Dynamics modeling and control method of double-star electromagnetic formation of port hamilton system

Abstract

The application relates to a kind of double-star electromagnetic formation dynamics modeling and control method of port Hamilton system, comprising: defining two slave spacecraft center-of-mass position coordinates in ECI coordinate system and LVLH coordinate system;According to the center-of-mass position coordinates of slave spacecraft, the kinetic energy and potential energy of slave spacecraft in ECI coordinate system are obtained;According to the center-of-mass position coordinates of slave spacecraft and the coordinate transformation matrix, the Lagrangian function L in LVLH coordinate system is obtained;The Euler-Lagrange equation is established, according to Legendre transformation, the generalized momentum coordinates p j are selected, the Hamilton function H is obtained, and the port Hamilton dynamics model is established;An electromagnetic formation controller is designed according to the trajectory tracking passive control method based on interconnection-damping distribution;The electromagnetic satellite formation trajectory tracking magnetic dipole strength control law is obtained according to the electromagnetic formation controller.The application retains the nonlinear term of the system, embodies the inherent dissipation characteristics of the system, can avoid the problem of reducing the accuracy of the system caused by linearization, and ensures the passivity of the system.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of near-earth dual-satellite electromagnetic formation flying dynamics modeling and control, and particularly relates to a dual-satellite electromagnetic formation flying dynamics modeling and control method of a port Hamilton system. BACKGROUND

[0002] Satellite formation flying involves multiple satellites working closely in a specific configuration. This method exhibits several significant advantages compared to traditional single large satellite systems, including cost-effectiveness, system redundancy enhancement, upgrade flexibility, and overall performance optimization. Due to these advantages, satellite formation flying has shown great potential and broad application prospects in fields such as Earth monitoring, celestial observation, and deep space exploration, becoming an important trend in the future development of space exploration.

[0003] However, to achieve precise formation flying configurations and maintain their stability, frequent maneuvering control is often required. Traditionally, such control mainly relies on control actuation systems using propellants, which not only consumes a large amount of valuable propellant resources, thereby limiting the duration of the mission, but also may cause plume pollution to sensitive payloads such as optical equipment carried by satellites, affecting observation effectiveness and satellite performance.

[0004] Electromagnetic formation flying (EMFF) technology does not rely on traditional propellants, but uses the interaction of electromagnetic forces between satellites for control. Its core idea is to install high-temperature superconducting coils on each satellite in the formation, and use the electromagnetic field interaction generated by the coils to produce the required electromagnetic force and electromagnetic torque, serving as the power source for formation control. However, current research on electromagnetic formation flying satellites mainly focuses on control strategies based on linear error dynamics, with little research on nonlinear controllers directly based on nonlinear dynamics. Traditional vector mechanics methods based on Newton-Euler equations mainly focus on the relative dynamics behavior between spacecraft, which may not fully reveal the working mechanism of the formation system under the action of inter-satellite internal forces, especially in highly nonlinear systems such as electromagnetic formation flying. The strong nonlinear characteristics of inter-satellite electromagnetic forces make it difficult to achieve the desired control accuracy by simply applying linearized dynamics or control methods. SUMMARY

[0005] To solve the technical problems existing in the prior art, the purpose of the present application is to provide a dual-satellite electromagnetic formation flying dynamics modeling and control method of a port Hamilton system, which models the near-earth orbit electromagnetic formation system as a port Hamilton system and designs a passive control method based on interconnection-damping distribution for trajectory tracking control.

[0006] To achieve the above object, the application provides a method for modeling and controlling electromagnetic formation dynamics of a double-star port Hamilton system, comprising the following steps:

[0007] Step S1, defining two slave spacecraft centroid position coordinates in an ECI coordinate system and an LVLH coordinate system;

[0008] Step S2, obtaining kinetic energy and potential energy of the slave spacecraft in the ECI coordinate system according to the slave spacecraft centroid position coordinates;

[0009] Step S3, obtaining a Lagrange function L in the LVLH coordinate system according to the slave spacecraft centroid position coordinates and a coordinate system conversion matrix;

[0010] Step S4, establishing an Euler-Lagrange equation, selecting a generalized momentum coordinate p j according to a Legendre transformation, obtaining a Hamilton function H, and establishing a port Hamilton dynamics model;

[0011] Step S5, designing an electromagnetic formation controller according to a trajectory tracking passive control method based on interconnection-damping distribution;

[0012] Step S6, obtaining an electromagnetic satellite formation trajectory tracking magnetic dipole strength control law according to the electromagnetic formation controller.

