Automatic Control of Hydraulic Systems 9781685071745, 9781685073466, 9781536198928, 9781536199062, 9798886977516, 9798886974553, 9798886975277, 9798886976199

Table of contents :
Contents
Preface
Chapter 1
Robust Controller Design for Uncertain Hydraulic Actuators Using Sine-Cosine Optimization Algorithms
Abstract
Introduction
Uncertain System Hydraulic Actuator Model
Robust Tracking Controller
Theory Notes
Computation of the Robust Position Tracking Controller
Robust Stabilization Algorithm
A Sine Cosine Algorithm for Optimizing Controller Parameters
Simulation Results
Conclusion
References
Chapter 2
Flatness-Based Control in Successive Loops for Robotic and Mechatronic Systems with Electro-Hydraulic Actuation
Chapter 3
Optimized Fuzzy Sliding Mode Tracking Control of an Electro-Hydraulic Actuator System: A Simulation Study
Abstract
Introduction
Methodology
Grey-Box Identification
The Fuzzy Logic Sliding Mode Control Design
Particle Swarm Optimization (PSO)
Performance Index
Result and Discussion
Conclusion
Acknowledgments
References
Chapter 4
An Advanced Neural-Disturbance-Observer Control Method for Constrained Hydraulic Systems
Abstract
Introduction
Nomenclature
System Modeling and Problem Statements
Neural-Disturbance-Observer Control Design with Output Constraints
A Simplified Output-Constraint Backstepping Controller (OCBS)
Disturbance Estimation for Pressure Dynamics
Disturbance Estimation for Force Dynamics
Closed-Loop Performance
Validation Results
Sinusoidal Test
Smooth-Multistep Test
Conclusion
References
Index
About the Editor
Blank Page

Citation preview

Systems Engineering Methods, Developments and Technology Mathematics Research Developments

No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

Systems Engineering Methods, Developments and Technology Channelizers and Reconstructors – A Design Guide Bradford S. Watson, MSc (Author) 2021. ISBN: 978-1-68507-174-5 (Hardcover) 2021. ISBN: 978-1-68507-346-6 (eBook) Networked Control Systems: Theory, Applications and Analysis Dianwei Qian, PhD (Editor) Shiwen Tong, PhD (Editor) 2021. ISBN: 978-1-53619-892-8 (Softcover) 2021. ISBN: 978-1-53619-906-2 (eBook) More information about this series can be found at https://novapublishers.com/product-category/series/systems-engineeringmethods-developments-and-technology/

Mathematics Research Developments Characterizations of Recently Introduced Univariate Continuous Distributions IV G. G. Hamedani (Editor) 2023. ISBN: 979-8-88697-751-6 (eBook) Future Relativity, Gravitation, Cosmolog Valeriy V. Dvoeglazov, PhD (Editor) M. de G. Caldera Cabral (Editor) J. A. Cázares Montes (Editor) J. L. Quintanar González (Editor) 2023. ISBN: 979-8-88697-455-3 (Hardcover) 2023. ISBN: 979-8-88697-527-7 (eBook) More information about this series can be found at https://novapublishers.com/product-category/series/mathematics-researchdevelopments/

Michael G. Skarpetis Editor

Automatic Control of Hydraulic Systems

Copyright © 2023 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. We have partnered with Copyright Clearance Center to make it easy for you to obtain permissions to reuse content from this publication. Please visit copyright.com and search by Title, ISBN, or ISSN. For further questions about using the service on copyright.com, please contact: Copyright Clearance Center Fax: +1-(978) 750-4470

Phone: +1-(978) 750-8400

E-mail: [email protected]

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the Publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regards to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS.

Library of Congress Cataloging-in-Publication Data

ISBN: 979-8-88697-619-9

Published by Nova Science Publishers, Inc. † New York

Contents

Preface

.......................................................................................... vii

Chapter 1

Robust Controller Design for Uncertain Hydraulic Actuators Using Sine-Cosine Optimization Algorithms ..................................................1 Michael G. Skarpetis and Fotis N. Koumboulis

Chapter 2

Flatness-Based Control in Successive Loops for Robotic and Mechatronic Systems with ElectroHydraulic Actuation ........................................................31 G. Rigatos, M. Abbaszadeh,J. Pomares and P. Wira

Chapter 3

Optimized Fuzzy Sliding Mode Tracking Control of an Electro-Hydraulic Actuator System: A Simulation Study ...........................................83 Muhamad Fadli Ghani, Rozaimi Ghazali, Hazriq Izzuan Jaafar, Chong Chee Soon and Zulfatman Has

Chapter 4

An Advanced Neural-Disturbance-Observer Control Method for Constrained Hydraulic Systems ...........................................................................101 Dang Xuan Ba and Kyoung Kwan Ahn

Index

.........................................................................................131

About the Editor .......................................................................................135

Preface

For industrial, automotive, and aeronautical applications, hydraulic actuators and hydraulic valves have an essential role to deliver and regulate forces and moments. The evolution of hydraulic systems to cover the demands of Industry 4.0 through the digitation of hydraulic valves makes the automatic control of hydraulic actuators crucial for production lines and processes. In this collection in Chapter 1 a robust asymptotic tracking controller for the accurate positioning of an industrial Hydraulic Actuator is presented. The nonlinear uncertain model of the actuator and the uncertain parameters occurring due to physical system uncertainties are analytically characterized. The uncertain vector involves supply pressure, load mass and stiffness, viscous damping, and fluid bulk modulus variations. Furthermore, unknown disturbances from the industrial environment are also considered. The nonlinear model of the actuator is linearized, and a robust asymptotic tracking controller is proposed to control the position of the actuator. Solvability conditions are derived and using a Hurwitz invariability algorithm, stability regions are characterized for the controller parameters. The best solution for the controller parameters for all uncertainties variation and load distribution is derived using a sine-cosine based swarm optimization algorithm. The combination of the Hurwitz invariability algorithm appropriately extended with a sine – cosine optimization algorithm seems to be a powerful tool for solving robust control problems. Simulation test results of the performance of the uncertain nonlinear closed-loop system, demonstrate the effectiveness of the robust controller over the whole uncertain domain and for various external disturbances. In Chapter 2, the control problem for the nonlinear dynamics of robotic and mechatronic systems with electrohydraulic actuation is solved with the use of flatness-based control approach which is implemented in successive loops. The state-space model of these systems is separated into a series of subsystems, which are connected between them in cascading loops. Each one of these subsystems can be viewed independently as a differentially flat

viii

Michael G. Skarpetis

system and control about it can be performed with inversion of its dynamics as in the case of input-output linearized flat systems. In this chain of subsystems, the state variables of the subsequent (i + 1-th) subsystem become virtual control inputs for the preceding (i-th) subsystem, and so on. In turn, exogenous control inputs are applied to the last subsystem and are computed by tracing backwards the virtual control inputs of the preceding N - 1 subsystems. The whole control method is implemented in successive loops and its global stability properties are also proven through Lyapunov stability analysis. The validity of the control method is confirmed in two case studies: (a) control of an electrohydraulic actuator, (ii) control of a multi-DOF robotic manipulator with electrohydraulic actuators. In Chapter 3, the performance of an electro-hydraulic actuator (EHA) system's trajectory tracking employing an optimized fuzzy sliding mode controller (FSMC) is presented. In simulations, the performance of the FSMC, which is developed based on the transfer function structure of a double-acting EHA system third-order model, is assessed using a chaotic trajectory. The particle swarm optimization (PSO) algorithm identifies the design gain variables of the control law, which is developed from the concept of the exponential reaching law. The Lyapunov theorem theoretically demonstrates the stability of the control system. Simulation findings indicate that the proposed controller is highly robust and capable of accommodating system parameter change during trajectory tracking control. It also demonstrates that the proposed controller is superior to conventional PID controllers. In Chapter 4, a new intelligent motion controller is presented for constrained hydraulic systems using a special combination of neural network and nonlinear disturbance observers. In this first stage, a simplified constrained sliding-mode-backstepping scheme is designed to drive the control objective to a vicinity around origin without any physical violations. In the second stage, uncertain nonlinearities inside the system dynamics are compensated by new neural networks with fast learning rules. The neural learning errors and external disturbances are in the third stage presented as extended nonautonomous models and are then approximated by nonlinear high-order disturbance observers. Robustness of the closed-loop control system is maintained by a proper Lyapunov theory. Effectiveness and feasibility of the proposed control system for an asymptotically tracking performance are then confirmed by comparative simulation results.

Chapter 1

Robust Controller Design for Uncertain Hydraulic Actuators Using Sine-Cosine Optimization Algorithms Michael G. Skarpetis1, and Fotis N. Koumboulis2 1Core

Department, National and Kapodistrian University of Athens, Athens, Greece of Digital Industry Technologies, National and Kapodistrian University of Athens, Athens, Greece 2Department

Abstract This chapter aims to present a robust asymptotic tracking controller for the accurate positioning of an industrial Hydraulic Actuator. The nonlinear uncertain model of the actuator is presented and the uncertain parameters occurring due to physical system uncertainties are analytically characterized. The uncertain vector involves supply pressure, load mass and stiffness, viscous damping, and fluid bulk modulus variations. Furthermore, unknown disturbances from the industrial environment are also considered. The nonlinear model of the actuator is linearized, and a robust asymptotic tracking controller is proposed to control the position of the actuator. Solvability conditions are derived and using a Hurwitz invariability algorithm, stability regions are characterized for the controller parameters. The best solution for the controller parameters for all uncertainties variation and load distribution is derived using a sine-cosine based swarm optimization algorithm. The combination of the Hurwitz invariability algorithm appropriately extended with a sine – cosine optimization algorithm seems to be a 

Corresponding Author’s Email: [email protected]

In: Automatic Control of Hydraulic Systems Editor: Michael G. Skarpetis ISBN: 979-8-88697-619-9 © 2023 Nova Science Publishers, Inc.

2

Michael G. Skarpetis and Fotis N. Koumboulis powerful tool for solving robust control problems. Simulation test results of the performance of the uncertain nonlinear closed-loop system, demonstrate the effectiveness of the robust controller over the whole uncertain domain and for various external disturbances.

Keywords: robust control, hydraulic actuators, uncertain systems, optimization

Introduction The critical role of hydraulic actuators in the operation of many industrial plants has provided valuable motivation for research and analysis of control systems. Hydraulic actuators are primarily used to actuate mechanical systems that require exceptionally large forces and moments. The main reason for using hydraulic control systems in many applications is their high capacity to store and deliver energy. The basic operating principle of hydraulic actuators is to convert the supplied flow into pressure and force, which is then used to induce or enhance linear or rotary motion. Hydraulic actuators take advantage of the properties of liquids to distribute the forces exerted on them to different positions in a production line. Another property is that the liquid is very incompressible. According to Pascal’s law, any liquid will transmit pressure applied from one surface to the other surface without damping. Hydraulic actuators operate at high pressures and are suitable for applications requiring very high forces. The main applications of hydraulic systems in industry are robotics (see e.g., [1-10]), handling unit production lines and lifting equipment (see e.g., [11-15]). Furthermore, they are widely used in aircraft flight systems (moving surfaces such as rudders, hydraulic brakes, see e.g., [16-25]) and vehicle safety and navigation systems (active suspension, vehicle brake, hydraulic clutch, etc., see e.g., [26-35]). Developing control algorithms for servo-hydraulic systems remains a challenging and active research area for fluid power and automation. The main problem to overcome is the physical non-linearity resulting from the properties of the fluid flowing through the hydraulic valve and the relationship between the pressure and the position of the piston within the cylinder chambers. In addition, variations in the system parameters such as the effective bulk modulus of the fluid, external forces, viscous damping of the actuators, and load mass render traditional control techniques ineffective. To overcome these problems, many robust approaches have been taken to design hydraulic

Robust Controller Design for Uncertain Hydraulic Actuators …

3

control systems. Over the past four years, numerous researchers have attempted to control hydraulic actuators using various control techniques, including robust adaptive techniques (see [36-44]), robust feedback linearization and robust sliding mode control (see [45-55]), robust nonlinear model-based control (see [56-61]), and robust position tracking control (see [62-75]). In this chapter, a robust asymptotic tracking controller for precise positioning of industrial hydraulic actuators is designed. First the nonlinear model is presented, and system uncertainties are characterized. The uncertainty vector involves variations in supply pressure, load mass and stiffness, viscous damping, and fluid bulk modulus. The nonlinear uncertain model is linearized, and a robust asymptotic command tracking controller is applied. Solvability conditions are presented using Hurwitz invariability results [73-75] and controller parameter regions are derived. Using a finite step algorithm proposed in [75] and [76] the controller parameters are computed. The proposed algorithm has been successfully used for controlling many hydraulic and pneumatic systems [75-84]. The algorithm is extended here using a sine-cosine swarm optimization technique and the controller parameters are finally tuned in better positions according to a performance index. Simulation test results to the nonlinear closed-loop system show the effectiveness of the robust controller over the uncertain region and for external disturbances.

Uncertain System Hydraulic Actuator Model The nonlinear equations describing the state variables of a hydraulic actuator connected to a hydraulic spool valve (see Figure 1) are [85] and [86] 𝑑𝑥𝑝 (𝑡) 𝑑𝑡 𝑑𝑣𝑝 (𝑡) 𝑑𝑡 𝑑𝑃1 (𝑡) 𝑑𝑡 𝑑𝑃2 (𝑡) 𝑑𝑡

= 𝑣𝑝 (𝑡) 1

= 𝑚 [𝐴𝑃1 (𝑡) − 𝐴𝑃2 (𝑡) − 𝑑𝑣𝑝 (𝑡) − 𝐹𝐿 (𝑡)] 𝛽

ℎ = 𝑉 (𝑡) [𝑄1 (𝑡) − 𝐴𝑣𝑝 (𝑡)] 1

𝛽

ℎ = 𝑉 (𝑡) [−𝑄2 (𝑡) + 𝐴𝑣𝑝 (𝑡)] 2

(1) (2) (3) (4)

4

Michael G. Skarpetis and Fotis N. Koumboulis

where 𝑥𝑝 (𝑡) is the actuator position, 𝑣𝑝 (𝑡) is the velocity of the actuator, 𝑚 is the piston load mass, 𝐴 is the piston area, 𝑑 is the viscus damping of the actuator and 𝛽ℎ is the bulk modulus of the fluid inside the chamber of the actuator. The variables 𝑃1 (𝑡), 𝑃2 (𝑡) are the hydraulic pressures in each actuator chamber. 𝑉1 (𝑡), 𝑉2 (𝑡) are the volumes of the actuator chambers and they are characterized by the following relations: 𝑉1 (𝑡) = 𝐴𝑥𝑝 (𝑡) + 𝑉1,0 , 𝑉2 (𝑡) = 𝐴[𝐿 − 𝑥𝑝 (𝑡)] + 𝑉2,0

(5)

where 𝑉1,0 and 𝑉2,0 are the connecting lines initial volumes.

Figure 1. Hydraulic actuator schematic.

The nonlinear equations presenting the fluid flows 𝑄1 (𝑡), 𝑄2 (𝑡) from the hydraulic valve that regulate the flow rate in the hydraulic actuator are [85]: 𝑃

𝑥 (𝑡)

𝑄1 (𝑡) = 𝐶𝑑 𝑤𝑥𝑣 (𝑡)√ 𝑠 + |𝑥𝑣 𝜌

𝑃

𝑣 (𝑡)|

𝑥 (𝑡)

𝑄2 (𝑡) = 𝐶𝑑 𝑤𝑥𝑣 (𝑡)√ 𝑠 + |𝑥𝑣 𝜌

𝑣 (𝑡)|

𝑃𝑠 −2𝑃1 (𝑡)

[

𝜌

]

(6)

−𝑃𝑠 +2𝑃2 (𝑡)

[

𝜌

]

(7)

Robust Controller Design for Uncertain Hydraulic Actuators …

5

where 𝐶𝑑 is the valve discharge coefficient, 𝑤 is the valve orifice area gradient, 𝑃𝑠 is the supply pressure, 𝜌 is the fluid density and 𝑥𝑣 (𝑡) is the displacement of the spool valve. The external force 𝐹𝐿 (𝑡) is the external load depending upon a variable stiffness and an unknown force and it is characterized by the following relation 𝐹𝐿 (𝑡) = 𝑘𝑥𝑝 (𝑡) + 𝐹0

(8)

where 𝑘 is the load stiffness and 𝐹0 is an unknown external disturbance. Introducing a new variable 𝑃𝐿 (𝑡) = 𝑃1 (𝑡) − 𝑃2 (𝑡), and since the following relations hold [85], 𝑄1 (𝑡) = 𝑄2 (𝑡) , 𝑃1(𝑡) =

𝑃𝑠 +𝑃𝐿 (𝑡) 2

, 𝑃2 (𝑡) =

𝑃𝑠 −𝑃𝐿 (𝑡) 2

, 𝑉1,0 = 𝑉2,0 = 𝑉0 (9)

equations (1) - (8) can be rewritten more compactly as follows 𝑑𝑥𝑝 (𝑡) 𝑑𝑡 𝑑𝑣𝑝 (𝑡) 𝑑𝑡

= 𝑣𝑝 (𝑡)

(10)

1

= 𝑚 [𝐴𝑃𝐿 (𝑡) − 𝑑𝑣𝑝 (𝑡) − 𝑘𝑥𝑝 (𝑡) − 𝐹0 ]

(11)

𝑥 (𝑡) 𝑃𝑠 −𝑃𝐿 (𝑡)|𝑥𝑣 𝑣 (𝑡)|] 𝜌

(𝐴𝐿+2𝑉0 )𝛽ℎ [𝐴𝑣𝑝 (𝑡)−𝐶𝑑 𝑤𝑥𝑣 (𝑡)√ 𝑑𝑃𝐿 (𝑡) 𝑑𝑡

=−

{𝑉0 +𝐴[𝐿−𝑥𝑝 (𝑡)]}[𝑉0 +𝐴𝑥𝑝 (𝑡)]

(12)

According to the uncertain nature of hydraulic systems the following parameters are considered here as uncertainties: 𝑞1 = 𝑃𝑠 , 𝑞2 = 𝑚 , 𝑞3 = 𝑑 , 𝑞4 = 𝛽ℎ , 𝑞5 = 𝑘

(13)

The vector 𝑞 = [𝑞1 , 𝑞2 , 𝑞3 , 𝑞4 , 𝑞5 ]𝜖𝒬 is the uncertain vector and 𝒬 is the uncertain domain given by the following expression 𝒬 = [𝑞1,𝑚𝑖𝑛 , 𝑞1,𝑚𝑎𝑥 ] × [𝑞2,𝑚𝑖𝑛 , 𝑞2,𝑚𝑎𝑥 ] × [𝑞3,𝑚𝑖𝑛 , 𝑞3,𝑚𝑎𝑥 ] × [𝑞4,𝑚𝑖𝑛 , 𝑞4,𝑚𝑎𝑥 ] × [𝑞5,𝑚𝑖𝑛 , 𝑞5,𝑚𝑎𝑥 ] Using the new variables 𝑞𝑖 (𝑖 = 1, … ,5) the nonlinear equations of the hydraulic actuator (10)-(12) can be rewritten as follows:

6

Michael G. Skarpetis and Fotis N. Koumboulis 𝑑𝑥𝑝 (𝑡) 𝑑𝑡 𝑑𝑣𝑝 (𝑡) 𝑑𝑡

= 𝑣𝑝 (𝑡)

(14)

1

= 𝑞 [𝐴𝑃𝐿 (𝑡) − 𝑞3 𝑣𝑝 (𝑡) − 𝑞5 𝑥𝑝 (𝑡) − 𝐹0 ]

(15)

2

𝑥 (𝑡) 𝑞1 −𝑃𝐿 (𝑡)|𝑥𝑣 𝑣 (𝑡)|] 𝜌

(𝐴𝐿+2𝑉0 )𝑞4 [𝐴𝑣𝑝 (𝑡)−𝐶𝑑 𝑤𝑥𝑣 (𝑡)√ 𝑑𝑃𝐿 (𝑡) 𝑑𝑡

=−

(16)

{𝑉0 +𝐴[𝐿−𝑥𝑝 (𝑡)]}[𝑉0 +𝐴𝑥𝑝 (𝑡)]

The nonlinear uncertain model (14)-(16) will be linearized around steady state initial conditions: 𝑥𝑝,0 =

−𝐹0,0 +𝐴𝑃𝐿,0 𝑞5

, 𝑥𝑣,0 = 0, 𝑣𝑝,0 = 0, 𝑃𝐿,0 = 0

(17)

where 𝐹0,0 is the initial value of the external force. Using the initial conditions (17) the uncertain linear model of the hydraulic actuator is presented by the following state space equations: 𝑑𝑥𝑙 (𝑡) 𝑑𝑡

= 𝐴𝑙 (𝑞)𝑥𝑙 (𝑡) + 𝐵𝑙 (𝑞)𝑢𝑙 (𝑡) + 𝐷𝑙 (𝑞)𝐹𝑙 (𝑡) , 𝑦𝑙 (𝑡) = 𝐶𝑙 𝑥𝑙 (𝑡)

(18)

𝑥𝑝 (𝑡) − 𝑥𝑝,0 where 𝑥𝑙 (𝑡) = [𝑣𝑝 (𝑡) − 𝑣𝑝,0 ] is the state space vector, 𝑢𝑙 (𝑡) = 𝑥𝑣 (𝑡) − 𝑥𝑣,0 𝑃𝐿 (𝑡) − 𝑃𝐿,0 is the input of the linear system, 𝑦𝑙 (𝑡) = 𝑥𝑝 (𝑡) − 𝑥𝑝,0 is the output of the system, 𝐹𝑙 (𝑡) = 𝐹0 − 𝐹0,0 and 𝐴𝑙 (𝑞), 𝐵𝑙 (𝑞) and 𝐶𝑙 (𝑞) are uncertain system matrices defined by the following relations:

𝐴𝑙 (𝑞) =

0 𝑞 − 5

1 𝑞 − 3

𝑞2

[

0

𝐴

𝑞2

𝑞2

𝐴𝑞4 𝑞52 (𝐴𝐿+2𝑉0 ) [−𝐴(𝐹0,0 +𝐿𝑞5 )−𝑞5 𝑉0 ](−𝐴𝐹0,0 +𝑞5 𝑉0 )

0 0 𝐵𝑙 (𝑞) =

0

𝑞 𝐶𝑑 𝑞4 (𝐴𝐿+2𝑉0 )𝑤√ 1 𝜌 𝐹0,0

[(𝐴[𝐿+ 𝑞5 ]+𝑉0 )(−

𝐴𝐹0,0 +𝑉0 ) 𝑞5

]

0

, ]

0 1 , 𝐷𝑙 (𝑞) = [− 𝑞 ] , 𝐶𝑙 (𝑞) = [1 0 0] 2 0 (19)

Robust Controller Design for Uncertain Hydraulic Actuators …

7

Also choosing 𝐹0,0 = 0 the uncertain system matrices (19) take on the form

𝐴𝑙 (𝑞) =

0 𝑞 − 𝑞5

1 𝑞 − 𝑞3

2

[ 0

2

𝐴𝑞4 𝑞52 (𝐴𝐿+2𝑉0 ) (−𝐴𝐿𝑞5 −𝑞5 𝑉0 )(𝑞5 𝑉0 )

0

0 0

𝐴

𝑞2

, 𝐵𝑙 (𝑞) =

0]

,

𝑞1 𝜌

𝐶𝑑 𝑞4 (𝐴𝐿+2𝑉0 )𝑤√

[

𝑉0 (𝐴𝐿+𝑉0 )

0 1 𝐷𝑙 (𝑞) = [− 𝑞 ] , 𝐶𝑙 (𝑞) = [1 0 0] 2

] (20)

0

Robust Tracking Controller Theory Notes Consider a step type reference signal 𝑟(𝑡) and consider that the external force disturbances 𝐹𝑙 (𝑡) are constant signals. For these signals holds that 𝑑𝑟(𝑡) 𝑑𝑡

=0,

𝑑𝐹𝑙 (𝑡) 𝑑𝑡

=0

(21)

Now, by definition, of the position error 𝑒(𝑡) = 𝑟(𝑡) − 𝑦𝑙 (𝑡)

(22)

and differentiating the error equation (22) the error dynamics are 𝑒̇ (𝑡) = −𝑦𝑙̇ (𝑡) = −𝐶𝑥𝑙̇ (𝑡)

(23)

After some algebraic manipulations, the open loop uncertain system (18) is augmented with error dynamics 𝑧̇ (𝑡) 𝐴 (𝑞) 0𝑛×1 𝑧(𝑡) 𝐵 (𝑞) [ ]=[ 𝑙 ][ ] + [ 𝑙 ] 𝑤𝑒 (𝑡) 𝑒̇ (𝑡) −𝐶𝑙 (𝑞) 01×1 𝑒(𝑡) 0

(24)

where 𝑧(𝑡) = 𝑥̇ 𝑙 (𝑡), 𝑤𝑒 (𝑡) = 𝑢̇ 𝑙 (𝑡)

(25)

8

Michael G. Skarpetis and Fotis N. Koumboulis

To the augmented system (24) apply an independent from the uncertainties static state feedback law of the form 𝑤𝑒 (𝑡) = −𝐾1 𝑧(𝑡) − 𝐾2 𝑒(𝑡)

(26)

where 𝐾1 ∈ ℝ1×𝑛 και 𝐾2 ∈ ℝ. The closed loop augmented system is of the form 𝑧̇ (𝑡) 𝐴 (𝑞) − 𝐵𝑙 (𝑞)𝐾1 [ ]=[ 𝑙 𝑒̇ (𝑡) −𝐶𝑙 (𝑞)

−𝐵𝑙 (𝑞)𝐾2 𝑧(𝑡) ][ ] 01×1 𝑒(𝑡)

(27)

According to [73] and [74] if the characteristic uncertain polynomial of the closed loop system (27) is Hurwitz invariant then the augmented closed loop system is asymptotically robustly stable i.e., ∃𝐾1 , 𝐾2 : 𝑑𝑒𝑡 [𝑠𝐼𝑛+1 − 𝐴 (𝑞) − 𝐵𝑙 (𝑞)𝐾1 [ 𝑙 −𝐶𝑙 (𝑞)

−𝐵𝑙 (𝑞)𝐾2 ]] 𝑖𝑠 𝐻𝑢𝑟𝑤𝑖𝑡𝑧 𝐼𝑛𝑣𝑎𝑟𝑖𝑎𝑛𝑡 01×1

and consequently 𝑙𝑖𝑚 (𝑧(𝑡)) = 0 , 𝑙𝑖𝑚 (𝑒(𝑡)) = 0

𝑡→∞

𝑡→∞

which certifies that the output of the system asymptotically tracks the reference signal for all the values of the uncertainties despite the existence of external disturbances. The original input signal that will be applied to the uncertain system is 𝑡

𝑢𝑙 (𝑡) = −𝐾2 ∫0 𝑒(𝜏)𝑑𝜏 − 𝐾1 𝑥𝑙 (𝑡)

(28)

and the closed loop transfer function of the uncertain closed loop system is 𝐻𝑐𝑙 (𝑠, 𝑞) = [𝐻𝑐𝑙,1 (𝑠, 𝑞) 𝐻𝑐𝑙,2 (𝑠, 𝑞)]

(29)

Robust Controller Design for Uncertain Hydraulic Actuators …

9

where 𝐾

𝐻𝑐𝑙,1 (𝑠, 𝑞) = −𝐶𝑙 (𝑞) (𝑠𝐼𝑛 − 𝐴𝑙 (𝑞) + 𝐵𝑙 (𝑞)𝐾1 − 𝐵𝑙 (𝑞)𝐶𝑙 (𝑞) 2)

−1

𝑠

𝐾

𝐻𝑐𝑙,2 (𝑠, 𝑞) = 𝐶𝑙 (𝑞) (𝑠𝐼𝑛 − 𝐴𝑙 (𝑞) + 𝐵𝑙 (𝑞)𝐾1 − 𝐵𝑙 (𝑞)𝐶𝑙 (𝑞) 2) 𝑠

−1

𝐵𝑙 (𝑞)

𝐾2 𝑠

𝐷𝑙 (𝑞)

The closed loop system block diagram is presented in Figure 2.

Figure 2. Closed loop system block diagram.

