Variable cycle engine component-level model solving method based on adaptive Newton method

Abstract

The invention provides a variable cycle engine component-level model solving method based on an adaptive Newton method, and belongs to the technical field of variable cycle engine mathematical modeling and simulation. When iteration variables of the Newton method are updated, regularization parameters are introduced, and the condition of iteration solution divergence caused by Jacobian matrix singularity is avoided; according to the method, the judgment parameters are calculated according to the residual errors output by the variable cycle engine component-level model, whether the Jacobian matrix needs to be updated or not is judged in a self-adaptive mode, and therefore the updating frequency of the Jacobian matrix is reduced; compared with a conventional Newton method, the method for solving the part-level model of the variable-cycle engine, provided by the invention, can greatly improve the real-time performance and the convergence range of iterative solution.

Description

Technical Field

[0001] The invention provides a method for solving a variable cycle engine component level model based on an adaptive Newton method, and belongs to the technical field of mathematical modeling and simulation of aero-engines. Background Art

[0002] Establishing an aircraft engine model with high accuracy, good real-time performance and strong adaptability is the basis for realizing model-based control. By using the aircraft engine model under airborne conditions, aircraft engine performance parameter estimation, fault diagnosis and health management, as well as the design and verification of advanced control algorithms can be realized. The nonlinear component-level model of the aircraft engine is established based on the aerodynamic thermodynamic principles followed by the various components of the aircraft engine during their working process. That is, the models of the various components of the aircraft engine are first constructed, and then combined into a complete engine model according to the common working conditions of the various components. It has the advantages of high accuracy and strong versatility and can be used as an airborne model.

[0003] Variable cycle engines are an important development direction for the next generation of aircraft engines. They introduce variable geometry components, change their own cycle parameters, and achieve optimal performance within the full envelope. Compared with traditional turbofan engines or turbojet engines, variable cycle engines have more components and more complex working relationships between components. Therefore, the component-level model of variable cycle engines has stronger nonlinearity and higher computational complexity. Specifically, when using the Newton iteration method to solve the equilibrium equations describing the working relationship between the components of the variable cycle engine, the number of iterations is greater. At the same time, due to the local convergence characteristics of the Newton iteration method, if the iterative variables are far away from the true value, the Jacobian matrix will be singular, which will lead to non-convergence or computational divergence in the solution process.

[0004] Existing patents show that the real-time performance of the Newton method iterative solution of the engine component-level model can be improved by improving the Newton method. The patent with publication number CN112507475A discloses a method for solving the aircraft engine component-level model based on the modified Newton method, and the convergence speed is improved by modifying the convergence direction of the Newton method. For the variable cycle engine component-level model, although the modified Newton method can improve the convergence speed, the improvement in real-time performance is limited. The main reason is that the Jacobian matrix still needs to be calculated by the difference algorithm in each iteration. In addition, due to the local quadratic convergence characteristics of the Newton method, when the Jacobian matrix is ​​singular, the iterative calculation will diverge, resulting in solution failure. Therefore, in view of the above problems, the present invention proposes a new variable cycle engine component-level model solution method which is of great significance. Summary of the invention

[0005] In view of the problems existing in the existing Newton iteration method for solving the variable cycle engine component level model, the present invention proposes a variable cycle engine component level model solving method based on the adaptive Newton method, which improves the real-time performance and convergence range of the component level model solution.

[0006] To achieve the above objectives, the concept and technical solution of the present invention are implemented as follows:

[0007] The basic concept of the present invention is to introduce a regularization parameter and change the diagonal elements of the Jacobian matrix to ensure the existence of an updated amount of iterative variables, taking into account the divergence of the solution caused by the singularity of the Jacobian matrix; considering that the calculation of the Jacobian matrix in the Newton method is very time-consuming and a differential algorithm is required to be used for the solution in each iteration, the present invention defines a judgment parameter to adaptively determine whether an update calculation of the Jacobian matrix is ​​required, thereby reducing the update frequency of the Jacobian matrix and improving the solution speed.

