Aircraft attitude control calculation method based on linear quadratic problem dual solution

Abstract

The application belongs to the field of aircraft control, and discloses an aircraft attitude control calculation method based on a linear quadratic problem dual solution. The control calculation method first establishes an aircraft attitude control loop linear model, and then sets up a finite time linear quadratic optimal control problem description with different configurations according to an attitude control target. Then, the original optimal control problem is converted into an unconstrained variation problem through linear quadratic optimal control problem duality processing, boundary conditions corresponding to the desired attitude motion of the aircraft in the problem configuration are introduced, and a linear homogeneous equation two-point boundary value problem is obtained. For the two-point boundary value problem, the optimal solution is obtained through a finite number of iteration calculations for attitude control. The control calculation method adopts a finite number of calculations to obtain the optimal solution, realizes efficient solution of attitude control instructions for different flight tasks, solves the problems of high initial value setting requirement, small convergence domain and easy failure of optimization iteration calculation of the shooting method, and has engineering practical value.

Description

Technical Field

[0001] This invention belongs to the field of aircraft flight control, specifically relating to an aircraft attitude control calculation method based on the dual solution of a linear quadratic problem. Background Technology

[0002] Flight control is a key technology in aircraft systems, crucial for ensuring safe flight and enabling the completion of complex missions. Linear quadratic optimal control, a cornerstone of modern control theory, has significant influence in the field of automatic control. Its core idea is to construct a quadratic cost function for linear control systems and design a state feedback controller based on optimal control theory. It not only provides optimal control strategies for linear systems but also demonstrates good performance in nonlinear systems as an efficient approximate optimal control method, thus finding widespread engineering applications in flight control. The finite-time linear quadratic optimal control problem for flight control is typically characterized as an optimal control problem with a free terminal state. However, for certain flight control tasks, such as finite-time attitude stabilization or adjustment, the terminal values ​​of the aircraft's attitude variables must be given. In such cases, the classical Riccati matrix differential equation solution method involves solving multiple matrix differential equations, making the solution cumbersome and complex, which is unfavorable for attitude control applications. Traditional boundary value problem-solving methods based on state variables, control variables, and costate variables suffer from small convergence regions and time-consuming computational iterations, affecting the efficient solution of optimal attitude control commands.

[0003] The duality principle, through "perspective transformation," reconstructs the problem and reveals the inherent symmetry of the optimal dynamic control structure. It demonstrates strong engineering application value in engineering optimization and machine learning, and also provides a new approach for the design and implementation of finite-time-domain linear quadratic optimal attitude control for aircraft. Therefore, it is essential to develop a computational method for aircraft attitude control based on the dual solution of linear quadratic problems. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide a calculation method for aircraft attitude control based on the dual solution of linear quadratic problems, so as to overcome the defects of the prior art.

[0005] The aircraft attitude control calculation method based on the dual solution of a linear quadratic problem of the present invention includes the following steps:

[0006] Step 1. For the aircraft object, establish a linearized dynamic model of the aircraft attitude control loop using the small perturbation principle linearization method;

[0007] The system dynamic equations obtained by linearization using the small perturbation principle are as follows:

[0008] ;

[0009] in, yes dimensional state vector, Indicates pitch angle, Indicates pitch attitude angular velocity, The first derivative of the state vector with respect to time. yes dimensional control vector, Indicates the elevator deflection angle. yes 3D matrix yes 3D matrix;

[0010] Step 2. Based on the aircraft attitude control objective, establish a finite-time linear quadratic optimal control problem description for the corresponding configuration, including a free terminal state, a partially determined terminal state, and setting terminal state performance indices. Specifically, given a partially determined terminal state, for a specified terminal attitude and angular velocity, the P1 configuration is defined as follows:

[0011] ;

[0012] in, and They are Wei and A positive definite symmetric matrix and These are the given initial time and terminal time, respectively. Given an initial state value, Given a terminal state value;

[0013] Step 3. Using the optimal control dual solution method of linear quadratic form, the corresponding two-point boundary value problem of linear homogeneous equation is derived, and the optimal solution is obtained by efficient calculation in 3 iterations;

[0014] Step 4. Apply the optimal solution to the aircraft attitude control.

