A Satellite Formation Configuration Preservation Method Based on Pipeline Model Predictive Control

Abstract

This invention discloses a satellite formation configuration maintenance method based on pipeline model predictive control, comprising: establishing a satellite relative motion model; determining the state and control input constraints of the satellite relative motion model; solving an optimization problem based on the determined state and control input constraints of the satellite relative motion model to obtain the optimal control quantity; and controlling the thruster ignition strategy of each satellite in the satellite formation according to the obtained optimal control quantity to achieve the optimal trajectory for each satellite to track the reference satellite. Based on the method described in this invention, the required orbital control force for each satellite in the satellite formation can be determined, ensuring both a safe separation distance and a relatively stable positional relationship between the satellites in the formation.

Description

Technical Field

[0001] This invention belongs to the field of satellite communication technology, and in particular relates to a satellite formation configuration maintenance method based on pipeline model predictive control. Background Technology

[0002] Geostationary orbit (GEO) is currently the most commonly used orbit for high-orbit communication satellites. Since GEO slots are limited, multiple satellites are often placed in the same or adjacent fixed-point areas to form a communication satellite constellation. To ensure the safety of satellites within the constellation and the stability of constellation operation, periodic configuration maintenance control is required to prevent constellation failure due to space environment perturbations and other factors.

[0003] There are two main methods for controlling satellite formations: geometric and algebraic. The geometric method is based on a relative motion model represented by orbital elements, analyzing the relative orbital elements between satellites in the formation to perform orbit design and perturbation analysis. The algebraic method is based on the position vectors between two satellites, describing the relative motion between them in the Hill coordinate system, and using this as a basis for related design. The most classic method is to use the CWH equations to establish a relative motion model; however, this model has certain errors, and the inherent uncertainties limit its applicability. Summary of the Invention

[0004] The technical problem solved by this invention is to overcome the shortcomings of the prior art and provide a satellite formation configuration maintenance method based on pipeline model predictive control. Based on this method, the orbital control force required for each satellite in the satellite formation can be determined, so that the satellites in the formation can maintain a safe distance and a relatively stable positional relationship.

[0005] To address the aforementioned technical problems, this invention discloses a satellite formation configuration maintenance method based on pipeline model predictive control, comprising:

[0006] Establish a satellite relative motion model;

[0007] Determine the state and control input constraints of the satellite's relative motion model;

[0008] Based on the determined state and control input constraints of the satellite relative motion model, an optimization problem is solved to obtain the optimal control quantity;

[0009] Based on the optimal control variables obtained from the solution, the thruster ignition strategy of each satellite in the satellite formation is controlled to achieve the optimal trajectory for the satellite to track the reference satellite.

[0010] In the above-mentioned satellite formation configuration maintenance method based on pipeline model predictive control, a satellite relative motion model is established, including:

[0011] Obtain the current velocity and position vectors of each satellite in the satellite formation in the geocentric equatorial coordinate system, and determine the reference orbit of each satellite based on the configuration requirements;

[0012] Represent the relative motion between each satellite in a satellite formation and a reference orbit using the CWH equations:

[0013]

[0014]

[0015]

[0016] Where m represents the satellite mass; [δxδyδz] T F represents the satellite's position vector in the Hill coordinate system relative to the reference satellite; x ,F y ,F z This represents the thrust of the satellite in the Hill coordinate system of the reference satellite; a p,x ,a p,y ,a p,z The variable represents the perturbation acceleration of the satellite in the Hill coordinate system of the reference satellite; n represents the average rotational speed of the orbit. μ E The reference satellite represents the Earth's gravitational coefficient, and r represents the distance from the satellite's center of mass to the Earth's center. For the primary satellite, the reference satellite is a hypothetical virtual satellite, while for the secondary satellite, the reference satellite is its neighboring satellite.

[0017] Let the satellite's nominal state relative to its orbit Satellite relative orbit nominal control thrust use Substitute Substitute The satellite's relative motion model can then be expressed as:

[0018]

[0019] Where A represents the system's state matrix, B represents the system's input matrix, D represents the system's disturbance matrix, and a p (t) represents the total perturbation acceleration.

