A linear system identification method for the thermoacoustic instability of Rijke tubes

Abstract

This invention provides a linear system identification method for the thermoacoustic instability of Rijke tubes, comprising: establishing a set of time-delay partial differential equations describing the thermoacoustic instability of Rijke tubes based on the physical mechanism of one-dimensional compressible gases, and transforming them into a transfer function model of pressure fluctuations at the boundary under mixed boundary conditions using the Galerkin and Taylor expansion method; inputting a frequency sweep identification signal to the thermoacoustically unstable Rijke tube system through a loudspeaker, and combining it with the sound pressure signal collected by a microphone to obtain the frequency characteristics of the system; estimating the unknown parameters in the system transfer function based on the frequency characteristics of the system; this invention helps to solve the technical problem of pressure oscillations caused by thermoacoustic instability in the combustion chamber of aero-engines.

Description

Technical Field

[0001] This invention relates to the field of active control technology for combustion instability in aero-engines, and more particularly to a linear system identification method for the thermoacoustic instability of Rijke tubes. Background Technology

[0002] Thermoacoustic instability has always been a thorny problem in gas turbines, aero engines, and rocket engines, and severe thermoacoustic instability can cause fatal damage to the combustion chamber structure. The Rijke tube, as a typical experimental device for thermoacoustic instability, has played a significant role in advancing research on this phenomenon. To date, modeling and analysis of thermoacoustic instability in Rijke tubes have mainly focused on first-principles modeling using nonlinear partial differential equations (PDEs) or ordinary differential equations (ODEs), which do not fully reflect the actual system. Active control based on PDEs or complex nonlinear ODE models often results in complex control algorithms due to the complexity of the models, leading to higher requirements for controller hardware. This may require high computational loads and complex real-time calculations to achieve control, posing many difficulties for controller design, adjustment, fault diagnosis, and recovery.

[0003] Existing research on the identification of thermoacoustic instability in Rijke tubes mainly involves obtaining the Bode plot (amplitude and phase frequency characteristics) of the system by identifying the Fourier transform of the signal, without involving the establishment of a specific mathematical model and parameter identification. Summary of the Invention

[0004] To address the aforementioned technical problem of pressure oscillations within the combustion chamber of aero-engines, this invention provides a linear system identification method for the thermoacoustic instability of Rijke tubes. This invention identifies the transfer function and specific parameters of the linearized open-loop system of the Rijke tube power unit, which reveals thermoacoustic instability, providing support and guidance for thermoacoustic unstable system model building, parameter estimation, controller design and adjustment, fault diagnosis, and recovery.

[0005] The technical means employed in this invention are as follows:

[0006] A method for identifying linear systems with thermoacoustic instability in Rijke tubes, characterized by comprising:

[0007] S1. Based on the physical mechanism, establish a model structure in the form of a transfer function to describe the thermoacoustic instability phenomenon;

[0008] S2. Input and detect the identification signal to identify the frequency characteristics of the Rijke tube thermoacoustic unstable system;

[0009] S3. Based on the identified frequency characteristics, estimate the unknown parameters in the transfer function model.

[0010] Furthermore, in step S1, the process of establishing a model structure in the form of a transfer function describing the thermoacoustic instability phenomenon includes:

[0011] S11. Based on the physical mechanism of mass, momentum and energy conservation of one-dimensional compressible gas, and considering the linearized Heckl heat release model, establish a set of partial differential equations describing the thermoacoustic instability of the Rijke tube.

[0012] S12. Considering mixed boundary conditions, the partial differential equations are simplified to time-delay ordinary differential equations with respect to the modes using the Galerkin method.

[0013] S13. Use Taylor expansion to handle the time delay term, and through the relationship between mode and pressure, convert it into a linear ordinary differential equation about sound pressure, and finally into the transfer function form of proportional element and second-order oscillatory element.

