The gain scheduling and the robust state feedback stabilization problems

We consider dynamical uncertain systems of the form x˙=a(x, w)+b(x, w)u where w(t)∈W is an unknown but bounded time-varying parameter. We consider the robust state feedback stabilization problem, in which we consider a control of the form u=Φ(x), and the gain-scheduling stabilization problem in which a control of the form u=Φ(x, w) (full information control) is admitted. We show that for convex processes, those systems in which for fixed x the set of all [a(x, w)b(x, w)] is convex, the two problems are equivalent; i.e., if there exists a (locally Lipschitz) gain scheduling stabilizing control then there exists a robustly stabilizing control (continuous everywhere except possibly at the origin). Thus, for convex process stabilization, knowledge of w(t) is not an advantage for the compensator. Then we consider the special class of polytopic LPV systems, and show that 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. We show that the existence of a dynamic controller always implies the existence of a static one. Finally we show that we can always find a linear gain-scheduling controller for a stabilizable system. This means that a possible advantage of the online measurement of the parameter w(t) is that this always allows for linear compensators, whose implementation can be easier than that of nonlinear ones.