Model-Free Nonlinear Feedback Optimization

TL;DR

The paper tackles autonomous operation of general nonlinear discrete-time plants by designing a model-free feedback controller that optimizes a nonconvex steady-state objective without requiring the plant model or steady-state sensitivities. It builds zeroth-order gradient estimates from current and past objective evaluations and real-time outputs, updating inputs through a descent-like scheme that is robust to disturbances. A Lyapunov-based analysis yields non-asymptotic performance certificates that quantify how optimality scales with problem dimension $p$, iteration count $T$, and the plant’s contraction rate $\mu$, and the method is extended to constrained inputs via Frank-Wolfe-type updates with corresponding performance guarantees. Numerical evaluations on a nonlinear 10-state, 5-input system show competitive performance with first-order controllers using exact or estimated sensitivities and illustrate the practical viability of fully model-free, gradient-free optimization in closed-loop settings. The work advances model-free optimization for nonconvex objectives under dynamics and constraints, with implications for robust, data-driven control in networks, power systems, and process control.

Abstract

Feedback optimization is a control paradigm that enables physical systems to autonomously reach efficient operating points. Its central idea is to interconnect optimization iterations in closed-loop with the physical plant. Since iterative gradient-based methods are extensively used to achieve optimality, feedback optimization controllers typically require the knowledge of the steady-state sensitivity of the plant, which may not be easily accessible in some applications. In contrast, in this paper, we develop a model-free feedback controller for efficient steady-state operation of general dynamical systems. The proposed design consists of updating control inputs via gradient estimates constructed from evaluations of the nonconvex objective at the current input and at the measured output. We study the dynamic interconnection of the proposed iterative controller with a stable nonlinear discrete-time plant. For this setup, we characterize the optimality and stability of the closed-loop behavior as functions of the problem dimension, the number of iterations, and the rate of convergence of the physical plant. To handle general constraints that affect multiple inputs, we enhance the controller with Frank-Wolfe-type updates.

Figures (5)

• Figure 1: An illustration of the interconnection of the physical plant and the model-free feedback optimization controller.
• Figure 2: Comparison of the proposed model-free feedback controller, the first-order counterparts, and a stochastic extremum seeking algorithm. Note that SE represents sensitivity estimation.
• Figure 3: Performance of the proposed model-free feedback controller with different $\delta$ and $\eta$.
• Figure 4: Comparison between the proposed model-free feedback controller and the first-order counterparts for solving the problem with the input constraint set. FW and PD refer to model-free FO based on Frank-Wolfe-type updates \ref{['eq:updateOPTCons']} and projected descent \ref{['eq:updateDTproj']}, respectively.
• Figure 5: Comparison of different methods in tracking the trajectory of time-varying optimal solutions.

Theorems & Definitions (27)

• Lemma 1: nesterov2017random
• Remark 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 5
• proof