[0015] Furthermore, the method for solving the duality of linear quadratic optimal control includes the following steps:

[0016] S10. Based on the duality principle, the finite-time linear quadratic optimal control problem of the aircraft attitude control loop dynamics model is transformed into an unconstrained variational problem; including the following steps:

[0017] S11. Based on the duality principle of convex optimization, we introduce costate variables corresponding to the flight state variables to obtain the maxima and minima problem of the Lagrange functional of the augmented aircraft dynamics equations.

[0018] Introduction of state variables corresponding Costate variables This leads to the minimax problem of Lagrange functionals:

[0019] ;

[0020] S12. For the maxima and minima problem of the Lagrange functional, solve the inner loop minimization problem to obtain the optimal flight state solution and the optimal flight control solution expressed by costate variables;

[0021] for First, solve the inner loop minimization problem. The derivation yields , The optimal solutions are as follows:

[0022] ;

[0023] S13. Substituting the optimal flight state solution and the optimal flight control solution, we obtain the unconstrained variational problem with respect to the costate variables;

[0024] Will , Substitute the optimal solution This yields the unconstrained variational problem:

[0025] ;

[0026] S20. Based on the variational principle, the Euler-Lagrange equation, which is only related to the costate variables, is derived. Boundary conditions corresponding to the aircraft state settings in the original optimal control problem configuration are introduced, and the problem is transformed into a two-point boundary value problem of linear homogeneous equations.

[0027] The Euler-Lagrange equation that maximizes the performance index is obtained:

[0028] ;

[0029] in, The second derivative of the costate variable with respect to time; The first derivative of the costate variable with respect to time; by The corresponding boundary conditions are derived as follows:

[0030] ;

[0031] S30. For the two-point boundary value problem of linear homogeneous equations, the aircraft state solution and control solution of the finite-time linear quadratic optimal control problem with different configurations are obtained through three iterations; including the following steps:

[0032] S31. Given three sets of initial guesses for costate variables, obtain the numerical solutions to the corresponding initial value problems based on different initial boundary conditions;

[0033] For the two-point boundary value problem of the transformed linear homogeneous equation, based on the principle of superposition of solutions to linear differential equations, three initial conjectures of costate variables are given: The number of initial guesses introduced is one more than the dimension of the costate variables; the corresponding initial boundary conditions are used to derive... initial value The corresponding solutions are respectively The final solution is:

[0034] ;

[0035] in, For the coefficients to be determined, satisfying Note the initial guess. The selection condition is: None exists. , making ;in, It is a scalar parameter;

[0036] S32. Calculate the coefficients of the solution based on the terminal boundary conditions to determine the final costate variable solution;

[0037] Based on the terminal boundary conditions, the coefficients are solved as follows:

[0038] ;

[0039] in, for A row vector with dimensions and elements all having a value of 1;

[0040] S33. Reconstruct the attitude state solution and attitude control solution based on costate variables.

[0041] get Then, using , Optimal solution and By understanding the relationship between the state and control solutions of the P1 configuration problem, the optimal solution can be applied to the attitude control of the aircraft.

[0042] The aircraft attitude control calculation method based on the dual solution of a linear quadratic problem, as proposed in this invention, first establishes a cyclic model of the aircraft attitude control loop and sets finite-time linear quadratic optimal control problem descriptions for different configurations according to the attitude control objective. Then, through dual processing of the linear quadratic optimal control problem, the original optimal control problem is transformed into an unconstrained variational problem. Boundary conditions corresponding to the desired attitude motion of the aircraft in the problem configuration are introduced, resulting in a two-point boundary value problem of linear homogeneous equations. For the two-point boundary value problem, the optimal solution is obtained through a finite number of iterations for attitude control.

