Revisiting disturbance decoupling with an optimization perspective

This paper presents an optimization-based perspective for incorporating disturbance decoupling constraints into controller synthesis, which paves the way for utilizing numerical optimization tools. We consider the constraints arising from the following sets of static state feedback: (i) The set of all disturbance decoupling controllers; (ii) The set of all disturbance decoupling and stabilizing controllers. To inner approximate these sets by means of matrix equations or inequalities, we provide a unifying review of the relevant results of the geometric control theory. The approximations build on the characterization of controlled invariant subspaces in terms of the solvability of a linear matrix equation (LME) involving the state feedback. The set (i) is inner approximated through the LME associated with any element of an upper semilattice generated by controlled invariant subspaces. The set (ii) is inner approximated through a bilinear matrix inequality (BMI) and the LME associated with any element of a different upper semilattice generated by internally stabilizable controlled invariant subspaces. However, the resulting inner approximations depend on the subspaces chosen from the semilattices. It is shown that a specific (internally stabilizable) self-bounded controlled invariant subspace, which is the best choice regarding eigenvalue assignment, yields the largest inner approximation for both of the sets among (internally stabilizable) self-bounded controlled invariant subspaces. The inner approximations exactly characterize the controller sets under particular structural conditions. We have been driven by two primary motivations in investigating inner approximations for the sets above: (i) Enable the formulation of a variety of equality (and inequality) constrained optimization problems, where cost functions, such as a norm of the state feedback, can be minimized over a large subset of the set of all disturbance decoupling (and stabilizing) controllers; (ii) Introduce the disturbance decoupling constraints to members of the control systems community who might not be quite familiar with the elegant geometric state-space theory, similar to the authors themselves. This can add another dimension to research endeavors in resilient control of networked multi-agent systems.