration Control of Half-Car EIS System In this section, we apply a half-car model to verify the effectiveness of vibration control with the EIS system at both frequency domain and time domain.

277

Interconnected Suspension Control

FIGURE 5.13  (a) Force-velocity graph of the EIS system in heave with RL 0 = 50 Ω; (b) force-velocity graph of the EIS system in heave with Re 0 = 5 Ω; (c) torque-angle graph of the EIS system in roll with Re 0 = 50 Ω; (d) torque-angle graph of the EIS system in roll with RL 0 = 5 Ω.

5.3.4.1 Half-Car Model with EIS System Figure 5.10 shows the half-car EIS system with EN Topology One, where ms and mu are the sprung and unsprung masses, respectively; I s is the roll moment of inertia about the centre of sprung mass; ks and kt are the suspension and tyre stiffness, respectively; zlr and zrr are the left and right terrain height displacements, respectively; zlr and zrr can synthesize to heave zr  and roll θ r vibrations; zlu and zru are the left and right unsprung mass displacements, respectively. In the practical application, the friction of the ball screw and the moment of inertia of the motor rotor can also affect the system dynamics. The friction in the left suspension is modelled as:

(

)

(

)

˙ ˙ ˙ ˙   frl = cs zls − zlu + sat  c f 0 zls − zlu     

(5.67)

where the damping cs is in the linear part of the friction model; in the nonlinear part, the saturation function sat() is in [ − f0  f0 ]; c f 0 and f0 are two parameters of the friction. The ball screw can amplify the rotation of the motor rotor and its output torque; the ball screw and the motor rotor are equivalent to an inerter device with inertance be = rg 2 I 0 , where I 0 is the moment of inertia of the rotor. Hence, the output force of the inerter in the left suspension is:

(

¨

¨

)

fil = be zls − zlu

(5.68)

For the convenient of model building, we define the disturbance force of the left suspension as:

278

Vibration Control of Vehicle Suspension Systems

(

)

˙ ˙   fl = fil + sat  c f 0 zls − zlu     

(5.69)

Similarly, we can define the corresponding forces on the right side, they are frr , fir , and  fr . The state variables of the system are selected

as

T

˙ ˙ ˙ ˙  X =  zls − zlu zls zlu − zlr zlu zrs − zru zrs zru − zrr zru  . Hence, the half-car suspension   model can be built as:

 = AX + B1d 0 + B2 I m + B3d1 X

where

 0  −a k 1 s   0  ks   mu A= 0  − a 2 ks   0   0 

1 − a1cs 0 cs mu 0 − a2cs 0

0 0 0 k − t mu 0 0 0

−1 a1cs 1 c − s mu 0 a2cs 0

0

0

0

0 − a2 ks 0

0 − a2cs 0

0 0 0

0 a2cs 0

0

0

0

0

0 − a1ks 0 ks mu

1 − a1cs 0 cs mu

0 0 0 k − t mu

−1 a1cs 1 c − s mu

  0 − a1ke B2 =    0 − a2 ke 

T

 0 0 −1 0 0 0 0 0  B1 =   ,  0 0 0 0 0 0 −1 0 

(5.70)

0

ke mu

0

0

       ˙   zlr  , d0 =  ˙   zrr     

 ,  

T

0 − a2 ke

0

0

− a1ke

0

ke mu

0

   ,   

T

  1 1 0 0 0 0 0   0 2 2 ms ms 1 l 1 l  , and d =  − fl − fr a1 = + , a2 = − , B3 =  1 2 2  ms I s ms I s l l  fl − fr 0 0 0 0 0   0 − I I s s   The left and right terrain height displacements can be represented as: d 0 = Dmd r

 . 

(5.71)

 ˙   1 −l  zr where d r =  ˙ , D m =  . zr is the heave vibration displacement, and θ r is the roll angle 1 l   θ  r  of vibration. According to (5.35) and (5.36), the generated voltages of the left and right electromagnetic suspensions are: E = CX

 0 ke where C =   0 0

0 − ke 0 0

0 0 0 ke

0 0 0 − ke

 . 

(5.72)

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Interconnected Suspension Control

 ˙  zs The sprung mass has two degrees of freedom which is represented as Z 0 =  ˙ . Hence, we  θ s  have: D mZ 0 = Ds X

 0 where D s =   0

1 0

0 0

0 0

0 0

0 1

0 0

(5.73)

0  . 0 

5.3.4.2  Vibration Transmissibility at Frequency Domain For investigating the frequency properties of the suspension, we only apply the linear part of the whole suspension model:  = AX + B1D m d r + B2 I m      X ˙

L m1 I m = R m1I m + E m1CX

Z 0 = D m −1D s X

(5.74) (5.75) (5.76)

According to the Laplace domain description of (5.74) and (5.75), we have:

−1

−1 X =  Is − A − B2 ( L m1s − R m1 ) E m1C  B1D m d r

(5.77)

where X and d r are the Laplace transforms of X and dr; I is an identity matrix Substituting (5.77) into the Laplace domain description of (5.76), we have:

Z 0 = Tm d r

(5.78)

−1

−1 where Tm = D m −1D s  Is − A − B2 ( L m1s − R m1 ) E m1C  B1D m; Z0 is the Laplace transform of Z0.  ˙   ˙  zs zr Tm is a 2 × 2 matrix, which contains the vibration transmissibility from  ˙  to  ˙ . Hence,  θ r   θ s  the transmissibility of the heave vibration to the heave of sprung mass is:

Th = Tm (1, 1)

(5.79)

where Tm (1, 1) represents the number in the first row and first column of the matrix Tm. The same representation method is used in the following content. The transmissibility of the roll vibration to the roll of sprung mass is:

Tr = Tm ( 2, 2 )

(5.80)

The transfer vibration transmissibilities, Tm (1, 2 ) and Tm ( 2, 1)are not presented in this chapter, as they are close to zeros in this decoupled suspension.

280

Vibration Control of Vehicle Suspension Systems

TABLE 5.4 Parameters of the Frequency Domain Simulation ks

20,000 N/m

mu

50 kg

cs kt

300 Ns/m 200,000 N/m

ms Is

600 kg 1222 kg⋅m 2

FIGURE 5.14  (a) Th with the variation of Re0 (RL 0 = 50 Ω); (b) Tr with the variation of RL 0 (Re 0 = 50 Ω); (c) roll acceleration; (d) vertical acceleration.