[0013] According to one technical solution of the application, in step S1, the two slave spacecraft are slave spacecraft A and slave spacecraft B, and the following is obtained:

[0014] In the ECI coordinate system, the coordinates of the slave spacecraft A are defined as Q a =[X a , Y a , Z a ] T , the coordinates of the slave spacecraft B are defined as Q b =[X b , Y b , Z b ] T , and the centroid coordinates of the spacecraft formation are defined as Q0=[X0, Y0, Z0] T ;

[0015] In the LVLH coordinate system, the centroid coordinates of the spacecraft formation are defined as q0=[q0, 0, 0] T , the vector between the centroid of the spacecraft formation and the slave spacecraft A is defined as q a =[x a , y a , z a ] T , and the vector between the centroid of the spacecraft formation and the slave spacecraft B is defined as q b =[x b, y b , z b ] T ;

[0016] In the LVLH coordinate system, define the vector of the geocentric direction from the spacecraft A .

[0017] According to one technical solution of the present application, in the step S2, kinetic energy and potential energy of the spacecraft A in the ECI coordinate system are calculated according to the mass center position and velocity of the spacecraft A, and the potential energy includes gravitational potential energy and magnetic potential energy;

[0018] The kinetic energy of the spacecraft A in the ECI coordinate system is represented as:

[0019]

[0020] The gravitational potential energy of the spacecraft A in the ECI coordinate system under the influence of J2 perturbation is represented as:

[0021]

[0022] Wherein, R a represents the distance from the spacecraft A to the center of the earth, μ, R e , J2 respectively represent the earth gravity constant, the earth equatorial average radius and the second order band harmonic coefficient;

[0023] The magnetic field generated by the spacecraft B is represented as:

[0024]

[0025] Wherein, μ b represents the magnetic dipole intensity on the spacecraft B, ρ represents the vector from the spacecraft A to B, and μ0 is the vacuum permeability;

[0026] Then the magnetic potential energy of the spacecraft A is represented as:

[0027]

[0028] In the formula, μ a represents the magnetic dipole intensity on the spacecraft A, and T represents the transpose matrix.

[0029] According to one technical solution of the present application, in the step S3, specifically comprising:

[0030] The coordinate transformation is carried out between the ECI coordinate system and the LVLH coordinate system, and the formula is:

[0031]

[0032] Wherein, s o= sin (o), c o = cos (o), Ω represents the right ascension of the ascending node, θ represents the argument of perigee, o represents the orbit inclination of the mass center of the spacecraft formation, T r is an orthogonal matrix, that is,

[0033] Since is an anti-symmetric matrix, the anti-symmetric matrix function S is defined as Then, the following is obtained:

[0034]

[0035] In the undisturbed orbit, the orbit parameters are constant, the last two terms of formula (6) are equal to zero, and the anti-symmetric matrix function S is represented as S;

[0036] The conversion relationship between the ECI coordinate system and the LVLH coordinate system is represented as:

[0037]

[0038] In the LVLH coordinate system, the kinetic energy of spacecraft A is represented as:

[0039]

[0040] In the LVLH coordinate system, the gravitational potential energy of spacecraft A is represented as:

[0041]

[0042] Then, in the LVLH coordinate system, the Lagrange equation is represented as:

[0043]

[0044] According to one technical solution of the present application, in step S4, the following is specifically included:

[0045] Select q a as the basic variable, according to the Legendre transformation and the Euler-Lagrange equation, the relative motion dynamics equation is obtained

[0046]

[0047] After arrangement, the following is obtained:

[0048]

[0049] Wherein, g = [O3, I3] T ,

[0050] Then, the Hamilton function H is obtained from the Lagrange equation:

[0051]

[0052] Formula (13) is the port Hamiltonian dynamics model of the binary star electromagnetic formation.

