Minimal inversion, command matching and disturbance decoupling in multivariable systems
Homayoun Seraji · International Journal of Control · 1989
This paper addresses the two basic and related problems of minimal inversion and perfect output control in linear multivariable systems. A simple analytical expression is obtained for the inverse of the transfer-function matrix of a square multivariable system. This expression is then used to construct a minimal-order inverse of the system. It is shown that the poles of the minimal-order inverse are the transmission zeros of the system. As a result, necessary and sufficient conditions for existence and stability of the inverse system are stated simply in terms of the zero polynomial of the original system. Furthermore, the minimal-order inverse is shown to be proper provided the original system has a full rank feedthrough matrix. The related problem of perfect output control, namely command matching and disturbance decoupling, in linear multivariable systems by means of feedforward controllers is also formulated and solved in a transfer-function setting. It is shown that a necessary and sufficient condition for existence of the required controllers is that the plant zero polynomial is not identical to zero or unstable. The order of the required controllers is equal to the number of plant transmission zeros. The control scheme proposed in this paper is composed of a feedback controller to enhance system stability and robustness, a feedforward controller to ensure command matching, and another feedforward controller to achieve disturbance decoupling. The three controllers have no effect on each other and can therefore be designed independently. A number of numerical examples are discussed for illustration.