Adaptive control method for thermoacoustic oscillation system with unknown input time delay

Abstract

The present application belongs to the technical field of adaptive control of thermoacoustic oscillation system, and discloses an adaptive control method for a thermoacoustic oscillation system with unknown input time delay. Through an adaptive backstepping control method, the exponential stability of a Rijke tube system can be ensured even in the case that the actuator time delay is unknown and time-varying, and the system instability problem caused by the fact that the existing control method does not consider the time delay or the time delay is unknown is solved. Full-state feedback control can be realized by measuring only part of the boundary information such as the sound pressure at the upper end of the tube and the sound velocity at the lower end, the sensor configuration cost and the system complexity are reduced, and the method is more suitable for engineering practice. Through adaptive time delay estimation and projection operator mechanism, the controller has good robustness to time delay variation and model uncertainty, and can quickly converge to the true time delay under different initial estimation values without affecting the system stability. The present application provides a new way for the stable control of an aero-engine combustion system.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of adaptive control of thermoacoustic oscillation systems, and in particular to an adaptive control method for a thermoacoustic oscillation system with unknown input time delay. BACKGROUND

[0002] In 2018, De Andrade G. A., Vazquez R., Pagano D. J. Backstepping stabilization of a linearized ODE-PDE Rijke tube model [J]. Automatica, 2018, 96: 98-109. designed a boundary stabilization controller for thermoacoustic oscillation in a PDE-ODE cascade Rijke tube using backstepping method. Although the existence of input time delay was verified to make the system unstable in simulation, the problem of time delay was not considered in the controller design. In 2022, Tao Y. Modeling and control of thermoacoustic instability in a PDE type Rijke tube [D]. Northeastern University, 2022. designed a boundary controller for the thermoacoustic instability model based on the PDE type Rijke tube model, without considering the input time delay, disturbance and thermal inertia in the control design. In 2023, Xing Y Q, Tao Y, Ma D. Boundary proportional control of thermoacoustic instability in a Rijke tube [J]. Journal of the Franklin Institute, 2024, 361: 46-59. designed a boundary proportional feedback control for the PDE type Rijke tube thermoacoustic instability model with known time delay, without considering the design of the controller when the input time delay of the actuator is unknown. SUMMARY

[0003] The purpose of the present application is to solve the problem of pressure oscillation in the combustion chamber of an aero-engine. For a thermoacoustic oscillation system with unknown input time delay, based on the partial differential equation (PDE) model, an adaptive control method for a thermoacoustic oscillation system with unknown input time delay is designed to effectively suppress the thermoacoustic instability and achieve exponential stability control of the system. This method overcomes the problem of system performance degradation or even instability caused by not considering the actuator time delay or unknown time delay in existing control strategies. Through the combination of backstepping transformation and adaptive time delay estimation law, online compensation of unknown time delay is realized, and the robustness and control accuracy of the system in actual engineering are improved.

[0004] The technical scheme of the present application is as follows: a self-adaptive control method for a thermoacoustic oscillation system with unknown input time delay, comprising the following steps:

[0005] Step 1: establishing a function model describing the thermoacoustic oscillation phenomenon;

[0006] Step 2: performing characteristic coordinate transformation on the function model obtained in step 1 and adding a state equation;

[0007] Step 3: representing the unknown boundary input time delay by using a transport PDE equation to obtain a system with unknown input time delay;

[0008] Step 4: setting the unknown boundary input time delay It is known that a target system represented by state is designed; by using a backstepping transformation, the system with unknown input time delay is mapped to the target system represented by state .

[0009] Step 5: solving the kernel function equation in the target system represented by state in step 4; according to the solution of the kernel function, a control law is obtained.

[0010] Step 6: when it is judged that the unknown boundary input time delay is unknown, a time delay estimation value is used to replace the unknown boundary input time delay to obtain a new control law.

[0011] Step 7: constructing an adaptive backstepping transformation for the kernel function solution replaced in the new control law; according to the adaptive backstepping transformation with the time delay estimation value and the replaced kernel function, the system with unknown input time delay in step 3 is mapped to obtain a stable new target system.

[0012] Step 8: designing a time delay updating law to obtain the change rule of the time delay estimation value and substituting it into the control law.

