Extremum seeking control of nonlinear dynamic systems using Lie bracket approximations

Grushkovskaya, Victoria; Ebenbauer, Christian

doi:10.1002/acs.3152

Cited by 21 publications

(41 citation statements)

References 45 publications

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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%

Extremum-seeking control for optimization of time-varying steady-state responses of nonlinear systems

2020

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Section: Introductionmentioning

confidence: 99%

Extremum-seeking control for optimization of time-varying steady-state responses of nonlinear systems

2020

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“…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%

Robust Hybrid Zero-Order Optimization Algorithms with Acceleration via Averaging in Time

Poveda

2019

Preprint

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We study novel robust zero-order algorithms with acceleration for the solution of real-time optimization problems. In particular, we propose a family of derivative-free dynamics that can be universally modeled as singularly perturbed hybrid dynamical systems with resetting mechanisms. From this family of dynamics, we synthesize four algorithms designed for convex, strongly convex, constrained, and unconstrained zero-order optimization problems. In each case, we establish semi-global practical asymptotic or exponential stability results, and we show how to obtain well-posed discretized algorithms that retain the main properties of the original hybrid dynamics. Given that existing averaging theorems for singularly perturbed hybrid systems are not directly applicable to our setting, we derive a new averaging theorem that relaxes some of the existing assumptions in the literature. This allows us to make a clear link between the KL bounds that characterize the rates of convergence of the hybrid dynamics and their average dynamics. We also show that our results are applicable to non-hybrid zero-order dynamics, thus providing a unifying framework for hybrid and non-hybrid zero-order algorithms based on averaging theory. The superior performance of the hybrid algorithms compared to the traditional schemes that do not incorporate acceleration is illustrated via numerical examples.

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“…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%

“…The main underlaying idea behind extremum seeking control is to induce multiple time scales in the dynamics of the closed-loop system in oder to emulate the behavior of a target nominal optimization algorithm chosen a priori to solve a particular type of optimization problem. Typical target nominal algorithms include gradient descent for convex optimization [8], [9], Newton-like methods [10], Riemannian gradient flows for constrained optimization [11], gradient descent with momentum [12], accelerated gradient descent [13], hybrid gradient descent on manifolds [14], gradient and Newton flows with delays [15], gradient descent with timevarying costs [16], distributed gradient systems for multiagent systems [17], and switched gradient flows modeled as hybrid systems [18]. Other ES approaches based on parameter estimation in adaptive control have also been considered in [19], [20], [21], [22].…”

Section: Introductionmentioning

confidence: 99%

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Fixed-Time Extremum Seeking

Poveda,

Krstic

2019

Preprint

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We introduce a new class of extremum seeking controllers able to achieve fixed time convergence to the solution of optimization problems defined by static and dynamical systems. Unlike existing approaches in the literature, the convergence time of the proposed algorithms does not depend on the initial conditions and it can be prescribed a priori by tuning the parameters of the controller. Specifically, our first contribution is a novel gradient-based extremum seeking algorithm for cost functions that satisfy the Polyak-Lojasiewicz (PL) inequality with some coefficient κ > 0, and for which the extremum seeking controller guarantees a fixed upper bound on the convergence time that is independent of the initial conditions but dependent on the coefficient κ. Second, in order to remove the dependence on κ, we introduce a novel Newton-based extremum seeking algorithm that guarantees a fully assignable fixed upper bound on the convergence time, thus paralleling existing asymptotic results in Newton-based extremum seeking where the rate of convergence is fully assignable. Finally, we study the problem of optimizing dynamical systems, where the cost function corresponds to the steady state input-to-output map of a stable but unknown dynamical system. In this case, after a time scale transformation is performed, the proposed extremum seeking controllers achieve the same fixed upper bound on the convergence time as in the static case. Our results exploit recent gradient flow structures proposed by Garg and Panagou in [3], and are established by using averaging theory and singular perturbation theory for dynamical systems that are not necessarily Lipschitz continuous. We confirm the validity of our results via numerical simulations that illustrate the key advantages of the extremum seeking controllers presented in this paper.

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Extremum seeking control of nonlinear dynamic systems using Lie bracket approximations

Cited by 21 publications

References 45 publications

Extremum-seeking control for optimization of time-varying steady-state responses of nonlinear systems

Extremum-seeking control for optimization of time-varying steady-state responses of nonlinear systems

Robust Hybrid Zero-Order Optimization Algorithms with Acceleration via Averaging in Time

Fixed-Time Extremum Seeking

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