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Advances in Robust Control and Applications
 9789819934621, 9789819934638

Table of contents :
Preface
Contents
An Integrated Exoskeleton-Upper Limb System: Modeling, Control, Stability Study and Robustness Analysis
1 Introduction
2 Modelling of the Integrated Exoskeleton-Upper Limb System
2.1 Human Upper Limb Anatomy
2.2 Modeling of the Integrated Exoskeleton-Upper Limb System
3 Control of the Integrated Exoskeleton-Upper Limb System
3.1 Control Strategy
3.2 Model Free Terminal Sliding Mode with Gravity Compensation (MFTSMGC)
4 Input-to-State Stability of the Integrated Exoskeleton-Upper Limb System
4.1 Preliminary
4.2 ISS Application to the Integrated Exoskeleton-Upper Limb System
5 Robustness Analysis
5.1 Disturbances and Parametric Uncertainties
6 Monte Carlo Analysis
7 Simulation and Results
8 Conclusion
References
Control of Lower Limb Rehabilitation Exoskeleton for Stroke Patients in Contact Environments
1 Introduction
2 Modeling of Lower Limb Rehabilitation Exoskeleton
2.1 Phases of Human Gait Cycle
3 Modeling of Lower Limb Rehabilitation Exoskeleton in Free Motion
3.1 Modeling of Lower Limb Exoskeleton in Constraint Motion
4 Proposed Control Solution
4.1 Classical Computed Torque Controller (CTC)
4.2 Robust Computed Torque Controller (RCTC)
5 Simulation Results
6 Conclusion
References
SOS-Based Robust Control Design Subject to Actuator Saturation for Maximum Power Point Tracking of Photovoltaic System
1 Introduction
2 PV System Modelling
2.1 Mathematical Modeling of PV Panel
2.2 Boost Converter Modeling
2.3 Reference Model
3 T-S Fuzzy Modeling
3.1 Sector Nonlinearity
3.2 TS Fuzzy Model of PV-Boost System
3.3 TS Fuzzy Model of Reference System
4 script upper H Subscript normal infinitymathcalHinfty Tracking Control Design Under Input Constraint
4.1 Actuator Saturation
5 LMI-Based Robust Control Design
5.1 Analysis and Reduction Techniques
5.2 Robust PDC Control Design for PV System
5.3 Robust SOF Control Design for PV System
6 SOS-Based Robust Control Design
6.1 Polynomial Fuzzy Model
6.2 Polynomial Model of the PV System
6.3 Sum of Squares
6.4 Robust PSF Control Design
6.5 Robust PSOF Control Design
7 Simulation
8 Conclusion
References
Nonlinear Optimal Control for Residential Microgrids with Wind Generators, Fuel Cells and PVs
1 Introduction
2 Dynamic Model of the Hybrid Residential Microgrid
2.1 The Components of the Residential Microgrid
2.2 State-Space Description of the Hybrid Residential Microgrid
2.3 A Concise Description of the PEM Fuel Cells' Dynamics
2.4 Dynamic Model of the Hybrid Residential Microgrid in Matrix Form
3 Approximate Linearization of the Dynamics of the Residential Microgrid
3.1 The Linearized State-Space Model of the Microgrid
3.2 Computation of Jacobian Matrices
3.3 Stabilizing Feedback Control
4 Lyapunov Stability Analysis
4.1 Stability Proof
4.2 Robust State Estimation with the Use of the ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (upper H Subscript normal infinity) /StPNE pdfmark [/StBMC pdfmarkto.ps: [/EMC pdfmark [/Artifact /BDC pdfmark Hinftyps: [/EMC pdfmark [/StBMC pdfmark ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark Kalman Filter
5 Differential Flatness Properties of the Hybrid Residential Microgrid
6 Simulation Tests
7 Conclusions
References
Nonlinear Optimal Control for VSI-fed Three-phase Asynchronous Motors
1 Introduction
2 Dynamic Model of the Induction Motor
2.1 Mathematical Model of the Induction Motor
2.2 Field Orientation
3 Dynamic Model of the Three-phase Voltage Source Inverter
4 Dynamic Model of the VSI-fed Induction Motor
4.1 State-space Description of the VSI-fed Induction Motor
4.2 Proof of Differential Flatness Properties of the VSI-fed Induction Motor
5 Linearization and Control of the VSI-fed Induction Motor
5.1 Linearized Model of the Induction Motor
5.2 Stabilizing Feedback Control
6 Lyapunov Stability Analysis
6.1 Global Stability Proof
6.2 Robust State Estimation with Kalman Filtering
7 Simulation Tests
8 Conclusions
References
Photovoltaic Energy Fed DC Motor for Water Pump
1 Introduction
2 Photovoltaic Pumping System
2.1 Design of PV Array and MPPT Approach
2.2 Modeling of Direct Current Motor
2.3 Modelling of Centrifugal Pump
3 Simulation Results and Discussions
4 Conclusion
References
A Robust Linear Feedback Control of PEMFC's Air Feed System
1 Introduction
2 Dynamic Modelling
2.1 Cathode Flow
2.2 Air Compressor
2.3 Supply Manifold
2.4 Full State Space Model
3 Fuel Cell Air Feed System Control
3.1 System Linearization
3.2 Linear Feedback Control
4 Simulation Result
5 Conclusion
References
Robust Saturated Control Based Static Output Feedback for Steering Control of the Autonomous Vehicle via Non Quadratic Lyapunov Function
1 Introduction
2 Notation
3 System Modeling
3.1 Vehicle Movements
3.2 Basic Principle of Autonomous Vehicle Control
3.3 Nonlinear Model of the Lateral Dynamics of a Vehicle
3.4 Bicycle Model: Structure and Hypotheses
3.5 Lane Keeping Dynamics
4 System Control Model
5 Takagi-Sugeno Fuzzy Model of Vehicle Lateral Dynamics
6 T-S Multi-model Validation
7 Reduction in the Number of Local Models
8 Robust upper H Subscript normal infinity Hinfty Static Output Feedback Stabilization
8.1 Simulation Results
9 Robust Saturated Nonlinear Output Feedback Control (SOF)
9.1 Saturated Effect Modeling
9.2 Simulation Results
10 Conclusion
References
Robust Constrained Gain-Scheduled Static Output Feedback Controller for NLPV Descriptor Systems Subject to State and Input Constraints
1 Introduction
2 System Modeling
3 Polytopic LPV System Descriptions
4 LMI Based Gain-Scheduled SOF Controller
5 Illustrative Examples
6 Conclusion
References
A Constrained Optimal Control Strategy for Switched Nonlinear Systems Based on Metaheuristic Algorithms
1 Introduction
2 Switched Nonlinear Systems: Problem Formulation
3 Constrained Optimal Control for Switched Nonlinear Systems
4 Overview: Metaheuristic Algorithms
4.1 Genetic algorithm
4.2 Particle Swarm Optimization
4.3 Cuckoo Search
4.4 Crow Search Algorithm
5 Optimal Control for Switched Nonlinear Systems: Proposed Optimization Algorithms
5.1 Proposed Genetic Algorithm
5.2 Proposed PSO Algorithm
5.3 Proposed Cuckoo Search Algorithm
5.4 Proposed Crow Search Algorithm
6 Illustrative Example
7 Conclusions
References
Robustness Analysis of a Discrete Integral Sliding Mode Controller for DC-DC Buck Converter Using Input-Output Measurement
1 Introduction
2 Mathematical Model of the DC-DC Buck Converter
3 Discrete-Time Integral Sliding Mode Control Using Input-Output Model
4 Simulation Results
5 Conclusion
References
Fault Tolerant Control for Uncertain Neutral Time-Delay System
1 Introduction
2 Preliminaries
3 Uncertain Neutral Time-Delay System with Not Delayed Input
3.1 Fault Detection
3.2 Fault Estimation
4 Uncertain Neutral Time-Delay Systems with Delayed Input
4.1 Robust Stabilization
4.2 Fault Compensation
5 Obtained Results
6 Conclusion
References
Observability Bound of Discrete-Time Linear Multivariable Singularly Perturbed System
1 Introduction
2 System Description
3 Preliminary
4 Stability Bound Analysis
5 Observability Bound Analysis
6 Numerical Example
7 Conclusion
References
Type-1 and Type-2 Fuzzy Techniques: Application to Robotic Systems
1 Introduction
2 Type-1 and Type-2 Fuzzy Logic Systems
2.1 Type-1 and Type-2 Fuzzy Sets
2.2 Type-1 and Type-2 Fuzzy Logic Controllers
3 Model of the Wheeled Mobile Robot
3.1 Coordinate Systems
3.2 Mobile Robot Parameters
4 Proposed Type-1 and Type-2 Robot Behaviours
4.1 Goal Seeking Task (GS)
4.2 Obstacle Avoidance Task (OA)
4.3 Wall-Following Task (WF)
5 Simulation Results
5.1 Results of the Goal Seeking Behavior
5.2 Results of the Obstacle Avoidance Behaviour
5.3 Results of the Wall-Following Behaviour
6 Conclusion
References
Design of Interval Type 2 Fuzzy Adaptive Sliding Mode Control
1 Introduction
2 Problem Formulation
3 Proposed Control Synthesis
4 Simulation Results
5 Conclusion
References
Robust Nonfragile Control Strategies for Bilinear Uncertain Fuzzy Systems Based on Proportional PDC Approach
1 Introduction
2 Problem Formulation and Preliminaries
3 Robust Nonfragile Stabilization Problem
4 Robust Nonfragile Tracking Control Design
5 Illustrative Example
5.1 Mathematical Model
5.2 Stabilization Result
5.3 Tracking Result
6 Conclusion
References

Citation preview

Studies in Systems, Decision and Control 474

Nabil Derbel Ahmed Said Nouri Quanmin Zhu   Editors

Advances in Robust Control and Applications

Studies in Systems, Decision and Control Volume 474

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

Nabil Derbel · Ahmed Said Nouri · Quanmin Zhu Editors

Advances in Robust Control and Applications

Editors Nabil Derbel Department of Electrical Engineering University of Sfax Sfax, Tunisia

Ahmed Said Nouri Department of Electrical Engineering University of Sfax Sfax, Tunisia

Quanmin Zhu School of Engineering University of the West of England Bristol, UK

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-981-99-3462-1 ISBN 978-981-99-3463-8 (eBook) https://doi.org/10.1007/978-981-99-3463-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

The current book “Advances in Robust Control and Applications” exhibits several recent developments and applications in the field of the control of industrial systems covering a wide range of modelling and feedback control using several robust approaches such as fuzzy systems, sliding mode control, H infinity... This book provides an overview of the theory, applications, and perspectives in the field of robotic systems, exoskeletons, power systems, photovoltaic systems, etc. as well as the general methodologies and paradigms around them. Each chapter enriches the understanding of a research topic and provides a balanced treatment of relevant theories, methods, or applications. It reports on the latest advances in the corresponding field. This book is a good reference for graduate students, researchers, educators, engineers and scientists, including 16 chapters structured into five parts as follows. The first part of this book proposes the control of lower and upper limb exoskeletons, and comprises two chapters: • The first chapter entitled “An Integrated Exoskeleton-Upper Limb System: Modeling, Control, Stability Study and Robustness Analysis” is dedicated to the modeling, the control and the robustness analysis of an integrated exoskeletonupper limb system, by developing a new dynamic model of the exoskeleton interacted with the human upper limb then to control the flexion/extension movement of the shoulder and the elbow and to study its robustness in presence of uncertainties and disturbances. • The second chapter entitled “Control of Lower Limb Rehabilitation Exoskeleton for Stroke Patients in Contact Environments” addresses a comparative study established between two computed torque controllers: Classical computed torque controller and Robust computed torque controller for controlling lower limb rehabilitation exoskeleton in case of free and contact with ground environments, using of learning techniques to monitor and detecting faults in biological systems resulting in a high computation cost in case of too large sets of training data.

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The second part is dedicated to the control of power systems, and comprises four chapters: • The third chapter entitled “SOS-Based Robust Control Design Subject to Actuator Saturation for Maximum Power Point Tracking of Photovoltaic System” provides a robust maximum power point tracking (MPPT) control design for a standalone photovoltaic system subject to actuator saturation via polynomial Tackagi-Seguno Fuzzy model based control approach. Then, an output feedback controller is designed for reducing the number of current sensors and required voltage. The stabilization conditions subject to H ∞ performance of the closed-loop system are derived and solved in terms of linear matrix inequalities. • The fourth chapter entitled “Nonlinear Optimal Control for Residential Microgrids with Wind Generators, Fuel Cells and PVs” proposes an implementation of a nonlinear optimal control method for a hybrid residential microgrid which comprises (i) a wind micro-turbine connected to a synchronous reluctance generator and a link to a DC bus through an AC/DC converter, (ii) a PEM fuel cells power unit which is connected to the same DC bus through a DC/DC converter, and (iii) a photovoltaic power unit which is connected to the DC bus again through a DC/ DC converter. A stabilizing H∞ feedback controller is designed for the linearized state-space model of the system. • The fifth chapter entitled “Nonlinear Optimal Control for VSI-fed Three-phase Asynchronous Motors” focuses on the design and implementation of a nonlinear optimal control for voltage source inverter-fed induction motors, using a H ∞ feedback controller for the approximately linearized model of the VSI-fed IM providing a solution for the nonlinear optimal control problem in the case of model uncertainties and external perturbations, and proving the global asymptotic stability properties of the control scheme using Lyapunov theory. • The sixth chapter entitled “Photovoltaic Energy Fed DC Motor for Water Pump” deals with the use of photovoltaic energy for direct current motor to drive water pump, optimizing the solar photovoltaic generated power by a maximum power point tracking method. The centrifugal pump is controlled by a fuzzy supervisor in order to get the optimum flow rate of the pump. The third part concerns several approaches for modeling and control of industrial processes, and comprises five chapters: • The seventh chapter entitled “A Robust Linear Feedback Control of PEMFC’s Air Feed System” suggests the design of linear feedback with integral action control for regulating the system to increase its efficiency. The dynamics of the fuel cell system are first modeled and linearized around its equilibrium point to design the linear feedback controller. • The eighth chapter entitled “Robust Saturated Control Based Static Output Feedback for Steering Control of the Autonomous Vehicle via Non Quadratic Lyapunov Function” presents the design of a robust nonlinear H ∞ Static Output Feedback control for steering control of the autonomous vehicle under actuator saturation, external disturbances, and taking into account the unavailability of the sideslip angle. The Takagi-Sugeno fuzzy systems is used to model both the

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lateral and the lane keeping dynamics of autonomous vehicles. Based on the non-quadratic Lyapunov approach, a static output Feedback controller based on a non-compensation parallel distributed technic the H∞ stabilization conditions are developed. • The ninth chapter entitled “Robust Constrained Gain-Scheduled Static Output Feedback Controller for NLPV Descriptor Systems Subject to State and Input Constraints” considers a new approach to design Gain-Scheduled Static Output Feedback (SOF) controller for constrained NLPV descriptor system, subject to input saturation and exogenous disturbances. Both state and input constraints are explicitly taken into account in the control design using set-invariance arguments. The new class of Parameter Dependent (PD) nonquadratic Lyapunov functions provides an effective solution to estimate the largest closed-loop Domain of Attraction (DoA), which can be nonconvex, formulating and solving an optimization problem under strict PDLMI constraints, using the L 2 gain performance to reduce the effect of external disturbances. • The tenth chapter entitled “A Constrained Optimal Control Strategy for Switched Nonlinear Systems Based on Metaheuristic Algorithms” considers the optimal control problem of switched nonlinear systems under mixed constraints, solved by determining two major factors (i) finding the optimal switched instants, as well as (ii) the optimal control input minimizing a performance criterion. Then, metaheuristic algorithms have been used and compared: Genetic Algorithm (AG), Particle Swarm Optimization (PSO), Cuckoo algorithm (CA) and Crow Search Algorithm (CSA). These approaches have been applied to a physical switched system that models a hydraulic system under constraints as an illustrative industrial process. • The eleventh chapter entitled “Robustness Analysis of a Discrete Integral Sliding Mode Controller for DC-DC Buck Converter Using Input-Output Measurement” develops a discrete-time integral sliding mode controller (DISMC) using only input-output measurement. It is a new method for voltage controller of electronic power systems, such as DC-DC buck converter. The proposed method uses an input-output model of a DC-DC buck converter. The fourth part details observers and fault tolerant control Systems, and comprises two chapters: • The twelth chapter entitled “Fault Tolerant Control for Uncertain Neutral Time-Delay System” presents the fault tolerant control for the uncertain neutral time-delay system with and without delayed input. the Adaptive observer for the fault detection and estimation steps have been considered. To guarantee the stability of the system in closed loop a robust stabilization approach has been developed. Then, the fault tolerant control law is the addition of two terms: (i) the first one is the nominal control, and (ii) the second one is an additive control to stabilize the system. • The thirteenth chapter entitled “Observability Bound of Discrete-Time Linear Multivariable Singularly Perturbed System” proposes the observability bound problem of discrete-time linear multivariable singularly perturbed systems. Using

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appropriate Lyapunov functionals, satisfactory conditions formulated in terms of Linear Matrix Inequalities (LMIs) have been elaborated to analyse the asymptotic stability of the system on one hand and on the other hand to prove the asymptotic stability of the observer system. The fifth and last part of this book focuses on the application of fuzzy control for robotic systems, and comprises three chapters: • The fourteenth chapter entitled “Type-1 and Type-2 Fuzzy Techniques: Application to Robotic Systems” addresses the importance of Type 1 and Type 2 fuzzy systems for the control of wheeled autonomous mobile robot systems. • The fifteenth chapter entitled “Design of Interval Type 2 Fuzzy Adaptive Sliding Mode Control” discusses theoretical and experimental investigations of the influence of Type 2 Fuzzy Adaptive Backstepping Sliding Mode Controller for nonlinear unknown systems. The proposed control ensures the asymptotic stability of closed loop systems. • The sixteenth chapter entitled “Robust Nonfragile Control Strategies for Bilinear Uncertain Fuzzy Systems Based on Proportional PDC Approach” focuses on the design of robust state-feedback controllers such that closed-loop systems are globally asymptotically stable with disturbance attenuation in spite of controllers gains variations using a quadratic Lyapunov function, a proportional parallel distributed compensation approach, H ∞ synthesis criterion, and linear matrix inequalities techniques, providing the design of sufficient stability conditions with decay rate. Sfax, Tunisia Sfax, Tunisia Bristol, UK April 2023

Nabil Derbel Ahmed Said Nouri Quanmin Zhu

Contents

An Integrated Exoskeleton-Upper Limb System: Modeling, Control, Stability Study and Robustness Analysis . . . . . . . . . . . . . . . . . . . . . Sana Bembli, Nahla Khraief Haddad, and Safya Belghith Control of Lower Limb Rehabilitation Exoskeleton for Stroke Patients in Contact Environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mohammad A. Faraj, Boutheina Maalej, Nabil Derbel, and Mohamed Deriche SOS-Based Robust Control Design Subject to Actuator Saturation for Maximum Power Point Tracking of Photovoltaic System . . . . . . . . . . . Noureddine Boubekri, Dounia Saifia, Sofiane Doudou, and Mohammed Chadli Nonlinear Optimal Control for Residential Microgrids with Wind Generators, Fuel Cells and PVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gerasimos Rigatos, Pierluigi Siano, Gennaro Cuccurullo, and Masoud Abbaszadeh

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Nonlinear Optimal Control for VSI-fed Three-phase Asynchronous Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Gerasimos Rigatos, Mohamed Assaad Hamida, Pierluigi Siano, Masoud Abbaszadeh, Godpromesse Kenné, and Patrice Wira Photovoltaic Energy Fed DC Motor for Water Pump . . . . . . . . . . . . . . . . . . 173 Hanen Abbes and Hafedh Abid A Robust Linear Feedback Control of PEMFC’s Air Feed System . . . . . . 191 Asma Rahmani, Mohamed Bougrine, Mohamed Benzoubir, and Atallah Benalia

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Robust Saturated Control Based Static Output Feedback for Steering Control of the Autonomous Vehicle via Non Quadratic Lyapunov Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Amine Kennouche, Dounia Saifia, Mohammed Chadli, and Mohamed Nasri Robust Constrained Gain-Scheduled Static Output Feedback Controller for NLPV Descriptor Systems Subject to State and Input Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Ines Righi, Sabrina Aouaouda, and Khaled Khelil A Constrained Optimal Control Strategy for Switched Nonlinear Systems Based on Metaheuristic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 253 Marwen Kermani and Anis Sakly Robustness Analysis of a Discrete Integral Sliding Mode Controller for DC-DC Buck Converter Using Input-Output Measurement . . . . . . . . 273 Zina Elhajji, Kadija Dehri, Zyad Bouchama, Ahmed Said Nouri, and Najib Essounbouli Fault Tolerant Control for Uncertain Neutral Time-Delay System . . . . . . 285 Rabeb Benjemaa, Aicha Elhsoumi, and Mohamed Naceur Abdelkrim Observability Bound of Discrete-Time Linear Multivariable Singularly Perturbed System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Marwa Ltifi, Nesrine Bahri, and Majda Ltaief Type-1 and Type-2 Fuzzy Techniques: Application to Robotic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Lakhmissi Cherroun, Mohamed Nadour, Abdellah Kouzou, and Mohamed Boumehraz Design of Interval Type 2 Fuzzy Adaptive Sliding Mode Control . . . . . . . 345 Khalissa Behih, Djamila Zehar, Aida Cherif, and Nabil Derbel Robust Nonfragile Control Strategies for Bilinear Uncertain Fuzzy Systems Based on Proportional PDC Approach . . . . . . . . . . . . . . . . . . . . . . . 361 Chekib Ghorbel and Naceur Benhadj Braiek

An Integrated Exoskeleton-Upper Limb System: Modeling, Control, Stability Study and Robustness Analysis Sana Bembli, Nahla Khraief Haddad, and Safya Belghith

Abstract This paper deals with the modeling, the control and the robustness analysis of an integrated exoskeleton-upper limb system. The considered system is a robot with two degrees of freedom controlled by a Model Free Terminal Sliding Mode with Gravity Compensation (MFTSMGC) algorithm. The objective is to develop a new dynamic model of the exoskeleton interacted with the human upper limb then to control the flexion/extension movement of the shoulder and the elbow and to study its robustness in presence of uncertainties and disturbances. Thus, in a first time, the dynamic model of the treated system was presented. In a second time, the development of a new control law was done. Then, an Input-to-State Stability (ISS) study is realized to demonstrate the stability of the considered system. Finally, a robustness analysis using Monte Carlo method is developed. Simulation results are provided to prove the performances and the efficiency of the Model Free Terminal Sliding mode with Gravity Compensation algorithm and the stability of the system in presence of disturbances and uncertainties. Keywords Integrated exoskeleton · Upper limb system · Modeling and control · Robustness analysis · Input-to-state stability (ISS) · Model Free Terminal Sliding mode with Gravity Compensation · Uncertainties and disturbances · Monte Carlo method

S. Bembli (B) · N. K. Haddad · S. Belghith RISC Laboratory, National Engineering School of Tunis, University of Tunis El-Manar, Tunis, Tunisia e-mail: [email protected] S. Belghith e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Derbel et al. (eds.), Advances in Robust Control and Applications, Studies in Systems, Decision and Control 474, https://doi.org/10.1007/978-981-99-3463-8_1

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1 Introduction Robotic systems, in general, suffer from two main components of uncertainties. The first one is that of parameter variations. The second one is coming from the external interaction forces on the suspended body which is generally unknown. So, a robustness test is important in order to identify the operating factors that are not necessarily studied during the method development phase, but which could have an influence on the results, and consequently to anticipate the problems that may occur at the control application. Robustness to parameter variations supple the precision constraints on system characterization, while resistance to external forces determines the dynamic stiffness of the suspension. If the dynamic equations of the studied system are not modeled in the suspension equations, the parameter variations and the external disturbance force terms appear in the same level of differentiation as the input of the system. In the literature, we find the development of different robustness analysis methods. In order to achieve robust stability and performance, design methods based on the structured singular value μ can be used. The structured singular value μ plays an important role in the robust stability and the designs robustness [31]. H∞ techniques can be used to minimize the closed-loop impact of a disturbance: depending on the problem formulation, the impact will either be measured in terms of stabilization or performance [22]. They are easily applicable to problems involving multivariate systems with cross coupling between channels. Different other methods are used, such as the Monte Carlo method [25, 27], which is often used to analyse the parameters effects on stability properties. It is used to solve a deterministic system incorporating uncertain parameters modelled by random variables. Also, we find Markov chains which make it possible to model in an elementary but robust way many random phenomena where a quantity future evolution depends on the past only through its present value. These models allow more complete modelling of the links between the hidden and observed processes. Markov chains [2] allow the estimation of variables of interest in the case of large data quantities and are widely used in the most diverse problems. Finally, the notion of L 2 gain is useful for quantifying the way in which the system rejects uncertainties and external disturbances w is used in [3, 19, 21]. The L 2 gain can be used to guarantee the robustness in stability of the treated system thanks to the small gain theorem. In this context, a stability study and a robustness analysis will be developed in this work when controlling an integrated exoskeleton-upper limb system. Thus, the development of a dynamic model of the exoskeleton interacted with the human upper limb is necessary. The human upper limb is characterized by its mobility and its ability to handle and grasp objects. The human joints movements result on the one hand from the interaction of complex bone surfaces and on the other hand from particular movements of the bones generated by the large number of distributed actuators that are the

An Integrated Exoskeleton-Upper Limb System: Modeling, Control, Stability …

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muscles. It is also very difficult to identify in a non-invasive way the real movements of these bone (and muscle) structures, because the only directly observable movements are those of the skin which are not representative of the structure movements. Therefore, it is currently impossible to find a kinematic model of the human arm that is consensually accepted in the biomedical literature [23]. The loss of the upper limb function due to a stroke presents a major obstacle. Therefore, scientists and researchers are becoming interested in the development of exoskeletons which can help to restore some of the functionality to the user and to give patients some kind of mobility and comfort. Thus, exoskeleton is a mechatronic system in contact with the human’s body [9, 10]. It acts as amplifier that augments, reinforces or restores human performances. The objective of controlling an exoskeleton is to follow the healthy user movements. To achieve this goal, it is necessary to apply an appropriate, performing and robust controller. So, in this paper, a Model Free Terminal Sliding Mode with Gravity Compensation algorithm will be presented. The contribution of this paper is in a first time to develop a new dynamic model of an integrated exoskeleton-upper limb system. In a second time, it consists in the control of the treated system using a new Model Free Terminal Sliding Mode with Gravity Compensation algorithm. The final contribution retains the study of the uncertainties and disturbances influence when controlling the system. So, an Inputto-State Stability study and a robustness analysis using Monte Carlo method were developed. The paper is structured as follows: the human upper limb anatomy as well as the modelling of the exoskeleton-upper limb system is presented in Sect. 2 (part 1 of annex 1). Section 3 deals with the control of the considered system (part 2 of annex 1). The Input-to-Sate stability study is given in Sect. 4. Section 5 presents the robustness analysis of the considered system using Monte Carlo method (part 3 of annex 1). In Sect. 6, simulation, results and discussions are given. Finally, Sect. 7 is kept for the conclusion and future work.

2 Modelling of the Integrated Exoskeleton-Upper Limb System In this section, we aim to develop a dynamic model of the integrated exoskeletonupper limb system ensuring the flexion/extension movement of the shoulder and the elbow joints. Thus, we started with an anatomy study of the human upper limb which will be useful to fully understand its morphology.

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Fig. 1 Modeling of the human upper limb

2.1 Human Upper Limb Anatomy The human upper limb is made up of three main joint areas modeled using seven degrees of freedom (DoF). We find 3 DoF at the level of the shoulder, a single degree at the level of the elbow and 3 DoF at the level of the wrist as shown in Fig. 1. These articulations are detailed as follows: • The shoulder is generally modeled by a three DoF allowing flexion/extension, adduction/abduction and internal/external rotation. From a kinematic point of view, this set of joints can be likened to a ball joint (Fig. 2a). • The elbow has only one degree of freedom ensuring the flexion/extension movement (Fig. 2b). • The wrist has three degrees of freedom (gimbal link) allowing movements of laterality, flexion/extension (or palmar and dorsiflexion) and pronation/supination (what is usually called hand rotation) (Fig. 2c). In this work, we are interested to the flexion/extension movement of the shoulder and the elbow as shown in Fig. 3.

2.2 Modeling of the Integrated Exoskeleton-Upper Limb System Exoskeletons are devices whose kinematics have generally been designed to reproduce that of the human limb to which they are attached. The considered system is an exoskeleton in interaction with a human upper limb controlling the flexion/extension movements of the shoulder and the elbow [5, 6] presented by Fig. 4. We extracted the dynamic equation of our system based on Euler Lagrange equation:

An Integrated Exoskeleton-Upper Limb System: Modeling, Control, Stability …

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Fig. 2 Different movements performed by: a the shoulder, b the elbow and c the wrist

M(q)q¨ + C(q, q) ˙ q˙ + G(q) + f v q˙ + k sign (q) ˙ = τ exo + τ ar m + τ ex

(1)

Letting: F(q, q) ˙ = f v q˙ + k sign (q) ˙ the dynamical model becomes: M(q)q¨ + C(q, q) ˙ q˙ + G(q) + F(q, q) ˙ = τ exo + τ ar m + τ ex

(2)

with: • q ∈ R2 ; q˙ ∈ R2 and q¨ ∈ R2 represent respectively the joint positions vector; the joint velocities vector and the joint accelerations vector; • M(q) ∈ R2×2 ; C(q, q) ˙ ∈ R2×2 are respectively the inertia matrix and the Coriolis matrix; • G(q) ∈ R2 represents the gravitational vector;

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Fig. 3 Kinematic presentation of the human upper limb: shoulder and elbow modeling

• F(q, q) ˙ ∈ R2 is the friction force; exo • τ ∈ R2 , τ ar m ∈ R2 and τ ex ∈ R2 represent respectively the exoskeleton control vector; the human torque and the external torque. • k sign (q) ˙ ∈ R2 corresponds to the resistive torque due to dry friction; • f v q˙ ∈ R2 represents the resistive torque due to the exoskeleton-human arm system viscous friction. [ M(q) =

] [ ] [ ] C11 C12 G1 M11 M12 , C(q, q) ˙ = , G(q) = M21 M22 C21 C22 G2

where: αd = q1 + q2 L 1 = O1 G 1 L 2 = O2 G 2 M11 = I1 + I11 + m 1 L 21 + m 11 L 211 + (m 2 + m 22 )L 22 M12 = M21 = 2L 2 (m 2 L 2 + m 22 L 22 ) cos q2 M22 = I2 + I2 + m 2 L 22 + m 22 L 222 C11 = 0 C12 = −(q˙1 + q˙2 )(m 2 L 2 + m 22 L 22 )L 2 sin q2

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C21 = 0 C22 = −q˙1 (m 2 L 2 + m 22 L 22 )L 2 sin q2 G 1 = [m 1 L 1 + m 11 L 11 + (m 2 + m 22 )L 2 ]g cos q1 G 2 = (m 2 + m 22 )L 2 g cos(q1 + q2 ) with (for i = 1, 2): • • • •

m i is the exoskeleton joint mass; m ii is the arm joint mass; L ii is the arm joint length; Iii is the arm joint inertia.

3 Control of the Integrated Exoskeleton-Upper Limb System In this section, we developed a control law used in order to force the exoskeletonupper limb system to follow the desired trajectories.

3.1 Control Strategy As mentioned in the previous section, the studied system is an integrated exoskeletonupper limb system dealing with the shoulder and the elbow flexion/extension movements. Our study was based on the work presented by the authors in references [20, 26] which treated the modelling and the control of the integrated orthosis-lower limb system. Figure 5 shows the control strategy of the considered system. In this part, we consider that the human arm is immobile (τ ar m = 0) at the simulation part in order to validate the proposed model and the applied control law.

3.2 Model Free Terminal Sliding Mode with Gravity Compensation (MFTSMGC) The proposed control law consists in combining the Model free (MF) with the terminal Sliding Mode (TSM) [11] and the Gravity Compensation (GC).

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Fig. 4 Dynamic model of the integrated exoskeleton-upper limb system

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Fig. 5 Control Strategy of the exoskeleton-upper limb system

The TSM has been developed by adding the non-linear fractional power element to the sliding surface to offer certain superior properties, such as the convergence in finite time of the state variables, faster and better tracking precision [4, 16, 32]. The Model Free control law consists of a PID controller supplemented by compensation conditions provided by the dynamics online estimation of the system [15, 17]. The general controller is also referred to as the i-PID (intelligent PID) controller. The comparison of such a controller with the classic PID controller can be found in [17]. The main advantage of this control strategy is that it does not require prior knowledge of the system dynamics, nor the adjustment of complex parameters. So, it is easy to build a controller for an unknown system. The model free control has been successfully applied to many academic control problems as well as to various industrial cases. Linear and nonlinear systems are studied in [15] as well as the mechanical ball and fire system for the stabilization trajectory and reference trajectories in [15]. The application to switched nonlinear systems, which is a generalization of hybrid systems, is studied in [17]. The gravity compensation applied to robotics could avoid some problems. It acts as a corrector which compensates only for all the forces which create the overshoot and the asymmetric transient behavior of the system. The control with gravity compensation makes it possible to achieve the control objective in global position for robots with n D.o.F. The use of gravity compensation is beneficial for the robotic system which can operate with relatively small actuators generating less torque [1]. The Model Free Terminal Sliding mode with Gravity Compensation (MFTSMGC) is obtained from the combination of the TSMC with the MF and the gravity compensation approach in order to remove the estimation error and to get better and faster desired trajectories tracking [8, 12, 14]. Thus, the MFTSMGC control law is given by:

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U = u M F + u T S M + u GC

(3)

where: • u T S M is the TSM controller; • u M F presents the MF controller; • u GC presents the gravity compensation approach; We start by determining the MF control law which is given by the following expression [17]: (4) q¨ = FM F (t) + α u M F (t) with: • α is a positive input gain; • FM F is an unknown nonlinear term; • u M F is the corresponding input signal. We use an intelligent proportional-derivative controller (iPD) to close the loop. Thus, we obtain: ͡M F − q¨d + u c F uMF = − (5) α with: • u c presents the feedback controller used to track the desired trajectories; • qd is a desired trajectory; • q¨d presents the desired acceleration. u c is chosen as a classic PD controller and defined as: u c = K d e˙ + K p e

(6)

with: • e = q − qd is the trajectory tracking error. • K d and K p are the PD’s gains matrices. • q˙d represents the desired velocity. Therefore, from Eqs. (5) and (6), we obtain: uMF = −

͡M F − q¨d + K d e˙ + K p e F α

(7)

By combining Eqs. (4) and (5), we obtain the following equation: ͡M F e¨ + u c = FM F − F

(8)

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According to the previous equations, we have: e¨ + K d e˙ + K p e = eest

(9)

͡M F . with: eest = FM F − F By determining the MF control law, we get: U =−

͡M F − q¨d + K d e˙ + K p e F + u T S M + u GC α

(10)

Then, we obtain the following equation: e¨ + K d e˙ + K p e = eest + u T S M + u GC

(11)

u T S M GC = u T S M + u GC

(12)

Letting:

Equation (10) will be rewritten in the following form: U =−

͡M F − q¨d + K d e˙ + K p e F + u T S M GC α

(13)

Consider the following state variables: {

x1 = e x2 = e˙

(14)

x˙1 = x2 x˙2 = −K p x1 − K d x2 + eest + αu T S M + αu GC

(15)

State equations of the system become: {

We pass now to the development of the TSM control law. So, to ensure a fast and a good desired trajectory tracking and to avoid the singularity problem, we choose the sliding surface as: 1 r (16) St = x1 + |x2 | s sign x2 β with: • β > 0; • r and s are positive odd integers satisfying the condition 1
0 γ(x) = θx T (A + A T )x − x T g(x)U > 0 Then: V˙ (x) ≤ −α3 (||x||)

(34)

As A = M − 1(q)C(q, q) ˙ is an invertible matrix, we get the following inequality: ||x|| > αx (||u||) =

1 ||A−1 g(x)|| ||u|| θ

(35)

Therefore, V˙ (x) ≤ −α3 (||x||) for all x and u and ||x|| ≥ αx (||u||). Moreover, αi ∈ K∞ for i = 1, 2, 3. Thus, function V is an ISS-Lyapunov function which proves that the system is input-to-state stable.

5 Robustness Analysis Robustness analysis provides an approach to the structuring problematic situations in which uncertainty is high and where decisions can or should be staged sequentially. The specific objective of robustness analysis is how the distinction between decisions and plans can be exploited to maintain flexibility.

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In literature, we find the use of different robustness analysis methods. In our work, we choose the use of Monte Carlo method. In this section, we present first the different uncertainties sources then we are interested in the robustness analysis of the integrated exoskeleton-upper limb system.

5.1 Disturbances and Parametric Uncertainties Disturbances and uncertainties can have several origins. In this part, we present the matched, unmatched disturbances and parametric uncertainties.

5.1.1

Matched Disturbances

The general form of a system affected by matched uncertainties is given by the following expression: q¨ = f (q, q, ˙ t) + g(q)[u(t) + δ1 ]

(36)

where: • • • •

f (q, q, ˙ t) = −M −1 (q)[C(q, q) ˙ q˙ + G(q) + F(q, q)] ˙ g(q) = M −1 (q) u is the control signal, δ1 is the unknown nonlinear disturbance.

Obviously, the uncertain term δ1 enters the state equation exactly at the point where the control variable comes into play. This structural property is said that the uncertain term satisfies the adaptation condition.

5.1.2

Unmatched Disturbances

Affected by unmatched uncertainties, the general form of the treated system is written as follows: q¨ = f (q, q, ˙ t) + g(q)u(t) + δ2 (37) with δ2 is an uncertain function known only to lie within certain limits. That is, ||δ2 || < Δ(q) with Δ(q) being a known non-negative continuous function. It represents the parametric uncertainties or external disturbances effect on the model.

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5.1.3

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Parametric Uncertainties

By applying parametric uncertainties to the considered system, the dynamic equation can be rewritten in the following form: q¨ = [ f (q, q, ˙ t) + Δ f ] + [g(q) + Δg ]u(t)

(38)

with Δ f and Δg present the parametric uncertainties. Parametric uncertainties can have several origins such as: • The identification techniques imprecision. • The exciting trajectory choice which cannot excite all the system parameters. • The difficulty of obtaining a satisfactory estimate of certain parameters experimentally. • The parameters variation from one subject to another.

6 Monte Carlo Analysis The name of these methods, which alludes to games of chance practiced in Monte Carlo, was coined in 1947 by Nicholas Metropolis, and first published in 1949 in an article co-authored with Stanislaw Ulam. This method designates a family of algorithmic methods aiming to calculate an approximate numerical value by using random methods, that is to say probabilistic techniques. Monte Carlo experiments are used to examine finite sample properties of estimators and test statistics. The term “Monte Carlo” is used in many disciplines and refers to procedures where quantities of interest are approximated by generating many random realizations of any stochastic process and calculating any average of their values. Since this is practically impossible to do without a powerful computer, the literature on Monte Carlo methods is quite recent. This method is used to study the performance and the effectiveness of the Model Free Terminal Sliding Mode with Gravity Compensation controller face to uncertainties and disturbances. It is a powerful and very general mathematical method which has earned it a wide range of applications. It is used to study the parameters effects on stability properties. It uses exhaustive and repeated simulations, where a specific value for each independent parameter of a model is drawn randomly from a given range of values, and then the output is computed [18, 28]. It constitutes a powerful and very general mathematical tool which has earned a wide range of applications. The Monte Carlo method is done according to the following steps [7]: • Identifying and characterizing the uncertain parameters in the model. • Sampling and randomly generation tests according to the identified probabilistic laws.

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• The propagation of the uncertainty defined by the dataset resulting from step 2 will be done. • The identification of the output set. • A statistical analysis of the set results corresponding to the data set. • Analyzing the convergence of the distribution of the model output.

7 Simulation and Results To prove the influence of uncertainties and disturbances on the integrated exoskeletonupper limb system, simulation results are provided using the Model Free Terminal Sliding Mode with Gravity Compensation controller. In these tests, we applied sine wave trajectories as reference trajectories. The desired trajectories are chosen as follows: q1d = sin 2πt q2d = sin 2πt The initial conditions are considered as follows: π T ] 4 q(0) ˙ = [0, 0]T q(0) = [0,

In these tests, we applied matched and unmatched disturbances as well as parametric uncertainties to the treated system defined as uniform random distributions. Thus, the dynamic equation of the system will be rewritten as follow: q¨ = [ f (q, q, ˙ t) + Δ f ] + [g(q) + Δg ][u(t) + δ1 ] + δ2

(39)

with: • Δ f and Δg ∈ [0, 0.05]; • δ1 and δ2 ∈ [0, 10]. In order to have a better appreciation of the obtained results during the system control, it is necessary to make a comparison of the control techniques statistical characteristics [13]. So, the Root-Mean-Square (RMS), the mean (Mean) and the standard deviation (Std) were calculated when tracking the desired trajectories. The Std can be expressed by: Σe =



E([(e − E(e)]2 ) =



E(e2 ) − E(e)2

(40)

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The sample mean is defined as: e=

m 1 ∑ ei m i=1

(41)

The RMS is given by the following expression:

eR M S

√ √ m √1 ∑ =√ e2 m i=1 i

(42)

with: • m is a whole positive integer representing the throws number. • i represents a whole positive integer counter that enumerates sum. The desired and measured trajectories as well as the position error tracking are given respectively by Fig. 7 in the nominal case and by Fig. 8 in the presence of uncertainties and disturbances. Using the Model Free Terminal Sliding Mode with Gravity Compensation Controller, we obtained very low values of the desired trajectory tracking errors around 10−4 . From the curves presented by Fig. 7, we notice that the error has a value equal to 4 × 10−3 Radian for the two joints then at t = 3 s, it reaches the value zero. From 2

q2 [rad]

q1 [rad]

2 0 -2 0

2

4

6

8

0 -2 0

10

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10 0 -10 0

2

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2

error q [rad] 1

error q [rad] 4

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10

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8

10

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10 -3

2

8

0 -10 0

10

0 -5 0

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4

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qp 1 [rad/s]

Time [sec]

8

10

5

10 -3

0 -5 0

2

4

6

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Fig. 7 Simulation results using the Model Free Terminal Sliding Mode with Gravity Compensation Controller in the nominal case

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q [rad]

0

0

2

q [rad] 1

2

-2 0

2

4

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8

-1 0

10

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error q [rad]

error q [rad] 4

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8

10

Time[sec]

10 -3

2

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0 -10 0

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4

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qp 1 [rad/s]

Time [sec]

8

10

5

10 -3

0 -5 0

2

4

6

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Fig. 8 Simulation results using the Model Free Terminal Sliding Mode with Gravity Compensation Controller in presence of uncertainties and disturbances Fig. 9 RMS calculation for each joint q1 and q2 using the tested controller in the nominal case and in the presence of disturbances

these curves, we can conclude the performance of the proposed dynamic model and the controller. Although the uncertainties and disturbances influence the system response, we can see from Figs. 8 and 9 the good performance of the controller while tracking the desired trajectories (RMS value around 10−4 ). Table 1 presents the RMS, the error average value and the standard deviation calculation for each joint q1 and q2 using the tested controller when tracking the desired trajectories in positions: simulation in the nominal case and in the presence of uncertainties and disturbances.

An Integrated Exoskeleton-Upper Limb System: Modeling, Control, Stability … Table 1 Model Free Terminal Sliding mode with Gravity Compensation Simulation case Criteria in 10−4 q1 (rad) Nominal case

In presence of uncertainties and disturbances

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q2

RMS Mean Std RMS

3.5 2.87 2.75 7.4

3.63 2.92 3.28 7.8

Mean Std

5.37 6.02

5.73 6.47

8 Conclusion In this work, the modeling, the control, the stability study and the robustness analysis of an integrated 2 D.o.F exoskeleton-upper limb system, used for rehabilitation in presence of uncertainties, were presented. A new dynamic model of the treated system was developed. Then, a Model Free Terminal Sliding mode with Gravity Compensation algorithm is used to control the system. The stability study was proved using the Input-to-State-Stability (ISS) and the robustness analysis was done to analyze the performance of the exoskeleton-upper limb system in presence of disturbances and uncertainties. Referring to the simulation results, an interpretation of the system responses is done in order to prove the performance of the proposed dynamic model and controller when tracking the desired trajectories. As a future work, the objective is to develop an upper limb exoskeleton prototype with two degrees of freedom acting at the shoulder and the elbow joints. Then, it will be considered to apply the work proposed in this paper to this prototype by considering the couple produced by the human upper limb. Finally, the development of other control laws and the use of biomedical signals (EMG, EEG, EOG, etc.) as desired trajectories will be one of the future work objectives.

Appendix See Fig. 10.

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Fig. 10 General diagram of the paper process

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Control of Lower Limb Rehabilitation Exoskeleton for Stroke Patients in Contact Environments Mohammad A. Faraj, Boutheina Maalej, Nabil Derbel, and Mohamed Deriche

Abstract The existence of a Lower limb Rehabilitation exoskeleton as an excellent solution for rehabilitation of lower limbs of stroke patients has been increased in last two decades. One of the fundamental issue in works of rehabilitation robotic devices is the modeling and controlling of lower limb Rehabilitation using suitable and efficient methods. In this chapter, a comparative study has been established between a two computed torque controllers: Classical computed torque controller (CTC) and Robust computed torque controller (RCTC) for controlling lower limb rehabilitation exoskeleton in case of free and contact with ground environments. A Lyapunov theorem has been used for the stability analysis for Robust computed torque controller. Simulation results and values of indices of tracking performances have proved the effectiveness and the robustness of Robust computed torque controller against system uncertainties and external disturbances as compared with classical computed torque controller. Keywords Rehabilitation robot · Lower limb exoskeleton · Robust control · Computed torque controller · Holonomic constraint

M. A. Faraj (B) College of Engineering, University of Anbar, Ramadi, Iraq e-mail: [email protected] B. Maalej · N. Derbel National Engineering School of Sfax (ENIS), Control & Energy Laboratory (CEMLab), University of Sfax, Sfax, Tunisia e-mail: [email protected] M. Deriche Artificial Intelligence Research Centre, AIRC, Ajman University, Ajman, UAE e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Derbel et al. (eds.), Advances in Robust Control and Applications, Studies in Systems, Decision and Control 474, https://doi.org/10.1007/978-981-99-3463-8_2

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1 Introduction The disability of lower and upper limbs of body of patients suffering from stroke illness is the fundamental complex problem for these patients, and has caused a serious challenge for the quality of their life. In the elderly people society, it is found that the stroke is the third dominant cause of death after cancer and heart diseases. Moreover, a variety of different forms of paralysis that result to Permanent disability can be caused by stroke [24]. So, rehabilitating the patients after stroke is still an essential and central issue for their daily life. the number of people that need program of rehabilitation has been increased for different reasons such as stroke, aging, work accidents [5]. These accidents often lead to a permanent disorder in the spinal cord and the central nervous system and may result of losing of upper or lower limb motor functions. To enhance a lower limb motion after stroke, the manual therapy approach is an effective way [14]. The traditional technique of rehabilitation depends on physiotherapists that focus on the rehabilitation of functional tasks and strengthening muscles for stroke patients [9]. The traditional techniques make therapists exhausted while assisting persons with daily rehabilitation training [17, 19]. Nowadays, lower limb rehabilitation exoskeletons (LLRE) are adopted as a suitable method used for augmenting the movement ability for stroke patients [8]. In recent years, various kinds of LLRE devices have been designed; for instance: LOKMAT [4], BLEEX [13] and ALEX [3]. A lot of works have been made for constructing an efficient dynamic model and implementing suitable control methods for tracking the human gait trajectory for LLRE in an accurate and an effective manner. The problem of trajectory tracking control of LLRE has received growing attention, and different control strategies have been proposed for solving this problem such as: adaptive control [1, 6], L 1 adaptive control [16], impedance control [2], fuzzy control [23] and sliding mode control SMC [22]. One of methods that gained a remarkable attention and has been explored by researchers and scholars in the field of modeling and controlling of robotic and lower limb rehabilitation exoskeleton devices is the computed torque control (CTC) strategy [7]. The theory of computed torque control approach (CTC) rely on eliminating the coupling effect in links of robotic devices and converting the nonlinear system into a linear one [25]. Owing to the presence of the external disturbances and the parameter variations in real time movements of lower limb rehabilitation exoskeleton, the formulating of the mathematical dynamic model of the LLRE system accurately is a complex issue. However, an effective configuration and an excellent improvement for classical computed torque controller (CTC) is an essential step for controlling LLRE in case of existing the outside perturbations and parametric uncertainties [15]. In the literature, most of the works has modeled LLRE in free motion that is when it doesn’t contact with the ground (swing phase). However, in stance walking phase, LLRE will be in contact with the ground (stance phase). In this case, the LLRE will be in constrained motion [20]. The main objective of this chapter is to model the LLRE in free and constraint motions for both phases of human gait walking cycle. A nominal model of LLRE

Control of Lower Limb Rehabilitation Exoskeleton for Stroke Patients …

27

will be controlled by the classical computed torque method (CTC). An additional robustifying term will be added to CTC for constructing a robust computed torque controller RCTC for overcoming the deviation driven by the uncertain terms such as: unmolded dynamics, parametric uncertainties, external disturbances. Depending on the Lyapunov theory, the stability analysis for RCTC will be performed. The chapter is organized as follows. In the second section, the dynamic model of the LLRE in swing and stance phases will be derived. In the third section, the derivation of control laws will be accomplished. The fourth section is dedicated to the simulation results. The conclusion of this study will be drawn in the fifth section.

2 Modeling of Lower Limb Rehabilitation Exoskeleton 2.1 Phases of Human Gait Cycle The basic human walking cycle includes two alternated Phases, a swing Phase (when no foot is grounded) which stands for 40% of the step duration, and the stance phase (when one leg is contact with grounded) which constitutes about 60% of the step time [10]. Figure 1 indicates the concept of a swing and stance phases of human gait cycle. LLRE can be considered as open chain manipulator in swing phase, while in stance phase LLRE will be in contacting with ground, that is LLRE will be considered as closed chain and will be in constrained motion.

Swing phase 40 %

+

Humain Gait Cycle

Fig. 1 Concept of swing and stance phases of human gait cycle

Stance phase 60 %

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M. A. Faraj et al.

3 Modeling of Lower Limb Rehabilitation Exoskeleton in Free Motion The dynamic model of the LLRE with three degree of freedom at the hip, knee and the ankle joints will be undertaken in this chapter. In swing phase, Motion of LLRE can be viewed as an open chain loop and hence Eq. (1) can be used to model the dynamic equation of LLRE. A lower limb exoskeleton with three link structure in case of free motion (swing phase) is shown in Fig. 2 where O is the coordinate origin and h represents a distance from the coordinate origin O at hip joint to the contacting point with the ground and φ(q) is an algebraic constraint equation in joint space [21] which don’t appear in swing phase because of contactless of LLRE with ground. The dynamic model of LLRE considering both the lower limb of stroke patient person and the exoskeleton can be performed using Euler-Lagrange formula [27] as follows: M(q) q¨ + C(q, q) ˙ q˙ + G(q) = τ (1) where: • q, q˙ and q¨ ∈ R3 stand for angular joint position, velocity and acceleration vectors, respectively. • M(q) ∈ R3×3 refers to uniformly bounded, symmetric, and positive definite inertia matrix. • C(q, q) ˙ q˙ ∈ R3 stands for Coriolis and centrifugal forces. • G(q) ∈ R3 denotes to the gravity torques. • τ ∈ R3 is the vector of applied torques at joints. The contents of M(q), C(q, q) ˙ and G(q) of three joints LLRE are expressed as follows:

Fig. 2 Motion of lower limb of human and exoskeleton in swing phase

x

O φ(q1 , q2 , q3 ) = 0

1

m1

h m2

y Hip Joint

2

m3

3

Ground / Floor Knee Joint

Ankle Joint

Control of Lower Limb Rehabilitation Exoskeleton for Stroke Patients …

29

M11 = a1 + a2 + a4 + 2a3 cos q2 + 2a5 cos(q2 + q3 ) + 2a6 cos q3 M12 = M21 = a2 + a4 + a3 cos q2 + a5 cos(q2 + q3 ) + 2a6 cos q3 M13 = M31 = a4 + a5 cos(q2 + q3 ) + 2a6 cos q3 M22 = a2 + a4 + 2a6 cos q3 M23 = M32 = a4 + 2a6 cos q3 M33 = a4

C11 = −a5 (q˙2 + q˙3 ) sin(q2 + q3 ) − a3 q˙2 sin q2 − a6 q˙3 sin q3 C12 = −a5 (q˙1 + q˙2 + q˙3 ) sin(q2 + q3 ) − a3 (q˙1 + q˙2 ) sin q2 − a6 q˙3 sin q3 C13 = −(q˙1 + q˙2 + q˙3 )[a5 sin(q2 + q3 ) + a6 sin q3 ] C21 = q˙1 [a5 sin(q2 + q3 ) + a3 sin q2 ] − a6 q˙3 sin q3 C22 = −a6 q˙3 sin q3 C23 = −a6 (q˙1 + q˙2 + q˙3 ) sin q3 C31 = a5 q˙1 sin(q2 + q3 ) + a6 (q˙1 + q˙2 ) sin q3 C32 = a6 (q˙1 + q˙2 ) sin q3 C33 = 0

G 1 = −b1 sin q1 − b2 sin(q1 + q2 ) − b3 sin(q1 + q2 + q3 ) G 2 = −b2 sin(q1 + q2 ) − b3 sin(q1 + q2 + q3 ) G 1 = −b3 sin(q1 + q2 + q3 )

with: b1 = [m 1 d1 + (m 2 + m 3 )l1 ]g, b2 = [m 2 d2 + m 3 l2 ]g, b3 = m 3 d3 g a1 = I1 + m 1 d12 + (m 2 + m 3 )l21 , a2 = I2 + m 2 d22 + m 3 l22 a3 = (m 2 d2 + m 3 l2 )l1 , a4 = I3 + m 3 d32 , a5 = m 3 d3 l1 , a6 = m 3 d3 l2 where: • m 1 , m 2 , m 3 , l1 , l2 and l3 , stands for masses and lengths of thigh, shank and foot links of the human and lower limb exoskeleton. • d1 , d2 , d3 represent a position of the center of mass of thigh, shank and foot of LLRE and human lower limb respectively. • I1 , I2 , I3 are the moments of inertia of thigh, shank and foot of the exoskeleton and human lower limb, respectively. • g is the gravity acceleration

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The dynamic equation (1) is useful under assumption that the mathematical model can be derived precisely. However, the error in system modeling will arise because of the presence the uncertainties and outside perturbations. The presence of external disturbances and uncertainties will result in system disability to perform the tracking control effectively and can’t ensure the stability of the closed-loop system. By undertaking the existence of error in lower limb exoskeleton parameters, friction effect and the external disturbance, the dynamic equation (1) can be re-written as: M(q) q¨ + C(q, q) ˙ q˙ + F(q) ˙ + G(q) + τd = τ

(2)

where: F(q) ˙ refers to forces of frictions and τd represents external disturbances at LLRE joints. A more compact form of (2) can be used for control purposes that is: M(q) q¨ + N (q, q) ˙ = τ

(3)

N (q, q) ˙ denotes to all nonlinear terms except the inertia matrix. Hence, the model uncertainties can be represented in the dynamic model of LLE as follows: M(q) = M0 (q) + ΔM(q) N (q, q) ˙ = N0 (q, q) ˙ + ΔN (q, q) ˙ where: M0 (q) and N0 (q, q) ˙ stands for the nominal model of the lower limb exoskele˙ denote to the ton system that can be determined easily and ΔM(q) and ΔN (q, q) uncertainty terms of the lower limb exoskeleton that cannot be found accurately.

