Localized LQR with adaptive constraint and performance guarantee

In previous work, we proposed the localized linear quadratic regulator (LLQR) method as a scalable way to synthesize and implement distributed controllers for large-scale systems. The idea is to impose an additional spatiotemporal constraint on the closed loop response, which limits the propagation of dynamics to user-specified subsets of the global network. This then allows the controller to be synthesized and implemented in a localized, distributed, parallel, and thus scalable way. Nevertheless, the additional spatiotemporal constraint also makes the LLQR controller sub-optimal to the traditional centralized one. The goal of this paper is to quantify and bound the sub-optimality of the LLQR controller introduced by the additional spatiotemporal constraint. Specifically, we propose an algorithm to compute a lower bound of the cost achieved by the centralized controller using only local plant model information. This allows us to determine the sub-optimality of the LLQR controller in a localized way, and adaptively update the LLQR constraint to exploit the tradeoff between controller complexity and closed loop performance. The algorithm is tested on a randomized heterogeneous network with 51200 states, where the LLQR controller achieves at least 99% optimality compared to the unconstrained centralized controller.