Singular Perturbation Methods in Control Theory

the usual theory of continuous dependance of the solutions with respect to the parameters cannot be applied. The analysis of such systems is achieved with the aid of the Singular Perturbation Theory. The purpose of Singular Perturbation Theory is to investigate the behaviour of solutions of (1) as e → 0 for 0 ≤ t ≤ T and also for 0 ≤ t < +∞. This chapter is organized as follows. In Section 2 we recall Tikhonov’s theorem on fast and slow systems, its extension to infinite time intervals, and Khalil’s theorem on exponential stability of the origin of a fast and slow system. In Section 2.4 we define the notion of practical stability in a system depending on a parameter and we show that the extension of Tikhonov’s theorem to infinite time intervals can be reformulated as a result of practical stability of the origin. In Section 3 we use the results of the preceeding section to reduce the dimension of systems in the problem of feedback stabilization. In Section 4 we discuss the peaking phenomenon in triangular systems of the form ẋ = f(x, y), ẏ = G(y, e). (2)