Robust Stabilization of Uncertain Systems

Stabilization of uncertain systems is addressed in this paper. We consider a controlled system with bounded dynamics uncertainty, the evolution law of which can be written as a differential inclusion. In order to stabilize the system, we consider a Lyapunov candidate function, and look for controls within the set of controls satisfying a universal stability condition. In order to meet this constraint, we differentiate the control law and derive the differential inclusion governing the invariant evolution of the control law inside the set of stabilizing controls. We show that, under regularity assumptions, a set-valued evolution law of controls can be found explicitly. This enables us to select the minimal norm velocity of control satisfying the universal stability condition, and finally, to propose a single valued explicit control scheme devoted to robust stabilization of nonlinear systems.