Regret Analysis of Distributed Online LQR Control for Unknown LTI Systems

Abstract

Online learning has recently opened avenues for rethinking classical optimal control beyond time-invariant cost metrics, and online controllers are designed when the performance criteria changes adversarially over time. Inspired by this line of research, we study the distributed online linear quadratic regulator (LQR) problem for linear time-invariant (LTI) systems with unknown dynamics. Consider a multi-agent network where each agent is modeled as a LTI system. The LTI systems are associated with time-varying quadratic costs that are revealed sequentially. The goal of the network is to collectively (i) estimate the unknown dynamics and (ii) compute local control sequences competitive to that of the best centralized policy in hindsight that minimizes the sum of costs for all agents. This problem is formulated as a regret minimization. We propose a distributed variant of the online LQR algorithm where each agent computes its system estimate during an exploration stage. The agent then applies distributed online gradient descent on a semi-definite programming (SDP) whose feasible set is based on the agent's system estimate. We prove that the regret bound of our proposed algorithm scales Õ(T 2/3 ), implying the consensus of the network over time. We also provide simulation results verifying our theoretical guarantee.

A Survey on Distributed Online Optimization and Game

“…In physical world, agents involved in real applications are often subject to some physical dynamics, such as bicycle dynamics for robots and Euler-Lagrange dynamics for manipulators, and thus, to better suit many applications in reality, control system dynamics has been integrated with the study of DOL recently [42], which can be viewed as sort of constraint for DOL. Time-Varying Scenarios.…”

“…Nevertheless, an agent often has its physical operating dynamics, such as bicycle dynamics for robots and Euler-Lagrange dynamics for manipulators, which should be appropriately considered and controlled, thought of as physical layer problems. Altough recent research [42] has integrated the control system dynamics into DOL, the related research is yet to be fully explored in order to smoothly apply decentralized online algorithms to real-world problems. 5) Continuous-time Algorithms.…”