A Design of Discrete-Time Indirect Multivariable MRACS with Structural Estimation of Interactor

Abstract

An interactor matrix introduced by Wolovich & Falb (1976) has an important role in the design of model reference adaptive control systems (MRACS) for a class of multi-input multi-output (MIMO) plants. In the early stage of the research, the interactor was supposed to be diagonal matrix and thus there were no unknown parameters (Goodwin et al., 1980). But, there exist many plants which require non-diagonal interactors (Chan & Goodwin, 1982). And the design of MIMO MRACS with non-diagonal interactors had been discussed, where all elements of the interactor are assumed to be known (Elliott & Wolovich, 1982; Goodwin & Long, 1980). However, this assumption is not adequate for adaptive control systems since the parameters of the interactor depend on the unknown parameters of the plant, i.e., the parameter values and the relative degree of each element of the plant must be used to determine the interactor. Furthermore, even we know all of these information, the structure of the interactor is not determined uniquely. In order to remove the assumption, the MRACS design has been proposed where the degree of diagonal elements and the upper bound of the highest degree of the lower triangular interactor matrix are assumed to be known (Elliott & Wolovich, 1984; Dugard et al., 1984). Under these assumptions, off-diagonal elements of the lower triangular interactor are estimated, and the method seemed suitable for adaptive controller design. However, it is not reasonable to assume the diagonal degrees in MRACS since the determination of the degrees depends on the relative degree and parameter values of each element of a transfer matrix of a given plant. From this view point, an interactor in generic sense was considered under the assumption that the relative degrees of all elements of the transfer function matrix are known (Kase & Tamura, 1990; Mutoh & Ortega, 1993). The method covers almost of all classes of MIMO plants having the same numbers of inputs and outputs generally. But there still exist some rare plants. By the way, there exists an idea of the certainty equivalence principle for the indirect MRACS design, i.e., estimate the unknown parameters of a plant first, then design the controllers on-line, using those estimated parameters. However, the design was seemed very difficult especially for MIMO plants, since large amount of calculation is needed to solve so-called Diophantine equation, beside the derivation of the interactor. In other words, there did not exist a suitable method to solve the Diophantine equation or to derive the interactor matrix. In this chapter, an indirect approach to MIMO MRACS will be shown. For