Active control of thermoacoustic oscillation based on the x-nlms algorithm

Thermoacoustic oscillations, caused by the coupling of unsteady heat release and pressure fluctuation, can result in severe damage to the combustion chamber of gas turbines. This paper presents a novel approach to active control of thermoacoustic oscillations, which is based on a Filtered-X normalized least mean square algorithm (X-NLMS). The controller employs the pressure magnitude at a predetermined axial position preceding the flame as the control variable, which is acquired through microphones installed around the combustor periphery. Control is accomplished by modifying the output frequency and phase of a loudspeaker. The performance of the control system is evaluated through simulations on a Rijke tube, with the classical PID method used as a comparison. The experimental results showcase the efficacy of the X-NLMS algorithm in actively controlling thermoacoustic oscillations. By adaptively estimating and attenuating pressure fluctuations, the algorithm effectively suppresses the undesired acoustic disturbances associated with thermoacoustic instabilities. Comparative analysis reveals that the X-NLMS algorithm outperforms PID control in terms of oscillation suppression and system stability. Its ability to adapt to time-varying dynamics and its low computational complexity make it a promising candidate for real-time thermoacoustic control applications.


Introduction
Thermoacoustic oscillations are a phenomenon that occurs in various combustion systems, leading to detrimental effects such as reduced efficiency, increased emissions, and even structural damage [1,2].These oscillations arise due to the coupling between heat release and pressure fluctuations within the combustion chamber, creating a feedback loop that sustains the instability.As a result, researchers have been exploring different control strategies to mitigate thermoacoustic oscillations and improve the performance and safety of combustion systems [3].
Other Passive control techniques, which involve modifications to the system's geometry or the addition of damping elements, have achieved some success in attenuating thermoacoustic oscillations.However, passive methods often lack the adaptability to effectively address the wide range of operating conditions experienced in practical combustion systems [4,5].Active control strategies, on the other hand, offer the potential to actively manipulate the acoustic field within the combustion chamber, providing a more versatile and adaptive approach [1].This paper proposes an active control approach for mitigating thermoacoustic oscillations based on the Filtered-X Normalized Least Mean Square (X-NLMS) algorithm.The X-NLMS algorithm is a variant of the widely used Normalized Least Mean Square (NLMS) algorithm, which has proven effective in various adaptive filtering applications [6].By adapting the X-NLMS algorithm to actively suppress thermoacoustic oscillations, this research aims to provide a robust and adaptable control strategy for combustion systems.The key advantage of the X-NLMS algorithm lies in its ability to estimate and track the oscillatory behavior of thermoacoustic instabilities in real-time.This estimation is achieved by processing the signals obtained from strategically placed sensors within the combustion chamber.By utilizing these estimates, the algorithm generates corrective signals that are introduced through actuators, effectively suppressing the undesired pressure fluctuations associated with thermoacoustic oscillations [7].
To evaluate the effectiveness of the proposed active control approach, a thermoacoustic coupling model of a Rijke tube was established.Thermoacoustic oscillations are induced within the system under various operating conditions, and the performance of the X-NLMS algorithm is assessed in terms of its ability to suppress these oscillations.In addition to evaluating the performance of the X-NLMS algorithm, this paper also compares its effectiveness against traditional control techniques, such as Proportional-Integral-Derivative (PID) control.This comparison provides valuable insights into the relative merits of the X-NLMS algorithm for active control of thermoacoustic oscillations.

Parameter settings for Rijke tube
The experimental setup utilized in this study is depicted in Figure 1.The establishment of the thermoacoustic coupling model is based on the Rijke tube.The acoustic pressure data required for the investigation were obtained from the Rijke tube, which forms a crucial component of the setup.The experimental device comprises three primary sections: a borosilicate glass duct, a flame burner, and the measurement system.The burner tube, responsible for mixing fuel (methane) and air, is constructed using stainless steel and measures 500 mm in length with an inner diameter of 22.5 mm.Positioned within the setup, the borosilicate glass duct extends to a length of l = 750 mm and features an inner diameter of d = 50 mm.The distance between the flame position and the bottom of the glass tube is precisely l1 = 60 mm.In order to capture pressure signals effectively, two holes are strategically placed along the glass tube to facilitate measurements within the combustion system.To ensure the optimal functioning of the sensors at appropriate temperatures, a semi-infinite pressure tube configuration, as illustrated in Figure 1, was implemented.Acoustic pressure data was collected using two CRYSOUND type 547 microphones.To enhance the accuracy of signal acquisition, a signal conditioner (PCB 482C16) was employed to amplify the minute voltage signals.Additionally, a 16-bit analog-to-digital conversion card (NI 6212) was utilized to acquire data at a sampling rate of 10 kHz.Considering that the oscillation frequency of the combustion system falls within the range of 1 kHz, the digital signal obtained after sampling retains all the essential information present in the original signal.
The Rijke tube burner operates under the following working conditions shown in the Table 1.The fuel used is methane (CH4), which is commonly employed in combustion systems due to its high energy content and clean burning characteristics.The equivalence ratio, defined as the ratio of the actual fuel-to-air ratio to the stoichiometric fuel-to-air ratio, is set at 0.7.This value indicates that there is a slight excess of air compared to the ideal amount required for complete combustion.This slightly lean mixture can help improve fuel efficiency and reduce emissions in certain applications.The thermal power of the burner is specified as 415 W, indicating the amount of heat released by the combustion process.

