Continuous-time proportional-integral distributed optimisation for networked systems

In this paper, we explore the relationship between dual decomposition and the consensus-based method for distributed optimisation. The relationship is developed by examining the similarities between the two approaches and their relationship to gradient-based constrained optimisation. By formulating each algorithm in continuous-time, it is seen that both approaches use a gradient method for optimisation with one using a proportional control term and the other using an integral control term to drive the system to the constraint set. Therefore, a significant contribution of this paper is to combine these methods to develop a continuous-time proportional-integral distributed optimisation method. Furthermore, we establish convergence using Lyapunov stability techniques and utilising properties from the network structure of the multi-agent system.

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

[2]  Yonina C. Eldar,et al.  Convex Optimization in Signal Processing and Communications , 2009 .

[3]  Angelia Nedic,et al.  Subgradient Methods for Saddle-Point Problems , 2009, J. Optimization Theory and Applications.

[4]  Douglas P. Looze.  Franklin, Powell and Emami-Naeini, Feedback Control of Dynamic Systems, 6 th Edition, Prentice-Hall, 2010. (referred to as FPE) References: Ogata, Modern Control Engineering, Prentice-Hall, 2009. Dorf, Modern Control Systems, Prentice-Hall, 2008. , 2013 .

[5]  R. Murray,et al.  Decentralized Multi-Agent Optimization via Dual Decomposition , 2011 .

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

[7]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[8]  M MurrayRichard,et al.  Distributed receding horizon control for multi-vehicle formation stabilization , 2006 .

[9]  Magnus Egerstedt,et al.  Graph Theoretic Methods in Multiagent Networks , 2010, Princeton Series in Applied Mathematics.

[10]  A. Ozdaglar,et al.  LIDS Report 1 Distributed Subgradient Methods over Random Networks , 2008 .

[11]  Lacra Pavel,et al.  Lyapunov analysis of a distributed optimization scheme , 2011, International Conference on NETwork Games, Control and Optimization (NetGCooP 2011).

[12]  Angelia Nedic,et al.  Distributed Stochastic Subgradient Projection Algorithms for Convex Optimization , 2008, J. Optim. Theory Appl..

[13]  Bahman Gharesifard,et al.  Continuous-time distributed convex optimization on weight-balanced digraphs , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[14]  Francesco Borrelli,et al.  Decentralized receding horizon control for large scale dynamically decoupled systems , 2009, Autom..

[15]  Fernando Paganini,et al.  Stability of primal-dual gradient dynamics and applications to network optimization , 2010, Autom..

[16]  Anders Rantzer,et al.  Dynamic dual decomposition for distributed control , 2009, 2009 American Control Conference.

[17]  Magnus Egerstedt,et al.  Distributed parameterized model predictive control of networked multi-agent systems , 2013, 2013 American Control Conference.

[18]  William B. Dunbar,et al.  Distributed receding horizon control for multi-vehicle formation stabilization , 2006, Autom..

[19]  Lacra Pavel,et al.  A continuous-time decentralized optimization scheme with positivity constraints , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

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

[21]  Anders Rantzer,et al.  Distributed Model Predictive Control with suboptimality and stability guarantees , 2010, 49th IEEE Conference on Decision and Control (CDC).

[22]  Bahman Gharesifard,et al.  Distributed Continuous-Time Convex Optimization on Weight-Balanced Digraphs , 2012, IEEE Transactions on Automatic Control.

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

[24]  Anders Rantzer,et al.  On Prize Mechanisms in linear quadratic team theory , 2007, 2007 46th IEEE Conference on Decision and Control.

[25]  David G. Luenberger,et al.  Linear and nonlinear programming , 1984 .

[26]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[27]  Gene F. Franklin,et al.  Feedback Control of Dynamic Systems , 1986 .

[28]  A. Ozdaglar,et al.  Convergence analysis of distributed subgradient methods over random networks , 2008, 2008 46th Annual Allerton Conference on Communication, Control, and Computing.

[29]  Reza Olfati-Saber,et al.  Distributed Kalman filtering for sensor networks , 2007, 2007 46th IEEE Conference on Decision and Control.

[30]  Jing Wang,et al.  A control perspective for centralized and distributed convex optimization , 2011, IEEE Conference on Decision and Control and European Control Conference.

[31]  Asuman E. Ozdaglar,et al.  Distributed Alternating Direction Method of Multipliers , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[32]  Jeff S. Shamma,et al.  Cooperative Control of Distributed Multi-Agent Systems: Shamma/Cooperative Control of Distributed Multi-Agent Systems , 2007 .

[33]  Marios M. Polycarpou,et al.  Cooperative Control of Distributed Multi-Agent Systems , 2001 .