Lyapunov-like equations and reachability/observabiliy Gramians for descriptor systems

Author

Bender, Douglas J.

Author_Institution

Hughes Aircraft Company, Los Angeles, CA, USA

Volume

Issue

fYear

1987

fDate

4/1/1987 12:00:00 AM

Firstpage

343

Lastpage

348

Abstract

Reachability and observability Gramians are defined for the descriptor systems $Ex_{k-1} = Ax_{k} + Bu_{k}, y_{k} = Cx_{k}$ and $E\\dot{x}(t) = Ax(t) + Bu(t), y(t) = Cx(t)$ . The pencils $(zE - A)$ and $(sE - A)$ are assumed to be regular. If $E$ is singular,these Gramians do not, in general, satisfy the "expected" Lyapunov-like (Sylvester) equations (for example, $EPA^{T} + APE^{T} = - BB^{T}$ if $P$ is the continuous-time teachability Gramian). This is shown by expressing the solution of the descriptor system in terms of the parameters of the Laurent expansion of $(zE - A)^{-1}$ or $(sE - A)^{-1}$ . We derive the Lyapunov-like equations which these Gramians do satisfy.

Keywords

Linear systems; Lyapunov matrix equations; Observability, linear systems; Singular systems; Control systems; Controllability; Eigenvalues and eigenfunctions; Equations; Military aircraft; Observability; Stability;

fLanguage

English

Journal_Title

Automatic Control, IEEE Transactions on

Publisher

ieee

ISSN

0018-9286

Type

jour

DOI

10.1109/TAC.1987.1104589

Filename

1104589

Link To Document

https://search.isc.ac/dl/search/defaultta.aspx?DTC=49&DC=855401