Aero-engine model linearization design method based on differential squirrel search algorithm

Abstract

The invention belongs to the technical field of aero-engine control and simulation, and discloses an aero-engine model linearization design method based on a differential squirrel search algorithm. A fusion algorithm of a differential evolution algorithm and a squirrel search algorithm is adopted. According to the differential squirrel search algorithm, on the basis of a differential evolution algorithm, random disturbance and local search strategies in the squirrel search algorithm are introduced, so that the search diversity and convergence of the algorithm are remarkably improved. By means of the combination, when the algorithm is used for solving a complex aero-engine state variable model, the global search capacity and the local search capacity can be better balanced, and therefore the search efficiency and the solution quality are improved. According to the method, the fusion algorithm is applied to establishment of the aero-engine state variable model, and the method is suitable for linearization of various types of aero-engine models and has high universality.

Description

Technical Field

[0001] The invention belongs to the technical field of aeroengine control and simulation, and provides an aeroengine model linearization design method based on a differential squirrel search algorithm. Background Art

[0002] In the early days of the development of the aviation industry, the design and control of aircraft engines were mainly based on experiments and experience. With the development of control theory, people began to realize the importance of establishing mathematical models for understanding and controlling engine performance. At this stage, linear models began to be introduced into the research of aircraft engines, but the complexity and accuracy of the models were relatively low. With the advancement of mathematical and physical theories, especially the development of control theory and system theory, researchers began to explore the linear modeling methods of aircraft engines in more depth. The research at this stage mainly focused on how to more accurately describe the dynamic and steady-state characteristics of the engine, and how to apply advanced control theory to the linear model of the engine. With the rapid development of computer technology and numerical methods, linear models are increasingly used in aircraft engine design, control and health management. At this stage, researchers began to focus on how to use linear models for engine fault diagnosis, performance optimization and health management. Despite great progress, the development of linear models for aircraft engines still faces some challenges. For example, the complexity of the engine gas path system makes the construction and maintenance of the model quite difficult; at the same time, the uncertainty factors in the actual operation of the engine will also affect the accuracy and reliability of the model. Therefore, it is necessary to improve the accuracy and completeness of the model, enhance the robustness and adaptability of the model, and introduce some advanced algorithms and tools such as optimization algorithms and machine learning into the construction and application of linear models, so as to better reflect the actual working status of the engine gas path system, cope with the uncertainty factors in the engine operation process, and further improve the efficiency and accuracy of gas path health management.

[0003] In 1997, Storn proposed in his paper Differential Evolution--A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces to identify the dynamic model of a turbofan engine as a multivariable piecewise linear model, and discussed the modeling uncertainty of the model in the turbofan engine control system. In his paper Regularization approach for real-time modeling of aero gasturbines, Breikin proposed a partial automatic regularization method for parameter identification of the state variable model. In his paper Method for Establishing Small Deviation State Variable Model of Aeroengine, Feng Zhengping et al. proposed a fitting method to calculate the state space system matrix parameters, and used this method to establish a model of small deviation state space description of a turbofan engine at a steady-state point. In his paper Linear Fitting Method for Establishing State Variable Model of Aeroengine, Lu Jun, Guo Yingqing et al. proposed that the linear function relationship between the system matrix and the system response constructed by using known states can transform nonlinear problems into linear least squares problems, thereby avoiding the initial value problem generated when establishing a state space model by nonlinear fitting. Zhang Haibo, Hu Yu and others proposed an improved fitting method in the article "An improved fitting method for obtaining engine state variable model", which can not only avoid the initial value problem, but also eliminate the influence of transient errors on the steady state and improve the calculation efficiency.

[0004] At present, the commonly used methods for establishing linearized models mainly include partial derivative method, fitting method, optimization algorithm, etc. The partial derivative method is to make a small disturbance to a certain control quantity or state quantity of the system when the engine is working in the selected steady state, and at the same time force other control quantities and state quantities to remain unchanged. The A and C matrix values ​​of the state variable model are obtained by calculating the rate of change of the state quantity and output quantity, and then the values ​​of the B and D matrices are obtained by using the equilibrium equation in the steady state. The fitting method is to collect the operating data of the aircraft engine under different conditions, and use mathematical methods (such as the least squares method) to fit the data to obtain a linearized model. Optimization algorithms such as particle swarm optimization, genetic algorithm, and self-optimization algorithm simulate the process of natural evolution or group behavior to find the optimal linearized model parameters in the search space. The partial derivative method and fitting method will make the analytical expression of the linear model very complicated as the order of the model increases. Traditional optimization algorithms have certain requirements for the selection of initial values, and there are also problems such as large amount of calculation, slow convergence speed, long calculation time, and falling into local optimal solutions.

