Unified bound on the order of controllers using matrix pencil characterisation

This study concerns the existence and design of reduced-order controllers for a rather general class of continuous- and discrete-time control design problems which can be characterised by the linear matrix inequality (LMI) framework. From a matrix pencil perspective, this paper presents a unified bound on the order of controllers in terms of the minimal rank of two matrix pencils valued at their generalised eigenvalues at infinity and in the unstable region (the closed right-half plane for continuous-time systems and the region excluding the open unit circle for discrete-time systems). When full-order controllers exist, such matrix pencil characterisation-based bound reveals the existence of reduced-order controllers if one of two subsystems of the generalised plant has infinite zeros or unstable invariant zeros. A numerical example of obtaining reduced-order controllers for the covariance upper bound control problem is given. This study provides a more complete view of the role of infinite zeros or unstable invariant zeros in feedback systems.