CN115114731A - A Dynamic Modeling Method of Aero-engine Based on Pseudo-Jacobian Matrix - Google Patents

Abstract

The invention discloses a pseudo Jacobian matrix-based aeroengine dynamic modeling method. Aiming at the problem of poor real-time performance of a component-level model of a bidirectional differential calculation Jacobian matrix, the method linearizes a common equation of a turboshaft engine based on a nonlinear model dynamic linearization strategy, constructs a pseudo Jacobian matrix evaluation function, calculates the Jacobian matrix approximately through optimal estimation, obtains a recursive calculation method of the pseudo Jacobian matrix, avoids repeated calling of the component model, and greatly improves the real-time performance of the dynamic model.

Description

A Dynamic Modeling Method of Aero-engine Based on Pseudo-Jacobian Matrix

technical field

The invention belongs to the field of system control and simulation in aerospace propulsion theory and engineering, and particularly relates to a dynamic real-time model modeling method for aero-engine.

Background technique

Aero-engine numerical simulation technology plays an extremely important role in aero-engine development and control system design and verification. Numerical simulation or semi-physical simulation can be carried out by replacing the real engine with a mathematical model, which can conduct preliminary verification of control algorithms, fault diagnosis technology, health management technology, etc., saving experimental costs and reducing experimental risks. Since the late 1980s, western countries have successively carried out research on aero-engine simulation technology and developed a variety of aero-engine numerical simulation systems. After years of research, development and application, the relevant theoretical technologies have matured, and the simulation confidence and accuracy have reached a very high level. With the development of model-based control technology, the requirements for model accuracy and real-time performance are further improved, and a lot of research work has also been carried out.

Based on the introduction of the basic parameters of the real engine components, the aero-engine mathematical model establishes the aerodynamic thermodynamic model of the engine components according to the characteristics of the components along the gas path of the engine. By solving the common working equation between the components, the components matching the current working state are obtained. parameters, and the convergence speed and accuracy of the common working equation will be directly related to the accuracy and real-time performance of the engine mathematical model.

There are usually Newton-Raphson (N-R) method and Broyden quasi-Newton method to solve the common working equation of aero-engine. Among them, the N-R method needs to repeatedly call the engine component model to calculate the Jacobian matrix, which seriously affects the real-time performance of the model; while the quasi-Newton method approximates the Jacobian matrix and obtains the approximate Jacobian matrix through recursion, which reduces the computational cost. However, the iterative convergence of the algorithm requires that the deviation of the initial solution is smaller, and is affected by the calculation step size, and the convergence ability is weaker than that of the N-R method.

Based on the above two methods, Chen Yuchun of Northwestern Polytechnical University et al. ^[1] improved the convergence problem of the solution by changing the calculation step size of the NR method and limiting the initial value, but still need to calculate the Jacobian matrix differentially. Liao Guanghuang et al. ^[2] of Nanjing University of Aeronautics and Astronautics proposed a neural network-based method for solving the engine co-working equation, collecting offline training data calculated based on the NR method, taking the residual as the input, and the co-working equation guessing correction as the output. The neural network trains it, avoiding the Jacobian matrix calculation and the iterative equation solving process, and effectively improving the real-time performance of the model. However, the neural network adopts an offline training method, and the model accuracy is affected by the generalization ability of the network. weak adaptability. Wang Yuan, Wu Qian and others of Nanjing University of Aeronautics and Astronautics ^[3-4] combined the square convergence of the NR method with the super-linear convergence characteristics of the quasi-Newton method, and proposed a self-correcting Broyden quasi-Newton method. The accuracy and speed have been improved, but at the working point with poor convergence trend, the initialization of the correction matrix still needs to be performed by difference, resulting in low real-time performance of individual working points. Pang Shuwei of Nanjing University of Aeronautics and Astronautics ^[5] proposed an engine component-level model building method based on accurate partial derivatives, which replaces differential calculation with accurate partial derivatives and constructs a Jacobian matrix, which improves the real-time performance of the model, but the algorithm calculation process is complicated and the realization is difficult. larger.

SUMMARY OF THE INVENTION

The technical problem to be solved by the present invention is to overcome the deficiencies of the prior art, and to provide an aero-engine dynamic modeling method based on a pseudo-Jacobian matrix, which can avoid repeated calls of the model by the differential algorithm and effectively improve the real-time performance of the model.