[0053] According to one technical solution of the present invention, in step S5, the electromagnetic formation controller is designed based on the passive control method for trajectory tracking based on interconnection-damping allocation, specifically including:

[0054] Electromagnetic formation satellite trajectory tracking control, the closed-loop system is designed as follows:

[0055]

[0056] The closed-loop Hamiltonian function is designed as follows:

[0057]

[0058] in,

[0059] According to a technical solution of the present invention, in step S5, the existence condition of the passive trajectory tracking controller based on interconnection-damping allocation is:

[0060] By designing the parameters of the closed-loop system K j L j This ensures that the closed-loop system meets the existence condition of the passive control method for trajectory tracking with interconnected damping distribution, while the closed-loop system contracts to the desired reference trajectory, thus completing the trajectory tracking controller design.

[0061] Design Matrix L j Design an ideal reference trajectory Then we have:

[0062]

[0063] Substituting into the closed-loop system, the following condition is satisfied at the desired equilibrium point:

[0064]

[0065] For a closed-loop system, the second-order partial derivative of its Hamiltonian function is expressed as:

[0066]

[0067] If there exist positive constants α1 and α2, and α1 < α2, then the condition is satisfied:

[0068]

[0069] For a given positive number ∈, the following matrix N has no eigenvalue on the imaginary axis, then:

[0070]

[0071] where, A d =J d -R d .

[0072] According to one technical scheme of the present application, in the step S5, the electromagnetic formation controller is designed based on the interconnection-damping distribution trajectory tracking passive control method, and the form is:

[0073]

[0074] The control system of the electromagnetic satellite formation is made to track the expected trajectory x * (t) at an exponential speed.

[0075] According to one technical scheme of the present application, in the step S6, the control force of the satellite in the double-satellite electromagnetic formation is coupled with the formation position and the satellite magnetic field strength, and the practical form of the electromagnetic formation satellite trajectory tracking passive controller is represented as:

[0076]

[0077] Compared with the prior art, the present application has the following beneficial effects:

[0078] The present application provides a double-satellite electromagnetic formation dynamics modeling and control method of a port Hamilton system, which models the near-earth electromagnetic formation dynamics through the port Hamilton system, retains the nonlinear terms of the system, embodies the inherent dissipation characteristics of the system, avoids the reduction of the precision of the system caused by linearization, and guarantees the passivity of the system.

[0079] The interconnection-damping distribution trajectory tracking passive controller is designed for the port Hamilton system near-earth double-satellite electromagnetic formation dynamics system, which is beneficial to solve the formation reconstruction problem in the near-earth electromagnetic satellite formation flight, and the interconnection-damping distribution trajectory tracking passive controller has higher precision in the presence of nonlinearities and disturbances.

[0080] The interconnection-damping distribution trajectory tracking passive controller designed in the present application can provide a reference for formation design and control law design.

[0081] The present application provides the practical form of the electromagnetic formation satellite trajectory tracking passive controller, which has good engineering implementation value. BRIEF DESCRIPTION OF DRAWINGS

[0082] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the following will briefly introduce the drawings needed in the embodiments. Obviously, the drawings in the following description only show some embodiments of the present application, and for those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0083] Figure 1 A flow chart of a method for modeling and controlling double-star electromagnetic formation dynamics of a port Hamilton system according to an embodiment of the present application is schematically shown in FIG. 1;

[0084] Figure 2 A schematic diagram of a simulation experiment coordinate system according to an embodiment of the present application is schematically shown in FIG. 2;

[0085] FIG. 3(a-c) schematically shows simulation experiment result graphs according to an embodiment of the present application. DETAILED DESCRIPTION

[0086] In order to make the objects, technical solutions and advantages of the embodiments of the present application clearer, the following will clearly and completely describe the technical solutions in the embodiments of the present application with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are some but not all of the embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative effort belong to the scope of the present application.

