A satellite model predictive control method based on convex mixed integer programming

Abstract

The application discloses a satellite model predictive control method based on convex mixed integer programming and belongs to the technical field of spacecraft orbit optimization and control, and solves the problems that traditional methods depend on continuous control input, are inconsistent with discrete characteristics of a satellite propulsion system, do not fully consider the discrete nature of a thruster, are difficult to consider accuracy and efficiency when dealing with complex maneuvers, and cannot effectively optimize fuel consumption. The application first establishes a non-convex optimal control model of a mission satellite, then convexes non-convex terms and introduces integer decision variables, then solves a benchmark trajectory of a convex optimization problem without collision avoidance constraints, finally solves a convex optimization problem with collision avoidance constraints to obtain an optimal control sequence, and realizes fuel optimization and collision avoidance. The application can uniformly deal with discrete switch decision of a satellite thruster and continuous trajectory optimization. The application can realize fuel optimization, reduce cost, accurately control the thruster and improve orbit adjustment accuracy.

Description

Technical Field

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

[0002] In the field of spacecraft orbit optimization and control technology, satellite model predictive control methods play a crucial 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. However, in real-world satellite operation scenarios, this reliance on continuous control inputs is inconsistent with the actual operating characteristics of satellite propulsion systems. Many satellites employ switch-propulsion systems, whose operating states are discrete, not continuously variable. Continuous control input methods cannot fully utilize the advantages of switch-propulsion systems, leading to difficulties in achieving optimal energy utilization during orbital adjustments and resulting in fuel waste.

[0004] Current technologies do not explicitly consider the discrete nature of thrusters. Because these discrete characteristics are ignored, it is impossible to precisely control the on / off states and operating times of the thrusters when designing control strategies. This not only affects the accuracy of satellite orbit adjustments but may also lead to unnecessary attitude changes during mission execution, increasing the complexity and risk of satellite control. In missions with extremely high orbital accuracy requirements, such as satellite rendezvous and docking, such precision deviations 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 space debris avoidance during missions. Traditional model predictive control methods, which do not fully consider the discrete characteristics of thrusters and the co-optimization of continuous-discrete variables, struggle to balance control accuracy with computational efficiency when dealing with these complex maneuvers. Excessive computational load may prevent the timely generation of control commands, affecting the satellite's response speed to complex situations and reducing the safety and reliability of satellite operation.

[0006] Fuel constraints are a critical issue that satellite operations must address. Current control technologies cannot effectively reduce propellant consumption by optimizing thruster operating modes when dealing with fuel constraints. In long-term space missions, fuel reserves are limited, and unreasonable fuel consumption can shorten the satellite's lifespan, increase mission costs, and even prematurely disable the satellite, preventing it from completing its intended mission. Therefore, to meet the demands of modern satellites operating efficiently and accurately in complex space environments, a novel satellite model predictive control method is urgently needed to address the challenges posed by traditional control technologies when handling complex maneuvers, fuel constraints, and the discrete characteristics of thrusters. This paper proposes a satellite model predictive control method based on convex mixed-integer programming, specifically developed to effectively address these issues. Summary of the Invention

[0007] This invention proposes a satellite model predictive control method based on convex hybrid integer programming. By integrating continuous and discrete variables, it solves the problem of integer variables in the prior art. This allows the satellite to optimize the ignition time of each thruster while considering the upper and lower limits of the thrust amplitude. By utilizing the rolling optimization and dynamic feedback characteristics of model predictive control, it solves the complex trajectory optimization problem with the least amount of 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, comprising the following steps:

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

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

[0011] S3. Based on the convexity result obtained in S2, solve the convex optimization problem without collision avoidance constraints as the baseline trajectory.

