What Does It Take to Control a Multi-Channel Linear System With Distributed Feedback?

This paper establishes the essential technique for exponentially stabilizing a jointly controllable, jointly observable, multi-channel linear system at any prescribed convergence rate using a time-invariant distributed linear control. It will be made clear that the well-known fixed spectrum (i.e., the set of fixed modes) of a multi-channel linear system is no longer an obstacle to the effective control of such a system provided that distributed rather than classical decentralized control is used. This is true for both continuous-time and discrete-time systems. Moreover, the effect of transmission delays is also investigated. It will be elucidated why in the face of such delays, exponential stabilization with an arbitrarily fast convergence rate can still be achieved with distributed control, at least for discrete-time multi-channel linear systems.