Optimality of POD for Data-Driven LQR With Low-Rank Structures

The optimal state-feedback gain for the Linear Quadratic Regulator (LQR) problem is computationally costly to compute for high-order systems. Reduced-order models (ROMs) can be used to compute feedback gains with reduced computational cost. However, the performance of this common practice is not fully understood. This letter studies this practice in the context of data-driven LQR problems. We show that, for a class of LQR problems with low-rank structures, the controllers designed via their ROM, based on the Proper Orthogonal Decomposition (POD), are indeed optimal. Experimental results not only validate our theory but also demonstrate that even with moderate perturbations on the low-rank structure, the incurred suboptimality is mild.