[0014] Furthermore, in step S11, the set of partial differential equations describing the thermoacoustic instability of the Rijke tube is established as follows:

[0015]

[0016]

[0017] in, This indicates the amount of gas flow velocity fluctuation. This indicates the amount of gas pressure fluctuation. This represents a constant gas density. Let ξ represent the constant gas pressure, ξ represent the damping coefficient, c represent the speed of sound, and L represent the tube length. y represents the adiabatic constant, and q(t, x) represents the heat release rate term. According to the linearized Heckl heat release model, it can be expanded as follows:

[0018]

[0019] Where A is the cross-sectional area of ​​the tube, δ(·) is the Dirac pulse, x0 is the position of the heat source in the tube, and L w T represents the length of the resistance wire. w The temperature of the heating wire is represented by T, the ambient air temperature is represented by k, and the thermal conductivity of air is represented by c. v For the specific heat capacity at constant pressure, d w The diameter of the resistance wire. This represents a constant gas flow rate, and τ is the time delay of heat release.

[0020] Furthermore, in step S12, the partial differential equation is simplified into an ordinary differential equation with respect to the mode using the Galerkin method, specifically as follows:

[0021] S121. Using the Galerkin method, the above partial differential equation is transformed into an ordinary differential equation. The pressure p and velocity v are superimposed using polynomial functions that satisfy the boundary conditions. The product function is selected as:

[0022]

[0023]

[0024] in N represents the number of acoustic modes, j represents the j-th acoustic mode in the system, and η j This represents the amplitude of the j-th acoustic mode as it varies with time. Indicates η j The derivative with respect to time;

[0025] S122. Substitute into the energy conservation equation:

[0026]

[0027] S123, Multiply both ends Then, integrating over the interval from 0 to L, we get:

[0028]

[0029] S124. Considering only the first mode, i.e., j=1, we can obtain:

[0030]

[0031]

[0032] S125, Order The ordinary differential equation with time delay is:

[0033]

[0034] Further, in step S13, the time delay term is processed using Taylor expansion, and through the relationship between mode and pressure, it is converted into a linear ordinary differential equation about sound pressure, and finally transformed into the transfer function form of a proportional element and a second-order oscillatory element. The specific process includes:

[0035] S131. Consider a first-order Taylor expansion. Handling time delays:

[0036]

[0037] S132. The simplified ordinary differential equation with coefficients is as follows:

[0038]

[0039] in,

[0040] S133. Considering the mixed boundary conditions at the upper opening, the pressure fluctuation expression at the upper opening is... Substituting the coefficients into the simplified ordinary differential equation, we get:

[0041]

[0042] S134. Differentiate and rearrange the parameters:

[0043]

[0044] S135, Considering the fluctuations of the system controller. As input:

[0045]

[0046] S136. The transfer function describing the pressure fluctuation from the lower opening to the upper opening in a thermoacoustic unstable Rijke tube system is:

[0047]

[0048] The input to this system is the sound pressure oscillation at the lower opening of the Rijke tube, and the output is the sound pressure oscillation at the upper opening.

[0049] S137. Consider a single-input single-output system. The input is the voltage signal from the loudspeaker, and the output is the voltage signal generated by the sound pressure oscillation at the upper opening of the Rijke tube, measured by a microphone. The loudspeaker is located at the lower opening of the Rijke tube. The process of converting the voltage signal into sound pressure oscillation is considered as a proportional element and a hysteresis element, with a proportional gain of k. a The time delay is τ1; similarly, the sound pressure oscillation at the upper opening of the Rijke tube is converted into a microphone voltage signal, which is considered as a proportional element k. m And a hysteresis element with a time delay of τ2; the sound pressure oscillation from the lower opening of the Rijke tube to the upper opening is considered as a single element G; the system between the loudspeaker voltage signal and the microphone voltage signal is denoted as G1, as follows:

[0050]

[0051] S138. Simplify system parameters and set the total system gain to k. G =k4k a k m If the total time delay is τ = τ1 + τ2, then:

[0052]

[0053] Further, step S2 includes applying a sweep frequency identification signal to the speaker, acquiring the microphone signal after passing through the Rijke tube and the microphone signal without passing through the Rijke tube, performing a Fourier transform on the two signals to obtain the identification results of the system's amplitude-frequency characteristics and phase-frequency characteristics. The specific implementation process is as follows:

[0054] S21. Applying the product of a sweep frequency signal and an exponentially decaying signal to the loudspeaker yields the following signal:

[0055] U a (t)=sin(2π·f(t)·t)·Ampitude

[0056] Where, Ampitude = 5·e -0.3t f(t) = 100 + 60t, t ∈ [0, 15]; the signal is scanned from 100 Hz to 1000 Hz within 15 seconds, which is the frequency range of interest in the Rijke tube experiment; the sound pressure oscillation of the Rijke tube is measured through a microphone, and the voltage signal U output by the microphone is obtained. m ;

