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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
An authoritative new exploration of the latest theoretical and applied advances in Linear Parameter-Varying systems
In Linear Parameter-Varying Control: Theory and Application to Automotive Systems, distinguished researcher Dr. Olivier Sename delivers a comprehensive and up-to-date discussion of the theoretical aspects and real applications of Linear Parameter-Varying (LPV) control, with a strong focus on systems theory and in real automotive systems. The author covers the primary methods used to model, control, and analyze LPV systems, and illustrates how to model those systems using examples.
This book covers developing adaptive LPV control using the provided recipes as guides and contextual aids as well as discovering effective methods to design LPV controllers that have already been validated through real applications.
Readers will also find:
- A thorough introduction to vehicle dynamics control in automated vehicles, as well as suspension control
- Comprehensive explorations of LPV systems modelling, including dynamical systems
- Practical discussions of the properties of LPV systems, including controllability, observability, and stability
- Complete treatments of LPV systems control, including state feedback control and dynamic output feedback LPV control
Perfect for researchers and students with an interest in vehicle dynamics, Linear Parameter-Varying Control will also benefit postgraduate and PhD students, control engineers, and academics teaching control theory and applications courses.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 444
Veröffentlichungsjahr: 2025
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
Dedication
About the Author
Preface
Bibliography
Acronyms
About the Companion Website
Introduction
Bibliography
Part I: Some Theoretical Aspects on LPV Systems: From Modeling to Control
1 Some Modeling Approaches for LPV and qLPV Systems
1.1 Introduction
1.2 Dynamical Systems
1.3 An Introduction to LPV Models
1.4 Specific Classes of LPV Systems
1.5 From a Nonlinear Model to an LPV Representation
1.6 An Introduction to Identification of LPV Systems
1.7 The Nonuniqueness Issue: A Control‐Oriented LPV Modeling Perspective
1.8 Illustrative Example 1: A Single Tank System
1.9 Illustrative Example 2: qLPV Modeling and Time‐Varying Characteristics
1.10 Conclusion
Bibliography
2 Properties of LPV Systems
2.1 Introduction
2.2 Controllability
2.3 Observability
2.4 Comments on State‐Space Realizations of LPV Systems
2.5 Stability
2.6 Performance Criteria: , , and Pole Placement
2.7 About Stabilizability and Detectability
2.8 The Case of Discrete‐Time LPV Systems
2.9 Conclusion
Bibliography
3 Control of LPV Systems
3.1 Introduction
3.2 LPV State‐Feedback Control
3.3 The LPV Dynamic Output Feedback Control
3.4 LPV Observer Design
3.5 About Control of Discrete‐Time LPV Systems
3.6 Conclusion
Bibliography
Part II: LPV Methods for Nonlinear Systems
4 Control and Observer Design for Nonlinear Systems Using Quasi‐LPV Models: An Illustration Through Examples
4.1 Introduction
4.2 Control of a Nonlinear System
4.3 An Observer of a Three‐Tank Nonlinear System
4.4 Conclusion
Bibliography
5 Observer Design for Semi‐active Suspension Systems: qLPV Approaches
5.1 Introduction
5.2 Illustrative Case Study: The INOVE Testbench, a Semi‐active Suspension System
5.3 Electro‐Rheological Dampers: Modeling Approaches
5.4 qLPV Quarter Car Semi‐active Suspension Models
5.5 Method 1: An / Observer for Suspension State Estimation
5.6 Method 2: A Filtering Approach for Damper Force Estimation
5.7 Method 3: A Nonlinear Parameter Varying Approach for State Estimation
5.8 Concluding Remarks
Bibliography
6 Lateral Control of Autonomous Vehicle
6.1 Introduction
6.2 Modeling
6.3 Control Design
6.4 Analysis of the Polytopic and Grid‐Based Design Methods
6.5 Simulation Results
6.6 Conclusion
Bibliography
Part III: LPV Adaptive‐Like Control Methods
7 Methods and Tools for LPV Adaptive‐Like Control
7.1 Introduction
7.2 The Framework: A Generic Tool for “Adaptive‐Like” Control
7.3 LPV Adaptive Control with Varying Closed‐Loop Performances (Function of External Parameters)
7.4 LPV Adaptive Control Function of Varying Endogeneous Parameters
7.5 Concluding Remarks
Bibliography
8 LPV Road Adaptive Suspension Control
8.1 Introduction
8.2 The Semi‐active Suspension Quarter‐Car Model
8.3 Road Roughness Estimator
8.4 Synthesis of a Semi‐active Suspension Control
8.5 Simulation Results
8.6 Conclusions
Bibliography
9 LPV Fault‐Tolerant Control Strategies for Suspension Systems
9.1 Introduction
9.2 Related Works
9.3 Fault Diagnosis Problem Formulation for Semi‐active ER Suspension Systems
9.4 Fault Estimation Using LPV PI Observers
9.5 FTC LPV Control of Semi‐active Suspension Systems
9.6 Conclusion
Bibliography
10 Lateral LPV Adaptive‐Like Control of Automated Vehicles Adapted to Driver Performance
10.1 Introduction
10.2 LPV Observer‐Based Control Structure for ADAS Systems
10.3 Driver Fault Estimation Using a Discrete‐Time LPV PI Observer
10.4 Robust ADAS Strategy
10.5 Simulation Results
10.6 Conclusion
Bibliography
Index
End User License Agreement
List of Tables
Chapter 5
Table 5.1 Parameter values of the extended Guo model of the ER damper.
Table 5.2 Parameter values of the INOVE quarter‐car model equipped with an ...
Table 5.3 Performance criteria – Method 1.
Table 5.4 Performance criteria – Method 2.
Table 5.5 Performance criteria – method 3.
Chapter 6
Table 6.1 Performance criteria: comparison of LPV strategies for the latera...
Chapter 7
Table 7.1 Performance criteria (RMS) – Scenario 1.
Table 7.2 Performance criteria – Scenario 2.
Chapter 8
Table 8.1 Parameter values of the quarter‐car model equipped with an MR dam...
Table 8.2 Parameter values of the control‐design MR damper model 8.2.
Table 8.3 Parameter values of the MR damper (Model 8.3 – for simulation)....
Table 8.4 Parameters of design in the observer.
