Control theory for linear systems Harry L. Trentelman Research Institute of Mathematics and Computer Science University of Groningen P.O. Box 800, 9700 AV Groningen The Netherlands Tel. +31-50-3633998 Fax. +31-50-3633976 E-mail. [email protected] Anton A. Stoorvogel Dept. of Mathematics and Computing Science Eindhoven Univ. of Technology P.O. Box 513, 5600 MB Eindhoven The Netherlands Tel. +31-40-2472378 Fax. +31-40-2442489 E-mail. [email protected] Malo Hautus Dept. of Mathematics and Computing Science Eindhoven Univ. of Technology P.O. Box 513, 5600 MB Eindhoven The Netherlands Tel. +31-40-2472628 Fax. +31-40-2442489 E-mail. [email protected] May 15, 2002 2 Preface This book originates from several editions of lecture notes that were used as teach- ing material for the course ‘Control Theory for Linear Systems’, given within the framework of the national Dutch graduate school of systems and control, in the pe- riod from 1987 to 1999. The aim of this course is to provide an extensive treatment of the theory of feedback control design for linear, finite-dimensional, time-invariant state space systems with inputs and outputs. One of the important themes of control is the design of controllers that, while achieving an internally stable closed system, make the influence of certain exogenous disturbance inputs on given to-be-controlled output variables as small as possible. In- deed, in the appropriate sense this theme is covered by the classical linear quadratic regulator problem and the linear quadratic Gaussian problem, as well as, more re- cently, by the H2 and H∞ control problems. Most of the research efforts on the linear quadratic regulator problem and the linear quadratic Gaussian problem took place in the period up to 1975, whereas in particular H ∞ control has been the important issue in the most recent period, starting around 1985. In, roughly, the intermediate period, from 1970 to 1985, much attention was at- tracted by control design problems that require to make the influence of the exoge- nous disturbances on the to-be-controlled outputs equal to zero. The static state feed- back versions of these control design problems, often called disturbance decoupling, or disturbance localization, problems were treated in the classical textbook ‘Linear Multivariable Control: A Geometric Approach’, by W.M. Wonham. Around 1980, a complete theory on the disturbance decoupling problem by dynamic measurement feedback became available. A central role in this theory is played by the geomet- ric (i.e., linear algebraic) properties of the coefficient matrices appearing in the sys- tem equations. In particular, the notions of (A, B)-invariant subspace and (C, A)- invariant subspace play an important role. These notions, and their generalizations, also turned out to be central in understanding and classifying the ‘fine structure’ of the system under consideration. For example, important dynamic properties such as system invertibility, strong observability, strong detectability, the minimum phase property, output stabilizability, etc., can be characterized in terms of these geometric concepts. The notions of (A, B)-invariance and (C, A)-invariance also turned out to be instrumental in other synthesis problems, like observer design, problems of track- ing and regulation, etc. vi Preface In this book, we will treat both the ‘pre-1975’ approach represented by the linear quadratic regulator problem and the H 2 control problem, as well as the ‘post-1985’ approach represented by the H∞ control problem and its applications to robust con- trol. However, we feel that a textbook dedicated to control theory for linear state space systems should also contain the central issues of the ‘geometric approach’, namely a treatment of the disturbance decoupling problem by dynamic measurement feedback, and the geometric concepts around this synthesis problem. Our motivation for this is three-fold. Firstly, in a context of making the influence of the exogenous disturbances on the to-be-controlled outputs as small as possible, it is natural to ask first under what conditions on the plant this influence can actually be made to vanish, i.e., under what conditions the closed loop transfer matrix can made zero by choosing an appropriate controller. Secondly, as also mentioned above, the notions of controlled invariance and con- ditioned invariance, and their generalizations of weakly unobservable subspace and strongly reachable subspace, play a very important role in studying the dynamic prop- erties of the system. As an example, the system property of strong observability holds if and only if the system coefficient matrices have the geometric property that the as- sociated weakly unobservable subspace is equal to zero. As another example, the system property of left-invertibility holds if and only if the intersection of the weakly unobservable subspace and the strongly reachable subspace is equal to zero. Also, the important notions of system transmission polynomials and system zeros can be given an interpretation in terms of the weakly unobservable subspace, etc. In other words, a