Gain scheduling Robust Design and Automated Tuning of Automotive Controllers Ir. G.J.L. Naus 2009.106 26th of October, 2009 WP3 - robust, low-complexity controller design Supervisors: Prof.dr.ir. M. Steinbuch (TU/e) Dr.ir. M.J.G. v.d. Molengraft (TU/e) Dr.ir. R. Huisman (DAF) Ir. J. Ploeg (TNO) TNO Science and Industry, Helmond (TNO) Business Unit Automotive Department of Advanced Chassis and Transport Systems DAF Trucks N.V., Eindhoven (DAF) Product Development Technical Analysis group University of Technology Eindhoven (TU/e) Department of Mechanical Engineering Division Dynamical Systems Design Control System Technology group 2 Contents 1 Introduction 1 1.1 Background and goals . 1 1.2 Outline . 2 2 Gain scheduling 3 2.1 Overview . 3 2.1.1 Classical gain scheduling . 3 2.1.2 LPV and LFT synthesis . 4 2.1.3 Fuzzy gain scheduling . 5 2.2 Gain scheduling vs Robust control . 5 3 Parameter-dependent plant models 7 3.1 linearization-based scheduling . 8 3.2 Off-equilibrium linearizations . 11 3.3 Quasi-LPV approach . 12 3.4 linear, parameter-dependent plant . 13 3.5 Linear fractional transformation description . 14 3.6 Conclusions / recap . 15 4 Classical gain scheduling 17 4.1 LTI controller design . 17 4.1.1 Direct gain scheduling controller design . 18 4.2 Gain scheduling controller design . 18 4.2.1 Linearization scheduling . 19 4.2.2 Interpolation methods . 21 4.2.3 Velocity-based scheduling . 23 4.3 Hidden coupling terms . 24 4.4 Stability properties . 25 5 LPV controller synthesis 27 5.1 General LPV controller synthesis setup . 28 5.1.1 Stability analysis . 29 5.1.2 Performance analysis . 30 5.1.3 General synthesis problem formulation . 31 5.2 Lyapunov-based LPV control synthesis . 32 5.2.1 Gridding . 32 5.2.2 Polytopic approximation . 32 5.3 Scaled small-gain or LFT synthesis . 33 5.3.1 LFT problem formulation . 33 5.3.2 Controller synthesis . 34 5.3.3 Pros and cons . 35 5.4 Mixed LPV-LFT approaches . 36 3 4 CONTENTS 5.4.1 Extended Kalman-Yakubovich-Popov (KYP) lemma . 36 5.4.2 LFT Lyapunov functions . 36 5.5 Conclusions . 36 6 Fuzzy gain-scheduling 39 6.1 Fuzzy modeling . 39 6.1.1 Weighting or scaling functions . 40 6.1.2 Approximation accuracy . 41 6.2 Fuzzy gain scheduling control . 43 6.2.1 Parallel Distributed Control . 43 7 Conclusions and future work 45 7.1 Conclusions . 45 7.2 Future work (tbd) . 47 Bibliography 52 A Longitudinal vehicle control 53 B Engine - idle governor 55 Chapter 1 Introduction In this literature survey, an overview of gain-scheduling syntheses is put together. Many different notions can be viewed as Gain Scheduling (GS). For example switching or blending of gain values of controllers or models or switching or blending of complete controllers or model dynamics according to different operating conditions, or according to preset times. As the terms switching and blending already indicate, GS may either involve continuous or discrete scheduling of controllers or model dynamics. In general, gain-scheduling encompasses the attenuation of nonlinear dynamics over a range of operations, the attenuation of environmental time-variations or the attenuation of parameter vari- ations and uncertainties. Classical gain scheduling involves offline linearization of nonlinear system dynamics at multiple operating conditions, which are parameterized by a so-called scheduling vari- able θ, and the design of corresponding linear controllers at each point. Next, online scheduling of controller gains is established, based on θ = θ(t) to reflect the nearby operating condition. The in- tention is to extend the reach of a single linearization-based control design over an entire operating envelope. In general, the scheduling variable is time-varying and may either be an internal plant variable (or a function of internal plant variables) called endogenous variables, or an externally prescribed exogenous variable. Consequently, difficulties regarding stability and performance requirements may arise when the LTI controller designs are implemented for θ = θ(t). More recent LPV and LFT approaches hence take into account the time-varying nature of θ = θ(t) in the controller design. Guaranteed global stability and performance requirements can thus be given a priori. This survey encompasses classical gain-scheduling syntheses as well as more recent LPV and LFT approaches. 