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Gradient-Based Stochastic Extremum-Seeking Control for Multivariable Systems with Distinct Input Delays

Paulo Cesar Souza Silva, Paulo Cesar Pellanda, Tiago Roux Oliveira

Abstract

This paper addresses the design and analysis of a multivariable gradient-based stochastic extremum-seeking control method for multi-input systems with arbitrary input delays. The approach accommodates systems with distinct time delays across input channels and achieves local exponential stability of the closed-loop system, guaranteeing convergence to a small neighborhood around the extremum point. By incorporating phase compensation for dither signals and a novel predictor-feedback mechanism with averaging-based estimates of the unknown gradient and Hessian, the proposed method overcomes traditional challenges associated with arbitrary, distinct input delays. Unlike previous work on deterministic multiparameter extremum-seeking with distinct input delays, this stability analysis is achieved without using backstepping transformations, simplifying the predictor design and enabling a more straightforward implementation. Specifically, the direct application of Artstein's reduction approach results in delay- and system-dimension-independent convergence rates, enhancing practical applicability. A numerical example illustrates the robust performance and advantages of the proposed delay-compensated stochastic extremum-seeking method.

Gradient-Based Stochastic Extremum-Seeking Control for Multivariable Systems with Distinct Input Delays

Abstract

This paper addresses the design and analysis of a multivariable gradient-based stochastic extremum-seeking control method for multi-input systems with arbitrary input delays. The approach accommodates systems with distinct time delays across input channels and achieves local exponential stability of the closed-loop system, guaranteeing convergence to a small neighborhood around the extremum point. By incorporating phase compensation for dither signals and a novel predictor-feedback mechanism with averaging-based estimates of the unknown gradient and Hessian, the proposed method overcomes traditional challenges associated with arbitrary, distinct input delays. Unlike previous work on deterministic multiparameter extremum-seeking with distinct input delays, this stability analysis is achieved without using backstepping transformations, simplifying the predictor design and enabling a more straightforward implementation. Specifically, the direct application of Artstein's reduction approach results in delay- and system-dimension-independent convergence rates, enhancing practical applicability. A numerical example illustrates the robust performance and advantages of the proposed delay-compensated stochastic extremum-seeking method.

Paper Structure

This paper contains 6 sections, 1 theorem, 55 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Simplified Predictor for Gradient-Based ESC with Multiple and Distinct Input Delays
  4. Stability Analysis
  5. Simulations
  6. Conclusion

Key Result

Theorem 1

Consider the closed-loop system in Figure Fig1 with multiple and distinct input delays as given in (eq:1) and a locally quadratic nonlinear map defined by (eq:3) and (TAChoje1). There exists $c^{\ast} > 0$ such that, $\forall c \geq c^{\ast}$, $\exists \omega^{\ast}(c)>0$ such that, $\forall \omega In particular, where $a= [a_1,\;a_2,\; ...\;,\; a_n ]^T$.

Figures (8)

  • Figure 1: Block diagram of the prediction scheme for compensating multiple and distinct input delays in the map $Q(\theta)$. The vector signals $\hat{G}$ and $\hat{H}$ represent the estimates for the gradient and Hessian of $Q(\theta)$, respectively. The multiple delays and gains are compactly represented by $D=$ diag$\{D_1,D_2,...,D_n\}$ and $K=$ diag$\{K_1,K_2,...,K_n\}$. The red block indicates the introduced delays, while the blue blocks show modifications to the classical stochastic gradient-based ESC algorithm LK:2012c16 for mitigating the effects of time delays. In particular, the prediction feedback law is governed by (\ref{['eq:14']}), while the stochastic perturbations $S(t)$ and $M(t)$ and the demodulation signal $N(t)$ are given by (\ref{['eq:4']}), (\ref{['eq:5']}), and (\ref{['eq:9']}), respectively.
  • Figure 2: System input $\theta(t)$ for the multivariable gradient-based stochastic ESC in the absence of delays.
  • Figure 3: System output $y(t)$ for the multivariable gradient-based stochastic ESC in the absence of delays.
  • Figure 4: System output $y(t)$ for the multivariable gradient-based stochastic ESC with delays, without predictor feedback.
  • Figure 5: System input $\theta(t)$ for the multivariable gradient-based stochastic ESC with delays and predictor feedback.
  • ...and 3 more figures

Theorems & Definitions (1)

  • Theorem 1