Linear quadratic regulator control of multi‐agent systems

Abstract

SummaryThis paper considers a collection of agents performing a shared task making use of relative information communicated over an information network. The designed suboptimal controllers are state feedback and static output feedback, which are guaranteed to provide a certain level of performance in terms of a linear quadratic regulator (LQR) cost. Because of the convexity of the LQR performance region, the suboptimal LQR control problem with state feedback is reduced to the solution of two inequalities, with the minimum and maximum eigenvalues of the Laplacian matrix as the coefficients. The advantage of the method is that the LQR control problem of network multi‐agent systems can be converted into the LQR control of a set of single‐agent systems, and the structure constraint on the feedback gain matrix can be eliminated. It can be shown that the size of the LQR control problem will not increase according to the number of the node in the fairly general framework. The method can be extended to the synthesis of the static output feedback, which is derived from the weighting matrices in LQR. Through some coordinate transformation and the augmentation of the output matrix, the LQR synthesis is provided on the basis of the output measurements of the adjacent agents. Numerical examples are presented to illustrate the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.

Distributed optimal consensus control for nonlinear multiagent systems based on DISOPE algorithm

“…In the work of Zhang et al, the classical linear quadratic regulator (LQR) design method was employed in the cooperative control of linear multiagent system on directed graphs, and an LQR‐based control gain was proposed to make the agents reach synchronization. In the work of Zhang et al, the optimal LQR problem with state feedback is reduced to the solution of two inequalities, which is connected with the minimum and maximum eigenvalues of the Laplacian matrix. In the work of Dong, the centralized optimal control of multiagent system on all connected graph was proposed, and distributed optimal control on general directed graph was designed to approximate the centralized optimal control, such that the multiagent system can reach near‐optimal.…”

Guaranteed cost consensus problems for second‐order multi‐agent systems

“…In [13][14][15], the objectives were to find the optimal estimated states for first-order multi-agent systems, but the control energy consumption was not considered. Differing from [13][14][15], [16][17][18][19] concentrated on the optimal protocols subject to the minimisation of certain cost functions. The optimal consensus problems were investigated in [16], but the analysis approach is only applied to a special case for first-order multi-agent systems.…”

“…The linear matrix inequality (LMI) criteria in [18] are dependent on the number of agents. Zhang et al [19] dealt with suboptimal problems for multi-agent systems, where only the stability problem was discussed. In summary, it is very difficult to achieve optimal consensus for multi-agent systems.…”

“…Secondly, an upper bound of the cost function is given. The optimisation problems with certain cost functions were studied in [16][17][18][19], but those results did not give the bound of the cost function.…”

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