Applying the parameters in Table 5.4 to Tm, the vibration transmissibility of the half-car EIS system can be obtained. The heave vibration transmissibility with different Re0 is presented in Figure 5.14a. The value of transmissibility at the first resonance frequency is decreased with the decrease of Re0, however, the lower Re0 will cause higher vibration transmissibility at high frequencies. Hence, by applying different resistances at different frequencies, the vibration transmissibility can keep at low values at the whole frequency range as shown in Figure 5.14a. The frequency simulation implies that an appropriately adopted control strategy can greatly improve the suspension’s performance in heave. In Figure 5.14b, the roll vibration transmissibility is determined by RL 0 , and an appropriately designed control strategy can also be applied in the roll vibration control to improve the performance. The first resonance frequency of the suspension is moved with the variation of RL 0 , which indicates the change of equivalent stiffness. Besides, simulation results show that the heave dynamics are unaffected by the RL 0 control, and the variation of Re0 does not influence the roll dynamics. Though this chapter claims that the proposed system is controllable, it can also be a passive EIS system by optimizing the branch circuits to keep the suspension with acceptable performance under most of the conditions.

281

Interconnected Suspension Control

5.3.4.3  Vibration Control in the Time Domain For this semi-active system, the controlled items are two resistors, RL and Re . We can reconstruct the system model such that many advanced control strategies can be adopted.  u1  −1 We define the system’s control input as U =   = L m1 R m1I m which contains two unknown  u2   =  X . Hence, the parameters RL and Re , and I m is measurable. The whole system states are X  Im  system model is rewritten as   +B  = AX 1w + B 2 U X

(5.81)

   d0    A B2  B3   = 1 =  B1 2 =  08 × 2 ; , B where A , w =  , B −1 0 0 d 2× 2   L m1 E m1C 0 2 × 2   I   2 × 2  1   In the suspension control, the ride comfort, suspension deflection limitation and road holding ability are generally taken into account [35]. Therefore, we define the control output as   + Dw  Z = CX

 ¨ ¨  =  zs θ s where Z 

where

zls − zlu z fmax

 ks  − ms   ksl   Is  1  z  = α  fmax C  0    0     0 

cs ms cs l Is

zrs − zru z fmax

0 0

kt ( zlu − zlr ) Ff cs ms cl − s Is

ks ms kl − s Is −

(5.82)

kt ( zru − zrr )  , Ff  cs ms cl − s Is

0 0

cs ms cs l Is

ke ms ke l Is

ke ms kl − e Is

0

0

0

0

0

0

0

0

0

0

0

0

1 z fmax

0

0

0

0

0

0

kt Ff

0

0

0

0

0

0

0

0

0

0

0

0

kt Ff

0

0

0

α = diag (α 1 ,α 2 ,α 3 ,α 4 ,α 5 ,α 6 )

is

a

weighting

matrix,

  = α  02× 2 D  0 4 × 2

D0 04× 2

         ,          , 

  1 0   ms ; 0 is a zero-matrix with specified dimensions; z D0 =  fmax is the maximum suspension  l  0   I s   deflection hard limits; Ff = ( ms + 2mu ) g / 2 is static tyre load.  where K is the For the system (5.82), the H ∞ state feedback control law is given by U = KX, feedback gain matrix to be designed. We can obtain K by solving

282

Vibration Control of Vehicle Suspension Systems

(

)

     P A + B2K + * PB1 *  −γ 2 I * *     −I C D 

   0 , γ is the desired level of disturbance attenuation [36]. Pre- and post-multiplying (5.83 by diag P −1 , I, I and its transposition, respectively, and defining Q = P −1, W = KQ, we have

(

 AQ  +B 2 W + *   *   CQ 

)

1 B −γ 2 I  D

  *  σ i l r −il + ir

(5.85)

RLmax                    −il + ir ≤ σ

 Li u1 + Li u2 − Ri              −il − ir > σ  −il − ir Red =                   Remax             −il − ir ≤ σ 

(5.86)

where σ is with a small positive value to avoid that zero is in the denominator. The real control inputs of RL and Re are

RL = sat ( RLd )    ∈[ RLmin    RLmax ]

(5.87)

Re = sat ( Red )    ∈[ Remin    Remax ]

(5.88)

For evaluating the performance of the EIS suspension, we apply a passive half-car model for comparison

 = A p X + B1d 0 X

(5.89)

where A p is the same with A except that cs is replaced by c p . The parameters in Table 5.5 is applied for simulation. With α = diag (1, 1.4, 1, 1,1, 1) and γ = 10, we  −4.1 −3.4 −4.3 6.7 −5.1 −8.3 −5.8 7.3 −0.07 −0.02  have the solution of K = 10 4  .  −5.1 −8.3 −5.8 7.3 −4.1 −3.4 −4.3 6.7 −0.02 −0.07  In the real-world scenarios, the rough road surface could cause the roll and heave vibrations of vehicles at the same time. A specially designed road profile is required to validate the effectiveness of the proposed EIS system at the roll and heave vibration control.

283

Interconnected Suspension Control

TABLE 5.5 Parameters of the Half-Car Suspension I0

1.34 × 10 −4 kg⋅m 2

RLmin

0.9 Ω

cp cf 0 σ

1500 Ns/m 2200 Ns/m

RLmax Remax Remin

50.9 Ω 50.2 Ω 0.2 Ω

z fmax

0.08 m

f0

1 × 10 −3 A 60 N

We assign a roll vibration profile as:

 0.01(1 − cos ( 2πt ))        t < 1  zlr1 =      0                              t > 0

  zrr1 =   

−0.01(1 − cos ( 2πt ))     t < 1 0

t 0 is a known diagonal matrix, then for any scalar  ε > 0, we have RΦS + S T ΦRT ≤ ε RVRT + ε −1S T VS (6.24)

Theorem 7.1

Consider the active suspension system in (6.13)–(6.16) with the proposed dynamic output feedback H ∞ controller in (6.9)–(6.10). For given positive scalar d,  γ , θ R   and ρ , if there exist matrices  Q11 Q=  *

 S11 Q12   > 0, S =  Q22   * 

 M i1 Aˆ d , Bˆ , Cˆ , M i =   M i 3

S12 S22

 R11   > 0, R =   * 

 N i1 Mi 2  , N i =  N M i 4  i3 

Ni 2 Ni 4

R12   > 0 , X > 0 , Y > 0, Aˆ , R22 

   ( i = 1,2,3,4 ) with appropriate dimen

sions and any positive scalars ε1, ε 2, ε 3 such that the following LMIs hold:

  Ψ11  *   *  *   *  * 

  Ψ11  *   *  *   *  * 

dM

Ψ13

Ψ14

ε1Ψ15

−R *

0 −I

0 0

0 ε1 D1

* * *

* * *

−Ψ 44 * *

ε1Ψ 45 − ε 1I *

dN

Ψ13

Ψ14

ε 2 Ψ15

−R *

0 −I

0 0

0 ε 2 D1

* * *

* * *

−Ψ 44 * *

ε 2 Ψ 45 −ε 2 I *

 −µ2 X max   *  *   * 

2 − µmax I

ρ Cˆ T λmT

2 − µmax Y * *

0 −I + ε 2I *

  0   0  < 0 (6.25) 0   0  −ε1I  Ψ16

   0  0  < 0 (6.26)  0  0  −ε 2 I  Ψ16

Cˆ T λ T   0  < 0, (6.27) 0  −ε 2 I 

293

Suspension Control for In-Wheel Motor Driven Electric Vehicle

{

 − z2 2 max   *   *   *  * 

}

q

{ }I − {z }Y − z22max

X

2 2 max

q

ρ ( X T C2T + Cˆ T λmT D2T )

