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Lie Bracket Approximation of Extremum Seeking Systems
Hans-Bernd Dürr, Miloš S. Stanković, Christian Ebenbauer, Karl H. Johansson
TL;DR
Extremum seeking must steer systems toward extrema of unknown maps, while its qualitative behavior can require advanced system-theoretic analysis. The paper rewrites perturbations as artificial inputs and uses Lie bracket systems to approximate trajectories and expose optimization. It proves that suitable Lie-bracket stability implies practical stability for extremum-seeking systems, including multi-agent vehicle systems.
Problem
Extremum seeking steers dynamical systems to extrema of partially or completely unknown maps, but understanding its qualitative behavior often requires advanced system-theoretic tools.
Method
The paper treats sinusoidal perturbations as artificial inputs, writes the system in input-affine form, and derives a trajectory-approximating Lie bracket system.
Results
The paper proves that local or global uniform asymptotic stability of the Lie bracket system implies local or semi-global practical uniform asymptotic stability of the corresponding system, and applies this to multi-agent vehicles.
Takeaways & Limitations
The Lie bracket viewpoint reveals the optimizing behavior of extremum seeking and supports stability analysis for multi-agent systems with single-integrator or unicycle dynamics.
Takeaways & Limitations
The results capture stability rather than performance and do not provide a systematic way to choose ω; the required choice depends on initial conditions, distance, and final time.
Abstract
from arXiv · showhide
Extremum seeking feedback is a powerful method to steer a dynamical system to an extremum of a partially or completely unknown map. It often requires advanced system-theoretic tools to understand the qualitative behavior of extremum seeking systems. In this paper, a novel interpretation of extremum seeking is introduced. We show that the trajectories of an extremum seeking system can be approximated by the trajectories of a system which involves certain Lie brackets of the vector fields of the extremum seeking system. It turns out that the Lie bracket system directly reveals the optimizing behavior of the extremum seeking system. Furthermore, we establish a theoretical foundation and prove that uniform asymptotic stability of the Lie bracket system implies practical uniform asymptotic stability of the corresponding extremum seeking system. We use the established results in order to prove local and semi-global practical uniform asymptotic stability of the extrema of a certain map for multi-agent extremum seeking systems.
1. Introduction
The paper introduces a Lie-bracket-based methodology for analyzing extremum seeking systems and applies it to multi-agent vehicles. Its theory links stability of Lie bracket systems to practical stability of the original systems.
- Motivation: Extremum seeking steers dynamical systems toward extrema of unknown maps using systematic feedback from measurements that provide no direct movement direction.The paper motivates this setting with a vehicle that only measures its distance to another vehicle.
- Methodology: The paper models sinusoidal perturbations as artificial inputs in input-affine form and derives an approximate system from the original trajectories.This interpretation differs from commonly used extremum-seeking analysis techniques.
- Theoretical foundation: Trajectories of a class of input-affine systems with certain inputs are approximated by trajectories of corresponding Lie bracket systems.The paper presents this result as a theoretical foundation for its proposed viewpoint.
- Stability results: Uniform asymptotic stability of a Lie bracket system implies practical uniform asymptotic stability of the corresponding input-affine system, locally or semi-globally.The local and semi-global forms follow from local and global stability assumptions, respectively.
- Applications: The theory is applied to multi-agent extremum seeking vehicles with single-integrator and unicycle dynamics, establishing practical stability of extrema of a potential function.The agents’ individual maps satisfy a relationship that assures existence of the common potential function.
2. Main Idea
The main idea is to reinterpret sinusoidal extremum-seeking feedback as an input-affine system and analyze its Lie bracket approximation. In a simple static-map example, the resulting bracket system reveals the optimizing behavior.
- Example setup: For a static map with a strict local maximum, a basic extremum-seeking feedback uses sinusoidal perturbations to guide the state.The example assumes f ∈ C2 and positive parameters α and ω.
- Input-affine representation: The sinusoidal signals are treated as artificial inputs, converting the extremum-seeking dynamics into input-affine form.The inputs are defined from cos(ωt) and sin(ωt).
- Lie bracket system: The corresponding Lie bracket system is constructed from the vector fields associated with the input-affine representation.In the example, the vector fields are b1(x) = α and b2(x) = f(x).
- Optimizing behavior: The Lie bracket system maximizes f, while sinusoidally driven trajectories of the original system can be approximated by its trajectories.This provides the basis for analyzing extremum-seeking behavior through the bracket system.
- General procedure: The proposed procedure is to write the extremum-seeking system in input-affine form, calculate its Lie bracket system, and prove its asymptotic stability.The resulting stability then implies practical asymptotic stability for the extremum-seeking system.
