Strong stability of discrete-time systems

The paper introduces a new notion of stability for internal autonomous system descriptions in discrete-time, referred to as ”strong stability”, which extends a parallel notion introduced in the continuous-time case. This is a stronger notion of stability compared to alternative definitions (asymptotic, Lyapunov), which prohibits systems described by natural coordinates to have overshooting responses for arbitrary initial conditions in state-space. Three finer notions of strong stability are introduced and necessary and sufficient conditions are established for each one of them. The invariance of strong stability under orthogonal transformations is also shown, and this enables the characterization of the property in terms of the invariants of the Schur form of the system’s state matrix. The class of discrete-time systems for which strong and asymptotic stability coincide is characterized and links between the skewness of the eigen-frame and the violation of strong stability property are obtained. Connections between the notions of strong stability in the continuous and discrete-domains are derived. Finally, as application, the strong stability property is studied in the context of balanced realizations, general similarity transformations and state/output-feedback stabilization problems. AMS classification: 15A18, 34A30, 65F15, 65F60