Nonlinear Control Sy.. - Free

4.6. PERTURBATION ANALYSIS 123
and related design issues is a difficult problem that has spurred a lot of research in recent
years. In our first look at this problem we consider a dynamical system of the form
i = f(x,t) + g(x, t) (4.33)
where g(x, t) is a perturbation term used to estimate the uncertainty between the true state
of the system and its estimate given by i = f (x, t). In our first pass we will seek for an
answer to the following question: Suppose that i = f (x, t) has an asymptotically stable
equilibrium point; what can it be said about the perturbed system ± = f (x, t) + g(x, t)?
Theorem 4.8 [41] Let x = 0 be an equilibrium point of the system (4.33) and assume that
there exist a differentiable function W(., ) : D x [0, oo) -p R such that
(i) k1IIxII2