Experimental Application of Predictive Cost Adaptive Control to Thermoacoustic Oscillations in a Rijke Tube

TL;DR

This paper addresses suppressing self-excited thermoacoustic oscillations in a Rijke-tube where analytic models are challenging. It adopts Predictive Cost Adaptive Control (PCAC), which performs online linear identification via Recursive Least Squares with variable-rate forgetting and uses a backward-propagating Riccati equation for receding-horizon optimization. A simple open-loop emulation guides hyperparameter selection, and PCAC is validated experimentally across multiple heater positions and voltages, demonstrating robust suppression of oscillations. PCAC outperforms retrospective cost adaptive control (RCAC) in speed of suppression while respecting actuator saturation, illustrating a practical data-driven framework for real-time control of complex thermoacoustic systems.

Abstract

Model predictive control (MPC) has been used successfully in diverse applications. As its name suggests, MPC requires a model for predictive optimization. The present paper focuses on the application of MPC to a Rijke tube, in which a heating source and acoustic dynamics interact to produce self-excited oscillations. Since the dynamics of a Rijke tube are difficult to model to a high level of accuracy, the implementation of MPC requires leveraging data from the physical setup as well as knowledge about thermoacoustics, which is labor intensive and requires domain expertise. With this motivation, the present paper uses predictive cost adaptive control (PCAC) for sampled-data control of an experimental Rijke-tube setup. PCAC performs online closed-loop linear model identification for receding-horizon optimization based on the backward propagating Riccati equation. In place of analytical modeling, open-loop experiments are used to create a simple emulation model, which is used for choosing PCAC hyperparameters. PCAC is applied to the Rijke-tube setup under various experimental scenarios.

Figures (8)

• Figure 1: Sampled-data implementation of predictive controller for stabilization of continuous-time system ${\mathcal{M}}.$ All sample-and-hold operations are synchronous. The predictive controller $G_{{\rm c},k}$ generates the requested discrete-time control $u_{{\rm req},k}\in{\mathbb R}^m$ at each step $k$. The implemented discrete-time control is $u_k=\sigma(u_{{\rm req},k})$, where $\sigma\colon{\mathbb R}^m\to{\mathbb R}^m$ represents control-magnitude saturation. The resulting continuous-time control $u(t)$ is generated by applying a zero-order-hold operation to $u_k$. For this work, ${\mathcal{M}}$ represents the Rijke-tube setup introduced in Section \ref{['sec:Rijke_Exp']} for physical experiments.
• Figure 2: Physical closed-loop Rijke-tube setup. The heating element can be raised or lowered by a DC motor (not shown) to vary the dynamics of the system.
• Figure 3: Pressure measurements from the open-loop experimental Rijke-tube setup obtained at the coil positions $x_{\rm us} \in \{0.3, 0.35, 0.4\}$ m and the AC voltage levels $V_{\rm RMS} \in \{75, 85, 95\}$ V, where $x_{\rm us}$ is the distance of the coil from the bottom of the tube, and $V_{\rm RMS}$ is the root-mean-square (RMS) voltage provided by the Variac.
• Figure 4: Amplitude spectra of the pressure measurements from the open-loop experiments at each setting considered in Figure \ref{['fig:rijke_tube_exp_OL_time']}.
• Figure 5: Pressure measurements $\tilde{p}_{\rm mic}$ from the closed-loop experiments using Retrospective Cost Adaptive Control (RCAC) from RijkeTCST are shown for $x_{\rm us} \in \{0.3, 0.35, 0.4\}$ m and $V_{\rm RMS} \in \{70, 80, 90\}$ V. Each experiment transitions from open-loop mode to closed-loop mode at the time indicated by the vertical red line. The same RCAC hyperparameters are used in all tests.