Table 8.5 Classification of road profiles (ISO 8608).
Table 8.6 controller parameters.
Table 8.7 Normalized Root Mean Square (NRMS) of the simulation scenarios – ...
Chapter 9
Table 9.1 Parameter values of the INOVE quarter‐car model with static/dynam...
Table 9.2 Root Mean Square (RMS) analysis of the simulation results.
Chapter 10
Table 10.1 Values of for different speeds.
Table 10.2 DM parameters.
List of Illustrations
8
Figure 1 LPV controller or observer scheme.
Chapter 1
Figure 1.1 LPV–LFT representation.
Figure 1.2 A grid‐based LPV model with two parameters.
Figure 1.3 A simple quarter‐vehicle model.
Figure 1.4 A simple nonlinear tank model.
Figure 1.5 Case 1: simulation with a slow‐varying parameter.
Figure 1.6 Case 2: simulation with a fast‐varying parameter.
Chapter 2
Figure 2.1 LMI regions in complex plane as illustrated.
Chapter 3
Figure 3.1 Eigenvalues of in the complex plan (imaginary vs. real part), f...
Figure 3.2 Eigenvalues of in the complex plan (imaginary vs real parts), f...
Figure 3.3 LFR representation with performance weights.
Figure 3.4 Generalized LPV plant and Controller.
Chapter 4
Figure 4.1 control scheme to solve the given control problem.
Figure 4.2 General control configuration corresponding to the control scheme...
Figure 4.3 Sensitivity functions LTI cases.
Figure 4.4 Sensitivity functions LPV case.
Figure 4.5 Cross‐validation of the LTI controllers ‐ sensitivity function ...
Figure 4.6 Simulation case 1. (a) and and (b) scheduling parameters.
Figure 4.7 Simulation case 2. (a) and and (b) scheduling parameters.
Figure 4.8 Three tanks system and system parameters (dimensions are given in...
Figure 4.9 Observer poles in the complex plane (real (x) versus imaginary (y...
Figure 4.10 Simulation LPV observer – three‐tank system. (a) Tank levels and...
Chapter 5
Figure 5.1 Scheme and types of suspension systems. (a) Schematic representat...
Figure 5.2 The experimental testbed INOVE at GIPSA‐lab. (a) 1/5 scaled vehic...
Figure 5.3 Force vs. piston velocity diagram of real passive (left), semi‐ac...
Figure 5.4 Force vs. Deflection plot for different control levels ‐ Left (Re...
Figure 5.5 Quarter vehicle (QoV) model [Pham, 2020; Pham et al., 2021].
Figure 5.6 Pole region in the complex plane (part).
Figure 5.7 / LPV Observer ‐Scenario 1.
Figure 5.8 / LPV Observer ‐Scenario 2.
Figure 5.9 Generalized LPV plant for the filtering problem.
Figure 5.10 Bode magnitude of the transfers from to the estimation error
Figure 5.11 filter ‐Scenario 1.
Figure 5.12 filter ‐Scenario 2.
Figure 5.13 NLPV observer ‐Scenario 1.
Figure 5.14 NLPV observer ‐Scenario 2.
Chapter 6
Figure 6.1 Two‐wheeled bicycle model representing the vehicle lateral dynami...
Figure 6.2 control scheme for lateral control.
Figure 6.3 Bode diagrams of the sensitivity functions – polytopic approach (...
Figure 6.4 Bode diagrams of the sensitivity functions – grid‐based approach ...
Figure 6.5 Bode magnitude of the polytopic and grid‐based vehicle model.
Figure 6.6 Poles of vehicle models in the complex plane (real () vs. imagin...
Figure 6.7 Bode magnitude of the controllers (a) Polytopic controller and (b...
Figure 6.8 Real two layers lateral control scheme [Atoui, 2022].
Figure 6.9 Satory track (up) and vehicle speed (down).
Figure 6.10 Lateral errors for the LTI, LPV polytopic, and LPV grid‐based co...
Figure 6.11 Yaw‐rate tracking performances for the LTI, LPV polytopic, and L...
Figure 6.12 Steering control input for the LTI, LPV polytopic and LPV grid‐b...
Figure 6.13 Derivative of the steering control input for the LTI, LPV polyto...
Chapter 7
Figure 7.1 LPV general control configuration.
Figure 7.2 LPV mixed sensitivity control problem.
Figure 7.3 Sensitivity functions .
Figure 7.6 Scenario 2: Settling time () and overshoot versus parameter valu...
Figure 7.4 Scenario 1 – Section 7.3.
Figure 7.5 Scenario 2: LPV with decreasing dynamic performances versus LTI c...
Figure 7.7 Flexible transmission system [Landau et al., 1995].
Figure 7.8 Bode diagrams (no load, half load, and full load models).
Figure 7.9 2 degrees of freedom control scheme.
Figure 7.10 Sensitivity functions LTI (top) and LPV (bottom) cases.
Figure 7.11 Nominal case.
Figure 7.12 LPV case, top output performance, bottom scheduling parameter.
Figure 7.13 LTI case (unstable), top output performance, bottom scheduling p...
Chapter 8
Figure 8.1 LPV road adaptive control scheme.
Figure 8.2
QoV
model for a semi‐active suspension in a vehicle ...
Figure 8.3 MR damper characteristics with different current values : force ...
Figure 8.4 Procedure for road profile estimation [Tudón...
Figure 8.5 observer design in a
QoV
....
Figure 8.6 Results in test #1: implemented road sequence (a), on‐line roughn...
Figure 8.7 LPV road adaptive suspension control scheme...
Figure 8.8 Frequency domain – polytopic approach – (a) – (b).
Figure 8.9 Frequency domain: grid‐based approach.
Figure 8.10 Scenario 1 – Comfort analysis – Sprung mass and scheduling param...
Figure 8.11 Scenario 1 – Force – Deflection velocity plot (a) Polytopic appr...
Figure 8.12 Scenario 3 – Road profile and spring mass position (comfort anal...
Chapter 9
Figure 9.1 LPV FTC scheme.
Figure 9.2 The quarter‐car model [Pham et al., 2021].
Figure 9.3 Oil leakage: force vs. velocity diagram – measurement vs. model....
Figure 9.4 Complete
FE
problem:
ER
damper faults.
