Hybrid Set-Seeking Systems: Model-Free Feedback Optimization via Hybrid Inclusions

This article aims to provide an accessible, tutorial-style introduction to hybrid extremum-seeking systems, which are model-free, feedback-optimization controllers that incorporate hybrid dynamics, meaning both continuous-time and discrete-time behaviors. Such systems arise when advanced control and optimization tools are needed to overcome the limitations of smooth feedback methods and to satisfy demanding transient and steady-state requirements in high-performance applications. They also appear when controllers must operate on plants that inherently exhibit hybrid behaviors, as is common in cyber-physical and autonomous systems that rely on digital sensing, computation, and actuation. To study hybrid extremum-seeking dynamics through control-theoretic methods, we first review the key concepts that support the development of perturbation theory for hybrid inclusions, forming the basis for averaging and singular perturbation analyses. We then show how these ideas apply to the design and evaluation of hybrid extremum-seeking algorithms for static and dynamic plants. Several examples are presented, including set-valued and switching algorithms under different switching regimes such as arbitrarily fast switching, dwell-time and average dwell-time constraints, and average activation time conditions. We also discuss state-based switching extremum seeking for obstacle-avoidance problems and gradient-Newton switching schemes. Additional topics include momentum-based and reset-type extremum seeking, intermittent updates, slowly varying parameters, hybrid filters, and safety-aware schemes that incorporate constraints. Across all these settings, we illustrate how perturbation-based methods traditionally used for extremum-seeking control naturally extend to hybrid systems when mild regularity assumptions are satisfied, and solutions are modeled on hybrid time domains.

💡 Research Summary

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This paper presents a comprehensive, tutorial‑style introduction to hybrid extremum‑seeking (ES) systems—model‑free feedback‑optimization controllers that combine continuous‑time and discrete‑time dynamics. The authors begin by reviewing classical ES theory, which relies on averaging and singular perturbation methods for smooth ordinary differential equations (ODEs). They point out that these tools assume Lipschitz continuity and smooth trajectories, which break down when the closed‑loop system exhibits jumps, resets, or logic‑based mode switches—features typical of hybrid dynamical systems found in cyber‑physical, robotic, and power‑electronics applications.

To bridge this gap, the paper reformulates ES algorithms as hybrid inclusions, i.e., differential inclusions for continuous flows and set‑valued difference inclusions for discrete jumps. Solutions are parameterized by a continuous time index (t) and a discrete jump index (j), yielding trajectories defined on hybrid time domains. Because standard uniform distance cannot capture closeness of trajectories that experience Zeno‑like behavior, the authors adopt graph convergence (set‑convergence of solution graphs) as the appropriate notion of ((T,\varepsilon))-closeness. Under mild regularity conditions (e.g., outer semicontinuity, locally boundedness), the perturbed hybrid system’s graph converges to that of the nominal system as the probing amplitude (\varepsilon) shrinks and the probing frequency (\omega) grows.

The core technical contributions are twofold:

1. Hybrid Averaging Theory – Extending classical averaging to hybrid time domains, the authors show that for sufficiently high (\omega) and small (\varepsilon), the original hybrid ES dynamics can be approximated by a set‑valued averaged vector field (\bar F(x)) together with a jump map (G(x)). This averaged system captures the essential gradient‑descent behavior of the unknown cost function while preserving the discrete switching logic.

2. Singular Perturbation for Hybrid Systems – By separating fast probing dynamics from slow optimization dynamics, the paper derives a multi‑time‑scale singular‑perturbation model for hybrid ES. Graph convergence results guarantee that stability properties (Lyapunov, LaSalle) of the reduced averaged system transfer to the full hybrid system, even in the presence of set‑valued dynamics.

With this theoretical foundation, the authors design a rich catalog of hybrid ES algorithms:

• Arbitrarily Fast Switching – No dwell‑time restriction; stability is ensured via average activation‑time constraints.
• Dwell‑Time and Average Dwell‑Time Switching – Minimum inter‑switch intervals prevent Zeno phenomena while preserving convergence.
• State‑Based Switching for Obstacle Avoidance – The controller switches modes based on the plant’s state, modeled as a set‑valued jump rule.
• Gradient–Newton Switching – Alternates between gradient flow and Newton‑type updates to accelerate convergence.
• Momentum, Reset, and Intermittent Update Schemes – Incorporate inertial terms, periodic state resets, or sporadic updates to improve transient performance.
• Safety‑Aware Schemes – Integrate projection or barrier‑function mechanisms into the hybrid inclusion framework to enforce constraints during optimization.

The paper treats both static maps (where the plant is a time‑invariant unknown function) and dynamic plants (where the plant itself follows ODEs or hybrid dynamics). For dynamic plants, a two‑level singular‑perturbation analysis separates the plant’s fast dynamics from the slower ES controller, yielding a hierarchical hybrid system whose overall stability follows from the composition of the two reduced models.

Numerical examples illustrate each algorithmic variant, demonstrating robustness to switching speed, parameter variations, and external disturbances. The authors emphasize that the hybrid ES framework retains the model‑free advantage of classical ES while extending its applicability to systems that require digital sampling, event‑triggered actions, or safety constraints—situations where purely smooth ES would either fail or be overly conservative.

In summary, the paper establishes a unified, mathematically rigorous framework for hybrid set‑seeking systems. By extending averaging and singular‑perturbation techniques to hybrid inclusions and by employing graph‑convergence as the metric of closeness, it provides designers with tools to analyze stability, convergence rate, and robustness of a wide spectrum of hybrid ES controllers. This work opens the door to high‑performance, real‑time, model‑free optimization in modern cyber‑physical and autonomous systems where hybrid dynamics are the norm.