D. Hinrichsen

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.

Stability radius for structured perturbations and the algebraic Riccati equation

Abstract In this paper we generalize the notion of stability radius introduced in [1] to allow for structured perturbations. We then relate the stability radius to the existence of Hermitian solutions of an algebraic Riccati equation and give some applications of this result.

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

Exponential estimates for time delay systems

Abstract In this paper, we demonstrate how Lyapunov–Krasovskii functionals can be used to obtain exponential bounds for the solutions of time-invariant linear delay systems.

An improved error estimate for reduced-order models of discrete-time systems

The authors derive, under weaker conditions, a discrete-time counterpart of K. Glovers (1984) error estimate for reduced-order models. Following the same lines, a simple proof of Glovers result for continuous-time systems can be given. >

Spectral value sets: a graphical tool for robustness analysis

In this note we present a method to determine and visualize the set of all complex numbers to which at least one eigenvalue of a matrix A can be shifted by real or complex perturbations of the form A→A(Δ) = A + DΔE, where D and E are fixed matrices and the unknown disturbance matrix Δ satisfies |Δ| 0. Both real and complex perturbations are studied. The graphical method may be considered as an extension of the classical root locus method to multivariable systems and multiparameter perturbations. The results are illustrated by various examples.

New robustness results for linear systems under real perturbations

Structured real perturbations of the system matrix A to A+ Delta A=BDC (with B, C given matrices) are considered. Robustness measures with respect to arbitrary stability domains C/sub g/ contained in/implied by C are introduced and characterized. These general formulas provide insight but are difficult to evaluate. Computable formulas are obtained when B is a column or C a row vector.<<ETX>>

Stability radii of higher order positive difference systems

In this paper we study stability radii of positive polynomial matrices under affine perturbations of the coefficient matrices. It is shown that the real and complex stability radii coincide. Moreover, explicit formulas are derived for these stability radii and illustrated by some examples.

Newton's method for a rational matrix equation occurring in stochastic control

We study a general class of rational matrix equations, which contains the continuous (CARE) and discrete (DARE) algebraic Riccati equations as special cases. Equations of this type were encountered in [SIAM J. Control and Optimization 36 (1998) 1504–1538; Stochastics and Stochastics Reports, 65 (1999) 255–297], where H∞-type problems of disturbance attenuation for stochastic linear systems were studied. We develop a unifying framework for the analysis of these equations based on the theory of (resolvent) positive operators and show that they can be solved by Newtons method starting at an arbitrary stabilizing matrix.