Satellite model prediction control method based on convex mixed integer programming

Abstract

The invention discloses a satellite model predictive control method based on convex mixed integer programming, belongs to the technical field of spacecraft orbit optimization and control, and solves the problems that a traditional method depends on continuous control input, does not conform to discrete characteristics of a satellite propulsion system, does not fully consider discrete characteristics of a propeller, and is difficult to consider precision and efficiency during complex maneuver processing. And the fuel consumption cannot be effectively optimized. The method comprises the following steps: firstly, establishing a task satellite non-convex optimal control model, then carrying out convexity on a non-convex item and introducing an integer decision variable, then solving a reference trajectory of a convex optimization problem without collision avoidance constraint, and finally solving the convex optimization problem containing the collision avoidance constraint to obtain an optimal control sequence, thereby realizing fuel optimization and collision avoidance. According to the invention, the discrete switching decision and continuous trajectory optimization of the satellite thruster can be processed uniformly. Fuel optimization can be achieved, and cost is reduced; the propeller is accurately controlled, and the track adjusting precision is improved.

Description

Technical Field

[0001] The present invention relates to the technical field of spacecraft orbit optimization and control, and in particular to a satellite model predictive control method based on convex mixed integer programming. Background Art

[0002] In the field of spacecraft orbit optimization and control technology, satellite model predictive control methods play a key role in ensuring that satellites can complete various complex space missions. However, existing model predictive control methods for satellites have many limitations.

[0003] Existing methods typically rely on continuous control inputs. In real-world satellite operations, this approach is inconsistent with the actual operating characteristics of satellite propulsion systems. Many satellites use on-off propulsion systems, whose operating states are discrete, not continuously variable. Continuous control input methods fail to fully exploit the advantages of on-off propulsion systems, making it difficult to achieve optimal energy utilization during orbit adjustments and resulting in fuel waste.

[0004] Existing technologies fail to explicitly consider the discrete nature of thrusters. This neglect makes it impossible to precisely control the on / off state and operating time of the thrusters when designing control strategies. This not only affects the accuracy of satellite orbit adjustments but can also cause unnecessary attitude changes during mission execution, increasing the complexity and risk of satellite control. In missions requiring extremely high orbital accuracy, such as satellite rendezvous and docking, this lack of accuracy can lead to mission failure.

[0005] Existing technologies also face challenges in handling complex maneuvers. Satellites may need to perform complex maneuvers such as rapid orbit changes and avoiding space debris during missions. Traditional model predictive control methods, which fail to fully consider the discrete characteristics of thrusters and the coordinated optimization of continuous and discrete variables, struggle to maintain both control accuracy and computational efficiency when handling these complex maneuvers. Excessive computational effort can prevent control commands from being generated in a timely manner, hindering the satellite's response to complex situations and reducing the safety and reliability of satellite operations.

[0006] Fuel limitation is an important issue that must be faced in satellite operation. Existing control technologies cannot effectively reduce propellant consumption by optimizing the working mode of the thrusters when dealing with fuel limitations. In long-term space missions, fuel reserves are limited. Unreasonable fuel consumption will shorten the service life of the satellite, increase mission costs, and may even cause the satellite to lose its working ability prematurely and be unable to complete the scheduled mission. In summary, in order to meet the needs of modern satellites for efficient and precise operation in complex space environments, a new satellite model predictive control method is urgently needed to address the challenges faced by traditional control technologies when dealing with complex maneuvers, fuel limitations and discrete characteristics of thrusters. This paper proposes a satellite model predictive control method based on convex mixed integer programming, which is developed to effectively address these problems. Summary of the Invention

[0007] The present invention proposes a satellite model predictive control method based on convex mixed integer programming. By integrating continuous variables and discrete variables, the problem of integer variables existing in the prior art is solved. The satellite optimizes the ignition time of each thruster while considering the upper and lower limits of the thrust amplitude of the thruster. By utilizing the rolling optimization and dynamic feedback characteristics of model predictive control, the satellite solves the complex trajectory optimization problem with the minimum ignition time and number of ignitions while ensuring the stability of satellite operation.

[0008] A satellite model predictive control method based on convex mixed integer programming, the satellite model predictive control method based on convex mixed integer programming comprising the following steps:

[0009] S1. Establish a non-convex optimal control model for the mission satellite based on the model predictive control method;

[0010] S2, convexify the non-convex terms in the non-convex optimal control model established in S1, obtain the convexified result, and introduce the corresponding integer decision variables according to the divided control interval;

[0011] S3, based on the convexification result obtained in S2, solve the convex optimization problem without collision avoidance constraints as the reference trajectory;

[0012] S4. Based on the convexification result of step S2 and the reference trajectory obtained in S3, solve the complete convex optimization problem including the collision avoidance constraint to obtain the optimal control sequence.

