Timescale Separation in Autonomous Optimization

Abstract

Autonomous optimization refers to the design of feedback controllers that steer a physical system to a steady state that solves a predefined, possibly constrained, optimization problem. As such, no exogenous control inputs such as setpoints or trajectories are required. Instead, these controllers are modeled after optimization algorithms that take the form of dynamical systems. The interconnection of this type of optimization dynamics with a physical system is however not guaranteed to be stable unless both dynamics act on sufficiently different timescales. In this paper, we quantify the required timescale separation and give prescriptions that can be directly used in the design of this type of feedback controllers. Using ideas from singular perturbation analysis we derive stability bounds for different feedback optimization schemes that are based on common continuoustime optimization schemes. In particular, we consider gradient descent and its variations, including projected gradient, and Newton gradient. We further give stability bounds for momentum methods and saddle-point flows interconnected with dynamical systems. Finally, we discuss how optimization algorithms like subgradient and accelerated gradient descent, while well-behaved in offline settings, are unsuitable for autonomous optimization due to their general lack of robustness.

Model-Free Nonlinear Feedback Optimization

“…Others take system dynamics into account and characterize sufficient conditions for the closedloop stability, including in continuous-time [21], [27]- [29] and sampled-data settings [30]. Specifically, among works that handle nonconvex objectives and nonlinear systems, [22], [26] address discrete-time systems represented by algebraic maps, while [28] tackles continuous-time systems. The results on nonconvex feedback optimization for discrete-time nonlinear dynamical systems are still lacking.…”

When can we safely return to normal? A novel method for identifying safe levels of NPIs in the context of COVID-19 vaccinations

“…In words, the controller adapts in real time the NPI so that the objectives set forth in (4) are met. The proposed method relies on feedback-based online optimization techniques for dynamical systems [13,14,15,16], and it is described next. We begin by defining the set of feasible NPIs, as described in the optimization problem (4), which, for each state x, is defined as:…”

“…The stability of the interconnection in Figure 9 can be investigated using arguments from singular perturbation theory [58]; in particular, when the set U x is constant, one can utilize the procedure explained in [15,16] to show that there exists a rage of values for the gain η that renders the strict optimizers of the optimization problems asymptotically stable [15,16].…”

“…We consider two variations of this model: a single-region model, which describes the behavior of the entire population at the aggregate-level neglecting regional diversities, and a more-comprehensive network model that takes into account structured coupling between regions. (ii) We leverage recent advances in feedback-based online optimization of dynamical systems [13,14,15,16,17] to synthesize a control algorithm that selects the level of transmission-relevant contact (hereafter referred to as contact level) needed under varying levels of tolerance in hospitalizations. Here, the objective of the control policy is to minimize the severity of the NPIs and behavioral modifications (i.e., to minimize the consequences on economic, as well as social, and individual-level health) while ensuring that a pre-specified hospitalization limit is not exceeded.…”

Sampled-Data Online Feedback Equilibrium Seeking: Stability and Tracking

“…Asymptotic stability results for linear time invariant systems (LTI) coupled with saddle-flow dynamics are provided in [16]. The interconnection of input-constrained nonlinear systems with gradient flow dynamics is studied in [17]. A design framework for unconstrained online optimization of LTI systems is presented in [18] and related work on low-gain integral controllers for nonlinear systems can be found in [19].…”