Computation of the Robust Position Tracking Controller Applying the dynamic controller presented in equation (28) to the open loop uncertain system model (18) and (19) the following closed loop uncertain characteristic polynomial is derived: 𝑝𝑐𝑙 (𝑠, 𝑞, 𝑓) = 𝑠𝑑𝑒𝑡 [𝑠𝐼3 − 𝐴𝑙 (𝑞) + 𝐵𝑙 (𝑞)𝐾1 (𝑞) − 𝐵𝑙 (𝑞)𝐶𝑙 (𝑞) = 𝑠 4 + 𝛼1 (𝑞, 𝑓)𝑠 3 + 𝛼2 (𝑞, 𝑓)𝑠 2 + 𝛼3 (𝑞, 𝑓)𝑠1 + 𝛼4 (𝑞, 𝑓)𝑠 0

𝐾2 ]= 𝑠 (30)

10

Michael G. Skarpetis and Fotis N. Koumboulis

where

𝑓 = [𝐾1

⏞ 𝐾2 ] = [𝑘1,1

𝐾1

𝐾2

𝑘1,2

⏞ 𝑘1,4 ]

𝑘1,3

𝑞

𝛼4 (𝑞, 𝑓) =

𝐴𝐶𝑑 𝑘1,4 𝑞4 𝑞25 (𝐴𝐿+2𝑉0 )𝑤√ 𝜌1 𝑞2 (𝐴𝐹0,0 −𝑞5 𝑉0 )(𝐴(𝐹0,0 +𝐿𝑞5 )+𝑞5 𝑉0 ) 𝑞

𝛼3 (𝑞, 𝑓) =

𝐶𝑑 𝑞4 𝑞25 (𝐴𝑘1,1 +𝑘1,3 𝑞5 )(𝐴𝐿+2𝑉0 )𝑤√ 𝜌1 𝑞2 (−𝐴𝐹0,0 +𝑞5 𝑉0 )(𝐴(𝐹0,0 +𝐿𝑞5 )+𝑞5 𝑉0 ) −𝑞5

𝛼2 (𝑞, 𝑓) = 𝑞

2 (𝐴𝐹0,0 −𝑞5 𝑉0 )(𝐴(𝐹0,0 +𝐿𝑞5 )+𝑞5 𝑉0 )

[𝐴3 𝐿𝑞4 𝑞5 − 𝐴2 (𝐹0,0 2 +

𝑞

𝐹0,0 𝐿𝑞5 − 𝑞4 𝑞5 (2𝑉0 + 𝐶𝑑 𝑘1,2 𝐿𝑤√ 𝜌1)) + 𝑞5 𝑉0 (𝑞5 𝑉0 + 𝑞

𝑞

2𝐶𝑑𝑘1,3 𝑞3 𝑞4 𝑤√ 𝜌1 ) + 𝐴𝑞5 (𝐿𝑞5 𝑉0 + 𝐶𝑑 𝑘1,3 𝐿𝑞3 𝑞4 𝑤√ 𝜌1 + 𝑞

2𝐶𝑑 𝑘1,2 𝑞4 𝑉0 𝑤√ 𝜌1)] 𝛼1 (𝑞, 𝑓) =

1 × 𝑞2 (𝐴𝐹0,0 −𝑞5 𝑉0 )(𝐴(𝐹0,0 +𝐿𝑞5 )+𝑞5 𝑉0 ) 𝑞

[𝐴2 𝐹0,0 𝑞3 (𝐹0,0 + 𝐿𝑞5 ) − 𝐴𝐿𝑞52 (𝑞3 𝑉0 + 𝐶𝑑 𝑘1,3 𝑞2 𝑞4 𝑤√ 𝜌1 ) − 𝑞

𝑞52 𝑉0 (𝑞3 𝑉0 + 2𝐶𝑑 𝑘1,3 𝑞2 𝑞4 𝑤√ 𝜌1 )] The characteristic closed loop polynomial (30) can be rewritten equivalently as follows 𝑝𝑐𝑙 (𝑠, 𝑞, 𝑓) = [𝑠 4 where

1 ⋯ 𝑠 0 ]𝑊 ∗∗ (𝑞) [ 𝑇 ] 𝑓

(31)

Robust Controller Design for Uncertain Hydraulic Actuators …

1 𝑤4 (𝑞)

0 0

0 0

𝑊 ∗∗ (𝑞) = 𝑤3 (𝑞) 0 𝑤0 (𝑞) 0 𝑤0 (𝑞) 0 0 0 [ 0

𝑤0 (𝑞) =

𝑤1 (𝑞) =

𝑤2 (𝑞) =

𝑤3 (𝑞) =

0 𝑤0 (𝑞)𝑞2 𝐴

𝑤2 (𝑞) 𝑤1 (𝑞) 0

11

0 0 0 0 −𝑤0(𝑞)]

𝑞1 𝜌

𝐴𝐶𝑑 𝑞4 (𝐴𝐿+2𝑉0 )𝑤√ 𝑞2 (𝐴[𝐿+

𝐹0,0 𝐴𝐹 ]+𝑉0 )(− 0,0 +𝑉0 ) 𝑞5 𝑞5

𝑞1 𝜌

𝐶𝑑 𝑞4 𝑞5 (𝐴𝐿+2𝑉0 )𝑤√ 𝑞2 (𝐴[𝐿+

𝐹0,0 𝐴𝐹 ]+𝑉0 )(− 0,0 +𝑉0 ) 𝑞5 𝑞5

𝑞1 𝜌

𝐶𝑑 𝑞4 𝑞3 (𝐴𝐿+2𝑉0 )𝑤√ 𝑞2 (𝐴[𝐿+

𝑞5 𝑞2

−

𝐹0,0 𝐴𝐹 ]+𝑉0 )(− 0,0 +𝑉0 ) 𝑞5 𝑞5

𝐴2 𝑞4 𝑞52 (𝐴𝐿+2𝑉0 ) 𝑞2(−𝐴[𝐹0,0 +𝐿𝑞5 ]−𝑞5 𝑉0 )(−𝐴𝐹0,0 +𝑞5 𝑉0 )

𝑞3

𝑤4 (𝑞) = 𝑞2 Consider the independent from the uncertainties invertible transformation 1 0 0 0 0 0 0 0 1 0 matrix 𝑇 = 0 0 1 0 0 . Using the transformation matrix, the closed 0 1 0 0 0 [0 0 0 0 −1] loop characteristic polynomial (31) can be rewritten as follows 𝑝𝑐𝑙 (𝑠, 𝑞, 𝑓) = [𝑠 4 where

1 ⋯ 𝑠 0 ]𝑊 ∗ (𝑞)𝑇 −1 [ 𝑇 ] 𝑓

12

Michael G. Skarpetis and Fotis N. Koumboulis 1 𝑤4 (𝑞) 𝑊 ∗ (𝑞) = 𝑊 ∗∗ (𝑞)𝑇 = 𝑤3 (𝑞) 0 [ 0

0

0

0

0

0

0

0

𝑤0 (𝑞) 0 0

0 𝑤0 (𝑞) 0

0 0 𝑤0 (𝑞)]

𝑤0 (𝑞)𝑞2 𝐴

𝑤2 (𝑞) 𝑤1 (𝑞) 0

(32)

According to [73] and [74] and since the matrix 𝑊 ∗ (𝑞) can be constructed by the following positive up augmentations starting from the core 𝑐̅(𝑞) = 𝑤0 (𝑞) > 0 ∀𝑞 ∈ 𝒬 . The core 𝑤0 (𝑞) is positive under the condition −𝐴𝐿𝑞5 −𝑞5 𝑉0 𝐴

< 𝐹0,0
0 for which 𝛷𝑖+1 (𝑞)[𝜀𝑖 𝜏𝑖 ]𝑇 is positive Hurwitz invariant): According to the form of the associated polynomial of 𝛷𝑖+1 (𝑞)[𝜀𝑖 𝜏𝑖 ]𝑇 , find a region of 𝜀𝑖 , let 𝑀𝑖 , where in the robust stability is guaranteed under the constrains of the stability regions 𝑀1 , . . . , 𝑀𝑖−1 . Let 𝜏𝑖+1 = [𝜀𝑖 1], and 𝑖 = 𝑖 + 1 Step 4: Repeat Step 3 until 𝑖 ≤ 4 Step 5: Using the metaheuristic finite step algorithm in the stability regions 𝑀𝑖 (𝑖 = 1, . . . ,4) and for all values of the uncertain parameters, compute the best values for 𝜀1 , . . . . , 𝜀4 under performance criteria. Step 6: (Derivation of the gain vector): Compute the robust tracking controller 𝑇 0 𝑇 from the relation: 𝑓 = { ([𝜀4 𝜀3 𝜀2 𝜀1 1])𝑇 } [ 1×4 ] 𝜀4 𝐼4

Figure 3. Algorithm flow chart.

14

Michael G. Skarpetis and Fotis N. Koumboulis

Using the values of the nominal parameters and the uncertainties with uncertain regions presented in Table 1 ([86]) the controller parameters are computed to be 𝜀1 = 0.00002, 𝜀2 = 𝑇 𝑓 = { ([𝜀4 𝜀4

𝜀3

3 1000000

𝜀2

𝜀1

, 𝜀3 = 𝑇

1 25000000

, 𝜀4 = 0.4505

(35)

0 1])𝑇 } [ 1×4 ] = 𝐼4 𝐾1

𝐾2

⏞ [ = 0.0000443951 6.65927 ∗ 10−6

8.87902 ∗ 10−8

⏞ −2.21976]

The integral of time weighted square error is computed for all possible uncertainties to be 10

𝐼𝑇𝑆𝐸 = ∫0 𝑡𝑒(𝑡)2 𝑑𝑡 = 0.04640270954476347 Table 1. Hydraulic actuator parameters

𝑃𝑠 (𝑃𝑎) 𝑚 (𝐾𝑔) 𝑁𝑠 𝑑 ( ) 𝑚 𝐴 (𝑚2 ) 𝑉1,0 , 𝑉2,0 (𝑚3 ) 𝜌 (𝐾𝑔/𝑚3 ) 𝛽ℎ (𝑃𝑎) 𝐶𝑑 𝑚2 𝑤( ) 𝑚 𝑁 𝑘 ( ) 𝑚 𝐿 (𝑚)

Nominal Value 17 ∗ 106 12.3

Uncertain regions 13.8 ∗ 106 − 18.6 ∗ 106 11 − 13.4

250

200 − 300

6.33 ∗ 10−4 88.7 ∗ (10−2 )3 847 689 ∗ 10^6 0.6

345 ∗ 106 − 1020 ∗ 106 -

20.75 ∗ 10−3

-

30 ∗ 103 0.5

1 ∗ 103 − 60 ∗ 103 -

A Sine Cosine Algorithm for Optimizing Controller Parameters Using the best values of the controller parameters computed in the previous section (relation (35)) a Sine-Cosine swarm optimizing algorithm will be used

Robust Controller Design for Uncertain Hydraulic Actuators …

15

in order to find the best solution for the controller ([87], [88]). Let 𝑉𝑖𝑡 = {𝜀1∗ , 𝜀2∗ , 𝜀3∗ , 𝜀4∗ } denote the current position of the controller parameters then according to [87] and [88] the next position is calculated by the following relations 𝑉𝑖𝑡+1 = 𝑉𝑖𝑡 + 𝑟1 sin(𝑟2 )|𝑟3 𝐵𝑖𝑡 − 𝑉𝑖𝑡 |

(36)

𝑉𝑖𝑡+1 = 𝑉𝑖𝑡 + 𝑟1 cos(𝑟2 )|𝑟3 𝐵𝑖𝑡 − 𝑉𝑖𝑡 |

(37)

where 𝑟1 , 𝑟2 and 𝑟3 are random real numbers and 𝐵𝑖𝑡 is the position of the destination point. Relation (36) and (37) can be combined as follows 𝑉𝑖𝑡+1

𝑉𝑖𝑡 + 𝑟1 sin(𝑟2 ) |𝑟3 𝐵𝑖𝑡 − 𝑉𝑖𝑡 | , 𝑟4 < 0.5 ={ 𝑡 𝑉𝑖 + 𝑟1 cos(𝑟2 ) |𝑟3 𝐵𝑖𝑡 − 𝑉𝑖𝑡 |, 𝑟4 ≥ 0.5

(38)

where 𝑟4 ∈ [0,1] is a random number. The number 𝑟1 is given by the relation 𝑟1 = 𝛼 −

𝑡𝛼 𝛵

where 𝛼 is a constant, 𝑡 is the present iteration and 𝑇 is the

maximum number of iterations. It is noted that 𝑟1 indicates the next position region, 𝑟2 is the length of the movement, 𝑟3 is a weighting factor and 𝑟4 defines the switch between (36) and (37). The proposed algorithm is modified by the relation 𝐵𝑖𝑡 = 𝑉𝑖𝑡 . The algorithm flow chart is summarized in Figure 4.

Figure 4. Sine-Cosine swarm optimizing algorithm flow chart.

16

Michael G. Skarpetis and Fotis N. Koumboulis

Using the values of the nominal parameters and the uncertainties with uncertain regions presented in Table 1 ([86]) and the aforementioned algorithm the controller parameters are finally tuned to the following values 𝜀1 = 0.0000228818, 𝜀2 = 3.43227 ∗ 10−6 , 𝜀3 = 4.57635 ∗ 10−8 , 𝜀4 = 0.515412 𝑇 0 𝑇 𝑓𝑡𝑢𝑛𝑒𝑑 = { ([𝜀4 𝜀3 𝜀2 𝜀1 1])𝑇 } [ 1×4 ] = 𝜀4 𝐼4 𝐾1

⏞ = [0.0000443952 6.65927 ∗ 10−6

𝐾2

8.87902 ∗ 10−8

⏞ −1.9402] (39)

The integral of time weighted square error is computed for all possible uncertainties to be 10

𝐼𝑇𝑆𝐸 = ∫0 𝑡𝑒(𝑡)2 𝑑𝑡 = 0.0216006 The values of the ITSE are presented in Figure 4 for 29 iterations

Figure 5. ITSE for 29 iterations.

Robust Controller Design for Uncertain Hydraulic Actuators …

17

Simulation Results Using the values of the nominal parameters and the uncertainties with uncertain regions presented in Table 1 ([86]), the nonlinear uncertain model 𝑑𝑥𝑝 (𝑡) 𝑑𝑡 𝑑𝑣𝑝 (𝑡) 𝑑𝑡

= 𝑣𝑝 (𝑡) 1

= 𝑚 [𝐴𝑃𝐿 (𝑡) − 𝑑𝑣𝑝 (𝑡) − 𝑘𝑥𝑝 (𝑡) − 𝐹0 ] 𝑥𝑣 (𝑡)

𝑃𝑠 −𝑃𝐿 (𝑡) |𝑥𝑣 (𝑡)| (𝐴𝐿+2𝑉0 )𝛽ℎ [𝐴𝑣𝑝 (𝑡)−𝐶𝑑 𝑤𝑥𝑣 (𝑡)√ ] 𝜌

𝑑𝑃𝐿 (𝑡) 𝑑𝑡

=−

{𝑉0 +𝐴[𝐿−𝑥𝑝 (𝑡)]}[𝑉0 +𝐴𝑥𝑝 (𝑡)]

and the controller 𝑥𝑝 (𝑡) 𝑡 𝑥𝑣 (𝑡) = −𝐾2 ∫0 (𝑟(𝑡) − 𝑥𝑝 (𝑡))𝑑𝜏 − 𝐾1 [𝑣𝑝 (𝑡)] 𝑃𝐿 (𝑡) with 𝐾1 and 𝐾2 as in (39) the following figures indicate the nonlinear closed loop system performance for a step reference signal with amplitude 0.1 (see Figure 6) and zero external disturbance. The simulation results present 162 different combinations of the uncertainty vector.

Figure 6. Actuator Position.

18

Michael G. Skarpetis and Fotis N. Koumboulis

Figure 7. Actuator Velocity.

Figure 8. Actuator Pressure.

Robust Controller Design for Uncertain Hydraulic Actuators …

19

Figure 9. Spool Valve Position.

Additionally, the nonlinear closed loop system performance for a step reference signal of amplitude 0.1 and 100𝑁 external force are presented in Figures 10-13.

Figure 10. Actuator Position.

20

Michael G. Skarpetis and Fotis N. Koumboulis

Figure 11. Actuator Velocity.

Figure 12. Actuator Pressure.

Robust Controller Design for Uncertain Hydraulic Actuators …

21

Figure 13. Spool Valve Position.

Additionally, the nonlinear closed loop system performance for a pulse reference signal see Figure 14 are presented in Figures 15-18.

Figure 14. Reference Signal.

22

Michael G. Skarpetis and Fotis N. Koumboulis

Figure 15. Actuator Position.

Figure 16. Actuator Velocity.

Robust Controller Design for Uncertain Hydraulic Actuators …

23

Figure 17. Actuator Pressure.

Figure 18. Spool Valve Position.

Conclusion The current paper has analytically proposed a robust control methodology for controlling the position of a hydraulic actuator via a hydraulic servo valve. The proposed robust controller is suitable for systems with physical uncertainties and external disturbances. Other types of reference signals, such as harmonic signals (sin-cos signals, triangular signals, ramp signals etc.), can

24

Michael G. Skarpetis and Fotis N. Koumboulis

be treated using the same analytic (solvability conditions) and algorithmic (computation of controller parameters via Sine-Cos swarm optimization) approach. Because of the robust control law's exceptional response to a wide range of system uncertainties, it is very simple to implement in an industrial environment. Together with the property that the control law properly regulates the flow from the valve to the power system, and with a primary focus on the integration of hydraulic valves for the industry 4 revolution, the overall design is powerful tool for real-world applications in a demanding industry environment.

References [1] [2]

[3] [4]

[5]

[6]

[7]

[8] [9]

[10]

[11]

Chen, G., Yang, X., Zhang, X., & Hu, H. (2021). Water hydraulic soft actuators for underwater autonomous robotic systems. Applied Ocean Research, 109, 102551. Wang, W., Tang, F., Zheng, C., Xie, T., Ma, C., & Zhang, Y. (2022). Prototyping a novel compact 3-DOF hydraulic robotic actuator via metallic additive manufacturing. Virtual and Physical Prototyping, 17(3), 617-630. Kittisares, S., Hirota, Y., Nabae, H., Endo, G., & Suzumori, K. (2022). Alternating pressure control system for hydraulic robots. Mechatronics, 85, 102822. Hussain, K., Omar, Z., Wang, X., Adewale, O. O., & Elnour, M. (2021). Analysis and Research of Quadruped Robot's Actuators: A Review. International Journal of Mechanical Engineering and Robotics Research, 10(8). Anh-My, C. (2021). Effective Solution to Integrate and Control a Heavy Robot Driven by Hydraulic Actuators. In Further Advances in Internet of Things in Biomedical and Cyber Physical Systems (pp. 331-345). Springer, Cham. Kellaris, N., Rothemund, P., Zeng, Y., Mitchell, S. K., Smith, G. M., Jayaram, K., & Keplinger, C. (2021). Spider‐Inspired Electrohydraulic Actuators for Fast, Soft‐ Actuated Joints. Advanced Science, 8(14), 2100916. Bai, X., Xue, Y., Xu, Y., Shang, J., Luo, Z., & Yang, J. (2022). Bio-Inspired New Hydraulic Actuator Imitating the Human Muscles for Mobile Robots. Frontiers in Bioengineering and Biotechnology, 10. Kittisares, S., Hirota, Y., Nabae, H., Endo, G., & Suzumori, K. (2022). Alternating pressure control system for hydraulic robots. Mechatronics, 85, 102822. Ntsiyin, T., Markus, E. D., & Masheane, L. (2022). A Survey of Energy-Efficient Electro-hydraulic Control System for Collaborative Humanoid Robots. In IOT with Smart Systems (pp. 65-77). Springer, Singapore. Glowinski, S., Obst, M., Majdanik, S., & Potocka-Banaś, B. (2021). Dynamic model of a humanoid exoskeleton of a lower limb with hydraulic actuators. Sensors, 21(10), 3432. Yang, F., Zhou, H., & Deng, W. (2022). Active Disturbance Rejection Adaptive Control for Hydraulic Lifting Systems with Valve Dead-Zone. Electronics, 11(11), 1788.

Robust Controller Design for Uncertain Hydraulic Actuators … [12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

25

Shi, L., Yu, X., & Bi, X. (2021, June). Application of Fuzzy Algorithm in Hydraulic Synchronous Lifting System Based on Internet. In 2021 International Wireless Communications and Mobile Computing (IWCMC) (pp. 754-758). IEEE. Wang, X., Zhang, H., Zhang, X., & Quan, L. (2021, October). Design and Efficiency Analysis of Closed Loop Pump Controlled Circuit Hydraulic Lifting System of Wheel Loaders Based on Gravity Self-Balancing Hydraulic Cylinder. In Fluid Power Systems Technology (Vol. 85239, p. V001T01A038). American Society of Mechanical Engineers. Kosucki, A., Stawiński, Ł., Morawiec, A., & Goszczak, J. (2021). Electro-Hydraulic Drive of the Variable Ratio Lifting Device under Active Load. Strojniski Vestnik/Journal of Mechanical Engineering, 67(11). Skarpetis, M. G., Koumboulis, F. N., Tzamtziou, K. A., & Michopoulos, P. (2021, October). A Robust Control Scheme for an Experimental Hydraulic System. In 2021 International Conference on Engineering and Emerging Technologies (ICEET) (pp. 1-6). IEEE. Lopes Jr, R. S., Nostrani, M. P., Carvalho, L. A., Dell’Amico, A., Krus, P., & De Negri, V. J. (2021, October). Modeling and Analysis of a Digital Hydraulic Actuator for Flight Control Surfaces. In Fluid Power Systems Technology (Vol. 85239, p. V001T01A043). American Society of Mechanical Engineers. Bertolino, A. C., De Martin, A., Jacazio, G., & Sorli, M. (2021, August). A Case Study on the Detection and Prognosis of Internal Leakages in Electro-Hydraulic Flight Control Actuators. In Actuators (Vol. 10, No. 9, p. 215). MDPI. Autin, S., De Martin, A., Jacazio, G., Socheleau, J., & Vachtsevanos, G. (2021). Results of a feasibility study of a Prognostic System for Electro-Hydraulic Flight Control Actuators. International Journal of Prognostics and Health Management, 12(3). Michela, E. (2022). Sizing and study of the regulation system of an electrohydrostatic actuator for primary flight controls in a regional transport aircraft (Doctoral dissertation, Politecnico di Torino). Juenemann, M., Kriewall, V., Bielsky, T., & Thielecke, F. (2022). Overall Systems Design Method for Evaluation of Electro-Hydraulic Power Supply Concepts for Modern Mid-Range Aircraft. In AIAA AVIATION 2022 Forum (p. 3953). Springmann, R., & Thielecke, F. (2021). Comparison of electro-hydraulic and electro-mechanic landing gear actuation for light VTOL aircraft. In Deutscher Luftund Raumfahrtkongress 2021. Berri, P. C., Dalla Vedova, M. D., & Santaera, S. (2022, February). Digital twins for prognostics of electro-hydraulic actuators: novel simplified fluid dynamic models for aerospace valves. In IOP Conference Series: Materials Science and Engineering (Vol. 1226, No. 1, p. 012106). IOP Publishing. Jordanov, D., Zagorski, N., & Tsholakov, S. (2022). Role of Power Reserve in Parallel Hydraulic Movements of Aircraft Controls. Aerospace Research in Bulgaria, 34, 95-105. Nisar, Z., & Munawar, H. (2021). System Identification and Controller Design for Hydraulic Actuator. arXiv preprint arXiv:2108.11756.

26 [25]

[26]

[27]

[28]

[29]

[30]

[31]

[32]

[33]

[34] [35]

[36] [37]

[38]

Michael G. Skarpetis and Fotis N. Koumboulis Gao, J., Du, S., Li, Y., Sun, T., & Wang, T. (2022). Research on Performance Degradation Characteristics of an Airborne Hydraulic Actuator. In Proceedings of the 5th China Aeronautical Science and Technology Conference (pp. 339-348). Springer, Singapore. Nguyen, T. A. (2021). Advance the stability of the vehicle by using the pneumatic suspension system integrated with the hydraulic actuator. Latin American Journal of Solids and Structures, 18. Nguyen, D. N., & Nguyen, T. A. (2022). Evaluate the stability of the vehicle when using the active suspension system with a hydraulic actuator controlled by the OSMC algorithm. Scientific Reports, 12(1), 1-15. Han, W., Xiong, L., Yu, Z., Zhuo, G., Leng, B., & Xu, S. (2022). Integrated Pressure Estimation and Control for Electro-hydraulic Brake Systems of Electric Vehicles Considering Actuator Characteristics and Vehicle Longitudinal Dynamics. IEEE/ASME Transactions on Mechatronics. Nguyen, D. N., Nguyen, T. A., & Dang, N. D. (2022). A Novel Sliding Mode Control Algorithm for an Active Suspension System Considering with the Hydraulic Actuator. Latin American Journal of Solids and Structures, 19. Othman, S. M., Sunar, N., Hassrizal, H. B., Ismail, A. H., Ayob, M. N., Azmi, M. M., & Hashim, M. S. M. (2021, November). Position Tracking Performance with Fine Tune Ziegler-Nichols PID Controller for Electro-Hydraulic Actuator in Aerospace Vehicle Model. In Journal of Physics: Conference Series (Vol. 2107, No. 1, p. 012064). IOP Publishing. Van Tan, V. (2021). Preventing rollover phenomenon with an active anti-roll bar system using electro-hydraulic actuators: a full car model. Journal of Applied Engineering Science, 19(1), 217-229. Nishanth, F. N. U., Bohach, G., Nahin, M. M., Van de Ven, J., & Severson, E. L. (2022). Development of an Integrated Electro-Hydraulic Machine to Electrify Offhighway Vehicles. IEEE Transactions on Industry Applications, 58(5), 6163-6174. Huang, Y., Hou, H., Na, J., He, H., Zhao, J., & Shi, Z. (2022). Output Feedback Control of Hydraulic Active Suspensions with Experimental Validation. IEEE Transactions on Circuits and Systems II: Express Briefs. Jo, T., Joo, K., & Lee, J. (2022). Control Strategy of Dual-Winding Motor for Vehicle Electro-Hydraulic Braking Systems. Energies, 15(14), 5090. Dinar, M. Q., Abid, H. J., & Khalaf, H. I. (2021). Enhanced the Damping Efficiency of Hydraulic Regenerative Suspension System Comparing with the Active and Passive Suspension Systems. Advances in Mechanics, 9(3), 108-125. Gu, W., Yao, J., Yao, Z., & Zheng, J. (2018). Robust adaptive control of hydraulic system with input saturation and valve dead-zone. IEEE Access, 6, 53521-53532. Luo, C., Yao, J., & Gu, J. (2019). Extended-state-observer-based output feedback adaptive control of hydraulic system with continuous friction compensation. Journal of the Franklin Institute, 356(15), 8414-8437. Chen, Q., Chen, H., Zhu, D., & Li, L. (2021, November). Design and analysis of an active disturbance rejection robust adaptive control system for electromechanical actuator. In Actuators (Vol. 10, No. 12, p. 307). MDPI.

Robust Controller Design for Uncertain Hydraulic Actuators … [39]

[40]

[41]

[42]

[43]

[44]

[45]

[46]

[47]

[48]

[49]

[50]

[51]

[52]

27

Dong, Y., Na, J., Wang, S., Gao, G., & He, H. (2019, July). Robust Adaptive Parameter Estimation and Tracking Control for Hydraulic Servo Systems. In 2019 Chinese Control Conference (CCC) (pp. 2468-2473). IEEE. Yao, Z., Yao, J., Yao, F., Xu, Q., Xu, M., & Deng, W. (2020). Model reference adaptive tracking control for hydraulic servo systems with nonlinear neuralnetworks. ISA transactions, 100, 396-404. Yang, G., & Yao, J. (2020). Nonlinear adaptive output feedback robust control of hydraulic actuators with largely unknown modeling uncertainties. Applied Mathematical Modelling, 79, 824-842. Yao, Z., Liang, X., Zhao, Q., & Yao, J. (2022). Adaptive disturbance observer-based control of hydraulic systems with asymptotic stability. Applied Mathematical Modelling, 105, 226-242. Guo, Q., Zuo, Z., & Ding, Z. (2020). Parametric adaptive control of single-rod electrohydraulic system with block-strict-feedback model. Automatica, 113, 108807. Song, Y., Hu, Z., & Ai, C. (2022). Fuzzy Compensation and Load Disturbance Adaptive Control Strategy for Electro-Hydraulic Servo Pump Control System. Electronics, 11(7), 1159. Vo, C. P., Dao, H. V., & Ahn, K. K. (2021). Actuator fault-tolerant control for an electro-hydraulic actuator using time delay estimation and feedback linearization. IEEE Access, 9, 107111-107123. Shen, W., Liu, S., & Liu, M. (2021). Adaptive sliding mode control of hydraulic systems with the event trigger and finite-time disturbance observer. Information Sciences, 569, 55-69. Dang, X., Zhao, X., Dang, C., Jiang, H., Wu, X., & Zha, L. (2021). Incomplete differentiation-based improved adaptive backstepping integral sliding mode control for position control of hydraulic system. ISA transactions, 109, 199-217. Abbas, M. J., Zad, H. S., Awais, M., & Ulasyar, A. (2019, July). Robust Sliding Mode Position Control of Electro-Hydraulic Servo System. In 2019 International Conference on Electrical, Communication, and Computer Engineering (ICECCE) (pp. 1-6). IEEE. Tony Thomas, A., Parameshwaran, R., Sathiyavathi, S., & Vimala Starbino, A. (2022). Improved position tracking performance of electro hydraulic actuator using PID and sliding mode controller. IETE Journal of Research, 68(3), 1683-1695. Chen, G., Jia, P., Yan, G., Liu, H., Chen, W., Jia, C., & Ai, C. (2021). Research on Feedback-Linearized Sliding Mode Control of Direct-Drive Volume Control Electro-Hydraulic Servo System. Processes, 9(9), 1676. Rahmat, M. F., Othman, S. M., Rozali, S. M., & Has, Z. (2018, October). Optimization of Modified Sliding Mode Control for an Electro-Hydraulic Actuator System with Mismatched Disturbance. In 2018 5th International Conference on Electrical Engineering, Computer Science and Informatics (EECSI) (pp. 1-6). IEEE. Temporelli, R., Boisvert, M., & Micheau, P. (2019). Control of an electromechanical clutch actuator using a dual sliding mode controller: Theory and experimental investigations. IEEE/ASME Transactions on Mechatronics, 24(4), 1674-1685.

