Electromagnetic docking Tube model predictive control method for on-orbit filling

Abstract

The invention discloses an on-orbit filling-oriented electromagnetic docking Tube model prediction control method, which comprises the following steps: S1, establishing a spacecraft orbit-electromagnetic-liquid coupling dynamic model, and converting the model into a discrete state space form; s2, designing a constraint tightening method based on a self-adaptive robust invariant set for interference terms existing in the kinetic model, and ensuring recursion feasibility under a disturbance bounded condition; s3, constructing an energy consumption-precision multi-objective optimization function, and adopting a rolling time domain optimization framework to realize dynamic balance of energy consumption and docking precision; s4, constructing a control Lyapunov function, ensuring the stability of the algorithm under the constraint condition based on a robust invariant set, and further obtaining an optimal controller; and S5, substituting the Tube model predictive controller into a closed-loop system, so that the relative position of the spacecraft reaches a stable state, and coupling control is realized. Under the coexistence condition of system time varying and external and storage tank liquid disturbance, accurate control over electromagnetic docking can be achieved, control energy consumption is low, and the application range is wide.

Description

Technical Field

[0001] The present invention belongs to the field of on-orbit servicing of spacecraft, and particularly relates to an electromagnetic docking Tube model predictive control method for on-orbit refueling. Background Art

[0002] The on-orbit refueling technology of spacecraft has become a research focus in the field of space operation due to its core role in extending mission cycles and reducing launch costs. The realization of this technology depends on high-precision and high-safety docking control. However, traditional service spacecraft carrying large-capacity propellant tanks face multiple challenges: First, the sloshing and distribution changes of liquid fuel cause the interaction force between liquid and solid, resulting in a sharp increase in the complexity of attitude-orbit coupling dynamics modeling; Second, the flammable and explosive characteristics of the propellant require strict avoidance of mechanical collisions and thermodynamic risks during the docking process; Third, the traditional docking method based on thrusters has inherent defects such as plume contamination, high fuel consumption, and high impact loads, making it difficult to meet the requirements of long-term on-orbit servicing.

[0003] Due to the non-linear attenuation characteristic of electromagnetic force with the increase of distance and its strong coupling with relative attitude, it is difficult to establish a high-precision model. At the same time, multi-source uncertainties such as space magnetic field interference and propellant sloshing disturbance significantly reduce the robustness of the controller, presenting many technical bottlenecks. Traditional spacecraft electromagnetic docking control systems mainly focus on linear time-invariant systems or single disturbance compensation, or use model-free control. These methods lack comprehensive consideration of the time-varying characteristics of elliptical orbits, multi-objective optimization requirements, and multi-source complex disturbances. Therefore, these methods have limitations in energy consumption optimization, constraint compatibility, and computational efficiency, and are difficult to meet the requirements of increasingly complex on-orbit servicing tasks.

[0004] Model predictive control has shown unique advantages in the control of complex systems such as vehicle fleets, unmanned aerial vehicles, and spacecraft due to its rolling optimization, explicit constraint handling, and multi-variable coordination capabilities. However, standard model predictive control is sensitive to disturbances and is difficult to directly apply to strongly uncertain scenarios. Tube model predictive control has become an efficient control strategy for dealing with bounded uncertainties by introducing a robust invariant set and a feedback compensation mechanism to confine the system state within the neighborhood of the nominal trajectory, while ensuring robustness and reducing computational complexity. However, general Tube model predictive control requires the controlled system to be a linear time-invariant system. Since the electromagnetic docking system of spacecraft in elliptical orbits is time-varying, traditional Tube model predictive control is difficult to directly apply. In addition, there is no relevant research on the electromagnetic docking Tube model predictive control for on-orbit refueling of spacecraft in elliptical orbits. Summary of the Invention

[0005] In view of this, the technical problem to be solved by the present invention is to provide an electromagnetic docking Tube model predictive control method for on-orbit refueling. Aiming at the above deficiencies in the prior art, the electromagnetic docking Tube model predictive control method for on-orbit refueling provided by the present invention establishes a liquid-solid-magnetic coupling dynamics model of an elliptical orbit spacecraft under external disturbances and liquid interference forces, and designs an improved Tube model predictive controller suitable for elliptical orbits, which is compatible with the time-varying characteristics of elliptical orbit parameters, and proposes an energy consumption-accuracy multi-objective optimization index, which can achieve the dynamic balance of control performance and energy consumption, with higher control accuracy, lower control energy consumption, and further improved control efficiency.

