Dynamic modeling and energy optimization control method of flexible ring-shaped space antenna

Abstract

The invention relates to a method for dynamic modeling and energy optimization control of a flexible annular space antenna, and belongs to the field of space antenna technology. The invention solves the problem that existing control methods are difficult to achieve a balance between high-precision pointing control, energy optimization, and vibration suppression under complex flexible structures and changeable operating environments. The invention is aimed at a space antenna system with a complex flexible structure, and adopts a modal simplification technique to simplify the dynamic model of the complex flexible structure. Then, a control method capable of dynamically switching control strategies is designed through an energy optimization switching model predictive control method, so as to optimize energy consumption during task switching, and ensure precise pointing and stable operation of the antenna system by effectively suppressing vibrations. It can achieve a dynamic balance between precise pointing control and vibration suppression, and optimize energy consumption. The method of the invention can be applied to the field of space antenna technology.

Description

Technical Field

[0001] The present invention belongs to the technical field of space antennas, and in particular relates to a method for dynamic modeling and energy optimization control of a flexible annular space antenna. Background Art

[0002] With the development of modern space technology, space antenna systems are playing an increasingly important role in fields such as Earth observation, deep space communications, and astronomical research. These space antennas are typically large, lightweight, and flexible, presenting numerous challenges during launch, deployment, and on-orbit operation. In particular, during mission switching, antennas must be quickly and precisely adjusted in their pointing direction while avoiding vibrations from the flexible structure that could affect system performance and stability.

[0003] Traditional space antenna control methods primarily rely on classical approaches such as proportional-derivative (PD) control. While simple and easy to implement, these methods often struggle to simultaneously balance control accuracy, system stability, and energy efficiency when dealing with flexible structures with complex dynamic characteristics. Furthermore, the high-order vibration modes and nonlinear characteristics of flexible antenna systems further complicate control strategy design, rendering traditional methods inadequate for complex space environments and multi-task switching.

[0004] In recent years, model predictive control (MPC), an advanced control method, has been increasingly applied to the control of complex spatial systems due to its ability to optimize control under multiple constraints. However, existing MPC methods still suffer from high model complexity, insufficient real-time performance, and sensitivity to parameter changes when addressing energy optimization and vibration suppression in flexible structures.

[0005] Faced with these challenges, there is an urgent need for an advanced control method that balances pointing control accuracy, energy efficiency, and vibration suppression for space antenna system operation. Existing proportional-derivative (PD) control methods, while simple in structure, struggle to maintain efficient control performance in complex flexible structures and changing operating environments. Model predictive control (MPC), as an advanced control strategy, possesses certain optimization capabilities. However, its application in the control of complex flexible structures still faces challenges such as high model complexity, insufficient real-time performance, and high parameter sensitivity. Summary of the Invention

[0006] The purpose of the present invention is to solve the problem that existing control methods are difficult to achieve high-precision pointing control, energy optimization and vibration suppression under complex flexible structures and changeable operating environments, and to propose a flexible annular space antenna dynamic modeling and energy optimization control method.

[0007] The technical solution adopted by the present invention to solve the above technical problems is: a method for dynamic modeling and energy optimization control of a flexible annular space antenna, which specifically includes the following steps:

[0008] Step 1: Establish the dynamic model of each flexible component in the annular space antenna satellite system;

[0009] Step 2: Based on the topological structure of the annular space antenna satellite system, construct the recursive format of the flexible body kinematic equations;

[0010] Step 3: Based on the kinematic equation constructed in step 2, the kinetic energy and potential energy of the annular space antenna satellite system are obtained, and then the dynamic equations of the flexible body are assembled according to the kinetic energy and potential energy to obtain the explicit dynamic equations of the entire annular space antenna satellite system;

[0011] Step 4: performing order reduction processing on the explicit dynamic equations of the annular space antenna satellite system to obtain the reduced-order dynamic equations;

[0012] Step 5: Designing an energy-optimized switching model predictive control method that meets the constraints, the energy-optimized switching model predictive control method comprising a prediction model, a first MPC controller, and a second MPC controller; determining a control objective of the annular space antenna satellite system, designing a cost function of the energy-optimized switching model predictive control method based on the control objective, and establishing a prediction model of the energy-optimized switching model predictive control method based on the reduced-order dynamic equation;

[0013] Calculating the prediction model output corresponding to the minimum cost function value J1 of the first MPC controller and the prediction model output corresponding to the minimum cost function value J2 of the second MPC controller; then comparing the energy consumption of the first MPC controller and the second MPC controller;

[0014] If the energy consumption of the first MPC controller is smaller than that of the second MPC controller, the first MPC controller is selected to control the annular space antenna, and the prediction model output corresponding to the minimum cost value J1 is used as the control input of the first MPC controller;

[0015] If the energy consumption of the second MPC controller is smaller than that of the first MPC controller, the second MPC controller is selected to control the annular space antenna, and the prediction model output corresponding to the minimum cost value J2 is used as the control input of the second MPC controller.

