Perfect Multivariable Feedback Theory

Abstract

The paper nicely combines the state-space description with the input-output description and elegantly formulates the multivariable feedback theory as well as obtains a number of useful results for modern network and control theory. In particular, it reveals various kinds of duality between a multivariable feedback network and its associated inverse network, such as structure duality, transfer function matrix (determinant) duality and duality on controllability (observability). It also thoroughly studies four pairs of the (null) return difference matrices of a multivariable feedback network and its associated inverse network, and presents not only the dual properties about these (null) return diferrence matrices and their respective determinants, but also treats the relationships among these determinants, the characteristic polynomials of a closed-loop network (the multivariable feedback network), its associated closed-loop inverse network, and their respective corresponding open-loop networks. Finally, the stability criteria, the testing criteria for a minimum-phase matrix and the sensitivity matrices are discussed. Although all of these results are obtained for a continuous system, they are also suitable for a discrete system provided that we use the z-transformation instead of the Laplace transformation.