The present invention specifically adopts the following technical solutions to solve the above-mentioned technical problems:

A dynamic modeling method of aero-engine based on pseudo-Jacobian matrix, by constructing and solving the dynamic co-working balance equation of the aero-engine during the working process of the aero-engine, the parameters of each component of the aero-engine matching the current working state are obtained; When the dynamic co-working balance equation of , the guess value of the dynamic co-working balance equation is iteratively revised by the following method, so that the current dynamic co-working balance equation satisfies the convergence condition:

Step 1. Linearize the dynamic co-working balance equation at time k;

(伪雅可比矩阵)：Step 2. Based on the following optimization objective function, calculate the optimal estimate of the Jacobian matrix J(k) of the linearized dynamic co-working equilibrium equation

(pseudo-Jacobian matrix):

表示k-1时刻的线性化动态共同工作平衡方程的雅可比矩阵的最优估计；Among them, ε(k) and ε(k-1) represent the residuals of the dynamic co-working balance equation at time k and time k-1, respectively, and Δx(k)=x(k)-x(k-1) represents time k The difference between the guessed values x(k) and x(k-1) of the dynamic co-working balance equation at time k-1, μ is the penalty factor,

the optimal estimate of the Jacobian matrix representing the linearized dynamic co-working equilibrium equation at time k-1;

Step 3. Update the guess value x(k+1) of the dynamic co-working balance equation at time k+1 by iterative calculation:

where λ∈(0,1] is the iteration step size.

通过下式计算得到：Preferably, the best estimate

It is calculated by the following formula:

Among them, η∈(0,1] is the step size factor, Δε(k)=ε(k)-ε(k-1) represents the residual ε(k) of the dynamic co-working balance equation at time k and time k-1 , ε(k-1) difference, the superscript "T" represents the matrix transpose, and "|| || ² " represents the square of the 2-norm of the vector.

Compared with the prior art, the technical solution of the present invention has the following beneficial effects:

(1) High real-time performance: The present invention replaces the Jacobian matrix calculation in the differential form with the recursive pseudo-Jacobian matrix calculation, which greatly reduces the number of component-level model calls and greatly improves the real-time performance of the dynamic model calculation.

(2) Strong versatility: The calculation of the pseudo Jacobian matrix is based on the dynamic linearization method, which is a data-driven method and has strong adaptability to changes in engine characteristics.

Description of drawings

FIG. 1 is a schematic structural diagram of a twin-rotor turboshaft engine;

Fig. 2 is a flow chart of dynamic modeling of turboshaft engine based on pseudo-Jacobian matrix;

Fig. 3 is the key engine output parameter comparison (1km height) of the two-way differential algorithm and the pseudo-Jacobian matrix algorithm of the present invention;

FIG. 4 is a comparison of key engine output parameters between the two-way differential algorithm and the pseudo-Jacobian matrix algorithm of the present invention (3km height).

Detailed ways

Aiming at the problem of poor real-time performance of Jacobian matrix calculation based on the existing difference method, the present invention combines the nonlinear model dynamic linearization method, and uses optimal estimation to calculate the pseudo-Jacobian matrix, thereby improving the real-time performance of the dynamic model.

The present invention specifically adopts the following technical solutions to solve the above-mentioned technical problems:

A dynamic modeling method of aero-engine based on pseudo-Jacobian matrix, by constructing and solving the dynamic co-working balance equation of the aero-engine during the working process of the aero-engine, the parameters of each component of the aero-engine matching the current working state are obtained; When the dynamic co-working balance equation of , the guess value of the dynamic co-working balance equation is iteratively revised by the following method, so that the current dynamic co-working balance equation satisfies the convergence condition:

Step 1. Linearize the dynamic co-working balance equation at time k;

(伪雅可比矩阵)：Step 2. Based on the following optimization objective function, calculate the optimal estimate of the Jacobian matrix J(k) of the linearized dynamic co-working equilibrium equation

(pseudo-Jacobian matrix):

表示k-1时刻的线性化动态共同工作平衡方程的雅可比矩阵的最优估计；Among them, ε(k) and ε(k-1) represent the residuals of the dynamic co-working balance equation at time k and time k-1, respectively, and Δx(k)=x(k)-x(k-1) represents time k The difference between the guessed values x(k) and x(k-1) of the dynamic co-working balance equation at time k-1, μ is the penalty factor,

the optimal estimate of the Jacobian matrix representing the linearized dynamic co-working equilibrium equation at time k-1;

Step 3. Update the guess value x(k+1) of the dynamic co-working balance equation at time k+1 by iterative calculation:

where λ∈(0,1] is the iteration step size.

The applicable objects of the dynamic modeling method of the present invention include, but are not limited to, turboshaft engines, turboprop engines, turbofan engines, variable cycle engines, combined engines, and the like.