[0013] Furthermore, in S1, the following steps are included:

[0014] S11. Establish a satellite reference coordinate system and describe the satellite relative motion model;

[0015] S12, dividing the control interval;

[0016] S13. Establish a non-convex optimal control model for the mission satellite based on the model predictive control method.

[0017] Furthermore, in S11,

[0018] First, define an RTN reference coordinate system, where the x direction is radial and consistent with the absolute position vector, the z direction is normal and consistent with the angular momentum of the orbit, and the y direction is tangential to complete the construction of the right-handed orthogonal basis. Assume that only the mission satellite is maneuvering and can provide control acceleration along the radial, tangential and normal directions of the RTN reference coordinate system. The relative motion of the slave star relative to the reference star is described by a set of quasi-nonsingular average relative orbital elements, δα=[δa,δλ,δe x ,δe y ,δi x ,δi y ] T , the dynamic equation of the motion of the slave star relative to the reference star in a near-circular orbit can be expressed as:

[0019]

[0020] in:

[0021]

[0022] is the Earth's gravitational parameter, is the equatorial radius, is the average angular velocity of the reference star, and the other parameters are P=3cos(i c ) 2 -1, Q = 5cos(i c ) 2 -1, S = sin(2i c ), T=sin(i c ) 2 , E=1+η, F=4+3η, Where the subscript c represents the orbital element of the reference star, a is the semi-major axis of the orbit, e is the orbital eccentricity, i is the orbital inclination, Ω is the right ascension of the ascending node, ω is the angular distance of perigee, M is the mean anomaly, and u is the orbital inclination. c The variable u represents the average latitude argument of the reference star at time t. c and t through u c =u0+W c (t-t0) linearly related, where W c =n+κQ+η c κP,u0=u c (t0).

[0023] Furthermore, in S12,

[0024] The average latitude angle range [u0,u T ] is divided into N d A fixed length of A finite number of subintervals, where u T =u c (t=t f ), t f is the task end time, and generates K d state points, where K d =2N d +1 and for each child

[0025] The interval is associated with a size of and m=1,...,N d The control acceleration is

[0026] In this way, the optimization variable is the control variable associated with the mth subinterval

[0027]

[0028] N (·) Indicates that in the interval [u0,u T ] along the axis (·), and and denote the mean latitude angle of the principal track at the beginning and end of the mth maneuver, respectively; and denote the half-angle duration and angular position of the m-th finite-time maneuver, respectively.

[0029] Furthermore, in S13, a non-convex optimal control model of the mission star is established based on the model predictive control method:

[0030]

[0031] Subject to

[0032]

[0033] δα(u0)=δα0,δα(u T )=δα f (5)

[0034]

[0035] In formula (3), J is the cost function of the mission star established based on the model predictive control method, where ||·|| pRepresents the p-norm, p=1. The first term of the cost function is the control term, which includes the control acceleration of each sub-interval; the second term represents the final state of the mission star. Relative to expected state The tracking error; the third term represents the relative state of the mission satellite at each time point Relative state to expectations The difference between them, where the matrix P and matrix Q are the weight matrices of the second and third items respectively,

[0036]

[0037] Defining variables is a set of task star state variables and decision variables, as shown below:

[0038]

[0039] Formula (4) represents the dynamic constraint of the slave satellite relative to the master satellite;

[0040] Equation (5) represents the initial and terminal constraints, δα0 and δα f Represent the initial and terminal states of the mission star respectively; through the matrix and From variable Extract the variables representing the initial state and the final state, where M = 6K d +12N d .

[0041]

[0042] Formula (6) represents the thruster amplitude constraint, f max is the upper limit of thruster amplitude, f min is the lower limit of thruster amplitude;

[0043] Equation (7) represents the collision avoidance constraint between the mission satellite and the obstacle, where ||·||2 represents the 2-norm, C represents the mapping function from the quasi-nonsingular mean relative orbital elements to the position state in the Cartesian system, and d safe is the minimum distance constraint; N is the number of obstacles.

[0044] Furthermore, in S2, the following steps are included:

[0045] S21, convexify the dynamic constraints;

[0046] S22, introduce mixed integer decision constraints;

[0047] S23. Convexify the collision avoidance constraints between the task star and obstacles at discrete points.

[0048] Furthermore, in S21, the dynamic constraints are linearized according to the reference trajectory, and then the linearized dynamic constraints are discretized using a zero-order hold method.