0

ρ C2T

0

q

 X  I 

 Θ11   * where Ψ11 =   *  * 

Θ12

Θ13

Θ 22

Θ 23

*

Θ33

*

*

I  > 0 (6.29) Y 

  Θ 24  , Θ34  −γ 2 I   Θ14

 AX + XAT + M + M T + Q + S 11 11 11 11   * 

 T A + Aˆ T + M12 + M13 + Q12 + S12 , T ˆ y + C yT Bˆ T + M14 + M14 YAT + AT Y + BC + Q22 + S22  

 T B2 λmCˆ + M 21 − M11 + N11 θ12 =  T  Y T B λ DC ˆ y + Aˆ d + M 22 − M13 + N13 2 m 

T T  − (1 − µ ) S11 + N 21 + N11 − M 21 − M 21   * 

T  M 31 − N11 θ13 =  T  M 32 − N13 

 N i1 N i 2 Ni =   N i 3 N i 4

T ˆ y + M 23 B2 λm DC − M12 + N12 T ˆ y + M 24 Y T B2 λm DC − M14 + N14

T T − M 33 − N 22 N 33 T T − M 34 − N 24 N 34

T T  B1 + M 41 −Q12 − N 33 − N 32   , θ14 =  T T −Q22 − N 34 − N 34   Y T B1 + M 42 

T   − N 41 , θ 34 =  T  − N 42 

   

T T − (1 − µ ) S12 + N 23 + N12 − M 23 − M 22  , T T − (1 − µ ) S22 + N 24 + N14 − M 24 − M 24  

T T T  N 31 − M 31 − N 21 M 33 − N12   , θ 23 =  T T T  N 32 − M 32 − N 23 M 34 − N14   

T  −Q11 − N 31 − N 31 θ 33 =   *  T T  − M 41 + N 41 θ 24 =  T T + N 42  − M 42

ε1 D2 −ε 3 I *

−I * *

* * *

Cˆ T λ T   0   0  < 0 , (6.28)  0  −ε 3 I 

  R , R =  11   *

 M i1 R12   , Mi =  R22   M i 3

 ,  

   Mi 2   Mi 4  

 T T  , M =  M1T M 2T M 3T M 4T  , N =  N1T N 2T N 3T N 4T          

 XC1T T T ψ 13 =  θ16T θ 26T 0 0  , ψ 14 =  θ17T θ 27T 0 θ 47T  , θ16 =       C1T

 , 

294

Vibration Control of Vehicle Suspension Systems

   θ =  17  , 

 ˆT T T θ 26 =  C λm D1  0

 θ 27 =  

 θ R2 R11 − 2θ R X ψ 44 =   *  Ψ 44

 =  

d XAT

d Aˆ1T

dAT

dAT Y + dC yT Bˆ T

 d Aˆ d  ,θ =  47   0

d Cˆ T λmT B2T 0

   

 d B1T Y , 

d B1T

 T  B2  , ψ 15 =  θ18T 0 0 0  ,θ18T =  T 2     θ R R22 − 2θ RY   Y B2

θ R2 R12 − 2θ R I

T   ˆT T T   , ψ 16 =  0 θ 29 0 0  and θ 29T =  C λ  0  

d B2 d Y T B2

 , 

 . 

Then, (1) the closed-loop system is asymptotically stable; (2) the H ∞ performance Tz1w ∞ < γ is minimized subject to the output constraints in (6.18) and maximum actuator force constraint in (6.17) with the disturbance energy under the bound wmax = ρ − V ( 0 ) / γ 2 . Proof. Considering the Lyapunov–Krasovskii function as follows: t

∫x

V ( t ) = x ( t ) Px ( t ) + T

0

t

T

∫x

( s ) Qx ( s ) ds +

T

( s ) Sx ( s ) ds +

t − d (t )

t−d

t

∫ ∫ x

T

( s ) Rx ( s ) dsdα ,, (6.30)

− d t +α

The derivative of V ( t ) along the solution of system (6.13)–(6.16) is expressed as

()

() ()

( )(

(

) ()

)

(

)

() ()

V t ≤ 2 x T t Px t + x T t Q + S x t − x T t − d Q x t − d + dx T t Sx t

(

) (

( ))

T

(

t

( ))

− 1− µ x t − d t Q x t − d t −

x t − d (t )

T

( s) Rx ( s) ds +

()

t−d t

() ()

(6.31)

x T s Rx s ds,

t−d

For any appropriately dimensioned matrices Mˆ and Nˆ , the following equalities hold directly according to the Newton–Leibniz formula

t   ˆ  2ξ ( t ) M × x ( t ) − x ( t − d ( t )) − x ( s ) ds  = 0, (6.32)   t − d (t )

t − d (t )   2ξ ( t ) Nˆ ×  x ( t − d ( t )) − x t − d − x ( s ) ds  = 0, (6.33)   t−d

T

(

T

(

 where ξ ( t ) =  x ( t ) x ( t − d ( t ))   x t − d 

Mˆ =  M1T 

M 2T

M 3T

)

)

w ( t )  ,  T

M 4T  ,  Nˆ =  N1T 

N 2T

N 3T

T

N 4T  .

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Suspension Control for In-Wheel Motor Driven Electric Vehicle

Adding the two equations (6.32) and (6.33) into the right-hand side of (6.31) and after some simple calculations, the following inequality is satisfied

() ()

() ()

()

()

()

(

()

) ()

T ˆ −1 Mˆ T + d − d t NR ˆ −1 Nˆ T  ξ t z1T t z1 t − γ 2ω t ω t + V x ≤ ξ T t  ψ + d t MR  

T

( ) ( )(

()

d t d −d t ˆ −1 Mˆ T + ˆ −1 Nˆ T ψ + dMR ψ + NR t  d  d

)

(

(6.34)

 ξ t , 

) ()

T T + Φ14 R −1Φ14 , where ψ = Φ11 + Φ13Φ13

 PA + PAT + M + M T + Q + S PBcl + M 2T − M1 + N1 M 3T − N1 PBc11 + M 4T 1 1 cl cl   − 1 − µ S + N 2T + N1 − M 2T − M 2 N 3T − M 3T − N 2 − M 4T + N 4T * Φ11 =   −Q − N 3T − N 3 − N 4T * *  −γ 2 I * * * 

(

)

T

T Φ13 = [Ccl1  Dcl1  0  0 ] , Φ14 =  d RAcl  d RBcl  0  d RBcl1  .  

