LQR-based optimal linear consensus algorithms

Laplacian matrices play an important role in linear consensus algorithms. This paper studies linear-quadratic regulator (LQR) based optimal linear consensus algorithms for multi-vehicle systems with single-integrator kinematics in a continuous-time setting. We propose two global cost functions, namely, interaction-free and interaction-related cost functions. With the interaction-free cost function, we derive the optimal (nonsymmetric) Laplacian matrix. It is shown that the optimal (nonsymmetric) Laplacian matrix corresponds to a complete directed graph. In addition, we show that any symmetric Laplacian matrix is inverse optimal with respect to a properly chosen cost function. With the interaction-related cost function, we derive the optimal scaling factor for a pre-specified symmetric Laplacian matrix associated with the interaction graph. Illustrative examples are given as a proof of concept.

[1]  Reza Olfati-Saber,et al.  Flocking for multi-agent dynamic systems: algorithms and theory , 2006, IEEE Transactions on Automatic Control.

[2]  George J. Pappas,et al.  Flocking in Fixed and Switching Networks , 2007, IEEE Transactions on Automatic Control.

[3]  Pavel Yu. Chebotarev,et al.  The Matrix of Maximum Out Forests of a Digraph and Its Applications , 2006, ArXiv.

[4]  Randal W. Beard,et al.  Consensus seeking in multiagent systems under dynamically changing interaction topologies , 2005, IEEE Transactions on Automatic Control.

[5]  Wei Ren,et al.  Information consensus in multivehicle cooperative control , 2007, IEEE Control Systems.

[6]  Stephen P. Boyd,et al.  Fast linear iterations for distributed averaging , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[7]  Mehran Mesbahi,et al.  On maximizing the second smallest eigenvalue of a state-dependent graph Laplacian , 2006, IEEE Transactions on Automatic Control.

[8]  U. Feige,et al.  Spectral Graph Theory , 2015 .

[9]  Jean-Charles Delvenne,et al.  Optimal strategies in the average consensus problem , 2007, 2007 46th IEEE Conference on Decision and Control.

[10]  Richard M. Murray,et al.  Consensus problems in networks of agents with switching topology and time-delays , 2004, IEEE Transactions on Automatic Control.

[11]  Richard M. Murray,et al.  Information flow and cooperative control of vehicle formations , 2004, IEEE Transactions on Automatic Control.

[12]  Jie Lin,et al.  Coordination of groups of mobile autonomous agents using nearest neighbor rules , 2003, IEEE Trans. Autom. Control..

[13]  Guangming Xie,et al.  Consensus control for a class of networks of dynamic agents , 2007 .

[14]  G. Alefeld,et al.  On square roots of M-matrices , 1982 .

[15]  Mireille E. Broucke,et al.  Local control strategies for groups of mobile autonomous agents , 2004, IEEE Transactions on Automatic Control.

[16]  Elham Semsar-Kazerooni,et al.  Optimal Control and Game Theoretic Approaches to Cooperative Control of a Team of Multi-Vehicle Unmanned Systems , 2007, 2007 IEEE International Conference on Networking, Sensing and Control.

[17]  Laura Giarré,et al.  Non-linear protocols for optimal distributed consensus in networks of dynamic agents , 2006, Syst. Control. Lett..

[18]  Gerardo Lafferriere,et al.  Decentralized control of vehicle formations , 2005, Syst. Control. Lett..

[19]  Reza Olfati-Saber,et al.  Consensus and Cooperation in Networked Multi-Agent Systems , 2007, Proceedings of the IEEE.

[20]  Ella M. Atkins,et al.  Distributed multi‐vehicle coordinated control via local information exchange , 2007 .

[21]  Dongjun Lee,et al.  Stable Flocking of Multiple Inertial Agents on Balanced Graphs , 2007, IEEE Transactions on Automatic Control.