A Fully Distributed Control Method for Heterogeneous Spacecraft Formation Considering Time-Varying Topology and Time-Varying Delay

Abstract

A fully distributed control method for heterogeneous spacecraft formations considering time-varying topology and time-varying time delays is proposed. The method includes establishing an orbital dynamics model of the accompanying satellite formation with distinct input time delays and obtaining its state-space equations; establishing a signal model of the master spacecraft to be tracked and obtaining its state-space equations; constructing explicit fully distributed observers for each accompanying satellite to obtain the master spacecraft's state information constrained by time-varying communication time delays; designing explicit linear state feedback gains and feedforward gains respectively; and using the state feedback gains, feedforward gains, and fully distributed observers, establishing a corresponding fully distributed control protocol for the accompanying satellite formation system affected by time-varying topology and time-varying time delays to ensure that each accompanying satellite completes its tracking mission of the master spacecraft. This invention exhibits good control performance, ensuring stable operation of the heterogeneous spacecraft formation system and achieving the task of accompanying satellites tracking the master spacecraft.

Description

Technical Field

[0001] This invention relates to a fully distributed control method for heterogeneous spacecraft formation systems, belonging to the field of spacecraft control technology, specifically to a fully distributed control method for heterogeneous spacecraft formations that considers time-varying topology and time-varying time delay. Background Technology

[0002] Heterogeneous spacecraft swarms refer to the collaborative work of spacecraft of different types or functions (such as satellites, probes, and spacecraft) within a single formation, forming a specific overall configuration to jointly achieve tasks such as communication, observation, and navigation. Compared to a single spacecraft, swarms of multiple smaller, heterogeneous spacecraft offer several significant advantages, such as simpler design, faster construction, lower replacement costs, and higher system redundancy. Furthermore, the combined operation of multiple satellites, each responsible for collecting data in a specific frequency band, enables comprehensive Earth observation coverage and often achieves higher target resolution. Therefore, the advantage of heterogeneous spacecraft swarms lies in their ability to fully utilize the characteristics of different spacecraft types, achieving more comprehensive and efficient mission execution. These advantages make heterogeneous spacecraft swarms a promising technology for future space missions, with significant applications in Earth observation, astronomical observation, and deep space exploration.

[0003] The control problem of heterogeneous spacecraft formation systems can be transformed into an output regulation problem. Traditional control methods often rely on incomplete system models that fail to consider real-world scenarios inherent in formation missions. Firstly, the communication topology may be dynamic, as communication links can fail or be reconfigured over time, rendering traditional control methods ineffective. Secondly, time delays are unavoidable in practice, significantly impacting control performance and potentially leading to system instability if not properly addressed in the control protocol design. Two main types of delays typically affect spacecraft formation systems: time-varying communication delays caused by inter-spacecraft communication and input delays resulting from the time required for each spacecraft to process data.

[0004] In summary, current control methods for heterogeneous spacecraft formation systems fail to consider time-varying topology and time-varying time delays, resulting in poor control performance, system instability, and difficulty in ensuring that each accompanying satellite completes its mission of tracking the main spacecraft. Summary of the Invention

[0005] To overcome existing technologies and realize heterogeneous spacecraft formation missions affected by time-varying topology and time-varying time delay, this invention proposes a fully distributed control method for heterogeneous spacecraft formation that considers time-varying topology and time-varying time delay.

[0006] The fully distributed control method for heterogeneous spacecraft formation considering time-varying topology and time-varying time delay in this application includes the following steps:

[0007] Step 1: Establish an orbital dynamics model of the accompanying satellite formation with mutually different input time delays and obtain its state-space equations; establish a signal model of the master spacecraft to be tracked and obtain its state-space equations.

[0008] Step 2: Construct explicit, fully distributed observers for each accompanying satellite, enabling each accompanying satellite to acquire the status information of the main spacecraft, which is constrained by time-varying communication delays.