[0049] Furthermore, in S22,

[0050] Will Divided into two subsets and where f max is the upper limit of thruster amplitude,

[0051]

[0052] and is the dimensionless positive and negative control acceleration in the mth interval, and The range of variation is 0-1. and is a binary integer variable in the range [0,1] that defines the sign of the control input along the (·)-axis and is defined as follows:

[0053]

[0054] Based on the determined integer decision variables, the thruster amplitude constraint needs to include the following inequality constraints:

[0055]

[0056] Among them, M f =f max / f min , formula (17) and formula (18) are used to define the quantity and Right now Only in is 1 and It is not zero when it is 0, and it is positive, otherwise Only in is 1 and It is not zero only when it is 0, and it is negative. Finally, formula (19) stipulates that in m intervals, or At least one must be non-zero, so only one force component is considered per interval, the amplitude of the positive and negative finite-time maneuvers and Respectively in [f min ,f max ] and [-f min ,-f max ] range, and f min >0.

[0057] Furthermore, in S23,

[0058] Convexify the collision avoidance constraints between the mission star and obstacles at discrete points;

[0059]

[0060] in represents the relative orbital element number of the ith obstacle at discrete point k; It represents the relative orbital elements of the reference trajectory of the mission satellite at the discrete point position k. Since the satellite orbit in this study is close to a circle, the linear mapping matrix can be used The instantaneous orbital elements are mapped to the position state in the Cartesian system, thereby introducing path collision avoidance constraints.

[0061]

[0062] Formula (22) is the convex collision avoidance constraint, where represents the relative track element number of the obstacle at each discrete point, The optimization variables obtained by solving the convex optimization problem without collision avoidance constraints are used as the reference trajectory; Represents the mapping matrix at each discrete point,

[0063] A computer device includes: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the above-mentioned satellite model predictive control method based on convex mixed integer programming.

[0064] Beneficial effects of the invention: This paper proposes a satellite model predictive control method based on convex mixed integer programming, which is used to generate fuel-optimized spacecraft rendezvous trajectories in the presence of mixed integer constraints. The convex mixed integer programming framework enables a set of discrete decision constraints to be effectively embedded in the convex optimization framework. Since the local optimal solution of the convex programming problem must be the global optimal solution, it can solve complex multi-constraint path planning and optimal control problems with higher efficiency.

[0065] Specifically, the proposed method optimizes the firing time and frequency of each thruster, as well as upper and lower thrust amplitude constraints. This method leverages the rolling optimization and dynamic feedback characteristics of model predictive control (MPC) to achieve orbit control with minimal firing time while ensuring satellite operational stability. This method, combining convex mixed integer programming with MPC, provides an efficient solution for satellite trajectory optimization that simultaneously meets the requirements of discrete thruster control and collision avoidance. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] Figure 1 Schematic diagram of the piecewise constant acceleration of the coordinate axis (·) in the RTN coordinate system;

[0067] Figure 2 This is a flow chart of the satellite model predictive control method based on convex mixed integer programming;

[0068] Figure 3 It is the mission satellite control force generated in the RTN coordinate system;

[0069] Figure 4 The relative motion trajectory is optimized in the reference coordinate system;

[0070] Figure 5 The projection of the relative motion trajectory to be optimized on the RT and RN planes;

[0071] Figure 6 is the change in the average relative orbital elements of the mission star from the initial state to the terminal state (marked with * and o, respectively);

[0072] Figure 7 The changes in the relative distance between the mission satellite and each sub-satellite in the constellation when there is no safety distance constraint and when there is a safety distance constraint. DETAILED DESCRIPTION

[0073] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without making any creative efforts shall fall within the scope of protection of the present invention.

[0074] Reference Figure 1-Figure 3 As shown, the present invention provides a satellite model predictive control method based on convex mixed integer programming, and its specific steps are as follows:

[0075] S1. Establish a non-convex optimal control model for the mission satellite based on the model predictive control method;

[0076] S2, convexify the non-convex terms in the non-convex optimal control model established in S1, and introduce corresponding integer decision variables according to the divided control intervals;

[0077] S3. Based on the convexification result of S2, solve the convex optimization problem without collision avoidance constraints as the reference trajectory;

[0078] S4, based on the convexification result of S2 and the reference trajectory obtained in S3, solve the complete convex optimization problem including the collision avoidance constraint and obtain the optimal control sequence;

