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On a class of generating vector fields for the extremum seeking problem: Lie bracket approximation and stability properties
Victoria Grushkovskaya, Alexander Zuyev, Christian Ebenbauer
TL;DR
Extremum seeking seeks to stabilize or track an unknown optimum, but existing Lie-bracket results often establish practical rather than Lyapunov stability. This paper develops a broad class of vector fields and a constructive frequency-selection approach. Under appropriate assumptions, it establishes practical stability and Lyapunov asymptotic and exponential stability, while also identifying a condition on the optimal cost value for the latter guarantees.
Problem
Extremum seeking addresses systems whose optimal operating point is unknown or may change, using measured responses to estimate descent directions of a steady-state map.
Source: §1 · p. 1Method
The paper constructs a broad class of Lie-bracket-based vector fields and gives a constructive procedure for selecting control frequencies.
Source: §1 · p. 2, §3.3 · pp. 7–8Results
Under appropriate assumptions, the proposed class supports practical stability and asymptotic and exponential Lyapunov stability, beyond the practical stability typically established in prior results.
Source: §1 · p. 2, §2.2 · p. 3, §3.3 · p. 7Takeaways & Limitations
The class unifies and generalizes existing controls and supports new designs for extremum seeking and vibrational stabilization.
Source: §1 · p. 2, §5 · p. 11Proving asymptotic and exponential Lyapunov stability requires knowing the optimal cost value; using an acceptable higher estimate supports only practical asymptotic stability.
Source: §3.3 · p. 8
Abstract
from arXiv · showhide
In this paper, we describe a broad class of control functions for extremum seeking problems. We show that it unifies and generalizes existing extremum seeking strategies which are based on Lie bracket approximations, and allows to design new controls with favorable properties in extremum seeking and vibrational stabilization tasks. The second result of this paper is a novel approach for studying the asymptotic behavior of extremum seeking systems. It provides a constructive procedure for defining frequencies of control functions to ensure the practical asymptotic and exponential stability. In contrast to many known results, we also prove asymptotic and exponential stability in the sense of Lyapunov for the proposed class of extremum seeking systems under appropriate assumptions on the vector fields.
1 Introduction
Extremum seeking addresses systems whose optimal operating point is unknown or may change, using control signals to estimate descent directions from system responses. The paper introduces a Lie-bracket-based class of vector fields and proves stability properties for this class under appropriate assumptions.
1 Introduction
- Unknown or changing optimal operating points motivate extremum seeking, which typically uses perturbed inputs and outputs to estimate descent directions of a system’s steady-state map.The shape of the control signals can affect convergence speed and may be subject to input constraints.Source: §1 · p. 1
- The proposed formula unifies and generalizes prior Lie-bracket-based controls, enabling new extremum seeking designs with bounded update rates and vanishing amplitudes.The paper also applies the formula to vibrational stabilization.Source: §1 · p. 2
- Under appropriate assumptions, the paper proves asymptotic and exponential Lyapunov stability for the proposed class, extending beyond the practical stability typically established in prior results.The proof extends existing techniques to a wider class of cost functions and vector fields that may be non-differentiable at the origin.Source: §1 · p. 2
2 Preliminaries
The preliminaries define notation and review how periodic controls can induce Lie-bracket dynamics. They then frame extremum seeking as approximating a cost function’s gradient flow, with trajectories approaching minima under stated conditions.
2.1 Notations
- The notation section defines the Lie derivative and Lie bracket, with the latter expressed as the difference of two Lie derivatives.It also specifies neighborhood and asymptotic-order notation.Source: §2.1 · p. 2
2.2 Lie bracket approximations
- Periodic control inputs generate trajectories that approximate a Lie-bracket system, whose stability can imply practical stability of the original system.If a compact set is locally or globally uniformly asymptotically stable for the bracket system, it is locally or semi-globally practically uniformly asymptotically stable for the controlled system.Source: §2.2 · p. 3
2.3 Extremum seeking problem
- The problem is framed as minimizing an unknown but measurable cost function, and Lie-bracket controls approximate its gradient flow.For convex costs with minima, trajectories approach an arbitrarily small neighborhood of the minima when the frequency is sufficiently large.Source: §2.3 · p. 3
3 Main results
The main results characterize vector fields whose Lie-bracket dynamics approximate gradient-like flows, then give stability conditions for the resulting extremum-seeking systems. The conditions include a constructive choice of control frequency and distinguish practical stability from asymptotic or exponential stability in the sense of Lyapunov.