[0008] Based on the above basic concept, the technical solution proposed by the present invention is a method for solving the variable cycle engine component level model based on the adaptive Newton method, comprising the following steps:

[0009] Step 1: Establish a variable cycle engine component-level model;

[0010] Step 2: Initialize various parameters and variables of the adaptive Newton method, including the iteration variable X k , Jacobian matrix J k , judgment parameter r k , number of iterations k, residual accuracy threshold δ, regularization parameter λ k and constants p1 and p2;

[0011] Step 3: Based on the iteration variable Substitute into the variable cycle engine component level model and calculate the model residual vector and the bi-norm of the model residual vector ||F(X k )||2, if||F(X k )||2 is less than the residual accuracy threshold δ, the iteration is terminated, otherwise it goes to step 4;

[0012] Step 4: Update the iteration variable X for the next iteration k+1 , specifically:

[0013] Step 4.1: Calculate the update amount ΔX of the iteration variable k =-(J k +λ k I) -1 F(X k );

[0014] Step 4.2: Calculate the iteration variable X for the next iterationk+1 =X k +ΔX k ;

[0015] Step 5: Calculate the judgment parameter r k , specifically:

[0016] Step 5.1: Calculate the residual prediction reduction pred k =||F(X k )||2-||F(X k )+J k ΔX k ||2;

[0017] Step 5.2: Set the iteration variable X for the next iteration k+1 , substitute back the variable cycle engine component level model and calculate the residual vector and the bi-norm of the model residual vector ||F(X k+1 )||2;

[0018] Step 5.2: Calculate the actual decrease in residuals k =||F(X k )||2-||F(X k+1 )||2;

[0019] Step 5.3: Calculate the judgment parameter r k =ared k / pred k ;

[0020] Step 6: According to the judgment parameter r k The size of the adaptive update Jacobian matrix J k+1 and the regularization parameter λ k+1 Specifically, if r k is not less than the constant p1, then the Jacobian matrix J of the current iteration is k As the Jacobian matrix J of the next iteration k +1 , the regularization parameter λ of the current iteration k As the regularization parameter λ for the next iteration k+1 ; if r k If it is less than the constant p1, the Jacobian matrix J is recalculated. k+1 and the regularization parameter λ k+1 ;

[0021] Step 7: Update the iteration variable X k+1 As the iteration variable for the next iteration, set k = k + 1 and return to step 3 until the residual accuracy requirement is met.

[0022] Furthermore, the step 6 recalculates the regularization parameter λ k+1 The specific method is as follows:

[0023] λ k+1 =p2||F(X k )||2

[0024] Where p2 is a constant, ||F(X k )||2 is the bi-norm of the model residual vector at the kth iteration.

[0025] Compared with the prior art, the present invention has the following beneficial effects:

[0026] 1. The present invention introduces a regularization parameter, which can avoid the divergence of iterative solution when the Jacobian matrix is ​​singular, and improves the convergence range of the conventional Newton iterative algorithm.

[0027] 2. The judgment parameter proposed in the present invention can determine the quality of the current iterative solution direction, and adaptively decide whether to update the Jacobian matrix, thereby avoiding the need to solve the Jacobian matrix in each iteration, thereby improving the real-time performance of the iterative solution. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] Figure 1 It is a flow chart of the variable cycle engine component level model solving method provided by the present invention;

[0029] Figure 2 It is a fuel quantity simulation diagram of a variable cycle engine component level model provided by the present invention;

[0030] Figure 3 It is a component call frequency diagram of a variable cycle engine component level model provided by the present invention;

[0031] Figure 4 It is a simulation time consumption diagram of the variable cycle engine component level model provided by the present invention; DETAILED DESCRIPTION

[0032] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0033] See also Figure 1 , which shows a flow chart of a variable cycle engine component level model solving method based on the adaptive Newton method provided by the present invention, which specifically includes the following steps:

[0034] Step 1: Establish a variable cycle engine component-level model;