3.1 Modeling of Lower Limb Exoskeleton in Constraint Motion During stance phase, the LLRE will be in contact with ground as shown in Fig. 3 and can be considered as a closed chain loop . Consequently, Eq. (1) of free motion cannot be used to model the dynamic equation of the LLRE when contacting with the ground. In fact, LLRE contacting with ground will constitute a holonomic constraint that can be expressed by an algebraic equation in joint space as [21]: φ(q) = l1 cos q1 + l2 cos(q1 + q2 ) + l3 cos(q1 + q2 + q3 ) − h

(4)

Hence, the force term τ = J T λ in the contact points with the ground will be added to dynamic equation (1) where the Lagrange multipliers λ regarded as the generalized forces that may be noticed at the contact points with the ground. By using the Euler-Lagrange equations including the generalized coordinates q and Lagrange multipliers λ, the constrained dynamic equation of LLRE can be obtained as [21]:

Control of Lower Limb Rehabilitation Exoskeleton for Stroke Patients … Fig. 3 Motion of lower limb of human and exoskeleton in stance phase

x

O 1

φ(q1 , q2 , q3 ) = 0

m1

h m2

y Hip Joint

2

m3

3

Ground / Floor Knee Joint

M(q) q¨ + C(q, q) ˙ q˙ + G(q) = τ + J (q)T λ where:

31

Ankle Joint

(5)

[ ] dφ ∂φ ∂φ ∂φ J (q) = = , , ∈ R1×3 dq ∂q1 ∂q2 ∂q3

J (q) is the Jacobian vector of function φ(q). The appearance of the Lagrange multipliers λ ∈ R and Jacobian vectors J (q) in Eq. (5) makes it complicated especially for control purpose. Hence, the Lagrange multipliers λ can be eliminated by differentiating the algebraic constraint Eq. (4) twice with respect to time as the following [21]: [ ] dφ ˙ = q˙ = J (q)q˙ = 0 φ(q) = 0 =⇒ φ(q) dq d J (q) ¨ q˙ = 0 φ(q) = J (q)q¨ + J˙(q)q˙ = J (q)q¨ + q˙ T dq

(6)

By substituting the joint accelerations from the dynamic model (5) into (6) and dropping dependencies, we get: J M −1 [J T λ + τ − C q˙ − G] + J˙q˙ = 0

(7)

Solving for multiplier vector λ, one can easily write: [ ( ) ] λ = (J M −1 J T )−1 J M −1 C q˙ + G − τ − J˙q˙

(8)

It can be noted that the Lagrange multiplier λ is uniquely determined by robot states and the input torques. By defining: )−1 ( Z = J M −1 , W = J M −1 J T

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M. A. Faraj et al.

and substituting these expressions in (8), we can obtain: [ ( ) ] λ = W Z C q˙ + G − τ − J˙q˙

(9)

Substituting (9) in (5), we can get: [ ( ) ] M q¨ + C q˙ + G = τ + J T W Z C q˙ + G − τ − J˙q˙

(10)

For control purpose, the input torque on the right hand side of this dynamic equation (10) must be separated, thus, the dynamic equation (10) can be written as: M q¨ + DC q˙ + DG = Dτ − J T W J˙q˙ where:

(11)

( )−1 J M −1 D = I − J T W Z = I − J T J M −1 J T

Pre-multiplying (11) by D −1 gives: D −1 M q¨ + C q˙ + G + D −1 J T W J˙q˙ = τ

(12)

Defining F = D −1 J T W J˙q, ˙ M * = D −1 M Equation (12) can be rewritten as: M * q¨ + C q˙ + G + F = τ

(13)

The vectors J and J˙ used in calculating the Lagrange multiplier λ and in calculating the dynamic equation (13) has been derived as follows: ] [ J (q) = J1 (q) J2 (q) J3 (q) with: J1 (q) = −l1 sin q1 − l2 sin(q1 + q2 ) − l3 sin(q1 + q2 + q3 ) J2 (q) = −l2 sin(q1 + q2 ) − l3 sin(q1 + q2 + q3 ) J3 (q) = −l3 sin(q1 + q2 + q3 ) and:

with:

dJ J˙ = q˙ T dq

(14)

Control of Lower Limb Rehabilitation Exoskeleton for Stroke Patients …

33

⎡ ⎤ H1 H2 H3 dJ H= = ⎣ H2 H2 H3 ⎦ dq H3 H3 H3 with: [ ] H1 = − l1 cos q1 + l2 cos(q1 + q2 ) + l3 cos(q1 + q2 + q3 ) [ ] H2 = − l2 cos(q1 + q2 ) + l3 cos(q1 + q2 + q3 ) H3 = −l3 cos(q1 + q2 + q3 ) Equation (13) represents the dynamic model of LLRE in case of contacting with ground with exact dynamic model. However, in real time system applications, acquiring the exact dynamic of LLRE is a complicated step because of the presence of model uncertainties and external disturbances. So, it is convenient to write Eq. (13) as follows: ˙ =τ (15) M * (q)q¨ + N (q, q) and the uncertainty terms can be expressed as: M * (q) = M0* (q) + ΔM * (q) ˙ + ΔN (q, q) ˙ N (q, q) ˙ = N0 (q, q)

(16) (17)

4 Proposed Control Solution 4.1 Classical Computed Torque Controller (CTC) Computed torque control is an effective and important motion control method owing to its global asymptotic stability. The control objective of classical computed torque controller CTC intends to design the control input to drive the joint trajectories of LLRE to the desired ones (Fig. 4). When there is no model uncertainty and external disturbances in LLRE, the dynamic model becomes: ˙ =τ M0 (q)q¨ + N0 (q, q)

(18)

during the swing phase. It becomes for the stance phase as: M0* (q)q¨ + N0 (q, q) ˙ =τ

(19)

Thus, for swing and stance phases, the classical computed torque controller (CTC) is expressed as, respectively:

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M. A. Faraj et al.

Fig. 4 Block diagram of classical computed torque controller

Fig. 5 Block diagram of Robust computed torque controller

τ = M0 [q¨d + K d e˙ + K p e] + N0 τ = M0* [q¨d + K d e˙ + K p e] + N0

(20) (21)

where qd , q˙d and q¨d : represent the desired angular joint position, velocity and acceleration vectors, respectively, e = qd − q and e˙ = q˙d − q˙ are the position error and the speed error, and K p and K d are diagonal positive definite gain matrices. The substitution of Eq. (20) into Eq. (18) in swing phase and Eq. (21) in Eq. (19) for stance phase will Lead to a closed-loop system equation as: e¨ + K d e˙ + K p e = 0

(22)

If K d > 0 and K p > 0 are definite positive matrices, then the trajectory tracking errors will go to zero asymptotically.

Control of Lower Limb Rehabilitation Exoskeleton for Stroke Patients …

35

4.2 Robust Computed Torque Controller (RCTC) The control law of Robust Computed torque controller (RCTC) for lower limb exoskeleton in both swing and stance phases will be formulated in this section (Fig. 5). The developing of the robust Computed torque controller inspired from [11, 12] in robotics manipulators filed. The control law of Robust Computed torque controller (RCTC) can be written as: τ = τ0 + τ R

(23)

where: τ0 is the torque provided by the classical CTC in Eqs. (20) and (21) and τ R is an additional robust control term that can be used to attenuate the system uncertainties and external disturbances and its structure can be expressed as: ˙ τ R = K R sign (σe + e)

(24)

where: K R denotes to the robust term gain, “sign” stand for the signum function and σ is a positive constant (it can be considered as a definite positive diagonal matrix). For the stability analysis, let’s define the following reference trajectories: q˙r = q˙d + σ(qd − q) ˙ q¨r = q¨d + σ(q˙d − q)

(25) (26)

Then, let’s define the following function: S = q˙r − q˙ = e˙ + σe

(27)

The time derivative of S can be written as: S˙ = q¨r − q¨ = σ e˙ + q¨d − q¨

(28)

For conducting the stability analysis, Lyapunov function V can be chosen as: V =

1 T S MS 2

(29)

The time derivative of V is: 1 V˙ = S T M S˙ + S T M˙ S = S T M S˙ + S T C S 2 ¨ + C(q˙r − q)] ˙ = S T [M q¨r + C q˙r − M q¨ − C q] ˙ = S T [M(q¨r − q) = S T [M q¨r + C q˙r + F + G − τ ] Substituting (23) into (30) yields:

(30)

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M. A. Faraj et al.

V˙ = S T [M q¨r + C q˙r + F + G − τ0 − τr ]

(31)

Substituting (20) and (24) into (31): V˙ = S T [M q¨r + C q˙r + F + G − M0 (q¨d + K d e˙ + K p e) − C0 − F0 − G 0 − K r sign S]

(32) Let: Ψ = M q¨r + C q˙r + F + G − M0 q¨d − C0 q˙ − F0 − G 0

(33)

V˙ = S T [ψ − M0 (K d e˙ + K p e) − K r sign S] = S T [ψ − M0 K d (e˙ + K d−1 K p e) − K r sign S]

(34)

Then:

If K d and K p are selected to be: K d−1 K p = σ

(35)

and substituting (35) into (34), we get : V˙ = S T [ψ − M0 K d S − K r sign S] = −S T M0 K d S − S T K r sign S + S T ψ ≤ −S T M0 K d S − ||K r S||1 + ||ψ||1 × ||S||1

(36)

Or: ||K r S||1 ≥ min K r ||S||1

(37)

− ||K r S||1 + ||ψ||1 × ||S||1 ≤ (||ψ||1 − min K r )||S||1 ≤ 0

(38)

Then:

If min K r is chosen too large in such a way: min K r > ||ψ||1

(39)

then the derivative of the Lyapunov function is: V˙ ≤ −S T M0 K d S ≤ 0

(40)

Consequently, it is guaranteed that the controlled system in (3) will be stable under using the control law in (23). It is essential to note that the stability proof in Eqs. (29–40) has been deriving for swing phase. In fact, the steps of guaranteeing the stability for LLRE in stance phase is the same as for swing phase with taking in consideration Eq. (15) and (21) instead of Eqs. (3) and (20) respectively.

Control of Lower Limb Rehabilitation Exoskeleton for Stroke Patients …

37

5 Simulation Results In this section, the effectiveness and robustness of RCTC controller is tested for nominal, uncertainties and perturbation rejection cases. Moreover, integral absolute error (I AE) and integral squared error (I S E) performance indices are used to examine the tracking error performances as follows: ∫ I AE = I SE =



||e||1 dt

(41)

e T e dt

(42)

The parameters of human and exoskeleton used in simulations are adopted from [26]. The desired trajectories of walking cycle have been developed using the method presented in [18]. Figures 6, 7 and 8 show the responses of position, position error, speed and torque control for a period of three gait cycles in nominal case for CTC. The desired trajectory of a healthy person is presented with a discontinuous line with blue color. The evolution of position, position error, speed and torque control for RCTC under nominal case are indicated in Figs. 9, 10 and 11. Table 1 and Figs. 12 and 13 exhibit the values of I AE and I S E for nominal cases for CTC and RCTC controllers. The tracking performance indices of three joints in the nominal case show the difference in tacking performance of the two controllers. It’s easily to notice that the performance Respnse of Position of hip Joint (rad) 1 0 −1

0

0.5

1

1.5

2

2.5

3

2.5

3

2.5

3

2.5

3

Position Tracking error of hip Joint (rad) 1 0 −1

0

0.5

1

1.5

2

Response of speed of hip joint (rad/sec) 10 0 −10

0

0.5

1

1.5

2

Torque control of hip Joint (Nm) 200 0 −200

0

0.5

1

1.5 Time (s)

2

Fig. 6 Hip joint with classical computed torque controller in nominal case

38

M. A. Faraj et al. Respnse of Position of Knee Joint (rad) 2 1 0

0

0.5

1

1.5

2

2.5

3

2.5

3

2.5

3

2.5

3

Position Tracking error of Knee Joint (rad) 0.2 0 −0.2

0

0.5

1

1.5

2

Response of speed of Knee joint (rad/sec) 10 0 −10

0

0.5

1

1.5

2

Torque control of Knee Joint (Nm) 50 0 −50

0

0.5

1

1.5 Time (s)

2

Fig. 7 Knee joint with classical computed torque controller in nominal case Respnse of Position of ankle Joint (rad) 0.5 0 −0.5

0

0.5

1

1.5

2

2.5

3

2.5

3

2.5

3

2.5

3

Position Tracking error of ankle Joint (rad) 0.5 0 −0.5

0

0.5

1

1.5

2

Response of speed of ankle joint (rad/sec) 10 0 −10

0

0.5

1

1.5

2

Torque control of ankle Joint (Nm) 5 0 −5

0

0.5

1

1.5 Time (s)

2

Fig. 8 Ankle joint with classical computed torque controller in nominal case

Control of Lower Limb Rehabilitation Exoskeleton for Stroke Patients …

39

Respnse of Position of hip Joint (rad) 1 0 −1

0

0.5

1

1.5

2

2.5

3

2.5

3

2.5

3

2.5

3

2.5

3

2.5

3

2.5

3

2.5

3

Position Tracking error of hip Joint (rad) 1 0 −1

0

0.5

1

1.5

2

Response of speed of hip joint (rad/sec) 10 0 −10

0

0.5

1

1.5

2

Torque control of hip Joint (Nm) 200 0 −200

0

0.5

1

1.5 Time (s)

2

Fig. 9 Hip joint with Robust computed torque controller in nominal case Respnse of Position of Knee Joint (rad) 2 1 0

0

0.5

1

1.5

2

Position Tracking error of Knee Joint (rad) 0.2 0 −0.2

0

0.5

1

1.5

2

Response of speed of Knee joint (rad/sec) 10 0 −10

0

0.5

1

1.5

2

Torque control of Knee Joint (Nm) 100 0 −100

0

0.5

1

1.5 Time (s)

2

Fig. 10 Knee joint with Robust computed torque controller in nominal case

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M. A. Faraj et al.

Table 1 Comparison of tracking performance indices for nominal case RCTC Performance index CTC Hip joint Knee joint Ankle joint Hip joint Knee joint Ankle joint I AE I SE

0.0357 0.0056

0.0713 0.0068

0.0271 0.0021

0.0215 0.0038

0.0357 0.0019

0.0153 0.0006

Respnse of Position of ankle Joint (rad) 0.5 0 −0.5

0

0.5

1

1.5

2

2.5

3

2.5

3

2.5

3

2.5

3

Position Tracking error of ankle Joint (rad) 0.2 0 −0.2

0

0.5

1

1.5

2

Response of speed of ankle joint (rad/sec) 10 0 −10

0

0.5

1

1.5

2

Torque control of ankle Joint (Nm) 5 0 −5

0

0.5

1

1.5 Time (s)

2

Fig. 11 Ankle joint with Robust computed torque controller in nominal case

of CTC is significantly changed (Figs. 14, 15 and 16) in case of varying the parameters of the LLRE by 20% of their original values. The RCTC controller has an excellent robustness against system uncertainty as compared with CTC as explained Figs. 17, 18 and 19.

Control of Lower Limb Rehabilitation Exoskeleton for Stroke Patients …

41

0.08 CTC RCTC 0.07

0.06

IAE

0.05

0.04

0.03

0.02

0.01

0

Hip joint

Knee joint

ankle joint

Fig. 12 Tracking performance (I AE) Index for two controllers in nominal case −3

7

x 10

CTC RCTC 6

5

ISE

4

3

2

1

0

Hip joint

Knee joint

ankle joint

Fig. 13 Tracking performance (I S E) Index for two controllers in nominal case

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M. A. Faraj et al. Respnse of Position of hip Joint (rad) 1 0 −1

0

0.5

1

1.5

2

2.5

3

2.5

3

2.5

3

2.5

3

Position Tracking error of hip Joint (rad) 1 0 −1

0

0.5

1

1.5

2

Response of speed of hip joint (rad/sec) 10 0 −10

0

0.5

1

1.5

2

Torque control of hip Joint (Nm) 400 200 0 −200 −400

0

0.5

1

1.5 Time (s)

2

Fig. 14 Hip joint with classical computed torque controller in uncertainty case Respnse of Position of Knee Joint (rad) 2 0 −2

0

0.5

1

1.5

2

2.5

3

2.5

3

2.5

3

2.5

3

Position Tracking error of Knee Joint (rad) 0.5 0 −0.5

0

0.5

1

1.5

2

Response of speed of Knee joint (rad/sec) 10 0 −10

0

0.5

1

1.5

2

Torque control of Knee Joint (Nm) 200 0 −200 0

0.5

1

1.5 Time (s)

2

Fig. 15 Knee joint with classical computed torque controller in uncertainty case

Control of Lower Limb Rehabilitation Exoskeleton for Stroke Patients …

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Respnse of Position of ankle Joint (rad) 0.5 0 −0.5

0

0.5

1

1.5

2

2.5

3

2.5

3

2.5

3

2.5

3

Position Tracking error of ankle Joint (rad) 0.5 0 −0.5

0

0.5

1

1.5

2

Response of speed of ankle joint (rad/sec) 20 0 −20

0

0.5

1

1.5

2

Torque control of ankle Joint (Nm) 40 20 0 −20 −40

0

0.5

1

1.5 Time (s)

2

Fig. 16 Ankle joint with classical computed torque controller in uncertainty case Respnse of Position of hip Joint (rad) 1 0 −1

0

0.5

1

1.5

2

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Fig. 17 Hip joint with Robust computed torque controller in uncertainty case

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Fig. 19 Ankle joint with Robust computed torque controller in uncertainty case

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Table 2 Comparison of tracking performance indices for uncertainty case RCTC Performance index CTC Hip joint Knee joint Ankle joint Hip joint Knee joint Ankle joint I AE I SE

0.0389 0.0062

0.0920 0.0113

0.0740 0.0110

0.0169 0.0036

0.0340 0.0018

0.0273 0.0017

0.1 CTC RCTC

0.09 0.08 0.07

IAE

0.06 0.05 0.04 0.03 0.02 0.01 0

Hip joint

Knee joint

ankle joint

Fig. 20 Tracking performance (I AE) Index for two controllers in uncertainty case

The superiority of RCTC over CTC in handling the parametric uncertainties in terms of tracking performance indices is indicated in Table 2 and Figs. 20 and 21. The robustness of RCTC and CTC controller against external disturbances are examined by applying a sinusoidal signal as an external perturbation for each joints. Figures 22, 23 and 24 show the tracking performance of three joints for CTC controller under external disturbances. In fact, the performance of the LLRE system towards external disturbance has witnessed a significant improvement by using RCTC controller as indicated by Table 3 and Figs. 25, 26, 27, 28 and 29, respectively.

6 Conclusion In this chapter, a mathematical model and an effective control law for a lower limb exoskeleton with three degrees of freedom used for rehabilitation of stroke patients has been established. Firstly, by using Lagrange equations, the dynamic equations for lower limb exoskeleton in free motion (swing phase) and in case of contacting with

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ISE

0.008

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Fig. 21 Tracking performance (I S E) Index for two controllers in uncertainty case

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Fig. 22 Hip joint with classical computed torque controller for disturbance case

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Fig. 24 Anklejoint with classical computed torque controller for disturbance case Table 3 Comparison of tracking performance indices for disturbance case Performance index

CTC

RCTC

Hip joint

Knee joint

Ankle joint

Hip joint

Knee joint

Ankle joint

I AE

0.0376

0.0701

0.0515

0.0170

0.0341

0.0271

I SE

0.0063

0.0074

0.0048

0.0036

0.0018

0.0016

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Fig. 25 Hip joint with Robust computed torque controller for disturbance case Respnse of Position of Knee Joint (rad) 2 1 0

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Fig. 26 Knee joint with Robust computed torque controller for disturbance case

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Fig. 28 Tracking performance (I AE) Index for two controllers for disturbance case

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Fig. 29 Tracking performance (I S E) Index for two controllers for disturbance case

ground (stance phase) have been derived. Then, a Classical computed torque controller CTC and Robust computed torque controller RCTC have been implemented in simulation environments for controlling lower limb exoskeleton in both phases. It is found from numerical simulations in three scenarios that the robust computed torque controller RCTC presents better results as compared with the classical computed torque controller CTC. Future work will be focused on extending our research in controlling lower limb exoskeleton in real time rehabilitation exoskeleton systems.

References 1. Ahmed, S., Wang, H., Tian, Y.: Modification to model reference adaptive control of 5-link exoskeleton with gravity compensation. In: 35th Chinese Control Conference (CCC), pp. 6115– 6120. IEEE (2016) 2. Azimi, V., Simon, D., Richter, H.: Stable robust adaptive impedance control of a prosthetic leg. In: Dynamic Systems and Control Conference, vol. 57243, pp. V001T09A003. American Society of Mechanical Engineers (2015) 3. Banala, S.K., Kim, S.H., Agrawal, S.K., Scholz, J.P.: Robot assisted gait training with active leg exoskeleton (alex). IEEE Trans. Neural Syst. Rehabil. Eng. 17(1), 2–8 (2008) 4. Bernhardt, M., Frey, M., Colombo, G., Riener, R.: Hybrid force position control yields cooperative behaviour of the rehabilitation robot lokomat. In: 9th International Conference on Rehabilitation Robotics, 2005. ICORR 2005, pp. 536–539. IEEE (2005) 5. Cenciarini, M., Dollar, A.M.: Biomechanical considerations in the design of lower limb exoskeletons. In: IEEE International Conference on Rehabilitation Robotics, pp. 1–6. IEEE (2011)

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6. Chen, X., Zhao, H., Zhen, S., Sun, H.: Adaptive robust control for a lower limbs rehabilitation robot running under passive training mode. IEEE/CAA J. Automatica Sinica 6(2), 493–502 (2019) 7. Daachi, M., Madani, T., Daachi, B., Djouani, K.: A radial basis function neural network adaptive controller to drive a powered lower limbknee joint orthosis. Appl. Soft Comput. 34, 324–336 (2015) 8. Fouad, K., Tetzlaff, W.: Rehabilitative training and plasticity following spinal cord injury. Exp. Neurol. 235(1), 91–99 (2012) 9. Hesse, S., Schmidt, H., Werner, C., Bardeleben, A.: Upper and lower extremity robotic devices for rehabilitation and for studying motor control. Curr. Opin. Neurol. 16(6), 705–710 (2003) 10. Hsieh, M.-H., Huang, Y.H., Chao, C.-L., Liu, C.-H., Hsu, W.-L., Shih, W.-P.: Single-actuatorbased lower-limb soft exoskeleton for preswing gait assistance. Appl. Bionics Biomech. (2020) 11. Jasim, H.H., Mary, A.H., Ahmed, M.S.: Robust computed torque control for uncertain robotic manipulators. Al-Khwarizmi Eng. J. 17(3) (2021) 12. Kara, T., Mary, A.H.: Adaptive pd-smc for nonlinear robotic manipulator tracking control. Stud. Inform. Control 26(1), 49–58 (2017) 13. Kazerooni, H., Steger, R., Huang, L.: Hybrid control of the berkeley lower extremity exoskeleton (bleex). Int. J. Robot. Res. 25(5–6), 561–573 (2006) 14. Langhorne, P., Coupar, F., Pollock, A.: Motor recovery after stroke: a systematic review. Lancet Neurol. 8(8), 741–754.12 (2009) 15. Lv, X., Yang, C., Li, X., Han, J., Jiang, F.: Passive training control for the lower limb rehabilitation robot. In: IEEE International Conference on Mechatronics and Automation (ICMA), pp. 904–909. IEEE (2017) 16. Maalej, B., Chemori, A., Derbel, N.: Intelligent tuning of augmented L1 adaptive control for cerebral palsy kids rehabilitation. In: 16th International Multi-conference on Systems, Signals & Devices (SSD), pp. 231–237. IEEE (2019) 17. Maalej, B., Chemori, A., Derbel, N.: Towards an effective robotic device for gait rehabilitation of children with cerebral palsy. In: International Multi-conference on Signal, Control and Communication (SCC), pp. 268–273. IEEE (2019) 18. Mefoued, S.: Commande robuste référencée intention d’une orthèse active pour l’assistance fonctionnelle aux mouvements du genou. PhD thesis, Paris Est (2012) 19. Meng, W., Liu, Q., Zhou, Z., Ai, Q., Sheng, B., Xie, S.S.: Recent development of mechanisms and control strategies for robot-assisted lower limb rehabilitation. Mechatronics 31, 132–145 (2015) 20. Mnif, F.: A robust feedback linearization control for constrained mechanical systems. Proc. Inst. Mech. Eng. Part I: J. Syst. Control Eng. 218(4), 299–310 (2004) 21. Murray, R.M., Li, Z., Sastry, S.S.: A Mathematical Introduction to Robotic Manipulation. CRC Press (2017) 22. Nair, A.S., Ezhilarasi, D.: Performance analysis of super twisting sliding mode controller by adams–matlab co-simulation in lower extremity exoskeleton. Int. J. Precis. Eng. Manufact. Green Technol. 7(3), 743–754 (2020) 23. Narayan, J., Dwivedy, S.K.: Robust lqr-based neural-fuzzy tracking control for a lower limb exoskeleton system with parametric uncertainties and external disturbances. Appl. Bionics Biomech. (2021) 24. Ozkul, F., Barkana, D.E.: Design and control of an upper limb exoskeleton robot rehabroby. In: Conference Towards Autonomous Robotic Systems, pp. 125–136. Springer (2011) 25. Peng, J., Liu, Y.: Adaptive robust quadratic stabilization tracking control for robotic system with uncertainties and external disturbances. J. Control Sci. Eng. (2014) 26. Soleimani Amiri, M., Ramli, R., Ibrahim, M.F., Abd Wahab, D., Aliman, N.: Adaptive particle swarm optimization of pid gain tuning for lower-limb human exoskeleton in virtual environment. Mathematics 8(11), 2040 (2020) 27. Spong, M.W., Hutchinson, S., Vidyasagar, M.: Robot Modeling and Control. Wiley (2020)

SOS-Based Robust Control Design Subject to Actuator Saturation for Maximum Power Point Tracking of Photovoltaic System Noureddine Boubekri, Dounia Saifia, Sofiane Doudou, and Mohammed Chadli Abstract This chapter presents a robust maximum power point tracking (MPPT) control design for a standalone photovoltaic (PV) system subject to actuator saturation via polynomial Tackagi-Seguno (T-S) Fuzzy model based control approach. In order to optimize the installation cost, an output feedback controller is designed which reduce the number of sensors of current and voltage required. In addition, the problem of input saturation (duty cycle) is addressed by using constrained control approach. The stabilization conditions subject to H∞ performance of the closed-loop system are derived and solved in terms of linear matrix inequalities (LMI’s) and sum of squares (SOS) optimization problems. Simulation results are provided to show the effectiveness of the proposed controllers. Keywords MPPT · Photovoltaic system · TS fuzzy model · Polynomial · Robust control · Linear matrix inequalities · Sum of squares

1 Introduction Energy demand continues to increase with population growth and industrial development. Total dependence on fossil fuels to secure energy supplies poses problems of depletion of sources and environmental damage resulting from greenhouse gas emissions. Solar photovoltaic as a clean, inexhaustible, free noise energy production resource has the advantages to be an alternative to traditional resources for the electricity production. N. Boubekri (B) · D. Saifia LAJ Laboratory, Faculty of Science and Technology, University of Jijel, Jijel, Algeria e-mail: [email protected] S. Doudou QUERE Laboratory, Faculty of Technology, University Setif 1, Setif, Algeria e-mail: [email protected] M. Chadli University Paris-Saclay, Univ Evry, IBISC, 91020 Evry, France © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Derbel et al. (eds.), Advances in Robust Control and Applications, Studies in Systems, Decision and Control 474, https://doi.org/10.1007/978-981-99-3463-8_3

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A photovoltaic panel exposed to light converts the solar energy (Irradiance) into electric current by photovoltaic effect. In practice, the PV panel is a nonlinear power source. The output power of a PV panel is affected by the connected load and unpredictable climatic conditions [7]. Indeed, a PV system has a maximum power point (MPP) at which the PV panel delivers the maximum power. In order to harvest maximum power from the PV panel, a DC-DC converter controlled by MPPT algorithm is used to adjust the load of the PV panel. In this context, several MPPT control techniques have been developed to improve photovoltaic systems efficiency, such as Perturb and Observe (P&O) [11, 16], Incremental Conductance (IC) [1, 8], proportional integral (PI) controller [18], fuzzy logic controller (FLC) [13, 29], sliding mode control [21], Takagi-Sugeno (T-S) fuzzy controllers [2–5, 7, 9, 17]. The T-S fuzzy model based control is a systematic approach for modeling and control design of complex nonlinear systems [24]. The main idea is to describe the nonlinear system by a collection of linear subsystems combined with weighting functions. Then, a parallel distributed compensation (PDC) controller is designed based of the TS fuzzy model of the nonlinear system. The gains of the feedback controller can be obtained by solving a set of stabilization conditions in terms of linear matrix inequalities. In this context, MPPT based on T-S fuzzy model has received a great deal of attention from researchers [2–4, 7, 9, 17]. Drawbacks in TS fuzzy model based control regarding the complexity of LMI constraints which increases with the number of local models and the number of variables to be searched for encourage researchers to exploit polynomial fuzzy model for modeling and control design of nonlinear systems [19, 20, 25]. This representation features polynomial models in the consequence part, which allows to represent a wider class of nonlinear systems, globally. This approach allows effectively to reduce the number of local models and increase the set of feasible solutions. The stabilization conditions of the polynomial fuzzy control are expressed as sum of squares (SOS) conditions that can be solved using third-party MATLAB SOSTOOLS [22]. This chapter introduces the design of MPPT control for a standalone PV system based on: firstly, the conventional TS fuzzy representation to design a nonlinear state feedback (PDC) controller and static output feedback (SOF) controller. Secondly, the polynomial representation is used to design a polynomial state feedback (PSF) controller and polynomial static output feedback (PSOF) controller. LMI and SOS approaches are used to derive the stabilization conditions based on H∞ performance criterion and Lyapunov’s stability theory. The input control saturation is taken into account by designing a low gain control law in order to to avoid the input saturation. This chapter is organized as follows: Sects. 2–3 present the nonlinear model of the PV system and its T-S fuzzy modeling. Section 4 discusses the problem of tracking control under input saturation. In Sect. 5, an MPPT based on LMI approach subject to actuator saturation is designed. The polynomial representation of the PV-boost system is exploited in Sects. 6 and 7 to the design an MPPT based on a polynomial state feedback controller and polynomial static output feedback controller. Section 7 gives the simulation results to illustrate the effectiveness of the proposed methods. Finally, the conclusion and prospects are presented in Sect. 8.

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Notation: We denote the set of the real numbers as R, the set of nonnegative integers as Z+ , Rm×n is the set of real m × n matrices. I and 0 denote identity and zeros matrices with appropriate dimensions. The symbol (*) ∑is used to represent the transposed elements in symmetric matrices. we denote SOS as the set of all sum of squares polynomials.

2 PV System Modelling The PV system considered in this work (Fig. 1), consists of a PV module connected to a load via a DC-DC boost converter. The boost converter is used to maintain the PV generator at its MPP and transfer the maximum power to the load.

2.1 Mathematical Modeling of PV Panel A photovoltaic module consists of a group of photovoltaic cells connected in series and in parallel to generate needed power. The current generated by a PV cell can be expressed by the following relation: ( (V + R I ) ) V +R I pv s pv pv s pv −1 − I pv = I ph − I0 exp AVT Rsh with:

Fig. 1 Structure of PV power generation system

(1)

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⎧ ) G ( ⎪ ⎪ = + k (T − T ) I I sc i r e f ph ⎪ ⎪ Gr e f ⎪ ⎪ ( ( ) ( )3 ⎪ ⎪ 1) 1 T ⎪ q Eg ⎪ ⎪ I = Ir s exp AK − ⎪ ⎨ 0 T Tr e f Tr e f Isc ⎪ Ir s = ( qV ) ⎪ ⎪ oc ⎪ ⎪ exp −1 ⎪ ⎪ AK T ⎪ ref ⎪ ⎪ ⎪ KT ⎪ ⎩ VT = q

(2)

where I ph is the photocurrent, I0 represents the cell revers saturation current, Ir s is the revere saturation current at the standard conditions (Tr e f = 298, 15 K and G r e f =1 kW/m2 ), VT is the thermal voltage, E g is the band-gap energy, q is the charge of an electron, A is the ideality factor, T and Tr e f are the ambient and the reference temperatures, respectively, G is the irradiance, K denotes the Boltzmann’s constant, ki is the temperature coefficient for short-circuit current, Rs and Rsh denotes the series and the shunt resistor, respectively. Since the shunt resistance Rsh is much greater than the series resistance Rs , the electric characteristic of the PV module can be determined by the following equation [7]: ) ] [ ( V pv −1 (3) I pv = n p I ph − n p I0 exp n s AVT where n s and n p denote the number of PV cells connected in series and parallel, respectively. Figure 2 shows the power-voltage (P-V) characteristics and the nonlinear behavior of the photovoltaic panel under different weather conditions.

2.2 Boost Converter Modeling In order to track the maximum power point, a DC/DC converter is contected to adjust the load of the PV panel. The boost converter is widely used in photovoltaic applications, its dynamic model can be described by the following state equations: ⎧ ˙ 1 1 ⎨ V pv = − Ca I L + Ca I pv R 1 I˙ = L V pv − LL I L − L1 Vc (1 − d(t)) ⎩ ˙L 1 Vc Vc = C1b I L (1 − d(t)) − RC b

(4)

where V pv and I pv denote the voltage and the current output of the PV module, respectively. I L represents the current on the inductance L, Vc is the voltage on the capacitance Cb , R L is the internal resistance of the inductance L and d(t) is the duty ratio. The model (4) can be rewritten in state-space as follows:

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120 25°C 45°C

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{

x(t) ˙ = A x(t) + B(x(t))u(t) + Bw w(t) y(t) = C x(t)

(5)

with ⎡ ⎤ 0 V pv (t) x(t) = ⎣ I L (t) ⎦ , A = ⎣ L1 Vc (t) 0 [ 0 C= 0 ⎡

−1 0 Ca −R L −1 L L 1 −1 Cb RCb

1 0

0 1

]





⎦, B = ⎣

0 Vc L −I L Cb





1 Ca



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u(t) = d(t), w(t) = I pv (t). where x(t) ∈ Rn is the state vector and y(t) ∈ R p is the vector of measurable output.

2.3 Reference Model The goal of this section is to find a reference model in which the photovoltaic panel operates at the MPP. If the system reaches its MPP, we have the following equality: d V pv d Ppv = V pv + I pv = 0 d I pv d I pv

(6)

Substituting (3) in (6), we obtain ( Iopt − (I ph − Iopt + I0 ) log

I ph − Iopt + I0 I0

) =0

(7)

where Iopt and Vopt are, respectively, the optimal values of the current I pv and the voltage V pv . According to [2], the solution of (7) gives: Iopt = 0.909I ph ( ) I ph − Iopt + I0 Vopt = nVt log I0

(8)

Consider the model of boost converter (4) and using (8), we have the following reference model: x˙r (t) = Ar xr (t) + Bw r (t) (9) yr (t) = C xr (t) with

⎡ ⎢ Ar = ⎢ ⎣

0

−1 Ca

1 L

−R L L

0

1 (1 Cb

− dopt (t)) /

dopt (t) = 1 −



⎡ ⎤ Vopt (t) ⎥ −1 ⎣ ⎦ (1 − dopt (t)) ⎥ L ⎦ , xr = I Lr (t) (t) V Cr −1 0

RCb

Vopt (t) − R L Iopt (t) and r (t) = Iopt (t). R Iopt (t)

where xr (t) ∈ Rn is the reference state vector, yr (t) ∈ R p is the reference output vector, and dopt (t) is the optimal control input.

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3 T-S Fuzzy Modeling The T-S fuzzy systems provide an effective approach for modeling and control of complex nonlinear systems. According to the T-S fuzzy model representation, nonlinear systems can be described as a weighted sum of linear sub-systems which locally describe the system dynamics. This approach has been widely used for the analysis of the stability and control of nonlinear systems in general and the control of PV systems in particular [3–5, 7, 9, 17]. The TS fuzzy model of a nonlinear system have the following representation: x(t) ˙ =

r ∑

( ) μi (ξ(t)) Ai x(t) + Bi u(t) + Bwi w(t)

i=1

z(t) =

r ∑

( ) μi (ξ(t)) Ci x(t) + D1i u(t) + D2i w(t)

(10)

i=1

where x ∈ Rn is the system state vector, z(t) ∈ Rnz is the controlled output variable, u(t) ∈ Rm is the control input, ξ(t) s the decision variable vector, w(t) is the disturbance variable, and μi (ξ(t)) are the normalized membership functions.

3.1 Sector Nonlinearity In general there are three approaches for constructing fuzzy models: Identification, linearization around different operating points or by transformation of sector nonlinearities. In identification method the structure of the the T-S fuzzy system is chosen, then, identification techniques are used to estimate the model’s parameters. This approach is suitable for systems that cannot be represented or are difficult to represent by analytical and/or physical models [14]. The second method is based on the linearization of the nonlinear system around different operating points. The membership functions are chosen a priori, then the T-S fuzzy model can be given by interpolation of linear local models. Sector nonlinearity approach is based on bounds of nonlinear functions, e.g, for a nonlinear system x˙ = f (x(t)) with f (0) = 0, the goal is to find the global sector such that x˙ = f (x(t)) ∈ [a1 a2 ]x(t) (see Fig. 3. This method gives an exact representation of a nonlinear system [24].

3.2 TS Fuzzy Model of PV-Boost System In this section, we present the T-S fuzzy model of the boost converter given in (5). By observing the matrices B(x(t)), we define the premise fuzzy variables ξk (t) = xk (t) ∈ [ξkmin ξkmax ] with k = 1, 2, as follows

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Fig. 3 Sector nonlinearity approach: a global sector nonlinearity, b local sector nonlinearity [14]

ξ1 (t) = Vc (t), ξ2 (t) = I L (t) By using sector nonlinearity approach [28], ξ1 (t) and ξ2 (t) can be represented as follows: ξk (t) = Mk,min (ξk (t))ξkmax + Mk,max (ξk (t))ξkmin where the fuzzy membership functions are derived as follows Mk,min (ξk (t)) =

ξ1 (t) − ξkmin , Mk,max (ξk (t)) = 1 − Mk,min (ξk (t)) ξkmax − ξkmin

The nonlinear system (5) is is described by the following T-S fuzzy rules: • Rule 1) : If ξ1 (t) is M1,min and ξ2 (t) is M2,min then x(t) ˙ = Ax(t) + B1 d(t) + Bw w(t) ˙ = • Rule 2) : If ξ1 (t) is M1,min and ξ2 (t) is M2,max then x(t) Ax(t) + B2 d(t) + Bw w(t) ˙ = • Rule 3) : If ξ1 (t) is M1,max and ξ2 (t) is M2,min then x(t) Ax(t) + B3 d(t) + Bw w(t) ˙ = • Rule 4) : If ξ1 (t) is M1,max and ξ2 (t) is M2,max then x(t) Ax(t) + B4 d(t) + Bw w(t) ⎡ B1 =

0



B3 = ⎣

0





(t) ⎦ ⎣ V cmin , L −I L min (t) Cb



V cmax (t) ⎦ , L −I L min (t) Cb

B2 =

0



(t) ⎦ ⎣ V cmin L −I L max (t) Cb

⎡ B4 = ⎣

0



V cmax (t) ⎦ L −I L max (t) Cb

(11)

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Therefore the equivalent fuzzy model of (5) can be given as follows x(t) ˙ =

4 ∑

μi (ξ(t))(Ax(t) + Bi d(t) + Bw w(t))

(12)

i=1

y(t) = C x(t) where: μi (ξ(t)) =

ωi (ξ(t)) ≥0 r ∑ ω j (ξ(t)) j=1

ωi (ξ1 (t), ξ2 (t)) = M1,a (ξ1 (t))M2,b (ξ2 (t)) with a, b ∈ [min, max].

3.3 TS Fuzzy Model of Reference System Using the sector nonlinearity approach [28], we can obtain a fuzzy model representing exactly the nonlinear reference model (9). Let’s define the premise variable ξ(t) = (1 − dopt (t)) ∈ [ξmin ξmax ], the reference model (9) can be described by the following T-S fuzzy rules [2]: • Rule 1) : If z(t) is Mmin then x˙r (t) = Ar 1 xr (t) + Bw r (t) • Rule 2) : If z(t) is Mmax then x˙r (t) = Ar2 xr (t) + Bw r (t) The membership functions can be defined as follows: Mmin = ⎡

0

Ar1 = ⎣ L1 0

ξ(t) − ξmin , Mmax = 1 − Mmin ξmax − ξmin

⎤ −1 0 Ca −R L −1 z ⎦, L min L 1 −1 z Cb rmin RCb



0

Ar2 = ⎣ L1 0

(13)

⎤ −1 0 Ca −R L −1 z ⎦ L max L 1 −1 z Cb max RCb

The global TS fuzzy reference model is inferred as: x˙r (t) =

2 ∑

( ) μi (z(t)) Ari xr (t) + Bw r (t)

i=1

y(t) = C xr (t) (Figs. 4 and 5).

(14)

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25

6

15

x2 (t)

x1 (t)

20

10

4 2

5 0 0

0.2

0.4

0.6

0.8

1

0 0

0.2

Time (s)

0.4

0.6

0.8

1

Time (s)

50

x3 (t)

40 30 20 10 0 0

0.2

0.4

0.6

0.8

1

Time (s)

Fig. 4 PV system responses, nonlinear model (red line), T-S fuzzy model (black line) 1

0.8

i

(t)

0.6

0.4

0.2

0

0

0.1

0.2

0.3

0.4

0.5

Time (s)

Fig. 5 Membership functions of the T-S fuzzy model

0.6

0.7

0.8

0.9

1

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4 H∞ Tracking Control Design Under Input Constraint H∞ control is well known as a robust control approach which ensures system stability and enhance control performance. The aim of an H∞ control problem is to design a controller such that the resulting closed-loop control system is stable with a prescribed level of attenuation from the exogenous disturbance input to the controlled output. In order to ensure a good MPPT, H∞ criterion performance is used. Throughout this paper, we assume that the disturbances signal w is norme-bounded by L2 norme, that satisfies: { ∞ 2 ||w||2 = w(t)T w(t)dt ≤ Q2 (15) 0

The robust H∞ tracking performance is given by: {



J∞ =

{ z(t)T Qz(t)dt < −γ 2

0



w T (t)w(t)dt

(16)

0

where, z(t) ∈ Rnz is the controlled output variable and w(t) is the bounded disturbance input. For a symmetric positive matrix P ∈ Rn×n , we define a quadratic Lyapunov function as follows: V (t) = x T (t)Px(t) (17) For a constant ρ define an ellipsoid | { } E(P, ρ) = x ∈ R n | x T (t)P x(t) ≤ ρ, ρ > 0

(18)

An ellipsoid E(P, ρ) is said to be contractively invariant set if V˙ (t) < 0, ∀x ∈ E(P, ρ)|{0}. Therefore, if an ellipsoid is contractively invariant, it is inside the domain of attraction (Ω).

4.1 Actuator Saturation In nonlinear systems, the stability and performance issues become more complex in the presence of saturation. Indeed, the stability of the system becomes local. One of the challenges is determining the largest domain of the state space within which any initialization of the system does not cause instability. In this context, the constrained control approach is one of the solutions to deal with this problem. In this approach, a low gain control law is designed in order to avoid the actuator saturation. This method requires that the non-linearity of the input saturation is symmetric. However, the duty cycle is limited between 0 and 1, which is considered as saturation with asymmetric nonlinearity. To solve this problem, we consider the following step:

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( ) x(t) ˙ = Ax(t) + Bμ d(t) − 0.45 + 0.45B1μ + Bw w(t) ˜ = Ax(t) + Bμ u φ (t) + B˜ w w(t)

(19)

[ ] ] [ w(t) ˜ with Bw = Bw 0.45Bμ and w(t) ˜ = . 1 From (19), we can see that duty-cycle asymmetric constraint (0 ≤ d(t) ≤ 0.9) is transformed into new symmetric saturation u φ (t) constraints by the following limits: ⎧ ⎨ −u i f u(t) ≤ −u (20) u φ (t) = u(t) i f − u ≤ u(t) ≤ u, u = 0.45 ⎩ i f u(t) ≥ u u The problem is solved by determining a limit of initial conditions of the system states which avoids the saturation of the control. The linear model of the command remains valid: sat (u(t), u) = u(t), such as x belonging to the polyhedral Ω defined as: { Ωj = x { Ωj = x { Ω(ξ) = x { Ω(ξ) = x

|| | } ∈ Rn | | K j x | ≤ u , in the case of a PDC controller. || | } ∈ Rn | | K j C x | ≤ u , in the case of a SOF controller. || | } ∈ Rn | |F(ξ) x | ≤ u , in the case of a PSF controller. || | } ∈ Rn | |G(ξ)C x | ≤ u , in the case of a PSOF controller.

(21)

5 LMI-Based Robust Control Design Definition 1 [6] A strict linear matrix inequality (LMI) has the form: F(x) = F(0) +

m ∑

xi Fi > 0

(22)

i=1

where x ∈ Rm is the vector of decision variables, and Fi = FiT ∈ Rn×n , i = 0, ..., m, are given symmetric matrices. The LMI (22) defines a convex constraint on x, i.e., the set {x| F(x) > 0} is convex. The convexity of an optimization problem has the advantages of globality, i.e., the result obtained corresponds to a unique global minimum. In addition the the computation demand is reasonable. The advantage of LMI’s method is demonstrated through its ability to formulate many control problems as a convex optimization problem that can be solved numerically and efficiently using semidefinite programming (SDP) tools.

SOS-Based Robust Control Design Subject to Actuator Saturation …

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5.1 Analysis and Reduction Techniques • Schur complement: frequently used as a direct method for converting convex nonlinearities to LMI’s. Lemma 1 [6] Consider the matrices Q(x) = Q(x)T , R(x) = R(x)T , and S(x) affine in the variable x, the following LMI’s are equivalent: ii. R(x) > 0, Q(x) − S(x)R(x)−1 S(x) > 0 [ i.

] Q(x) S(x) >0 S(x)T R(x)

(23) (24)

• Variable change: allows to transform a billinear matrix inequality (BMI) into a linear matrix equality. A BMI constraint appears when there is a product of two variables, the trick is just to replace the product with a new variable.

5.2 Robust PDC Control Design for PV System The aim of control design is to derive the system states (12) to follows the states of the reference model (14). This mean that, the following tracking error converge to zero under climatic condition changes and presence of duty-cycle saturation (Fig. 6). e(t) = x(t) − xr (t)

(25)

The control of PV system based on fuzzy model generally uses a nonlinear state feedback of the following form: u(t) =

r =4 ∑

μ j (ξ(t))K j e(t)

j=1

Fig. 6 Diagram of control system with PDC controller

(26)

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We denote: K μ =

r =4 ∑

μ j (ξ(t))K j .

j=1

Consider now the fuzzy augmented system expressed as follows: ˙ = Aμ x(t) + B μ u(t) + B w w(t) x(t)

(27)

where: ] [ [ ] [ [ [ ] ] ] A 0 Bμ x(t) w(t) ˜ B˜ w 0 Aμ = , Bw = , w(t) = , Bμ = , x(t) = 0 0 Aμr e f xr (t) r (t) 0 Bw where Aμr e f =

2 ∑

μk (ξ(t))Ari and Bμ =

i=1

4 ∑

μi (ξ(t))Bi , 0 denotes a zeros matrix

i=1

with appropriate dimensions. Then, the fuzzy controller (26) can be rewritten as follows u(t) = K μ x(t), with K μ = [K μ − K μ ].

(28)

Finally, the fuzzy augmented system can be expressed as follows: ˙ = x(t)

4 ∑ 2 4 ∑ ∑

( ) μi μ j μk Ai jk x(t) + B w w(t)

(29)

i=1 j=1 k=1

with: Ai jk = (Aμ + B μ K μ ). Consider the H∞ performance related to the tracking error given by the objective function: (30) J∞ = V˙ (t) + z(t)T z(t) − γ 2 w T (t)w(t) < 0 where z(t) = e(t) = C x(t), C = [I − I ]. For all e(0) ∈ E(P, 1) ⊂ E(P, Q), if J∞ < 0 then, ∀ T > 0: {

T

{

T

J∞ =

0

V˙ (t) + z(t)T z(t) − γ 2 w(t)T w(t) dt < 0

0

thus: T • if w(t) = 0, then : V (T ) − V (0) < −z(t) { z(t) ≤ 0 ∞

• if w(t) /= 0, then : V (T ) ≤ V (0) + γ 2

w(t)T w(t) dt ≤ 1 + γ 2 Q2

0

denote: ρ = 1 + γ Q then, E(P, ρ) is the domaine of attraction. 2 2

SOS-Based Robust Control Design Subject to Actuator Saturation …

67

If T → ∞: {



{



z(t)T z(t) dt < −γ 2

0

w(t)T w(t) dt + V (0)

0

and under the zero-initial conditions, we get: { J∞ =



{



z(t)T z(t)dt < −γ 2

0

w(t)T w(t) dt

0

Then the H∞ disturbance attenuation level γ is guaranteed. Lemma 2 [10] The ellipsoid E(P, ρ) is inside the polyhedral Ω if and only if: (K μ )T

( )−1 P Kμ ≤ u2 ρ

(31)

In the following, we give the LMI formulation of the problem of MPPT control based on H∞ performance criterion taking into account the actuator saturation. Theorem 1 [23] The closed-loop system (29) is asymptotically stable with H∞ performance if there exist matrices Q ∈ Rn×n and F j ∈ Rm×n solution for the following optimization problem: min γ Subject to: ⎤ Ai Q + B i F j + (*) * * T ⎣ −γ 2 I * ⎦ < 0 ∀i ∈ Ir and ∀ j ∈ Ir Bw CQ O −I −1

(32)

⎤ u i2 *⎦ ⎣ ρ ≥ 0 ∀i ∈ Im and ∀ j ∈ Ir T (F j ) Q

(33)





The closed-loop system (29) is stabilized by the nonlinear state feedback controller: K j = F j Q−1 Proof Consider the following quadratic Lyapunov function candidate: V (t) = x T (t)Px(t) where P ∈ Rn×n a symmetric positive define matrix.

(34)

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The time-derivative of Lyapunov function (34) is given by: ˙ T Px(t) + x(t)T Px(t) ˙ V˙ (t) = x(t) (( )T ) = Aμ + B μ K μ x(t) + B w w(t) Px(t) ) (( ) +x(t)T P Aμ + B μ K μ x(t) + B w w(t)

(35)

Then, the inequality (30) can be written as: ( )T [ T ] )) Aμ + B μ K μ P + P Aμ + B μ K μ x(t) + w(t)T B w P x(t) [ ] (36) +x T (t) PB w w(t) + z T (t)z(t) − γ 2 w(t)T w(t) < 0 x(t)T

((

Or, in matrix form ] [( ) ] [( T ]T [ ] ]T [ ( [ P Aμ + B μ K μ + (*) * x(t) x(t) C) C)T + < 0 (37) T w(t) w(t) O O −γ 2 I BwP Using Schur compliment:

with:

]T [ ] [ x(t) x(t) T 2 T ˙ 0 such that [27]: f (x(t)) = z T (x)Qz(x) where z(x) is a vector of monomials in x with degree no greater than d. Proposition 2 Let L(x) be an N × N symmetric matrix of degree 2d in x ∈ R. Furthermore, let z(x) be a column vector whose entries are all monomials in x with degree no greater than d, which satisfy the following conditions: (a) L(x) > 0 for all x ∈ R. (b) v(t)T L(x)v(t) is SOS, where v(t) ∈ Rn . (c) There exists a positive semidefinite matrix Q(x) such that v(t)T L(x)v(t) = (v(t) ⊗ z(x))T Q(x)(v(t) ⊗ z(x)), where ⊗ denotes the Kronecker product. Then (a) ⇐ (b) and (b) ⇐ (c).

6.4 Robust PSF Control Design A polynomial state feedback controller is defined as follows: u φ (t) = F(ξ)e(t)

(56)

where ξ = (x, xr ), e(t) = x − xr is the state error and F(ξ) ∈ Rm×n is the polynomial feedback gain. By substituting controller (56) into (5) we get the following control system: ( ) x(t) ˙ = A + B(x)F(ξ) e(t) + Axr (t) + B˜ w w(t)

(57)

From (5) and (14) we have the error dynamic as: [ ] e(t) ˙ = x(t) ˙ − x˙r (t) = AP(ξ) + B(x)K(ξ) P−1 (ξ)e(t) +

2 ∑ k=1

( ) μk A − Ar k xr (t) + B˜ w δ(t)

(58)

SOS-Based Robust Control Design Subject to Actuator Saturation …

73

where δ(t) = w(t) − r (t) and F(ξ) is defined as follows: F(ξ) = K(ξ)P−1 (ξ)

(59)

The Eq. (58) can be rewritten in the following format: e(t) ˙ =

2 ∑

μk Θk (ξ)(t)

(60)

k=1

where [

(

)

Θk (ξ) = AP(ξ) + B(x)K(ξ) A − Ar k B˜ w with:

]

⎤ ⎡ ⎤ η1 (t) η1 (t) , η(t) = ⎣η2 (t)⎦ = ⎣ xr (t)⎦ η3 (t) δ(t) (61) ⎡

η1 (t) = P−1 (ξ)e(t)

The following theorem gives a solution of MPPT control problem of PV system in terms of SOS optimization problem. Theorem 3 The closed-loop system formed by the polynomial model of boost converter (5), the TS fuzzy model of the reference model (9), and the polynomial controller (76) satisfies the H∞ tracking performance, if there exist matrices Q(ξ) = Q(ξ)T ∈ Rn×n and K(ξ) ∈ Rm× p and nonnegative matrices ε1 (ξ) and ε(ξ)2 such that the following optimization problem: min γ1 , γ2 subject to:

( ) ∑ υ1T Q(ξ) − ε1 (ξ) υ1 ∈ SOS

(62)

⎞ ⎤ * * Φk (ξ) ∑ − ω T ⎝⎣(A − Ar k )T −γ1 I * ⎦ + ε2 (ξ)⎠ ω ∈ SOS B˜ wT 0 −γ2 I

(63)

⎛⎡

⎞ ⎤ u i2 ∑ * ⎦ υ2T ⎝⎣ ρ − ε2 (ξ)⎠ υ2T ∈ SOS K(ξ)T Q(ξ) ⎛⎡

(64)

has zero optimum, where υ1 ∈ Rn , υ2 ∈ Rn+1 , and ω = [ω1 ω2 ω3 ]T ∈ R2n+m are arbitrary vectors independent of x and xr , λ1 . The feedback gain control can be obtained by F(ξ) = K(ξ)P−1 (ξ).

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Proof Consider the Lyapunov function candidate: V (t) = e T (t)P−1 (ξ)e(t)

(65)

∂P−1 (ξ) ˙ + e T (t) e(t) V˙ (t) = e˙ T (t)P−1 (ξ)e(t) + e T (t)P−1 (ξ)e(t) ∂t

(66)

it’s time derivative is:

Remark 1 To simplify the stability analysis, the Lyaponov matrix is considered as a constant matrix. )T ) ( ) AP + B(x)K(ξ) P−1 e(t) + A − Ar μ xr (t) + B˜ w δ(t) P−1 e(t) ) (( ( ) ) (67) + e T P−1 AP + B(x)K(ξ) P−1 e(t) + A − Ar μ xr (t) + B˜ w δ(t) ((

V˙ (t) =

using (60) and (61), (67) can be written as follows: V˙ (t) =

2 ∑

μi η T (t)Φk (ξ)η(t) − η1T (t)η1 (t) + γ1 η2T (t)η2 (t) + γ2 η3T (t)η3 (t) (68)

k=1

⎡(

) ⎤ AP + B1 (x)K(ξ) + (*) + I * * ( )T Φk (ξ) = ⎣ A − Ar k −γ1 I * ⎦ T B˜ w 0 −γ2 I

where

(69)

If the following inequality is hold: Φμ (ξ) =

2 ∑

μk Φk (ξ) < 0

(70)

k=1

Then, from (68) we have:

Define

V˙ (t) ≤ −η1T (t)η1 (t) + γ1 η2T (t)η2 (t) + γ2 η3T (t)η3 (t)

(71)

J = V˙ (t) + η1T (t)η1 (t) − γ1 η2T (t)η2 (t) − γ2 η3T (t)η3 (t)

(72)

For all initial conditions e(0) ∈ E(P, 1) ⊂ E(P, ρ), if J < 0, then ∀ T > 0: { 0

T

V˙ (t) + η1T (t)η1 (t) − γ1 η2T (t)η2 (t) − γ2 η3T (t)η3 (t) dt < 0

(73)

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75

Thus: • If η2 (t) = η3 (t) = 0, then: V (t f ) − V (0) < −

{ T 0

η1T (t)η1 (t) d(t) ≤ 0

• If η2 (t) / = 0, η3 (t) / = 0, then: V (t f ) ≤ V (0) + γ12

{ T 0

η2T (t)η2 (t) d(t) + γ22

{ T 0

η3T (t)η3 (t) dt ≤ 1 + γ12 Q21 + γ22 Q22

Denote: ρ = 1 + γ12 Q21 + γ22 Q22 then, E(P, ρ) is the attraction domain, meaning that the trajectories of the error system (58) converge and do not leave the set E(P, ρ). If T → ∞: { T { tf { ∞ η1T (t)η1 (t) dt < γ12 η2T (t)η2 (t) d(t) + γ22 η3T (t)η3 (t) dt + V (0) 0

0

0

and under the zero-initial conditions, the system achieves H∞ performance with attenuation levels γ1 , γ2 . To demonstrate that the SOS condition (64) guarantees the set inclusion E(P, ρ) ∈ Ω(ξ), The inequality condition (31) can be rewritten as follows: u2 − F(ξ)T P−1 (ξ)F(ξ) > 0 ρ

(74)

Multiplying both side of the last inequality by P−1 (ξ) and using Schur complement we get: ⎤ ⎡ 2 u * ⎦>0 ⎣ ρ (75) T K(ξ) Q(ξ) and this is equivalent to the SOS condition (64).

