CN115114731A - Aircraft engine dynamic modeling method based on pseudo Jacobian matrix - Google Patents

Aircraft engine dynamic modeling method based on pseudo Jacobian matrix Download PDF

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CN115114731A
CN115114731A CN202210810361.7A CN202210810361A CN115114731A CN 115114731 A CN115114731 A CN 115114731A CN 202210810361 A CN202210810361 A CN 202210810361A CN 115114731 A CN115114731 A CN 115114731A
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徐嘉伸
李秋红
庞淑伟
周文祥
刘鑫洋
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Nanjing University of Aeronautics and Astronautics
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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

Aircraft engine dynamic modeling method 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 an aero-engine.
Background
The aeroengine numerical simulation technology plays an extremely important role in the process of developing an aeroengine and designing and verifying a control system. The mathematical model is used for replacing a real engine to carry out numerical simulation or semi-physical simulation, preliminary verification can be carried out on a control algorithm, a fault diagnosis technology, a health management technology and the like, the experiment cost is saved, and the experiment risk is reduced. Since the end of the 80 s of the last century, western countries have been working on aircraft engine simulation techniques and developed various aircraft engine numerical simulation systems. Through years of research development and application, the related theoretical technology is mature at present, and the simulation confidence and precision reach a quite high level. With the development of model-based control technology, the requirements on the precision and the real-time performance of the model are further improved, and a great deal of research work is also carried out.
The aeroengine mathematical model establishes the aerothermal model of each component of the engine according to the characteristics of the components along the gas path flow of the engine on the basis of introducing the basic parameters of each component of the real engine, obtains the parameters of each component matching the current working state by solving the common working equation among the components, and the convergence speed and precision of the common working equation are directly related to the accuracy and the real-time performance of the engine mathematical model.
Common methods for solving the common operating equations of aircraft engines are newton-raphson (N-R) and Broyden newton. The N-R method seriously influences the real-time performance of the model because the model of the engine part needs to be repeatedly called to calculate the Jacobian matrix; the quasi-Newton method is to approximate the Jacobian matrix and obtain the approximate Jacobian matrix by recursion, so that the computational complexity is reduced, but the iterative convergence of the algorithm requires that the deviation degree of the initial solution is smaller and is influenced by the computational step length, and the convergence capability is weaker than that of the N-R method.
Based on the two methods, Chenyuchun et al, northwest industrial university [1] The method for calculating the step length and limiting the initial value by changing the N-R method improves the convergence problem of the solution, but the jacobian matrix needs to be calculated differentially. Liao Guang Huang of Nanjing aerospace university [2] The method comprises collecting off-line training data calculated based on N-R method, using residual error as input, and guessing value of common working equationThe correction is used as output, the neural network is used for training the correction, the Jacobian matrix calculation and equation iteration solving processes are avoided, the real-time performance of the model is effectively improved, but the neural network adopts an off-line training mode, the model precision is influenced by the generalization capability of the network, and the adaptability to uncertainty is weak. Wanyuan, wumodest et al of Nanjing aerospace university [3-4] The N-R method square convergence and the quasi-Newton method super-linear convergence characteristic are combined, a self-correcting Broyden quasi-Newton method is provided, the calculation step length is adjusted by judging the convergence trend of the model, the convergence precision and the convergence speed are improved, but at the working point with poor convergence trend, the initialization of a correction matrix is required to be carried out through difference, and the real-time performance of the individual working point is low. Great gentlewoman of Nanjing aerospace university [5] The method for establishing the engine component level model based on the accurate partial derivative is provided, the accurate partial derivative replaces difference calculation, a Jacobian matrix is constructed, the real-time performance of the model is improved, but the algorithm calculation process is complex, and the realization difficulty is high.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide a dynamic modeling method of an aircraft engine based on a pseudo Jacobian matrix, which can avoid repeated calling of a model by a differential algorithm and effectively improve the real-time property of the model.