According to inequalities (6.25), (6.26), and Schur complement, the following inequalities can be obtained

ˆ −1 Mˆ T < 0, (6.35) ψ + dMR

ˆ −1 Nˆ T < 0, (6.36) ψ + NR According to inequalities (6.34)–(6.36), z1T ( t ) z1 ( t ) − γ 2ω T ( t ) w ( t ) + V ( t ) < 0 can be obtained for all nonzero w ∈ L2 [ 0,∞ ). Therefore, the H ∞ performance z1 ( t )2 ≤ γ w ( t )2 are guaranteed for all nonzero w ∈ L2 [ 0,∞ ) in the presence of actuator faults and delay. In addition, when ˙ w ( t ) = 0 ,  V ( t ) < 0 is obtained, which means the system (6.13)–(6.16) is asymptotically stable for the actuator delay and faults. By evaluating

Acl = A, Bcl1 = B1 , Bcl = B + HN 0 E , Ccl1 = C1 , Ccl 2 = C2 ,  Dcl1 = D1 + D1 N 0 E , Cu =  0 The equation in (6.35) can be written as

λCk  .

   ,   

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Vibration Control of Vehicle Suspension Systems

  Ψ11 ˆ −1 Mˆ T =  * Ψ + dMR  *   *

d Mˆ −R * *      N 0T       

 Ψ16  0 +  0  0 

Ψ13 0 −I * Ψ15 0 D1 dRH

Ψ14 0 0 −R

    +      

Ψ15 0 D1 dRH

  Ψ 16   N  0  0 0     0

     

T

T

(6.37)

   ,   

According to inequalities (6.35)–(6.36), then the following LMIs hold:

  Ψ11  *  *   *   *  * 

d Mˆ −R *

Ψ13 0 −I

Ψ14 0 0

Ψ15 0 ε1 D1

* * *

* * *

−R * *

ε1 d RH − ε 1I *

  Ψ11  *  *   *   *  * 

d Nˆ −R *

Ψ13 0 −I

Ψ14 0 0

Ψ 25 0 ε 2 D1

* * *

* * *

−R * *

ε 2 d RH −ε 2 I *

Θ12 Θ 22 *

Θ13 Θ 23 Θ33

*

*

   T T  , θ11 = PA + PA + M1 + M1 + Q + S + θ12 ,,  −γ 2 I  

     < 0, (6.38) 0   0  −ε1I  Ψ16 0 0

     < 0, (6.39)  0  0  −ε 2 I  Ψ 26 0 0

where

 Θ11   * Ψ11 =  *   *

Θ14 Θ 24 Θ34

θ13 = M 3T − N1 ,θ14 = PB1 + M 4T , θ 22 = − (1 − µ ) S + N 2T + N1 − M 2T − M 2 , θ 23 = N 3T − M 3T − N 2 , θ 24 = − M 4T + N 4T , θ 33 = −Q − N 3T − N 3 , θ 34 = − N 4T ,

T

Ψ13 =  C1 D1 0 0  , Ψ14 =     T

d RA T

d RB 0

 d RB1   T

T

T

Ψ15 =  ε1H T P 0 0 0  , Ψ 25 =  ε 2 H T P 0 0 0  , Ψ16 =  0 E 0 0  , Ψ 26 =  0 E 0 0  .      

297

Suspension Control for In-Wheel Motor Driven Electric Vehicle

There are several nonlinear variables presented in the above equations that cannot be eliminated by the change in the variables. By using the variable substitution method proposed in [5]. Since the matrix P is nonsingular, we partition matrix P and its inverse as follows  Y N   X M  −1 P= T ,   P =  T . N W Z     M

where X ,Y ∈  n × n are symmetric matrices. The following equation is achieved  X I   I Y P T = T  M 0   0 N

 X I   I Y By defining F1 =  T  ,   F2 =  T  M 0   0 N can be obtained:

 . 

  , then PF1 = F2, and the following equations 

 B λ Cˆ 0   2 m ,  , F1T PBF1 =   Aˆ d  0  

 AX A F1T PAF1 = F2T AF1 =  ˆ T ˆ y  A Y A + BC

 Q11 Q12  T  S11 S12  T  R11 R12  F1T QF1 =  , F1 SF1 =  , F1 RF1 =  ,  * Q22   * R22   * S22 

 M i1 M i 2 F1T M i F1 =   M i 3 M i 4

 B2 F1T PH =  T  Y B2

 N i1 N i 2  T , F1 N i F1 =   N i 3 N i 4 

  B1 , F1T PB1 =  T   Y B1

 T  X I  T T  Cˆ T λ − , F1 PF1 = F2T F1 =  , F1 E =    0  I Y 

 ,   , 

C1F1 =  C1X C1  and D1F1 =  D1λmCˆ 0 .     We define the variables

Aˆ = Y T AX + NBκ Cy X + NAκ M T , (6.40)

Aˆ d = Y T B2 λmCκ M T + NAκ d M T , (6.41)

Bˆ = NBκ , (6.42)

Cˆ = Cκ M T , (6.43)

298

Vibration Control of Vehicle Suspension Systems

{

(

Γ1 = diag F1 , F1 , F1 , I , F1 , I , PR −1

)

T

}

F1 , I , I . (6.44)

Given matrices X, Y , M and N , we can determine Aκ , Aκ d , Bκ and Cκ from Aˆ , Aˆ d , Bˆ and Cˆ . Performing congruence transformation by multiplying full rank matrices Γ1T on the left and Γ1 on the right, respectively. The inequalities (6.38)–(6.39) are equivalent to the inequalities (6.25)–(6.26) in Theorem 7.1. In what follows, we will show that hard constraints in (6.27)– (6.28) are guaranteed. From the definition of the Lyapunov function in (6.30), we know that x T ( t ) Px ( t ) < ρ with V = γ 2 wmax + V ( 0 ). Similarly, the following inequalities hold:

max z2 (t )q ≤ max x T ( t ) {Ccl 2 + Dcl 2 }q {Ccl 2 + Dcl 2 }q x ( t )2

1  −1 −  T < ρθ max  P 2 {Ccl 2 + Dcl 2 }q {Ccl 2 + Dcl 2 }q P 2  , q = 1,2,3, (6.45)  

2

t >0

T

t >0

max | u ( t ) |2 ≤ max χ T ( t ) {Cu } {Cu } χ ( t )2 T

t >0

t >0

(6.46)

1  −1 −  T < ρθ max  P 2 {Cu } {Cu } P 2  ,  

where θ max (·) represents maximal eigenvalue. Consequently, the output feedback controller parameters are obtained

−1 Cκ = Cˆ ( M T ) , (6.47)

Bκ = N −1 Bˆ ,  (6.48)

(

)( M ) AX − NB C X ) ( M )

Aκ d = N −1 Aˆ d − Y T B2 λmCκ M T

Aκ = N −1 Aˆ − Y T

(

κ

y

T −1

, (6.49)

T −1

. (6.50)