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

[0013] Furthermore, S1 includes the following steps:

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

[0015] S12. Divide the control zone;

[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 coincides with the absolute position vector, the z-direction is normal and coincides with the orbital angular momentum, and the y-direction is tangential, thus completing the construction of the right-handed orthogonal basis. Assume that only the mission satellite performs maneuvers and can provide control acceleration along the radial, tangential, and normal directions of the RTN reference coordinate system. The relative motion of a star to a reference star is described by a set of quasi-nonsingular mean relative orbital elements, δα=[δa,δλ,δe]. x ,δe y ,δi x ,δi y ] T The dynamic equations for the motion of a star relative to a reference star in a near-circular orbit are expressed as follows:

[0019]

[0020] in:

[0021]

[0022] It is Earth's gravitational parameter. It is the equatorial radius. It is the average angular velocity of the reference star; 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 elements 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 from perigee, M is the mean perigee angle, and u... 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) are linearly correlated, where W c =n+κQ+η c κP, u0=u c (t0).

[0023] Furthermore, in S12,

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

[0025] Interval association with a size of And m = 1,...,N d The control acceleration, in which

[0026] In this approach, the optimization variable is the manipulation amount associated with the m-th subinterval.

[0027]

[0028] N (·) This indicates that in the interval [u0, u] T The number of finite-time maneuvers along the inner axis (·), while and represent the average latitude angles of the chief trajectory at the start and end of the m-th maneuver, respectively; and Let represent 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 for the mission satellite 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 equation (3), J is the cost function of the mission satellite established based on the model predictive control method, where ||·|| pRepresenting 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 satellite. Relative to the expected relative state The tracking error; the third term represents the relative state of the mission satellite at each time point. Relative state to expectations The differences between them are shown, where matrices P and Q are the weight matrices for the second and third terms, respectively.

[0036]

[0037] Define variables The set of mission star state variables and decision variables is as follows:

[0038]

[0039] Equation (4) represents the dynamic constraints of the slave star relative to the master star;

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

[0041]

[0042] Equation (6) represents the thrust amplitude constraint, f max f is the upper limit of the thrust amplitude. min This is the lower limit of the thrust amplitude;

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

[0044] Furthermore, S2 includes the following steps:

[0045] S21. Convexify the dynamic constraints;

[0046] S22. Introduce mixed integer decision constraints;

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

[0048] Furthermore, in S21, the dynamic constraints are linearized based on the reference trajectory, and then the linearized dynamic constraints are discretized using the zero-order preservation method.

[0049] Furthermore, in S22,

[0050] Will Divided into two subsets and Where f max This is the upper limit of the thrust amplitude.

[0051]

[0052] and It represents the dimensionless positive and negative control acceleration in the m-th interval. and The range of variation is 0-1. and It is a binary integer variable in the range [0,1], used to define the sign of the control input along the (·) axis, as defined below:

[0053]

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

[0055]

[0056] Where M f =f max / f min Formulas (17) and (18) are used to enforce the definition of quantities. and Right now Only =1 and It is only non-zero when it is 0, and is positive; otherwise... Only =1 and It is only non-zero when it is 0, and it is negative. Finally, formula (19) stipulates that in m intervals, or At least one of them must be non-zero; therefore, only one force component is considered for each interval, with the amplitude of the positive and negative finite-time maneuvers. and In [f] min ,f max ] and [-f min ,-f max The range varies, 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 This represents the relative orbital elements of the i-th obstacle at discrete point k; This represents the relative orbital elements of the mission satellite's reference trajectory at discrete point k at time k. Since the satellite orbit in this study is nearly circular, a linear mapping matrix can be used. The instantaneous orbital elements are mapped to the positional state in the Cartesian system, thereby introducing path collision avoidance constraints.

[0061]

[0062] Equation (22) is the collision avoidance constraint after convexity shaping, where This represents the relative orbital elements of the obstacle at each discrete point. The optimization variables obtained from solving a convex optimization problem without collision avoidance constraints are used as the baseline trajectory. This 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-described satellite model predictive control method based on convex mixed integer programming.

[0064] The beneficial effects of this invention are as follows: 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 under mixed integer constraints. The convex mixed integer programming framework enables the effective embedding of a set of discrete decision constraints into the convex optimization framework. Since the local optimal solution of the convex programming problem is necessarily the global optimal solution, it can solve complex multi-constraint path planning and optimal control problems with high efficiency.