[0013] The step 1 is specifically: the thermoacoustic oscillation phenomenon in a Rijke tube is described by a set of one-dimensional nonlinear gas dynamics equations, which are a set of partial differential equations describing mass conservation, momentum conservation and energy conservation:

[0014] Mass conservation:

[0015]

[0016] Momentum conservation:

[0017]

[0018] Energy conservation:

[0019]

[0020] where, denotes the wave time, denotes the location in the Rijke tube, denotes the Rijke tube length, denotes the gas density, denotes the gas pressure, denotes the gas flow rate, denotes the heat release from the heat source, , denotes the adiabatic constant;

[0021] The Heckl heat release law for the heat source in the Rijke tube to the fluid is as follows:

[0022]

[0023] where, is the heat source length, is the heat source diameter, is the heat source temperature, is the gas temperature, is the cross-sectional area of the Rijke tube, is the air thermal conductivity, is the constant-pressure specific heat capacity, denotes the steady-state density, denotes the steady-state pressure, denotes the steady-state velocity, denotes the linearization time, denotes the fluid velocity, denotes the thermal inertia between the heat release and the velocity;

[0024] The one-dimensional nonlinear gas dynamics equation is linearized by a small deviation linearization method, and the small deviation linearization formula is substituted into the one-dimensional nonlinear gas dynamics equation, and the second-order and higher-order terms in the wave quantity are discarded; a gas dynamics model after linearization and with an unknown input delay of the actuator is obtained:

[0025]

[0026]

[0027] denotes the Dirac impulse, denotes the heat release position of the heat source in the Rijke tube, denotes , denotes , denotes , represents , represents the heat release function;

[0028] Linearized boundary conditions:

[0029]

[0030]

[0031] represents the loudspeaker input with unknown time delay, represents the time delay error, represents the acoustic impedance constant;

[0032] A dimensionless system model is introduced;

[0033]

[0034]

[0035] wherein, is the Mach number; is the steady-state sound speed;

[0036] Let , the gas dynamics model after linearization and with the actuator unknown input time delay is converted into a dimensionless form system:

[0037]

[0038]

[0039] Boundary conditions:

[0040]

[0041]

[0042] represents the position of the heat source in the Rijke tube; represents , represents , represents , represents ;

[0043] is the control input with time delay.

[0044] Said step 2 is specifically: according to the dimensionless form system, and combining the backstepping transformation method of the infinite-dimensional system, a controller is designed;

[0045] The first step in controller design using the backstep transformation method is to use state variables. , Riemann coordinate transformation as represented:

[0046]

[0047]

[0048] Substituting the Riemann coordinate transformation into the dimensionless system yields a Rijke tube system with domain-coupled variables. Since the state of the dimensionless system varies along the characteristic lines of the transport equations, There are discontinuous equations at that point;

[0049] For in To remove point source terms caused by the Dirac function, new state variables and boundary conditions are added to the equations where there are discontinuities.

[0050] Define new state variables :

[0051]

[0052]

[0053]

[0054]

[0055] Rescaling space variables :

[0056] .

[0057] Step 3 specifically involves: delaying the input of unknown boundaries. Expressed in the form of cascaded first-order transport equations:

[0058]

[0059]

[0060] express , express ;

[0061] Modify the Rijke tube system with in-domain coupling variables into the following system with unknown input delay:

[0062]

[0063]

[0064]

[0065]

[0066]

[0067] wherein, , , denotes , denotes ; denotes , denotes ; denotes , denotes ; denotes , denotes ;

[0068] The boundary conditions for the system with unknown input time delay are as follows:

[0069]

[0070]

[0071]

[0072]

[0073]

[0074] wherein , , , .

[0075] The step 4 is specifically as follows:

[0076] To design the control law of the system with unknown input time delay in step 3, the unknown boundary input time delay is set as It is known that the target system represented by the state is obtained as follows:

[0077]

[0078]

[0079]

[0080]

[0081]

[0082] The boundary conditions are:

[0083]

[0084]

[0085]

[0086]

[0087]

[0088] where represents , represents ; represents , represents ; represents , represents ; represents , represents ; all represent different state variables; the domain-internal structure of the target system represented by the state is consistent with the system with unknown input time delay, but removes the boundary feedback coupling term between the states , , and ;

[0089] The following backstepping transformation is adopted:

[0090]

[0091] where , , , , is a pending kernel function; is a kernel function of Volterra type integral transformation defined on the triangular domain ; is a kernel function of Volterra type integral transformation defined on the rectangular domain The kernel function of the Fredholm-type integral transform on;

[0092] Backstep transformation with respect to time Differentiation and space After differentiation, substituting the boundary conditions, we obtain the equation concerning the kernel function:

[0093] .

[0094] Step 5 specifically involves: for step 4, the state... The kernel function equation in the target system is represented, and the solution of the corresponding kernel function is found using the method of characteristics:

[0095]

[0096] The control law is obtained by solving the kernel function. :

[0097] .

