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Multivariable Gradient-Based Extremum Seeking Control with Saturation Constraints

Enzo Ferreira Tomaz Silva, Pedro Henrique Silva Coutinho, Tiago Roux Oliveira, Miroslav Krstić, Sophie Tarbouriech

Abstract

This paper addresses the multivariable gradient-based extremum seeking control (ESC) subject to saturation. Two distinct saturation scenarios are investigated here: saturation acting on the input of the function to be optimized, which is addressed using an anti-windup compensation strategy, and saturation affecting the gradient estimate. In both cases, the unknown Hessian matrix is represented using a polytopic uncertainty description, and sufficient conditions in the form of linear matrix inequalities (LMIs) are derived to design a stabilizing control gain. The proposed conditions guarantee exponential stability of the origin for the average closed-loop system under saturation constraints. With the proposed design conditions, non-diagonal control gain matrices can be obtained, generalizing conventional ESC designs that typically rely on diagonal structures. Stability and convergence are rigorously proven using the Averaging Theory for dynamical systems with Lipschitz continuous right-hand sides. Numerical simulations illustrate the effectiveness of the proposed ESC algorithms, confirming the convergence even in the presence of saturation.

Multivariable Gradient-Based Extremum Seeking Control with Saturation Constraints

Abstract

This paper addresses the multivariable gradient-based extremum seeking control (ESC) subject to saturation. Two distinct saturation scenarios are investigated here: saturation acting on the input of the function to be optimized, which is addressed using an anti-windup compensation strategy, and saturation affecting the gradient estimate. In both cases, the unknown Hessian matrix is represented using a polytopic uncertainty description, and sufficient conditions in the form of linear matrix inequalities (LMIs) are derived to design a stabilizing control gain. The proposed conditions guarantee exponential stability of the origin for the average closed-loop system under saturation constraints. With the proposed design conditions, non-diagonal control gain matrices can be obtained, generalizing conventional ESC designs that typically rely on diagonal structures. Stability and convergence are rigorously proven using the Averaging Theory for dynamical systems with Lipschitz continuous right-hand sides. Numerical simulations illustrate the effectiveness of the proposed ESC algorithms, confirming the convergence even in the presence of saturation.

Paper Structure

This paper contains 23 sections, 2 theorems, 118 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Extremum Seeking Control under Actuator Saturation
  3. Probing and Demodulation Signals
  4. Extremum Seeking with Anti-Windup Design
  5. Reshaping the Gradient Estimate
  6. Closed-Loop System
  7. Rescaling of Time
  8. Average Closed-Loop System
  9. Polytopic Embedding of the Hessian Matrix
  10. Stability Analysis
  11. Stabilization of the Average Closed-Loop System
  12. Asymptotic Convergence to a Neighborhood of the Extremum
  13. Extremum Seeking Control under Gradient Saturation
  14. Extremum Seeking with Bounded Update Rates
  15. Defining a New Time Scale for Averaging
  16. ...and 8 more sections

Key Result

Theorem 1

Consider the ESC system in Fig. fig:diag_ESC_input_constrained with locally quadratic nonlinear map (eq:map_static_input_constrained)--(VitoriaVascao) subject to input saturation and the corresponding average closed-loop dynamics eq:theta_til_medio under Assumptions assump:theta*, assump:probing_w, with $a = \sqrt{\sum_{i=1}^n a_i^2}$.

Figures (6)

  • Figure 1: Extremum seeking control system with saturation in the input map.
  • Figure 2: Extremum seeking control system under gradient saturation.
  • Figure 3: Trajectories of the closed-loop ESC system under input saturation with anti-windup controller \ref{['eq:control_law_1']} designed according to Lemma \ref{['thm:1_input_constrained']} -- Example 1.
  • Figure 4: Trajectory of the output $y(t)$ of the closed-loop ESC system under input saturation with the anti-windup controller \ref{['eq:control_law_1']} designed according to Lemma \ref{['thm:1_input_constrained']} -- Example 1.
  • Figure 5: Trajectories of the closed-loop ESC system under input saturation without anti-windup compensation -- Example 1.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2