${cal H}$ -Representation and Applications to Generalized Lyapunov Equations and Linear Stochastic Systems

Author

Zhang, Weihai ; Chen, Bor-Sen

Author_Institution

Qingdao Econ. & Tech. Dev. Zone, Shandong Univ. of Sci. & Technol., Qingdao, China

Volume

57

Issue

12

fYear

2012

Firstpage

3009

Lastpage

3022

Abstract

This paper introduces an H-representation method to express an n² × 1 vector X^→ as X^→=H[(X)tilde]. Based on the introduced H-representation approach, several topics are extensively discussed, including the generalized Lyapunov equations (GLEs) arising from stochastic control, stochastic observability, generalized D-stability and D-stabilization, weak stability, and stabilization. A necessary and sufficient condition for the existence and uniqueness of the symmetric and skew-symmetric solutions of GLEs is presented, respectively. Moreover, the solution structure of GLEs is also clarified. Through the H-representation method, several necessary and sufficient conditions are also obtained for stochastic observability, generalized D-stability and D-stabilization, weak stability, and stabilization.

Keywords

Lyapunov methods; linear systems; stability; stochastic systems; vectors; D-stabilization; GLE; H-representation method; Linear Stochastic Systems; generalized D-stability; generalized Lyapunov equations; skew-symmetric solutions; stochastic control; stochastic observability; symmetric solutions; weak stability; weak stabilization; Asymptotic stability; Equations; Observability; Stability criteria; Symmetric matrices; Vectors; ${cal H}$-representation; Complete observability; generalized ${cal D}$-stability and stabilization; generalized Lyapunov equations; weak stability and stabilization;

fLanguage

English

Journal_Title

Automatic Control, IEEE Transactions on

Publisher

ieee

ISSN

0018-9286

Type

jour

DOI

10.1109/TAC.2012.2197074

Filename

6193413