The gain scheduling and the robust state feedback stabilization problems

In this paper we consider dynamical uncertain systems of the form x˙=a(x,w)+b(x,w)u, where w(t) ε W is a unknown-but bounded uncertain time-varying parameter. For these systems we consider two problems: the robust state feedback stabilization problem, in which we consider a control of the form u=Φ(x), and the gain-scheduling (often referred to as full information) stabilization problem in which a control of the form u=Φ(x,w) is admitted. We show that for convex processes, namely those systems in which for fixed x the set of all [a(x,w) b(x,w)], w(t) ε W is convex (including the class of linear parameter varying (LPV) systems as special case) the two problems are equivalent. We mean that if there exists a (Locally Lipschitz) gain scheduling stabilizing control then there 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 x(t) is not an advantage for the compensator. Then we consider the special class of LPV systems for which it is known that in the robust stabilization problem nonlinear controllers can outperform linear (even dynamic) ones. We show that for the gain scheduling stabilization problem the situation is different since we can always find a linear gain-scheduling controller for a stabilizable system. This means that a possible advantage of the on-line measurement of the parameter x(t) is that this always allows for linear compensators, whose implementation can be easier than that of nonlinear ones