Modifications of Finitely-Converging Algorithms of Adaptive Control

Abstract

Finitely-converging algorithms were originally proposed by V.A.Yakubovich to solve problems of pattern recognition. Then, in his seminal paper “To The Theory Of Adaptive Systems,” published in 1968 in Russian Mathematics Doklady, V.A.Yakubovich applied these algorithms for robot adaptive control. To our knowledge it was the first usage of the word “robot” in scientific literature. The proposed approach consists in reformulation of an adaptive control problem as the solution problem for some countable system of inequalities. A procedure of their solution plays a role of adaptation algorithm. It should converge in a finite time in a closed loop. Initially V.A.Yakubovich developed algorithms to solve a problem of adaptive control of discrete linear time-invariant systems. With the expansion of a set of the systems under consideration those basic algorithms were modified, but theirs main idea remained the same. Namely, each correction of the vector of estimated parameters ensures decreasing of the distance between current vector and unknown vector of true parameters by the positive value separated from zero. Thus there may be only finite number of these corrections. This article describes Modifications of basic V.A.Yakubovich's algorithms in relation to adaptive control of several classes of the plants. Among them are hybrid continuous/discrete systems, nonlinear systems, infinite-dimensional systems, and so on.