[0065] Specifically, the method proposed in this invention optimizes the ignition time and number of times for each thruster, as well as the upper and lower limits of thrust amplitude constraints, to achieve orbit control with minimal ignition time while ensuring satellite operational stability. This method, combining convex mixed-integer programming and model predictive control, provides an efficient solution to the satellite trajectory optimization problem that simultaneously satisfies the requirements of discrete thruster control and collision avoidance. Attached Figure Description

[0066] Figure 1 This is a schematic diagram of the piecewise constant acceleration along the coordinate axis (·) in the RTN coordinate system.

[0067] Figure 2 The flowchart shows a satellite model predictive control method based on convex mixed integer programming.

[0068] Figure 3 The control force for the mission star generated in the RTN coordinate system;

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

[0070] Figure 5 To optimize the projection of the solved relative motion trajectory onto the RT and RN planes;

[0071] Figure 6 The average relative orbital elements change of the mission satellite from its initial state to its final state (marked with * and o respectively);

[0072] Figure 7 Changes in the relative distance between the mission satellite and each sub-satellite of the constellation with and without safety distance constraints. Detailed Implementation

[0073] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0074] Reference Figures 1-3 As shown, this invention provides a satellite model predictive control method based on convex mixed-integer programming, the specific steps of which are as follows:

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

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

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

[0078] S4. Based on the convexity result of S2 and the baseline trajectory obtained in S3, solve the complete convex optimization problem including collision avoidance constraints to obtain the optimal control sequence.

[0079] Specifically, the satellite model predictive control method based on convex mixed-integer programming described in this invention unifies the discrete switching decisions and continuous trajectory optimization of satellite thrusters through convex mixed-integer programming. This overcomes the drawbacks of traditional methods relying on continuous control inputs, fully leveraging the advantages of satellite switching propulsion systems, avoiding fuel waste, and effectively reducing propellant consumption, which is of great significance in fuel-constrained satellite operation scenarios. Simultaneously, it explicitly considers the discrete nature of the thrusters, enabling precise control of the thruster switching state and operating time, greatly improving the accuracy of satellite orbit adjustment, reducing unnecessary attitude changes, lowering the complexity and risk of satellite control, and ensuring the success rate of satellites performing high-precision missions. When dealing with complex maneuvering missions, by considering the discrete characteristics of the thrusters and continuous-discrete variable co-optimization, it balances control accuracy and computational efficiency, avoiding excessive computational load leading to command generation delays, improving the satellite's response speed to complex situations, enhancing the safety and reliability of satellite operation, and providing strong support for the efficient and precise operation of satellites 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's relative motion model:

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

[0083]

[0084] in:

[0085]

[0086] It is Earth's gravitational parameter. It is the equatorial radius. It is the average angular velocity of the reference star; 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 elements 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 from perigee, M is the mean perigee angle, and u... 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) are linearly correlated, where W c =n+κQ+η c κP, u0=u c (t0).

[0087] Specifically, this embodiment describes the satellite's relative motion model by defining an RTN reference coordinate system. In this coordinate system, the x, y, and z axes are clearly defined, with the x-direction consistent with the absolute position vector, the z-direction consistent with the orbital angular momentum, and the y-direction completing a right-handed orthogonal basis construction. This provides a clear and unified framework for accurately describing the satellite's spatial position and direction of motion. Based on this, under the assumption of mission-only maneuvering, a set of quasi-nonsingular average relative orbital elements describes the relative motion of the satellite relative to the reference satellite, and provides the dynamic equations for a near-circular orbit. These equations encompass many key parameters such as Earth's gravitational parameters, equatorial radius, and the reference satellite's average angular velocity, as well as various orbital elements of the reference satellite. This precise description method in this embodiment allows for full consideration of the influence of various practical factors in subsequent satellite orbit optimization and control, more accurately simulating the satellite's motion state, thus laying a solid foundation for establishing a more accurate control model and improving the accuracy and reliability of satellite orbit control.

[0088] S12. Divide the control zone:

[0089] The controlled average latitude angle range [u0, u T (where u) T =u c (t=t f ), t f Divide the task into N groups based on the task completion time. d A fixed length is A finite number of subintervals, and generate K d There are K state points, where K d =2N d +1, and associate each sub-interval with a size of And m = 1,...,N d The control acceleration, in which In this approach, the optimization variable is the manipulation amount associated with the m-th subinterval.