6.5 Robust PSOF Control Design A polynomial output feedback controller is defined as follows: u(t) = G(ξ)e y (t) = G(ξ)Ce(t)

(76)

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Fig. 7 Diagram of control system with PSOF controller

where G(ξ) ∈ Rm×n is the polynomial feedback gain, e y (t) = y(t) − yr (t) = C(x(t) − xr (t)) is the output error (Fig. 7). By substituting controller (76) into (5) we get the following control system: ) ( x(t) ˙ = A + B(x)G(ξ)C e(t) + Axr + B˜ w w(t)

(77)

From (5) and (14) we have 2 ∑ ( ) ] [ μi A − Ar k xr (t) + B˜ w δ(t) e˙ = A + B1 (x)G(ξ)C e(t) +

(78)

k=1

The following lemma will be used in the proof of the result Lemma 3 Define [19]: [ ] Υ = C T (CC T )−1 art(C T ) ∈ Rn×n

(79)

where art(C T ) denotes the orthogonal complement of C T . Assuming that Υ be nonsingular, we have ] [ (80) CΥ = I p 0 where I p is an p × p identity matrix. Now, the error vector will be augmented as follows: ˙˜ = Υ −1 e(t) ˙ e(t) ) ( −1 = Υ AΥ P−1 (ξ) + Υ −1 B(x)G(ξ)CΥ P−1 (ξ) P(ξ)Υ −1 e(t) +Υ −1

2 ∑

( ) μk A − Ar k xr (t) + Υ −1 B˜ w δ(t)

(81)

k=1

To study the stability of the augmented tracking error system (81), we consider the following Lyapunov function candidate: ˜ V (t) = e˜ T (t)P(ξ)e(t)

(82)

SOS-Based Robust Control Design Subject to Actuator Saturation …

77

with P(ξ) is a symmetric positive definite matrix. We consider the polynomial Lyapunov matrix P(ξ) in the following form [12]: [

P (ξ) 0 P(ξ) = 11 0 P22 (ξ)

] (83)

where P11 (ξ) ∈ R p× p and P22 (ξ) ∈ Rn− p×n− p . And let define the polynomial output feedback gain as follows: G(ξ) = M(ξ)P11 (ξ)

(84)

where M(ξ) ∈ Rm× p . Using (80), (83) and (84) we get: [ ] G(ξ)CΥ P−1 = M(ξ) 0

(85)

which allows us to rewrite (81) as follows: ( [ ]) ˙˜ = Υ −1 AΥ P−1 (ξ) + Υ −1 B1 (x) M(ξ) 0 P(ξ)e(t) ˜ e(t) +Υ −1

2 ∑

( ) μk A − Ar k xr (t) + Υ −1 B˜ w δ(t)

(86)

k=1

˙˜ = e(t)

2 ∑

μk Θk (ξ)(t)

(87)

k=1

where ] [ ( ) [ ] Θk (ξ) = Υ −1 AΥ P−1 (ξ) + Υ −1 B(x) M(ξ) 0 Υ −1 A − Ar k Υ −1 B˜ w , (88) ⎡

⎤ ⎡ ⎤ η1 (t) P(ξ)e(t) ˜ η(t) = ⎣η2 (t)⎦ = ⎣ xr (t) ⎦ η3 (t) δ(t) Theorem 4 The closed-loop system formed by the polynomial model of boost converter (5), the TS fuzzy model of the reference model (9), and the polynomial controller (76) satisfies the H∞ tracking performance, if there exist matrices Q(ξ) ∈ Rn×n and M(ξ) ∈ Rm× p and nonnegative matrices ε1 (ξ) and ε(ξ)2 such that the following optimisation problem: min γ1 , γ2 subject to:

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( ) ∑ υ1T Q(ξ) − ε1 (ξ) υ1 ∈ SOS

(89)

⎞ ⎤ * * Φk (ξ) ∑ − ω T ⎝⎣Υ −1 ( A − Ari )T −γ1 I * ⎦ + ε2 (ξ)⎠ ω ∈ SOS −1 T 0 −γ2 I Υ B2

(90)

⎞ ⎤ u i2 ∑ * ⎦ − ε2 (ξ)⎠ υ2 ∈ υ2T ⎝⎣ ρ ] SOS [ T M(ξ) 0 Q(ξ)

(91)

⎛⎡

⎛⎡

has zero optimum, where υ1 ∈ Rn , υ2 ∈ Rn+1 , and ω = [ω1 ω2 ω3 ]T ∈ R2n+m are arbitrary vectors independent of x. The output feedback gain control can be obtained by G(ξ) = M(ξ)P11 (ξ). Proof Consider the Lyapunov function (82), the time derivative is: V˙ (t) =

[(

[ ]) ˜ Υ −1 AΥ P−1 (ξ) + Υ −1 B(x) M(ξ) 0 Pe(t)



−1

2 ∑

]T ( ) μk A − Ar k xr (t) + Υ −1 B˜ w δ(t) P(ξ) e(t) ˜

k=1

+e˜ (t) P(ξ) T

+Υ −1

2 ∑

[(

[ ]) Υ −1 AΥ P−1 (ξ) + Υ −1 B(x) M(ξ) 0 P(ξ)e(t) ˜

] ( ) μk A − Ar k xr + Υ −1 B˜ w δ(t)

(92)

k=1

using (86) and (88), (92) can be written as follows: V˙ (t) =

2 ∑

μk η T Φk (ξ)η − η1T η1 + γ1 η2T η2 + γ2 η3T η3

(93)

k=1

where ⎡(

[ ]) ⎤ Υ −1 AΥ P−1 (ξ) + Υ −1 B(x) M(ξ) 0 + (*) + I * * ( ) T Φk (ξ) = ⎣ Υ −1 A − Ar k −γ1 I * ⎦ −1 ˜ T 0 −γ2 I Υ Bw (94) where γ1 and γ2 are positive scalars. After this point, the steps for proving the H∞ performance are identical to steps (70) to (73). The inequality (31) guarantees the inclusion of the ellipsoid E(P, ρ) in the polyhedral Ω(ξ) defined in (21). The inequality (31) can be rewritten as follows:

SOS-Based Robust Control Design Subject to Actuator Saturation …

)T ( ) u2 ( − K(ξ)CΥ P−1 (ξ) G(ξ)CΥ > 0 ρ

79

(95)

by using Schur complement we get ⎡

⎤ u2 * ⎣ ⎦>0 ρ ) ( T G(ξ)CΥ P(ξ) [ pre-and post multiplying the last inequality by Δ =

(96)

] I * with Q(ξ) = P−1 (ξ) 0 Q(ξ)

and using (85) we get SOS condition (91). This completes the proof of the theorem.

7 Simulation In this section, we present the simulation results under MATLAB/Simulink environment to demonstrate the effectiveness of the proposed methods. Lorentz LC120-12P is the PV module used, whose specifications are listed in Table 1. The parameters of the boost converter are choosen as Ca = 500 µF, Cb = 100 µF, L = 10 mH, R L = 0.01 Ω, and R = 20 Ω. Solving the LMI conditions given in Theorem 1 via LMI toolbox give the following feedback gain: [ ] K 1 = 3.4461 −5.4409 0.8445 ; K 2 [ ] K 3 = 3.5101 −5.6108 0.8710 ; K 4

[ ] = 3.6078 −5.5009 0.8103 ; [ ] = 3.2489 −5.2101 0.7928 .

Solving the SOS conditions given in Theorems 2 and 4 via SOSTOOLS toolbox give the following polynomial feedback gains:

Table 1 The LC120-12P PV module Parameters Parameters Symbol Series cells Maximum power Current at MPP Voltage at MPP Open-circuit voltage Short-circuit current

Ns PMPP IMPP VMPP Voc Isc

Value 36 120 W 7A 17.1 V 21.8 V 7.7 A

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Table 2 Comparison of minimized H∞ parameters Controller γi 0.0001 Infeasibility 0.01 0.01

50

1000

T (°C)

G (w/m2 )

PDC SOF Polynomial state feedback Polynomial output feedback

800 600 400 0

0.2

0.4

0.6

0.8

1

40 30 20 0

0.2

0.4

0.6

0.8

1

Time (s)

Time (s)

Fig. 8 Irradiation and temperature variations [ F(ξ) =

0.0001x12 x2 + 0.00013x23 +0.0001x2 x32 + 0.0002x2 ⎡

0.0004x12 x2 + 0.0004x23 +0.0004x2 x32 + 0.0008x2

−3.0201x23 − 0.0090x22 x3 G(ξ) = ⎣−2.2648x2 xr23 − 5.1584x2 −0.0016x33 + 0.3861x3

0.0001x12 x2 + 0.00013x23 +0.0001x2 x32 + 0.0001x2

]

⎤ 0.3141x23 − 0.0002x22 x3 + 0.1812x2 x32 +0.2357x2 xr23 + 0.2357x2 xr22 + 0.5461x2 ⎦ −0.0006x33 + 0.0077x3

where the degree of K(ξ) and M(ξ) was chosen as 3, γ1 = γ2 = 0.01, and the values of ε1 (ξ), ε2 (ξ) have been fixed at 0.001. Table 2 shows a comparison of H∞ parameters obtained by Theorems 1–4. For the scenario of simulation, the changes in weather conditions (irradiation, temperature) are considered as in Fig. 8. Figures 9 and 10 illustrate the MPPT results based on PDC controller, polynomial state feedback (PSF) controller, polynomial static output feedback (PSOF) controller, and the Perturb and Observe (P&O) method. It should be noted that the P&O method is widely used in the field and belongs to the group of direct methods. These methods are considered as a reference and any improvements to the MPPT control strategy should be compared to them [3]. Figure 11 shows the Power-Voltage curves of the PV power system with different control approaches and under a dynamic irradiation profile that start from 100 W/m2 and ends with 1000 W/m2 in 5 s. From the simulation results, It is clear that the controllers based of TS fuzzy model and polynomial model give better MPP tracking performance in terms of convergence speed and tracking accuracy compared to P&O method. The P&O method provides a poor performance under dynamic and constant climatic conditions. In contrast, PDC, PSF, and PSF present a good dynamic performance characterized by a very fast response (about 0.01 s), and an exact MPP tracking without ripples or oscillations.

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81

P&O P opt

100

PDC PPDC PSOF

(a)

Power (w)

80 80

60

60 45

40

40

20

40

0

0.01

0.02 35

20

30 0.8

0

0

0.1

0.2

0.3

0.4

0.81

0.5

0.6

0.82

0.7

0.8

0.9

1

Time (s)

0.9 0.8 0.7

(b)

Duty cycle

0.6 0.5 0.8

0.4

0.6

0.3

P&O PDC PSF PSOF

0.2

0.2

0

0.1 0

P opt

0.4

0

0

0.1

0.005

0.2

0.3

0.01

0.4

0.015

0.02

0.5

0.6

Time (sec)

Fig. 9 MPPT control response: a PV power Ppv , b Duty cycle d(t)

0.7

0.8

0.9

1

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(a)

reference

PDC

PFC

PSOF

Voltage (V)

20

15 16

10

14

5

0

12 10 0.79

0

0.1

0.2

0.795

0.3

0.8

0.4

0.805

0.81

0.5

0.6

0.815

0.82

0.7

0.8

0.9

1

0.7

0.8

0.9

1

0.8

0.9

1

Time (s)

(b)

8 7 6

current (A)

5 4

7.5

3 7

2 6.5

1 0

0.24

0

0.25

0.1

0.2

0.26

0.3

0.27

0.28

0.4

0.29

0.3

0.5

0.6

Time (s)

(c)

50

Voltage (V)

40

30

40 30

20

20 10

10

0 0

0

0

0.1

0.005

0.2

0.3

0.01

0.4

0.015

0.5

0.6

0.02

0.7

Time (s)

Fig. 10 Tracking of reference system: a Panel voltage (V pv ), b Inductor current (I L ), c Voltage across the capacitor Cb

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Fig. 11 P-V curve response under dynamic insolation: a PDC controller, b P&O method, c PSF controller, d PSOF controller

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8 Conclusion This chapter has presented the MPPT control design for a standalone PV system based on the conventional TS fuzzy model and the recently developed polynomial fuzzy model. The problem of duty cycle saturation was addressed by exploiting a constrained control approach. Based on Lyapunov approach, the stabilization conditions subject to H∞ performance have been derived in terms of LMI and SOS optimization problems and solved numerically by using LMI toolbox and SOSTOOLS in MATLAB environment. The simulation and the comparison results with the P&O method clearly proved the advantage of the presented controllers regarding constancy and tracking precision. Future work is aimed at the real-time implementation of the control system, dispensing the reference model by exploiting the partial derivative of power as the controlled output.

References 1. Ali, M.N., Mahmoud, K., Lehtonen, M., Darwish, M.M.: An efficient fuzzy-logic based variable-step incremental conductance mppt method for grid-connected pv systems. IEEE Access 9, 26420–26430 (2021) 2. Allouche, M., Dahech, K., Chaabane, M.: Multiobjective maximum power tracking control of photovoltaic systems: TS fuzzy model-based approach. Soft Comput. 22(7), 2121–2132 (2018) 3. Allouche, M., Dahech, K., Chaabane, M., Mehdi, D.: Fuzzy observer-based control for maximum power-point tracking of a photovoltaic system. Int. J. Syst. Sci. 49(5), 1061–1073 (2018) 4. Boubekri, N., Doudou, S., Saifia, D., Chadli, M.: Robust mixed H2 /H∞ fuzzy tracking control of photovoltaic system subject to asymmetric actuator saturation. Trans. Inst. Meas. Control 44(7), 1528–1541 (2022) 5. Boubekri, N., Saifia, D., Doudou, S., Chadli, M.: Robust H∞ fuzzy saturated control of photovoltaic system. Procedia Comput. Sci. 186, 95–108 (2021) 6. Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM (1994) 7. Chiu, C.-S.: TS fuzzy maximum power point tracking control of solar power generation systems. IEEE Trans. Energy Convers. 25(4), 1123–1132 (2010) 8. Hebchi, M., Kouzou, A., Choucha, A.: Improved incremental conductance algorithm for mppt in photovoltaic system. In: 2021 18th International Multi-Conference on Systems, Signals & Devices (SSD), pp 1271–1278. IEEE (2021) 9. Houda, K., Saifia, D., Chadli, M., Labiod, S.: H∞ fuzzy proportional integral state feedback controller of photovoltaic systems under asymmetric actuator constraints. Trans. Inst. Meas. Control 43(1), 34–46 (2021) 10. Hu, T., Lin, Z., Chen, B.M.: Analysis and design for discrete-time linear systems subject to actuator saturation. Syst. Control Lett. 45(2), 97–112 (2002) 11. Kamran, M., Mudassar, M., Fazal, M.R., Asghar, M.U., Bilal, M., Asghar, R.: Implementation of improved perturb & observe mppt technique with confined search space for standalone photovoltaic system. J. King Saud Univ. Eng. Sci. 32(7), 432–441 (2020) 12. Lam, H.-K.: Output-feedback tracking control for polynomial fuzzy model-based control systems. In: Polynomial Fuzzy Model-Based Control Systems, pp 175–196. Springer (2016) 13. Li, S., Liao, H., Yuan, H., Ai, Q., Chen, K.: A MPPT strategy with variable weather parameters through analyzing the effect of the DC/DC converter to the MPP of PV system. Solar Energy 144, 175–184 (2017)

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14. Mehran, K.: Takagi-sugeno fuzzy modeling for process control. Ind. Autom. Robot. Artif. Intell. (EEE8005) 262, 1–31 (2008) 15. Meng, F., Wang, D., Yang, P., Xie, G., Guo, F.: Application of sum-of-squares method in estimation of region of attraction for nonlinear polynomial systems. IEEE Access 8, 14234– 14243 (2020) 16. Mohammadinodoushan, M., Abbassi, R., Jerbi, H., Ahmed, F.W., Rezvani, A., et al.: A new mppt design using variable step size perturb and observe method for pv system under partially shaded conditions by modified shuffled frog leaping algorithm-smc controller. Sustain. Energy Technol. Assess. 45, 101056 (2021) 17. Nasri, M., Saifia, D., Chadli, M., Labiod, S.: H∞ switching fuzzy control of solar power generation systems with asymmetric input constraint. Asian J. Control 21(4), 1869–1880 (2019) 18. Oshaba, A., Ali, E.S., Abd Elazim, S.M.: PI controller design for MPPT of photovoltaic system supplying SRM via BAT search algorithm. Neural Comput. Appl. 28(4), 651–667 (2017) 19. Pakkhesal, S., Mohammadzaman, I.: Less conservative output-feedback tracking control design for polynomial fuzzy systems. IET Control Theory Appl. 12(13), 1843–1852 (2018) 20. Pakkhesal, S., Mohammadzaman, I.: An improved sum-of-squares based approach to fuzzy tracking control design of nonlinear systems. Asian J. Control 22(4), 1447–1457 (2020) 21. Pradhan, R., Subudhi, B.: Double integral sliding mode MPPT control of a photovoltaic system. IEEE Trans. Control Syst. Technol. 24(1), 285–292 (2015) 22. Prajna, S., Papachristodoulou, A., Parrilo, P.A.: Sostools: sum of squares optimization toolbox for matlab-user’s guide. Control and Dynamical Systems, California Institute of Technology, Pasadena, CA, 91125 (2004) 23. Saifia, D., Chadli, M., Labiod, S., Guerra, T.M.: Robust H∞ static output feedback stabilization of TS fuzzy systems subject to actuator saturation. Int. J. Control Autom. Syst. 10(3), 613–622 (2012) 24. Takagi, T., Michio, S.: Fuzzy identification of systems and its applications to modeling and control. Syst. Man Cybern. 1 SMC-15(1), 116–132 (1985) 25. Tanaka, K., Yoshida, H., Ohtake, H., Wang, H.O.: Stabilization of polynomial fuzzy systems via a sum of squares approach. In: 2007 IEEE 22nd International Symposium on Intelligent Control, pp 160–165. IEEE (2007) 26. Tanaka, K., Yoshida, H., Ohtake, H., Wang, H.O.: A sum of squares approach to stability analysis of polynomial fuzzy systems. In: 2007 American Control Conference, pp 4071–4076. IEEE (2007) 27. Tanaka, K., Yoshida, H., Ohtake, H., Wang, H.O.: A sum-of-squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems. IEEE Trans. Fuzzy Syst. 17(4), 911–922 (2008) 28. Taniguchi, T., Tanaka, K., Ohtake, H., Wang, H.O.: Model construction, rule reduction, and robust compensation for generalized form of Takagi-Sugeno fuzzy systems. IEEE Trans. Fuzzy Syst. 9(4), 525–538 (2001) 29. Yilmaz, U., Kircay, A., Borekci, S.: PV system fuzzy logic MPPT method and PI control as a charge controller. Renew. Sustain. Energy Rev. 81, 994–1001 (2018)

Nonlinear Optimal Control for Residential Microgrids with Wind Generators, Fuel Cells and PVs Gerasimos Rigatos, Pierluigi Siano, Gennaro Cuccurullo, and Masoud Abbaszadeh

Abstract The progressive shift towards green energy and the implementation of policies for limiting power generation that causes harmful gas emissions has fostered the development of hybrid microgrids that consist of renewable energy sources of both the AC and the DC type. To make such microgrids exploitable the related control, stabilization and synchronization problems have to be treated. The chapter proposes a nonlinear optimal control method for a hybrid residential microgrid which comprises (i) a wind micro-turbine connected to a synchronous reluctance generator and a link to a DC bus through an AC/DC converter, (ii) a PEM fuel cells power unit which is connected to the same DC bus through a DC/DC converter, and (iii) a photovoltaic power unit which is connected to the DC bus again through a DC/DC converter. To implement the proposed nonlinear optimal control method, the dynamic model of the residential microgrid undergoes linearization through firstorder Taylor-series expansion around a temporary operating point which is updated at each sampling instance. At a next stage, a stabilizing H.∞ feedback controller is designed for the linearized state-space model of the system. To select the feedback gains of this controller an algebraic Riccati equation has to be solved at each timestep of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis. Finally, the differential flatness properties of the residential microgrid are proven, thus confirming the system’s controllability G. Rigatos (B) Unit of Industrial Automation, Industrial Systems Institute, Rion Patras, Greece e-mail: [email protected] P. Siano Department of Management and Innovation Systems, University of Salerno, 84084 Fisciano, Italy e-mail: [email protected] G. Cuccurullo Department of Industrial Engineering, University of Salerno, 84084 Fisciano, Italy e-mail: [email protected] M. Abbaszadeh Department of ECS Engineering, Rensselaer Polytechnic Institute, New York 12605, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Derbel et al. (eds.), Advances in Robust Control and Applications, Studies in Systems, Decision and Control 474, https://doi.org/10.1007/978-981-99-3463-8_4

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and allowing to solve the setpoints definition problem for this system. The nonlinear optimal control method achieves fast and accurate tracking of reference setpoints by the state variables of the residential microgrid, under moderate variations of the control inputs. Keywords Residential microgrids · Hybrid microgrids · Wind power generators · AC/DC converter · PEM fuel cells · DC/DC converter · Photovoltaics · Nonlinear H.∞ control · Taylor series expansion · Jacobian matrices · Riccati equation · Global stability · Differential flatness properties

1 Introduction The chapter treats the problem of nonlinear optimal control for hybrid residential microgrids. The objective is to synchronize and stabilize the functioning of distributed AC and DC power sources that constitute a residential microgrid [9, 16, 21, 27]. Actually, it is aimed to assure quality of the produced electric power, uninterrupted power supply and complete coverage of the residence’s power needs under variable operating and environmental conditions [7, 14, 23, 25]. The considered residential microgrid comprises heterogeneous power sources, such as (i) A wind micro-turbine which provides rotational motion to a synchronous reluctance generator, while the generator is connected to an AC/DC converter. Thus finally, the AC power of the generator is turned into DC power that is further distributed through a DC bus, (ii) A stack of PEM fuel cells which provide DC power to a DC/DC converter. In turn the converter is connected to the previously noted DC bus, (iii) photovoltaic arrays that are also connected to the aforementioned DC bus through another DC/DC converter. The aggregate DC voltage of the individual renewable power sources that constitute the residential microgrid can be finally turned into AC voltage with the intervention of a DC/AC inverter. The use of fuel cells in residential microgrids is a research topic that has attracted much interest during the last years [6, 8, 28]. The distributed residential microgrid exhibits nonlinear and multivariable dynamics [2, 11, 12]. As a result, the solution of the associated nonlinear control and estimation problem is a non-trivial task [3, 29, 32]. Actually, it is aimed to ensure the undistorted power supply from the microgrid despite changes in the operating conditions of the individual renewable power sources and despite weather variations (for instance changes in wind power, or solar energy intensity). To this end there is need to establish an orchestrated management of the functioning of the individual power sources in a manner that renewable energy units contribute more power at time intervals when weather conditions are favorable [4, 10, 15, 30]. On the other side, it is anticipated to get the fuel cells stack contribute more power at those periods when wind and solar power supply gets constrained [1, 13, 24, 31, 33]. To achieve these goals, a nonlinear optimal control method is developed [21]. First the dynamic model of the residential microgrid undergoes approximate linearization with the use of first-order Taylor series expansion and through the

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computation of the associated Jacobian matrices [5, 17, 18]. The linearization process takes place at each sampling instance, around a temporary operating point which is defined by the present value of the system’s state vector and by the last sampled value of the control inputs vector. The modelling inaccuracies which are due to truncation of higher-order terms from the Taylor series expansion or which are caused by exogenous perturbations are compensated by the robustness of the control method. For the approximately linearized model of the residential microgrid the related optimal (. H∞ ) control problem is solved. An. H∞ feedback controller is designed and a min-max differential game is treated, in which (i) the control inputs try to minimize a cost function that contains a quadratic term of the state vector’s tracking error, (ii) the model uncertainty and external disturbance terms try to maximize this cost function [21]. To select the . H∞ controller’s feedback gains an algebraic Riccati equation is repetitively solved at each time-step of the control method [19, 22]. The global stability properties of the control scheme are proven through Lyapunov analysis [21, 22]. Actually, it is demonstrated first that the control loop satisfies the . H∞ tracking performance criterion [20, 26]. At a second stage (and under moderate conditions) global asymptotic stability properties are proven. Furthermore, to implement state estimation-based control without the need to measure the entire state vector of the residential microgrid, the . H∞ Kalman Filter is introduced as a robust state estimator [21, 22]. Additionally, differential flatness properties are proven for the microgrid. These are an implicit proof of the system’s controllability and allow to compute through a well-defined procedure setpoints for all state variables of the microgrid [19, 21]. The present chapter provides one of the few existing solutions to the nonlinear optimal control problem of residential AC/DC microgrids which is of proven global stability while also remaining computationally efficient [22]. The proposed nonlinear optimal control method is novel comparing to past attempts for solving the optimal control problem for nonlinear dynamical systems. Unlike past approaches, in the new nonlinear optimal control method linearization is performed around a temporary operating point which is defined by the present value of the system’s state vector and by the last sampled value of the control inputs vector and not at points that belong to the desirable trajectory (setpoint). Besides the Riccati equation which is used for computing the feedback gains of the controller is new, and so is the global stability proof for this control method. Comparing to NMPC (Nonlinear Model Predictive Control) which is a popular approach for treating the optimal control problem in industry, the new nonlinear optimal (. H∞ ) control scheme is of proven global stability and the convergence of its iterative search for the optimum does depend on initial conditions and trials with multiple sets of controller parameters. It is also noteworthy that the nonlinear optimal control method is applicable to a wider class of dynamical systems than approaches based on the solution of State Dependent Riccati Equations (SDRE). The SDRE approaches can be applied only to dynamical systems which can be transformed to the Linear Parameter Varying (LPV) form. Besides, the nonlinear optimal control method performs better than nonlinear

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optimal control schemes which use approximation of the solution of the HamiltonJacobi-Bellman equation by Galerkin series expansions. The stability properties of the Galerkin series expansion -based optimal control approaches are still unproven. The chapter has a meaningful contribution because it provides one of the few algorithmically simple and computationally efficient solutions for the nonlinear optimal control problem of residential microgrids [21, 22]. The proposed nonlinear optimal control method exhibits specific advantages comparing to other nonlinear control schemes one can consider for the dynamics of microgrids. (i) its successful use is subject to less constraints when compared against other nonlinear control schemes (such as Lie algebra-based control, differential flatness theory-based control, Modelbased Predictive Control, Nonlinear Model-based Predictive Control, Sliding-mode control, Backstepping control, etc.), (ii) it achieves fast and accurate tracking of all reference setpoints for the residential microgrid under moderate variations of the control inputs, (iii) it minimizes the dispersion of energy in the implementation of control and in the functioning of the residential microgrid. The chapter’s contribution is also in proving differential flatness properties for the residential microgrid. The differential flatness properties confirm the system’s controllability and allow for solving the setpoints definition problem for this system.

2 Dynamic Model of the Hybrid Residential Microgrid 2.1 The Components of the Residential Microgrid The considered residential microgrid comprises (a) a wind microturbine-driven synchronous reluctance generator (SRG) which is connected to a DC bus through a three-phase AC/DC converter, (b) a stack of PEM fuel cells which is connected to the above-noted DC bus through a DC/DC converter and (c) a photovoltaics (PV) power unit which is again connected to the DC bus through another DC/DC converter [21, 22]. This is shown in Fig. 1. It is aimed to establish an orchestrated management of the functioning of the individual power sources in a manner that renewable energy units contribute more power at time intervals when weather conditions are favorable. Dynamic model of the wind microturbine-driven power unit By considering a synchronously rotating .dq reference frame, the dynamic model of the wind microturbine-driven Synchronous Reluctance Generator (SRG), which is a main component of the residential microgrid, is given by [21]:

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Fig. 1 Diagram of the residential microgrid which comprises: a a wind power generation unit consisting of a micro-turbine connected to a synchronous reluctance generator while the generator is connected in turn to an AC /DC converter, b a fuel-cells power unit which is connected to a DC converter, c a photovoltaic power unit which is connected again to DC/DC converter. The outputs of the converters are serially connected to the same DC bus

⎧ dθg ⎪ ⎪ = ωg ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ 3P(L d − L q ) dωg ⎪ ⎪ = i sd i sq − ⎨ dt 4J Lq di sd Rs ⎪ ⎪ ωg i sq + = − i sd + ⎪ ⎪ dt L L ⎪ d d ⎪ ⎪ ⎪ di sq Ld Rs ⎪ ⎪ ωg i sd + = − i sq − ⎩ dt Lq Lq

Bm β ωg − Tm J 2J 1 vsd Ld 1 vsq Ld

(1)

where . Rs is the stator’s resistance, . L d is the direct axis inductance, . L q is the quadrature axis inductance, . P is the number of poles of the machine, . Bm is a friction coefficient and .Tm is the mechanical torque. The blades of the wind turbine have variable pitch angle, and this is denoted by coefficient .β which modifies the effective mechanical torque which is finally applied by the wind to the turbine. The electromagnetic torque which is developed by the synchronous reluctance machine is given by

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Te = K (ψd i q − ψq i d ) = =

3P (i d L d i q − i q L q i d ) 22

3P (L d − L q )i d i q 4

(2)

The dynamic model of the AC/DC converter that connects the stator of the Synchronous Reluctance Generator to the microgrid is given by the following equation, after considering that the control inputs .η1 and .η2 are proportional to the stator currents of the SRG this is .η1 = k1 i sd and .η2 = k2 i sq [21]: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

di d = − R2 i d + L 2 ωdq i q + vd − 21 η1 Vdc dt di q L = −L 2 ωdq i d − R1 i q + vq − 21 η2 Vdc ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎩ C2 d Vdc2 = − 1 Vdc2 + 3 i d η1 + 3 i q η2 4 4 Rc dt L

(3)

Dynamic model of the PEM fuel cells power unit A nonlinear model of the PEM fuel cells system is presented [21]. Focusing on the cathode, the state vector of the model is defined as .x = [ p O2 , p N2 , ωcp , psm ]T , where . p O2 is the oxygen pressure at the cathode, . p N2 is the nitrogen pressure at the cathode, .ωcp is the compressor’s rotational speed (r/min), and . psm is the supply manifold pressure [21]. By applying the ideal gas law and by considering that the volume of the cathode is known one has ⎧ RT dp ⎪ ⎪ O2 = (W O2,in − W O2 ,out − W O2,r eact ) ⎨ dt M O2 Vca (4) RT dp N2 ⎪ ⎪ ⎩ = (W N2,in − W N2 ,out ) M N2 Vca dt where .V is the volume of the cathode, . R is the universal gas constant, and . M O2 , . M N2 are the mass concentrations (in mole) of oxygen and nitrogen. The incoming flow rates of oxygen and nitrogen are given by ⎧ .

W O2 ,in = x O2 Wca,in W N2 ,in = (1 − x O2 )Wca,in

(5)

where .x O2 is the oxygen mass fraction of the inlet air, .1 − x O2 is the nitrogen mass fraction of the inlet air, and .Wca,in is the mass flow rate entering the cathode which is given by 1 kca,in ( psm − pca ) (6) . Wca,in = 1 + ωatm

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where .ωatm is the humidity ratio ωatm =

.

.

φca psat (Tatm ) Mv Ma,ca,in patm − φca psat (Tatm )

(7)

Mv is the mass of the vapor in mole, . Ma,ca,in is the mass of the air in mole, .φca is the relative humidity in ambient conditions, . psat (Tatm ) is the saturation pressure in ambient temperature, . patm is the atmospheric pressure and .kca,in is the cathode inlet orifice constant. The outlet flow rates of oxygen and nitrogen .W O2 ,out and .W N2 ,out are calculated from the mass fraction of oxygen and nitrogen in the stack after reaction ⎧ ⎪ ⎪ ⎨ W O2 ,out = .

⎪ ⎪ ⎩ W N2 ,out =

M O2 p O2 M O2 p O2

M O2 p O2 Wca,out + M N2 p N2 + Mv psat M N2 p N2 Wca,out + M N2 p N2 + Mv psat

(8)

The flow rate at the cathode’s exit .Wca,out is calculated by the nozzle flow equation | )1 ( | C D A T pca patm T | | Wca,out = | RT pca [ .| γ ⎧ ( ) ( ) γ−1 ]) ( ) γ−1 | patm 2γ 2 patm γ | | if > 1− | pca γ+1 γ−1 pca

(9)

where .γ is the ratio of the specific heat capacities of the air, . pca = p O2 + p N2 + Psat . The mass flow rate of oxygen is expressed as .

W O2 ,r eact =

n Ist M O2 4F

(10)

where .n is the number of cells in the stack, . F is the Faraday number and . Ist is the stack current. The compressor’s turn speed is related to the associated mechanical torque dωcp 1 = . (τcm − τcp ) (11) dt Jcp where .τcm is the mechanical input torque, .τcp is the load torque [21] ⎧ ⎪ τ = η K v (v )k ω ⎪ cm cm v cp ⎪ ⎨ cm Rcm ] [ ( ) γ−1 . C p Tatm patm γ ⎪ ⎪ ⎪ − 1 Wcp ⎩ τcp = ω pca cp ηcp

(12)

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where .kt , . Rcm and .kv are motor constants, .ηcm is a coefficient that denotes the motor’s mechanical efficiency..C p is the specific heat capacity of air and.Wcp is the compressor mass flow rate. The dynamics of the air pressure in the supply manifold depend on the compressor flow into the supply manifold .Wcp = Aωcp , on the flow out of the supply manifold into the cathode .Wco,in and on the compressor flow temperature .Tcp [21] RTcp dpsm = . [Wcp − kca,in ( psm − pca )] (13) dt Ma Vsm where .Vsm is the supply manifold volume and .Tcp is the temperature of the air leaving the compressor ] [( ) γ−1 psm γ Tatm . Tcp = Tatm + −1 (14) ηcp patm The nonlinear state-space model of the PEM fuel-cells model is based om Eqs. (4), (11) and (13) [21] .

P˙O2 = c1 ( psm − PO2 − PN2 − c2 ) −

.

(15)

c3 PN2 Wco,out c4 PO2 + c5 PN2 + c6

(16)

P˙N2 = c8 ( psm − PO2 − PN2 − c2 ) − [( ω˙ cp = −c9 ωcp − c10

.

.

c3 PO2 Wco,out − c7 ζ c4 PO2 + c5 PN2 + c6

psm c11

)c12

] − 1 + c13 u

(17)

⎧ [( ) ]) psm c12 p˙ sm = c14 1 + c15 − 1 [Wcp − c16 ( psm − PO2 − PN2 − c2 )] (18) c11

where the coefficients .c1 , .c2 , .. . ., .c16 are constants. The control input .u depends the motor’s current. The control input .ζ is the stack current (which can be considered as an external perturbation to the model). The dynamic model of the DC/DC converter that connects the PEM fuel cells stack, to the microgrid is given by [21]: ⎧ d V dc1 1 1 ⎪ ⎪ = Vdc1 + I1 ⎨ dt R3 C 3 C3 . Vdc1 dI1 1 ⎪ ⎪ ⎩ =− + u dt L3 L3

(19)

About the fuel cells’ power unit and the associated DC/DC converter one has that . L 3 is the inductance at the circuit of the converter, .C3 is the capacitance at the circuit of the converter, . R3 is the resistance at the circuit of the converter, while .u = δ is the PWM control input of the DC/DC converter.

Nonlinear Optimal Control for Residential Microgrids with Wind Generators …

95

Dynamic model of the photovoltaics power unit The dynamic model of the DC/DC converter that connects DC Photovoltaic power unit, to the microgrid is given by [21]: ⎧ 1 1 d V dc2 ⎪ ⎪ Vdc2 + I2 = ⎨ dt R4 C 4 C4 . dI2 1 1 ⎪ ⎪ ⎩ u = − Vdc2 + dt L4 L4

.

(20)

About the photovoltaic power unit and the associated DC/DC converter one has that L 4 is the inductance at the circuit of the converter, .C4 is the capacitance at the circuit of the converter, . R4 is the resistance at the circuit of the converter, while .u = δ is the PWM control input of the DC/DC converter.

2.2 State-Space Description of the Hybrid Residential Microgrid As noted above, the dynamic model of the hybrid residential microgrid comprises: (a) a wind power generation unit consisting of a micro-turbine connected to a synchronous reluctance generator while the generator is connected in turn to a n AC /DC converter, (b) a fuel-cells power unit which is connected to a DC converter, (c) a photovoltaics power unit which is connected again to DC/DC converter. The outputs of the converters are serially connected to the same DC bus. By denoting the state variables .x1 = θ, .x2 = ω, .x3 = i d and .x4 = i q the dynamic model of the synchronous reluctance power generator is given by x˙1 = x2 3P(L d − L q ) β Bm x2 − Tm x˙2 = x3 x4 − 4J J 2J Lq Rs 1 x2 x4 + vs x˙3 = − x3 + Ld Ld Ld d Rs Ld 1 x˙4 = − x4 − x2 x3 + vs Lq Lq q Lq

(21)

By denoting the state variables of the AC/DC converter as .x5 = i d , .x6 = i q , .x7 = Vdc the state-space model of this converter is as follows

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R1 x7 x5 + ωdq x6 + vd − u 1c L1 2L 1 vq x7 R1 x6 + − u 2c x˙6 = −ωdq x5 − L1 L1 2L 1 3 3 1 x6 u 2c x7 + x5 u 1c + x˙7 = − Rc1 Cdc1 4Cdc1 4Cdc1 x˙5 = −

(22)

By denoting the state variables of PEM Fuel Cells as .x8 = PO2 , .x9 = PN2 , .x10 = ωcp and .x11 = p S M the state-space model of this power unit is c3 x8 Wco,out c4 x8 + c5 x9 + c6 c3 x9 Wco,out x˙9 = c8 (x11 − x8 − x9 − c2 ) − c x + c5 x9 + c6 [( )c12 4 ]8 x11 − 1 + c13 u FC x˙10 = −c9 x10 − c10 c11 ⎧ [( )c12 ]) x11 − 1 [Wcp − c16 (x11 − x8 − x9 − c2 )] (23) x˙11 = c14 1 + c15 c11 x˙8 = c1 (x11 − x8 − x9 − c2 ) −

By denoting the state variables of the DC/DC converter which is connected to the PEM fuel cells as .x12 = Vdc2 , .x13 = I2 the associated state-space model is 1 1 x12 + x13 R3 C 3 C3 1 1 = − x12 + u3 L3 L3

x˙12 = − x˙13

(24)

By denoting the state variables of the DC/DC converter which is connected to the photovoltaics power unit as .x14 = Vdc3 , .x15 = I3 the associated state-space model is 1 1 x14 + x15 R4 C 4 C4 1 1 = − x14 + u4 L4 L4

x˙14 = − x˙15

(25)

Moreover, by considering that the aggregate power which is provided to the microgrid by the wind-power unit, the PEM fuel cells power unit and the PV power unit is 2 . P = (Vdc1 + Vdc2 + Vdc3 ) /R L , the dynamics of the power at the load which is connected to the DC microgrid is given by 2 P˙ = (Vdc1 + Vdc2 + Vdc3 )(V˙dc1 + V˙dc2 + V˙dc3 )=⇒ RL 2 x˙16 = (x7 + x13 + x15 )(x˙7 + x˙13 + x˙15 ) RL

(26)

Nonlinear Optimal Control for Residential Microgrids with Wind Generators …

97

Next considering that .u 1c = k1 x3 , .u 2c = k2 x4 , .u 3 = k3 x8 , the aggregate state-space model of the residential hybrid power generation system is formulated.

2.3 A Concise Description of the PEM Fuel Cells’ Dynamics By proving that the PEM fuel-cells system is differentially flat it can be also assured that it can be transformed into an equivalent input-output linearized form [19, 21]. From Eq. (15), and after omitting the disturbance term (unknown stack current) one has c3 x8 Wco,out (27) .x ˙8 = c1 (x11 − x8 − x9 − c2 ) − c4 x8 + c5 x9 + c6 By differentiating with respect to time one gets x¨8 = c1 (x˙11 − x˙8 − x˙9 ) (c3 x˙8 Wco,out )(c4 x11 + c5 x9 + c6 ) − (c3 x8 Wca,out )(c4 x˙8 + c5 x˙9 ) − (c4 x8 + c5 x9 + c6 )2

(28)

By substituting in Eq. (28) the derivatives .x˙8 from Eq. (15), .x˙9 from Eq. (16) and .x˙4 from Eq. (18) one gets ( ⎧ [( )c12 ])) x4 −1 [Ax7 − c16 (x11 − x8 − x9 − c2 )] x¨8 = c1 c14 1 + c15 c11 c3 x8 Wca,out −c1 (x11 − x8 − x9 − c2 ) − (c4 x8 + c5 x9 + c6 ) c3 x9 Wca,out −c8 (x11 − x8 − x9 − c2 ) − − (c4 x8 + c5 x9 + c6 ) c3 c1 (x11 − x8 − x9 − c2 ) c3 x8 Wca ,out − Wca ,out − (c4 x8 + c5 x9 + c6 )2 c4 x8 + c5 x9 + c6 [ c3 x8 Wca ,out c3 x8 Wca,out c4 c1 (x11 − x8 − x9 − c) − + 2 (c4 x8 + c5 x9 + c6 ) c4 x8 + c5 x9 + c6 ] c3 x2 Wca ,out +c5 c8 (x11 − x8 − x9 − c2 ) − (29) c4 x8 + c5 x9 + c6 By differentiating the previous relation once more with respect to time one gets an input-output linearized description of the system’s dynamics is obtained in the form x (3) = f˜(x) + g(x)u ˜

. 8

where function . f˜(x) is given by:

(30)

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] ⎧ [( )c12 x4 f˜(x) = c1 c14 c15 − 1 x˙4 [Ax3 − c16 (x4 − x1 − x2 − c2 )] c11 [ ⎧( ) )] x4 −1 × +c14 1 + c15 c11 [ ( )) ] ⎧( )c12 x4 A −c9 x3 − c10 −1 − c16 (x˙4 − x˙1 − x˙2 ) c11 −c1 (x˙4 − x˙1 − x˙2 ) (c3 x˙1 Wca,out )(c4 x1 + c5 x2 + c6 ) − (c3 x1 Wca,out )(c4 x˙1 + c5 x˙2 ) − (c4 x1 + c5 x2 + c6 )2 −c8 (x˙4 − x˙1 − x˙2 ) ) (c3 x˙2 Wca,out )(c4 x1 + c5 x2 + c6 ) − (c3 x2 Wca,out )(c4 x˙1 + c5 x˙2 ) − (c x + c5 x2 + c6 )2 ⎧ 4 1 Wca,out − c3 (c1 (x˙4 − x˙1 − x˙2 ) − (c4 x1 + c5 x2 + c6 )2 ) (c2 x˙1 )Wca,out (c4 x1 + c5 x2 + c6 ) − (c3 x1 Wca,out )(c4 x˙1 + c5 x˙2 ) − (c4 x1 + c5 x2 + c6 )2 [ ( ) (c3 x˙1 Wca,out ) c3 x1 Wca,out + c4 c1 (x4 − x1 − x2 − c2 ) − c4 x1 + c5 x2 + c6 (c4 x1 + c5 x2 + c6 )2 )] ( c3 x2 Wca,out −c5 c8 (x4 − x1 − x2 − c2 ) − c4 x1 + c5 x2 + c6 [ (c3 x1 Wca,out ) + c4 (c1 (x˙4 − x˙1 − x˙2 ) (c4 x1 + c5 x2 + c6 )2 (c3 x˙1 Wca,out )(c4 x1 + c5 x2 + c6 ) − (c3 x1 Wca,out )(c4 x˙1 + c5 x˙2 ) − (c4 x1 + c5 x2 + c6 )2 ( +c5 c8 (x˙4 − x˙1 − x˙2 ) )] (c3 x˙2 Wca,out )(c4 x1 + c5 x2 + c6 ) − (c3 x2 Wca,out )(c4 x˙1 + c5 x˙2 ) (c4 x1 + c5 x2 + c6 )2 ] [ (c3 x1 Wca,out ) c3 x1 Wca,out − c4 c1 (x4 − x1 − x2 − c2 ) − (c4 x1 + c5 x2 + c6 )2 (c4 x1 + c5 x2 + c6 ) ( ) c3 x2 Wca,out +c5 c8 (x4 − x1 − x2 − c2 ) − (31) 2(c4 x˙1 + c5 x˙2 ) c4 x1 + c5 x2 + c6



Nonlinear Optimal Control for Residential Microgrids with Wind Generators …

99

and function .g(x) ˜ is given by ( .



g(x) ˜ = c1 c14 1 + c15

[(

x11 c11

)c12

]) ) − 1 Ac13

(32)

The above analysis demonstrates that the entire dynamics of the PEM fuel cells is included in the variations of state variable .x8 (oxygen pressure at the cathode), which is also the only differentially flat output for this power source. Consequently, the voltage and current magnitude at the output of the fuel cells is related with the values of state variable .x8 . By controlling .x8 one can also control the voltage and current outputs of the fuel cells.

2.4 Dynamic Model of the Hybrid Residential Microgrid in Matrix Form The aggregate state vector of the hybrid residential microgrid is defined as x = [x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 , x9 , x10 , x11 , x12 , x13 , x14 , x15 , x16 ]T x = [θ, ω, i sd , i sq , i d , i q , Vdc , PO2 , PN2 , ωcp , psm , Vdca , Ia , Vdcb , Ib , Pw ]T

(33)

According to the previous analysis, the state-space model of the hybrid residential microgrid comprises the following equations: x˙1 = x2 3P(L d − L q ) Bm Tm x3 x4 − u1 x˙2 = x2 − 4Jm J 2J

(34) (35)

x˙3 = −

Lq 1 Rs x3 + x2 x4 + u2 Lq Lq Ld

(36)

x˙4 = −

Rs Ld 1 x4 − x2 x3 + u3 Lq Lq Lq

(37)

x˙5 = −

1 x7 Rc1 x5 + ωdq x6 + vd − K 1 x3 L c1 L c1 2L c1

(38)

Rc1 1 x7 x6 + vq − K 2 x4 2L c1 L c1 L c1

(39)

x˙6 = −ωdq x5 −

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G. Rigatos et al.

x˙7 = −

1 3 3 x7 + x5 K 1 x3 + x6 K 2 x4 Rc1 Cdc1 4Cdc1 4Cdc1

(40)

x˙8 = c1 (x11 − x8 − x9 − c2 ) −

c3 x8 Wca,out c4 x8c5 x9 + c6

(41)

x˙9 = c8 (x11 − x8 − x9 − c2 ) −

c3 x9 Wca,out c4 x8c5 x9 + c6

(42)

] ) x11 c12 = −c9 x10 − c10 − 1 + c13 u 4 c11 ]) ⎧ [( )c12 x11 = c14 1 + c15 − 1 [Wcp − c16 (x11 − x8 − x9 − c2 )] c11 [(

x˙10 x˙11

(43)

(44)

x˙12 = −

1 1 x12 + x13 R3 C 3 C3

(45)

x˙13 = −

1 1 x12 + K 3 x8 L3 L3

(46)

x˙14 =

1 1 x14 + x15 R4 C 4 C4

(47)

1 1 x14 + u5 L4 L4

(48)

x˙15 = − x˙16

[ ] 1 1 3 3 = (x7 + x12 + x4 ) × − x7 + x5 K 1 x3 + x6 K 2 x4 RL Rc1 Cc1 4Cdc1 4Cdc1 ] [ ] [ x13 x15 x14 x12 + + + − (49) + − R3 C 3 C3 R4 C 4 C3

The state-space model of the system can be also written in the following matrix form (affine-in-the-input state-space form) .

x˙ = f (x) + g(x)u

where .x∈R16×1 , . f (x)∈R16×1 , .g(x)∈R16×5 , and .u∈R5×1 , or analytically

(50)

Nonlinear Optimal Control for Residential Microgrids with Wind Generators …



x2 3P(L d −L q ) Bm x 3 x4 − J x2 4Jm L − LRqs x3 + L qq x2 x4 Ld Rs − L q x4 − L q x2 x3

101



⎟ ⎞ ⎜ ⎟ ⎜ x˙1 ⎟ ⎜ ⎟ ⎜ ⎜ x˙2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ x˙3 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ R ⎟ c x 1 ⎜ x˙4 ⎟ ⎜ − L c1 x5 + ωdq x6 + L c vd − 2L7c K 1 x3 ⎟ ⎜ ⎟ ⎜ 1 1 1 ⎟ ⎜ x˙5 ⎟ ⎜ R c1 x7 1 ⎟ ⎜ ⎟ ⎜ −ωdq x5 − L c x6 + L c vq − 2L c K 2 x4 ⎟ ⎜ x˙6 ⎟ ⎜ 1 1 1 ⎟ 3 3 1 ⎜ ⎟ ⎜ ⎟ x + x K x + x K x − ⎜ x˙7 ⎟ ⎜ Rc1 Cdc1 7 4Cdc1 5 1 3 4Cdc1 6 2 4 ⎟ ⎜ ⎟ ⎜ c3 x8 Wca,out ⎟ ⎜ x˙8 ⎟ = ⎜ c1 (x11 − x8 − x9 − c2 ) − c4 x8 c x9 +c6 ⎟ ⎜ ⎟ ⎜ 5 ⎟ ⎜x˙10 ⎟ ⎜ 9 Wca,out ⎟ c8 (x11 − x8 − x9 − c2 ) − c4cx38x+c ⎜ ⎟ ⎜ x +c ⎟ 5 9 6 ⎜x˙11 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ −c9 x10 − c10 [( cx1111 )c12 − 1] ⎟ ⎜x˙12 ⎟ ⎜ ⎜ ⎟ ⎜c14 {1 + c15 [( xc11 )c12 − 1]}[Wcp − c16 (x11 − x8 − x9 − c2 )]⎟ ⎟ 11 ⎜x˙13 ⎟ ⎜ ⎟ 1 1 ⎜ ⎟ ⎜ − x + x ⎟ 12 13 ⎜x˙14 ⎟ ⎜ R3 C 3 C3 ⎟ 1 1 ⎜ ⎟ ⎜ ⎟ x + K x − ⎝x˙15 ⎠ ⎜ L 3 12 L3 3 8 ⎟ 1 1 ⎟ ⎜ x + C4 x15 x˙16 R4 C4 14 ⎟ ⎜ 1 ⎠ ⎝ − x14 ⎛

L4



0 ⎜ Tm ⎜ 2J ⎜0 ⎜ ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 +⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎝0 0

0 0 0 0 0 0 1 0 0 Ld 0 L1q 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

f 16 ⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ ⎟ 0⎟ ⎟ 0 ⎟ ⎛u ⎞ ⎟ 1 0 ⎟ ⎜u ⎟ ⎟ ⎜ 2⎟ 0 ⎟ ⎜u ⎟ ⎟ 3⎟ 0 ⎟⎜ ⎟ ⎝u 4 ⎠ 0⎟ u ⎟ 5 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 1 ⎠ L4 0

(51)

where: f 16

[ ] 1 3 3 1 (x7 + x12 + x4 )× − x7 + x5 K 1 x3 + x6 K 2 x4 = RL Rc C c 4Cdc1 4Cdc1 ] [ 1 1 ] [ x13 x15 x14 x12 + + (52) + − + − R3 C 3 C3 R4 C 4 C3

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3 Approximate Linearization of the Dynamics of the Residential Microgrid 3.1 The Linearized State-Space Model of the Microgrid The dynamic model of the hybrid residential microgrid undergoes approximate linearization around the temporary operating point.(x ∗ , u ∗ ) where.x ∗ is the present value of the system’s state vector and .u ∗ is the last sampled value of the control inputs vector [21, 22]. The linearization process is repeated at each sampling instance and is based on first-order Taylor series expansion and the computation of the associated Jacobian matrices. The initial nonlinear state-space description of the system being in the affine-in-the-input nonlinear state-space form .

x˙ = f (x) + g(x)u

(53)

.

x˙ = Ax + Bu + d˜

(54)

after linearization is written as

where . A and . B are the Jacobian matrices of the linearization process which are given by . A = ∇ x [ f (x) + g(x)u] |(x ∗ ,u ∗ ) ⇒A = ∇ x f (x) |(x ∗ ,u ∗ ) (55) .

B = ∇u [ f (x) + g(x)u] |(x ∗ ,u ∗ ) ⇒B = g(x) |(x ∗ ,u ∗ )

(56)

while .d˜ is the cumulative disturbance term which is due to: (i) model inaccuracies, (ii) external perturbations, (iii) sensors noise of any distribution. This linearization approach which has been followed for implementing the nonlinear optimal control scheme results into a quite accurate model of the system’s dynamics. Consider for instance the following affine-in-the-input state-space model x˙ = f (x) + g(x)u⇒ = [ f (x ∗ ) + ∇x f (x) |x ∗ (x − x ∗ )] + [g(x ∗ ) + ∇x g(x) |x ∗ (x − x ∗ )]u ∗ +g(x ∗ )u ∗ + g(x ∗ )(u − u ∗ ) + d˜1 ⇒ = [∇x f (x) |x ∗ +∇x g(x) |x ∗ u ∗ ]x + g(x ∗ )u − [∇x f (x) |x ∗ +∇x g(x) |x ∗ u ∗ ]x ∗ + f (x ∗ ) + g(x ∗ )u ∗ + d˜1 (57) where .d˜1 is the modelling error due to truncation of higher order terms in the Taylor series expansion of . f (x) and .g(x). Next, by defining A = [∇x f (x) |x ∗ +∇x g(x) |x ∗ u ∗ ] B = g(x ∗ )

Nonlinear Optimal Control for Residential Microgrids with Wind Generators …

one obtains .

103

x˙ = Ax + Bu − Ax ∗ + f (x ∗ ) + g(x ∗ )u ∗ + d˜1

(58)

Moreover by denoting d˜ = −Ax ∗ + f (x ∗ ) + g(x ∗ )u ∗ + d˜1 about the cumulative modelling error term in the Taylor series expansion procedure one has .x ˙ = Ax + Bu + d˜ (59) which is the approximately linearized model of the dynamics of the system of Eq. (54). The term . f (x ∗ ) + g(x ∗ )u ∗ is the derivative of the state vector at .(x ∗ , u ∗ ) which is almost annihilated by .−Ax ∗ .

3.2 Computation of Jacobian Matrices Next, the elements of the system’s Jacobian matrices are computed. f 1 (x) f 1 (x) f 1 (x) = 0,. ∂ ∂x = 1,. ∂ ∂x = First row of the Jacobian matrix. A = ∇x f (x) |(x ∗ ,u ∗ ) :. ∂ ∂x 1 2 3

f 1 (x) f 1 (x) f 1 (x) f 1 (x) f 1 (x) f 1 (x) f 1 (x) = 0, . ∂ ∂x = 0, . ∂ ∂x = 0, . ∂ ∂x = 0, . ∂ ∂x = 0, . ∂ ∂x = 0, . ∂∂x = 0, 0, . ∂ ∂x 4 5 6 7 8 9 10 .

