Relative Stability of Linear Dynamic Systems with Bounded Uncertain Coefficients

Abstract

In this paper, the stability problems, of linear dynamic systems with bounded uncertain coefficients or nonlinear time-varying elements are investigated in the time domain relating to the transition matrices of the nominal systems. Sufficient conditions for the relative stability of such systems are given in L₁ and L_∞ spaces, respectively.
It is shown that these two conditions are equivalent and described by a simpler form when the nominal systems are time-invariant.
The method of analysis is as follows. First, integral inequality systems are given from the integral equations, which themselves are obtained from the state equations, and then the conditions of boundedness of norms are derived by using the properties of the M-matrix. The principle of this idea is quite simple. But the stability criteria developed here will be perhaps more favourable than any other results which have been already obtained for nonhomogeneous systems involving nonlinear time-invariant ones.