[0090]

[0091] N (·) This indicates that in the interval [u0, u] T The number of finite-time maneuvers along the inner axis (·), while and represent the average latitude angles of the chief trajectory at the start and end of the m-th maneuver, respectively; and Let represent 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 the control into a finite number of sub-intervals and associates control acceleration with each sub-interval. By setting sub-intervals of fixed length and generating corresponding state points, the optimization variables are closely related to the sub-intervals, enabling refined control. Parameters related to maneuvers are clearly defined, such as the number of maneuvers, the average latitude angle at the start and end of the maneuver, and the half-angle duration and angular position of the maneuver, making the parameters in the control process clearer and more explicit. This not only helps to more accurately describe the satellite's motion state at different stages when building the control model, but also allows for more precise planning of the satellite's trajectory. Simultaneously, this division method facilitates the subsequent processing of complex constraints and optimization objectives, enabling more efficient consideration of various factors when solving for the optimal control sequence, thereby improving the accuracy of satellite orbit control and ensuring that the satellite completes its mission in the optimal manner while satisfying 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 equation (3), J is the cost function of the mission satellite established based on the model predictive control method, where ||·|| pThis represents the p-norm, where p = 1. The first term of the cost function is the control term, which includes the control acceleration for each sub-interval; the second term represents the final state of the mission satellite. Relative to the expected relative state The tracking error; the third term represents the relative state of the mission satellite at each time point. Relative state to expectations The differences between them. Matrix P and matrix Q are the weight matrices for the second and third terms, respectively.

[0100]

[0101] Define variables The set of mission star state variables and decision variables is as follows:

[0102]

[0103] Equation (4) represents the dynamic constraints of the slave star relative to the master star;

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

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

[0106]

[0107] Equation (6) represents the thrust amplitude constraint, f max f is the upper limit of the thrust amplitude. min This is the lower limit of the thrust amplitude;

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

[0109] Specifically, the non-convex optimal control model established in this embodiment, by constructing a cost function that includes control terms, terminal state tracking errors, and tracking errors at each state point, comprehensively considers fuel consumption, terminal state accuracy, and overall state tracking performance during satellite control. The priority of different control objectives can be flexibly adjusted through the setting of the weight matrix, providing a unified framework for multi-objective optimization. The dynamic constraints incorporated into the model ensure that satellite motion conforms to actual physical laws, initial and terminal state constraints strictly define the start and end conditions of the mission, and thrust amplitude constraints accurately reflect the physical limitations of the propulsion system, avoiding a disconnect between theoretical control input and actual execution capability. In particular, collision avoidance constraints, by introducing position mapping and minimum distance constraints under the Cartesian system, effectively address the safety distance problem between the satellite and obstacles in complex space environments, ensuring the safety of the orbit control process. This model integrates state variables and decision variables, forming a complete optimization system that includes continuous control input, discrete thrust states, and spatial constraints. This lays the foundation for subsequently transforming complex non-convex problems into efficiently solvable convex optimization problems through convexification processing, ultimately achieving fuel-optimal, trajectory-accurate, and safety-constrained autonomous satellite orbit control.

[0110] S21. Convexify the dynamic constraints;

[0111] According to the definition of convex optimization problems, the equality constraints in convex optimization problems must be affine. Therefore, the dynamic constraints are linearized based on the baseline trajectory, and then the zero-order preservation method is used to discretize the linearized dynamic constraints.

[0112] Specifically, this embodiment employs convexification of dynamic constraints by linearizing them based on a reference trajectory and then discretizing them using the zero-order preservation method. This approach has several significant implications. First, it aligns with the requirement that equality constraints in convex optimization problems must be affine, enabling the application of convex optimization theories and methods to solve the problem, thus providing an effective approach to addressing complex satellite control issues. Linearization based on the reference trajectory simplifies the form of dynamic constraints, reduces computational complexity, and improves solution efficiency. The discretization using the zero-order preservation method transforms 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 allows for efficient solutions to the originally complex satellite dynamics problem within the framework of convex optimization, facilitating the more accurate acquisition of satellite orbit control schemes that satisfy various constraints. This, in turn, enables precise satellite control and optimized fuel utilization, improving the stability and reliability of the satellite during orbit control.