[0057] S22. The voltage signal output from the microphone is passed through a speaker and applied directly to the microphone without passing through the Rijke tube, thus obtaining a signal:

[0058] U m ′=k a k m ·e -τs ·U a

[0059] Where, k a k represents the loudspeaker proportional gain. m U represents the microphone ratio gain. a This indicates the signal applied to the speaker;

[0060] S23. The response and amplitude-phase frequency characteristics of system G are obtained by comparing the output signal directly acting on the microphone with the output signal passing through the Rijke tube:

[0061]

[0062] Among them, G(e jω ) represents the frequency response of system G, k G =k4k a k m U m This represents the acquired signal U after passing through the Rijke tube. m Frequency response, U m ′(e jω ) represents the acquisition signal U without passing through a Rijke tube. mThe frequency response of ′.

[0063] Furthermore, the specific process of step S3 includes:

[0064] Based on the system's amplitude-frequency and phase-frequency characteristics, the specific parameters of the transfer function of the system, which is a proportional element and a second-order oscillating element, are identified as follows:

[0065]

[0066] in, The identification results for k1′ and k2′ are given; the oscillation frequency is the peak value of the amplitude-frequency characteristic curve near the first mode. When Tω << 1, the amplitude-frequency characteristic A(ω) ≈ K, and the logarithmic amplitude-frequency characteristic L(ω) ≈ 20logK. The value of k can be obtained from this. The system has an unstable second-order oscillatory element, so the damping ratio ξ = 0 is taken.

[0067] Compared with the prior art, the present invention has the following advantages:

[0068] 1. The linear system identification method for the thermoacoustic instability of Rijke tubes provided by this invention establishes a linearized transfer function model of the thermoacoustic unstable Rijke tube system, modeling its transfer function as a proportional element and a second-order oscillatory element. In the Rijke tube experimental system, a swept frequency identification signal is used as input through a loudspeaker to collect the output signal of a microphone, obtain the response and amplitude-phase frequency characteristics of the system G, and identify the specific parameters in the transfer function model. This makes the controller design based on this model more realistic and is of great significance for controller design and system analysis.

[0069] 2. The linear system identification method for the thermoacoustic instability of Rijke tubes provided by this invention simplifies the system description to a certain extent compared with complex partial differential equations or nonlinear ordinary differential equations. The system parameters that conform to reality are obtained through identification, which has important guiding significance and practical application value for the analysis and active control of thermoacoustic unstable systems.

[0070] Based on the above reasons, this invention can be widely applied in fields such as active control of combustion instability in aero-engines. Attached Figure Description

[0071] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0072] Figure 1 This is a schematic diagram of the Rijke tube linearization system identification system provided in an embodiment of the present invention.

[0073] Figure 2 This is a flowchart of the linearization system identification process for the Rijke tube thermoacoustic unstable system of the present invention.

[0074] Figure 3 This is a schematic diagram of the G-system provided in an embodiment of the present invention.

[0075] Figure 4 A block diagram of a modeled Rijke pipe system provided for an embodiment of the present invention.

[0076] Figure 5 This is a sweep frequency signal with exponentially decaying amplitude provided in an embodiment of the present invention.

[0077] Figure 6 This is a schematic diagram showing a signal acting directly on a microphone through a speaker, as provided in an embodiment of the present invention.

[0078] Figure 7 The sweep frequency signal provided in this embodiment of the invention is obtained by directly applying the speaker to the microphone. m 'Signal.

[0079] Figure 8 The sweep frequency signal provided in this embodiment of the invention acts on the lower boundary of the Rijke tube and is obtained by the microphone at the upper boundary. m Signal.

[0080] Figure 9 The amplitude-frequency characteristics of the identified system G Bode plot (considering only the first mode) provided in the embodiments of the present invention.

[0081] Figure 10 The phase-frequency characteristics of the identified system G Bode diagram (considering only the first mode) provided in the embodiments of the present invention.

[0082] Figure 11 The system established based on the identified transfer function parameters provided in this embodiment of the invention Bode plot (amplitude-frequency response).

[0083] Figure 12 The system established based on the identified transfer function parameters provided in this embodiment of the invention Bode plot (phase frequency response).

