Distributed optimisation for multi-agent systems with the first-order integrals under Markovian switching topologies

ABSTRACT This paper studies the distributed optimisation problem for multi-agent systems with the first-order dynamics over Markovian switching topologies. The interaction topology among agents’ switches following a Markov process and each topology is modelled as a state of the Markov process. The aim is to minimise the global cost functions and make the agents converge to the optimal point through the network communication between the agents, where each agent has a local convex cost function only known by itself. Utilising the knowledge of convex analysis and graph theory, we establish a distributed algorithm for the optimisation problem with randomly switching topologies. A sufficient condition for the existence of such algorithm is obtained by using the Lyapunov method. Besides, the result is also extended to the cases of a Markov process with partially unknown transition rates. Finally, numerical simulations are given to validate the proposed algorithm.

[1]  Jie Chen,et al.  Distributed discrete-time coordinated tracking with Markovian switching topologies , 2012, Syst. Control. Lett..

[2]  Jun Hu,et al.  A variance-constrained approach to recursive state estimation for time-varying complex networks with missing measurements , 2016, Autom..

[3]  Qingshan Liu,et al.  A Second-Order Multi-Agent Network for Bound-Constrained Distributed Optimization , 2015, IEEE Transactions on Automatic Control.

[4]  Debasish Chatterjee,et al.  Stabilizing Randomly Switched Systems , 2008, SIAM J. Control. Optim..

[5]  Hongwei Zhang,et al.  Cooperative output feedback adaptive control of uncertain nonlinear multi-agent systems with a dynamic leader , 2015, Neurocomputing.

[6]  Jing Wang,et al.  Control approach to distributed optimization , 2010, 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

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

[8]  Weisheng Chen,et al.  Distributed convex optimisation with event-triggered communication in networked systems , 2016, Int. J. Syst. Sci..

[9]  Zidong Wang,et al.  H∞H∞ consensus control for multi-agent systems with missing measurements: The finite-horizon case , 2013, Syst. Control. Lett..

[10]  Donghua Zhou,et al.  State estimation for networked systems with randomly occurring quantisations , 2013, Int. J. Syst. Sci..

[11]  Zheng-Guang Wu,et al.  Robust output synchronisation of non-identical linear agents via internal model principle , 2015 .

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

[13]  Yutao Tang,et al.  Coordination of multi-agent systems under switching topologies via disturbance observer-based approach , 2016, Int. J. Syst. Sci..

[14]  E. Boukas,et al.  Mode-dependent Hºº filtering for discrete-time Markovian jump linear systems with partly unknown transition probabilities. , 2007 .

[15]  Samuel Martin,et al.  Multi-agent flocking under topological interactions , 2013, Syst. Control. Lett..

[16]  Daniel W. C. Ho,et al.  Observer-Based Event-Triggering Consensus Control for Multiagent Systems With Lossy Sensors and Cyber-Attacks , 2017, IEEE Transactions on Cybernetics.

[17]  Long Cheng,et al.  A Mean Square Consensus Protocol for Linear Multi-Agent Systems With Communication Noises and Fixed Topologies , 2014, IEEE Transactions on Automatic Control.

[18]  Xinghu Wang,et al.  Distributed optimisation design with triggers for disturbed continuous-time multi-agent systems , 2017 .

[19]  James Lam,et al.  Semi-Global Leader-Following Consensus of Linear Multi-Agent Systems With Input Saturation via Low Gain Feedback , 2013, IEEE Transactions on Circuits and Systems I: Regular Papers.

[20]  Sonia Martínez,et al.  Distributed convex optimization via continuous-time coordination algorithms with discrete-time communication , 2014, Autom..

[21]  John S. Baras,et al.  Convergence Results for the Linear Consensus Problem under Markovian Random Graphs , 2013, SIAM J. Control. Optim..

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

[23]  James Lam,et al.  Analysis and Synthesis of Markov Jump Linear Systems With Time-Varying Delays and Partially Known Transition Probabilities , 2008, IEEE Transactions on Automatic Control.

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

[25]  Long Cheng,et al.  Containment Control of Multiagent Systems With Dynamic Leaders Based on a $PI^{n}$ -Type Approach , 2014, IEEE Transactions on Cybernetics.

[26]  Guanghui Wen,et al.  Containment of Higher-Order Multi-Leader Multi-Agent Systems: A Dynamic Output Approach , 2016, IEEE Transactions on Automatic Control.

[27]  Gerard J. M. Smit,et al.  Management and Control of Domestic Smart Grid Technology , 2010, IEEE Transactions on Smart Grid.

[28]  Lixian Zhang,et al.  Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities , 2009, Autom..

[29]  Jiangping Hu,et al.  Tracking control for multi-agent consensus with an active leader and variable topology , 2006, Autom..

[30]  Marcelo D. Fragoso,et al.  A Unified Approach for Stochastic and Mean Square Stability of Continuous-Time Linear Systems with Markovian Jumping Parameters and Additive Disturbances , 2005, SIAM J. Control. Optim..

[31]  Yiguang Hong,et al.  Distributed Optimization for Continuous-Time Multi-Agent Systems with External Disturbance and Discrete-Time Communication , 2016 .

[32]  Asuman E. Ozdaglar,et al.  Constrained Consensus and Optimization in Multi-Agent Networks , 2008, IEEE Transactions on Automatic Control.

[33]  Zhang Ren,et al.  Time-varying group formation analysis and design for second-order multi-agent systems with directed topologies , 2016, Neurocomputing.

[34]  Yiguang Hong,et al.  Distributed optimization design for high-order multi-agent systems , 2015, 2015 34th Chinese Control Conference (CCC).

[35]  Asuman E. Ozdaglar,et al.  Distributed Subgradient Methods for Multi-Agent Optimization , 2009, IEEE Transactions on Automatic Control.

[36]  Jinde Cao,et al.  Distributed node-to-node consensus of multi-agent systems with time-varying pinning links , 2015, Neurocomputing.

[37]  Lihua Xie,et al.  Consensus condition for linear multi-agent systems over randomly switching topologies , 2013, Autom..

[38]  Guoqiang Hu,et al.  Pinning Synchronization of Directed Networks With Switching Topologies: A Multiple Lyapunov Functions Approach , 2015, IEEE Transactions on Neural Networks and Learning Systems.