28 [53]

[54]

[55]

[56]

[57]

[58]

[59]

[60]

[61]

[62]

[63]

[64]

[65]

[66]

Michael G. Skarpetis and Fotis N. Koumboulis Cheng, L., Zhu, Z. C., Shen, G., Wang, S., Li, X., & Tang, Y. (2019). Real-time force tracking control of an electro-hydraulic system using a novel robust adaptive sliding mode controller. IEEE Access, 8, 13315-13328. Tran, D. T., Do, T. C., & Ahn, K. K. (2019). Extended high gain observer-based sliding mode control for an electro-hydraulic system with a variant payload. International Journal of Precision Engineering and Manufacturing, 20(12), 20892100. Zhu, R., Yang, Q., Liu, Y., Dong, R., Jiang, C., & Song, J. (2022). Sliding Mode Robust Control of Hydraulic Drive Unit of Hydraulic Quadruped Robot. International Journal of Control, Automation and Systems, 20(4), 1336-1350. Rybarczyk, D., & Milecki, A. (2022). The Use of a Model-Based Controller for Dynamics Improvement of the Hydraulic Drive with Proportional Valve and Synchronous Motor. Energies, 15(9), 3111. Yao, Z., Liang, X., Jiang, G. P., & Yao, J. (2022). Model-Based Reinforcement Learning Control of Electrohydraulic Position Servo Systems. IEEE/ASME Transactions on Mechatronics. Zhang, Z., Yang, Z., Liu, S., Chen, S., & Zhang, X. (2022). A multi-model based adaptive reconfiguration control scheme for an electro-hydraulic position servo system. International Journal of Applied Mathematics and Computer Science, 32(2). Yan, H., Li, J., Nouri, H., & Xu, L. (2020). U-model-based finite-time control for nonlinear valve-controlled hydraulic servosystem. Mathematical Problems in Engineering, 2020. Sun, H., Sun, Z., & Li, S. (2022). Robust Tracking Control for Electrohydraulic System Using an Internal Model-Based Sliding Surface. Journal of Dynamic Systems, Measurement, and Control, 144(6), 061006. Ye, N., Song, J., & Ren, G. (2020). Model-based adaptive command filtering control of an electrohydraulic actuator with input saturation and friction. IEEE Access, 8, 48252-48263. Nguyen, M. H., Dao, H. V., & Ahn, K. K. (2022). Adaptive Robust Position Control of Electro-Hydraulic Servo Systems with Large Uncertainties and Disturbances. Applied Sciences, 12(2), 794. Do, T. C., Tran, D. T., Dinh, T. Q., & Ahn, K. K. (2020). Tracking control for an Electro-hydraulic rotary actuator using fractional order fuzzy PID controller. Electronics, 9(6), 926. Liang, X., Wan, Y., Zhang, C., Kou, Y., Xin, Q., & Yi, W. (2018). Robust position control of hydraulic manipulators using time delay estimation and nonsingular fast terminal sliding mode. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 232(1), 50-61. Yuan, H. B., Na, H. C., & Kim, Y. B. (2018). System identification and robust position control for electro-hydraulic servo system using hybrid model predictive control. Journal of Vibration and Control, 24(18), 4145-4159. Dang, X., Zhao, X., Dang, C., Jiang, H., Wu, X., & Zha, L. (2021). Incomplete differentiation-based improved adaptive backstepping integral sliding mode control for position control of hydraulic system. ISA transactions, 109, 199-217.

Robust Controller Design for Uncertain Hydraulic Actuators … [67]

[68]

[69]

[70]

[71]

[72]

[73]

[74] [75]

[76]

[77]

[78]

[79]

[80]

29

Deng, W., Yao, J., Wang, Y., Yang, X., & Chen, J. (2021). Output feedback backstepping control of hydraulic actuators with valve dynamics compensation. Mechanical Systems and Signal Processing, 158, 107769. Cai, Y., Ren, G., Song, J., & Sepehri, N. (2020). High precision position control of electro-hydrostatic actuators in the presence of parametric uncertainties and uncertain nonlinearities. Mechatronics, 68, 102363. Zheng, J., & Yao, J. (2019). Robust adaptive tracking control of hydraulic actuators with unmodeled dynamics. Transactions of the Institute of Measurement and Control, 41(14), 3887-3898. Yao, Z., Yao, J., Yao, F., Xu, Q., Xu, M., & Deng, W. (2020). Model reference adaptive tracking control for hydraulic servo systems with nonlinear neuralnetworks. ISA transactions, 100, 396-404. Yang, G., & Yao, J. (2020). High-precision motion servo control of double-rod electro-hydraulic actuators with exact tracking performance. ISA transactions, 103, 266-279. Guo, K., Li, M., Shi, W., & Pan, Y. (2021). Adaptive tracking control of hydraulic systems with improved parameter convergence. IEEE Transactions on Industrial Electronics, 69(7), 7140-7150. Wei, K. H., & Barmish, B. R. (1985, December). On making a polynomial Hurwitz invariant by choice of feedback gains. In 1985 24th IEEE Conference on Decision and Control (pp. 679-685). IEEE. Koumboulis, F. N., & Skarpetis, M. G. (2000). Robust triangular decoupling with application to 4WS cars. IEEE Transactions on Automatic Control, 45(2), 344-352. Skarpetis, M. G., Koumboulis, F. N., & Ntellis, A. S. (2014). A Heuristic Control Algorithm for Robust Internal Model Control with Arbitrary Reference Model. In Informatics in Control, Automation and Robotics (pp. 67-81). Springer, Cham. Skarpetis, M. G., Koumboulis, F. N., & Ntellis, A. S. (2013, June). Robust control of pneumatic clutch actuators using simulated annealing techniques. In 21st Mediterranean Conference on Control and Automation (pp. 1069-1075). IEEE. Skarpetis, M. G., Koumboulis, F. N., & Ntellis, A. S. (2012, September). Robust control algorithms for a hydraulic actuator with variable displacement vane pump. In Proceedings of 2012 IEEE 17th International Conference on Emerging Technologies & Factory Automation (ETFA 2012) (pp. 1-7). IEEE. Skarpetis, M. G., & Koumboulis, F. N. (2013, September). Robust PID controller for electro—Hydraulic actuators. In 2013 IEEE 18th Conference on Emerging Technologies & Factory Automation (ETFA) (pp. 1-5). IEEE. Koumboulis, F. N., Skarpetis, M. G., & Tzamtzi, M. P. (2006, June). Robust PI controllers for command following with application to an electropneumatic actuator. In 2006 14th Mediterranean Conference on Control and Automation (pp. 1-6). IEEE. Skarpetis, M. G., Koumboulis, F. N., & Ntellis, A. S. (2014, June). Robust position tracking for a hydraulic servo system. In 22nd Mediterranean Conference on Control and Automation (pp. 1553-1558). IEEE.

30 [81]

[82]

[83]

[84]

[85] [86]

[87] [88]

Michael G. Skarpetis and Fotis N. Koumboulis Skarpetis, M. G., Koumboulis, F. N., Panagiotakis, G., & Kouvakas, N. D. (2016). Robust PID Controller for a Pneumatic Actuator. In MATEC Web of Conferences (Vol. 41, p. 02001). EDP Sciences. Skarpetis, M. G., & Koumboulis, F. N. (2015, May). Robust stabilizability algorithms with application to force control of a hydraulic actuator. In 2015 6th International Conference on Modeling, Simulation, and Applied Optimization (ICMSAO) (pp. 1-6). IEEE. Koumboulis, F. N., Skarpetis, M. G., & Mertzios, B. G. (1998). Robust regional stabilisation of an electropneumatic actuator. IEE Proceedings-Control Theory and Applications, 145(2), 226-230. Skarpetis, M. G., Koumboulis, F. N., Ntellis, A. S., & Sarris, A. (2007, September). Robust controller design for active hydraulic suspension. In 2007 IEEE Conference on Emerging Technologies and Factory Automation (EFTA 2007) (pp. 1433-1436). IEEE. Merritt, H. E. (1967). Hydraulic Control Systems, John Wiley & Sons. New York, New York. Karpenko, M., & Sepehri, N. (2010). On quantitative feedback design for robust position control of hydraulic actuators. Control Engineering Practice, 18(3), 289299. Mirjalili, S. (2016). SCA: a sine cosine algorithm for solving optimization problems. Knowledge-based systems, 96, 120-133. Tuncer, T. (2018). SCSO: A novel sine-cosine based swarm optimization algorithm for numerical function optimization. Periodicals of Engineering and Natural Sciences (PEN), 6(2), 1-9.

Chapter 2

Flatness-Based Control in Successive Loops for Robotic and Mechatronic Systems with Electro-Hydraulic Actuation G. Rigatos1 ,∗ M. Abbaszadeh2 ,† J. Pomares3 ,‡ P. Wira4 ,§ 1

Unit of Industrial Automation, Industrial Systems Institute, Rion Patras, Greece 2 Department of ECS Eng., Rensselaer Polytechnic Institute, Troy, New York, USA 3 Department of Systems Engineering, University of Alicante, Alicante, Spain 4 IRIMAS, Universit´e de Haute Alsace, Mulhouse, France

Abstract The control problem for the nonlinear dynamics of robotic and mechatronic systems with electrohydraulic actuation is solved with the use of

∗

Corresponding Author’s Email: [email protected] Email: [email protected] ‡ Email: [email protected] § Email: [email protected] †

In: Automatic Control of Hydraulic Systems Editor: Michael G. Skarpetis ISBN: 979-8-88697-619-9 © 2023 Nova Science Publishers, Inc.

32

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira a flatness-based control approach which is implemented in successive loops. The state-space model of these systems is separated into a series of subsystems, which are connected between them in cascading loops. Each one of these subsystems can be viewed independently as a differentially flat system and control about it can be performed with inversion of its dynamics as in the case of input-output linearized flat systems. In this chain of i = 1, 2, · · · , N subsystems, the state variables of the subsequent (i+1-th) subsystem become virtual control inputs for the preceding (i-th) subsystem, and so on. In turn, exogenous control inputs are applied to the last subsystem and are computed by tracing backwards the virtual control inputs of the preceding N − 1 subsystems. The whole control method is implemented in successive loops and its global stability properties are also proven through Lyapunov stability analysis. The validity of the control method is confirmed in two case studies: (a) control of an electrohydraulic actuator, (ii) control of a multi-DOF robotic manipulator with electrohydraulic actuators.

PACS 07.05.Dz, 02.30.Yy Control theory Keywords: control systems, control theory, robotic systems, mechatronic systems, electrohydraulic actuation, nonlinear control, differential flatness properties, flatness-based control in successive loops, global stability, Lyapunov analysis AMS Subject Classification: 37N35, 70E60, 93C10, 93C35, 93D05.

1

Introduction

Control of electrohydraulic actuators, as well as of electrohydraulically actuated robots is a topic with high importance for industry that has attracted much research interest during the last years [Rigatos, 2016], [Rigatos and Busawon, 2018], [Rigatos, Abbaszadeh and Siano, 2022]. Electro-hydraulic actuation systems have been widely used in robotic manipulators, in construction machinery and in aircrafts [Kim et al., 2019], [Nguyen, 2018]. Nowadays, manipulators and vehicle receiving hydraulic actuation constitute a huge global industrial sector [Mohanty and Yao, 2011], [Taylor and Robertson, 2013], [Ding et al., 2019]. Hydraulic manipulators are favored in industry, mining, agriculture and construction works where large actuation forces are required [Liang et al., 2018], [Guo et al., 2016], [Lu and Yao, 2014]. For instance in 2016 the construction business alone sold 0.7 million units of construction machines,

Flatness-Based Control in Successive Loops for Robotic ...

33

while in 2019 the sales of robotic manipulators, including also hydraulic manipulators, reached the level of 0.4 million units. Hydraulic actuation in robotic manipulators exhibits specific technical advantages, such as large force and torque, fast response, small size-to-power ratio, reliability and low manufacturing cost [Sirouspour and Salcudean, 2001], [Mohanty and Yao, 2011], [Tafazoli et al., 2022]. Hydraulic heavy-duty manipulators are irreplaceable for heavy workpiece handling. They improve the efficiency of fine manipulation of heavy loads, thus ensuring production, safety and reducing labor costs [Koivumaki and Mattila, 2015a], [Li et al., 2019], [Wang et al., 2018]. Comparing against electric actuators, hydraulic actuators have the advantage of producing high power while having a smaller size [Henikl et al., 2016], [Wind et al., 2019]. Minimization of energy consumption is an issue that has to be taken into account in the development of such robotic systems [Koivumaki et al., 2019], [Maddani et al., 2016]. Furthermore high precision motion is a vital functionality for electro-hydraulic robotic systems [Ranjan et al., 2020], [Koivumaki and Mattila, 2015b]. Research on nonlinear control for electrohydraulic actuators is growing and several noteworthy results have been obtained during the last years. Control for electrohydraulic actuation in robotic systems can be found in [Zhum et al., 2022], [Kittissares et al., 2022], [Dao and Ahn., 2022] . Control for electrohydraulic actuation in mechatronic systems can be found in [Pham et al., 2021a], [Pham et al., 2021b], [Shen et al., 2022], [Guo et al., 2022], [Taheri et al., 2022]. Moreover, control for electrohydraulic actuation in vehicles can be found in [Cho and You, 2021], [Du et al., 2021]. The manuscript introduces a solution to the problem of nonlinear control of the aforementioned electrohydraulic actuators and of robotic manipulators with electrohydraulic actuation without the need to apply changes of state variables (diffeomorphisms) and complicated state-space transformations. A system is considered to be differentially flat if all its state variables and its control inputs can be expressed as functions of one single algebraic variable which is the so-called flat output, and also as functions of the flat-output’s derivatives [Levine, 2009], [Fliess and Mounier, 1999], [Villagra et al., 2007], [Boouden et al., 2011], [Menhour et al., 2014], [Nicolau et al., 2022], [Letelier and Barbot, 2021]. The differential flatness property enables the transformation of the nonlinear system’s dynamics in the linear canonical form [Sira-Ramirez and Agrawal, 2004], [L´evine et al., 2011], [Nicolau et al., 2021], [Limaverde Filho et al., 2021], [Barbot et al., 2007]. The latter description is controllable and observable thus allowing to treat effectively control and estimation prob-

34

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira

lems [Khalil et al., 1996], [Rigatos and Tzafestas, 2007], [Basseville and Nikiforov, 1993], [Rigatos and Zhang, 2009]. The demonstration of differential flatness properties for the dynamic model of an electrohydraulically actuated system is an important finding for two reasons (i) It is an implicit proof of the controllability of this nonlinear dynamical system thus signifying that for any targeted setpoints one can compute control inputs which enable the state variables’ convergence to these setpoints. Thus the differential flatness properties confirm that the control problem for this nonlinear dynamical system has a solution (ii) It allows to compute feasible setpoints for all state variables of the system. First, one selects in an unconstraint manner setpoints for the flat outputs of the system. Next, setpoints for the rest of the state variables are chosen subject to the constraint that these state vector elements are differential functions of the flat outputs. By assuring convergence of the flat outputs to their targeted values one can also ascertain that the rest of the state vector elements of the system will reach their desirable values. The manuscript develops a flatness-based control approach implemented in successive loops. In this method the dynamic model of the nonlinear system is separated into subsystems which are connected in a cascading manner. This control method is directly applicable to dynamical systems of the triangular form and to nonlinear systems which can be transformed into such a form. The state-space model of the initial nonlinear system is decomposed into cascading subsystems which satisfy differential flatness properties. For each subsystem of the state-space model a virtual control input is computed, capable of inverting the subsystem’s dynamics and of eliminating the subsystem’s tracking error. The control input which is actually applied to the initial nonlinear system is computed from the last row of the state-space description. This control input incorporates in a recursive manner all virtual control inputs which were computed from the individual subsystems included in the initial state-space equation. The control input that should be applied to the nonlinear system so as to assure that all state vector elements will converge to the desirable setpoints is obtained at each iteration of the control algorithm by tracing backwards the subsystems of the state-space model. The flatness-based control method which is implemented in successive loops is being developed during the last years and has been successfully applied and tested in several nonlinear dynamical systems [Rigatos et al., 2015], [Rigatos et al., 2018], [Rigatos, 2015], [Rigatos and Melkikh, 2016], [Rigatos et al., 2022a], [Rigatos et al., 2022b], [Rigatos and Siano, 2017].

Flatness-Based Control in Successive Loops for Robotic ...

35

The structure of the manuscript is as follows: In Section 2 the concept of flatness-based control in successive loops is analyzed. Stability and convergence properties for such a control scheme are proven. In Section 3 flatnessbased control in consecutive loops is implemented in the dynamic model of an electrohydraulic actuator. It is proven that this control method ensures global stability of the closed-loop of the actuator and elimination of the state variables’ tracking error. In Section 4 flatness-based control in consecutive loops is implemented for the dynamic model a multi-DOF electrohydraulically actuated robot. It is proven that this control method ensures global stability of the closed-loop of the robot and elimination of the state variables tracking error. In Section 5 the setpoints tracking performance of the flatness-based control approach which is implemented in successive loops is tested through simulation experiments. The presented results show the fast and accurate tracking of time-varying setpoints by the state variables of both the electrohydraulic actuator and of the electrohydraulically actuated robot. Finally, in Section 5.2 concluding remarks are stated.

2

2.1

Flatness-Based Control of Nonlinear Dynamical Systems in Cascading Loops Decomposition of the State-Space Model into Cascading Differentially Flat Subsystems

The following nonlinear dynamical system is now examined:

x˙ = f (x) + g(x)u x∈Rm u∈Rq y = h(x)

(1)

Moreover, it is considered that the system can be decomposed into n subsystems which have the so-called triangular form:

36

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira

x˙ n−1

x˙ 1 = f1 (x1 ) + g1 (x1 )x2 x˙ 2 = f2 (x1 , x2 ) + g2 (x1 , x2 )x3 x˙ 3 = f3 (x1 , x2 , x3 ) + g3 (x1 , x2 , x3 )x4 ··· x˙ i = fi (x1 , x2 , · · · , xi ) + gi (x1 , x2 , · · · , xi )xi+1 ··· = fn−1 (x1 , x2 , · · · , xn−1 ) + gn−1 (x1 , x2, · · · , xn−1 )xn x˙ n = fn (x1 , x2 , · · · , xn ) + gn (x1 , x2 , · · · , xn )u

(2)

The following virtual control inputs αi = xi+1 are defined for the i=th subsystem of the state-space model of Eq. (2)

x˙ n−1

x˙ 1 = f1 (x1 ) + g1 (x1 )α1 x˙ 2 = f2 (x1 , x2 ) + g2 (x1 , x2 )α2 x˙ 3 = f3 (x1 , x2 , x3 ) + g3 (x1 , x2 , x3 )α3 ··· x˙ i = fi (x1 , x2, · · · , xi) + gi (x1 , x2 , · · · , xi)αi ··· = fn−1 (x1 , x2 , · · · , xn−1 ) + gn−1 (x1 , x2 , · · · , xn−1 )αn−1 x˙ n = fn (x1 , x2 , · · · , xn) + gn (x1 , x2 , · · · , xn )u

(3)

The system of Eq. (3) is a differentially flat one. It is considered that y = x1 is the flat output of system. It can be easily shown that each virtual control input αi = xi+1 , i = 1, 2, · · · can be expressed as a function of the flat output and its derivatives, since it holds (4) For i = 1 one has (5) which means that α1 is a function of the flat output and its derivative. For i = 2 one has α2 =

1 (x˙ g2 (x1 ,x2 ) 2

− f2 (x1 , x2 ))

(6)

Flatness-Based Control in Successive Loops for Robotic ...

37

which means that α2 = x3 is a function of the flat output y = x1 and its derivatives. Continuing in a similar manner one has that αn−1 = xn and consequently αn = u is also a function of the flat output y = x1 and its derivatives. According to the above, one has a nonlinear dynamical system in which, all its state variables and the control input can be written as functions of the flat output and its derivatives. Therefore, such a system is differentially flat. Additionally, by considering each subsystem of the model of Eq. (3), one has a set of n subsystems of the form x˙ i = fi (x1 , x2 , · · · , xi ) + gi (x1 , x2 , · · · , xi )αi

(7)

where each subsystem describes the dynamics of the single state variable xi . For each one of these subsystems one can consider the state variable xi as the flat output. Obviously, the virtual control input αi is a function of this flat output and its derivatives. Therefore, each local subsystem is also differentially flat. Next, one can compute the virtual inputs which are applied to each subsystem. For the first subsystem, which is associated with the first row of Eq. (2), and by defining zi = xi − x∗i = x1 − αi−1 , the virtual control input is given by

From the second row of Eq. (2), and using that z2 = x2 − x∗2 = x2 − α1 one has α2 = x∗3 =

1 (x˙ ∗ g2 (x1,x2 ) 2

− f2 (x1 , x2 ) − K12 (x2 − x∗2 )) ⇒ (9)

α2 = x∗3 =

1 ˙1 g2 (x1 ,x2 ) (α

− f2 (x1 , x2 ) − K12 z2 )

From the third row of Eq. (2), and using that z3 = x3 − x∗3 = x3 − α2 one has

α3 = x∗4 =

1 ˙ ∗3 g3 (x1 ,x2 ,x3 ) (x

− f3 (x1 , x2 , x3 ) − K13 (x3 − x∗3 )) ⇒ (10)

α3 = x∗4 =

1 ˙2 g3 (x1,x2 ,x3 ) (α

− f3 (x1 , x2 , x3 ) − K13 z3 )

38

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira

Continuing in a similar manner and from the i-th row of the state-space description of the system given in Eq. (2), and while also using that zi = xi − x∗i = xi − αi−1 one obtains αi = x∗i+1 =

1 ˙ ∗i gi (x1 ,x2 ,··· ,xi ) (x

αi = x∗i+1 =

− fi (x1 , x2, · · · , xi) − K1i (xi − x∗i )) ⇒

1 ˙ i−1 gi (x1 ,x2 ,··· ,xi ) (α

− fi (x1 , x2 , · · · , x3 ) − K1i zi )

(11) Equivalently, from the (n − 1)-th row of the state-space model of Eq. (2) and using that zn−1 = xn−1 − x∗n−1 = xn−1 − αn−2 one has

1 ˙ ∗n−1 gn−1 (x1 ,x2 ,··· ,xn−1 ) (x

αn−1 = x∗n = − fn−1 (x1 , x2 , · · · , xn−1) − K1n−1 (xn−1 − x∗n−1 )) ⇒

1 ˙ n−2 gn−1 (x1 ,x2 ,··· ,xn−1 ) (α

αn−1 = x∗n = − fn−1 (x1 , x2 , · · · , xn−1) − K1n−1 zn−1 ) (12)

Finally, from the n-th row of the state-space model of Eq. (2) and using that zn = xn − x∗n = xn − αn−1 one has

αn = u =

1 (x˙ ∗ gn (x1 ,x2 ,··· ,xn ) n

αn = u =

− fn (x1 , x2, · · · , xn) − K1n (xn − x∗n )) ⇒

1 ˙ n−1 gn (x1 ,x2 ,··· ,xn ) (α

− fn (x1 , x2 , · · · , xn) − K1n zn )

(13) The computation of the control input u that should be actually applied to the nonlinear system is performed in a recursive manner by processing backwards the virtual control inputs described in Eq. (8) to Eq. (13). Thus, from the last subsystem of the state-space description the control input that is actually applied to the nonlinear system is found. This control input contains recursively all virtual control inputs which were computed for the individual subsystems associated with the state-space equation. Thus, by tracing the rows of the state-space model backwards, at each iteration of the control algorithm, one can finally obtain the control input that should be applied to the

Flatness-Based Control in Successive Loops for Robotic ...

39

nonlinear system so as to assure that all its state vector elements will converge to the desirable setpoints.

2.2

Tracking Error Dynamics for Flatness-Based Control in Successive Loops

By substituting Eq. (13) into the last row of the state space model of Eq. (2), and using the definition xn − an−1 = zn , one obtains: x˙ n = a˙ n−1 − K1n (xn − αn−1 )⇒ (x˙ n − a˙ n−1 ) + K1n (xn − αn−1 ) = 0⇒ z˙n + K1n zn = 0

(14)

By substituting Eq. (12) into the last row of the state space model of Eq. (2), and using the definition xn−1 − an−2 = zn−1 , one obtains: x˙ n−1 = a˙ n−2 − K1n−1 (xn−1 − αn−2 )⇒ (x˙ n−1 − a˙ n−2 ) + K1n−1 (xn−1 − αn−2 ) = 0⇒ z˙ n−1 + K1n−1 zn−1 = 0

(15)

By substituting Eq. (11) into the last row of the state space model of Eq. (2), and using the definition xi − ai−1 = zi , one obtains: x˙ i = a˙ i−1 − K1n−1 (xi − αi−1 )⇒ (x˙ i − a˙ i−1 ) + K1i (xi − αi−1 ) = 0⇒ z˙i + K1i zi = 0

(16)

while continuing backwards and by substituting Eq. (9) into the second row of the state space model of Eq. (2), and using the definition x2 − a1 = z2 , one gets: x˙ 2 = a˙ 1 − K12 (x2 − α1 )⇒ (x˙ 2 − a˙ 1 ) + K12 (x2 − α1 ) = 0⇒ z˙2 + K12 z2 = 0

(17)

Finally, by substituting Eq. (8) into the first row of the state space model of Eq. (2), one has: x˙ 1 = x˙ d1 − K11 (x1 − xd1 )⇒ (x˙ 1 − x˙ d1 ) + K11 (x1 − xd1 ) = 0⇒ z˙1 + K11 z1 = 0

(18)

40

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira

Therefore, after the application of the feedback control law, the closed-loop dynamics becomes z˙1 + K11 z1 = 0, z˙2 + K12 z2 = 0, z˙3 + K13 z3 = 0, · · · , z˙i + K1i zi = 0, · · · , z˙n−1 + K1n−1 zn−1 = 0, z˙n + K1n zn = 0. In matrix form, the closed-loop dynamics is written as 0

B B B B B B B B B @z˙

z˙1 z˙2 z˙3 ··· z˙i ···

n−1

z˙n

1 C C C C C C C C C A

0

B B B B B =B B B B B @

−K11 0 0 ··· 0 ··· 0 0

0 −K12 0 ··· 0 ··· 0 0

0 0 −K13 ··· 0 ··· 0 0

··· ··· ··· ··· ··· ··· ··· ···

0 0 0 ··· −K1i ··· 0 0

··· ··· ···

0 0 0

···

0

··· ···

−K1n−1 0

0 0 0

10

1 z1 C B z2 C CB C B z3 C C CB CB ··· C C CB C 0 C zi C CB C CB · · · B C C@ A A z 0 n−1 n zn −K1

(19)

or equivalently Z˙ = KZ

(20)

By selecting the eigenvalues of matrix K to be in the left complex semiplane, one has that limt→∞ Z = 0n×1

(21)

which also implies that limt→∞ x1 = xd1 , limt→∞ x2 = α1 = xd2 , limt→∞ x3 = α2 = xd3 , · · · , limt→∞ xi = αi−1 = xdi , · · · , limt→∞ xn−1 = αn−2 = xdn−1 , and limt→∞ xn = αn−1 = xdn . To prove asymptotic stability for the proposed control scheme the following Lyapunov function can be defined V =

PN

1 2 i=1 2 zi

(22)

The time derivative of the aforementioned Lyapunov function is V˙ =

PN

P ˙ = − N K i z 2 ⇒V˙ < 0 i=1 1 i

i=1 zi z˙i ⇒V

(23)

By selecting the feedback control gains K1i , i = 1, · · · , n to be K1i > 0, the asymptotic stability of the control loop is assured.

Flatness-Based Control in Successive Loops for Robotic ...

3

3.1

41

Flatness-Based Control in Successive Loops for Electrohydraulic Actuators Dynamic Model of the Electrohydraulic Actuator

The first test-case in the present manuscript is concerned with electrohydraulic actuators under flatness-based control in successive loops. Electrohydraulic actuators are extensively used in industry, in robotics and in avionics, because of their capability to produce high power and torque [Guo et al., 2018], [Guo, Wang and Liu, 2018], [Guo et al., 2016], [Lyn et al., 2019]. Electrohydraulic actuators have a small size-to-power ratio, large force output and high power to weight ratio. In particular, they are used in positioning, transfer of heavy loads and lifting tasks [Kim et al., 2012], [Guo et al., 2017]. The dynamic model of electrohydraulic actuators comprises three parts, namely the electrical, the hydraulic and the mechanical part [Kaddissi et al., 2011], [Mintsa et al., 2012], [Ding et al., 2017]. Their control problem exhibits particular difficulties due to nonlinearities of the related state-space model, as well as due to underactuation [Yang et al., 2016], [Sofiane et al., 2015], [Dinh et al., 2018]. So far several control methods have been proposed for this type of actuators, such as model predictive control, sliding-mode control and Lyapunov theorybased adaptive control [Yang and Yao, 2019], [Guo et al., 2013], [Pisaturo and Senatore, 2015], [Shi et al., 2015]. Another problem that arises is the difficulty in measuring the complete set of state variables of the actuator, as for instance pressures in its forward and return chamber [Ali et al., 2016], [Guo et al., 2016], [Kim et al., 2013], [Sakain and Tsuji, 2017]. Precision in the positioning of the load as well as fast convergence to the targeted positions are the primary objectives in the design of the actuator’s control loop [Ren et al., 2019], [Li et al., 2018], [Li and Wang, 2019], [Karpenko and Sepehri, 2009]. The diagram of a typical electrohydraulic actuator is given in Fig. 1. With the use of a pump, hydraulic fluid flows under pressure into the cylinder’s chambers through cylinder ports which are controlled by servovalves. The dynamic model of the electrohydraulic actuator comprises the electrical, the mechanical and the hydraulic part: (a) the electrical part comprises the servovalves which control the fluid dynamics into the cylinder’s chambers, Each servovalve is driven by the electrical input current. The displacement of the valve, together with the load’s pressure control the fluid’s dynamics in the two chambers of the cylinder (b) the hydraulic part of the actuator is associated with the dynamics of

42

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira

the fluid in the chambers of the cylinder, (c) the mechanical part of the actuator is associated with the moving piston and with the load which is attached to it.

Figure 1: Diagram of the electrohydraulic actuator which is controlled by a servo-valve The fluid flow through the servovalve is given by ρ

where PL is the load pressure of the cylinder, xv is the position of the servovalve, Ps is the supply pressure of the pump, Cd is the discharge coefficient, w is the area gradient of the valve’s sppols, ρ is the viscosity of the flowing fluid. The continuous-time flow-pressure of the hydraulic cylinder is QL = Ap dxp dt + Ctl PL +

Vt dPL 4βe dt

(25)

where xp is the position of the cylinder’s piston, Ctl is the coefficient of the total leakage of the cylinder, βe is the effective bulk modulus, AP is the effective area of the cylinder’s chamber and Vt is the half volume of the cylinder. A relation about the motion of the mechanical load is:

Flatness-Based Control in Successive Loops for Robotic ...

