Methods for stability evaluation for linear time varying, discrete-time systems on finite time horizon

Abstract

This paper develops a method for determining stability of discrete-time (DT), linear time-varying (LTV) systems defined on a finite time horizon (FTH). In our considerations we use a collection of stability definitions for linear time invariant (LTI) and LTV systems which are defined on an infinite time horizon (ITH). Based on the analysis carried out and the introduced operator-matrix notation we define four stability functions. These functions allow examination of the stability of a system described by an LTV state space model. Moreover, using these functions we introduce the time stability margin and making use of the operator-matrix notation we propose a method of determining the stability on a finite time horizon. The theoretical considerations are numerically verified on two examples of LTV systems with a variable degree of non-stationarity depending on a parameter ε. Several examples illustrate the application of the introduced concepts and definitions to examination of the system stability also depending on parameter ε.