[0113] S22. Introduce mixed integer decision constraints;

[0114] Will Divided into two subsets and Where f max This represents the upper limit of the thrust amplitude.

[0115]

[0116] and These represent the dimensionless positive and negative control accelerations within the m-th interval. They can vary from 0 to 1. and It is a binary integer variable in the range [0,1], used to define the sign of the control input along the (·) axis, as defined below:

[0117]

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

[0119]

[0120] Where M f =f max / f min Formulas (17) and (18) are used to enforce the definition of quantities. and Right now Only =1 and It is only non-zero (and positive) when it is 0, otherwise... Only =1 and It is only non-zero (and negative) when it is 0. Finally, formula (19) specifies that in m intervals, or At least one of them must be non-zero. Therefore, each interval considers only one force component (positive or negative), with the amplitude of the positive and negative finite-time maneuvers being... and In [f] min ,f max ] and [-f min ,-f max The range varies, and f min >0.

[0121] Specifically, this embodiment achieves several significant advantages by subsetting the thruster amplitude-related variables and introducing binary integer variables to define the control input symbols, combined with a series of inequality constraints. In satellite propulsion system control, this approach makes thruster amplitude control more precise and flexible. By setting the range of dimensionless positive and negative control accelerations and determining the control input symbols using binary integer variables, the operating state of the thruster 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 discrete characteristics of the thruster. In terms of fuel optimization, precise thruster control helps reduce unnecessary thrust output, thereby reducing fuel consumption and significantly extending the satellite's lifespan and mission execution capabilities when fuel is limited. From the perspective of orbit control accuracy, precise control of thruster amplitude and direction allows for more accurate adjustment of the satellite's orbit, reducing orbital deviations and meeting the precision requirements of demanding missions such as satellite rendezvous and docking.

[0122] S23. Convexify the collision avoidance constraints between the mission star and obstacles at discrete points;

[0123]

[0124] in This represents the relative orbital elements of the i-th obstacle at discrete point k; This represents the relative orbital elements of the mission satellite's reference trajectory at discrete point k at time k. Since the satellite orbit in this study is nearly circular, a linear mapping matrix can be used. The instantaneous orbital elements are mapped to the positional state in the Cartesian system, thereby introducing path collision avoidance constraints.

[0125]

[0126]

[0127] Equation (22) is the collision avoidance constraint after convexity shaping, where This represents the relative orbital elements of the obstacle at each discrete point. The optimization variables obtained from solving a convex optimization problem without collision avoidance constraints are used as the baseline trajectory. This represents the mapping matrix at each discrete point.

[0128] Specifically, this embodiment convexizes the collision avoidance constraints between the mission satellite and obstacles at discrete points. In the complex space environment where satellites operate, numerous obstacles exist, and satellites must effectively avoid collisions to ensure safe and stable operation. This claim utilizes the near-circular nature of satellite orbits, using a linear mapping matrix to map instantaneous orbital elements to Cartesian position states, thereby introducing path collision avoidance constraints and making the collision avoidance constraints convex. This convexization process not only transforms the complex collision avoidance problem into a more manageable mathematical form but also integrates it into the overall convex optimization framework, solving it together with other constraints and optimization objectives. In practical applications, by solving the optimization problem including convex collision avoidance constraints, the safe distance from obstacles can be fully considered during the satellite trajectory planning stage, resulting in the optimal control sequence that meets the collision avoidance requirements. This ensures that the satellite maintains a safe distance from obstacles throughout its operation, effectively reducing the risk of collisions and greatly improving the safety and reliability of satellite operation, providing crucial assurance for the successful completion of various complex space missions.

[0129] A computer device, characterized in that it comprises: 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-described satellite model predictive control method based on convex mixed integer programming.