[0035] Step 2: Initialize various parameters and variables of the adaptive Newton method, including the iteration variable X k , Jacobian matrix J k , judgment parameter r k , number of iterations k, residual accuracy threshold δ, regularization parameter λ k and constants p1 and p2;

[0036] Step 3: Based on the iteration variable Substitute into the variable cycle engine component level model and calculate the model residual vector and the bi-norm of the model residual vector ||F(X k )||2, if||F(X k )||2 is less than the residual accuracy threshold δ, the iteration is terminated, otherwise it goes to step 4;

[0037] Step 4: Update the iteration variable X for the next iteration k+1 , specifically:

[0038] Step 4.1: Calculate the update amount ΔX of the iteration variable k =-(J k +λ k I) -1 F(X k );

[0039] Step 4.2: Calculate the iteration variable X for the next iteration k+1 =X k +ΔX k ;

[0040] Step 5: Calculate the judgment parameter r k , specifically:

[0041] Step 5.1: Calculate the residual prediction reduction pred k =||F(X k )||2-||F(X k )+J k ΔX k ||2;

[0042] Step 5.2: Set the iteration variable X for the next iteration k+1 , substitute back the variable cycle engine component level model and calculate the residual vector and the bi-norm of the model residual vector ||F(X k+1 )||2;

[0043] Step 5.2: Calculate the actual decrease in residuals k =||F(X k )||2-||F(Xk+1 )||2;

[0044] Step 5.3: Calculate the judgment parameter r k =ared k / pred k ;

[0045] Step 6: According to the judgment parameter r k The size of the adaptive update Jacobian matrix J k+1 and the regularization parameter λ k+1 Specifically, if r k is not less than the constant p1, then the Jacobian matrix J of the current iteration is k As the Jacobian matrix J of the next iteration k +1 , the regularization parameter λ of the current iteration k As the regularization parameter λ for the next iteration k+1 ; if r k If it is less than the constant p1, the Jacobian matrix J is recalculated. k+1 and the regularization parameter λ k+1 ;

[0046] Step 7: Update the iteration variable X k+1 As the iteration variable for the next iteration, set k = k + 1 and return to step 3 until the residual accuracy requirement is met.

[0047] Furthermore, the step 6 recalculates the regularization parameter λ k+1 The specific method is as follows:

[0048] λ k+1 =p2||F(X k )||2

[0049] Where p2 is a constant, ||F(X k )||2 is the bi-norm of the model residual vector at the kth iteration.

[0050] In order to verify the effectiveness of the variable cycle engine component-level model solving method based on the adaptive Newton method designed in the present invention, the solving efficiency of the Newton method and the adaptive Newton method for two sets of nonlinear equations and the variable cycle engine component-level model is compared.

[0051] Two sets of nonlinear equations are provided below, and the solution efficiency is compared.

[0052] Example 1: Solving a nonlinear system of equations

[0053]

[0054] Take the initial value of the iteration variable as [2, 4] T, its exact solution is x1=1; x2=2

[0055] Example 2: Solving a nonlinear system of equations

[0056]

[0057] Take the initial value of the iteration variable as [-2, 1.5, 1.5] T , the exact solution is x1 = -0.196; x2 = 0.982; x3 = -0.274

[0058] As shown in Table 1 below, the number of iterations of the adaptive Newton method and the conventional Newton method for solving nonlinear equations is compared. For solving simple nonlinear equations, the number of iterations of the adaptive Newton method is reduced by nearly 57%. For complex nonlinear equations, when the initial value of the iteration variable is far from the exact solution, the conventional Newton method cannot complete the calculation, but the adaptive Newton method can find the exact solution. As shown in Table 2 below, when the calculation is repeated 100,000 times, the adaptive Newton method and the conventional Newton method are compared in the iterative events of the nonlinear equation group in Example 1. For solving simple nonlinear equations, the adaptive Newton method reduces the number of events by 48.25% compared with the conventional Newton method.