Remark 7.1 Theorem 7.1 presents an output feedback controller for an active suspension system with control delay and faults. When there are only control faults in the quarter car model, the fault-tolerant controller is presented for the active suspension system of electric vehicle with DVA based on the output feedback H ∞ control method. The active suspension system can be described by the following state-space equations:

x ( t ) = Ax ( t ) + B1w ( t ) + B2u ( t ) , (6.51)

z2 ( t ) = C2 x ( t ) + D2u ( t ) , (6.52)

z1 ( t ) = C1 x ( t ) + D1u ( t ) , (6.53)

y ( t ) = C y x ( t ) , (6.54)

Suspension Control for In-Wheel Motor Driven Electric Vehicle

299

where A,  B1 , B2 , C1 , C2 , C y , D1 are defined in (6.6)–(6.8) and (6.11). Considering the actuator faults, the output feedback H ∞ controller can be defined by the following state-space equation: ˙

xˆ ( t ) = Aκ xˆ ( t ) + Bκ y ( t ) , (6.55)

u ( t ) = λ ud ( t ) = λCk xˆ ( t ) = λm + N 0 λ Ck xˆ ( t ) , (6.56)

(

)

where Ak , Bk, Ck , λ , λm, and λ are defined in (6.9)–(6.10) and (6.12). Substituting (6.54)–(6.56) into (6.51)–(6.53), the closed-loop system is obtained ˙

x ( t ) = Acl x ( t ) + Bcl1w ( t ) , (6.57)

z2 ( t ) = Ccl 2 x ( t ) , (6.58)

z1 ( t ) = Ccl1 x ( t ) , (6.59)

u ( t ) = Cu x ( t ) , (6.60) T

where x =  x xˆ  ,  Acl = A + HN 0 E ,  Bcl1 = B , Ccl1 = C1 + D1 N 0 E ,  Ccl 2 = C2 + D2 N 0 E with  A B2 λmCκ A= Aκ  Bκ Cy

  B   B  , B =  1 , H =  2 , E =  0 λ Cκ  , C1 =  C1 D2 λmCκ  ,       0   0 

C2 =  C2 D2 λmCκ  , Cu =  0 λCκ .     Employing a similar method to which is proposed in Theorem 7.1, the following Corollary 7.1 is obtained for the active suspension system with only actuator faults. Corollary 7.1 Given positive scalars γ and ρ , a dynamic output feedback controller in the form of (6.55)– (6.56) exists, such that the closed-loop system in (6.57)–(6.60) is asymptotically stable with w ( t ) = 0, and H ∞ performance z1 ( t )2 ≤ γ w ( t )2 are guaranteed for all nonzero w ∈ L2 [ 0  ∞ ), while the control output constraints in (6.17)–(6.18) are guaranteed with the disturbance energy under the bound wmax = ρ − V ( 0 ) / γ 2 , if there exist symmetric matrices X   >  0, Y   >  0, positive scalars ε1, ε 2, ε 3 , and Aˆ , Bˆ , Cˆ with appropriate dimensions satisfying the following LMIs:

 Λ11   *  *   *  * 

Λ12

Λ13

Λ14

−γ 2 I * * *

0 −I * *

0 ε1 D12 − ε 1I *

  0   < 0, (6.61) 0  0  −ε1I    Λ15

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Vibration Control of Vehicle Suspension Systems

 −µ2 X max   *  *   * 

{

 − z2 2 max   *   *   *  * 

}

q

2 − µmax I

ρ Cˆ T λmT

2 − µmax Y * *

0 −I + ε 2I *

{ }I − {z }Y − z22max

X

2 2 max

q

ρ ( X T C2T + Cˆ T λmT D2T )

0

ρ C2T

0

q

 X  I 

(

(

 ˆ   C1 X + D12 λmC ,  Λ13 =   C1T 

)

)

T

Aˆ T + A

( )

ˆ y + BC ˆ y AT Y + Y T A + BC

  ,  Λ =  ε1 B2  14  ε1Y T B2  

T

Cˆ T λ T   0   0  < 0, (6.63)  0  −ε 3 I 

I  > 0, (6.64) Y 

 T ˆ ˆ  AX + XA + B2 λmC + B2 λmC Λ11 =   * 

 B1 Λ12 =  T  Y B1

ε1 D2 −ε 3 I *

−I * *

* * *

Cˆ T λ T   0  < 0, (6.62) 0  −ε 2 I 

T

  ,  

  ˆT T  ,  Λ15 =  C λ .   0 

Corollary 7.1 can be easily proved following the proof of Theorem 7.1. In this case, the output feedback controller parameters are obtained

−1 Cκ = Cˆ ( M T ) , (6.65)

Bκ = N −1 Bˆ ,  (6.66)

−1 −1 Aκ = N −1 Aˆ − Y T AX ( M T ) − Bκ C y X ( M T ) − N −1Y T B2 λmCκ . (6.67)

(

)

Substitute (6.65)–(6.67) into (6.55)–(6.56), the active suspension control force can be obtained.

6.5  SIMULATION RESULTS 6.5.1 Parameter Optimization Results PSO results for vehicle suspension and DVA and the effectiveness of DVA are illustrated in this section. The parameters of the quarter-car model with DVA and without DVA are listed in Table 6.1. In the PSO algorithm, the value of win, wfn, c1, c2, and iter is defined as follows: win = 0.9, w fn = 0.1, c1 = 1.3, c2 = 1.7, iter = 20. Since the effect of sprung mass acceleration is more important for the

Suspension Control for In-Wheel Motor Driven Electric Vehicle

301

vehicle ride performance than the other three hard constraints, so we choose four performance indexes as: p1 = 0.4 , p2 = 0.2, p3 = 0.2, p4 = 0.2 . The optimized parameters are listed as follows: md = 30 kg, ks = 42,288 N/m, cs = 1686 Ns/m, ka 2 = 30,542 N/m, ca 2 = 1157 Ns/m. Random road excitation is used to demonstrate the effectiveness of DVA configuration. The class B road profile with constant vehicle speed of 40 km/h is used in this section. The “IWM-EV” represents the inwheel motor-driven electric vehicle without DVA, while the “DVA-EV” denotes the in-wheel motordriven electric vehicle with DVA. Figure 6.2a shows the random responses of sprung mass acceleration and tyre dynamic force, from which it can be seen that both sprung mass acceleration and tyre dynamic force of the DVA-EV are smaller than those of IWM-EV. Frequency responses of the sprung mass acceleration and tyre deflection are shown in Figure 6.2b, it can be seen in the figure that the DVA-EV decreases the sprung mass acceleration by around 10 Hz. Furthermore, the suspension deflection of DVA-EV is greatly reduced around 10 Hz compared to the IWM-EV, which shows that the DVA-EV performs better than the IWM-EV, especially in the range of unsprung mass resonance. As a result, this kind of DVA-EV configuration has the ability of improving vehicle ride comfort performance and road-holding ability. Table 6.2 shows the root mean square (RMS) comparison of vehicle dynamic responses in terms of body acceleration, suspension stroke, and tyre dynamic load under random

FIGURE 6.2  (a) Random response of sprung mass acceleration and tyre dynamic force; (b) frequency responses of sprung mass acceleration and tyre deflection.