The invention specifically adopts the following technical scheme to solve the technical problems:
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; when solving the dynamic common working balance equation of the aircraft engine, iteratively correcting the guessed value of the dynamic common working balance equation by the following method so as to enable the current dynamic common working balance equation to meet the convergence condition:
step 1, linearizing a dynamic common working balance equation at the moment k;
step 2, calculating a linearized dynamic common based on the following optimization objective functionOptimal estimation of the Jacobian matrix J (k) of the operating balance equation
Figure BDA0003738711370000021
(pseudo jacobian matrix):
Figure BDA0003738711370000022
wherein ∈ (k) and ∈ (k-1) respectively represent the dynamic cooperative work balance equation residuals at time k and time k-1, Δ x (k) ═ x (k) — x (k-1) represents the difference between guesses x (k) and x (k-1) of the dynamic cooperative work balance equations at time k and time k-1, and μ is a penalty factor,
Figure BDA0003738711370000023
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 co-working balance equation at the moment of k +1 through iterative calculation:
Figure BDA0003738711370000031
where λ ∈ (0, 1) is the iteration step size.
Preferably, the optimal estimation
Figure BDA0003738711370000032
Calculated by the following formula:
Figure BDA0003738711370000033
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.
Compared with the prior art, the technical scheme of the invention has the following beneficial effects:
(1) the real-time performance is high: the invention replaces the differential Jacobian matrix calculation with the recursion pseudo Jacobian matrix calculation, thereby greatly reducing the calling times of the component level model and greatly improving the real-time performance of the dynamic model calculation.
(2) The universality is strong: the calculation of the pseudo Jacobian matrix is based on a dynamic linearization method, is a data-driven method, has strong adaptability to the characteristic change of the engine, and can be used for the dynamic real-time model calculation of the engines of various models.
Drawings
FIG. 1 is a schematic structural view of a twin rotor turboshaft engine;
FIG. 2 is a flow chart of dynamic modeling of a turboshaft engine based on a pseudo Jacobian matrix;
FIG. 3 is a comparison of key engine output parameters (1km height) for a two-way difference algorithm and a pseudo Jacobian matrix algorithm of the present invention;
FIG. 4 is a comparison of key engine output parameters (3km height) for the two-way difference algorithm and the pseudo Jacobian matrix algorithm of the present invention.
Detailed Description
Aiming at the problem that the real-time performance of the jacobian matrix calculated by the existing difference-based method is poor, the invention combines a nonlinear model dynamic linearization method, utilizes optimal estimation to calculate the pseudo jacobian matrix, and improves the real-time performance of the dynamic model.
The invention specifically adopts the following technical scheme to solve the technical problems:
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; when solving the dynamic common working balance equation of the aircraft engine, iteratively correcting the guessed value of the dynamic common working balance equation by the following method so as to enable the current dynamic common working balance equation to meet 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
Figure BDA0003738711370000041
(pseudo jacobian matrix):
Figure BDA0003738711370000042
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,
Figure BDA0003738711370000043
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:
Figure BDA0003738711370000044
where λ ∈ (0, 1) is the iteration step size.
Suitable objects for the dynamic modeling method of the present invention include, but are not limited to, turboshaft engines, turboprop engines, turbofan engines, variable cycle engines, combination engines, and the like.
For the public understanding, the technical scheme of the invention is explained in detail by a specific embodiment and the accompanying drawings:
in the embodiment, a dual-rotor turboshaft engine shown in fig. 1 is taken as a research object, a gas turbine and a gas compressor of the dual-rotor turboshaft engine are coaxial, and a power turbine and a main rotor are coaxial. In the working process, air enters from the air inlet channel, flows to the combustion chamber through the air compressor, is mixed with fuel oil for combustion to generate high-temperature and high-pressure fuel gas, part of energy is extracted to drive the air compressor when the fuel gas flows through the gas turbine, the gas leaving the gas turbine flows through the power turbine, and the power turbine extracts residual energy to drive the main rotor wing, so that the working requirement of the helicopter is met.