∂ f 1 (x) ∂x11

f 1 (x) f 1 (x) f 1 (x) f 1 (x) f 1 (x) = 0, . ∂∂x = 0, . ∂∂x = 0, . ∂∂x = 0, . ∂∂x = 0 and . ∂∂x = 0. 12 13 14 15 16

f 2 (x) f 2 (x) Second row of the Jacobian matrix . A = ∇x f (x) |(x ∗ ,u ∗ ) : . ∂ ∂x = 0, . ∂ ∂x = − BJm , 1 2

∂ f 2 (x) ∂x3 ∂ f 2 (x) . ∂x8 ∂ f 2 (x) . ∂x15

.

= = =

3P(L d −L q ) 3P(L d −L q ) f 2 (x) f 2 (x) f 2 (x) x4 , . ∂ ∂x = x3 , . ∂ ∂x = 0, . ∂ ∂x = 0, 4J 4J 4 5 6 ∂ f 2 (x) ∂ f 2 (x) ∂ f 2 (x) ∂ f 2 (x) ∂ f 2 (x) 0, . ∂x9 = 0, . ∂x10 = 0, . ∂x11 = 0, . ∂x12 = 0, . ∂x13 = 0, f 2 (x) 0 and . ∂∂x = 0. 16

∂ f 2 (x) ∂x7 ∂ f 2 (x) . ∂x14

.

= 0, = 0,

f 3 (x) f 3 (x) Third row of the Jacobian matrix . A = ∇x f (x) |(x ∗ ,u ∗ ) : . ∂ ∂x = 0, . ∂ ∂x = 1 2

Lq x , Ld 4 R L ∂ f 3 (x) ∂ f 3 (x) ∂ f 3 (x) ∂ f 3 (x) ∂ f 3 (x) ∂ f 3 (x) f q . = − L d , . ∂x4 = L d x2 , . ∂x5 = 0, . ∂x6 = 0, . ∂x7 = 0, . ∂x8 = 0, ∂x3 f 3 (x) f 3 (x) f 3 (x) f 3 (x) f 3 (x) f 3 (x) ∂ f 3 (x) = 0, . ∂∂x = 0, . ∂∂x = 0, . ∂∂x = 0, . ∂∂x = 0, . ∂∂x = 0, . ∂∂x =0 . ∂x9 10 11 12 13 14 15 ∂ f 3 (x) and . ∂x16 = 0. f 4 (x) f 4 (x) Fourth row of the Jacobian matrix . A = ∇x f (x) |(x ∗ ,u ∗ ) : . ∂ ∂x = 0, . ∂ ∂x = − LL qd x3 , 1 2

∂ f 4 (x) ∂x3 ∂ f 4 (x) . ∂x9 ∂ f 4 (x) . ∂x16

.

∂ f 4 (x) f 4 (x) = − LRqs , . ∂ ∂x ∂x4 5 f 4 (x) ∂ f 4 (x) ∂ f 4 (x) 0, . ∂∂x = 0, . = 0, . ∂x11 ∂x12 10

= − LL qd x2 , =

.

= 0.

∂ f 4 (x) f 4 (x) f 4 (x) = 0, . ∂ ∂x = 0, . ∂ ∂x = 0, ∂x6 7 8 ∂ f 4 (x) ∂ f 4 (x) ∂ f 4 (x) 0, . ∂x13 = 0, . ∂x14 = 0, . ∂x15 = 0 and

= 0, =

.

Fifth row of the Jacobian matrix . A = ∇x f (x) |(x ∗ ,u ∗ ) :

∂ f 5 (x) . ∂x3

=

f 5 (x) x7 − 2L k1 , . ∂ ∂x 1 4

= 0,

∂ f 5 (x) . ∂x5

=

f 5 (x) − LR11 , . ∂ ∂x 6

= ωdq ,

∂ f 5 (x) ∂x1 ∂ f 5 (x) . ∂x7

.

= 0, = 0,

∂ f 5 (x) ∂x2 ∂ f 5 (x) . ∂x8

.

= 0, = 0,

104 ∂ f 5 (x) ∂x9 ∂ f 5 (x) . ∂x16 .

G. Rigatos et al. f 5 (x) f 5 (x) f 5 (x) f 5 (x) f 5 (x) f 5 (x) = 0, . ∂∂x = 0, . ∂∂x = 0, . ∂∂x = 0, . ∂∂x = 0, . ∂∂x = 0, . ∂∂x = 0 and 10 11 12 13 14 15

= 0.

∂ f 6 (x) f 6 (x) = 0, . ∂ ∂x = 0, ∂x1 2 ∂ f 6 (x) R1 ∂ f 6 (x) − L 1 , . ∂x7 = 0, . ∂x8 = 0, f 6 (x) f 6 (x) 0, . ∂∂x = 0, . ∂∂x = 0 and 14 15

Sixth row of the Jacobian matrix . A = ∇x f (x) |(x ∗ ,u ∗ ) :

∂ f 6 (x) ∂x3 ∂ f 6 (x) . ∂x9 ∂ f 6 (x) . ∂x16 .

f 6 (x) f 6 (x) f 6 (x) x7 = 0, . ∂ ∂x = − 2L k2 , . ∂ ∂x = −ωdq , . ∂ ∂x = 4 1 5 6

f 6 (x) f 6 (x) f 6 (x) f 6 (x) = 0, . ∂∂x = 0, . ∂∂x = 0, . ∂∂x = 0, . ∂∂x = 10 11 12 13

= 0.

.

f 7 (x) f 7 (x) Seventh row of the Jacobian matrix . A = ∇x f (x) |(x ∗ ,u ∗ ) : . ∂ ∂x = 0, . ∂ ∂x = 0, 1 2

∂ f 7 (x) f 7 (x) f 7 (x) f 7 (x) f 7 (x) 1 2 1 2 = 4C3kdc1 x5 , . ∂ ∂x = 4C3kdc1 x6 , . ∂ ∂x = 4C3kdc1 x3 , . ∂ ∂x = 4C3kdc1 x4 , . ∂ ∂x ∂x3 4 5 6 7 f 7 (x) ∂ f 7 (x) ∂ f 7 (x) ∂ f 7 (x) ∂ f 7 (x) ∂ f 7 (x) = 0, . = 0, . = 0, . = 0, . = 0, . = − Rc11Cdc1 , . ∂ ∂x ∂x9 ∂x10 ∂x11 ∂x12 ∂x13 8 ∂ f 7 (x) ∂ f 7 (x) ∂ f 7 (x) . = 0, . = 0 and . = 0. ∂x14 ∂x15 ∂x16 .

∂ f 8 (x) = ∂x2 ∂ f 8 (x) ∂ f 8 (x) ∂ f 8 (x) ∂ f 8 (x) ∂ f 8 (x) ∂ f 8 (x) . = 0, . ∂x4 = 0, . ∂x5 = 0, . ∂x6 = 0, . ∂x7 = 0, . ∂x8 = −c1 ∂x3 ∂ f 8 (x) c3 Wca,out (c4 x8 + c5 x9 + c4 ) − c3 x8 Wca,out c4 /(c4 x8 + c5 x9 + c6 )2 , . ∂x9 c3 x8 Wca,out c5 f 8 (x) ∂ f 8 (x) ∂ f 8 (x) ∂ f 8 (x) ∂ f 8 (x) = 0, . = c , . = 0, . = 0, . = −c1 + (c4 x8 +c5 x9 +c6 )2 , . ∂∂x 1 ∂x11 ∂x12 ∂x13 ∂x14 10 ∂ f 8 (x) ∂ f 8 (x) . = 0 and . ∂x16 = 0. ∂x15

Eighth row of the Jacobian matrix . A = ∇x f (x) |(x ∗ ,u ∗ ) :

Ninth row of the Jacobian matrix . A = ∇x f (x) |(x ∗ ,u ∗ ) :

.

.

∂ f 8 (x) ∂x1

∂ f 9 (x) ∂x1

= 0,

.

∂ f 9 (x) = ∂x2 ∂ f 9 (x) . = −c8 ∂x8

= 0,

.

∂ f 9 (x) f 9 (x) f 9 (x) f 9 (x) f 9 (x) . = 0, . ∂ ∂x = 0, . ∂ ∂x = 0, . ∂ ∂x = 0, . ∂ ∂x = 0, ∂x3 4 5 6 7 c3 x8 Wca,out ∂ f 9 (x) , . ∂x9 = −c8 − (c3 Wca,out (c4 x8 + c5 x9 + c4 )− .c3 x9 Wca,out c5 ) (c4 x8 +c5 x9 +c6 )2 f 9 (x) f 9 (x) f 9 (x) f 9 (x) f 9 (x) = 0, . ∂∂x = 0, . ∂∂x = 0, . ∂∂x = 0, . ∂∂x = 1/(c4 x8 + c5 x9 + c6 )2 , . ∂∂x 10 11 12 13 14 ∂ f 9 (x) ∂ f 9 (x) . = 0 and . ∂x16 = 0. ∂x15 ∂ f 10 (x) 10 (x) = 0, . ∂ f∂x ∂x1 2 10 (x) 10 (x) 0, . ∂ f∂x = 0, . ∂ f∂x 8 9 ∂ f 10 (x) ∂ f 10 (x) . = 0, . ∂x13 ∂x14

Tenth row of the Jacobian matrix . A = ∇x f (x) |(x ∗ ,u ∗ ) :

∂ f 10 (x) ∂x3 ∂ f 10 (x) . ∂x10 ∂ f 10 (x) . ∂x15 .

10 (x) 10 (x) 10 (x) 10 (x) = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, . ∂ f∂x = 4 5 6 7

= =

f 10 (x) −c9 , . ∂ ∂x 11 f 10 (x) 0 and . ∂ ∂x 16

=

f 10 (x) −c10 c12 ( xc1111 c12 −1 ), . ∂ ∂x 12

= 0.

= 0,

.

= 0,

0, − = 0,

0, + × 0,

= 0, = 0, = 0,

11 (x) 11 (x) Eleventh row of the Jacobian matrix . A = ∇x f (x) |(x ∗ ,u ∗ ) : . ∂ f∂x = 0, . ∂ f∂x = 0, 1 2

∂ f 11 (x) 11 (x) 11 (x) 11 (x) = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, . ∂ f∂x = ∂x5 6 7 8 ∂ f 11 (x) x11 c12 11 (x) −c14 {c15 [( xc1111 ) − 1]Wcp x16 },. ∂ f∂x = c {1 − c [( ) − 1]W C }, . = 14 15 c11 cp 16 ∂x10 9 ∂ f 11 (x) x11 c12−1 1 x11 c12 0, . ∂x11 = c14 {c15 c12 ( c11 ) c11 [Wcp c16 (x11 − x8 − x9 − c2 )] + c15 c15 [( c11 ) − f 11 (x) f 11 (x) f 11 (x) f 11 (x) 1][Wcp x16 (x11 − x8 − x9 − c2 )]}, . ∂ ∂x = 0, . ∂ ∂x = 0, . ∂ ∂x = 0, . ∂ ∂x =0 12 13 14 15 ∂ f 11 (x) and . ∂x16 = 0.

.

∂ f 11 (x) ∂x3

= 0,

∂ f 11 (x) ∂x4 c12 .

= 0,

.

Nonlinear Optimal Control for Residential Microgrids with Wind Generators …

105

12 (x) 12 (x) Twelfth row of the Jacobian matrix . A = ∇x f (x) |(x ∗ ,u ∗ ) : . ∂ f∂x = 0, . ∂ f∂x = 0, 1 2

.

∂ f 12 (x) ∂x3

∂ f 12 (x) ∂x10 ∂ f 12 (x) . ∂x16 .

12 (x) 12 (x) 12 (x) 12 (x) 12 (x) 12 (x) = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, 4 5 6 7 8 9

= 0,

.

= 0.

∂ f 12 (x) ∂x11

= 0,

.

∂ f 12 (x) ∂x12 =1

R3 C 3

,

.

∂ f 12 (x) ∂x13

=

1 , C3

.

∂ f 12 (x) ∂x14

= 0,

.

∂ f 12 (x) ∂x15

= 0 and

13 (x) 13 (x) Thirteenth row of the Jacobian matrix . A = ∇x f (x) |(x ∗ ,u ∗ ) : . ∂ f∂x = 0, . ∂ f∂x = 0, 1 2

∂ f 13 (x) ∂x3 ∂ f 13 (x) . ∂x10 ∂ f 13 (x) . ∂x16 .

13 (x) 13 (x) 13 (x) 13 (x) 13 (x) 13 (x) = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, 4 5 6 7 8 9

= 0,

.

= 0.

∂ f 13 (x) ∂x11

= 0,

.

∂ f 13 (x) ∂x12

= − L13 ,

.

∂ f 13 (x) ∂x13

= 0,

.

∂ f 13 (x) ∂x14

= 0,

.

∂ f 13 (x) ∂x15

= 0 and

14 (x) 14 (x) Fourteenth row of the Jacobian matrix . A = ∇x f (x) |(x ∗ ,u ∗ ) : . ∂ f∂x = 0, . ∂ f∂x = 0, 1 2

∂ f 14 (x) ∂x3 ∂ f 14 (x) . ∂x10 ∂ f 14 (x) . ∂x16 .

14 (x) 14 (x) 14 (x) 14 (x) 14 (x) 14 (x) = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, 4 5 6 7 8 9

= 0,

.

= 0.

∂ f 14 (x) ∂x11

= 0,

.

∂ f 14 (x) ∂x12

= 0,

.

∂ f 14 (x) ∂x13

= 0,

.

∂ f 14 (x) ∂x14

=

f 14 (x) 1 , . ∂ ∂x R4 C 4 15

=

1 C4

and

15 (x) 15 (x) Fifteenth row of the Jacobian matrix . A = ∇x f (x) |(x ∗ ,u ∗ ) : . ∂ f∂x = 0, . ∂ f∂x = 0, 1 2

∂ f 15 (x) ∂x3 ∂ f 15 (x) . ∂x10 ∂ f 15 (x) . ∂x16 .

15 (x) 15 (x) 15 (x) 15 (x) 15 (x) 15 (x) = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, . ∂ f∂x = 0, 4 5 6 7 8 9

= 0, = 0.

.

∂ f 15 (x) ∂x11

= 0,

.

∂ f 15 (x) ∂x12

= 0,

.

∂ f 15 (x) ∂x13

= 0,

.

∂ f 15 (x) ∂x14

= − L14 ,

.

∂ f 15 (x) ∂x15

= 0 and

16 (x) 16 (x) Sixteenth row of the Jacobian matrix . A = ∇x f (x) |(x ∗ ,u ∗ ) : . ∂ f∂x = 0, . ∂ f∂x = 0, 1 2

∂ f 16 (x) ∂ f 16 (x) 1 2 = R1L (x7 + x12 + x14 )( 4C3kdc1 x5 ), . = R1L (x7 + x12 + x14 )( 4C3kdc1 x6 ), ∂x3 ∂x4 ∂ f 16 (x) ∂ f 16 (x) 3k1 3k2 1 1 . = R L (x7 + x12 + x14 )( 4Cdc1 x3 ), . = R L (x7 + x12 + x14 )( 4Cdc1 x4 ), ∂x5 ∂x6 ∂ f 16 (x) 3k1 3k2 1 1 . = R L {[− Rc1 Cdc1 x7 + 4Cdc1 x5 x3 + 4Cdc1 x6 x4 ] + [− Rx312C3 + xC133 ] + [− Rx414C4 + ∂x7 f 16 (x) f 16 (x) x15 16 (x) 16 (x) ]} + R1L (x7 + x12 + x14 ) Rc11Cdc1 , . ∂ f∂x = 0, . ∂ f∂x = 0, . ∂ ∂x = 0, . ∂ ∂x = 0, C4 8 9 10 11 ∂ f 16 (x) 3k1 3k2 x12 x13 x14 1 1 . = R L {[− Rc1 Cdc1 x7 + 4Cdc1 x5 x3 + 4Cdc1 x6 )x4 ] + [− R3 C3 + C3 ] + [− R4 C4 + ∂x12 f 16 (x) f 16 (x) x15 ]} + R1L (x7 + x12 + x14 )(− R31C3 ), . ∂ ∂x = R1L (x7 + x12 + x14 ) C13 , . ∂ ∂x = C4 13 14 3k1 3k2 x12 x13 x14 x15 1 1 {[− Rc1 Cdc1 x7 + 4Cdc1 x5 x3 + 4Cdc1 x6 )x4 ] + [− R3 C3 + C3 ] + [− R4 C4 + C4 ]} + RL f 16 (x) f 16 (x) = R1L (x7 + x12 + x14 ) C14 and . ∂ ∂x = 0. +(x7 + x12 + x14 )−( R41C4 ), . ∂ ∂x 15 16 .

It is noted that by using the concise formulation of the fuel cells dynamics in Eq. (30), the associated part of the state-space model of the residential microgrid (rows 8 to 12) is simplified and the complexity of the computation of the elements of rows 8 to 11 of the Jacobian matrix of the system is also alleviated.

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3.3 Stabilizing Feedback Control After linearization around its current operating point, the hybrid residential microgrid is written in the form [21] .x ˙ = Ax + Bu + d1 (60) Parameter .d1 stands for the linearization error in the hybrid residential microgrid appearing above in Eq. (60). The reference setpoints for the hybrid residential microd ]. Tracking of this trajectory is grid’s state vector are denoted by .xd = [x1d , . . . , x16 achieved after applying the control input .u d . At every time instant the control input .u d is assumed to differ from the control input .u appearing in Eq. (53) by an amount equal to .Δu, that is .u d = u + Δu x˙ = Axd + Bu d + d2

(61)

. d

The dynamics of the controlled system described in Eq. (60) can be also written as .

x˙ = Ax + Bu + Bu d − Bu d + d1

(62)

and by denoting .d3 = −Bu d + d1 as an aggregate disturbance term one obtains .

x˙ = Ax + Bu + Bu d + d3

(63)

By subtracting Eq. (61) from Eq. (63) one has .

x˙ − x˙d = A(x − xd ) + Bu + d3 − d2

(64)

By denoting the tracking error as .e = x − xd and the aggregate disturbance term as d˜ = d3 − d2 , the tracking error dynamics becomes

.

e˙ = Ae + Bu + L d˜

(65)

.

where . L is the disturbance inputs gain matrix. For the approximately linearized model of the system a stabilizing feedback controller is developed. The controller has the form .u(t) = −K e(t) (66) with . K = r1 B T P where . P is a positive definite symmetric matrix which is obtained from the solution of the Riccati equation [21, 22] ( .

AT P + P A + Q − P

2 1 B BT − 2 L LT r ρ

) P=0

(67)

Nonlinear Optimal Control for Residential Microgrids with Wind Generators …

107

Fig. 2 Diagram of the nonlinear optimal (H-infinity) control loop for the hybrid residential microgrid

where . Q is a positive semi-definite symmetric matrix. The diagram of the considered control loop is depicted in Fig. 2. It is also mentioned that the solution of the . H∞ feedback control problem for the hybrid residential microgrid and the computation of the worst case disturbance that this controller can sustain, comes from superposition of Bellman’s optimality principle when considering that the micrigrid is affected by two separate inputs (i) the ˜ Solving the optimal control control input .u (ii) the cumulative disturbance input .d(t). problem for .u that is for the minimum variation (optimal) control input that achieves elimination of the state vector’s tracking error gives .u = − r1 B T Pe. Equivalently, ˜ that is for the worst case disturbance that solving the optimal control problem for .d, the control loop can sustain gives .d˜ = ρ12 L T Pe.

4 Lyapunov Stability Analysis 4.1 Stability Proof Through Lyapunov stability analysis it will be shown that the proposed nonlinear control scheme assures . H∞ tracking performance for the hybrid residential microgrid, and that in case of bounded disturbance terms asymptotic convergence to the reference setpoints is achieved. The tracking error dynamics for the microgrid is written in the form [21, 22]

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G. Rigatos et al.

e˙ = Ae + Bu + L d˜

.

(68)

where in the residential microgrid’s case . L∈R16×16 is the disturbance inputs gain matrix. Variable .d˜ denotes model uncertainties and external disturbances of the microgrid’s model. The following Lyapunov equation is considered .

V =

1 T e Pe 2

(69)

where .e = x − xd is the tracking error. By differentiating with respect to time one obtains 1 1 V˙ = e˙ T Pe + e T P e˙ 2 2 1 ˜ ˜ T Pe + 1 e T P[ Ae + Bu + L d] (70) = [ Ae + Bu + L d] 2 2 1 1 ˜ = [e T A T + u T B T + d˜ T L T ]Pe + e T P[ Ae + Bu + L d] (71) 2 2 1 1 1 1 1 1 = e T A T Pe + u T B T Pe + d˜ T L T Pe + e T P Ae + e T P Bu + e T P L d˜ 2 2 2 2 2 2 (72) The previous equation is rewritten as .

1 1 1 1 1 ˜ V˙ = e T (A T P + P A)e + ( u T B T Pe + e T P Bu) + ( d˜ T L T Pe + e T P L d) 2 2 2 2 2 (73)

Assumption For given positive definite matrix . Q and coefficients .r and .ρ there exists a positive definite matrix . P, which is the solution of the following matrix equation ( ) 2 1 T T T . A P + P A = −Q + P (74) BB − 2 LL P r ρ Moreover, the following feedback control law is applied to the system 1 u = − B T Pe r

.

(75)

By substituting Eqs. (74) and (75) one obtains 1 2 1 1 V˙ = e T [−Q + P ( B B T − 2 L L T )P]e + e T P B(− B T Pe) + e T P L d˜ 2 r ρ r 1 1 1 1 = − e T Qe + ( e T P B B T Pe − 2 e T P L L T Pe) − e T P B B T Pe + e T P L d˜ 2 r 2ρ r

(76) (77)

Nonlinear Optimal Control for Residential Microgrids with Wind Generators …

109

which after intermediate operations gives 1 1 V˙ = − e T Qe − 2 e T P L L T Pe + e T P L d˜ 2 2ρ

(78)

1 1 1 1 V˙ = − e T Qe − 2 e T P L L T Pe + e T P L d˜ + d˜ T L T Pe 2 2ρ 2 2

(79)

.

or, equivalently .

Lemma The following inequality holds [21, 22] .

1 T ˜ 1˜ T 1 1 e L d + d L Pe − 2 e T P L L T Pe≤ ρ2 d˜ T d˜ 2 2 2ρ 2

(80)

Proof The binomial .(ρα − ρ1 b)2 is considered. Expanding the left part of the above inequality one gets successively 1 2 b − 2ab ≥ 0 ρ2 1 1 2 2 ρ a + 2 b2 − ab ≥ 0 2 2ρ 1 2 1 2 2 ab − 2 b ≤ ρ a 2ρ 2 1 1 1 1 ab + ab − 2 b2 ≤ ρ2 a 2 2ρ 2 2 2 ρ2 a 2 +

(81)

The following substitutions are carried out: .a = d˜ and .b = e T P L and the previous relation becomes .

1 ˜T T 1 1 1 d L Pe + e T P L d˜ − 2 e T P L L T Pe≤ ρ2 d˜ T d˜ 2 2 2ρ 2

(82)

Equation (82) is substituted in Eq. (79) and the inequality is enforced, thus giving [21, 22] ˙ ≤ − 1 e T Qe + 1 ρ2 d˜ T d˜ .V (83) 2 2 Equation (83) shows that the . H∞ tracking performance criterion is satisfied. The integration of .V˙ from .0 to .T gives {

T 0

1 V˙ (t )dt≤ − 2

{

T 0

1 ||e||2Q dt + ρ2 2

{ 0

T

||d˜ ||2 dt

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G. Rigatos et al.

Then:

{ 2V (T ) +

T

.

0

{ ||e||2Q dt≤2V (0) + ρ2

T

˜ 2 dt ||d||

(84)

0

Moreover, if there exists a positive constant . Md > 0 such that {



˜ 2 dt ≤ Md ||d||

(85)

||e||2Q dt ≤ 2V (0) + ρ2 Md

(86)

.

0

then one gets

{



.

0

{∞ Thus, the integral . 0 ||e||2Q dt is bounded. Moreover, .V (T ) is bounded and from the definition of the Lyapunov function .V in Eq. (69) it becomes clear that .e(t) will be also bounded since .e(t) ∈ Ωe = {e|e T Pe≤2V (0) + ρ2 Md }. According to the above and with the use of Barbalat’s Lemma one obtains .lim t→∞ e(t) = 0. After following the stages of the stability proof one arrives at Eq. (83) which shows that the . H∞ tracking performance criterion holds. By selecting the attenuation ˜ 2 one coefficient .ρ to be sufficiently small and in particular to satisfy .ρ2 < ||e||2Q /||d|| has that the first derivative of the Lyapunov function is upper bounded by 0. This condition holds at each sampling instance and consequently global stability for the control loop can be concluded.

4.2 Robust State Estimation with the Use of the . H∞ Kalman Filter The control loop has to be implemented with the use of information provided by a small number of sensors and by processing only a small number of state variables. To reconstruct the missing information about the state vector of the hybrid residential microgrid it is proposed to use a filtering scheme and based on it to apply state estimation-based control [19, 22]. By denoting as . A(k), . B(k), .C(k) the discrete-time equivalents of matrices . A, . B, .C which constitute the linearized state-space model of Eq. (54), the recursion of the . H∞ Kalman Filter, for the model of the microgrid, can be formulated in terms of a measurement update and a time update part Measurement update: D(k) = [I − θW (k)P − (k) + C T (k)R(k)−1 C(k)P − (k)]−1 − T −1 . K (k) = P (k)D(k)C (k)R(k) − x(k) ˆ = xˆ (k) + K (k)[y(k) − C xˆ − (k)]

(87)

Nonlinear Optimal Control for Residential Microgrids with Wind Generators …

Time update: .

xˆ − (k + 1) = A(k)x(k) + B(k)u(k) P − (k + 1) = A(k)P − (k)D(k)A T (k) + Q(k)

111

(88)

where it is assumed that parameter .θ is sufficiently small to assure that the covari−1 ance matrix . P − (k) − θW (k) + C T (k)R(k)−1 C(k) will be positive definite. When .θ = 0 the . H∞ Kalman Filter becomes equivalent to the standard Kalman Filter. One can measure only a part of the state vector of the hybrid residential microgrid, and can estimate through filtering the rest of the state vector elements (.x2 , .x9 and .x10 ). Moreover, the proposed Kalman filtering method can be used for sensor fusion purposes.

5 Differential Flatness Properties of the Hybrid Residential Microgrid It will be proven that the hybrid residential microgrid is a differentially flat system with flat outputs vector: y = [y1 , y2 , y3 , y4 , y5 ]T ]T [ Cdc 2 3L c1 2 2 = x1 , (x5 + x6 ) + x , x6 , x12 , x14 4 2 7 ]T [ Cdc 2 3L c1 2 (i sd + i s2q ) + Vdc , i sq , Vdca , Vdcb = θ, 4 2

(89)

The proof of differential flatness for the hybrid residential microgrid is an implicit confirmation about the system’s controllability. It also confirms that the system can be transformed into an input-output linearized form. Moreover, it allows to solve the setpoints definition problem for this system. Actually, one defines freely setpoints for the flat outputs and based on them computes next setpoints for the state variables of the microgrids which are connected to the flat outputs through the previously given differential relations. From Eq. (34) it holds: x = x˙1 ⇒x2 = h 2 (y, y˙ )

. 2

(90)

Consequently, .x2 is a differential function of the flat outputs of the system. Next the flat output . y2 is considered. By differentiating it in time one gets y˙ =

. 2

3L c1 (x5 x˙5 + x6 x˙6 ) + Cdc1 x7 x˙7 2

(91)

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G. Rigatos et al.

Using that .vd is constant and .vq = 0 it holds that 3Rc1 2 3 1 2 x (x5 + x62 ) + x5 vd − 2 2 Rc1 7 3Rc1 2 3Rc1 2 3vd 1 2 y + =− x − x5 − x 2 5 2 3 2 Rc1 7

y˙2 = −

(92)

Besides, one has that 3L c1 2 Cdc1 2 (x5 + x62 ) + x 4 2 7 2 3L c1 2 x72 = y2 − (x + x62 ) Cdc1 2Cdc1 5 2 3L c1 2 = y2 − (x + y32 ) Cdc1 2Cdc1 5 y2 =

(93)

By substituting Eq. (93) into Eq. (92) one obtains y˙ = −

. 2

3Rc1 2 3Rc1 2 3vd 2 3L c1 y1 + (x 2 + y32 ) x5 − y3 + x5 − 2 2 2 Rc1 Cdc1 2Rc1 Cdc1 5

(94)

and after regrouping terms and intermediate operations one arrives at the relation [ ] ] 3vd 3Rc1 2 2 3L c1 3L c1 3Rc1 2 . y1 − − x5 + y˙2 + y + x − =0 2 2Rc1 Cdc1 5 2 2 3 Rc1 Cdc1 2Rc1 Cdc1 (95) By defining: 3Rc1 3L c1 a= − 2Rc1 Cdc1 2 [

b= c = y˙2 +

3vd 2

3Rc1 2 2 3L c1 y1 − y3 + 2 Rc1 Cdc1 2Rc1 Cdc1

one has a binomial of the form ax52 + bx5 + c = 0

.

(96)

which is solved for .x5 thus giving √ −b± b2 − 4ac .x5 = 2a

(97)

Nonlinear Optimal Control for Residential Microgrids with Wind Generators …

113

By setting .x5 to be equal to the largest of the roots of the binomial one has that x = h 5 (y, y˙ , y¨ )

(98)

. 5

which signifies that state variable .x5 is a differential function of the system’s flat outputs. Besides, from Eq. (93) 3L c1 2 2 y2 − (x + y32 ) Cdc1 2Cdc1 5 / 2 3L c1 2 x7 = ± y2 − (x + y32 ) = h 7 (y, y˙ ) Cdc1 2Cdc1 5

x72 =

(99)

That us .x7 is a differential function of the system’s flat outputs. Additionally, from Eq. (38) one solves for .x3 . This gives [ ] 2L c1 Rc1 1 .x3 = − vd x˙5 + x5 − ωdq x6 − K 1 x7 L c1 L c1

(100)

Since .x5 , .x6 , .x7 are differential functions of the flat outputs of the system it is inferred that .x3 is also a differential function of the system’s flat outputs, that is x = h 3 (y, y˙ , y¨ )

(101)

. 3

Next, from Eq. (39) one solves for .x4 . This gives x =−

. 4

[ ] 2L c1 Rc 1 vq x˙6 + ωdq x5 + 1 x6 − K 2 x7 L c1 L c1

(102)

Since .x5 , .x6 , .x7 are differential functions of the flat outputs of the system it is concluded that .x4 is also a differential function of the flat outputs, or x = h 4 (y, y˙ , y¨ )

(103)

. 4

From Eq. (41), using that . y¯4 = x8 one has an equation which can be solved for .x11 . This gives x11

1 = c1



y˙¯4 − c1 (− y¯4 − x9 − c2 ) −

c3 x8 Wca,out c4 y¯4 + c5 x9 + c6

= q11 ( y¯4 , y˙¯4 , x9 )

)

(104)

By differentiating Eq. (104) one obtains x˙

. 11

= q¯11 ( y¯4 , y˙¯4 , y¨¯4 , x9 , x˙9 )

(105)

114

G. Rigatos et al.

By substituting Eq. (104) into Eq. (105). and using Eq. (42) one obtains c3 x9 Wca ,out c4 x8c5 x9 + c6 c3 x9 Wca ,out = c8 [q11 ( y¯4 , y¯˙4 , x9 ) − x8 − x9 − c2 ] − c4 x8c5 x9 + c6 = q9 ( y¯4 , y˙¯4 , x9 )

x˙9 = c8 (x11 − x8 − x9 − c2 ) −

(106)

Moreover, by differentiating Eq. (41) one obtains x¨8 = c1 (x˙11 − x˙8 − x˙9 ) (c3 x˙9 Wca,out )(c4 x8 + c5 x9 + c6 ) − (c3 x9 Wca,out )(c4 x˙8 + c5 x˙9 ) − (c4 x8 + c5 x9 + c6 )2 y¨¯4 = c1 (q¯11 ( y¯4 , y˙¯4 , x9 , q9 ( y¯4 , y˙¯4 , x9 )) − y˙¯4 − q9 ( y¯4 , y˙¯4 , x9 )) (c3 q9 ( y¯4 , y˙¯4 , x9 )Wca,out ) (c3 x9 Wca,out )(c4 y˙¯4 + c5 q9 ( y¯4 , y˙¯4 , x9 )) − − c4 y¯4 + c5 x9 + c6 (c4 y¯4 c5 x9 + c6 )2 (107) The above relation of Eq. (107) can be solved for .x9 , thus allowing to express .x9 as a differential function of . y¯4 or x = h 9 ( y¯4 , y˙¯4 , y¨¯4 )

. 9

(108)

By substituting Eq. (108) into Eq. (105) one obtains x

. 11

= h 11 ( y¯4 , y˙¯4 , y¨¯4 )

Moreover, it holds that .x10 = ωcp = that Wcp =

A Wcp

(109)

[21, 22]. Besides, from Eq. (44) one has

x˙11 + c16 (x11 − x8 − x9 − c2 ) {1 + c15 [( xc1111 )c12 − 1]}

= qW cp ( y¯4 , y˙¯4 , y¨¯4 )

(110)

Consequently, it holds that x

. 10

=

A A = Wcp qW cp ( y¯4 , y˙¯4 , y¨¯4 )

(111)

Moreover, from Eq. (45) and using that .x12 = y4 is a flat output of the system one has that

Nonlinear Optimal Control for Residential Microgrids with Wind Generators …

] [ ] 1 1 x12 = c3 y˙4 − y3 R3 C 3 R3 C 3 = h 13 (y4 , y˙4 ) = h 13 (y, y˙ )

115

[ x13 = c3 x˙12 −

(112)

Thus, .x13 is a differential function of the flat outputs vector of the system, Additionally, from Eq. (46) one has that [ ] 1 L3 x12 .x8 = − x˙12 + K3 L3

(113)

Knowing that .x12 is the flat output . y4 and that .x13 = h 13 (y, y˙ ) one obtains also that x is a differential function of the system’s flat outputs, or

. 13

y¯ = x8 = h 8 (y, y˙ )

. 4

(114)

Consequently, from Eqs. (114) and (108), Eqs. (111) and (109) one has also that state variables .x9 , .x10 , .x11 are differential functions of the system’s flat outputs. Besides, from Eq. (47) and using that .x14 = y5 which is the last flat output of the system, one has that [ ] 1 x14 x15 = C4 x˙14 + R1 C 1 [ ] 1 = C4 y˙5 + y5 R1 C 1

(115)

Consequently, .x15 is also a differential function of the system’s flat outputs, or x

. 15

= h 15 (y, y˙ )

(116)

Moreover, about the last state vector element one has 1 (Vdc1 + Vdca + Vdcb )2 RL 1 = (x7 + x12 + x14 )2 RL

x16 =

(117)

where .x7 , .x12 , .x14 are differential functions of the flat outputs of the system. Consequently . x 16 = h 16 (y, y ˙) (118) which signifies that .x16 is a differential function of the system’s flat outputs. This comes to complete the conclusion that all state variables of the hybrid residential microgrid .x1 , x2 , . . . , x16 are differential functions of the system’s flat outputs.

116

G. Rigatos et al.

Next, from Eq. (35) one solves for the control input .u 1 . This gives [ ] 3P(L d − L q ) 2J Bm x˙− x3 x4 + x2 .u 1 = − Tm 4J J

(119)

Thus, .u 1 is a differential function of the system’s flat outputs, or u = h u 1 (y, y˙ )

. 1

(120)

Additionally, from Eq. (35) one has [ ] Lq Rs u = L d x˙3 + x3 − x2 x4 = h 2 (y, y˙ ) Ld Ld

. 2

(121)

which signifies that control input .u 2 is a differential function of the system’s flat outputs. Moreover, from Eq. (37) one solves for control input .u 3 . This gives [ u = Lq

. 3

] Rs Ld x˙4 + x4 + x2 x3 = h u 3 (y, y˙ ) Lq Lq

(122)

Besides, from Eq. (43) one solves for control input .u 4 . This gives ⎧ [( )c12 ]) x11 1 x˙10 + c9 x10 + c10 .u 4 = − 1 = h u 4 (y, y˙ ) c13 c11

(123)

Thus, control input .u 4 is also a differential function of the system’s flat outputs. Finally, from Eq. (48) one solves for control input .u 5 . This gives [ ] 1 u = L 4 x˙15 + x14 = h u 5 (y, y˙ ) L4

. 5

(124)

Thus, control input .u 5 is also a differential function of the system’s flat outputs. Therefore, all control inputs .u 1 , . . . , u 5 are also differential functions of the system’s flat outputs and the hybrid residential microgrid is a differentially flat system. It is noted that differential flatness properties also hold when using the concise formulation of the fuel cells dynamics in Eq. (30). In such a case the equations relating state variables .x9 to .x10 with the flat output .x8 are simplified and the complexity of the computation of the associated setpoints is again alleviated.

Nonlinear Optimal Control for Residential Microgrids with Wind Generators …

117

6 Simulation Tests The performance of the nonlinear optimal (H-infinity) control method for the dynamic model of the hybrid residential microgrid has been further tested and confirmed through simulation experiments. The simulation code was written in Matlab. The sampling period was set to .Ts = 0.01 s. To implement the nonlinear optimal control scheme, the algebraic Riccati equation which appears in Eq. (74) had to be solved at each time-step of the control algorithm with the use of Matlab’s aresolv() function. The obtained results are depicted in Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 and 20. The values of the plotted variables are scaled, after describing the functioning of the power sources and of the associated converters in a per unit (p.u.) system. It can be noticed that in all cases fast and accurate tracking of reference setpoints was achieved under moderate variations of the control inputs. The real values of the state vector of the hybrid residential microgrid are printed in blue, the estimated state variables provided by the . H∞ Kalman Filter are plotted in green, while the associated setpoints are shown in red colour. Despite its algorithmic simplicity, the nonlinear optimal control method is very efficient and achieves fast and accurate tracking of reference setpoints under moderate variations of the control inputs. The transient performance of the nonlinear optimal control scheme depends primarily on the values of the parameters which appear in the method’s algebraic Riccati equation. These are parameters .r , .ρ and . Q which appear in the Riccati equation of Eq. (74). Actually, relatively small values of .r result in elimination of the tracking

1 x5

x2

0.5 0 −0.5

0

5

10

15

20 25 time (sec)

30

35

0

40

x6

x3

1

0

5

10

15

20 25 time (sec)

30

35

10

15

25 20 time (sec)

30

35

40

0

5

10

15

25 20 time (sec)

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35

40

0

5

10

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20 25 time (sec)

30

35

40

1 7

0

x

x4

5

0 −1

40

0.5

−0.5

0

1

2

0

0.5

0

5

10

15

20 25 time (sec)

(a)

30

35

40

0.5 0

(b)

Fig. 3 Tracking of setpoint 1 for the hybrid residential microgrid a convergence of state variables and.x4 of the synchronous reluctance generator to their reference setpoints (red line: setpoint, blue line: real value, green line: estimated value), b convergence of state variables .x5 , .x6 .x7 of the AC/DC converter to their reference setpoints

.x 2 ,.x 3

118

G. Rigatos et al. 2 x14

x8

0.5 0 −0.5

0 −2

0

5

10

15

25 20 time (sec)

30

35

40

15

x12

0

5

10

15

20 25 time (sec)

30

35

−5

40

40

0

20

P

10 13

10

15

20 25 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

0

x

0 −2

x

5

5

2

−10

0

0

5

10

15

20 25 time (sec)

30

35

0

40

(a)

(b)

Fig. 4 Tracking of setpoint 1 for the hybrid residential microgrid a convergence of state variables of the fuel cells and .x12 , .x13 of the associated DC/DC converter to their reference setpoints (red line: setpoint, blue line: real value, green line: estimated value), b convergence of state variables . x 14 , . x 15 of the DC/DC converter connected to the PV power unit and of the aggregate power . x 16 at the DC bus to their reference setpoints .x8

50 u4

u

1

5 0 −5

0

5

10

15

20 25 time (sec)

30

35

0 −50

40

0 −5

0

5

10

15

20 25 time (sec)

30

35

40

−20

15

25 20 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

5 VL

3

u

10

0

5 0 −5

5

20 u5

u2

5

0

0

5

10

15

25 20 time (sec)

(a)

30

35

40

0 −5

(b)

Fig. 5 Tracking of setpoint 1 for the hybrid residential microgrid a control inputs .u 1 to .u 3 applied to the microgrid (at the synchronous reluctance generator), b control input .u 4 of the microgrid (at the fuel cells) and control input .u 5 (at the PV power unit), as well as aggregate voltage .VL at the load

Nonlinear Optimal Control for Residential Microgrids with Wind Generators … 2 x5

x2

0.5 0 −0.5

0

5

10

15

25 20 time (sec)

30

35

6

3

x

x

5

10

15

25 20 time (sec)

30

35

40

0

5

10

15

25 20 time (sec)

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35

40

0

5

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25 20 time (sec)

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35

40

0

0

5

10

15

25 20 time (sec)

30

35

−1

40

2 x7

2 x4

0

1

0

0 −2

1 0

40

10

−10

119

0

5

10

15

20 25 time (sec)

30

35

1 0

40

(a)

(b)

Fig. 6 Tracking of setpoint 2 for the hybrid residential microgrid a convergence of state variables and.x4 of the synchronous reluctance generator to their reference setpoints (red line: setpoint, blue line: real value, green line: estimated value), b convergence of state variables .x5 , .x6 , .x7 of the AC/DC converter to their reference setpoints

.x 2 ,.x 3

2 14

8

0.5

−0.5

x

x

0

0

5

10

15

25 20 time (sec)

30

35

40

15

x

x12

0 −2 0

5

10

15

25 20 time (sec)

30

35

40

5

10

15

25 20 time (sec)

30

35

0

5

10

15

20 25 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

0 −5

40

40

0

20

P

10

−10

0

5

2

x13

0 −2

0

5

10

15

20 25 time (sec)

(a)

30

35

40

0

(b)

Fig. 7 Tracking of setpoint 2 for the hybrid residential microgrid a convergence of state variables of the fuel cells and .x12 , .x13 of the associated DC/DC converter to their reference setpoints (red line: setpoint, blue line: real value, green line: estimated value), b convergence of state variables . x 14 , . x 15 of the DC/DC converter connected to the PV power unit and of the aggregate power . x 16 at the DC bus to their reference setpoints .x8

120

G. Rigatos et al. 50 u4

1

5 u

0 −5

0

5

10

15

20 25 time (sec)

30

35

0 −50

40

0 −5

0

5

10

15

20 25 time (sec)

30

35

−20

40

15

20 25 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

5 VL

u3

10

0

5 0 −5

5

20 u5

u2

5

0

0

5

10

15

20 25 time (sec)

30

35

0 −5

40

(a)

(b)

Fig. 8 Tracking of setpoint 2 for the hybrid residential microgrid a control inputs .u 1 to .u 3 applied to the microgrid (at the synchronous reluctance generator), b control input .u 4 of the microgrid (at the fuel cells) and control input .u 5 (at the PV power unit), as well as aggregate voltage .VL at the load 2 5

x2

0.5

−0.5

x

0

0

5

10

15

20 25 time (sec)

30

35

0

40

0

5

10

15

20 25 time (sec)

30

35

−1

40

5

10

15

20 25 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

7

2

0

x

x4

5

0

x

x

0

−5

0

1 6

3

10

−10

1

0

5

10

15

20 25 time (sec)

(a)

30

35

40

1 0

(b)

Fig. 9 Tracking of setpoint 3 for the hybrid residential microgrid a convergence of state variables and.x4 of the synchronous reluctance generator to their reference setpoints (red line: setpoint, blue line: real value, green line: estimated value), b convergence of state variables .x5 , .x6 .x7 of the AC/DC converter to their reference setpoints

.x 2 ,.x 3

Nonlinear Optimal Control for Residential Microgrids with Wind Generators … 2 x14

x8

0.5 0 −0.5

0 −2

0

5

10

15

25 20 time (sec)

30

35

40

15

x12

5

0

10

15

25 20 time (sec)

30

35

−5

40

40

0

20

P

10 13

5

10

15

20 25 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

0

5

10

15

25 20 time (sec)

30

35

40

0

x

0 −2

x

0

5

2

−10

121

0

5

10

15

20 25 time (sec)

30

35

0

40

(a)

(b)

Fig. 10 Tracking of setpoint 3 for the hybrid residential microgrid a convergence of state variables of the fuel cells and .x12 , .x13 of the associated DC/DC converter to their reference setpoints (red line: setpoint, blue line: real value, green line: estimated value), b convergence of state variables . x 14 , . x 15 of the DC/DC converter connected to the PV power unit and of the aggregate power . x 16 at the DC bus to their reference setpoints .x8

4

50

0 −5

0

u

u

1

5

0

5

10

15

20 25 time (sec)

30

35

−50

40

0 −5

0

5

10

15

20 25 time (sec)

30

35

40

−20

15

20 25 time (sec)

30

35

40

0

5

10

15

25 20 time (sec)

30

35

40

0

5

10

15

25 20 time (sec)

30

35

40

5 VL

3

u

10

0

5 0 −5

5

20 u5

u2

5

0

0

5

10

15

20 25 time (sec)

(a)

30

35

40

0 −5

(b)

Fig. 11 Tracking of setpoint 3 for the hybrid residential microgrid a control inputs.u 1 to.u 3 applied to the microgrid (at the synchronous reluctance generator), b control input .u 4 of the microgrid (at the fuel cells) and control input .u 5 (at the PV power unit), as well as aggregate voltage .VL at the load

122

G. Rigatos et al. 2 x5

x2

0.5 0 −0.5

0

5

10

15

25 20 time (sec)

30

35

0

40

6

3

0

5

10

15

25 20 time (sec)

30

35

−1

40

10

15

25 20 time (sec)

30

35

40

0

5

10

15

25 20 time (sec)

30

35

40

0

5

10

15

25 20 time (sec)

30

35

40

2 x7

5 x4

5

0

x

x

0

0 −5

0

1

10

−10

1

0

5

10

15

20 25 time (sec)

30

35

1 0

40

(a)

(b)

Fig. 12 Tracking of setpoint 4 for the hybrid residential microgrid a convergence of state variables and.x4 of the synchronous reluctance generator to their reference setpoints (red line: setpoint, blue line: real value, green line: estimated value), b convergence of state variables .x5 , .x6 .x7 of the AC/DC converter to their reference setpoints

.x 2 ,.x 3

2 x14

8

0.5 x

0 −0.5

0

5

10

15

25 20 time (sec)

30

35

40

x15

x12

0 −2

5

10

15

25 20 time (sec)

30

35

40

5

10

15

25 20 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

0 −10

10

40

0

20

P

13

0

−10

0 10

2

x

0 −2

0

5

10

15

20 25 time (sec)

(a)

30

35

40

0

(b)

Fig. 13 Tracking of setpoint 4 for the hybrid residential microgrid a convergence of state variables of the fuel cells and .x12 , .x13 of the associated DC/DC converter to their reference setpoints (red line: setpoint, blue line: real value, green line: estimated value), b convergence of state variables . x 14 , . x 15 of the DC/DC converter connected to the PV power unit and of the aggregate power . x 16 at the DC bus to their reference setpoints .x8

Nonlinear Optimal Control for Residential Microgrids with Wind Generators … 50 u4

1

5 u

0 −5

0

5

10

15

20 25 time (sec)

30

35

0 −50

40

0

0

5

10

15

20 25 time (sec)

30

35

40

−20

10

15

20 25 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

5 VL

u3

5

0

5 0 −5

0

20 u5

u2

5

−5

123

10

5

0

15

25 20 time (sec)

30

35

0 −5

40

(a)

(b)

Fig. 14 Tracking of setpoint 4 for the hybrid residential microgrid a control inputs.u 1 to.u 3 applied to the microgrid (at the synchronous reluctance generator), b control input .u 4 of the microgrid (at the fuel cells) and control input .u 5 (at the PV power unit), as well as aggregate voltage .VL at the load 2 5

x2

0.5

−0.5

x

0

0

5

10

15

20 25 time (sec)

30

35

0

40

x6

3

x

0

0

5

10

15

20 25 time (sec)

30

35

−1

40

10

15

25 20 time (sec)

30

35

40

0

5

10

15

25 20 time (sec)

30

35

40

0

5

10

15

25 20 time (sec)

30

35

40

2 7

0

x

x4

5

0

5

−5

0

1

10

−10

1

0

5

10

15

25 20 time (sec)

(a)

30

35

40

1 0

(b)

Fig. 15 Tracking of setpoint 5 for the hybrid residential microgrid a convergence of state variables and.x4 of the synchronous reluctance generator to their reference setpoints (red line: setpoint, blue line: real value, green line: estimated value), b convergence of state variables .x5 , .x6 .x7 of the AC/DC converter to their reference setpoints

.x 2 ,.x 3

124

G. Rigatos et al. 2 x14

8

0.5 x

0

0 −2

−0.5

0

5

10

15

20 time (sec)

25

30

35

40

x15

12

5

10

15

25 20 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

5

2 x

0

0

0

−2 5

10

15

20 time (sec)

25

30

35

−5

40

10

40

0

20

−10

P

x

13

0

0

5

10

15

20 time (sec)

25

30

35

0

40

(a)

(b)

Fig. 16 Tracking of setpoint 5 for the hybrid residential microgrid a convergence of state variables of the fuel cells and .x12 , .x13 of the associated DC/DC converter to their reference setpoints (red line: setpoint, blue line: real value, green line: estimated value), b convergence of state variables . x 14 , . x 15 of the DC/DC converter connected to the PV power unit and of the aggregate power . x 16 at the DC bus to their reference setpoints .x8

4

50

0 −5

0

u

u

1

5

0

5

10

15

20 25 time (sec)

30

35

−50

40

0 −5

0

5

10

15

25 20 time (sec)

30

35

40

−20

15

20 25 time (sec)

30

35

40

0

5

10

15

20 25 time (sec)

30

35

40

0

5

10

15

25 20 time (sec)

30

35

40

5 VL

3

u

10

0

5 0 −5

5

20 u5

u2

5

0

0

5

10

15

20 25 time (sec)

(a)

30

35

40

0 −5

(b)

Fig. 17 Tracking of setpoint 5 for the hybrid residential microgrid a control inputs.u 1 to.u 3 applied to the microgrid (at the synchronous reluctance generator), b control input .u 4 of the microgrid (at the fuel cells) and control input .u 5 (at the PV power unit), as well as aggregate voltage .VL at the load

Nonlinear Optimal Control for Residential Microgrids with Wind Generators … 2 x5

x2

0.5 0 −0.5

0

5

10

15

25 20 time (sec)

30

35

6

3

x

x

5

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25 20 time (sec)

30

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40

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25 20 time (sec)

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25 20 time (sec)

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35

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0

0

5

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25 20 time (sec)

30

35

−1

40

2 x7

5 x4

0

1

0

0 −5

1 0

40

10

−10

125

0

5

10

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25 20 time (sec)

30

35

1 0

40

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error while relatively large values of matrix . Q result in fast convergence to the reference setpoints. Moreover, coefficient .ρ affects the robustness of the control loop. The smallest value of .ρ for which one can obtain a valid solution from the algebraic Riccati equation of Eq. (74) is the one that provides the control loop with maximum robustness. It is also pointed out that the use of the . H∞ Kalman Filter as a robust state estimator has allowed to implement feedback control by using measurement of a subset of the state vector elements of the residential AC/DC microgrid. There is a clear contribution of this research work comparing to past approaches for solving the nonlinear optimal (H-infinity) control problem. One can point out the advantages of the nonlinear optimal control method against Nonlinear Model Predictive Control (NMPC). In NMPC the stability properties of the control scheme remain unproven and the convergence of the iterative search for an optimum often depends on initialization and parameter values’ selection. It is also noteworthy that the nonlinear optimal control method is applicable to a wider class of dynamical systems than approaches based on the solution of State Dependent Riccati Equations (SDRE). The SDRE approaches can be applied only to dynamical systems which can be transformed to the Linear Parameter Varying (LPV) form. Besides, the nonlinear optimal control method performs better than nonlinear optimal control schemes which use approximation of the solution of the Hamilton-Jacobi-Bellman equation by Galerkin series expansions. The stability properties of the Galerkin series expansion -based optimal control approaches are still unproven. A comparison of the proposed nonlinear optimal (. H∞ ) control method against other linear and nonlinear control schemes for the hybrid residential microgrid, shows the following: (1) unlike global linearization-based control approaches, such as Lie

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algebra-based control and differential flatness theory-based control, the optimal control approach does not rely on complicated transformations (diffeomorphisms) of the system’s state variables. Besides, the computed control inputs are applied directly on the initial nonlinear model of the residential AC/DC microgrid’s dynamics and not on its linearized equivalent. The inverse transformations which are met in global linearization-based control are avoided and consequently one does not come against the related singularity problems. (2) unlike Model Predictive Control (MPC) and Nonlinear Model Predictive control (NMPC), the proposed control method is of proven global stability. It is known that MPC is a linear control approach that if applied to the nonlinear dynamics of the residential AC/DC microgrid the stability of the control loop will be lost. Besides, in NMPC the convergence of its iterative search for an optimum depends on initialization and parameter values selection and consequently the global stability of this control method cannot be always assured. (3) unlike sliding-mode control and backstepping control the proposed optimal control method does not require the state-space description of the system to be found in a specific form. About sliding-mode control it is known that when the controlled system is not found in the input-output linearized form the definition of the sliding surface can be an intuitive procedure. About backstepping control it is known that it can not be directly applied to a dynamical system if the related state-space model is not found in the triangular (backstepping integral) form. (4) unlike PID control, the proposed nonlinear optimal control method is of proven global stability, the selection of the controller’s parameters does not rely on a heuristic tuning procedure, and the stability of the control loop is assured in the case of changes of operating points (5) unlike multiple local models-based control the nonlinear optimal control method uses only one linearization point and needs the solution of only one Riccati equation so as to compute the stabilizing feedback gains of the controller. Consequently, in terms of computation load the proposed control method for the residential AC/DC microgrid’s dynamics is much more efficient.

7 Conclusions The chapter has analysed a novel solution to the nonlinear optimal control problem of residential microgrids. Such microgrids comprise heterogenous power sources of the AC and DC type and the associated power electronics such as converters and inverters. The objective is to synchronize and stabilize the functioning of distributed AC and DC power sources that constitute a residential microgrid, so as to assure quality of the produced electric power, uninterrupted power supply and complete coverage of the residence’s power needs under variable operating and environmental conditions. The considered residential microgrid comprises heterogeneous power sources, such as (i) A wind micro-turbine which provides rotational motion to a synchronous reluctance generator, while the generator is connected to an AC/DC converter. Thus finally, the AC power of the generator is turned into DC power that is further distributed to a DC bus, (ii) A stack of PEM fuel cells which provide DC

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power to a DC/DC converter. In turn the converter is connected to the previously noted DC bus, (iii) photovoltaic arrays that are also connected to the aforementioned DC bus through another DC/DC converter. The dynamic model of the integrated system has a complicated multi-variable and nonlinear state-space description and the solution of the related nonlinear control problem is a nontrivial task. To apply the proposed nonlinear optimal control scheme the dynamic model of the microgrid has undergone first approximate linearization with the use of first-order Taylor series expansion and through the computation of the system’s Jacobian matrices. The linearization process was performed at each sampling instance around a time varying operating point that was defined by the present value of the system’s state vector and by the last sample value of the control inputs vector. For the linearized model of the microgrid an . H∞ (optimal) feedback controller has been designed. The proposed control scheme represents a min-max differential game taking place between the control inputs of the system and the exogenous perturbations or model uncertainty terms. To compute the stabilizing feedback gains of the controller an algebraic Riccati equation had to be repetitively solved at each sampling period. The global stability properties of the control method have been proven through Lyapunov analysis. Despite its algorithmic and conceptual complexity the new nonlinear optimal control method achieves fast and accurate tracking of reference setpoints under moderate variations of the control inputs. Furthermore, differential flatness properties have been proven for the residential microgrid. These properties come to confirm implicitly the system’s controllability and allow for defining feasible setpoints for all state variables of the microgrid.

