Structural synthesis of multivariable controllers

Abstract

In this paper the problem of achieving a desired transfer function matrix H d( s) between external inputs and controlled outputs in a linear multivariable system by connecting proper, stabilizing controllers between measured outputs and control inputs is solved in both transfer function and state space settings. The class H d of achievable transfer functions is directly and constructively characterized via the theory of transfer function valuations. For each H d( s) ϵH d, the class of synthesizing controllers is determined. Similar valuation conditions are given for the asymptotic tracking and disturbance rejection problem in which H d( s) is only partially specified. These results extend and complement earlier results. The state space geometric solution of the problem of achieving a desired H d( s) is obtained by formulating it as an equivalent output feedback disturbance rejection problem. A constructive solvability condition in terms of a pair of measurement and control invariant subspaces is given. This requires a nontrivial generalization of the notion of ( C, A, B) pairs. An H d( s) is shown to be admissible if and only if it induces an appropriate pair of invariant subspaces. The signal flow structure and certain factorizability conditions for a robust synthesis of the output feedback disturbance rejection problem are also given which extend earlier results on robust state feedback disturbance rejection. It is shown that every compensator corresponds to a state feedback control law implemented by an unknown input observer. The results of this paper are expected to be useful in the development of parameter optimization or other computer aided design algorithms where response specifications are to be traded against other design criteria such as sensitivity or stability margins since they explicitly characterize the algebraic variety H d in which the response transfer function may lie.