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On Delay-robustness of Extremum Seeking of Nonlinear Static Maps with Small Disturbance

Jianzhong Li, Yang Zhu, Hongye Su

arXiv:2609.16523v1eess.SYcs.SY

TL;DR

Delays can threaten the stability of real-time extremum seeking, raising the question of how much delay an ES system can tolerate. This paper analyzes predictor-free ES for nonlinear static maps under delay uncertainty and measurement disturbance. It derives conditions for practical stability, including for large known constant delays.

  • Problem

    Because delays can destabilize controlled systems and particularly affect real-time ES, the paper asks how much delay an ES system can withstand.

    Source: §1 · p. 1

  • Method

    The paper designs dithers for delay uncertainty and uses a Lyapunov-based LMI framework to analyze predictor-free ES on non-quadratic static maps.

    Source: §1 · p. 2, §2 · p. 5

  • Results

    Appropriately selected ES parameters achieve practical stability for any large known constant delay.

    Source: §5 · p. 18

  • Takeaways & Limitations

    Under assumptions on the nonlinear maps, the LMI conditions provide quantitative upper bounds on tolerable delay and dither period.

    Source: §1 · p. 2

    The time-varying delay is not compensated, and its known upper bound is used without requiring its specific form.

    Source: §2 · p. 3

Abstract

from arXiv · show

Extremum seeking (ES) is a real-time optimization strategy, thus transmission delays in the feedback loop of ES have big impact on its stability. How big delay that ES control systems are able to withstand? This paper provides a potential answer to this problem. We focus on gradient-based ES for nonlinear static maps subject to known constant delays plus a small time-varying delay uncertainty. We also consider the measurement to be subject to a small disturbance. Different from a majority of existing literature addressing quadratic maps with delays by predictor feedback, this paper deals with a wider class of non-quadratic maps without any predictor or observer for delay compensation. Dither signals in modulation and demodulation are carefully designed to handle constant delays and time-varying delay uncertainties. When the nonlinear map is unknown, we offer a rigorously analytical framework of ES convergence and delay-robustness. When some a prior knowledge of nonlinear maps is available, we are able to provide a quantitative estimation on upper bounds of time delay and dither periods to keep ES systems to remain stable. A suitable choice of ES parameters guarantees practical stability for any large known constant delay.

1. Introduction

ES is a real-time optimization method, and delays can destabilize controlled systems, making delay tolerance a central design question for ES. This paper addresses that question for non-quadratic static maps with uncertain delays.

1. Introduction

  • ES steers an unknown map’s output toward a local optimum, and delays are a particularly serious concern for this real-time optimization method.When disregarded, delays can render controlled systems unstable.Source: §1 · p. 1
  • Predictor-based ES can tolerate large delays but requires more complicated controllers, while predictor-free controllers are simpler when delays are not large [27].The paper asks how much delay ES can withstand before predictor-based compensation is needed.Source: §1 · p. 1
  • The paper develops a predictor-free framework for analyzing delay robustness of continuous-time n-dimensional ES on non-quadratic static maps with time-varying uncertain delays and small measurement bias.It transforms the system into a retarded differential equation with distributed delays and uses Lyapunov-based linear matrix inequalities.Source: §1 · p. 2
  • Under assumptions on the nonlinear maps, feasible LMIs yield quantitative upper bounds on the dither period and tolerable delay, with practical stability for large known constant delays.The framework addresses a gap in delay-robustness analysis without delay compensation.Source: §1 · p. 2

2. Extremum Seeking of Scalar Case

The scalar-case analysis designs dither-based gradient extremum seeking for nonlinear static maps with known constant delay, bounded time-varying delay uncertainty, and measurement disturbance. It develops both a non-approximate stability analysis and quantitative delay bounds under additional map knowledge.

2. Extremum Seeking of Scalar Case

  • With unknown map bounds, the paper gives qualitative analysis; when derivative bounds are known, it quantifies delay and dither-period limits and estimation-error bounds.The search region must contain the unique extremum, and its known radius is important for selecting the initial condition.Source: §2 · p. 3
  • The controller holds its estimate fixed during the initial interval of length equal to the largest transmission delay, then begins updating after the output measurement arrives.The held estimate starts within the specified neighborhood of the extremum.Source: §2 · p. 4
  • Designed dither signals address constant delay and time-varying delay uncertainty, with the dominant stabilizing term averaging nonzero and perturbation terms averaging to zero.For the described signals, the largest allowed delay uncertainty is d̄=ε/4.Source: §2 · pp. 5–6
  • For non-quadratic maps, suitable parameter scaling can retain stability for arbitrarily large known constant delay, but the adaptation gain and convergence rate shrink as delay grows.For a given gain, the analysis therefore seeks a finite upper delay bound that preserves a useful real-time convergence rate.Source: §2 · p. 6
  • The time-delay transformation avoids averaging approximations and explicitly represents disturbance terms, allowing stability of the original ES system to follow from stability of the transformed system.Unlike the averaging approach, it does not freeze map terms or discard the Taylor remainder.Source: §2 · p. 7
  • Under the stated assumptions and feasible LMIs, the estimation error converges exponentially at rate δ to a tunable set for d̄≤d̄* and 4d̄<ε≤ε*.The set is adjustable through ε and a; the LMIs are feasible for sufficiently small d̄*, ε*, ν*, a, and k.Source: §2 · p. 9

3. Extremum Seeking of Vector Case

The paper extends its scalar delay-robustness analysis to vector extremum seeking for nonlinear static maps with measurement disturbance and delays. With suitable LMI conditions, the estimation error is exponentially attracted to a tunable set under bounded delay uncertainty.