Figure 9.5 Simulation results – LPV PI observer – Approach 1.
Figure 9.6 Analysis: LPV PI observer – Approach 2. (a) Frequency domain anal...
Figure 9.7 Simulation results – LPV PI observer – Approach 2.
Figure 9.8 Dissipative force domain in presence of fault [Nguyen, 2016].
Figure 9.9 Generalized FTC LPV control design scheme.
Figure 9.10 Transfer – ‐bode diagram.
Figure 9.11 Scheduling parameter function .
Figure 9.12 Road profile.
Figure 9.13 Suspension deflection.
Figure 9.14 Sprung mass acceleration.
Chapter 10
Figure 10.1 Combined driver error detection/ADAS controller scheme [Medero e...
Figure 10.2 Integrated driver‐vehicle control model [Medero et al., 2022].
Figure 10.3 LPV/ generalized control configuration LPV/ADAS problem.
Figure 10.4 Input sensitivity functions. (a) Steering control and (b) brakin...
Figure 10.5 Scheduling parameters. (a) Scheduling parameters and . (b) Lo...
Figure 10.6 Driving comparison with and w/o ADAS during DLC maneuver.
Figure 10.7 Scheduling signals and actuator's commands for all driver cases....
Figure 10.8 Driver fault estimation: without ADAS (a) – with ADAS (b).
Figure 10.9 Scheduling signals and actuator's commands for the case of worst...
Figure 10.10 Scheduling signals and actuator's commands for the case of best...
Guide
Cover
Table of Contents
Title Page
Copyright
Dedication
About the Author
Preface
Acronyms
About the Companion Website
Begin Reading
Index
End User License Agreement
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Linear Parameter-Varying Control
Theory and Application to Automotive Systems
Olivier Sename
Université Grenoble Alpes, Grenoble INP, CNRS, GIPSA‐lab, France
Copyright © 2025 by John Wiley & Sons, Inc. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
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Library of Congress Cataloging‐in‐Publication Data:
Names: Sename, Olivier, author.
Title: Linear Parameter‐Varying Control : Theory and Application to Automotive Systems / Olivier Sename, Université Grenoble Alpes, Grenoble INP, CNRS, GIPSA-lab, France.
Description: Hoboken, NJ : Wiley, [2025] | Includes biblographical references and index.
Identifiers: LCCN 2024061703 (print) | LCCN 2024061704 (ebook) | ISBN 9781394285952 (hardback) | ISBN 9781394285969 (adobe pdf) | ISBN 9781394285976 (epub)
Subjects: LCSH: Motor vehicles–Electronic equipment. | Robust control. | Motor vehicles–Dynamics. | Linear control systems.
Classification: LCC TL272.5 .S43 2025 (print) | LCC TL272.5 (ebook) | DDC 629.8/32–dc23/eng/20250224
LC record available at https://lccn.loc.gov/2024061703
LC ebook record available at https://lccn.loc.gov/2024061704
Cover Design: Wiley
Cover Image: © Just_Super/Getty Images
To my wife Isabelle, my sons Corentin and Grégoire, and my parents Danièle and Bernard.
Thank you for your unwavering support and encouragement, and for always being there.
– Olivier Sename
About the Author
Olivier Sename received an engineering (1991) and a PhD degree (1994) from École Centrale de Nantes. He is now a Professor at Grenoble INP – UGA within GIPSA‐lab. His main research interests include linear parameter‐varying systems and automotive applications. He is the (co‐)author of 4 books, around 100 international journal papers, more than 280 international conference papers, and 6 patents. He was General Chair of the IFAC Joint Conference SSSC‐TDS‐FDA 2013, of the 1st IFAC Workshop on Linear Parameter‐Varying Systems 2015, and of the 12th International Conference on Mechatronics and Control Engineering 2024. He was the IPC Chair of the 2nd and 4th IFAC Workshops on LPVS 2018 and 2022 and the Program Chair of the 10th/11th ICMCE Conferences in 2021 and 2023. He presented several plenary talks (IFAC SSSC 2019, IFAC LPVS 2021, ICMCE 2021, ICSTCC 2015). He is the Section Editor‐in‐Chief for MDPI Electronics, is the Associated Editor of the European Control Conferences, and has been AE of the IEEE CSS Letters for 5 years. He is a member of the IFAC TC Linear Control Systems, Robust Control, and Automotive Control. He has led several industrial (Renault, Volvo Trucks, JTEKT, Delphi) and international (Mexico, Italy, Hungary, Spain) collaboration projects. He has supervised 34 PhD students.
Preface
This book is the culmination of over 20 years of experience with linear parameter‐varying (LPV) systems and control, particularly in automotive applications. We started working on this topic during the PhD thesis of Alessandro Zin to design a suspension controller that could account for the nonlinearities of the suspension stiffness [Zin et al., 2004; Zin, 2005] to improve passenger comfort. From this study, numerous projects have been conducted across various academic and industrial settings, such as in engine control [Gauthier, 2007; Rivas Caicedo, 2012; Ngo, 2014; Dubuc, 2018], energy systems [Hernandez‐Torres, 2011; Nwesaty, 2015; Wang, 2013], network‐controlled systems [Robert, 2007; Roche, 2011], time‐delay systems [Briat, 2008], model predictive control [Morato, 2023], aerospace [Vinco, 2024], and automotive vehicle dynamics [Poussot‐Vassal, 2008; Aubouet, 2010; Do, 2011; Lozoya‐Santos, 2012; Fergani, 2014; Tudon Martinez, 2013; Nguyen, 2016; Vu, 2017; Yamamoto, 2017; Pham, 2020; Atoui, 2022; Kapsalis, 2022; Medero Borrell, 2023].
The inspiration behind this book is to present a holistic approach to studying LPV systems, both in theory and in practical applications, building upon concepts introduced in my master's/PhD course [Sename, 2024]. It aims to cover the key aspects of system theory – from modeling to analysis and control design – and their application to real‐world automotive systems.
On the theoretical side, the book offers methods for modeling, analyzing, and controlling LPV systems, providing an integrated approach to their study. Throughout this part, several examples illustrate how to model LPV systems from nonlinear ones; analyze properties such as controllability, observability, and stability; and develop LPV control and observation methods, particularly emphasizing the adaptive nature of LPV control. The ultimate objective is to offer guidelines for developing and synthesizing control and observation approaches within the LPV context.
