Lyapunov-Based Output-Feedback Adaptive Stabilization of Minimum Phase Second-Order Systems

We consider output feedback adaptive stabilization for second-order systems in the presence of bounded exogenous disturbances. This case is of particular interest as it has been shown that, in the presence of exogenous disturbances, direct adaptive control schemes for minimum phase plants with relative degree 1 exhibit parameter divergence eventually leading to instability. We present controllers that guarantee convergence of the measured output and boundedness of all controller parameters and signals. The controller has the form of a 7th-order dynamic compensator for the relative-degree 1 case. The proof of convergence is based on a variant of Lyapunov's method in which the Lyapunov derivative is shown to be asymptotically nonpositive. 1 Robust Adaptive Control It has been shown in [9] that several direct adaptive control schemes for minimum phase plants with relative degree 1 (such as those in [1, 2]), exhibit parameter divergence eventually leading to instability. In general, non-identi er based adaptive controllers based on high gain feedback tend to su er from lack of robustness with respect to bounded disturbances. This research was supported in part by the Air Force OÆce of Scienti c Research under grant F49620-98-1-0037 and the University of Michigan OÆce of the Vice President for Research. Harshad S Sane is with IntelliSense Corp., 36 Jonspin Rd., Wilmington, MA01887, Dennis S Bernstein is with the Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI 48105-2140 and H ector J. Sussman is with the Department of Mathematics, Rutgers, The State University of New Jersey, New Brunswick, NJ 08903 118 A minimum-phase system with relative degree 1 can be stabilized by output feedback u = ky, provided that k > ks, with ks suÆciently large. To ensure that k grows beyond ks the controller takes the form [1, 2, 6] u = ky; (1) _ k = y: (2) With y ! 0 as t!1 it can be proved [1, 6] that k converges. However, there is a practical limitation in implementing this controller. In the presence of sustained plant disturbance or measurement noise, k diverges eventually leading to instability. There have been several xes to amend this problem in the literature and some of them are summarized below. To avoid divergence of the gain k, (2) is replaced by [3]