Sampled-Data Extremum Seeking With Constant Delay: A Time-Delay Approach

Zhu, Yang; Fridman, Emilia; Oliveira, Tiago Roux

doi:10.1109/tac.2022.3140259

Cited by 17 publications

(12 citation statements)

References 20 publications

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“…For the stability analysis of the ES control system (5), several methods are proposed in the existing literature including the classical averaging approach (see [1], [9], [16]), Lie brackets approximation (see [5], [17], [25]) and the recent timedelay approach to averaging (see [33], [34]). The classical averaging approach usually resorts to the averaged system via the averaging theorem [15].…”

Section: A1mentioning

confidence: 99%

“…This approach allows, for the first time, to derive efficient linear matrix inequality (LMI)-based conditions for finding the upper bound of the small parameter that ensures the stability. Later on, the time-delay approach to averaging was successfully applied to the quantitative stability analysis of continuous-time ES algorithms (see [33]) and sampled-data ES algorithms (see [34]) in the case of static maps by constructing appropriate Lyapunov-Krasovskii (L-K) functionals. However, the analysis via L-K method is complicated and the results are conservative, since only small uncertainties in Hessian and initial conditions are available.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we suggest a robust time-delay approach to ES via ISS analysis of the averaged system both in the continuous and the discrete domains. After transforming the ES dynamics into a time-delay neutral type model as in [33], [34], we further transform it into an averaged ODE perturbed model, and then use the variation of constants formula instead of L-K method to quantitatively analyze the practical stability of the ODE system (and thus of the original ES system). Explicit conditions in terms of simple inequalities are established to guarantee the practical stability of the ES control systems.…”

Section: Introductionmentioning

confidence: 99%

“…Compared with the existing results, the main contribution of this paper and the significance of the obtained results can be stated as follows. First, comparatively to the considered continuous-time ES systems of one and two input variables in [33], [34], in the present paper we consider the Nvariable case with arbitrary positive integer N, which is more general. Second, we develop, for the first time, the time-delay approach to ES control for discrete-time systems, and provide a quantitative analysis on the control parameters and the ultimate bound of seeking error.…”

Section: Introductionmentioning

confidence: 99%

“…Second, we develop, for the first time, the time-delay approach to ES control for discrete-time systems, and provide a quantitative analysis on the control parameters and the ultimate bound of seeking error. Third, comparatively to the L-K method utilized for neutral type systems in [33], [34], here we adopt the variation of constants formula for the ODE systems. This greatly simplifies the stability analysis process along with the stability conditions, and improve the quantitative bounds as well as the permissible range of the extremum value and the Hessian matrix.…”

Section: Introductionmentioning

confidence: 99%

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A Robust Time-Delay Approach to Extremum Seeking via ISS Analysis of the Averaged System

Yang¹,

Fridman²

2023

Preprint

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For N-dimensional (ND) static quadratic map, we present a time-delay approach to gradient-based extremum seeking (ES) both, in the continuous and, for the first time, the discrete domains. As in the recently introduced (for 2D maps in the continuous domain), we transform the system to the time-delay one (neutral type system in the form of Hale in the continuous case). This system is O(ε)-perturbation of the averaged linear ODE system, where ε is a period of averaging. We further explicitly present the neutral system as the linear ODE, where O(ε)-terms are considered as disturbances with distributed delays of the length of the small parameter ε. Regional input-tostate stability (ISS) analysis is provided by employing a variation of constants formula that greatly simplifies the previously used analysis via Lyapunov-Krasovskii (L-K) method, simplifies the conditions and improves the results. Examples from the literature illustrate the efficiency of the new approach, allowing essentially large uncertainty of the Hessian matrix with bounds on ε that are not too small.

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Section: A1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Robust Time-Delay Approach to Extremum Seeking via ISS Analysis of the Averaged System

Yang¹,

Fridman²

2023

Preprint

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Stabilization under unknown control directions and disturbed measurements via a time‐delay approach to extremum seeking

Zhang,

Fridman

2024

Intl J Robust & Nonlinear

Self Cite

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We study stabilization of linear uncertain systems under unknown control directions using a bounded extremum seeking controller in the presence of a small time‐varying measurement delay. We assume that the measurements are subject to discontinuous disturbances. The main novelty is that these disturbances possess not only the constant part as in the existing results, but also small discontinuous part that may appear due to quantization. We consider two types of measurements: the state measurements and the state quadratic norm ones. In the latter case the constant part of the disturbances may be arbitrary large. By using the recently proposed time‐delay approach to Lie‐brackets‐based averaging, we transform the closed‐loop system to a time‐delay (neutral type) one with no terms depending on the disturbance derivative, which has a form of perturbed Lie brackets system. The input‐to‐state stability (ISS) of the time‐delay system guarantees the same for the original one. We further transform the neutral system to an ordinary differential equation (ODE) with delayed perturbations and employ variation of constants formula leading to explicit conditions in terms of simple inequalities with less conservative results in the most of numerical examples. Two numerical examples are provided to illustrate the efficiency of the method.

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Lie-brackets-based averaging of affine systems via a time-delay approach

Zhang

Fridman

2023

Automatica

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Sampled-Data Extremum Seeking With Constant Delay: A Time-Delay Approach

Cited by 17 publications

References 20 publications

A Robust Time-Delay Approach to Extremum Seeking via ISS Analysis of the Averaged System

A Robust Time-Delay Approach to Extremum Seeking via ISS Analysis of the Averaged System

Stabilization under unknown control directions and disturbed measurements via a time‐delay approach to extremum seeking

Lie-brackets-based averaging of affine systems via a time-delay approach

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