Aero-engine multivariable robust controller optimization method based on cooperative game

Abstract

The application discloses an aero-engine multivariable robust controller optimization method based on cooperative game, which comprises the following steps: step 1), establishing an aero-engine state space model and designing a controller; step 2), analyzing the performance of the control system based on the controller; step 3), constructing a cooperative game framework based on the performance analysis; step 4), establishing a controller parameter optimization problem based on the cooperative game; and step 5), solving the Pareto optimal problem by using a numerical method to obtain optimal controller parameters. The application introduces the cooperative game theory into the aero-engine robust controller optimization method, designs optimal parameters of the multivariable robust controller, and proposes a parameter optimization framework based on the cooperative game and a scalarization solving method of the Pareto optimal solution.

Description

Technical Field

[0001] This invention belongs to the field of aerospace combined engine control technology, specifically relating to an optimization method for aero-engine multivariable robust controller based on cooperative game theory. Background Technology

[0002] Aero engines play a crucial role in the advancement and development of aircraft and the aviation industry, often referred to as the crown jewel of industry. In recent years, with the continuous improvement of aero engine performance—such as excellent stability and dynamic qualities, sufficiently high steady-state control accuracy, high thrust-to-weight ratio, and high reliability—higher demands have been placed on aero engine control systems. Currently, several classic modern control methods commonly used in aero engines include linear quadratic optimal control and robust control. Linear quadratic optimal control is based on linear system theory, but an aero engine is a highly complex aerodynamic and thermodynamic system with strong nonlinearity. Therefore, when applying modern control theory to aero engines, it is first necessary to linearize the engine's nonlinear model at the steady-state operating point, establishing a state-space model (or state-variable model, SVM), and designing the controller according to the state-space model at each point. However, due to modeling errors, the established state-space model often differs from the actual engine system, and this uncertainty leads to a decline in controller performance. Robust control can maintain good stability and dynamic qualities when the system is subject to uncertainty and external disturbances, and is therefore frequently used in practical aero engine control systems.

[0003] In controller design, multiple adjustable parameters are often used. These parameters are optimized based on engine performance and cost requirements to achieve optimal performance. Multi-objective, multi-parameter optimization problems refer to selecting appropriate design parameters under certain constraints to optimize multiple objectives simultaneously as much as possible. However, the sub-objectives of multi-objective optimization are often contradictory; improving one objective may lead to a decrease in the performance of another or several sub-objectives. In other words, it is impossible for all sub-objectives to reach their optimal values ​​simultaneously; only compromises can be made. Therefore, the most fundamental difference between multi-objective, multi-parameter optimization problems and single-parameter optimization problems is that the solution to a multi-objective, multi-parameter optimization problem is not unique, but rather a set of optimal solutions composed of numerous Pareto optimal solutions. Traditional multi-objective optimization methods based on objective weight allocation, such as the evaluation function method, not only require a lot of prior knowledge and have low computational efficiency, but also cannot converge to the Pareto optimal front. While multi-objective evolutionary algorithms can obtain a large number of Pareto optimal solutions to multi-objective optimization problems at once, they cannot prove whether the solutions are meaningful. Using game theory to solve multi-objective, multi-parameter optimization problems not only gives the cost function a concrete physical meaning, but also provides a clear significance to the calculated Pareto optimal solution. Any solution deviating from the Pareto Optimal will inevitably lead to an increase in the cost for at least one player. Therefore, solutions at Pareto Optimal points can guarantee that the costs of each player remain in a dynamic equilibrium where the costs are relatively minimized. Therefore, this invention proposes an optimization method for a multivariate robust controller of aero-engines based on cooperative game theory. Summary of the Invention

[0004] Objective: To address the multi-parameter optimization problem of aero-engine controllers, this invention proposes a robust multivariable controller optimization method for aero-engines based on cooperative game theory. Considering uncertainties such as modeling errors and parameter perturbations, a state-space model of the aero-engine is established, and fuzzy sets are used to describe these uncertainties, constructing steady-state and transient performance functions of the system. Based on the steady-state and transient performance functions, a controller parameter optimization cost function oriented towards cooperative game theory is constructed. A controller parameter optimization problem based on cooperative game theory is established. The optimal controller parameter settings are obtained by solving the Pareto optimality problem using numerical methods.