For the dual-rotor turboshaft engine shown in fig. 1, the dynamic modeling method provided by the invention is shown in fig. 2, wherein T is the dynamic model simulation time, k is the iteration number of solving the balance equation, and T is the simulation step length. The method specifically comprises the following steps:
step A, obtaining flight conditions (the flight height H and the forward flight speed V of the helicopter) at the moment t x ) And fuel flow W f ；
B, sequentially calculating thermodynamic parameters of all parts and the load power demand of the rotor wing from an air inlet channel to a tail nozzle along an air flow;
step C, constructing a dynamic common working balance equation of the turboshaft engine;
under the conditions of known flight conditions and known fuel flow, the analysis is carried out according to the conditions of continuous flow and pressure balance required to be met in the dynamic working process of the double-rotor turboshaft engine, usually 3 common equations are selected, and epsilon is recorded i I is the dynamic co-working equation residual, including
(1) Gas turbine inlet flow equation of continuity phi 1 (x)：
φ 1 (x)＝(W 41xs -Q 41xs )/Q 41xs ＝ε 1 (1)
Wherein x is a guess value in a dynamic common working equation, W 41xs For gas turbine inlet similar flow, Q, at current operating conditions calculated from gas turbine component characteristics 41xs For similar flows, ε, from the gas turbine nozzle into the gas turbine as calculated by gas path flow 1 Is an equation phi 1 (x) Residual errors;
(2) inlet flow equation phi of power turbine 2 (x)：
φ 2 (x)＝(W 44xs -Q 44xs )/Q 44xs ＝ε 2 (2)
Wherein, W 44xs For similar power turbine inlet flow at current operating conditions, calculated from power turbine component characteristics, Q 44xs For similar flow rates, ε, from the power turbine nozzle into the power turbine calculated according to the gas path flow 2 Is an equation phi 2 (x) Residual errors;
(3) exhaust nozzle outlet pressure balance equation phi 3 (x)：
φ 3 (x)＝(p c7 -p 7 )/p 7 ＝ε 3 (3)
Wherein p is c7 Total pressure of air flow entering the tail nozzle outlet, p 7 Is the back pressure of the nozzle, ∈ 3 Is an equation phi 3 (x) Residual errors;
in the common working balance equation of the component-level models, a guess value x is selected as [ Z ═ Z% C Z G Z P ] T (ii) a Wherein Z is C Is the compressor pressure ratio coefficient, Z G Is the gas turbine pressure ratio coefficient, Z P Is the power turbine pressure ratio coefficient; with Z c For example, the pressure ratio coefficient is defined as:
Figure BDA0003738711370000061
wherein, pi c Subscripts min and max represent the minimum value and the maximum value of the pressure ratio on the current rotating speed line, respectively, for the pressure ratio of the compressor at the current working point.
D, judging whether the dynamic co-working balance equation meets a convergence condition, if so, jumping to the step F, and if not, executing the step E, if not, changing the flag to 0;
Figure BDA0003738711370000062
wherein k is max Is the maximum allowed number of iterations.