[0084] Figure 13 The system established based on the identified transfer function parameters provided in this embodiment of the invention Step response (time domain signal).

[0085] Figure 14 The system established based on the identified transfer function parameters provided in this embodiment of the invention Step response (spectral analysis).

[0086] Figure 15 The output signal spectrum characteristics of the actual system provided in the embodiments of the present invention. Detailed Implementation

[0087] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other. The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0088] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The following description of at least one exemplary embodiment is merely illustrative and is in no way intended to limit the present invention or its application or use. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0089] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the scope of exemplary embodiments according to the invention. As used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0090] Unless otherwise specifically stated, the relative arrangement, numerical expressions, and values ​​of the components and steps described in these embodiments do not limit the scope of the invention. It should also be understood that, for ease of description, the dimensions of the various parts shown in the drawings are not drawn to actual scale. Techniques, methods, and devices known to those skilled in the art may not be discussed in detail, but where appropriate, such techniques, methods, and devices should be considered part of the specification. In all examples shown and discussed herein, any specific values ​​should be interpreted as merely exemplary and not as limitations. Therefore, other examples of exemplary embodiments may have different values. It should be noted that similar reference numerals and letters in the following figures denote similar items; therefore, once an item is defined in one figure, it need not be further discussed in subsequent figures.

[0091] This invention provides a linear system identification method for the thermoacoustic instability of Rijke tubes. This method identifies linearized systems based on typical Rijke tube models that reveal thermoacoustic instability phenomena, such as... Figure 1 The figure shows a schematic diagram of the identification system for the Rijke tube linearization system.

[0092] like Figure 2 The diagram shows a flowchart of a linear system identification method for the thermoacoustic instability of Rijke tubes provided by the present invention, including:

[0093] S1. Based on the physical mechanism, establish a model structure in the form of a transfer function to describe the thermoacoustic instability phenomenon;

[0094] S2. Input and detect the identification signal to identify the frequency characteristics of the Rijke tube thermoacoustic unstable system;

[0095] S3. Based on the identified frequency characteristics, estimate the unknown parameters in the transfer function model.

[0096] In a specific implementation, as a preferred embodiment of the present invention, step S1, the process of establishing a model describing the thermoacoustic instability phenomenon, includes:

[0097] S11. Based on the physical mechanism of mass, momentum and energy conservation of one-dimensional compressible gas, and considering the linearized Heckl heat release model, establish a set of partial differential equations describing the thermoacoustic instability of the Rijke tube.

[0098] S12. Considering mixed boundary conditions, the partial differential equations are simplified to time-delay ordinary differential equations with respect to the modes using the Galerkin method.

[0099] S13. Use Taylor expansion to handle the time delay term, and through the relationship between mode and pressure, convert it into a linear ordinary differential equation about sound pressure, and finally into the transfer function form of proportional element and second-order oscillatory element.

[0100] In a specific implementation, as a preferred embodiment of the present invention, the partial differential equations describing the thermoacoustic instability of the Rijke tube established in step S11 are as follows:

[0101]

[0102]

[0103] in, This indicates the amount of gas flow velocity fluctuation. This indicates the amount of gas pressure fluctuation. This represents a constant gas density. Let ξ represent the constant gas pressure, ξ represent the damping coefficient, c represent the speed of sound, and L represent the tube length. y represents the adiabatic constant, and q(t, x) represents the heat release rate term. According to the linearized Heckl heat release model, it can be expanded as follows:

[0104]

[0105] Where A is the cross-sectional area of ​​the tube, δ(·) is the Dirac pulse, x0 is the position of the heat source in the tube, and L w T represents the length of the resistance wire. w The temperature of the heating wire is represented by T, the ambient air temperature is represented by k, and the thermal conductivity of air is represented by c. v For the specific heat capacity at constant pressure, d w The diameter of the resistance wire. This represents a constant gas flow rate, and τ is the time delay of heat release.

[0106] In this embodiment, since the mathematical model revealing the thermoacoustic instability phenomenon is the basis for further research, this invention establishes a partial differential model that is closer to the physical phenomenon based on the equations described by combining the one-dimensional gas dynamic equation of the Rijke tube and the linearized Heckl heat release model.

