Two singular value inequalities and their implications in H∞approach to control system design

Author

Foo, Yung Kuan

Author_Institution

Nanyang Technological Institute, Singapore, Republic of Singapore

Volume

Issue

fYear

1987

fDate

2/1/1987 12:00:00 AM

Firstpage

156

Lastpage

157

Abstract

In this note we prove that if $A$ and $B$ are both nonnegative definite Hermitian matrices and $A - B$ is also nonnegative definite, then the singular values of A and B satisfy the inequalities $\\sigma _{i}(A)\\geq \\sigma _{i}(B)$ , where $\\bar{\\sigma}(\\cdot) = \\sigma_{1}(\\cdot) \\geq \\sigma_{2}(\\cdot) \\geq \\cdots \\geq \\sigma_{m}(\\cdot) = \\underline{\\sigma}(\\cdot)$ denote the singular values of a matrix. A consequence of this property is that, in a nonsquare H^{infty} optimization problem, if $\\sup_{\\omega } \\bar{\\sigma }[Z(j\\sigma )] {\\underline {\\underline \\Delta }} \\sup_{\\omega } \\bar{\\sigma }[x(j\\omega )^{T}/ Y(j\\omega )^{T}]^{T} = \\lambda$ , then the singular values of $X$ and $Y$ satisfy the inequality $\\lambda ^{2} \\geq \\max _{i} \\sup_{\\omega } [\\sigma _{i}^{2}(X) + \\sigma _{m-i-1}^{2}(Y)]$ where $m$ is the number of columns of the matrix $Z$ .

Keywords

H∞ optimization; Linear systems; Minimax control, linear systems; Control systems; Frequency; Functional analysis; Linear matrix inequalities; Minimax techniques; Transfer functions;

fLanguage

English

Journal_Title

Automatic Control, IEEE Transactions on

Publisher

ieee

ISSN

0018-9286

Type

jour

DOI

10.1109/TAC.1987.1104529

Filename

1104529

Link To Document

https://search.isc.ac/dl/search/defaultta.aspx?DTC=49&DC=854781