Recursive Approach to System Identification and Control

A unified recursive approach to i) identification for systems like ARMAX, nonlinear ARX, and others, and to ii) adaptive regulation of Hammerstein and Wiener systems is presented. By this method the problem under consideration is transformed to a root-seeking problem, to which the Robbins-Monro (RM) algorithm, the classical stochastic approximation (SA) algorithm, aiming at seeking for roots of an unknown regression function, may be applied. However, this may not lead to a satisfactory result, because for convergence of the RM algorithm, rather restrictive conditions are usually required and they are hardly satisfied for problems considered here. Thus, the SA algorithm with expanding truncations (SAAWET), a modification of the RM algorithm, and its general convergence theorem (GCT) are introduced to serve as the main too[. All identification and adaptive regulation problems discussed in the paper are transformed to root-seeking problems, to which applying SAAWET yields recursive identification or adaptive regulation algorithms. For each system under discussion reasonable conditions are demonstrated under which the strong consistency of estimates or the optimality of adaptive regulators are derived. This approach is possible to be applied to other problems from systems, control, signal processing, and other areas, but it by no means gives an automatic solution, it only points out the solution route for problems of interest. According to the route the main effort is supposed to be devoted to i) transforming the problem in question to an adequate root-seeking problem and to ii) proving satisfaction of conditions required by GCT.