In order to facilitate the public's understanding, the technical solutions of the present invention will be described in detail below through a specific embodiment and in conjunction with the accompanying drawings:

This embodiment takes the twin-rotor turboshaft engine shown in FIG. 1 as the research object. The gas turbine and the compressor of the twin-rotor turboshaft engine are coaxial, and the power turbine and the main rotor are coaxial. During the working process, the air enters from the intake port, flows through the compressor to the combustion chamber, mixes with the fuel oil, and generates high temperature and high pressure gas. When the gas flows through the gas turbine, a part of the energy is extracted to drive the compressor, and the gas leaving the gas turbine flows through Power turbine, the power turbine extracts the remaining energy to drive the main rotor to meet the working needs of the helicopter.

For the dual-rotor turboshaft engine shown in Figure 1, the dynamic modeling method proposed by the present invention is shown in Figure 2, where t is the simulation time of the dynamic model, k is the number of iterations for solving the balance equation, and T is the simulation step size. The method specifically includes the following steps:

Step A, obtain the flight conditions (helicopter flight height H, forward flight speed V _x ) and fuel flow W _f at time t;

Step B, calculating the thermal parameters of each component and the rotor load power requirement in turn from the air inlet to the tail nozzle along the airflow process;

Step C, constructing the dynamic co-working balance equation of the turboshaft engine;

Under the conditions of known flight conditions and fuel flow, the analysis is carried out according to the continuous flow and pressure balance conditions that need to be satisfied in the dynamic working process of the twin-rotor turboshaft engine. Usually, three common equations are selected, denoted ε _i , i=1,2 , 3 is the residual of the dynamic co-working equation, including

(1) The gas turbine inlet flow continuous equation φ ₁ (x):

φ ₁ (x)=(W _41xs -Q _41xs )/Q _41xs =ε ₁ (1)

Among them, x is the guess value in the dynamic joint working equation, W _41xs is the similar flow rate of the gas turbine inlet under the current working conditions calculated from the gas turbine component characteristic curve, Q _41xs is calculated according to the gas path flow from the gas turbine guide Similar flow into the gas turbine, ε ₁ is the residual of equation φ ₁ (x);

(2) Continuity equation of power turbine inlet flow φ ₂ (x):

φ ₂ (x)=(W _44xs -Q _44xs )/Q _44xs =ε ₂ (2)

Among them, W _44xs is the similar flow at the inlet of the power turbine under the current working conditions calculated from the characteristic curve of the power turbine components, Q _44xs is the similar flow from the power turbine guide to the power turbine calculated according to the gas flow process, and ε ₂ is the equation φ ₂ (x) residual;

(3) The pressure balance equation φ ₃ (x) at the outlet of the tail nozzle:

φ ₃ (x)=(p _c7 -p ₇ )/p ₇ =ε ₃ (3)

Among them, p _c7 is the total pressure of the airflow entering the exit of the tail nozzle, p ₇ is the back pressure of the nozzle, and ε ₃ is the residual of the equation φ ₃ (x);

In the joint work balance equation of the component-level model, select the guess value x=[Z _C Z _G Z _P ] ^T ; wherein, Z _C is the compressor pressure ratio coefficient, Z _G is the gas turbine pressure ratio coefficient, and Z _P is the power Turbine pressure ratio coefficient; taking Z _c as an example, the pressure ratio coefficient is defined as:

Among them, π _c is the compressor pressure ratio at the current operating point, and the subscripts min and max represent the minimum and maximum pressure ratios on the current rotational speed line, respectively.

Step D, judge whether the dynamic joint work balance equation satisfies the convergence condition, if it is flag=1, then jump to step F, otherwise flag=0 and execute step E;

where k _max is the maximum allowed number of iterations.

The convergence conditions in this embodiment are as follows:

ε _i <10 ⁻⁵ , i=1, 2, 3, or k>km _max .

Step E, calculate the pseudo Jacobian matrix, correct the guess value, and return to step B; specifically, the following sub-steps are included:

Step E1, linearize the dynamic co-working balance equation at time k:

ε(k)=φ(x(k)) (6)

Assuming that the partial derivatives of φ(x(k)) with respect to x are continuous and satisfy ||ε(k)-ε(k-1)||≤b||x(k)-x(k-1)||, That is, if the change of the guess value is limited, the change of the residual of the equation is also limited, and this condition is satisfied when the model converges;

Linearize equation (6) to get the linearized equation of the following form:

为伪雅可比矩阵；式(7)为线性化的平衡方程；Among them, Δε(k)=ε(k)-ε(k-1), Δx(k)=x(k)-x(k-1),

is the pseudo-Jacobi matrix; formula (7) is the linearized balance equation;