[0043] The aircraft attitude control calculation method based on the dual solution of linear quadratic problems of the present invention obtains the optimal solution by performing a finite number of calculations, realizing the efficient solution of attitude control commands for different flight missions. In particular, it calculates the independence between the values ​​based on a finite number of calculations and adopts a parallel calculation scheme, thereby improving the efficiency of flight control command solution. It avoids the problems of high requirements for initial value setting, small convergence domain and easy failure of optimization iteration calculation in the target shooting method, and has practical engineering value. Attached Figure Description

[0044] Figure 1 This is a flowchart illustrating the specific implementation process of the present invention;

[0045] Figure 2a The calculation results obtained from the aircraft attitude control P1 configuration in Example 1 are compared with the calculation results from the target shooting method. Curve comparison chart;

[0046] Figure 2b The calculation results obtained from the aircraft attitude control P1 configuration in Example 1 are compared with the calculation results from the target shooting method. Curve comparison chart;

[0047] Figure 2c The calculation results obtained from the aircraft attitude control P1 configuration in Example 1 are compared with the calculation results from the target shooting method. Curve comparison chart;

[0048] Figure 3a The calculation results obtained from the P2 configuration of the aircraft attitude control in Example 2 are compared with the calculation results from the target shooting method. Curve comparison chart;

[0049] Figure 3b The calculation results obtained from the P2 configuration of the aircraft attitude control in Example 2 are compared with the calculation results from the target shooting method. Curve comparison chart;

[0050] Figure 3c The calculation results obtained from the P2 configuration of the aircraft attitude control in Example 2 are compared with the calculation results from the target shooting method. Curve comparison chart;

[0051] Figure 4a The calculation results obtained from the P3 configuration of the aircraft attitude control in Example 3 are compared with the calculation results from the target shooting method. Curve comparison chart;

[0052] Figure 4b The calculation results obtained from the P3 configuration of the aircraft attitude control in Example 3 are compared with the calculation results from the target shooting method. Curve comparison chart;

[0053] Figure 4c The calculation results obtained from the P3 configuration of the aircraft attitude control in Example 3 are compared with the calculation results from the target shooting method. Curve comparison chart;

[0054] Figure 5a The calculation results obtained from the P4 configuration of the aircraft attitude control in Example 4 are compared with the calculation results from the target shooting method. Curve comparison chart;

[0055] Figure 5b The calculation results obtained from the P4 configuration of the aircraft attitude control in Example 4 are compared with the calculation results from the target shooting method. Curve comparison chart;

[0056] Figure 5c The calculation results obtained from the P4 configuration of the aircraft attitude control in Example 4 are compared with the calculation results from the target shooting method. Curve comparison chart.

[0057] In the picture, Indicates pitch angle, Indicates pitch attitude angular velocity, Indicates the elevator deflection angle. Indicates time. Detailed Implementation

[0058] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0059] like Figure 1 As shown, the aircraft attitude control calculation method based on the dual solution of a linear quadratic problem of the present invention includes the following steps:

[0060] Step 1. For the aircraft object, establish a linearized dynamic model of the aircraft attitude control loop using the small perturbation principle linearization method;

[0061] The system dynamic equations obtained by linearization using the small perturbation principle are as follows:

[0062] ;

[0063] in, yes dimensional state vector, Indicates pitch angle, Indicates pitch attitude angular velocity, The first derivative of the state vector with respect to time. yes dimensional control vector, Indicates the elevator deflection angle. yes 3D matrix yes 3D matrix;

[0064] Step 2. Based on the aircraft attitude control objective, establish a finite-time linear quadratic optimal control problem description for the corresponding configuration, including a free terminal state, a partially determined terminal state, and setting terminal state performance indices. Specifically, given a partially determined terminal state, for a specified terminal attitude and angular velocity, the P1 configuration is defined as follows:

[0065] ;

[0066] in, and They are Wei and A positive definite symmetric matrix and These are the given initial time and terminal time, respectively. Given an initial state value, Given a terminal state value;

[0067] Step 3. Using the optimal control dual solution method of linear quadratic form, the corresponding two-point boundary value problem of linear homogeneous equation is derived, and the optimal solution is obtained by efficient calculation in 3 iterations;

[0068] Step 4. Apply the optimal solution to the aircraft attitude control.

