Global Uniform Asymptotic Stability of Cascaded Non-autonomous Non-linear Systems: Application to Stabilisation of a Diesel Engine

In this paper we deal with the stability analysis problem of cascaded non autonomous nonlinear systems. In particular we answer to the following questions: (i) What happens with the solutions of a time-varying nonlinear system which is globally uniformly stable, when it is perturbed by the output of a globally exponentially stable system (GES), in particular, when both systems form a cascade?. (ii) If a time varying nonlinear system is globally uniformly asymptotically stable (GUAS), is this stability property preserved when it is perturbed by an exponentially decaying input?. Our proofs are based on a standard \delta-epsilon" Lyapunov analysis. Finally, we show the utility of our results by applying our theorems to the problem of stabilisation of a turbo charged diesel engine. Notation. In this paper the solution of a diierential equation _ x = f(t; x) where f : IR 0 IR n ! IR n , with initial conditions 1 (t 0 ; x 0) 2 IR 0 IR n , x(t 0) = x 0 , is denoted x(t; t 0 ; x 0) or simply x(t). We say that the system _ x = f(t; x), is uniformly stable (resp. GUAS, GES) if the trivial solution x(t) = 0 is uniformly stable (resp. GUAS, GES). A continuous function : IR 0 ! IR 0 is said to be of class K, 2 K, if (x) is strictly increasing and (0) = 0; 2 K 1 if in addition (x) ! 1 as x ! 1. A continuous function (t; x) : IR 0 IR 0 ! IR 0 is of class KL if (t;) 2 K for each xed t 0 and (t; x) ! 0 as t ! 1 for each x 0. We use n 4 = 1; : : :; n]. kkk stands for the usual Euclidean norm of vectors. _ V (#) (t; x) is the time derivative of Lyapunov function V (t; x) along the solutions of the diierential equation (#). 1 For the sake of simplicity we consider, as 24], that the solutions x(t; t0; x0) are deened for all t0 0 instead of for all t0 ?1, as considered by other authors.