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An alternating direction method of multipliers algorithm for symmetric model predictive control

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Abstract

This article presents an alternating direction method of multipliers (ADMM) algorithm for solving large‐scale model predictive control (MPC) problems that are invariant under the symmetric‐group. Symmetry was used to find transformations of the inputs, states, and constraints of the MPC problem that decompose the dynamics and cost. We prove an important property of the symmetric decomposition for the symmetric‐group that allows us to efficiently transform between the original and decomposed symmetric domains. This allows us to solve different subproblems of a baseline ADMM algorithm in different domains where the computations are less expensive. This reduces the computational cost of each iteration from quadratic to linear in the number of repetitions in the system. In addition, we show that the memory complexity for our ADMM algorithm is also linear in number of repetitions in the system, rather than the typical quadratic complexity. We demonstrate our algorithm for two case studies; battery balancing and heating, ventilation, and air conditioning. In both case studies, the symmetric algorithm reduced the computation‐time from minutes to seconds and memory usage from tens of megabytes to tens or hundreds of kilobytes, allowing the previously nonviable MPCs to be implemented in real time on embedded computers with limited computational and memory resources.

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... An efficient method has been proposed in [12], [13] where an equivalent simple affine function has been employed to remove the regions whose associated control laws attain a saturated value, leading to significant reduction in the storage requirement. In another proposed approach [14], [15], symmetries of the MPC problem have been computed as a mathematical problem in order to reduce inherent complexity of explicit MPC with no change on optimality of the original representation. Identification of the set of all controller symmetries plays an important role in efficiency of the method of eliminating the symmetric regions [14]. ...
... Identification of the set of all controller symmetries plays an important role in efficiency of the method of eliminating the symmetric regions [14]. To identify these controller symmetries, however, [15], [16] has shown the fact that the symmetries corresponding to control regions can be identified by using graph theory. For this objective, it is shown that the problem of finding controller symmetries is converted to graph automorphism problem which can easily be solved by the standard graph automorphism software packages [17], [19]. ...
... Ref. [34] exploited symmetries in interior-point algorithms involved in the solution of optimization problems. References [35], [36], [37] used symmetry to reduce the computational complexity required by the solution of model predictive control problems. While all these papers have exploited symmetry to reduce the computational complexity involved in the solution of problems of different nature, in this paper we recognize that sometimes dimensionality reductions are possible even in systems that do not possess symmetry. ...
... As an example of this, the reader may consider the distinction between 'orbit partitions' and 'equitable partitions' in graph theory [38], where the former is generated by symmetry, while the latter is not. References [36], [26], [32], [39], [40], [41] have focused on a decomposition based on the symmetries of the system and, if applicable, of the objective function, which requires calculation of the irreducible representations (IRR) of the symmetry group. The goal of this paper is to compare the symmetry approach with an alternative symmetry-independent approach, which we will show yields better and faster decompositions. ...
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