[0016] Furthermore, the flexible components in the annular space antenna system include solar panels and antennas. A dynamic model of each flexible component is established. The dynamic model is specifically as follows:

[0017]

[0018] Among them, M II,i and M IZ,i is the mass matrix of the internal node of the i-th flexible body, M ZI,i and M ZZ,i is the mass matrix of the i-th flexible body interface node, K II,i and K IZ,i is the stiffness matrix of the internal node of the i-th flexible body, K ZI,i and K ZZ,i is the stiffness matrix of the i-th flexible body interface node;

[0019] x i ′=[x′ I,i T x′ Z,i T ] T (2)

[0020] Where x′ I,i is the displacement vector of the ith flexible body interface node, x′ Z,i is the displacement vector of the internal node of the i-th flexible body, x i ′ is the displacement vector of the i-th flexible body, and the superscript T represents the transpose. is x I ' ,i The second derivative of is x′ Z,i The second derivative of .

[0021] Furthermore, the displacement vector x of the i-th flexible body i ′ uses the modal reduction vector η i Expressed as:

[0022]

[0023] in, is the modal matrix of the first D modes of the i-th flexible body, is the constraint modal matrix of the i-th flexible body, the superscript -1 represents the inverse of the matrix, η D,i is the displacement vector of the first D modes of the i-th flexible body, η Z,i is the displacement vector of the constrained mode, Φ i is the intermediate variable, I Z is the identity matrix of the constraint mode of the i-th flexible body.

[0024] Furthermore, the modal reduction vector η i The mass matrix and stiffness matrix are:

[0025]

[0026] Among them, m i is the modal reduction vector η i The mass matrix, k i is the modal reduction vector η i The stiffness matrix, M i is the mass matrix of the i-th flexible body without modal reduction, K i is the stiffness matrix of the i-th flexible body without modal reduction, I D,i is the identity matrix of the first D modes of the i-th flexible body, m DZ,i is the mass coupling matrix of the interaction between the i-th flexible body modes, m ZD,i It is m DZ,i The transposed matrix, m ZZ,i is the mass matrix of the i-th flexible body interface node, k DD,i is the stiffness matrix of the i-th flexible body mode, k ZZ,i is the stiffness matrix of the ith flexible body interface node.

[0027] Furthermore, the specific process of step 2 is as follows:

[0028] The connection relationship between adjacent flexible bodies in the annular space antenna system is:

[0029]

[0030] in, is η Z,i The first derivative of , β i is the rotation axis selection matrix of the i-th individual; q i is the joint rotation angle array of the i-th individual; It's q i The first derivative of C i-1 is the coordinate transformation matrix of the joint position and posture before and after rotation of the i-1th individual; S i-1 is the coordinate installation matrix of the front and rear interface nodes of the i-1th individual; is the modal matrix of the first D modes of the i-1th flexible body; η D,i-1 is the displacement vector of the first D modes of the i-1th flexible body, is η D,i-1 The first derivative of ;

[0031] Through the recursive relationship of kinematics, the kinematic equation of each flexible body is obtained:

[0032]

[0033] Among them, q j is the joint rotation angle array of the jth individual, It's qj The first derivative of C j-1 is the coordinate transformation matrix of the position and posture before and after the joint rotation of the j-1th individual; S j-1 is the coordinate installation matrix of the front and rear interface nodes of the j-1th individual; C k-1 is the coordinate transformation matrix of the k-1th individual’s joint position and posture before and after rotation; q k is the joint rotation angle array of the kth individual, It's q k The first derivative of S k-1 is the coordinate installation matrix of the front and rear interface nodes of the k-1th individual; β j is the rotation axis selection matrix of the jth individual; q k-1 is the joint rotation angle array of the k-1th individual, It's q k-1 The first derivative of q j+1 is the joint rotation angle array of the j+1th individual; C j is the coordinate transformation matrix of the position and posture before and after the joint rotation of the jth individual; is the modal matrix of the first D modes of the j-th flexible body; is the modal matrix of the first D modes of the i-1th flexible body; C i-1 is the coordinate transformation matrix of the position and posture before and after the joint rotation of the i-1th individual; q i-1 is the joint rotation angle array of the i-1th individual, It's q i-1 The first derivative of ; Represents the kinematic information of the initial body.

[0034] Furthermore, the specific process of step three is:

[0035] Step 3.1: Based on the kinematic equations constructed in step 2 and the modal reduction vector η i The mass matrix and stiffness matrix of the annular space antenna system are used to obtain the kinetic energy T and potential energy V:

[0036]

[0037] Among them, R represents the rigid body set, F represents the flexible body set, v0 is the velocity of the satellite in the donut-shaped space antenna system, ω0 is the angular velocity of the satellite in the donut-shaped space antenna satellite system, is the joint angle array, n represents the number of bodies, I′ is the total number of flexible bodies, is η DThe first derivative of , M(q) is the total mass matrix of the annular space antenna satellite system, and K is the total stiffness matrix;

[0038] Step 32: Establish the Lagrangian equation based on the kinetic energy and potential energy of the annular space antenna satellite system, assemble the dynamic equation of the flexible body according to the Lagrangian equation, and obtain the explicit dynamic equation of the entire annular space antenna satellite system.

[0039] Furthermore, in step 32, the dynamic equations of the flexible body are assembled according to the Lagrange equation to obtain the explicit dynamic equations of the entire annular space antenna satellite system, which are specifically:

[0040]

[0041] in, is the first-order derivative of M(q), is the first-order derivative of κ, M 1～6 are the first six rows of the mass matrix M(q), τ is the force generated by the torque on each revolute joint, F D It is the force generated by the piezoelectric material on each rotating joint. F0 represents the three-axis thrust of the satellite, and T0 represents the three-axis torque of the satellite. The antisymmetric matrix representing the satellite angular velocity ω0, The antisymmetric matrix representing the satellite velocity v0.