Step E2, using the optimal estimation method to calculate the pseudo Jacobian matrix

Specifically, the following evaluation indicators are used:

Among them, μ is the penalty factor, and J(k) is the Jacobian matrix;

即线性化方程(6)成立，且对雅可比矩阵的变化进行约束；Equation (8) shows that when L(J(k))=0, ε(k)-ε(k-1)=J(k)Δx(k),

That is, the linearization equation (6) is established, and the change of the Jacobian matrix is constrained;

The Jacobian matrix is shown in formula (9):

Since the working process of aero-engine is a complex aero-thermodynamic process with strong nonlinearity, its mathematical modeling relies on the interpolation and iterative operation of the component characteristic map and aero-thermodynamic parameters, so φ _i (x), i=1, 2, 3 is a The implicit equations of the calculation process of each component of the engine cannot be directly calculated for partial derivatives, but can only be obtained by numerical difference method. In order to improve the accuracy of the model, the intermediate difference method is usually used to calculate the partial derivatives in the Jacobian matrix, namely:

In the calculation process of the partial derivative of formula (10), the guess value x _i needs to be perturbed upward and downward respectively, and the component-level mathematical model is called for differential calculation. There are three guess values, so a total of 6 calculations are required. , which seriously affects the real-time performance of the dynamic model. Therefore, the present invention adopts a recursive way to calculate the pseudo Jacobian matrix.

有：Find the minimum value of J(k) for equation (8) to obtain the optimal estimate of J(k)

Have:

Move items have

Subtract both sides of the equation

F

where η∈(0,1] is the step size factor;

通过差分进行初始化，否则，

此处下标t代表动态模型仿真时刻，即当前仿真时刻的伪雅可比矩阵初始值为上一仿真时刻迭代结束时候的伪雅可比矩阵值；When t=0,

Initialize by differencing, otherwise,

Here, the subscript t represents the simulation time of the dynamic model, that is, the initial value of the pseudo Jacobian matrix at the current simulation time is the pseudo Jacobian matrix value at the end of the iteration at the previous simulation time;

Equation (14) calculates the pseudo-Jacobian matrix in a recursive way. In the calculation process, only the increment of the residual of the current equation and the increment of the guess value are needed, and there is no need to additionally call the component model for differential calculation, and the calculation process is greatly simplified ;

Step E3, update the guess value by iterative calculation:

where λ∈(0,1] is the iteration step size.

Step F, calculate rotor acceleration, update rotational speed;

where n _G , n _P are the rotational speeds of the gas turbine and power turbine, J _G , J _P are the moments of inertia of the gas turbine rotor and the power turbine rotor, η _G , η _P are the mechanical efficiencies of the gas turbine rotor and the power turbine rotor, W _GT , W _PT are gas turbine and power turbine power, _WC , _WR are compressor and helicopter rotor power.

In order to verify the advantages of the pseudo-Jacobian matrix algorithm adopted in the present invention in the process of solving the dynamic co-working equation of the turboshaft engine, the simulation was carried out on the component-level model of the T700 turboshaft engine, and the initial flight height of the turboshaft engine was set as H=1km, and the initial The forward flying speed is V _x =20m/s. During the simulation, the dynamic working process of the engine is simulated, and the models of the two algorithms are called 10,000 times each. The model is developed based on VC++6.0, the computer system is Windows 10 Home Edition, CPU i5-8250U1.60GHz, 8GB RAM. The average computation time of 10,000 calls of the pseudo-Jacobi matrix algorithm model is 3.19s, while the average computation time of the two-way difference calculation Jacobian matrix model is 6.97s. It can be seen that the time used by the pseudo-Jacobi matrix algorithm is about 3/7 of the two-way difference algorithm, which verifies the superiority of the algorithm in real-time performance.

The accuracy of the model is further verified within the envelope range. Two sets of simulation results are given here, as shown in Figure 3 and Figure 4. The simulation results in other flight areas and engine working conditions are similar. In the figure, PJ (Pseudo-Jacobian) represents the simulation effect of the component model using the pseudo-Jacobian matrix algorithm; BDD (Bi-directional differential) represents the model simulation effect of the bidirectional differential calculation Jacobian matrix algorithm. The figure shows the change curve of the key parameters of the turboshaft engine, where sfc is the fuel consumption rate, T ₄₁ is the total inlet temperature of the gas turbine, e represents the output deviation between the PPD algorithm and the differential algorithm model, and the deviation calculation formula is

(1) Simulation test at a flight altitude of 1km

Set the helicopter flight height H = 1km, the forward flight speed V _x is increased from 20m/s to 50m/s, and the output parameter change curve is shown in Figure 3.