[0130] Specifically, this embodiment involves a computer device including a memory, 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 can run programs to execute each step of the satellite model predictive control method, achieving fully automated operation from establishing a non-convex optimal control model to convexification processing, solving for the reference trajectory, and obtaining the optimal control sequence including collision avoidance constraints. This significantly improves the efficiency and accuracy of satellite orbit control. On one hand, the powerful computing capabilities of the computer device can quickly process large amounts of complex data, effectively solving the instruction generation delay problem caused by the excessive computational load of traditional methods, enabling the satellite to respond promptly to complex space conditions and enhancing the safety and reliability of satellite operation. On the other hand, this integrated device transforms 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 platform 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 is for method demonstration only and is not an actual flight mission. The simulation solution is obtained using the optimized control scheme of this patent. It is assumed that the target constellation contains four sub-stars, and the mission spacecraft can provide continuous, 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 mean 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 satellite are shown in Table 3. Each item of the quasi-nonsingular mean orbital elements of the mission satellite has changed.

[0138]

[0139] Table 3

[0140] The key simulation parameters for optimizing the solution settings are shown in Table 4:

[0141]

[0142] Table 4

[0143] According to the steps described in this invention, the trajectory of the mission satellite is optimized and controlled, and the solution obtained is as follows: Figures 4 to 7 As shown, simulation examples verify that the satellite model predictive control method based on convex mixed integer programming of this invention can effectively consider the duration of thrust and the minimum thrust constraint to obtain the trajectory under collision avoidance.

[0144] This invention proposes a satellite model predictive control method based on convex mixed-integer programming. By integrating continuous and discrete variables (such as thruster switching states), it effectively addresses the challenges faced by traditional control techniques in handling complex maneuvers, fuel constraints, and the discrete characteristics of thrusters. This optimized control framework combines mixed-integer optimization and rolling time-domain control. Its core lies in modeling the satellite's continuous state variables and discrete integer decision variables as a unified convex mixed-integer programming problem. Utilizing the predictive optimization capabilities of model predictive control, it simultaneously optimizes thruster scheduling and collision avoidance strategies while ensuring computational efficiency. This reduces propellant consumption or the number of switching operations, achieving efficient coordination between discrete and continuous decision-making in satellite control and providing a solution for complex space missions that combines real-time performance, optimality, and reliability.

[0145] The specific embodiments of the invention have been described in detail above, but they are only examples, and the invention is not limited to the specific embodiments described above. For those skilled in the art, any equivalent modifications and substitutions to the invention are also within the scope of this invention. Therefore, all equivalent changes and modifications made without departing from the spirit and scope of this invention should be covered within the scope of this 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 includes the following steps: S1. Establish a non-convex optimal control model for the mission satellite based on the model predictive control method. S1 includes the following steps: S11. Establish a satellite reference coordinate system and describe the satellite's relative motion model; S12. Divide the control zone; S13. Establishing a non-convex optimal control model for the mission satellite based on model predictive control methods. In S13, a non-convex optimal control model for the mission satellite is established based on model predictive control methods: （3） （4） （5） （6） （7） In formula (3) Let be the cost function of the mission satellite established based on the model predictive control method, where represent Norm, The first term of the cost function is the control term, which includes the control acceleration for each sub-interval; the second term represents the final state of the mission satellite. Relative to the expected relative state The tracking error; the third term represents the relative state of the mission satellite at each time point. Relative state to expectations The differences between them, where the matrix sum matrix The weight matrices for the second and third terms are respectively. , , Define variables The set of mission star state variables and decision variables is as follows: （8） （9） （10） （11） Equation (4) represents the dynamic constraints of the slave star relative to the master star; Equation (5) represents the initial and terminal constraints. and These represent the initial and final states of the mission satellite, respectively; through the matrix and From variables Extract the variables representing the initial state and the final state, where , （12） （13） Equation (6) represents the thrust amplitude constraint. This is the upper limit of the thrust amplitude. This is the lower limit of the thrust amplitude; Equation (7) represents the collision avoidance constraint between the mission star and the obstacle, where Represents the 2-norm. This represents the mapping function from the quasi-nonsingular average relative orbital elements to the position state in the Cartesian frame. Minimum distance constraint; The number of obstacles; S2. Convexify the non-convex terms in the non-convex optimal control model established in S1 to obtain the convexification result, and introduce the corresponding integer decision variables according to the divided control interval; S3. Based on the convexity result obtained in S2, solve the convex optimization problem without collision avoidance constraints as the baseline trajectory. S4. Based on the convexity result of step S2 and the baseline trajectory obtained in S3, solve the complete convex optimization problem including collision avoidance constraints 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 S11, First, define an RTN reference coordinate system, where the x-direction is radial and coincides with the absolute position vector, the z-direction is normal and coincides with the orbital angular momentum, and the y-direction is tangential, thus completing the construction of the right-handed orthogonal basis. Assume that only the mission satellite performs maneuvers and can provide control acceleration along the radial, tangential, and normal directions of the RTN reference coordinate system. The relative motion of a star to a reference star is described by a set of quasi-nonsingular mean relative orbital elements. The dynamic equations for the motion of a star relative to a reference star in a near-circular orbit are expressed as follows: （1） in: It is Earth's gravitational parameter. It is the equatorial radius. It is the average angular velocity of the reference star; other parameters are... , , , , , , , , , , , subscript Indicates the orbital elements of the reference star. For the semi-major axis of the track, For orbital eccentricity, For the track inclination angle, Right ascension of the ascending node, Angular distance from perigee For the angle of near point, Indicates the reference star is The average latitudinal argument at any given time, variable and pass Linear correlation, where , .