[0079] Specifically, the satellite model predictive control method based on convex mixed integer programming described in this invention utilizes convex mixed integer programming to uniformly handle discrete satellite thruster switching decisions and continuous trajectory optimization. This overcomes the drawbacks of traditional methods that rely on continuous control inputs, fully leveraging the advantages of satellite switch propulsion systems, avoiding fuel waste and effectively reducing propellant consumption. This is of great significance in fuel-constrained satellite operation scenarios. Furthermore, it explicitly considers the discrete nature of thrusters, enabling precise control of thruster switching states and operating times. This significantly improves the accuracy of satellite orbit adjustment, reduces unnecessary attitude changes, reduces satellite control complexity and risk, and ensures the success rate of satellites performing high-precision missions. When handling complex maneuvers, by considering the discrete nature of thrusters and co-optimizing continuous-discrete variables, it balances control accuracy and computational efficiency, avoids delays in command generation due to excessive computational effort, improves the satellite's response to complex situations, and enhances the safety and reliability of satellite operations, providing a strong guarantee for efficient and precise satellite operation in complex space environments.

[0080] The specific process of S1 is as follows:

[0081] S11. Establish a satellite reference coordinate system and describe the satellite relative motion model:

[0082] First, define an RTN reference coordinate system, where the x-direction is radial and consistent with the absolute position vector, the z-direction is normal and consistent with the angular momentum of the orbit, and the y-direction is tangential to complete the construction of the right-handed orthogonal basis. Assume that only the mission satellite can maneuver and can provide control acceleration along the radial, tangential and normal directions of the RTN reference coordinate system. The relative motion of the slave star relative to the reference star is described by a set of quasi-nonsingular average relative orbital elements, δα=[δa,δλ,δe x ,δe y ,δi x ,δi y ] T The dynamic equation of the motion of the slave star relative to the reference star in a near-circular orbit is expressed as:

[0083]

[0084] in:

[0085]

[0086] is the Earth's gravitational parameter, is the equatorial radius, is the average angular velocity of the reference star, and the other parameters are P=3cos(i c ) 2 -1, Q = 5cos(i c )2 -1, S = sin(2i c ), T=sin(i c ) 2 , E=1+η, F=4+3η, Where the subscript c represents the orbital element of the reference star, a is the semi-major axis of the orbit, e is the orbital eccentricity, i is the orbital inclination, Ω is the right ascension of the ascending node, ω is the angular distance of perigee, M is the mean anomaly, and u is the orbital inclination. c The variable u represents the average latitude argument of the reference star at time t. c and t through u c =u0+W c (t-t0) linearly related, where W c =n+κQ+η c κP,u0=u c (t0).

[0087] Specifically, this embodiment describes the satellite relative motion model by defining an RTN reference coordinate system. Within this coordinate system, the x, y, and z axis directions are clearly defined. The x direction corresponds to the absolute position vector, the z direction corresponds to the orbital angular momentum, and the y direction is constructed using a right-handed orthogonal basis. This provides a clear and unified framework for accurately describing the satellite's spatial position and motion direction. Based on this, under the assumption that only the mission satellite is maneuvering, a set of quasi-nonsingular average relative orbital elements is used to describe the relative motion of the slave satellite relative to the reference satellite. The dynamic equations for a near-circular orbit are then derived, accounting for key parameters such as Earth's gravitational parameters, the equatorial radius, the average angular velocity of the reference satellite, and various orbital elements of the reference satellite. This precise description allows the full consideration of various practical factors in subsequent satellite orbit optimization and control, enabling more accurate simulation of the satellite's motion state. This, in turn, lays a solid foundation for establishing more precise control models, improving the accuracy and reliability of satellite orbit control.

[0088] S12. Divide the control interval:

[0089] The average latitude angle range [u0,u T ](where u T =u c (t=t f ), t f is the task end time) is divided into N d A fixed length of A finite number of subintervals of d state points, where K d =2N d +1, and associate a size of and m=1,...,N d The control acceleration is In this way, the optimization variable is the control variable associated with the mth subinterval

[0090]

[0091] N (·) Indicates that in the interval [u0,u T ] along the axis (·), and and denote the mean latitude angle of the principal track at the beginning and end of the mth maneuver, respectively; and denote the half-angle duration and angular position of the m-th finite-time maneuver, respectively.

[0092] Specifically, this embodiment divides the average latitude angle interval of control into a finite number of sub-intervals, and associates a control acceleration with each sub-interval. By setting sub-intervals of fixed length and generating corresponding state points, the optimization variables are closely associated with the sub-intervals, which makes it possible to fine-tune the control. The parameters related to maneuvers are clarified, such as the number of maneuvers, the average latitude angle at the beginning and end of the maneuver, and the half-angle duration and angular position of the maneuver, so that the various parameters in the control process are clearer and more specific. This not only helps to more accurately describe the motion state of the satellite at different stages when establishing the control model, but also can more accurately plan the trajectory of the satellite. At the same time, this division method facilitates the subsequent processing of complex constraints and optimization goals, so that various factors can be considered more efficiently when solving the optimal control sequence, thereby improving the accuracy of satellite orbit control and ensuring that the satellite completes the mission in the best way while meeting multiple constraints.