Modeling of flame-acoustic coupling
The Rijke tube is a classic device used to study thermoacoustic oscillations.We establish a thermoacoustic coupling model for the Rijke tube in Figure 1.The system is assumed to have axial symmetry, enabling the use of one-dimensional wave equations along the direction of acoustic propagation.This assumption implies that variations in the cross-section are negligible compared to the axial length of the system.The thermoacoustic coupling mechanism model typically begins with the Navier-Stokes equation and utilizes linearization and Newton decomposition to express the relationship between pressure pulsation and heat release rate pulsation in the form of a wave equation [8]: In the equation,  represents the velocity of sound, and  and  represent the pressure pulsation and heat release rate pulsation, respectively. is the gas constant, here  = 1.4.The left side of the equation describes the propagation of the sound field, while the right side represents the active term, indicating the influence of unstable combustion heat  on the sound field.
The flame transfer function is based on an  - model, where  represents the interaction index and  denotes the time delay.By utilizing this model, the heat release transfer function of the flame can be derived as follows [9]: where   ( 1 ) represents the heat release,   denotes the time constant of the low pass element, and  1 represents the acoustic velocity upstream of the flame.
The governing equation of the wave equation are: The reflection coefficient for the choked boundary condition, R  , was defined as 0.9.The reflection coefficient for the open-ended downstream boundary can be expressed as follows: The flame front and back regions can be modeled using the principles of mass, momentum, and energy conservation equations: Based on the above equations, the transfer matrix for flame-acoustic coupling can be obtained [8]: A thermoacoustic coupling model of the Rijke tube was constructed using the Simulink module in MATLAB shown in

X-NLMS algorithm
The NLMS algorithm is widely used in Acoustic Echo Cancellation (AEC) systems.The NLMS algorithm uses adaptive filtering to identify the characteristics of the acoustic path.It estimates the impulse response of the acoustic path by correlating the received signal with the known transmitted signal.The NLMS algorithm adapts the coefficients of the filter based on the error between the estimated echo and the actual echo in the received signal.It updates the filter coefficients iteratively, adjusting them to minimize the mean square error between the estimated and actual echoes.To ensure stability and prevent large coefficient updates, the NLMS algorithm normalizes the update step based on the power of the input signal.This allows for a more controlled adaptation of the filter coefficients.
As shown in Figure 1, microphone 1 receives a reference sound source, the speaker acts as a sound transmitter, emitting a certain phase and amplitude of sound waves, and microphone 2 obtains the residual signal after cancellation.There is a propagation process between the echo canceller and the error microphone.This propagation from the loudspeaker to the error microphone is referred to as the secondary path, and its transfer function is denoted as ().The control circuit diagram can be clearly displayed through Figure 3.The LMS algorithm comprises three main components: filtering, error estimation, and weight coefficient vector update.Following the derivation of the steepest gradient descent method, the algorithm follows this process [6,7]: Here, w represents a column vector that can be viewed as the coefficients of an FIR filter. represents a column vector, which is a reference signal for a particular segment.d represent the desired signal.y represent the predicted value of the desired signal.w ´ represents the filter coefficients for the next segment (usually obtained by shifting the current segment as a whole by one sample).μ is a constant step size parameter.
NLMS algorithm improves upon the LMS algorithm by considering variations in the signal level at the filter input.It incorporates a normalized step size parameter, resulting in a stable and quickly converging adaptive algorithm.the algorithm follows this process [6,7]: and μ ̃ is a normal number that requires appropriate selection.The sufficient condition for the convergence of the filter weight vector w in the mean sense is that the step size of the algorithm should meet [10]: The excess means square error  and steady-state misalignment  of the algorithm can be approximated as [10]: Where λ  is the maximum eigenvalue of  = {()  ()}.
A schematic of the X-NLMS control system for thermoacoustic oscillation was developed using the Simulink module in MATLAB shown in Figure 4.By interconnecting these components and setting appropriate parameters, the Simulink model facilitated the analysis and optimization of the X-NLMS control system's ability to mitigate and suppress thermoacoustic oscillations.