[0020] In the aforementioned satellite formation configuration preservation method based on pipeline model predictive control,

[0021]

[0022] In the aforementioned satellite formation configuration maintenance method based on pipeline model predictive control, there are N satellites in the formation. a The satellites include: 1 primary satellite and N satellites.a -1 follower satellite; the primary satellite's reference orbit is a standard circular orbit; the reference orbits of each follower satellite are determined by the orbits of its adjacent satellites.

[0023] In the aforementioned satellite formation configuration maintenance method based on pipeline model predictive control, the state and control input constraints of the satellite relative motion model are determined, including:

[0024] Discretizing equation (1) yields the discretized expression for the satellite's relative motion model:

[0025]

[0026] Among them, A d B d D d Let A, B, and D represent the discretized forms of matrices A, B, and D, respectively.

[0027] The errors in the satellite's relative motion model, along with various external uncertainties, are considered as additive uncertainties w of the system; where,

[0028] Therefore, the actual relative motion model of the satellite is represented as follows:

[0029] x(k+1)=A d x(k)+B d u(k)+D d a p (k)+w···(3)

[0030] Where x(k) represents the actual state of the satellite relative to its orbit, and u(k) represents the actual control thrust of the satellite relative to its orbit;

[0031] The system state and system input constraints are described as follows:

[0032]

[0033]

[0034] Where x is the abbreviation for x(k), x represents the set of actual states of a satellite relative to its orbit. lb x represents the boundary of the satellite's actual state relative to its orbit. ub This represents the upper bound of the satellite's actual state relative to its orbit, where u is abbreviation for u(k). u represents the set of actual control thrusts relative to the satellite's orbit. lb u represents the lower bound of the actual control thrust of the satellite relative to its orbit. ub This indicates the upper limit of the actual control thrust of the satellite relative to its orbit;

[0035] Consider the following feedback controller:

[0036]

[0037] Where K represents the feedback matrix; This represents the actual relative position of the satellite, x(k), compared to the position predicted by the model. The error between them

[0038] Therefore, the corresponding error system is:

[0039]

[0040] Then we have:

[0041]

[0042] Where, φ k This represents the robust positive invariant set of the error system;

[0043] Then, at any time k, the true state of the error system satisfies:

[0044]

[0045] in, Denote the Minkowski sum between two sets;

[0046] Therefore, the state and control input constraints of the satellite relative motion model are expressed as follows:

[0047]

[0048] in, This represents the Minkowski difference between two sets.

[0049] In the aforementioned satellite formation configuration maintenance method based on pipeline model predictive control, an optimization problem is solved according to the determined state of the satellite relative motion model and control input constraints to obtain the optimal control variables, including:

[0050] Define the cost function J:

[0051]

[0052] Where, N p This represents the prediction step size at time k. Indicates the predicted state. The terminal state is represented by Q, the state weight matrix, R, and P.

[0053] For all satellites in the satellite formation, solve the following optimization problem individually for each satellite:

[0054]

[0055] in, and Let them represent the optimal control quantity and optimal state in the finite time domain obtained by solving equation (12), respectively;

[0056] The secondary controller defined by equation (6) is expressed as:

[0057]

[0058] The optimal control quantity u is obtained by solving equation (13). * (k).

[0059] The present invention has the following advantages:

[0060] This invention discloses a satellite formation configuration maintenance method based on pipeline model predictive control. Based on this method, the orbital control force required for each satellite in the satellite formation can be determined, so that the satellites in the formation can maintain a safe distance and a relatively stable positional relationship. Attached Figure Description

[0061] Figure 1 This is a schematic flowchart of a satellite formation configuration maintenance method based on pipeline model predictive control in an embodiment of the present invention. Detailed Implementation

[0062] To make the objectives, technical solutions, and advantages of the present invention clearer, the embodiments disclosed in the present invention will be described in further detail below with reference to the accompanying drawings.

