Control Theory for Linear Systems

Abstract

PrefaceThis book originates from several editions of lecture notes that were used as teaching 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 period 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. Indeed, 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 recently, by the H 2 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 attracted by control design problems that require to make the influence of the exogenous disturbances on the to-be-controlled outputs equal to zero. The static state feedback 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 geometric (i.e., linear algebraic) properties of the coefficient matrices appearing in the system 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 tracking and regulation, etc. 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 control. 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...

Linear passive systems and maximal monotone mappings

“…[34]) which would return the minimal subspace in a finite number of steps for a given triple (A, V, W).…”

The role of the generalised continuous algebraic Riccati equation in impulse-free continuous-time singular LQ optimal control

“…Since Π = Π T ≥ 0 and H = H T ≥ 0, from [8, Corollary 2.4] we conclude that both (15) and (17) admit a unique solution defined in (−∞, T ]. It is easy to see that P T (t) = O O O P 22 (t) , where P 22 (t), t ∈ (−∞, T ], is the solution of (17-18), solves (15) and (16). We can therefore conclude that P T (t) is the unique solution of (15)(16).…”

Dissipativity preserving model reduction by retention of trajectories of minimal dissipation

“…Proof Let V * be the weakly unobservable subspace of (A, B, C, D) (see [22], Section 7.3). By [22] (A, B, C, D) ext , the pair (A, B) is controllable.…”