good understanding of the fine, structural, dynamic properties of the system goes hand in hand with an understanding of the basic geometric properties associated with the system parameter matrices. Thirdly, also in the linear quadratic regulator problem, in the H 2 control problem, and in the H∞ control problem, the idea of disturbance decoupling and its associ- ated geometric concepts play an important role. For example, the notion of output stabilizability, and the associated output stabilizable subspace of the system, turn out to be relevant in establishing necessary and sufficient conditions for the existence of a positive semi-definite solution of the LQ algebraic Riccati equation. Also, by an appropriate transformation of the system parameter matrices, the H 2 control problem can be transformed into a disturbance decoupling problem. In fact, any controller that achieves disturbance decoupling for the transformed system turns out to be an optimal controller for the original H2 problem. The same holds for the H∞ con- trol problem: by an appropriate transformation of the system parameter matrices, the original problem of making the H∞ norm of the closed loop transfer matrix strictly less than a given tolerance, is transformed into a disturbance decoupling problem. Any controller that achieves disturbance decoupling for the transformed system turns out to achieve the required strict upper bound on H ∞-norm of the closed loop transfer matrix. The outline of this book is as follows. After a general introduction in chapter 1, and a summary of the mathematical prerequisites in chapter 2, chapter 3 of this book Preface vii deals with the basic material on linear state space systems. We review controllability and observability, the notions of controllable eigenvalues and observable eigenvalues, and basis transformations in state space. Then we treat the problem of stabilization by dynamic measurement feedback. As intermediate steps in this synthesis problem, we discuss state observers, detectability, the problem of pole placement by static state feedback, and the notion of stabilizability. The central issue of chapters 4 to 6 is the problem of disturbance decoupling by dynamic measurement feedback. First, in chapter 4, we introduce the notion of con- trolled invariance, or (A, B)-invariance. As an immediate application, we treat the problem of disturbance decoupling by static state feedback. Next, we introduce con- trollability subspaces, and stabilizability subspaces. These are used to treat the static state feedback versions of the disturbance decoupling problem with internal stability, and the problem of external stabilization. In chapter 5, we introduce the central notion of conditioned invariance, or (C, A)-invariance. Next, we discuss detectability sub- spaces, and their application to the problem of designing estimators in the presence of external disturbances. In chapter 6, we combine the notions of controlled invari- ance and conditioned invariance into the notion of (C, A, B)-pair of subspaces. As an immediate, straightforward, application we treat the dynamic measurement feedback version of the disturbance decoupling problem. Next, we take stability issues into consideration, and consider (C, A, B)-pairs of subspaces consisting of a detectability subspace and a stabilizability subspace. This structure is applied to resolve the dy- namic measurement feedback version of the problem of disturbance decoupling with internal stability. The final subject of chapter 6 is the application of the idea of pairs of (C, A, B)-pairs to the problem of external stabilization by dynamic measurement feedback. Chapters 7 and 8 of this book deal with system structure. In chapter 7, we first give a review of some basic material on polynomial matrices, elementary operations, Smith form, and left- and right-unimodularity. Then we introduce the notions of transmission polynomials and zeros, in terms of the system matrix associated with the system. We then discuss the weakly unobservable subspace, and the related no- tion of strong observability, and finally give a characterization of the transmission polynomials and zeros in terms of a linear map associated with the weakly unobserv- able subspace. In chapter 8 we discuss the idea of distributions as inputs. Allowing distributions (instead of just functions) as inputs gives rise to some new concepts in state space, such as the strongly reachable subspace and the distributionally weakly unobservable subspace. The notions of system left- and right-invertibility are intro- duced, and characterized in terms of these new subspaces. The basic material on distributions that is used in chapter 8 is treated in appendix A of this book. In chapter 9 we treat the problem of tracking and regulation. In this problem, certain variables of the plant are required to track an a priori given signal, regardless of the disturbance input and the initial state of the plant. Both the signal to be tracked as well as the disturbance input are modeled as being generated by an additional finite-dimensional linear system, called the exosystem.