On the application side, the book presents effective methods to develop LPV controllers and observers that have been validated in real‐world applications. The primary focus is on the automotive domain, where extensive control studies have been conducted over the past 30 years. However, applying robust control to automotive systems has been limited by relying purely on linear approaches. The LPV approach addresses this limitation by extending robust control theory to nonlinear systems through the representation of these systems as LPV models using scheduling parameters. We illustrate this with several classical and important cases related to comfort and road safety, such as vehicle dynamics control for automated vehicles, and suspension control and fault diagnosis.
This book would not have seen the light of day without the collaborative efforts of our PhD students. I am deeply grateful for their dedication, teamwork, and the innovative contributions they made throughout their research. Special thanks to Charles, Lam, Jorge, Juan-Carlos, Soheib, Quan, Donatien, Hussam, Phong, Ariel, and Marcelo, whose results are prominently featured in this book.
I extend my heartfelt thanks to my international collaborators, whose contributions have been invaluable to the projects featured in this book. Special thanks to Peter Gaspar, Jozsef Bokor, and Zoltan Szabo from Hungary; Ricardo Ramirez Mendoza and Ruben Morales Menendez from Mexico; Sergio Savaresi from Italy; and Vicenc Puig from Spain.
On the other hand, a number of the results presented in this book would not have been possible without the support of the automotive industrial collaborations I have nurtured over the years Vicente Milanes, Vincent Talon, and Hubert Béchard from Renault; Benjamin Talon from Soben; Christophe Gauthier from Volvo Trucks; and Pascal Moulaire from JTEKT.
I am equally grateful to my close colleagues from our research team, Emmanuel Witrant, Delphine Bresch‐Pietri, Damien Koenig, and John‐Jairo Martinez‐Molina, for their collaboration on these automotive projects.
Last but not least, I cannot thank my friend and colleague Luc Dugard enough. He warmly welcomed me to the Laboratoire d'Automatique de Grenoble when I arrived as a Maître de Conférences in 1995 and supported me for many years in the studies that led to this achievement.
Now, I invite you to delve into this book and explore its comprehensive approach to understanding LPV systems, both theoretically and in practical applications for control. You'll discover a range of methods and recipes for developing LPV controllers and observers, presented in an integrated and accessible manner. Additionally, we illustrate these concepts with real‐world applications in automotive systems, bringing the theory to life. I hope you enjoy the journey and find the information as exciting and valuable as I do.
September 30, 2024
Olivier Sename
GrenobleFrance
Bibliography
Hussam Atoui.
Switching/Interpolating LPV Control based on Youla‐Kucera Parameterization: Application to Autonomous Vehicles
. PhD thesis, Université Grenoble Alpes, 2022.
Sébastien Aubouet.
Semi‐Active SOBEN Suspensions Modeling and Control
. PhD thesis, Institut National Polytechnique de Grenoble‐INPG, 2010.
Corentin Briat.
Robust Control and Observation of LPV Time‐Delay Systems
. PhD thesis, Grenoble Institute of Technology, GIPSA‐lab/Control Systems dpt, 2008.
Anh‐Lam Do.
Approche LPV pour la commande robuste de la dynamique des véhicules: amélioration conjointe du confort et de la sécurité
. PhD thesis, Université de Grenoble, 2011.
Donatien Dubuc.
Observation and Diagnosis for Trucks
. PhD thesis, Université Grenoble Alpes (GIPSA‐lab), France, 2018.
Soheib Fergani.
Robust Multivariable Control for Vehicle Dynamics
. PhD thesis, Université de Grenoble, 2014.
Christophe Gauthier.
Modélisation et commande d'un système common rail
. PhD thesis, INPG, Laboratoire d'Automatique de Grenoble (new GIPSA‐lab) and Delphi Diesel Systems, Grenoble, France, 2007.
David Hernandez‐Torres.
Robust Control of Hybrid Electro‐Chemical Generators
. PhD thesis, Université de Grenoble, 2011.
Dimitrios Kapsalis.
LPV/Gain‐Scheduled Lateral Control Architectures for Autonomous Vehicles
. PhD thesis, Université Grenoble Alpes, 2022.
Jorge de Jesús Lozoya‐Santos.
Control of Automotive Semi‐Active Suspensions
. PhD thesis, Instituto Tecnologico y de Estudios Superiores de Monterrey, Mexico, 2012.
Ariel Medero Borrell.
LPV Lateral Control of Autonomous and Automated Vehicles
. PhD thesis, Université Grenoble Alpes, 2023.
Marcelo Menezes Morato.
Predictive Control Methods for Linear Parameter Varying Systems
. PhD thesis, Université Grenoble Alpes, 2023.
Caroline Ngo.
Surveillance du système de post‐traitement essence et contrôle de chaîne d'air suralimentée
. PhD thesis, Université de Grenoble, 2014.
Manh‐Quan Nguyen.
LPV Approaches for Modeling and Control of Vehicle Dynamics: Application to a Small Car Pilot Plant with ER Dampers
. PhD thesis, Université de Grenoble, 2016.
Waleed Nwesaty.
LPV/Hinf Control Design of On‐Board Energy Management Systems for Electric Vehicles
. PhD thesis, Université de Grenoble, 2015.
Thanh‐Phong Pham.
LPV Observer and Fault‐Tolerant Control of Vehicle Dynamics: Application to An Automotive Semi‐Active Suspension System
. PhD thesis, Université de Grenoble, 2020.
Charles Poussot‐Vassal.
Robust Multivariable Linear Parameter Varying Control of Automotive Chassis
. PhD thesis, Université de Grenoble, 2008.
Maria Adelina Rivas Caicedo.
Modeling and Control for Euro VI Spark Ignited (SI) Engine
. PhD thesis, Université de Grenoble, 2012.
David Robert.
Contribution à l'interaction commande/ordonnancement
. PhD thesis, INP Grenoble (Laboratoire d'Automatique de Grenoble), France, January 2007.
Emilie Roche.
Commande à échantillonnage variable pour les systèmes LPV: application à un sous‐marin autonome
. PhD thesis, Université de Grenoble (GIPSA‐lab), France, October 2011.
