A robust indirect discrete adaptive-control approach based on passitivity results for nonlinear systems (Q1115393)

This paper considers the problem of designing a robust adaptive algorithm for discrete systems. This algorithm employs the usual method of an on- line adjustment of the gain matrix so as to reduce the error signal to zero, while other signals remain bounded. It is well known that usually the adaptive systems present problems and thus the theoretical results fail. The problems are due to poor transient performances and poor behaviour which implies instability under unmodelled dynamics. In this paper the problem of instability is treated as a robustness problem, so as to make the closed-loop system stable under variations of the parameters. A passitivity result taken from \textit{C. A. Desoer} and \textit{M. Vidyasagar} [Feedback systems: Input-output properties (1975; Zbl 0327.93009)] is used to formulate a global convergence criterion in terms of positive realness of a transfer matrix. The main result is presented in Theorem 3.2 in the text of the paper. The robustness is provided in terms of the strictly discrete positive realness of a marix termed \(W(z)Gv(z)\) which is easily computable. It is noted that the results are valid for any initial conditions of the adaptive algorithm. Basically, the results in this paper extend the work by \textit{M. A. L. Thathachar} and \textit{F. Gajendran} [Int. J. Control 34, 127--142 (1981; Zbl 0467.93045)] in the case that additive noise is present (together with unmodelled dynamics) while stability is maintained for the closed- loop adaptive system.