Closed-form adaptive tracking control of heat equations aided by Fourier regularization and bi-orthogonal series

Abstract

This article proposes a closed-form adaptive tracking control approach for linear heat equations with unknown parameters to achieve full temperature profile tracking by leveraging Fourier regularization and bi-orthogonal series. A state predictor which copies the plant with state partial derivatives and unknown parameters replaced by their estimates is built and an adaptive law is designed to estimate the unknown parameters. The state predictor is decomposed into two subsystems for tracking control synthesis: the first subsystem involves terms from the original heat equation, while the second subsystem is simpler and can be reformulated as a standard heat equation. Specifically, the first subsystem is regarded as an unforced PDE whose terminal states always follow the desired temperature profile such that its initial condition can be calculated by solving the backward heat equation at every time step. To address the blow-up issue in backward calculation, a Fourier regularization scheme is explored to cut off the higher-order Fourier modes and an appropriate tradeoff between approximation accuracy and robustness is achieved. Given the solutions from the first subsystem, the initial condition for the second subsystem can be subsequently calculated. We propose a numerical algorithm to calculate a set of bi-orthogonal series online and employ them to compute the boundary control function that drives the second subsystem to zero at every time step. Combining these two subsystems, it guarantees that the overall system follows the desired temperature profile. We demonstrate that the proposed closed-form adaptive tracking control algorithm achieves full temperature profile tracking with around 5% error averaged over the entire space, that is, $L_2$ norm over time.