Performance evaluation in identification and adaptive control of time varying systems

The main goal of this paper has been to characterize the performance of identification and adaptive control algorithms for time-varying systems. We have considered the Kalman filter estimator, and shown that its performance has a lower bound with several interesting properties. In particular, the bound has an infinite slope at the origin, and appears to be accurate for small values of the parameter Q. We combined this identification bound with a new stability result to obtain performance bounds for a scalar adaptive control problem. Once again, the performance shows a steep degradation as parameter variation increases. The accuracy of the theoretical bounds in both the identification and adaptive control problems is borne out by the simulation results that have been provided. We note that in the multidimensional adaptive control problem, at present the only known method for obtaining bounds is to reduce the problem to the scalar case by employing a somewhat conservative lower bound. It would be of interest to avoid this approximation to obtain tighter bounds in the general multivariate case. Further, currently in the case of adaptive control, we do not know how to analyze the closed loop system unless the system zeros are fixed and known. Relaxing this assumption appears to be a far more difficult problem. One approach to this problem is the cautious control of Astrom and Wittenmark (1989). Research in this direction is currently in progress