3. Lie Bracket Approximation for a Class of Input-Affine Systems
The paper develops general results connecting input-affine systems with parameterized periodic inputs to corresponding Lie bracket systems. Trajectory approximation and stability-transfer theorems establish practical asymptotic stability of the original systems from uniform asymptotic stability of their Lie bracket systems.
- Framework: The framework considers input-affine systems depending on a parameter ω and defines practical stability notions for compact invariant sets.Practical stability is formulated for differential equations whose vector fields depend on ω.
- Assumptions: The input assumptions include measurable, bounded, Lipschitz-in-time, T-periodic, zero-average signals and sufficiently regular vector fields.The vector fields satisfy C2 regularity and uniform boundedness conditions on compact state sets.
- Approximation: For large ω, trajectories of the input-affine system are approximated by trajectories of its corresponding Lie bracket system over arbitrarily long finite time intervals.The approximation holds for bounded initial-condition sets B and sufficiently large ω.
- Stability transfer: Local uniform asymptotic stability of a compact set for the Lie bracket system implies local practical uniform asymptotic stability for the input-affine system.This is the local stability-transfer result stated in Theorem 2.
- Scope: The results establish stability transfer but do not provide performance guarantees or a systematic choice of ω.The required ω depends on the initial-condition set K, distance D, and approximation horizon tf.
4. Lie Bracket Approximation of Extremum Seeking Systems
The paper applies Lie bracket approximation to multi-agent extremum seeking systems whose individual objectives share a potential function. For both single-integrator and unicycle agents, the resulting extrema sets are proved locally or semi-globally practically uniformly asymptotically stable under stated assumptions.
- Method: The analysis writes the extremum seeking system in input-affine form, computes its Lie bracket system, and transfers stability using the general theorems.This procedure is applied to both single-integrator and unicycle dynamics.
- Multi-agent framework: Each agent uses an individual map f_i whose gradient with respect to its state matches the corresponding gradient of a common potential function F.Thus, coordinated optimization of individual maps is linked to optimization of F.
- Single-integrator dynamics: For single-integrator agents, Sloc × ESloc is locally practically uniformly asymptotically stable under Assumptions B1–B3 and B5.Theorem 4 gives the local result, while the filter-state set is included in the stable set.
- Method: Different agent parameters make some Lie brackets vanish, supporting the multi-agent Lie bracket construction.The frequency condition ω_i = a_iω with distinct rational a_i is used for this purpose.
- Single-integrator dynamics: For single-integrator agents, Sloc × ESglob is semi-globally practically uniformly asymptotically stable under Assumptions B1, B2, B4, and B5.The global-potential assumptions support the corresponding global Lie bracket stability used in the transfer.
- Unicycle dynamics: For unicycle agents, Sloc × ESloc is locally practically uniformly asymptotically stable, and Sloc × ESglob is semi-globally practically uniformly asymptotically stable under the stated assumptions.These conclusions are given by Theorems 6 and 7.
5. Discussion
The discussion connects the Lie bracket system to averaging while emphasizing amplitude and frequency choices, broader periodic inputs, and practical extensions of the stability results.
- Relationship to Averaging Methods: The Lie bracket system can be viewed as the averaged system of the input-affine extremum seeking model.
- Relationship to Averaging Methods: Standard averaging cannot be applied directly because time rescaling introduces a 1/√ϵ term that prevents the required twice-continuous differentiability.
- Relationship to Averaging Methods: The paper instead connects the systems through integral and partial-integration arguments, with the remainder vanishing for bounded trajectories as ω increases.
- Discussion: Choosing perturbation amplitudes √ω with frequency ω is crucial for obtaining the stated Lie bracket approximation; other amplitude choices produce different averaged systems.
- Discussion: The results cover single-agent systems and establish local or semi-global practical uniform asymptotic stability for single-integrator and unicycle multi-agent systems.
- Discussion: The theoretical results also apply when sinusoidal perturbations are replaced by appropriately defined periodic, discontinuous, or non-differentiable signals.
6. Examples
Numerical examples compare original and Lie bracket trajectories for three-agent single-integrator and unicycle systems, illustrating approximation, convergence, and characteristic trajectory features.
- Single-Integrator Examples: The examples compare original and Lie bracket trajectories for three-agent single-integrator systems at ω = 10 and ω = 100.
- Multi-Agent Setup: The quadratic potential has maximum at x̄* = [1, 1, −1, −1, −1, 1]ᵀ, supporting the predicted stability of the extremum-seeking systems.
- Single-Integrator Examples: For the three-agent single-integrator setup, the Lie bracket trajectories capture the qualitative evolution of the original trajectories.
- Single-Integrator Examples: As ω increases, the original trajectories approach those of the Lie bracket system, consistent with the trajectory-approximation theorem.
- Single-Integrator Examples: Despite highly nonlinear inter-agent maps, the overall system practically converges to the expected extremum even for small ω.
