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Stability tests for constrained linear systems. (English) Zbl 0997.93086
Moheimani, S. O. Reza (ed.), Perspectives in robust control. Papers of the robust control workshop, Newcastle, Australia, December 6-8, 2000. London: Springer. Lect. Notes Control Inf. Sci. 268, 241-257 (2001).

The publication offers stability tests that extend some standard tests for linear time-invariant systems. A standard approach to verifying the stability of $\stackrel{˙}{x}=Ax$ is to look for a symmetric positive definite matrix $P$ that satisfies the equation ${A}^{T}P+PA<0$ where $B stands for $C-B$ being positive definite. The extension offered in the paper is the existence of a symmetric positive definite matrix $P$ such that

$\left[\begin{array}{c}{A}^{T}{F}^{T}+FA\\ GA-{F}^{T}+P\end{array}\begin{array}{c}{A}^{T}{G}^{T}-F+P\\ -G-{G}^{T}\end{array}\right]<0$

for some matrices $F$ and $G$. Although the new inequalities concern an enlarged system, the additional free parameters may enable an easier location of the matrix $P$. The claim is based on the Finsler lemma. Some variants of the above mentioned test are also displayed and adaptations are offered covering cases like discrete-time systems, input output relations and systems described by transfer functions. The method is described and explained very clearly, although concrete examples for using the technique are not offered. Further possibilities for applying the methodology are pointed out.

MSC:
 93D20 Asymptotic stability of control systems 93D15 Stabilization of systems by feedback 93D25 Input-output approaches to stability of control systems