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Vibration Control of Vehicle Suspension Systems

TABLE 6.2 RMS Comparison of Vehicle Dynamic Responses Vehicle Type IWM-EV DVA-EV

Body Acceleration (m/s2)

Suspension Stroke (mm)

0.1806 0.1570

0.9382 0.8806

Tyre Dynamic Force (N) 132.5611 98.6614

road excitation. It is clear that the DVA-EV has better performance than IWMEV, which indicate that the DVA-EV structure has the potential to improve ride quality and road-holding performance.

6.5.2 Proposed Control Methods Validation In this section, bump road excitation and random road excitation are used to illustrate the effectiveness of the proposed controller. The quarter vehicle with in-wheel DVA model is used and vehicle parameters are listed in Table 6.1. We assume that the maximum suspension deflection zmax = 50 mm, the maximum control force umax = 2000 N and the dynamic force applied on the bearings Fmax = 3000 N. The “Passive” denotes quarter-car passive suspension system with in-wheel DVA structure. “Output feedback controller I” is the output feedback controller for the active suspension with only actuator faults while the “output feedback controller II” is the output feedback controller for the active suspension with actuator faults and time delay. The dynamic output feedback controller I for the active suspension systems in (6.57)–(6.60) with only control faults can be derived by using Corollary 7.1. In addition, we can obtain the minimum guaranteed closed-loop H ∞ performance index γ is 4.52.  −0.0007  0.0023  0.001 5 Ac = 1 × 10  −0.0219   −0.0124  −4.513

−0.0001 0.0004 −0.0001 −0.0118 −0.0265 −0.1491

 0  0  0 Bc = 1 × 10 6   0.0004   0.0002  0.0497

0 0 −0.0001 0.001 0.0011 −0.0134

0 0 0.0002 −0.0018 −0.0013 −0.1267

0 0 0.0001 0.0017 0.0048 −0.0463

0 0 0 0.0037 0.0003 0.7373

0 0 0 0 0 0

0 0 −0.0001 −0.001 −0.0026 0.0301 0 0 0 0.0039 −0.0071 6.0597

0 0 0 0 0 −0.0087 0 0 0 0 0 0

   ,    

   ,    

Cc =  −214.04 63.12 409.22 − 144.05 36.56 − 5.09  .  

We compare the proposed output feedback controller I with those in references [6] and [7]. In [6], a finite-frequency H ∞ state-feedback controller was proposed for active suspension equipped in inwheel motor-driven electric ground vehicle. Based on Theorem 2 in [6], the control gain matrix is obtained as follows:

K finite   = 10 4 ×  3.895 1.004 −0.195 0.018 −4.352 0.073  .

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Suspension Control for In-Wheel Motor Driven Electric Vehicle

In [7], A fault-tolerant fuzzy H ∞ control design approach was proposed for active suspension of in-wheel drive electric vehicle. Based on Corollary 3 in [7], the control gain matrix is obtained as follows:

K c = 10 4 ×  −0.681 0.487 −0.012 0.025 −6.885 −0.029  .

Furthermore, the dynamic output feedback controller for the active suspension systems with control delay and faults can be derived by using Theorem 7.1. In this chapter, ρ = 1, θ R = 1, upper bound d of actuator delay d ( t ) it d = 5 ms. The dynamic output feedback controller in (6.9)–(6.10) for the upper bound d = 5 ms is obtained, and it can be found that the minimum guaranteed closed-loop H ∞ performance index γ is (6.36).

   Ac = 1 × 10 4     

−0.0037 −0.0371 0.0919 1.0097 −2.0433 −4.0055

 0.0243  −7.6245  −17.1847 Ad =   39.2780   169.6451  154.0348

−0.0002 −0.0059 −0.0208 −0.0488 −0.244 −1.0079 0.0021 −0.6448 −1.4625 3.3321 14.3083 13.0703

 0  0.0006  0.0015 6 Bc = 1 × 10  −0.0022   0.2548  −0.2908

0 0 −0.0178 0.0002 −0.0005 −0.0073 0.0091 −2.8443 −6.4091 14.6446 63.2485 57.4290

0 0 −0.0027 0.0006 0.0166 −0.0027

0 0 0 −0.0178 −0.0001 −0.0052 −0.0023 0.7191 1.6194 −3.6969 −15.9698 −14.4991

0 0.0004 −0.0003 −0.0124 0.0137 0.0117

0 0 0 0 −0.0179 −0.0012 0.0003 −0.0818 −0.1847 0.4227 1.7480 1.6982

0 0 0 0 0 0

0 0 0 0 −0.0001 −0.0192 −0.0001 0.0253 0.0569 −0.1301 −0.5525 −0.4929

0 −0.0035 0.0028 0.0148 −0.0472 −1.2374

0 0 0 0 0 0

   ,    

   ,    

   ,    

Cc =  −971.85 − 82.16 − 362.48 91.64 − 10.43 3.22  .  

The following bump road profile is used to validate the effectiveness of the proposed dynamic output feedback controllers:

 a l  2πv0    1 − cos  t   ,     0 ≤ t ≤ ,      v0 l  2 x (t ) =  (6.68)  0,                                              t > l ,  v0 

where a is the height of the bump and l is the length of the bump. Here we choose a = 0.1 m, l = 2 m and the vehicle forward velocity of v0 = 25 km/h. Figure 6.3a shows the bump responses for four suspensions, i.e., passive suspension, active suspension with state feedback H ∞ controller, active suspension with finite-frequency H ∞ controller and

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Vibration Control of Vehicle Suspension Systems

FIGURE 6.3  Suspension with 60% actuator thrust loss: (a) bump responses; (b) active forces; (c) frequency responses.

active suspension with the proposed fault-tolerant output feedback controller when 60% actuator thrust loss occurs. The responses, including the body acceleration, suspension deflection, dynamic tyre load, and motor dynamic force, are plotted. It can be seen from Figure 6.3a that the proposed fault-tolerant output feedback controller reduces the body acceleration significantly compare to the passive suspension when 60% actuator thrust loss occurs. The active suspension with the proposed controller also achieves better suspension performance when compared to the other two active suspensions. In addition, the suspension deflection, tyre dynamic force and maximum control force constraints are guaranteed simultaneously. Figure 6.3b shows the active forces with 60% actuator

