Structure exploiting optimization methods for model predictive control
Abstract: This thesis considers optimization methods for Model Predictive Control (MPC). MPC is the preferred control technique in a growing set of applications due to its ﬂexibility and to the natural way in which constraints can be incorporated in the control policy. Its applicability is, however, limited by the high computational burden associated with the solution of the underlying optimization problems. To alleviate this drawback we study structures in the MPC problems, which can enhance their solution. The ﬁrst topic of the thesis is numerical structures in matrices arising in gradient-based optimization methods for MPC. The idea is that due to the nu-merical structures, dense matrices can be approximated by sparse matrices to reduce the computational cost per iteration, and also for the overall solution of the MPC problem. The second topic of the thesis is parallelizable optimization methods for multi-stage MPC. Multi-stage MPC is a popular framework used to increase the robustness of MPC schemes. One major drawback, however, is that the under-lying optimization problems become very large. In this context, we consider parallel implementations of two diﬀerent classes of optimization methods. First, we propose a parallelizable linear algebra for a primal-dual interior point method for two-stage MPC problems, i.e. for multi-stage MPC problems where the sce-nario tree is restricted to only branch in its root node. Secondly, we consider Newton’s method to solve the Lagrange dual problem of multi-stage MPC prob-lems. We show that the Hessian of the dual function is permutation similar to a block-tridiagonal matrix, propose a strategy for reducing the need for regularization, and reduce the cost of globalization strategies for problems with simple constraints and a diagonal cost. The third topic of the thesis is optimization methods for solving distributed MPC problems in a distributed fashion using dual decomposition. Dual decomposition is commonly used with gradient-based methods to achieve a completely distributed method. In this thesis, however, we use dual decomposition together with Newton’s method to achieve semi-distributed methods with a fast practical convergence. We study the occurence of a singular dual Hessian and pro-pose a constraint relaxation to prevent it. Additionally, we propose a distributed dual Newton strategy which can be viewed as a distributed primal-dual interior point method, and study the numerical structure of the dual Hessian for problems stemming from MPC deployed on multi-agent systems that are interacting via non-delayed couplings.
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