Robust stabilization of nonlinear systems with pointwise norm-bounded uncertainties: a control Lyapunov function approach

The authors give a necessary and sufficient condition for globally stabilizing a nonlinear system, robustly with respect to unstructured uncertainties Φ˜(u,x,t), norm-bounded for each fixed x and u. This condition requires one to find a smooth, proper, and positive definite solution V(x) of a suitable partial differential inequality depending only on the system data. A procedure, based on the knowledge of V(x), is outlined for constructing almost smooth robustly stabilizing controllers. Our approach, based on Lyapunov functions, generalizes previous results for linear uncertain systems and establishes a precise connection between robust stabilization, on one hand, and H _∞-control sector conditions and input-to-state stabilization on the other.