[0063] One of the core ideas of this invention is to superimpose uncertainties such as the space environment and orbit determination errors onto a standard motion reference model in a pipeline manner to determine the error pipeline control model for each satellite in the formation. For the primary satellite of the satellite formation (usually one), its standard motion reference model is its own ideal orbit. The standard motion reference models of the secondary satellites (the remaining satellites) can be obtained based on their relative positions to the primary satellite. Then, the control model is optimized according to the set constraints to determine the orbital control force, so that each satellite tracks its reference trajectory, ultimately achieving configuration maintenance of the satellite orbit formation. The satellite formation configuration maintenance method based on pipeline model predictive control described in this invention includes: based on the autonomous navigation results, the satellites in the formation obtain their current velocity and position vectors in the geocentric equatorial coordinate system (with the Earth's center as the origin, the x-axis pointing to the vertex, the z-axis perpendicular to the equatorial plane and consistent with the Earth's rotational angular velocity vector, and the y-axis forming a right-handed rectangular coordinate system). According to the formation configuration requirements, the reference trajectory of each satellite is determined. Using the CWH equations, a relative motion model between the satellite and its reference trajectory is constructed, and after discretization, an iterative model of relative motion is obtained. Due to inherent uncertainties in the iterative relative motion model constructed using the CWH equations, as well as uncertainties caused by navigation errors, the actual motion state of the satellite deviates from the state predicted by the model. This deviation is considered as additive uncertainty in the model, and a set of satellite state centers obtained by solving the model, with its maximum error as the boundary, is constructed, called the error pipeline. The error pipeline is incorporated into the design of the subsequent controller optimization problem. Model predictive controller. Model predictive control is a control method based on rolling time-domain optimization, iteratively solving the optimization problem in each control cycle. This invention designs a controller for maintaining the relative position of stationary satellites based on model predictive control. It uses a discretized iterative model to predict the system's state over the next N steps and solves the optimization problem while considering system constraints including the error pipeline. By solving for the optimal control sequence, the control input for each satellite in the formation at the current moment is obtained. Based on the control input, it is converted into the corresponding thruster operating strategy.

[0064] like Figure 1 In this embodiment, the satellite formation configuration maintenance method based on pipeline model predictive control includes:

[0065] Step 1: Establish a model of the relative motion of the satellite.

[0066] In this embodiment, let there be N satellites in the formation. a There are 1 satellite, including 1 primary satellite and N satellites. a-1 follower satellite. The primary satellite's reference orbit is a standard circular orbit; the reference orbits of each follower satellite, excluding the primary satellite, are determined by the orbits of its adjacent satellites. Obtain the current velocity and position vectors of each satellite in the formation within the geocentric equatorial coordinate system (with the Earth's center as the origin, the x-axis pointing to the vernal equinox, the z-axis perpendicular to the equatorial plane and aligned with the Earth's rotational angular velocity vector, and the y-axis forming a right-handed rectangular coordinate system), and determine its reference trajectory according to the configuration requirements. Define the Hill coordinate system (with the satellite's center of mass as the origin, the x-axis pointing outwards along the line connecting the Earth's center and the satellite; the y-axis along the direction of the satellite's linear velocity; and the z-axis forming a right-handed orthogonal coordinate system). For any satellite in the formation and its reference orbit, use the CWH equations to represent the relative motion between each satellite and its reference trajectory:

[0067]

[0068]

[0069]

[0070] Where m represents the satellite mass; [δxδyδz] T F represents the satellite's position vector in the Hill coordinate system relative to the reference satellite; x ,F y ,F z This represents the thrust of the satellite in the Hill coordinate system of the reference satellite; a p,x ,a p,y ,a p,z The variable represents the perturbation acceleration of the satellite in the Hill coordinate system of the reference satellite; n represents the average rotational speed of the orbit. μ E The reference satellite represents the Earth's gravitational coefficient, and r represents the distance from the satellite's center of mass to the Earth's center. For the primary satellite, the reference satellite is a hypothetical virtual satellite, while for the secondary satellite, the reference satellite is its neighboring satellite.

[0071] Let the satellite's nominal state relative to its orbit Satellite relative orbit nominal control thrust use Substitute Substitute The satellite's relative motion model can then be expressed as:

[0072]

[0073]

[0074] Where A represents the system's state matrix, B represents the system's input matrix, D represents the system's disturbance matrix, and a p (t) represents the total perturbation acceleration.