[0042] Furthermore, the process of establishing the prediction model is specifically as follows:

[0043] The reduced-order dynamic equations of the annular space antenna satellite system are expressed as follows through local linearization:

[0044]

[0045] in, and is the Jacobian matrix of the state space variable x(t), and is the Jacobian matrix of the control input u(t), is the first derivative of x(t), and y(t) is the output of the annular space antenna satellite system;

[0046] The prediction model is:

[0047]

[0048] Where X is the stacked vector of the predicted state for the next N steps; x k+i″is the predicted state for the k+i″th step in the future, i″=1,2,…,N; U is the stacked vector of control inputs for the next N steps, u k+i″ is the control input for the next k+i″ step, x k is the current state, Φ and Γ are both intermediate variable matrices;

[0049]

[0050] Among them, A d represent The discretized state matrix, B d Represents the input matrix after discretization of the system.

[0051] Furthermore, the cost function is specifically:

[0052]

[0053] Where J is the cost function, is the tracking error cost, y k+i″ represents the output variable of the annular space antenna satellite system at step k+i″, r k+i″ represents the value of the k+i″th step in the reference trajectory r, Q represents the state error weight matrix, is the cost of the control input change, and R represents the control input weight matrix.

[0054] Furthermore, the cost function is solved as follows:

[0055]

[0056] Among them, y min is the minimum value of the output variable of the annular space antenna satellite system, y max is the maximum value of the output variable of the annular space antenna satellite system, y k+i″ is the output variable of the annular space antenna satellite system in the next k+i″ step, u min is the lower limit of the control input, u max is the upper limit of the control input, C d is the discretized output matrix of the annular space antenna satellite system.

[0057] The beneficial effects of the present invention are:

[0058] For space antenna systems with complex flexible structures, the present invention uses modal simplification technology to simplify the dynamic model of the complex flexible structure, and then designs a control method that can dynamically switch control strategies through the energy optimization switching model predictive control method to optimize energy consumption during task switching. By effectively suppressing vibration, the precise pointing and stable operation of the antenna system are ensured, and a dynamic balance can be achieved between precise pointing control and vibration suppression, thereby optimizing energy consumption.

[0059] The present invention has the following advantages:

[0060] 1. Improve system control accuracy and stability. This invention adopts the Energy Optimized Switching Model Predictive Control (EOS-MPC) method to dynamically adjust the control strategy under different operating conditions, effectively improving the control accuracy and stability of the system and ensuring that the antenna always maintains precise pointing during the target switching process.

[0061] 2. Significantly Reduced Energy Consumption. This invention minimizes system energy consumption by optimizing control algorithms and rationally allocating controller resources. Compared to traditional control methods, this invention significantly reduces energy consumption during operation and extends the operational life of the satellite.

[0062] 3. Effectively suppress vibration and improve system reliability. While achieving precise control, this invention can effectively suppress the vibration of the antenna system, reduce mechanical stress and fatigue, and significantly improve the long-term reliability and safety of the system, making it suitable for mission execution in complex space environments.

[0063] 4. Improve computational efficiency and achieve real-time control. By introducing modal simplification and order reduction, the present invention significantly improves computational efficiency while ensuring the integrity of the system's dynamic characteristics, making real-time control of complex flexible structures possible and adapting to rapidly changing task requirements. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1 Schematic diagram of the structure of the annular space antenna system of the method of the present invention;

[0065] Figure 2 is a frequency diagram of antenna modal of the method of the present invention;

[0066] Figure 3 This is a block diagram of the energy optimization switching model predictive control of the method of the present invention;

[0067] Figure 4a is a graph showing the change of joint angle q3 over time under PD control and EOS-MPC control in the method of the present invention;

[0068] For the graphs containing two curves, the curves connecting circles represent the results of the PD algorithm, and the curves connecting squares represent the results of the EOS-MPC algorithm;

[0069] Figure 4b is a graph showing the change in angle error of the joint angle q3 over time under PD control and EOS-MPC control in the method of the present invention;

[0070] Figure 4c is a graph showing the change of joint torque τ3 over time under PD control in the method of the present invention;

[0071] Figure 4d is a graph showing the change of joint torque τ3 over time under EOS-MPC control in the method of the present invention;

[0072] Figure 4e is a graph showing the change of joint angle q4 over time under PD control and EOS-MPC control in the method of the present invention;

[0073] Figure 4f is a graph showing the angle error of the joint angle q4 changing with time under PD control and EOS-MPC control in the method of the present invention;

[0074] Figure 4g is a graph showing the change of joint torque τ4 over time under PD control in the method of the present invention;

[0075] Figure 4h is a graph showing the change of joint torque τ4 over time under EOS-MPC control in the method of the present invention;

[0076] Figure 4i is a graph showing the change of the joint angle q5 over time under PD control and EOS-MPC control in the method of the present invention;

[0077] Figure 4j is a graph showing the angle error of the joint angle q5 changing with time under PD control and EOS-MPC control in the method of the present invention;

[0078] Figure 4k is a graph showing the change of joint torque τ5 over time under PD control in the method of the present invention;

[0079] Figure 4l It is a graph showing the change of joint torque τ5 over time under EOS-MPC control in the method of the present invention. DETAILED DESCRIPTION

