Control and Estimation of Piecewise Affine Systems

Jun Xu, Lihua Xie

• Publisher: Woodhead Publishing
• Year: 2014
• Print ISBN:
  9781782421610
• eBook ISBN:
  9781782421627
• Topic: Technology & Engineering
• Subtopic: Mechanical Engineering
• Index: Technology & Engineering

As a powerful tool to study nonlinear systems and hybrid systems, piecewise affine (PWA) systems have been widely applied to mechanical systems. Control and Estimation of Piecewise Affine Systems presents several research findings relating to the control and estimation of PWA systems in one unified view. Chapters in this title discuss stability results of PWA systems, using piecewise quadratic Lyapunov functions and piecewise homogeneous polynomial Lyapunov functions. Explicit necessary and sufficient conditions for the controllability and reachability of a class of PWA systems are considered along with controller and estimator design methods for PWA systems using linear matrix inequality (LMI) and bilinear matrix inequality (BMI) techniques. A PWA approach to a class of Takagi-Sugeno fuzzy system is discussed in depth. The book uses a number of mechanical systems, such as disk servo systems to illustrate the advantages of the proposed methods. - Provides new insights on properties of PWA systems, including stability, stabilizability, reachability and controllability - Presents a unified framework for analysis and synthesis of both continuous-time and discrete-time PWA systems - Presents novel approaches for stability analysis and control design based on the promising SOS techniques

1

Introduction

Abstract:

The chapter provides the background of P-WA systems. We show that the wide applications of PWA systems motivate the intensive study of various properties. We also list the research objectives, contribution and organization of this book in this chapter.

Key words

Piecewise affine piecewise linear

1.1 Motivations

Piecewise affine/linear (PWA or PWL) systems have been studied for a long time.^{[37,46,55,68]} The first study of PWA systems in the control literature may date back to Andronov’s investigation on oscillations in nonlinear systems in the 1940’s.^[3] An early practical example of PWA systems can be found in piecewise linear servomechanisms.^[62] A concrete work on qualitative understanding of piecewise linear systems was due to Kalman who treated a saturated system by a series of polyhedral regions in the state space, separated by switching boundaries in the 1950’s.^[42]

However, it was not until the 1970’s that PWA systems were perceived as a class of system models by the circuit community, where efficient simulations and analysis of large-scale circuits with diodes and other piecewise linear elements were desired, which led to a considerable research effort on efficient representation of PWA systems.^{[11,12,63,64]} As one of the pioneering works, Sontag analyzed the discrete-time PWA systems in the 1980’s. His idea was still employed in some recent works, say,.^[37,43,59]

In recent years, there have been increasing interests in PWA systems due to their applications in the following aspects.

Firstly, PWA maps have universal approximation properties^[48,49] for some class of nonlinear systems. For example, in,^[48] the canonical representation of piecewise-linear functions is considered as a universal approximation scheme of multivariate functions. Meanwhile, two universal approximation schemes in terms of combinations of univariate canonical piecewise-linear functions are proposed. In,^[37] Johansson shows how approximation errors can be accounted for in the analysis to yield rigorous results for the underlying smooth nonlinear system, and develops a converse theorem which states that a piecewise quadratic Lyapunov function (PQLF) suffices to prove the exponential stability of the smooth system. Lown and Zohdy^[50,51] discuss a robust linear estimation technique on nonlinear systems with multiple piecewise models. Note that linear time-invariant (LTI) systems combined with PWL sector-bounded nonlinearity^[25,30] can be considered as a kind of PWA system naturally. For example, the system consisting of the feedback interconnection of a linear system and a PWL sector-bounded nonlinearity is a PWA system.^[30]

Secondly, PWA systems can be used to model many types of switched and hybrid systems. Heemels et al.^[31] establish an equivalence between PWA systems and some classes of hybrid systems: mixed logical dynamical (MLD) systems, linear complementarity (LC) systems, extended linear complementarity (ELC) systems, and max-min-plus-scaling (MMPS) systems.^[9,11] Thus PWA systems provide a powerful means for analysis and design of switched and hybrid systems.

Thirdly, PWA systems can be employed to formulate a large class of intelligent systems, especially, fuzzy systems.^[17,19,22] The intelligent controllers are commonly rule-based. Linear controllers can be designed at the main operating points and switched when required. For instance, Johansson^[37] analyzes continuous-time Takagi-Sugeno (T-S) fuzzy systems using a differential inclusion that involves all consequent dynamics via operating regime based models. Feng et al. use uncertain PWA system as the underlying model of T-S fuzzy system for stability analysis and controller design.^{[17,19,22,76]}

Fourthly, some special cases of sampled-data systems can be treated as PWA systems. In^[33,34], Imura proposes a so-called sampled-data PWA system, where the switching action of the discrete state is determined at each sampling time according to a condition on the continuous state.