Olivier Sename. The linear parameter varying approach: theory and application to vehicle dynamic, 2024. URL
https://oliviersename.fr
.
Juan‐Carlos Tudon Martinez.
Fault Tolerant Control in Automotive Semi‐Active Suspensions
. PhD thesis, Instituto Tecnologico y de Estudios Superiores de Monterrey, Mexico, 2013.
Gian Marco Vinco.
Flight Dynamics Modeling and Autopilot Design for Guided Projectiles via Linear Parameter‐Varying Techniques
. PhD thesis, Université Grenoble Alpes, 2024.
Van Tan Vu.
Enhancing the Roll Stability of Heavy Vehicles by Using An Active Anti‐Roll Bar System
. PhD thesis, Université Grenoble Alpes, 2017.
Tinghong Wang.
Robust Control Approach to Battery Health Accommodation of EMS in HEV
. PhD thesis, Université de Grenoble, 2013.
Kazusa Yamamoto.
Control of Electromechanical Systems, Application on Electric Power Steering Systems
. PhD thesis, Université Grenoble Alpes, 2017.
Alessandro Zin.
Robust Automotive Suspension Control Toward Global Chassis Control (in French)
. PhD thesis, INP Grenoble (Laboratoire d'Automatique de Grenoble), France, October 2005.
A. Zin, O. Sename, and L. Dugard. Switched LPV/ control strategy of passenger car active suspensions. In
Vehicle System Dynamics, Identification and Anomalies Budapest, Hungary
, November 8–10, 2004.
Acronyms
BRL
Bounded Real Lemma
ER
Electro Rheological
FDI
Fault Detection and Isolation
FDD
Fault Detection and Diagnosis
FTC
Fault‐Tolerant Control
IQC
Integral Quadratic Constraint
LPV
Linear Parameter Varying
qLPV
quasi Linear Parameter Varying
LDI
Linear Differential Inclusion
LMI
Linear Matrix Inequality
LTI
Linear Time Invariant
LFR
Linear Fractional Representation
LFT
Linear Fractional Transformation
MIMO
Multi Input Multi Output
MPC
Model Predictive Control
MR
Magneto Rheological
NLPV
Nonlinear Parameter Varying
NRMS
Normalized Root Mean Square
NRMSE
Normalized Root Mean Square Error
ODE
Ordinary Differential Equation
PDLF
Parameter‐Dependent Lyapunov Function
PI
Proportional Integral
PID
Proportional Integral Derivative
QoV
Quarter of Vehicle
RMS
Root Mean Square
RMSE
Root Mean Square Error
SA
Semi‐active
SISO
Single Input Single Output
Mathematical Symbols
set of all real numbers
the ‐dimensional space
set of real matrices with rows and columns
the set of all complex numbers
the conjugate of the complex number
(resp. )
the minimal (resp. maximal) value of
set of continuous functions from to
set of continuously differentiable functions from to
set of square integrable signals
set of bounded signals
vector norm
the 2‐norm of a vector‐valued signal (also denoted the norm)
the ‐norm of a vector‐valued signal (also denoted the norm)
set of all proper and real rational stable transfer matrices
the Laplace variable
the norm of a system
induced ‐norm
vector value at time
(or )
vector value at the sampling instant
the time derivative of the vector
the identity matrix
0
the zero element (scalar or matrix)
the identity matrix of size
the transpose of the matrix
the inverse of the matrix
the matrix is positive definite
the matrix is negative definite
interval of values between and
the system matrix of a linear system state‐space representation
About the Companion Website
This book is accompanied by a companion website:
www.wiley.com/go/sename/lpvcontrol
The website includes GitHub Repository.
Introduction
The 21st century has seen a significant increase in the development of control approaches for linear parameter‐varying (LPV) systems. Indeed, the theoretical framework of such control problems, being an almost direct extension of robust control theory, makes them very attractive to control a large class of dynamical systems.
The LPV approach, as illustrated in the control scheme Figure 1, is actually a very appropriate tool not only to handle nonlinearities in physical dynamical systems, but also to provide real‐time varying performances, through an ad hoc definition of varying parameters.
Moreover, the synthesis of LPV controllers (sometimes referred to as gain‐scheduling) and observers can ensure stability and performance when parameters are time‐varying, therefore for a larger domain of systems operation.
Figure 1 LPV controller or observer scheme.
However, the study of LPV systems in view of control/observer design needs to pay attention to theoretical properties and to the optimization problems to be solved. First, the way to provide (quasi) LPV models from given nonlinear systems is not unique, and it is important to understand the different methods that can be used in that context. Then, because LPV systems (and quasi‐LPV ones) are an intermediate class of systems between linear and nonlinear ones, the usual properties of dynamical systems (controllability, observability, and stability) must be defined carefully. Finally, while the formulation of control/observer design problems can be seen as a direct extension of what has been developed in robust control theory (in particular for , , , or multiobjective frameworks), the corresponding optimization problems are more complex since an infinite number of linear matrix inequalities (LMIs) are to be solved due to the parameter space. This has led to several methods that can be used to reduce the problem to a finite number of LMIs (referred to as polytopic, grid‐based, or linear fractional transformation [LFT] approaches).
These issues will be studied in detail in Part 1, where three chapters will concern the modeling, analysis, and control design methods for LPV systems.
Besides theoretical studies, the LPV approach has been applied to a wide range of applications [Shamma, 2012]. It started and is still widely considered in aerospace applications (from gain‐scheduling approaches), as shown in Pfifer et al. 2015, Biannic et al. 1997, Biannic [1996, 2013], Theodoulis 2008, Navarro‐Tapia et al. 2022, and Preda et al. 2018, but is also used in several applications such as mechatronics [Dettori, 2001], robotics [Rotondo et al., 2015; Incremona et al., 2022], energy systems [Mercère et al., 2011; Nwesaty et al., 2020], and battery health [Colmegna et al., 2016]; see the interesting survey paper by Hoffmann and Werner 2015.
This book is devoted to automotive applications and, mainly vehicle dynamic control (for automated and autonomous vehicles). We will present different control/observer design methods for two subsystems: the semi‐active suspension system and the lateral dynamics.
Since the two main mentioned interests of the LPV approach are, on one hand, to tackle nonlinear systems and, on the other hand, to provide an adaptive‐like control through parameter scheduling, the remaining parts of the book will be divided into two.
