Hamiltonian Decomposition for Online Implementation of Model Predictive Control

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Abstract

Reliable and optimal decision-making tools are essential in industry to achieve high performance. One of these tools is Model Predictive Control (MPC), which is an advanced control technique that generates an action that affects the controlled variables of a given process with respect to a performance criteria, while satisfying the process' physical and operational restrictions. At the core of the MPC algorithm lies an optimization problem that is solved by a numerical method at every sample time. New demand for more self-contained and autonomous modular processes has seen MPC embedded in small-scale platforms, such as Programmable Logic Controllers (PLCs). This has prompted a need for custom-made numerical methods that help to efficiently run the computationally demanding optimization algorithms. In this thesis, we design effective optimization solvers for PLCs by proposing several approaches that factorize the Newton system of the interior-point method (IPM). These approaches are based on the two-point boundary-value (TPBV) problem structure, rarely explored in MPC, called the Hamiltonian system. One of the main proposals is that, once the augmented system is in the Hamiltonian form, it can be reduced to an incomplete LU factorization in which two possible options are available to compute the solution of the system: (i) A direct method called the Hamiltonian recursion method, and (ii) an iterative method called the Hamiltonian/GMRES method. Regarding the former, a forward substitution of a sequence of matrices is carried out, whereas with the latter, a Krylov method is used. We prove that the convergence of the iterative method is bounded and its rate is quantified. Numerical experiments demonstrate that both methods are feasible and efficient compared to the state-of-the-art methods.

Keyword(s)

Direct Methods; Iterative Methods; Model Predictive Control; Online Implementation; Optimization Algorithms; Real-time Optimization Algorithms

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