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An Alternating Direction Method of Multipliers Algorithm for Symmetric MPC
Abstract
This paper presents an alternating-direction method of multipliers (admm) algorithm for solving large-scale symmetric model predictive control (MPC) problems in real-time on embedded computers with limited computational and memory resources. Symmetry was used to find transformations of the states, inputs, and constraints of the mpc problem that decompose the dynamics and cost. We prove a key-property of the symmetric group that allows us to efficiently transform between the original and decomposed symmetric domains. This allows us to solve different sub-problems of a baseline admm algorithm in different domains where the computations are less expensive. This reduces the computational cost of each iteration from quadratic in problem size to linear. In addition, we show that our admm algorithm requires a constant amount of memory regardless of the problem size. We demonstrate our algorithm for a battery balancing problem which results in a reduction of computation-times from hours to seconds and a reduction in memory from hundreds of megabytes to tens of kilobytes.
... In particular, convex optimization algorithms have been considered as a conductive solution for their computational efficiency and parallelizable characteristic. Gradient-based convex optimization techniques, such as the alternating direction method of multipliers (ADMM) [17][18][19], primal-dual interior point method (PD-IPM), parallel quadratic programming (PQP) [20,21], and active set method (ASM) [22][23][24], are employed. In this work, PD-IPM [24,25], which is the most commonly used technique for convex optimization, is applied. ...
This paper addresses the problem of real-time model predictive control (MPC) in the integrated guidance and control (IGC) of missile systems. When the primal-dual interior point method (PD-IPM), which is a convex optimization method, is used as an optimization solution for the MPC, the real-time performance of PD-IPM degenerates due to the elevated computation time in checking the Karush–Kuhn–Tucker (KKT) conditions in PD-IPM. This paper proposes a graphics processing unit (GPU)-based method to parallelize and accelerate PD-IPM for real-time MPC. The real-time performance of the proposed method was tested and analyzed on a widely-used embedded system. The comparison results with the conventional PD-IPM and other methods showed that the proposed method improved the real-time performance by reducing the computation time significantly.
... Decoupling the system dynamics allows for a parallel implementation of the algorithm. In [120], another ADMM based solver for MPC that decouples the system dynamics is presented, in this case by exploiting the symmetry of the system. ...
This Ph.D. dissertation contains results in two different but related fields: the implementation of model predictive control (MPC) in embedded systems using first order methods, and restart schemes for accelerated first order methods (AFOM). We start by presenting three novel restart schemes for AFOM. These schemes can improve the convergence of the AFOM by suppressing the undesirable oscillations that they are prone to present. The schemes we develop have theoretical guarantees and do not require knowledge of difficult-to-obtain parameters of the optimization problem. Next, we present sparse solvers for various MPC formulations which take advantage of the structures of the optimization problems. The solvers have been made available in an open-source toolbox for Matlab called SPCIES (https://github.com/GepocUS/Spcies). Finally, we present a novel MPC formulation that displays a larger domain of attraction and better performance than other MPC formulations, especially when using small prediction horizons. This, along with its recursive feasibility and asymptotic stability, makes it especially suitable for its implementation in embedded systems.
... The alternating direction method of multipliers (ADMM) was developed in the 1970s [5] and belongs to the family of augmented Lagrangian techniques [6]. The ADMM has been applied in many areas, including signal and image processing [7][8][9], statistics and machine learning [10], and system control [11]. In 2011, Afonso et al. [6] developed a fast method for solving constrained optimization problems using variable splitting [12] and the ADMM, which is known as the constrained split augmented Lagrangian shrinkage algorithm (C-SALSA). ...
We investigate the effects of the regularization parameter for the norm () and penalty parameter () in the alternating direction method of multipliers (ADMM) on the quality of restored medical images. Simulation studies are performed using images degraded by a point spread function (PSF) and Gaussian noise. The j-th column of the system matrix () is calculated by convolving the image with unity at pixel j and zero at all other pixels and the PSF. The simulation studies show that the mean structural similarity index is maximal when is approximately 10 to 20, where , with and being the transpose of A and the observed data, respectively. The restored image became blurred with a decrease in . This study will be useful for identifying optimal parameter values in the ADMM when applied to medical image restoration.
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