Part 2 will be dedicated to control/observer design for qLPV systems, with three chapters: Chapter 4 illustrates this case with two academic examples, Chapter 5 presents different methods to design LPV observers for a semi‐active suspension system, and Chapter 6 illustrates the polytopic and grid‐based methods to control the lateral motion of autonomous vehicles.
Finally, Part 3 will concern the development of adaptive‐like control approaches in the LPV framework, with four chapters: Chapter 7 illustrates how we can provide performance adaptation on academic examples, Chapter 8 demonstrates an LPV road‐adaptive suspension control coordinated with a road roughness estimation algorithm, Chapter 9 describes a fault‐tolerant control strategy for a semi‐active suspension system, including LPV observers (for fault estimation) and LPV control reconfiguration, and finally, Chapter 10 proposes an LPV driver assistance system to guarantee road safety, thanks to driver ability monitoring used to schedule the needed lateral steering/braking control actions.
Bibliography
Jean‐Marc Biannic.
Robust Control of Parameter Varying Systems: Aerospace Applications
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Jean‐Marc Biannic.
Linear Parameter‐Varying Control Strategies for Aerospace Applications
, pages 347–373. Springer‐Verlag, Berlin, Heidelberg, 2013. ISBN 978‐3‐642‐36110‐4. doi:
https://doi.org/10.1007/978-3-642-36110-414
.
Jean‐Marc Biannic, Pierre Apkarian, and William L. Garrard. Parameter varying control of a high‐performance aircraft.
Journal of Guidance, Control, and Dynamics
, 20(2):225–231, 1997. doi:
https://doi.org/10.2514/2.4045
.
Patricio H. Colmegna, Ricardo S. Sánchez‐Pea, Ravi Gondhalekar, Eyal Dassau, and Frank J. Doyle. Switched LPV glucose control in type 1 diabetes.
IEEE Transactions on Biomedical Engineering
, 63(6):1192–1200, 2016. doi:
https://doi.org/10.1109/TBME.2015.2487043
.
Marco Dettori.
LMI Techniques for Control, With Application to a Compact Disc Player Mechanism
. PhD thesis,Delft University, 2001.
Christian Hoffmann and Herbert Werner. A survey of linear parameter‐varying control applications validated by experiments or high‐fidelity simulations.
IEEE Transactions on Control Systems Technology
, 23(2):416–433, 2015.
Gian Paolo Incremona, Antonella Ferrara, and Vadim I. Utkin. Sliding mode optimization in robot dynamics with LPV controller design.
IEEE Control Systems Letters
, 6:1760–1765, 2022. doi:
https://doi.org/10.1109/LCSYS.2021.3133362
.
Guillaume Mercère, Halldor Palsson, and Thierry Poinot. Continuous‐time linear parameter‐varying identification of a cross flow heat exchanger: a local approach.
IEEE Transactions on Control Systems Technology
, 19(1):64–76, 2011. doi:
https://doi.org/10.1109/TCST.2010.2071874
.
Diego Navarro‐Tapia, Andrés Marcos, and Samir Bennani. The VEGA launcher atmospheric control problem: a case for linear parameter‐varying synthesis.
Journal of the Franklin Institute
, 359(2):899–927, 2022. ISSN 0016‐0032. doi:
https://doi.org/10.1016/j.jfranklin.2021.07.057
.
Jean‐Marc Nwesaty, Antoneta Iuliana Bratcu, Alexandre Ravey, David Bouquain, and Olivier Sename. Robust energy management system for multi‐source dc energy systems–real‐time setup and validation.
IEEE Transactions on Control Systems Technology
, 28(6):2591–2599, 2020. doi:
https://doi.org/10.1109/TCST.2019.2937931
.
Harald Pfifer, Claudia P. Moreno, Julian Theis, Aditya Kotikapuldi, Abhineet Gupta, Bela Takarics, and Peter Seiler. Linear parameter varying techniques applied to aeroservoelastic aircraft: in memory of Gary Balas.
IFAC‐PapersOnLine
, 48(26):103–108, 2015. ISSN 2405‐8963. doi:
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Valentin Preda, Jérôme Cieslak, David Henry, Samir Bennani, and Alexandre Falcoz. Robust microvibration mitigation and pointing performance analysis for high stability spacecraft.
International Journal of Robust and Nonlinear Control
, 28(18):5688–5716, 2018. doi:
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.
Damiano Rotondo, Vicenç Puig, Fatiha Nejjari, and Juli Romera. A fault‐hiding approach for the switching quasi‐LPV fault‐tolerant control of a four‐wheeled omnidirectional mobile robot.
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, 62(6):3932–3944, 2015. doi:
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.
Jeff S. Shamma.
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, pages 3–26. Springer US, Boston, MA, 2012. doi:
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.
Spilios Theodoulis.
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. PhD thesis,Université Paris Sud – Paris XI, 2008.
Part ISome Theoretical Aspects on LPV Systems: From Modeling to Control
This part is dedicated to the theoretical background on LPV systems. It will be divided into three chapters.
In Chapter 1 some definitions will be given, different types of LPV systems will be presented, and it will be shown how to model LPV systems from nonlinear ones. Some illustrative academic examples will allow the reader to understand the main notions in LPV modeling. Note that, as said in Chapter 1, we will not be interested in data‐driven methods.
In Chapter 2 the classical properties of dynamical systems, namely controllability, observability, and stability, will be defined for LPV systems. A specific attention will be paid to the differences between the LTI and LPV cases, so as to illustrate what specific issues are due to time‐varying parameters. The definition of several performance criteria, namely , and pole placement, will be presented.
Chapter 3 will be devoted to control and observer design. The reader will find classical or generic methods that can be used in most of the cases encountered. Note that the main focus of the book will be on the polytopic and grid‐based approaches.
1Some Modeling Approaches for LPV and qLPV Systems
1.1 Introduction
The modeling step is crucial for control and diagnosis purposes. In general, system models could be data‐based (system identification) or knowledge‐based (written from physical laws, with differential and/or algebraic equations). In the later case, models can be very involved, including many state variables and parameters, in particular, if obtained using professional software. In this case, Identification is related to parameter identification.