[0005] In order to overcome the challenges of initial value sensitivity, easy to fall into local optimal solutions and slow convergence speed that are common in traditional optimization algorithms, such as genetic algorithms and particle swarm algorithms, when constructing aero-engine linear models. The present invention proposes a fusion algorithm based on differential evolution algorithm and squirrel search algorithm to establish aero-engine linear models. The initialization population of the algorithm is generated by random allocation, which gets rid of the dependence of traditional algorithms on initial value selection. The differential evolution algorithm has high computational efficiency and shows good robustness and convergence in dealing with complex aero-engine system models. On this basis, the algorithm incorporates the powerful global search capability of the squirrel search algorithm, effectively avoiding falling into local optimal solutions, and provides a reliable solution for the establishment of aero-engine linear models. Summary of the invention

[0006] In order to overcome the challenges of traditional optimization algorithms, such as genetic algorithms and particle swarm algorithms, when constructing aero-engine linear models, such as initial value sensitivity, easy to fall into local optimal solutions, and slow convergence speed, the present invention proposes a fusion algorithm based on differential evolution algorithm and squirrel search algorithm to establish aero-engine linear models.

[0007] The technical solution of the present invention:

[0008] A linear design method for an aero-engine model based on a differential squirrel search algorithm, the steps are as follows:

[0009] S1: Establishing the state variable model of turbofan engine

[0010] S1.1: The input variables of the state variable model of the turbofan engine are the high-pressure rotor speed and the low-pressure rotor speed; the control input is the main fuel flow rate; the output includes the high-pressure rotor speed, the low-pressure rotor speed, the high-pressure compressor outlet total pressure, the low-pressure turbine outlet total pressure and the low-pressure turbine outlet total temperature;

[0011] S1.2: The state variable model of the turbofan engine is as follows:

[0012]

[0013]

[0014] Where n1 is the low-pressure rotor speed, r / min; n2 is the high-pressure rotor speed, r / min; W f is the main fuel flow, kg / h; P3 is the total pressure at the high pressure compressor outlet, MPa; P5 is the total pressure at the low pressure turbine outlet, MPa; T5 is the total temperature at the low pressure turbine outlet, K; is the rate of change of low-pressure rotor speed; is the rate of change of the high-pressure rotor speed; A is the state matrix; B is the input matrix; C is the output matrix; D is the direct transfer matrix;

[0015] S1.3: The steps to establish a model for extracting small deviation state variables of a turbofan engine using the differential squirrel search algorithm are as follows:

[0016] The research object of this patent is the JT9D high bypass ratio turbofan engine, the control input W f When a certain steady-state point remains unchanged, the steady-state model corresponding to the engine can be obtained. The control input quantity W near the steady-state point f By making a step input within ±10%, the corresponding dynamic model of the engine can be obtained. The output of the dynamic model is subtracted from the output of the steady-state model to obtain the deviation signal of the dynamic response of the nonlinear model of the turbofan engine Δy(t) = [Δn1(t)Δn2(t)ΔP3(t)ΔP5(t)ΔT5(t)] T ;

[0017] S1.4: Randomly initialize the population through the differential squirrel search algorithm to obtain A0, B0, C0, and D0; where A0 is the initial value of the state matrix A, B0 is the initial value of the input matrix B, C0 is the initial value of the output matrix C, and D0 is the initial value of the direct transfer matrix D. The upper and lower limits of each element in the matrix are set to [-50, 50]; the initial value is randomly selected within the upper and lower limits;

[0018] S1.5: Substituting the A0, B0, C0, D0 matrices into the state variable model can obtain its preliminary state variable model. The control input W of the preliminary state variable model of the turbofan engine and the nonlinear model of the turbofan engine f The same step response is made to obtain the dynamic response deviation signal of the preliminary linear model of the turbofan engine:

[0019]

[0020] S1.6: According to the dynamic response data Δy(t) and Determine the optimization objective function using the differential squirrel search algorithm The differential squirrel search algorithm is used to calculate the minimum sum of squares of the dynamic response data deviations, thereby obtaining the optimal solution;

[0021] S2: Design optimization algorithm

[0022] By comparing different optimization algorithms, the differential evolution algorithm and the squirrel search algorithm were selected. By consulting materials and deriving formulas, the two algorithms were integrated to obtain the differential squirrel search algorithm.

