A Suboptimality Approach to Distributed Linear Quadratic Optimal Control

Abstract

This paper is concerned with the distributed linear quadratic optimal control problem. In particular, we consider a suboptimal version of the distributed optimal control problem for undirected multi-agent networks. Given a multi-agent system with identical agent dynamics and an associated global quadratic cost functional, our objective is to design suboptimal distributed control laws that guarantee the controlled network to reach consensus and the associated cost to be smaller than an a priori given upper bound. We first analyze the suboptimality for a given linear system and then apply the results to linear multi-agent systems. Two design methods are then provided to compute such suboptimal distributed controllers, involving the solution of a single Riccati inequality of dimension equal to the dimension of the agent dynamics, and the smallest nonzero and the largest eigenvalue of the graph Laplacian. Furthermore, we relax the requirement of exact knowledge of the smallest nonzero and largest eigenvalue of the graph Laplacian by using only lower and upper bounds on these eigenvalues. Finally, a simulation example is provided to illustrate our design method.

Distributed Linear Quadratic Optimal Control: Compute Locally and Act Globally

“…As an illustration, we will now provide two important special cases of LQ cost functionals. The first one was studied before in [1] and [2]:…”

“…We are now ready to present the main result of this paper. Theorem 7: Consider the multi-agent system (1). Let T > 0 be a sampling period, α > 0 a discount factor, and let q, r > 0 be given weights.…”

“…In the case that the agent dynamics is given by a general state space system, this optimal control problem is nonconvex and difficult to solve, and it is unclear whether a solution exists in general, see [1]. In contrast, for the case of single integrator dynamics it is fairly easy to find an explicit expression for the optimal distributed diffusive control law, see, for example, [2].…”

“…In [9] a game theoretic approach was considered to obtain a suboptimal solution. Also, [1] considers a suboptimal version of this problem. In [10], a suboptimal consensus controller design was developed by employing a hierarchical LQ control approach for an appropriately chosen global performance index, and a similar idea for constructing a particular cost functional was employed in [11] to design a reduced order distributed controller.…”

A Suboptimality Approach to Distributed H Optimal Control

“…We are interested in the problem of finding a controller of the form (12) such that the controlled system (13) is internally stable and the associated cost (14) is smaller than an a priori given upper bound. Before we proceed, we first review a well-known result that provides necessary and sufficient conditions such that a closed loop system is H 2 suboptimal, see e.g.…”