[0080] Specific embodiment 1: This embodiment describes a method for dynamic modeling and energy optimization control of a flexible annular space antenna, the method specifically comprising the following steps:

[0081] Step 1: Establish the dynamic model of each flexible component in the annular space antenna satellite system to ensure that the dynamic model can accurately reflect the inherent dynamic characteristics of the flexible body. The schematic diagram of the annular space antenna satellite system is shown in the figure. Figure 1 As shown;

[0082] Step 2: Based on the topological structure of the annular space antenna satellite system, construct the recursive format of the flexible body kinematic equations to ensure that the motion state of all flexible components can be accurately captured;

[0083] Step 3: Based on the kinematic equations constructed in Step 2, the kinetic energy and potential energy of the annular space antenna satellite system are obtained. Then, the dynamic equations of the flexible body are assembled according to the kinetic energy and potential energy to obtain the explicit dynamic equations of the entire annular space antenna satellite system. The explicit dynamic equations of the system can accurately describe the dynamic response of the system under different working conditions and have high computational efficiency.

[0084] Step 4: Using a modal simplification method to reduce the order of the explicit dynamic equations of the annular space antenna satellite system, to obtain the reduced-order dynamic equations;

[0085] Due to the complex structure of the ring column antenna, in order to improve computational efficiency and facilitate real-time control, the system's dynamic equations are simplified by selecting important modes and discarding high-order modes with less influence, while retaining the system's key dynamic characteristics.

[0086] Step 5: Design an energy-optimized switching model predictive control method (EOS-MPC) that satisfies the constraints (including the maximum allowable displacement, angle change rate, and actuator force and torque limits), such as Figure 3 As shown, the energy-optimized switching model predictive control method includes a predictive model, a first MPC controller, and a second MPC controller; determining the control target of the annular space antenna satellite system (the control target is to maintain the precise pointing direction of the antenna and suppress vibration during the target switching period), then designing the cost function of the energy-optimized switching model predictive control method (EOS-MPC) based on the control target, and establishing a predictive model of the energy-optimized switching model predictive control method based on the reduced-order dynamic equation;

[0087] Calculate the prediction model output corresponding to the minimum cost function value J1 of the first MPC controller and the prediction model output corresponding to the minimum cost function value J2 of the second MPC controller;

[0088] Then, the energy consumption of the first MPC controller and the second MPC controller are compared based on the calculation results (because the cost function designed by the present invention includes an energy consumption term, the energy consumption used for subsequent comparison is obtained in the process of calculating the minimum cost function value);

[0089] If the energy consumption of the first MPC controller is smaller than that of the second MPC controller, the first MPC controller is selected to control the annular space antenna, and the prediction model output corresponding to the minimum cost value J1 is used as the control input of the first MPC controller;

[0090] If the energy consumption of the second MPC controller is smaller than that of the first MPC controller, the second MPC controller is selected to control the annular space antenna, and the prediction model output corresponding to the minimum cost value J2 is used as the control input of the second MPC controller.

[0091] Due to the complex dynamic characteristics of the ring-column space antenna system, the control method of the present invention is selected to optimize energy consumption under dynamic conditions and maintain the stability of the system.

[0092] Specific embodiment 2: This embodiment differs from specific embodiment 1 in that the flexible components in the annular space antenna system include solar panels and antennas, and a dynamic model is established for each flexible component. The dynamic model is specifically as follows:

[0093]

[0094] Among them, M II,i and M IZ,i is the mass matrix of the internal node of the i-th flexible body, M ZI,i and M ZZ,i is the mass matrix of the i-th flexible body interface node (the connection point on the flexible body is defined as the interface node, and the other nodes are internal nodes), K II,i and K IZ,i is the stiffness matrix of the internal node of the i-th flexible body, K ZI,i and K ZZ,i is the stiffness matrix of the i-th flexible body interface node;

[0095] x i ′=[x I ' ,i T x′ Z,i T ] T (2)

[0096] Among them, x I ' ,i is the displacement vector of the ith flexible body interface node, x′ Z,i is the displacement vector of the internal node of the i-th flexible body, x i ′ is the displacement vector of the i-th flexible body, and the superscript T represents the transpose. is x I ',i The second derivative of is x′ Z,i The second derivative of .

[0097] Other steps and parameters are the same as those in the first embodiment.

[0098] Specific embodiment 3: This embodiment differs from specific embodiment 1 or 2 in that the displacement vector x of the i-th flexible body is i ′ uses the modal reduction vector η i Expressed as:

[0099]

[0100] in, is the modal matrix of the first D modes of the i-th flexible body (D<<I, I is the total number of modes of the i-th flexible body), is the constraint mode matrix of the i-th flexible body, The superscript -1 represents the inverse of the matrix, η D,i is the displacement vector of the first D modes of the i-th flexible body, η Z,i is the displacement vector of the constrained mode, Z is the number of constrained modes, Φ i is the intermediate variable, I Z is the identity matrix of the constraint mode of the i-th flexible body.

[0101] Other steps and parameters are the same as those in the first or second embodiment.