[0006] In order to achieve the above invention purpose, the technical solution adopted by the present invention is: an electromagnetic docking Tube model predictive control method based on on-orbit refueling, including the following steps:

[0007] S1. Establish a spacecraft orbit-electromagnetic-liquid coupling dynamics model and convert it into a discrete state space form;

[0008] S2. For the interference terms existing in the dynamics model, design a constraint tightening method based on an adaptive robust invariant set to ensure the recursive feasibility under the condition of bounded disturbances;

[0009] S3. Construct an energy consumption-accuracy multi-objective optimization function and adopt a rolling horizon optimization framework to achieve the dynamic trade-off between energy consumption and docking accuracy;

[0010] S4. Construct a control Lyapunov function, and ensure the stability of the algorithm under the constraint conditions based on a robust invariant set, and then obtain an optimal controller;

[0011] S5. Substitute the Tube model predictive controller into the closed-loop system to make the relative position of the spacecraft reach a stable state and achieve coupled control.

[0012] Further, in the step S1, the continuous state space form of the spacecraft orbit-electromagnetic-liquid coupling dynamics model is:

[0013]

[0014] In the formula, represents the system state variable, u(t) = [F Cx F Cy F Cz T represents the system control variable, F Cx 、F Cy 、F Cz represent the control forces in the three-axis directions; w(t) = B c w0(t) represents the disturbance variable received by the system, where w0(t) = [d​x d y d z T , d x , d y , d z represent the interference forces in the three-axis directions, which are composed of external interference and liquid acting forces. A c represents the steady part of the electromagnetic docking system matrix, ΔA c is the time-varying part of the isolated electromagnetic docking system matrix, B c is the control matrix of the electromagnetic docking system.

[0015] Further, the calculation method of the electromagnetic force received by the spacecraft is:

[0016]

[0017] In the formula, μ T and μ C are the total combined magnetic moments of the two spacecrafts, which are synthesized from the magnetic moments generated by multiple coils on the spacecrafts. N i i i and N j i j are respectively the ampere-turns of the i-th coil of the target spacecraft and the j-th coil of the tracking spacecraft, μ0 is the magnetic permeability of vacuum, and r is the relative position of the spacecrafts.

[0018] Further, the calculation method of the interference variables received by the system is

[0019] w0(t) = ε + F L

[0020] F L = Ne + F b

[0021]

[0022] In the formula: ε is the random bounded interference, F L is the liquid acting force, N is the magnitude of the normal component of the liquid acting force, F b is the tangential liquid acting force, m s is the equivalent pulsating sphere mass of the liquid, r S is the position vector of the pulsating sphere centroid in the spacecraft body coordinate system, r S is its modulus, e is its unit vector, Ω is the rotational angular velocity of the spacecraft in the inertial system, V C is the translational velocity of the spacecraft in the inertial system, R t is the equivalent radius of the storage tank, ω S is the rotational angular velocity of the pulsating sphere in the body system, σ is the liquid surface tension, μ​l is the kinematic viscosity coefficient of the liquid, V u is the relative translational velocity of the pulsating sphere at the contact point in the tangential direction, V u =-e×(e×V S +Lω s ), V S is the translational velocity of the pulsating sphere relative to this system, and L is the radius of the pulsating sphere. When the radius of the pulsating sphere reaches the minimum value, the radial velocity component e·(e·V S ) reverses. At this time, there is not only a force between the pulsating sphere and the spacecraft, but also a momentum exchange will occur:

[0023]

[0024] In the formula: represents the translational velocity of the spacecraft's center of mass immediately after the collision; κ∈[0,1] is the collision coefficient.

[0025] Furthermore, in the step S1, the discrete state space form of the spacecraft orbit-attitude vibration coupling dynamics model is

[0026] x(k + 1)=(A + ΔA(k))x(k)+Bu(k)+w(k)

[0027] In the formula, is the state transition matrix of the system at time t k ,

[0028] Furthermore, in the step S2, the designed constraint tightening method for the adaptive robust invariant set includes the following constraint C(k):

[0029]

[0030] X t ={x∈R n |Yx≤q}, Y∈R r×n , q∈R r

[0031]

[0032] w * (k)∈(A K +ΔA(k + 1))(A K +ΔA(k)) N-1 W

[0033]

[0034] In the formula, X is a closed convex set, U and W are compact convex sets containing the origin, and X t is the terminal constraint set, x* (k) and u * (k) is the predicted system state and input at time k, is the state sequence obtained by the prediction system at the current time k with a prediction horizon N, is the control sequence obtained by the prediction system at the current time k with a prediction horizon N, A K = A + BK, matrix A K + ΔA(k) is Schur, ~ represents the Pontryagin difference of two sets, X i represents the state admissible set of the prediction system at time k + i, U i represents the input admissible set of the prediction system at time k + i, X f for the system x(k + 1) = (A K + ΔA(k + 1))x(k) + w * (k) is a robust invariant set.

