Distributed MPC Via Dual Decomposition and Alternative Direction Method of Multipliers
A conventional way to handle model predictive control (MPC) problems distributedly is to solve them via dual decomposition and gradient ascent. However, at each time-step, it might not be feasible to wait for the dual algorithm to converge. As a result, t
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Distributed MPC Via Dual Decomposition and Alternative Direction Method of Multipliers F. Farokhi, I. Shames and K. H. Johansson
Abstract A conventional way to handle model predictive control (MPC) problems distributedly is to solve them via dual decomposition and gradient ascent. However, at each time-step, it might not be feasible to wait for the dual algorithm to converge. As a result, the algorithm might be needed to be terminated prematurely. One is then interested to see if the solution at the point of termination is close to the optimal solution and when one should terminate the algorithm if a certain distance to optimality is to be guaranteed. In this chapter, we look at this problem for distributed systems under general dynamical and performance couplings, then, we make a statement on validity of similar results where the problem is solved using alternative direction method of multipliers.
7.1 Introduction Model predictive control (MPC) can be used to control dynamical systems with input and output constraints while ensuring the optimality of the performance of the system with respect to cost functions [5, 15, 19]. Typically, the way that the This work was supported in part by the Swedish Research Council, the Swedish Foundation for Strategic Research, and the Knut and Alice Wallenberg Foundation. F. Farokhi (B) · K. H. Johansson School of Electrical Engineering, ACCESS Linnaeus Centre, KTH Royal Institute of Technology, Stockholm, Sweden e-mail: [email protected] K. H. Johansson e-mail: [email protected] I. Shames Department of Electrical and Electronic Engineering, University of Melbourne, Melbourne, Australia e-mail: [email protected]
J. M. Maestre and R. R. Negenborn (eds.), Distributed Model Predictive Control Made Easy, Intelligent Systems, Control and Automation: Science and Engineering 69, DOI: 10.1007/978-94-007-7006-5_7, © Springer Science+Business Media Dordrecht 2014
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control input is calculated at each time-step is via applying the first control in a sequence obtained from solving an optimal control problem over a finite or infinite horizon. The optimal problem is reformulated at each time step based on the available measurements at that time step. Traditionally, a full model of the system is required to solve the MPC problem and all the control inputs are calculated centrally. However, in large-scale interconnected systems, such as power systems [12, 18], water distribution systems [18], transport systems [17], manufacturing systems [8], biological systems [10], and irrigation systems [11], the assumption on knowing the whole model and calculating all the inputs centrally is often not realistic. Recently, much attention has been paid to solve MPC problems in a distributed way [6, 9, 16, 22, 23, 25]. The problem of distributed model predictive control using dual decomposition was considered in [25]. However, in solving any optimization problem when using dual decomposition methods the convergence behaviors of dual iterations does not necessarily coincid
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