References 1. Abdalla, S.A., Abdullah, S.S., Kassem, A.M.: Performance enhancement and power management strategy of an autonomous hybrid fuel cell/wind power system based on adaptive neurofuzzy inference system. Ain Shams Eng. J. Elsevier (2022) 2. Armghan, H., Yang, M., Armghan, A., Ali, N., Wang, M.O., Ahmad, I.: Design of integral sliding-mode controller for the hybrid AC/DC microgrids involving renewables and energy storage systems. Electric Power Energy Syst. Elsevier 119, 105857–105871 (2020) 3. Baghaee, H.R., Mirsalim, M., Gharehpetian, G.B., Taleb, H.A.: Decentralized sliding-mode control of WG/PV/FC microgrids under unbalanced and nonlinear load conditions for On and Off-grid modes. IEEE Syst. J. 12, 3108–3119 (2018) 4. Baghaee, H.R., Mirsalim, M., Gharehpetian, G.B., Talebi, H.A.: A decentralized power management and sliding-mode control strategy for hybrid AC/DC microgrids including renewable energy sources. IEEE Trans. Ind. Inform. (2022) 5. Basseville, M., Nikiforov, I.: Detection of Abrupt Changes: Theory and Applications. PrenticeHall (1993) 6. Chen, W., Tai, K., Lau, M., Abdelhakim, A., Chan, R.C., Adnanes, A.K., Tjahjawidodo, T.: DC distributed power system modelling and hardware in-the-loop (HIL) evaluation of fuel cell-powered marine vessel. IEEE J. Emerg. Sel. Top. Ind. Electron. (2022)

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7. Fagundes, T.A., Favaro Fuzato, G.H., Bueno Ferreira, P.G., Biczkowski, M., Mochada, R.Q.: Fuzzy controller for energy management and SoC equalization in DC microgrids powered by fuel cell and energy storage units. IEEE J. Emerg. Sel. Top. Ind. Electron. 3, 90–100 (2022) 8. Ferahtia, S., Djerioui, A., Zeglache, S., Houari, A.: A hybrid power system based on fuel cell, photovoltaic source and supercapacitors. J. Appl. Sci. Springer 2, 1–11 (2020) 9. Garcia-Tores, F., Valverde, L., Bordons, C.: Optimal load-sharing of hydrogen-based microgrids with hybrid storage using model-predictive control. IEEE Trans. Ind. Electron. 63, 4919– 4928 (2016) 10. Ghiasi, M.I., Golkan, M.A., Hajizadeh, A.: Lyapunov-based distributed fuzzy-sliding mode control for building integrated-DC microgrid with plug-in electric vehicle. IEEE Access 5, 7746–7752 (2017) 11. Iovine, A., Corrizosa, M.J., Damn, G., Alou, P.: Nonlinear control for DC microgrids enabling efficient renewable power integration and ancillary services for AC grids. IEEE Trans. Power Syst. 34, 5136–5146 (2019) 12. Jain, S., Hote, Y.V.: Generalized active disturbance rejection controller design: application to hybrid microgrid with communication delay. Intl. J. Electr. Power Energy Syst. Elsevier 132, 107166–107180 (2021) 13. Li, Q., Li, R., Pu, Y., Li, S., Sun, C., Chen, W.: Coordinated control of electric-hydrogen hybrid energy storage for multi-microgrid with fuel cell/electrolyzer/PV/battery. J. Energy Storage, Elsevier 42, 103110–103121 (2021) 14. Liaqat, K., Rehman, Z., Ahmad, I.: Nonlinear controller for fuel cell photovoltaic cell and battery-based hybrid energy management system. J. Energy Storage, Elsevier 32, 101796– 101805 (2020) 15. Mahmud, M.A., Roy, T.K., Islam, S.N., Saha, S., Haque, M.E.: Nonlinear decentralized feedback linearizing controller design for islanded DC microgrids. Electric Power Compon. Syst. Taylor and Francis 45, 1747–1761 (2017) 16. Mane, S., Mejani, M., Kazi, F., Singh, N.: Improving life-time of fuel cell in hybrid energy management system by Lure-Lyapunov-based control formulation. IEEE Trans. Ind. Electron. 64, 6671–6679 (2017) 17. Rigatos, G.G., Tzafestas, S.G.: Extended Kalman filtering for fuzzy modelling and multi-sensor fusion. Math. Comput. Model. Dyn. Syst. Taylor & Francis 13, 251–266 (2007) 18. Rigatos, G., Zhang, Q.: Fuzzy model validation using the local statistical approach. Fuzzy Sets Syst. Elsevier 60, 882–904 (2009) 19. Rigatos, G.G.: Nonlinear Control and Filtering Using Differential Flatness Approaches: Applications to Electromechanical Systems. Springer (2015) 20. Rigatos, G., Siano, P., Cecati, C.: A new nonlinear . H∞ feedback control approach for threephase voltage source converters. Electric Power Compon. Syst. Taylor and Francis 44, 302–312 (2015) 21. Rigatos, G.: Intelligent Renewable Energy Systems: Modelling and Control. Springer (2016) 22. Rigatos, G., Karapanou, E.: Advances in Applied Nonlinear Optimal Control. Cambridge Scholars Publishing (2020) 23. Sabahi, K., Hajizadeh, A., Tavan, M.: Lyapunov-Krasovskii based FPID controller for LFC in a time-delayed micro-grid system with fuel cell power units. COMPEL - Int. J. Comput. Math. Electr. Electron. Eng. Emerald Publishing 40, 488–501 (2021) 24. Sekhar, P.C., Mishra, S.: Storage-free smart energy management for frequency control in a Diesel-PV-fuel cell-based hybrid AC microgrid. IEEE Trans. Neural Netw. Learn. Syst. 27, 1657–1661 (2016) 25. Thounthong, P., Mungporn, P., Guilbert, D., Takorabat, N., Pierfederici, S., Nahid-Mobarakeh, S., Yu, H., Bizon, M., Huamgfei, Y., Kumam, P.: Design and control of multiphase interleaved boost converter based on differential flatness theory for PEM fuel cell multi-task applications. Electric Power and Energy Syst., Elevier 124, 106346–106358 (2021) 26. Toussaint, G.J., Basar, T., Bullo, F.: . H∞ optimal tracking control techniques for nonlinear underactuated systems. In: Proceedings of the IEEE CDC 2000, 39th IEEE Conference on Decision and Control, Sydney Australia (2000)

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Nonlinear Optimal Control for VSI-fed Three-phase Asynchronous Motors Gerasimos Rigatos, Mohamed Assaad Hamida, Pierluigi Siano, Masoud Abbaszadeh, Godpromesse Kenné, and Patrice Wira

Abstract The use of voltage-source inverter-fed induction (asynchronous) motors is widely met in industry (for instance for actuation in tasks that require high torque and power as well as in transportation systems (for the traction of trains and electric vehicles). In this chapter a nonlinear optimal control method is proposed for voltage source inverter-fed asynchronous (induction) motors (VSI-fed IM). Approximate linearization is performed on the nonlinear dynamic model of VSI-fed induction motors around a temporary operating point which is recomputed at each iteration of the control method. This time-varying operating point is determined by the present value of the VSI-fed IM state vector and by the last sampled value of the system’s control inputs vector. The linearization process is based on Taylor series expansion and on the computation of the system’s Jacobian matrices. Next, an H∞ feedback controller is designed for the approximately linearized model of the VSI-fed IM. This control approach provides a solution for the nonlinear optimal control problem of the VSI-fed IM in the case of model uncertainty and external perturbations. To compute G. Rigatos (B) Unit of Industrial Automation, Industrial Systems Institute, Rion Patras, Greece e-mail: [email protected] M. A. Hamida LS2N, CNRS, UMR 6004, Ecole Centrale de Nantes, France e-mail: [email protected] P. Siano Department of Management and Innovation Systems, University of Salerno, 84084 Fisciano, Italy e-mail: [email protected] M. Abbaszadeh Department of ECS Engineering, Rensselaer Polytechnic Institute, 12605 New York, USA e-mail: [email protected] G. Kenné Unité de Recherche AIA, Université de Dschang, BP 134, Dschang, Cameroon P. Wira IRIMAS, Université de Haute Alsace, 68093 Mulhouse, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Derbel et al. (eds.), Advances in Robust Control and Applications, Studies in Systems, Decision and Control 474, https://doi.org/10.1007/978-981-99-3463-8_5

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the controller’s feedback gains an algebraic Riccati equation is solved at each timestep of the control method. Lyapunov analysis is used to prove the global asymptotic stability properties of the control scheme. Moreover, to apply state estimation-based control for this system the H∞ Kalman Filter is used as a state observer. Keywords Voltage source inverters · Induction (Asynchronous) machines · Nonlinear optimal control · Differential flatness properties · H∞ control · Approximate linearization · Jacobian matrices · Riccati equations · Lyapunov analysis · Global asymptotic stability

1 Introduction Induction motors are widely used in electric traction and propulsion applications because of their improved torque and power characteristics as well as because of their reliable performance under harsh and variable operating conditions [25, 28, 37]. Due to the nonlinear and multivariable structure of induction motors the associated nonlinear control problem is a nontrivial one [19, 20, 32, 38]. Besides, the implementation of feedback control for induction motors depends on estimation of some non-measurable state variables such as the rotor’s magnetic flux [2, 9, 17, 18]. On the one side, state estimation methods enable to identify several parameters in the induction motor’s dynamic model [12, 15, 16, 36, 40, 42]. On the other side, parametric changes and external perturbations may always emerge and consequently robustness is a prerequisite for the induction motors control loop [3, 11, 14, 34, 35, 39]. Among nonlinear control methods for induction motors one can primarily distinguish those based on global linearization transformations (such as Lie algebra-based control), backstepping control and sliding control ([6, 8, 10, 21, 25]. To treat the associated optimal control problem common solutions have been based on ModelPredictive Control under a linear model assumption, as well as on Nonlinear Model Predictive Control [1, 5, 13, 41]. Improving the performance of induction motors control loop remains an open challenge in the case of Electric Vehicles and trains electric traction systems [26, 27, 29, 31]. In the present chapter a novel nonlinear optimal (H∞ ) control method is proposed for the integrated system which comprises an induction motor being fed by a threephase voltage source inverter. The method is based on (a) approximate linearization of the dynamic model of the VSI-fed asynchronous motor with the use of first-order Taylor series expansion and (b) computation of the associated Jacobian matrices [7, 22, 23]. The linearization point is updated at each iteration of the control algorithm and is defined by the present value of the VSI-fed motor’s state vector and by the last sampled value of the control inputs vector. The modelling error which is due to truncation of higher-order terms in the Taylor series expansion is a perturbation which is asymptotically compensated by the robustness of the control scheme. For the approximately linearized model of the VSI-fed induction motor a stabilizing H∞ feedback controller is designed.

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The proposed H∞ controller provides a solution to the optimal control problem of the VSI-fed induction motor under model nonlinearities, parametric uncertainties and external perturbations. Actually, it implements a min-max differential game taking place between (i) the control inputs which try to minimize a quadratic cost function of the state vector’s tracking error, (ii) model uncertainty and external perturbation terms which try to maximize this cost function. The control method has a double objective (a) to eliminate the deviation of the state vector from the associated reference values, (b) to minimize the variation of the control inputs and the associated dispersion of energy. To select the feedback gains of the H∞ controller an algebraic Riccati equation is being repetitively solved at each time-step of the control method [24, 30]. The global stability properties of the control method are proven through Lyapunov analysis. First, it is demonstrated that the control scheme satisfies the H∞ tracking performance criterion which denotes robustness against model uncertainties and external perturbations [28, 30, 33]. Next, under moderate conditions it is also proven that the control loop is globally asymptotically stable. To implement state estimationbased control the H∞ Kalman Filter is used as a robust state estimator. The significance of the proposed control method is outlined as follows: (i) it offers a solution to the nonlinear optimal control problem which is of proven global stability while also remaining computationally tractable, (ii) it retains the advantages of linear optimal control that is fast and accurate tracking of reference setpoints under moderate variations of the control inputs, (iii) it minimizes the amount of energy that is dispersed by the control system of the VSI-fed induction motor. All other control schemes (Lie algebra-based control, flatness-based control, sliding-mode control, backstepping control, PID control, multiple linear local models-based control) are non-optimal or sub-optimal since their objective is to minimize only the tracking error of the state variables of the VSI-fed induction motor without suppressing the variations of the control inputs [28]. Without achieving any alleviation of algorithmic complexity and without reducing computational effort, the non-optimal or suboptimal control schemes leave also aside the problem of minimizing the consumption of energy by the VSI-fed induction motor. One can also point out the advantages of the chapter’s nonlinear optimal control method against Nonlinear Model Predictive Control (NMPC). In NMPC the stability properties of the control scheme remain unproven and the convergence of the iterative search for an optimum often depends on initialization and parameter values’ selection. It is also noteworthy that the nonlinear optimal control method is applicable to a wider class of dynamical systems than approaches based on the solution of State Dependent Riccati Equations (SDRE). The SDRE approaches can be applied only to dynamical systems which can be transformed to the Linear Parameter Varying (LPV) form. Besides, the nonlinear optimal control method performs better than nonlinear optimal control schemes which use approximation of the solution of the HamiltonJacobi-Bellman equation by Galerkin series expansions. The stability properties of the Galerkin series expansion -based optimal control approaches are still unproven. The present chapter provides a novel solution to the nonlinear optimal control problem of the VSI-fed induction motor which is of proven global stability while also remaining computationally efficient. Preceding results on the use of H∞ control

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in nonlinear dynamical systems were limited to the case of affine-in-the-input systems with drift-only dynamics and considered that the control inputs gain matrix is not dependent on the values of the system’s state vector. Moreover, in these approaches the linearization was performed around points of the desirable trajectory whereas in the present chapter’s control method the linearization points are related with the value of the state vector at each sampling instant as well as with the last sampled value of the control inputs vector. The Riccati equation which has been proposed for computing the feedback gains of the controller is novel, so is the presented global stability proof through Lyapunov analysis.

2 Dynamic Model of the Induction Motor 2.1 Mathematical Model of the Induction Motor To derive the dynamic model of an induction motor the three-phase variables are first transformed to two-phase ones [25]. This two-phase system can be described in the stator-coordinates frame α − b, and the associated voltages are denoted as vs α and vs b , while the currents of the stator are i s α and i s b , respectively (see Fig. 1). Then, the rotation angle of the rotor with respect to the stator is denoted by δ. Next, the rotating reference frame d − q on rotor, is defined. The currents of the rotor are decomposed into d − q coordinates, thus resulting into ird and irq . Since the frame d − q of the rotor aligns with the frame α − b of the stator after rotation by an angle δ it holds that )( ) ( ) ( cos δ − sin δ i rd i rα (1) = i rb i rq sin δ cos δ The voltage developed along frame α of the stator is given by Rs i s α +

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where the magnetic flux ψs α is the result of the magnetic flux that is generated by current i s α of the stator (self-inductance) and of the magnetic flux which is generated by current ir α of the rotor (mutual inductance), that is ψs α = L s i s α + Mir α

(3)

The voltage developed along frame b of the stator is Rs i s b +

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(4)

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'

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Fig. 1 Induction Motor circuit, with the a − b stator reference frame and the d − q rotor reference frame

where the magnetic flux ψs b is the result of the magnetic flux that is generated by currrent i s b of the stator (self-inductance) and of the magnetic flux which is generated by current ir b of the rotor (mutual inductance), that is ψs b = L s i s b + Mir b

(5)

Similarly the voltage along frames d and q of the rotor is calculated as follows Rr ird +

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(6)

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d ψr = 0 dt q

(7)

After intermediate computations the equations of the induction motor are found to be: ( ) M d M2 d Rs i s α + (8) ψr α + L s − i s α = vs α L r dt L r dt ( ) M2 d M d Rs i s b + i s b = vs b ψr + L s − L r dt b L r dt

(9)

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Rr Rr M d ψr α − i s α + ψr α + n p ωψr b = 0 Lr Lr dt

(10)

Rr Rr M d ψr b − i s b + ψr b + n p ωψr α = 0 Lr Lr dt

(11)

The torque that is applied to the rotor is developed according to the principle of energy preservation and is given by T =

npM (ψr α i s b − ψr b i s α ) Lr

(12)

If the motor has to move a load of torque TL it holds: J ω˙ = T − TL Then: ω˙ =

npM TL (ψr α i s b − ψr b i s α ) − J Lr J

(13)

Denoting: σ =1−

M2 Ls Lr

the equations of the induction motor are finally written as: θ˙ = ω

(14)

npM dω TL (ψr α i s b − ψr b i s α ) − = dt J Lr J

(15)

dψr α Rr RL Mi s α = − ψr α − n p ωψr b + dt Lr Lr

(16)

dψr b Rr RL Mi s b = − ψr b + n p ωψr α + dt Lr Lr

(17)

npM d M Rr M 2 Rr + L r2 Rs 1 ψr α + ωψr b − ( )i s α + vs α is α = 2 dt σ Ls Lr σ Ls Lr σ L s L r2 σ Ls

(18)

npM d M Rr M 2 Rr + L r2 Rs 1 ωψr α + ψ − ( )i s b + vs b is b = − r b 2 2 σ Ls dt σ Ls Lr σ Ls Lr σ Ls Lr

(19)

Therefore one can define the state vector x = [θ, ω, ψr α , ψr b , i s α , i s b ]T . Uncertainty can be associated with the value of the load torque TL , or the value of the

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components of the electric circuits of the stator and the rotor. The following parameters are also defined: α1 =

Rr Lr

M σ Ls Lr M 2 Rr Rs γ1 = ( + ) σ L s L r2 σ Ls npM μ1 = J Lr β1 =

Therefore, the dynamic model of the induction motor can be written as: x˙ = f (x) + gα u s α + gb u s b

(20)

In state equations form, the dynamic model of the motor can be written as ⎤ ⎡ 0 x2 TL ⎥ ⎢ ⎢ μ (x x − x x ) − 1 3 6 4 5 ⎥ ⎢ ⎢ 0 J ⎢ α1 x3 − n p x2 x4 + α1 M x5 ⎥ ⎢ 0 ⎥ ⎢ f (x) = ⎢ ⎢ n p x2 x3 − α1 x4 + α1 M x6 ⎥ , gα = ⎢ 0 ⎥ ⎢ ⎢ 1 ⎣ α1 β1 x3 + n p β1 x2 x4 − γ x5 ⎦ ⎣ σ Ls −n p β1 x2 x3 + α1 β1 x4 − γ1 x6 0 ⎡





⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ , gb = ⎢ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣

0 0 0 0 0

⎤ ⎥ ⎥ ⎥ ⎥ (21) ⎥ ⎥ ⎦

1 σ Ls

2.2 Field Orientation The classical method for induction motors control was introduced by Blascke (1971) and is based on a transformation of the stator’s currents (i s α ) and (i s b ) and of the magnetic flux of the rotor (ψr α and ψr b ) to the reference frame d − q which rotates together with the rotor [25]. Thus the controller’s design uses the currents i s d and i s q and the fluxes ψr d and ψr q . The angle of the vectors that describe magnetic flux ψr α and ψr b is first defined, i.e. ( ρ¯ = arctan

ψrb ψra

) (22)

The angle between the inertial reference frame of the stator and the rotating reference frame of the rotor is taken to be equal to ρ¯ . The transition from (i s α , i s b ) to (i s d , i s q ) is given by

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( ) ( )( ) is d cos ρ¯ sin ρ¯ is α = is q is b − sin ρ¯ cos ρ¯

(23)

The transition from (ψr α , ψr b ) to (ψr d , ψr q ) is given by (

ψr d ψr q

)

( =

cos ρ¯ sin ρ¯ − sin ρ¯ cos ρ¯

)(

ψr α ψr b

) (24)

Moreover, it holds that: ψra ||ψ|| ψrb sin ρ¯ = ||ψ|| √ ||ψ|| = ψr2α + ψr2b

cos ρ¯ =

Using the above transformation ones obtains ⎧

is d = is q =

ψr α i s α +ψr b i s b ||ψ|| ψr α i s b −ψr b i s α ||ψ||

⎫ ,

ψr d = ||ψ|| ψr q = 0

(25)

Therefore, in the rotating frame d − q of the motor there will be only one nonzero component of the magnetic flux ψrd , while the component of the flux along the q axis equals 0. The new inputs of the system are considered to be vs d , vs q , which are connect to vs a , vs b according to the relation (

vs α vs b

) = ||ψ||·

( )−1 ( ) ψra ψrb vs d ψrb ψra vs q

(26)

In the new coordinates the induction motor model is written as: d θ =ω dt

(27)

TL d ω = μψr d i s q − dt J

(28)

d ψr = −αψr d + α Mi s d dt d

(29)

α Mi s q 2 d 1 + vs d i s d = −γ i s d + αβψr d + n p ωi s q + dt ψr d σ Ls

(30)

Nonlinear Optimal Control for VSI-fed Three-phase Asynchronous Motors

139

α Mi s q i s d d 1 + vs q i s q = −γ i s q − βn p ωψr d − n p ωi s d − dt ψd σ Ls

(31)

α Mi s q d ρ¯ = n p ω + dt ψr d

(32)

Defining the state vector of the motor dynamics in the dq reference frame as x = [θ, ω, ψrd , i sd , i sq , ρ¯ ] the associated state-space model becomes x˙ = f (x) + gd vsd + gq vsq , where ⎡

x2 TL ⎢ μx x 3 5− J ⎢ ⎢ −αx3 + α M x4 ⎢ f (x) = ⎢ αMx2 ⎢ −γ x4 + αβx3 + n p x2 x5 + x3 5 ⎢ ⎣ −γ x5 − βn p x2 x3 − n p x2 x4 − α Mxx34 x5 −n p x2 + α Mx3x5



⎡ 0 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎥ , gd = ⎢ ⎢ 0 ⎥ ⎢ 1 ⎥ ⎣ ⎦ σ Ls 0





⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ , gq = ⎢ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦



0 0 0 0 0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

1 σ Ls

(33)

Next, the following nonlinear feedback control law is defined (

vs d vs q

(

) = σ Ls

−n p ωi s q −

α Mi s q 2 ψr d

− αbψr d + vd

n p ωi s d + bn p ωψr d +

α Mi s q i s d ψr d

) (34)

+ vq

The terms in Eq. (34) have been selected so as to linearize Eqs. (30) and (31) and to produce first-order linear ODE. The control signal in the inertial coordinates system a − b will be (

vs α vs b

)

( ) ) α Mi s q 2 ψs α ψs b −1 −n p ωi s q − ψr d − αβψr d + vd · α Mi s q i s d −ψs b ψs α n p ωi s d + βn p ωψr d + + vq

( = ||ψ||σ L s

ψr d

(35)

Substituting Eqs. (34) into (30) and Eq. (31) one obtains [25]: θ˙ = ω

(36)

d TL ω = μψr d i s q − dt J

(37)

d i s q = −γ i s q + vq dt

(38)

d ψr = −αψr d + α Mi s d dt d

(39)

d i s d = −γ i s d + vd dt

(40)

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is q d ρ¯ = n p ω + α M dt ψr d

(41)

The system of Eqs. (37)–(41) consists of two linear subsystems, where the first one has as output the magnetic flux ψr d and the second has as output the rotation speed ω, i.e. d (42) ψr = −αψr d + α Mi s d dt d d i s d = −γ i s d + vd dt

(43)

TL d ω = μψr d i s q − dt J

(44)

d i s q = −γ i s q + vq dt

(45)

If ψr d →ψr ref d , i.e. the transient phenomena for ψr d have been eliminated and therefore ψr d has converged to a steady state value, then the two subsystems described by Eqs. (42)–(45) are decoupled. The subsystem that is described by Eqs. (42) and (43) is linear with control input vs d , and can be controlled using methods of linear control, such as optimal control, or PID control [28].

3 Dynamic Model of the Three-phase Voltage Source Inverter The inverter’s (DC to AC converter’s) circuit is depicted in Fig. 2. By applying Kirchoff’s voltage and current laws one obtains d i = L1f VI − L1f VL dt I d V = C1f i I − C1f i L dt L

(46)

For the representation of the voltage and current variables, denoted as X = {I, V } in the ab static reference frame one has X ab = X a e j0 + X b e

j2π 3

+ Xce

j4π 3

(47)

which finally gives a complex variable of the form X ab = X a + j X b

(48)

Nonlinear Optimal Control for VSI-fed Three-phase Asynchronous Motors

141

Fig. 2 Circuit of the three-phase voltage inverter

Next, the voltage and current variables are represented in the rotating dq reference frame [28]. It holds that X dq = xd + j xq and X dq = X ab e−∫jθ ⇒X ab = X dq e jθ t θ (t) = 0 ω(t)dt + θ0

(49)

By differentiating with respect to time one obtains the following description of the system’s dynamics (50) X˙ ab = dtd X dq + j ωX dq Thus, one has for the current and voltage variables respectively,

d dt

d i = dtd i I,dq + ( j ω)i I,dq dt I,ab VL ,ab = dtd VL ,dq + ( jω)VL ,dq

(51)

By substituting Eqs. (51) into (46) one obtains d i + j ωi I,dq = L1f VI,dq − L1f VL ,dq dt I,dq d V + j ωVL ,dq = C1f i I,dq − C1f i L ,dq dt L ,dq

(52)

Using Eq. (52) and by rearranging rows one finally obtains the inverter’s dynamic model expressed in the dq reference frame:

142

G. Rigatos et al. d V = ωVL ,q + C1f i I,d − C1f i L ,d dt L ,d d V = −ωVL ,d + C1f i I,q − C1f i L ,q dt L ,q d i = ωi I,q + L1f VI,d − L1f VL ,d dt I,d d i = −ωi I,d + L1f VI,q − L1f VL ,q dt I,q

(53)

X = [VL d , VL q , i I,d , i I,q ]T while the The state vector of the inverter is taken to be ~ ~ = [VI,d , VI,q ]. The load currents i L ,d and control input is taken to be the vector U i L ,q are taken to be unknown parameters which can be considered as perturbation terms. Alternatively, these currents can be expressed as functions of the inverter’s active and reactive power. In the latter approach one has that the active power of the inverter is [28] p f = VL d i L d + VL q i L q (54) while the reactive power, consisting of reactive power at the load, reactive power at the capacitor and reactive power at the inductance is given by 2 2 q f = VL q i L d − VL d i L q − ωC f (VL2d + VL2q ) + ωL f (i I,d + i I,q )

(55)

By solving Eqs. (54) and (55), with respect to the currents i L d and i L q one obtains i Ld =

p f VL d +q f VL q VL2 +VL2q

+ ωC f VL q −

p f VL q −q f VL d VL2 +VL2q

− ωC f VL d +

d

i Lq =

ωL f VL q (i I2 +i I2q )

(56)

d

(VL2 +VL2q ) d

d

ωL f VL d (i I2 +i I2q )

(57)

d

(VL2 +VL2q ) d

Using Eqs. (53), (56) and (57) one obtains the state-space description of the inverter’s dynamics ⎡

[

p f VL d +q f VL q +ωC f VL2 +VL2q

ωL f VL q (i I2 +i I2q )

] ⎤

VL q − ⎤ ⎢ VL d [ d ]⎥ ⎢ ⎥ 2 2 ωL V (i +i ) ⎥ ⎢ ⎥ p V −q V f L f L f L I I d ⎢ d q q d d ⎥ ⎢ VL q ⎥ = ⎢ −ωVL d+C1f i Iq−C1f −ωC V + 2 2 f L 2 2 d VL +VL q (VL +VL q ) ⎢ ⎥ ⎦ ⎣ d d dt i Id ⎢ ⎥ 1 ωi V − ⎣ ⎦ I L i Iq q d Lf 1 −ωi Id − L f VL q ⎡ ⎤ 0 0 ] [ ⎢ 0 0 ⎥ V Id ⎥ 1 (58) +⎢ ⎣ L f 0 ⎦ V Iq 0 L1f ⎡

ωVL q+C1f i Id−C1f

while the measurement equation of the inverter’s model is

d (VL2 +VL2q ) d

Nonlinear Optimal Control for VSI-fed Three-phase Asynchronous Motors

⎛ ⎞ ( ) ( ) ( ) VL d ⎟ y1 VL d 1000 ⎜ ⎜ VL q ⎟ = = ⎝ y2 VL q i Id ⎠ 0100 i Iq

143

(59)

and by using the state variables notation x1 = VL d , x2 = VL q , x3 = i Id and x4 = i Iq one has ⎤ ⎡ ωL f x2 (x32 +x42 ) 1 1 p f x1 +q f x2 ⎡ ⎤ x − [ + ωC x − ] + ωx 2 3 2 2 f 2 2 2 C C x1 +x2 (x1 +x2 ) f f x1 ⎥ ⎢ ⎥ ⎢ −ωx1 + 1 x4 − 1 [ p f x22 −q2f x1 − ωC f x1 + ωL f x21 (x322+x42 ) ] ⎥ d ⎢ x 2 ⎢ ⎥=⎢ ⎥ Cf Cf x1 +x2 (x1 +x2 ) ⎥ dt ⎣ x3 ⎦ ⎢ ωx4 − L1f x1 ⎦ ⎣ x4 −ωx3 − L1f x2 ⎤ ⎡ 0 0 [ ] ⎢ 0 0 ⎥ u1 ⎥ ⎢ +⎣ 1 0 ⎦ (60) u2 Lf 1 0 Lf while the measurement equation of the inverter’s model is ⎛ ⎞ ( ) ( ) ( ) x1 ⎟ y1 VL d 1000 ⎜ ⎜x2 ⎟ = = y2 VL q 0 1 0 0 ⎝x3 ⎠ x4

(61)

thus, the inverter’s model is written in the nonlinear state-space form x˙ = f (x) + G(x)u y = h(x)

(62)

where f (x)∈R4×1 , G(x)∈R4×1 and h(x)∈R2×4 .

4 Dynamic Model of the VSI-fed Induction Motor 4.1 State-space Description of the VSI-fed Induction Motor The diagram of the voltage Source Inverter-fed Induction (Asynchronous) Machine is shown in Fig. 3. As it was previously analyzed, the state vector of the induction motor is defined as x = [θ, ω, ψsd , ird , irq , ρ]T . The equations that constitute the state-space model of the motor are given by [28]

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Fig. 3 Voltage source inverter-fed induction (asynchronous) motor (VSI-IM)

⎞ ⎛ ⎛ ⎞ ⎛ x2 0 x˙1 T ⎟ ⎜ 0 μx3 x5 − JL ⎜x˙2 ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ −ax3 + a M x4 ⎜x˙3 ⎟ ⎜ 0 ⎟ ⎜ ⎜ ⎟=⎜ 2 +⎜ ⎜x˙4 ⎟ ⎜ −γ x4 + αβx3 + n p x2 x5 + α M x5 ⎟ ⎜ σ 1L ⎟ x3 ⎜ ⎟ ⎜ ⎜ s ⎝x˙5 ⎠ ⎝−γ x5 − βn p x2 x3 − n p x2 x4 − α M x4 x5 ⎟ ⎠ ⎝ 0 x3 x˙6 0 −n p x2 + a M x5 x3

0 0 0 0 1 σ Ls

⎞ ⎟ ⎟( ) ⎟ vsd ⎟ ⎟ vsq ⎟ ⎠

(63)

0

The state vector of the inverter is defined as x = [x7 , x8 , x9 , x10 ]T and the control inputs vector is [u 1 , u 2 ]T = [VId , VIq ]T . The equations that constitute the state-space model of the inverter are given by ⎡ ⎤ p x +q x ωL x (x 2 +x 2 ) ⎤ ωx8 + C1f x9 − C1f [ f x 27+x 2f 8 + ωC f x8 − f x82 +x9 2 10 ] x˙7 7 8 7 8 ⎢ ⎥ 2 10 ) ⎥ ⎢ x˙8 ⎥ ⎢ −ωx7 + 1 x10 − 1 [ p f x28 −q2f x7 − ωC f x7 + ωL f x72(x92 +x ]⎥ ⎢ ⎥=⎢ Cf Cf x7 +x8 x7 +x82 ⎥ ⎣ x˙9 ⎦ ⎢ ωx10 − L1f x7 ⎣ ⎦ x˙10 −ωx9 − L1f x8 ⎡ ⎤ 0 0 [ ] ⎢ 0 0 ⎥ u1 ⎥ 1 (64) +⎢ ⎣ L f 0 ⎦ u2 0 L1f ⎡

To arrive at the model of the VSI-fed induction motor, the control inputs of the motor which are the stator voltage variables vsd and vsq are substituted by the inverter’s

Nonlinear Optimal Control for VSI-fed Three-phase Asynchronous Motors

145

output voltage variables VL ,d and VL ,q respectively. Thus the dynamic model of the VSI-fed induction motor comprises the following equations: x˙1 = x2

(65)

x˙2 = μx3 x5 −

TL J

(66)

x˙3 = −ax3 + a M x4 x˙4 = −γ x4 + αβx3 + n p x2 x5 +

α M x52 x3

x˙5 = −γ x5 − βn p x2 x3 − n p x2 x4 − x˙6 = −n p x2 + x˙7 = ωx8 + x˙8 = −ωx7 +

1 Cf

x9 −

1 1 x10 − Cf Cf

[

1 Cf

[

p f x7 +q f x8 x72 +x82

(67) +

α M x4 x5 x3

1 x σ Ls 7

(68)

+

(69)

1 x σ Ls 8

a M x5 x3 + ωC f x8 −

(70) 2 ωL f x8 (x92 +x10 ) x72 +x82

]

2 ωL f x7 (x92 + x10 ) p f x8 − q f x7 − ωC x + f 7 2 2 2 2 x7 + x8 x7 + x8

x˙9 = ωx10 − x˙10 = −ωx9 −

1 1 x7 + u1 Lf Lf 1 1 x8 + u2 Lf Lf

(71) ] (72)

(73)

(74)

4.2 Proof of Differential Flatness Properties of the VSI-fed Induction Motor It can be proven that the VSI-fed induction motor is a differentially flat system [25]. The flat output vector of the VSI-fed induction motor is defined as Y = [y1 , y2 ]T = [x1 , x6 ]T where x1 = θ and x6 = ρ. From Eq. (65) one has x2 = x˙1 ⇒x2 = h 2 (y, y˙ )

(75)

that is x2 is a differential function of the flat outputs of the system. Next, considering that the load torque of the motor is given by TL = mgl sin x1 , from Eq. (66) one obtains mgl x˙2 = μx3 x5 − sin x1 J

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G. Rigatos et al.

Then: x3 x5 =

1 mgl [x˙2 − sin x1 ] μ J

(76)

Next, from Eq. (70) one has: x˙6 = −n p x2 + Then: x5 =

a M x5 x3

1 [x˙6 + n p x2 ]x3 aM

(77)

By substituting the second row of Eqs. (77) in (76) one has [ ] 1 1 mgl 2 [x˙6 + n p x2 ]x3 = x˙2 − sin x1 aM μ J Then: x32 =

aM J x˙2 − mgl sin x1 × μJ x˙6 + n p x2

(78)

Equation (78) signifies that x3 is a differential function of the flat outputs of the system, that is (79) x3 = h 3 (y, y˙ ) Next, from Eq. (79) and the second row of Eq. (77) it can be also concluded that state variable x5 is also a differential function of the flat outputs of the system, that is (80) x5 = h 5 (y, y˙ ) Next, from Eq. (67) one has that x˙3 = −ax3 + a M x4 Then: x4 =

1 [x˙3 + ax3 ] = h 4 (y, y˙ ) aM

(81)

which signifies that state variable x4 is also a differential function of the flat outputs of the system. From Eq. (68) one can solve for state variable x7 . This gives [ x7 = σ L s

] x52 x˙4 + γ x4 − αβx3 − n p x2 x5 − a M = h 7 (y, y˙ ) x2

(82)

Nonlinear Optimal Control for VSI-fed Three-phase Asynchronous Motors

147

Consequently, x7 is a differential function of the system’s flat outputs. From Eq. (69) one can solve for x8 . This gives [ x8 = σ L s

] x4 x5 −γ x5 − βn p x2 x3 − n p x2 x4 − α M = h 8 (y, y˙ ) x3

(83)

Consequently. x8 is a differential function of the system’s flat outputs. Next, one multiplies Eq. (71) with x7 and Eq. (72) with x8 . This gives 1 x7 x9 Cf

x7 x˙7 = ωx7 x8 + −

1 Cf

[

2 ωL f x7 x8 (x92 + x10 p f x7 2 + q f x7 x8 ) + ωC f x7 x8 − 2 2 2 2 x7 + x8 x7 + x8

x8 x˙8 = −ωx7 x8 + −

1 Cf

[

1 x8 x10 Cf

2 ωL f x7 x8 (x92 + x10 ) p f x8 2 − q f x7 x8 − ωC f x7 x8 + 2 2 2 2 x7 + x8 x7 + x8

] (84)

] (85)

By adding Eqs. (84) and (85) and after intermediate operations one obtains x7 x˙7 + x8 x˙8 =

pf 1 (x7 x9 + x8 x10 ) − Cf Cf

(86)

Equation (86) allows to solve for x10 thus giving x10 =

C f (x7 x˙7 + x8 x˙8 ) + p f − x7 x9 x8

(87)

Using that state variables x7 and x8 are differential functions of the system’s flat outputs one has (88) x10 = h a10 (y, y˙ ) + h b10 (y, y˙ )x9 where: C f (x7 x˙7 + x8 x˙8 ) + p f x8 x7 b = h 10 (y, y˙ ) = − x8

h a10 = h a10 (y, y˙ ) = h b10

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G. Rigatos et al.

By substituting Eqs. (88) into (71) one gets 1 1 x˙7 = ωx8 + x9 − Cf Cf

[

p f x7 + q f x8 x72 + x82

+ ωC f x8 −

ωL f x8 (x92 + [h a10 + h b10 x9 ]2 ) x72 + x82

]

(89)

After intermediate operations one obtains a binomial of the form ax92 + bx9 + c = 0

(90)

where: 2

1 ω0 L f x8 + h b10 (y, y˙ ) Cf x72 + x82 1 1 2ωL f x8 a b= − h (y, y˙ )h b10 (y, y˙ ) Cf C f x72 + x82 10 [ ] ωL f x8 a 1 p f x7 + q f x8 2 − ωC x + h (y, y ˙ ) c = −x˙7 + ωx8 − f 8 Cf x72 + x82 x72 + x82 10

a=−

This binomial can be solved with respect to x9 , thus proving that x9 is a differential function of the flat outputs, or equivalently x9 = h 9 (y, y˙ )

(91)

By substituting Eqs. (91) into (88) one has that x10 is also a differential function of the flat outputs of the system, that is x10 = h 10 (y, y˙ )

(92)

Next, from Eqs. (73) and (74) one solves for u 1 and u 2 respectively which gives u 1 = L f [x˙9 − ωx10 +

1 x7 ] Lf

(93)

u 2 = L f [x˙10 + ωx9 +

1 x8 ] Lf

(94)

From Eq. (93) one has that control input u 1 is a differential function of the flat outputs of the system, that is (95) u 1 = h u 1 (y, y˙ )

Nonlinear Optimal Control for VSI-fed Three-phase Asynchronous Motors

149

From Eq. (94) one has that control input u 2 is a differential function of the flat outputs of the system, that is u 2 = h u 2 (y, y˙ ) (96) Consequently, it has been proven that all state variables and the control inputs of the VSI-fed induction motor can be written as differential functions of the system’s flat outputs. Therefore, this system is differentially flat. The differential flatness property signifies that: (i) the system can be transformed into the input-output linearized form through successive differentiations of its flat outputs, that is of x1 = θ and x2 = ρ, (2) one can compute setpoints for all state variables of the system after defining setpoints for the above-noted flat outputs.

5 Linearization and Control of the VSI-fed Induction Motor 5.1 Linearized Model of the Induction Motor The dynamic model of the VSI-fed induction motor undergoes approximate linearization around the temporary operating point (x ∗ , u ∗ ) where x ∗ is the present value of the system’s state vector and u ∗ is the last sampled value of the control inputs vector. The linearization takes place at each time-step of the control algorithm and which is based on Taylor series expansion and on the computation of the system’s Jacobian matrices. The initial state-space form of the VSI-fed induction motor is x˙ = f (x) + g(x)u

(97)

while the linearized model is in the form x˙ = Ax + Bu + d~

(98)

The Jacobian matrices of the linearized state-space model of the system are given by A = ∇x [ f (x) + g(x)u] |(x ∗ ,u ∗ ) = ∇x [ f (x)] |(x ∗ ,u ∗ )

(99)

B = ∇u [ f (x) + g(x)u] |(x ∗ ,u ∗ ) = g(x) |(x ∗ ,u ∗ )

(100)

The linearization approach which has been followed for implementing the nonlinear optimal control scheme results into a quite accurate model of the system’s dynamics. Consider for instance the following affine-in-the-input state-space model

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G. Rigatos et al.

x˙ = f (x) + g(x)u = [ f (x ∗ ) + ∇x f (x) |x ∗ (x − x ∗ )] + [g(x ∗ ) + ∇x g(x) |x ∗ (x − x ∗ )]u ∗ +g(x ∗ )u ∗ + g(x ∗ )(u − u ∗ ) + d~1 = [∇x f (x) |x ∗ +∇x g(x) |x ∗ u ∗ ]x + g(x ∗ )u −[∇x f (x) |x ∗ +∇x g(x) |x ∗ u ∗ ]x ∗ + f (x ∗ ) + g(x ∗ )u ∗ + d~1 (101) where d~1 is the modelling error due to truncation of higher order terms in the Taylor series expansion of f (x) and g(x). Next, by defining A = [∇x f (x) |x ∗ +∇x g(x) |x ∗ u ∗ ], B = g(x ∗ ) one obtains x˙ = Ax + Bu − Ax ∗ + f (x ∗ ) + g(x ∗ )u ∗ + d~1

(102)

Moreover by denoting d~ = −Ax ∗ + f (x ∗ ) + g(x ∗ )u ∗ + d~1 about the cumulative modelling error term in the Taylor series expansion procedure one has x˙ = Ax + Bu + d~

(103)

which is the approximately linearized model of the dynamics of the system of Eq. (98). The term f (x ∗ ) + g(x ∗ )u ∗ is the derivative of the state vector at (x ∗ , u ∗ ) which is almost annihilated by −Ax ∗ . Computation of the Jacobian matrix A = ∇x [ f (x)] |(x ∗ ,u ∗ ) ∂ f1 ∂ x4

First row of the Jacobian matrix A = ∇x [ f (x)] |(x ∗ ,u ∗ ) : ∂∂ xf11 = 0, ∂∂ xf12 = 1, ∂∂ xf13 = 0, = 0, ∂∂ xf15 = 0, ∂∂ xf16 = 0, ∂∂ xf17 = 0, ∂∂ xf18 = 0, ∂∂ xf19 = 0, ∂∂xf101 = 0.

Second row of the Jacobian matrix A = ∇x [ f (x)] |(x ∗ ,u ∗ ) : ∂∂ xf21 = − mgl cos x1 , J ∂ f2 ∂ f2 ∂ f2 ∂ f2 ∂ f2 ∂ f2 ∂ f2 ∂ f2 ∂ f2 = 0, = μx , = 0, = μx , = 0, = 0, = 0, = 0, = 5 ∂ x4 3 ∂ x6 ∂ x2 ∂ x3 ∂ x3 ∂ x7 ∂ x8 ∂ x9 ∂ x10 0. Third row of the Jacobian matrix A = ∇x [ f (x)] |(x ∗ ,u ∗ ) : ∂∂ xf31 = 0, ∂∂ xf32 = 0, −α, ∂∂ xf34 = α M, ∂∂ xf35 = 0, ∂∂ xf36 = 0, ∂∂ xf37 = 0, ∂∂ xf38 = 0, ∂∂ xf39 = 0, ∂∂xf103 = 0.

∂ f3 ∂ x3

=

∂ f4 = 0, ∂∂ xf42 = n p x5 , ∂ x1 1 , ∂ f4 = 0, ∂∂ xf49 = 0, σ L s ∂ x8

Fourth row of the Jacobian matrix A = ∇x [ f (x)] |(x ∗ ,u ∗ ) : ∂ f4 = αβ ∂ x3 ∂ f4 = 0. ∂ x10



α M x52 x32

,

∂ f4 ∂ x4

= −γ ,

∂ f4 ∂ x5

= n p x2 ,

∂ f4 ∂ x6

= 0,

∂ f4 ∂ x7

=

Fifth row of the Jacobian matrix A = ∇x [ f (x)] |(x ∗ ,u ∗ ) : ∂∂ xf51 = 0, ∂∂ xf52 = −βn p x3 , ∂ f5 x5 ∂ f 5 = βn p x2 + α Mxx24 x5 , ∂∂ xf54 = −n p x2 − α M , ∂ x5 = −γ − α Mx3x4 , ∂∂ xf56 = 0. ∂∂ xf57 = 0, ∂ x3 x2 ∂ f5 ∂ x8

=

1 , ∂ f5 σ L s ∂ x9

3

= 0,

∂ f5 ∂ x10

= 0.

3

Nonlinear Optimal Control for VSI-fed Three-phase Asynchronous Motors

Sixth row of the Jacobian matrix A = ∇x [ f (x)] |(x ∗ ,u ∗ ) : ∂ f6 = − a Mx 2x5 , ∂∂ xf64 = 0, ∂∂ xf65 = axM3 , ∂∂ xf66 = 0, ∂∂ xf67 = 0, ∂∂ xf68 = 0, ∂ x3 3

∂ f7 ∂ x3

∂ f6 ∂ x1 ∂ f6 ∂ x9

Seventh row of the Jacobian matrix A = ∇x [ f (x)] |(x ∗ ,u ∗ ) : = 0, ∂∂ xf74 = 0, ∂∂ xf75 = 0, ∂∂ xf76 = 0,

151

= 0, ∂∂ xf62 = −n p , = 0, ∂∂xf106 = 0. ∂ f7 ∂ x1

= 0,

∂ f7 ∂ x2

= 0,

2 )(2x7 ) 1 p f (x72 + x82 ) − ( p f x7 + q f x9 )(2x7 ) ωL f x8 (x92 + x10 ∂ f7 =− + 2 2 2 2 2 2 ∂ x7 Cf (x7 + x8 ) (x7 + x8 )

[

q f (x72 + x82 ) − ( p f x7 + q f x8 )(2x8 ) + ωC f (x72 + x82 )2 ] 2 ωL f (x92 + x10 )(x72 + x82 ) − 2ωL f x8 (x92 + 2x10 )x8 − (x72 + x82 )2

∂ f7 1 =ω− ∂ x8 Cf

2ωL f x8 x9 ∂ f 7 2ωL f x8 x10 1 ∂ f7 =− − 2 , =− 2 2 ∂ x9 Cf ∂ x10 x7 + x8 x7 + x82 Eighth row of the Jacobian matrix A = ∇x [ f (x)] |(x ∗ ,u ∗ ) : ∂ f8 = 0, ∂∂ xf85 = 0, ∂∂ xf86 = 0, ∂ x4

∂ f8 ∂ x1

= 0,

∂ f8 ∂ x2

= 0,

∂ f8 ∂ x3

= 0,

[

−q f (x72 + x82 ) − 2( p f x8 − q f x7 )x7 − ωC f (x72 + x82 )2 ] 2 2 ωL f (x92 + x10 )(x72 + x82 ) − 2ωL f x7 (x92 + x10 )x7 + (x72 + x82 )2

1 ∂ f8 = −ω − ∂ x7 Cf

2 2ωL f x7 (x92 + x10 )x8 ∂ f8 1 p f (x72 + x82 ) − 2( p f x8 − q f x7 )x8 − =− 2 2 2 2 2 2 ∂ x8 Cf (x7 + x8 ) (x7 + x8 )

2ωL f x7 x9 ∂ f 8 2ωL f x7 x10 ∂ f8 1 = 2 , = + ∂ x9 ∂ x10 Cf x72 + x82 x7 + x82 Ninth row of the Jacobian matrix A = ∇x [ f (x)] |(x ∗ ,u ∗ ) : ∂∂ xf91 = 0, 0, ∂∂ xf94 = 0, ∂∂ xf95 = 0, ∂∂ xf96 = 0, ∂∂ xf97 = L1f , ∂∂ xf98 = 0, ∂∂ xf99 = 0, ∂∂xf109 = ω.

∂ f9 ∂ x2

= 0,

∂ f9 ∂ x3

=

Tenth row of the Jacobian matrix A = ∇x [ f (x)] |(x ∗ ,u ∗ ) : ∂∂fx101 = 0, ∂∂fx102 = 0, ∂∂fx103 = 0, ∂∂fx104 = 0, ∂∂fx105 = 0, ∂∂fx106 = 0, ∂∂fx107 = 0, ∂∂fx108 = − L1f , ∂∂fx109 = −ω, ∂∂ xf1010 = 0.

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5.2 Stabilizing Feedback Control After linearization around its current operating point (x ∗ , u ∗ ), the dynamic model of the VSI-IM system is written as [28] x˙ = Ax + Bu + d1

(104)

Parameter d1 stands for the linearization error in the VSI-IM’s model appearing previously in Eq. (104). The reference setpoints for the VSI-IM’s state vector are d ]. Tracking of this trajectory is achieved after applying denoted by xd = [x1d , · · · , x10 ∗ the control input u . At every time instant the control input u ∗ is assumed to differ from the control input u appearing in Eq. (104) by an amount equal to Δu, that is u ∗ = u + Δu (105) x˙d = Axd + Bu ∗ + d2 The dynamics of the controlled system described in Eq. (104) can be also written as

x˙ = Ax + Bu + Bu ∗ − Bu ∗ + d1

(106)

and by denoting d3 = −Bu ∗ + d1 as an aggregate disturbance term one obtains x˙ = Ax + Bu + Bu ∗ + d3

(107)

By subtracting Eqs. (105) from (107) one has x˙ − x˙d = A(x − xd ) + Bu + d3 − d2

(108)

By denoting the tracking error as e = x − xd and the aggregate disturbance term as d~ = d3 − d2 , the tracking error dynamics becomes e˙ = Ae + Bu + d~

(109)

For the approximately linearized model of the system a stabilizing feedback controller is developed. The controller has the form u(t) = −K e(t)

(110)

with K = r1 B T P where P is a positive definite symmetric matrix which is obtained from the solution of the Riccati equation [28] ( AT P + P A + Q − P

2 1 B BT − 2 L LT ρ r

) P=0

(111)

Nonlinear Optimal Control for VSI-fed Three-phase Asynchronous Motors

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Fig. 4 Diagram of the control scheme for the voltage source inverter-fed induction motor (with potential application to an electric vehicle)

where Q is a positive semi-definite symmetric matrix. The previously analyzed concept about the nonlinear optimal control loop for the VSI-IM system is given in Fig. 4. Whereas the Linear Quadratic Regulator (LQR) is the solution of the quadratic optimal control problem using Bellman’s optimality principle, H∞ control is the solution of the optimal control problem under model uncertainty and external perturbations. The cost function that is subject to minimization in the case of LQR comprises a quadratic term of the state vector’s tracking error, as well as a quadratic term of the variations of the control inputs. In the case of H∞ control the cost function is extended with the inclusion of a quadratic term of the cumulative disturbance and model uncertainty inputs that affect the model of the controlled system. In the case of the VSI-IM, the model is nonlinear and is also affected by uncertainty and external perturbations. By applying approximate linearization to the VSI-IM’s dynamics one obtains a linear state-space description which is subject to modelling imprecision and exogenous disturbances. Therefore, one can arrive at a solution of the related optimal control problem only by applying the H∞ control approach. It is also noted that the solution of the H∞ feedback control problem for the VSIIM and the computation of the worst case disturbance that this controller can sustain, comes from superposition of Bellman’s optimality principle when considering that the VSI-IM system is affected by two separate inputs (i) the control input u (ii) the ~ Solving the optimal control problem for u that is cumulative disturbance input d(t). for the minimum variation (optimal) control input that achieves elimination of the

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state vector’s tracking error gives u = − r1 B T Pe. Equivalently, solving the optimal ~ that is for the worst case disturbance that the control loop can control problem for d, sustain gives d~ = ρ12 L T Pe.

6 Lyapunov Stability Analysis 6.1 Global Stability Proof Through Lyapunov stability analysis it will be shown that the proposed nonlinear control scheme assures H∞ tracking performance for the VSI-fed IM, and that in case of bounded disturbance terms asymptotic convergence to the reference setpoints is achieved [28]. The tracking error dynamics for the VSI-IM system is written in the form e˙ = Ae + Bu + L d~ (112) where in the VSI-fed IM’s case L∈R10×10 is the disturbance inputs gain matrix. Variable d~ denotes model uncertainties and external disturbances of the VSI-fed IM’s model. The following Lyapunov equation is considered V =

1 T e Pe 2

(113)

where e = x − xd is the tracking error. By differentiating with respect to time one obtains 1 1 V˙ = e˙ T Pe + e T P e˙ 2 2 1 ~ ~ T Pe + 1 e T P[ Ae + Bu + L d] = [Ae + Bu + L d] 2 2 1 1 ~ = [e T A T + u T B T + d~T L T ]Pe + e T P[ Ae + Bu + L d] 2 2 1 1 1 1 1 1 = e T A T Pe + u T B T Pe + d~T L T Pe + e T P Ae + e T P Bu + e T P L d~ 2 2 2 2 2 2 (114) The previous equation is rewritten as 1 1 1 ~ V˙ = e T ( A T P + P A)e + (u T B T Pe + e T P Bu) + (d~T L T Pe + e T P L d) 2 2 2 (115) Assumption: For given positive definite matrix Q and coefficients r and ρ there exists a positive definite matrix P, which is the solution of the following matrix equation

Nonlinear Optimal Control for VSI-fed Three-phase Asynchronous Motors

( A T P + P A = −Q + P

1 2 B BT − 2 L LT r ρ

155

) P

(116)

Moreover, the following feedback control law is applied to the system 1 u = − B T Pe r

(117)

By substituting Eqs. (116) and (117) into Eq. (115) and by performing intermediate operations one obtains 2 1 1 1 V˙ = e T [−Q + P( B B T − 2 L L T )P]e + e T P B(− B T Pe) + e T P L d~ 2 r 2ρ r [ ] 1 1 1 1 = − e T Qe+ e T P B B T Pe− 2 e T P L L T Pe − e T P B B T Pe+e T P L d~ 2 r 2ρ r 1 1 = = − e T Qe − 2 e T P L L T Pe + e T P L d~ 2 2ρ 1 T 1 T 1 1 = − e Qe − 2 e P L L T Pe + e T P L d~ + d~T L T Pe (118) 2 2ρ 2 2 Lemma: The following inequality holds 1 T e P L d~ + 2

1~ T 1 1 d L Pe − 2 e T P L L T Pe≤ ρ 2 d~T d~ 2 2ρ 2

(119)

Proof : The binomial (ρα − ρ1 b)2 is considered. Expanding the left part of the above inequality one gets successively: 1 2 b − 2ab ≥ 0 ρ2 1 2 2 1 ρ a + 2 b2 − ab ≥ 0 2 2ρ 1 2 1 2 2 ab − 2 b ≤ ρ a 2ρ 2 1 1 1 1 ab + ab − 2 b2 ≤ ρ 2 a 2 2 2 2ρ 2 ρ2a2 +

(120)

The following substitutions are carried out: a = d~and b = e T P L and the previous relation becomes 1 1 1 1 ~T T d L Pe + e T P L d~ − 2 e T P L L T Pe≤ ρ 2 d~T d~ 2 2 2 2ρ

(121)

Equation (121) is substituted in the last row of Eq. (118) and the inequality is enforced, thus giving

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1 1 V˙ ≤ − e T Qe + ρ 2 d~T d~ 2 2

(122)

Equation (122) shows that the H∞ tracking performance criterion is satisfied. The integration of V˙ from 0 to T gives succesively: ∫ 0

T

1 V˙ (t)dt ≤ − 2

Then:



T

2V (T ) + 0



T 0

1 ||e||2Q dt + ρ 2 2



T 0

∫ ||e||2Q dt

≤ 2V (0) + ρ

~ 2 dt ||d||

T

2

~ 2 dt ||d||

(123)

0

Moreover, if there exists a positive constant Md > 0 such that: ∫



~ 2 dt ≤ Md ||d||

0

then one gets



∞ 0

||e||2Q dt ≤ 2V (0) + ρ 2 Md

(124)

∫∞ Thus, the integral 0 ||e||2Q dt is bounded. Moreover, V (T ) is bounded and from the definition of the Lyapunov function V in Eq. (113) it becomes clear that e(t) will be also bounded since e(t) ∈ Ωe : Ωe = {e|e T Pe≤2V (0) + ρ 2 Md } According to the above and with the use of Barbalat’s Lemma one obtains: lim e(t) = 0

t−→+∞

Elaborating on the stability analysis up to Eq. (124), it can be noted that the proof of global asymptotic stability for the control loop of the VSI-fed IM is based on Eq. (122) and on the application of Barbalat’s Lemma. It uses the condition about the boundedness of the square of the aggregate disturbance and modelling error term d~ that affects the model. However, as explained in the previous stability analysis, the proof of global asymptotic stability is not restricted by this condition. By selecting the attenuation coefficient ρ to be sufficiently small and in particular to ~ 2 one has that the first derivative of the Lyapunov function satisfy ρ 2 < ||e||2Q /||d|| is upper bounded by 0. Therefore for the i-th time interval it is proven that the Lyapunov function defined in Eq. (113) is a decreasing one. This also assures that the Lyapunov function of the system defined in Eq. (29) will always have a negative first-order derivative.

