Distributed Algorithm for Discrete-Time Lyapunov Equations

This paper investigates the problem of solving a unique solution to discrete-time Lyapunov equations (DTLE) using multi-agent networks. We propose a distributed algorithm where each agent only uses partial information of the matrices. The agents of the algorithm reach a consensus by exchanging information with their neighbors over an undirected connected graph. We provide convergence analysis and the convergence rate estimate for the proposed algorithm. Finally, convergence performance is verified by numerical simulations.

[1]  Angelia Nedic,et al.  On the rate of convergence of distributed subgradient methods for multi-agent optimization , 2007, 2007 46th IEEE Conference on Decision and Control.

[2]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.

[3]  Asuman E. Ozdaglar,et al.  A distributed Newton method for Network Utility Maximization , 2010, 49th IEEE Conference on Decision and Control (CDC).

[4]  Isak Nielsen,et al.  Distributed primal–dual interior-point methods for solving tree-structured coupled convex problems using message-passing , 2017, Optim. Methods Softw..

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

[6]  James Lam,et al.  Toward solution of matrix equation X = Af (X)B + C , 2011, 1211.0346.

[7]  Enrico Pontelli,et al.  Distributed Constraint Optimization Problems and Applications: A Survey , 2016, J. Artif. Intell. Res..

[8]  Dimitri P. Bertsekas,et al.  Incremental Gradient, Subgradient, and Proximal Methods for Convex Optimization: A Survey , 2015, ArXiv.

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

[10]  Wen-Wei Lin,et al.  Large-scale Stein and Lyapunov equations, Smith method, and applications , 2013, Numerical Algorithms.

[11]  Asuman E. Ozdaglar,et al.  Distributed Subgradient Methods for Convex Optimization Over Random Networks , 2011, IEEE Transactions on Automatic Control.

[12]  Anna Scaglione,et al.  Distributed Constrained Optimization by Consensus-Based Primal-Dual Perturbation Method , 2013, IEEE Transactions on Automatic Control.

[13]  Wen-Wei Lin,et al.  Solving large-scale continuous-time algebraic Riccati equations by doubling , 2013, J. Comput. Appl. Math..

[14]  James Lam,et al.  H∞ filtering of discrete-time fuzzy systems via basis-dependent Lyapunov function approach , 2007, Fuzzy Sets Syst..

[15]  M. Sadkane.  A low-rank Krylov squared Smith method for large-scale discrete-time Lyapunov equations , 2012 .

[16]  Vivek S. Borkar,et al.  Distributed Asynchronous Incremental Subgradient Methods , 2001 .

[17]  Shu Liang,et al.  Distributed Computation of Linear Matrix Equations: An Optimization Perspective , 2017, IEEE Transactions on Automatic Control.

[18]  Zhi-Quan Luo,et al.  On the linear convergence of the alternating direction method of multipliers , 2012, Mathematical Programming.

[19]  J.N. Tsitsiklis,et al.  Convergence Rates in Distributed Consensus and Averaging , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.