Smooth stabilization implies coprime factorization. (English) Zbl 0682.93045

Sontag, Eduardo D.

IEEE Trans. Autom. Control 34, No. 4, 435-443 (1989).

The main theorem states that if a nonlinear system \(\dot x=f(x)+G(x)u\) can be made globally asymptotically stable by a smooth feedback \(u=K(x)\), then there also exists a smooth feedback \(u=K'(x)+v\) such that the feedback modified system is input-to-state stable. The construction of \(K'(x)\) is given explicitly for feedback linearizable systems, which are trivially smoothly stabilizable. Based upon this main result it is also shown that smoothly stabilizable systems admit coprime factorizations. Finally some results about input-to-output stability are given.

Reviewer: A.van der Schaft

MSC:

93D15	Stabilization of systems by feedback
93B28	Operator-theoretic methods
47A68	Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
93C10	Nonlinear systems in control theory
93D25	Input-output approaches in control theory

Keywords:

globally asymptotically stable; input-to-state stable; smoothly stabilizable; coprime factorizations