Decompositie Methoden voor Gemengd-Gehele Optimalisatieproblemen met Wachttijdbeperkingen

Control of dynamical systems is central to many real-world applications. Many of these systems involve discrete-valued inputs. Examples include systems operated by switches or valves, automobiles with discrete gear ratios, hybrid systems such as heating-cooling units, robots performing discrete tasks, and, more generally, systems with multiple modes of operation. Designing effective controllers for such systems often requires optimization-based approaches, collectively referred to as optimal control. This optimization is commonly modeled as a Mixed-Integer Optimal Control Problem (MIOCP), which is considerably more challenging than its counterparts with continuous-valued control inputs. The difficulty arises because the feasible set of inputs is disjoint, rendering continuous optimization tools inapplicable, while brute-force enumeration is impractical. Physical systems further impose constraints on how quickly control inputs can switch between discrete values. For instance, a switch cannot be toggled arbitrarily fast. These so-called Minimum Dwell Time (MDT) constraints deepen the divide between continuous and discrete inputs, revealing the tension between the practical importance of solving MIOCP with MDT constraints and their inherent computational complexity. This dissertation addresses these challenges by developing efficient solution methods for general MIOCPs under MDT constraints, making them viable for real-time implementation. The focus is on decomposition methods, which have shown promise in the literature. Such methods break the original problem into subproblems that are easier to solve, hence reducing computation costs. Two decompositions are considered. The first builds on the idea that fast switching of discrete inputs can approximate the effect of continuous inputs. By relaxing the discreteness, one can apply continuous optimization techniques and approximate the resulting behavior with fast switching while respecting MDT constraints. The Combinatorial Integral Approximation (CIA) decomposition is built on this foundation. The second builds on the observation that a discrete input can be represented as a sequence of values together with dwell times for each value. With this distinction, the problem decomposes into finding the optimal sequence of values, a Sequence Optimization (SO) that captures the combinatorial challenge, and the optimal dwell times, a Switching Time Optimization (STO) that reduces to continuous optimization. Three methods are developed from these decompositions. Chapter 3 introduces Dynamic Programming-inspired Approximation (DPA), an approximation strategy that overcomes shortcomings of existing CIA-based techniques. It advances the state of the art by accommodating more general MDT constraints and providing improved approximations for a wider class of problems. Building on the STO/SO decomposition, Chapter 4 presents a novel formulation for SO. Combined with STO, this results in Mixed-Integer Sequence Decomposition (MISD), which outperforms approximation-based methods by optimizing switching times without being set back by exponential tree search in SO. Finally, Chapter 5 introduces Iterative Switching Time Optimization (ISTO), a heuristic method for solving SO. ISTO achieves near-optimal solutions comparable to MISD while maintaining substantially lower computational costs, bridging the gap between the efficiency of DPA and the optimality of MISD. The claims are supported by numerous examples and a consistent treatment of five benchmark MIOCP problems. While inherent limitations are discussed in the following chapters, the presented methods achieve low computation costs, paving the way for the real-time solution of MIOCP with MDT constraints.