Optimization in Control Applications

Mathematical optimization is the selection of the best element in a set with respect to a given criterion. Optimization has become one of the most used tools in modern control theory for computing the control law, adjusting the controller parameters (tuning), model fitting, finding suitable conditions in order to fulfill a given closed-loop property, etc. In the simplest case, optimization consists of maximizing or minimizing a function by systematically choosing input values from a valid input set and computing the function value. To solve optimization problems, researchers can use algorithms that end in a finite number of steps, or iterative methods that converge to a solution (in some specific class of problems), or heuristics that can provide approximate solutions to some problems (although their iterations do not necessarily converge). In practice, real-world control systems need to comply with several conditions and physical and product-quality constraints that have to be taken into account in the problem formulation. These represent challenges in the application/implementation of the optimization algorithms, particularly when the solutions of these optimization problems have to be computed in a constrained time window and/or in an embedded platform.