Hybrid State Space and Frequency Domain System Level Synthesis for Sparsity-Promoting H₂/H_∞ Control Design

Design of optimal linear feedback controllers is a challenging but important problem in many applications. The main difficulties arise from nonconvexity and infinite dimensionality of the associated optimization problem for the design. A promising recent approach to address these challenges is to first use system level synthesis to render the problem convex using a clever reparameterization, and then to apply an approximation by simple poles to obtain a finite dimensional problem. However, when computing ${\mathcal{H}}_{2}$ and ${\mathcal{H}}_{\infty}$ norms, this prior approach requires an additional approximation of a finite time horizon for the closed-loop impulse response. This finite horizon results in increased suboptimality, degraded performance, and increased problem size and memory requirements. To address these limitations, we present a novel control design framework that combines the frequency domain system level synthesis constraints with a state space formulation of the ${\mathcal{H}}_{2}$ and ${\mathcal{H}}_{\infty}$ norms using linear matrix inequalities. This state space formulation eliminates the need for a finite time horizon approximation, and results in a convex and tractable semidefinite program for the control design. To preserve robustness, in practice it is important that controllers only contain a relatively small number of poles. Therefore, we propose to make an optimal sparse selection of simple poles from a large initial collection to maintain robustness while improving performance. As this sparsity constraint is nonconvex, we use group lasso regularization to enforce sparsity while maintaining convexity for the control design. Finally, the superior performance of the proposed method is illustrated on an example of power converter control design.

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