On common quadratic Lyapunov functions for pairs of stable LTI systems whose system matrices are in companion form

Author

Shorten, Robert N. ; Narendra, Kumpati S.

Author_Institution

Hamilton Inst., Nat. Univ. of Ireland, Maynooth, Ireland

Volume

48

Issue

4

fYear

2003

fDate

4/1/2003 12:00:00 AM

Firstpage

618

Lastpage

621

Abstract

In this note, the problem of determining necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for a pair of stable linear time-invariant systems whose system matrices A₁ and A₂ are in companion form is considered. It is shown that a necessary and sufficient condition for the existence of such a function is that the matrix product A₁A₂ does not have an eigenvalue that is real and negative. Examples are presented to illustrate the result.

Keywords

Lyapunov methods; linear systems; matrix algebra; stability; time-varying systems; common quadratic Lyapunov functions; necessary and sufficient conditions; positive-definite real symmetric matrix; quadratic stability; stable LTI systems; stable linear time-invariant systems; switched linear systems; Eigenvalues and eigenfunctions; Linear algebra; Linear matrix inequalities; Linear systems; Lyapunov method; Polynomials; Stability; Sufficient conditions; Symmetric matrices; Time domain analysis;

fLanguage

English

Journal_Title

Automatic Control, IEEE Transactions on

Publisher

ieee

ISSN

0018-9286

Type

jour

DOI

10.1109/TAC.2003.809795

Filename

1193741