[0107] In a specific implementation, as a preferred embodiment of the present invention, the Galerkin method is used to simplify the partial differential equations into ordinary differential equations with respect to the modes, specifically:

[0108] S121. Using the Galerkin method, the above partial differential equation is transformed into an ordinary differential equation. The pressure p and velocity v are superimposed using polynomial functions that satisfy the boundary conditions. The product function is selected as:

[0109]

[0110]

[0111] in N represents the number of acoustic modes, j represents the j-th order acoustic mode in the system, and η j This represents the amplitude of the j-th acoustic mode as it varies with time. Indicates η j The derivative with respect to time.

[0112] S122. Substitute into the energy conservation equation:

[0113]

[0114] S123, Multiply both ends Then, integrating over the interval from 0 to L, we get:

[0115]

[0116] S124. Considering only the first mode, i.e., j=1, we can obtain:

[0117]

[0118]

[0119] S125, Order The ordinary differential equation with time delay is:

[0120]

[0121] In a preferred embodiment of the present invention, in step S13, the time delay term is processed using Taylor expansion, and through the relationship between mode and pressure, it is converted into a linear ordinary differential equation about sound pressure, and finally into the transfer function form of a proportional element and a second-order oscillatory element. The specific process includes:

[0122] S131. Consider a first-order Taylor expansion. Handling time delays:

[0123]

[0124] S132. The simplified ordinary differential equation with coefficients is as follows:

[0125]

[0126] in,

[0127] S133. Considering the mixed boundary conditions at the upper opening, the pressure fluctuation expression at the upper opening is... Substituting the coefficients into the simplified ordinary differential equation, we get:

[0128]

[0129] S134. Differentiate and rearrange the parameters:

[0130]

[0131] S135, Considering the fluctuations of the system controller. As input:

[0132]

[0133] S136. The transfer function describing the pressure fluctuation from the lower opening to the upper opening in a thermoacoustic unstable Rijke tube system is:

[0134]

[0135] The input to this system is the sound pressure oscillation at the lower opening of the Rijke tube, and the output is the sound pressure oscillation at the upper opening.

[0136] S137. Consider a single-input single-output system. The input is the voltage signal from the loudspeaker, and the output is the voltage signal generated by the sound pressure oscillation at the upper opening of the Rijke tube, measured by a microphone. The loudspeaker is located at the lower opening of the Rijke tube. The process of converting the voltage signal into sound pressure oscillation is considered as a proportional element and a hysteresis element, with a proportional gain of k. a The time delay is τ1; similarly, the sound pressure oscillation at the upper opening of the Rijke tube is converted into a microphone voltage signal, which is considered as a proportional element k. m And a hysteresis element with a time delay of τ2; the sound pressure oscillation from the lower opening of the Rijke tube to the upper opening is considered as a single element G; the system between the loudspeaker voltage signal and the microphone voltage signal is denoted as G1, as follows:

[0137]

[0138] S138. Simplify system parameters and set the total system gain to k. G =k4k a k m If the total time delay is τ = τ1 + τ2, then:

[0139]

[0140] In a preferred embodiment of the present invention, step S2 includes applying a sweep frequency identification signal to the speaker, acquiring the microphone signal after passing through the Rijke tube and the microphone signal without passing through the Rijke tube, performing a Fourier transform on the two signals to obtain the identification results of the system's amplitude-frequency characteristics and phase-frequency characteristics. The specific implementation process is as follows:

[0141] S21. Applying the product of a sweep frequency signal and an exponentially decaying signal to the loudspeaker yields the following signal:

[0142] U a (t)=sin(2π·f(t)·t)·Ampitude

[0143] Where, Ampitude = 5·e -0.3t f(t) = 100 + 60t, t ∈ [0, 15]; the signal is scanned from 100 Hz to 1000 Hz within 15 seconds, which is the frequency range of interest in the Rijke tube experiment; the sound pressure oscillation of the Rijke tube is measured through a microphone, and the voltage signal U output by the microphone is obtained. m .

[0144] S22. The voltage signal output from the microphone is passed through a speaker and applied directly to the microphone without passing through the Rijke tube, thus obtaining a signal:

[0145] U m ′=k a k m ·e -τs ·U a

[0146] Where, k a k represents the loudspeaker proportional gain. m U represents the microphone ratio gain. a This indicates the signal applied to the speaker.

