Averaging of linear systems with almost periodic coefficients: A time-delay approach

Abstract We study stability of linear systems with fast almost periodic coefficients that are piecewise-continuous in time. The classical averaging method guarantees the stability of such systems for small enough values of parameter provided the corresponding averaged system is stable. However, it is difficult to find an upper bound on the small parameter by using classical tools for asymptotic analysis. In this paper we introduce an efficient constructive method for finding an upper bound on the value of the small parameter that guarantees a desired exponential decay rate. We transform the system into a model with time-delays of the length of the small parameter. The resulting time-delay system is a perturbation of the averaged system. The averaged system is supposed to be exponentially stable. The stability of the time-delay system guarantees the stability of the original one. We construct an appropriate Lyapunov functional for finding sufficient stability conditions in the form of linear matrix inequalities (LMIs). The upper bound on the small parameter that preserves the exponential stability is found from the LMIs. Two numerical examples (stabilization by vibrational control and by time-dependent switching) illustrate the efficiency of the method. Moreover, we apply the time-delay approach to persistently excited systems that leads to a novel quadratic time-independent Lyapunov functional for such systems. We further extend our method to input-to-state stability (ISS) analysis. Finally the results are extended to linear fast-varying systems with time-varying delays.