[0075] Discretizing equation (1) yields the discretized expression for the satellite's relative motion model:

[0076]

[0077] Among them, A d B d D d Let A, B, and D represent the discretized forms of matrices A, B, and D, respectively.

[0078] Step 2: Determine the state and control input constraints of the satellite relative motion model.

[0079] In this embodiment, the state and control input constraints of the satellite relative motion model are determined, that is, an error pipeline describing the uncertainty is constructed. The error of the satellite relative motion model, along with various external uncertainties, is considered as the additive uncertainty w of the system. Therefore, the actual relative motion model of the satellite can be expressed as:

[0080] x(k+1)=A d x(k)+B d u(k)+D d a p (k)+w···(3)

[0081] Where x(k) represents the actual state of the satellite relative to its orbit, and u(k) represents the actual control thrust of the satellite relative to its orbit.

[0082] Meanwhile, the system state and system input constraints can be described as follows:

[0083]

[0084]

[0085] Where x is the abbreviation for x(k), x represents the set of actual states of a satellite relative to its orbit. lb x represents the boundary of the satellite's actual state relative to its orbit. ub This represents the upper bound of the satellite's actual state relative to its orbit, where u is abbreviation for u(k). u represents the set of actual control thrusts relative to the satellite's orbit. lb u represents the lower bound of the actual control thrust of the satellite relative to its orbit. ub This indicates the upper limit of the actual control thrust of the satellite relative to its orbit.

[0086] Define the error between the actual relative position of the satellite and the position predicted by the model:

[0087]

[0088] Consider the following feedback controller:

[0089]

[0090] Where K represents the feedback matrix; This represents the actual relative position of the satellite, x(k), compared to the position predicted by the model. The error between them

[0091] Then we can obtain the corresponding error system:

[0092]

[0093] For an error system, its robust positive invariant set is defined as φ. k Then we have:

[0094]

[0095] Therefore, at any time k, the true state of the error system satisfies:

[0096]

[0097] in, Let represent the Minkowski sum between two sets.

[0098] Therefore, the state and control input constraints of the satellite relative motion model are expressed as follows:

[0099]

[0100] in, This represents the Minkowski difference between two sets.

[0101] Step 3: Based on the determined state and control input constraints of the satellite relative motion model, solve the optimization problem to obtain the optimal control quantity.

[0102] In this embodiment, step 3 is also the solution problem for the pipeline-based model predictive controller optimization problem. Let the current time be k, and the prediction step size at time k be N. p Then the cost function J is defined as follows:

[0103]

[0104] in, Indicates the predicted state. The terminal state is represented by Q, the state weight matrix, R, and P.

[0105] For all satellites in the satellite formation, solve the following optimization problem individually for each satellite:

[0106]

[0107] in, and Let represent the optimal control quantity and optimal state in the finite time domain obtained by solving equation (12), respectively. Then the secondary controller defined by equation (6) is expressed as:

[0108]

[0109] The optimal control quantity u is obtained by solving equation (13). * (k).

[0110] Step 4: Based on the optimal control variables obtained from the solution, control the thruster ignition strategy of each satellite in the satellite formation to achieve the optimal trajectory for the satellite to track the reference satellite.

[0111] In this embodiment, the optimal control quantity u is solved. * (k) represents the control force of the satellite in the Hill coordinate system. Actual geostationary satellites generally use onboard electric thrusters. Depending on the distribution of the onboard thrusters, and considering practical considerations, the control parameters need to be calculated to obtain the current actual thruster ignition strategy. Simulation verification shows that the method of this invention can achieve high-precision satellite formation maintenance missions.

[0112] Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make possible changes and modifications to the technical solutions of the present invention by utilizing the methods and techniques disclosed above without departing from the spirit and scope of the present invention. Therefore, any simple modifications, equivalent changes and alterations made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solutions of the present invention shall fall within the protection scope of the technical solutions of the present invention.

[0113] The contents not described in detail in this specification are common knowledge to those skilled in the art.