[0023] S2.1: Briefly introduce the differential evolution algorithm, clarify the advantages and disadvantages of the algorithm and the conditions for use, and lay the foundation for the integration and improvement of subsequent algorithms. Differential evolution (DE) is a heuristic random search algorithm based on population differences. Its idea is to generate temporary individuals based on the individual differences within the population, and then randomly recombine to achieve population evolution. The differential evolution algorithm includes initialization, mutation operation, crossover operation and selection operation.

[0024] S2.2: Understand the operating principle of the squirrel search algorithm. The squirrel search algorithm (SSA) is inspired by the dynamic foraging behavior and effective sliding mechanism of a certain type of squirrel. Before modeling the algorithm, some basic assumptions need to be made: there are n squirrels in the forest, and each squirrel is on a tree. There are only three types of trees in the forest, namely ordinary trees, acorn trees, and hickory trees. Ordinary trees do not provide food for squirrels; acorn trees provide normal food for squirrels; and hickory trees provide high-quality food for squirrels. Generally speaking, most squirrels live on ordinary trees, a small number of squirrels live on acorn trees, and a very small number of squirrels live on hickory trees. The position of the squirrel is represented by a vector, so when the algorithm is used to solve the problem, the problem can be expanded into multiple dimensions, that is, the squirrel can change its position in the multidimensional search space. After completing the initialization of the position, sort them according to the fitness value of each squirrel. The squirrel with the best fitness value is on the hickory tree, the next three squirrels are on the acorn tree, and the rest of the squirrels are on the ordinary tree. When foraging, squirrels generate new solutions through the sliding mechanism. There are three sliding situations, and the specific one that occurs is determined by the probability of the appearance of natural enemies.

[0025] S2.3: The differential evolution algorithm is combined with the squirrel search algorithm to obtain the differential squirrel search algorithm. The differential squirrel search algorithm (DSSA) is a hybrid intelligent optimization algorithm. The algorithm combines the search method of the squirrel search algorithm with the optimization process of the differential evolution algorithm, making full use of the advantages of both. The squirrel search algorithm is a group intelligence optimization algorithm inspired by nature, which is inspired by the survival strategy of the animal squirrel; the crossover characteristics of the differential evolution algorithm enhance the exploration ability of squirrels in the foraging process, thereby reducing the chance of falling into local minima. In addition, the update rule of the squirrel position is modified in each iteration, so that the squirrel can reach the best food source in a shorter time before the arrival of winter. This combined optimization algorithm has the characteristics of strong evolution ability, fast search speed, and strong optimization ability. DSSA combines the differential evolution algorithm (DE) and the squirrel search algorithm (SSA). In SSA, squirrels update their positions by following the positions of squirrels in hickory trees or acorn trees. Therefore, the update rules of these leading squirrels have been modified to improve their search strategies. The detection capability of the algorithm is further improved by introducing a DE-inspired crossover operation.

[0026] S3: Algorithm simulation verification

[0027] By adding a small disturbance to the nonlinear model of the turbofan engine at a certain steady-state point, the output curve of the model after the disturbance can be obtained. The coefficients of the state variable model can be obtained by fitting the output curve with the optimization algorithm.

[0028] S3.1: Nonlinear model and state variable linear model of turbofan engine At the steady-state point of H=0; Ma=0; W_f=0.0187kg / s, the main fuel flow is injected with appropriate step disturbance, and the appropriate simulation step and simulation time are set. (The main fuel flow increases and decreases by 6% steps, respectively, and is optimized using the differential squirrel algorithm. Among them, the population size is set to 100; the scaling factor is set to 0.95; the crossover rate is set to 0.9; the maximum number of iterations is set to 5000; the simulation step is set to 0.04s; and the simulation time is set to 20s.)

[0029] S4: Evaluation of fitting effect

[0030] In order to evaluate the fitting effect of the algorithm, the maximum error or peak error in the dynamic process, the steady-state process error, the determination coefficient, the mean absolute error and the root mean square error were used for evaluation, and some results were plotted as bar graphs and scatter plots.

[0031] Beneficial effects of the present invention:

[0032] (1) The present invention adopts a fusion algorithm of the differential evolution algorithm and the squirrel search algorithm. Based on the differential evolution algorithm, the differential squirrel search algorithm significantly increases the search diversity and convergence of the algorithm by introducing strategies such as random perturbation and local search in the squirrel search algorithm. This combination enables the algorithm to better balance the capabilities of global search and local search when solving complex aircraft engine state variable models, thereby improving search efficiency and solution quality.