Ap PL = m¨ xp + bxp + kxp + FL

43

(26)

This comes by simplifying the main viscous function of the hydraulic oil and of Coulomb friction. In this relation m is the mass of the load, k is the load-spring constant, b is the viscous function coefficient and FL is the force of the load. The motion of the electrovalve is described by a relation which connects the valve’s spool position and the associated input control voltage. This takes the form Tsw x˙ v + xv = Ksv u

(27)

where Ksv is the gain of the control input voltage u and Tsv is a response-time constant of the servovalve. Next, by defining the state-vector of the electro-hydraulic actuator as x = [x1 , x2 , x3 , x4 ]T = [xp, x˙ p, PL , xv ]T , the state-space model of the electrohydraulic actuator is written as

x˙ 3 =

x˙ 1 = x2 1 x˙ 2 = m [−kx1 − bx2 + App x3 − FL ] 4βe Ctl 4βe C P dw √ − 4βVeA x − x + Ps − sign(x4 )x3 x4 2 3 Vt Vt ρ t 1 Ksv x˙ 4 = − Tsv x4 + Tsv u

(28)

where FL is a force due to the load that is moved by the actuator. In matrix form, the state-space model of the electrohydraulic actuator is written as 1 0 0 1 0 1 x2 0 x˙ 1 1 B C [−kx1 − bx2 + Ap x3 − FL ] Bx˙ 2 C B 0 C m C B B C = B 4βe AP Cu wp +B e Ctl @x˙ 3 A @− V √d x2 − 4βV x3 + 4βVe C Ps − sign(x4 )x3 x4 C A @ 0 A t t t ρ K sv x˙ 4 − 1 x4 T Tsv

(29)

sv

Equivalently, the dynamic model of the electrohydraulic actuator can be written in the nonlinear affine-in-the-input state-space form x˙ = f (x) + g(x)u where x∈R4×1 , f (x)∈R4×1 , g(x)∈R4×1 and u∈R.

(30)

44

3.2

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira

Proof of Differential Flatness Properties

It will be shown that the dynamic model of the electrohydraulic actuator is differentially flat, with flat output Y = x1 The state-space model of the electrohydraulic actuator that was previously defined in Eq. (29) is now re-defined as follows: x˙ 1 = f1 (x1 ) + g1 (x1 )x2 x˙ 2 = f2 (x1 , x2 ) + g2 (x1 , x2 )x3 x˙ 3 = f3 (x1 , x2 , x3 ) + g3 (x1 , x2 , x3 )x4 x˙ 4 = f4 (x1 , x2 , x3 , x4 ) + g4 (x1 , x2 , x3 , x4 )u

(31)

1 f1 (x1 ) = 0, g1 (x1 ) = 1. f2 (x1 , x2 ) = m [−kx1 − bx2 − FL ], Ap 4βe AP 4βe Ctl g2 (x1 , x2 ) = m , f3 (x1 , x2 , x3 ) = − Vt x2 − Vt x3 , g3 (x1 , x2 .x3 ) = p 4βe C √d w Ps − sign(x4 )x3 , f4 (x1 , x2, x3 , x4 ) = 0, g4 (x1 , x2 , x3 , x4) = − T1sv Vt ρ

From the first row of the state-space model in the form of Eq. (31) one solves for x2 . This gives x2 = x˙ 1 therefore x2 is a differential function of the flat output of the system or x2 = h2 (Y, Y˙ )

(32)

From the second row of the state-space model in the form of Eq. (31) one solves for x3 . This gives x3 = g2 (x1 , x2 )−1 (x˙ 2 − f2 (x1 , x2 )) therefore x3 is a differential function of the flat output of the system or x3 = h3 (Y, Y˙ )

(33)

From the third row of the state-space model in the form of Eq. (31) one solves for x4 . This gives x4 = g2 (x1 , x2, x3 )−1 (x˙ 3 − f3 (x1 , x2 , x3)) therefore x4 is a differential function of the flat output of the system or x4 = h4 (Y, Y˙ )

(34)

From the fourth row of the state-space model in the form of Eq. (31) one solves for u. This gives u = g4 (x1 , x2 , x3, x4 )−1 (x˙ 4 − f4 (x1 , x2 , x3 , x4 )) therefore the control input u is a differential function of the flat output of the system or u = hu (Y, Y˙ )

(35)

Flatness-Based Control in Successive Loops for Robotic ...

45

Consequently, all state variables and the control inputs of the dynamic model of the electrohydraulic actuator are differential functions of the flat output Y = x1 and the system is differentially flat. Next, it will be shown that the dynamic model of the electrohydraulic actuator can undergo a per-row decomposition, where each one of its rows is also a differentially flat subsystem. In this decomposition, the i-th subsystem which is associated with state variable xi is considered to have the virtual control input xi+1 . In the subsystem of the first row of the state-space model of Eq. (31) the flat output is x1 and the virtual control input is x2 . Obviously, x2 = x˙ 1 and thus x2 is a differential function of the flat output x1 . Consequently, this subsystem is differentially flat. In the subsystem of the second row of the state-space model of Eq. (31) the flat output is x2 and the virtual control input is x3 , while x1 is viewed as a coefficient. Obviously, x3 = g2 (x1 , x2 )−1 (x˙ 2 − f (x1 , x2 )) and thus x3 is a differential function of the flat output x2 . Consequently, this subsystem is differentially flat. In the subsystem of the third row of the state-space model of Eq. (31) the flat output is x3 and the virtual control input is x4 , while x1 , x2 are viewed as coefficients. Obviously, x4 = g3 (x1 , x2 , x3 )−1 (x˙ 3 − f (x1 , x2 , x3 , x4 )) and thus x4 is a differential function of the flat output x3 . Consequently, this subsystem is differentially flat. In the subsystem of the third row of the state-space model of Eq. (31) the flat output is x4 and the real control input is u, while x1 , x2 , x3 are viewed as coefficients. Obviously, u = g4 (x1 , x2, x3 , x4 )−1 (x˙ 4 − f (x1 , x2 , x3 , x4 )) and thus u is a differential function of the flat output x4 . Consequently, this subsystem is differentially flat.

3.3

Design of a Flatness-Based Controller in Successive Loops

Next, for each one of the differentially flat subsystems of Eq. (31) one can design a stabilizing feedback controller following the controller definition stages

46

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira

for input-output linearized state-space descriptions. For the subsystem of the first row of Eq. (31) the setpoint is xd1 and the virtual control input that satisfies convergence to this setpoint is x∗2 = x˙ d1 − K11 (x1 − xd1 )

(36)

where K11 > 0. By substituting Eq. (36) into the first row of Eq. (31) one obtains the closed subsystem error dynamics (x˙ 1 − x˙ d1 ) + K11 (x1 − xd1 ) = 0⇒e˙ 1 + K11 e1 = 0 ⇒limt→∞ e1 (t) = 0⇒limt→∞ x1 (t) = xd1 (t)

(37)

For the subsystem of the second row of Eq. (31) the setpoint is x∗2 , that is xd2 = x∗3 and the virtual control input that satisfies convergence to this setpoint is x∗3 = g2 (x1 , x2 )−1 [x˙ d2 − f2 (x1 , x2 ) − K12 (x2 − xd2 )]

(38)

where K12 > 0, By substituting Eq. (38) into the second row of Eq. (31) one obtains the closed subsystem error dynamics (x˙ 2 − x˙ d2 ) + K12 (x2 − xd2 ) = 0⇒e˙ 2 + K12 e2 = 0 ⇒limt→∞ e2 (t) = 0⇒limt→∞ x2 (t) = xd2 (t)

(39)

For the subsystem of the third row of Eq. (31) the setpoint is x∗3 , that is xd3 = x∗3 and the virtual control input that satisfies convergence to this setpoint is x∗4 = g3 (x1 , x2 , x3 )−1 [x˙ d3 − f3 (x1 , x2 , x3 ) − K13 (x3 − xd3 )]

(40)

where K13 > 0. By substituting Eq. (40) into the third row of Eq. (31) one obtains the closed subsystem error dynamics (x˙ 3 − x˙ d3 ) + K13 (x3 − xd3 ) = 0⇒e˙ 3 + K13 e3 = 0 ⇒limt→∞ e3 (t) = 0⇒limt→∞ x3 (t) = xd3 (t)

(41)

For the subsystem of the fourth row of Eq. (31) the setpoint is x∗4 , that is xd4 = x∗4 and the real control input that satisfies convergence to this setpoint is u = g4 (x1 , x2, x3 , x4 )−1 [x˙ d4 − f4 (x1 , x2 , x3 .x4 ) − K14 (x4 − xd4 )]

(42)

Flatness-Based Control in Successive Loops for Robotic ...

47

where K14 > 0 By substituting Eq. (42) into the fourth row of Eq. (31) one obtains the closed subsystem error dynamics (x˙ 4 − x˙ d4 ) + K14 (x4 − xd4 ) = 0⇒e˙ 4 + K14 e4 = 0 ⇒limt→∞ e4 (t) = 0⇒limt→∞ x4 (t) = xd4 (t)

(43)

Consequently, all state variables xi , i = 1, 2, 3, 4 converge to the associated setpoints, the tracking error is eliminated and the control system of the electrohydraulic actuator is globally asymptotically stable. Global asymptotic stability properties can be also proven through Lyapunov analysis. The following Lyapunov function is chosen V =

1 P4 2 i=1 ei 2

(44)

By differentiating the above Lyapunov function with respect to time one gets V˙ =

P 1 P4 ˙ i ⇒V˙ = 4i=1 ei (−Ki ei ) i=1 2eie 2 P ⇒V˙ = − 4i=1 Ki e2i ⇒V˙ < 0

(45)

It holds that V˙ is strictly negative ∀ei 6=0 i = 1, 2, 3, 4. It becomes 0 only when ei = 0 for i = 1, 2, 3, 4. Therefore, the Lyapunov function of the system is strictly diminishing and finally converges to 0, no matter what the initial conditions of the electrohydraulic actuator are. Thus, the control loop of the electrohydraulic actuator is globally asymptotically stable.

4 4.1

Flatness-Based Control in Successive Loops for Electrohydraulic Robots Dynamic Model of the Electrohydraulic Robot

The second test-case in this manuscript is concerned with electrohydraulically actuated robotic manipulators under flatness-based control in successive loops. The control of electro-hydraulic robotic manipulators remains a challenging problem that exhibits a higher degree of complexity than the control of electrically actuated manipulators. This is because electro-hydraulic manipulators have complicated nonlinear dynamics, and are subject to modeling uncertainties (such as friction, internal leakage, deadbands in valves operation and control saturation). Besides, the functioning of electro-hydraulic manipulators is subject to several parametric variations, such as uncertain or varying loads.

.

48

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira

Heavy loads lead to significant rigid-flexible coupling characteristics and mechanical deformation in multi-DOF hydraulic manipulators. For this reason the design of reliable controllers for hydraulic manipulators is acknowledged to be a demanding task. Various control methods have been developed for electrohydraulic robots. Most of the results on the control of these manipulators exclude actuator dynamics from the robots’ dynamic behavior. However, actuator dynamics affects also the dynamics of the entire robotic system and the stability properties of the robots’ control loop [Hera et al., 2009], [Lee and Chung, 2019]. The design of control and actuation schemes for hydraulic manipulators can be improved if rotary hydraulic actuators are used in place of linear hydraulic actuators [Heidtmann and Brucker, 2005], [Brucker and Heidtmann, 2004], [Liang et al., 2019]. Rotary vane actuators, functioning as rotational drives, provide rotational movements directly because they are constructed as joints and actuators in one. Nevertheless, these actuators may exhibit internal leakage and high friction. Therefore, the controller design for hydraulic manipulators should compensate for such perturbations too. Indicative results about the control of electrohydraulically actuated multi-DOF robotic manipulators can be found in [Hyan et al., 2020]. [Gao et al., 2020], [Li et al., 2019], [Liang et al., 2018], [Kalmari et al., 2017]. The diagram of the multi-DOF robotic manipulator with electrohydraulic actuators is given in Fig. 2. Under the assumption that masses are concentrated, the dynamic model of the 2-DOF robotic manipulator is given by ˙ + G(θ) = τ M (θ)θ¨ + C(θ, θ)

(46)

where M (θ) is the symmetric inertia matrix of the robot

M (θ) =



M11 M12 M21 M22



=

  (m1 + m2 )l12 + m1 l22 + 2m2 l1 l2 cos(θ2 ) m2 l22 + m2 l1 l2 cos(θ) = m2 l22 + m2 l1 l2 cos(θ) m2 l2 (l1 + l2 )cos(θ2 ) (47) The Coriolis forces vector is given by ˙ = C(θ, θ)

  −m2 l12 sin(θ2 )θ˙22 − 2m2 l12 sin(θ2 )θ˙1 θ˙2 m2 l12 sin(θ2 )θ˙12

(48)

Flatness-Based Control in Successive Loops for Robotic ...

49

The gravitational forces vector is given by G(θ) =

  (m1 + m2 )gl1cos(θ1 ) + m2 gl2cos(θ1 + θ2 ) m2 gl2cos(θ1 + θ2 )

(49)

Figure 2: Diagram of the multi-DOF electrohydraulic robotic manipulator In Eq. (46) τ (t) is the torques (control inputs) vector which are generated by rotary electrohydraulic actuators mounted on the joints of the robot. It holds that τ (t) = [τ1 , τ2 ]T and by considering elasticity in the second joint one has τ (t) = [τ1 , τ2 + k(θ1 − θ2 )]T . 2 and the The determinant of the inertia matrix is detM = M11 M22 − M12 inverse of the inertial matrix is given by M −1

=

1 detM



M22 −M12 −M12 M11



(50)

and next the dynamics of the two-DOF electrohydraulic robotic manipulator is written as

50

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira

˙ − M −1 (θ)G(θ) + M −1 (θ)τ θ¨ = −M −1 (θ)C(θ, θ)

(51)

which is also written as

   −M2 2(C1 +G1 )+M12 (C2 +G2 )   M2 2 θ¨1 detM   detM  =  + M1 2(C1+G1 )−M11 (C2 +G2 ) ¨ − M12 θ2

M12 − detM

  τ1   M11 τ2 detM detM detM (52) The torques which are generated by the electropneumatic actuators are given by

τ1 = KP1 AP1 PL1

τ2 = KP2 AP2 PL2

(53)

where KPi , i = 1, 2 are force to torque conversion coefficients for the i-th actuator, Api i = 1, 2 is the efficient area of the cylinder’s chamber of the i-th actuator, PLi , i = 1, 2 is the differential load pressure of the cylinder of the i-th actuator. Moreover, the dynamics of pressure in the two actuators is given by

4βe A P˙ L1 = − V1t P1 r1 w1 +

4βe1 Ctl1 PL1 V t1

+

4βe A P˙ L2 = − V2t P2 r2 w2 +

4βe2 Ctl2 PL2 V t2

+

1

4βe1 Cd1 w1 p √ Ps1 Vt1 ρ1

4βe2 Cd2 w2 p √ Ps2 Vt2 ρ2

− sign(xv1 )PL1 xv1 (54)

− sign(xv2 )PL2 xv2 (55) Next, by defining the state variables x1 = θ1 , x2 = θ˙1 , x3 = θ2 , x4 = θ˙2 , x5 = PL1 , x6 = xv1 , x7 = PL2 , x8 = xv2 one arrives at the following statespace description 2

Flatness-Based Control in Successive Loops for Robotic ...

51

x˙ 1 = x2 M22 M12 + detM KP1 AP1 x5 − detM KP2 AP2 x7 x˙ 2 = x˙ 3 = x4 M (C +G )−M (C +G ) M12 M11 x˙ 4 = 12 1 1detM 11 2 2 − detM KP1 AP1 x5 + detM KP2 AP2 x7 p 4βe1 Ctl1 4βe1 Cd1 x2 4βe1 AP1 Ps1 − sign(x6 )x5 x6 x˙ 5 = − Vt r1 x2 + Vt x5 + Vt √ρ1 −M22 (C1 +G1 )+M12 (C2 +G2 ) detM

1

x˙ 7 =

1

1

Ksv1 Tsv1 u1 1 4βe2 Ctl2 4βe2 Cd2 x4 p 4βe2 AP2 − Vt r2 x4 + Vt x7 + Vt √ρ2 Ps2 2 2 2 K sv2 1 x˙ 8 = − Tsv x8 + Tsv u2 2 2 1 x˙ 6 = − Tsv x6 +

− sign(x8 )x7 x8

(56) The dynamic model of the electrohydraulic robot can be also written in matrix form 0

0

1

˙ C Bx B 1C B C B C B C B C B C B C B C B C B C Bx C B ˙ 2C B C B C B C B C B C B C B C B C B C Bx C ˙ B 3C B C B C B C B C B C B C B C B C B C B C Bx ˙ C 4 B C B C B C B C B C B C B C B C B C B C Bx C B ˙ 5C B C B C B C B C B C B C B C B C B C Bx C B ˙ 6C B C B C B C B C B C B C B C B C B C B C Bx ˙ C B 7C B C B C B C B C B C B C B C @ A

=

x ˙8

B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @

1

x2

C C C C C C C C C C C −M22 (C1 +G1 )+M12 (C2 +G2 ) M22 M12 + KP AP x5 − KP AP x7 C C detM detM detM C 1 1 2 2 C C C C C C C C C C x4 C C C C C C C C C C M12 (C1 +G1 )−M11 (C2 +G2 ) M12 M11 − KP AP x5 + KP AP x7 C C detM detM detM 1 1 2 2 C C C C C C C C 4βe1 AP 4βe1 Ctl 4βe1 Cd x2 q C C 1 r x + 1x + 1 − √ P − sign(x )x x s 1 2 5 6 5 6 C 1 Vt Vt Vt ρ1 C C 1 1 1 C C C C C C C C C − 1 x6 C Tsv1 C C C C C C C C C 4βe2 AP 4βe2 Ctl 4βe2 Cd x4 q C 2 r x + 2x + 2 C √ − P − sign(x )x x s2 C 2 4 7 8 7 8 C Vt Vt Vt ρ2 2 2 2 C C C C C C C C A 1

−

Tsv2

x8

0

+

B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @

0

0

0

0

0

0

0

0

0

0

Ksv1 Tsv1

0

0

0

0

Ksv2 Tsv2

1 C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C A

0

1

Bu C B 1C B C B C B C B C B C B C B C B C @ A

u2

(57)

Equivalently, the dynamic model of the electrohydraulic robotic manipulator can be written in the nonlinear affine-in-the-input state-space form x˙ = f (x) + g(x)u where

x∈R8×1 ,

f (x)∈R8×1 ,

g(x)∈R8×2 ,

and u∈R.

(58)

52

4.2

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira

Proof of Differential Flatness Properties

First, a redefinition of the state variables of the electrohydraulic robot is performed as follows: z1 = x1 , z2 = x3 , z2 = x2 , x4 = x4 , z5 = x5 , z6 = x7 , z7 = x6 and z8 = x8 . Using the redefinition of the state variables of the robot, its state equations are re-written as: z˙1 = z2 z˙2 = z4 −M22 (C1 +G1 )+M12 (C2 +G2 ) M22 M12 z˙3 = + detM KP1 AP1 z5 − detM KP2 AP2 z6 detM M12 (C1 +G1 )−M11 (C2 +G2 ) M11 M12 z˙4 = − detM KP1 AP1 z5 + detM KP2 AP2 z6 detM 4βe1 AP1 4βe1 Ctl1 4βe C z p z˙5 = − Vt r1 z3 + Vt z5 + Vt1 √dρ11 3 Ps1 − sign(z7 )z5 z7 1 1 1 4βe A 4βe C 4βe C z p z˙6 = − V2t P2 r2 z4 + V2t tl2 z6 + Vt2 √dρ22 4 Ps2 − sign(x8 )z6 z8 2

2

1 z˙7 = − Tsv z7 + 1

1 z˙8 = − Tsv z8 + 2

2

Ksv1 Tsv1 u1 Ksv2 Tsv2 u2

(59) Equivalently, the state-space model of the electrohydraulic robot can written in the form of four serially interconnected subsystems        z˙1 0 1 0 z3 = + (60) z˙2 0 0 1 z4 „ « z˙ 3 = z˙ 4

−M22 (C1 +G1 )+M12 (C2 +G2 ) detM M12 (C1 +G1 )−M11 (C2 +G2 ) detM

!

+

M22 detM KP1 AP1 M12 − detM KP1 AP1

„

M12 − detM KP2 AP2 M11 K P2 AP2 detM

«„ « z5 z6

(61)

  4βe1 Ctl1 4βe1 AP1   r z + z − 1 3 5 z˙5 V V t1 + =  4βe tA1 P 4βe2 Ctl2 2 2 z˙6 − Vt r2 z4 + Vt z6 2 2   4βe1 Cd1 z3 p   √ P − sign(z )z 0 s 7 5 1 V ρ t 1 1  z7 + 4βe2 Cd2 z4 p z8 √ 0 Ps2 − sign(x8 )z6 Vt2 ρ2 (62)   !   Ksv1   1 − Tsv z7 0 z˙ 7 u1 Tsv1 1   = + (63) 1 Ksv2 z˙ 8 − Tsv z8 u2 0 2

Tsv2

Flatness-Based Control in Successive Loops for Robotic ...

53

In the above noted subsystems of Eq. (60) to Eq. (63) the following vectors and matrices are defined z12 =

  z1 z2

f12 =

  0 0

z34 =

f34 =

−M22 (C1 +G1 )+M12 (C2 +G2 ) detM M12 (C1 +G1 )−M11 (C2 +G2 ) detM

!

„

z2 z4

g12 =



 1 0 0 1

(64)

«

g34 =

M22 K A detM P1 P1 − M12 K A detM P1 P1

M12 KP2 AP2 − detM M11 K A detM

P2

P2

!

(65)

g56

  4βe C 4βe A   − V1t P1 r1 z3 + V1t tl1 z5 z5 1  z56 = f56 =  4βe A1 P 4βe2 Ctl2 2 2 z6 − Vt r2 z4 + Vt z6 2 2   4βe1 Cd1 z3 p √ P − sign(z )z 0 s 7 5 1 V ρ  =  t1 1 4βe2 Cd2 z4 p √ P − sign(x )z 0 s2 8 6 Vt2 ρ2 (66)

z78

  z = 7 z8

f78 =

1 − Tsv z7 1 1 − Tsv z8 2

!

g78 =



Ksv1  Tsv1

0

0 Ksv2 Tsv2

 

(67)

Using the previous notation one arrives at a concise state-space description of the form: z˙12 = f12 + g12 z34

(68)

z˙34 = f34 + g34 z56

(69)

z˙56 = f56 + g56 z78

(70)

z˙78 = f78 + g78 u

(71)

54

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira

Next, it will be proven that the dynamic model of the electrohydraulic manipT = [z , z ]T = [θ , θ ]T . ulator is differentially flat, with flat output Y = z12 ! 2 1 2 From Eq. (68) it holds that −1 z34 = g12 (z˙12 − f12 )

(72)

where f12 and g12 have constant elements. Thus, z34 is a differential function of the flat outputs vector or z34 = h34 (Y, Y˙ )

(73)

From Eq. (69) one solves for z56 which gives −1 z56 = g34 (z˙34 − f34 )

(74)

It holds that f34 , g34 are functions of z12 and z34 , and consequently they are finally differential functions of the flat outputs vector z12 . Thus, z56 is a differential function of the flat outputs vector z12 . Therefore, z56 is a differential function of the flat outputs vector z12 or z56 = h56 (Y, Y˙ )

(75)

Additionally, from Eq. (70) one solves for z78 . Thus. one obtains −1 z78 = g56 (z˙56 − f56 )

(76)

It holds that f56 , g56 are functions of z12 , z34 , z56 and therefore they are finally differential functions of the flat output z12 . Consequently z78 is also a differential function of the flat outputs vector z12 or z78 = h78 (Y, Y˙ )

(77)

Finally, from Eq. (71) one solves for the control input u. This gives −1 u = g78 (z˙ 78 − f78 )

(78)

where f78 , g78 are functions of z12 , z34 , z56 and z78 and thus they are also differential functions of the flat outputs vector z12 . Consequently, the control inputs vector u is also a differential function of the flat outputs vector z12 or u = hu (Y, Y˙ )

(79)

Flatness-Based Control in Successive Loops for Robotic ...

55

As a result of the above, all state variables and the control inputs of the dynamic model of the electrohydraulic robotic manipulator are differential ’functions of the flat outputs vector z12 and the system is differentially flat. Next, it will be proven that each one of the subsystems of Eq. (68) to Eq. (71) can be viewed independently as a differentially flat subsystem. In the subsystem of Eq. (68) the flat output is taken to be z12 while z34 is a virtual control input. Besides, f12 , g12 have constant elements. Solving for z34 gives Eq. (73) which demonstrates that z34 is a differential function of the flat output z12 and consequently the subsystem of Eq. (68) is differentially flat. In the subsystem of Eq. (69) the flat output is taken to be z34 while z56 is a virtual control input. Besides, f34 , g34 are functions of z12 , z34 where z12 is viewed as a coefficients vector and z34 is the local flat outputs vector. Solving for z56 gives Eq. (75) which demonstrates that z56 is a differential function of the flat output z34 and consequently the subsystem of Eq. (69) is differentially flat. In the subsystem of Eq. (70) the flat output is taken to be z56 while z78 is a virtual control input. Besides, f56 , g56 are functions of z12 , z34 , z56 where z12 , z34 can be viewed as coefficients vectors and z56 is the flat outputs vector. Solving for z78 gives Eq. (77) which demonstrates that z78 is a differential function of the flat output z56 and consequently the subsystem of Eq. (70) is differentially flat. In the subsystem of Eq. (71) the flat output is taken to be z78 while u is the real control input. Besides, f78 , g78 are functions of z78 where z78 is the flat outputs vector. Solving for u gives Eq. (79) which demonstrates that u is a differential function of the flat output z78 and consequently the subsystem of Eq. (70) is differentially flat.

4.3

Design of a Flatness-Based Controller in Successive Loops

Next, for each one of the differentially flat subsystems of Eq. (68) to Eq. (71) one can design a stabilizing feedback controller following the controller definition stages for input-output linearized state-space descriptions. d . The For the subsystem of Eq. (68) the desirable setpoint is taken to be z12

56

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira

value of the virtual control input which achieves convergence to this setpoint is ∗ = g −1 [z˙ d − f − K 1 (z − z d )] z34 12 1 12 12 12 12

(80)

1 ], where where matrix K11 is a diagonal matrix of the form K11 = diag[K1i 1 K1i > 0 for i = 1, 2. By substituting Eq. (80) into Eq. 68 one obtains the closed subsystem dynamics d ) + K 1 (z − z d ) = 0⇒e˙ + K 1 e (z˙12 − z˙12 12 1 12 12 1 12 = 0 d ⇒limt→∞ e12 (t) = 0⇒limt→∞ z12 (t) = z12 (t)

(81)

∗ For the subsystem of Eq. (68) the control input z34 becomes the subsystem’s d ∗ setpoint. Thus z34 = z34 . The value of the virtual control input which achieves convergence to this setpoint is −1 d ∗ d z56 = g34 [z˙ 34 − f34 − K12 (z34 − z34 )]

(82)

2 ], where where matrix K12 is a diagonal matrix of the form K12 = diag[K1i 2 K1i > 0 for i = 1, 2. By substituting Eq. (82) into Eq. 69 one obtains the closed subsystem dynamics d ) + K 2 (z − z d ) = 0⇒e˙ + K 2 e (z˙34 − z˙34 34 1 34 34 1 34 = 0 d ⇒limt→∞ e34 (t) = 0⇒limt→∞ z34 (t) = z34 (t)

(83)

∗ becomes the subsystem’s For the subsystem of Eq. (70) the control input z56 d ∗ setpoint. Thus z56 = z56 . The value of the virtual control input which achieves convergence to this setpoint is ∗ = g −1 [z˙ d − f − K 3 (z − z d )] z78 56 1 56 56 56 56

(84)

3 where matrix K13 is a diagonal matrix of the form K13 = diag[K1i ], where 3 K1i > 0 for i = 1, 2. By substituting Eq. (84) into Eq. (70) one obtains the closed subsystem dynamics d d (z˙56 − z˙56 ) + K13 (z56 − z56 ) = 0⇒e˙ 56 + K13 e56 = 0 d (t) ⇒limt→∞ e56 (t) = 0⇒limt→∞ z56 (t) = z56

(85)

∗ becomes the subsystem’s For the subsystem of Eq. (71) the control input z78 d ∗ setpoint. Thus z78 = z78 . The value of the real control input which achieves convergence to this setpoint is

Flatness-Based Control in Successive Loops for Robotic ... −1 d d )] u = g78 [z˙ 78 − f78 − K14 (z78 − z78

57 (86)

4 where matrix K14 is a diagonal matrix of the form K14 = diag[K1i ], where 4 K1i > 0 for i = 1, 2. By substituting Eq. (86) into Eq. (71) one obtains the closed subsystem dynamics d ) + K 4 (z − z d ) = 0⇒e˙ + K 4 e (z˙78 − z˙78 78 1 78 78 1 78 = 0 d ⇒limt→∞ e78 (t) = 0⇒limt→∞ z78 (t) = z78 (t)

(87)

Therefore, the concept of flatness-based control in successive loops achieves elimination of the tracking error for all state vector elements and the closedloop system is globally asymptotically stable. Global asymptotic stability can be also proven through Lyapunov analysis. The following Lyapunov function is defined V = 21 [eT12 e12 + eT34 e34 + eT56 e56 + eT78 e78 ]

(88)

By differentiating in time one obtains V˙ = 21 [2eT12 e˙ 12 + 2eT34 e˙ 34 + 2eT56 e˙ 56 + 2eT78 e˙ 78 ]

(89)

By substituting in Eq. (89) the previously given relations about the tracking error dynamics one obtains V˙ = [eT12 (−K11 e12 ) + eT34 (−K12 e34 ) + eT56 (−K13 e56 ) + eT78 (−K14 e78 )] (90) Thus one finally gets V˙ = −[eT12 K11 e12 + eT34 K12 e34 + eT56 K13 e56 + eT78 K14 e78 ]

(91)

which signifies that V˙ < 0 for K11 > 0, K12 > 0, K13 > 0, K14 > 0

(92)

It holds that V˙ is strictly negative ∀e12 , e34 , e56, e78 6=0. Therefore, V˙ becomes 0 when e12 = 0, e34 = 0, e56 = 0, e78 = 0. Consequently, the Lyapunov function of the system is strictly diminishing and finally converges to 0, no matter what the initial conditions of the electrohydraulic robot are. Thus, the control loop of the electrohydraulic robotic manipulator is globally asymptotically stable.