1.1 Background and goals Background This report is based on the conclusions of (Naus 2007b, Naus 2007a), which discuss the controller design and tuning at Integrated Safety, Business Unit Automotive, TNO Helmond, and DAF Trucks N.V. respectively. Essentially, the main problems regarding controller design at DAF and TNO comprise i) a lack of proper system identification and corresponding modeling and ii) the lack of appropriate performance specifications. In some cases, application of system identification, subsequent modeling and corresponding local controller design is applied. However, scheduling of the resulting controller gains or switching between various controllers is commonly applied. This is done by experience and insight in the system. Stability issues as well as performance and robustness specifications are not considered. Hence, insight in gain scheduling and related control syntheses is demanded. At DAF, gain scheduling is adopted in many practical applications. In all cases, the scheduling is employed in an ad hoc manner, not considering stability issues or performance and robustness specifications. Besides employing gain scheduling as a suitable control synthesis for nonlinear 1 2 CHAPTER 1. INTRODUCTION systems, gain scheduling is also adopted for closed-loop performance improvements of linear plants by scheduling the controller gains on the basis of e.g. a tracking error. At TNO, discrete scheduling or switching as well as smooth scheduling is employed in an ad hoc fashion. The tuning and design of the controller (gains) corresponding to specific working conditions involves much trial-and-error. Furthermore, recent application of Model Predictive Control (MPC) in an explicit fashion (Naus, van den Bleek, Ploeg, Scheepers, van de Molengraft & Steinbuch 2008) yields a gain-scheduled controller. The interpretation of this controller and corresponding tuning require more insight in gain scheduling and related control syntheses. Goals The main goals of this report involve two issues. • The report provides an overview of gain scheduling and related controller design syntheses. Available gain scheduling techniques, and guidelines for appropriate application of these techniques are discussed. • Based on the conclusions regarding the application of gain scheduling, goals for further research have to be defined within the project robust design and automated tuning of auto- motive controllers. Appropriate syntheses and research directions to elaborate on have to be defined, focusing on generic design and tuning methodologies for applications at DAF and TNO. 1.2 Outline To start with, an overview of gain scheduling techniques is given (Chapter 2). Parameter- dependent plant models, which serve as a basis for gain scheduling controller design, are discussed in Chapter 3. Next, each of the main classes of gain scheduling techniques is discussed in detail; classical gain scheduling (Chapter 4), LPV and LFT control synthesis (Chapter 5) and fuzzy gain scheduling (Chapter 6). In Chapter ??, application examples, focusing on the automotive industry are discussed. Finally, conclusions and future work regarding the applicability of gain scheduling techniques in the remainder of this project are given (Chapter 7). The future work will be deter- mined in cooperation with DAF and TNO. Throughout this report, two example case studies are used; the problem of designing a longitudinal vehicle speed controller, i.e. cruise controller, and an engine example involving the design of a low idle governor. These case studies are defined in Appendix A and B respectively. Chapter 2 Gain scheduling 2.1 Overview Different gain scheduling approaches can be distinguished, which may be classified in different ways. To start with, gain scheduling methods can be divided into i) methods decomposing non- linear design problems into linear sub-problems and ii) methods decomposing nonlinear design problems into simpler nonlinear sub-problems or affine sub-problems. (Leith & Leithead 2000) give an overview of the main theoretical results and design procedures relating to (continuous) gain-scheduling control in the sense of decomposing nonlinear design problems into linear sub- problems. Focus will lie on the former one, as linear system and corresponding control theory is well developed. Furthermore, gain scheduling methods decompose into i) continuous gain-scheduling methods and ii) discrete, i.e. hybrid or switched gain-scheduling methods. Discrete or hybrid in this sense involves the switching of a system or controller between dynamical regimes. E.g. friction models that have a clear distinction between stick and slip phases, backlash in gears, dead zones in cog wheels; phenomena like saturation, hysteresis, sensor and actuator failures. Hybrid systems in the sense of combining digital controllers with physical processes is not meant here (Schutter & Heemels 2004). Finally, three main approaches can be distinguished regarding gain-scheduling; i) so-called divide and conquer techniques, or the classical gain-scheduling approach, more recent LPV and LFT syntheses and thirdly, fuzzy techniques. A brief overview of each of these approaches is given next. 2.1.1 Classical gain scheduling Generally, classical