[0098] Step 6 specifically involves: based on the control law and the deterministic equivalence principle in step 5, defining an adaptive controller that stabilizes the system with unknown input delay, resulting in the following new control law:

[0099]

[0100] Using time delay estimates Replace unknown boundary input delay The solution for the replaced kernel function is as follows:

[0101] .

[0102] Step 7 specifically involves designing an adaptive backstepping transform based on the kernel function solution obtained in step 6, as follows:

[0103]

[0104] The adaptive backstep transformation is applied to time. Differentiation, with respect to space Differentiate and simplify to obtain and The implicit form of the function is as follows, which yields:

[0105]

[0106] Based on the time delay estimate The adaptive backstepping transform and the replaced kernel function map the system with unknown input delay in step 3 to the following new target system:

[0107]

[0108]

[0109]

[0110]

[0111]

[0112] The boundary conditions are:

[0113]

[0114]

[0115]

[0116]

[0117]

[0118] denotes the derivative of the delay estimate .

[0119] The step 8 is specifically:

[0120] In the new target system and The specific expression of the function, the design delay estimate The update law is:

[0121]

[0122]

[0123] denotes the lower bound of the unknown boundary input delay, denotes the upper bound of the unknown boundary input delay; Given by the following formula:

[0124]

[0125] Where the standard projection operator Given by the following formula:

[0126]

[0127] The delay estimate The update law is brought into the control law The final adaptive control law is obtained.

[0128] Compared with the prior art, the present application has the beneficial effects that:

[0129] 1. The thermoacoustic unstable system with unknown boundary input delay is effectively stabilized: through the adaptive backstepping control method, the exponential stability of the Rijke tube system can be ensured even if the actuator delay is unknown and time-varying, and the system instability problem caused by the fact that the existing control method does not consider the delay or the delay is unknown is solved;

[0130] 2. The control precision and response speed are significantly improved: simulation results show that under the condition of unknown delay (such as D=1.2s), the system pressure and speed fluctuations can quickly converge to zero in a short time, the control precision is high, and the dynamic response is fast, which is superior to the traditional proportional feedback or fixed delay compensation method;

[0131] 3. The dependence of the system on full state measurement is reduced: only partial boundary information such as the sound pressure at the upper end of the tube and the sound velocity at the lower end needs to be measured to realize full state feedback control, which reduces the sensor configuration cost and system complexity, and is more suitable for engineering practice;

[0132] 4. Strong robustness and adaptability are possessed: through adaptive delay estimation and projection operator mechanism, the controller has good robustness to delay variation and model uncertainty, and can quickly converge to the true delay under different initial estimation values, and the system stability is not affected;

[0133] 5. A new way is provided for the stable control of the combustion system of an aero-engine: the control framework proposed in the present application can be extended to combustion systems such as aero-engines and gas turbines which have similar thermoacoustic instability characteristics, and provides a theoretical basis and technical support for improving combustion efficiency, prolonging equipment life, and reducing noise and vibration. BRIEF DESCRIPTION OF DRAWINGS

[0134] Figure 1 is a control structure diagram of the Rijke tube with boundary input delay of the present application;

[0135] Figure 2 is a flow chart of the adaptive control method of the thermoacoustic oscillation system with unknown input delay of the present application;

[0136] Figure 3 is the pressure fluctuation of the Rijke tube under the initial state and without applying control;

[0137] Figure 4 is the speed fluctuation of the Rijke tube under the initial state and without applying control;

[0138] Figure 5 is the pressure fluctuation of the Rijke tube under the initial state and without applying control;

[0139] Figure 6 is the velocity fluctuation under the control of the Rijke tube. DETAILED DESCRIPTION

[0140] The focus of the method is as follows: a partial differential equation model of the Rijke tube thermoacoustic instability system based on the Heckl heat release model is established, the nonlinear heat transfer mechanism between the heat source and the fluid is accurately described, and a transport equation representation of the boundary input time delay is introduced;

[0141] Through characteristic coordinate transformation and state reconstruction, the original thermoacoustic instability system is converted into a 5*5 first-order hyperbolic function system, the boundary coupling term is retained, and a basis is provided for backstepping control design;

[0142] A backstepping transformation containing Volterra type and Fredholm type integral transformation is designed to map the original system with thermoacoustic oscillation to a target system with ideal stability, and an explicit kernel function is solved by the method of characteristics;

[0143] An adaptive time delay estimation mechanism is used to estimate the real time delay using the time delay estimation value, and a time delay update law is designed to ensure the boundedness of the estimation value combined with the standard projection operator;

[0144] The stability control of the system state is realized by integral weighted form, and the effectiveness and robustness under unknown time delay are verified by numerical simulation.

