Strong stability of internal system descriptions

Abstract

This article introduces a new notion of stability for internal autonomous system descriptions that is referred to as ‘strong stability’ and characterises the case of avoiding overshoots for all initial conditions within a given sphere. This is a stronger notion of stability compared with alternative definitions (asymptotic, Lyapunov), which allows the analysis and design of control systems described by natural coordinates to have no overshooting response for arbitrary initial conditions. For the case of linear time-invariant systems necessary and sufficient conditions for strong stability are established in terms of the negative definiteness of the symmetric part of the state matrix. The invariance of strong stability under orthogonal transformations is established and this enables the characterisation of the property in terms of the invariants of the Schur form of the state matrix. Some interesting relations between strong stability and special forms of coordinate frames are examined and links between the skewness of the eigenframe and the violation of strong stability are derived. The characterisation of properties of strong stability given here provides the basis for the development of feedback theory for strong stabilisation.