Real and Complex Stability Radii: A Survey

Hinrichsen, D.; Pritchard, A. J.

doi:10.1007/978-1-4757-2108-9_7

Cited by 153 publications

(100 citation statements)

References 18 publications

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“…Our proof of theorem 12.4 worked for simplicity with a static and complex uncertainty system. The effect of allowing for dynamic, time-varying or nonlinear and the difference between real and complex uncertainty systems are described in [79] by Hinrichsen and Pritchard.…”

Section: Notes and Referencesmentioning

confidence: 99%

Control Theory for Linear Systems

Trentelman

Stoorvogel

Hautus

2001

Communications and Control Engineering

490

462

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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...

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Section: Notes and Referencesmentioning

confidence: 99%

Control Theory for Linear Systems

Trentelman

Stoorvogel

Hautus

2001

Communications and Control Engineering

490

462

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“…However, these results and techniques are not sufficient to uncover phenomena like those described by the above theorems. Perturbed Toeplitz matrices arise in a variety of settings in applied mathematics and physics, including nonHermitian quantum mechanics [8,10,15,29] (also see [9]), population biology [23], linear systems theory [17,18], small world networks [21], and eigenvalue perturbation theory for general matrices [22]. In particular, single-entry perturbations are discussed in [10,21].…”

Section: Thus Sp (Jk) ωmentioning

confidence: 99%

The spectra of large Toeplitz band matrices with a randomly perturbed entry

2003

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Abstract. This paper is concerned with the union sp (j,k) Ω Tn(a) of all possible spectra that may emerge when perturbing a large n × n Toeplitz band matrix Tn(a) in the (j, k) site by a number randomly chosen from some set Ω. The main results give descriptive bounds and, in several interesting situations, even provide complete identifications of the limit of spTn(a) as n → ∞. Also discussed are the cases of small and large sets Ω as well as the "discontinuity of the infinite volume case", which means that in general sp

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“…Let e A : {(t, τ ) ∈ T × T : t ≥ τ } → K d×d denote the transition matrix corresponding to (3), that is, x(t) = e A (t, τ )ξ solves the initial value problem (3) with initial condition x(τ ) = ξ for ξ ∈ K d and t, τ ∈ T with t ≥ τ . The subsequent notions are recalled from [9].…”

Section: Preliminariesmentioning

confidence: 99%

“…We also study the case of structured perturbations A A + B∆C for given structure operators B and C. For continuous-or discrete-time systems stability radii are well-investigated notions, see [4]. A discussion on the differences between the complex and the real stability radius can be found in [3]. The complex stability radius is more easily analysed and computed than the real one.…”

Section: Introductionmentioning

confidence: 99%