[0033] (2) The present invention applies the fusion algorithm to the construction of the aircraft engine state variable model, which is suitable for the linearization of various types of aircraft engine models and has strong universality. The differential squirrel search algorithm not only inherits the global search capability and fast convergence characteristics of the differential evolution algorithm, but also integrates the search diversity and strong adaptability of the squirrel search algorithm, so that the algorithm can perform well when facing different types of aircraft engine linearization problems and show strong robustness. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 This is the flow chart of the differential squirrel search algorithm.

[0035] Figure 2 The state variable model fitting effect diagram of the fuel flow rate rising by 6%, where (a) is the main fuel flow rate W f , (b) is the main fuel flow difference ΔW f , (c) is the low-pressure rotor speed difference Δn1, (d) is the high-pressure rotor speed difference Δn2, (e) is the total pressure difference ΔP3 at the high-pressure compressor outlet, (f) is the total pressure difference ΔP5 at the low-pressure turbine outlet, and (g) is the total temperature difference ΔT5 at the low-pressure turbine outlet. In the figure, red represents real data and blue represents linear model fitting data.

[0036] Figure 3 The following is the effect diagram of the state variable model fitting when the fuel flow rate drops by 6%, where (a) is the main fuel flow rate W f , (b) is the main fuel flow difference ΔW f , (c) is the low-pressure rotor speed difference Δn1, (d) is the high-pressure rotor speed difference Δn2, (e) is the total pressure difference ΔP3 at the high-pressure compressor outlet, (f) is the total pressure difference ΔP5 at the low-pressure turbine outlet, and (g) is the total temperature difference ΔT5 at the low-pressure turbine outlet. In the figure, red represents real data and blue represents linear model fitting data.

[0037] Figure 4It is a scatter plot of the true value and predicted value of each physical quantity, where (a) is the low-pressure rotor speed (rising stage), (b) is the low-pressure rotor speed (falling stage), (c) is the high-pressure rotor speed (rising stage), (d) is the high-pressure rotor speed (falling stage), (e) is the high-pressure compressor outlet total pressure (rising stage), (f) is the high-pressure compressor outlet total pressure (falling stage), (g) is the low-pressure turbine outlet total pressure (rising stage), (h) is the low-pressure turbine outlet total pressure (falling stage), (i) is the low-pressure turbine outlet total temperature (rising stage), and (j) is the low-pressure turbine outlet total temperature (falling stage). DETAILED DESCRIPTION

[0038] The specific implementation of the present invention is further described below in conjunction with the accompanying drawings and technical solutions.

[0039] A linear design method for an aero-engine model based on a differential squirrel search algorithm comprises the following steps:

[0040] Step 1: Establish a turbofan engine state variable model

[0041] The operation steps are consistent with the technical solution and will not be repeated here.

[0042] Step 2: Design an optimization algorithm

[0043] Step 2.1: Differential Evolution Algorithm

[0044] Differential Evolution (DE) is a heuristic random search algorithm based on population differences. Its idea is to generate temporary individuals based on the individual differences within the population, and then randomly recombine to achieve population evolution. The differential evolution algorithm includes initialization, mutation operation, crossover operation and selection operation.

[0045] For optimization problems: minf(x1,x2,…,x D ) The optimization goal is to make each parameter x1, x2, ..., x D , the corresponding calculation result f is as small as possible. The main calculation steps of the differential evolution algorithm are as follows:

[0046] 1) Initialize the population. The initial population is randomly generated:

[0047]

[0048] Among them, x i (0) represents the i-th "chromosome" (or individual) in the 0th generation of the population, x j,i (0) represents the jth "gene" of the i-th "chromosome" of generation 0. NP represents the population size, and rand(0,1) represents a random number uniformly distributed in the interval (0,1).

[0049] 2) Mutation operation. In the differential evolution algorithm, a common differential strategy is to randomly select two different individuals in the population, scale their vector difference, and then synthesize the vector with the individual to be mutated, that is,

[0050] v i (g+1)=x r1 (g)+F*(x r2 (g)-x r3 (g)),

[0051] i≠r1≠r2≠r3,

[0052] Where F is the scaling factor, x i (g) represents the i-th individual in the g-th generation population. In the process of evolution, in order to ensure the validity of the solution, it is necessary to determine whether each "gene" in the "chromosome" meets the boundary conditions. If the boundary conditions are not met, the "gene" is regenerated by random method (the same method as the generation of the initial population).

