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On the Gap between System Level Synthesis and Structured Controller Design: the Case of Relative Feedback

We consider the optimal distributed control design problem with subcontroller communication constraints, by designing controllers with state-space realizations composed of matrices with specified sparsity patterns. The recently developed System Level Synthesis (SLS) framework provides a computationally tractable method for optimizing over a convex subset of these structured-realizable controllers, which we refer to as SLS-structured. The performance of the optimal SLS-structured controller will thus provide an upper bound on the performance of the optimal structured-realizable controller. We take a first step toward quantifying this bound by considering the design of controllers with access to only relative system measurements. Our main result demonstrates through an example, that, when relative feedback constraints are imposed, the gap in performance between the optimal SLS-structured controller and optimal controller with structured realization may be infinite.