[0102] Specific embodiment 4: This embodiment differs from any one of specific embodiments 1 to 3 in that the modal reduction vector η i The mass matrix and stiffness matrix are:

[0103]

[0104] Among them, m i is the modal reduction vector η i The mass matrix, k i is the modal reduction vector η i The stiffness matrix, M i is the mass matrix of the i-th flexible body without modal reduction, K i is the stiffness matrix of the i-th flexible body without modal reduction, I D,i is the identity matrix of the first D modes of the i-th flexible body, m DZ,i is the mass coupling matrix of the interaction between the i-th flexible body modes, m ZD,i It is m DZ,i The transposed matrix, m ZZ,iis the mass matrix of the i-th flexible body interface node, k DD,i is the stiffness matrix of the i-th flexible body mode, k ZZ,i is the stiffness matrix of the ith flexible body interface node,

[0105] The other steps and parameters are the same as those in the first to third embodiments.

[0106] Specific embodiment 5: This embodiment differs from specific embodiments 1 to 4 in that the specific process of step 2 is as follows:

[0107] The connection relationship between adjacent flexible bodies in the annular space antenna system is:

[0108]

[0109] in, is η Z,i The first derivative of , β i is the rotation axis selection matrix of the i-th body (rigid body or flexible body), q i is the joint rotation angle array of the i-th body (rigid body or flexible body), It's q i The first derivative of C i-1 is the coordinate transformation matrix of the position and posture before and after the joint rotation of the i-1th individual (rigid body or flexible body), S i-1 is the coordinate installation matrix of the front and rear two interface nodes of the i-1th body (rigid body or flexible body) (indicates the next body B i The movement of the previous body B i-1 The rigid motion, rotation at the interface point and flexible vibration at the interface point are composed of. When the previous one is a rigid body, This item is 0), is the modal matrix of the first D modes of the i-1th flexible body; η D,i-1 is the displacement vector of the first D modes of the i-1th flexible body, is η D,i-1 The first derivative of ;

[0110]

[0111] Among them, the x-axis, y-axis, and z-axis are the three coordinate axes of the joint coordinate system, and the directions of the coordinate axes are consistent with the directions of the coordinate axes of the i-1th individual coordinate system;

[0112]

[0113] Among them, Ri-1 The rotation matrix of the i-1th body (rigid body or flexible body) is used to describe the joint rotation angle array q i The rotation of the lower coordinate system, I is the unit matrix, r is the displacement vector from the previous body to the next body, is the antisymmetric matrix of r;

[0114] Through the recursive relationship of kinematics, the kinematic equation of each flexible body is obtained:

[0115]

[0116] Among them, q j is the joint rotation angle array of the j-th individual (rigid body or flexible body), It's q j The first derivative of C j-1 is the coordinate transformation matrix of the position and posture before and after the joint rotation of the j-1th individual (rigid body or flexible body); S j-1 is the coordinate installation matrix of the front and rear interface nodes of the j-1th body (rigid body or flexible body); C k-1 is the coordinate transformation matrix of the position and posture before and after the joint rotation of the k-1th individual (rigid body or flexible body); q k is the joint rotation angle array of the kth individual (rigid body or flexible body), It's q k The first derivative of S k-1 is the coordinate installation matrix of the front and rear interface nodes of the k-1th body (rigid body or flexible body); β j is the rotation axis selection matrix of the jth individual (rigid body or flexible body); q k-1 is the joint rotation angle array of the k-1th individual (rigid body or flexible body), It's q k-1 The first derivative of q j+1 is the joint rotation angle array of the j+1th individual (rigid body or flexible body); C j is the coordinate transformation matrix of the position and posture before and after the joint rotation of the jth individual (rigid body or flexible body); is the modal matrix of the first D modes of the j-th flexible body; is the modal matrix of the first D modes of the i-1th flexible body; C i-1 is the coordinate transformation matrix of the position and posture before and after the joint rotation of the i-1th individual (rigid body or flexible body); q i-1 is the joint rotation angle array of the i-1th body (rigid body or flexible body), It's q i-1 The first derivative of ; Represents the kinematic information of the initial body (including velocity and angular velocity, etc.).

[0117] The other steps and parameters are the same as those in the first to fourth embodiments.

[0118] Specific embodiment 6: This embodiment differs from any one of specific embodiments 1 to 5 in that the specific process of step 3 is as follows:

[0119] Step 3.1: Based on the kinematic equations constructed in step 2 and the modal reduction vector η i The mass matrix and stiffness matrix of the annular space antenna system are used to obtain the kinetic energy T and potential energy V:

[0120]

[0121] Among them, R represents the rigid body set, F represents the flexible body set, v0 is the velocity of the satellite in the donut-shaped space antenna system, ω0 is the angular velocity of the satellite in the donut-shaped space antenna satellite system, is the joint angle array, n represents the number of bodies, I′ is the total number of flexible bodies, is η D The first derivative of , M(q) is the total mass matrix of the annular space antenna satellite system, and K is the total stiffness matrix;

[0122] Step 32: Establish the Lagrangian equation based on the kinetic energy and potential energy of the annular space antenna satellite system, assemble the dynamic equation of the flexible body according to the Lagrangian equation, and obtain the explicit dynamic equation of the entire annular space antenna satellite system.

[0123] The other steps and parameters are the same as those in the first to fifth embodiments.

[0124] Specific embodiment seven: This embodiment differs from any one of specific embodiments one to six in that, in step three-two, the dynamic equations of the flexible body are assembled according to the Lagrange equation to obtain the explicit dynamic equations of the entire annular space antenna satellite system, specifically:

[0125]

[0126] in, Contains the motion state of the satellite, the speed of each joint and the modal coordinates of the flexible body. is the first-order derivative of M(q), is the first-order derivative of κ, M 1～6 are the first six rows of the mass matrix M(q), τ is the force generated by the torque on each revolute joint, F DIt is the force generated by the piezoelectric material on each rotating joint. F0 represents the three-axis thrust of the satellite, and T0 represents the three-axis torque of the satellite. The antisymmetric matrix representing the satellite angular velocity ω0, The antisymmetric matrix representing the satellite velocity v0.

