A.J. Pritchard

Stability radii of linear systems

Abstract In this note the concepts of real and complex stability radii are introduced and some of their properties discussed.

Mathematical Systems Theory I

The state–space approach to control-systems theory seems to have originated in the 1950s with the study of necessary conditions for the existence of optimal controls. Viewed as a natural outgrowth of the calculus of variations—but with more problematic constraints—optimal control theory received widespread recognition with the publication of [6]. Contemporaneously with these initial investigations into optimal control theory, engineers were developing (sometimes ad-hoc) techniques for the design of control systems based on the input–output (or “frequency domain”) approach to the modeling of physical systems. The first attempt to reconcile the state–space approach with the input–output approach is generally attributed to R. E. Kalman as set forth in his seminal papers [2] and [3], which appeared in the early 1960s. Particularly significant was Kalman’s enunciation of the axiomatic definition of a (controlled) dynamical system in [3]. From these fruitful beginnings, research in controlled dynamical systems has experienced explosive growth in the intervening 40-plus years, resulting in a mature and well developed intellectual discipline with myriad and wide-ranging applications. Accompanying the maturation of the discipline is the increasing availability of monographs, textbooks, and research journals that specialize in controlled dynamical systems. Books on the subject are now available for audiences with widely diverse backgrounds, interests, and levels of mathematical preparation. As a result, reviewers of books in the subject area are obliged to place books in the context of a ever expanding sphere of literature. The ensuing discussion will therefore address not only the contents of the book under review, but also how it relates to a (small) sample of other existing books with similar objectives.

H∞-type control for discrete-time stochastic systems

In this paper we consider discrete-time, linear stochastic systems with random state and input matrices which are subjected to stochastic disturbances and controlled by dynamic output feedback. The aim is to develop an H∞-type theory for such systems. For this class of systems a stochastic bounded real lemma is derived which provides the basis for a linear matrix inequality (LMI) approach similar to, but more general than the one presented in Reference 1 for stochastic differential systems. Necessary and sufficient conditions are derived for the existence of a stabilizing controller which reduces the norm of the closed-loop perturbation operator to a level below a given threshold γ. These conditions take the form of coupled nonlinear matrix inequalities. In the absence of the stochastic terms they get reduced to the linear matrix inequalities of deterministic H∞-theory for discrete time systems. Copyright

The linear quadratic control problem for infinite dimensional systems with unbounded input and output operators

This paper establishes a general semigroup framework for solving quadratic control problems with infinite dimensional state space and unbounded input and output operators.

Sensors and actuators in distributed systems

The concepts of strategic sensors and actuators are introduced for a class of distributed parameter systems. This emphasizes the spatial structure and location of the sensors and controls in order that observability and controllability can be achieved.

Robustness of linear systems

Abstract In this paper we consider linear systems which are stable and examine the robustness of this property. The perturbations are assumed to have unbounded structure and we determine the smallest perturbation which destroys stability. We begin with an abstract analysis of the problem, but in later sections specialize to three types of systems: those whose mild solutions are given in terms of 1. (a) strongly continuous semigroups, 2. (b) resolvent operators-integrodifferential systems, 3. (c) evolution operators-time varying systems. For B a Banach space, L2(t0, T; B) denotes the space of all square integrable functions defined on [t0, T] with values in B. For B1, B2 Banach spaces, [t0, T] denotes the space of strongly measurable functions f(·):[t0, T]→L(B1, B2) with ess sup ∥f(·)∥⩽∞.B−1(t0, ∞;L(B1, B2)) denotes the space of strongly measurable functions f(·): [t0, ∞) → L(B1, B2) with ∥f(t)∥ 0 and k(·) ϵ L1(t0, ∞).

Regional controllability of distributed parameter systems

The purpose of this paper is to show how one can achieve regional controllability for distributed systems. First, we give a definition and some properties of this new concept, then we concentrate on the determination of a control achieving regional controllability with minimum energy. A direct method is developed and is illustrated by a hyperbolic system with point control and a parabolic system with zone control. We also give an important remark that converts the problem into a constrained optimization one.

A note on some differences between real and complex stability radii

Abstract Real and complex stability radii measure the robustness of stable linear systems under real and complex parameter perturbations, respectively. In this note we point out some basic differences between them. In particular, we investigate their different behaviour with respect to system interconnections and changes in system data.

Stability radii of linear systems with respect to stochastic perturbations

Abstract Real and complex stability radii for stable linear systems with respect to structured stochastic multiperturbations are introduced and completely characterized.

Brief Stability radii of discrete-time stochastic systems with respect to blockdiagonal perturbations

We consider stochastic discrete-time systems with multiplicative noise which are controlled by dynamic output feedback and subjected to blockdiagonal stochastic parameter perturbations. Stability radii for these systems are characterized via scaling techniques and it is shown that for real data, the real and the complex stability radii coincide. In a second part of the paper we investigate the problem of maximizing the stability radii by dynamic output feedback. Necessary and sufficient conditions are derived for the existence of a stabilizing compensator which ensures that the stability radius is above a prespecified level. These conditions consist of parametrized matrix inequalities and a coupling condition.