3. The satellite model predictive control method based on convex mixed-integer programming according to claim 2, characterized in that, In S12, Control the average latitude angle range Divided into A fixed length is A finite number of subintervals, where , Set the task end time and generate There are state points, among which And associate each sub-interval with a size of and The control acceleration, in which In this approach, the optimization variable is the same as the first... Control volume related to each sub-range , （2） Indicates the interval inner axis The limited number of maneuvers within a given time, and and They represent the first The average latitude angle of the primary trajectory at the start and end of the secondary maneuver; and They represent the first The half-angle duration and angular position of a finite-time maneuver.

4. The satellite model predictive control method based on convex mixed-integer programming according to claim 1, characterized in that, S2 includes the following steps: S21. Convexify the dynamic constraints; S22. Introduce mixed integer decision constraints; S23. Convexify the collision avoidance constraints between the mission star and obstacles at discrete points.

5. The satellite model predictive control method based on convex mixed-integer programming according to claim 4, characterized in that, In S21, the dynamic constraints are linearized based on the reference trajectory, and then the linearized dynamic constraints are discretized using the zero-order preservation method.

6. The satellite model predictive control method based on convex mixed-integer programming according to claim 5, characterized in that, In S22, Will Divided into two subsets and ,in This is the upper limit of the thrust amplitude. （14） （15） and It means the first Dimensionless positive and negative control acceleration within each interval and The range of variation is 0-1. and The range is A binary integer variable used to define along The symbols for axis control inputs are defined as follows: （16） Based on the determined integer decision variables, the thrust amplitude constraint needs to include the following inequality constraints: （17） （18） （19） in Formulas (17) and (18) are used to enforce the definition of quantities. and ,Right now Only =1 and It is only non-zero when it is 0, and is positive; otherwise... Only =1 and It is only non-zero when it is 0, and it is negative. Finally, formula (19) stipulates that... In each interval, or At least one of them must be non-zero; therefore, only one force component is considered for each interval, with the amplitude of the positive and negative finite-time maneuvers. and In respectively and Variations within the range, and .

7. The satellite model predictive control method based on convex mixed-integer programming according to claim 6, characterized in that, In S23, Convexify the collision avoidance constraints between the mission star and obstacles at discrete points; （20） in Indicates the first An obstacle at a discrete point The relative orbital elements at the location; This indicates the location of the reference trajectory of the mission satellite at discrete points. The relative orbital elements at time points are determined using a linear mapping matrix, since the satellite orbit in this study is nearly circular. By mapping instantaneous orbital elements to position states in the Cartesian coordinate system, path collision avoidance constraints are introduced. （21） （22） （23） （24） （25） （26） Equation (22) is the collision avoidance constraint after convexity shaping, where This represents the relative orbital elements of the obstacle at each discrete point. ; The optimization variables obtained from solving a convex optimization problem without collision avoidance constraints are used as the baseline trajectory. This represents the mapping matrix at each discrete point. .

8. A computer device, characterized in that, include: The memory, the processor, and the computer program stored in the memory and executable on the processor, the processor executing the program to implement the satellite model predictive control method based on convex hybrid integer programming as described in any one of claims 1-7.

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