[0053] 3) Crossover operation. For the g-th generation population {x i (g)} and its variant intermediates {v i (g+1)} performs crossover operation between individuals:

[0054]

[0055] Among them, CR is the crossover probability, j rand A random integer in the range [1,2,…,D]. When the random number generated by rand(0,1) is less than or equal to CR or j=j rand , the v in the current generation (g generation) j,i , assigned to the next generation (g+1 generation) u j,i Otherwise, x j,i Assign to u j,i .

[0056] 4) Selection operation. The selection operation uses a greedy algorithm to select individuals to enter the next generation population:

[0057]

[0058] By calculating the fitness function, individuals with higher fitness are selected to enter the next generation population.

[0059] Step 2.2: Squirrel Search Algorithm

[0060] The Squirrel Search Algorithm (SSA) is inspired by the dynamic foraging behavior and effective sliding mechanism of a certain type of squirrel. Before modeling the algorithm, some basic assumptions need to be made: there are n squirrels in the forest, and each squirrel is on a tree. There are only three types of trees in the forest, namely ordinary trees, acorn trees, and hickory trees. Ordinary trees do not provide food for squirrels; acorn trees provide normal food for squirrels; and hickory trees provide high-quality food for squirrels. Generally speaking, most squirrels live on ordinary trees, a small number of squirrels live on acorn trees, and a very small number of squirrels live on hickory trees. The position of the squirrel is represented by a vector, so when the algorithm is used to solve the problem, the problem can be expanded into multiple dimensions, that is, the squirrel can change its position in the multidimensional search space. After the position is initialized, the squirrels are sorted according to their fitness values. The squirrel with the best fitness value is on the hickory tree, the next three squirrels are on the acorn tree, and the rest of the squirrels are on the ordinary tree. During the foraging process, squirrels generate new solutions through the sliding mechanism. There are three sliding situations, and the specific occurrence is determined by the probability of the appearance of natural enemies. The three sliding situations are as follows:

[0061] 1) The squirrel in the acorn tree glides toward the hickory tree

[0062]

[0063] in represents the i+1 generation of squirrels on the acorn tree, represents the i-th generation of squirrels in the hickory tree, d g is the random sliding distance, G c is the sliding constant, rand(0,1) is a random number between 0 and 1, P dp It represents the probability of the presence of natural enemies. When natural enemies exist, squirrels will only move in a small range.

[0064] 2) Squirrels in ordinary trees move to acorn trees

[0065]

[0066] in Represents the i+1th generation of squirrels in a common tree.

[0067] 3) Squirrels in regular trees move to hickory trees

[0068]

[0069] The foraging behavior of squirrels is affected by weather changes, and squirrels become less active in winter. Incorporating this seasonal behavior into the algorithm when designing it is called seasonal monitoring. Since seasonal detection is not involved in this article, it will not be introduced in detail.

[0070] Step 2.3: Differential Squirrel Search Algorithm

[0071] The differential squirrel search algorithm (DSSA) is a hybrid intelligent optimization algorithm. The algorithm combines the search method of the squirrel search algorithm with the optimization process of the differential evolution algorithm, making full use of the advantages of both. The squirrel search algorithm is a swarm intelligence optimization algorithm inspired by nature, which is inspired by the survival strategy of the animal squirrel; the crossover feature of the differential evolution algorithm enhances the exploration ability of squirrels during foraging, thereby reducing the chance of falling into local minima. In addition, the update rule of the squirrel position is modified in each iteration, so that the squirrel can reach the best food source in a shorter time before the arrival of winter. This combined optimization algorithm has the characteristics of strong evolutionary ability, fast search speed and strong optimization ability. DSSA combines the differential evolution algorithm (DE) and the squirrel search algorithm (SSA). In SSA, squirrels update their positions by following the positions of squirrels in hickory trees or acorn trees. Therefore, the update rules of these leading squirrels have been modified to improve their search strategies. The detection ability of the algorithm is further improved by introducing the crossover operation inspired by DE.

