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Contribution to Extremum Seeking Control: First and Second Order Lie Bracket Approximation Approaches

par Labar, Christophe
Président du jury Gyselinck, Johan
Promoteur Garone, Emanuele
Co-Promoteur Kinnaert, Michel
Publication Non publié, 2019-10-04

Thèse de doctorat

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Résumé :

This thesis focuses on a class of real-time model-free optimization methods, the so-called extremum seeking systems. Those algorithms typically steer the input of a cost function towards an optimizer by combining time-periodic signals with the on-line measurement of the cost. This makes them particularly powerful, since they do not require either the mathematical expression of the cost function, or the value of its gradient, to perform the optimization. Using trajectory approximation techniques, the Lie bracket approximation approach and the singular perturbation theory, we propose novel extremum seeking systems, addressing constrained and unconstrained optimization problems. More precisely, we start by presenting first and second order Lie bracket approximation results. We propose then a singular perturbation framework to account for the case where the cost function is the output of a stable dynamical system. Based on those approximation results, four types of extremum seeking systems are developed. Firstly, we design extremum seeking systems approximat- ing a filtered gradient-descent law. Namely, a law in which the gradient is low-pass filtered, before being fed in the descent law. We show that the presence of this low-pass filter allows to design novel extremum seeking systems, able to tune both the transient and steady-state performances. Secondly, we present extremum seeking systems approximating a Newton-based optimization law. This ensures similar convergence rates for the cost inputs when considering multi-input quadratic cost functions. It also improves the robustness with respect to changes in the Hessian matrix of the cost function. Thirdly, we consider the case where a given reserve has to be kept with respect to the maximal cost. This reserve aims at providing a buffer than can be used to counteract fast changes in the operating conditions of an associated system. A so-called sub-optimum seeking system, able to steer the cost towards a given percentage of its estimated maximum, is proposed. Fourthly, we consider the case where the cost inputs are subject to some limitations, formulated as constraints. Namely, we address the problem of minimizing a cost function subject to constraints, while ensuring that the constraints are not violated of more than a user-defined value during the transient phase. Combining a modified-barrier function with a saddle point seeking, we propose a constrained ex- tremum seeking system enforcing this objective. The stability properties of the proposed schemes are proved and their efficiency is shown via the simulation of various case studies.