2020
DOI: 10.1002/acs.3152
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Extremum seeking control of nonlinear dynamic systems using Lie bracket approximations

Abstract: Summary In this article, we consider extremum seeking problems for a general class of nonlinear dynamic control systems. The main result of the article is a broad family of control laws which optimize the steady‐state performance of the system. We prove practical asymptotic stability of the optimal steady‐state and, moreover, propose sufficient conditions for the asymptotic stability in the sense of Lyapunov. The results generalize and extend existing results which are based on Lie bracket approximations. In p… Show more

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Cited by 21 publications
(41 citation statements)
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“…ESC for slowly time-varying performance maps is considered in, e.g., Cao, Dürr, Ebenbauer, Allgöwer, and Gao (2017), Fu andÖzgüner (2011), Grushkovskaya, Dürr, Ebenbauer, andZuyev (2017), Rušiti, Evangelisti, Oliveira, Gerdts, and Krstić (2019) and Sahneh, Hu, and Xie (2012). In here, optimal plant performance is obtained by tracking optimal time-varying plant parameters.…”
Section: Introductionmentioning
confidence: 99%
“…ESC for slowly time-varying performance maps is considered in, e.g., Cao, Dürr, Ebenbauer, Allgöwer, and Gao (2017), Fu andÖzgüner (2011), Grushkovskaya, Dürr, Ebenbauer, andZuyev (2017), Rušiti, Evangelisti, Oliveira, Gerdts, and Krstić (2019) and Sahneh, Hu, and Xie (2012). In here, optimal plant performance is obtained by tracking optimal time-varying plant parameters.…”
Section: Introductionmentioning
confidence: 99%
“…Email addresses: jorge.poveda@colorado.edu (Jorge I. Poveda), nali@seas.harvard.edu (Na Li). seeking (ES) dynamics, a class of zero-order optimization algorithms based on averaging theory, have also been extensively studied in [27,2,3,18,41,19] and [30] for ODEs, and in [34,32] for hybrid inclusions. One of the main designing goals in feedback-based optimization algorithms is to generate dynamical systems that induce fast rates of convergence for a general class of cost functions.…”
Section: Introductionmentioning
confidence: 99%
“…From the proof of Theorem 1 we can observe that the fixed time convergence property, established in Step 1, is achieved by selecting admissible parameters q 1 and q 2 that also guarantee continuity of the vector field 1 , and by an appropriate orderly tuning of the parameters (ε 2 , a, ε 1 ). Moreover, since only positivity is required for the gain k, any fixed time T * G > 0 can be assigned a priori by using equation (16). Note, however, that T * G depends on the parameter κ that characterizes the cost function in Assumption 1, which is assumed to be unknown.…”
Section: Discussion and Numerical Examplesmentioning
confidence: 99%
“…In this case, for all cost functions φ κ the FTGES dynamics are tuned to guarantee that the convergence time is upper bounded by 1. In order to achieve this property, the parameters were selected as a = 0.1, ε 1 = 0.001, ε 2 = 0.05, q 1 = 3, q 2 = 1.5, and the gain k was obtained via equation (16). Figure 3 shows the trajectories of u for κ ∈ {0.25, 1, 2}.…”
Section: Discussion and Numerical Examplesmentioning
confidence: 99%
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