arXiv subjects

Pontus Giselsson

Publications and source records attributed to Pontus Giselsson.

40 records · Page 3Linked to original sources

arXiv2014

Improving Fast Dual Ascent for MPC - Part I: The Distributed Case

In dual decomposition, the dual to an optimization problem with a specific structure is solved in distributed fashion using (sub)gradient and recently also fast gradient methods. The traditional dual decomposition suffers from two main short-comings. The first is that the convergence is often slow, although fast gradient methods have significantly improved the situation. The second is that computation of the optimal step-size requires centralized computations, which hinders a fully distributed implementation of the algorithm. In this paper, the first issue is addressed by providing a tighter characterization of the dual function than what has previously been reported in the literature. Then a distributed and a parallel algorithm are presented in which the provided dual function approximation is minimized in each step. Since the approximation is more accurate than the approximation used in standard and fast dual decomposition, the convergence properties are improved. For the second issue, we extend a recent result to allow for a fully distributed parameter selection in the algorithm. Further, we show how to apply the proposed algorithms to optimization problems arising in distributed model predictive control (DMPC) and show that the proposed distributed algorithm enjoys distributed reconfiguration, i.e. plug-and-play, in the DMPC context.

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arXiv2014

Improving Fast Dual Ascent for MPC - Part II: The Embedded Case

Recently, several authors have suggested the use of first order methods, such as fast dual ascent and the alternating direction method of multipliers, for embedded model predictive control. The main reason is that they can be implemented using simple arithmetic operations only. However, a known limitation of gradient-based methods is that they are sensitive to ill-conditioning of the problem data. In this paper, we present a fast dual gradient method for which the sensitivity to ill-conditioning is greatly reduced. This is achieved by approximating the negative dual function with a quadratic upper bound with different curvature in different directions in the algorithm, as opposed to having the same curvature in all directions as in standard fast gradient methods. The main contribution of this paper is a characterization of the set of matrices that can be used to form such a quadratic upper bound to the negative dual function. We also describe how to choose a matrix from this set to get an improved approximation of the dual function, especially if it is ill-conditioned, compared to the approximation used in standard fast dual gradient methods. This can give a significantly improved performance as illustrated by a numerical evaluation on an ill-conditioned AFTI-16 aircraft model.

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arXiv2013

On feasibility, stability and performance in distributed model predictive control

In distributed model predictive control (DMPC), where a centralized optimization problem is solved in distributed fashion using dual decomposition, it is important to keep the number of iterations in the solution algorithm, i.e. the amount of communication between subsystems, as small as possible. At the same time, the number of iterations must be enough to give a feasible solution to the optimization problem and to guarantee stability of the closed loop system. In this paper, a stopping condition to the distributed optimization algorithm that guarantees these properties, is presented. The stopping condition is based on two theoretical contributions. First, since the optimization problem is solved using dual decomposition, standard techniques to prove stability in model predictive control (MPC), i.e. with a terminal cost and a terminal constraint set that involve all state variables, do not apply. For the case without a terminal cost or a terminal constraint set, we present a new method to quantify the control horizon needed to ensure stability and a prespecified performance. Second, the stopping condition is based on a novel adaptive constraint tightening approach. Using this adaptive constraint tightening approach, we guarantee that a primal feasible solution to the optimization problem is found and that closed loop stability and performance is obtained. Numerical examples show that the number of iterations needed to guarantee feasibility of the optimization problem, stability and a prespecified performance of the closed-loop system can be reduced significantly using the proposed stopping condition.

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arXiv2013

A distributed accelerated gradient algorithm for distributed model predictive control of a hydro power valley

A distributed model predictive control (DMPC) approach based on distributed optimization is applied to the power reference tracking problem of a hydro power valley (HPV) system. The applied optimization algorithm is based on accelerated gradient methods and achieves a convergence rate of O(1/k^2), where k is the iteration number. Major challenges in the control of the HPV include a nonlinear and large-scale model, nonsmoothness in the power-production functions, and a globally coupled cost function that prevents distributed schemes to be applied directly. We propose a linearization and approximation approach that accommodates the proposed the DMPC framework and provides very similar performance compared to a centralized solution in simulations. The provided numerical studies also suggest that for the sparsely interconnected system at hand, the distributed algorithm we propose is faster than a centralized state-of-the-art solver such as CPLEX.

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