[0093] S13. Establish a non-convex optimal control model for the mission satellite based on the model predictive control method:

[0094]

[0095] Subject to

[0096]

[0097] δα(u0)=δα0,δα(u T )=δα f (5)

[0098]

[0099] In formula (3), J is the cost function of the mission star established based on the model predictive control method, where ||·|| pRepresents the p-norm, p = 1. The first term of the cost function is the control term, which includes the control acceleration of each sub-interval; the second term represents the final state of the mission star Relative to expected state The tracking error; the third term represents the relative state of the mission satellite at each time point Relative state to expectations The difference between them. The matrix P and matrix Q are the weight matrices of the second and third items respectively.

[0100]

[0101] Defining variables is a set of task star state variables and decision variables, as shown below:

[0102]

[0103] Formula (4) represents the dynamic constraint of the slave satellite relative to the master satellite;

[0104] Equation (5) represents the initial and terminal constraints, δα0 and δα f Represent the initial and terminal states of the mission star respectively; through the matrix

[0105] and From the variable Extract the variables representing the initial state and the final state, where M = 6K d +12N d .

[0106]

[0107] Formula (6) represents the thruster amplitude constraint, f max is the upper limit of thruster amplitude, f min is the lower limit of thruster amplitude;

[0108] Equation (7) represents the collision avoidance constraint between the mission satellite and the obstacle, where ||·||2 represents the 2-norm, C represents the mapping function from the quasi-nonsingular mean relative orbital elements to the position state in the Cartesian system, and d safe is the minimum distance constraint; N is the number of obstacles.

[0109] Specifically, the non-convex optimal control model established in this embodiment comprehensively considers fuel consumption, terminal state accuracy, and overall state tracking performance during satellite control by constructing a cost function that includes control terms, terminal state tracking error, and tracking error at each state point. The weight matrix allows for flexible adjustment of the priorities of different control objectives, providing a unified framework for multi-objective optimization. The dynamic constraints incorporated into the model ensure that satellite motion conforms to actual physical laws. The initial and terminal state constraints strictly define the start and end conditions of the mission, and the thruster amplitude constraints accurately reflect the physical limitations of the propulsion system, avoiding a disconnect between theoretical control inputs and actual execution capabilities. In particular, the collision avoidance constraints, by introducing Cartesian position mapping and minimum distance constraints, effectively address the issue of maintaining safe separation between the satellite and obstacles in complex space environments, ensuring the safety of the orbital control process. This model integrates state variables and decision variables to form a complete optimization system that includes continuous control inputs, discrete thrust states, and spatial constraints. This lays the foundation for the subsequent transformation of complex non-convex problems into efficiently solvable convex optimization problems through convexification, ultimately achieving autonomous satellite orbital control with optimal fuel, precise trajectory, and safety constraints.

[0110] S21, convexify the dynamic constraints;

[0111] According to the definition of convex optimization problems, equality constraints in convex optimization problems must be affine. Therefore, the dynamic constraints are linearized according to the reference trajectory and then discretized using the zero-order hold method.

[0112] Specifically, this embodiment utilizes a convexification process for the dynamic constraints, linearizing them based on a reference trajectory and then discretizing them using the zero-order hold method. This approach has several significant implications. First, this approach complies with the requirement that equality constraints in convex optimization problems must be affine, enabling the subsequent application of convex optimization theories and methods to solve the problem, providing an effective approach for solving complex satellite control problems. Linearization based on the reference trajectory simplifies the form of the dynamic constraints, reduces computational complexity, and improves solution efficiency. Discretization using the zero-order hold method converts the continuous dynamic process into discrete time points for analysis and calculation, making it easier for computers to process and implement. This series of operations enables the efficient solution of previously complex satellite dynamics problems within the convex optimization framework, helping to more accurately obtain satellite orbit control solutions that meet various constraints. This, in turn, enables precise satellite control and optimized fuel utilization, improving satellite stability and reliability during orbit control.

[0113] S22, introduce mixed integer decision constraints;

[0114] Will Divided into two subsets and where f max is the upper limit of thruster amplitude.