[0087] As shown in FIG. 1, a method for modeling and controlling double-star electromagnetic formation dynamics of a port Hamilton system according to an embodiment of the present application includes the following steps: Figure 1 and Figure 2 As shown in FIG. 1, a method for modeling and controlling double-star electromagnetic formation dynamics of a port Hamilton system according to an embodiment of the present application includes the following steps:

[0088] Step S1, defining two slave spacecraft center-of-mass position coordinates in an ECI coordinate system and an LVLH coordinate system.

[0089] Suppose that there are two electromagnetic satellites in orbit, named slave spacecraft A and slave spacecraft B, as shown in FIG. 2. Figure 2 As shown in FIG. 2, in the ECI coordinate system, the slave spacecraft A coordinate is defined as Q a = [X a , Y a , Z a ] T , the slave spacecraft B coordinate is defined as Q b = [X b , Y b , Z b ] T , and the spacecraft formation center-of-mass coordinate is defined as Q0= [X0, Y0, Z0] T .

[0090] In the LVLH coordinate system, the mass center coordinate of the spacecraft formation is defined as q0=[q0, 0, 0] T , and the vector between the mass center of the spacecraft formation and spacecraft A is defined as q a =[x a , u a , z a ] T , and the vector between the mass center of the spacecraft formation and spacecraft B is defined as q b =[x b , y b , z b ] T .

[0091] Meanwhile, for the convenience of description, the vector pointing from the spacecraft A to the Earth center is defined as

[0092] In step S2, kinetic energy and potential energy of the spacecraft in the ECI coordinate system are obtained according to the mass center position coordinates of the spacecraft.

[0093] The potential energy includes gravitational potential energy and magnetic potential energy.

[0094] The kinetic energy of the spacecraft A in the ECI coordinate system is represented as:

[0095]

[0096] The gravitational potential energy of the spacecraft A in the ECI coordinate system under the influence of J2 perturbation is represented as:

[0097]

[0098] wherein R a represents the distance between the spacecraft A and the Earth center, μ, R e , J2 represent the Earth gravitational constant, the Earth equatorial average radius and the second-order zonal harmonic coefficient respectively;

[0099] By regarding the electromagnetic coil on each satellite as a controllable magnetic dipole, the magnetic field generated by the spacecraft B is represented as:

[0100]

[0101] wherein μ b represents the magnetic dipole strength on the spacecraft B, ρ represents the vector pointing from the spacecraft A to B, and μ0 is the vacuum permeability;

[0102] The magnetic potential energy of the spacecraft A is represented as:

[0103]

[0104] In the formula μ a Let T represent the magnetic dipole strength from spacecraft A, and let T represent the transpose matrix.

[0105] R a The distance between spacecraft A and the Earth's center, μ, is the Earth's gravitational constant, which is 3.98603 × 10⁻⁶. 14 m 3 / s 2 R e The average radius of the Earth's equator is taken as 6878.137 km, and the second-order zonal harmonic coefficient of J2 is taken as 1.08263 × 10⁻⁶. -3 μ b Designed for strength, always 5×10 5 [1, 1, 1] T .

[0106] Step S3: Based on the coordinates of the spacecraft's center of mass and the coordinate system transformation matrix, obtain the Lagrangian function L in the LVLH coordinate system.

[0107] To describe the dynamics of a spacecraft following another in the LVLH coordinate system, the following coordinate transformation is required between the LVLH and ECI frames, as shown in the formula:

[0108]

[0109] Among them, s o =sin(o), c o =cos(o), where Ω represents the right ascension of the ascending node, θ represents the argument of perigee, and i represents the orbital inclination of the center of mass of the spacecraft formation;

[0110] In the formula T r It is an orthogonal matrix, that is By differentiating both sides of the formula, we can obtain Right now Since it is an antisymmetric matrix, we can define an antisymmetric matrix function. Then we have:

[0111]

[0112] In an undisturbed orbit, the orbital parameters are constant, and the last two terms of formula (6) are equal to zero. The antisymmetric matrix function is then expressed as S.