Concerning dynamical systems, models could be written in two main structures: the input–output one, where the output behavior is related to the input one through a recurrence equation, or the state‐space representation, i.e., a “matrix‐form” representation of the dynamics of an ‐order differential equation system into a first order differential equation in a vector form of size , which is called the state.
While input–output forms are directly related to transfer functions and interesting for single input single output (SISO) systems and simple control design (PID controllers), state‐space representation has many advantages:
Physical laws are mathematically well defined in many fields: electrical engineering, mechanical engineering, aerospace engineering, microsystems, process plants, biology, etc.
State‐space equations are easily exported from advanced modeling softwares.
State‐space models include physical parameters: they are then easy to use when parameters are to be updated or changed for design.
State variables have physical meaning: position, velocity, current, voltage, species, etc.
The extension to multi‐input multi‐output (MIMO) systems, in particular in control theory, is almost direct [Kailath,
1980
].
Nonlinearities (state constraints and input saturation) can easily be formulated in the same context.
Most advanced control design methods today are based on state‐space representations (model predictive control, sliding modes, robust control).
Reliable numerical optimization tools (convex optimization [Boyd and Vandenbergue,
2004
]) can be used for control design, such as Linear Matrix Inequalities that are associated with matrices and state‐space representations [Boyd et al.,
1994
; Scherer and Weiland,
2004
].
State space is inherently related to what is referred to as the system state.
Definition 1.1 The state of a dynamical system is the set of variables, known as state variables, that fully describes the system and its response to any given set of inputs.
Mathematically, the knowledge of the initial values of the state variables at (namely, ), together with the knowledge of the system inputs for time , are sufficient to predict the behavior of the future system state and output variables (for ).
Section 1.2 gives some definitions of nonlinear and linear time‐invariant (LTI) models. Section 1.3 presents and defines the LPV systems, and Section 1.4 gives different classes of LPV systems. In Section 1.5, the model of an LPV system from a nonlinear one is discussed. Finally, Sections 1.8 and 1.9 present two illustrative examples of LPV modeling.
1.2 Dynamical Systems
1.2.1 Nonlinear Dynamical Systems
Dynamical systems are usually modeled using a set of nonlinear ordinary differential equations (ODE) to represent most of the physical phenomena. Such a model is generally referred to as a knowledge‐based model and is represented as follows.
Definition 1.2 (Nonlinear dynamical system) Given (nonlinear) functions and , a nonlinear dynamical system () can be described as:
where is the state (vector of state or internal variables), which takes values in a state space , is the input taking values in the input space and is the output that belongs to the output space .
Remark 1.1 It is worth noticing that this book does not consider infinite dimensional systems, such as linear partial differential equations, time‐delay systems.
While nonlinear equations are interesting since they provide a mathematical description closer to the (physical) reality, they may be difficult to handle for analysis and design purposes. Up to now, they still lack of generic methods for system identification, observer, and control design, even if recent advances have been performed in, among others, model predictive control.
This is one of the reasons why linear representations are still preferred in the synthesis of control laws for industrial applications.
1.2.2 Linear Time‐Invariant (LTI) Dynamical Systems
Linear systems can be obtained when the system representation is described by a set of linear ODEs, resulting from a linearization step of the nonlinear equations around a given equilibrium point.
Definition 1.3 (LTI dynamical system) Given matrices , , and , a LTI dynamical system () can be described as:
where is the state that takes values in a state space , is the input taking values in the input space , and is the output that belongs to the output space .
Remark 1.2 A linear system only gives a local approximation of the real system behavior, which is restrictive. However, control theory of such systems is well established [Kailath, 1980; Callier and Desoer, 1991]. Properties such as stability, controllability, and observability are defined uniquely. Observer and control design methods are available for SISO and MIMO systems. Moreover, robustness can be analyzed using efficient tools (e.g. analysis).
1.3 An Introduction to LPV Models
LPV systems appeared first in gain‐scheduling control, where a Jacobian linearization of a nonlinear plant about a family of equilibrium points is achieved and used to find a set of LTI controllers, then interpolated function of the operating conditions [Shamma and Athans, 1990; Rugh and Shamma, 2000; Gáspár et al., 2017]. However, since the LPV paradigm has evolved to propose tools able to provide stability and performance guarantees, a more abundant literature has been dedicated to the modeling of LPV systems from a nonlinear one.
The modeling of LPV systems is considered as a key step in view of control/diagnosis purpose. Indeed, as it will be seen along the book chapters, such a modeling step is not trivial. In particular, the choice of scheduling parameters is not unique and may lead to different (and not equivalent) state‐space representations [Bruzelius et al., 2004; Gáspár et al., 2017; Vinco et al., 2024; Vinco, 2024].
1.3.1 Definition
A dynamical LPV system can be expressed by the following state‐space equations:
LPV dynamical system
Definition 1.4 Given , the linear matrix functions , , and , a linear parameter varying (LPV) dynamical system () can be described as:
where is the state that takes values in a state space , is the input taking values in the input space , and is the output that belongs to the output space . Then, is the vector collecting the varying parameters. It varies in the set of continuously differentiable parameter curves and takes values in the compact set (parameter space) such that,
where is the number of varying parameters.
Remark 1.3 Below are a few remarks to be reminded about all along the book.
As the matrix functions , , , and are continuous functions of parameter they are in fact norm‐bounded on the compact set .
It is important to note that the state‐space representation (
1.3
) is linear in the state and input space but may be nonlinear in the parameter space.
Note that will be referred to as the system matrix of the LPV system (
1.3
).
For the sake of readability, will be denoted as when there is no confusion.
1.3.2 About the Time‐Varying Parameters
From a general viewpoint, the selection of the parameter characteristics produces different classes of systems, as mentioned in the PhD theses [Biannic, 1996; Briat, 2008] and in the book [Briat, 2015] (Chapter 1).
Type of parameters and classes of systems
, a constant value, (
1.3
) is a LTI system.
, an unknown uncertain (constant or time‐varying) parameter, (
1.3
) is an uncertain LTI system
, an explicitly given time trajectory, (
1.3
) is a linear‐time varying (LTV) system.
, a state‐dependent parameter, (
1.3
) is a quasi‐LPV (qLPV) system.
, a known parameter, (
1.3
) is an LPV system.
