Control Effort Considerations in the Stabilization of Uncertain Dynamical Systems

This paper investigates the problem of designing a state feedback control to stabilize an uncertain nonlinear system. We focus attention on the amplitude (norm) of the controller which is used to achieve this end. The uncertain system is described by a state equation which contains uncertain parameters which are unknown but bounded. A Lyapunov function is used to establish the stability of the closed loop system. The paper gives necessary and sufficient conditions for the uncertain system to be stabilizable with a given Lyapunov function. Furthermore, a procedure is indicated for the construction of the desired feedback control law. The control scheme presented is referred to as the minimum effort control. This terminology refers to the fact that at each state and time, the resulting control vector is the smallest allowable control vector which will ensure the desired degree of negativity of the Lyapunov derivative. Implementation of this control scheme requires the repeated solution of a mathematical program.