[0145] The adaptive control method for a thermoacoustic oscillation system with unknown input time delay proposed by the application comprises the following steps:

[0146] Step 1, a function model describing the thermoacoustic oscillation phenomenon is established;

[0147] Step 2, the function model obtained in step 1 is subjected to characteristic coordinate transformation and state equation is added;

[0148] Step 3, the unknown boundary input time delay is represented by a transport PDE equation, and a system with unknown input time delay is obtained;

[0149] Step 4, the unknown boundary input time delay is set is known, a target system represented by state is designed; a backstepping transformation is used to map the system with unknown input time delay to a target system represented by state ;

[0150] Step 5, the kernel function equation in the target system represented by state in step 4 is solved; and a control law is obtained according to the solution of the kernel function;

[0151] Step 6: When the input delay is unknown due to the unknown boundary condition, use the estimated delay value. Replacement of unknown boundary input delay A new control law is obtained;

[0152] Step 7: Construct an adaptive backstepping transform for the replaced kernel function in the new control law; based on the time delay estimate... The adaptive backstepping transform and the replaced kernel function map the system with unknown input delay in step 3 to obtain a stable new target system;

[0153] Step 8: Design the delay update law to obtain the delay estimate. The changing patterns are then substituted into the control law.

[0154] The following is a specific embodiment that uses a resistance wire as a heat source to demonstrate the numerical simulation of a system with unknown input delay and a new target system;

[0155] The system parameters are given in Table 1:

[0156] Table 1 System Parameters

[0157]

[0158] Given a system with unknown input delay and the initial state of a new target system:

[0159]

[0160] Unknown boundary input delay setting Adaptive gain is set to The initial value of the time delay estimate is... .Depend on Figure 3 and Figure 4 Before a new control law is applied, the pressure and velocity at 0 to 1 meter in the Rijke tube, which has an unknown input delay, exhibit large oscillations, and the system is unstable at this time. Figure 5 and Figure 6 The simulation results show that the system's pressure and velocity converge rapidly to zero after the new control law is added, proving that the designed control law can achieve stable control of a system with unknown input time delay. The known upper and lower limits are defined as follows: and .

Claims

1. An adaptive control method for a thermoacoustic oscillation system with unknown input time delay, characterized in that, The steps include the following: Step 1: Establish a functional model describing the thermoacoustic oscillation phenomenon; Step 2: Perform characteristic coordinate transformation on the function model obtained in Step 1 and add state equations; Step 3: Express the unknown boundary input delay using the transport PDE equation to obtain the system with unknown input delay; Step 4: Set the input delay for unknown boundaries Given that the design is based on the state The target system represented; By employing a backstepping transformation, a system with unknown input time delay is mapped to a system defined by states. The target system represented; Step 5: Solve the problem from the state in step 4. The kernel function equation in the target system is represented; the control law is obtained from the solution of the kernel function. : Step 6: When the input delay is unknown due to the unknown boundary condition, use the estimated delay value. Replacement of unknown boundary input delay A new control law is obtained; Based on the control law in step 5 and the deterministic equivalence principle, an adaptive controller is defined to stabilize a system with unknown input delay, resulting in the following new control law: ； Using time delay estimates Replace unknown boundary input delay The solution for the replaced kernel function is as follows: ； Step 7: Construct an adaptive backstepping transform for the replaced kernel function in the new control law; based on the time delay estimate... The adaptive backstepping transform and the replaced kernel function map the system with unknown input delay in step 3 to obtain a stable new target system; Based on the kernel function solution obtained from the replacement in step 6, the adaptive backstepping transform is designed as follows: ； The adaptive backstep transformation is applied to time. Differentiate, differentiate with respect to space and simplify, to obtain and The implicit form of the function is as follows, which yields: ； Based on the time delay estimate The adaptive backstepping transform and the replaced kernel function map the system with unknown input delay in step 3 to the following new target system: ； The boundary conditions are: ； Indicates the time delay estimate The derivative; Step 8: Design the delay update law to obtain the delay estimate. The changing patterns are then substituted into the control law.