[0005] Technical solution: To achieve the above objectives, the technical solution adopted by this invention is as follows:

[0006] A robust multivariable controller optimization method for aero-engines based on cooperative game theory includes the following steps:

[0007] Step 1), establish the state-space model of the aero-engine and design a multivariable robust controller;

[0008] Step 2), construct the steady-state and transient performance functions of the system;

[0009] Step 3) Based on the steady-state and transient performance functions, construct the controller parameter optimization cost function for cooperative game;

[0010] Step 4) Establish a controller parameter optimization problem based on cooperative game theory;

[0011] Step 5) Solve the Pareto optimality problem using numerical methods to obtain the optimal controller parameters.

[0012] Furthermore, the specific steps in step 1) are as follows:

[0013] Step 1.1), considering uncertainties such as modeling errors, parameter perturbations, and external disturbances, the uncertain nonlinear system of the aero-engine is as follows:

[0014]

[0015] in, Indicates time, Indicate the state, x0 is the initial state. Indicates control, This represents an unknown time-varying parameter that includes system uncertainties and input disturbances. Let be a known constant matrix, and ΔA(x,σ,t) and ΔB(x,σ,t) be a matrix and a vector, respectively, depending on x, t, and the unknown time-varying parameter σ. ΔA(·,t) and ΔB(·,t) are continuous and Lebesgue measurable.

[0016] Step 1.2) uses fuzzy sets to describe the time-varying uncertain parameter σ, that is, the uncertain parameters are all bounded and lie within the fuzzy set.

[0017] S i ={(σ i ,μ i (σ i ))|σ i ∈∑ i}, i = 1, 2, ..., p (24)

[0018] Within the range, where ∑ i Given a closed compact set, μ i Let μ be the membership function, and let μ be the membership function. i :Σ i →[0,1];

[0019] Step 1.3): Based on the component-level model of the aero-engine, a fuzzy dynamic system model of the engine is established using the small perturbation method to obtain the coefficient matrices A and B of system (1), and (A,B) is stable. Consider the Riccati equation.

[0020] A T P+PA-2PBR -1 B T P+Q=0 (25)

[0021] Where Q and R are matrices of appropriate dimensions greater than 0. Since (A,B) is stable, there exists a matrix P such that equation (3) holds.

[0022] Construct matrices D(x,σ,t) and E(x,σ,t) such that

[0023]

[0024] Constructing fuzzy number ρ D , satisfy

[0025]

[0026] Where, λ m (λ M ) represents the minimum (maximum) eigenvalue of the corresponding matrix.

[0027] Step 1.4): Design a robust controller for the established fuzzy dynamic system (1) of the aero-engine.

[0028] u(t) = -R -1 B T Px(t)-γ||x(t)|| η R -1 B T Px(t) (28)

[0029] Where γ and η are adjustable real parameters, and their adjustment ranges are γ∈(0,+∞) and η∈[2,+∞), respectively. P is the solution of the Riccati equation (3).

[0030] Furthermore, the specific steps in step 2) are as follows:

[0031] Step 2.1), consider the Lyapunov function.

[0032] V = x T Px (29)

[0033] Differentiating the Lyapunov function with respect to time t yields...

[0034]

[0035] definition Combining equations (4) and (5), define It is possible to obtain any All

[0036]

[0037] Step 2.2), according to Rayleigh's criterion, we have

[0038] λ m (P)||x|| 2 ≤x T Px≤λ M (P)||x|| 2 (32)

[0039] Combining equation (9) with the analysis of system performance, we can obtain the differential inequality.

[0040]

[0041] in, For any t s and τ≥t s Solving equation (11) yields the solution to the corresponding differential inequality.

[0042]

[0043] in, t s It is the system time when the controller starts controlling the system.

[0044] Step 2.3), for any τ>t s Construct the system according to equation (12)

[0045]

[0046]

[0047] Furthermore, the specific steps in step 3) are as follows:

[0048] Step 3.1): Construct an optimization cost function that includes system control variables and system performance based on equations (13) and (14):

[0049]

[0050]

[0051] Where a and b are any chosen positive real numbers; H1(γ,η) contains aγ 2 and Two parts, aγ 2 Indicates the control input quantity. Related to time τ, it represents the overall transient performance of the system; in H2(γ,η), bη 2 Indicates the control input quantity. It is independent of time τ, representing the steady-state performance of the system;

[0052] Step 3.2), construct The cost function is obtained by defuzzifying equations (15) and (16) using the D-operation method in fuzzy set theory.