Claims

1. A satellite formation configuration maintenance method based on pipeline model predictive control, characterized in that, include: Establish a model of the relative motion of the satellite; Determine the state and control input constraints of the satellite's relative motion model; Based on the determined state and control input constraints of the satellite relative motion model, an optimization problem is solved to obtain the optimal control quantity; Based on the optimal control variables obtained from the solution, the thruster ignition strategy of each satellite in the satellite formation is controlled to achieve the optimal trajectory for the satellite to track the reference satellite. Establish a satellite relative motion model, including: Obtain the current velocity and position vectors of each satellite in the satellite formation in the geocentric equatorial coordinate system, and determine the reference orbit of each satellite based on the configuration requirements; Represent the relative motion between each satellite in a satellite formation and a reference orbit using the CWH equations: Where m represents the satellite mass; [δxδyδz] T F represents the satellite's position vector in the Hill coordinate system relative to the reference satellite; x ,F y ,F z This represents the thrust of the satellite in the Hill coordinate system of the reference satellite; a p,x ,a p,y ,a p,z The variable represents the perturbation acceleration of the satellite in the Hill coordinate system of the reference satellite; n represents the average rotational speed of the orbit. μ E The reference satellite represents the Earth's gravitational coefficient, and r represents the distance from the satellite's center of mass to the Earth's center. For the primary satellite, the reference satellite is a hypothetical virtual satellite, while for the secondary satellite, the reference satellite is its neighboring satellite. Let the satellite's nominal state relative to its orbit Satellite relative orbit nominal control thrust use Substitute Substitute The satellite's relative motion model can then be expressed as: Where A represents the system's state matrix, B represents the system's input matrix, D represents the system's disturbance matrix, and a p (t) represents the total perturbation acceleration; Determine the state and control input constraints of the satellite relative motion model, including: Discretizing equation (1) yields the discretized expression for the satellite's relative motion model: Among them, A d B d D d Let A, B, and D represent the discretized forms of matrices A, B, and D, respectively. The error in the satellite's relative motion model, along with various external uncertainties, is considered as the additive uncertainty w of the system; where, Therefore, the actual relative motion model of the satellite is represented as follows: x(k+1)=A d x(k)+B d u(k)+D d a p (k)+w···(3) Where x(k) represents the actual state of the satellite relative to its orbit, and u(k) represents the actual control thrust of the satellite relative to its orbit; The system state and system input constraints are described as follows: Where x is the abbreviation for x(k), x represents the set of actual states of a satellite relative to its orbit. lb x represents the boundary of the satellite's actual state relative to its orbit. ub This represents the upper bound of the satellite's actual state relative to its orbit, where u is abbreviation for u(k). u represents the set of actual control thrusts relative to the satellite's orbit. lb u represents the lower bound of the actual control thrust of the satellite relative to its orbit. ub This indicates the upper limit of the actual control thrust of the satellite relative to its orbit; Define the actual satellite relative position x(k) and the position predicted by the model. Error between Consider the following feedback controller: Where K represents the feedback matrix; Therefore, the corresponding error system is: Then we have: Where, φ k This represents the robust positive invariant set of the error system; Then, at any time k, the true state of the error system satisfies: in, Denote the Minkowski sum between two sets; Therefore, the state and control input constraints of the satellite relative motion model are expressed as follows: in, Denotes the Minkowski difference between two sets; Based on the determined state and control input constraints of the satellite relative motion model, an optimization problem is solved to obtain the optimal control variables, including: Define the cost function J: Where, N p This represents the prediction step size at time k. Indicates the predicted state. The terminal state is represented by Q, the state weight matrix, R, and P. For all satellites in the satellite formation, solve the following optimization problem individually for each satellite: in, and Let them represent the optimal control quantity and optimal state in the finite time domain obtained by solving equation (12), respectively; The secondary controller defined by equation (6) is expressed as: The optimal control quantity u is obtained by solving equation (13). * (k).

2. The satellite formation configuration maintenance method based on pipeline model predictive control according to claim 1, characterized in that, 3. The satellite formation configuration maintenance method based on pipeline model predictive control according to claim 1, characterized in that, There are N satellites in the formation. a The satellites include: 1 primary satellite and N satellites. a -1 follower satellite; the primary satellite's reference orbit is a standard circular orbit; the reference orbits of each follower satellite are determined by the orbits of its adjacent satellites.

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