[0115]

[0116] and are dimensionless positive and negative control accelerations in the mth interval. They can vary from 0 to 1. and is a binary integer variable in the range [0,1] that defines the sign of the control input along the (·)-axis and is defined as follows:

[0117]

[0118] Based on the determined integer decision variables, the thruster amplitude constraint needs to include the following inequality constraints:

[0119]

[0120] Among them, M f =f max / f min , formula (17) and formula (18) are used to define the quantity and Right now Only in is 1 and It is not zero (and positive) only when it is 0, otherwise Only in is 1 and It is non-zero (and negative) only when it is 0. Finally, formula (19) stipulates that in m intervals, or At least one of them must be non-zero. Therefore, only one force component (positive or negative) is considered per interval, and the amplitude of the positive and negative finite-time maneuvers and Respectively in [f min ,f max ] and [-f min ,-f max ] range, and f min >0.

[0121] Specifically, this embodiment achieves numerous significant advantages by subsetting the thruster amplitude-related variables, introducing binary integer variables to define the control input signs, and combining them with a series of inequality constraints. This approach makes thruster amplitude control more precise and flexible in satellite propulsion system control. By setting the range of dimensionless positive and negative control accelerations and determining the control input signs using binary integer variables, the thruster operating state within each interval can be clearly defined, ensuring that only one force component is considered in each interval. This avoids confusion in thruster control and effectively utilizes the thruster's discrete characteristics. Regarding fuel optimization, precise thruster control helps reduce unnecessary thrust output, thereby lowering fuel consumption. Given limited satellite fuel, this significantly extends the satellite's service life and mission performance. From the perspective of orbital control accuracy, precise control of thruster amplitude and direction enables more accurate adjustment of the satellite's orbit, reduces orbital deviation, and meets the precision requirements of demanding missions such as satellite rendezvous and docking.

[0122] S23, convexifying the collision avoidance constraints between the task star and the obstacle at the discrete points;

[0123]

[0124] in represents the relative orbital element number of the ith obstacle at discrete point k; It represents the relative orbital elements of the reference trajectory of the mission satellite at the discrete point position k. Since the satellite orbit in this study is close to a circle, the linear mapping matrix can be used The instantaneous orbital elements are mapped to the position state in the Cartesian system, thereby introducing path collision avoidance constraints.

[0125]

[0126]

[0127] Formula (22) is the convex collision avoidance constraint, where represents the relative track element number of the obstacle at each discrete point, The optimization variables obtained by solving the convex optimization problem without collision avoidance constraints are used as the reference trajectory; Represents the mapping matrix at each discrete point,

[0128] Specifically, this embodiment convexifies the collision avoidance constraints between the mission satellite and obstacles at discrete points. In the complex space environment in which satellites operate, numerous obstacles exist, and satellites must effectively avoid collisions to ensure safe and stable operation. This claim utilizes the characteristic of satellite orbits being nearly circular, mapping the instantaneous orbital elements into position states in a Cartesian system using a linear mapping matrix. This then introduces path collision avoidance constraints, allowing the collision avoidance constraints to be convexified. This convexification process not only transforms the complex collision avoidance problem into a more manageable mathematical form, but also allows it to be integrated into the overall convex optimization framework and solved together with other constraints and optimization objectives. In practical applications, by solving the optimization problem containing the convexified collision avoidance constraints, the safe distance from obstacles can be fully considered during the satellite trajectory planning stage, thereby obtaining the optimal control sequence that meets the collision avoidance requirements, ensuring that the satellite maintains a safe distance from obstacles throughout its operation, effectively reducing the risk of collision between the satellite and obstacles, greatly improving the safety and reliability of satellite operations, and providing an important guarantee for the satellite to successfully complete various complex space missions.

[0129] A computer device, characterized in that it includes: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the above-mentioned satellite model predictive control method based on convex mixed integer programming.

[0130] Specifically, this embodiment involves a computer device comprising a memory, a processor, and related computer programs. Its significance lies in the practical implementation of a satellite model predictive control method based on convex mixed-integer programming. This computer device is capable of running a program to execute each step of the satellite model predictive control method, automating the entire process from establishing a non-convex optimal control model to convexifying it, solving the reference trajectory, and obtaining the optimal control sequence with collision avoidance constraints. This significantly improves the efficiency and accuracy of satellite orbit control. On the one hand, the computer device's powerful computing power enables rapid processing of large amounts of complex data, effectively resolving the instruction generation delays caused by the excessive computational effort of traditional methods. This enables satellites to respond promptly to complex space conditions, enhancing the safety and reliability of satellite operations. On the other hand, this integrated device converts the control method into an executable program, avoiding errors that may occur during manual operation and ensuring the stability and consistency of the control process. Furthermore, this computer device provides a vehicle for the practical application of the satellite model predictive control method, helping to promote the widespread application of related technologies in the field of spacecraft orbit optimization and control, and providing strong technical support for the efficient execution of various complex space missions.