[0147] S23. The response and amplitude-phase frequency characteristics of system G are obtained by comparing the output signal directly acting on the microphone with the output signal passing through the Rijke tube:

[0148]

[0149] Among them, G(e jω ) represents the frequency response of system G, k G =k4k a k m U m This represents the acquired signal U after passing through the Rijke tube. m Frequency response, U m ′(e jω ) represents the acquisition signal U without passing through a Rijke tube. m The frequency response of ′.

[0150] In a specific implementation, as a preferred embodiment of the present invention, the specific process of step S3 includes:

[0151] Based on the system's amplitude-frequency and phase-frequency characteristics, the specific parameters of the transfer function of the system, which is a proportional element and a second-order oscillating element, are identified as follows:

[0152]

[0153]

[0154] in, The identification results for k1′ and k2′ are given; the oscillation frequency is the peak value of the amplitude-frequency characteristic curve near the first mode. When Tω << 1, the amplitude-frequency response A(ω) ≈ K, and the logarithmic amplitude-frequency response L(ω) ≈ 20logK. The value of k can be calculated from these values. The system contains an unstable second-order oscillatory element, so the damping ratio ξ = 0 is taken. The system parameter identification results are as follows

[0155] Example

[0156] To verify the effectiveness of the method of this invention, a simulation program for the identified transfer function model was written using Python to obtain the system. Bode plot, such as Figure 11 , 12 As shown. Calculate the step response of the system, as... Figure 13 , 14 As shown, the signal is a periodic signal oscillating at a frequency of 412Hz.

[0157] The signal acquired by the actual experimental setup, i.e., its frequency domain analysis, is as follows: Figure 15 As shown, the amplitude-phase-frequency characteristics and response characteristics of the system model obtained by the identification method of the present invention are consistent with the actual system, and the results can be used as the basis for further analysis and controller design of thermoacoustic unstable systems.

[0158] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for identifying linear systems with thermoacoustic instability in Rijke tubes, characterized in that, include: S1. Based on physical mechanisms, establish a model structure in the form of a transfer function to describe thermoacoustic instability phenomena, including: S11. Based on the physical mechanism of mass, momentum and energy conservation of one-dimensional compressible gas, and considering the linearized Heckl heat release model, establish a set of partial differential equations describing the thermoacoustic instability of the Rijke tube. S12. Considering mixed boundary conditions, the partial differential equations are simplified to time-delay ordinary differential equations with respect to the modes using the Galerkin method. S13. Use Taylor expansion to deal with the time delay term, and through the relationship between mode and pressure, convert it into a linear ordinary differential equation about sound pressure, and finally convert it into the transfer function form of proportional element and second-order oscillatory element. S2. Input and detect the identification signal to the system to identify the frequency characteristics of the Rijke tube thermoacoustic unstable system; apply a frequency sweep identification signal to the loudspeaker, and collect the microphone signal after passing through the Rijke tube and the microphone signal without passing through the Rijke tube respectively. Perform Fourier transform on the two signals to obtain the identification results of the system's amplitude frequency characteristics and phase frequency characteristics. The specific implementation process is as follows: S21. Applying the product of a sweep frequency signal and an exponentially decaying signal to the loudspeaker yields the following signal: U a (t)=sin(2π·f(t)·t)·Ampitude Where, Ampitude = 5·e -0.3t f(t) = 100 + 60t, t ∈ [0, 15]; the signal is scanned from 100 Hz to 1000 Hz within 15 seconds, which is the frequency range of interest in the Rijke tube experiment; the sound pressure oscillation of the Rijke tube is measured through a microphone, and the voltage signal U output by the microphone is obtained. m ; S22. The voltage signal output from the microphone is passed through a speaker and applied directly to the microphone without passing through the Rijke tube, thus obtaining a signal: YOU m ′=k a k m ·e -τs ·YOU a Where, k a k represents the loudspeaker proportional gain. m U represents the microphone ratio gain. a This indicates the signal applied to the speaker; S23. The response and amplitude-phase frequency characteristics of system G are obtained by comparing the output signal directly acting on the microphone with the output signal passing through the Rijke tube: Among them, G(e jω ) represents the frequency response of system G, k G =k4k a k m U m This represents the acquired signal U after passing through the Rijke tube. m Frequency response, U m ′(e jω ) represents the acquisition signal U without passing through a Rijke tube. m Frequency response; S3. Based on the identified frequency characteristics, estimate the unknown parameters in the transfer function model, including: Based on the system's amplitude-frequency and phase-frequency characteristics, the specific parameters of the transfer function of the system, which is a proportional element and a second-order oscillating element, are identified as follows: in The identification results for k1′ and k2′ are given; the oscillation frequency is the peak value of the amplitude-frequency characteristic curve near the first mode. When Tω << 1, the amplitude-frequency characteristic A(ω) ≈ K, and the logarithmic amplitude-frequency characteristic L(ω) ≈ 20logK. The value of k can be obtained from this. The system has an unstable second-order oscillatory element, so the damping ratio ξ = 0 is taken.

