(545a) Near-Optimal Distributed Linear-Quadratic Regulator for Networked Systems

Due to the increasing complexity of networked systems of practical interest, distributed control has gained substantial attention in the literature. Distributed control aims to design a set of local controllers that cooperatively optimize the systemwide performance while having access only to certain local information. This architecture has advantages over its centralized counterpart in robustness (failure in a component does not cause system failure), computation (online computation load is small), privacy (global information sharing is not needed), and implementation (less and shorter communication) and has advantages over its decentralized counterpart in performance (communication between agents mitigates the impact of decentralization).

However, the synthesis of an optimal distributed controller is a challenging problem, proven to be intractable [1]. Several works have studied making the problem tractable by imposing additional structural assumptions, such as finite-dimensional linear policy [2], quadratic invariance [3], and locality in the system-level synthesis [4]. However, even such optimal distributed controllers may have significantly worse performance compared to their centralized counterparts due to structural constraints imposed on the controllers. An important gap in the literature is the limited understanding of how much performance loss is incurred by the decentralization; to be specific, the trade-off relationship between the degree of decentralization and the distributed controller's performance has not been adequately characterized.

We seek to fill this gap by analyzing the performance of a κ-distributed linear-quadratic regulator (LQR). The system under study is a linear system composed of agents interconnected over a graph, which appears in a wide range of applications [5-7]. The graph allows us to construct a limited-range communication structure based on the distance on the graph. In particular, κ-distributed LQR implements truncated state feedback, which considers only the agents within a prescribed distance κ, while ignoring the agents beyond that distance. This controller becomes more decentralized for small κ and less decentralized for large κ, and thus, allows the user to tune the degree of decentralization using the user-defined parameter κ. We show that under standard assumptions in linear system theory (stabilizability and detectability) and a mild assumption on the graph topology (polynomial growth condition), the nominal performance of the κ-distributed policy, measured by a quadratic performance criterion, becomes exponentially close to that of the centralized optimal policy as κ is increased. Thus, one can indeed improve the performance of the distributed controller by enforcing a less restrictive information exchange structure. Moreover, the exponential relationship reveals that distributed control can achieve near-optimal performance with a moderate degree of decentralization. This result demonstrates the effectiveness of the distributed control architecture for networked system control. The technical results are built on the recently established exponential decay of sensitivity in graph-structured optimization problems [8,9].

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