[0053]

[0054]

[0055] Where, κ1=D[V s ],

[0056] Furthermore, the specific steps in step 4) are as follows:

[0057] Step 4.1) Based on equations (17) and (18), and combining equations (1) and (6), determine the optimization problem of controller parameters γ and η based on cooperative game, that is, the two players in the cooperative game are the two adjustable real parameters γ and η in the controller; according to the adjustment range of the two parameters, determine their decision sets as D1=(0,+∞) and D2=[2,+∞), and the cost functions of the two players γ and η are taken as equations (17) and (18), respectively;

[0058] Step 4.2) Establish the controller parameter optimization problem based on cooperative game theory. First, let the optimization cost function...

[0059] J(γ,η)=α1J1(γ,η)+α2J2(γ,η) (41)

[0060] Where α1+α2=1, γ∈(0,+∞), η∈[2,+∞). Secondly, calculate to satisfy equation (19).

[0061] J(γ * ,η * )≤J(γ,η) (42)

[0062] Pareto optimal solution (γ) * ,η * ), which is the optimal parameter of controller (6).

[0063] Furthermore, the specific steps in step 5) are as follows:

[0064] Step 5.1): Solve for the Pareto optimal solution that satisfies inequality (20) using a numerical method. First, select a set of α1 and α2, calculate the partial derivative of equation (20), and set the partial derivative to 0, i.e.

[0065]

[0066] Find all possible solutions that may contain extreme points;

[0067] Step 5.2), solve the inequality.

[0068]

[0069] Find the minimum point that satisfies equation (21) and inequality (22), which is the Pareto optimal solution.

[0070] Step 5.3): Select a new set of α1 and α2, and repeat steps 5.1)-5.1). Select the optimal γ based on the actual performance requirements of the engine. * η * .

[0071] The beneficial effects of this invention are as follows:

[0072] (1) In response to uncertainties such as modeling errors and external disturbances of aero-engines, a new robust control method is proposed by making full use of the application of fuzzy set theory in the field of uncertainty. Under the condition of bounded uncertainty, the deterministic performance of the system is guaranteed, namely, uniformly bounded and always eventually bounded, which makes up for the technical shortcomings and can guarantee the good working performance of the system.

[0073] (2) For the optimal design of robust control with dual objectives and dual control design parameters, an optimization framework guided by cooperative game theory was developed, providing a new approach to the handling of multi-objective and multi-parameter optimization problems.

[0074] (3) The existence of Pareto optimal solution in the established game was proved, and a scalarization solution method for Pareto optimal solution was proposed, providing an optimization solution idea for similar problems that may exist in the future. Attached Figure Description

[0075] Figure 1 A schematic diagram of the method flow of this invention.

[0076] Figure 2 This is a schematic diagram of the structure and cross-section of the aero-engine in this invention. ① Inlet, ② Fan, ③ Compressor, ④ Combustion chamber, ⑤ Low-pressure turbine, ⑥ High-pressure turbine, ⑦ Outer bypass duct, ⑧ Mixing chamber, ⑨ Afterburner, ⑩ Tail nozzle.

[0077] Figure 3 This refers to the engine's flight envelope.

[0078] Figure 4 The relationship between the Pareto optimal solution and the cost function when α1=α2=0.5.

[0079] Figure 5The game theory-optimized control method and sliding mode control method of this invention are based on the following n... L A diagram showing the comparison of response results.

[0080] Figure 6 The game theory-optimized control method and sliding mode control method of this invention are based on the following n... H A diagram showing the comparison of response results.

[0081] Figure 7 This diagram illustrates a comparison of the ||x|| response results under the control method of this invention and the control method optimized by game theory. Detailed Implementation

[0082] This invention discloses an optimization method for a multivariable robust controller of an aero-engine based on cooperative game theory, comprising the following steps: Step 1), establishing a state-space model of the aero-engine and designing a multivariable robust controller; Step 2), constructing the steady-state and transient performance functions of the system; Step 3), constructing a controller parameter optimization cost function oriented towards cooperative game theory based on the steady-state and transient performance functions; Step 4), establishing a controller parameter optimization problem based on cooperative game theory; Step 5), using numerical methods to solve the Pareto optimality problem to obtain the optimal controller parameters.