Nonlinear Optimal Control for VSI-fed Three-phase Asynchronous Motors

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6.2 Robust State Estimation with Kalman Filtering The control loop has to be implemented with the use of information provided by a small number of sensors and by processing only a small number of state variables. For instance, one can implement feedback control without measuring state variables x3 , that is the d-axis component of the magnetic flux of the rotor. To reconstruct the missing information about the state vector of the VSI-fed IM it is proposed to use a filtering scheme and based on it to apply state estimation-based control [28]. The recursion of the H∞ Kalman Filter, for the VSI-fed IM’s model, can be formulated in terms of a measurement update and a time update part. Measurement update: ⎧ ⎨ D(k) = [I − θ W (k)P − (k) + C T (k)R(k)−1 C(k)P − (k)]−1 K (k) = P − (k)D(k)C T (k)R(k)−1 ⎩ ⌃ x (k) = ⌃ x − (k) + K (k)[y(k) − C⌃ x − (k)] Time update:



⌃ x − (k + 1) = A(k)x(k) + B(k)u(k) P − (k + 1) = A(k)P − (k)D(k) A T (k) + Q(k)

(125)

(126)

where it is assumed that parameter θ is sufficiently small to assure that the covariance −1 matrix P − (k) − θ W (k) + C T (k)R(k)−1 C(k) will be positive definite. When θ = 0 the H∞ Kalman Filter becomes equivalent to the standard Kalman Filter. One can measure only a part of the state vector of the VSI-fed IM, and can estimate through filtering the rest of the state vector elements.

7 Simulation Tests The performance of the proposed nonlinear optimal (H∞ ) control scheme for VSI-fed induction machines (IM) has been further confirmed through simulation experiments. To compute the stabilizing feedback gains of the H∞ controller, the algebraic Riccati equation of Eq. (116) had to be solved at each time-step of the control algorithm. The obtained results are depicted in Figs. 5–16. In these diagrams, the real value of the state vector elements is printed in blue colour, the estimated value which is generated by the H∞ Kalman Filter is plotted in green colour while the associated setpoints are marked in red colour. From the above noted results it can be observed that the nonlinear optimal control scheme enables the reliable functioning of the VSI-fed induction motor under variable operating conditions. Actually, in all test cases fast and accurate tracking of reference setpoints can be noticed. The performed experiments demonstrate tracking of time-varying setpoints as well as of setpoints that exhibit abrupt changes. Thus these experiments come to confirm the global stability properties of the nonlinear optimal control method. The

158 0.8

0.4

0.6 x (ψ )

0.3 rd

0.2

0.1

0 −0.2

10 30 20 time (iterations x Ts)

0

40

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30 10 20 time (iterations x Ts)

40

0

10 20 30 time (iterations x Ts)

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40

0.3

0.6

0.2

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x (i )

sd

0

5

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x2 (ω)

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x (i )

Fig. 5 Tracking of setpoint 1 for the VSI-IM system a convergence of state variables x2 to x5 to their reference setpoints (red line: setpoint, blue line: real value, green line: estimated value), b convergence of state variables x7 to x10 to their reference setpoints (red line: setpoint, blue line: real value, green line: estimated value)

G. Rigatos et al.

0.2

0 −0.1

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10 20 30 time (iterations x Ts)

−0.2

40

8 6 x8 (VLq)

2 0 Ld

x (V )

(a)

7

−2

2

−4

0

30 20 10 time (iterations x Ts)

0

40

20

10

10 Iq

(I )

20

10

0

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x

9

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20 10 30 time (iterations x Ts)

40

−20

(b)

transient performance of the control scheme depends on the values of the parameters of the method’s Riccati equation r , ρ and Q. Actually, by assigning relatively small values to r , the steady-state tracking error is eliminated. Moreover, the smallest value of ρ for which a valid solution of the aforementioned Riccati equation is obtained (a positive definite matrix P) is the one that provides the control loop with maximum robustness. Moreover, for relatively high values of the elements of the diagonal matrix Q faster convergence of the state variables to the associated setpoints is achieved. Indicative values for this parameters are r = 0.01, ρ = 0.1 and for the diagonal elements of matrix Q one has qii ∈[1 40] A comparison of the proposed nonlinear optimal (H∞ ) control method against other linear and nonlinear control schemes for VSI-fed induction motors, shows

Nonlinear Optimal Control for VSI-fed Three-phase Asynchronous Motors

e

1

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−0.5

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10

15 25 20 time (iterations x Ts)

30

35

40

0

5

10

20 15 25 time (iterations x Ts)

30

35

40

0.5

e2

Fig. 6 Tracking of setpoint 1 for the VSI-IM system a tracking error for state variables x2 (turn speed of the IM) and x3 (magnetic flux of the IM’s rotor), b control inputs u 1 (inverter’s voltage VI,d ) and u 2 (inverter’s voltage VI,q )

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u1 (V )

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15 25 20 time (iterations x Ts)

30

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40

0

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20 15 25 time (iterations x Ts)

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35

40

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−5 −10

(b)

the following: (1) unlike global linearization-based control approaches, such as Lie algebra-based control and differential flatness theory-based control, the optimal control approach does not rely on complicated transformations (diffeomorphisms) of the system’s state variables. Besides, the computed control inputs are applied directly on the initial nonlinear model of the VSI-fed induction motor and not on its linearized equivalent. The inverse transformations which are met in global linearization-based control are avoided and consequently one does not come against the related singularity problems. (2) unlike Model Predictive Control (MPC) and Nonlinear Model Predictive control (NMPC), the proposed control method is of proven global stability. It is known that MPC is a linear control approach that if applied to the nonlinear dynamics of the VSI-fed induction motor the stability of the control loop will be lost.

G. Rigatos et al. 0.6

0.4

0.4

0.3 rd

x3 (ψ )

x2 (ω)

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0.2 0

0.1

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10 20 30 time (iterations x Ts)

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40

0

10 20 30 time (iterations x Ts)

40

0

10 20 30 time (iterations x Ts)

40

0

10 20 30 time (iterations x Ts)

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30 20 10 time (iterations x Ts)

40

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0.2

0.4 sq

x (i )

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sd

x4 (i )

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0.2

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10 20 30 time (iterations x Ts)

−0.2

40

(a) 8 6

−5

−10

2

0

10 20 30 time (iterations x Ts)

20

10

10 (IIq)

20

10

0

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x

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0

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Id

x (I )

4

8

x (V ) Lq

0

7

x (V ) Ld

5

−10

−10 −20

0

30 20 10 time (iterations x Ts)

−20

40

(b)

Fig. 7 Tracking of setpoint 2 for the VSI-IM system a convergence of state variables x2 to x5 to their reference setpoints (red line: setpoint, blue line: real value, green line: estimated value), b convergence of state variables x7 to x10 to their reference setpoints (red line: setpoint, blue line: real value, green line: estimated value)

Nonlinear Optimal Control for VSI-fed Three-phase Asynchronous Motors

e

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−0.5

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10

25 20 15 time (iterations x Ts)

30

35

40

0

5

10

15 20 25 time (iterations x Ts)

30

35

40

0.5

e3

Fig. 8 Tracking of setpoint 2 for the VSI-IM system a tracking error for state variables x2 (turn speed of the IM) and x3 (magnetic flux of the IM’s rotor), b control inputs u 1 (inverter’s voltage VI,d ) and u 2 (inverter’s voltage VI,q )

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u1 (V )

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15 20 25 time (iterations x Ts)

30

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40

0

5

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15 20 25 time (iterations x Ts)

30

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2

u (VIq)

5

−5 −10

(b)

Besides, in NMPC the convergence of its iterative search for an optimum depends on initialization and parameter values selection and consequently the global stability of this control method cannot be always assured. (3) unlike sliding-mode control and backstepping control the proposed optimal control method does not require the statespace description of the system to be found in a specific form. About sliding-mode control it is known that when the controlled system is not found in the input-output linearized form the definition of the sliding surface can be an intuitive procedure. About backstepping control it is known that it can not be directly applied to a dynamical system if the related state-space model is not found in the triangular (backstepping integral) form. (4) unlike PID control, the proposed nonlinear optimal control method is of proven global stability, the selection of the controller’s parameters does not rely

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x (ω)

rd

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0.8

0.4 0.2

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10 20 30 time (iterations x Ts)

0

40

0

10 20 30 time (iterations x Ts)

40

0

10 20 30 time (iterations x Ts)

40

0

10 20 30 time (iterations x Ts)

40

0

10 20 30 time (iterations x Ts)

40

0.3

0.6

0.2

0.4 sq

x (i )

sd

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(a)

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10 (IIq)

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x

9

0

40

Id

x (I )

−6

4

−10 −20

−10

0

10 20 30 time (iterations x Ts)

−20

40

(b)

Fig. 9 Tracking of setpoint 3 for the VSI-IM system a convergence of state variables x2 to x5 to their reference setpoints (red line: setpoint, blue line: real value, green line: estimated value), b convergence of state variables x7 to x10 to their reference setpoints (red line: setpoint, blue line: real value, green line: estimated value)

Nonlinear Optimal Control for VSI-fed Three-phase Asynchronous Motors

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10

15 25 20 time (iterations x Ts)

30

35

40

0

5

10

15 25 20 time (iterations x Ts)

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35

40

0.5

e3

Fig. 10 Tracking of setpoint 3 for the VSI-IM system a tracking error for state variables x2 (turn speed of the IM) and x3 (magnetic flux of the IM’s rotor), b control inputs u 1 (inverter’s voltage VI,d ) and u 2 (inverter’s voltage VI,q )

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(b)

on a heuristic tuning procedure, and the stability of the control loop is assured in the case of changes of operating points (5) unlike multiple local models-based control the nonlinear optimal control method uses only one linearization point and needs the solution of only one Riccati equation so as to compute the stabilizing feedback gains of the controller. Consequently, in terms of computation load the proposed control method for the VSI-fed induction motor is much more efficient (Figs. 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and 16).

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0.2 0

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6 x (V ) Lq

2

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10 20 30 time (iterations x Ts)

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10 (IIq)

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10

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x

9

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Id

x (I )

4

8

−2

7

x (V ) Ld

(a)

−10

−10 −20

0

30 20 10 time (iterations x Ts)

−20

40

(b)

Fig. 11 Tracking of setpoint 4 for the VSI-IM system a convergence of state variables x2 to x5 to their reference setpoints (red line: setpoint, blue line: real value, green line: estimated value), b convergence of state variables x7 to x10 to their reference setpoints (red line: setpoint, blue line: real value, green line: estimated value)

Nonlinear Optimal Control for VSI-fed Three-phase Asynchronous Motors

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25 20 15 time (iterations x Ts)

30

35

40

0

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0.5

e3

Fig. 12 Tracking of setpoint 4 for the VSI-IM system a tracking error for state variables x2 (turn speed of the IM) and x3 (magnetic flux of the IM’s rotor), b control inputs u 1 (inverter’s voltage VI,d ) and u 2 (inverter’s voltage VI,q )

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8 Conclusions The chapter has proposed a new nonlinear optimal control method for VSI-fed induction motors, with potential use in the electric traction system of trains and electric vehicles. The method relies on approximate linearization of the state-space model of the VSI-fed induction motor with the use of first-order Taylor series expansion and through the computation of the associated Jacobian matrices. This linearization takes place at each sampling instance around a temporary operating points which is also recomputed at each time-step of the control algorithm. For the approximately linearized model of the VSI-fed induction motor a stabilizing H∞ feedback controller

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has been designed. To select the controller’s gains an algebraic Riccati equation had to be solved repetitively at each time-step of the control algorithm. To implement state esitmation-based control without the need to measure the entire state vector of the VSI-fed induction motor, the H∞ Kalman Filter has been used as a robust state estimator. The proposed control approach achieves a solution for the nonlinear optimal control problem of VSI-fed induction motors under model uncertainty and external perturbations. The global stability properties of the control scheme have been proven with the use of Lyapunov analysis. The new control method retains the advantages of linear optimal control, that is fast and accurate tracking of reference setpoints under moderate variations of the control inputs. Unlike most control approaches in the area of electric motors and power electronics which treat

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separately the problem of controller design for an induction motor and for the associated three-phase inverter, the chapter’s developments allow for solving the nonlinear optimal control problem for the jointsystem of the VSI-fed induction motor.

References 1. Abdelai, R., Faouzi, M.M.: Optimal control strategy of an induction motor for loss minimization using Pontryagin principle. Eur. J. Control 49, 94–106 (2011) 2. Alanis, A.Y., Sanchez, E.N., Loukianov, A.G.: Real-time discrete backstepping neural control for induction motors. IEEE Trans. Control Syst. Technol. 19, 259–267 (2011)

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24. Rigatos, G.: Modelling and Control for Intelligent Industrial Systems: Adaptive Algorithms in Robotics and Industrial Engineering. Springer (2011) 25. Rigatos, G.: Nonlinear Control and Filtering using Differential Flatness Approaches: Applications to Electromechanical Systems. Springer (2018) 26. Rigatos, G., Siano, P., Cecati, C.: A new non-linear H∞ feedback control approach for threephase voltage source converters, electric power components and systems. Taylor Francis 44, 302–312 (2015) 27. Rigatos, G., Siano, P., Wira, P., Profumo, F.: Nonlinear H∞ Feedback Control for Asynchronous Motors of Electric Trains. Journal of Intelligent Industrial Systems, Springer (2015) 28. Rigatos, G.: Intelligent Renewable Energy Systems: Modelling and Control. Springer (2016) 29. Rigatos, G., Siano, P., Jovanovic, M., Ademi, S., Wira, P., Tir, Z.: Nonlinear Optimal Control for Synchronous Reluctance Machines. IEEE CPE-PowerEng 2017, 2017 11th IEEE International Conference on Compatibility, Power Electronics and Power Engineering. Cadiz, Spain (2017) 30. Rigatos, G., Busawon, K.: Robotic Manipulators and Vehicles: Control. Springer, Estimation and Filtering (2018) 31. Rigatos, G., Siano, P., Marignetti, F., Gros, I.: A Nonlinear Optimal Control Aporoach for PM Linear Synchronous Motors. IEEE INDIN 2018, IEEE 16th Intl. Conference on Industrial Informatics. Porto, Portugal (2018) 32. Tilli, A., Conficoni, C.: Semiglobal uniform asymptotic stability of an easy-to-implement PLLlike sensorless observer for induction motors. IEEE Trans. Autom. Control 64, 3612–3618 (2016) 33. Toussaint, G.J., Basar, T., Bullo, F.: l H∞ Optimal Tracking Control Techniques for Nonlinear Underactuated Systems. In: Proc. IEEE CDC 2000, 39th IEEE Conference on Decision and Control. Sydney Australia (2000) 34. Traoré, D., de Leon, J., Glumineau, A.: Adaptive interconnected observer-based backstepping control design for sensorless induction motor. Automatica 48, 682–687 (2012) 35. Yan, L., Wang, F., Dou, M., Zhang, Z., Kennel, R., Rodriguez, J.: Active disturbance rejectionbased sliding-mode control in Model Predictive Control for induction machines. IEEE Trans. Ind. Electron. 67, 2574–2584 (2020) 36. Yildiz, R., Murat, B., Zerdali, E.: A comprehensive comparison of Extended and Unscented Kalman Filter for speed-sensorless control application of Induction motors. IEEE Trans. Ind. Inf. 16, 6423–6438 (2020) 37. Yin, Z., Li, G., Zhang, Y., Liu, J.: Symmetric strong-tracking Extended Kalman Filter-based sensorless control of induction motor drives for modeling error reduction. IEEE Trans. Ind. Inf. 15, 650–662 (2019) 38. Xu, D., Wang, B., Zhang, G., Wang, G., Yu, Y.: A review of sensorless control methods for AC motor drives. CES Trans. Electr. Mach. Syst. 2, 104–115 (2018) 39. Zhan, Z., Yu, J., Zhao, L., Yu, H., Liu, C.: Adaptive fuzzy control of induction motors stochastic nonlinear systems with input saturation based on command filtering. Inf. Sci. 463–464, 186– 195 (2018) 40. Zhang, L., Zhang, H., Obeid, H., Lagrouche, S.: Time-varying state observer-based twisting control of linear induction motor considering dynamic end effects with unknown local torque. ISA Trans. 93, 290–301 (2019) 41. Zhang, Y., Bai, Y., Yang, H., Zhang, B.: Low switching frequency model predictive control of three-level inverter-fed IM drives with speed sensorless and field weakening operations. IEEE Trans. Ind. Electron. 66, 4262–4272 (2019) 42. Zhao, Q., Yang, Z., Sun, X., Ding, Q.: Speed-sensorless control system of a bearingless onduction motor based on iterative control difference Kalman Filter. Int. J. Electorn. Taylor Francis 107, 1524–1542 (2020) 43. Zerdali, E., Barut, M.: The comparison of optimized Extended Kalman Filter for speedsensorless control of induction motors. IEEE Trans. Ind. Electron. 64, 4340–4351 (2017)

Photovoltaic Energy Fed DC Motor for Water Pump Hanen Abbes and Hafedh Abid

Abstract This chapter deals with the use of photovoltaic energy for direct current motor to drive water pump. The resort to clean renewable energy, instead of fossil fuels, is step up day by day. The contribution is to set up a water pump system based on the solar energy. To optimize solar photovoltaic generated power, maximum power point tracking method is usually required. Proposed system is made up an arrangement of solar panels, two DC-DC converters, and DC motor followed by a pump. In fact, the presence of the DC-DC converter in the set is very beneficial; it operates as an interface that aims to enhance the efficiency of photovoltaic array and ensure smooth start-up of the DC motor with definite control.The first DC-DC converter gives constant DC output. Then, the second converter supplies the DC motor. The centrifugal pump is controlled by a fuzzy supervisor so as to get the optimum flow rate of the pump. The designed water pump system is simulated and its diverse performances are evaluated by the means of MATLAB/simulink. Then, the system operation and its efficiency are verified. Keywords DC motor · Fuzzy system · Photovoltaic system · MPPT · Water pump

1 Introduction Solar energy is a freely available clean source in the world. As a renewable energy, solar power not only reduces dependence on foreign oil and fossil fuels but also causes no greenhouse gazs. Tunisia presents a good potential in renewable energy. H. Abbes (B) Laboratory of Computer and Embedded Systems (Lab-CES), National School of Engineering of Sfax, University of Sfax, Sfax, Tunisia e-mail: [email protected] H. Abid Laboratory of Sciences and Techniques of Automatic Control and Computer Engineering (Lab-STA), National School of Engineering of Sfax, University of Sfax, Sfax, Tunisia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Derbel et al. (eds.), Advances in Robust Control and Applications, Studies in Systems, Decision and Control 474, https://doi.org/10.1007/978-981-99-3463-8_6

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Thus, the government is trying to warrant a safe and secure energy future. The country potential of solar radiation ranges, on year, from 1800 kWh/m2 in the North to 2600 kWh/m2 in the South. Tunisian government, in 2009, undertook “Plan Solaire Tunisien” or Tunisia Solar Plan to reach 4.7 GW renewable energy capacity by 2030, using for photovoltaic systems, water heating systems and concentrated power units. Photovoltaic (PV) technology is an encouraging solution to harness the solar energy. An array of photovoltaic cells called also PV panels are widely used for electric power generation. In the last decade, grid associated with solar PV is a system which is broadly set up in household applications and industrial purposes with control techniques. Excess power from SPV may be injected to the grid and utilized later by other consumers. IRENA: International Renewable Energy Agency has estimated a 59% cost reduction of electricity provided by solar PV by 2025. In addition, IRENA specifies that solar PV module prices have dropped roughly 80% since 2009. As the solar panels cost has dramatically decreased, its use is widespread in various sectors. Photovoltaic energy affords electric energy in several cases, mainly in regions without electricity grid. A usefulness study for the establishment of 400 MW of a PumpedStorage Power Plant is now ongoing in the North of Tunisia Melah amount place. Yet, the electrical power coming from PV system is conditional on climate change such as temperature and insolation variations. By this way, tracking the maximum power point seems to be essential to get a maximum amount of power from photovoltaic installation [11]. Several researches relating to the MPPT algorithms study and analysis have been conducted in the literature, such as in [4]. They are examined while taking into account a variety of parameters; including easiness, implementation type, cost, time-response and accuracy. Classical approaches as well as intelligents approaches are often applied [3]. The serious problem is a scarcity of electric grid in isolated locations, in the majority of regions of the world. Hence, photovoltaic pumping application based on solar energy seems to be a promising key. The technology is comparable to the traditional pumping system, with the exception that it is powered by solar energy [12]. PV water pumping has grown in popularity in recent years because of the lack of energy and the rise in diesel prices. Pumped water flow rates are determined by incident solar energy and PV array size. A well-designed solar system ensures conspicuous long-term efficiency gains with regard to traditional pumping systems. Additionally, instead of using batteries for electricity storage, tanks can be used to store water. Likewise, in the current energy crisis, an irrigation system fed by solar power could be a viable option for peasants. In developing countries, agricultural production is extremely influenced by rains and water availability during summers. Nonetheless, sunshine is abundant in summers, so, much water can be pumped to fulfill water requirements [8–10]. To meet this challenge, researchers are blessed with pumping systems fed by solar energy in order to supply remote and industry establishments [24, 25]. A well run standalone photovoltaic pumping system is obtained by the set up of converters and motor drives with sophisticated power electronics [6]. A variety of motors types can be an essential item of this system. Amongst these motors, induction motor is distinguished by cheap cost and weight, as well as reliability and, most importantly,

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low maintenance requirements [16, 27]. Whereas, DC motors provide highly controllable speed [28]. By varying the armature or field voltage, it is possible to reach wide speed variation and together with this level of controllability, DC motors give the precision required by a broad range of industry applications. In addition, it offers high starting torque [23]. For these reasons, this chapter proposes an accurate model of water pump system, which is fed by a photovoltaic energy source, then, the focusing of the work, is to highly take advantage from the solar energy and to effectively monitor the system. Thereafter, this chapter is arranged as following: Sect. 2 touches at modelling PV pumping system. Section 2’s first part handles the model of photovoltaic panel and the MPPT algorithm. The second part gives a detailed design of the DC motor and the third part presents a clear model of the centrifugal pump. In Sect. 3, discussions and simulation results show the performances of proposed system and T-S fuzzy supervisor. Finally, the chapter is closed by a conclusion.

2 Photovoltaic Pumping System The main components of photovoltaic pumping system are shown by Fig. 1. The model schema includes PV energy generator, DC-DC converter, monitoring by MPPT item so as to have the maximum amount of energy, assembled with a second DC-DC converter. This latest feeds a direct current motor coupled to water pump.

Fig. 1 Pattern schema of proposed pump system

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2.1 Design of PV Array and MPPT Approach This section comprises two parts. The first is dedicated to PV array model and converter model, the second concerns the description of the MPPT method. Needless to say, the elementary component of photovoltaic panel is the photovoltaic cell. A photovoltaic array is composed of a big number of panels electrically related in series. In the literature, two basic mathematical models dominate: the model based on one diode and the model based on two diodes describing the PV operation under temperature and irradiation parameters variations [1]. As the cells are joined in series-parallels arrangement on a panel to obtain high power, the corresponding circuit of the PV array is depicted in Fig. 2.

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Equations describing the behavior of PV array, following temperature and irradiation changes, are summed up by Eqs. (1) and (2). The final current and voltage expression of PV array is equal to I P V : ⎡

I P V = N P I ph



⎞ ⎤ Rs I P V VP V N P VP V + + Rs I P V ⎢ ⎜ NS ⎟ ⎥ NP ⎟ Ns ⎜ ⎥− − N P Is ⎢ exp − 1 ⎣ ⎝ ⎠ ⎦ nVT Rsh

(1)

I ph current depends mostly on the sunshine and cell’s operating temperature, its expression is: I ph = (Ish + K I ΔT )

Fig. 2 Equivalent electric circuit of photovoltaic array

E En

(2)

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Fig. 3 Electrical circuit of BOOST converter

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Ish : current of short circuit of the cell at 25 ◦ C and 1 kW/m2 , K I : temperature coefficient of short-circuit current of the cell, E: solar sunlight amount (kW/m2 ), Is : cell’s saturation current depends on the cell temperature, VT =

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MPPT Approach

Boost Converter The role of the power converter is to allow the adjustment of PV generator to the load so as to acquire the peak power from PV source. DC-DC converter type is conditional on the intended purpose of the photovoltaic system. Numerous types of DC-DC converter are available [14]. We consider here the Boost type (Fig. 3). For t ∈ [0, αT ], the transistor switch is on. In this case, converter equations are: 1 di L = V1 L dt d V2 1 =− V2 R L Cs dt

(4) (5)

For t ∈ [αT , T ], the transistor switch is off. In this case, converter equations are: di L 1 = (V1 − V2 ) L dt d V2 1 1 = iL − V2 Cs R L Cs dt

(6) (7)

where, K , L and C S represent respectively the IGBT transistor switch, the inductance and the capacitor.

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Fig. 4 Organization chart of PandO approach

MPPT Perturb and Observe approach Perturb and Observe is the easiest among all MPPT algorithms, it is broadly used and it is inexpensive control method [2, 18, 19]. Owing to its simple implementation, we adopt this method to follow optimal operating point of PV panel. Figure 4 illustrates P and O’s flowchart. The process principle is to do a perturbation on the PV panel voltage whilst varying the duty ratio α. Subsequent to this perturbation, we compute the power given by the generator at instant k, then, it is compared to the previous one k − 1 instant. On condition that the power gets bigger, we go toward the peak power and the change on the duty ratio is uphold in the same direction. In the opposite side, if power becomes less, we go away from the MPP. So, the duty cycle variation should be reserved.

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2.2 Modeling of Direct Current Motor DC motors are used for multi-purpose applications [13, 15, 21]. Modeling a dynamic process is often necessary before the determination of a control law. In order to get an accurate model, you need to apply a suitable method among various methods. Design a numerical model is done through the determination of the discrete or continuous transfer function whose structure is often determined by empirical methods. To ensure the determination of a classic control technique, a such model is usually sufficient [5, 20]. The operating equations of the separately excited direct current motor are as following: Electrical equation V = RI + L

dI +E dt

(8)

where: V : Voltage supply of the armature R: Total resistance of the armature (with compensation winding) I : armature current L: Total leakage inductance of the armature E : Back emf developed. Mechanical equation C m − Cr = J

dω + fω dt

(9)

where: Cm : motor torque Cr : Resistant torque J : Overall moment of inertia (motor and driven machine) f : Viscous coefficient of friction ω : Angular speed of rotation of the motor shaft. Electromechanical equations E = Km ω Cm = K m I

(10) (11)

Output power is equal to Pu drawn by Eq. (12) Pu = Cu ω

(12)

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Direct current motor is considered as a second order system of the hyper damped type (τe , (≥, 0.

(36)

i=1

For accomplishing asymptotically steering tracking with H∞ performance, the following optimization problem holds: ⎧

minβ Rμ = ΔV (k) + z υT (k)z υ (k) − β 2 ω T (k)ω(k) < 0

(37)

such us: ΔV (k) = V (x v (k + 1)) − V (xv (k)) = xv (k + 1)T Pμ xv (k + 1) − xv (k)T Pμ xv (k) (38) /∑ ∞ T (k)w(k) the set of initial conditions V (0) < ω(k) is L2 norm: ||w(k)||2 = w i=0 1 and the attraction set ε(Pμ , κ) = xvT (k)Pμ xv (k) < κ. For the control design the closed-loop system can be rewritten from (32), and (34) as follows: ⎧ ⎨ xv (k + 1) = Avμ xv (k) + Bμvu K μ Hμ−1 C1 xv (k) + Bμvω ω(k) (39) y (k) = C1 xv (k) ⎩ v z υ (k) = C2 xv (k) Theorem 1 For a given positive scalar β the closed loop system (39) is asymptotically stable with H∞ performance, if there exist a symmetric positive definite matrices Q i ∈ Rn xv n xv , and matrices K i ∈ Rn u n xv , Hi ∈ Rn xv n xv , G i ∈ Rn u n u , solution of the following LMI problem: ⎤ −X Tj − X j + Q i 0 ∗ ∗ ⎢ 0 −β 2 I ∗ ∗ ⎥ ⎥ ⎢ vu v ⎣ A X j + B K i C1 B vω −Q i ∗ ⎦ < 0 i i i 0 0 −I C2 X j ⎡

(40)

(i, j) ∈ Ir2 , s = 1, ..., 2m Hi C1 = C1 X i , i ∈ Ir

(41)

Proof ∀J : • if ω = 0, then, V (k + 1) − V (k) < −z υT (k)z υ (k) ≤ 0 ∑ J =∞ T • if ω /= 0, then, V (J ) ≤ V (0) + β 2 i=0 ω (k)ω(k) ≤ 1 + β 2 ω 2 = κ ∑ J =∞ T z υ (k)z υ (k) < then, ε(Pμ , κ) is attraction set and if J goes to the infinity, i=0 ∑ J =∞ T 2 β i=0 ω (k)ω(k) + V (0)

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then, the H ∞ performance will be ensured under the zero-initial conditions with a β disturbance attenuation level. Considering the optimization problem (37) and via a Schur complement, we obtain: [ ]T ([ T ] ⌃ [ ] x A ⌃⌃ P V (k + 1) − V (k) + z υT (k)z υ (k) − β 2 ω T (k)ω(k) = A B i T ⌃ B [ T] ]) [ ω] [ (42) ] C2 [ x Pi 0 C2 0 − + 0 ω 0 β2 I ⌃ = Avμ + Bμvu K μ Hμ−1 C1 , ⌃ with: A B = Bμvω . From the last inequality the closed loop system is stable if: [

] [ ][ ]T [ ] ⌃ ⌃ ⌃ ⌃ Pμ 0 −Pμ 0 A B A B + u max if u min ≤ u(t) ≤ u max (49) σ(t) = sat (u(t)) = u(t) ⎩ −u max if u(t) < −u max where u max denotes the saturation level. Lemma 1 Let E be the set of m × m diagonal matrices whose diagonal elements are 1 or 0. Suppose that |δi | ≤ u i for all i ∈ Im where δi and u i denotes the i th element of δ ∈ Rm and u ∈ Rm , respectively [5]. if i ∈ Ψ (G i ) for x ∈ Rn , then

A. Kennouche et al. × 10-3

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4 2 0

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× 10-3 0 -2 -4 5

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1000 500 0

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Fig. 10 Vehicle states responses under wind force disturbance

Fig. 11 Saturation function

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Robust Saturated Control Based Static Output Feedback . . .

⎧ 2m ∑ ⎪ ⎪ = αs (E s u(k) + E s δ(k)) u) sat(u(k), ⎪ ⎪ ⎪ s=1 ⎪ ⎪ ⎪ ⎨∑ 2m αs = 1, 0 ≤ αs ≤ 1 ⎪ ⎪ s=1 ⎪ ⎪ ⎪ r ⎪ ∑ ⎪ ⎪ ⎩ δ(k) = μi (Φ)G i xv (k)

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i=1

| | { } | j | Ψ (G i ) = x(k) ∈ Rn , |gi x(k)| ≤ u i where E s denotes all the components of E, and E s = 1 − E s . G j is r × n matrix j and gi is the i th row of G j . Let us consider the Lyapunov function: V (k) = xvT (k)Pμ xv (k), Pμ =

r ∑

μi (Φ)Pi , Pi = PiT > 0

(51)

i=1

For accomplishing asymptotically saturated steering tracking with H∞ performance, the following optimization problem (37) holds: Lemma 2 The ellipsoid ε(Pμ , κ) is included in the polyhedron Ω(u), if and only if [5]: ( )−1 Pμ jT j ki ≤ u i2 (52) ∀i ∈ Im and j ∈ Ir : ki κ For the control design the closed-loop system can be rewritten from (32), (34) and (50) as follows: ⎧ ( ) ⎨ xv (k + 1) = Avμ xv (k) + Bμvu E s K μ (Hμ−1 )C1 + E s G μ C1 xv (k) + Bμvω ω(k) y (k) = C1 xv (k) ⎩ v z υ (k) = C2 xv (k) (53) Theorem 2 For a given positive scalar β the closed loop system (53) is asymptotically stable with H∞ performance, if there exist a symmetric positive definite matrices Q i ∈ Rn xv n xv , and matrices K i ∈ Rn u n xv , Hi ∈ Rn xv n xv , G i ∈ Rn u n u , solution of the following LMI problem: ⎡

⎤ 0 ∗ ∗ −X Tj − X j + Q i ⎢ 0 −β 2 I ∗ ∗ ⎥ ⎢ ⎥ ⎣ Av X j + B vu (E s K i C1 + E s N j C1 ) B vω −Q i ∗ ⎦ < 0 i i i 0 0 −I C2 X j

(54)

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(i, j ) ∈ Ir2 , s = 1, ..., 2m ⎡ 2 ⎤ u max ∗ ⎢ κ ⎥ ⎣ ⎦ ≥ 0, j ∈ Ir , i = 1...m jT Z i Pi

(55)

Hi C1 = C1 X i , i ∈ Ir

(56)

with: κ = 1 + β 2 ω 2 , such as β is minimum as possible. Proof For all ε(Pμ , κ) ⊂ Ω(u), if Rμ < 0 we get, ∀J : • if ω = 0, then, V (k + 1) − V (k) < −z υT (k)z υ (k) ≤ 0 ∑ J =∞ T • if ω /= 0, then, V (J ) ≤ V (0) + β 2 i=0 ω (k)ω(k) ≤ 1 + β 2 ω 2 = κ then, ε(Pμ , κ) is attraction set and if J goes to the infinity, J∑ =∞

z υT (k)z υ (k) < β 2

i=0

J∑ =∞

ω T (k)ω(k) + V (0)

i=0

then, the H ∞ performance will be ensured under the zero-initial conditions with a β disturbance attenuation level. Considering the optimization problem (37) and via a Schur complement, we obtain: [ ]T ([ T ] ⌃ [ ] x A ⌃⌃ Pi A V (k + 1) − V (k) + z υT (k)z υ (k) − β 2 ω T (k)ω(k) = B T ⌃ B [ T] ]) [ ω] [ (57) ] C2 [ x Pi 0 C 0 + − 2 0 ω 0 β2 I with:

⌃ = Avμ + Bμvu (E s K μ Hμ−1 + E s G μ )C1 , ⌃ A B = Bμvω

From the last inequality the closed loop system is stable if: [

] [ ][ ]T [ ] ⌃ ⌃ ⌃ ⌃ −Pμ 0 Pμ 0 A B A B + 0 (10) Wδ = ωk : R −→ R , k=1

where δ is the energy is bounded exogenous signal, known. Remark 1 Using Finsler’s Lemma, and Schur complement a better result have been proposed making it possible to decouple the PD gain matrices of control law with the Lyapunov matrix system, as well as to introduce free decision variables, helping to reduce the conservatism, and with progressively more relaxed results via controllers with nested convex sum, see [5].

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3 Polytopic LPV System Descriptions For generality, we consider the LPV system (7) in its general state-space form as: [

E(ρk )xk+ = A(ρk )xk + B(ρk )sat (u k ) + Bw wk yk = C xk

(11)

where xk ∈ Rn x is the state of the system, sat (u k ) ∈ Rn u is the saturated control input, ωk ∈ Rn ω is the disturbance input, yk ∈ Rn y is the system output, and ρk ∈ Rn nl is the vector of varying parameters, whose measurement is available in real time for ]T [ gain scheduling control. Assume that the parameter ρk = ρ1,k . . . ρnl ,k is smooth and valued in the hypercube: ] } { [ Θ = (ρ1,k , . . . , ρnl ,k )T : ρ j ∈ ρ j,min , ρ j,max , j ∈ Ωnl

(12)

We assume that the PD state-space matrices Π (ρk ) of (11), with ∏ = {A, B, C, D} are continuous in the hypercube Θ, using the sector nonlinearity approach in [23], the state space matrices can be equivalently represented by: Π (ρk ) =

r ∑ i=1

μi (ρk )Πi /

re ∑

vk (ρk )Πk

k=1

r = 2nl , re = 2nl . The matrices of appropriate dimensions ∏i = {Ai , Bi } ∏k = {E k } represent the i-th and k-th local model. For i ∈ Ir and k ∈ Ire represent the set of r local linear sub-models in the left hand side, and re sub-models in the right hand side of NLPV descriptor model (11). The matrix C involved in (11) is assumed to be a full row rank, k is a current samples. It’s assumed that all parameters ρk , where n l is a number of nonlinearities in the left/right hand side of system (12), which are bounded, measurable and valued and continuous in the domain of an hypercube such that, ρimin and ρimax are known lower and upper bounds of ρk , where: ρk ∈ Rnl = {ρi | ρimin ≤ ρi,k ≤ ρimax , ∀k > 0, ∀ρk ∈ Θ}

(13)

The NLPV descriptor system (12) with bounded parameter can be represented by a polytopic form, where the polytopic coordinates are denotes μi (ρk ), vk (ρk ) and vary within the convex sets Ω1 and Ω2 respectively: Ω1 = μi (ρk ) ∈ Rr ; [μ1 (ρk ) , . . . . . . ., μr (ρk )]T

(14)

Robust Constrained Gain-Scheduled Static Output Feedback Controller for NLPV …

Ω2 = vk (ρk ) Rre ; [v1 (ρk ) . . . . . . ., vre (ρk )]T

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where the membership functions (MFs) μi (.) and vk (.) satisfy the convex sum property. ⎧ ⎧ ⎨vk (ρk ) ≥ 0 ⎨μi (ρk ) ≥ 0 re r ∑ ∑ (16) and μi (ρk ) = 1 vk (ρk ) = 1 ⎩ ⎩ i=1

k=1

So, Ai = Ai + Δ Ai ; Bi = Bi + ΔBi ; Ek = E k + ΔE k where, the uncertain matrices; ΔAi ∈ Rn x ×n x , ΔBi ∈ Rn x ×n u , ΔE k ∈ Rn x ×n x ,corresponding to the i-th and k-th sub-system contains the bounded uncertain terms which can be rewritten as: ΔAi,k = Hai Dai,k Nai ; ΔBi,k = Hbi Dbi,k Nbi and ΔE k,k = Hek Dek,k Nek , with Hai , Hbi , Hek , Nai , Nbi and Nek are known constant matrices and, Dai,k , Dbi,k and Dek,k are unknown matrices functions bounded, for all index ε = a, b or e and θ = i or k, T i ∈ Ir and k ∈ Ire , one has Dεθ,k .Dεθ,k ≤ I . The NLPV model (11) can be equivalently rewritten in the polytopic form: [ ∑re

vk (ρk )Ek = yk = C xk k=1

∑r i=1

μi (ρk ) (Ai xk = Bi sat(u k )) + Bw wk

(17)

Assumption 3 ([22]) The control input vector u k(l) is subject to symmetric amplitude limitation, that is: − u max(l) ≤ u k(l) ≤ u max l)l ∈ Il

(18)

where the control bound u max(l) > 0 is given, and l represents the l-th component of input control. The vector valued saturation function: sat (.) ∈ Rn u → Rn u is defined as: sat (u l ) = sign (u l ) min (|u l | , u max ) ; l ∈ Il (19) where u max > 0 denotes the given bound of the l-th input. Let us consider a nonlinear unconstrained Gain Scheduled SOF controller of the form: u k = Fμv (ρk )Hμv (ρk )−1 yk

(20)

where the matrices of appropriate dimensions Fμv (ρk ), Hμv (ρk ) are to be designed. These MF-dependent matrices are defined as: [

re r ∑ ] ∑ ] [ Fμv (ρ) Hμv (ρ) = μi (ρ) vk (ρ) Fik Hik . i=1 k=1

The nonsingularity of matrix Hμv (ρ) in (20) is examined in Theorem 1. We define the dead-zone nonlinearity ψ (.) : Rn u → Rn u as:

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Fig. 2 The proposed SOF constrained controller design scheme

) ( ψ u k(l) = u k(l) − sat (u k(l) )

(21)

( ) Assumption 4 DF Fi j , u max(l) denotes the polyhedral region associated with a matrix Fi j ∈ Rn u ×n x and a vector u max (l) ∈ Rn u defined by: ( ) DF Fi j , u max(l) = xk ∈ Rn x : |Fi j (l) xk | ≤ u max (l) ; ∀ l ∈ Il

(22)

( ) Fi j (l) is a component of vector Fi j and DF Fi j , u max(l) is polyhedral set consisting of states for which saturation does not occur. The architecture of the proposed SOF controller scheme is depicted on Fig. 2. For the control design of system (17), we propose a new PD nonquadratic Lyapunov function associated to the NLPV descriptor system as: V (xk , ρk ) = xkT E vT Pμ−1 (ρk )E v xk

(23)

∑ where Pμ = ri=1 μi (ρk )Pi , and Pi > 0, for i ∈ Ir . The level set associated with the Lyapunov function V (xk , ρk ) is defined as: Definition 1 ([22]) Let Pμ (ρk ) ∈ R n x ×n x be a symmetric PD Lyapunov matrix, and E vT Pμ−1 (ρk )E v ≥ 0, let us define the ellipsoids (Lyapunov surfaces) as Parameter Dependent Level Set (PDLS) associated to V (xk , ρk ) as follows: [

Ls = xk ∈ Rn x : V (xk , ρk ) = xkT E vT Pμ−1 (ρk )E v xk ≤ λ Ls0 = xk ∈ Rn x : V (xk , ρk ) = xkT E vT Pμ−1 (ρk )E v xk ≤ 1

(24)

The PDLS is contractively invariant set, because the estimate of Ls is the intersection of the ellipsoids, and is contained in the DoA which is nonconvex ∩due to its associated to the nonquadratic Lyapunov function, and is included in Dx Du . Proposition ([16]) A simple estimation of Ls and Ls0 are the intersection of the ellipsoids, then: [ ∩ ∩ E(E vT Pμ−1 (ρk )E v , λ) = ri=1 (k=1 re )E(E kT Pi−1 E k , λ) ⊆ Ls ∩ re ∩ (25) E(E vT Pμ−1 (ρk )E v , 1) = ri=1 Ek=1 E(E kT Pi−1 E k , 1) ⊆ Ls0

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The concept of invariant set is crucial to characterize the domain of attraction of the input-saturated system (17). This paper proposes a constructive LMI-based solution for the following control problem: Control Problem 1. Determine the nonlinear Gain scheduled SOF controller with the PD gain matrices Fμv(ρ) and Hμv(ρ) and a contractively invariant set Ls defined in (24), as large as possible inside ∩ Dx , of the closed-loop system , with any ρk ∈ Θ, in order to verifies Ls ⊆ Dx Du . Control problem 2. (P1 ) [Local stability]. Given a scalar α > 0, any closed-loop trajectory of the undisturbed system (i.e. wk = 0) starting from E(E kT Pi−1 E k , 1) converges exponentially to the origin with a decay rate α. (P2 ) [L2 -gain performance]. When wk /= 0 or wk ∈ Wδ2 , there exist positive scalar λ and γ such that: ∀xk ∈ Ls \{0}, the closed-loop trajectory remains inside the validity domain Dx defined in (9). Moreover the L2 -gain of the output vector yk is bounded as follows: ||yk ||22 < γ 2 ||wk ||22 + λ, ∀k > 0

(26)

And ∀xk ∈ Rn x \Ls , the trajectory of the closed-loop system converge toward Ls . The following technical lemma is useful for the subsequent theoretical development: Lemma 1 Consider matrices H jk ∈ Rn x ×n x , F jk ∈ Rn u ×n x , W jk ∈ Rn u ×n x , for all (i, j) ∈ (Ir × Ir ), k ∈ Ire . Let us define the following set: Du = {xk ∈ Rn x ×n x : |u k(l) − vk(l) | ≤ u max(l) }

(27)

If xk , u k(l) and vk(l) ∈ Du , then the inequality of the dead-zone nonlinearity ψ(u k(l) ), where u k(l) is defined in (21) satisfy the following PD generalized sector condition: ψ(u k(l) )T S j~(l) (ρk )−1 [ψ(u k(l) ) − vk(l) ] ≤ 0

(28)

Holds for any diagonal matrix S jk ∈ Rn u ×n u , and for any scalar functions: μi (ρk ) and vk (ρk ) satisfying the convex sum property, and vk(l) = [W jk (ρk )H jk (ρk )−1 ]l xk

(29)

4 LMI Based Gain-Scheduled SOF Controller The following theorem provides a numerically tractable solution to design a nonlinear SOF controller (20), that can stabilize exponentially the NLPV polytopic descriptor system (17) under an optimization problem.

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Theorem 1 Consider the uncertain and saturated NLPV descriptor system (17) with the validity domain Dx defined in (9). If there exist a symmetric positive definite matrices Pi = PiT , diagonal positive definite matrices S jk ∈ Rn u ×n u and H jk ∈ Rn x ×n x , F jk ∈ Rn u ×n x , W jk ∈ Rn u ×n x , for (i, j ) ∈ (Ir × Ir ), k ∈ Ire and a positive scalars γ > 0, λ > 0, η > 0, ε1 > 0, ε2 > 0, ∈3 > 0 such that the optimization problem (30) is feasible. Then, the SOF controller (20) solves Problem 1 with the guaranteed contractively invariant set Ls of the closed-loop system: [

( ) min η + γ 2 Pi,R

(30)

E kT X i E k > 0 s.t. inequalities λ + γ2δ < 1 [

(31)

Tiik (ρk ) < 0 ; (i, j) ∈ (Ir × Ir ), k ∈ Ire and i /= j 2 T (ρ ) + Ti jk (ρk ) + T jik (ρk ) < 0) r−1 iik k (32) ⎤ ⎡ −E kT X i E k ∗ ⎦ ⎣ (33) −1 T Nm λ ⎡ ⎢ ⎢ ⎣

T T E k + Pi −E k H jk − H jk

Ti jk = ⎣ ⎡

Πi jk

2 −(u max (l) )

F jk(l) − W jk(l) ⎡

∗ Πi(1,1) jk





⎥ ⎥ 0 ↔ H jk Pi H jk > 0, ∀H jk /= 0. The above inequality holds: T −1 T T Pi H jk ≥ H jk + H jk − Pi , Pi > 0 then: H jk + H jk > 0, this guarantee that H jk −1 H jk is nonsingular, there the existence of H jk .

Proof To study the local asymptotic stability of the closed-loop system, the polytopic PD nonquadratic Lyapunov function defined in (23) is considered, a new LMI conditions for the design of controller to maintain stability performance with disturbance attenuation are designed. Specifically, the following definition will have to be satisfied for xk ∈ Dx , and ∀xk ∈ Ls \0, it comes easily that any solution of the closed-loop system, remains in the admissible set Dx defined in (9) if: ΔV (xk , ρk ) + 2αV (xk , ρk ) + ykT yk − γ 2 wkT wk −2ψ(u k )T Sμv (ρk )−1 [ψ(u k ) − Wμv (ρk )Hμv (ρk )−1 xk ] < 0

(42)

ΔV (xk , ρk ) = V (xk+ , ρk+ ) − V (xk , ρk )

(43)

with

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and ρk+ = ρ(k + 1)

(44)

After using Schur complement and Finsler’s lemma, with some computational calculations the strict LMIs of theorem are found. The design method will be formulated as a multi-objective optimization problem. Two objectives are considered: the first one is maximization of the disturbance rejection by augmenting the level energy disturbances δ, and/or minimizing the L2 attenuation level γ, the second is the estimation of the size of the DoA. In fact, both objectives could be unified by minimizing (β −2 + γ 2 ) [22]. The domain of initial condition E(E vT Pμ (ρk )−1 E v , 1) is included in the DoA E(E vT Pμ (ρk )−1 E v , λ), we can maximize the latter by optimizing the set E(E vT Pμ (ρk )−1 E v , 1). In order to obtain a sufficiently large estimated DoA, the typical type of the reference shape XR is considered. Consider an ellipsoid X R = {xk ∈ Rn x : xkT Rxk ≤ 1}, where XR ⊆ R2n x is a prescribed bounded convex ellipsoid set containing the origin, and R > 0 is diagonal matrix with compatible dimension. The following optimization problem gives: [

a) sup β ∩ ∩ re e T −1 b)βXR ⊆ ri=1 k=1 E(E k Pi E k , 1)

(45)

Constraint (b) is equivalent to: (βxkT )E kT Pi−1 E k (βxk ) ≤ xkT Rxk and β −2 R − E kT Pi−1 E k ≥ 0, setting β −2 = η, the LMI (39) is found. Ends the proof.

5 Illustrative Examples In the sequel, we apply the SOF control approach presented in the previous section of the 2-DoF robot arm. For stabilizing control purposes. In this paper, the workspace of the studied manipulator is defined as: Dx = |q1 | ≤ π(rad), |q2 | ≤ π(rad), |q˙1 | ≤ 5(rad/s), |q˙2 | ≤ 5(rad/s)

(46)

From the physical parameters given in Table 1, we can obtain the following bounds on the varying parameter: ρ1max = −ρ1min = 1, ρ2max = ρ3max = 4.79, ρ2min = −103.01, ρ3min = −22.07, ρ4max = ρ5max = −ρ4min = −ρ5min = 16.87, an exact descriptor representation for the 2-DoF robot can be easily derived with 25 = 32 local linear subsystems. Some nonlinearity will be considered as modeling uncertainties to reduce the number of premise variables, thus the rule number and conservatism. We present here an example when deriving a 8-rules from the NLPV descriptor system (17). To this end, we consider: ρ4,k = ρ4,max f 1 , ρ5,k = ρ5,max f 2 with | f i | ≤ 1, i = 1, 2, and w = u f si + u f vi . Then the system (17) can be represented in the form:

Robust Constrained Gain-Scheduled Static Output Feedback Controller for NLPV …

⎧ r r e ∑ ⎨∑ vk (ρk )E k = (μi (ρk ))( Ai xk + Bi satu k ) + Bw wk i=1 ⎩ k=1 yk = C xk ⎡

c1 + 2c2 ρ2,k c3 + c2 ρ2,k ⎢ c3 + c2 ρ2,k c3 Ek = ⎢ ⎣ 0 0 0 0 ⎡ ⎤ 10 [ ⎢0 1⎥ 2ρ4,max ⎢ ⎥ Hai = ⎣ ;N = ρ5,max 0 0 ⎦ ai 00

0 0 1 0

⎡ ⎤ 0 − f v1 0 ⎢ 0 − f v2 0⎥ ⎥; A = ⎢ 0 0⎦ i ⎣ 1 0 1 1

ρ2,k ρ3,k 0 0

⎤ ρ3,k ρ3,k ⎥ ⎥; 0 ⎦ 0

]

247

(47)

(48)

( ) ρ4,max 0 0 ; Dai = diag [ f 1 f 2 ] 0 00

Observe that the nonlinear uncertain descriptor system (17) has now only 3 varying parameter ρ1,k , ρ2,k and ρ3,k . Hence, using the sector nonlinearity approach, the statespace matrices are given as: E 1 (ρ1,min , ρ1,min ); E 2 (ρ1,max , ρ1,max ).A1 (ρ1,min , ρ3,min ); A2 (ρ2,min , ρ3,max ); A3 (ρ2,max , ρ3,min ); A4 (ρ2,max , ρ3,max )

(49)

Let us define the membership function in the left hand side: v1 (ρ1,k ) = (ρ1,k − ρ1,min )/(ρ1,max − ρ1,min ); v2 (ρ1,k ) = 1 − v2 (ρ1,k ).

(50)

Let us define the membership function in the right hand side: ( ) ( 2,k −ρ2,min ) ( ρ3,k −ρ3,min ) μ1 ρ2,k , ρ3,k = ρρ2,max −ρ2,min ρ3,max −ρ3,min ( ) ( 2,k −ρ2,min ) ( ρ3,max −ρ3,k ) μ2 ρ2,k , ρ3,k = ρρ2,max −ρ2,min ρ3,max −ρ3,min ( ) ( ρ2,max −ρ2,k ) ( ρ3,k −ρ3,min ) μ3 ρ2,k , ρ3,k = ρ2,max −ρ2,min ρ3,max −ρ3,min ( ) ( 2,max −ρ2,k ) ( ρ3,max −ρ3,k ) μ4 ρ2,k , ρ3,k = ρρ2,max −ρ2,min ρ3,max −ρ3,min

(51)

Notice that the decay rate α in P2 is related to the time performance of the closed-loop system. Solving the optimization problem of Theorem 1 with a decay rate α = 0.1 leads to: [ F11 = [ F12 =

] ] [ 0.0001 0.0146 0.0000 −0.0144 0.0000 0.0089 0.0017 −0.0151 ; F21 = −0.0005 −0.0696 0.0001 0.0132 0.0009 0.1955 0.0024 −0.0022

] ] [ −0.0001 0.0176 0.0020 −0.0198 0.0000 0.0089 0.0018 −0.0151 ; F22 = 0.0003 −0.0854 −0.0018 0.0237 0.0010 0.1960 0.0030 −0.0022

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F31 =

] [ ] −0.0003 −0.0553 −0.0057 0.0580 0.0002 −0.0556 −0.0057 0.0580 ; F32 = −0.0001 −0.0152 0.0061 −0.0537 0.0001 −0.0153 0.0061 −0.0537

[ F41 =

] ] [ 0.0020 0.0735 −0.0081 0.0635 0.0010 0.0738 −0.0081 0.0634 ; F42 = −0.0010 0.0745 −0.0069 0.0644 −0.0017 0.0748 −0.0066 0.0644

[ ] H11 = diag 0.0101 0.0094 −0.0004 −0.0049 ; [ ] H12 = diag 0.0100 0.0110 −0.0006 −0.0059 [ ] H21 = diag 0.0106 0.0099 −0.0002 −0.0427 ; [ ] H22 = diag 0.0105 0.0116 −0.0002 −0.0019 [ ] H31 = diag 0.0106 0.0100 −0.0047 −0.0026 ; [ ] H32 = diag 0.0106 0.0116 −0.0047 −0.0022 [ ] H41 = diag 0.0106 0.0100 −0.0597 −0.0591 ; [ ] H42 = diag 0.0105 0.0116 −0.0597 −0.0591 . √ An L2 −gain is 2.2813 and λ = 6 the parameters uncertainties are ε1 = 0.937, ε2 = ε3 = 1.8008 and η = 0.7451. For simulation, the closed-loop response corresponding to the initial condition x0T = [0.25 0.25 0 0)] is depicted in Fig. 2. Observe that the closed-loop system is stabilized despite an important input saturation with maximal level u max = 2 Nm/s in Fig. 2; the persistent disturbance is well attenuated. This clearly shows the importance of considering the system constraints into the robust control design in industrial applications. For the maximization of E(E vT Pμ−1 (ρk )E v , 1), (red set) the optimization problem is performed leading to ∩ desired performance E(E vT Pμ−1 (ρk )E v , 1) is maximized along Dx Du which is included inside Du = |u k(l) − vk(l) | ≤ u max(l) (move set), and Dx Blue set. Thus, the set inclusion property is clearly verified (Figs. 3, 4, 5, 6 and 7).

