Retrospective Cost-based Extremum Seeking Control with Vanishing Perturbation for Online Output Minimization

TL;DR

This paper develops a retrospective cost-based ESC (RC/ESC) technique for online output minimization with a vanishing perturbation, that is, a perturbation that becomes zero as time increases independently from the state of the controller or the controlled system.

Abstract

Extremum seeking control (ESC) constitutes a powerful technique for online optimization with theoretical guarantees for convergence to the neighborhood of the optimizer under well-understood conditions. However, ESC requires a nonconstant perturbation signal to provide persistent excitation to the target system to yield convergent results, which usually results in steady state oscillations. While certain techniques have been proposed to eliminate perturbations once the neighborhood of the minimizer is reached, system modifications and environmental perturbations can suddenly change the minimizer and nonconstant perturbations would once more be required to convergence to the new minimizer. Hence, this paper develops a retrospective cost-based ESC(RC/ESC) technique for online output minimization with a vanishing perturbation, that is, a perturbation that becomes zero as time increases independently from the state of the controller or the controlled system. The performance of the proposed algorithm is illustrated via numerical examples.

Figures (13)

• Figure 1: Sampled-data implementation of adaptive controller for control of the continuous-time system ${\mathcal{M}}.$ All sample-and-hold operations are synchronous. The adaptive controller uses $J_k$ as an input and generates the discrete-time control $u_k$ at each step $k$. Note that all components of $J_k$ are nonnegative. 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 yield an input signal that minimizes the output of the continuous-time system, that is, yield $u(t)$ such that $\lim_{t \to \infty} J(t) = 0.$
• Figure 2: Continuous-time extremum seeking control (ESC) of the continuous-time system ${\mathcal{M}}.$ ESC uses $J$ as an input and generates the control $u_k$ at each step $k$. Note that all components of $J$ are nonnegative. The objective of the controller is yield an input signal that minimizes the output of the continuous-time system, that is, yield $u(t)$ such that $\lim_{t \to \infty} J(t) = 0.$
• Figure 3: RC/ESC block diagram.
• Figure 4: Example \ref{['ex:static_siso']}:SISO Static map. Controller output $u$ for the static map given by \ref{['SISO_static_example']} with ESC and RC/ESC. Note that the ESC response is shown in blue and the RC/ESC response is shown in red.
• Figure 5: Example \ref{['ex:static_siso']}:SISO Static map. Output error with respect to the optimal value $J=0$ in log scale with ESC and RC/ESC. Note that the error by ESC is shown in blue and the error by RC/ESC is shown in red.

Theorems & Definitions (3)

• Example V.1
• Example V.2
• Example V.3