Extremum Seeking Control for Scalar Maps with Distributed Diffusion PDEs

TL;DR

This paper develops an extremum seeking control method for scalar maps governed by distributed diffusion PDEs, using backstepping transformation and averaging theory to ensure convergence to the optimal point.

Contribution

It introduces a novel control design combining backstepping and motion planning for distributed PDEs, with stability analysis and convergence guarantees.

Findings

The proposed controller achieves local exponential stability.

Trajectory converges to a neighborhood of the optimal point.

Numerical simulations demonstrate effectiveness in cascade nonlinear maps.

Abstract

This paper deals with the gradient extremum seeking control for static scalar maps with actuators governed by distributed diffusion partial differential equations (PDEs). To achieve the real-time optimization objective, we design a compensation controller for the distributed diffusion PDE via backstepping transformation in infinite dimensions. A further contribution of this paper is the appropriate motion planning design of the so-called probing (or perturbation) signal, which is more involved than in the non-distributed counterpart. Hence, with these two design ingredients, we provide an averaging-based methodology that can be implemented using the gradient and Hessian estimates. Local exponential stability for the closed-loop equilibrium of the average error dynamics is guaranteed through a Lyapunov-based analysis. By employing the averaging theory for infinite-dimensional systems, we prove that the trajectory converges to a small neighborhood surrounding the optimal point. The effectiveness of the proposed extremum seeking controller for distributed diffusion PDEs in cascade of nonlinear maps to be optimized is illustrated by means of numerical simulations.