[0083] A cooperative game-based optimization method for multivariable robust controllers of aero-engines, such as... Figure 1 As shown, it includes the following steps:

[0084] Step 1), establish the state-space model of the aero-engine and design a multivariable robust controller;

[0085] Step 1.1), considering uncertainties such as modeling errors, parameter perturbations, and external disturbances, the uncertain nonlinear system of the aero-engine is as follows:

[0086]

[0087] in, Indicates time, Indicate the state, x0 is the initial state. Indicates control, This represents an unknown time-varying parameter that includes system uncertainties and input disturbances. Let be a known constant matrix, and ΔA(x,σ,t) and ΔB(x,σ,t) be a matrix and a vector, respectively, depending on x, t, and the unknown time-varying parameter σ. ΔA(·,t) and ΔB(·,t) are continuous and Lebesgue measurable.

[0088] Step 1.2) uses fuzzy sets to describe the time-varying uncertain parameter σ, that is, the uncertain parameters are all bounded and lie within the fuzzy set.

[0089] S i ={(σ i ,μ i (σ i ))|σ i ∈∑ i}, i = 1, 2, ..., p (46)

[0090] Within the range, where ∑ i Given a closed compact set, μ i Let μ be the membership function, and let μ be the membership function. i :Σ i →[0,1];

[0091] Step 1.3): Based on the component-level model of the aero-engine, a fuzzy dynamic system model of the engine is established using the small perturbation method to obtain the coefficient matrices A and B of system (1), and (A,B) is stable. Consider the Riccati equation.

[0092] A T P+PA-2PBR -1 B T P+Q=0 (47)

[0093] Where Q and R are matrices of appropriate dimensions greater than 0. Since (A,B) is stable, there exists a matrix P such that equation (3) holds.

[0094] Construct matrices D(x,σ,t) and E(x,σ,t) such that

[0095]

[0096] Constructing fuzzy number ρ D , satisfy

[0097]

[0098] Where, λ m (λ M ) represents the minimum (maximum) eigenvalue of the corresponding matrix.

[0099] Step 1.4): Design a robust controller for the established fuzzy dynamic system (1) of the aero-engine.

[0100] u(t) = -R -1 B T Px(t)-γ||x(t)|| η R -1 B T Px(t) (50)

[0101] Where γ and η are adjustable real parameters, and their adjustment ranges are γ∈(0,+∞) and η∈[2,+∞), respectively. P is the solution of the Riccati equation (3).

[0102] Step 2), construct the steady-state and transient performance functions of the system;

[0103] Step 2.1), consider the Lyapunov function.

[0104] V = x T Px (51)

[0105] Differentiating the Lyapunov function with respect to time t yields...

[0106]

[0107] definition Combining equations (4) and (5), define It is possible to obtain any All

[0108]

[0109] Step 2.2), according to Rayleigh's criterion, we have

[0110] λ m (P)||x|| 2 ≤x T Px≤λ M (P)||x|| 2 (54)

[0111] Combining equation (9) with the analysis of system performance, we can obtain the differential inequality.

[0112]

[0113] in, For any t s and τ≥t s Solving equation (11) yields the solution to the corresponding differential inequality.

[0114]

[0115] in, t s It is the system time when the controller starts controlling the system.

[0116] Step 2.3), for any τ>t s Construct the system according to equation (12)

[0117]

[0118]

[0119] Step 3) Based on the transient and steady-state performance functions, construct the controller parameter optimization cost function for cooperative game;

[0120] Step 3.1): Construct an optimization cost function that includes system control variables and system performance based on equations (13) and (14):

[0121]

[0122]

[0123] Where a and b are any chosen positive real numbers; H1(γ,η) contains aγ 2 and Two parts, aγ 2 Indicates the control input quantity. Related to time τ, it represents the overall transient performance of the system; in H2(γ,η), bη 2 Indicates the control input quantity. It is independent of time τ, representing the steady-state performance of the system;

[0124] Step 3.2), construct The cost function is obtained by defuzzifying equations (15) and (16) using the D-operation method in fuzzy set theory.

