Algoritmische vooruitgang in haalbaar-iteratie- en SQP-methoden voor directe optimale controle

In the modern world, technical systems need to perform increasingly complex tasks in an automated fashion. For mechatronic systems, adequate performance often requires the use of advanced control strategies. One such strategy is Model Predictive Control (MPC), which effectively addresses these complex tasks by considering constraints while minimizing specific performance objectives, such as time or energy consumption. MPC depends on the iterative solution of Nonlinear Optimization Problems (NLP), which are derived from optimal control problems. These NLPs must be solved within a designated sampling period to ensure the system operates smoothly. The key challenges are: first to reliably find a solution for the NLPs and second to do so within the given time limit. This thesis focuses on the first challenge by enhancing the robustness of state-of-the-art optimization methods. This is achieved by introducing a mechanism that enables solvers to find solutions from arbitrary starting points, ensuring global convergence. Moreover, it is crucial to address challenges related to finite precision arithmetic and numerical instabilities. Firstly, we examine a Feasible Sequential Linear Programming (FSLP) algorithm tailored for time-optimal control problems. The main advantage of this method is that all its iterates remain feasible, allowing the solver to be stopped at suboptimal points. Unlike standard methods, these intermediate iterates can still benefit MPC applications, as all constraints would be satisfied. We introduce and analyze a novel method to guarantee the feasibility of all iterates. The FSLP algorithm is compared against the state-of-the-art solver ipopt on a time-optimal control problem. In a second version of the algorithm, we enhance these feasibility iterations using Anderson acceleration, which results in a 40% performance improvement in CPU time of the overall algorithm. However, the algorithm reveals several impracticalities, which lead to the development of a more general sequential linear programming algorithm inspired by a tolerance-tube method. The tolerance-tube is a relaxation of the feasible set that gives the iterates more flexibility which allows for faster convergence and faster execution time. This superior performance is demonstrated in numerical experiments. Secondly, we investigate the funnel method to promote global convergence of Sequential Quadratic Programming (SQP) methods. The funnel method generalizes the tolerance-tube approach and closely relates to filter methods. We implement the funnel method in the uno solver and compare it against the filterSQP preset of uno. Our algorithm demonstrates superior performance on the CUTEst test set due to its straightforward implementation and slightly faster convergence when compared to the filter method. Thirdly, insights gained from the previous developments facilitate the creation of a numerically robust SQP solver within the acados high-performance optimization framework. We implement a line-search funnel-penalty method to guarantee global convergence of the solver. To avoid numerical problems, we employ proper scaling of the Quadratic Problems (QPs). In cases where QPs are infeasible, the solver switches to a phase that constructs a search direction that retains the objective gradient information by solving two feasible subproblems. The increased numerical robustness and improved convergence is illustrated through challenging NLPs and motion planning problems. This thesis concludes with the development of a structure-exploiting method for finding feasible trajectories. If the solver fails to find a solution within the given sampling time, it can switch to the feasibility solver, which efficiently seeks feasible trajectories. We formulate a feasibility problem that possesses favorable theoretical properties. Building on this formulation, we develop and implement a Differential Dynamic Programming (DDP) algorithm in acados. The solver is robust and achieves rapid local convergence to feasible points.