2. The linear system identification method for the thermoacoustic instability of Rijke tubes according to claim 1, characterized in that, In step S11, the set of partial differential equations describing the thermoacoustic instability of the Rijke tube is established as follows: in, This indicates the amount of gas flow velocity fluctuation. This indicates the amount of gas pressure fluctuation. This represents a constant gas density. Let ξ represent the constant gas pressure, ξ represent the damping coefficient, c represent the speed of sound, and L represent the tube length. γ represents the adiabatic constant, and q(t,x) represents the heat release rate term. According to the linearized Heckl heat release model, it can be expanded as follows: Where A is the cross-sectional area of ​​the tube, δ(·) is the Dirac pulse, x0 is the position of the heat source in the tube, and L w T represents the length of the resistance wire. w The temperature of the heating wire is represented by T, the ambient air temperature is represented by k, and the thermal conductivity of air is represented by c. v For the specific heat capacity at constant pressure, d w The diameter of the resistance wire. This represents a constant gas flow rate, and τ is the time delay of heat release.

3. The linear system identification method for the thermoacoustic instability of Rijke tubes according to claim 1, characterized in that, In step S12, the partial differential equation is simplified into an ordinary differential equation with respect to the mode using the Galerkin method, specifically as follows: S121. Using the Galerkin method, the above partial differential equation is transformed into an ordinary differential equation. The pressure p and velocity v are superimposed using polynomial functions that satisfy the boundary conditions. The product function is selected as: in N represents the number of acoustic modes, j represents the j-th acoustic mode in the system, and η j This represents the amplitude of the j-th acoustic mode as it varies with time. Indicates η j The derivative with respect to time; S122. Substitute into the energy conservation equation: S123, Multiply both ends Then, integrating over the interval from 0 to L, we get: S124. Considering only the first mode, i.e., j=1, we can obtain: S125, Order The ordinary differential equation with time delay is:

4. The linear system identification method for the thermoacoustic instability of Rijke tubes according to claim 1, characterized in that, In step S13, the time delay term is processed using Taylor expansion, and through the relationship between mode and pressure, it is transformed into a linear ordinary differential equation about sound pressure, and finally into the transfer function form of a proportional element and a second-order oscillatory element. The specific process includes: S131. Consider a first-order Taylor expansion. Handling time delays: S132. The simplified ordinary differential equation with coefficients is as follows: in, S133. Considering the mixed boundary conditions at the upper opening, the pressure fluctuation expression at the upper opening is... Substituting the coefficients into the simplified ordinary differential equation, we get: S134. Differentiate and rearrange the parameters: S135. Consider the fluctuations of the system controller. As input: S136. The transfer function describing the pressure fluctuation from the lower opening to the upper opening in a thermoacoustic unstable Rijke tube system is: The input to this system is the sound pressure oscillation at the lower opening of the Rijke tube, and the output is the sound pressure oscillation at the upper opening. S137. Consider a single-input single-output system. The input is the voltage signal from the loudspeaker, and the output is the voltage signal generated by the sound pressure oscillation at the upper opening of the Rijke tube, measured by a microphone. The loudspeaker is located at the lower opening of the Rijke tube. The process of converting the voltage signal into sound pressure oscillation is considered as a proportional element and a hysteresis element, with a proportional gain of k. a The time delay is τ1; similarly, the sound pressure oscillation at the upper opening of the Rijke tube is converted into a microphone voltage signal, which is considered as a proportional element k. m And a hysteresis element with a time delay of τ2; the sound pressure oscillation from the lower opening of the Rijke tube to the upper opening is considered as a single element G; the system between the loudspeaker voltage signal and the microphone voltage signal is denoted as G1, as follows: S138. Simplify system parameters and set the total system gain to k. G =k4k a k m If the total time delay is τ = τ1 + τ2, then:

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