[0035] Furthermore, in the step S3, the energy consumption-accuracy multi-objective optimization function is:

[0036]

[0037] where Q and R are given positive definite and semi-positive definite matrices respectively.

[0038] Furthermore, in the step S4, the control Lyapunov function is expressed as:

[0039]

[0040] where d represents the distance in the normed space (R n , ||·|| p ), and the norm ||·|| p is defined as

[0041] Furthermore, in the step S4, the designed optimal controller is expressed as:

[0042]

[0043] where represents the first element of the control sequence of the prediction system when the problems described in claims 6 and 7 are solvable.

[0044] The beneficial effects of the present invention are:

[0045] (1) The present invention proposes an electromagnetic docking Tube model predictive control method for on-orbit refueling, which is applicable to the electromagnetic docking control of spacecraft in elliptical orbits considering external disturbances and liquid disturbances, and can achieve the dynamic balance of control performance and energy consumption, with low control energy consumption and high precision.

[0046] (2) The present invention has a wide range of applications and can be applied not only to the electromagnetic docking control for on-orbit refueling of spacecraft, but also to the electromagnetic docking control for on-orbit construction and assembly of future ultra-large space infrastructures. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below in conjunction with the accompanying drawings, where:

[0048] Figure 1 It is a flowchart of the electromagnetic docking Tube model predictive control method for on-orbit refueling.

[0049] Figure 2 It is the relative position change curve of the spacecraft. x, y and z represent the change amounts of the relative positions of the three axes, and m represents the unit of the relative position in meters.

[0050] Figure 3 It is the relative velocity change curve of the spacecraft, v x , v y and v z represent the change amounts of the relative velocities of the three axes, and m / s represents the unit of the relative velocity in meters per second.

[0051] Figure 4 It is the required electromagnetic force change curve, u x , u y and u z represent the change amounts of the electromagnetic forces of the three axes, and N represents the unit of the electromagnetic force in Newtons.

[0052] Figure 5 It is the required magnetic moment of the tracking spacecraft, μ x , μ x and μ x represent the change amounts of the magnetic moments of the three axes of the tracking spacecraft, and Am 2 represents the unit of the magnetic moment in ampere square meters. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0053] The specific embodiments of the present invention will be described below in conjunction with the accompanying drawings to facilitate those skilled in the art to understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, as long as various changes are within the spirit and scope of the present invention defined and determined by the appended claims, these changes are obvious, and all inventions created using the concept of the present invention are within the scope of protection.

[0054] The embodiments of the present invention provide an electromagnetic docking Tube model predictive control method for on-orbit refueling, including the following steps:

[0055] S1. Establish a spacecraft orbit - electromagnetic - liquid coupling dynamics model and convert it into a discrete state - space form;

[0056] S2. For the disturbance terms in the spacecraft orbit - electromagnetic - liquid coupling dynamics model, design a constraint tightening method based on an adaptive robust invariant set, and adjust the contraction strategy to be compatible with the time - varying characteristics of the system to ensure the recursive feasibility under bounded disturbance conditions;

[0057] S3. Construct an energy - consumption - accuracy multi - objective optimization function and use a receding - horizon optimization framework to achieve the dynamic trade - off between energy consumption and docking accuracy;

[0058] S4. Construct a control Lyapunov function, and based on the robust invariant set, ensure the stability of the algorithm under constraint conditions, and then obtain the optimal controller;

[0059] S5. Substitute the Tube model predictive controller into the closed - loop system to make the relative position of the spacecraft reach a stable state and achieve coupled control.

[0060] In step S1 of the embodiment of the present invention, the continuous state - space form of the spacecraft orbit - electromagnetic - liquid coupling dynamics model is:

[0061]

[0062] In the formula, represents the system state variable, u(t)=[F Cx F Cy F Cz T represents the system control variable, F Cx 、F Cy 、F Cz represent the control forces in the three - axis directions; w(t)=B c w0(t) represents the disturbance variable received by the system, where w0(t)=[d x d y d z T , d x 、d y 、d z represent the disturbance forces in the three - axis directions, which are composed of external disturbances and liquid forces, A c represents the constant part of the electromagnetic docking system matrix, ΔA c is the time - varying part of the separated electromagnetic docking system matrix, B c is the control matrix of the electromagnetic docking system.