[0072] 1) Initialize the population. There are n squirrels in the forest, and the position of the i-th squirrel can be represented by a vector. The positions of all squirrels can be represented by a matrix:

[0073]

[0074] PS i,j Denotes the jth dimension of the ith squirrel. Assign the initial position of each squirrel in the forest:

[0075] PS i =PS L +U(0,1)×(PS U -PS L )

[0076] PS L and PS U are the lower and upper bounds of the i-th squirrel in the j-th dimension, and U(0,1) is a uniformly distributed random number in the range [0,1].

[0077] 2) Position update. In the absence of natural enemies, the squirrels in the acorn tree will update their position and follow the current best PS htMove toward the best source. Squirrels in common trees update their location by following squirrels in acorn or hickory trees. Squirrels randomly change paths when foraging in the presence of predators. Squirrels in acorn trees update their location based on the locations of other squirrels.

[0078]

[0079] Where P avg is the average position of all squirrels in the current population.

[0080] 3) Differential evolution algorithm crossover operation. In order to avoid the possibility of falling into a local minimum, the crossover mechanism of the differential evolution algorithm is used to increase the diversity of squirrels. The current position and new position of the squirrel are crossovered.

[0081]

[0082] Where i = 1, 2, 3, .... NP, NP is the total size. is the position of the squirrel after the crossover operation. and are the current positions of the squirrel in the acorn tree, new and old. Q is the dimension of the problem, and Cr is the crossover rate. rand ∈[1,Q] is a randomly generated index. rand j Represents the j-th evaluation of a uniformly generated random number in a given range.

[0083] The movement of squirrels on ordinary trees to acorn trees and hickory trees is the same as the original squirrel search algorithm, and the crossover operation is consistent with the above formula. Finally, the optimal solution for squirrels on hickory trees is obtained.

[0084]

[0085] in is the average of all squirrel positions on the acorn tree. The algorithm flow is as follows Figure 1 shown.

[0086] The differential squirrel search algorithm significantly increases the search diversity and convergence of the algorithm by introducing random perturbations and local search strategies in the squirrel search algorithm on the basis of the differential evolution algorithm. This combination enables the algorithm to better balance the capabilities of global search and local search when solving complex aircraft engine state variable models, thereby improving search efficiency and solution quality. The differential squirrel search algorithm not only inherits the global search capability and fast convergence characteristics of the differential evolution algorithm, but also integrates the search diversity and strong adaptability of the squirrel search algorithm, so that the algorithm can perform well when facing different types of aircraft engine linearization problems and show strong robustness. In the population update process, the differential squirrel search algorithm adopts a selection mechanism based on fitness evaluation. By comparing the fitness values ​​of the newly generated candidate solutions with those of the individuals in the current population, individuals with higher fitness are selected for replacement. This mechanism effectively maintains the diversity of the population and further improves the search efficiency of the algorithm.

[0087] Step 3: Algorithm simulation verification

[0088] The nonlinear model and state variable linear model of the turbofan engine are optimized using the differential squirrel algorithm at the steady-state point of H = 0; Ma = 0; W_f = 0.0187 kg / s, with the main fuel flow rate rising and falling by 6% steps respectively. Among them, the population size is set to 100; the scaling factor is set to 0.95; the crossover rate is set to 0.9; the maximum number of iterations is set to 5000; the simulation step size is set to 0.04s; and the simulation time is set to 20s (in the fitting curve, red represents real data and blue represents linear model fitting data).

[0089] The optimal solution for the ascending step is obtained by the differential squirrel search algorithm:

[0090]

[0091] Table 1 Analysis of fuel flow rate increase results

[0092]

[0093]

[0094] The optimal solution for the descending step is obtained by the differential squirrel search algorithm:

[0095]

[0096] Table 2 Analysis of fuel flow rate drop results

[0097]

[0098] Step 4: Evaluate the fitting effect

[0099] In order to evaluate the fitting effect of the algorithm, the maximum error or peak error in the dynamic process, the steady-state process error, the determination coefficient, the mean absolute error and the root mean square error were used for evaluation, and some results were plotted as bar graphs and scatter plots.

[0100] Mean Absolute Error (MAE) is the average of the absolute errors between the predicted value and the true value. It has good anti-interference ability for outliers in the data. The calculation formula is as follows:

[0101]

[0102] Root Mean Square Error (RMSE) is the square root of the ratio of the sum of the squares of the deviations between the predicted value and the true value to the number of sample points in the test set n, which is used to measure the deviation between the predicted value and the true value. The calculation formula is as follows:

[0103]

[0104] Coefficient of Determination (R 2 ) is used to describe the correlation between the predicted value and the true value, R 2 The value range of is between 0 and 1. The closer it is to 1, the better the fit. The calculation formula is as follows:

[0105]

[0106] Among them, y i Represents the true value of the i-th point in the test set sample, represents the predicted value of the i-th point in the test set sample, It represents the average of the true value of the test set samples, and n represents the number of sample points in the test set.