[0125]

[0126]

[0127] Where, κ1=D[V s ],

[0128] Step 4) Establish a controller parameter optimization problem based on cooperative game theory;

[0129] Step 4.1) Based on equations (17) and (18), and combining equations (1) and (6), determine the optimization problem of controller parameters γ and η based on cooperative game, that is, the two players in the cooperative game are the two adjustable real parameters γ and η in the controller; according to the adjustment range of the two parameters, determine their decision sets as D1=(0,+∞) and D2=[2,+∞), and the cost functions of the two players γ and η are taken as equations (17) and (18), respectively;

[0130] Step 4.2) Establish the controller parameter optimization problem based on cooperative game theory. First, let the optimization cost function...

[0131] J(γ,η)=α1J1(γ,η)+α2J2(γ,η) (63)

[0132] Where α1+α2=1, γ∈(0,+∞), η∈[2,+∞). Secondly, calculate to satisfy equation (19).

[0133] J(γ * ,η * )≤J(γ,η) (64)

[0134] Pareto optimal solution (γ) * ,η * ), which is the optimal parameter of controller (6).

[0135] Step 5) Solve the Pareto optimality problem using numerical methods to obtain the optimal controller parameters;

[0136] Step 5.1): Solve for the Pareto optimal solution that satisfies inequality (20) using a numerical method. First, select a set of α1 and α2, calculate the partial derivative of equation (20), and set the partial derivative to 0, i.e.

[0137]

[0138] Find all possible solutions that may contain extreme points;

[0139] Step 5.2), solve the inequality.

[0140]

[0141] Find the minimum point that satisfies equation (21) and inequality (22), which is the Pareto optimal solution.

[0142] Step 5.3): Select a new set of α1 and α2, and repeat steps 5.1)-5.1). Select the optimal γ based on the actual performance requirements of the engine. * η * .

[0143] Example

[0144] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0145] The present invention can be better understood from the following embodiments. However, those skilled in the art will readily understand that the specific material ratios, process conditions, and results described in the embodiments are for illustrative purposes only and should not, and will not, limit the invention as described in detail in the claims.

[0146] by Figure 2Taking a certain type of turbofan engine as an example, based on the component-level model of turbofan engine aerodynamics and thermodynamics, the state variable is selected as the low-pressure rotor speed n. L and high-pressure rotor speed n H The main control variable is the fuel flow rate W in the combustion chamber. f And the area of ​​the tailpipe throat is A8. For example... Figure 3 As shown, 18 operating points are selected within the engine's operating envelope. Each point can be written as...

[0147]

[0148] according to Figure 3 The average value of the 18 working points determines the coefficient matrix A and B of the nominal point, i.e.

[0149]

[0150] The calculation result is

[0151]

[0152] Fuzzy sets w1 and w2 are used to characterize the degree to which each point within the envelope deviates from the nominal point, i.e., the magnitude of the system uncertainty.

[0153] ΔA=w1A,ΔB=w1B (4)

[0154] The fuzzy sets w1 and w2 are described using triangular membership functions, i.e.

[0155]

[0156]

[0157] Where σ1 and σ2 are determined by the elements of the coefficient matrix at each operating point, i.e.

[0158]

[0159]

[0160] Calculation h 1 = -1, h 2 = -0.8.

[0161] Select The calculated values ​​are κ1 = 0.1387, κ2 = 0.0215, and κ3 = 0.1190. Based on these parameters, the Pareto optimal solution is calculated by selecting α1 = 0.1, 0.2, ..., 0.9 respectively. Figure 4 The relationship between the Pareto optimal solution and the cost function is shown when α1 = 0.5.

[0162] Figures 4-7 The response curves of the state and control variables after applying an initial disturbance to the system are shown, and a sliding mode controller is compared with the controller designed in this invention. Simulation results show that when the system is disturbed, the system state under the sliding mode controller exhibits overshoot and a slower response time, while under the game-theoretic optimized controller, the system has no overshoot and a faster response time, entering the eventually bounded range in approximately 0.2 seconds. Figure 7 It can be seen that the cumulative error of the controller after game-theoretic optimization is less than that of the controller with randomly given parameters. Therefore, it can be concluded that for the nonlinear system of uncertain fuzzy aero-engines, the controller designed in this invention based on the cooperative game-theoretic optimization method outperforms the sliding mode controller.