[0063] Specifically, the calculation method of the electromagnetic force received by the spacecraft is:

[0064] ​​

[0065]

[0066] Wherein, μ T and μ C are the total combined magnetic moments of the two spacecrafts, which are synthesized by the magnetic moments generated by multiple coils on the spacecrafts. N i i i and N j i j are respectively the ampere-turns of the i-th coil of the target spacecraft and the j-th coil of the tracking spacecraft. μ0 is the magnetic permeability of vacuum, and r is the relative position of the spacecrafts.

[0067] Specifically, the calculation method of the interference variables received by the system is

[0068] w0(t) = ε + F L

[0069] F L = Ne + F b

[0070]

[0071] Wherein: ε is the random bounded interference, F L is the liquid acting force, N is the magnitude of the normal component of the liquid acting force, F b is the tangential acting force of the liquid, m s is the equivalent pulsating sphere mass of the liquid, r S is the position vector of the pulsating sphere centroid in the spacecraft body coordinate system, r S is its modulus, e is its unit vector, Ω is the rotational angular velocity of the spacecraft in the inertial system, V C is the translational velocity of the spacecraft in the inertial system, R t is the equivalent radius of the storage tank, ω S is the rotational angular velocity of the pulsating sphere in the body system, σ is the liquid surface tension, μ l is the kinematic viscosity coefficient of the liquid; V u is the relative translational velocity of the pulsating sphere at the contact point in the tangential direction, V u = -e × (e × V S + Lω s ), V S is the translational velocity of the pulsating sphere relative to the body system, and L is the radius of the pulsating sphere. When the radius of the pulsating sphere reaches the minimum value, the radial velocity component e·(e·V S ) is reversed. At this time, there is not only an acting force between the pulsating sphere and the spacecraft, but also a momentum exchange will occur:

[0072]

[0073] In the formula: represents the translational velocity of the spacecraft's center of mass at the instant after the collision; κ ∈ [0, 1] is the collision coefficient.

[0074] Furthermore, through the zero-order hold method, the continuous state-space form of the orbit-electromagnetic-liquid coupling dynamics model is transformed into a discrete state-space form:

[0075] x(k + 1) = (A + ΔA(k))x(k) + Bu(k) + w(k)

[0076] In the formula, is the state transition matrix of the system at time t k moment,

[0077] In step S2 of the embodiment of the present invention, under the condition of considering the time-varying nature of the system brought about by the elliptical orbit, a method for tightening the constraints of the adaptive robust invariant set is designed; specifically, the introduced constraint C(k) includes:

[0078]

[0079] X t = {x ∈ R n | Yx ≤ q}, Y ∈ R r×n , q ∈ R r

[0080]

[0081] w * (k) ∈ (A K + ΔA(k + 1))(A K + ΔA(k)) N-1 W

[0082]

[0083] In the formula, X is a closed convex set, U and W are compact convex sets containing the origin, X t is the terminal constraint set, x * (k) and u * (k) are the predicted system states and inputs at time k, is the state sequence of the predicted system at the current time k with a prediction horizon of N, is the control sequence of the predicted system at the current time k with a prediction horizon of N, A K = A + BK, the matrix A K + ΔA(k) is Schur, ~ represents the Pontryagin difference of two sets, X i represents the state allowable set of the predicted system at time k + i, U iDenote the input admissible set of the prediction system at time \(k + i\), \(X\). f For the system \(x(k + 1)=(A K +\Delta A(k + 1))x(k)+w * (k)\), it is a robust invariant set.

[0084] Based on the above constraints, by adjusting the contraction strategy to be compatible with the time-varying characteristics of the angular velocity of the elliptical orbit, the recursive feasibility under bounded disturbances can be guaranteed.

[0085] In step S3 of the embodiment of the present invention, to achieve the dynamic trade-off between energy consumption and docking accuracy, the following energy consumption-accuracy multi-objective optimization function is designed:

[0086]

[0087] In the formula, \(Q\) and \(R\) are given positive definite and semi-positive definite matrices respectively.