[0107] Table 3 Evaluation indicators (rising stage)

[0108]

[0109]

[0110] Table 4 Evaluation indicators (declining stage)

[0111]

[0112] From the simulation results and error analysis, it can be seen that the linearized model has a good tracking effect in the dynamic process, the steady-state error is less than 1%, the average absolute value error in the rising and falling stages is small, the determination coefficient is greater than 0.95 and generally higher than 0.99, and the fitting effect is excellent. Draw the scatter plots of each physical quantity in the rising and falling stages, with the horizontal axis as the true value and the vertical axis as the predicted value, and get the fitting line through the least squares method to observe the fitting effect of the data obtained through the state variable model.

[0113] Depend on Figure 3 and Figure 4 From the response results, we can see that the simulation time is 20 seconds, and both the true value and the predicted value tend to a stable state at about 8 seconds. Based on this observation, analyzing the distribution characteristics of the scatter plot, we noticed that most of the data points are concentrated in the final steady-state stage, which reflects that the fitting curve has a significant influence in the steady-state stage. The linear fitting results obtained by the least squares method show that, except for a few scattered points reflecting the dynamic process that deviate far, overall, the fitting effect between the predicted value and the true value is good.

[0114] From the simulation results, the linear model optimized by the differential squirrel search algorithm not only matches well with the dynamic process of the nonlinear model, but also ensures the consistency of the final stable value of the engine, with high accuracy. The multiple simulation results of the random initialization of the differential squirrel search algorithm and the fitting effect of the nonlinear model have achieved high accuracy, overcoming the dependence of the traditional genetic algorithm on the selection of initial values. When the differential evolution algorithm and the differential squirrel search algorithm are compared, when the same key parameters such as the population size and the maximum number of iterations are set for both, the differential squirrel search algorithm shows a significant advantage in simulation efficiency, and its simulation time is greatly improved compared with the differential evolution algorithm.

[0115] After in-depth research and experimental verification, this paper applies the differential squirrel search algorithm to the linearization problem of small disturbances at the steady-state point of a turbofan engine. Through the differential squirrel search algorithm, the linearization problem is effectively transformed into the optimization problem of the deviation function between the nonlinear model and the linear model, thereby realizing the accurate linearization modeling of the engine steady-state point. The obtained linearized model and the output of the nonlinear model are fitted in detail. The steady-state error of each physical quantity in the rising and falling processes is less than 1%, the average absolute value error in the rising and falling stages is small, the determination coefficient is greater than 0.95 and generally higher than 0.99, and most of the data points in the scatter plot fit well with the straight line obtained by the least squares method. The results show that the linear model has high accuracy and can accurately reflect the dynamic characteristics of the engine under small disturbances at the steady-state point. At the same time, the powerful global search capability of the squirrel search algorithm and the high computational efficiency of the differential evolution algorithm effectively avoid falling into the local optimal solution in principle, and show good robustness and convergence in dealing with complex aero-engine system models.

