High-Order Lie Bracket ESC

• The paper demonstrates that high-order Lie bracket ESC restores exponential convergence in model-free control for objectives with flat minima.
• It leverages iterated Lie bracket averaging and rigorous sequential expansion to isolate higher-order derivatives of the cost function.
• Experimental results in mobile robotics confirm its robust performance and faster convergence compared to traditional ESC methods.

High-Order Lie Bracket Extremum Seeking Control (ESC) refers to extremum seeking algorithms that employ the selective excitation and exploitation of iterated Lie brackets (of order greater than one) of control vector fields to govern the slow, average behavior of a system. This design enables exponential convergence of model-free controllers for objective functions with flat, higher-order minima—where classic (first-order) Lie bracket or gradient-based ESC is fundamentally limited to sub-exponential convergence. Recent developments have unified the mathematical basis of high-order Lie bracket ESC and provided theoretically rigorous, experimentally validated control laws for challenging robotics and control applications.

1. Classical ESC, Lie Bracket Averaging, and Motivation

Most classical ESC schemes are built on first-order Lie bracket averaging: by adding periodic dither signals to input channels of a control-affine system, one induces averaged dynamics that approximate a gradient flow for locally quadratic objectives. Mathematically, for a system of the form

$$\dot{x} = b_0(x) + \sum_{i=1}^{m} b_i(x) u_i(t, \omega t)$$

where $u_i$ are high-frequency, zero-mean dither signals, the corresponding averaged (Lie bracket) system is

$$\dot{z} = b_0(z) + \sum_{i<j} [b_i, b_j](z) \nu_{ji}$$

with $\nu_{ji}$ determined by the dither waveforms (Dürr et al., 2011). For model-free ESC, this produces a gradient flow toward the optimizer when $J(x)$ is locally quadratic. However, for objectives $J(x) \sim (x-x^*)^m$ with $m > 2$, the averaged dynamics become highly nonlinear in $(x-x^*)$, leading only to polynomial convergence (Grushkovskaya et al., 1 Apr 2025).

Practical applications such as source seeking with mobile robots often confront "flat" extrema, where the function near the optimum behaves quartically (or has higher local order), making first-order ESC impractically slow. High-order Lie bracket ESC was introduced to resolve this, leveraging the system’s capacity to make the averaged dynamics align with higher-order derivatives of $J$, restoring exponential rates.

2. High-Order Lie Bracket ESC: Theoretical Basis and System Structure

The essential principle is that for a cost function locally of degree $N$, the ESC system must excite the $(N-1)$-th order Lie bracket to generate an averaged dynamic involving $J^{(N-1)}(x)$ (Palanikumar et al., 2 Nov 2025, Grushkovskaya et al., 1 Apr 2025, Pokhrel et al., 2023). The general control-affine system is recast as

$$\dot{x} = \sum_{k=1}^{n_u} g_k(J(x)) u_k^\varepsilon(t)$$

where $u_k^\varepsilon$ are carefully designed periodic signals with frequency and amplitude scaling, and the $g_k$ are vector fields engineered so that their $N-1$-fold nested Lie bracket gives $-c_N J^{(N-1)}(x)$ (Grushkovskaya et al., 1 Apr 2025). The high-frequency structure and relative phase of $u_k$ ensure that all lower-order averaged Lie brackets vanish, isolating the desired term.

Unicycle Example: For source seeking with a unicycle robot and a quartic objective,

$J(x, y) = C_1 (x - x_d)^4 + C_2 (y - y_d)^4$

the velocity command is synthesized as

$$v(t) = 2 (2\pi/\varepsilon)^{3/4} \left( 3c J(x, y) \sin\left(\frac{6\pi t}{\varepsilon}\right) + a \cos\left(\frac{2\pi t}{\varepsilon}\right) \right)$$

exciting the third-order Lie bracket necessary for quartic functions, with all lower-order averages canceled (Palanikumar et al., 2 Nov 2025).

3. Mathematical Foundations and Averaging Theory

The core mathematical tool is the sequential expansion of the system flow using chronological calculus, which expresses the solution as a Volterra series indexed by iterated Lie brackets (Pokhrel et al., 2023). It is formally established that the $n$-th order Lie bracket system (LBS) is exactly the $(n+1)$-th order average of the system's dynamics. For example, the leading term in the averaged system after appropriate high-frequency scaling is

$g_{I_N}(J(x)) = -c_N J^{(N-1)}(x)$

where $g_{I_N}$ denotes the repeated application of the commutator with the chosen vector fields according to the structure required by $J$'s degree at the minimum (Grushkovskaya et al., 1 Apr 2025, Pokhrel et al., 2023).