58

5 5.1

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira

Simulation Tests Control of Electrohydraulic Actuators

Results about the tracking accuracy and the speed of convergence to setpoints of the successive-loops flatness-based control method, in the case of the electrohydraylic actuator of Section 3, are shown in Fig. 3 to Fig. 10. It can be noticed, that under this control scheme one achieves fast and precise tracking of reference setpoints for all state variables of the autonomous actuator. It is noteworthy, that through the stages of this method one solves also the setpoints definition problem for all state variables of the electrohydraulic actuator. Actually, the selection of setpoints for state variable x1 is unconstrained. On the other side by defining state variables x2 , x3 , x4 as virtual control inputs for the subsystem of x1 one can find the setpoints for x2 , x3 , x4 as functions of the setpoints for x1 . The speed of convergence of the state variables of the electrohydraulic actuator under flatness-based control implemented in successive loops depends on the selection of positive values for the feedback gains K11 , K12, K13 , K14 .

5.2

Control of the Multi-DOF Electrohydraulically Actuated Robot

Results about the tracking accuracy and the speed of convergence to setpoints of the successive-loops flatness-based control method, in the case of the multiDOF electrohydraulically actuated robot of Section 4, are shown in Fig. 11 to Fig. 26. It can be noticed again, that under this control scheme one achieves fast and precise tracking of reference setpoints for all state variables of the dynamic model of the electrohydraulic robotic manipulator. It is noteworthy, that through the stages of this method one solves also the setpoints definition problem for all state variables of the electrohydraulic robot. Actually, the selection of setpoints for state variables x1 , x3 is unconstrained. On the other side by defining state variables x2 , x4 , x5 , x7 , and x6 , x8 as virtual control inputs for the subsystem of x1 , x3 , and for the rest of the subsystems in which the state-space model of the robot is decomposed one can find the setpoints for x2 , x4 , x5 , x7 , and x6 , x8 as functions of the setpoints for x1 , x3 . The speed of convergence of the state variables of the electrohydraulic robot under flatness-based control implemented in successive loops is determined by the selection of values for the diagonal gain matrices K11 , K12 , K13, K14 .

4

4

2

2

2

0 −2 −4

0 −2

0

10

20 time

30

−4

40

10 8

e3

4

2

e2

4

x2

x1

Flatness-Based Control in Successive Loops for Robotic ...

0 −2

0

10

20 time

30

−4

40

59

0 −2

0

10

20 30 time (sec)

−4

40

4

0.2

4

2

0.15

2

0

10

20 30 time (sec)

40

0

10

20 30 time (sec)

40

e4

0

0.1

0

u

x4

x3

6 4 −2

2 0

0

10

20 time

30

−4

40

0.05

0

10

20 time

30

0

40

−2

0

10

20 30 time (sec)

(a)

−4

40

(b)

4

4

2

2

2

0 −2 −4

0 −2

0

10

20 time

30

−4

40

10 8

e3

4

2

e2

4

x2

x1

Figure 3: Control for the 1-DOF electrohydraulic actuator when tracking setpoint 1 (a) convergence of state variables x1 to x4 to their reference setpoints (red line: setpoint, blue line: real value), (b) tracking errors for state variables x2 to x4 (red line) and variations of the control input u (blue line: real)

0 −2

0

10

20 time

30

−4

40

0 −2

0

10

20 30 time (sec)

−4

40

4

0.2

4

2

0.15

2

0

10

20 30 time (sec)

40

0

10

20 30 time (sec)

40

0

0.1

0

u

e4

x4

x3

6 4 −2

2 0

0

10

20 time

30

40

−4

(a)

0.05

0

10

20 time

30

40

0

−2

0

10

20 30 time (sec)

40

−4

(b)

Figure 4: Control for the 1-DOF electrohydraulic actuator when tracking setpoint 2 (a) convergence of state variables x1 to x4 to their reference setpoints (red line: setpoint, blue line: real value), (b) tracking errors for state variables x2 to x4 (red line) and variations of the control input u (blue line)

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira 4

4

2

2

2

0 −2 −4

0 −2

0

10

20 time

30

−4

40

10 8

e3

4

2

e2

4

x2

x1

60

0 −2

0

10

20 time

30

−4

40

0 −2

0

10

20 30 time (sec)

−4

40

4

0.2

4

2

0.15

2

0

10

20 30 time (sec)

40

0

10

20 30 time (sec)

40

e4

0

0.1

0

u

x4

x3

6 4 −2

2 0

0

10

20 time

30

−4

40

0.05

0

10

20 time

30

0

40

−2

0

10

20 30 time (sec)

(a)

−4

40

(b)

4

4

2

2

2

0 −2 −4

0 −2

0

10

20 time

30

−4

40

10 8

e3

4

2

e2

4

x2

x1

Figure 5: Control for the 1-DOF electrohydraulic actuator when tracking setpoint 3 (a) convergence of state variables x1 to x4 to their reference setpoints (red line: setpoint, blue line: real value), (b) tracking errors for state variables x2 to x4 (red line) and variations of the control input u (blue line)

0 −2

0

10

20 time

30

−4

40

0 −2

0

10

20 30 time (sec)

−4

40

4

0.2

4

2

0.15

2

0

10

20 30 time (sec)

40

0

10

20 30 time (sec)

40

0

0.1

0

u

e4

x4

x3

6 4 −2

2 0

0

10

20 time

30

40

−4

(a)

0.05

0

10

20 time

30

40

0

−2

0

10

20 30 time (sec)

40

−4

(b)

Figure 6: Control for the 1-DOF electrohydraulic actuator when tracking setpoint 4 (a) convergence of state variables x1 to x4 to their reference setpoints (red line: setpoint, blue line: real value), (b) tracking errors for state variables x2 to x4 (red line) and variations of the control input u (blue line)

4

4

2

2

2

0 −2 −4

0 −2

0

10

20 time

30

−4

40

10 8

e3

4

2

e2

4

x2

x1

Flatness-Based Control in Successive Loops for Robotic ...

0 −2

0

10

20 time

30

−4

40

61

0 −2

0

10

20 30 time (sec)

−4

40

4

0.2

4

2

0.15

2

0

10

20 30 time (sec)

40

0

10

20 30 time (sec)

40

e4

0

0.1

0

u

x4

x3

6 4 −2

2 0

0

10

20 time

30

−4

40

0.05

0

10

20 time

30

0

40

−2

0

10

20 30 time (sec)

(a)

−4

40

(b)

4

4

2

2

2

0 −2 −4

0 −2

0

10

20 time

30

−4

40

10 8

e3

4

2

e2

4

x2

x1

Figure 7: Control for the 1-DOF electrohydraulic actuator when tracking setpoint 5 (a) convergence of state variables x1 to x4 to their reference setpoints (red line: setpoint, blue line: real value), (b) tracking errors for state variables x2 to x4 (red line) and variations of the control input u (blue line)

0 −2

0

10

20 time

30

−4

40

0 −2

0

10

20 30 time (sec)

−4

40

4

0.2

4

2

0.15

2

0

10

20 30 time (sec)

40

0

10

20 30 time (sec)

40

0

0.1

0

u

e4

x4

x3

6 4 −2

2 0

0

10

20 time

30

40

−4

(a)

0.05

0

10

20 time

30

40

0

−2

0

10

20 30 time (sec)

40

−4

(b)

Figure 8: Control for the 1-DOF electrohydraulic actuator when tracking setpoint 6 (a) convergence of state variables x1 to x4 to their reference setpoints (red line: setpoint, blue line: real value), (b) tracking errors for state variables x2 to x4 (red line) and variations of the control input u (blue line)

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira 4

4

2

2

2

0 −2 −4

0 −2

0

10

20 time

30

−4

40

10 8

e3

4

2

e2

4

x2

x1

62

0 −2

0

10

20 time

30

−4

40

0 −2

0

10

20 30 time (sec)

−4

40

4

0.2

4

2

0.15

2

0

10

20 30 time (sec)

40

0

10

20 30 time (sec)

40

e4

0

0.1

0

u

x4

x3

6 4 −2

2 0

0

10

20 time

30

−4

40

0.05

0

10

20 time

30

0

40

−2

0

10

20 30 time (sec)

(a)

−4

40

(b)

4

4

2

2

2

0 −2 −4

0 −2

0

10

20 time

30

−4

40

10 8

e3

4

2

e2

4

x2

x1

Figure 9: Control for the 1-DOF electrohydraulic actuator when tracking setpoint 7 (a) convergence of state variables x1 to x4 to their reference setpoints (red line: setpoint, blue line: real value), (b) tracking errors for state variables x2 to x4 (red line) and variations of the control input u (blue line)

0 −2

0

10

20 time

30

−4

40

0 −2

0

10

20 30 time (sec)

−4

40

4

0.2

4

2

0.15

2

0

10

20 30 time (sec)

40

0

10

20 30 time (sec)

40

0

0.1

0

u

e4

x4

x3

6 4 −2

2 0

0

10

20 time

30

40

−4

(a)

0.05

0

10

20 time

30

40

0

−2

0

10

20 30 time (sec)

40

−4

(b)

Figure 10: Control for the 1-DOF electrohydraulic actuator when tracking setpoint 8 (a) convergence of state variables x1 to x4 to their reference setpoints (red line: setpoint, blue line: real value), (b) tracking errors for state variables x2 to x4 (red line) and variations of the control input u (blue line)

Flatness-Based Control in Successive Loops for Robotic ...

10

20 time

30

−3

40

2

10

20 time

30

3

50

2

40

20 time

30

−3

40

10

20 time

30

−10

40

0

10

20 time

30

40

0

10

20 time

30

40

10 5

x7

x4

10

0

30

0

0

20 −5

10

−2 0

x6

0

40

−1

1

0 −5

10 0

1

1.5

5

20

−2

0

10

30

0

2.5

x3

40

−1

1

0.5

50

2

x5

1.5

0.5

3

1 x2

x1

2

x8

2.5

63

0

10

20 time

30

0

40

0

10

20 time

30

40

(a)

−10

(b)

1

100

0.5

50

0 −0.5

0

0

10

20 time

30

−1

40

0 −50

−0.5

0

10

20 time

30

−100

40

2

100

10

1

50

0 −10 −20

u2

20

e6

e5

−1

u1

1 0.5 e3

e1

Figure 11: Control for the 2-DOF electrohydraulic robot when tracking setpoint 1 (a) convergence of state variables x1 to x4 to their reference setpoints (red line: setpoint, blue line: real value), (b) convergence of state variables x5 to x8 to their reference setpoints (red line: setpoint, blue line: real value)

0 −1

0

10

20 time

30

40

(a)

−2

0

5

10

15

20 25 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

0 −50

0

10

20 time

30

40

−100

(b)

Figure 12: Control for the 2-DOF electrohydraulic robot when tracking setpoint 1 (a) tracking error of state variables x1 , x2 , x5 , x6 (red line), (b) variation of the control inputs u1 to u2 (blue line)

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira

10

20 time

30

−3

40

2

10

20 time

30

0

40

3

50

2

40

20 time

30

−3

40

20 time

30

−10

40

0

10

20 time

30

40

0

10

20 time

30

40

10 5

x7

x4

10

10

0

20 −5

10

−2 0

0

30

0 −1

1

0 −5

10 0

1

1.5

5

20

−2

0

10

30

0

2.5

x3

40

−1

1

0.5

50

2

x5

1.5

0.5

3

1 x2

x1

2

x6

2.5

x8

64

0

10

20 time

30

0

40

0

10

20 time

30

40

(a)

−10

(b)

1

100

0.5

50

0 −0.5

0

0

10

20 time

30

−1

40

0 −50

−0.5

0

10

20 time

30

−100

40

2

100

10

1

50

0 −10 −20

u2

20

e6

e5

−1

u1

1 0.5 e3

e1

Figure 13: Control for the 2-DOF electrohydraulic robot when tracking setpoint 2 (a) convergence of state variables x1 to x4 to their reference setpoints (red line: setpoint, blue line: real value), (b) convergence of state variables x5 to x8 to their reference setpoints (red line: setpoint, blue line: real value)

0 −1

0

10

20 time

30

40

(a)

−2

0

5

10

15

20 25 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

0 −50

0

10

20 time

30

40

−100

(b)

Figure 14: Control for the 2-DOF electrohydraulic robot when tracking setpoint 2 (a) tracking error of state variables x1 , x2 , x5 , x6 (red line), (b) variation of the control inputs u1 to u2 (blue line)

Flatness-Based Control in Successive Loops for Robotic ...

10

20 time

30

−3

40

2

10

20 time

30

3

50

2

40

20 time

30

−3

40

10

20 time

30

−10

40

0

10

20 time

30

40

0

10

20 time

30

40

10 5

x7

x4

10

0

30

0

0

20 −5

10

−2 0

x6

0

40

−1

1

0 −5

10 0

1

1.5

5

20

−2

0

10

30

0

2.5

x3

40

−1

1

0.5

50

2

x5

1.5

0.5

3

1 x2

x1

2

x8

2.5

65

0

10

20 time

30

0

40

0

10

20 time

30

40

(a)

−10

(b)

1

100

0.5

50

0 −0.5

0

0

10

20 time

30

−1

40

0 −50

−0.5

0

10

20 time

30

−100

40

2

100

10

1

50

0 −10 −20

u2

20

e6

e5

−1

u1

1 0.5 e3

e1

Figure 15: Control for the 2-DOF electrohydraulic robot when tracking setpoint 3 (a) convergence of state variables x1 to x4 to their reference setpoints (red line: setpoint, blue line: real value), (b) convergence of state variables x5 to x8 to their reference setpoints (red line: setpoint, blue line: real value)

0 −1

0

10

20 time

30

40

(a)

−2

0

5

10

15

20 25 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

0 −50

0

10

20 time

30

40

−100

(b)

Figure 16: Control for the 2-DOF electrohydraulic robot when tracking setpoint 3 (a) tracking error of state variables x1 , x2 , x5 , x6 (red line), (b) variation of the control inputs u1 to u2 (blue line)

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira

10

20 time

30

−3

40

2

10

20 time

30

0

40

3

50

2

40

20 time

30

−3

40

20 time

30

−10

40

0

10

20 time

30

40

0

10

20 time

30

40

10 5

x7

x4

10

10

0

20 −5

10

−2 0

0

30

0 −1

1

0 −5

10 0

1

1.5

5

20

−2

0

10

30

0

2.5

x3

40

−1

1

0.5

50

2

x5

1.5

0.5

3

1 x2

x1

2

x6

2.5

x8

66

0

10

20 time

30

0

40

0

10

20 time

30

40

(a)

−10

(b)

1

100

0.5

50

0 −0.5

0

0

10

20 time

30

−1

40

0 −50

−0.5

0

10

20 time

30

−100

40

2

100

10

1

50

0 −10 −20

u2

20

e6

e5

−1

u1

1 0.5 e3

e1

Figure 17: Control for the 2-DOF electrohydraulic robot when tracking setpoint 4 (a) convergence of state variables x1 to x4 to their reference setpoints (red line: setpoint, blue line: real value), (b) convergence of state variables x5 to x8 to their reference setpoints (red line: setpoint, blue line: real value)

0 −1

0

10

20 time

30

40

(a)

−2

0

5

10

15

20 25 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

0 −50

0

10

20 time

30

40

−100

(b)

Figure 18: Control for the 2-DOF electrohydraulic robot when tracking setpoint 4 (a) tracking error of state variables x1 , x2 , x5 , x6 (red line), (b) variation of the control inputs u1 to u2 (blue line)

Flatness-Based Control in Successive Loops for Robotic ...

10

20 time

30

−3

40

2

10

20 time

30

3

50

2

40

20 time

30

−3

40

10

20 time

30

−10

40

0

10

20 time

30

40

0

10

20 time

30

40

10 5

x7

x4

10

0

30

0

0

20 −5

10

−2 0

x6

0

40

−1

1

0 −5

10 0

1

1.5

5

20

−2

0

10

30

0

2.5

x3

40

−1

1

0.5

50

2

x5

1.5

0.5

3

1 x2

x1

2

x8

2.5

67

0

10

20 time

30

0

40

0

10

20 time

30

40

(a)

−10

(b)

1

100

0.5

50

0 −0.5

0

0

10

20 time

30

−1

40

0 −50

−0.5

0

10

20 time

30

−100

40

2

100

10

1

50

0 −10 −20

u2

20

e6

e5

−1

u1

1 0.5 e3

e1

Figure 19: Control for the 2-DOF electrohydraulic robot when tracking setpoint 5 (a) convergence of state variables x1 to x4 to their reference setpoints (red line: setpoint, blue line: real value), (b) convergence of state variables x5 to x8 to their reference setpoints (red line: setpoint, blue line: real value)

0 −1

0

10

20 time

30

40

(a)

−2

0

5

10

15

20 25 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

0 −50

0

10

20 time

30

40

−100

(b)

Figure 20: Control for the 2-DOF electrohydraulic robot when tracking setpoint 5 (a) tracking error of state variables x1 , x2 , x5 , x6 (red line), (b) variation of the control inputs u1 to u2 (blue line)

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira

10

20 time

30

−3

40

2

10

20 time

30

0

40

3

50

2

40

20 time

30

−3

40

20 time

30

−10

40

0

10

20 time

30

40

0

10

20 time

30

40

10 5

x7

x4

10

10

0

20 −5

10

−2 0

0

30

0 −1

1

0 −5

10 0

1

1.5

5

20

−2

0

10

30

0

2.5

x3

40

−1

1

0.5

50

2

x5

1.5

0.5

3

1 x2

x1

2

x6

2.5

x8

68

0

10

20 time

30

0

40

0

10

20 time

30

40

(a)

−10

(b)

1

100

0.5

50

0 −0.5

0

0

10

20 time

30

−1

40

0 −50

−0.5

0

10

20 time

30

−100

40

2

100

10

1

50

0 −10 −20

u2

20

e6

e5

−1

u1

1 0.5 e3

e1

Figure 21: Control for the 2-DOF electrohydraulic robot when tracking setpoint 6 (a) convergence of state variables x1 to x4 to their reference setpoints (red line: setpoint, blue line: real value), (b) convergence of state variables x5 to x8 to their reference setpoints (red line: setpoint, blue line: real value)

0 −1

0

10

20 time

30

40

(a)

−2

0

5

10

15

20 25 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

0 −50

0

10

20 time

30

40

−100

(b)

Figure 22: Control for the 2-DOF electrohydraulic robot when tracking setpoint 6 (a) tracking error of state variables x1 , x2 , x5 , x6 (red line), (b) variation of the control inputs u1 to u2 (blue line)

Flatness-Based Control in Successive Loops for Robotic ...

10

20 time

30

−3

40

2

10

20 time

30

3

50

2

40

20 time

30

−3

40

10

20 time

30

−10

40

0

10

20 time

30

40

0

10

20 time

30

40

10 5

x7

x4

10

0

30

0

0

20 −5

10

−2 0

x6

0

40

−1

1

0 −5

10 0

1

1.5

5

20

−2

0

10

30

0

2.5

x3

40

−1

1

0.5

50

2

x5

1.5

0.5

3

1 x2

x1

2

x8

2.5

69

0

10

20 time

30

0

40

0

10

20 time

30

40

(a)

−10

(b)

1

100

0.5

50

0 −0.5

0

0

10

20 time

30

−1

40

0 −50

−0.5

0

10

20 time

30

−100

40

2

100

10

1

50

0 −10 −20

u2

20

e6

e5

−1

u1

1 0.5 e3

e1

Figure 23: Control for the 2-DOF electrohydraulic robot when tracking setpoint 7 (a) convergence of state variables x1 to x4 to their reference setpoints (red line: setpoint, blue line: real value), (b) convergence of state variables x5 to x8 to their reference setpoints (red line: setpoint, blue line: real value)

0 −1

0

10

20 time

30

40

(a)

−2

0

5

10

15

20 25 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

0 −50

0

10

20 time

30

40

−100

(b)

Figure 24: Control for the 2-DOF electrohydraulic robot when tracking setpoint 7 (a) tracking error of state variables x1 , x2 , x5 , x6 (red line), (b) variation of the control inputs u1 to u2 (blue line)

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira

10

20 time

30

−3

40

2

10

20 time

30

0

40

3

50

2

40

20 time

30

−3

40

20 time

30

−10

40

0

10

20 time

30

40

0

10

20 time

30

40

10 5

x7

x4

10

10

0

20 −5

10

−2 0

0

30

0 −1

1

0 −5

10 0

1

1.5

5

20

−2

0

10

30

0

2.5

x3

40

−1

1

0.5

50

2

x5

1.5

0.5

3

1 x2

x1

2

x6

2.5

x8

70

0

10

20 time

30

0

40

0

10

20 time

30

(a)

40

−10

(b)

1

100

0.5

50

0 −0.5

0

0

10

20 time

30

−1

40

0 −50

−0.5

0

10

20 time

30

−100

40

2

100

10

1

50

0 −10 −20

u2

20

e6

e5

−1

u1

1 0.5 e3

e1

Figure 25: Control for the 2-DOF electrohydraulic robot when tracking setpoint 8 (a) convergence of state variables x1 to x4 to their reference setpoints (red line: setpoint, blue line: real value), (b) convergence of state variables x5 to x8 to their reference setpoints (red line: setpoint, blue line: real value)

0 −1

0

10

20 time

30

40

−2

(a)

0

5

10

15

20 25 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

0 −50

0

10

20 time

30

40

−100

(b)

Figure 26: Control for the 2-DOF electrohydraulic robot when tracking setpoint 8 (a) tracking error of state variables x1 , x2 , x5 , x6 (red line), (b) variation of the control inputs u1 to u2 (blue line)

Flatness-Based Control in Successive Loops for Robotic ...

71

Conclusion The present manuscript has proposed a flatness-based control method in successive loops for (i) the dynamic model of an electrohydraulic actuator, (ii) the dynamic model of a multi-DOF robotic manipulator with electrohydraulic actuators. These methods are of high interest for the industry, where hydraulic actuation is preferred in place of electric actuation for tasks that require to develop high forces and torques. In flatness-based control in successive loops there is no need to apply changes of state variables (diffeomorphisms) and complicated state-space transformations. In this method the dynamic model of the nonlinear system is separated into subsystems which are connected in a cascading manner. This control approach is directly applicable to dynamical systems of the triangular form and to nonlinear systems which can be transformed into such a form. The state-space model of the initial nonlinear system is decomposed into cascading subsystems which satisfy differential flatness properties. For each subsystem of the state-space model a virtual control input is computed, capable of inverting the subsystem’s dynamics and of eliminating the subsystem’s tracking error. The control input which is actually applied to the initial nonlinear system is computed from the last row of the state-space description. This control input incorporates in a recursive manner all virtual control inputs which were computed from the individual subsystems included in the initial state-space equation. The control input that should be applied to the nonlinear system so as to assure that all state vector elements will converge to the desirable setpoints is obtained at each iteration of the control algorithm by tracing backwards the subsystems of the state-space model. The first case study of the manuscript was concerned with the control of the 1-DOF electrohydraulic actuator. The state-space model of the system has undergone a per-row decomposition. Each one of the rows of this system i = 1, 2, · · · , 4 was viewed independently as a differentially flat system, in which state variable xi was the flat output and state variable xi+1 was the virtual control input. Stabilizing feedback control for each one of these subsystems was computed after following a dynamic inversion procedure as is usually done for input-output linearized systems. From the fourth row of the state-space model the real control input of the actuator was found and this contained recursively the virtual control inputs which were computed for the three preceding subsystems. The second case study of the manuscript was concerned with the control of the 2-DOF robotic manipulator with electrohydraulic actuators. The

72

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira

state-space model of the system has undergone a decomposition into four subsystems, comprising state variables x1 and x3 , x2 and x4 , x5 and x7 as well as x6 and x8 respectively. Stabilizing feedback control for each one of these subsystems was computed again after following a dynamic inversion procedure as is usually done for input-output linearized systems. From the fourth subsystem of the state-space model, the real control input of the robot was found and this contained recursively the virtual control inputs which were computed for the three preceding subsystems. In conclusion, the performance of the flatnessbased control method in successive loops was excellent in both test cases. The method achieved fast convergence to setpoints with smooth variations of the control inputs.

References Ali S.A., Christen A., Begg S. and Langlois N. (2016). Continuous-discrete time observer design for state and disturbance estimator of electro-hydraulic actuator systems, IEEE Transactions on Industrial Electronics, 63(7): 43144324. Barbot J.P., Fliess M. and Floquet T. (2007). An algebraic framework for the design of nonlinear observers with uknown inputs. IEEE CDC 2007, IEEE 46th Intl. Conference on Decision and Control, New Orleans, USA. Basseville M. and Nikiforov I. (1993). Detection of abrupt changes: Theory and Applications, Prentice-Hall. Bououden S., Boutat D., Zheng G., Barbot J.P. and Kratz F. (2011). A triangular canonical form for a class of 0-flat nonlinear systems, International Journal of Control, Taylor and Francis, 84(2): 61-269. Brucker M. and Heidtmann P. (2004). Flatness-based control of a rotary vane actuator, in Proc of 16th Intl. Conf. on Mathematical Theory of Networks and Systems, IEEE MTNS 2004, Leuven, Belgiumm. Cho H.D. and You S.H. (2021). Fuzzy finite memory state estimation for electrohydraulic active suspension systems, IEEE Access, 9: 99364-99373.

Flatness-Based Control in Successive Loops for Robotic ...

73

Dao H.V. and Ahn K.K. (2022). Extended sliding-mode observer-based admittance control for hydraulic robots, IEEE Robotics and Automation Letters, 7(2): 3992-3999. Ding W.H., Deng H., Xin Y.M. and Duan X.G. (2017). Tracking control of electro-hydraulic servo multi-closed-chain mechanisms with the use of an approximate nonlinear internal model. Control Engineering Practice, Elsevier, 58: 225-241. Dinh T.X., Thien T.D., Ahn T.H.V. and Ahn K.K. (2018). Disturbance observerbased finite-time trajectory tracking control for a 3-DOF hydraulic manipulator including actuator dynamics. IEEE Access, 6: 36798-36809. Dinh T.X., Thien T.D., Anh T.H.V. and Ahn K.K. (2018). Disturbance observerbased finite-time trajectory tracking control for a 3-DOF hydraulic manipulator including actuator dynamics, IEEE Access, 6: 36798-36809. Du H., Shi J., Chou J., Zhang Z. and Feng X. (2021). High-gain observer-based integral sliding-mode tracking control for heavy vehicle electrohydraulic steering system, Mechatronics, Elsevier, 74: 102484-102498. Fliess M. and Mounier H. (1999). Tracking control and π-freeness of infinite dimensional linear systems, In: G. Picci and D.S. Gilliam Eds., Dynamical Systems, Control, Coding and Computer Vision, Birkha¨user, 258: 41-68. Gao Q., Liu Y. and Jiang D. (2020). Nonlinear control method of electrohydraulic system driving two-DOF robot arm with output constraint. IOP Conf. Series on Materials Science and Engineering, 715: 012090. Guo Q., Sun P., Yin J.M., Yu T. and Jiang D. (2016). Parametric adaptive estimation and backstepping control of electro-hydraulic actuator with delayed memory filter. ISA Transactions, Elsevier, 62: 202-214. Guo Q., Sun P., Yin J.M., Yu T. and Jiang D. (2016). Parametric adaptive estimation and backstepping control of electro-hydraulic actuator with delayed memory filter. ISA Transactions, Elsevier, 62: 202-214. Guo K., Wei J., Fang J., Feng R. and Wang X. (2016). Position tracking control of electro-hydraulic actuator based on extended disturbance observer. Mechatronics, Elsevier, 27: 47-56.

74

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira

Guo Q., Yin J., Yu T. and Jiang D. (2017). Saturated adaptive control of an electrohydraulic actuator with parametric uncertainty and load disturbance. IEEE Transactions on Industrial Electronics, 67(10): 7930-7941. Guo Q., Zhang Y., Keller B.G. and Wu S.W. (2018). State-constrained control of single-rod electrohydraulic actuator with parametric uncertainty and load disturbance. IEEE Transactions on Control Systems Technology, 26(6): 2942-2950. Guo Q., Wang Q. and Liu Y. (2018). Anti-windup control of an electrohydraulic system with load disturbance and modeling uncertainty. IEEE Transactions on Industrial Informatics, 14(7): 3097-3108. Guo Q., Wang Q. and Li X. (2013). Finite-time convergence control of eectrohydraulic velocity servo-system under uncertaint parameter and external load. IEEE Transactions on Industrial Electronics, 68(6): 4513-4523. Guo Q., Chen Z., Shi Y, Li X., Yan Y., Guo F. and Li S. (2022). Synchronous control for multiple electrohydraulic actuators with feedback linearization, Mechanical Systems and Signal Processing, Elsevier, 178: 109280109295. Heidtmann P. and Brucker M. (2005). Nonlinear modelling and tracking control of a hydraulic rotary vane actuator. PAMM Proceeding on Applied Mathematics and Mechanics, 5: 161-162. Henikl J., Kemmetmuller W., Meurer T. and Kugi A. (2016). Infinite dimensional decentralized damping control of large-scale manipulators with hydeaulic actuation. Automatica, Elsevier, 63: 101-115. Hyan S.H., Taniai Y., Hianum K., Yosunaga K. and Mizui H. (2020). Overpressure compensation for hydraulic hybrid servo booster applied to hydraulic manipulators. IEEE Robotics and Automation Letters, 4(2): 942-949. Kaddissi C., Kenn´e G.P. and Saad M. (2011). Indirect adaptive control of an electrohydraulic servo system based on nonlinear backstepping. IEEE Transactions on Mechatronics, 16(6): 1171-1177. Kalmari J., Backman J. and Visala A. (2017). Coordinated motion of an hydraulic forestry crane and vehicle using nonlinear model predictive control. Computers and Electronics in Agriculture, Elsevier, 133: 119-127.

Flatness-Based Control in Successive Loops for Robotic ...

