Distributed Feedback Control of Multichannel Linear Systems

Abstract

In this paper it is established that any jointly controllable, jointly observable, multi-channel, discrete or continuous time linear system with a strongly connected neighbor communication graph can be exponentially stabilized with any pre-specified convergence rate using a time-invariant distributed linear control. As an illustration of how this finding can be used to deal with certain distributed tracking problems, a solution is given to a distributed set-point control problem for a continuous-time, multi-channel linear system in which each and every agent with access to the system is able to independently adjust its scalar-valued, controlled output to any desired set-point value. To better understand the constraints on controller design, the distributed control problem is recast as a classical decentralized control problem. Armed with the tools of decentralized control, including the notion of a “fixed spectrum”, it is possible to show quite surprisingly that if the only information each agent is allowed to share with its neighbors is its measured output, then distributed stabilization in some cases is impossible. Using well-known decentralized control concepts, lower bounds are derived on the dimensions of the shared sub-states of local controllers which, if satisfied, guarantee that there will be no fixed closed-loop system eigenvalues to contend with. The decentralized control perspective also enables one to assert definitively that without imposing a partitioning constraint, the closed-loop spectrum of any jointly controllable, jointly observable multi-channel linear system with a strongly connected neighbor graph, can be freely assigned with distributed feedback control. The paper then turns to the important and often overlooked design problem of dealing with the effects of transmission delays across a network. It is explained why in the face of finite delays, exponential stabilization at any prescribed convergence rate can still be achieved with distributed control, at least for discrete-time multi-channel linear systems.