The convergence conditions in this example are as follows:
ε i ＜10 -5 ,i＝1,2,3，or k＞k max 。
e, calculating a pseudo Jacobian matrix, correcting a guess value, and returning to the step B; the method specifically comprises the following substeps:
step E1, linearizing the dynamic co-working balance equation at the moment k:
ε(k)＝φ(x(k)) (6)
assuming that the partial derivative of phi (x (k)) to x is continuous and satisfies | | | epsilon (k) -epsilon (k-1) | | < b | | | | x (k) -x (k-1) | |, if the guess value change is limited, the change of equation residual error is also limited, and the condition is satisfied when the model converges;
linearizing equation (6) yields a linearized equation of the form:
Figure BDA0003738711370000063
wherein Δ ∈ (k) — ∈ (k-1), Δ x (k) ═ x (k) — x (k-1),
Figure BDA0003738711370000064
is a pseudo Jacobian matrix; equation (7) is a linearized equilibrium equation;
step E2, calculating the pseudo Jacobian matrix by adopting the optimal estimation method
Figure BDA0003738711370000065
Specifically, the following evaluation indexes are adopted:
Figure BDA0003738711370000066
wherein mu is a penalty factor, and J (k) is a Jacobian matrix;
formula (8) indicates that when L (j) (k) ═ 0, epsilon (k) -epsilon (k-1) ═ j (k) Δ x (k),
Figure BDA0003738711370000071
namely, the linearized equation (6) is established, and the change of the Jacobian matrix is restrained;
the jacobian matrix is shown as equation (9):
Figure BDA0003738711370000072
since the working process of the aircraft engine is a strong nonlinear complex pneumatic thermodynamic process, and the mathematical modeling of the aircraft engine depends on the interpolation and iterative operation of a component characteristic diagram and pneumatic thermodynamic parameters, phi is i (x) And i is an implicit equation containing calculation processes of various components of the engine, and the partial derivative calculation cannot be directly carried out, and can only be obtained by a numerical difference method. To improve the model accuracy, the partial derivatives in the jacobian matrix are usually calculated by using an intermediate difference method, that is:
Figure BDA0003738711370000073
in the calculation of the partial derivative of the formula (10), the guess value x is needed i Small disturbances are respectively carried out upwards and downwards, a component-level mathematical model is called to carry out differential calculation, and three guess values are total, so 6 times of calculation are needed, and the real-time performance of the dynamic model is seriously influenced. For this purpose, the invention adopts a recursion mode to calculate the pseudo Jacobian matrix.
Minimizing equation (8) with respect to J (k) to obtain an optimal estimate of J (k)
Figure BDA0003738711370000076
Comprises the following steps:
Figure BDA0003738711370000074
the shift arrangement is provided with
Figure BDA0003738711370000075
Subtracting at both sides of the equation simultaneously
Figure BDA0003738711370000081
Figure BDA0003738711370000082
So that there are
Figure BDA0003738711370000083
Where η ∈ (0,1] is the stepsize factor;
when the t is equal to 0, the second phase is,
Figure BDA0003738711370000084
the initialization is performed by a difference, otherwise,
Figure BDA0003738711370000085
the subscript t represents the simulation time of the dynamic model, namely the initial value of the pseudo Jacobian matrix at the current simulation time is the value of the pseudo Jacobian matrix at the end of iteration at the previous simulation time;
the formula (14) is used for calculating the pseudo Jacobian matrix in a recursion mode, only the increment of the residual error of the current equation and the increment of a guess value are needed in the calculation process, a component model is not needed to be additionally called for differential calculation, and the calculation process is greatly simplified;
and E3, updating guess values through iterative calculation:
Figure BDA0003738711370000086
where λ ∈ (0, 1) is the iteration step size.
Step F, calculating the acceleration of the rotor, and updating the rotating speed;
Figure BDA0003738711370000087
wherein n is G 、n P Is the gas and power turbine speed, J G 、J P Is the moment of inertia, η, of the gas turbine rotor and the power turbine rotor G 、η P Is the mechanical efficiency, W, of gas turbine rotors and power turbine rotors GT 、W PT Is the gas turbine and power turbine power, W C 、W R Is the power of the air compressor and the helicopter rotor.
In order to verify the advantages of the pseudo Jacobian matrix algorithm adopted by the invention in the process of solving the dynamic common working equation of the turboshaft engine, simulation is carried out on a T700 turboshaft engine component level model, the initial flying height H of the turboshaft engine is set to be 1km, and the initial forward flying speed V is set to be V x And (5) simulating the dynamic working process of the engine in the simulation process, and calling the models of the two algorithms 1 ten thousand times respectively. The model is developed based on VC + +6.0, and the computer system is Windows 10 family edition, CPU i5-8250U 1.60GHz,8GB RAM. The average calculation time of the pseudo Jacobian matrix algorithm model 1 ten thousand times of calling is 3.19s, and the average calculation time of the bidirectional difference calculation Jacobian matrix model is 6.97 s. As can be seen, the time used by the pseudo Jacobian matrix algorithm is about 3/7 of the bidirectional difference algorithm, and the superiority of the algorithm in real time is verified.