[0113] The transformation relationship between the ECI coordinate system and the LVLH coordinate system is expressed as follows:

[0114]

[0115] In the LVLH coordinate system, the kinetic energy of spacecraft A can be expressed as:

[0116]

[0117] In the LVLH coordinate system, the gravitational potential energy of spacecraft A can be expressed as:

[0118]

[0119] In the LVLH coordinate system, the Lagrange equation is expressed as:

[0120]

[0121] For example, the orbital parameters can be set as r0 = 6878.137 km, e = 0, i = 60°, ω = 90°, Ω = 0°, υ = 0°.

[0122] Step S4: Establish the Euler-Lagrange equations and, based on the Legendre transformation, select the generalized momentum coordinate p. j We obtain the Hamiltonian function H and establish a port Hamiltonian dynamic model.

[0123] Choose q a As a basis variable, according to the Legendre transformation And from the Euler-Lagrange equations, we obtain the equations of relative motion dynamics.

[0124]

[0125] The results were:

[0126]

[0127]

[0128] Where g = [O3, I3] T ,

[0129] The Hamiltonian function H is derived from the Lagrange equation:

[0130]

[0131] Formula (13) is the port Hamiltonian dynamics model of the binary star electromagnetic formation.

[0132] In the context of near-Earth electromagnetic satellite formation dynamics, let x represent the function variable, i.e., refer to (q) a p a ).

[0133] Step S5: Design an electromagnetic formation controller based on the passive control method for trajectory tracking based on interconnection-damping allocation.

[0134] According to the trajectory tracking passive control method based on interconnection-damping assignment, an electromagnetic formation controller is designed, and specifically includes:

[0135] The electromagnetic formation satellite trajectory tracking control is designed as a closed loop system:

[0136]

[0137] The closed loop Hamilton function is designed as:

[0138]

[0139] Wherein,

[0140] According to one of the technical solutions of the present application, in the step S5, the trajectory tracking passive controller based on interconnection-damping assignment exists under the condition that:

[0141] In order to complete the design of the trajectory tracking controller, it is necessary to design the closed loop system parameter term K j , L j , so that the closed loop system satisfies the existence condition of the controller of the trajectory tracking passive control method based on interconnection-damping assignment, and the closed loop system is contracted to the desired reference trajectory;

[0142] The design matrix L j , the ideal reference trajectory is designed, and then:

[0143]

[0144] The closed loop system is brought in, and the condition at the desired equilibrium point is satisfied:

[0145]

[0146] For the closed loop system, the second order partial derivative of the Hamilton function is expressed as:

[0147]

[0148] If there are normal numbers α1, α2, and α1<α2, then the condition is satisfied:

[0149]

[0150] For a given normal number ∈, the following matrix N has no eigenvalue on the imaginary axis, then:

[0151]

[0152] Wherein, A d = J d -Rd .

[0153] This satisfies the existence condition of controller u in the passive control method for trajectory tracking with interconnected damping distribution, i.e., the existence of a controller.

[0154] The electromagnetic formation controller designed based on the passive control method for trajectory tracking using interconnection-damping allocation is as follows:

[0155]

[0156] The control system of the electromagnetic satellite formation tracks the desired trajectory x at an exponential rate. * (t).

[0157] In step 5, the system reference trajectory is set as follows:

[0158]

[0159] Where, α 0j and β 0j These are the normally equal in-plane and out-of-plane phase angles, with parameter c. 1j c 2j and c 3j It is the integral constant that determines the trajectory configuration. In the PCO formation configuration, it can be taken as... c 2j =0, α 0j =0,β 0j =0.

[0160] K j =10 -4 diag(2.4, 4, 6) can be used to verify that conditions (19) and (20) are met, thus satisfying the condition that the controller exists.

[0161] Step S6: Obtain the electromagnetic satellite formation trajectory tracking magnetic dipole intensity control law based on the electromagnetic formation controller.

[0162] For electromagnetic formation satellites, the change in control force is achieved by adjusting the strength of their magnetic dipoles. This adjustment is coupled to the formation position and the satellite's magnetic field strength. Therefore, the control force is not simply applied directly to the satellite; instead, it is based on the current magnetic field strength, altering the satellite's own magnetic dipole strength to match the required control force. Thus, a practical form of a passive controller for electromagnetic formation satellite trajectory tracking should be as follows:

[0163]

[0164] Simulation Experiment

[0165] Simulation experiments are carried out with the above parameters to obtain the satellite formation trajectory and control input.