[0127] The other steps and parameters are the same as those in the first to sixth embodiments.

[0128] Specific embodiment eight: This embodiment differs from any one of specific embodiments one to seven in that the process of establishing the prediction model is as follows:

[0129] The reduced-order dynamic equations of the annular space antenna satellite system are expressed as follows through local linearization:

[0130]

[0131] in, and is the Jacobian matrix of the state space variable x(t) obtained by transforming the dynamic equation into the state space equation, and is the Jacobian matrix of the control input u(t), is the first derivative of x(t), and y(t) is the output of the annular space antenna satellite system;

[0132] The prediction model is:

[0133]

[0134] Where X is the stacked vector of the predicted state for the next N steps; x k+i″ is the predicted state for the k+i″th step in the future, i″=1,2,…,N; U is the stacked vector of control inputs for the next N steps, u k+i″ is the control input for the next k+i″ step, x k is the current state, Φ and Γ are both intermediate variable matrices;

[0135] The matrices Φ and Γ describe the effects of state transitions and control inputs on future states, respectively, and are in the following form:

[0136]

[0137] Among them, A d represent The discretized state matrix, B d Represents the input matrix after discretization of the system.

[0138] The other steps and parameters are the same as those in the first to seventh embodiments.

[0139] The prediction model of the present invention should be able to accurately predict the state changes of the system in the future and adapt to the nonlinear changes that may occur in the system during operation, and be used to calculate the optimal control input in real time.

[0140] Specific embodiment 9: This embodiment differs from any one of specific embodiments 1 to 8 in that the cost function is specifically:

[0141]

[0142] Where J is the cost function, is the tracking error cost, y k+i″ represents the output variable of the annular space antenna satellite system at step k+i″, r k+i″ represents the value of the k+i″th step in the reference trajectory r, Q represents the state error weight matrix, is the cost of the control input change, which is used to measure the output variable y of the annular space antenna satellite system k+i″ and the reference trajectory r k+i″ The error cost between, R represents the control input weight matrix, which is used to measure the control input u k+i″ The price.

[0143] The other steps and parameters are the same as those in Specific Embodiments 1 to 8.

[0144] Specific embodiment 10: This embodiment differs from any one of specific embodiments 1 to 9 in that the cost function is solved as follows:

[0145] Assuming that the state is available at each time step t, the following optimization problem can be solved at time step t by minimizing J and satisfying the following constraints:

[0146]

[0147] Among them, y min is the minimum value of the output variable (physical quantity controlled and monitored) of the annular space antenna satellite system, y max is the maximum value of the output variable (physical quantity controlled and monitored) of the annular space antenna satellite system, y k+i″ is the output variable of the annular space antenna satellite system in the next k+i″ step, u min is the lower limit of the control input, u max is the upper limit of the control input, C d is the discretized output matrix of the annular space antenna satellite system.

[0148] The other steps and parameters are the same as those in Specific Embodiments 1 to 9.

[0149] The cost function of the present invention consists of a tracking error component and a control input variation component. The tracking error component measures the difference between the system output and the reference trajectory, while the control input variation component ensures smoothness and energy efficiency during the control process, thereby minimizing the control input. By optimizing the cost function, an optimal control strategy that meets operational requirements can be obtained, achieving energy optimization.

[0150] At each sampling moment, the optimization problem is solved to obtain the optimal control input. To improve system efficiency, the controller can be dynamically switched according to the operation requirements, that is, the controller corresponding to the smallest cost function value is selected to ensure that the global optimal control is achieved at every moment. The EOS-MPC method takes advantage of the advantages of two MPC controllers, such as Figure 3 As shown in Figure 2 . During target switching operations, system characteristics change dynamically, necessitating the adoption of different control strategies for optimal performance. The EOS-MPC approach ensures the use of the predictive controller with the lowest energy consumption by dynamically selecting between these MPC controllers. The first MPC controller is used for precise angle adjustment, with a weight matrix designed to prioritize accurate tracking of key state variables and control effort. The second MPC2 controller is used for effective vibration suppression, with a weight matrix set to minimize vibration while maintaining overall control stability.

[0151] Experimental part

[0152] To verify the effectiveness of the proposed dynamic modeling and energy optimization control method for a space loop antenna system, the algorithm was tested and evaluated in detail in a simulation environment. The experiment mainly consists of two parts: model verification and control performance evaluation.

[0153] First, to verify the accuracy of the dynamic model, the antenna system was modeled using standard dynamic simulation software, and the simulation results were compared with the response of the actual physical model. Figure 2 The modal frequency distribution of the system under different control methods is demonstrated. The results show that the method of the present invention can effectively reduce the occurrence of repeated modes and improve computational efficiency.

[0154] Subsequently, the performance of the control algorithm of the present invention was evaluated. In the experiment, the energy optimization and controller switching parts of the energy optimized switching model predictive control method (EOS-MPC) were ablated respectively and trained using the same simulation parameters. Figures 4a to 41A comparison of energy and time consumption between an energy-optimized switching model predictive control and a PD control method is presented. The results show that the proposed control method significantly reduces energy and time consumption while ensuring successful completion of the target switching task. In contrast, solutions that only utilize energy optimization or controller switching components perform poorly in both energy and time consumption, while the baseline method without any optimization fails to ensure system stability and successful task completion.

