Adaptive Numerical Differentiation for Extremum Seeking with Sensor Noise

TL;DR

This work tackles sensitivity of extremum-seeking control (ESC) to sensor noise by substituting the ESC high-pass differentiator with adaptive input and state estimation (AISE) in discrete-time, yielding ESC/AISE for SISO systems. AISE provides real-time numerical differentiation through a Kalman-based forecast, input estimation, and data assimilation, with adaptive tuning of noise covariances. Numerical results on a quadratic cost and an antilock braking system show ESC/AISE significantly reduces noise-induced disturbances and improves stopping performance under sensor noise, compared with standard ESC. The approach offers a practical, noise-robust alternative for ESC implementation with potential extensions to MIMO and Newton-based variants.

Abstract

Extremum-seeking control (ESC) is widely used to optimize performance when the system dynamics are uncertain. However, sensitivity to sensor noise is an important 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 adaptive input and state estimation (AISE) for numerical differentiation. In particular, this paper develops extremum-seeking control with adaptive input and state estimation (ESC/AISE), where the high-pass filter of ESC is replaced by AISE to improve performance under sensor noise. The effectiveness of ESC/AISE is illustrated via numerical examples.

Figures (13)

• Figure 1: Sampled-data implementation of the discrete-time controller $G_{\rm c}$ for controlling the continuous-time system $M$ with input $u,$ output $y,$ and sensor noise $v.$ All sample-and-hold operations are synchronous, and the sampling time is given by $T_{\rm s}>0.$ The discrete-time controller uses the sampled noisy measurement $y_{{\rm n}, k} \stackrel{\triangle}{=} y(k T_{\rm s}) + v(k T_{\rm s})$ as the input and generates the discrete-time control $u_k$ at each step $k$. The resulting continuous-time control $u(t)$ is generated by applying a zero-order-hold operation to $u_k$. The objective of the controller is to provide an input $u(t)$ that converges to a neighborhood of an input value that either locally minimizes or locally maximizes the output $y(t).$
• Figure 2: Discrete-time extremum-seeking control.
• Figure 3: Block diagram of AISE.
• Figure 4: Discrete-time extremum-seeking control with adaptive input and state estimation (ESC/AISE).
• Figure 5: Example \ref{['ex:static_quadratic']}: Quadratic Cost. Sensor noise $v$ defined in \ref{['ex1:v']} added to $y$ for $k \in [0, 6000]$.

Theorems & Definitions (3)

• Proposition IV.1
• Example V.1
• Example V.2