Robust Adaptive Boundary Control of a Thermal Process with Thermoelectric Actuators: Theory and Experimental Validation

TL;DR

The paper tackles robust stabilization of an uncertain diffusion-reaction PDE with boundary actuation by introducing an adaptive sliding-mode boundary controller that tunes the discontinuous gain online. It formalizes two adaptation schemes—monodirectional and bidirectional—and proves, via Lyapunov analysis, that monodirectional adaptation yields global asymptotic stability in $L_2$, while bidirectional adaptation delivers global uniformly ultimately bounded behavior with an explicit bound. The theoretical results are complemented by experimental validation on a metal beam actuated by thermoelectric modules, showing that bidirectional adaptation reduces chattering and yields better disturbance rejection. The work advances boundary control for distributed parameter systems with unknown spatially varying coefficients and demonstrates practical applicability to thermally actuated engineering systems.

Abstract

A sliding-mode-based adaptive boundary control law is proposed for a class of uncertain thermal reaction-diffusion processes subject to matched disturbances. The disturbances are assumed to be bounded, but the corresponding bounds are unknown, thus motivating the use of adaptive control strategies. A boundary control law comprising a proportional and discontinuous term is proposed, wherein the magnitude of the discontinuous relay term is adjusted via a gradient-based adaptation algorithm. Depending on how the adaptation algorithm is parameterized, the adaptive gain can be either a nondecreasing function of time (monodirectional adaptation) or it can both increase and decrease (bidirectional adaptation). The convergence and stability properties of these two solutions are investigated by Lyapunov analyses, and two distinct stability results are derived, namely, asymptotic stability for the monodirectional adaptation and globally uniformly ultimately bounded solutions for the bidirectional adaptation. The proposed algorithms are then specified to address the control problem of stabilizing a desired temperature profile in a metal beam equipped with thermoelectric boundary actuators. Experiments are conducted to investigate the real-world performance of the proposed sliding-mode-based adaptive control, with a particular focus on comparing the monodirectional and bidirectional adaptation laws.

Figures (11)

• Figure 1: A thermal imaging camera measures the temperature of the aluminium beam. The bottom sides of the TEMs are kept at ambient temperature by a water cooling system.
• Figure 2: The aluminium beam is cooled/heated by the outermost TEMs, used as boundary actuators, whereas the inner TEMs are inactive during all conducted experiments. The TEMs highlighted by orange dashed lines are used for imposing matched boundary disturbances.
• Figure 3: Validation of the model by comparing an experiment with a corresponding simulation.
• Figure 4: There are two possibilities for the current to achieve a certain heat flux $\sigma_i u_i$. The current is plotted over $\sigma_i u_i$ instead of $u_i$ for better physical interpretation.
• Figure 5: The feedback loop stabilizes the equilibrium $(T_\text{e}, u_{i,\text{e}})$ and is robust against external, matched disturbances $\psi_i(t)$, $i \in \{0,1\}$.

Theorems & Definitions (3)

• Lemma 1
• Theorem 1
• Remark 1