[0069] Example 1: This example addresses a finite-time control task for the inner-loop pitch attitude angle of an aircraft. First, the system dynamics equations are obtained using the linearization method based on the small disturbance principle:

[0070] ;

[0071] in, yes dimensional state vector, Indicates pitch angle, Indicates pitch attitude angular velocity, The first derivative of the state vector with respect to time. yes dimensional control vector, Indicates the elevator deflection angle. yes 3D matrix yes 3D matrix;

[0072] The finite-time linear quadratic optimal control problem is described based on the aircraft attitude control objective, with the P1 configuration defined as follows:

[0073] ;

[0074] in, and They are Wei and A positive definite symmetric matrix and These are the given initial time and terminal time, respectively. Given an initial state value, Given a terminal state value.

[0075] Based on the duality principle of convex optimization, we introduce the state variables... corresponding Costate variables This leads to the minimax problem of Lagrange functionals:

[0076] ;

[0077] for First, solve the inner loop minimization problem. The derivation yields , The optimal solutions are as follows:

[0078] ;

[0079] Will , Substitute the optimal solution This yields the unconstrained variational problem:

[0080] ;

[0081] According to the variational principle, the Euler-Lagrange equation that maximizes the performance index is obtained:

[0082] ;

[0083] in, The second derivative of the costate variable with respect to time; The first derivative of the costate variable with respect to time; by The corresponding initial boundary conditions are derived as follows:

[0084] ;

[0085] For the two-point boundary value problem of the transformed linear homogeneous equation, based on the principle of superposition of solutions to linear differential equations, three sets of costate variables are given. Initial guess: Note that the number of initial guesses introduced is one more than the dimension of the costate variables; derive the corresponding values ​​using the initial boundary conditions. initial value The corresponding solutions are respectively The final solution is:

[0086] ;

[0087] in, For the coefficients to be determined, satisfying Note the initial guess. The selection must meet the following condition: None , making ;in, Let be a scalar parameter. Combining the terminal boundary conditions, the coefficients are solved as follows:

[0088] ;

[0089] in, for A row vector with dimensions and elements all having a value of 1;

[0090] get Then, using , Optimal solution and By understanding the relationship between the state and control solutions of the P1 configuration problem, the optimal solution can be applied to the attitude control of the aircraft.

[0091] This embodiment only requires three initial value integrations. The initial value guesses for the costate variables in these three initial value integrations are as follows: , , The corresponding coefficient calculation results are as follows: , , , Figures 2a-2c The state and control solutions for the P1 configuration are given. For comparison, calculation results obtained based on the target shooting method are also presented. It can be seen that the solution obtained by the aircraft attitude control calculation method based on the dual solution of a linear quadratic problem in this invention is completely consistent with the solution obtained by the target shooting method. The specified terminal attitude and angular velocity boundary conditions were achieved. .

[0092] Example 2: Also for the finite-time control task of the inner loop pitch attitude angle loop of a certain aircraft in Example 1.

[0093] Similarly, we establish the linear dynamic equations of the system and, based on the aircraft attitude control objective, define the finite-time linear quadratic optimal control problem description for the corresponding configuration. Here, the P2 configuration is defined as follows:

[0094] ;

[0095] Here, yes dimensional state vector, The first derivative of the state vector with respect to time. yes dimensional control vector, yes 3D matrix yes 3D matrix and They are Wei and A positive definite symmetric matrix and The initial time and the terminal time are given, respectively. The initial value is given, and the terminal state is free.

[0096] For the P2 configuration, the two-point boundary value problem of the linear homogeneous equation obtained based on the duality principle is:

[0097] ;

[0098] in, The second boundary condition, representing the second derivative of the costate variable with respect to time, is derived from the Lagrange functional through the existence of the dual solution. Similarly, given three sets of costate variables... Initial guess: The corresponding initial boundary conditions are used to derive the... initial value The corresponding solutions are respectively ,coefficient for:

[0099] ;

[0100] but The final solution is:

[0101] ;

[0102] get Then, using , Optimal solution and By understanding the relationship between the state and control solutions of the P2 configuration problem, the optimal solution can be applied to attitude control.

[0103] This embodiment only requires three initial value integrations. The initial value guesses for the costate variables in these three initial value integrations are as follows: , , . Figures 3a-3c The state and control solutions for the P2 configuration are given. For comparison, the calculation results obtained based on the shooting method are also presented. It can be seen that the solution obtained by the aircraft attitude control calculation method based on the dual solution of the linear quadratic problem in this invention is completely consistent with the solution obtained by the shooting method.