3. Extremum Seeking of Vector Case

  • The vector analysis extends the scalar framework to nonlinear static maps with measurement-bias uncertainty and transmission delays in both input and output channels.The measurement uncertainty is denoted ν(t), and the map is subject to the stated assumptions.Source: §3 · p. 9
  • Phase-shifted dithers M(t−D) address the known constant delay, while specially designed dsq signals address unknown time-varying delay uncertainty.When d̄ is known, choosing ε > 2n d̄ establishes stability of the delay-free averaged system; the analysis uses ε ≥ 2n+1 d̄ for computational simplicity.Source: §3 · p. 11
  • For vector maps, Theorem 2 gives LMI conditions under which the estimation error is exponentially attracted to a set when d̄ ≤ d̄* and ε lies in (2n+1 d̄, ε*].The theorem assumes the initial estimation error is bounded within σ for the initial interval [0, h̄].Source: §3 · pp. 14–15
  • The ultimate estimation-error set is adjustable through ε and a, and the LMI conditions are feasible for sufficiently small d̄*, ε*, ν*, a, and k.The theorem's conditions also depend on the stated map assumptions and tuning parameters.Source: §3 · pp. 14–15

4. Examples

The examples examine scalar and vector extremum seeking under transmission delays. The scalar results show worsening stability as the constant delay grows, while the vector example demonstrates operation on a two-input nonlinear map with a maximum at (0, 0).

4.1. Scalar Case

  • As D increases, the scalar system becomes increasingly unstable, and reducing the controller gain can partially compensate for significant delays.The reported simulation uses a = 0.04, ε = 0.005, d̄ = 1.25 × 10−3, and ν* = 0.0002.Source: §4.1 · p. 16

4.2. Vector Case

  • For the two-input nonlinear map with a maximum at (0, 0), the vector regulator seeks the extremum under delay, with input and output trajectories shown in the simulations.The reported setup uses K = 0.16 I, a = 0.02, ε = 0.01, D = 1.5, and d̄ = 0.0012.Source: §4.2 · pp. 16–17, Figure 8 · p. 17, Figure 9 · p. 17

5. Conclusion

The paper develops delay-robustness analysis for extremum seeking on nonlinear static maps with small measurement disturbance, and establishes practical stability under suitable parameter choices.

5. Conclusion

  • The proposed framework analyzes delay robustness for extremum seeking on non-quadratic static maps subject to small measurement disturbance.The dither signals are selected to handle time-varying delay uncertainty.Source: §5 · p. 18
  • The Lyapunov functional and resulting LMIs provide quantitative upper bounds on tolerable delay and dither period.Source: §5 · p. 18
  • Appropriate extremum-seeking parameter choices guarantee practical stability for any large known constant-delay component.Source: §5 · p. 18

Appendix. Proof of Theorem 2

The proof uses a Lyapunov-functional and LMI argument to establish practical stability and verify the required bounds, then shows feasibility under sufficiently small uncertainties and suitable parameter choices.

Appendix. Proof of Theorem 2

  • The feasibility argument uses the dissipativity inequality from Assumption 3 and Jensen’s inequality in bounding the Lyapunov-functional derivative.Source: Appendix. Proof of Theorem 2 · p. 18
  • A Lyapunov functional and S-procedure yield an inequality used to establish practical stability of system (61).The functional uses decision variables P, Q, and R; the proof applies the S-procedure with positive parameters.Source: Appendix. Proof of Theorem 2 · pp. 18–19
  • The proof verifies practical stability of system (49) and that the proposed overall bound (63) is not violated, given the stated initial-condition bound.The argument concludes that condition (67) implies bound (63).Source: Appendix. Proof of Theorem 2 · pp. 19–20
  • Feasibility of condition (67) is established for given σ and D when the delay, disturbance, and selected parameters are sufficiently small.The proof also relates the exponential decay rate δ to controller gain k through 2kμ(σ) − 2δ > 0.Source: Appendix. Proof of Theorem 2 · p. 20

CRediT authorship contribution statement

The contribution statement assigns investigation, methodology, drafting, and validation roles to Jianzhong Li, while Yang Zhu and Hongye Su are credited with methodology and supervision.

CRediT authorship contribution statement

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