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

A.A. Stoorvogel, H.L. Trentelman

### 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 language English Eindhoven Technische Universiteit Eindhoven 27 Published - 1989

### Publication series

Name Memorandum COSOR 8903 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).
title = "The quadratic matrix inequality in singular $H_\infty$ control with state feedback",
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.",
author = "A.A. Stoorvogel and H.L. Trentelman",
year = "1989",
language = "English",
series = "Memorandum COSOR",
publisher = "Technische Universiteit Eindhoven",

}

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).

TY - BOOK

T1 - The quadratic matrix inequality in singular $H_\infty$ control with state feedback

AU - Stoorvogel, A.A.

AU - Trentelman, H.L.

PY - 1989

Y1 - 1989

N2 - 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.

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.

M3 - Report

T3 - Memorandum COSOR

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

PB - Technische Universiteit Eindhoven

CY - Eindhoven

ER -

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).