[0131] This case study serves only as a method demonstration and is not an actual flight mission. It is solved through simulation using the optimized control scheme of this patent. It is assumed that the target constellation consists of four satellites, and the mission spacecraft can provide continuous and spaced thrust in the radial, tangential, and normal directions of its RTN coordinate system. The initial quasi-nonsingular mean orbital elements of the target constellation are shown in Table 1:

[0132]

[0133] Table 1

[0134] The initial average orbital elements of the reference star are shown in Table 2:

[0135]

[0136] Table 2

[0137] The initial and expected quasi-nonsingular mean orbital elements of the mission star are shown in Table 3. Each item of the quasi-nonsingular mean orbital elements of the mission star has changed.

[0138]

[0139] Table 3

[0140] Some key simulation parameters for optimization solution setting are shown in Table 4:

[0141]

[0142] Table 4

[0143] According to the steps of the present invention, the trajectory of the mission satellite is optimized and controlled, and the results obtained are as follows: Figures 4 to 7 The simulation case verifies that the satellite model predictive control method based on convex mixed integer programming can effectively consider the duration of thrust and the minimum thrust constraint to obtain the trajectory under collision avoidance.

[0144] This paper proposes a satellite model predictive control method based on convex mixed-integer programming. By integrating continuous and discrete variables (such as thruster on / off states), it effectively addresses the challenges faced by traditional control technologies when dealing with complex maneuvers, fuel limitations, and the discrete characteristics of thrusters. This optimization control framework combines mixed-integer optimization with rolling-horizon control. Its core lies in modeling the satellite's continuous state variables and discrete integer decision variables as a convex mixed-integer programming problem. It leverages the predictive optimization capabilities of model predictive control to simultaneously optimize thruster scheduling and collision avoidance strategies while ensuring computational efficiency, thereby reducing propellant consumption or the number of on / off cycles. This achieves efficient coordination of discrete-continuous decision-making in satellite control, providing a real-time, optimal, and reliable solution for complex space missions.

[0145] While the specific embodiments of the present invention have been described in detail above, these are intended to be exemplary only, and the present invention is not limited thereto. Any equivalent modifications or substitutions to the present invention that would be apparent to those skilled in the art are also within the scope of the present invention. Therefore, any equivalent modifications or substitutions made without departing from the spirit and scope of the present invention are intended to be encompassed within the scope of the present invention.

Claims

1. A satellite model predictive control method based on convex mixed integer programming, characterized in that: The satellite model predictive control method based on convex mixed integer programming comprises the following steps: S1. Establish a non-convex optimal control model for the mission satellite based on the model predictive control method; S2, convexify the non-convex terms in the non-convex optimal control model established in S1, obtain the convexified result, and introduce the corresponding integer decision variables according to the divided control interval; S3, based on the convexification result obtained in S2, solve the convex optimization problem without collision avoidance constraints as the reference trajectory; S4. Based on the convexification result of step S2 and the reference trajectory obtained in S3, solve the complete convex optimization problem including the collision avoidance constraint to obtain the optimal control sequence.

2. The satellite model predictive control method based on convex mixed integer programming according to claim 1, characterized in that: In S1, the following steps are included: S11. Establish a satellite reference coordinate system and describe the satellite relative motion model; S12, dividing the control interval; S13. Establish a non-convex optimal control model for the mission satellite based on the model predictive control method.

3. The satellite model predictive control method based on convex mixed integer programming according to claim 2, characterized in that: In S11, First, define an RTN reference coordinate system, where the x direction is radial and consistent with the absolute position vector, the z direction is normal and consistent with the angular momentum of the orbit, and the y direction is tangential to complete the construction of the right-handed orthogonal basis. Assume that only the mission satellite is maneuvering and can provide control acceleration along the radial, tangential and normal directions of the RTN reference coordinate system. The relative motion of the slave star relative to the reference star is described by a set of quasi-nonsingular average relative orbital elements, δα=[δa,δλ,δe x ,δe y ,δi x ,δi y ] T , the dynamic equation of the motion of the slave star relative to the reference star in a near-circular orbit can be expressed as: in: is the Earth's gravitational parameter, is the equatorial radius, is the average angular velocity of the reference star, and the other parameters are P=3cos(i c ) 2 -1, Q = 5cos(i c ) 2 -1, S = sin(2i c ), T=sin(i c ) 2 , E=1+η, F=4+3η, Where the subscript c represents the orbital element of the reference star, a is the semi-major axis of the orbit, e is the orbital eccentricity, i is the orbital inclination, Ω is the right ascension of the ascending node, ω is the angular distance of perigee, M is the mean anomaly, and u is the orbital inclination. c The variable u represents the average latitude argument of the reference star at time t. c and t through u c =u0+W c (t-t0) linearly related, where W c =n+κQ+η c κP,u0=u c (t0).

