Extremum Seeking Control in Nonlinear Dynamical Systems

Extremum seeking control (ESC) is a model-free adaptive technique for driving a nonlinear dynamical system towards an optimal operating point in real time. Rather than relying on detailed mathematical models, ESC introduces small perturbations—often sinusoidal dither signals—to the system input and uses measurements of the resulting output fluctuations to estimate the local gradient of an unknown performance map. Through iterative adjustment of the input parameters, the system converges toward a steady-state extremum, typically a maximum or minimum of a cost or utility function. Over the past century, ESC has progressed from early heuristic schemes to a mature framework underpinned by averaging theory, Lyapunov stability analysis and Lie bracket approximations. Modern developments address practical issues such as input and output constraints, time-varying delays and safety requirements, expanding the applicability of ESC across domains as diverse as chemical reactors, renewable energy systems, robotics and aerospace. The global significance of ESC lies in its ability to optimise complex processes without exhaustive system identification, enabling robust and efficient operation under uncertainty. Recent advances have focused on sampled-data implementations that honour actuator limits, barrier-based methods for guaranteed constraint satisfaction and safety-filtered designs that preserve viability through control barrier functions.

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A comprehensive survey published in 2024 synthesises a century of ESC literature, tracing its evolution from initial engineering inventions to modern analytical frameworks. This work categorises algorithmic variants, outlines convergence proofs for broad classes of nonlinear systems and highlights industrial and laboratory applications, demonstrating the method’s enduring relevance and adaptability. In parallel, a 2022 study introduced a sampled-data ESC framework that explicitly incorporates unknown input and output constraints via barrier-function methods. The proposed scheme employs output-only measurements of both performance and constraint functions, embedding them into a closed-loop design that guarantees stability, constraint satisfaction at each iteration and convergence to the constrained optimum. This development addresses the practical necessity of operating within safety margins and actuator limits, as illustrated through optimisation of light generation in lithography systems. More recently, a safety-filtered ESC algorithm has been devised to ensure practical safety throughout the optimisation process. By integrating control barrier function filters into each perturbation cycle, the method maintains an unknown but measurable safety metric above a predetermined threshold—even during transients—while steering the system towards the global optimum. Nonsmooth analysis and Lyapunov arguments establish semiglobal practical asymptotic stability of both the safety condition and the optimisation objective, marking a significant step toward reliable deployment of ESC in safety-critical applications.

Extremum Seeking Control in Nonlinear Dynamical Systems publication trend

The graph below shows the total number of articles in extremum seeking control in nonlinear dynamical systems across all publications each year (not limited to Nature Index journals).

Technical terms

Extremum seeking control (ESC): An adaptive optimisation method that perturbs inputs and uses output measurements to locate and maintain a system’s optimal operating point without a precise model.

Nonlinear dynamical system: A system governed by nonlinear differential or difference equations, whose behaviour cannot be expressed as a simple linear combination of its states.

Barrier function: A mathematical construct added to an optimisation scheme to penalise or prevent violation of constraints, ensuring the solution remains within admissible bounds.

Control barrier function (CBF): A real-time safety filter that modifies control inputs to guarantee the system state remains within a safe set, often formulated as a quadratic programme.

Sampled-data approach: A control implementation in which continuous-time measurements and inputs are processed at discrete sampling instants, accommodating digital computation and communication limits.

References

1. 100 years of extremum seeking: A survey. Automatica (2024).
2. Sampled-data extremum-seeking framework for constrained optimization of nonlinear dynamical systems. Automatica (2022).
3. Extremum seeking control of nonlinear dynamic systems using Lie bracket approximations. International Journal of Adaptive Control and Signal Processing (2020).
4. Semiglobal Safety-Filtered Extremum Seeking With Unknown CBFs. IEEE Transactions on Automatic Control (2024).
5. Gradient Extremum Seeking With Nonconstant Delays. IEEE Access (2020).