[0104] Example 3: Also for the finite-time control task of the inner loop pitch attitude angle loop of a certain aircraft in Example 1.

[0105] Similarly, we establish the linear dynamic equations of the system, and describe the finite-time linear quadratic optimal control problem of the corresponding configuration based on the aircraft attitude control objective. Here, the P3 configuration is defined as follows:

[0106] ;

[0107] in, yes dimensional state vector, The first derivative of the state vector with respect to time. yes dimensional control vector, yes 3D matrix yes 3D matrix and They are Wei and A positive definite symmetric matrix for 3D positive definite matrix and These are the given initial time and terminal time, respectively. Given initial values, the terminal state is free.

[0108] For the P3 configuration, the two-point boundary value problem of the linear homogeneous equation obtained based on the duality principle is:

[0109] ;

[0110] in, Let be the second derivative of the costate variable with respect to time; similarly, given 3 sets of costate variables... Initial guess: The corresponding initial boundary conditions are used to derive the... initial value The corresponding solutions are respectively ,coefficient for:

[0111] ;

[0112] in, for identity matrix for A row vector with dimensions and elements all having a value of 1; The final solution is:

[0113] ;

[0114] use , Optimal solution and By understanding the relationship between the state and control solutions of the P3 configuration problem, the optimal solution can be applied to the attitude control of the aircraft.

[0115] Similarly, the initial value guesses for the costate variables in the three initial value solutions are as follows: , , . Figures 4a-4c The state and control solutions for the P3 configuration are given. For comparison, the calculation results obtained based on the target shooting method are also presented. It can be seen that the solution obtained by the aircraft attitude control calculation method based on the dual solution of the linear quadratic problem in this invention is completely consistent with the solution obtained by the target shooting method.

[0116] Example 4: Also for the finite-time control task of the inner loop pitch attitude angle loop of a certain aircraft in Example 1.

[0117] Similarly, we establish the linear dynamic equations of the system and, based on the aircraft attitude control objective, define the finite-time linear quadratic optimal control problem description for the corresponding configuration. Here, the P4 configuration is defined as follows:

[0118] ;

[0119] in, yes dimensional state vector, The first derivative of the state vector with respect to time. yes dimensional control vector, yes 3D matrix yes 3D matrix and They are Wei and A positive definite symmetric matrix for 3D positive definite matrix and These are the given initial time and terminal time, respectively. Given an initial state value, the pitch attitude angle at the terminal time is specified as... The terminal pitch angular velocity state is free.

[0120] For the P4 configuration, the two-point boundary value problem of the linear homogeneous equation obtained based on the duality principle is:

[0121] ;

[0122] in, The second derivative of the costate variable with respect to time; It is a matrix The matrix formed in the first row, It is a matrix The matrix formed in the second row, i.e. , identity matrix The matrix formed in the second row. Similarly, given 3 sets of costate variables. Initial guess: The corresponding initial boundary conditions are used to derive the... initial value The corresponding solutions are respectively ,coefficient for:

[0123] ;

[0124] The final solution is:

[0125] ;

[0126] use , Optimal solution and By understanding the relationship between the state and control solutions of the P4 configuration problem, the optimal solution is then applied to the aircraft attitude control.

[0127] Similarly, the initial value guesses for the costate variables in the three initial value solutions are as follows: , , . Figures 5a-5c The state and control solutions for the P4 configuration are given. For comparison, results obtained based on the target shooting method are also presented. It can be seen that the solution obtained by the aircraft attitude control calculation method based on the dual solution of a linear quadratic problem in this invention is completely consistent with the solution obtained by the target shooting method. The specified pitch angle terminal boundary conditions were implemented. .

[0128] Examples 1 to 4 demonstrate that the aircraft attitude control calculation method based on the dual solution of linear quadratic problems of the present invention efficiently solves the problem, and the method has good effectiveness and practicality.

[0129] Although the embodiments of the present invention have been disclosed above, they are not limited to the applications listed in the specification and embodiments. For those skilled in the art, all features disclosed in the present invention, or all steps in all methods or processes disclosed, except for mutually exclusive features and / or steps, can be combined in any way without departing from the principles of the present invention. The present invention is not limited to the specific details and illustrations shown and described herein.