[0166] Figure 3(a) shows the motion of the spacecraft A and the spacecraft B relative to the reference trajectory, and it can be seen that the system completes convergence within 5200s from the condition that the two electromagnetic satellites have initial errors relative to the reference trajectory, and each satellite gradually approaches the target point under the action of the thrust, and finally tracks the ideal reference trajectory. At the terminal time, the three-axis accuracy is less than 0.001m.

[0167] Figure 3(b) shows the error change of the spacecraft A in the x, y and z axes.

[0168] Figure 3(c) shows the magnetic dipole intensity change of the spacecraft A in the x, y and z axes, which conforms to the magnetic dipole intensity range.

[0169] In the port Hamilton system framework, the near-earth orbit electromagnetic formation satellite dynamics is modeled, the non-linear link of the system can be effectively reserved, the passive characteristics of the system are reflected, and the modeling accuracy is improved. More importantly, the electromagnetic formation is also a typical Euler-Lagrange system, which can be modeled in the port Hamilton system framework, and the passive control method is applied to the energy-based control of the dynamic model, and the control accuracy is improved.

[0170] Therefore, the above-mentioned port Hamilton system-based double-satellite electromagnetic formation dynamics modeling and control method is adopted, the near-earth electromagnetic formation dynamics is modeled by the port Hamilton system, the non-linear term of the system is reserved, the inherent dissipation characteristics of the system are reflected, the precision of the system is not reduced due to linearization, and the passivity of the system is ensured. For the port Hamilton system-based near-earth double-satellite electromagnetic formation dynamics system, an interconnection-damping distribution trajectory tracking passive controller is designed, which is beneficial to solve the formation reconstruction problem in near-earth electromagnetic satellite formation flight, and the interconnection-damping distribution trajectory tracking passive controller has higher precision in the case of nonlinearity and disturbance.

[0171] Further, a practical form of the electromagnetic formation satellite trajectory tracking passive controller is given, and the electromagnetic formation satellite trajectory tracking passive controller has good engineering implementation value.

[0172] It should be noted that although the above embodiments of the present application are illustrative, this is not a limitation of the present application, and therefore the present application is not limited to the above specific embodiments. Any other embodiments obtained by those skilled in the art under the inspiration of the present application without departing from the principles of the present application are considered to be within the protection scope of the present application.

Claims

1. A method for dynamic modeling and control of binary star electromagnetic formations in a port Hamiltonian system, characterized in that, Includes the following steps: Step S1: Define two coordinates for the position of the spacecraft's center of mass in the ECI coordinate system and the LVLH coordinate system; Step S2: Based on the coordinates of the spacecraft's center of mass, obtain the kinetic and potential energy of the spacecraft in the ECI coordinate system; Step S3: Based on the coordinates of the spacecraft's center of mass and the coordinate system transformation matrix, obtain the Lagrangian function in the LVLH coordinate system. ; Step S4: Establish the Euler-Lagrange equations and select the generalized momentum coordinates according to the Legendre transformation. The Hamiltonian function is obtained. Establish a port Hamiltonian dynamics model; Step S5: Design an electromagnetic formation controller based on the passive control method for trajectory tracking based on interconnection-damping allocation; Step S6: Obtain the electromagnetic satellite formation trajectory tracking magnetic dipole intensity control law based on the electromagnetic formation controller; In step S1, the two slave spacecraft are slave spacecraft A and slave spacecraft B, then: In the ECI coordinate system, the coordinates from spacecraft A are defined as follows: From the coordinates of spacecraft B: The coordinates of the center of mass of the spacecraft formation are ; In the LVLH coordinate system, the coordinates of the center of mass of a spacecraft formation are defined as follows: Define the vector between the centroid of the spacecraft formation and spacecraft A as follows: The vector between the center of mass of the spacecraft formation and spacecraft B is... ; In the LVLH coordinate system, define the vector pointing from the geocenter to spacecraft A. ; Step S4 specifically includes: choose As a basis variable, according to the Legendre transformation And the Euler-Lagrange equations, yielding the equations of relative motion dynamics. （11） The results were: （12） （13） in, , , ; Then Hamiltonian function From the Lagrange equations, we can derive: （14） Formula (13) is the port Hamiltonian dynamics model of the binary star electromagnetic formation; In step S5, an electromagnetic formation controller is designed based on the passive control method for trajectory tracking using interconnection-damping allocation, specifically including: Electromagnetic formation satellite trajectory tracking control, the closed-loop system is designed as follows: （15） The closed-loop Hamiltonian function is designed as follows: （16） in, ; In step S5, the existence condition of the passive trajectory tracking controller based on interconnection-damping allocation is: By designing the parameters of the closed-loop system This ensures that the closed-loop system meets the existence condition of the passive control method for trajectory tracking with interconnected damping distribution, while the closed-loop system contracts to the desired reference trajectory, thus completing the trajectory tracking controller design. Design Matrix Design an ideal reference trajectory Then we have: （17） Substituting this into the closed-loop system, the following condition is satisfied at the desired equilibrium point: ； For a closed-loop system, the second-order partial derivative of its Hamiltonian function is expressed as: （18） If positive numbers exist ,and Then the condition is met: （19） For a given positive constant The following matrix If there are no eigenvalues ​​on the imaginary axis, then: （20） in, , ; In step S5, the electromagnetic formation controller is designed based on the passive control method for trajectory tracking using interconnection-damping allocation as follows: （21） The control system of the electromagnetic satellite formation tracks the desired trajectory at an exponential rate. .

