# Structure Exploiting Numerical Algorithms for Optimal Control

Abstract: Efficient numerical algorithms for solving optimal control problems are important for commonly used advanced control strategies such as model predictive control (MPC), but can also be useful for advanced estimation techniques such as moving horizon estimation (MHE).In MPC the control input is computed by solving a constrained finite-time optimal control (CFTOC) problem on-line, and in MHE the estimated states are obtained by solving an optimization problem that often can be formulated as a CFTOC problem. Two common types of optimization methods for solving CFTOC problems are interior-point (IP) methods and active-set (AS) methods. In both IP and AS methods, the main computational effort is often the computation of the second-order search directions, which boils down to solving a sequence of systems of equations that correspond to unconstrained finite-time optimal control (UFTOC) problems. Hence, high-performing IP and AS methods for CFTOC problems rely on efficient numerical algorithms for solving UFTOC problems.When the solution to a CFTOC problem is computed using an AS type method, the system of equations that is solved in each iteration to compute the search direction is only changed by a low-rank modification between two AS iterations. In this thesis, it is shown how to exploit these structured modifications while still exploiting the UFTOC problem structure using the Riccati recursion. Furthermore, in the thesis direct (non-iterative) parallel algorithms for computing the solution to the search directions in both IP and AS methods are presented. These algorithms exploit, and retain, the sparse structure of the UFTOC problem such that no dense system of equations needs to be solved serially as in many other algorithms, and the proposed algorithms can be applied recursively to obtain logarithmic computational complexity growth in the prediction horizon length. Another approach to solve linear MPC problems is to use multiparametric quadratic programming (mp-QP), where the corresponding CFTOC problem can be solved explicitly off-line. This is referred to as explicit MPC. One of the main limitations with mp-QP is the memory requirement for storing the parametric solution. In this thesis, an approach for decreasing the required memory is presented, with the aim of making mp-QP and explicit MPC more useful in practical applications such as embedded systems with limited memory resources. The proposed approach exploits the structure from the QP problem in the parametric solution to reduce the memory footprint for explicit MPC and general mp-QP solutions. The algorithm can be used directly in mp-QP solvers or as a post-processing step.

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