2. The adaptive control method for a thermoacoustic oscillation system with unknown input time delay according to claim 1, characterized in that, Step 1 specifically involves describing the thermoacoustic oscillation phenomenon in the Rijke tube using a set of one-dimensional nonlinear gas dynamics equations, which are a set of partial differential equations describing the conservation of mass, momentum, and energy: Conservation of mass: ； Conservation of momentum: ； Energy conservation: ； in, Indicates the time of fluctuation. This represents the position within the Rijke tube. This represents the length of the Rijke tube. Represents gas density, Represents gas pressure. Represents gas flow rate, Represents the heat release generated by the heat source. , Represents the adiabatic constant; Heckl's law of heat release from a heat source to a fluid in a Rijke tube is as follows: ； in, For the length of the heat source, The diameter of the heat source, The temperature of the heat source. For gas temperature, Let be the cross-sectional area of ​​the Rijke tube. For the thermal conductivity of air, For isobaric specific heat capacity, Represents steady-state density, Represents steady-state pressure, Represents steady-state velocity, Represents linearization time, Represents fluid velocity, Thermal inertia represents the relationship between heat release and velocity; The one-dimensional nonlinear gas dynamics equations are linearized using the small-deviation linearization method, and the small-deviation linearization formula is substituted into the one-dimensional nonlinear gas dynamics equations, discarding second-order and higher-order terms in the fluctuation quantities; thus, a linearized gas dynamics model with unknown actuator input delay is obtained. ； ； Represents the Dirac pulse. This indicates the location of the heat source dissipating heat within the Rijke tube. express , express , express , express , Represents the heat release function; Linearization boundary conditions: ； ； Indicates speaker input with unknown time delay, Indicates time delay error, , representing the acoustic impedance constant; Introduce a dimensionless system model; ； ； in, It is the Mach number; The steady-state speed of sound; make The linearized gas dynamics model with unknown actuator input delay is transformed into a dimensionless system: ； ； Boundary conditions: ； ； This indicates the location of the heat source within the Rijke tube; express , express , express , express ; It is a control input with a time delay.

3. The adaptive control method for a thermoacoustic oscillation system with unknown input time delay according to claim 2, characterized in that, Step 2 specifically involves designing a controller based on the dimensionless form system and the backstep transformation method for infinite-dimensional systems. The first step in controller design using the backstep transformation method is to use state variables. , Riemann coordinate transformation as represented: ； ； Substituting the Riemann coordinate transformation into the dimensionless system yields a Rijke tube system with domain-coupled variables. Since the state of the dimensionless system varies along the characteristic lines of the transport equations, There are discontinuous equations at that point; For in To remove point source terms caused by the Dirac function, new state variables and boundary conditions are added to the equations where there are discontinuities. Define new state variables : ； ； ； ； Rescaling space variables : 。 4. The adaptive control method for a thermoacoustic oscillation system with unknown input time delay according to claim 3, characterized in that, Step 3 specifically involves: delaying the input of unknown boundaries. Expressed in the form of cascaded first-order transport equations: ； ； express , express ; Modify the Rijke tube system with in-domain coupling variables into the following system with unknown input delay: ； ； ； ； ； in, , , express , express ; express , express ; express , express ; express , express ; The system boundary conditions with unknown input delay are as follows: ； ； ； ； ； in , , , .

5. The adaptive control method for a thermoacoustic oscillation system with unknown input time delay according to claim 4, characterized in that, Step 4 specifically involves: To design the control law for the system with unknown input delay in step 3, define the unknown boundary input delay. Given that the following is obtained from the state The target system represented: ； ； ； ； ； The boundary conditions are: ； ； ； ； ； in express , express ; express , express ; express , express ; express , express ; Each represents a different state variable; the state is described as... The domain structure of the target system is consistent with that of a system with unknown input delay, but the states in the boundary conditions are removed. , , and Boundary feedback coupling terms between; The following backstep transformation is adopted: ； in, , , , , The kernel function is yet to be determined. It is defined in the triangular domain The kernel function of the Volterra-type integral transform on; It is defined in a rectangular field The kernel function of the Fredholm-type integral transform on; Backstep transformation with respect to time Differentiation and space After differentiation, substituting the boundary conditions, we obtain the equation concerning the kernel function: 。 6. The adaptive control method for a thermoacoustic oscillation system with unknown input time delay according to claim 5, characterized in that, Step 5 specifically involves: for step 4, the state... The kernel function equation in the target system is represented, and the solution of the corresponding kernel function is found using the method of characteristics: ； The control law is obtained by solving the kernel function. : 。 7. The adaptive control method for a thermoacoustic oscillation system with unknown input time delay according to claim 6, characterized in that, Step 8 specifically involves: New target system and The specific expression of the function, and the estimated design delay. The update law is: ； ； This represents the lower bound of the input delay at unknown boundaries. This represents the upper bound of the input delay at unknown boundaries; It is given by the following formula: ； The standard projection operator It is given by the following formula: ； Delay estimate The update law is applied to the control law. The final adaptive control law is obtained from this.

Images