Optimization Algorithms as Robust Feedback Controllers

Abstract

Mathematical optimization is one of the cornerstones of modern engineering research and practice. Yet, throughout all application domains, mathematical optimization is, for the most part, considered to be a numerical discipline. Optimization problems are formulated to be solved numerically with specific algorithms running on microprocessors. An emerging alternative is to view optimization algorithms as dynamical systems. While this new perspective is insightful in itself, liberating optimization methods from specific numerical and algorithmic aspects opens up new possibilities to endow complex real-world systems with sophisticated self-optimizing behavior. Towards this goal, it is necessary to understand how numerical optimization algorithms can be converted into feedback controllers to enable robust "closed-loop optimization". In this article, we review several research streams that have been pursued in this direction, including extremum seeking and pertinent methods from model predictive and process control. However, our primary focus lies on recent methods under the name of "feedback-based optimization". This research stream studies control designs that directly implement optimization algorithms in closed loop with physical systems. Such ideas are finding widespread application in the design and retrofit of control protocols for communication networks and electricity grids. In addition to an overview over continuous-time dynamical systems for optimization, our particular emphasis in this survey lies on closed-loop stability as well as the enforcement of physical and operational constraints in closed-loop implementations. We further illustrate these methods in the context of classical problems, namely congestion control in communication networks and optimal frequency control in electricity grids, and we highlight one potential future application in the form of autonomous reserve dispatch in power systems.

Model-Free Nonlinear Feedback Optimization

“…Then, with the interconnection of the plant and the controller, the coupled decay of the above two quantities are synthesized to obtain a bound on the convergence measure. This path of analysis is different from those took by the related literature, e.g., singular perturbation analysis [10], [27], [28], LMI stability certificates [21], and invariance principles [26]. We also elaborate on the extension to tackle coupling constraints on the inputs via Frank-Wolfe type updates.…”

“…As an emerging paradigm, feedback optimization (FO) [10], [11] offers a promising approach to drive general systems to efficient operating points, which constitute the optimal solutions of problems involving steady-state inputs and outputs. The key idea of FO is to implement optimization algorithms as feedback controllers, which are connected with physical plants to form a closed loop.…”

“…Various approaches also share this closed-loop optimization viewpoint, including modifier adaptation [12], real-time iteration schemes [13], and realtime model predictive control [14]. Compared to these approaches, FO requires limited model information and less computational effort (see the review in [10]). By utilizing real-time measurements of system outputs, FO circumvents the need of accessing input-output models to evaluate the gradient of the objective function at the current operating point.…”

“…where v k , v k−1 are standard normal random vectors, and δ is a smoothing parameter as in (8). The update (10) describes the actions performed by the controller at time k + 1. First, a new candidate solution w k+1 is computed based on φk , which is the gradient estimate at the previous candidate solution w k .…”

“…In this section, we analyze the performance of implementing the proposed feedback optimization controller (10) in closed loop with the system (1). We first establish some supporting lemmas on the bounds and recursive inequalities of certain important variables.…”