75

Karpenko M. and Sepehri N. (2009). Hardware-in-the-loop simulator for research on fault tolerant control of electrohydraulic actuators in a flight control application. Mechatronics, Elsevier, 19: 1067-1077. Khalil H. (1996). Nonlinear Systems, Prentice Hall. Kim W., Won D., Shin D. and Chung C. (2017). Output feedback nonlinear control for electro-hydraulic systems, Mechatronics, 22: 766-777. Kim W., Shin D., Won D. and Chung C. (2013). Disturbance-observer-based position tracking controller in the presence of biased sinusoidal disturbance for electrohydraulic actuators. IEEE Transactions on Control Systems Technology, 11(6): 2290-2298. Kim J., Choi N.W. and Lee J. (2019). Discrete-time delay control for hydraulic excavator motion control with terminal sliding-mode control. Mechatronics, Elsevier, 60: 15-25. Kittissares S., Hirota Y., Nabae H., Endo G. and Suzumori K. (2022). Alternating pressure control system for hydraulic robots. Mechatronics, Elsevier, 85: 102822-102831. Koivumaki J. and Mattila J. (2015). Stability-generated force-sensorless contact force-motion control of heavy duty hydraulic manipulators. IEEE Transactions on Robotics, 31(4): 916-936. Koivumaki J. and Mattila J. (2015). High performance nonlinear motion/force controller design for redunant hydraulic construction crane automation, Automation in Construction, Elsevier, 51: 59-97. Koivumaki J., Zhu W.H. and Mattila J. (2019). Energy-efficient and highprecision control of hydraulic robots, Control Engineering Practice, Elsevier, 85: 176-193. t La Hera P.M., Mettin U., Westenberg S. and Shiriaev A.S. (2009). Modelling and control of hydraulic rotary actuators used in forestry cranes. IEEE ICRA 2009, IEEE 2009 Intl. Conference on Robotics and Automation, Kobe, Japan. Lee W. and Chung W.K. (2019). Disturbance observer-based compliance control of electo-hydraulic actuators with backdrivability. IEEE Robotics and Automation Letters, 4(2): 1722-1730.

76

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira

Letelier C. and Barbot J.P. (2021). Optimal flatness placement of sensors and actuators for controlling chaotic systems, Chaos, AIP Publications, 31(10), article No 103114. Levine J.(2009). Analysis and Control of Nonlinear Systems: A flatness-based approach, Springer. L´evine J. (2011). On necessary and sufficient conditions for differential flatness. Applicable Algebra in Engineering, Communications and Computing, Springer, vol. 22, 1: 47-90. Liang L.H., Wang L.Y.and Wang J.F. (2019). Energy-saving nonlinear friction compensation method for direct drive volume control flange-type rotaryvane steering gear. IEEE Access, 7: 148351-148362. Li Z., Bai Z. and Xu S. (2018). Design and operating mode analysis of hybrid actuation system based on EHA/SHA, IET Journal of Engineering, 18(13): 385-391. Li S. and Wang W. (2019). Adaptive robust H∞ control for double support bsalance systems, Information Sciences, Elsevier. Li L., Xin L., Luo X. and Wang Z. (2019). Compliance control using hydraulic heavy-duty manipulator. IEEE Transactions on Industrial Informatics, 15 (2): 1193-12-2. Li L., Xie L., Luo X. and Wang Z. (2019). Compliance control using hydraulic heavy-duty manipulator. IEEE Transactions on Industrial Informatics, 15(2): 1193-1204. Liang X., Wan Y., Zhang C., Kou Y., Xin Q. and Yi W. (2018). Robust position control of hydraulic manipulators using time-delay estimation and non-singular fast terminal sliding-mode. Proc. IMeche Part I: Journal of Systems and Control Engineering, Sage Publications, 232(1): 50-61. Liang X., Wan Y., Zhang C., Kim Y,, Xin O. and Yi W. (2018). Robust position control of hydraulic manipulators using time-delay estimation and nonsingular fast terminal sliding-mode. Proc. IMehE, Journal of Systems and Control Engineering, 232(1): 50-61.

Flatness-Based Control in Successive Loops for Robotic ...

77

Limaverde Filho J.O., Fortaleza E.C.R. and Campos M.C.M. (2021). A derivative-free nonlinear Kalman Filtering approach using flat inputs, International Journal of Control, Taylor and Francis. Lu L. and Yao B. (2014). Energy-saving adaptive robust contol of a hydraulic manipulator using five cartridge valves with an accumulator. IEEE Transactions on Industrial Electronics, 61(12): 7046-7055. Lyn L., Chen Z. and Yao B. (2008). Nonlinear adaptive robust control of singlerod electro-hydraulic actuator with unknown nonlinear parameters. IEEE Transactions on Control Systems Technology, 16(3): 434-445. Maddahi Y., Liao S., Fung W.K. and Sepehri N. (2016). Position referenced force augmentation in teleoperated hydraulic manipulators operating under delayed and lossy networks: A pilot survey. Robotics and Autonomus Systems, Elsevier, 83: 231-242. Menhour L., d’Andre’a-Novel B., Fliess M. and Mounier H. (2014). Coupled nonlinear vehicle control: Flatness-based setting with algebraic estimation techniques, Control Engineering Practice, Elsevier, 22: 135146. Mintsa H.A., Venugopal R., Kenn´e J.P. and Belleau C. (2012). Feedback linearization-based position control of an electrohydraulic servosystem with supply pressure uncertainty, IEEE Transactions on Control Systems Technology, 20(4): 1092-1099. Mohanty A. and Yao B. (2011). Indirect adaptive robust control of hydraaulic manipulators with accurate parameter estimates. IEEE Transactions on Control Systems Technology, vol. 19(3): 567-576. Mohanty A. and Yao B. (2011). Integrated direct/indirect adaptive robust control of hydraulic manipulators with valve deadband, IEEE Transactions on Mechatronics, 16(6): 707-716. Nicolau F., Respondek W. and Barbot J.P. (2021). Construction of flat inputs for mechanical systems. 7th IFAC Workshop on Lagrangian and Hamiltonian methods for nonlinear control, Berlin, Germany. Nicolau F., Respondek W. and Barbot J.P. (2022). How to minimally modify a dynamical system when constructing flat inputs, International Journal of Robust and Nonlinear Control, J. Wiley.

78

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira

Nguyen M.N., Tran D.T. and Ahn K.K. (2018). Robust position and vibration control of an electrohydraulic series elastic manipulator against disturbance generated by a variable stiffness actuator. Mechatronics, Elsevier, 52: 22-35. Pham V.D., Vo C.P., Dao H.V. and Ahn K.K. (2021). Actuator fault-tolerant control for an electrohydraulic actuator using time-delay estimation and feedback linearization. IEEE Access, 9: 107111-107122. Pham V.D., Vo C.P., Dao H.V. and Ahn K.K. (2021). Robust fault tolerant control of an electrohydraulic actuator with a novel nonlinear Unknown Input Observer. IEEE Access, 9: 30750-30760. Pisaturo M. and Senatore A. (2015). Improvement of traction through mMPC in linear vehicle dynamics based on electrohydraulic dry clutch. Journal of Intelligent Industrial Systems, Springer, 1: 153-161. Ranjan P., Wrat G., Bhala M., Mishara S.K. and Das J. (2020). A novel approach for the energy recovery and position control of a hybrid hydraulic excavator. ISA Transactions, Elsevier. Ren H., Deng G., Hou B., Wang S. and Zhou G. (2019). Finite-time command filtered backstepping algorithm-based pitch angle tracking control for wind-turbine pitch systems. IEEE Access, 5: 135514-135534. Rigatos G.G. and Tzafestas S.G. (2007). Extended Kalman Filtering for Fuzzy Modelling and Multi-Sensor Fusion. Mathematical and Computer Modelling of Dynamical Systems, Taylor & Francis), 13: 251-266. Rigatos G. and Zhang Q. (2009). Fuzzy model validation using the local statistical approach, Fuzzy Sets and Systems, Elsevier, 60(7): 882-904. Rigatos G. (2016). Nonlinear control and filtering using differential flatnesss theory approaches: Applications to electromechanical systems. Springer. Rigatos G., Siano P. and Zervos N. (2015). A new concept of flatness-based control of nonlinear dynamical systems. IEEE INDIN 2015, 13th IEEE Intl. Conf. on Industrial Informatics, Cambridge, UK. Rigatos G. (2015). Flatness-based embedded control in successive loops for spark ignited engines. Journal of Physics, IOP Publications, Conference

Flatness-Based Control in Successive Loops for Robotic ...

79

Series No 659, 012019, IFAC ACD 2015, Proc. of the 12th European Workshop on Advanced Control and Diagnosis. Rigatos G. and Melkikh A. (2016). Flatness-based control in successive loops for stabilization of heart’s electrical activity. In Proc. ICNAAM 2016, AIP Conf. Proceedings No 1790 060004, Intl. Conf. on Numerical Analysis and Applied Mathematics. Rigatos G. and Siano P. (2017). Flatness-based control in successive loops for Business Cycles of Finance Agents. Journal of Intelligent Industrial Systems, Springer, 3: 77-89. Rigatos G., Siano P., Ademi S. and Wira P. (2018). Flatness-based control of DC-DC converters implemented in successive loops. Electric Power Components and Systems, Taylor and Francis, 46(6): 673-687. Rigatos G. and Busawon K. (2018). Robotic manipulators and vehicles: Control, estimation and filtering. Springer. Rigatos G., Abbaszadeh M., and Siano P. (2022). Control and estimation of dynamical nonlinear and partial differential equation systems: Theory and Applications. IET Publications. Rigatos G., Wira P., Abbaszadeh M. and Pomares J. (2022). Flatness-based control in successive loops for industrial and mobile robots. IEEE IECON 2022, IEEE 48th Annual Conference of the Industrial Electronics Society, Brussels, Belgium. Rigatos G., Wira P., Abbaszadeh M. and Siano P. (2022). A nonlinear optimal control approach for the Lotka-Volterra dynamical system, IEEE IECON 2022, IEEE 48th Annual Conference of the Industrial Electronics Society, Brussels, Belgium. Sakain S. and Tsuji T. (2017). Development of friction-free controller for electro-hydrostatic actuator using feedback modulator and disturbance observer. Robomech, Springer, 4(1): 1-10. Shen W., Zhao H., Zhang G. and Lin Y.J. (2022). An event trigerred robust control for electrohydraulic servo machines with active disturbances rejection. Journal of the Franklin Institute, Elsevier, 359: 2857-2885.

80

G. Rigatos, M. Abbaszadeh, J. Pomares and P. Wira

Shi C., Wang X., Wang S., Wang J. and Tomovic M.T. (2015). Adaptive decoupling synchronous control of dissimilar redundant actuator system for large civil aircraft. Aerospace Science and Technology, Elsevier, 47: 114-124. Sira-Ramirez H. and Agrawal S. (2004). Differentially Flat Systems. Marcel Dekker, New York. Sirouspour M.R. and Salcudean S.E. (2001). Nonlinear control of hydraulic robots. IEEE Transactions on Robotics and Automation, 17(2): 173-182. Sofiane A.A. (2015). Sample-data observer-based intersample output predictor for electrohydraulic actuators, ISA Transactions, Elsevier, 58: 421-433. Tafazoli S., Salcudean S.F., Hashtradi-Zaad K. and Lawrence P.D. (2002). Impedance control of a teleoperated excavator. IEEE Transsactions on Control Systems Technology, 9(5): 355-369. Taheri A., Gustafsson P., R¨osth M., Ghabeheloo R. and Pajarinen J. (2022). Nonlinear model learning for compensation and feedforward control of real-world hydraulic actuators using Gaussian processes, IEEE Robotics and Automation Letters, 7(4): 3525-3592. Taylor C.J. and Robertson D. (2013). State-dependent control of a hydraulically actuated nuclear decomissioning robot. Control Engineering Pracrice, Elsevier, 21: 1716-1725. Villagra J., d’Andrea-Novel B., Mounier H. and Pengov M. (2007). Flatnessbased vehicle steering control strategy with SDRE feedback gains tuned via a sensitivity approach. IEEE Transactions on Control Systems Technology, 15: 554-565. Wang W., Chi H., Zhao S. and Du Z. (2018). A control method for hydraulic manipulators in automatic emulsion filling. Automation in Construction, Elsevier, 91: 92-99. Wind H., Renner A., Schmit S., Albrecht S. and Sawodny O. (2019). Comparioson of joint angle, velocity and acceleration estimates for hydraulically actuated manipuators to a novel dynamical approach. Control Engineering Practice, Elsevier, 91: 10411.

Flatness-Based Control in Successive Loops for Robotic ...

81

Yang G., Yao J., Le G. and Ma D. (2016). Asymptotic output tracking control of electrohydraulic systems with unmatched disturbances. IET Control Theory and Applications, 10(19): 2543-2551. Yang G. and Yao J. (2019). Output feedback control of electrohydraulic servo actuators with matched and mismatched disturbances rejection. Journal of the Franklin Institute, Elsevier, 356: 3152-3179. Zhum Q., Huang D., Yu B., Ba K., Kang X. and Wang S. (2022). An improved method combined SMC and MLESO for impendance control of legged robot’s electrohydraulic servo system, ISA Transactions, Elsevier, 130: 598-609.

Chapter 3

Optimized Fuzzy Sliding Mode Tracking Control of an Electro-Hydraulic Actuator System: A Simulation Study Muhamad Fadli Ghani1,2 Rozaimi Ghazali2,* Hazriq Izzuan Jaafar2 Chong Chee Soon3 and Zulfatman Has4 1 Malaysian

Institute of Marine Engineering Technology (MIMET), Universiti Kuala Lumpur, Perak, Malaysia 2 Centre for Robotics and Industrial Automation (CeRIA), Fakulti Kejuruteraan Elektrik, Universiti Teknikal Malaysia Melaka, Melaka, Malaysia 3 Department of Engineering and Built Environment, Tunku Abdul Rahman University College, Penang Branch Campus, Pulau Pinang, Malaysia 4 Electrical Engineering Department, University of Muhammadiyah Malang, Malang, Indonesia

Abstract The performance of an electro-hydraulic actuator (EHA) system’s trajectory tracking employing an optimized fuzzy sliding mode controller (FSMC) is presented in this paper. In simulations, the performance of the FSMC, which is developed based on the transfer function structure of a double-acting EHA system third-order model, is assessed using a chaotic

*

Corresponding Author’s Email: [email protected].

In: Automatic Control of Hydraulic Systems Editor: Michael G. Skarpetis ISBN: 979-8-88697-619-9 © 2023 Nova Science Publishers, Inc.

84

Muhamad Fadli Ghani, Rozaimi Ghazali, Hazriq Izzuan Jaafar et al. trajectory. The particle swarm optimization (PSO) algorithm identifies the design gain variables of the control law, which is developed from the concept of the exponential reaching law. The Lyapunov theorem theoretically demonstrates the stability of the control system. Simulation findings indicate that the proposed controller is highly robust and capable of accommodating system parameter change during trajectory tracking control. It also demonstrates that the proposed controller is superior to conventional PID controllers.

Keywords: parameters variation, electro-hydraulic actuator system, optimized fuzzy sliding mode controller, particle swarm optimization

Introduction The electro-hydraulic actuator (EHA) system arises from the development of fluid power technology in the early twentieth century, which was first described by the French scientist Blaise Pascal in 1663. In 1967, Merritt presented a detailed overview of fluid power technology, also known as hydraulic systems, followed by Maskrey and Thayer in 1978. In the initial generation of hydraulic systems, a flow control device powered a hydraulic actuator with an open loop configuration. The electro-hydraulic actuator (EHA) system, which was developed in response to the technological revolution, consists of a hydraulic system and an electrical system that control pressurized fluid flows from the motor pump to the electro-hydraulic control valve via an electrical signal. This electrical signal is then sent to the hydraulic actuator to produce a precise trajectory. The EHA systems are advantageous and dependable due to their small size in relation to their power, vast forceproducing capabilities, and quick reaction times (Kou et al., 2017; Cheng et al., 2020). This makes it a superior option for mobile hydraulic equipment used in construction (Qu et al., 2021). It is generally known that the EHA system is a nonlinear, highly uncertain system due to the high-frequency behavior of the servo valve and external disturbances. Moreover, errors in tracking and phase lag may arise during trajectory tracking due to nonlinearities, uncertainties, and disturbances in the EHA system, making controller design more complicated (Won et al., 2020; Du et al., 2020; Ghani et al., 2021). As a result of these problems, researchers are attempting to enhance the EHA control system. These problems have been addressed with an emphasis on the development of robust control systems to deal with uncertainty, system parameter changes, and potential disturbances. The EHA

Optimized Fuzzy Sliding Mode Tracking Control ...

85

system’s complexity and issues will necessitate a sophisticated and efficient control mechanism. In the last few decades, various control strategies for the application of issues in the tracking control of EHA systems have been documented. The number of initiatives for EHA system control, ranging from linear and nonlinear control to intelligent control strategies, has increased such as generalized predictive control (GPC) (Wang et al., 2018; Huang et al., 2020), model reference adaptive control (MRAC) (Kireçci et al., 2003; Yifei Zhao and Zongxia Jiao, 2016), sliding mode control (SMC) (Chen et al., 2019; Wang et al., 2020), self-tuning fuzzy proportional-integral-derivative (PID) (Quan et al., 2014), and neural network control (NN) (Guo and Chen, 2021). The SMC control strategy has demonstrated a great deal of promise and has been widely implemented in the EHA system. The SMC, a model-based nonlinear control technique with variable order, has been found to be useful for regulating nonlinear and uncertain systems, among other applications. It was utilized regularly and shown to be extraordinarily successful in the control of nonlinear, complicated systems. The SMC’s basic concept is derived from the discipline of variable structure control (VSC). The development of an SMC begins with the design of the sliding surface, which must be capable of responding to the control demand. It is anticipated that the sliding surface will retain the received control signal and return to its initial position. If the sliding surface criteria is satisfied, system performance can be predicted from the SMC. Several technical applications have utilized the capabilities of the SMC including active pneumatic systems (Iskandar Putra et al., 2020), suspension systems (Liu et al., 2020), and active magnetic bearing systems (Amrr and Alturki, 2021). Therefore, the simulation work required to develop an optimal fuzzy sliding mode controller (FSMC) for an EHA system is discussed in this research. The development of a third-order FSMC is anticipated. A particle swarm optimization (PSO) technique is utilized to simplify the control design approach for determining the optimal values for design variables. It is also anticipated that the proposed controller will be applicable to a range of future applications employing electrohydraulic actuators with double-acting pistons. To demonstrate the usefulness of the proposed control method, its performance is simulated and compared to the performance of a PID controller. The paper’s contributions are summarized as follows: a) For trajectory tracking on an EHA system, the FSMC was developed and proved to be an effective control method and b) Optimized variables design for the proposed controller is accomplished by employing the PSO algorithm.

86

Muhamad Fadli Ghani, Rozaimi Ghazali, Hazriq Izzuan Jaafar et al.

Methodology This work’s methodology began with the identification of a linear EHA system model using the Grey-box identification technique, followed by the construction of the FSMC controller. The optimized FSMC controller variables were determined using a PSO approach that required fewer mathematical formulations. Based on the mean square error (MSE) index, the effectiveness of the FSMC controller in trajectory tracking was tested by simulation employing the established linear model. For performance comparisons with the proposed controller, a PID controller scheme employing the same PSO algorithm was also built.

Grey-Box Identification The linear model of an EHA system is often valid across a fairly wide range, and it is well known that a linear model is sufficient for tracking control with advanced control techniques to obtain excellent performance of an EHA system in trajectory tracking applications (Kireçci et al., 2003). This study establishes a linear model for the EHA system using a parametric Grey-box technique, which can lead to more precise parameter predictions by accounting for prior knowledge of the system (Chen et al., 2019). This approach is used to physically model the EHA system using the fundamental law of physics. By neglecting nonlinearities and creating a model structure based on a continuous transfer function in (1), Knohl and Unbehauen (2000) discussed about physical-based modeling for the EHA system shown in Figure 1. Furthermore, the parameters 𝑎𝑖 (i = 1,2,3) are system parameters that will be computed using the MATLAB System Identification toolbox. 𝑋𝑝 (𝑠) 𝑈(𝑠)

𝑎

= 𝑠(𝑠2 +𝑎1𝑠+𝑎 2

3)

(1)

𝑋𝑝 (𝑠) and 𝑈(𝑠) are the Laplace Transformation of the actuator position xp and the input signal u, respectively. To collect time-domain input-output data for the continuous transfer function, the experimental hardware and software configuration depicted in Figure 2 is employed. The configuration includes a proportional valve, hydraulic plant, and hydraulic actuator, and MATLAB and Simulink are

Optimized Fuzzy Sliding Mode Tracking Control ...

87

utilized to operate the system. In most cases, the proportional directional valve is positioned close enough to the double-acting hydraulic actuator that pipeline dynamics can be neglected. Due to pipeline effects, low-frequency operation has little effect on input-output behavior. Meanwhile, the power supply unit is designed such that the systems maintain a consistent power supply pressure over a specific operating range. In addition, a well-designed proportional directional valve was implemented to decrease nonlinearities and inconsistencies in the dynamic performance. Due to the greater rate of dynamic reaction of the proportional directional valve in comparison to the dynamic response of the actuator, the dynamics of the proportional directional valve are never considered (Jelali and Kroll, 2003).

Figure 1. Schematic diagram of the EHA system.

Figure 2. Configuration of hardware and software for system identification.

88

Muhamad Fadli Ghani, Rozaimi Ghazali, Hazriq Izzuan Jaafar et al.

Figure 3. Open-loop performance.

For the system parameters identification process, the obtained input and output data were loaded into the MATLAB System Identification toolbox and as a result, the equation in (1) can be update as 𝑋𝑝 (𝑠) 𝑈(𝑠)

144400

= 𝑠(𝑠2 +372.3𝑠+7855)

(2)

where the values for 𝑎1 , 𝑎2 , and 𝑎3 are 144400, 372.3, and 7855, respectively. Furthermore, Figure 3 illustrates the simulation open loop system performance of the obtained system model in (2) by using a sinusoidal signal as the input.

The Fuzzy Logic Sliding Mode Control Design The SMC technique is a variable-structure control developed in the 1960s (Utkin, 1977). Construction of the sliding surface mathematical equation is widely regarded as the most crucial stage of SMC design (Eker, 2006). As illustrated in Figure 4, the state trajectories are directed toward the sliding surface.

Optimized Fuzzy Sliding Mode Tracking Control ...

89

Figure 4. Phase illustration of a sliding motion in SMC.

Figure 5 shows the error signal (e), and equation in (3) identifies e, where r and xp are the reference input and piston displacement signals, respectively. 𝑒 = 𝑟 − 𝑥𝑝

(3)

The sliding surface (s) is represented by equation (4) in the FSMC design, where 𝜆 is the control gain coefficient factor of the sliding surface and n is the order of the EHA system. It has been determined that the EHA system is a third-order structure for the current FSMC development effort. Furthermore, differentiating equation (4) the sliding surface satisfies equation (5).

Figure 5. The controller design’s block diagram.

90

Muhamad Fadli Ghani, Rozaimi Ghazali, Hazriq Izzuan Jaafar et al.

𝑠 = (𝜆 +

𝑑

𝑛−1

) 𝑑𝑡

𝑒

2

= 𝜆 𝑒 + 2𝜆𝑒̇ + 𝑒̈

(4)

𝑠̇ = 𝜆2 𝑒̇ + 2𝜆𝑒̈ + 𝑒⃛

(5)

When a reaching law is applied, the output of the system is compelled to follow the given surface. To maintain the stability of a closed-loop system, it is necessary that the reaching law be formulated in compliance with specific conditions. The exponential law (Liu and Wang, 2012), given by equation (6), is used in this work. 𝑠̇ = −𝜖𝑠𝑖𝑔𝑛(𝑠) − 𝑘𝑠; 1, 𝑠 > 0 𝜖 > 0, 𝑘 > 0, 𝑠𝑖𝑔𝑛(𝑠) = { 0, 𝑠 = 0 −1, 𝑠 < 0

(6)

The primary objective of the controller design is to ensure that the operation of the feedback control system is always stable. The Lyapunov stability theorem predicts that when the specified condition sṡ < 0 is met, the whole system will be stable and approach the sliding surface (Polyakov and Poznyak, 2012; Detiček and Kastrevc, 2016; Rezkallah et al., 2017). In the current study, the Lyapunov function is represented as in (7) with 𝑉(0) = 0 and 𝑉(𝑡) > 0 for 𝑠 ≠ 0. To ensure the transition from the reaching phase to the sliding phase and to maintain the trajectory’s stability, it is essential to adhere to the reaching condition designated as in (8). 1

𝑉 = 𝑠2

(7)

𝑉̇ = 𝑠𝑠̇ < 0 for 𝑠 ≠ 0

(8)

2

By substituting (6) into (8), 𝑉̇ = 𝑠(−𝜖𝑠𝑖𝑔𝑛(𝑠) − 𝑘𝑠) = −𝑠(𝜖𝑠𝑖𝑔𝑛(𝑠) + 𝑘𝑠)

(9)

Optimized Fuzzy Sliding Mode Tracking Control ...

91

when 𝑠 > 0, 𝑠𝑖𝑔𝑛(𝑠) = 1, 𝜖 > 0, 𝑘 > 0, 𝑉̇ = −𝑠(𝜖 + 𝑘𝑠) < 0

(10)

when 𝑠 < 0, 𝑠𝑖𝑔𝑛(𝑠) = −1, 𝜖 > 0, 𝑘 > 0, 𝑉̇ = −𝑠(−𝜖 + 𝑘𝑠) = 𝑠𝜖 − 𝑘𝑠 2 < 0

(11)

During control action, a discontinuity in the signum function causes the SMC controller to create rapid oscillations at the controller output (Utkin and Lee, 2006; Husain et al., 2008; Soon et al., 2022). In place of the signum function, the fuzzy logic (FL) function denoted in as fuzz is used in this work. As a result, equation (6) can be updated as 𝑠̇ = −𝜖𝑓𝑢𝑧𝑧(𝑠) − 𝑘𝑠;

(12)

Therefore, the overall control signal of the FSMC is represented in (13), where 𝑟⃛ is the third order derivative of reference signal.

𝑢𝐹𝑆𝑀𝐶 =

𝜆2 𝑒̇ +2𝜆𝑒̈ +𝑟⃛+𝑎2 𝑥̈ 𝑝 +𝑎3 𝑥̇ 𝑝 +𝜖𝑓𝑢𝑧𝑧(𝑠)+𝑘𝑠 𝑎1

(13)

Figure 6. Input membership functions of FL function.

A single-input, single-output FL function receives its input from the sliding surface. As shown in Figure 6, negative input (NI), zero input (ZI), and

92

Muhamad Fadli Ghani, Rozaimi Ghazali, Hazriq Izzuan Jaafar et al.

positive input (PI) were employed as triangle input membership functions (MFs).

Figure 7. Output membership functions of FL function.

As illustrated in Figure 7, three output MFs were utilized: negative output (NO), zero output (ZO), and positive output (PO). The employed rule base considered for this inquiry is displayed in Table 1. As a result of the rule-based design, Figure 8 displays the control action of the input output profile. Table 1. Rule Base for Input-Output Control in FL function Input Output

NI NO

ZI ZO

Figure 8. Input-output surface plot of FL function.

PI PO

Optimized Fuzzy Sliding Mode Tracking Control ...

93

Particle Swarm Optimization (PSO) It was invented by Kennedy and Eberhart in 1995 and expanded in 1998 with the addition of the word inertia weight. The PSO algorithm mimics the hunting behavior of fish schools and bird flocks. The solutions are obtained by modifying the particle’s, i position and adjusting based on its velocity data. Every particle in the swarm exhibits an extraordinary response and follows a fundamental rule by repeating its previous successes. The particle’s personal best position also influences its location in the surrounding area, with the global best position, abbreviated 𝑔𝑏𝑒𝑠𝑡𝑗, being the best option when compared to the individual best, abbreviated 𝑝𝑏𝑒𝑠𝑡𝑖𝑗 locations (Poli et al., 2007). Two initial particle parameters, namely position (𝑋𝑖𝑗𝑘 ) and velocity (𝑉𝑖𝑗𝑘 ), are defined for the search process, where k is the number of iterations. The new particle velocity, 𝑉𝑖𝑗𝑘+1 , and particle position, 𝑋𝑖𝑗𝑘+1 , are modified and updated according to equation (14) and (15). The particle population size, maximum iterations, social component, and cognitive component in this study had corresponding values of 20, 50, 2, and 2, respectively. 𝑉𝑖𝑗𝑘+1 = 𝑤𝑉𝑖𝑗𝑘 + 𝑐1 𝑟1 (𝑝𝑏𝑒𝑠𝑡𝑖𝑗 − 𝑋𝑖𝑗𝑘 ) + 𝑐2 𝑟2 (𝑔𝑏𝑒𝑠𝑡𝑗 − 𝑋𝑖𝑗𝑘 )

(14)

𝑋𝑖𝑗𝑘+1 = 𝑋𝑖𝑗𝑘 + 𝑉𝑖𝑗𝑘+1

(15)

In relation to the FSMC design, the parameters of 𝜆, 𝜖, and 𝑘 are aimed to be optimized. As a result, the position, 𝑋𝑖𝑗𝑘 can be defined as 𝑋𝑖𝑗𝑘 = [𝜆𝑘𝑖1

𝑘 𝜖𝑖2

𝑘] 𝑘𝑖3

(16)

In addition, the fitness function for the searching optimization procedure provided in this study is represented as 𝐽 = ∑|𝑒|

(17)

94

Muhamad Fadli Ghani, Rozaimi Ghazali, Hazriq Izzuan Jaafar et al.