6 Conclusion We presented a new convex design conditions for a Gain-Scheduled SOF controller for NLPV uncertain descriptor system subject to state and input constraints. These conditions ensure the local stabilization of parameter dependent processes. To reduce further the design conservatism, we proposed a new type of nonquadratic Lyapunov function. We illustrated the proposal approach with the application to a NLPV robot arm system. The optimization problem of Theorem 1 showed the effectiveness of

Robust Constrained Gain-Scheduled Static Output Feedback Controller for NLPV … 0.4

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1

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the design approach that also ensures a performance index, given by contractively invariant set of the nonquadratic Lyapunov function. An example of robot arm is provided to demonstrate the effectiveness of the new control approach.

References 1. Aouaouda, S., Chadli, M., Karimi, H.R.: Robust static output-feedback controller design against sensor failure for vehicle dynamics. IET Control Theory Appl. 8(9), 728–737 (2014) 2. Aouaouda, S., Moussaoui, L., Righi, I.: Input-constrained controller design for nonlinear systems. In: International Conference on the Sciences of Electronics, Technologies of Information and Telecommunications, pp. 240–253. Springer, Cham (2018) 3. Blesa, J., Jiménez, P., Rotondo, D., Nejjari, F., Puig, V.: An interval NLPV parity equations approach for fault detection and isolation of a wind farm. IEEE Trans. Ind. Electron. 62(6), 3794–3805 (2014) 4. Binazadeh, T., Bahmani, M.: Design of robust controller for a class of uncertain discrete-time systems subject to actuator saturation. IEEE Trans. Autom. Control 62(3), 1505–1510 (2016) 5. Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM (1994) 6. Briat, C.: Linear parameter-varying and time-delay systems. Anal. Obs. Filter. Control 3, 5–7 (2014) 7. Coutinho, D., de Souza, C.E., Barbosa, K.A.: Robust H∞ control of discrete-time descriptor systems. In: 2014 European Control Conference (ECC), pp. 1915–1920 (2014) 8. Fu, R., Sun, H., Zeng, J.: Exponential stabilisation of nonlinear parameter-varying systems with applications to conversion flight control of a tilt rotor aircraft. Int. J. Control 92(11), 2473–2483 (2019) 9. Grimble, M.J.: Three degrees of freedom restricted structure optimal control for quasi-LPV systems. In: 2018 IEEE Conference on Decision and Control (CDC), pp. 2470–2477 (2018)

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10. Hamdi, H., Rodrigues, M., Mechmeche, C., Theilliol, D., Braiek, N.B.: State estimation for polytopic LPV descriptor systems: application to fault diagnosis. IFAC Proc. Vol. 42(8), 438– 443 (2009) 11. Leite, V.J.S., Silva, L.F.P.: On the integral action of discrete-time fuzzy TS control under saturated actuator. In: 2018 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), pp. 1–8 (2018) 12. Mohammadpour, J., Scherer, C.W. (eds.).: Control of Linear Parameter Varying Systems with Applications. Springer (2012) 13. Righi, I., Aouaouda, S., Chadli, M., Khelil, K.: Robust L2 control for uncertain TS descriptor model with input saturation. In: 2020 28th Mediterranean Conference on Control and Automation (MED), pp. 428–433 (2020) 14. Righi, I., Aouaouda, S., Chadli, M., Khelil, K.: Robust controllers design for constrained nonlinear parameter varying descriptor systems. Int. J. Robust Nonlinear Control 31(17), 8295– 8328 (2021) 15. Rodríguez, C., Barbosa, K.A., Coutinho, D.: Robust H∞ state-feedback design for discretetime descriptor systems. IFAC-PapersOnLine 51(25), 78–83 (2018) 16. Saifia, D., Chadli, M., Labiod, S., Guerra, T.M.: Robust H∞ static output-feedback control for discrete-time fuzzy systems with actuator saturation via fuzzy Lyapunov functions. Asian J. Control 22(2), 611–623 (2020) 17. Sala, A., Ariño, C., Robles, R.: Gain-scheduled control via convex nonlinear parameter varying models. IFAC-PapersOnLine 52(28), 70–75 (2019) 18. Sename, O., Dugard, L., Gáspár, P.: H∞ /LPV controller design for an active anti-roll bar system of heavy vehicles using parameter dependent weighting functions. Heliyon 5(6), e01827 (2019) 19. Shamma, J.S.: An Overview of LPV Systems. Control of Linear Parameter Varying Systems with Applications, pp. 3–26 (2012) 20. Spong, M.W.: Modeling and Control of Elastic Joint Robots (1987) 21. Spong, M.W., Vidyasagar, M.: Robot Dynamics and Control. Wiley (2008) 22. Tarbouriech, S., Garcia, G., da Silva Jr., J.M.G., Queinnec, I.: Stability and Stabilization of Linear Systems with Saturating Actuators. Springer (2011) 23. Wang, H.O., Tanaka, K.: Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach. Wiley (2004)

A Constrained Optimal Control Strategy for Switched Nonlinear Systems Based on Metaheuristic Algorithms Marwen Kermani and Anis Sakly

Abstract In this study, we investigate the optimal control problem of switched nonlinear systems under mixed constraints. This issue can be solved by determining two major factors. Indeed, we need to find the optimal switched instants sequence as well as the optimal control input, minimizing a performance criterion. For this, the proposed approach consists of dividing the problem into two stages. In the first one, the Lagrange multipliers method through the Karush-Kuhn-Tucker (KKT) conditions will be used. In the second one, the metaheuristic algorithms will be employed to obtain the optimal switching instants sequence. In this framework, Genetic Algorithm (AG), Particle Swarm Optimization (PSO) Cuckoo algorithm (CA), and Crow Search Algorithm (CSA) have been suggested to be used and compared. Finally, the effectiveness of this proposed approach has been illustrated through a physical switched system that models a hydraulic system under constraints. Keywords Switched nonlinear system · Constrained optimal control · KKT conditions · Metaheuristic algorithms

M. Kermani (B) · A. Sakly Laboratory of Automation, Electrical Systems and Environment, National Engineering School of Monastir, University of Monastir, Monastir, Tunisia e-mail: [email protected] M. Kermani National School of Advanced Sciences and Technology of Borj Cedria, University of Carthage, BP 122 Hammam-Chott, 1164 Carthage, Tunisia © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Derbel et al. (eds.), Advances in Robust Control and Applications, Studies in Systems, Decision and Control 474, https://doi.org/10.1007/978-981-99-3463-8_10

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1 Introduction Switched systems are a particular class of hybrid systems which are formed by several subsystems and a switching rule orchestrating the switching between the subsystems [17, 18]. Many engineering systems such as electrical circuit systems, chemical processes, automotive industry, flexible manufacturing systems, air traffic control, communication networks and other real processes can be modeled as switched systems [19, 22]. Up to now, switched nonlinear systems have attracted considerable attention, and some valuable results have been achieved [1, 2, 5, 6, 8–18, 21, 23, 24]. Among these research topics, the optimal control problem is fundamental in the studying of these systems. However, this problem is still open because the analytical techniques are not sufficient to determine an optimal control policy. Generally, this problem can be solved by determining both continuous control inputs, and the switched instants sequence. As one of the most existing approaches for this issue is presented in [5, 6, 11, 12, 15, 16, 21, 23, 24]. Indeed, this approach consists of decomposing the problem into two stages. Stage (a) deals with a conventional optimal control problem that minimizes a performance index. Stage (b) is a nonlinear optimization problem that allows to obtain the optimal switched sequence. The conventional methods such as the gradient algorithm have been used to solve this matter. However, those methods present may get stuck in a local optimum. Hence, they are unable to give the global optimum. By associating conventional approaches, metaheuristic techniques are recently used to solve the optimal control problems of switched systems. Thus, the main advantage of those methods is illustrated in the fact that they allow us to obtain the global optimum. In this work the Genetic Algorithm (AG) [4, 21], the Particle Swarm Optimization (PSO) [6, 7], the Cuckoo algorithm (CA) [11, 12, 26], and the Crow Search Algorithm (CSA) [3, 15, 16] are suggested to be used. On the other hand, practical switched systems are usually considered under imposed constraints which can be presented on both the state variable and the optimal control. Generally, the existence of the constraints leads to some restrictions on the systems operation such as a speed limitation, a minimal or maximal tank fluid level ext. This kind of problem has been considered in numerous research [6, 24]. In this chapter, the optimal control problem of switched nonlinear systems under constraints will be addressed. Thus, a hybrid approach composed of a conventional method and metaheuristic algorithms has been proposed to solve this matter. Indeed, the problem has been divided into two stages. In the first one, the optimal control law minimizing a performance criterion will be determined by using the Pontryagin maximum principle (PMP). Moreover, the Karush-Kuhn-Tucker optimality conditions (KKT) [20] and the bang-bang control [25] are employed respectively to insure the state constraints feasibility and the bounded control input. In the second stage, the AG, the PSO, the CA, and the CSA algorithms have been used to obtain the best switched sequence corresponding to the global fitness under considered constraints.

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By the end, the effectiveness of the proposed approach is proved by considering a hydraulic two tank switched system under constraint. This chapter is organized as follows. In Sect. 2, we present the problem formulation. In Sect. 3, we give a general overview of the used proposed metaheuristic algorithm. The optimal control problem of constrained switched nonlinear systems is given in Sect. 4. The proposed metaheuristic algorithms minimizing the cost function are introduced in Sect. 5. In Sect. 6, a simulation example is presented to illustrate the efficiency of the proposed approach. Finally, some conclusions are given in Sect. 7.

2 Switched Nonlinear Systems: Problem Formulation We consider the following switched nonlinear systems given by: η˙ = h l (η, v)

(1)

where η is the state vector, η˙ is its time derivative, h l is an indexed field of vectors and N is a set of L subsystems which have configurations. To control system (1), we need to choose a continuous-time input and a switching sequence ζ defined for t ∈ [t0 , t f ] such as: ( ) ζ = (t0 , l0 ), (t1 , l1 ), (t2 , l2 ), . . . , (ts , ls )

(2)

where (tk , lk ) denotes that at time tk , the system will switch from sub-model lk−1 to sub model lk and during the time interval [tk , tk+1 ], sub-model lk will be active [17].

3 Constrained Optimal Control for Switched Nonlinear Systems This section, we will define the constrained optimal control problem for switched nonlinear systems. We consider the switched nonlinear system (1) and the switching sequence (2). Now, we assume that the system (1) is subject to the following constraints [6]: ⎧ ⎨ m(η) = 0 n(η) ≤ 0 ⎩ L V ≤ v(t) ≤ V H where:

(3)

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⎡ ⎤ ⎤ n 1 (η) m 1 (η) ⎢ n 2 (η) ⎥ ⎢ m 2 (η) ⎥ ⎢ ⎢ ⎥ ⎥ m(η) = ⎢ . ⎥ , n(η) = ⎢ . ⎥ ⎣ .. ⎦ ⎣ .. ⎦ ⎡

m i (η)

n j (η)

are two vectors that represent the i equality type constraints as well as the j inequality type constraints to be imposed on the state vector. V L and V H are the lower and the upper order limits. Such optimal control problem amounts to minimizing a performance criterion J which is be defined as the following [23, 24]: ∫ J = Ψ [η(t f )] +

tf

L[η(t), v(t), t] dt

(4)

t0

The optimal control problem of the contracted system (1) involves two steps that consists of determining the optimal switching sequence given in (2) and the optimal control guarantees the minimization of the performance criterion (4) while satisfying the constraints (3) imposed on the system. In what follows, the Lagrangian of the system is given as [6]: Γ [η(t), v(t), λ, μ, t] = L[η(t), v(t), t] +

i ∑

λw m w (η(t), t) +

w=1

where:





j ∑ l=1



μl n l (η(t), t) (5)



μ1 λ1 ⎢ μ2 ⎥ ⎢ λ2 ⎥ ⎢ ⎥ ⎢ ⎥ λ = ⎢ . ⎥, μ = ⎢ . ⎥ ⎣ .. ⎦ ⎣ .. ⎦ λi μj

represent the vectors of Lagrange multipliers satisfying the KKT conditions. In the sequel, we will present the definition of the KKT conditions. Definition 1 Karush-Kuhn-Tucker Conditions If we assume that the state η * is a local optimum of the performance criterion J , then there are two vectors of multipliers λ and μ satisfying the following conditions [20]: ⎧ ∇η Γ (η * , v, λ, μ, t) ⎪ ⎪ ⎪ ⎪ * ⎪ ⎪ ⎨ ∇λ Γ (η , v, λ, μ, t) ∇μ Γ (η * , v, λ, μ, t) ⎪ ⎪ ⎪ λ*w n w ⎪ ⎪ ⎪ ⎩ λ*

= 0 = n w (η * ) ≤ 0 = m l (η * ) = 0 λ = 0 = 0 ≥ 0

where w = 1, 2, . . . , i and l = 1, 2, . . . , j.

(6)

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Now, we will construct the augmented Hamiltonian as [6]: Hl (η, γ, v, t) = Γ (η, γ, v, t) + γ h l (η, v, t) + T

i ∑

λw m w (η, t) +

w=1

j ∑

μs h s (η, t)

s=1

(7) where γ represents the conjoint state vector of the system for any time t ∈ [ts , ts+1 ] (if s = L, t ∈ [ts , t f ]), Thus, the state and the costate vectors are given by: ∂ Hl (η, γ, v, t) ∂γ ∂ Hl γ˙ = − (η, γ, v, t) ∂η η˙ =

(8) (9)

For t = t f , the conjoint state satisfies the following boundary condition [23]: γ(t f ) =

∂Ψ ∂η(t f )

(10)

The continuity condition for t = ts , s = 1, 2, . . . , L is given by [23]: γ(ts− ) = γ(ts+ )

(11)

The determination of the optimal control law is based on the following stationarity equation: ∂ Hl (η, γ, v, t) = 0 ∂v

(12)

Finally, by applying the Bang-Bang principle for the control, we obtain the optimal control signal ~ v (t) under constraints [25]: | L | V if v* < V L | | * ~ v (t) = | v if v L ≤ v * ≤ V H | H | V if v* > V H

(13)

The stage (b) consists of solving the constrained nonlinear optimization problem [24]: t) (14) min J (⌃ i

with:

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t1 ⎢ t2 ⎢ ⌃ t =⎢ . ⎣ ..

⎤ ⎥ ⎥ ⎥ ⎦

tK where: t0 < t1 < t2 < · · · < t K < t f The mechanistic algorithms allow us to obtain high performance [6, 15, 16, 21]. Ended, Genetic Algorithm (AG), Particle Swarm Optimization (PSO) Cuckoo algorithm (CA), and Crow Search Algorithm (CSA) are suggested to solve problem (14).

4 Overview: Metaheuristic Algorithms In this section, we will present the used metaheuristic algorithms for solving the optimal control problem of switched nonlinear systems. In the sequel, the GA, the PSO, the CA and the CSA algorithms, will be presented.

4.1 Genetic algorithm The GA [4] is an optimization algorithm based on a natural selection mechanism inspired by genetics. Indeed, in cases when the solution is unknown, a set of possible solutions will be created randomly. This set is represented as a population. Thus, the characteristics (variables to be determined) are represented by the genes which will be combined with other genes to form chromosomes of other individuals. Each solution represents an individual which will be evaluated and ranked according to its similarity with the best individual. The basic concept of the GA consists of evolving a population composed of a set of individuals through a set of generations until the verification of a stopping criterion. Certain specific operations ensure the transition from one population to another. These operations are the evaluation, the selection, cross and mutation. The first step in this algorithm is to randomly create a set of individuals. Then, we will select for each iteration, the best individuals dedicated to surviving and reproducing. The operation of individual selection is based on their qualities measured from the objective function. Subsequently, the crossover and the mutation operations undergone for the parents. Then, a new population of individuals called children is generated. The individuals of the new generation will be evaluated to substitute some individuals belonging to the current population.

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4.2 Particle Swarm Optimization The PSO method [7] has been inspired from the behavior of bee colonies and fish spawn in the search for food. Several improvements have been made compared to the original version of this algorithm. The basic algorithms are developed using a large number of candidate solutions. The best solution for a given problem corresponds to the optimal level which will be measured by the fitness function.

4.3 Cuckoo Search The CS algorithm is a recent metaheuristic method [26]. This method was inspired by the mode of reproduction of certain species of cuckoos. Indeed, their reproductive strategy is characterized by the fact that females lay their eggs in the nests of other species. These eggs can be hatched by surrogate parents. Moreover, when the cuckoo eggs manage to hatch in the host nest (they hatch more quickly), the chick’s cuckoo has the reflex to eject the eggs of the host species outside the nest and even imitate the cry of the host chicks to be fed by the host species. The solution in the CS algorithm is represented by an egg in a nest, and the new solution will be given by a cuckoo egg. The rules in the cuckoo search are summary as follows: • Every cuckoo lays a single egg and throws it in a random nest. • The best solution will be retained to the next generations. • The host nests number is initially defined. The host can detect a stranger egg with probability . In such a situation, the host bird skips the egg, and the nest builds a new nest in another place.

4.4 Crow Search Algorithm The crows are considered as the most intelligent birds. Their brain is slightly inferior to the human brain. Crows can use tools to communicate in complex ways and remember their food stash for months. After the owner left, other birds hid the food and stole it. If the crow is stolen, it will be very careful in these hiding places to avoid becoming a future victim. They use their experience of thieves to predict the behavior of the thief and can determine the safest way to protect against thieves. The principle of CSA is summarized by [3]: • Crows live in groups. • Crow remembered the location of his hiding place. • Crows remember his flocks.

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The crow protects its hiding place with a possibility of being robbed. We assume that there is an environment that contains many crows. The number of crows is N and the position of the crow in the search space is defined by the vector x i,iter , (i = 1, 2, . . . , N , iter = 1, 2, . . . , itermax ), where: ⎤ x1i,iter ⎢ x i,iter ⎥ ⎥ ⎢ 2 =⎢ . ⎥ ⎣ .. ⎦ ⎡

x i,iter

xdi,iter

and itermax is the maximum number of iterations. Each crow memorizes its hidden position. The position of the crow is displayed in the iteration. This is the best crow position Then, in the memory of every crow, the place of his best experience has been recalled. Crows move about and look for better food sources (hiding places).

5 Optimal Control for Switched Nonlinear Systems: Proposed Optimization Algorithms In this section, we will present the proposed optimization mechanistic algorithms for optimal control problem of switched nonlinear systems. These proposed algorithms allow us to obtain numerical cost of each switching sequence and return the best sequence (optimal switching sequence) that corresponding to the minimal value of the cost function. For more details, we present, the following proposed algorithms.

5.1 Proposed Genetic Algorithm In this section, the principle of the used GA for determining the optimal switching instants and the optimal control law will be presented. The proposed GA is summarized by the following steps of Algorithm 1 [21]:

Algorithm 1 1. Step 1: Initialize the sizing parameters (population size, mutation probability, mutation probability · · · ). 2. Step 2: Create an initial population corresponding to the switching instants (t1 , . . . , tn ). 3. Step 3: Evaluate the individuals based on the cost function J (t1 , . . . , tn ). 4. Step 4: Apply selection, crossover, and mutation operators to obtain a new population.

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5. Step 5: Insert new individuals into the population. 6. Step 6: Check the termination criterion. Otherwise retune to step 4.

5.2 Proposed PSO Algorithm The different steps of the PSO algorithm will be presented in the Algorithm 2 [6].

Algorithm 2 1. Step 1: Initializing the population size (number of particles), the search space and the dimension which corresponds to the number of switching instants to be optimized: Randomly initialize the parameters c1 , c2 , wmax , wmin and max I . 2. Step 2: Generate a generation of switching instants in the search space randomly. 3. Step 3: Evaluate each initial particle according to the cost function J relating to the switching sequence. The best will be assigned to pbestk . 4. Step 4: Assign the best sequence among pbestk which corresponds to the best sequence of global switching instants pgbestk . 5. Step 5: Update weight and speed. 6. Step 6: Evaluate each particle according to its identified cost function and the corresponding switching sequence. The best will be assigned to pbestk . 7. Step 7: If the stopping condition is verified, then stop. Otherwise, go to Step 2.

n k+1 = wk n k + c1r1 ( pbestk − pk ) + c2 r2 ( pgbestk − pk ) pk+1 = pk + vk+1

(15) (16)

where k denotes the iteration index. pk represents the particle position at iteration k. vk denotes the particle velocity at iteration . c1 and c2 represent the attraction strength. r1 and r2 are two numbers in the interval [0,1] which are randomly generated. pbestk represents the best position discovered by the particle until the iteration index k. pgbestk denotes the global best particle position of the population. The weight is updated according to the following equation: wk = wmax −

wmax − wmin k max I

(17)

where max I is maximal iterations number. wmax and wmin are the maximum, and the minimum values of w, respectively.

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5.3 Proposed Cuckoo Search Algorithm In this algorithm, we will first define the number of switching instants sequence to be optimized. Then, the cost J that corresponds to each sequence will be calculated. Thus, we keep the best value as the initial value and the best elements. For the rest of the elements, we will reproduce them by regenerating the cuckoo. If the cuckoos have good values of cost J , the nest will be replaced by the cuckoo; otherwise, there is no change. Until the last generation. The proposed Cuckoo Search is illustrated by the following steps of Algorithm 3 [11, 12].

Algorithm 3 1. Step 1: Create a nest according to the switching sequence and coding the population nest. 2. Step 2: Randomly generate the initial population that corresponding to the switching sequence in the search space. 3. Step 3: Calculate the values of the cost function J and the switching sequence that minimizes J . Keep the best value (minimal value). 4. Step 4: The probability of finding eggs in nests is pa ; Choose a random nest with fitness Jn For l = 1: number of cuckoos discovered. Generate a random cuckoo with fitness Fc If Jc < Jn Then 5. Step 5: Replace nest with cuckoo %. Keep the best individual 6. Step 6: Sort the solutions and its associated nests keep a percentage pa of the population 7. Step 7: Go to the next generation: the stopping criterion is given by the maximum number of generations.

5.4 Proposed Crow Search Algorithm In this subsection, we will present the proposed CSA. Indeed, we assume that we have N crows in a d-dimensional space. Thus, the position of crow ith at the iteration iter is represented by x i,iter , where i = 1, 2, . . . , N and iter = 1, 2, . . . , itermax ; itermax represents the iteration maximum number. The positions and the memories of each crow are randomly initialized. Thus, the fitness value J of each position will be calculated. Therefore, a new position of the crow ith at iteration iter will be created, the crow j will be randomly selected and the crow i follows the crow j to discover its hidden place. Thus, the new position will be obtained according to the next equation:

A Constrained Optimal Control Strategy for Switched Nonlinear Systems …

x

i,iter +1

| | x i,iter + r × (m i,iter − x i,iter ); r ≥ A P i,iter i i | =| | a −→ randomposition

263

(18)

where ri is a random selected in [0,1]. The different steps of the CSA will be presented by the Algorithm 4 as below [15, 16]. Algorithm 4 1. Step 1: Problem initialization by fixing the parameters: the flock size (N ), the maximum number of iterations (itermax ), the flight length ( f l ) and the awareness probability (A P). 2. Step 2: Initialization randomly the positions that corresponding to the switching sequences and the memory of N crows in a d-dimensional search space. Each crow has a feasible solution given by: ⎡

x11 ⎢ x12 ⎢ Cr ows = ⎢ . ⎣ ..

x21 x22 .. .

··· ··· .. .

xd1 xd2 .. .

⎤ ⎥ ⎥ ⎥ ⎦

(19)

x1N x2N · · · xdN

The crow memory will be initialized as: ⎡

m 11 m 12 · · · ⎢ m 21 m 22 · · · ⎢ Memor y = ⎢ . . . ⎣ .. .. .. m 1N m 2N · · ·

m 1d m 2d .. .

⎤ ⎥ ⎥ ⎥ ⎦

(20)

m dN

3. Step 3: Evaluate cost function J of each crow position. 4. Step 4: Create a new position of the ith crow at iteration iter , the crow j will be randomly selected. Ended, the crow i follows the crow j to discover its hidden food place. 5. Step 5: Check the new positions feasibility. 6. Step 6: Evaluate the cost function J corresponding to the new positions. 7. Step 7: Memory update. The memory of the crow will be given such as: m i,iter +1 = x i,iter +1 , J (x i,iter +1 ) is better than J (m i,iter )

(21)

where J is the cost function value, and m i,iter is the hidden place of the ith crow at iteration iter . 8. Step 8: Check the stopping criterion. If the current iteration is less than itermax return to step 4.

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6 Illustrative Example To validate the proposed approach, we illustrate the following example that modeling a constrained nonlinear switched system [6, 15, 16]. We consider the hydraulic system represented by Fig. 1. This system is composed of a motor pump, two tanks, and two valves V1 and V2 . The switched system is represented by three subsystems defined in the time interval [0, 60]s. • For t ∈ [0, t1 ]: The tank 1 is filled with liquid from the motor pump and the two valves V1 and V2 are closed. • For t ∈ [t1 , t2 ]: The valve is opened and remains closed until t2 . • For t ∈ [t2 , t f ]: Both the valves V1 and V2 will be open. Based on the hybrid behavior, the resulting switched system will be described by the following system of equations:

Fig. 1 Hydraulic system description

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• For t ∈ [0, t1 ]: 1 dh 1 = qe S dt dh 1 =0 dt

(22) (23)

• For t ∈ [t1 , t2 ]: √ ) 1 ( dh 1 = qe − k 1 h 1 S dt k1 √ dh 1 = h1 S dt

(24) (25)

• For t ∈ [t2 , t f ]: √ ) 1 ( dh 1 = qe − k 1 h 1 S dt √ ) 1 ( √ dh 1 = k1 h 1 − k2 h 2 S dt

(26) (27)

where S: the section of the two tanks; h 1 and h 2 : liquid levels; k1 and k2 : hydraulic resistances; qe : the flow input. With the following numerical values: S = 1m2 , k1 = 0.02m 2 s−1 , k2 = 0.03m 2 s−1 5

5

At t = t0 , the initial conditions are h 1 (t0 ) = 3 m and h 2 (t0 ) = 0.5 m. In this application, we want to find the optimal control input qe* and the best switching * * and t2opt minimizing the following functional cost: instants t1opt J=

1 2



60

[(

] h 1 (t) − 5)2 + (h 2 (t) − 1)2 + qe (t)2 dt

(28)

0

Subject to the following constraints: ⎧

h 1 (t) ≤ 4.3 0 ≤ qe (t) ≤ 1

(29)

To solve this problem, we begin by contracting Lagrangian associate to the system

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H=

] 1 [( h 1 − 5)2 + (h 2 − 1)2 + qe2 + λ(h 1 − 4.3) 2

(30)

The Lagrange multiplier verifying the KKT conditions is given by λ = 0.07. Finally, the augmented Hamiltonian in each time interval is given such as follows. • For t ∈ [0, t1 ]: H1 =

] 1 1 [( h 1 − 5)2 + (h 2 − 1)2 + qe2 + γ1 qe + 0.7(h 1 − 4.3) 2 S

(31)

According to the stationarity condition, the optimal control rate will be given by: | * |1 ≥1 if ⌃ qe1 | | * if ⌃ qe1 ≤ 0 ⌃ qe1 = | 0 | 1 | − γ1 elsewhere S

(32)

The expressions of state and the costate are: ⎧ dh ⎪ ⎪ 1 ⎪ ⎪ dt ⎪ ⎪ ⎪ dh 2 ⎪ ⎪ ⎨ dt dγ1 ⎪ ⎪ ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ dγ ⎪ ⎩ 2 dt

=

1 ⌃ qe1 S

=0 (33) = −(h 1 − 5) − 0.7 = −(h 2 − 1)

• For t ∈ [t1 , t2 ]: H2 =

] 1 [( h 1 − 5)2 + (h 2 − 1)2 + qe2 + 0.7(h 1 − 4.3) 2 √ ) k1 √ 1 ( + γ1 qe − k1 h 1 + γ2 h 1 S S

(34)

The expression for the optimal control rate is given by: | * |1 ≥1 if ⌃ qe2 | | * if ⌃ qe2 ≤ 0 ⌃ qe2 = | 0 | 1 | − γ1 elsewhere S The expressions of state and the costate are presented by:

(35)

A Constrained Optimal Control Strategy for Switched Nonlinear Systems …

⎧ ⎪ dh 1 ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ dh 2 ⎪ ⎪ ⎨ dt dγ1 ⎪ ⎪ ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎩ dγ2 dt

√ ) 1( ⌃ qe3 − k1 h 1 S √ ) 1( √ k1 h 1 − k2 h 2 = S k1 k1 = −(h 1 − 5) − √ γ1 + √ γ2 − 0.7 2S h 1 2S h 1

267

=

(36)

= −(h 2 − 1)

• For t ∈ [t2 , t f ]: H3 =

] 1 [( h 1 − 5)2 + (h 2 − 1)2 + qe2 + 0.7(h 1 − 4.3) 2 √ ) √ √ ) 1 ( √ 1 ( k1 h 1 − k2 h 2 γ2 h 1 + γ1 qe − k1 h 1 + S S

(37)

The optimal control rate is given by the following equation: | * |1 ≥1 if ⌃ qe3 | | * if ⌃ qe3 ≤ 0 ⌃ qe3 = | 0 | 1 | − γ1 elsewhere S

(38)

The expressions of state and the costate are: ⎧ dh 1 ⎪ ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ dh ⎪ 2 ⎪ ⎪ ⎨ dt ⎪ dγ1 ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ dγ ⎪ 2 ⎪ ⎩ dt

√ ) 1( ⌃ qe3 − k1 h 1 S √ ) 1( √ k1 h 1 − k2 h 2 = S k1 k1 = −(h 1 − 5) − √ γ1 + √ γ2 − 0.7 2S h 1 2S h 1 k2 = −(h 2 − 1) + √ γ2 2S h 2 =

(39)

By applying the GA, the PSO, the CA and the CSA algorithms with the following parameters values: • For the GA population size N = 40; and the maximum number of generations is fixed to 15. • For the PSO algorithm: c1 = c2 = 0.75; r1 and r2 are two random numbers with values in the range [0, 1]. wmax = 0.9, wmin = 0.4, the swarm size is N = 40 and the maximum number of iterations 15. • For the CA, the parameters are the Number of nests N = 40; and maximum number of iterations is fixed to 15.

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Table 1 Simulation results Optimal values Algorithm Instant * t1opt (s) GA PSO CA CSA

0.7806 0.1293 0.0815 0.0536

Instant * t2opt (s)

Cost Jopt

Execution time tex (min)

2.3609 1.0730 2.3314 0.592

18.3910 18.3804 18.3596 18.3192

19.78 3.64 5.095 6.73

4.4

4.2

4

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3.4

3.2

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Fig. 2 Trajectory evolution of h 1 (t)

• For the CSA algorithm: the population size N = 40; the Awareness probability A P = 0.1, the flight length f l = 2 and the maximum number of iterations (itermax = 15). From the simulation results given in Table 1, it can be verified that among that the CA and the CSA have a larger execution time, they converge to a best solution compared to the PSO. The simulation results are given by Figs. 2, 3 and 4 which represent the evolutions qe , respectively. of the liquid level h 1 , the liquid h 2 and the optimal flow input ⌃

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1.3 1.2 1.1

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7 Conclusions In this chapter, a hybrid approach for the optimal control problem of constrained switched nonlinear systems has been proposed. This approach is composed into two stages. The first one focuses on the optimal control problem, and it has been solved by using the PMP, the KKT conditions and the bang-bang control. In the second one, the metaheuristic algorithms are used to obtain the optimal switching instants sequence. Thus, the GA, the PSO, the CA and CSA allow to obtain a global minimum cost. The proposed metaheuristic algorithms were tested for constrained switched nonlinear systems with no convex cost function that meddling a hydraulic system and satisfactory results were obtained. Particularly, although all the used metaheuristic algorithms return the global optimum, the result obtained from the CSA was suggested better than all other algorithms. The future work consists of including the approach’s effectiveness, it would be interesting to carry out a comparative study with other methods applied in the optimal control of constrained switched nonlinear systems.

References 1. Aoun, N., Kermani, M., Sakly, A.: Vector norms-based stability of a class of discrete-time Takagi-Sugeno fuzzy switched time-delay systems. In: International Multi-Conference on Systems, Signals and Devices SSD 2018, Hammamet, Tunisia, Proceedings, pp. 965–970 (2018a) 2. Aoun, N., Kermani, M., Sakly, A.: Simple algebraic stability criteria for a class of discretetime model-based switched multiple time delay systems. In: International Conference on Control, Decision and Information Technologies, Thessaloniki, Greece, Proceedings, pp. 476–481 (2018b) 3. Askarzadeh, A.: A novel metaheuristic method for solving constrained engineering optimization problems: crow search algorithm. Comput. Struct. 169, 1–12 (2016) 4. Holland, J.H.: Adaptation in Natural and Artificial Systems. Ann Arbor, 105 University of Michigan Press (1975) 5. Kahloul, A.A., Kermani, M., Sakly, A.: Optimal control of switched based on continuous Hopfield neural network. Int. Rev. Autom. Control 7(5), 506–516 (2014) 6. Kahloul, A.A., Sakly, A.: Hybrid approach for constrained optimal control of nonlinear switched systems. J. Control, Autom. Electr. Syst. 31, 865–873 (2020) 7. Kennedy, J.: Particle swarm optimization. In: International Conference on Neural Networks. Proceedings, pp. 1942–1948 (1995) 8. Kermani, M., Sakly, A., M’Sahli, F.: A new stability analysis and stabilization of uncertain switched linear systems based on vector norms approach. In: 10th International MultiConference on Systems, Signals & Devices (SSD) Hammamet, Tunisia (2013) 9. Kermani, M., Sakly, A.: A new robust pole placement stabilization for a class of time-varying polytopic uncertain switched nonlinear systems under arbitrary switching. Control Eng. Appl. Inform. 17(2), 20–31 (2015a) 10. Kermani, M., Sakly, A.: Delay-independent stability criteria under arbitrary switching of a class of switched nonlinear time-delay systems. Adv. Differ. Equ. 2015(127) (2015b). https:// doi.org/10.1186/s13662-015-0560-1

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11. Kermani, M., Kahloul, A.A., Sakly, A.: Cuckoo search algorithm based an optimal control strategy for switched nonlinear systems. In: International Conference on Sciences and Techniques of Automatic Control & Computer Engineering STA 2020, Sfax, Tunisia, Proceedings, pp. 13–17 (2020) 12. Kermani, M., Sakly, A.: Pole assignment stabilization for a class of switched nonlinear timevarying delay systems. ISA Trans. 106, 138–151 (2020) 13. Kermani, M., Sakly, A.: M-matrix based stability analysis switched nonlinear time-varying delay systems. J. Dyn. Control Syst. 9(3), 1236–1249 (2021a) 14. Kermani, M., Sakly, A.: M-matrix-based robust stability and stabilization criteria for uncertain switched nonlinear systems with multiple time-varying delays. Math. Probl. Eng. (2021b). https://doi.org/10.1155/2021/8871603 15. Kermani, M., Sakly, A.: Robust stability and memory state feedback control for a class of switched nonlinear systems with multiple time-varying delays. J. Vib. Control 28(7–8), 932– 951 (2022a) 16. Kermani, M., Sakly, A.: Crow search algorithm based an optimal control for switched nonlinear systems. In: International Multi-Conference on Systems, Signals and Devices, SSD 2022. Sétif, Algeria (2022b) 17. Liberzon, D.: Switching in Systems and Control. Birkhauser, Boston (2003) 18. Liberzon, D.: Switching in Systems and Control. Springer Science & Business Media (2012) 19. Morse, A.S.: Control Using Logic-based Switching. Springer, London (1997) 20. Naidu, D.S.: Optimal control systems. Electrical Engineering Textbook Series. CRC Press LLC, Boca Raton (2003) 21. Sakly, M., Sakly, A., Mejdoub, N., Benrejeb, M.: Optimization of switching instants for optimal control of linear switched systems based on genetic algorithms. In: 2nd IFAC International Conference on Intelligent Control Systems and Signa,l Proceedings, pp. 249–253 (2009) 22. Varaiya, P.: Smart cars on smart roads: problems of control. IEEE Trans. Autom. Control 38(2), 195–207 (1993) 23. Xu, X., Antsaklis, P.J.: An optimal control of switching systems based on parameterization of the switching instants. IEEE Trans. Autom. Control 49(1), 2–16 (2004) 24. Xu, X., Antsaklis, P.J.: On time optimal control of integrator switched systems with state constraints. Nonlinear Anal. 62, 1453–1465 (2005) 25. Yan, H., Zhu, Y.: Bang-bang control model for uncertain switched systems. Appl. Math. Model. 39, 2994–3002 (2015) 26. Yang, X.S., Deb, S.: Engineering optimisation by Cuckoo search. Int. J. Math. Model. Numer. Optim 1(4), 330–343 (2010)

Robustness Analysis of a Discrete Integral Sliding Mode Controller for DC-DC Buck Converter Using Input-Output Measurement Zina Elhajji, Kadija Dehri, Zyad Bouchama, Ahmed Said Nouri, and Najib Essounbouli Abstract The present paper deals with the design of a discrete-time integral sliding mode controller (DISMC) using only input-output measurement. The proposed strategy of control is developed and considered as a new method for voltage controller of electronic power systems, such as DC-DC buck converter. An input-output model of DC-DC buck converter is given. Simulations results are presented to demonstrate the good effectiveness and the efficiency of the proposed DISMC in terms of reduction of chattering phenomenon and robustness against external disturbances and parameters variations. Keywords Discrete-time · Integral sliding mode control · Input-output representation · DC-DC buck converter · Robustness · Parameters variations

1 Introduction DC-DC converters are considered as nonlinear and time-varying power systems. They are widely used in many electronic applications, such as AC and DC motor drivers [1, 2], renewable energy systems, full cells and hybrid electronic vehicles [3, 4]. There are three modes of DC-DC converter; buck, boost and buck-boost. The buck is used to minimize the output voltage and the boost can increase it. However, in the third mode the output voltage can be maintained higher or lower than the Z. Elhajji (B) · K. Dehri University of Gabes, Gabes, Tunisia e-mail: [email protected] Z. Bouchama University of Bordj Bou Arreridj, Bordj Bou Arreridj, Algeria A. S. Nouri University of Sfax, Sfax, Tunisia N. Essounbouli University of Reims Champagne Ardennes, Reims, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Derbel et al. (eds.), Advances in Robust Control and Applications, Studies in Systems, Decision and Control 474, https://doi.org/10.1007/978-981-99-3463-8_11

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source and in opposite polarity. It is crucial to design an adequate control technique to improve the efficiency of converters. In this context, many control methods have been proposed in the literature such as Proportional Integral Derivative (PID) [5], fuzzy logic control [6], an adaptative fuzzy logic controller [7] and Sliding Mode Control (SMC) [8–10]. The main objective of SMC is to drive the system states trajectory to attain a specified sliding surface and to keep moving along it. Two steps are considered: the first is defining a sliding function along which the state of system trajectory slide to find its final value. The second is synthesizing a sliding mode control law by forcing the system moving along a sliding surface [11]. Thanks to its robustness and implementation simplicity, the SMC is considered as the most robust nonlinear control technique for DC-DC converters, especially for the buck in the literature [10, 18]. For this reason, many researchers have focused on the control design of the DC-DC buck converter using an SMC in which the output voltage is smaller than the input source [12–14]. This has been mainly developed in continuous time. For example, a new sliding mode control method for the DC-DC buck converter is proposed in [13]. An adaptative sliding mode controller of a DC DC converter is designed and developed in [14] and a second order sliding mode control is developed in [21]. Moreover, the discrete version of SMC has been widely developed in the literature because of the expensive use of computer and digital controllers in implementation [3, 19]. The difference between the continuous time and the discrete time SMC is that the sliding conditions in the former are verified at any time. However, in the latter, which presents more sensitivity to the external disturbances and uncertainties, these conditions are only verified at the sampling instant [11]. The remaining sections of this article is organized as follows. In Sect. 2, we give a schematic diagram and a mathematical model of the DC-DC buck converter. Then, the Sect. 3 presents a new discrete integral sliding mode control using input-output representation. After that, a simulation results are given in Sect. 4 in order to show the effectiveness of the proposed approach. Finally, some conclusion remarks are drawn in the last section.

2 Mathematical Model of the DC-DC Buck Converter Figure 1 shows the schematic diagram of the DC-DC buck converter. This circuit consists of a DC input voltage (Vin), a switch (S), a diode (D), an inductor (L), a capacitor (C) and a load resistor (R). The output voltage of the buck converter is always less than or equal to the source voltage. This converter can be described by the following state space model in continuous time [12]:

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Fig. 1 DC-DC buck converter

⎧ ⎨

x˙1 = x2 1 1 x1 − RC x2 + x˙2 = − LC ⎩ y = x1

Vin u LC

(1)

where x1 = V0d and x2 = x˙1 = C1 i C are the state variables. u is a switching control law equal to the value of pulse width modulation (PWM) output of this converter. Based on the state space model described by (1), the transfer function of this system can be given as: H ( p) =

y( p) = 2 u( p) p +

Vin LC 1 p RC

+

1 LC

(2)

with y( p) = x1 ( p) = V0d ( p). The input output model of the buck converter can be given as: y(k) =

z −1 B(z −1 ) u(k) A(z −1 )

where A(z −1 ) and B(z −1 ) are two polynomials expressed as follows: A(z −1 ) = 1 + a1 z −1 + a2 z −2 B(z −1 ) = b0 + b1 z −1 such that

where

a1 = −2e−ζ w0 Te cos(wTe ) w0 Te a2 = e−2ζ ( ( )) e) b0 = η 1 − e−ζ w0 Te cos(wTe ) + ζ w0 sin(wT w ( )) ( e) b1 = η e−2ζ w0 Te + e−ζ w0 Te ζ w0 sin(wT − cos(wT ) e w

(3)

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η = V/ in 1 w0 = LC 1 ζ = 2RCw0 √ w = w0 1 − ζ 2

3 Discrete-Time Integral Sliding Mode Control Using Input-Output Model Generally, there are two modes for the SMC: the sliding mode and the reaching mode. In the former, an equivalent control law is applied to lead the system states to attain the sliding function from the origin. This mode is characterized by the existence of a high frequency known as chattering phenomenon. Many solutions have been proposed, such as replacing the signum function by a saturation function and developing a higher and a second order sliding mode control [15]. In the reaching mode, the system is more sensitive to the uncertainties like unmodeled dynamics and external disturbances. To solve this problem, a new discrete-time integral sliding mode controller using input-output model is proposed and developed in [15]. In this paper, we apply this technique to a DC-DC buck converter in order to demonstrate its effectiveness to power electronic system application. Consider the input output model of DC-DC buck converter described by Eq. (3). The integral sliding surface is defined as [15]: S(k) = αS(k − 1) + C(z −1 )e(k)

(4)

with e(k) = y(k) − yd (k) is a tracking error where yd (k) is the desired output voltage. α is an integral coefficient. C(z −1 ) is a stable polynomial defined as: C(z −1 ) = 1 + c1 z −1 + .......cnC z −nC Consider the two polynomials E(z −1 ) and F(z −1 ) solutions of the following diophantine equation: C(z −1 ) = A(z −1 )G(z −1 )E(z −1 ) + z −1 F(z −1 ) with

⎧ E(z −1 ) = 1 ⎪ ⎪ ⎨ F(z −1 ) = 1 + f 1 z −1 + ....... f n F z −n F n F = sup(n C − 1, 2) ⎪ ⎪ ⎩ G(z −1 ) = 1 − z −1 : is a differential operator

The integral sliding mode control is given as:

(5)

Robustness Analysis of a Discrete Integral Sliding Mode Controller …

u(k) = u eq (k) + u dis (k)

277

(6)

where • u dis (k) = −M sign(S(k)) is a discontinuous term added to ensure the robustness of the discrete integral sliding mode control against disturbances. M is selected as the following quasi-sliding condition verifying [11]: |S(k + 1)| < |S(k)| and sign is the signum function expressed by: sign(S(k)) =

S(k) if S(k) /= 0 |S(k)|

• u eq (k) is the equivalent control law obtained to verify the following condition: S(k + 1) = S(k) = 0 Based on (1), (3)–(5), the integral sliding surface at instance (k + 1) becomes: S(k + 1) = αS(k) + C(z −1 )(y(k + 1) − yd (k + 1)) (7) = αS(k) + G(z −1 )B(z −1 )u(k) + F(z −1 )y(k) − C(z −1 )yd (k + 1) Based on the condition of existence of the quasi SMC S(k + 1) = S(k) = 0, the equivalent control law is calculated as: [ ]−1 [ ] −F(z −1 )y(k) + C(z −1 )yd (k + 1) u eq (k) = G(z −1 )B(z −1 )

(8)

Thus, the global discrete integral sliding mode control law is given as follows: u(k) = u eq (k) + u dis (k) ]−1 ] ] ] −F(z −1 )y(k) + C(z −1 )yd (k + 1) − M sign(S(k)) = G(z −1 )B(z −1 )

(9)

4 Simulation Results In this part of the paper, a discrete integral sliding mode control (DISMC) is applied to the DC-DC buck converter. This converter is described by the input-output model (3). Parameter values of converter are shown in Table 1. The synthesis parameters of the integral sliding surface are chosen as:

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Table 1 Parameter values of DC-DC buck converter

Parameter values of DC-DC buck 24 V 40 µ F 1.5 mH 250 Ω 12 V

Vin C L R V0

Refer to the parameter values of this buck given in Table 1, the two polynomials A(z −1 ) and B(z −1 ) are given as follows: A(z −1 ) = 1 + 0.1634z −1 + 0.0067z −2 B(z −1 ) = 25.9589 + 2.1252z −1 Three cases are considered: DC-DC buck converter without parameter variations, with parameter variations and with external disturbances and parameter variations. Case 1: Without parameter variations In this case, the schematic diagram blocks of the considered DC-DC buck converter and the discrete integral sliding mode controller are shown in Figs. 2 and 3, respectively. Simulation results based on MATLAB-SIMULINK of our technique are illustrated in Figs. 4, 5 and 6. These figures show respectively the evolution of the output

Discre te , Ts = 5e -06 s. powe rgui iL t

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G oto4 d

C onsta nt

T o Workspa ce5

P WM B lock

12 y ref (k) c

d

u(k) y (k)

V out V out

IS MC 1

Fig. 2 Schematic diagram of the DC-DC buck converter with DISMC (without parameter variations)

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Sc To Workspace1 S(k)

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yref(k)

num(z)

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u(k)

den(z) Discrete Transfer Fcn2 2 y(k)

num(z) den(z) Discrete Transfer Fcn1

Fig. 3 Schematic diagram of discrete integral sliding mode controller (DISMC) Fig. 4 Evolution of the output voltage y(k) of the DC-DC buck converter (without parameter variations)

Fig. 5 Evolution of the Sliding surface S(k) of the DC-DC buck converter (without parameter variations)

voltage y(k), the evolution of the sliding surface S(k) and the evolution of the control law u(k) for the DC-DC buck (Fig. 7). All these figures explain that the proposed control method is applicable to digital power converters. For instance, it can be seen from Fig. 4 that the output voltage y(k) follows its desired value yd (k). Furthermore, it can be seen that the DISMC can eliminate the negative effect of the chattering phenomenon. Case 2: With parameters variations Rmin < R < Rmax To prove the robustness of our proposed DISMC, we suppose that the considered DC-DC buck converter is subjected to parameter variations. Furthermore, we change

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Fig. 6 Evolution of the control law u(k) of the DC-DC buck converter (without parameter variations)

Fig. 7 Schematic diagram of the DC-DC buck converter with parameter variations

the resistance between two values; a minimum load resistance Rmin and a maximum load resistance Rmax . The following figure presents the schematic diagram of the DC-DC buck converter with variable resistance. The evolutions of the output voltage y(k), the sliding surface S(k) and the control u(k) are shown respectively in Figs. 8, 9 and 10. From these figures, the output voltage y(k) converge to the desired reference yd (k) despite the presence of a parameter variations. It can be concluded that the proposed control achieves better robustness against uncertainties. Case 3: With disturbances and parameter variations In this case, we suppose that the DC-DC buck converter is subjected simultaneously to external disturbances and parameter variations.

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Fig. 8 Evolution of the output voltage y(k) of the DC-DC buck converter with parameter variations

Fig. 9 Evolution of the integral sliding surface S(k) of the DC-DC buck converter with parameter variations

Fig. 10 Evolution of the control law u(k) of the DC-DC buck converter with parameter variations

The parameter variations are the same given in case 2 and the schematic diagram of the DC-DC buck converter in this case is shown by Fig. 11. Figure 12 illustrate the external disturbance which is chosen as a variable signal. The evolution of the output voltage y(k) and the integral sliding function S(k) are shown respectively by Figs. 13 and 14. Figure 15 gives the evolution of the control law u(k). The application of our proposed discrete integral sliding mode control (see Fig. 13) allows to track reference trajectory (Fig. 14) and to have a quasi-sliding mode (Fig. 15) despite the presence of external and internal disturbances.

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Fig. 11 Schematic diagram of the DC-DC buck converter with parameter variations Fig. 12 Evolution of the external disturbances d(k)

Fig. 13 Evolution of the output voltage y(k) of the DC-DC buck converter with disturbances and parameter variations

Fig. 14 Evolution of the integral sliding surface S(k) of the DC-DC buck converter with disturbances and parameter variations

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Fig. 15 Evolution of the control law u(k) of the DC-DC buck converter with disturbances and parameter variations

5 Conclusion In this paper, a discrete integral sliding mode controller was proposed. It is synthesized using only the input-output measurement. The proposed strategy is very appropriate for time-varying power systems (DC-DC Buck converter). It ensures a high output voltage in the presence of internal and external disturbances. A simulation example, conducted using SIMULINK-MATLAB, was applied to a DC-DC buck converter. Simulation results showed good performances in terms of the reduction of chattering phenomenon and the robustness against parameter variations and external disturbances. Acknowledgements This work was supported by the Ministry of the Higher Education and Scientific Research in Tunisia and the University of Reims, Champagne Ardenne in France.

References 1. Sira-Ramirez, H., Oliver-Salazar, M.A.: On the robust control of buck-converter DC-motor combinations. IEEE Trans. Power Electron. 28(8), 3912–3922 (2012) 2. Krishnan, R.: Permanent Magnet Synchronous and Brushless DC Motor Drives. CRC Press (2017) 3. Qamar, M.A., Feng, J., Ur Rehman, A., Raza, A.: Discrete time sliding mode control of DCDC buck converter. In: IEEE Conference on Systems, Process and Control (ICSPC), pp. 91–95 (2015) 4. Rashid, M.: Power Electronics: Circuits, Devices, and Applications. Prentice-Hall (1993) 5. Prodic, A., Maksimovic, D.: Design of a digital PID regulator based on look-up tables for control of high-frequency DC-DC converters. In: IEEE Workshop on Proceedings in Computers in Power Electronics, pp. 18–22 (2002) 6. Ugale, C.P., Dixit, V.V.: Dc-dc converter using fuzzy logic controller. Int. Res. J. Eng. Technol. (IRJET) 2(4), 593–596 (2015) 7. Elmas, C., Deperlioglub, O., Sayan, H.: Adaptive fuzzy logic controller for DC-DC converters. Expert Syst. Appl. 36(2), 1540–1548 (2009) 8. Al-Baidhani, H., Kazimierczuk, M.K.: PWM based proportional integral sliding mode current control of DC-DC boost converter. In: IEEE Conference in Texas Power and Energy(TPEC), pp. 1–6 (2018) 9. Ling, R., Shu, Z., Hu, Q., Song, Y.D.: Second-order sliding-mode controlled three-level buck DC-DC converters. IEEE Trans. Ind. Electron. 65(1), 898–906 (2018)

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10. Utkin, V.: Sliding mode control of DC/DC converters. J. Frankl. Inst. 2146–2165 (2013) 11. Nouri, A.S.: Sur les regimes glissants continu et discret. Habilitation universitaire, National Engineering School of Sfax, Tunisia (2008) 12. Mitic, D., Antic, D., Peric, S.: Input-output based quasi-sliding mode control of DC-DC converter. Facta Univ.-Ser.: Electron. Energ. 25(1), 69–80 (2012) 13. Perry, A.G., Feng, G., Yan-Fei, L., Sen, P.C.: A new sliding mode like control method for buck converter. IEEE Annu. Power Electron. Spec. Conf. 4, 3688–3693 (2004) 14. Shen, L., Lu, D.D.-C., Chengwei, L.: Adaptive sliding mode control method for DC DCconverters. IET Power Electron. 8(9), 1723–1732 (2015) 15. Elhajji, Z., Dehri, K., Nouri, A.S.: Stability analysis of discrete integral sliding mode control for input-output model. J. Dyn. Syst., Meas., Control 139(3) (2017) 16. Babazadeh, A., Maksimovic, D.: Hybrid digital adaptive control for fast transient response in synchronous buck DC–DC converters. IEEE Trans. Power Electron. 24(11), 2625–2638 (2009) 17. Yang, Y., Siew-Chong, T., Hui, S.Y.R.: Adaptive reference model predictive control with improved performance for voltage-source inverters. IEEE Trans. Control Syst. Technol. 26(2), 724–731 (2017) 18. Belkaid, A., Colak, I., Kayisli, K., Bayindir R.: Indirect sliding mode voltage control of buck converter. In: 8th International Conference on Smart Grid (icSmartGrid), pp. 90–95 (2020) 19. Zhang, J., Shi, P., Xia, Y., Yang, H.: Discrete-time sliding mode control with disturbance rejection. IEEE Trans. Ind. Electron. 66(10), 7967–7975 (2018) 20. Li, H., Ye, X.: Sliding-mode PID control of DC-DC converter. In: 5th IEEE Conference on Industrial Electronics and Applications, pp. 730–734 (2010) 21. Ling, R., Maksimovic, D., Leyva, R.: Second-order sliding-mode controlled synchronous buck DC-DC converter. IEEE Trans. Power Electron. 31(3), 2539–2549 (2015)

Fault Tolerant Control for Uncertain Neutral Time-Delay System Rabeb Benjemaa, Aicha Elhsoumi, and Mohamed Naceur Abdelkrim

Abstract The main goal of this work is the fault tolerant control (FTC) for the uncertain neutral time-delay system with and without delayed input. For the uncertain neutral time-delay system with not delayed input, we consider the adaptive observer for the fault detection and estimation steps. For the case of delayed input system, we consider the robust stabilization to guarantee the stability of system in closed loop. Then, for the fault compensation step, the fault tolerant control law is the addition of two terms, the first one is the nominal control and the second is the additive control. Finally, simulation results are presented to prove the theoretical development. Keywords Delayed input · Uncertain system · Neutral time-delay · Adaptive observer · FTC control · Robust stabilization · Nominal control · Additive control

1 Introduction Uncertainties and neutral time-delay occur in the dynamic response of many physical systems [1, 8, 10, 11]. The presence of delay causes degradation of the performance and instability in the control system. This type of delay can be a property inherent in real systems such as: chemical reactor, transmission line,... For this reason, it is important to take these delays into consideration. The stability study for this type of system is a complex problem on which many work is done [2, 3, 5, 6]. The complexity of industrial systems is increasing in the last decades, so R. Benjemaa (B) · M. N. Abdelkrim National Engineering School of Gabes, MACS LR16ES22, Gabes University, Gabes, Tunisia e-mail: [email protected] M. N. Abdelkrim e-mail: [email protected] A. Elhsoumi Higher Institute of Informatics of Medenine ISIMed, MACS LR16ES22, Gabes University, Gabes, Tunisia © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Derbel et al. (eds.), Advances in Robust Control and Applications, Studies in Systems, Decision and Control 474, https://doi.org/10.1007/978-981-99-3463-8_12

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they have need to guarantee the reliability, and the desired performances. It is which increased interest in the diagnosis. Diagnosis based on the observer allow respectively to the fault detection and estimation, so a good decision of the presence or the absence of the fault. The adaptive observer used in this work allow both fault detection therefore residual generation and fault estimation. If a malfunction appears whatever the process level, actuators or sensors, so it becomes necessary to intervene a strategy of control to ensure reliability, maintain performance and react quickly if an anomaly is detected, we are therefore talking about “Fault Tolerant Control” (FTC) which can compensate for the effect of the fault on the system in order to guarantee the desired performance. There are generally, two main classes of FTC: passive and active approaches. A large number of works deals with the stability analysis [2, 3, 5, 6] and the control synthesis of uncertain time delay systems. Soltani and El Harabi [8] developed a H∞ Fault tolerant control for uncertain state time-delay systems. An adaptive robust control scheme was proposed in Sun and Zhao [9] to stabilize uncertain time-delay systems. In [7] , the authors proposed an adaptive robust control of uncertain neutral timedelay systems. As well as [4] developed an active fault tolerant control law for uncertain neutral time delay system. The contribution of the presence paper concerns the development of an active fault tolerant control law for uncertain neutral time delay with delayed and not delayed input. Also, the development of robust stabilization for the uncertain neutral time delay in the presence of delayed input. The main idea of the proposed control law is based on adding an additional term to the nominal law that is considered like a state feedback controller to stabilize the closed-loop system. This paper is organized as follows: Sect. 2 presents the preliminaries for this work. In Sect. 3, we present the uncertain neutral time-delay system with not delayed input, two steps of active fault tolerant control are present in this section: Fault detection and Fault estimation. Section 4 presents the uncertain neutral time-delay system with delayed input. The robust stabilization is developed in this section to stabilize the system with delayed input and the fault compensation is present to compensate the fault effect for two types of system with and without delayed input. This latter developed a control law, based on adding an additional term to the nominal control to compensate the fault effect. Section 5 gives simulation results of numerical example and the last part is a conclusion.