Finally, besides the various applications listed above, some identification techniques for PWA systems have been developed. These include the Gausss least squares regression, the multi-variable Taylor series method, the point-to-point estimation techniques,^[50,51] the statistical clustering technique,^[28] the K-means clustering-based procedure,^[24] the Bayesian procedure,^[41] the mixed-integer linear or quadratic programming,^[61] a bounded-error approach,^[6] the polynomial factorization (algebraic approach)^[66] and so on. Thus many physical systems can be tackled as PWA models, such as a fermentation process^[14] and a pick-and-place machine.^[41]

There have been many works done on PWA systems, ranging from the modelling,^{[5,17,37,38,60]} characteristic analysis^{[5,16,32,35,72,74]} to controller^{[15,17−20,22,37,75]} and estimator^{[1,2,21,39,40,69−71,73]} design. However, many properties of PWA systems have not been fully explored and their design approaches are generally conservative.

For example, the results for estimation problems of PWA systems^[1,2] are conservative because some important characteristics of PWA models, such as the partition information, are not taken into account. Recently, some researchers explored ways of integrating the partition information into observer design.^[39,40] However, the results are only for bimodal systems.

A similar problem also appears in control of PWA systems. In,^[13,18,37] controllers are designed for PWA systems without considering the partition information. In,^[17] Feng includes the partition information for a fuzzy controller design based on continuous-time PWA systems. Nevertheless, it is difficult to solve this problem for discrete-time systems due to the non-convex nature of optimization involved. In this book, we shall address this problem.

Moreover, many properties of PWA systems remain unknown and many existing results are generally conservative. For example, although much research has been carried out on stability analysis^{[16,23,37,57]} based on piecewise Lyapunov functions which are more suitable than common quadratic Lyapunov functions (CQLFs), it remains challenging to verify the stability of some simple systems.^[70] Furthermore, conditions for well-posedness of general PWA systems appear to be very difficult to derive.^[37] Imura and Schaft^[35] provide necessary and sufficient conditions for bimodal systems to be well-posed, based on the lexicographic inequalities and the smooth continuity of solutions. Extensions to multi-modal cases have also been reported.^[32,35] Other important properties, such as controllability and observability, also lack of investigation. In fact, due to the declaration of NP-hardness,^[7] most researches on the controllability of discrete-time PWA systems either focus on developing efficient computational algorithms,^[44,45] or on deriving conditions for special/simplified models.^[67]

Many of these analysis and synthesis results above are related to the so-called Lyapunov functions, which also play a significant role in our study. Although CQLF has advantages in terms of simple structure and highly efficient computation, several facts make CQLF conservative. Firstly, CQLF does not allow affine terms in the dynamics so that simple systems such as those with saturation can not be analyzed.^[13,37] Secondly, it does not consider the partition information in the analysis, which leads to the problem that regional dynamics are forced into global dynamics. Thus several non-quadratic Lyapunov functions are proposed.^{[1,2,10,16,36,37,54]} Note that we use the term of non-quadratic Lyapunov functions to refer to all types of Lyapunov functions that are not commonly quadratic. PQLFs are widely employed in many recent researches,^{[1,2,16,37,54]} because they are relatively simple and highly efficient. There is no doubt that other types of non-quadratic Lyapunov functions are worth studying though they are more complex, such as bi-quadratic Lyapunov functions,^[4,65] parameterized Lyapunov functions^[52] and homogeneous Lyapunov functions.^[10,36] Exploring the advantages of these functions for PWA systems, therefore, becomes one of the topics of this book.

In addition, due to the amazing advances in computer technologies, it is very promising to develop analysis and design methods by using numerical computation. In the past years, linear matrix inequality (LMI)^[8]and bilinear matrix inequality (BMI)^[26,27] techniques have been widely applied in control engineering, which are also adopted in our research. A brief introduction to LMI and BMI is available in Ap...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright
• Dedication
• List of Figures
• List of Tables
• Preface
• Symbols and Acronyms
• 1. Introduction
• 2. Piecewise affine systems
• 3. Stability conditions based on PQLFs
• 4. Stability conditions based on SOS polynomials
• 5. Stability conditions based on vertex representation
• 6. Controllability and reachability
• 7. H∞ and generalized H2 controller design for PWA systems
• 8. H∞ and generalized H2 estimator design for PWA systems
• 9. A PWA approach to Takagi-Sugeno fuzzy logic systems
• 10. Control and estimation of mechanical systems
• Semidefinite programming
• Some proofs
• Index