3.1 Vector fields for approximating gradient flows
- Formula (7) characterizes vector fields whose Lie-bracket system approximates a gradient-like flow, including a multivariable extension with practical stability inherited from the corresponding bracket system.For the system form (6), this is the whole class of solutions yielding the stated bracket system; the extension in (10) inherits local or semi-global practical uniform asymptotic stability under the stated conditions.Source: §3.1 · pp. 4–5
3.2 Examples
- Formula (7) unifies and generalizes earlier controls and permits controls combining bounded update rates with perturbation amplitudes that vanish at the origin.Such vanishing controls are useful when the minimum value of J is known, even if the extremum point is not.Source: §3.2 · pp. 5–6
3.3 Stability conditions
- Under Theorem 3, the system is practically exponentially stable when m1 = 1 and practically asymptotically stable when m1 > 1; under A4, it is exponentially stable if m̃ = 1 and asymptotically stable if m̃ > 0.The second result uses controls satisfying A3 and the stated assumptions on J; A4 is not required for practical asymptotic stability.Source: §3.3 · p. 7
- A constructive procedure chooses ε so that, for initial states near x*, the system satisfies the stability bound with ρ = 0.The theorem specifies sufficiently small ε and λ; the proof gives a procedure for choosing the ε bound.Source: §3.3 · pp. 7–8
- Lyapunov asymptotic and exponential stability require the minimum value J* to be known so the control functions can be chosen appropriately.The theorem gives explicit frequency and decay-rate estimates in terms of constants whose existence is required, but whose exact values need not be known in the stability proof.Source: §3.3 · p. 8
4 Examples
The examples illustrate extremum-seeking controls with bounded update rates and vanishing magnitudes, then extend the construction to vibrational stabilization using control-Lyapunov-function values. They also show how cost-function shape affects convergence.
4.1 Extremum seeking
- The earlier strategies do not vanish at the extremum and therefore produce oscillatory behavior; vanishing oscillations are possible when the extremum value is known.The extremum point itself need not be known for this setting.Source: §4.1 · pp. 8–9
- For J1, controls satisfying A4 decrease in magnitude as the cost approaches its minimum, while the reported decay-rate estimate matches the trajectories well.The examples include a proposed strategy with bounded update rates and vanishing amplitudes.Source: §4.1 · p. 9
- The higher-order cost J2 converges more slowly than quadratic J1, and the decay rate can be increased by an appropriate choice of control parameters.The J2 comparison uses systems (16) and (18).Source: §4.1 · p. 9
4.2 Vibrational stabilization
- The constructed vibrational control laws approximate LgV-control laws and stabilize systems of form (19) using only values of the control Lyapunov function V.The controls generalize the vibrational law while retaining its reliance on V rather than g or ∇V.Source: §4.2 · p. 10
- A scalar example uses an unknown parameter with |µ| ≥ 1 and compares trajectories under controls corresponding to (17) and (18) with the law (20).The initial condition is x(0)=1; the figure also includes the control corresponding to (15).Source: §4.2 · p. 11, Figure 3 · p. 11
5 Conclusions
The paper proposes a Lie-bracket-based formula for vector fields that approximate gradient-like flows, yielding controls for extremum seeking and vibrational stabilization. It also presents constructive stability analysis, including Lyapunov asymptotic and exponential stability under additional conditions.
5 Conclusions
- The proposed formula constructs vector fields whose Lie-bracket approximation follows gradient-like flows and yields controls for extremum seeking and vibrational stabilization.It unifies and generalizes some existing results while enabling new control-function designs.Source: §5 · p. 11
- The stability proofs provide a constructive procedure for choosing control frequencies to ensure practical asymptotic stability, with practical exponential stability proven for certain cost functions..Source: §5 · pp. 11–12
- Under additional vector-field requirements, the paper proves asymptotic and exponential Lyapunov stability and derives novel decay-rate estimates for the cost function.Auxiliary lemmas also extend prior results and may apply to nonholonomic stabilization when exponential stabilization is not possible.Source: §5 · p. 12
A.1 Preliminary results
The preliminary results develop estimates used to analyze controlled trajectories: a Volterra-series representation, a bound on its remainder, and an estimate for a Lyapunov-like function under stated regularity and domain conditions.
A.1 Preliminary results
- Solutions of system (24) in D admit a Volterra-series representation, with a remainder term in the expansion.The proof treats trajectories in the common-zero set separately, where the vector fields and relevant Lie derivatives vanish.Source: §A.1 · p. 14
- Lemma 4 bounds the Volterra-expansion remainder over [0,τ] under its stated assumptions, including ντ < 1.The result follows from the preceding trajectory estimate and the expansion.Source: §A.1 · p. 15
- Lemma 5 gives an estimate for V under stated assumptions on a bounded convex domain and extends [24, Lemma 3.2] from m̃=0 to arbitrary m̃≥0.Its proof bounds the variation along the segment between the initial and terminal points.Source: §A.1 · pp. 15–16
A.2 Proof of Theorem 3
The proof establishes Theorem 3 through local existence and stability estimates, then iterates sampled-time bounds to show decay and practical convergence. It treats practical stability separately from stability in the sense of Lyapunov.
A.2 Proof of Theorem 3
- The proof follows an approach from [24] but uses continuous controls, Carathéodory solutions, and more general assumptions on the vector fields and cost function.Source: §A.2 · p. 17
Proof of exponential and asymptotic stability in the sense of Lyapunov
- For the Lyapunov-stability branch, the proof first ensures solutions stay in D over an initial interval and introduces nested cost level sets.It then derives estimates for the solutions from those sets.Source: §A.2 · p. 17
- Iterating the sampled-time inequality gives J(x(t)) ≤ J(x0)e^−λt at t = 0, ε, 2ε, … .The proof subsequently combines its estimates to conclude the second assertion of Theorem 3.Source: §A.2 · pp. 18–19
Proof of practical exponential and asymptotic stability
- After a finite number of ε-steps, trajectories enter Bρ(x∗) and remain there, completing the practical-stability argument.The proof also uses a decreasing bound that eventually falls below ρ0.Source: §A.2 · p. 20