2. The method for modeling and controlling the dynamics of binary star electromagnetic formations in a port Hamiltonian system according to claim 1, characterized in that, In step S2, the kinetic and potential energy of spacecraft A in the ECI coordinate system are calculated based on the position and velocity of the center of mass of spacecraft A. The potential energy includes gravitational potential energy and magnetic potential energy. The kinetic energy of spacecraft A in the ECI coordinate system is expressed as follows: （1） The spacecraft A is subjected to in the ECI coordinate system The gravitational potential energy under the influence of perturbation is expressed as: （2） in, This represents the distance from spacecraft A to the Earth's center. These represent the Earth's gravitational constant, the Earth's average equatorial radius, and the second-order zonal harmonic coefficients, respectively. The magnetic field generated from spacecraft B is represented as follows: （3） in, This indicates the strength of the magnetic dipole on spacecraft B. This represents the vector pointing from spacecraft A to spacecraft B. It is the vacuum permeability; The magnetic potential energy of spacecraft A can then be expressed as: （4） In the formula Let T represent the magnetic dipole strength from spacecraft A, and let T represent the transpose matrix.

3. The method for modeling and controlling the dynamics of binary star electromagnetic formations in a port Hamiltonian system according to claim 2, characterized in that, Step S3 specifically includes: The formula for coordinate transformation between the ECI coordinate system and the LVLH coordinate system is as follows: （5） in, , Indicates the right ascension of the ascending node. Indicates the argument of perigee. The orbital inclination that represents the center of mass of a spacecraft formation. It is an orthogonal matrix, that is ; because Define the antisymmetric matrix function for an antisymmetric matrix. Then we have: （6） In an undisturbed orbit, the orbital parameters are constant, and the last two terms of equation (6) are equal to zero. Therefore, the antisymmetric matrix function is expressed as: ; The transformation relationship between the ECI coordinate system and the LVLH coordinate system is expressed as follows: （7） In the LVLH coordinate system, the kinetic energy of spacecraft A can be expressed as: （8） In the LVLH coordinate system, the gravitational potential energy of spacecraft A can be expressed as: （9） In the LVLH coordinate system, the Lagrange equation is expressed as: （10）。 4. The method for modeling and controlling the dynamics of binary star electromagnetic formations in a port Hamiltonian system according to claim 3, characterized in that, In step S6, the control force of the satellites in the dual-satellite electromagnetic formation is coupled with the formation position and the satellite magnetic field strength. Therefore, the practical form of the passive controller for tracking the satellite trajectory of the electromagnetic formation is expressed as follows: （22）。

Images