[0059] Table 1 Comparison of the number of iterations of adaptive Newton method and conventional Newton method for iterative solution of conventional equations (p1 = 0.5, p2 = 10 -2 )

[0060] Adaptive Newton method iteration number Conventional Newton method iteration number Example 1 12 28 Example 2 36 Divergence

[0061] Table 2 Comparison of iteration time of adaptive Newton method and conventional Newton method for iterative solution of nonlinear equations (repeated 100,000 times, p1 = 0.5, p2 = 10 -2 )

[0062] Adaptive Newton method iteration time Conventional Newton's method iteration time Example 1 0.475s 0.918s Example 2 1.364s Divergence

[0063] For solving the variable cycle engine component level model, given the variable cycle engine component level model fuel quantity change curve as follows Figure 2 As shown in the figure, the adaptive Newton method and the conventional Newton method are used to solve the variable cycle engine component level model, and the corresponding component call times are compared. Figure 3 As shown, the corresponding time consumption is Figure 4 As shown in the figure, the adaptive Newton method calls components 6 times on average, and the conventional Newton method calls components 44 times on average. The adaptive Newton method reduces the number of component calls by 86.36% on average. The adaptive Newton method takes an average of 3.7981ms, and the conventional Newton method takes an average of 26.3892ms. The adaptive Newton method reduces the time consumption by 85.61% on average.

[0064] The above description is only a preferred specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any technician familiar with the technical field can make equivalent replacements or changes according to the technical scheme and inventive concept of the present invention within the technical scope disclosed by the present invention, which should be covered by the protection scope of the present invention.

Claims

1. A method for solving a variable cycle engine component level model based on an adaptive Newton method, characterized in that: The specific steps include: Step 1: Establish a variable cycle engine component-level model; Step 2: Initialize various parameters and variables of the adaptive Newton method, including the iteration variable X k , Jacobian matrix J k , judgment parameter r k , number of iterations k, residual accuracy threshold δ, regularization parameter λ k and constants p1 and p2; Step 3: Based on the iteration variable Substitute into the variable cycle engine component level model and calculate the model residual vector and the bi-norm of the model residual vector ||F(X k )||2, if||F(X k )||2 is less than the residual accuracy threshold δ, the iteration is terminated, otherwise it goes to step 4; Step 4: Update the iteration variable X for the next iteration k+1 , specifically: Step 4.1: Calculate the update amount ΔX of the iteration variable k =-(J k +λ k I) -1 F(X k ); Step 4.2: Calculate the iteration variable X for the next iteration k+1 =X k +ΔX k ; Step 5: Calculate the judgment parameter r k , specifically: Step 5.1: Calculate the residual prediction reduction pred k =||F(X k )||2-||F(X k )+J k ΔX k ||2; Step 5.2: Set the iteration variable X for the next iteration k+1 , substitute back the variable cycle engine component level model and calculate the residual vector and the bi-norm of the model residual vector ||F(X k+1 )||2; Step 5.2: Calculate the actual decrease in residuals k =||F(X k )||2-||F(X k+1 )||2; Step 5.3: Calculate the judgment parameter r k =ared k / pred k ; Step 6: According to the judgment parameter r k The size of the adaptive update Jacobian matrix J k+1 and the regularization parameter λ k+1 Specifically, if r k is not less than the constant p1, then the Jacobian matrix J of the current iteration is k As the Jacobian matrix J of the next iteration k+1 , the regularization parameter λ of the current iteration k As the regularization parameter λ for the next iteration k+1 ; if r k If it is less than the constant p1, the Jacobian matrix J is recalculated. k+1 and the regularization parameter λ k+1 ; Step 7: Update the iteration variable X k+1 As the iteration variable for the next iteration, set k = k + 1 and return to step 3 until the residual accuracy requirement is met.

2. A variable cycle engine component level model solving method based on adaptive Newton method according to claim 1, characterized in that: The step 6 recalculates the regularization parameter λ k+1 The specific method is as follows: k+1 =p2||F(X k )||2 Where p2 is a constant, ||F(X k )||2 is the bi-norm of the model residual vector at the kth iteration.

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