[0155] Experimental results show that the method of the present invention can not only achieve high-precision pointing and stable vibration suppression of the antenna system, but also significantly improve energy utilization efficiency and reduce task completion time, and has high practical value and promotion prospects.

[0156] By testing and validating the control strategy designed in this paper in a simulation environment, controller parameters were adjusted to meet actual requirements. Performance under different operating conditions, such as control accuracy and energy consumption during different target switching tasks, was emphasized. Through repeated iterations and optimization, the robustness and reliability of the control strategy in practical applications were ensured.

[0157] The above examples are merely illustrative of the calculation model and process of the present invention and are not intended to limit the embodiments of the present invention. Persons skilled in the art will readily appreciate that other variations or modifications based on the above description are possible. This list of embodiments is not exhaustive; however, any obvious variations or modifications derived from the technical solution of the present invention remain within the scope of protection of the present invention.

Claims

1. Flexible annular space antenna dynamics modeling and energy optimization control method, characterized by: The method specifically comprises the following steps: Step 1: Establish the dynamic model of each flexible component in the annular space antenna satellite system; Step 2: Based on the topological structure of the annular space antenna satellite system, construct the recursive format of the flexible body kinematic equations; Step 3: Based on the kinematic equation constructed in step 2, the kinetic energy and potential energy of the annular space antenna satellite system are obtained, and then the dynamic equations of the flexible body are assembled according to the kinetic energy and potential energy to obtain the explicit dynamic equations of the entire annular space antenna satellite system; Step 4: performing order reduction processing on the explicit dynamic equations of the annular space antenna satellite system to obtain the reduced-order dynamic equations; Step 5: Designing an energy-optimized switching model predictive control method that meets the constraints, the energy-optimized switching model predictive control method comprising a prediction model, a first MPC controller, and a second MPC controller; determining a control objective of the annular space antenna satellite system, designing a cost function of the energy-optimized switching model predictive control method based on the control objective, and establishing a prediction model of the energy-optimized switching model predictive control method based on the reduced-order dynamic equation; Calculating the prediction model output corresponding to the minimum cost function value J1 of the first MPC controller and the prediction model output corresponding to the minimum cost function value J2 of the second MPC controller; then comparing the energy consumption of the first MPC controller and the second MPC controller; If the energy consumption of the first MPC controller is smaller than that of the second MPC controller, the first MPC controller is selected to control the annular space antenna, and the prediction model output corresponding to the minimum cost value J1 is used as the control input of the first MPC controller; If the energy consumption of the second MPC controller is smaller than that of the first MPC controller, the second MPC controller is selected to control the annular space antenna, and the prediction model output corresponding to the minimum cost value J2 is used as the control input of the second MPC controller.

2. The method for dynamic modeling and energy optimization control of a flexible annular space antenna according to claim 1 is characterized in that: The flexible components in the annular space antenna system include solar panels and antennas. A dynamic model is established for each flexible component. The dynamic model is specifically as follows: Among them, M II,i and M IZ,i is the mass matrix of the internal node of the i-th flexible body, M ZI,i and M ZZ,i is the mass matrix of the i-th flexible body interface node, K II,i and K IZ,i is the stiffness matrix of the internal node of the i-th flexible body, K ZI,i and K ZZ,i is the stiffness matrix of the i-th flexible body interface node; Where x′ I,i is the displacement vector of the ith flexible body interface node, x′ Z,i is the displacement vector of the internal node of the i-th flexible body, x′ i is the displacement vector of the i-th flexible body, and the superscript T represents the transpose. is x′ I,i The second derivative of is x′ Z,i The second derivative of .

3. The method for dynamic modeling and energy optimization control of a flexible annular space antenna according to claim 2, characterized in that: The displacement vector x of the i-th flexible body i ′ uses the modal reduction vector η i Expressed as: in, is the modal matrix of the first D modes of the i-th flexible body, is the constraint modal matrix of the i-th flexible body, the superscript -1 represents the inverse of the matrix, η D,i is the displacement vector of the first D modes of the i-th flexible body, η Z,i is the displacement vector of the constrained mode, Φ i is the intermediate variable, I Z is the identity matrix of the constraint mode of the i-th flexible body.

4. The method for dynamic modeling and energy optimization control of a flexible annular space antenna according to claim 3 is characterized in that: The modal reduction vector η i The mass matrix and stiffness matrix are: Among them, m i is the modal reduction vector η i The mass matrix, k i is the modal reduction vector η i The stiffness matrix, M i is the mass matrix of the i-th flexible body without modal reduction, K i is the stiffness matrix of the i-th flexible body without modal reduction, I D,i is the identity matrix of the first D modes of the i-th flexible body, m DZ,i is the mass coupling matrix of the interaction between the i-th flexible body modes, m ZD,i It is m DZ,i The transposed matrix, m ZZ,i is the mass matrix of the i-th flexible body interface node, k DD,i is the stiffness matrix of the i-th flexible body mode, k ZZ,i is the stiffness matrix of the ith flexible body interface node.

