Control Lyapunov Functions :

A control design method for nonlinear systems based on control Lyapunov functions and inverse optimality is analyzed. This method is shown to recover the LQ optimal control when applied to linear systems. More generally, it is shown to recover the optimal control whenever the level sets of the control Lyapunov function match those of the optimal value function. The method can be readily applied to feedback linearizable systems, and the resulting inverse optimal control law is generally much diierent from the linearizing control law. Examples in two dimensions are given to illustrate both the strengths and the weaknesses of the method. 1 Control Lyapunov functions We consider single-input, control-aane nonlinear systems of the form _ x = f(x) + g(x) u (1) where x 2 IR n is the state vector, u 2 IR is the control input, and f and g are known continuous functions. Our goal is to construct a continuous state feedback law u = k(x) such that x = 0 is a globally asymptot-ically stable equilibrium point of the resulting closed-loop system. Our control design will be based on knowledge of a control Lyapunov function (clf), that is, a C 1 , proper, positive deenite function V : IR n ! IR + such that inf u h L f V (x) + L g V (x) u i < 0 (2) for all x 6 = 0 1, 2]. The existence of a clf for the system (1) is equivalent to the existence of a globally asymptotically stabilizing control law u = k(x) which is smooth everywhere except possibly at x = 0 1]. Moreover , one can calculate such a control law k explicitly from f, g, and V 3]. We will say that V as above is a weak clf when the inequality (2) is non-strict, namely, when inf u h L f V (x) + L g V (x) u i 0 (3) for all x. The existence of a weak clf does not guarantee global stabilizability as does the existence of a clf. Nevertheless, in many cases a weak clf can indeed be used to design a globally stabilizing control law as we will see in Section 4 below. Given a general system of the form (1), it may be dif-cult to nd a clf or even to determine whether or not one exists. Fortunately, there are signiicant classes of systems …