Worst-case system identification in H∞: validation of apriori information, essentially optimal algorithms, and error bounds

Abstract

This paper is concerned with a particular control-oriented system identification problem recently considered by several authors. This problem has been referred to as the problem of worst-case system identification in H∞ in the literature. The formulation of this problem is worst-case/deterministic in nature. The available apriori information consists of a lower bound on the relative stability of the plant, an upper bound on a certain gain associated with the plant, and an upper bound on the noise level. The available aposteriori information consists-of a finite number of noisy plant point frequency response samples. The objective is to identify the plant transfer function in H∞ using the available apriori and aposteriori information. In this paper we resolve several important open issues pertaining to this problem. First, a method is presented for developing confidence that the available apriori information is correct. This method requires the solution of a certain nondifferentiable convex programming problem. Second, an essentially optimal identification algorithm is given for this problem. This algorithm is (worst-case strongy) optimal to within a factor of two. Finally, new upper and lower bounds on the optimal identification error for this problem are derived and used to estimate the identification error associated with the algorithm presented here. Interestingly, the development of each of the results described above draws heavily upon the classical Nevanlinna-Pick optimal interpolation theory. As such, the results of this paper establish a clear link between the areas of system identification and optimal interpolation theory.