Suspension Control for In-Wheel Motor Driven Electric Vehicle

305

thrust loss. The time-domain responses show that the proposed output feedback controller provides better vehicle performance than the passive suspension and the other two active controllers with partial faults in the actuator. Figure 6.3c shows the frequency responses of passive and active suspension systems with 60% actuator thrust loss. The natural frequencies of vehicle sprung mass and unsprung mass are about 1.8 and 10 Hz, respectively. In the frequency domain, it can be seen that the active suspension with the designed fault-tolerant controller performs significantly better in the natural frequencies of vehicle sprung mass. This confirms that the proposed control method can realize good vehicle ride performance despite the change of the actuator thrust loss. Figure 6.4 shows the bump responses of passive suspension and active suspensions with two different output feedback controllers in the presence of control delay and actuator faults. The upper bound of actuator delay is 5 ms. Figure 6.5 shows the active forces of two output feedback controllers under 30% and 60% actuator thrust losses. As observed in these diagrams, the proposed dynamic output feedback controller II achieves better suspension and motor performance than those of the passive suspension and the output feedback controller I when actuator faults and delay occur. This clearly demonstrates the effectiveness of the output feedback controller II in the presence of control delay. Moreover, with actuator time delay, the output feedback controller I achieves the worst performance among the three types of suspensions. This is because the output feedback controller I become instability when actuator delay occurs in the system. Vehicle performance of the output feedback controller I becomes worse when the actuator time delay occurs, which shows that controller I is not robust to actuator delay. Furthermore, with an increase in the actuator thrust loss, the output feedback controller II reveals marginally better performance than the output feedback controller I and the passive suspension in spite of actuator faults and time delay under bump road excitation. Figure 6.6 illustrates the frequency responses for passive suspension and active suspension with output feedback controller II under 60% actuator thrust loss and 5 ms actuator delay, from which we can observe that the ride performance of the active suspension is much better than that of the passive one around the resonant frequency of sprung mass, demonstrating better robustness of the proposed output feedback controller II. Furthermore, the random road excitation is used to demonstrate the effectiveness of the proposed control system. The Power Spectral Density (PSD) of the random road excitation can be expressed as the following equation:

 n Gq ( n ) = Gq ( n0 )    n0 

−ω

. (6.69)

The PSD of the road velocity can be expressed as the following equation:

Gq ( n ) = (2πn)2 Gq ( n ) . (6.70)

The random road excitation is determined by feeding a white noise through a linear first-order filter:

x g + 2πf0 x g = 2π Gq ( n0 ) vw. (6.71)

where n is the spatial frequency and n0 is the reference spatial frequency. Gq ( n0 ) is PSD for the reference spatial frequency. ω is frequency index, usually ω = 2. v is the vehicle speed, and w stands for the white noise disturbance of the road. f0 is cut-off frequency, which could avoid overestimating the low frequency component of the road. The class B road profile with constant vehicle speed of 40 km/h is used to test the system. The RMS comparison of the vehicle dynamic responses under random road excitation is shown in Table 6.3. The active suspension with the fault-tolerant controller achieves marginally better performance than the passive suspension in the presence of 60% actuator thrust loss. Suspension deflection, actuator force maximum force applied to the motor bearing and

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Vibration Control of Vehicle Suspension Systems

FIGURE 6.4  Bump responses with: (a) 30% actuator thrust loss and 5 ms time delay; (b) 60% actuator thrust loss and 5 ms time delay.

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(b)

6000 Output-feedback controller I Output-feedback controller II

2000 0 –2000 –4000

Output-feedback controller I Output-feedback controller II

2000 Active force (N)

Active force (N)

4000

3000

1000 0 –1000

0

0.5

1

1.5 Time (s)

2

2.5

3

–2000

0

0.5

1

1.5 Time (s)

2

2.5

3

FIGURE 6.5  Active forces: (a) 30% actuator thrust loss; (b) 60% actuator thrust loss.

FIGURE 6.6  Frequency responses of passive and active suspensions with 60%.

tyre dynamic force are guaranteed simultaneously. The RMS comparison of the passive suspension, the active suspensions with the output feedback controller I and the output feedback controller II in the presence of control delay and different actuator faults are shown in Table 6.4. Suspension performance of the output feedback controller I is impaired when the actuator delay occurs. The active suspension with the output feedback controller II reveals much better performance than those with the output feedback controller I and the passive suspension; which shows that the output feedback controller II is able to guarantee better suspension and motor performance in spite of actuator faults and delays under random road excitation.

6.6 CONCLUSIONS In this chapter, the problem of output feedback H ∞ control for active suspensions deployed in inwheel motor-driven electric vehicles with actuator faults and time delay was investigated. A quartercar active suspension system with in-wheel motor served as DVA was established, and this kind of configuration was demonstrated to improve ride performance and road-holding ability around

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Vibration Control of Vehicle Suspension Systems

TABLE 6.3 The RMS Comparison of Vehicle Dynamic Responses with 60% Actuator Thrust Loss Suspension Types Passive State-feedback controller Finite-frequency H ∞ controller Output feedback controller I

Body Acceleration (m/s2)

Suspension Deflection (m)

Motor Dynamic Force (N)

Active Force (N)

0.162 0.131

8.996e-4 7.351e-4

110.726 103.454

75.012 66.017

– 13.455

0.129

7.024e-4

104.133

64.891

15.285

0.129

7.143e-4

111.552

69.267

16.559

TABLE 6.4 The RMS Comparison of Vehicle Dynamic Responses in the Presence of Control delay and different actuator faults. Suspension Types Passive controller I with 30% actuator thrust loss controller II with 30% actuator thrust loss controller I with 60% actuator thrust loss controller II with 60% actuator thrust loss

Body Acceleration (m/s2)

Suspension Deflection (m)

Motor Dynamic force (N)

0.1617 0.2539

8.9964e-04 0.0016

0.0282 0.0312

75.0124 174.7807

– 102.1168

0.1301

7.0113e-04

0.0293

68.5019

21.1690

0.2204

0.0013

0.0299

114.5852

53.0063

0.1369

7.2105e-04

0.0285

69.3987

15.2151

Active Force (N)

10 Hz. Parameters of vehicle suspension and DVA were optimized based on the PSO method. To achieve better ride comfort and reduce the force applied on the in-wheel motor bearing, a robust H ∞ dynamic output feedback controller was derived such that the closed-loop system was asymptotic stable and simultaneously satisfied the constraint performances such as road holding, suspension stroke, dynamic load applied to the bearings and actuator limitation. Finally, the simulation results demonstrated the effectiveness of the proposed output feedback controllers in improving suspension performance in spite of actuator faults and time delay. Meanwhile, the proposed fault-tolerant output feedback H ∞ controller achieved a better vehicle and motor performance than those of the passive suspension for different actuator thrust losses. When different actuator thrust losses and time delay occurred, the proposed output feedback controller II revealed much better performance than the output feedback controller I and the passive suspension.

REFERENCES [1] Kaveh, A., Advances in Metaheuristic Algorithms for Optimal Design of Structures. 2014, Wien: Springer. [2] Shi, Y. and R. Eberhart, A modified particle swarm optimizer, IEEE. doi: 10.1109/icec.1998.699146. [Online].