The accuracy of the model is further verified in an envelope range, and 2 sets of simulation results are given, as shown in fig. 3 and 4, and the simulation results in other flight areas and engine working states are similar. In the figure, PJ (Pseudo-Jacobian) represents the simulation effect of a component model adopting a Pseudo Jacobian matrix algorithm; BDD (Bi-directional differential) represents the model simulation effect of the bidirectional differential computation Jacobian matrix algorithm. The graph shows the variation curve of the key parameter of the turboshaft engine, wherein sfc is the fuel consumption and T 41 E represents the output deviation of the PPD algorithm compared with a differential algorithm model, and the deviation calculation formula is
Figure BDA0003738711370000091
(1) Simulation test at flying height of 1km
Setting the flying height H of the helicopter to be 1km and the front flying speed V x The change curve of the output parameter is shown in figure 3 when the speed is increased from 20m/s to 50 m/s.
As can be seen from fig. 3, compared with the bidirectional difference algorithm, the rotation speed deviation output by the component-level model based on the pseudo-jacobian matrix is less than 0.15%, the temperature deviation is less than 0.25%, the fuel flow is smaller in order of magnitude, the maximum deviation in closed-loop control is less than 3%, the fuel consumption calculation deviation caused by the fuel flow deviation is less than 1.2%, the detailed error information is shown in table 1, as can be seen from table 1 and fig. 3, the turboshaft engine mathematical model based on the pseudo-jacobian matrix obtains higher accuracy, the average error is less than 0.14%, and the validity of the algorithm in the aspect of the dynamic model accuracy is verified.
TABLE 1 errors of pseudo Jacobian matrix model versus bidirectional differential model
Figure BDA0003738711370000092
(2) Simulation test at flight altitude of 3km
Setting the flying height H of the helicopter to be 3km and the front flying speed V x The change curve of the output parameter is shown in FIG. 4 when the speed is increased from 20m/s to 50 m/s.
As can be seen from fig. 4, compared with the bidirectional difference algorithm, the rotation speed deviation output by the component-level model based on the pseudo-jacobian matrix at the height of 3km is less than 0.3%, the temperature deviation is less than 0.4%, the fuel flow is smaller in order of magnitude, the maximum deviation in closed-loop control is less than 1.1%, the fuel consumption calculation deviation caused by the fuel flow deviation is less than 1.1%, and the maximum modeling error is better than the height of 1km, so that the applicability of the model in the envelope is verified.
Reference documents:
[1] chenyuchun, Xunxiyuan, Yangyinkai, and the like, a method for improving the calculation convergence of characteristics of an aeroengine [ J ]. an aeronautical dynamics report, 2008,23(12): 2242-.
[2] A turboshaft engine high-precision real-time part-class model research (J) is carried out, 2016, 37(01):25-33.
[3] Wangyuan, Li autumn Red, Huang Daohua, calculation of values of an aeroengine model based on a self-correcting Broyden quasi-Newton method [ J ]. aeronautical dynamics, 2016,31(1): 249-.
[4] Modesty, lie red, mono-bin, aero-engine pneumatic thermodynamic system model solving methods study [ J ] computer simulations, 2019, 36(01):76-81.
[5]Pang,S.；Li,Q.；Ni,B.Improved nonlinear MPC for aircraft gas turbine engine 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
Figure FDA0003738711360000011
Figure FDA0003738711360000012
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,
Figure FDA0003738711360000013
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:
Figure FDA0003738711360000014
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
Figure FDA0003738711360000015
Calculated by the following formula:
Figure FDA0003738711360000016
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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CN118089819A (en) * 2024-04-23 2024-05-28 南京市计量监督检测院 Online temperature and humidity testing system and method

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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
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