Optimization Algorithms for Regulation of Dynamic Processes

In this tutorial we consider the problem of designing optimization algorithms for settings where the solution is used to regulate a physical plant. This scenario is ubiquitous in engineering applications e.g. model predictive controllers for regulating industrial processes, safety filters in robotics, and online feedback optimization for power systems. In these applications, the optimization problem often incorporates constraints which, when violated, would threaten the safe operation of the system. Thus, we require that the algorithm is anytime, meaning that it is guaranteed to return a feasible point even when terminated before it converges to a solution. To solve this problem, rather than adopting the dominant view of an optimization algorithm as simply a computational method, we instead approach its design from a systems theoretic perspective. This is because the optimization algorithm, along with the physical process it regulates, can be viewed as a pair of interconnected dynamical systems, which is naturally analyzed using control theoretic tools. Our solution begins by considering a "static" nonlinear optimization problem and introducing an anytime solver in the form of a continuous time dynamical system – derived using techniques from safety critical control – called the safe gradient flow. Next, we "close the loop" and analyze the interconnection of the safe gradient flow with a dynamical system modeling the plant. Finally, to address the practical implementation of these methods on digital computers, we tackle the challenge of discretizing the safe gradient flow.