Convex method for localized control design in spatially invariant systems

A method is presented to impose localization in controller design for distributed arrays with underlying spatial invariance. The method applies to problems where the performance objective (e.g., stabilization, H₂ or H_∞ control) can be stated in terms of the search for a suitable Lyapunov matrix over spatial frequency. By restricting this matrix to be constant across frequency, controller localization can be naturally imposed. Thus we obtain sufficient conditions for the existence of a controller with the desired localization and performance, which take the form of linear matrix inequalities (LMIs) over spatial frequency. For one-dimensional arrays, we further show how to convert these conditions exactly to finite dimensional LMIs by means of the KYP lemma