Performance Index Mean Square Error (MSE): A lower MSE value suggests a greater potential for successful tracking control in the EHA system using the proposed control strategy. The MSE is formulated in (18), where n is the size of the sample. 1

2 𝑀𝑆𝐸 = 𝑛 ∑𝑡=20 𝑡=0 (𝑒)

(18)

Result and Discussion This section presents the results of the evaluation of the tracking control performance of the FSMC and PID controllers using MATLAB and Simulink (R2021b) with a 1 ms sample time and PSO-optimized parameters. In addition, chaotic signals are used to generate the required trajectory. Using both sinusoidal and point-to-point tracking, a chaotic trajectory is constructed. In a simulation exercise, the FSMC and PID controllers’ capacity to follow 100 seconds chaotic and sinusoidal trajectories set points during the actuator stroke is examined. Table 2 lists the obtained values for the parameters and optimized control variables. Table 2. Parameters and optimized control variables Parameter 𝑎1 𝑎2 𝑎3 𝑘𝑝 𝑘𝑖 𝑘𝑑 𝜆 𝜖 𝑘

Value 144400 372.3 7855 20.57 15.07 2.22 259.30 3.08 10.97

By employing the parameters and optimized control variable, the control algorithm for the PID and FMSC controller can be written as in (19) and (20), respectively.

Optimized Fuzzy Sliding Mode Tracking Control ...

95

𝑡

𝑢𝑃𝐼𝐷 = 𝑘𝑝 𝑒 + 𝑘𝑖 ∫0 𝑒𝑑𝑡 + 𝑘𝑑 𝑒̇ 𝑡

= 20.57𝑒 + 15.07 ∫0 𝑒𝑑𝑡 + 2.22𝑒̇ 𝑢𝐹𝑆𝑀𝐶 =

(19)

67236.49𝑒̇ +518.6𝑒̈ +𝑟⃛+372.3𝑥̈ 𝑝 +7855𝑥̇ 𝑝 +3.08𝑓𝑢𝑧𝑧(𝑠)+10.97𝑠 144400

(20)

The MSE values of PID and FSMC controllers are shown in Table 3 as the result of an exhaustive analysis. Figures 9 and 10 illustrate the controllers’ tracking effort signal for the chaotic and sinusoidal trajectories, respectively, demonstrating that the FSMC controller is more effective than PID.

Figure 9. Output performance of chaotic trajectory tracking; (A) Origin and (B) zoom.

Table 3. MSE performances Trajectory Chaotic Sinusoidal

Controller PID FSMC PID FSMC

MSE (mm2) 0.0138 0.0027 0.0099 0.0062

96

Muhamad Fadli Ghani, Rozaimi Ghazali, Hazriq Izzuan Jaafar et al.

In the meantime, as depicted in Figures 11 and 12, the control signal patterns between PID and FSMC appear to be nearly incomparable, with significant variances. This indicates that the PID and FSMC controllers can continue to function throughout a trajectory operation. In comparison to the FMSC controller, the PID controller delivered a larger control signal magnitude.

Figure 10. Output performance of sinusoidal trajectory tracking; (A) Origin and (B) zoom.

Figure 11. Control signal of chaotic trajectory tracking.

Optimized Fuzzy Sliding Mode Tracking Control ...

97

Figure12. Control signal of sinusoidal trajectory tracking.

Conclusion Because it has a high power-to-load ratio, can move quickly, and creates a substantial amount of torque, the EHA system is commonly utilized in modern industrial applications. In contrary, the EHA system is difficult to control, particularly in terms of trajectory tracking. This is due to the fact that it is a system with unknown variables, such as leakage, friction, pressure, and fluid temperature. In addition, an SMC-based controller design for trajectory tracking has been implemented in the EHA system, and it has been discovered that SMC is an effective method for complicated nonlinear systems. However, the EHA system’s SMC controller is influenced by the high-frequency servo valve action, resulting in damage to the system’s final control element. The FSMC controller is therefore proposed to resolve this issue. The results of this investigation reveal that the proposed controller is significantly more effective than the PID controller at tracking trajectory. Particle swarm optimization (PSO) is also used to optimize the variables for the proposed and PID controllers. In the future, it will be necessary to test the proposed controller on a real-time hardware platform and to experiment with various SMC versions, including hybrid fuzzy PID and fractional order SMCs controllers.

Acknowledgments The assistance of Universiti Teknikal Malaysia Melaka (UTeM), Universiti Kuala Lumpur Malaysian Institute of Marine Engineering Technology

98

Muhamad Fadli Ghani, Rozaimi Ghazali, Hazriq Izzuan Jaafar et al.

(UniKL-MIMET), and Ministry of Higher Education (MoHE) are sincerely appreciated and greatly acknowledged. The research was funded by Fundamental Research Grant Scheme (FRGS) Grant No. FRGS/1/2021/FKE/F00468.

References Amrr, S. M., and Alturki, A., 2021. Robust Control Design for an Active Magnetic Bearing System Using Advanced Adaptive SMC Technique. IEEE Access, IEEE, 9, pp. 155662–155672. Chen, Z., Yuan, X., Yuan, Y., Lei, X., and Zhang, B., 2019. Parameter Estimation of Fuzzy Sliding Mode Controller for Hydraulic Turbine Regulating System Based on HICA Algorithm. Renewable Energy, Elsevier Ltd, 133, pp. 551–565. Cheng, L., Zhu, Z. C., Shen, G., Wang, S., Li, X., and Tang, Y., 2020. Real-Time Force Tracking Control of an Electro-Hydraulic System Using a Novel Robust Adaptive Sliding Mode Controller. IEEE Access, 8, pp. 13315–13328. Detiček, E., and Kastrevc, M., 2016. Design of Lyapunov Based Nonlinear Position Control of Electrohydraulic Servo Systems. Strojniški Vestnik - Journal of Mechanical Engineering, 62(3), pp. 163–170. Du, M., Zhao, D., Ni, T., Ma, L., and Du, S., 2020. Output Feedback Control for Active Suspension Electro-hydraulic Actuator Systems with a Novel Sampled-data Nonlinear Extended State Observer. IEEE Access, 8, pp. 128741–128756. Eker, İ., 2006. Sliding Mode Control with PID Sliding Surface and Experimental Application to An Electromechanical Plant. ISA Transactions, 45(1), pp. 109–118. Ghani, M. F., Ghazali, R., Jaafar, H. I., Soon, C. C., Shern, C. M., and Has, Z., 2021. The Effects of Mass Variation on Closed-loop EHA System under High Leakage Flow Condition. 2021 11th IEEE International Conference on Control System, Computing and Engineering (ICCSCE), IEEE, pp. 206–209. Guo, Q., and Chen, Z., 2021. Neural Adaptive Control of Single-Rod Electrohydraulic System with Lumped Uncertainty. Mechanical Systems and Signal Processing, Elsevier Ltd, 146, p. 106869. Huang, J., An, H., Yang, Y., Wu, C., Wei, Q., and Ma, H., 2020. Model Predictive Trajectory Tracking Control of Electro-Hydraulic Actuator in Legged Robot With Multi-Scale Online Estimator. IEEE Access, 8, pp. 95918–95933. Husain, A. R., Ahmad, M. N., Halim, A., and Yatim, M., 2008. Chattering-free Sliding Mode Control for an Active Magnetic Bearing System. International Journal of Mechanical, Aerospace, Industrial, Mechatronics and Manufacturing Engineering, 2(3), pp. 300–306. Iskandar Putra, M., Irawan, A., and Mohd Taufika, R., 2020. Fuzzy Self-Adaptive Sliding Mode Control for Pneumatic Cylinder Rod-Piston Motion Precision Control. Journal of Physics: Conference Series, 1532(1), pp. 1–12. Jelali, M., and Kroll, A., 2003. Hydraulic Servo-Systems: Modelling, Identification and Control, Springer Science & Business Media.

Optimized Fuzzy Sliding Mode Tracking Control ...

99

Kireçci, A., Topalbekiroglu, M., and Eker, I., 2003. Experimental Evaluation of A Model Reference Adaptive Control for A Hydraulic Robot: A Case Study. Robotica, 21, pp. 71–78. Knohl, T., and Unbehauen, H., 2000. Adaptive Position Control of Electrohydraulic Servo Systems Using ANN. Mechatronics, 10(1–2), pp. 127–143. Kou, F., Wang, Z., Du, J., Li, D., and Fan, E., 2017. Study On Force Tracking Control of Electro-Hydraulic Active Suspension. 2017 IEEE 3rd Information Technology and Mechatronics Engineering Conference (ITOEC). Vol. 6, IEEE, pp. 1078–1082. Liu, J., and Wang, X., 2012. Advanced Sliding Mode Control for Mechanical Systems, Springer Berlin Heidelberg, Berlin, Heidelberg. Liu, S., Hao, R., Zhao, D., and Tian, Z., 2020. Adaptive Dynamic Surface Control for Active Suspension With Electro-Hydraulic Actuator Parameter Uncertainty and External Disturbance. IEEE Access, 8, pp. 156645–156653. Poli, R., Kennedy, J., and Blackwell, T., 2007. Particle Swarm Optimization. Swarm Intelligence, 1(1), pp. 33–57. Polyakov, A., and Poznyak, A., 2012. Unified Lyapunov Function for A Finite-Time Stability Analysis of Relay Second-Order Sliding Mode Control Systems. IMA Journal of Mathematical Control and Information, 29(4), pp. 529–550. Qu, S., Fassbender, D., Vacca, A., and Busquets, E., 2021. A High-Efficient Solution for Electro-Hydraulic Actuators with Energy Regeneration Capability. Energy, Elsevier Ltd, 216, pp. 1–16. Quan, Z., Quan, L., and Zhang, J., 2014. Review of Energy Efficient Direct Pump Controlled Cylinder Electro-Hydraulic Technology. Renewable and Sustainable Energy Reviews, 35, pp. 336–346. Rezkallah, M., Sharma, S. K., Chandra, A., Singh, B., and Rousse, D. R., 2017. Lyapunov Function and Sliding Mode Control Approach for the Solar-PV Grid Interface System. IEEE Transactions on Industrial Electronics, IEEE, 64(1), pp. 785–795. Soon, C. C., Ghazali, R., Ghani, M. F., Shern, C. M., Sam, Y., and Has, Z., 2022. Chattering Analysis of an Optimized Sliding Mode Controller for an Electro-Hydraulic Actuator System. Journal of Robotics and Control, 2(1), pp. 7–12. Utkin, V., 1977. Variable Structure Systems with Sliding Modes. IEEE Transactions on Automatic Control, 22(2), pp. 212–222. Utkin, V., and Lee, H., 2006. Chattering Problem in Sliding Mode Control Systems. IFAC Proceedings Volumes. Vol. 39, IFAC, p. 1. Wang, D., Zhao, D., Gong, M., and Yang, B., 2018. Research on Robust Model Predictive Control for Electro-Hydraulic Servo Active Suspension Systems. IEEE Access, IEEE, 6, pp. 3231–3240. Wang, M., Wang, Y., Yang, R., Fu, Y., and Zhu, D., 2020. A Sliding Mode Control Strategy for an ElectroHydrostatic Actuator with Damping Variable Sliding Surface. Actuators, 10(1), pp. 1–17.

100

Muhamad Fadli Ghani, Rozaimi Ghazali, Hazriq Izzuan Jaafar et al.

Won, D., Kim, W., and Tomizuka, M., 2020. Nonlinear Control With High-Gain Extended State Observer for Position Tracking of Electro-Hydraulic Systems. IEEE/ASME Transactions on Mechatronics, IEEE, 25(6), pp. 2610–2621. Zhao, Y. and Zongxia Jiao, 2016. Fractional Model Reference Adaptive Control for Electro-Hydraulic Servo System. 2016 IEEE Chinese Guidance, Navigation and Control Conference (CGNCC), IEEE, pp. 2048–2052.

Chapter 4

An Advanced Neural-Disturbance-Observer Control Method for Constrained Hydraulic Systems Dang Xuan Ba1, and Kyoung Kwan Ahn2 1Department

of Automatic Control, Hochiminh City University of Technology and Education (HCMUTE), Ho Chi Minh City, Vietnam 2Department of Mechanical Engineering, University of Ulsan (UoU), Ulsan City, Korea

Abstract Nowadays, hydraulic systems play a key role in aerospace applications and heavy industry. New research for improvements of such the systems in terms of control accuracy and energy efficiency shows no signs of stopping. Complicated systematic dynamics, unpredictable working conditions and physical constraints are however big practical challenges restricting the expected control performances. In this chapter, a new intelligent motion controller is presented for constrained hydraulic systems using a special combination of neural network and nonlinear disturbance observers. In this first stage, a simplified constrained slidingmode-backstepping scheme is designed to drive the control objective to a vicinity around origin without any physical violations. In the second stage, uncertain nonlinearities inside the system dynamics are compensated by new neural networks with fast learning rules. The neural learning errors and external disturbances are in the third stage presented as extended nonautonomous models and are then approximated by nonlinear high-order disturbance observers. Robustness of the closed

Corresponding Author’s Email: [email protected].

In: Automatic Control of Hydraulic Systems Editor: Michael G. Skarpetis ISBN: 979-8-88697-619-9 © 2023 Nova Science Publishers, Inc.

102

Dang Xuan Ba and Kyoung Kwan Ahn loop control system is maintained by a proper Lyapunov theory. Effectiveness and feasibility of the proposed control system for an asymptotically tracking performance are then confirmed by comparative simulation results.

Keywords: hydraulic system, constraint control, sliding mode control, backstepping control, neural network, disturbance observer, nonlinear control

Introduction Hydraulic actuators (HAs) have been widespread employed in heavy industries and airplane manufacturing, thanks to capacities of a high powerto-weight ratio and a large force generation. Typical applications of the HAs in market can be listed as municipal water management systems [1], excavators [2], civil machines, press machines, or truck cranes [3]. Between the HAs, pump-controlled hydraulic actuators (PHAs) have been favorite used in practical systems that require higher energy efficiency, less leakage, and lower overall weight [4]-[6]. Despite holding unique advantages, it challenges to design high-precision controllers for such the system due to their complicated dynamics with uncertain nonlinearities and unpredictable external disturbances. Many control methods have been recently successfully developed for the position or force control of HA systems. Proportionalintegral-derivative (PID) approaches [7-9] were first studied to realize the control mission by using a conventional mode or combining with genetic algorithm, fuzzy machine, or analyzing the system model for advanced modes. Promising control outcomes were exhibited, but the good performances were difficult to maintain under various working conditions due to the unpredictable external disturbances and high nonlinearities existing during the working process. To tackle these limitations, model-based linear methods [10-12] and linearized techniques [13, 14] were proposed. Uncertain parameters could be estimated by an indirect adaptive controller under a linear pole placement design [15]. This method delivered better control performances than suboptimal PID ones. Nevertheless, manipulating the control algorithm based on linear models, which in fact contain large numbers of complexities and nonlinearities [16], obviously cut down the control efficiency. As a result, adaptive nonlinear-based controllers have been developed using

An Advanced Neural-Disturbance-Observer Control Method …

103

comprehensive mathematical nonlinear models of the plants [17-19]. Parametric uncertainties and the nonlinearities could be coped with by adaptation rules and robust nonlinear designs using the adaptive sliding mode [20], virtual decomposition control (VDC) [21] and backstepping [22-27] control schemes. Higher control accuracies were presented, but influence of unknown terms (i.e., external disturbances, and/or unmodeled dynamics) in the system models bring to a new challenge. To boost the control quality of the HAs in both transient responses and steady-state phases, integrated nonlinear control approaches [28-29] were studied with the following improvements: the estimation-convergence speed was enhanced by integration learning rules; and offset values of the lumped disturbances were eliminated by integral functions of state control errors. However, problems of time-varying disturbances were till opened. To terminate influence of disturbances on the force dynamics, a disturbance-rejection nonlinear control method [30] was developed. Though outstanding tracking control precision was achieved, a few notes in adoption of this control approach can be remarked here: 1) Total disturbances in pressure dynamics of the systems were not considered; 2) Use of large control gains in robust control signals could degrade transient control performances. As a sequence, disturbance-observer-based nonlinear control methodologies [31, 32] were researched. The linear disturbance observer (DO) versions were used with high-gain feedback correction terms [33-35]. A high-order DO model [36] was also studied to approximate both disturbances and their time derivatives. The total disturbances (or unknown terms) were then well eliminated by the advanced DO designs in which the approximation errors could be maintained inside a ball with a finite radius. The side of the ball could be moderated by the observer bandwidth selected. A new sliding mode DO version which adopted a switching gain rule [37] was synthesized to deal with time-varying disturbances. The asymptotic performance could be provided for a class of disturbances that are modelled as linear autonomous systems. Approximation performances of dynamical disturbances have been further accreted by high-order sliding mode differentiators (or disturbance observers) [38, 39]. Excellent learning outcome could be theoretically proven using induction-based discussions. Besides, a nonlinear disturbance-observer based control method has been proposed for the asymptotic stability of hydraulic systems [49]. Its effectiveness was confirmed by analytical proofs and realtime experiments. However, it still requires mathematical models to apply such the new learning algorithm.

104

Dang Xuan Ba and Kyoung Kwan Ahn

In this chapter, we discuss about an intensive study, referred to as an advanced neural disturbance-observer-based nonlinear constrained (NDC) controller, for position tracking of hydraulic systems. Such the systems are normally employed in automation applications in heavy industries and flight control that require reliability, high output power, high energy efficiency, and precision positioning control [1], [4], [40], [41]. The complications in system modeling, control and design however could result in an undesired performance. To obtain a high tracking performance for this system, the control strategy possesses the following natures: 1) To drive the system output track to a desired profile safely inside physical bounds, a new constrained control algorithm is developed based on a modified sliding-mode-backstepping framework. 2) To improve the control performance by removing internal functionalities and external disturbances inside the system dynamics in the control process, two versions of nonlinear neural disturbance observers are studied using special learning rules. 3) Asymptotical stabilities of the estimation subsystems and the closedloop system are concretely proven by Lyapunov approaches. 4) Effectiveness and feasibility of the designed controller were intensively investigated by comparative simulations. The following Nomenclature table declares parameters using throughout this paper. The chapter content is outlined as follows: the system modeling is reviewed in Section II; the constrained control design and the nonlinear neural disturbance observers are designed in Section III; the comparative validation results are extensively discussed in Section IV; finally, the paper is concluded in Section V.

Nomenclature

 ˆ   ˆ  

Time derivative of  .

, , 

Maximum, minimum, and nominal values of  .

  sup(|  |)

Maximum absolute value of  .

Estimate of  . Estimation error of  .

An Advanced Neural-Disturbance-Observer Control Method …

P1 , P2

Displacement of the cylinder or system position. Desired position or prescribed profile of the control system. Pressures inside chambers 1 and 2 of the cylinder.

A1 , A2

Bore- and rod- side areas of the cylinder.

m

Total mass affecting to the system motion.

V10 ,V20

Original volumes of the chambers 1 and 2.

V1t  V10  A1 x

Active volume of the chamber 1.

x

xd

V2t  V20  A2 x

Active volume of the chamber 2.

e

Effective bulk modulus of the hydraulic fluid.

CLi

Coefficient of internal leakages.

D

Displacement of the pump.

Kdr

Control gain of the motor driver.

V

Volumetric efficiency of the pump.

u

Driving voltage of the motor driver.

b1 , b2 , 2 , 3

f2

Positive constants in the system model. Lumped disturbances of force dynamics and pressure dynamics. Virtual bounded disturbances of force dynamics and pressure dynamics. Nonlinear function in force dynamics.

f 3i|i 1,2,3.

Nonlinear function in pressure dynamics.

 2 , 3 h2 , h3

k1 , k2 , k3 ,  2 , 3 l2i|i 1..4. ,  2 l3i|i 1,2,3. ,  3

 2 , 3  21 ,  22 , ,  2 , 1 , 2

Positive control gains. Positive estimation gains of the disturbance observer for force dynamics. Positive estimation gains of the disturbance observer for pressure dynamics. Positive functions chosen temporarily for theoretical proofs. Positive constants chosen temporarily for theoretical proofs.

105

106

Dang Xuan Ba and Kyoung Kwan Ahn

System Modeling and Problem Statements A double-acting single-rod hydraulic system, as sketched in Figure 1, is studied in this chapter. The main components of the system are listed in Table I. The system motion is the displacement of the cylinder (01) which is driven by a fixed-displacement gear pump (07) [42] through a control hydraulic circuit (10). The pump is actuated by an AC servo motor (08) [43] regulated by a proper driver (09) [44]. Flow differences between chambers A1 and A2 for the system movement are automatically compensated by check valves (04.1 and 04.2). A directional control valve (05) is set up as a hydraulic locker of the valve (04.1). The positive direction of the cylinder motion is chosen as shown in Figure 1.

Figure 1. Working principle of the studied system.

An Advanced Neural-Disturbance-Observer Control Method …

107

Table 1. Component List of the Studied System Number 01 02 03 04.1, 04.2 05

Device Hydraulic Cylinder Relief valves Reservoir Check valves Directional valve

Number 06 07 08 09 10

Device Pilot check valve Hydraulic pump AC motor AC motor driver Hydraulic circuit

By applying Newton’s second law, the motion of the system is governed by the force dynamics: mx  P1 A1  P2 A2  f Fr   2

(1)

where f Fr denotes a lumped friction, and  2 is a combination of the external force, hard-to-model forces and modeling errors. As referred to in [16], [26], and [28], pressure dynamics is expressed by the continuity laws of the hydraulic system

V1t P1  e V DK dr u  A1 x  CLi  P1  P2   Qcv1    31   V2t P2  e  V DK dr u  A1 x  CLi  P1  P2   Qcv 2    32

(2)

where Qcv1 , Qcv 2 are flows throughout the check valves (04.1 and 04.2), respectively, and 31 and 32 present pressure disturbances which include external leakages, un-modeled elements and modeling errors. Assumption 1: a) In fact, the term f Fr is a function of the system velocity ( x) and is differentiable everywhere except at the zero value because of the existence of the Coulomb friction. Hence, its nominal value can be approximated as a differentiable function [30], [33], as follows: f Fr  b1 x

(3)

108

Dang Xuan Ba and Kyoung Kwan Ahn

b) Qcv1 and Qcv 2 are discharged and supplemental flows supporting for the negative- and positive-direction motions, respectively. Based on their functionalities, the nominal terms of Qcv1 , Qcv 2 can be modelled as  Qcv1   x  A1  A2  sm   x   Qcv 2  x  A1  A2  sm  x   1 sm  x2   1  e 3 x2 



(4)



Note that, the function sm can work as smooth switching elements. Their parameters can be chosen to be large enough for this study. The modeling errors can be included into the force and pressure disturbances ( 2 , 31 and 32 ) [30-34]. Defining state variables as

 x1 , x2 , x3    x, x,  P1 A1  P2 A2   , T

T

and

combining Eqs. (1)-(4), the total mathematical model of the hydraulic system is presented in the following state-space form:  x1  x2   mx2  x3  f 2   2 x  f  g u   3 3 3  3

(5)

where the detailed dynamical functions are summarized as  f 2  f Fr   f31   e A1V2t x2   A1  ( A1  A2 ) sm( x2 )     e A2V1t x2  A2 + ( A1  A2 ) sm( x2 )      e CLi ( P1  P2 )  A1V2t  A2V1t     f32   eV DK dr ( A2V10  A1V20 )  f V V 1t 2t  33   A  V 1  A  V 1 1 31 1t 2 32 2t  3  1  f3  f31 f33  1   g3  f32 f33

(6)

An Advanced Neural-Disturbance-Observer Control Method …

109

Remark 1: For practical hydraulic systems, the system output is limited in a certain constraint: x1  x1  x1

(7)

where x1 , x1 are the upper and lower bounds of the system output x1 . Remark 2: The studied system is an uncertain nonlinear system. Indeed, the parameters CLi , 2 , 3 , b1 , b2 are hard to determine exactly. V10 ,V20 , m are also uncertain. Meanwhile, V and e may change in nonlinear fashions during the working process [42], [45], [46]. Moreover,  2 and  3 are timevarying disturbances. Hence, designing a highly accurate position-tracking controller for the system is a challenging task.

Neural-Disturbance-Observer Control Design with Output Constraints In this section, the intelligent controller is designed to drive the system output x to track the desired trajectory xd as closely as possible based on the model derived from Sections II. The structure of the controller is a simplified combination of the sliding mode and backstepping techniques. Two modified disturbance estimation mechanisms are then employed for the obtained system model to approximate the unknown terms which include the deviations of uncertainties, external disturbances, and hard-to-model elements. Assumption 2: a) The system variables x1 , x2 , P1 ,and P2 are measurable. b) Nominal values of system uncertainties can be determined using the model-based identification method [28]. Hence, their deviations are implicitly included into the disturbance terms. c) Total disturbances ( 2 and 3 ) of the force and pressure dynamics, and the desired reference input ( xd ) are bounded and their time derivatives are also bounded up to the second, first, and third orders [30], [33], respectively. There always exist virtual bounded disturbances satisfying:

110

Dang Xuan Ba and Kyoung Kwan Ahn

hi|i 2.3  t    i  i i

(8)

A Simplified Output-Constraint Backstepping Controller (OCBS) The control objective is first represented by the main tracking error: e  x1  xd

(9)

Note that since the output is restricted in a given range (7), the error (9) is limited in e  e  e  e  x1  xd e  x  x 1 d 

(10)

where e , e are the upper and lower bounds of the control error e . To tackle the constraint (10) in control design process, a new free control variable is employed using the following transformation:

  e e   ( e)  z   1  (e)  e e e e e     h e, e ,e     (e)  0 if e  0  1 otherwise

(11)

where h and (e) are called as transformation function and step function respectively. We next define a nonlinear sliding surface as s  k1 z  me

where k1 is a positive constant.

(12)

An Advanced Neural-Disturbance-Observer Control Method …

111

Differentiating the surface with respect to time and combining with the systems (5) and (11) lead to s  k1 z  mxd  f 2  x3   2 .

(13)

The constraint (11) reveals that once the free variable z is bounded, the main control objective e will be inside of its defined bounds. Furthermore, according to the definition (11), a good control outcome will be achieved from a small value of the surface. To this end, a virtual control input is selected as

x3d  k1 z  mxd  f 2  k2 s  ˆ2

(14)

where k1 is a positive gain. By integrating the virtual signal (14) into (13), it can be seen that if

(ˆ2 and x3d ) respectively approach to the true values of ( 2

and x3 ) , then the

surface s in (12) will approach to zero. Hence, it is worth considering a new state control error as

e3  x3  x3d .

(15)

Its time derivative is e3  f3  g3u  3  x3d .

(16)

To realize the control error e3 as the aforementioned analysis, the final control input is proposed as follows: u

   1   f3   1 s  x3d  ˆ3   k3 g3e3  g3   2  

(17)

Remark 2: The control inputs (14) and (17) always contain estimates of the system disturbances which mainly degrade the system control performance. However, the closed control system will be asymptotically

112

Dang Xuan Ba and Kyoung Kwan Ahn

stable once the estimation errors of the disturbances converge to zeros in infinite time. In the following sections, nonlinear disturbance estimation observers are going to be designed for an excellent control accuracy.

Disturbance Estimation for Pressure Dynamics A non-autonomous model is adopted to represent the pressure dynamics in (5), as follows [31], [34], [37]: T   x3  g3u  W3 r3   3    3   3 3  3

(18)

where r3  n1 and W3  n1 are respectively regression vector and optimal weight vector, that are employed to model the total uncertain nonlinearities of the pressure dynamics. All elements of the regression vector r3 are positive functions. Note that, the disturbance ( 3 ) and its time derivative are bounded for any bounded virtual disturbance (3 ) . An estimation model is then used from the known functions of the pressure dynamics and the estimation variables:  xˆ3  g3u  Wˆ3T r3  ˆ3  l30 sgn  x3   l31 x3  ˆ  3   3ˆ3  l32 x3  l33 sgn  x3   Wˆ3  3diag(r3 )Wˆ3  l34 x3 r3  l35 sgn  x3  r3 

(19)

To verify the effectiveness of the proposed observer, the following lemma is studied: Lemma 1: Given a bounded system (14) under Assumption 2, if employing an estimation model as Eq. (19), the following properties hold: a) The estimation system is stable for any positive-bounded estimation gains (l3i|i=0..5). b) The system is asymptotically stable with the gain l33 chosen to be

An Advanced Neural-Disturbance-Observer Control Method … 1 l35l32l34  l33  0  l33  o3 (t )  0   1  3  1 1 1  l33  o3 (t )  l30  0.25 33l32l34 r3 W3  I n13 l34l32 o3 (t )



113

(20)



2

0

Proof: After combining (18) and (19), the estimation-error dynamics are  x3  W3T r3   3  l30 sgn  x3   l31 x3    3   3 3  l32 x3  l33 sgn  x3   o3  t   W3  3diag(r3 )W3  l34 x3 r3  l35 sgn  x3  r3  3diag(r3 )W3

(21)

We then consider a Lyapunov function:

L11  0.5l32 x32  0.5 32  0.5l32l341W3T W3

(22)

Differentiating (22) with respect to time and using (21) can result in L11  l32 x3  l30 sgn  x3   l31 x3 



  3  3 3  l33 sgn  x3   o3  t 



  x32 x32 1 T  l32l34 W3  3 diag( r ) W  l sgn x r   diag(r3 )W3    3 3 35 3 3 3 2 2 2   1  x3 1  x3 1  x3   x32

 l32l30 x3  l32l31 x32   3 32  l33 sgn  x3   3   3o3  t  1  3l32l34

 3

x32

W T diag(r3 )W3  l35 2 3

1  x3

x32 1  x32

x32 1

l l 1W T 2 32 34 3 x3

sgn  x3  r3

1 T l32l34 W3 diag(r3 )W3

(23) By applying Young inequality, the system (23) can be rewritten as

114

Dang Xuan Ba and Kyoung Kwan Ahn

  l  o 2  L11  l32l30 x3  l32l31 x32  3   3   32   33 3   4  3    x32

1 3l32l34

1   w3 W3T diag(r3 )W3  3 2

1  x3

2

l  1 l l 1r T 35 I n1  W3  2 32 34 3  1  x3  3  4  w3 (24) x32

where  3 and w3 are arbitrary positive constants, and I n1  [1;1;.....;1] is a one-full vector. It can be seen that there always exit the positive constants (  3 and w3 ) to ensure T

3   3  0  1   w3  0

(25)

It leads to the proof of the first statement of Lemma 1. Now, we continue a new integral positive function [49] as follows: 1 T L11  0.5l32 x32  0.5 32  0.5l32l34 W3 W3 t

L1  L11  L10 

  x l 3

33 sgn

 x3   o3 ( )   d

0

1 T  0.5l32 x32  0.5 32  0.5l32 l34 W3 W3 t



  x l 3

33 sgn

(26)

 x3   o3 ( )   d  L10

0

where L10 is a positive constant that could be selected as in previous work [49]. The time derivative of the new Lyapunov function is

An Advanced Neural-Disturbance-Observer Control Method …

115

L1  l32l30 x3  l32l31 x32   3 32  l33 sgn  x3   3   3o3  t  1 T 1 T  3l32l34 W3 diag(r3 )W3  l35l32l34 W3 sgn  x3  r3

1 T  3l32l34 W3 diag(r3 )W3  x3  l33 sgn  x3   o3 ( ) 

  l32l30 x3  l32l31 x32   3 32



  l33 sgn  x3   o3  t   x3  W3T r3  l30 sgn  x3   l31 x3



1 T 1 T  3l32l34 W3 diag(r3 )W3  l35l32l34 W3 sgn  x3  r3

1 T  3l32l34 W3 diag(r3 )W3  x3  l33 sgn  x3   o3 ( ) 

 l32l30 x3  l32l31 x32   3 32   l33 sgn  x3   o3  t    l30 sgn  x3   l31 x3 





1 T 1  3l32l34 W3 diag(r3 )W3  l35l32l34  l33 W3T sgn  x3  r3



1 T 1  3l32l34 W3 diag(r3 ) W3  I n131l34l32 o3  t 

 (27)

It follows L1  l32l30 x3  l32l31 x32  3 32   l33 sgn  x3   o3  t    l30 sgn  x3   l31 x3 





1 T 1  3l32l34 W3 diag(r3 )W3  l35l32l34  l33 W3T sgn  x3  r3



1 T 1  3l32l34 W3 diag(r3 ) W3  I n131l34l32 o3  t 

 r W  I

 o (t ) 

1 1  0.25 33l32l34 r3 W3  I n131l34l32 o3 (t )

2

1  0.25 33l32 l34 3

2

3

1 1 n13 l34 l32



3

(28) Now, if we recall the constraint (20), the second statement of Lemma 1 has been proven. Remark 3: By employing the designed observer, it can confirm that the lumped unknown term  3 of the pressure dynamics is definitely determined with proper selections of the learning gains

( 3 , l3i|i 0..5 ).