Accuracy of the flame-acoustic coupling model
The accuracy of a control algorithm is dependent on the precision of the underlying model.Hence, this chapter aims to validate the accuracy of the thermoacoustic coupling model employed.
Figure 5 presents a comparison of time-domain and frequency-domain results between the model and experimental data.Analyzing Figure 5 (a) and Figure 5 (d), it is evident that combustion oscillation comprises three distinct stages: the stable stage, oscillation development stage, and thermoacoustic oscillation stage.Notably, both the model and experimental data exhibit consistency in their behavior across these stages.
Examining Figure 5 (b) and Figure 5 (e), it becomes apparent that during the oscillation stage, the thermoacoustic coupling model approximates a sinusoidal curve, signifying a limit cycle oscillation.Remarkably, the experimental data exhibits similar characteristics.However, the experimental data may appear less smooth due to environmental interferences that influence the measurements.
Further analysis of Figure 5 (c) and Figure 5 (f) reveals that both sets of data display a single prominent peak in the frequency domain.The simulation data indicates a frequency of 151Hz, while the experimental data records a frequency of 139Hz, resulting in an error of 7.95%.This discrepancy could stem from the inherent frequency response characteristics of the sensor utilized in the experiments.In conclusion, based on the observations made, it can be affirmed that the thermoacoustic coupling model demonstrates reliability and possesses adequate accuracy for controller design purposes.The alignment observed between the model predictions and the experimental data across various stages, oscillatory behavior, and frequency characteristics validates the effectiveness of the model.Minor discrepancies can be attributed to environmental factors and inherent limitations of measurement equipment.Nonetheless, the overall agreement between the model and experimental data affirms the utility and accuracy of the thermoacoustic coupling model in guiding controller design endeavors.

Comparison of control effects between PID and X-NLMS
Figure 6 shows the performance of the PID algorithm (a) and X-NLMS algorithm (b) in feedback control of thermoacoustic oscillations.Upon analyzing the control results, it is observed that the application of the PID control signal leads to a 68% reduction in the amplitude of thermoacoustic oscillation.In contrast, when the X-NLMS control signal is applied, the amplitude of thermoacoustic oscillation decreases by 96%.This indicates that the X-NLMS algorithm outperforms the PID controller in terms of control effectiveness.
Examining the control responses more closely, it is evident that the amplitude of thermoacoustic oscillation rapidly decreases upon applying the PID control signal.Conversely, when the X-NLMS control signal is implemented, the amplitude exhibits a gradual decrease at a certain slope.However, it is worth noting that the control speed of the X-NLMS algorithm is not as rapid as that of the PID controller.This disparity in control speed can be attributed to the significant computational time required for updating the filter weights within the X-NLMS algorithm.
The superior control effectiveness of the X-NLMS algorithm can be attributed to its adaptive nature, which enables the algorithm to continuously refine the filter coefficients based on the error estimation.This adaptability allows the X-NLMS algorithm to better track and mitigate the thermoacoustic oscillations compared to the PID controller, which relies on fixed parameters and may struggle to respond accurately to changing conditions.While the PID controller offers faster control speed in this particular scenario, the X-NLMS algorithm's ability to achieve a significantly higher reduction in thermoacoustic oscillation amplitude suggests its suitability for robust and precise control applications.The trade-off with slower control speed is acceptable considering the substantial improvement in control effectiveness achieved through the X-NLMS approach.
In summary, Figure 6 demonstrates that the X-NLMS algorithm surpasses the PID controller in controlling thermoacoustic oscillations.By achieving a 96% reduction in amplitude compared to the PID controller's 68%, the X-NLMS algorithm showcases superior control effectiveness.Although the X-NLMS algorithm may exhibit slower control speed due to computational overhead, its adaptive nature and ability to continuously update filter weights contribute to its enhanced control performance.

Conclusions
This study highlights the successful application of the X-NLMS algorithm for active control of thermoacoustic oscillations in gas turbines.The developed thermoacoustic coupling model accurately captures the system behavior, providing a reliable foundation for control analysis.Through comparison with experimental data, the model's accuracy is confirmed.The X-NLMS algorithm demonstrates superior control effectiveness compared to the traditional PID controller, achieving a remarkable reduction in thermoacoustic oscillation amplitude.Despite slightly slower control speed due to computational requirements, the adaptability and accuracy of the X-NLMS algorithm make it a highly promising approach for mitigating thermoacoustic oscillations in gas turbines.Future research may focus on optimizing the control parameters and exploring real-world implementation possibilities.

Figure 1 .
Figure 1.Schematic drawing of the Rijke tube setup.

Figure 2 .
The model encompassed the geometrical and boundary conditions of the Rijke tube, along with the governing equations of fluid dynamics and energy transfer.

Figure 3 .
Figure 3. X-NLMS Control System with secondary path.

Figure 5 .
Figure 5.Comparison of model and experimental data: (a) (b) (c) is the data obtained from the flame-acoustic coupling model, and (d) (e) (f) is the experimental data obtained from the Rijke tube.

Figure 6 .
Figure 6.Performance of the PID algorithm (a) and X-NLMS algorithm (b) in feedback control of thermoacoustic oscillations.

Table 1 .
Experimental conditions of the Rijke tube.