IEEE Transactions on Automatic Control, Vol.65, No.7, 3176-3183, 2020
Novel All-Pass Factorization, All Solutions to Rational Matrix Equation and Control Application
For an arbitrary rational matrix (RM), a new descriptor realization of its all-pass factorization is presented. The coefficients of the factors are expressed explicitly in the solution of a linear matrix inequality, which is well known in control theory. Using the all-pass factorization, necessary and sufficient conditions for existence of a solution of an equation with RMs where the unknown RM has its poles in the left closed complex half plane and it can be improper are presented, as well as all solutions. As an application of the latter result, the notion of almost disturbance decoupling, i.e., when the infimum of the H-infinity norm of the transfer matrix from the disturbance input to the controlled output over the stabilizing controllers is zero, is generalized. The generalization is such that instead of the whole frequency axis, the frequency axis with excluded intervals of small lengths, is considered. The controller consists of output feedback plus disturbance feedforward designed independently. A necessary and sufficient solvability condition is presented, which is expressed in terms of the given data. The three main results are illustrated by an example.