5. The method for dynamic modeling and energy optimization control of a flexible annular space antenna according to claim 4, characterized in that: The specific process of step 2 is as follows: The connection relationship between adjacent flexible bodies in the annular space antenna system is: in, is η Z,i The first derivative of β i is the rotation axis selection matrix of the i-th individual; q i is the joint rotation angle array of the i-th individual; It is q i The first derivative of C i-1 is the coordinate transformation matrix of the joint position and posture before and after rotation of the i-1th individual; S i-1 is the coordinate installation matrix of the front and rear interface nodes of the i-1th individual; is the modal matrix of the first D modes of the i-1th flexible body; η D,i-1 is the displacement vector of the first D modes of the i-1th flexible body, is η D,i-1 The first derivative of ; Through the recursive relationship of kinematics, the kinematic equation of each flexible body is obtained: Among them, q j is the joint rotation angle array of the jth individual, It is q j The first derivative of C j-1 is the coordinate transformation matrix of the position and posture before and after the joint rotation of the j-1th individual; S j-1 is the coordinate installation matrix of the front and rear interface nodes of the j-1th individual; C k-1 is the coordinate transformation matrix of the k-1th individual’s joint position and posture before and after rotation; q k is the joint rotation angle array of the kth individual, It is q k The first derivative of S k-1 is the coordinate installation matrix of the front and rear interface nodes of the k-1th individual; β j is the rotation axis selection matrix of the jth individual; q k-1 is the joint rotation angle array of the k-1th individual, It is q k-1 The first derivative of q j+1 is the joint rotation angle array of the j+1th individual; C j is the coordinate transformation matrix of the position and posture before and after the joint rotation of the jth individual; is the modal matrix of the first D modes of the j-th flexible body; is the modal matrix of the first D modes of the i-1th flexible body; C i-1 is the coordinate transformation matrix of the position and posture before and after the joint rotation of the i-1th individual; q i-1 is the joint rotation angle array of the i-1th individual, It is q i-1 The first derivative of ; Represents the kinematic information of the initial body.

6. The method for dynamic modeling and energy optimization control of a flexible annular space antenna according to claim 5, characterized in that: The specific process of step three is: Step 3.1: Based on the kinematic equations constructed in step 2 and the modal reduction vector η i The mass matrix and stiffness matrix of the annular space antenna system are used to obtain the kinetic energy T and potential energy V: Among them, R represents the rigid body set, F represents the flexible body set, v0 is the velocity of the satellite in the donut-shaped space antenna system, ω0 is the angular velocity of the satellite in the donut-shaped space antenna satellite system, is the joint angle array, n represents the number of bodies, I′ is the total number of flexible bodies, is η D The first derivative of , M(q) is the total mass matrix of the annular space antenna satellite system, and K is the total stiffness matrix; Step 32: Establish the Lagrangian equation based on the kinetic energy and potential energy of the annular space antenna satellite system, assemble the dynamic equation of the flexible body according to the Lagrangian equation, and obtain the explicit dynamic equation of the entire annular space antenna satellite system.

7. The method for dynamic modeling and energy optimization control of a flexible annular space antenna according to claim 6, characterized in that: In step 32, the dynamic equations of the flexible body are assembled according to the Lagrange equation to obtain the explicit dynamic equations of the entire annular space antenna satellite system, which are specifically: in, is the first-order derivative of M(q), is the first-order derivative of κ, M 1～6 are the first six rows of the mass matrix M(q), τ is the force generated by the torque on each revolute joint, F D It is the force generated by the piezoelectric material on each rotating joint. F0 represents the three-axis thrust of the satellite, and T0 represents the three-axis torque of the satellite. The antisymmetric matrix representing the satellite angular velocity ω0, The antisymmetric matrix representing the satellite velocity v0.

8. The method for dynamic modeling and energy optimization control of a flexible annular space antenna according to claim 7, characterized in that: The specific process of establishing the prediction model is as follows: The reduced-order dynamic equations of the annular space antenna satellite system are expressed as follows through local linearization: in, and is the Jacobian matrix of the state space variable x(t), and is the Jacobian matrix of the control input u(t), is the first derivative of x(t), and y(t) is the output of the annular space antenna satellite system; The prediction model is: Where X is the stacked vector of the predicted state for the next N steps; x k+i″ is the predicted state for the k+i″th step in the future, i″=1,2,…,N; U is the stacked vector of control inputs for the next N steps, u k+i″ is the control input for the next k+i″ step, x k is the current state, Φ and Γ are both intermediate variable matrices; Among them, A d represent The discretized state matrix, B d Represents the input matrix after discretization of the system.

9. The method for dynamic modeling and energy optimization control of a flexible annular space antenna according to claim 8, characterized in that: The cost function is specifically: Where J is the cost function, is the tracking error cost, y k+i″ represents the output variable of the annular space antenna satellite system at step k+i″, r k+i″ represents the value of the k+i″th step in the reference trajectory r, Q represents the state error weight matrix, is the cost of the control input change, and R represents the control input weight matrix.

10. The method for dynamic modeling and energy optimization control of a flexible annular space antenna according to claim 9, characterized in that: The solution method of the cost function is: Among them, y min is the minimum value of the output variable of the annular space antenna satellite system, y max is the maximum value of the output variable of the annular space antenna satellite system, y k+i″ is the output variable of the annular space antenna satellite system in the next k+i″ step, u min is the lower limit of the control input, u max is the upper limit of the control input, C d is the discretized output matrix of the annular space antenna satellite system.

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