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2 Preliminaries Notation: R n and R n×m denote, respectively, the n-dimensional Euclidean space and the set of n×m real matrices. The superscript “T” denotes the transpose and LMI denote Linear Matrix Inequality. I is the identity matrix of appropriate dimension. Noting the “zero-initial” condition of x(t); V (x0 ) = 0 and V (x∞ ) > 0, then: ∫



Φ= ≤

∫0 ∞ 0

y T (t)y(t)dt (

) y T (t)y(t) + V˙ (t, xt ) dt −





x T (t)C T C x(t) + V˙ (t, xt ) dt

0

(1)

Lemma 1 For given matrices Q = Q T , H , E, Ψ1 , Ψ2 , φ1 , φ2 and T with appropriate dimensions. Q + H T E + E T T T H T + Ψ1 T Ψ2 + Ψ2T T T Ψ1T + φ1 T φ2 + φ2T T T φ1T < 0 For all T satisfies T T T ≤ I , if there are positive numbers γ1 > 0, γ2 > 0 and γ3 > 0 such that: Q + γ1 H H T + γ1−1 E T E + γ2 Ψ1 Ψ1T + γ2−1 Ψ2T Ψ2 + γ3 φ1 φ1T + γ3−1 φ2T φ2 < 0

3 Uncertain Neutral Time-Delay System with Not Delayed Input Consider a class of linear uncertain neutral time delay systems with not delayed input described by Eq. (2): ⎧ x(t) ˙ = (A0 + ΔA0 ) x(t) + (A1 + Δ A1 ) x (t − d) ⎪ ⎪ ⎨ + ( A2 + ΔA2 ) x˙ (t − h) + (B + ΔB) u(t) y(t) = C x(t) ⎪ ⎪ ⎩ x(t) = ϕ(t); t ∈ [−τ , 0]

(2)

where x(t) ∈ R n is the state vector,u(t) ∈ R m is the input vector, y(t) ∈ R p is the output vector. d > 0 and h > 0 are respectively state and its derivative delay. A0 ,A1 ,A2 ,B and C are constant matrix with appropriate dimensions.with appropriate dimensions. ΔA0 ,ΔA1 ,Δ A2 and ΔB are matrix-valued functions representing time-varying parameter uncertainties in the system model; and ϕ(t) is a given continuous vector valued initial function.

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Fig. 1 The steps of the active fault tolerant control

The parameter uncertainties considered here are described as following: [

[ ] ] ΔA0 ΔA1 ΔA2 ΔB = Z ∗ T∗ E 1 E 2 E 3 E 4

(3)

where Z ∗⎡, E 1 ,⎤E 2 , E 3 and E 4 are constant matrix with appropriate dimensions. T1 ⎢ T2 ⎥ ⎥ T∗ = ⎢ ⎣ T3 ⎦ is a matrix known with Lebesgue elements measurable and satisfied: T4 T∗T T∗ ≤ I . ¯ 1 = A1 + ΔA1 ; A ¯ 2 = A2 + ΔA2 and B¯ = B + ΔB. ¯ 0 = A0 + ΔA0 ; A Noting: A Considering the uncertain system (2) affected by actuator fault as following: ⎧ ¯ 1 x (t − d) + A ¯ 2 x˙ (t − h) + Bu(t) ¯ ¯ 0 x(t) + A + Fa f a (t) ˙ =A ⎨ x(t) y(t) = C x(t) ⎩ x (0) = x0 ; t ≤ 0

(4)

where f a (t) is actuator fault and Fa is constant matrix with appropriate dimension. For the concept of active fault tolerant control, we follow the steps as described in Fig. 1.

3.1 Fault Detection Considering the adaptive observer for the system (4) to detect the actuator fault as following: ⎧ x˙obs (t) = A0 xobs (t) + A1 xobs (t − d) + A2 x˙obs (t − h) + Bu(t) ⎪ ⎪ ⎪ ⎪ ⎨ +O1 (y(t) − yobs (t)) + O2 (y (t − d) − yobs (t − d)) yobs (t) = C xobs (t) ⎪ ⎪ x (0) = x0 ; t ≤ 0 ⎪ ⎪ ⎩ r (t) = L (y(t) − yobs (t))

(5)

where xobs (t) ∈ R n and yobs (t) ∈ R p are respectively the state vector of the observer and its output. L is the residue weight. The error and the residual model are described as follows:

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⎧ e(t) ˙ = ( A0 − O1 C) e(t) + ( A1 − O2 C) e (t − d) + A2 e˙ (t − h) ⎪ ⎪ ⎨ +Fa f a (t) + ΔA0 x(t) + Δ A1 x (t − d) + ΔA2 x˙ (t − h) + ΔBu(t) r (t) = Le(t) ⎪ ⎪ ⎩ e (0) = e0 ; t ≤ 0 (6) The following LMI developed by [3] to calculate O1 and O2 : Theorem 1 ([3]) If there exist symmetric positive definite matrices P1 , P2 , P3 , P4 , P5 , P6 , P7 ∈ R n×n and Q 1 , Q 2 ∈ R n× p , Wi , Z i ∈ R n×n , i = 1, . . . , 4 such that: ⎡

N11 ⎢ ∗ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎣ ∗ ∗

N12 N22 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

N13 N14 N15 N23 N24 N25 N33 A2T Z 4T 0 ∗ −P4 −A2T P1 Fa ∗ ∗ N55 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

hW1 hW2 hW3 hW4 0 −h P5 ∗ ∗ ∗ ∗



h Z1 N18 N19 N10 h Z2 N28 N29 N20 ⎥ ⎥ h Z3 0 0 0 ⎥ ⎥ h N4 A2T P1 h A2T P1 0 ⎥ T T T 0 Fa P1 h Fa P1 h Fa P1 ⎥ ⎥ 0, G 2 > 0 and G 3 > 0, such that: G 2 = O2 .G 1 and V is an arbitraries matrix with appropriate dimensions and satisfied the following LMI: ⎡

G 1 A2 −G 1 Fa 2G 1 A0 − 2G 2 C + C T G 4 C + G 3 ⎢ ∗ − (1 − β) G 3 ⎢ ⎢ ∗ ∗ ⎢ ⎣ ∗ ∗ ∗ ∗

+ C T V T Fa 0 Γ2 ∗ ∗

⎤ 00 0 0⎥ ⎥ 0 0⎥ ⎥ 0, if there exist a positive definite symmetric matrices Ω1 , Ω2 , Ω3 , Ω4 and matrices Ω5 , Ω6 , Ω7 and γ1 > 0, γ2 > 0 and γ3 > 0 such that it satisfies the matrix equation and the LMI as following: ][ ]T [ ] [ ] [ Ω1 + Ω5 Ω3 = Ω6 Ω7 K 2 = Ω6 Ω7 Ω1 + Ω5 Ω3 ⎡

Ω11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ Ω=⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

Ω12 Ω22 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

⎤ Ω13 Ω14 Ω4T Ω17 Ω18 Ω19 Ω10 0 0 0 A1 Ω3 0 0 0 ⎥ ⎥ 0 A2 Ω3 0 0 0 ⎥ Ω33 0 ⎥ 0 0 ⎥ ∗ −γ 2 I 0 CΩ3 0 ⎥ ∗ ∗ −I 0 0 0 0 ⎥ ⎥ 0. Λ < 0 is equivalent to Ω˜ < 0 such that Ω˜ satisfies the following LMI:

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Ω˜ 11 ⎢ ∗ ⎢ ⎢ ∗ ˜ Ω=⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

Ω˜ 12 Ω˜ 22 ∗ ∗ ∗ ∗

Ω˜ 13 Ω˜ 23 Ω˜ 33 ∗ ∗ ∗

Ω4T 0 0 −I ∗ ∗

CT 0 0 0 −I ∗

⎤ A0T Ω3 A1T Ω3 ⎥ ⎥ A2T Ω3 ⎥ ⎥ 0, if there are symmetric matrices defined positive Ω1 , Ω2 , Ω3 and Ω4 and matrices Ω5 , Ω6 , Ω7 such that the following LMI is verified: ⎤ π11 π12 π13 π14 Ω4T A0 Ω3 + BΩ7 ⎥ ⎢ ∗ −Ω2 0 0 0 A1 Ω3 ⎥ ⎢ ⎥ ⎢ ∗ ∗ −Ω3 − Ω T Ω4 0 0 A2 Ω3 4 ⎥ 0, γ2 > 0 and γ3 > 0 such that: π + γ1 Θ1 Θ1T + γ1−1 Θ2T Θ2 + γ2 δ1 δ1T + γ2−1 δ2T + γ3 ρ1 ρ1T + γ3−1 ρ2T ρ2 < 0 which is equal to Eq. (14) under condition of Eq. (13). Fin Proof.

4.2 Fault Compensation We propose to synthesis a fault tolerant control u F T C (t). The main idea of this type of control is in the adding of new control “additive control” to the nominal one to maintain the desired performance and to compensate the fault effect. The FTC control is described as follows: u F T C (t) = u nom (t) + u add (t)

(23)

For the case of the system (2), the nominal control is described as follows: u nom1 (t) = K 1 x(t) + ε(t) ∫

where: ε(t) =

0

t

(yr (θ) − y (θ)) dθ

(24)

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and ε(t) ˙ = yr (t) − y(t) where: ε is the static error and yr is the desired output. K 1 is the feedback matrix gain obtained by LMI resolution in the following theorem. Theorem 4 ([4]) For given scalars τ > 0 and μ < 1, and the constant matrices A0 , A1 , and A2 , the system (2) is asymptotically stabilizable by the state feedback, if there exists P = P T > 0, Q = Q T > 0, Z z = Z z T > 0 and the matrix with appropriate dimensions Ni (i = 1, 2, 3), Ss and Tt such that satisfy the following LMI: ⎡

∗ Γ11 ⎢ ∗ Γ∗ = ⎢ ⎣ ∗ ∗

where:

∗ Γ12 ∗ Γ22 ∗ ∗

∗ Γ13 ∗ Γ23 ∗ Γ33 ∗

⎤ τ N1 τ N2 ⎥ ⎥ 0 , if the following conditions are satisfied: Z3 ≥ 0 ε02 Z 3 + ε0 Z 2 + Z 1 < 0 Z1 < 0 Then, ∀ ε ∈ [0 ε0 ]:

ε2 Z 3 + ε Z 2 + Z 1 < 0

Lemma 2 (Schur complement [1, 15]) For the matrices N , M and S , if N and M are symmetric matrices, then the following LMIs are equivalents: (

and

N S ST M

) 0, the multivariable MIMO SPS given by Eq. (7) is asymptotically stable for all ε ∈ [ 0 ε∗ ], if there exist a matrix Q = Q T > 0 satisfying the followings LMIs: ( ) −Q A T (0)Q ' φ (0) = L, yl in (12) can be also rewritten as: L ∑

yl =

i=1

i

f yli +

L ∑ i=1

i

f +

M ∑ i=L+1 M ∑

f i yli = fi

L ∑

q li yli +

i=1

M ∑

q li yli

i=L+1

i=L+1

= ξlT θl

(13)

∑L ∑M i i f + i=L+1 f i . In the meanwhere: q li = f i /Dl , q li = f /Dl and Dl = i=1 [ ] [ [ ] ] time, we have Q l = q l1 , . . . , q lL , Q l = q l1 , . . . , q lL , ξlT = Q l Q l and [ ] θlT = yl yl . The defuzzified crisp value from an interval T-2 FLS is obtained as: y (x) = ξ T θ [ 1

(14)

] T

where: ξ T = 2 ξrT , ξl and θ T = [θr , θl ] The unknown system dynamics given by (2) and the switching control are approximates. ⎧ f (x) = ⌃ f (x) + Δ f (x) = θ Tf ξ f (x) + Δ f (x) ⎪ ⎪ ⎨ g (x) = ⌃ g (x) + Δ g (x) = θgT ξg (x) + Δ g (x) ⎪ ⎪ ⎩ h (x) + Δ h (x) = θhT ξh (x) + Δ h (x) u sw (x) = ⌃

(15)

Such that: w =Δ f (x) + Δ g (x) u is the approximation error.

3 Proposed Control Synthesis The recursive nature of the suggested control design is the same as in the standard Backstepping methodology. However, the control design proposed here uses Backstepping to design controllers with a zero order sliding surface at each stage [7, 12, 15]. The design proceeds is as follows: For the first step we consider zero-order sliding surface: (16) s1 = x1 − x1d

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Let the first Lyapunov function candidate: V1 (s) =

1 2 s 2 1

(17)

The time derivation of (17) is given by: V˙1 (s) = s1 s˙1 = s1 (x2 + ϕ1 (x1 ) − x˙1d ) = −c1 s12 + s1 s2

(18)

The stabilization of s1 can be obtained by introducing a new virtual control x2d , such that: (19) c1 > 0 x2d = x˙1d − c1 s1 − ϕ1 (x1 ) , where c1 is the feedback gain. For the second step, we consider the following zeroorder sliding surface: s2 = x2 − x2d = x2 − x˙1d + c1 s1 + ϕ1 (x1 )

(20)

The augmented Lyapunov function is given by: V2 (s1 , s2 ) = V1 +

1 2 1 ~T ~ 1 ~T ~ θf θf + θ θg s2 + 2 2μ f 2μg g

(21)

with: ~ θf = θf −⌃ θ f and ~ θg = θ g − ⌃ θg , θ f and θg are the parameter vectors respecθf,⌃ θg , are their estimations. The time derivatively of functions f (x) and g (x), and ⌃ tive of V2 (s1 , s2 ) is then: 1 ~ ~ 1 ~~ V˙2 (s1 , s2 ) = s1 s˙1 + s2 s˙2 + θ f θ˙ f + θg θ˙ g μf μg

(22)

We have: ~ θ˙ f = −⌃ θ˙ f and ~ θ˙ g = −⌃ θ˙ g . Then: [ ] V˙2 (s1 , s2 ) = −c1 s12 + s2 s1 + ~ θ Tf ξ f (x) + ~ θgT ξg (x)u − x˙2d + w + d(t) [ ] [ ] 1 ⌃ 1 ⌃ +~ θ Tf s2 ξ f (x) − θgT s2 ξg (x)u − (23) θ˙ f + ~ θ˙ g μf μg with: x˙2d = −c1 s2 + c12 s1 + x¨1d − ϕ˙ 1 (x1 )

(24)

The negativity of the differential of the Lyapunov function, allows getting the following control law:

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u = u eq −

1 u sw ⌃ g

(25)

u sw = −k sign (s2 )

(26)

u sw is the switching control. Then, V˙2 (s1 , s2 ) =

(

[

−c1 s12

+

s2 s1 + ⌃ θ Tf ξ f

(x)

+⌃ θgT ξg

(x) u eq −

1 ⌃ θgT ξg (x)

) u sw

] +c1 s2 − c12 s2 − x¨1d + ϕ˙ 1 (x1 ) + w + d (t) [ ] [ ] 1 ⌃ 1 ⌃ +~ θ Tf s2 ξ f (x) − θgT s2 ξg (x) u − θ˙ f + ~ θ˙ g μf μg

(27)

w is the approximation errors of functions f (x) and g (x). The equivalent control is then: u eq =

[

1 ⌃ θgT ξg

(x)

] −c2 s2 − s1 − ⌃ θ Tf ξ f (x) + x¨1d − c1 s2 + c12 s1 − ϕ˙ 1 (x1 )

(28)

The control law becomes: u=

[ ] 1 θ Tf ξ f (x) + x¨1d − c1 s2 + c12 s1 − ϕ˙ 1 (x1 ) − u sw −c2 s2 − s1 − ⌃ T ⌃ θg ξg (x) (29) ⎧ ⌃ θ˙ f = μ f s2 ξ f (x) (30) ⌃ θ˙ g = μg s2 ξg (x) u

The Eq. (27) is developed to: [ ] V˙2 = −c1 s12 − c2 s22 − s2 k sign (s2 ) − d (t) − w

(31)

Introducing the norm, we get: V˙2 ≤ −c1 s12 − c2 s22 − |s2 | (k − γ)

(32)

We have: |d (t) + w| ≤ γ, {c2 , k} are positive constants, with k > γ. The value of the constant k depends on the upper bound of the structural uncertainties and external disturbances, which are unknown. In order to resolve this problem, we modify the previous control law, using a fuzzy adaptive system ⌃ h (s) in (15). The derivative of the sliding surface given in (20), is as follows: h ∗ (s) + d (t) − x˙2d s˙2 (x, t) = θ Tf ξ f (x) + θgT ξg (x) u + θhT ξh (s) + w ' − ⌃

(33)

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where: ⌃ h (s) = θhT ξh (s) and w ' =Δ f (x) + Δ g (x) u− Δ h (x). We consider the following Lyapunov function: V2 (s1 , s2 ) = V1 +

1 2 1 ~T ~ 1 ~T ~ 1 ~T ~ θ θf + θ θg + θ θh s + 2 2 2μ f f 2μg g 2μh h

(34)

The derivative of the latter introducing the control law (35), is given by: [ V˙2 (s1 , s2 ) = −c1 s12 + s2 s1 − c12 s2 + c1 s2 − x¨1d + ϕ˙ 1 (x1 ) + ⌃ θ Tf ξ f (x) [ ] 1 ⌃ +⌃ θgT ξg (x)u eq − u sw − ⌃ h ∗ (s) +w + d(t) ] + ~ θ Tf s2 ξ f (x) − θ˙ f μf [ ] [ ] 1 ⌃ 1 ⌃ +~ θgT s2 ξg (x)u − θhT s2 ξh (x) − (35) θ˙ g + ~ θ˙ h μg μh θhT ξh (s2 ) and u sw = ⌃ θhT ξh (s2 ). where: u = u eq − ⌃θ T ξ1 (x) ⌃ g g Consequently, the equivalent control law is given by: u eq =

[

1 ⌃ θgT ξg (x)

] −c2 s2 − s1 − ⌃ θ Tf ξ f (x) + ⌃ θhT ξh (s2 ) + x¨1d + c1 (c1 s1 − s2 ) (36)

To ensure the negativity of the Lyapunov function derivative, we choose the following control law: u=

[ ] 1 −c2 s2 −s1 −⌃ θ Tf ξ f (x)+⌃ θhT ξh (s2 )+ x¨1d +c1 (c1 s1 −s2 )−u sw (37) T ⌃ θg ξg (x)

The adaptation laws are given as follows: ⎧ ⌃ ⎪ θ˙ f = μ f s2 ξ f (x) ⎪ ⎪ ⎨ ⌃ θ˙ g = μg s2 ξg (x)u ⎪ ⎪ ⎪ ⎩⌃ θ˙ h = μh s2 ξh (s2 )

(38)

| ∗ | | '| The optimal value of ⌃ h (s) is such that: |⌃ h (s)| ≥ |w | + |d (t)| From Eq. (38), we have: | ∗ |) (| | h (s)| < 0 V˙2 ≤ −c1 s12 − c2 s22 + |s2 | |w' | + |d| − |⌃ which implies that: V˙2 ≤ 0, so the closed loop system is stable and robust.

(39)

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4 Simulation Results In order to verify the performance and the robustness of the proposed control law, we apply the latter on the inverted pendulum, simulations made considering different cases and conditions [18]. The Cart-Pendulum is a non-linear system with two degrees of freedom. ⎧ ⎨ x˙1 (t) = x2 x˙2 (t) = f (x) + g(x)u(t) + d(t), [g(x) /= 0] (40) ⎩ y(t) = x1 where: u (t) and y (t) are respectively, the input and the output of the system. f (x) and g (x) are nonlinear unknown continuous smooth functions [18]. The [ state-space ] variables are the position and the velocity of the pendulum [x1 x2 ] = θ θ˙ and the output is y (t). To show the robustness of the system, we choose the disturbances and uncertainties as: Δm = ±0.1 m ΔM = ±0.4 M d(t) = 0.1 sin 2t The membership functions for f (x) and g (x) are given as in [18]. To construct the fuzzy system for the signal h (s), which approximates the switching control, the following membership functions are associated: 1 1 + 8. exp (si + 0.1) [ ( ] si )2 μZero (si ) = exp − 0.5 1 μPositive (si ) = 1 + 8. exp (0.1 − si )

μNegative (si ) =

Where three fuzzy rules are used to deduce the signal: (

)

(

) ⌃ R : if si is Negative then h (si ) = −C ( ) ( ) 2 ⌃ R : if si is Zero then h (si ) = 0 ( ) ( ) 3 ⌃ R : if si is Positive then h (si ) = +C 1

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Fig. 1 Pendulum angle tracking

Fig. 2 Tracking error

To verify the effectiveness and benefits of T2FS, we make a comparative study between the two algorithms already proposed (T1FABSMC and T2FABSMC). The results of the Fuzzy Adaptive Backstepping Sliding Mode Control are shown in Figs. 1, 2 and 3. The innovation that will be given to this new controller is the function of type-2 memberships as following: [ ( xi + μ Fi1 (xi ) = exp − π 20

π 6

)2 ]

[ ( xi + , μ F 1 (xi ) = exp − π i

30

π 6

)2 ]

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Fig. 3 Control signal

[ ( xi + μ Fi2 (xi ) = exp − π

π 12

)2 ]

[ ( xi + , μ F 2 (xi ) = exp − π i

24

π 12

)2 ]

30

[ ( [ ( ] ] π )2 π )2 μ Fi3 (xi ) = exp − xi + , μ F 3 (xi ) = exp − xi + i 24 30 [ ( xi − μ Fi4 (xi ) = exp − π

π 12

)2 ]

20

π 12

[ ( xi − , μ F 5 (xi ) = exp − π

π 6

i

24

[ ( xi − μ Fi5 (xi ) = exp − π

[ ( xi − , μ F 4 (xi ) = exp − π

π 6

)2 ]

i

)2 ]

30

)2 ]

30

Simulation results were carried out in the presence of uncertainties. We can observe in the figures (Figs. 4, 5 and 6) that the dynamic response of the position obtained by the Type-1 fuzzy regulator is better than the one obtained using Type-2 fuzzy regulator (response time ≈ 0.15s and ≈ 0.2s). However, the permanent response obtained by the Type-2 fuzzy regulator is more accurate than the one obtained using Type-1 fuzzy regulator. We can clearly see that the control efforts generated by the Type-2 regulator are less than those generated using the Type-1 fuzzy regulator, which is an advantage of our proposed algorithm (Type-2 fuzzy).

Design of Interval Type 2 Fuzzy Adaptive Sliding Mode Control Fig. 4 Pendulum angle tracking

Fig. 5 Tracking error

Fig. 6 Control signal

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5 Conclusion In this paper, we have presented an interval type 2 fuzzy Adaptive Backstep-ping Sliding Mode Controller for a class of nonlinear perturbed uncertain systems, which integrate the interval type 2 fuzzy logic control, the sliding mode control and the Backstepping control. The combined strategies are used in order to have the advantages of these approaches. The IT2AF systems are introduced to ap-proximate on one hand the unknown dynamics of the system, and on the other hand the switching control. In order to overcome some restrictive constraints and to decrease efficiently the chattering phenomenon, the stability and the robust-ness of the closed loop system are proved analytically. The updates of adjustable parameters of the Type-2 fuzzy systems are ensured by adaptation laws derived using the Lyapunov theory. Simulation results show that the IT2FABSMC achieves the best tracking performances in comparison with the T1ABSMC. It has been shown that the proposed approach performs good tracking performanc-es and guarantees the control objective for tracking problems. It guarantees the convergence of the tracking errors to a small area of the origin.

References 1. Abdelhedi, F., Bouteraa, Y., Derbel, N.: Distributed second order sliding mode control for synchronization of robot manipulator. In: Second International Conference on Automation, Control, Engineering and Computer Science (2014) 2. Abdelhedi, F., Derbel, N.: Adaptive second order sliding mode control based cross coupling concept for synchronization of robotic systems accepted in advances in systems signals & devices. In: Issues on Systems, Analysis & Automatic Control (2016) 3. Ayadi, M., Naifar, O., Derbel, N.: High-order sliding mode control for variable speed pmsgwind turbine-based disturbance observer. Int. J. Model. Ident. Control 32(1), 85–92 (2019) 4. Bartolini, G., Ferrara, A., Usai, E.: Chattering avoidance by second-order sliding mode control. IEEE Trans. Autom. Control 43(2), 241–246 (1998) 5. Castro, J.L.: Fuzzy logic controllers are universal approximators. IEEE Trans. Syst. Man Cybern. 25(4), 629–635 (1995) 6. Hamzaoui, A., Essounbouli, N., Zaytoon, J.: Fuzzy sliding mode control for uncertain siso systems. In: IFAC Conf. on Intelligent Control Systems and Signal Processing (ICONS’03), pp. 233–238 (2003) 7. Han, S.-I., Lee, J.-M.: Backstepping sliding mode control with fwnn for strict output feedback non-smooth nonlinear dynamic system. Int. J. Control Autom. Syst. 11(2), 398–409 (2013) 8. Karnik, N.N., Mendel, J.M., Liang, Q.: Type-2 fuzzy logic systems. IEEE Trans. Fuzzy Syst. 7(6), 643–658 (1999) 9. Lee, C.-H., Pan, H.-Y., Chang, H.-H., Wang, B.-H.: Decoupled adaptive type-2 fuzzy controller (dat2fc) design for nonlinear tora systems. In: 2006 IEEE International Conference on Fuzzy Systems, pp. 506–512, IEEE (2006) 10. Levant, A., Livne, M.: Uncertain disturbances’ attenuation by homogeneous multi-input multioutput sliding mode control and its discretisation. IET Control Theory Appl. 9(4), 515–525 (2015) 11. Lin, W.-S., Chen, C.-S.: Robust adaptive sliding mode control using fuzzy modelling for a class of uncertain mimo nonlinear systems. IEE Proceed. Control Theory Appl. 149(3), 193–202 (2002)

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12. Lu, C.-H., Hwang, Y.-R., Shen, Y.-T.: Backstepping sliding mode tracking control of a vanetype air motor x-y table motion system. ISA Trans. 50(2), 278–286 (2011) 13. Manceur, M., Essounbouli, N., Hamzaoui, A.: Commande floue type-2 par modes glissants d’ordre deux a gains adaptatifs d’un systeme multivariableincertain. In: Les rencontres Francophones sur la Logique Floue et ses Applications (2010) 14. Slotine, J.-J.E.: Sliding controller design for non-linear systems. Int. J. Control 40(2), 421–434 (1984) 15. Song, Z., Sun, K.: Adaptive backstepping sliding mode control with fuzzy monitoring strategy for a kind of mechanical system. ISA Trans. 53(1), 125–133 (2014) 16. Utkin, V.: Variable structure systems with sliding modes. IEEE Trans. Autom. Control 22(2), 212–222 (1977) 17. Wang, L.-X.: Fuzzy systems are universal approximators. In: [1992 Proceedings] IEEE International Conference on Fuzzy Systems, pp. 1163–1170, IEEE (1992) 18. Wang, L.-X.: Stable adaptive fuzzy controllers with application to inverted pendulum tracking. IEEE Trans. Syst. Man Cybern. Part B (Cybern.) 26(5), 677–691 (1996) 19. Wang, L.-X.: A Course in Fuzzy Systems and Control Prentice Hall. Facsimile edition (1997) 20. Wang, L.-X., Mendel, J.M., et al.: Fuzzy basis functions, universal approximation, and orthogonal least-squares learning. IEEE Trans. Neural Networks 3(5), 807–814 (1992)

Robust Nonfragile Control Strategies for Bilinear Uncertain Fuzzy Systems Based on Proportional PDC Approach Chekib Ghorbel and Naceur Benhadj Braiek

Abstract This chapter deals with robust nonfragile stabilization and tracking control problems of nonlinear systems described as bilinear Takagi-Sugeno fuzzy systems with parametric uncertainties and external disturbances. Besides, the objective is focused on the design of robust state-feedback controllers such that closed-loop systems are globally asymptotically stable with disturbance attenuation in spite of controllers gains variations. However, the quadratic Lyapunov function, proportional parallel distributed compensation approach, H∞ synthesis criterion, and linear matrix inequalities techniques are provided to design sufficient stability conditions with decay rate. Finally, a numerical example based on nonlinear mass-spring-damper mechanical system is considered to demonstrate the effectiveness and the performances of the proposed control schemes. Keywords Robust control · Nonfragile control · H∞ criterion · Bilinear fuzzy systems · Stabilization · Tracking

1 Introduction In the last decades, fuzzy logic control techniques have received considerable attention in numerous fields as in robotics, power systems, communications, networking, image processing, chemical engineering, sensor technology, consumer electronics, C. Ghorbel (B) National Engineering School of Carthage, University of Carthage, Carthage, Tunisia e-mail: [email protected] C. Ghorbel · N. B. Braiek Advanced Systems Laboratory, Polytechnic School of Tunisia, University of Carthage, Carthage, Tunisia e-mail: [email protected] N. B. Braiek Polytechnic School of Tunisia, University of Carthage, Carthage, Tunisia © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Derbel et al. (eds.), Advances in Robust Control and Applications, Studies in Systems, Decision and Control 474, https://doi.org/10.1007/978-981-99-3463-8_16

361

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and automobiles [1, 6, 15]. In particular, in the area of nonlinear control systems design, Takagi-Sugeno (T-S) fuzzy model representation is an important approach that describes nonlinear dynamic systems by a set of fuzzy If-then rules. However, the resulting T-S system is achieved by interpolating local fuzzy models through nonlinear fuzzy membership functions by using the sector nonlinearity technique [16, 17]. More specifically, a parallel distributed compensation (PDC) approach has been employed to overcome different stability problems [13, 22]. As a result, the obtained stability conditions of the controlled systems are in general established in terms of linear matrix inequalities (LMI) and can be efficiently solved with the convex programming tools [5, 9, 18]. Besides, it is known on the one hand that many real-world nonlinear dynamic systems can be adequately approximated by bilinear models [8, 12]. A good process example is the population of biological species which is described by a bilinear mathematical model [2, 19]. As well, there are many research works related to modeling and robust control problems for bilinear fuzzy systems as are provided in references [19–21]. On the other hand, in practical applications, inaccuracies, parametric uncertainties, external disturbances, modeling errors, measurement errors, and controllers gains variations are frequently sources of the system’s instability. Besides that, most control systems necessitate accurate controllers since actuators may be of malfunction and/or round-off errors in numerical computations [11, 24]. Thus, there are considerable studies on the robust nonfragile control and filtering problems of fuzzy systems such as in references [4, 7, 23]. In spite of this, there are no studies explored on the robust nonfragile control problem for bilinear fuzzy systems subject to control objectives as speed of response, attenuation of disturbances effect, and measure of controllers gains variations. In this context, the main goal of this chapter is to purpose fuzzy stabilization and tracking control schemes for bilinear T-S fuzzy systems containing parametric uncertainties and external disturbances. Besides, sufficient stability conditions with decay rate are established by using the proportional PDC (PPDC) approach. Subsequently, for r −rules, m−inputs, and n−state variables, the total number of LMI constraints is reduced to r + mn, compared by r m n with the normal PDC concept [3, 10, 23]. Following the Introduction, this chapter is structured as follows: The problem formulation and preliminaries are exposed in Sect. 2. The robust nonfragile stabilization problem for bilinear T-S fuzzy systems is presented in Sect. 3. The nonfragile H∞ tracking control scheme for such systems is studied in Sect. 4. Simulation studies are provided in Sect. 5 to illustrate the merit of the proposed control strategies. Finally, conclusion is drawn in the last section. Throughout(this chapter, M T denotes the)transpose of a matrix M, M + (∗) stands ( ) A ∗ A BT T for M + M , represents , and χ (z) = χ (z (t)). BC B C

Robust Nonfragile Control Strategies for Bilinear Uncertain Fuzzy Systems …

363

2 Problem Formulation and Preliminaries There are many real-world nonlinear dynamic systems that can be adequately approximated by bilinear fuzzy models of the following form If (z 1 is M1i ) and (z 2 is M2i ) and . . . and (z k is Mki ) then x˙ = Ai x + Bi u + N i x u + w, y = Ci x + Di u,

(1)

for i = 1, 2, . . . , r and k ≤ n where w is the disturbances M ji is the fuzzy ( vector, ) set, z 1 , z 2 , . . . , z k are the known premise variables, M ji z j is the degree of membership of z j , and r is the number of If-then rules. Matrices Ai and N i are defined as Ai = Ai + ΔAi (t) = Ai + Hi ψ (t) E Ai (2) N i = Ni + ΔNi (t) = Ni + Hi ψ (t) E N i , where Hi , E Ai , and E N i are known real matrices of appropriate dimensions and ψ (t) is a varying matrix satisfying ψ (t)T ψ (t) ≤ I . Based on the fuzzy inference method [16, 17], the overall state x˙ and the final output z of the T-S fuzzy bilinear model is represented by ⎧ r ) ( ∑ ⎪ ⎪ h i (x) Ai x + Bi u + N i x u + D1i w ⎨ x˙ = i=1

r ∑ ⎪ ⎪ h i (x) (Ci x + D2i w), ⎩z =

(3)

i=1

where: h i (z) =

wi (z) r ∑ w j (z) j=1

wi (z) =

k ∏

( ) M ji z j

j=1

)T ( and z = z 1 z 2 . . . z k . The membership functions verify the convex conditions r ∑

h i (z) = 1

i=1

and h i (z) ≥ 0. In the following, this chapter presented robust nonfragile stabilization and tracking control schemes for T-S fuzzy bilinear systems with parametric uncertainties and external disturbances. Besides, new sufficient stability conditions with decay rate

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and disturbance attenuation using the PPDC concept, H∞ synthesis criterion, and LMI techniques are discussed. First of all, the following technical lemmas that will be needed throughout the proof of proposed theorems are recalled. Lemma 1 For matrices A and B with appropriate dimensions and a positive scalar τ , the following inequality holds [22, 23] A T B + B T A ≤ τ A T A + τ −1 B T B.

(4)

Lemma 2 Given matrices M = M T , Q = Q T , and L with appropriate dimensions, these statements are equivalent [14, 19] ) M ∗ 0 are positive scalars to be designed. K is the common matrix verifying [ ] K = K + ΔK (t) = I + β φ (t) K , (7) such that φ (t)T φ (t) ≤ I . The overall fuzzy controller is then defined as

Robust Nonfragile Control Strategies for Bilinear Uncertain Fuzzy Systems … r ∑

u=

h i (z) ρ /

i=1 r ∑

= =

i=1 r ∑

365

ki K x T

1 + ki2 x T K K x (8)

h i (z) ρ sin θi h i (z) ρ ki K x cos θi

i=1

Substituting (8) into (3), the closed-loop fuzzy system is expressed by ⎧ r r ∑ ) ( ∑ ⎪ ⎪ h i (z) h j (x) G i j x + D1i w ⎨ x˙ = i=1 j=1

(9)

r ∑ ⎪ ⎪ h i (z) (Ci x + D2i w) , ⎩z = i=1

where G i j = G i j + ΔG i j (t) G i j = Ai + ρ k j Bi K cos θ j + ρ Ni sin θ j ΔG i j (t) = ΔAi (t) + ρ k j Bi ΔK (t) cos θ j In order to converge the states of the controlled system (9) into the origin, sufficient stability conditions are established in Theorem 1. Such conditions must guarantee the global asymptotic stability and achieve a prescribed level of disturbance attenuation γ for all admissible uncertainties such that ||z||2 < γ, ||w||2 /=0 ||w||2

||Hz w ||∞ = sup

(10)

where ||Hz w ||∞ denotes the H∞ norm, and: { ||z||2 = 0



{ z T z dt, ||w||2 =



w T w dt

0

Theorem 1 The origin of the controlled fuzzy system (9) is globally asymptotically stable with decay rate α and satisfying the H∞ performance objective (10) if there exist a common positive definite matrix P and positive scalars ρ and ki , for i = 1, 2, . . . , r , such that maximize α P, K , γ

subject to : ⎛ ⎞ ∗ P G i j + (∗) + 2 α P ∗ ⎝ −γ 2 I ∗ ⎠ < 0, for i, j = 1, 2, . . . , r, D1iT P D2i −I Ci

(11)

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Proof In order to prove Theorem 1, we used the quadratic Lyapunov function V (x) = x T P x subject to

V˙ (x) ≤ − 2 α V (x)

where

V˙ (x) = x˙ T P x + x T P x˙

P T = P > 0, and α > 0. Moreover, the global asymptotic stability of the controlled T-S fuzzy system (9) is satisfied if V˙ (x) + 2 αV (x) + z T z − γ 2 w T w < 0. ( By choosing xw =

(12)

) x , inequality (12) can be rewritten as w r r ∑ ∑

h i (z) h j (z) xwT wi j xw < 0,

(13)

i=1 j=1

(

where wi j =

T

wi11j



)

D 1i P + D2iT Ci wi22

wi11j = P G i j + (∗) + 2 α P + CiT Ci wi22 = D2iT D2i − γ 2 I As h i (z) ∈ [0, 1], it becomes (

∗ P G i j + (∗) + 2 α P + CiT Ci T D 1i P + D2iT Ci D2iT D2i − γ 2 I

) < 0.

(14)

Then by applying Lemma 2, the nonlinear matrix inequalities (11) are demonstrated. Next, after some LMI manipulations, we presented in Theorem 2 the LMI constraints result. Theorem 2 The origin of the controlled fuzzy system (9) is globally asymptotically stable with decay rate α and satisfying the H∞ performance objective (10) if there exist a common positive definite matrix P and positive scalars ki , ρ, δ=



1 + ρ2

Robust Nonfragile Control Strategies for Bilinear Uncertain Fuzzy Systems …

/ υi j =

1 τ1i j

+

367

β2 τ2i j

τ1i j and τ2i j for i, j = 1, 2, . . . , r , such that: maximize α P, K , γ

subject to : ) ( L M1 ∗ < 0, for i, j = 1, 2, . . . , r, L M2 L M3 where

⎛(

P Ai + (∗) + 2 αP+ ( ) ⎜ τ1i j N T Ni + τ2i j E T E 1i + E T E 2i i 1i 2i ⎜ • L M1 = ⎝ D1iT P C ⎛ ⎞i βk j Bi K 0 0 ⎜ βk j K 0 0 ⎟ ⎟ • L M2 = ⎜ ⎝ δ HiT P 0 0 ⎠ ; ρP 0 0 ⎛ −1 ⎞ −τ1i j I 0 0 0 ⎜ 0 −τ2i−1j I 0 0 ⎟ ⎟. • L M3 = ⎜ ⎝ 0 0 −τ −1 I 0 ⎠ 2i j

0

0

0

)

(15)

⎞ ∗

∗ ⎟ ⎟; −γ I ∗ ⎠ D2i −I 2

−υi−1 j I

Proof It is clear that the nonlinear matrix inequalities (14) contained certain parts Ψi j and uncertain ΔΨi j parts. This leads to transform it of the form Ψi j + ΔΨi j (t) < 0, for i, j = 1, 2, . . . , r, where



P G i j + (∗) + 2 α P D1iT P • Ψi j = ⎝ Ci ⎛ P ΔG i j (t) + (∗) 0 • ΔΨi j = ⎝ 0

(16)

⎞ ∗ ∗ −γ 2 I ∗ ⎠ ; D2i −I ⎞ ∗ ∗ 0 ∗ ⎠. 0 0

Then, it is worth pointing out that quantities Ψi j and ΔΨi j (t) contained anti-diagonal terms. In order to transform them into diagonal terms, we used Lemma 1. As a result, we obtained ⎞ ⎛ 11 θi j ∗ ∗ ⎝ D T P −γ 2 I (17) ∗ ⎠ < 0, for i, j = 1, 2, . . . , r, 1i D2i −I Ci

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where θi11j = P Ai + (∗) + 2αP + τ1i j NiT Ni + τ1i j β 2 k 2j K T BiT Bi K + θi11,bis j ( ) δ2 θi11,bis = τ2i j E 1iT E 1i + E 2iT E 2i + τ2i j β 2 k 2j K T K + P Hi HiT P + υi2j ρ2 P 2 j τ2i j By using Lemma 2, we obtained the LMI constraints formulation (15).

4 Robust Nonfragile Tracking Control Design In this study, the objective is to determine H∞ nonfragile controllers that ensure a good tracking between the state variables of the nonlinear system and the desired state variables of the following reference model x˙e = Ae xe + Be e,

(18)

where Ae is a specific asymptotically stable matrix, Be is an input matrix, xe is a reference state, and e is a reference input. However, the fuzzy controller is employed: u=

r ∑

h i (z) ρ /

i=1

= =

r ∑ i=1 r ∑

li L ε T

1 + li2 εT L L ε

h i (z) ρ sin θi h i (z) ρ li L ε cos θi

(19)

i=1

where: li L ε sin θi = / T 1 + li2 εT L L ε cos θi = /

1 T

1 + li2 εT L L ε

] [ for θi ∈ − π2 , π2 . li and ρ > 0 are positive scalars to be designed. L is the common matrix verifying (20) L = L + ΔL (t) = (I + η φ (t)) L , such that φ (t)T φ (t) ≤ I .

Robust Nonfragile Control Strategies for Bilinear Uncertain Fuzzy Systems …

369

Substituting then (19) into (3) and (18), the dynamic of the tracking error is expressed as ε˙ =

r r ∑ ∑

) ) ( ( h i (z) h j (z) F i j ε + Ae − G i j xe − w + Be e,

(21)

i=1 j=1

where: F i j = Ai − ρ Bi l j L cos θ j + ρ N i sin θ j G i j = Ai + ρ N i sin θ j Then, Eqs. (18) and (21) lead to the controlled augmented system x˙aug =

r r ∑ ∑

) ( h i (z) h j (z) Υ i j xaug + Φ ϕ ,

(22)

i=1 j=1

where: ( xaug =

ε xe

)

( , ϕ=

w e

(

) , Υ ij =

F i j Ae − G i j 0 Ae

)

( , Φ=

−I Be 0 Be

)

Mainly, the control objective is to determine the tracking gains that guarantee the global asymptotic stability of the controlled fuzzy system (22) and ensure the H∞ control performance || || Hx

aug

|| || ϕ



= sup

||ϕ||2 /=0

|| || ||xaug ||

2

||ϕ||2

< γ,

(23)

where γ > 0 is a prescribed disturbances attenuation level. As a consequence, Theorem 3 presented the result. Theorem 3 The controlled fuzzy augmented system (22) is globally asymptotically stable with a decay rate α > 0 and a guaranteed H∞ control performance (23) if there exist common symmetric positive definite matrices P1 and P2 and positive scalars κ1 , κ2 , κ3 , κ4 , 1 1 + δ2 = κ1 κ2 δ3 = ρ, and η, for i = 1, 2, . . . , r , such that

1 1 + κ3 κ4

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maximize α P1 , P2 , L , γ

subject to : ) ( L M4 ∗ < 0, for i, j = 1, 2, . . . , r, L M5 L M6 where • • •





(24)



⎞ ∗ ∗ ∗ L M4a ⎜ (Ae − Ai )T P1 L M b ∗ ∗ ⎟ 4 ⎟; L M4 = ⎜ ⎝ 0 −γ 2 I ∗ ⎠ −P1 BeT P2 0 −γ 2 I BeT P1 ( ) P1 Ai + (∗)( + Q + 2 αP1 + κ1)NiT Ni + ; L M4a = κ3 E TAi E Ai + E NT i E N i ( ) P2 Ae + ((∗) + 2 αP2 + κ2 Ni)T Ni + ; L M4b = κ4 E TAi E Ai + E NT i E N i ⎞ ⎛ 0 0 0 l j Bi L ⎜ 0 0 0 ⎟ ρP 1 ⎟ ⎜ L 0 0 0 ⎟ l L M5 = ⎜ j ⎟; ⎜√ ⎝ 1 + ρ2 H T P1 0 0 0 ⎠ i 0 0 0 ρ η BiT P1 ⎞ ⎛ −1 0 0 0 0 −κ1 I ⎜ 0 −δ −1 I 0 0 0 ⎟ 2 ⎟ ⎜ −1 ⎜ L M6 = ⎜ 0 0 −κ3 I 0 0 ⎟ ⎟. ⎝ 0 0 0 −δ3−1 I 0 ⎠ 0 0 0 0 −κ3 I

Proof To prove the above theorem, we used the quadratic Lyapunov function ) ( T P xaug V xaug = xaug subject to

) ( ) ( V˙ xaug ≤ − 2 α V xaug

where P = P T > 0 and α > 0. As well, the controlled fuzzy system (22) is globally asymptotically stable with decay rate α > 0 if ) ( ) ( T Q + xaug − γ 2 ϕT ϕ < 0. V˙ xaug + 2 α V xaug + xaug

(25)

The development of inequality (25) gives r r ∑ ∑ i=1 j=1

T h i (z) h j (z) xaug Δi j xaug < 0,

(26)

Robust Nonfragile Control Strategies for Bilinear Uncertain Fuzzy Systems …

371

(

) P Υ i j + (∗) + 2 α P + Q + ∗ . As h i (z) ∈ [0, 1], inequality −γ 2 I ΦT P (26) allows to write where Δi j = (

P Υ i j + (∗) + 2 α P + Q + ∗ −γ 2 I ΦT P

) < 0, for i, j = 1, 2, . . . , r.

(27)

⎞ χi11j + Δχi11j (t) ∗ ∗ ∗ ⎜ χ21 + Δχ21 (t) P2 Ae + (∗) + 2 αP2 ∗ ∗ ⎟ ij ⎟ < 0, ⎜ ij 2 ⎝ − P1 0 −γ I ∗ ⎠ BeT P2 0 −γ 2 I BeT P1

(28)

Assuming that P = diag (P1 P2 ), it becomes ⎛

where • • • •

) ( χi11j = P1 Ai − ρP1 Bi l j L cos θ j + ρP1 Ni sin θ j + (∗) + 2αP1 + Q; ] [ Δχi11j (t)= P1 Δ Ai (t) − ρ P1 Bi l j ΔL (t) cos θ j + ρ P1 ΔNi (t) sin θ j + (∗); ) ( χi21j = AeT − AiT P1 − ρ NiT P1 sin θ j ; Δχi21j (t)=− Δ AiT (t) P1 − ρ ΔNiT (t) P1 sin θ j .

Furthermore, it is pointing out that inequality (28) contained certain parts Θi j and uncertain parts ΔΘi j (t). This leads to transform them of the form Θi j + ΔΘi j (t) < 0, for i, j = 1, 2, . . . , r, where



χi11j ∗ ∗ 22 ⎜ χ21 χ ∗ i • Θi j = ⎜ ⎝ − P1 0 −γ 2 I T T 0 Be P1 Be P2 ⎛ Δχi11j (t) ∗ ⎜ Δχ21 (t) 0 ij • ΔΘi j (t) = ⎜ ⎝ 0 0 0 0

(29)

⎞ ∗ ∗ ⎟ ⎟; ∗ ⎠ −γ 2 I ⎞ ∗ ∗ ∗ ∗ ⎟ ⎟. 0 ∗ ⎠ 0 0

Next, by using Lemma 1 for positive scalars κ1 , κ2 , κ3 , κ4 , δ2 and δ3 , it becomes ⎞ ∗ ∗ ∗ Θiaj ⎜ (Ae − Ai )T P1 Θ b ∗ ∗ ⎟ ij ⎟, Θi j ≤ ⎜ 2 ⎝ −P1 0 −γ I ∗ ⎠ BeT P2 0 −γ 2 I BeT P1 ⎛

(30)

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ΔΘiaj (t) ∗ b ⎜ 0 ΔΘ i j (t) ΔΘi j (t) ≤ ⎜ ⎝ 0 0 0 0

∗ ∗ 0 0

⎞ ∗ ∗ ⎟ ⎟, ∗ ⎠ 0

(31)

where • Θiaj = P1 Ai + (∗) + 2 αP1 + Q + κ1 NiT Ni + κ1l 2j L T BiT Bi L + δ2 ρ2 P12 ; • Θibj = P2 Ae + (∗) + 2 αP2 + κ2 NiT Ni ; ( ) ) ( κ3 E TAi E Ai + κ3 E NT i E N i + δ3 1 + ρ2 P1 Hi HiT P1 + • ΔΘiaj (t) = ; T 2 2 κ−1 3 ρ η P1 Bi Bi P1 • ΔΘibj (t) = κ4 E TAi E Ai + κ4 E NT i E N i . From (30) and (31), inequality (29) is equivalent to ⎛

⎞ Δiaj ) ∗ ∗ ∗ ⎜ A T − A T P1 Δb ∗ ∗ ⎟ i ij ⎜ e ⎟ < 0, ⎝ −P1 0 −γ 2 I ∗ ⎠ BeT P2 0 −γ 2 I BeT P1 where

(

(32)

( ) ) P1 Ai + (∗) + 2 αP1 + Q + κ1 NiT Ni + κ3 E TAi E Ai E NT i E N i + • = ; T 2 κ1l 2j L T BiT Bi L + δ2 ρ2 P12 + κ3l 2j L T L + κ−1 3 ρ P1 Bi Bi) P1 ( • Δibj = P2 Ae + (∗) + 2 αP2 + κ2 NiT Ni + κ4 E TAi E Ai + E NT i E N i . (

Δiaj

Using again Lemma 2, the nonlinear inequalities (32) are transformed in linear constraints (24).

5 Illustrative Example This section presented simulation studies that illustrate the merit of the proposed stabilization and tracking control strategies.

5.1 Mathematical Model A bilinear T-S fuzzy system is designed for a nonlinear mass-spring-damper mechanical system which is described by [3] y¨ + g (y, y˙ ) = h (y, y˙ ) u, where:

(33)

Robust Nonfragile Control Strategies for Bilinear Uncertain Fuzzy Systems … Fig. 1 Evolutions of the membership functions

373

1.2 M11 M21

Membership functions

1

0.8

0.6

0.4

0.2

0 −0.6

−0.4

−0.2

0 x2

0.2

0.4

0.6

g (y, y˙ ) = 3.7y + 2.8 sin y˙ − 0.85 y˙ 2 sin y˙ h (y, y˙ ) = 1 + 3 y˙ − 0.65 y˙ 2 sin y˙ ( ) Thereafter, for x1 = y, x2 = y˙ , x2 ∈ [−x20 , x20 ], v2 = max x2 2 , and |ψ (t)| < 1, the nonlinearities x2 |→ sin x2 and x2 |→ x2 2 are decomposed as sin x2 = M11 (x2 ) x2 sin x20 + M12 (x2 ) x2 and x2 2 = 0 + v2 ψ (t) where: x20 x20 sin x2 − x2 sin x20 x2 (x20 − sin x20 ) x20 x2 − x20 sin x2 M12 (x2 ) = x2 (x20 − sin x20 ) M11 (x2 ) =

] [ For x2 ∈ − π4 , π4 , the evolutions of membership functions M11 (x2 ) and M12 (x2 ) are depicted in Fig. 1. The studied system is then described by two fuzzy rules ⎧

If (x1 is M11 (x2 )) then x˙ = A1 x + B1 u + N 1 x u + w If (x1 is M12 (x2 )) then x˙ = A2 x + B2 u + N 2 x u + w,

( ) ( ) ( ) 0 1 0 1 0 where A1 = , A2 = , B1 = B2 = , −3.5 −2.8 −3.5 −2.52 1 ( ) ( ) ) ) ( ( 0 0 ψ (t) 0 v2 , ΔA2 (t) = ψ (t) 0 v2 , ΔA1 (t) = 0.65 0.01 ( ) ) ( 0 ψ (t) 0 −0.85 v2 , ΔN1 (t) = 0.65

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Fig. 2 Responses of the state variables

1 x1

0.8

x2

Position and velocity signals

0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0

2

4

6

8

10

time (sec)

(

) ) ( 0 ψ (t) 0 −0.015 v2 , 0.01 ψ (t) = sin (0.6 t), and v2 = 0.68. ΔN2 (t) =

5.2 Stabilization Result Using γ = 0.8, ρ = 0.64, k1(= 0.85, and k2 =)0.65, the solutions satisfying ( ) 32.0576 4.1288 Theorem 2 are α = 0.73, P = , and K = 25.5074 19.3960 . 4.1288 21.3686 The responses of the state variables x1 and x2 of the controlled nonlinear system are shown in Fig. 2 whereas the control signal is depicted in Fig. 3. From the simulation results, the state variables of the controlled nonlinear mechanical system converge in a shorter time into the origin. However, the designed robust nonfragile fuzzy controller stabilizes the system in spite of parametric uncertainties and additive external disturbances on the states.

5.3 Tracking Result ( Using γ = 0.8, ρ = 0.65, l1 = 0.7, and l2 = 0.85, Ae = ( ) 0 , the solutions satisfying Theorem 3 are α = 0.674, 1

) 0 1 , and Be = −8 −6

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Fig. 3 Evolution of the control signal

u

6 5

Control signal

4 3 2 1 0 −1 0

2

4

6

8

10

time (sec)

Fig. 4 Position tracking trajectory

2 x1 xe1

1.5

Position tracking

1 0.5 0 −0.5 −1 −1.5 0

2

4

6

8

10

12

14

time (sec)

( P1 = and:

) ) ( 43.8726 9.0648 27.6344 −6.3284 , P2 = , 9.0648 27.6441 −6.3284 16.4501 ( ) L = 15.3287 −27.3410

The tracking trajectories of the position and the velocity of the nonlinear system are depicted respectfully in Figs. 4 and 5 whereas the control signal is given in Fig. 6. As a result, the state variables of the controlled nonlinear system followed perfectly the desired ones despite the presence of uncertainties, controller nonfragility variations, and external disturbances.

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Fig. 5 Velocity tracking trajectory

1 x

2

x

0.8

e2

Velocity tracking

0.6 0.4 0.2 0 −0.2 −0.4 −0.6 15

10

5

0

time (sec)

Fig. 6 Evolution of the control signal

15 u

Control signal

10

5

0

−5

−10

0

2

4

6

8

10

12

14

time (sec)

6 Conclusion This chapter has presented nonfragile H∞ controllers for nonlinear systems as described by bilinear T-S fuzzy models containing parameter uncertainties and external disturbances. Based on the proportional PDC concept and Lyapunov direct approach with decay rate, sufficient conditions in terms of LMI that ensure good control objectives have been provided. Consequently, designed controllers guaranteed the robust asymptotic stability of controlled systems despite the presence of parametric uncertainties, external disturbances, and controllers gains variations. Finally, the simulation results have shown the applicability and effectiveness of the proposed control strategies.

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