Claims

1. A computational method for aircraft attitude control based on the dual solution of a linear quadratic problem, characterized in that, Includes the following steps: Step 1. For the aircraft object, establish a linearized dynamic model of the aircraft attitude control loop using the small perturbation principle linearization method; The system dynamic equations obtained by linearization using the small perturbation principle are as follows: ； in, yes dimensional state vector, Indicates pitch angle, Indicates pitch attitude angular velocity, The first derivative of the state vector with respect to time. yes dimensional control vector, Indicates the elevator deflection angle. yes 3D matrix yes 3D matrix; Step 2. Based on the aircraft attitude control objective, establish a finite-time linear quadratic optimal control problem description for the corresponding configuration, including a free terminal state, a partially determined terminal state, and setting terminal state performance indices. Specifically, given a partially determined terminal state, for a specified terminal attitude and angular velocity, the P1 configuration is defined as follows: ； in, and They are Wei and A positive definite symmetric matrix and These are the given initial time and terminal time, respectively. Given an initial state value, Given a terminal state value; Step 3. Using the optimal control dual solution method of linear quadratic form, the corresponding two-point boundary value problem of linear homogeneous equation is derived, and the optimal solution is obtained by efficient calculation in 3 iterations; Step 4. Apply the optimal solution to the aircraft attitude control.

2. The aircraft attitude control calculation method based on the dual solution of a linear quadratic problem according to claim 1, characterized in that, The method for solving the dual problem of linear quadratic optimal control includes the following steps: S10. Based on the duality principle, the finite-time linear quadratic optimal control problem of the aircraft attitude control loop dynamics model is transformed into an unconstrained variational problem; including the following steps: S11. Based on the duality principle of convex optimization, we introduce costate variables corresponding to the flight state variables to obtain the maxima and minima problem of the Lagrange functional of the augmented aircraft dynamics equations. Introduction of state variables corresponding Costate variables This leads to the minimax problem of Lagrange functionals: ； S12. For the maxima and minima problem of the Lagrange functional, solve the inner loop minimization problem to obtain the optimal flight state solution and the optimal flight control solution expressed by costate variables; for First, solve the inner loop minimization problem. The derivation yields , The optimal solutions are as follows: ； S13. Substituting the optimal flight state solution and the optimal flight control solution, we obtain the unconstrained variational problem with respect to the costate variables; Will , Substitute the optimal solution This leads to the unconstrained variational problem: ； S20. Based on the variational principle, the Euler-Lagrange equation, which is only related to the costate variables, is derived. Boundary conditions corresponding to the aircraft state settings in the original optimal control problem configuration are introduced, and the problem is transformed into a two-point boundary value problem of linear homogeneous equations. The Euler-Lagrange equation that maximizes the performance index is obtained: ； in, The second derivative of the costate variable with respect to time; The first derivative of the costate variable with respect to time; by The corresponding boundary conditions are derived as follows: ； S30. For the two-point boundary value problem of linear homogeneous equations, the aircraft state solution and control solution of the finite-time linear quadratic optimal control problem with different configurations are obtained through three iterations; including the following steps: S31. Given three sets of initial guesses for costate variables, obtain the numerical solutions to the corresponding initial value problems based on different initial boundary conditions; For the two-point boundary value problem of the transformed linear homogeneous equation, based on the principle of superposition of solutions to linear differential equations, three sets of costate variables are given. Initial guess: The number of initial guesses introduced is one more than the dimension of the costate variables; the corresponding initial boundary conditions are used to derive... initial value The corresponding solutions are respectively The final solution is: ； in, For the coefficients to be determined, satisfying Note the initial guess. The selection condition is: None exists. , making ;in, It is a scalar parameter; S32. Calculate the coefficients of the solution based on the terminal boundary conditions to determine the final costate variable solution; Based on the terminal boundary conditions, the coefficients are solved as follows: ； in, for A row vector with dimensions and elements all having a value of 1; S33. Reconstructing attitude state solution and attitude control solution based on costate variable solution; get Then, using , Optimal solution and By understanding the relationship between the state and control solutions of the P1 configuration problem, the optimal solution can be applied to the attitude control of the aircraft.

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