Claims

1. A linear design method for an aero-engine model based on a differential squirrel search algorithm, characterized in that: Here are the steps: S1: Establishing the state variable model of turbofan engine S1.1: The input variables of the state variable model of the turbofan engine are the high-pressure rotor speed and the low-pressure rotor speed; the control input is the main fuel flow rate; the output includes the high-pressure rotor speed, the low-pressure rotor speed, the high-pressure compressor outlet total pressure, the low-pressure turbine outlet total pressure and the low-pressure turbine outlet total temperature; S1.2: The state variable model of the turbofan engine is as follows: Where n1 is the low-pressure rotor speed, r / min; n2 is the high-pressure rotor speed, r / min; W f is the main fuel flow, kg / h; P3 is the total pressure at the high pressure compressor outlet, MPa; P5 is the total pressure at the low pressure turbine outlet, MPa; T5 is the total temperature at the low pressure turbine outlet, K; is the rate of change of low-pressure rotor speed; is the rate of change of the high-pressure rotor speed; A is the state matrix; B is the input matrix; C is the output matrix; D is the direct transfer matrix; S1.3: The steps to establish a model for extracting small deviation state variables of a turbofan engine using the differential squirrel search algorithm are as follows: Control input W of high bypass ratio turbofan engine f Keep the control input quantity W at a certain steady-state point and obtain the corresponding steady-state model of the engine. f Make a step input in the range of -10% to 10% to obtain the dynamic model corresponding to the engine; subtract the output of the dynamic model from the steady-state model to obtain the deviation signal Δy(t)=[Δn1(t)Δn2(t)ΔP3(t)ΔP5(t)ΔT5(t)] of the dynamic response of the nonlinear model of the turbofan engine T ; S1.4: Randomly initialize the population through the differential squirrel search algorithm to obtain the A0, B0, C0, D0 matrices, where A0 is the initial value of the state matrix A, B0 is the initial value of the input matrix B, C0 is the initial value of the output matrix C, and D0 is the initial value of the direct transfer matrix D; set the upper and lower limits of each element in the above matrices to [-50, 50], and the initial values ​​of the above matrices are randomly selected within the upper and lower limits; S1.5: Substitute the A0, B0, C0, D0 matrices into the state variable model to obtain the preliminary state variable model of the turbofan engine. The control input W of the preliminary state variable model of the turbofan engine and the nonlinear model of the turbofan engine f The same step response is made to obtain the dynamic response deviation signal of the preliminary linear model of the turbofan engine: S1.6: Based on the dynamic response data Δy(t) and Determine the optimization objective function using the differential squirrel search algorithm The differential squirrel search algorithm is used to calculate the minimum sum of squares of the dynamic response data deviations, thereby obtaining the optimal solution; S2: Design optimization algorithm The differential evolution algorithm and the squirrel search algorithm are integrated to obtain the differential squirrel search algorithm; The Differential Squirrel Search Algorithm (DSSA) is a hybrid intelligent optimization algorithm that combines the search method of the Squirrel Search Algorithm with the optimization process of the Differential Evolution Algorithm. 1) Initialize the population; there are n squirrels in the forest, the position of the i-th squirrel is represented by a vector; the positions of all squirrels are represented by a matrix: Among them, PS i,j represents the jth dimension of the ith squirrel; assigning the initial position of each squirrel in the forest: PS i =PS L +U(0,1)×(PS U -PS L ) Among them, PS L and PS U are the lower and upper bounds of the i-th squirrel in the j-th dimension, respectively. U(0,1) is a uniformly distributed random number in the range [0,1]. 2) Position update: In the absence of natural enemies, the squirrels in the acorn tree will update their position and follow the current best PS ht Move toward the best source; Squirrels in common trees update their location by following squirrels in acorn or hickory trees; Squirrels randomly change paths when foraging in the presence of predators; Squirrels in acorn trees update their location based on the locations of other squirrels; Among them, P avg is the average position of all squirrels in the current population, represents the i+1 generation of squirrels on the acorn tree, represents the i-th generation of squirrels in the hickory tree, d g is the random sliding distance, G c is the sliding constant, rand(0,1) is a random number between 0 and 1, P dp It represents the probability of the presence of natural enemies. If natural enemies exist, squirrels will only move in a small area. 3) Crossover operation of differential evolution algorithm: In order to avoid the possibility of falling into local minimum, the crossover mechanism of differential evolution algorithm is used to increase the diversity of squirrels; crossover operation is performed on the current position and new position of the squirrels; Where i = 1, 2, 3, .... NP, NP is the population size; is the position of the squirrel after the crossover operation; and are the current positions of the squirrel in the acorn tree, new and old; Q is the dimension of the problem, Cr is the crossover rate; j rand ∈[1,Q] is a randomly generated index; rand j represents the jth evaluation of a random number uniformly generated in a given range; The movement of squirrels on ordinary trees to acorn trees or hickory trees is the same as the original squirrel search algorithm, and the crossover operation is consistent with the above formula; finally, the optimal solution for squirrels on the updated hickory tree is obtained; in, is the average of all squirrel positions on the acorn tree; S3: Algorithm simulation verification A small disturbance is added to the nonlinear model of the turbofan engine at a certain steady-state point to obtain the output curve of the nonlinear model of the turbofan engine after the disturbance. The coefficients of the state variable model are obtained by fitting the output curve through the differential squirrel search algorithm. The nonlinear model and state variable model of the turbofan engine are at the steady-state point of H=0, Ma=0, W_f=0.0187kg / s, and the main fuel flow is injected with an appropriate step disturbance. At the same time, the simulation step size and simulation time are set for verification.

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