Input–output-to-state stability for discrete-time systems

• Published in: Automatica (Oxford) Vol. 44; no. 2; pp. 326 - 336
• Main Authors: Cai, Chaohong, Teel, Andrew R.
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.02.2008; Elsevier
• Subjects: Applied sciences; Computer science; control theory; systems; Control system analysis; Control theory. Systems; Discrete-time systems; Exact sciences and technology; Lyapunov functions; Modelling and identification; Norm estimators; Stability; Discrete-time systems; Stability; Lyapunov functions; Norm estimators; Non linear control; Optimization; Input to state stability; Smooth function; Observer; Discrete time; System identification; State estimation; Lyapunov function
• ISSN: 0005-1098; 1873-2836
• DOI: 10.1016/j.automatica.2007.05.022

This paper presents Lyapunov characterizations of uniform output-to-state stability and uniform input–output-to-state stability (IOSS) (with respect to disturbances) for discrete-time (DT) nonlinear systems. We show the equivalence of the following three properties for DT systems: uniform IOSS, existence of a smooth Lyapunov function for uniform IOSS, and existence of a (state-)norm estimator. This equivalence result is a DT counterpart of Krichman et al. [2001. Input–output-to-state stability. SIAM Journal on Control and Optimization 39, 1874–1928. Theorem 2.4] and a generalization of Jiang and Wang [2002. A converse Lyapunov theorem for discrete-time systems with disturbances. Systems and Control Letters 45, 49–58. Theorem 1.1] and Jiand and Wang [2001. Input-to-state stability for discrete-time nonlinear systems. Automatica 37, 857–869. Theorem 1, 1 ⇔ 4 ].