Claims

1. A method for optimizing a multivariable robust controller for aero-engines based on cooperative game theory, characterized in that: Includes the following steps: Step 1), establish the state-space model of the aero-engine and design a multivariable robust controller; Step 2), construct the steady-state and transient performance functions of the system; Step 3) Based on the steady-state and transient performance functions, construct the controller parameter optimization cost function for cooperative game; Step 4), establish a controller parameter optimization problem based on cooperative game theory; Step 5) Solve the Pareto optimality problem using numerical methods to obtain the optimal controller parameters; The specific steps for establishing the state-space model of the aero-engine and designing the controller in step 1) are as follows: Step 1.1), considering uncertainties such as modeling errors, parameter perturbations, and external disturbances, the uncertain nonlinear system of the aero-engine is... ； in, Indicates time, Indicates state, This is the initial state. Indicates control, This represents an unknown time-varying parameter that includes system uncertainties and input disturbances; , Given a constant matrix, , They depend on , and unknown time-varying parameters Matrix and vector; , It is continuous and , It is measurable by Lebesgue; Step 1.2) uses fuzzy sets to describe time-varying uncertain parameters. That is, the uncertain parameters are all bounded and lie in the fuzzy set. ； Within the range, Given a closed compact set, Let be the membership function, and have ; Step 1.3): Based on the component-level model of the aero-engine, a fuzzy dynamic system model of the engine is established using the small perturbation method to obtain the system... coefficient matrix , ,and It is stable; consider the Riccati equation. ； in , For a matrix of appropriate dimension that is greater than 0, since Stable, with a matrix. Make the formula Established; Constructing a matrix , Make ； Constructing fuzzy numbers , , satisfy ； in, The minimum or maximum eigenvalue of the corresponding matrix; Step 1.4): Design a robust controller for the established fuzzy dynamic system of the aero-engine. ； in , These are adjustable real parameters, and their adjustment ranges are respectively , , This is a solution to the Riccati equation; The specific steps for constructing the steady-state and dynamic performance functions of the system in step 2) are as follows: Step 2.1), consider the Lyapunov function. ； Plot the Lyapunov function against time Taking the derivative, we get ； definition Combined -Mode ,definition , , to obtain any All have Step 2.2), according to Rayleigh's criterion, we have ； Combined The system performance was analyzed, and the differential inequality was obtained. ； in, For any as well as Solving equation The solution to the corresponding differential inequality is: ； in, , , , It is the system time when the controller starts controlling the system; Step 2.3), for any According to the formula Construction System ； ； The specific steps for constructing the controller parameter optimization function for cooperative game based on transient and steady-state performance functions in step 3) are as follows: Step 3.1), according to formula -Mode Construct an optimal cost function that includes system control variables and system performance: ； ； in, , Let be any positive real number selected; Include and Two parts, Indicates the control input quantity. With time Correlation indicates the overall transient performance of the system; middle Indicates the control input quantity. With time Irrelevant indicates the steady-state performance of the system; Step 3.2), Construction The D-operation method in fuzzy set theory is used to apply the formula... -Mode Defuzzification yields the cost function, i.e. ； ； in, , , .

2. The optimization method for a multivariable robust controller of an aero-engine based on cooperative game theory as described in claim 1, characterized in that: The specific steps for establishing the controller parameter optimization problem based on cooperative game theory in step 4) are as follows: Step 4.1), based on formula , Combined with formula Determine the controller parameters based on cooperative game theory. and The optimization problem, i.e., a cooperative game in which two players are the controllers... , These two adjustable real parameters; their decision sets are determined based on the adjustment ranges of the two parameters. , , and The cost functions for the two players are respectively taken as equations. , ; Step 4.2), establish the controller parameter optimization problem based on cooperative game theory; first, let the optimization cost function... ； in , , Secondly, the calculation makes the formula satisfy ； Pareto optimal solution This refers to the optimal parameters of the controller.

3. The optimization method for a multivariable robust controller of an aero-engine based on cooperative game theory according to claim 2, characterized in that: The specific steps for obtaining the optimal controller parameters by numerically solving the Pareto optimality problem in step 5) are as follows: Step 5.1) Solve the inequality using numerical methods. The Pareto optimal solution is satisfied; first, select a set and , for example Find the partial differential and set the partial differential to 0, i.e. ； Find all possible solutions that may contain extreme points; Step 5.2), solve the inequality. ； Find the satisfying formula Sum of inequalities The minimum point is the Pareto optimal solution; Step 5.3), select a new group. and Repeat steps 5.1)-5.2); select the optimal one based on the actual performance requirements of the engine. , .

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