Assumption 1.1 To cope with the system definition and with the framework of continuous‐time LPV systems, classical assumptions on are:
the parameters are assumed to be known or measurable, even if their variation is not known in advance,
varies in the set of continuously differentiable parameter curves,
is bounded, i.e. ,
The system matrices , etc., are continuous functions on .
Assumption 1.2 It is sometimes required that the derivative of the parameters are bounded, i.e.:
defined by the minimal and maximal values of
This corresponds to the case of slow varying parameters. Note that when such an assumption is not written, then the parameter derivatives could be infinite (i.e. ).
It is also important to distinguish between the two main types of parameters used in LPV system representations:
First, parameters are
exogenous
if they are external variables. The system is in that case
non‐stationary
. As a simple example, let us consider the damped mass spring system, presented in [Scherer,
2012
]: where is a position, (resp. ) is the damping (resp. stiffness) coefficient. Defining , which is the only time‐varying parameter, the first‐order state‐space representation is as follows:
with , , , , .
The other important type are the
endogenous
parameters, when they are a function of the state variables, as . As mentioned above, in that case, the LPV system is referred to as a
quasi‐LPV system
or
qLPV system
.
Classical in gain‐scheduling approaches, and as seen later in detail, this case is encountered when approximating nonlinear systems. Let us consider the nonlinear system:
Denoting , the system can be rewritten as:
Remark 1.4 It is worth denoting that the implicit assumption in the above example is that should be known or measurable, which we shall discuss later in the book.
In a more general way, may depend on state and input vectors as well, as . Note also that when the system is in a closed‐loop, the dependency on becomes one on some state variables as well, which does take part of qLPV representations.
Remark 1.5 An interesting discussion is provided in Briat [2008, 2015] about the mathematical classification of the parameters. Indeed, in a general case, the parameter values could be continuous, piecewise‐continuous, or discontinuous functions (for instance, when for switched systems), eventually non differentiable. This may induce additional difficulties when dealing with stability analysis.
1.3.3 Is an LPV System Linear or Nonlinear?
As we will explain in Section 1.5, what is often referred to as gain‐scheduling control corresponds to Jacobian linearization of the nonlinear plant about a family of equilibrium points [Shamma and Athans, 1990; Rugh and Shamma, 2000; Gáspár et al., 2017]. The LPV system representation is then made of a set of LTI systems. In terms of control design, this means that the steps to be carried out are: linearization around operating conditions, design (at each operating point) of an LTI controller, and interpolation of the LTI controllers in between operating conditions. Such a method has been often used in the aerospace and automotive industries. Pros and cons are the simplicity of control design for a nonlinear system and the lack of a priori guarantee of stability or robustness, respectively.
As already seen earlier, the gain‐scheduling approach differs from the quasi‐LPV one, for which nonlinearities are hidden (or embedded) in the parameter definitions.
On the other hand, when a specific parameter trajectory is given as a function of time, the LPV system becomes a linear time‐varying system. Therefore, as explained in Chapter 2, theoretical analysis of LPV system properties (stability, controllability, and observability) often falls into the framework of LTV systems.
1.3.4 Discrete‐Time LPV Systems
This book is mainly dedicated to continuous‐time systems. However, several tools and approaches that will be presented here can be directly extended to the discrete‐time case.
As for the continuous‐time case, we can consider two different types of discrete‐time systems: the ones written in the class of polytopic systems as in Pandey and de Oliveira [2017], De Caigny et al. [2012], and Heemels et al. [2010] and the ones given in a general form as in de Souza et al. [2006] and Borrell et al. [2024].
In discrete‐time, an LPV system can be written as
where is the state vector, is the vector of inputs, and is the vector of outputs. For simplicity, (resp. , , ) denotes the instantaneous value of the state vector (resp. , , ) at the time instant ( being the sampling period). Note that may also be denoted as . The vector of scheduling parameters is defined as:
As for continuous‐time model, the time‐varying (scheduling) parameter vector – i.e., in (1.9) – is considered to be known (measured or estimated) and to satisfy several assumptions. The most typical ones (Assumptions 1.3 and 1.4) are written in this context:
Assumption 1.3 Each varying parameter value is known and is bounded by extremal values and such that . This defines the admissible space of the varying parameter vector, such that , .
Assumption 1.4 The rate of variation for each varying parameter between two consecutive sampling times and is bounded by and such that .
Remark 1.6 A key issue to be considered when providing a discrete‐time representation from a continuous one is that the discretization of leads to [Apkarian, 1997]. Therefore, this makes the mathematical dependency of the state matrix on the parameter vector more complex. As an illustration, when is affine in , is not. Such an issue is of interest when control design is performed in continuous time and followed by a discretization step before implementation.
The interested reader is referred to Chapter 6 in Toth [2010] dedicated to the discretization of LPV systems. Some of those issues are also discussed in Robert et al. [2010], Roche [2011], and Roche et al. [2012] in view of sampling‐dependent control design.
1.4 Specific Classes of LPV Systems
This section is devoted to the definition of the main types of LPV systems representations that one can find in the literature. Some examples illustrate the way to get such types. Note that even if the continuous‐time case is illustrated only, such classes do exist for the discrete‐time one.
1.4.1 Class 1: Affine Parameter Dependence
This case arises when the system matrices are affine functions of the parameter vector , as
Definition 1.5(Affine LPV system) An LPV system is said to be an affine LPV system if the parameter dependency on of its state‐space matrices , and is affine, i.e.:
where , and are real constant matrices.
Example 1.1 A class 1 model
Let us consider the case of the damped mass‐spring example represented by the linear system (1.26). We can write (1.26) as:
The varying parameter being , we therefore have an affine LPV system (1.11) where:
1.4.2 Class 2: Polytopic Representations
This representation is often used to rewrite an affine LPV system and is probably the most used approach to design LPV controllers due to its simplicity.
Matrix polytope
Definition 1.6 [Boyd et al., 1994; Apkarian et al., 1995] A matrix polytope is defined as the convex hull of a finite number of matrices such that:
Now, let us consider an LPV system (1.3) where the system matrices are defined with an affine parameter dependence as stated in Definition 1.5. Assuming that the parameters are bounded , the vector of parameters evolves inside a hypercube (a polytope) represented by vertices , as