- Unicycle Examples: For the three-agent unicycle system at ω = 80, both systems practically converge to the extremum and share characteristic trajectory points.
7. Conclusion
The conclusion presents the Lie bracket viewpoint as a methodology for interpreting extremum seeking, approximating trajectories, and deriving practical stability results for multi-agent systems.
- The method rewrites sinusoidally perturbed extremum seeking systems in input-affine form and relates them to Lie bracket systems that reveal optimizing behavior.
- Trajectories of the considered input-affine systems are approximated by trajectories of their corresponding Lie bracket systems.
- Global or local uniform asymptotic stability of a Lie bracket system implies semi-global or local practical uniform asymptotic stability of the input-affine system.
- The results are applied to multi-agent extremum seeking with single-integrator or unicycle dynamics and illustrated numerically.
Appendix A. Existence and Uniqueness
The appendix states existence and uniqueness conditions for solutions of the considered differential equations under regularity and measurability assumptions.
- A solution is an absolutely continuous function satisfying the differential equation almost everywhere on its interval of definition.
- The solution is defined on an interval [t0, t0 + te) for some positive te.
- Under the stated compact-set conditions, every initial condition admits a unique local solution.
Appendix B. Preliminary Lemmas
The preliminary lemmas establish periodicity, zero-mean properties, and bounds for oscillatory inputs and their integral combinations. These results support the later decomposition and uniform estimates used in the approximation proof.
- Lemma 2: Lemma 2 bounds integrals of bounded, periodic inputs over intervals by separating complete periods from a leftover segment.The leftover length satisfies 0 ≤ δ < T, and the bound vanishes when the interval is an integer multiple of T.
- Uniform bounds: The resulting estimates provide constants k1 and k2, with k2 = 0 when ω(t−t0) is an integer multiple of T.This special case follows because the leftover interval has length δ = 0.
- Lemma 3: Lemma 3 proves that the combined input ˜uij is T-periodic and establishes additional zero-mean and boundedness properties.The proof repeatedly uses T-periodicity and zero average to cancel complete periods.
- Lemma 4: Lemma 4 applies the preceding properties to bound the integral expressions involving pairs of oscillatory inputs.The proof decomposes the integration interval into R1, R2, and R3 and treats each part separately.
- Integral decomposition: The proof handles R1, R2, and R3 using periodicity, zero averages, changes of variables, and bounds on the input functions.For R3, the integrand is bounded using |I(q,p)| ≤ Ljω|q−p| and |ui(t,θ)| ≤ Mi.
Appendix C. Proof of Theorem 1
The proof of Theorem 1 compares the original oscillatory system with its averaged Lie-bracket system on finite intervals. Under the stated assumptions, the trajectories become uniformly close as the frequency ω increases.
- Remainder estimates: The estimates rely on boundedness and regularity of the vector fields, including Lipschitz bounds and bounded Lie-bracket derivatives.These properties follow from Assumptions A1–A3 on the relevant compact set.
- Solution comparison: The proof establishes existence and uniqueness of the original trajectory and uses the bounded Lie-bracket trajectory as the comparison solution.The comparison solution z is assumed bounded for all future times from initial conditions in B.
- Uniform approximation: The proof confines both trajectories to bounded sets and defines a tubular neighborhood O(t) around z(t) to control their separation.A contradiction argument shows the original trajectory cannot leave this neighborhood before the prescribed finite horizon.
- Remainder estimates: The distance estimate is obtained by decomposing the remainder into five terms and showing their sum is bounded by k/√ω on the finite interval.The individual terms R1 through R5 decay uniformly as ω tends to infinity on compact sets.
- Uniform approximation: For any bounded initial set K, distance D, and finite horizon tf, a sufficiently large ω keeps x(t) and z(t) within D uniformly over t0 and x0.The frequency threshold is independent of the initial time and initial state within K.
Appendix D. Proof of Theorem 2
The proof of Theorem 2 transfers stability properties from the Lie-bracket system to the extremum-seeking system. It combines finite-time trajectory approximation with stability and attractivity of the target set.
- Practical uniform stability: Practical uniform stability is obtained by selecting neighborhoods and a finite horizon from uniform stability and attractivity of the Lie-bracket system.Theorem 1 then transfers the corresponding bounds to the oscillatory system for sufficiently large ω.
- Practical uniform stability: The resulting stability estimates can be iterated by restarting the comparison solution at the end of each finite interval.This repeated application extends the finite-time approximation argument across the required stability construction.
- Practical uniform attractivity: Practical uniform attractivity follows by first obtaining practical stability and then applying Lie-bracket attractivity over a suitable finite time.The proof chooses nested bounds B3 and B4 before applying Theorem 1.
- Conclusion: The proof concludes the required property after combining the transferred estimates with the corresponding bounds for the Lie-bracket system.The final step is identified as the last property needed for the theorem.