4. The satellite model predictive control method based on convex mixed integer programming according to claim 3, characterized in that: In S12, The average latitude angle range [u0,u T ] is divided into N d A fixed length of A finite number of subintervals, where u T =u c (t=t f ), t f is the task end time, and generates K d state points, where K d =2N d +1, and associate a size of and m=1,...,N d The control acceleration is In this way, the optimization variable is the control variable associated with the mth subinterval N (·) Indicates that in the interval [u0,u T ] along the axis (·), and and denote the mean latitude angle of the principal track at the beginning and end of the mth maneuver, respectively; and denote the half-angle duration and angular position of the m-th finite-time maneuver, respectively.

5. The satellite model predictive control method based on convex mixed integer programming according to claim 4, characterized in that: In S13, a non-convex optimal control model of the mission satellite is established based on the model predictive control method: δα(u0)=δα0,δα(u T )=da f (5) In formula (3), J is the cost function of the mission star established based on the model predictive control method, where ||·|| p Represents the p-norm, p=1. The first term of the cost function is the control term, which includes the control acceleration of each sub-interval; the second term represents the final state of the mission star. Relative to expected state The tracking error; the third term represents the relative state of the mission satellite at each time point Relative state to expectations The difference between them, where the matrix P and matrix Q are the weight matrices of the second and third items respectively, Defining variables is a set of task star state variables and decision variables, as shown below: Formula (4) represents the dynamic constraint of the slave satellite relative to the master satellite; Equation (5) represents the initial and terminal constraints, δα0 and δα f Represent the initial and terminal states of the mission star respectively; through the matrix and From variable Extract the variables representing the initial state and the final state, where M = 6K d +12N d , Formula (6) represents the thruster amplitude constraint, f max is the upper limit of thruster amplitude, f min is the lower limit of thruster amplitude; Equation (7) represents the collision avoidance constraint between the mission satellite and the obstacle, where ||·||2 represents the 2-norm, C represents the mapping function from the quasi-nonsingular mean relative orbital elements to the position state in the Cartesian system, and d safe is the minimum distance constraint; N is the number of obstacles.

6. The satellite model predictive control method based on convex mixed integer programming according to claim 5, characterized in that: In S2, the following steps are included: S21, convexify the dynamic constraints; S22, introduce mixed integer decision constraints; S23. Convexify the collision avoidance constraints between the task star and obstacles at discrete points.

7. The satellite model predictive control method based on convex mixed integer programming according to claim 6, characterized in that: In S21, the dynamic constraints are linearized according to the reference trajectory, and then the linearized dynamic constraints are discretized using the zero-order hold method.

8. The satellite model predictive control method based on convex mixed integer programming according to claim 7, characterized in that: In S22, Will Divided into two subsets and where f max is the upper limit of thruster amplitude, and is the dimensionless positive and negative control acceleration in the mth interval, and The range of variation is 0-1. and is a binary integer variable in the range [0,1] that defines the sign of the control input along the (·)-axis and is defined as follows: Based on the determined integer decision variables, the thruster amplitude constraint needs to include the following inequality constraints: Among them, M f =f max / f min , formula (17) and formula (18) are used to define the quantity and Right now Only in is 1 and It is not zero when it is 0, and it is positive, otherwise Only in is 1 and It is not zero only when it is 0, and it is negative. Finally, formula (19) stipulates that in m intervals, or At least one must be non-zero, so only one force component is considered per interval, the amplitude of the positive and negative finite-time maneuvers and Respectively in [f min ,f max ] and [-f min ,-f max ] range, and f min >0.

9. The satellite model predictive control method based on convex mixed integer programming according to claim 8, characterized in that: In S23, Convexify the collision avoidance constraints between the mission star and obstacles at discrete points; in represents the relative orbital element number of the ith obstacle at discrete point k; It represents the relative orbital elements of the reference trajectory of the mission satellite at the discrete point position k. Since the satellite orbit in this study is close to a circle, the linear mapping matrix can be used Map the instantaneous orbital elements to the position state in the Cartesian system, thereby introducing the path collision avoidance constraint. Formula (22) is the convex collision avoidance constraint, where represents the relative track element number of the obstacle at each discrete point, The optimization variables obtained by solving the convex optimization problem without collision avoidance constraints are used as the reference trajectory; Represents the mapping matrix at each discrete point, 10. A computer device, characterized in that: include: A memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the satellite model predictive control method based on convex mixed integer programming according to any one of claims 1 to 9.

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