It is also possible to

apply the estimation results in the control phase. Nonetheless, employing this observer for the force dynamics may generate the chattering control input.

116

Dang Xuan Ba and Kyoung Kwan Ahn

Disturbance Estimation for Force Dynamics From the force dynamics (5) and Assumption 2, an equivalent disturbance model is proposed as follows  x21  x3  f 2   2 with x21  mx2   2   2 2  o2

(29)

Looking back to the control rule in (14) and (17) reminds us that the time derivative of the estimation disturbance (ˆ2 ) is adopted in the final control signal. A new estimation model for the dynamics is then designed as  xˆ21  x3  f 2  ˆ2  l21 x21  t ˆ ˆ  2   2 2  l22 x21   l23 x21  l24 sgn  x21   d 0 



(30)

To investigate the convergence of the proposed observer, a new lemma is investigated: Lemma 2: Considering a bounded system (29) under Assumption 2, if employing an estimation model as (30) with some positive constants ( 21 ,  22 , , and 2 ), the following results are obtained: a) The estimation system is stable if the used gains are calculated from the condition  2     21  l22   21   22   2 l     l     22 21  21 2  23

(31)

b) The system is asymptotically stable if the gains satisfy the condition Eq. (25) and:

l24  h

2

(32)

An Advanced Neural-Disturbance-Observer Control Method …

117

Proof: The estimation-error dynamics of the force system can be presented by combining (29) and (30), as follows  x21   2  l21 x21  t   2   2 2  l22 x21   l23 x21  l24 sgn  x21   d  o2  0



(33)

We synthesize a composited variable as

 2   2   21 2   22 x21

(34)

A new Lyapunov function is verified as



2 L21  0.5  2 22 x21   2 22   22



(35)

The time derivative of the new function with employing (33) and (34) is 2 L21    2 22l21 x21   2 21 22   2  l24 sgn  x21   o2 



  2  2  2   21    2  l22   22   2 



  2  x21  l22l21   22l21  l23   . (36) Applying the given condition (31) leads to completion of the first statement of Lemma 2: 2 L21    2 22l21 x21   2 21 22   22

  2  l24 sgn  x21   o2 

(37)

118

Dang Xuan Ba and Kyoung Kwan Ahn

The Lyapunov function is upgraded to be t

L2  L21  L20 

   l 2

24



sgn  x21   o2  d

(38)

0

where L20 is a positive constant that could be selected as



L20  2 l24   h

2



2

  21 x3



(39)

The proofs of the positive function (38) and the second statement of Lemma 2 are respectively discussed in similar ways that are presented in previous work [49] and in Lemma 1, in which the following time derivative is resulted in: 2 L2    2 22l21 x21   2 21 22   22  0

(40)

Remark 4: It can be remarked that the design and implementation of the new observer is more complicated than those of the pressure-based observer. However, with the new design, not only the unknown terms ( 2 ) of the force dynamics can be determined, but also the applicability of the estimation results into the control phase is provided.

Closed-Loop Performance In industrial applications, the closed-loop stability is a very important criterion. For this reason, the overall performance of the designed system is investigated again through the following theorem. Theorem 1: Consider a nonlinear bounded system (5) under Assumption 2, if employing the control laws as (9)-(17) incorporated with the estimation models ((19) and (30)) then the following properties hold:

An Advanced Neural-Disturbance-Observer Control Method …

119

a) The closed-loop system is stable for all bounded positive control gains and the bounded disturbances estimated. b) The system is asymptotically stable for a group of the positive control gains in constraints with given positive constants (1 and 2 ) as the following condition:

k2 13  1  k3 g3 2  2 21   2

(41)

Proof: Since the observers are stable, we simply formulate the following Lyapunov function for the closed-loop performance:

L3  0.51s 2  0.5 2 e32

(42)

Substituting (9)-(17) into its time derivative, we have    L3  1s e3  k2 s   2   2 e3   1 s   3  k3 g3e3   2 





 1k2 s 2  1s 2   2 k3 g3e32   2 e3 3  1k2 s  1 s    2 k3 g3 e3   2 e3 

(43)

2

2

2

3

which leads to the proof of the first statement of Theorem 1. As seen in Eq. (37) and discussed in Remark 2, once the asymptotic estimation errors are obtained, the control error ideally converges to zero. Although the desired steady-state control objective can be accomplished, the transient response sometimes is under control as unexpected. As a result, to more intensively verify the close-loop performance, the following comprehensive Lyapunov function is considered: L4  1L1  2 L2  L3

Applying (28), (40), and (45), and the condition (20) yields

(44)

120

Dang Xuan Ba and Kyoung Kwan Ahn



L4   1 l32l30 x3  l32l31 x32   3 32   l33 sgn  x3   o3  t    l30 sgn  x3   l31 x3 





1 T 1 T 1  1 3l32 l34 W3 diag(r3 )W3  3l32l34 W3 diag(r3 ) W3  I n131l34l32 o3  t 

 r W

 o (t ) 

1 1  1 0.25 33l32 l34 r3 W3  I n131l34l32 o3 (t )

2

1  1 0.25 33l32 l34 3

2



3

1  I n131l34l32







3

2  2  2 22 l21 x21   2 21 22   22  1k2 s 2  1s 2   2 k3 g3 e32   2 e3 3

(45) By further noting the constraint (41), we have



L4   1 l32l30 x3  l32l31 x32   3 32   l33 sgn  x3   o3  t    l30 sgn  x3   l31 x3 





1 T 1 T 1  1 3l32 l34 W3 diag(r3 )W3  3l32l34 W3 diag(r3 ) W3  I n131l34l32 o3  t 

 W

 o (t ) 

1 1  1 0.25 33l32 l34 r3 W3  I n131l34l32 o3 (t )

2

1  1 0.25 33l32 l34 r3

2



3



1 I n131l34l32







3

2  2  2 22 l21 x21   2 21 22   22  1k2 s 2  1 s 2   2 k3 g3 e32   2 e3 3





1   1 l32l30 x3  l32l31 x32  0.5 3 32  13l32 l34





 3 1 T W3 diag(r3 )W3 3

2  2  2 22 l21 x21  0.5 2 21 22   22  0.51k2 s 2  0.5 2 k3 g3e32  0

(46) Here, the second statement of Theorem 1 has been proven. Remark 5: Theorem 1 implies that there exist proper control and learning gains ensuring the stability of the closed-loop system. Furthermore, it does not need to exactly derive the internal dynamics ( f3 ,  2 and 3 ) of the system. The designed neural network and disturbance observers can be incorporated to deal with the modeling burden.

Validation Results This section discusses verification results of the developed control algorithm in a comparative manner. The validation system used is presented in Figure 2. To create a complicated external force for the main hydraulic system, another hydraulic system was set up in the opposite side.

An Advanced Neural-Disturbance-Observer Control Method …

121

Figure 2. Structure of the validation system.

A PID controller [47] and an adaptive nonlinear controller (ANC) [23] were applied to this hydraulic system as benchmarking of the designed controller (neural disturbance controller - NDC). The control parameters of the PID controller were selected as follows: K P  6000, K I  20, K D  50 . The learning and control gains of the ANC were inherited chosen as   10, k1  0.5, k2  0.2, k3  0.1, k4  0.1,  6 Qn  diag ([1;1;10;0.01]), Qg  6  10  25 20 20 4 20 20 Q f  diag ([0.1;5 10 ; 2 10 ; 2 10 ;1.4 10 ; 4 10 ;7.6 10 ])  10 9 9 9 Qh  diag ([10 ; 2  10 ;10 ;1.3 10 ]).

The gains of the NDC were manually tuned and obtained as

122

Dang Xuan Ba and Kyoung Kwan Ahn

1  k1  5000, k2  10,   1000, k3  0.5, 2  l  l  l  l  5000, l  5000,   0.2, 22 2  21 22 23 24 l  1, l  2 104 , l  8 104 , l  2 105 , l  105 , 31 32 33 34  30 3  0.1,  3  0.2.

Sinusoidal Test In this verification case, the system output was controlled to track a sinusoidal signal of xd  0.1sin(0.2 t ) , with a low load. Control performances of the validating controllers obtained for this system are presented in Figures 3-7. It can be seen from Figure 3 that the PID control provided an acceptable control accuracy while better ones were resulted in by the ANC and NDC methods. However, employing previous nonlinear control approaches could make the system output violating its physical bounds. As observed in Figure 3, it can confirm that this shortcoming was effectively tackled by the constrained control algorithm designed. To reach to excellent control precision, advanced model-based control approaches, such as ANC, are perfect solutions, but they also require much effort to derive the exact mathematical representation of the system dynamics. To alleviate the analytical work with the same control results, as shown in Figure 3, the intelligent control method developed is a good alternative solution. In such the new controller, as shown in Figures 4 and 5, the nonlinear disturbance observer was successfully adopted to accurately estimate the force disturbance. The observer designed could be upgraded with a neural network to learn both the internal dynamical functions and external disturbances in the pressure dynamics. Figures 6 and 7 reveal that the estimation error was stabilized around zero, and the total dynamics were approximated well. These learn outcomes provided adequate information for outstanding control results illustrated in Figure 3.

An Advanced Neural-Disturbance-Observer Control Method …

123

Figure 3. System responses obtained by the comparative controllers in the first test.

Figure 4. Estimation performance of the nonlinear disturbance observer designed for force dynamics in the first test.

Figure 5. Estimation result of the nonlinear disturbance observer designed for force dynamics in the second test.

124

Dang Xuan Ba and Kyoung Kwan Ahn

Figure 6. Estimation performance of the neural disturbance observer designed for pressure dynamics in the first test.

a) Estimation of the internal dynamics by the neural network system developed.

b) Estimation of the lumped disturbance. Figure 7. Estimation result of the nonlinear disturbance observers designed for force dynamics in the first test.

An Advanced Neural-Disturbance-Observer Control Method …

125

Smooth-Multistep Test In this test, the controllers were challenged with a smooth-multistep signal and a hard loading condition. Various loads could be set by changing cracking pressures of relief valves (r3 and r4 in Figure 2). By using the same control and learning parameters as the previous test, the new control results obtained are depicted in Figures 8 – 12. The control results in Figure 8 show that the PID could achieve high control precision with some setpoints. The ANC always delivered outstanding steady-state control outcomes, but its transient response was inacceptable in cases of large initial control errors. To exhibit good control performances in both transient and steady-state phases, the intelligent controller was employed special design in tracking control and nonlinear neural-disturbance observers. As shown in Figures 9 -12, the observers were caught well the variations of the system dynamics in the new testing conditions. Even though in the transient phase, the learning performances were not high, the constrained control laws developed could attenuate these results to force the system output track the desired profile as well as possible. Once the observers were back to their natures, the excellent control results were accomplished. Hence, the model-free characteristics and robustness of the designed controllers are remarkably confirmed.

Figure 8. System responses obtained by the comparative controllers in the second test.

126

Dang Xuan Ba and Kyoung Kwan Ahn

Figure 9. Estimation performance of the nonlinear disturbance observer designed for force dynamics in the second test.

Figure 10. Estimation result of the nonlinear disturbance observer designed for force dynamics in the second test.

Figure 11. Estimation performance of the neural disturbance observer designed for pressure dynamics in the second test.

An Advanced Neural-Disturbance-Observer Control Method …

127

a) Estimation of the internal dynamics by the neural network system developed.

b) Estimation of the lumped disturbance. Figure 12. Estimation result of the nonlinear disturbance observers designed for force dynamics in the second test.

Conclusion In this chapter, an advanced position-tracking control method is studied for constrained hydraulic systems. The controller is structured from a new constrained control rule and nonlinear intelligent disturbance observers. The control law is designed in a special way to ensure the system output tracking the desired signal as accurately as possible and avoiding violation of the output constraints. To eliminate the internal and external disturbances without using model-based information in the control process, a nonlinear disturbance observer is used for the force dynamics and another nonlinear neuraldisturbance observer version is upgraded for the pressure dynamics. Learning performances of the observers and the stability of the closed-loop system are thoroughly investigated by using Lyapunov methods. Effectiveness of the developed control method was intensively verified by extensive simulation tests.

128

Dang Xuan Ba and Kyoung Kwan Ahn

References [1] [2] [3] [4] [5]

[6] [7]

[8]

[9]

[10]

[11] [12] [13]

[14]

[15]

[16] [17]

Butterfly Valves in municipal water management, VAG-Armaturen (2017). [Online]. Available: http://www.vag-armaturen.com. Volvo EC380E Excavator, Volvo Contruction Equipment, Volvo Group. (2014). [Online]. Available:http://www.volvoce.com. Truck Cranes: QY100 Package, Sany Group Co., Ltd. (2014). [Online]. Available: http://www.sanygroup.com. Moir and A. Seabridge, Aircraft System: Mechanical, electrical, and avionics subsystems integration, Third Edition, England: Wiley, 2008. Alle N., S. S. Hiremath, S. Makaram, K. Subramanian, and A. Talukdar, “Review on electro hydrostatic actuator for flight control,” Int. J. Fluid Power, vol. 17, no. 2, pp. 125-145. 2016. Habibi S. and A. Goldenberg, “Design of a new high-performance electrohydraulic actuator,” IEEE/ASME Trans. Mechatronics, vol. 5, no. 2, pp. 158-164, 2000. Ayman and Aly, “PID parameters optimization using genetic algorithm technique for electrohydraulic servo control system,” Intelligent Control and Automation, vol. 2, pp. 69-76, 2011. Truong D. Q. and K.K. Ahn, “Force control for press machines using an online smart tuning fuzzy PID based on a robust extended Kalman filter,” Expert Systems with Applications, vol. 38, no. 5, pp. 5879-5894, 2011. Jiangbo Z., W. Junzheng and W. Shoukun, “Fractional order control to the electrohydraulic system in insulator fatigue test device,” Mechatronics, vol. 23, no. 7, pp. 828-839, 2013. Tsao T. C. and M. Tomizuka, “Robust adaptive and repetitive digital control and application to hydraulic servo for noncircular machining,” Trans. ASME J. Dyn. Syst., Meas., Control, vol. 116, pp. 24–32, 1994. Plummer R. and N. D. Vaughan, “Robust adaptive control for hydraulic servo systems,” ASME J. Dynam. Syst., Meas., Contr., vol. 118, no. 2, pp. 237–244, 1996. Bobrow J. E. and K. Lum, “Adaptive, high bandwidth control of a hydraulic actuator,” ASME J. Dynam. Syst., Meas., Contr., vol. 118, no. 4, pp. 714–720, 1996. Vossoughi R. and M. Donath, “Dynamic feedback linearization for electrohydraulically actuated control systems,” ASME J. Dynam. Syst., Meas., Contr., vol. 117, no. 4, pp. 468–477, 1995. Re L. D. and A. Isidori, “Performance enhancement of nonlinear drives by feedback linearization of linear-bilinear cascade models,” IEEE Trans. Contr. Syst. Technol., vol. 3, pp. 299–308, 1995. Yu W. S. and T. S. Kuo, “Continuous-time indirect adaptive control of the electrohydraulic servo systems,” IEEE Trans. Control Syst. Technol., vol. 5, no. 2, pp. 163–177, 1997. Merritt H. E., Hydraulic Control Systems, New York, NY, USA: Wiley, 1967. Bech M. M., T. O. Andersen, H. C. Pedersen, and L. Schmidt, “Experimental evaluation of control strategies for hydraulic servo robot,” in Proc. IEEE Int. Conf. Mechatronics Autom., 2013, pp. 342-347.

An Advanced Neural-Disturbance-Observer Control Method … [18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[28]

[29]

[30]

[31]

[32]

129

Koivumaki J. and J. Mattila, “The automation of multi degree of freedom hydraulic crane by using virtual decomposition control,” in Proc. IEEE/ASME Int. Conf. Adv. Intell. Mechatronics., 2013, pp. 912-919. Koivumaki and J. Mattila, “High performance nonlinear motion/force controller design for redundant hydraulic construction crane automation,” Autom. Construction, vol. 51, pp. 59-77, 2015. Lin A. Y., Y. Shi, and R. Burton, “Modeling and robust discrete-time slidingmode control design for a fluid power electrohydraulic actuator (EHA) system,” IEEE/ASME Trans. Mechatronics, vol. 18, no. 1, pp. 1–10, 2013. Zhu W.-H. and J. Piedboeuf, “Adaptive output force tracking control of hydraulic cylinders with applications to robot manipulators,” J. Dyn. Syst. Meas. Control, vol. 127, no. 2, pp. 206-217, 2005. Mohanty and B. Yao,” Intergrated direct/indirect adaptive robust control of hydraulic manipulators with valve deadband, ” IEEE/ASME Trans. Mechatronics, vol. 16, no. 4, pp. 707-715, 2011. Mohanty and B. Yao,”Indirect adaptive robust control of hydraulic manipulators with accurate parameter estimates, ” IEEE Trans. Control Syst. Technol., vol. 19, no. 3, pp. 567-575, 2011. Xu, M. Cheng, H. Yang, and C. Sun, “A hybrid displacement/pressure control scheme for an electrohydraulic flow matching system,” IEEE/ASME Trans. Mechatronics, vol. 20, no. 6, pp. 2771-2782, Dec 2015. Tri N. M., D.N.C. Nam, H.G. Park and K.K. Ahn, “Trajectory control of an electro hydraulic actuator using an iterative backstepping control scheme,” Mechatronics, vol. 29, pp. 96-102, 2015. Ahn K. K., D.N.C. Nam and M. Jin, “Adaptive backstepping control of an electrohydraulic actuator,” IEEE/ASME Trans. Mechatronics, vol. 19, no. 3, pp. 987-995, 2014. Bu F. and B. Yao, “Desired compensation adaptive robust control of single-rod electro-hydraulic actuator,” in Proc. Amer. Control Conf., 2001, vol. 5, pp. 39263931. Ba X., K. K. Ahn, D. Q. Truong, and H. G. Park, “Integrated model-based backstepping control for an electro-hydraulic system,” Int. J. Prec. Eng. Manufacturing, vol. 17, no. 5, pp. 1-13, 2016. Ba X., T. Q. Dinh, and K. K. Ahn, “An integrated intelligent nonlinear control method for a pneumatic artificial muscle,” IEEE/ASME Trans. Mechatronics, vol. 21, no. 4, pp. 1835-1845, 2016. J. Yao, Z. Jiao, and D. Ma, “High-accuracy tracking control of hydraulic rotary actuators with modeling uncertainties,” IEEE/ASME Trans. Mechatronics, vol. 19, no. 2, pp. 633-641, 2014. Yao J., Z. Jiao, and D. Ma, “Extended-state-observer-based output feedback nonlinear robust control of hydraulic systems with backstepping,” IEEE Trans. Ind. Electron., vol. 61, no. 11, pp. 6285-6293, 2014. Guo K., J. Wei, J. Fang, R. Feng, and X. Wang, “Position tracking control of electrohydraulic single-rod actuator based on an extended disturbance observer,” Mechatronics, vol. 27, pp. 47-56, 2015.

130 [33]

[34]

[35]

[36]

[37]

[38]

[39] [40]

[41]

[42] [43] [44] [45] [46] [47]

[48]

[49]

Dang Xuan Ba and Kyoung Kwan Ahn Won, W. Kim, D. Shin, and C. C. Chung, “High-gain disturbance observer-based backstepping control with output tracking error constraint for electro-hydraulic systems,” IEEE Trans. Control Syst. Technol., vol. 32, no. 2, pp. 787-795, 2015. Ali S. A., A. Christen, S. Begg, and N. Langlois, “Continuous-discrete timeobserver design for state and disturbance estimation of electro-hydraulic actuator systems,” IEEE Trans. Ind. Electron., vol. 63, no. 07, pp. 4314-4324, 2016. Zhang J., X. Liu, Y. Xia, Z. Zuo, and Y. Wang, “Disturbance observer-based integral sliding-mode control for systems with mismatched disturbances,” IEEE Trans. Ind. Electron., vol. 63, no. 11, pp. 7040-7048, 2016. Ginoya D., P. D. Shendge, and S. B. Phadke, “Sliding mode control for mismatched uncertain systems using an extended disturbance observer,” IEEE Trans. Ind. Electron., vol. 61, no. 04, pp. 1983-1992, 2014. Lu Y. S., “Sliding-mode disturbance observer with switching-gain adaptation and its application to optical disk drives,” IEEE Trans. Ind. Electron., vol. 56, no. 09, pp. 3743-3750, 2009. Yang A. J., J. Su, S. Li, and X. Yu, “High-order mismatched disturbance compensation for motion control systems via a continuous dynamics sliding-mode approach,” IEEE Trans. Ind. Info., vol. 10, no. 01, pp. 604-614, 2014. Levant, “Higher-order sliding modes, differentiation and output-feedback control,” Int. J. Control, vol. 76, no. 6/10, pp. 924-941, 2003. Grabbel J. and M. Ivantysynova, “An investigation of swash plate control concepts for displacement controlled actuators,” International Journal of Fluid Power, vol. 6, no. 2, pp. 19-36, 2005. Mattila J., J. Koivumaki, D. G. Caldwell, and C. Semini, “A survey on control of hydraulic robotic manipulators with projection to future trends,” IEEE/ASME Trans. Mechatronics, vol. 22, no. 2, pp. 669-680, 2017. Gear Pump, Galtech Hydrocontrol, (2017). [Online]. Available: http:// www.galtech.it. Higen A C Servo Motor, Higen Motor Co., Ltd, (2017). [Online]. Available: http://www.higenmotor.com. Higen A C Servo Drive, Higen Motor Co., Ltd, (2017). [Online]. Available: http://www.higenmotor.com. Grandall D. R., The performance and efficiency of hydraulic pumps and motors, Master Thesis, The University of Minnesota, USA. 2010, Technical Report, Bulk modulus investigation – hydraulic fluid stiffness, The Boeing Company, 1967. Thanh T. D. C, and K. K Ahn, “Nonlinear PID control to improve the control performance of 2 axes pneumatic artificial muscle manipulator using neural network,” Mechatronics, vol. 16, no. 9, pp. 577-587, 2006. Bonsignorio and A. del Pobil, “Toward replicable and measurable robotics research [from the guest editors],” IEEE Robot. Autom. Mag., vol. 22, no. 3, pp. 32–32, Sep. 2015. Ba D. X., T. Q. Dinh, J. B. Bae, and K. K. Ahn, “An effective disturbance observer based nonlinear controller for a pump-controlled hydraulic system,” IEEE/ASME Trans. Mechatronics, vol. 25, no. 1, pp. 32-43, 2020.

Index

A

E

actuator(s), vii, viii, 1, 2, 3, 4, 5, 6, 14, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 35, 41, 42, 43, 44, 45, 47, 48, 49, 50, 58, 59, 60, 61, 62, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 83, 84, 85, 86, 94, 102, 128, 129, 130 affine-in-the-input state-space form, 43, 51

electrohydraulic actuation, vii, 31, 32, 33 electro-hydraulic actuator (EHA) system (EHA system), viii, 76, 83, 84, 85, 86, 87, 89, 94, 97, 98, 129 electrohydraulic actuator(s), viii, 24, 28, 32, 33, 35, 41, 42, 43, 44, 45, 47, 48, 49, 58, 59, 60, 61, 62, 71, 74, 75, 78, 80, 85, 128, 129 electrohydraulic robotic manipulator, 49, 51, 55, 57, 58 electrohydraulically actuated, 32, 34, 35, 48, 58 electropneumatic actuators, 50

B backstepping control, 29, 73, 102, 110, 129, 130

C closed-loop system, vii, 2, 3, 90, 104, 119, 120, 127 constraint control, 102 Coriolis forces vector, 48 Coulomb friction, 43, 107 cylinder ports, 41

D differential flatness properties, 32, 34, 44, 52, 71 differential function of the flat output, 44, 45, 54, 55 disturbance observer, viii, 27, 73, 101, 102, 103, 104, 105, 120, 122, 123, 124, 125, 126, 127, 129, 130 dynamic model of the electrohydraulic robot, 51, 55, 58

F feedback control gains, 40 flat output(s), 33, 34, 36, 37, 44, 45, 54, 55, 71 fuzzy sliding mode controller (FSMC), viii, 83, 84, 85, 86, 89, 91, 93, 94, 95, 96, 97, 98

G generalized predictive control (GPC), 85 global stability, viii, 32, 35 globally asymptotically stable, 47, 57 gravitational forces vector, 49

H Hurwitz invariability results, 3 hydraulic actuator(s), v, vii, viii, 1, 2, 3, 4, 5, 6, 23, 24, 25, 26, 27, 29, 30, 33, 43,

132

Index

44, 47, 48, 73, 75, 77, 80, 83, 84, 86, 98, 99, 102, 128, 129, 130 hydraulic system, iii, v, vii, viii, 2, 5, 25, 26, 27, 28, 29, 31, 73, 75, 84, 98, 100, 101, 102, 103, 104, 106, 107, 108, 109, 120, 121, 127, 128, 129, 130

I inertia matrix of the robot, 48

L linear canonical form, 33 Lyapunov analysis, 32, 47, 57 Lyapunov function, 40, 47, 57, 90, 99, 113, 114, 117, 118, 119 Lyapunov stability analysis, viii, 32 Lyapunov stability theorem, 90

M MATLAB System Identification toolbox, 86, 88 Mean Square Error (MSE), 86, 94, 95 mechatronic systems, v, vii, 31, 32, 33 model reference adaptive control (MRAC), 85, 99, 100 multi-DOF robotic manipulator, viii

N negative output (NO), 25, 26, 76, 79, 92, 98 neural network (NN), viii, 85, 101, 102, 120, 122, 124, 127, 130 neural network control, 85 nonlinear control, 32, 33, 77, 85, 100, 102, 103, 121, 122, 129, 130

O optimization, v, vii, viii, 1, 2, 3, 24, 27, 30, 84, 85, 93, 97, 99, 128

optimized fuzzy sliding mode controller, viii, 83, 84

P parameters variation, 84 particle swarm optimization (PSO), viii, 84, 85, 86, 93, 94, 97, 99 positive output (PO), 92 proportional-integral-derivative (PID), viii, 26, 27, 28, 29, 30, 84, 85, 86, 94, 95, 96, 97, 98, 102, 121, 122, 125, 128, 130

R robotic systems, 24, 32, 33 robust control, v, vii, 1, 2, 3, 23, 25, 27, 28, 77, 84, 98, 103, 129, 135

S self-tuning fuzzy proportional-integralderivative, 85 servovalve, 41, 42, 43 Sine-Cosine swarm optimizing algorithm, 14, 15 sinusoidal test, 122 sliding mode control (SMC), viii, 3, 26, 27, 28, 81, 83, 84, 85, 88, 89, 91, 97, 98, 99, 102 smooth-multistep signal, 125 stabilizing feedback control, 45, 55 state-space model, vii supply pressure of the pump, 42

T triangular form, 35, 71

U uncertain systems, 2, 85, 130

Index

133

V

Z

variable structure control (VSC), 85 virtual decomposition control (VDC), 103, 129

zero output (ZO), 92

About the Editor

Michael G. Skarpetis, PhD Associate Professor Core Department, National and Kapodistrian University of Athens, Athens, Greece Email: [email protected]

Michael G. Skarpetis is an Associate Professor of Automatic Control Systems - Hydraulic and Pneumatic Control Systems at the Core Department of the National and Kapodistrian University of Athens. He serves as Director of the Post Graduate Program “Advanced Control Systems and Robotics”. He has published over 100 scholarly articles in journals, conferences, and edited books. His current interests include robust control, automatic flight control systems, vehicle control systems, hydraulic and pneumatic control systems, and industrial supervisory control algorithms. He has received the 2002/03 IEE Coaless Award. He is included in the Academic and Scientific Excellence Hub of the Ministry of Education, Research and Religious Affairs.