Analytical error bounds show that by increasing the excitation frequency (reducing $\varepsilon$), the originally oscillatory system's trajectory shadows the Lie bracket system with arbitrarily small residual error (practical exponential stability) (Palanikumar et al., 2 Nov 2025, Weber et al., 2024).

4. Convergence Properties and Exponential Stability

The key theoretical result is that exciting the correct high-order Lie bracket eliminates slow (power-law) convergence for flat objectives:

• Classical/first-order ESC: For $J(x) \sim (x-x^*)^m$, averaged system has $\dot{x} = -m(x-x^*)^{m-1}$, yielding only $O(t^{-1/(m-2)})$ convergence for $m > 2$.
• High-order Lie bracket ESC: For $N$-degree $J$, using $(N-1)$-th order Lie bracket, averaged system is linear: $\dot{x} = -c_N (x-x^*)$, restoring exponential convergence (Grushkovskaya et al., 1 Apr 2025, Palanikumar et al., 2 Nov 2025).

Rigorous Lyapunov analysis, typically with time-varying coordinate changes and specially crafted Lyapunov functions (e.g.,

$$V(\xi, \eta) = \frac{1}{2}\xi^2 + \frac{1}{2}\eta^2 + \gamma \xi \eta$$

), provides exponential convergence estimates and explicit design constraints on controller parameters and persistent excitation frequencies (Palanikumar et al., 2 Nov 2025).

5. Design and Implementation Methodologies

Dither Signal Construction: The design mandates trigonometric dithers with carefully selected frequencies and amplitudes; these are tuned to satisfy resonant and cancelation conditions so that only the targeted high-order Lie bracket survives upon averaging, as per generalized Chen–Fliess expansions (Pokhrel et al., 2023).

Vector Field Choice: Control vector fields $g_k$ (functions of measured $J$) are shaped so that their nested Lie brackets yield the desired order derivative. For robust, model-free implementation, no derivative or model terms are used; only scalar measurements of $J$ (possibly after filtering) are required (Palanikumar et al., 2 Nov 2025, Grushkovskaya et al., 1 Apr 2025).

Attenuation and Filtering: Practical deployments, as in mobile robotics, include online high-pass filters to remove sensor bias and dither-induced offsets. Adaptive amplitude modulation and nonlinear filtering are used to further attenuate oscillations at the extremum (Palanikumar et al., 2 Nov 2025, Pokhrel et al., 2021).

6. Experimental Validation and Applications

Extensive real-world validations, notably with TurtleBot3 robots, confirm the practical advantage of high-order schemes. For quartic objectives, high-order Lie bracket ESC drives the system to the true extremum exponentially faster—seconds rather than hundreds of seconds—while classic ESC struggles or fails outright, missing the optimal point due to exceedingly slow local dynamics (Palanikumar et al., 2 Nov 2025). Notably, these designs require only access to scalar sensor measurements (e.g., from light sensors in model-free source seeking), not explicit function forms or gradients.

Broader applications include robust source-seeking in robotics, environmental monitoring under flat signal distributions, and optimization for underactuated or nonholonomic systems with polynomial-like cost structures.

7. Limitations, Theoretical Extensions, and Outlook

Assumptions and Limitations: The convergence and stability results are contingent on high-frequency dithers, sufficient smoothness and growth conditions on $J$, and, in some cases, matching or bounded controller gains. Explicit global exponential stability of the full non-averaged system can require stringent global Lipschitz-type bounds (Weber et al., 2024). The selection of appropriate excitation frequencies requires careful analysis (analytic bounds are typically conservative).

Theoretical Extensions: Unified mathematical frameworks now rigorously connect high-order Lie bracket averaging and classical high-order averaging (volterra/chronological expansion), allowing systematic, order-reconfigurable controller synthesis (Pokhrel et al., 2023, Grushkovskaya et al., 1 Apr 2025). Future research will address adaptive order selection (for unknown flatness degree), improved amplitude attenuation, multi-agent extensions, and further experimental generalizations.

Summary Table: Convergence Profiles

Objective Local Form	Classic (1st Order ESC)	High-Order Lie Bracket ESC
Quadratic ($m=2$)	Exponential	Exponential
Quartic or higher ($m>2$)	Polynomial	Exponential

The high-order Lie bracket ESC paradigm bridges a major performance gap in model-free optimization and adaptive control for systems encountering flat non-quadratic optima, and has transitioned from theoretical proposal to experimentally validated methodology in mobile robotics and beyond.