Stability and stabilizability

Abstract

In this chapter stable linear systems are characterized in terms of associated characteristic polynomials and Lyapunov equations. A proof of the Routh theorem on stable polynomials is given as well as a complete description of completely stabilizable systems. Luenberger’s observer is introduced and used to illustrate the concept of detectability.

Bibliographical notes

   In the proof of the Routh theorem we basically follow F.R. Gantmacher [40]. There exist proofs which do not use analytic function theory. In particular, in P.C. Parks [75] the proof is based on Theorem 2.7 from §2.4. For numerous modifications of the Routh algorithm we refer to W.M. Wonham [101]. The proof of Theorem 2.9 is due to W.M. Wonham [100]. The Hautus lemma was discovered by V.M. Popov, see [79]. Its applicability was a subject of several papers by M.L.J. Hautus, see, e.g. [43], [44]. The main aim of §2.7 was to illustrate the concept of detectability.