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[3] Rajeswari, K. and P. Lakshmi, PSO Optimized Fuzzy Logic Controller for Active Suspension System, in 2010 International Conference on Advances in Recent Technologies in Communication and Computing, 16–17 October 2010, pp. 278–283, doi: 10.1109/ARTCom.2010.22. [4] Zhao, Y., L. Zhao, and H. Gao, Vibration control of seat suspension using $H∞$ reliable control. jvc, 2010. 16(12): 1859–1879. [5] Li, H., X. Jing, and H. R. Karimi, Output-feedback-based $H∞$ control for vehicle suspension systems with control delay. IEEE Transactions on Industrial Electronics, 2014. 61(1): 436–446. doi: 10.1109/ TIE.2013.2242418. [6] Wang, R., H. Jing, F. Yan, H. Reza Karimi, and N. Chen, Optimization and finite-frequency H∞ control of active suspensions in in-wheel motor driven electric ground vehicles. Journal of the Franklin Institute, 2015. 352(2): 468–484. doi: 10.1016/j.jfranklin.2014.05.005. [7] Shao, X., F. Naghdy, and H. Du, Reliable fuzzy H∞ control for active suspension of in-wheel motor driven electric vehicles with dynamic damping. Mechanical Systems and Signal Processing, 2017. 87: 365–383. doi: 10.1016/j.ymssp.2016.10.032.

Index Note: Bold page numbers refer to tables and Italic page numbers refer to figures. 4-degree-of-freedom (DOF) driver body model 187 4-DOF half-car model 4, 85 7-DOF full-car model 4, 243, 249 active suspension control 1, 9 active suspension system 1, 9 actuator faults 286 actuator input delays 66 actuator saturation 127, 195 actuator time delay 305 actuator time delays 48 adaptive control law 163 adaptive inerter control 162 admittance 269 bicycle model 175, 176 body-wheel motion-mode 256 Bouc-wen phenomenological model 122 bounce motion-mode 251 bump road excitation 6, 16 cone complementarity linearization (CCL) algorithms 53, 93 control input constraint 121 controllable branch circuits 272 controllable electrical network (EN) 248 direct voltage control 121 driver head acceleration 187 driving point impedances 272 dynamic vibration absorber 286 electrical energy 268 electrically interconnected suspension (EIS) 258, 267, 268 electrical network 122 electric vehicles 286 electrohydraulic active suspensions 187 electromagnetic suspensions 268 electrorheological damper 2, 140 energy intersection 248 EN topologies 271 evolution process (figure) 15 extended state observer design 113 finite-frequency output feedback control 31 force-current analogy 122, 267 force-velocity graph 277 frequency domain inequality (FDI) 65 frequency responses 14, 15, 16 full-car suspension model with a driver seat (figure) 225 fuzzy finite frequency SOF controller 34, 39 fuzzy PD controller 113 fuzzy rules 63 fuzzy state feedback control strategy 187 fuzzy weighting functions 63, 64

genetic algorithms (GA) 14, 53 hydraulic energy 267 hydraulic interconnected suspension (HIS) 267 impedance 269 integrated suspension control 2 integrated suspension control methods with lateral dynamics and vertical dynamics 187 integrated with seat suspension and driver model and cabin model 224 interconnected suspension 248 interconnected suspension control 2 in-wheel motor-driven electric vehicle 2, 286 iterative linear matrix inequality (ILMI) 92 kinetic energy 256 lateral acceleration 179 lateral vehicle dynamics 175 linear matrix inequalities (LMI) 11 linear parameter-varying controller 105 linear time-invariant (LTI) system 47 LPV control 98 magnetorheological damper 2, 122 maximum suspension travel 26 mechanical-electrical analogies 267 mechanical VESI network 122 membership functions 28, 63 motion-based active disturbance rejection control (ADRC) controllers 111 motion decoupling 248 motion mode control 110 motion mode control strategy 248 motion-mode detection 248, 256 motion-modes 251 motor bearing wear 286 multi-objective control with uncertainties 22 multi-objective disturbance attenuation 91 multi-objective static output feedback /G controllers 9, 10 multi-objective velocity-dependent controller 102 natural frequency 160 non-fragile controller 17 non-fragile fuzzy control criterion 66 non-fragile fuzzy controller 66 nonlinear function 124 nonlinear uncertain electrohydraulic active suspensions 208 observer-based gain-scheduling integrated control strategy 187 observer-based multi-objective integrated control 187 observer-based robust gain-scheduling integrated control strategy 187

311

312 parallel distributed compensation (PDC) 127 parallel-distributed compensation (PDC) scheme 187 parameter-dependent control 98 parameter-dependent control law 48 parameter-dependent controller 48, 98 parameter optimization 290 particle swarm optimization (PSO) 286, 290 passive suspension 2 passive suspension system 1 PD controller 113 phenomenological model 122 pitch motion-mode 251 polynomial parameter-dependent controller (HPPD) 103 population-based search algorithms 290 potential energy 256 power spectral density (PSD) 6, 182 preview control 85 random road excitation 6 ride comfort 2, 7, 9, 26 road holding ability 26 road-holding properties 187 robust control methods 9 robust multi-objective controller 232 roll motion-mode 252 root-mean-square (RMS) gain 47 Schur complement 51 seat suspension 187 seat suspension dynamics with driver model and cabin model 187 semi-active suspension control 1, 121 semi-active suspension system 1 semi-active variable equivalent stiffness and inertance (VESI) device 122 servo motors 273 sliding mode controller 179 sprung mass accelerations 11 sprung mass heave acceleration 221 sprung mass pitch acceleration 221 sprung mass roll acceleration 221 state feedback controller 227 steering angle 184 stop block 2 suspension deflection 11

Index suspension parameter uncertainties 121 suspension stroke limitation 187 switchable interconnected suspensions 248 switch control strategy 187 switched control 248 Takagi-Sugeno (T-S) fuzzy model 64, 125 Takagi-Sugeno (T-S) model-based fuzzy control design approach 187 Tire cornering stiffness 187 Tire deflection 11 Torque-angle graph 277 transfer function 21, 47 transfer impedances 272 transmission devices 2 TS fuzzy observer 132 two-degree-of-freedom (2-DOF) quarter-car model 3, 123, 143 two-port impedance network 269 two-port network method 248 two-terminal element 121 two-terminal mechanical and electrical element 169 unsprung mass velocity 286, 289 variable damping device 122 variable electrical network 167 variable inertance semi-active suspension 121 variable mechanical networks 168 varying damping damper 122 varying equivalent stiffness and inerter 167 varying inerter 158 vehicle chassis 1 vehicle handling stability 2 vehicle inertial properties 48 vehicle longitudinal velocity 187 vehicle rollover 179 vehicle sideslip angle 187 vehicle wheel motion-mode 256 velocity-dependent controller design 102 vibration control 2 vibration energy 267 vibration transmissibility 279 voltage source 269 wheelbase preview strategy 88