[0088] In step S4 of the embodiment of the present invention, the designed control Lyapunov function is expressed as:

[0089]

[0090] In the formula, \(d\) represents the distance in the canonical space (\(R n ,\|\cdot\| p ), and the norm \(\|\cdot\| p is defined as

[0091] Based on the designed control Lyapunov function, the designed robust inverse optimal controller is obtained and expressed as:

[0092]

[0093] In the formula, represents the first element of the control sequence of the prediction system when the problems described in claims 6 and 7 are solvable.

[0094] In step S5 of the embodiment of the present invention, substituting the Tube model predictive controller into the spacecraft closed-loop dynamics system to make the relative position and velocity of the spacecraft reach a stable state can effectively achieve the electromagnetic docking control of the spacecraft.

[0095] The specific working process of this embodiment is as follows:

[0096] Step 1: Given the semi-major axis \(a\), eccentricity \(e\) of the orbit where the center of mass of the spacecraft electromagnetic docking system is located, and the masses \(m C \), \(m T \) of the two spacecraft, set the prediction horizon \(N\), sampling time \(T\), set the initial state \(x(0)\) and the initial true anomaly \(\theta(0)\), and set the time \(k = 0\);

[0097] Step 2: Solve the optimization problem

[0098]

[0099] Obtain the current optimal control input

[0100] Step 3: Update the spacecraft state x(k) and the current true anomaly θ(k). If the docking of the two spacecrafts is completed, the algorithm terminates; otherwise, update k to k + 1, return to Step 2, and solve iteratively.

[0101] Finally, obtain the optimal control input at each moment, and achieve high-precision control of the electromagnetic docking of the spacecraft under external disturbances and liquid forces.

[0102] In a specific example of the present invention, the control results obtained based on the above method are as Figure 2-5 shown, Figure 2 representing the relative position change curve of the spacecraft, Figure 3 representing the relative velocity change curve of the spacecraft, Figure 4 representing the required electromagnetic force change curve, Figure 5 representing the required magnetic moment change curve of the tracking spacecraft, indicating that the control method has good control effects.

[0103] It should be noted that the method of the present invention has a wide range of applications, and can not only be applied to the electromagnetic docking control of on-orbit refueling of spacecrafts, but also be used for the electromagnetic docking control of on-orbit construction and assembly of future ultra-large space infrastructures.

[0104] Specific embodiments are applied in the present invention to elaborate the principles and implementation manners of the present invention. The descriptions of the above embodiments are only used to help understand the method of the present invention and its core idea; at the same time, for those of ordinary skill in the art, according to the idea of the present invention, there will be changes in the specific implementation manners and application scopes. In summary, the content of this specification should not be construed as a limitation to the present invention.

[0105] Those of ordinary skill in the art will realize that the embodiments described herein are for helping the reader understand the principles of the present invention, and should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those of ordinary skill in the art can make various specific deformations and combinations that do not depart from the essence of the present invention according to the technical revelations disclosed in the present invention, and these deformations and combinations are still within the protection scope of the present invention.

Claims

1. An electromagnetic docking Tube model predictive control method for on-orbit refueling, characterized in that It includes the following steps: S1. Establish a spacecraft orbit - electromagnetic - liquid coupling dynamics model and convert it into a discrete state - space form; S2. For the disturbance terms existing in the dynamics model, design a constraint tightening method based on an adaptive robust invariant set to ensure recursive feasibility under the condition of bounded disturbances; S3. Construct an energy consumption - accuracy multi - objective optimization function and adopt a receding - horizon optimization framework to achieve dynamic trade - off between energy consumption and docking accuracy; S4. Construct a control Lyapunov function and ensure the stability of the algorithm under constraint conditions based on a robust invariant set, and then obtain an optimal controller; S5. Substitute the Tube model predictive controller into the closed - loop system to make the relative position of the spacecraft reach a stable state and achieve coupling control.

