Optimal Spatially-Invariant Controllers with Locality Constraints: A System Level Approach

Abstract

We consider the problem of designing optimal controllers for distributed systems with a priori constraints on the spatial spread of a closed-loop response. We employ the recently developed framework of the “system level approach”. This guarantees a certain degree of controller interaction locality given a prescribed spatial locality of a closed-loop response, with the latter being a convex problem. We use the setting of spatially invariant systems to obtain special, but sharp results. We show that for infinite extent spatially invariant systems, in addition to convexity, further interesting properties of the optimal controller design problem emerge. In this case, when finite spread constraints on the closed-loop responses are imposed, the infinite-dimensional $\mathcal{H}_{2}$ problem reduces to a standard finite-dimensional $\mathcal{H}_{2}$ problem. We present numerical examples to illustrate the effect of the spatial spread constraint on the best achievable performance.