The quadratic matrix inequality in singular $H_\infty$ control with state feedback

A.A. Stoorvogel, H.L. Trentelman

Research output: Book/ReportReportAcademic

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Abstract

In this paper we consider the standard $H_\infty$ control problem using state feedback. Given a linear, time-invariant, finite-dimensional system this problem consists of finding a static state feedback such that the resulting closed loop transfer matrix has $H_\infty$ norm smaller than some a priori given upper bound. In addition it is required that the closed loop system is internally stable. Conditions for the existence of a suitable state feedback are formulated in terms of a quadratic matrix inequality, reminiscent of the dissipation inequality of singular linear quadratic optimal control. In case that the direct feedthrough matrix of the control input is injective our results specialize to known results in terms of solvability of a certain indefinite algebraic Riccati equation. Keywords: $H_\infty$ control, state feedback, quadratic matrix inequality, strong controllability, almost disturbance decoupling.
Original languageEnglish
Place of PublicationEindhoven
PublisherTechnische Universiteit Eindhoven
Number of pages27
Publication statusPublished - 1989

Publication series

NameMemorandum COSOR
Volume8903
ISSN (Print)0926-4493

Fingerprint

State Feedback
Matrix Inequality
Algebraic Riccati Equation
State Feedback Control
Transfer Matrix
Decoupling
Injective
Controllability
Closed-loop
Closed-loop System
Solvability
Linear Time
Dissipation
Control Problem
Optimal Control
Disturbance
Upper bound
Norm
Invariant

Cite this

Stoorvogel, A. A., & Trentelman, H. L. (1989). The quadratic matrix inequality in singular $H_\infty$ control with state feedback. (Memorandum COSOR; Vol. 8903). Eindhoven: Technische Universiteit Eindhoven.
Stoorvogel, A.A. ; Trentelman, H.L. / The quadratic matrix inequality in singular $H_\infty$ control with state feedback. Eindhoven : Technische Universiteit Eindhoven, 1989. 27 p. (Memorandum COSOR).
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Stoorvogel, AA & Trentelman, HL 1989, The quadratic matrix inequality in singular $H_\infty$ control with state feedback. Memorandum COSOR, vol. 8903, Technische Universiteit Eindhoven, Eindhoven.

The quadratic matrix inequality in singular $H_\infty$ control with state feedback. / Stoorvogel, A.A.; Trentelman, H.L.

Eindhoven : Technische Universiteit Eindhoven, 1989. 27 p. (Memorandum COSOR; Vol. 8903).

Research output: Book/ReportReportAcademic

TY - BOOK

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AU - Trentelman, H.L.

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AB - In this paper we consider the standard $H_\infty$ control problem using state feedback. Given a linear, time-invariant, finite-dimensional system this problem consists of finding a static state feedback such that the resulting closed loop transfer matrix has $H_\infty$ norm smaller than some a priori given upper bound. In addition it is required that the closed loop system is internally stable. Conditions for the existence of a suitable state feedback are formulated in terms of a quadratic matrix inequality, reminiscent of the dissipation inequality of singular linear quadratic optimal control. In case that the direct feedthrough matrix of the control input is injective our results specialize to known results in terms of solvability of a certain indefinite algebraic Riccati equation. Keywords: $H_\infty$ control, state feedback, quadratic matrix inequality, strong controllability, almost disturbance decoupling.

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Stoorvogel AA, Trentelman HL. The quadratic matrix inequality in singular $H_\infty$ control with state feedback. Eindhoven: Technische Universiteit Eindhoven, 1989. 27 p. (Memorandum COSOR).