Sensor-Noise Mitigation in Extremum Seeking Control Using Adaptive Numerical Differentiation

Extremum-seeking control (ESC) is widely used to optimize performance when the system dynamics are uncertain. However, sensitivity to sensor noise is a crucial issue in ESC implementation due to the use of high-pass filters or gradient estimators. To reduce the sensitivity of ESC to noise, this paper investigates the use of the recently developed adaptive input and state estimation (AISE) technique for numerical differentiation. In particular, this paper develops extremum-seeking control with adaptive input and state estimation (ESC/AISE), where AISE replaces the high-pass filter of ESC to improve performance under sensor noise. The effectiveness of ESC/AISE is illustrated via numerical examples.

💡 Research Summary

The paper addresses a well‑known drawback of extremum‑seeking control (ESC): the amplification of sensor noise caused by the high‑pass filter or numerical differentiation step that extracts the gradient of the performance map. Classical ESC injects a small sinusoidal dither into the control input, multiplies the measured output by the dither, and then passes the product through a high‑pass filter to obtain an estimate of the local gradient. Because the high‑pass operation emphasizes high‑frequency components, any measurement noise present in the sensor signal is also amplified, leading to degraded convergence speed, larger steady‑state error, and possible instability.

To mitigate this issue, the authors propose replacing the high‑pass filter with the Adaptive Input and State Estimation (AISE) algorithm, a recently developed real‑time numerical differentiator. AISE treats the sampled plant output as the output of a discrete‑time linear integrator model xₖ₊₁ = A xₖ + B dₖ, yₖ = C xₖ + D vₖ, where dₖ represents the unknown derivative of yₖ. By jointly estimating the hidden state xₖ and the unknown input dₖ, AISE produces an adaptive estimate \hat dₖ that serves as a noise‑robust gradient estimate. The algorithm incorporates several tunable matrices (R_θ, R_z, R_d) to control adaptation aggressiveness, a covariance‑reset matrix R_∞ to prevent divergence, and forgetting‑factor parameters (τ_n, τ_d, η, α) that allow the estimator to track slowly varying noise statistics. The estimator also uses a data‑window length n_e and FIR order n_f to improve accuracy when the signal‑to‑noise ratio is low.

The resulting ESC/AISE scheme retains the standard low‑pass filter, integrator, and dither generation of conventional ESC, but substitutes the high‑pass filter update (y_h,k) with the AISE‑derived derivative \hat dₖ. This substitution eliminates the direct high‑pass operation, thereby preventing the usual noise amplification while still providing a reliable gradient signal for the outer loop.

Two simulation studies illustrate the benefits. In Example 5.1, a static quadratic cost y = ¼ u² is minimized under additive Gaussian sensor noise that increases in variance after step k = 1500. Both ESC and ESC/AISE converge to the optimal input u = 0, but ESC exhibits large oscillations after the noise level rises, whereas ESC/AISE maintains a smooth trajectory. Over 200 Monte‑Carlo runs, the average root‑mean‑square error (RMSE) drops from 0.476 (ESC) to 0.240 (ESC/AISE), a reduction of roughly 50 %.

Example 5.2 applies the methods to an anti‑lock braking system (ABS) model, where the goal is to maximize the friction force by regulating wheel slip λ. The plant dynamics involve coupled longitudinal and rotational equations, and the friction coefficient μ(λ) is a nonlinear function of slip. Sensor noise corrupts the measured velocity and slip signals. Again, ESC/AISE achieves tighter tracking of the desired slip and smoother brake‑torque commands compared with conventional ESC, confirming robustness in a more complex, dynamic scenario.

The paper also provides a detailed parameter‑tuning guide for AISE, specifying typical ranges for the estimation window n_e, FIR order n_f, regularization matrices, and forgetting‑factor settings. This makes the approach readily implementable on digital controllers where sampling and zero‑order‑hold are already assumed.

In conclusion, ESC/AISE offers a practical, model‑free solution that substantially reduces the sensitivity of extremum‑seeking control to measurement noise without sacrificing the simplicity of the original algorithm. By leveraging an adaptive, causal numerical differentiator, the method can be deployed across the many domains where ESC is popular—robotics, energy management, combustion control, and even nuclear‑fusion experiments—enhancing both convergence speed and steady‑state accuracy. Future work suggested includes extending the framework to multi‑input‑multi‑output (MIMO) systems, hardware‑in‑the‑loop validation, and comparative studies with other learning‑based differentiation techniques.