As can be seen in Figure 3, compared with the bidirectional differential algorithm, the output speed deviation of the component-level model based on the pseudo-Jacobian matrix is less than 0.15%, the temperature deviation is less than 0.25%, and the maximum deviation in the closed-loop control is less than 3 due to the small order of magnitude of the fuel flow itself. %, the calculation deviation of fuel consumption rate caused by the deviation of fuel flow rate is less than 1.2%. The detailed error information is shown in Table 1. It can be seen from Table 1 and Figure 3 that the mathematical model of the turboshaft engine based on the pseudo-Jacobian matrix has achieved high accuracy. The average error is less than 0.14%, which verifies the effectiveness of the proposed algorithm in terms of dynamic model accuracy.

Table 1 Errors of the pseudo-Jacobian matrix model compared to the two-way difference model

(2) Simulation test at a flight altitude of 3km

Set the helicopter flight height H = 3km, the forward flight speed V _x is increased from 20m/s to 50m/s, and the output parameter change curve is shown in Figure 4.

It can be seen from Figure 4 that, compared with the two-way differential algorithm, the component-level model based on the pseudo-Jacobian matrix has an output speed deviation of less than 0.3% and a temperature deviation of less than 0.4% at an altitude of 3km. The maximum deviation is less than 1.1%, the calculation deviation of fuel consumption rate due to fuel flow deviation is also less than 1.1%, and the maximum modeling error is better than the case of height 1km, which verifies its applicability within the envelope.

references:

[1] Chen Yuchun, Xu Siyuan, Yang Yunkai, et al. Methods to improve the convergence of aero-engine characteristics calculation [J]. Journal of Aerodynamics, 2008, 23(12): 2242-2248.

[2] Liao Guanghuang, Jiao Yang, Li Qiuhong, et al. Research on high-precision real-time component-level model of turboshaft engine [J]. Propulsion Technology, 2016, 37(01): 25-33.

[3] Wang Yuan, Li Qiuhong, Huang Xianghua. Numerical calculation of aero-engine model based on self-correcting Broyden quasi-Newton method [J]. Journal of Aerodynamics, 2016,31(1):249-256.

[4] Wu Qian, Li Qiuhong, Shan Ruibin. Research on the solution method of aero-engine aero-thermal system model [J]. Computer Simulation, 2019, 36(01):76-81.

[5]Pang,S.;Li,Q.;Ni,B.Improved nonlinear MPC for aircraft gas turbineengine based on semi-alternative optimization strategy[J].Aerosp.Sci.Technol.2021,118,106983.

Claims (2)

1. A dynamic modeling method of an aero-engine based on a pseudo Jacobian matrix is characterized in that parameters of all parts of the aero-engine matched with the current working state are obtained by constructing and solving a dynamic common working balance equation of the aero-engine in the working process of the aero-engine; the method is characterized in that when the dynamic common working balance equation of the aero-engine is solved, the guess value of the dynamic common working balance equation is iteratively corrected by the following method, so that the current dynamic common working balance equation meets the convergence condition:

step 1, linearizing a dynamic common working balance equation at the moment k;

step 2, calculating the optimal estimation of a Jacobian matrix J (k) of a linearized dynamic joint working balance equation based on the following optimization objective function

Wherein epsilon (k) and epsilon (k-1) respectively represent dynamic cooperative work balance equation residuals at time k and time k-1, delta x (k) ═ x (k) -x (k-1) represents the difference between guesses x (k) and x (k-1) of the dynamic cooperative work balance equation at time k and time k-1, and mu is a penalty factor,

an optimal estimate of a Jacobian matrix representing a linearized dynamic co-working equilibrium equation at time k-1;

step 3, updating the guess value x (k +1) of the dynamic common working balance equation at the moment of k +1 through iterative calculation:

where λ ∈ (0, 1) is the iteration step size.

2. The method of claim 1 wherein the optimal estimate is based on a pseudo-jacobian matrix for dynamically modeling an aircraft engine

Calculated by the following formula:

wherein, eta ∈ (0, 1)]Is a step factor, where Δ ∈ (k) — ∈ (k) - ∈ (k-1) represents the difference between the dynamic cooperative equilibrium equation residuals, epsilon (k), and epsilon (k-1), at the time of k and k-1, the superscript "T" represents the matrix transposition, and "| | | survival |) ² "denotes the square of the vector 2 norm.

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