The gain scheduling and the robust state feedback stabilization problems

In this paper we consider dynamical uncertain systems of the form (x)over dot = a (x, w) + b(x, w)u where w(t) is an element of W is an unknown but bounded uncertain time-vary ing parameter. For these systems we consider two problems: the robust state feedback stabilization problem, in which we consider a control of the form u = Phi (a), and the gain-scheduling stabilization problem in which a cont rol of the form u = Phi (x, w) (often referred to as full information contr ol) is admitted, We show that for convex processes, namely those systems in which for fixed a: the set of all [a(x, w)b(x, w)], w(t) is an element of W is convex (including the class of convex linear parameter varying (LPV) s ystems as special case) the two problems are equivalent. We mean that if th ere exists a (locally Lipschitz) gain scheduling stabilizing control then t here exists a robustly stabilizing control (which is continuous everywhere possibly except at the origin). In few words, for convex processes, as far as it concerns stabilization capability, the knowledge of w(t) is not an ad vantage for the compensator. Then we consider the special class of polytopic LPV systems, and we show th at there is no loss of regularity as in the general case, if we pass from a gain-scheduling controller to a state feedback controller. In particular, no discontinuity at the origin may occur. Then we show that the existence o f a dynamic controller always implies the existence of a static one. Finally we show that, differently from the robust stabilization problem in which it is known that nonlinear controllers can outperform linear (even dy namic) ones, we can always find a linear gain-scheduling controller for a s tabilizable system, This means that a possible advantage of the online meas urement of the parameter w(t) is that this always allows for linear compens ators, whose implementation can be easier than that of nonlinear ones.