2. The electromagnetic docking Tube model predictive control method for on-orbit refueling according to claim 1, characterized in that, In the step S1, the continuous state - space form of the spacecraft orbit - electromagnetic - liquid coupling dynamics model is: In the formula, represents the system state variable, u(t) = [F Cx F Cy F Cz T represents the system control variable, F Cx , F Cy , F Cz represent the control forces in the three-axis directions; w(t) = B c w0(t) represents the disturbance variable acting on the system, where w0(t) = [d x d y d z T , d x , d y , d z represent the disturbance forces in the three-axis directions, which are composed of external disturbances and liquid forces, A c represents the constant part of the electromagnetic docking system matrix, ΔA c is the time-varying part of the isolated electromagnetic docking system matrix, B c is the control matrix of the electromagnetic docking system.​​ 3. The electromagnetic docking Tube model predictive control method for on-orbit refueling according to claim 2, characterized in that, The calculation method of the electromagnetic force received by the spacecraft is: where μ T and μ C are the total combined magnetic moments of the two spacecrafts, which are synthesized from the magnetic moments generated by multiple coils on the spacecrafts. N i i i and N j j j are the ampere-turns of the i-th coil of the target spacecraft and the j-th coil of the chaser spacecraft respectively. μ0 is the magnetic permeability of vacuum, and r is the relative position of the spacecrafts.

4. The continuous state space form based on the track-electromagnetic-liquid coupling dynamics model according to claim 2, wherein The calculation method of the disturbance variable received by the system is w0(t) = ε + F L F L = Ne + F b where ε is a randomly bounded disturbance, F L is the liquid force, N is the magnitude of the normal component of the liquid force, F b is the tangential liquid force, m s is the equivalent pulsating sphere mass of the liquid, r S is the position vector of the pulsating sphere centroid in the spacecraft body coordinate system, r S is its modulus, e is its unit vector, Ω is the rotational angular velocity of the spacecraft in the inertial system, V C is the translational velocity of the spacecraft in the inertial system, R t is the equivalent radius of the storage tank, ω S is the rotational angular velocity of the pulsating sphere in the body system, σ is the liquid surface tension, μ l is the kinematic viscosity coefficient of the liquid, V u is the relative translational velocity of the pulsating sphere along the tangential direction at the contact point, V u =-e×(e×V S +Lω s ), V S is the translational velocity of the pulsating sphere relative to the body system, L is the radius of the pulsating sphere. When the radius of the pulsating sphere reaches the minimum value, the radial velocity component e·(e·V S ) reverses. At this time, there is not only a force between the pulsating sphere and the spacecraft, but also a momentum exchange will occur: where: represents the translational velocity of the spacecraft's center of mass at the instant after the collision; κ ∈ [0, 1] is the collision coefficient.

5. The continuous state space form based on the track-electromagnetic-liquid coupling dynamics model according to claim 2, characterized in that In the step S1, the discrete state - space form is expressed as: x(k + 1)=(A+ΔA(k))x(k)+Bu(k)+w(k) In the formula, is the state transition matrix of the system at time t k , 6. The electromagnetic docking Tube model predictive control method for on-orbit refueling according to claim 1, characterized in that In the step S2, the designed constraint tightening method of the adaptive robust invariant set includes the following constraint C(k): X t ={x ∈ R n | Yx ≤ q}, Y ∈ R r×n , q ∈ R r w * (k)∈(A K +ΔA(k + 1))(A K +ΔA(k)) N-1 W where \(X\) is a closed convex set, \(U\) and \(W\) are compact convex sets containing the origin, \(X\) t is the terminal constraint set, \(x\) * (k) and \(u\) * (k) are the predicted system state and input at time \(k\), is the state sequence of the prediction system obtained at the current time \(k\) with a prediction horizon \(N\), is the control sequence of the prediction system obtained at the current time \(k\) with a prediction horizon \(N\), \(A\) K = \(A + BK\), the matrix \(A\) K +\(\Delta A(k)\) is Schur, \(\sim\) represents the Pontryagin difference of two sets, \(X\) i represents the state admissible set of the prediction system at time \(k + i\), \(U\) i represents the input admissible set of the prediction system at time \(k + i\), \(X\) f is a robust invariant set of the system \(x(k + 1)=(A\) K +\(\Delta A(k + 1))x(k)+w\) * (k).

7. The electromagnetic docking Tube model predictive control method for on-orbit refueling according to claim 1, wherein In the step S3, the energy consumption - accuracy multi - objective optimization function is: In the formula, Q and R are given positive - definite and semi - positive - definite matrices respectively.

8. The electromagnetic docking Tube model predictive control method for on-orbit refueling according to claim 1, wherein In the step S4, the control Lyapunov function is expressed as: where d represents the distance in the normed space (R n , ||·|| p ), and the norm ||·|| p is defined as 9. The electromagnetic docking Tube model predictive control method for on-orbit refueling according to claim 1, wherein In the step S4, the designed optimal controller is expressed as: In the formula, represents the first element of the predicted system control sequence when the problems according to claims 6 and 7 are solvable.

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