Nonlinear and Adaptive Control: An Abbreviated Status Report∗

We briefly review selected results in nonlinear and adaptive control, focusing on constructive concepts and design procedures: control Lyapunov functions, feedback passivation, robust and adaptive backstepping. For a more extensive discussion see Kokotović and Arcak (2001). 1 Control Lyapunov Functions Early nonlinear concepts have been descriptive. Their ’feedback activation’ began only recently, when some local properties were replaced with new concepts applicable to large regions of the state space. The main effort of activation is to make new concepts dependent on, and transformable by feedback control. A prominent example is the concept of control Lyapunov function (CLF) whose derivative depends on the control and can be made negative by feedback. This seemingly obvious concept, introduced by Artstein (1983) and Sontag (1983), made a tremendous impact on stabilization theory, which, at the end of the 1970’s was stagnant. It converted stability descriptions into tools for solving stabilization tasks. One way to stabilize a nonlinear system is to select a Lyapunov function V (x) and then try to find a feedback control u(x) that renders V̇ (x, u(x)) negative definite. With an arbitrary choice of V (x) this attempt may fail, but if V (x) is a CLF, we can find a stabilizing control law u(x). For the nonlinear system ẋ = f(x) + g(x)u , (1) V (x) is a CLF if, for all x = 0, LgV (x) = 0 ⇒ LfV (x) < 0 , (2) where LgV denotes ∂V ∂x g(x). By standard converse theorems, if (1) is stabilizable, a CLF exists. From (2), we see that the set where LgV (x) = 0 is significant, because in this set the uncontrolled ∗Research supported in part by the National Science Foundation under grant ECS-9812346 and the Air Force Office of Scientific Research under grant F49620-00-1-0358. †P. Kokotović is with the Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106-9560. E-mail: [email protected]. ‡Murat Arcak is with the Department of Electrical, Computer and Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590. E-Mail: [email protected].