Rate Analysis of Inexact Dual Fast Gradient Method for Distributed MPC

In this chapter we propose a dual decomposition method based on inexact dual gradient information and constraint tightening for solving distributed model predictive control (MPC) problems for network systems with state-input constraints. The coupling constraints are tightened and moved in the cost using the Lagrange multipliers. The dual problem is solved by a fast gradient method based on approximate gradients for which we prove sublinear rate of convergence. We also provide estimates on the primal and dual suboptimality of the generated approximate primal and dual solutions and we show that primal feasibility is ensured by our method. Our analysis relies on the Lipschitz property of the dual MPC function and inexact dual gradients. We obtain a distributed control strategy that has the following features: state and input constraints are satisfied, stability of the plant is guaranteed, whilst the number of iterations for the suboptimal solution can be precisely determined.

[1]  Manfred Morari,et al.  Towards computational complexity certification for constrained MPC based on Lagrange Relaxation and the fast gradient method , 2011, IEEE Conference on Decision and Control and European Control Conference.

[2]  Stephen J. Wright,et al.  Cooperative distributed model predictive control , 2010, Syst. Control. Lett..

[3]  Marcello Farina,et al.  Distributed predictive control: A non-cooperative algorithm with neighbor-to-neighbor communication for linear systems , 2012, Autom..

[4]  Yurii Nesterov,et al.  Efficiency of Coordinate Descent Methods on Huge-Scale Optimization Problems , 2012, SIAM J. Optim..

[5]  Bart De Schutter,et al.  Multi-agent model predictive control for transportation networks: Serial versus parallel schemes , 2008, Eng. Appl. Artif. Intell..

[6]  Jonathan P. How,et al.  Distributed Robust Receding Horizon Control for Multivehicle Guidance , 2007, IEEE Transactions on Control Systems Technology.

[7]  Bart De Schutter,et al.  A distributed optimization-based approach for hierarchical MPC of large-scale systems with coupled dynamics and constraints , 2011, IEEE Conference on Decision and Control and European Control Conference.

[8]  Yurii Nesterov,et al.  Smooth minimization of non-smooth functions , 2005, Math. Program..

[9]  Asuman E. Ozdaglar,et al.  Approximate Primal Solutions and Rate Analysis for Dual Subgradient Methods , 2008, SIAM J. Optim..

[10]  Ion Necoara,et al.  Parallel and distributed optimization methods for estimation and control in networks , 2011, 1302.3103.

[11]  Eduardo Camponogara,et al.  Distributed Optimization for Model Predictive Control of Linear Dynamic Networks With Control-Input and Output Constraints , 2011, IEEE Transactions on Automation Science and Engineering.

[12]  Jairo Espinosa,et al.  A comparative analysis of distributed MPC techniques applied to the HD-MPC four-tank benchmark , 2011 .

[13]  Bart De Schutter,et al.  A distributed optimization-based approach for hierarchical model predictive control of large-scale systems with coupled dynamics and constraints , 2011, ArXiv.

[14]  Ion Necoara,et al.  Efficient parallel coordinate descent algorithm for convex optimization problems with separable constraints: Application to distributed MPC , 2013, 1302.3092.

[15]  Jonathan P. How,et al.  Robust distributed model predictive control , 2007, Int. J. Control.

[16]  D. Q. Mayne,et al.  Suboptimal model predictive control (feasibility implies stability) , 1999, IEEE Trans. Autom. Control..

[17]  M. Patriksson,et al.  Ergodic convergence in subgradient optimization , 1998 .

[18]  Yurii Nesterov,et al.  First-order methods of smooth convex optimization with inexact oracle , 2013, Mathematical Programming.

[19]  Johan A. K. Suykens,et al.  Application of a Smoothing Technique to Decomposition in Convex Optimization , 2008, IEEE Transactions on Automatic Control.

[20]  Torbjörn Larsson,et al.  Lagrangian Relaxation via Ballstep Subgradient Methods , 2007, Math. Oper. Res..