Synthesis of reduced order prefilters for multivariable tracking

Abstract

A primary disadvantage of using an internal model to achieve multivariable tracking is the high order of the internal model. In situations where it is known that each output is to track only its associated reference input, the internal model formulation is shown to be unnecessary for certain classes of problems. Using the notion of orthogonal operators, a method is presented through which a prefilter may be constructed to achieve asymptotic tracking of only the required reference inputs. It is shown that obtaining the prefilter requires the solution of a polynomial matrix equation. Conditions for existence of a solution to this equation as well as an algorithm for its construction are presented. Since existence of a solution implies an infinite number of solutions, the algorithm provides a means of parameterizing all solutions of a given order. Unlike prefilter techniques such as plant inversion, the method presented may be applied to non-minimum phase systems and results in proper, physically realizable systems. Since an infinite number of solutions exist, criteria for defining and obtaining the optimal solution are presented. In fact, it is shown that obtaining the optimal prefilter reduces to solving a set of linear equations. A minimum phase multivariable system and a non-minimum phase single input, single output system are used to demonstrate the effectiveness of the optimization procedure.