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Extremum Seeking for Linear Time-Varying Systems with Unknown Control Directions

Frederic Mazenc, Michael Malisoff, Emilia Fridman

TL;DR

This paper tackles bounded extremum seeking for time-varying linear systems with unknown control directions and measurement disturbances by introducing a delay-free change of variables, a Lyapunov-based analysis, and a comparison principle to derive practical exponential stability bounds that hold for all $t\ge0$. The main contribution is a rigorous framework that yields explicit bounds on the state via a transformed Lyapunov function $V(t,R)=R^T P(t)R$ and a scalar comparison system, accommodating coefficient-uncertainty in $A(t)$ and $B(t)$ and measurement delay through a bound on $\delta(t)$. It advances the theory by providing sufficient conditions under which these bounds hold, including persistence-of-excitation premises and constructions of $P(t)$, as well as a reduction-model approach to quantify the impact of delays on the closed-loop performance. The illustrated examples demonstrate practical applicability to multidimensional systems with unstable drifts and show how choosing design parameters like $\varepsilon$ and the weighting constants can significantly reduce the ultimate bound, supporting potential use in aerial and networked systems with uncertain dynamics.

Abstract

We consider bounded extremum seeking controls for time-varying linear systems with uncertain coefficient matrices and measurement uncertainty. Using a new change of variables, Lyapunov functions, and a comparison principle, we provide practical exponential stability bounds for the states of the closed loop systems that hold for all nonnegative times. For the first time for linear time-varying systems with unknown control directions, we consider bounded extremum seeking controls in the presence of uncertain time-varying input delays with small time-varying delay uncertainties, and we provide reduction model controllers to compensate for the constant part of the delays.

Extremum Seeking for Linear Time-Varying Systems with Unknown Control Directions

TL;DR

This paper tackles bounded extremum seeking for time-varying linear systems with unknown control directions and measurement disturbances by introducing a delay-free change of variables, a Lyapunov-based analysis, and a comparison principle to derive practical exponential stability bounds that hold for all . The main contribution is a rigorous framework that yields explicit bounds on the state via a transformed Lyapunov function and a scalar comparison system, accommodating coefficient-uncertainty in and and measurement delay through a bound on . It advances the theory by providing sufficient conditions under which these bounds hold, including persistence-of-excitation premises and constructions of , as well as a reduction-model approach to quantify the impact of delays on the closed-loop performance. The illustrated examples demonstrate practical applicability to multidimensional systems with unstable drifts and show how choosing design parameters like and the weighting constants can significantly reduce the ultimate bound, supporting potential use in aerial and networked systems with uncertain dynamics.

Abstract

We consider bounded extremum seeking controls for time-varying linear systems with uncertain coefficient matrices and measurement uncertainty. Using a new change of variables, Lyapunov functions, and a comparison principle, we provide practical exponential stability bounds for the states of the closed loop systems that hold for all nonnegative times. For the first time for linear time-varying systems with unknown control directions, we consider bounded extremum seeking controls in the presence of uncertain time-varying input delays with small time-varying delay uncertainties, and we provide reduction model controllers to compensate for the constant part of the delays.
Paper Structure (11 sections, 5 theorems, 137 equations, 1 figure, 6 tables)

This paper contains 11 sections, 5 theorems, 137 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Class of Systems and Main Result
  3. Sufficient Conditions for Assumption \ref{['as2']}
  4. Proof of Theorem \ref{['thm1']}
  5. Using $\delta$ to Represent Measurement Delays
  6. Reduction Model Approach
  7. Illustrations
  8. Two Dimensional System
  9. One Dimensional System
  10. Illustration of Corollary \ref{['cor1']}
  11. Conclusions

Key Result

Theorem 1

Let Assumptions as1-as3 hold. Then for any constant $\xi_0$ such that and for any initial state $x(0)$ for (a2v) satisfying $|x(0)|\le \sigma_0$, the solution $x: [0,+\infty)\to \mathbb R^n$ of (a2v) satisfies for all $t\ge 0$. $\square$

Figures (1)

  • Figure 1: Closed Loop Solutions for (\ref{['f50']}) with Initial States $x(0)=(1/\sqrt{2},-1/\sqrt{2})$, $x(0)=(0.6/\sqrt{2},-0.6/\sqrt{2})$, and $x(0)=(0.3/\sqrt{2},-0.3/\sqrt{2})$ Showing $x_1(t)$ (Solid Red) and $x_2(t)$ (Dashed Blue) on $0.1$ Second Time Horizon

Theorems & Definitions (13)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 1
  • Remark 5
  • Remark 6
  • Theorem 2
  • Proposition 1
  • ...and 3 more