Solutions to disturbance decoupling problem with constant measurement feedback for linear systems

We study in this paper the problem of disturbance decoupling with constant (i.e., static) measurement feedback (DDPCM) for linear systems. For a class of systems which have a left invertible transfer function from the control input to the controlled output and/or a right invertible transfer function from the disturbance input to the measurement output, we obtain a complete characterization of all solutions to the DDPCM. For a system that does not satisfies the above invertibility condition, we use the special coordinate basis to obtain a reduced-order system. Then a complete characterization of all possible solutions to the DDPCM for the given system can be explicitly obtained, if the obtained reduced-order system itself satisfies the invertibility condition. The main contribution of these solutions is that the solutions are given in a set of linear equations. This resolves the well known difficulty in solving nonlinear equations associated with the DDPCM. When the invertibility condition is not satisfied, the solutions are characterized by a set of polynomial equations related to the obtained reduced-order system. This reduced-order characterization significantly simplifies the problem and reduces the computational cost in finding solutions to the DDPCM.