Robustness of systems with uncertainties in the input

Abstract

In B. R. Barmish (IEEE Trans. Automat. ControlAC-22, No. 7 (1977) 123, 124; AC-24, No. 6 (1979), 921-926) and B. R. Barmish and Y. H. Lin ("Proceedings of the 7th IFAC World Congress, Helsinki 1978") a new notion of "robustness" was defined for a class of dynamical systems having uncertainty in the input-output relationship. This paper generalizes the results in the above-mentioned references in two fundamental ways: (i) We make significantly less restrictive hypotheses about the manner in which the uncertain parameters enter the system model. Unlike the multiplicative structure assumed in previous work, we study a far more general class of nonlinear integral flows, (ii) We remove the restriction that the admissible input set be compact. The appropriate notion to investigate in this framework is seen to be that of approximate robustness. Roughly speaking, an approximately robust system is one for which the output can be guaranteed to lie "[var epsilon]-close" to a prespecified set at some future time T > 0. This guarantee must hold for all admissible (possibly time-varying) variations in the values of the uncertain parameters. The principal result of this paper is a necessary and sufficient condition for approximate robustness. To "test" this condition, one must solve a finite-dimensional optimization problem over a compact domain, the unit simplex. Such a result is tantamount to a major reduction in the complexity of the problem; i.e., the original robustness problem which is infinite-dimensional admits a finite-dimensional parameterization. It is also shown how this theory specializes to the existing theory of Barmish and Barmish and Lin under the imposition of additional assumptions. A number of illustrative examples and special cases are presented. A detailed computer implementation of the theory is also discussed.