[0009] Step 3: Based on the positive definite solution of the parametric Lyapunov equation and the solution of the time-delay regulation equation, design explicit linear state feedback gain and feedforward gain respectively. Using the state feedback gain, feedforward gain and the fully distributed observer, establish a corresponding fully distributed control protocol for the accompanying satellite formation system affected by time-varying topology and time-varying time delay, so as to ensure that each accompanying satellite completes the task of tracking the main spacecraft.

[0010] The advantages of this invention compared to the prior art are:

[0011] The most significant advantages of the fully distributed control method proposed in this invention for heterogeneous spacecraft formation systems affected by time-varying topology and time-varying time delays are threefold. First, the system considered is more comprehensive, allowing for heterogeneous spacecraft, time-varying communication topologies, and arbitrarily large but bounded time-varying communication and input delays. Second, the proposed control protocol is fully distributed, requiring no global communication topology information between the accompanying satellite and the host spacecraft, indicating stronger robustness. Finally, simulations of a heterogeneous spacecraft formation system with time-varying topology and time-varying time delays demonstrate that the designed fully distributed control protocol achieves the task of the accompanying satellite tracking the host spacecraft.

[0012] The method proposed in this application has good control performance and stable operation of heterogeneous spacecraft formation systems. Therefore, designing a fully distributed control method for heterogeneous spacecraft formations with time-varying topology and time-varying time delay has important theoretical and engineering significance.

[0013] The present invention will be further described below with reference to the accompanying drawings and embodiments: Attached Figure Description

[0014] Figure 1 This is the first scenario of the time-varying communication topology diagram between the accompanying satellite and the main spacecraft in the embodiment;

[0015] Figure 2 This is the second scenario of the time-varying communication topology diagram between the accompanying satellite and the main spacecraft in the embodiment;

[0016] Figure 3 These are the controlled trajectories of the four companion satellites relative to the main spacecraft in the embodiment;

[0017] Figure 4 These are the control input signals for the four accompanying satellites in the embodiment. Detailed Implementation

[0018] The embodiments of the technical solution of the present invention will now be described in detail with reference to the accompanying drawings. Unless otherwise stated, the technical or scientific terms used in this application have the ordinary meaning as understood by those skilled in the art.

[0019] In this embodiment, heterogeneous spacecraft refers to spacecraft in a spacecraft formation that are of different types. For example, multiple satellites orbit a main spacecraft, but the satellites and the main spacecraft are not the same type of spacecraft, have different functions, and have different equations of motion.

[0020] The heterogeneous spacecraft formation in this embodiment refers to spacecraft of different types or functions (such as satellites, detectors, and spacecraft) working together in a specific overall configuration within the same formation to jointly achieve tasks such as communication, observation, and navigation.

[0021] Specific Implementation Method 1: This implementation method, which considers time-varying topology and time-varying time delay, is a fully distributed control method for heterogeneous spacecraft formations and includes the following steps:

[0022] Step 1: Establish an orbital dynamics model of the accompanying satellite formation with mutually different input time delays and obtain its state-space equations; establish a signal model of the master spacecraft to be tracked and obtain its state-space equations.

[0023] Step 2: Construct explicit, fully distributed observers for each accompanying satellite to obtain the status information of the master spacecraft, which is constrained by time-varying communication delays;

[0024] Step 3: Based on the positive definite solution of the parametric Lyapunov equation and the solution of the time-delay regulation equation, design explicit linear state feedback gain and feedforward gain respectively. Using the state feedback gain, the feedforward gain and the fully distributed observer, establish a corresponding fully distributed control protocol for the accompanying satellite formation system affected by time-varying topology and time-varying time delay, so as to ensure that each accompanying satellite completes the task of tracking the main spacecraft.

[0025] Specific Implementation Method Two: This implementation method differs from Specific Implementation Method One in that: the specific process of establishing the orbital dynamics model of the accompanying satellite formation with mutually different input time delays and obtaining its state-space equation, and establishing the signal model of the master spacecraft to be tracked and obtaining its state-space equation in step one is as follows:

[0026] Step 1: Establish the mathematical model and state-space equations of the accompanying satellite formation flight system;

[0027] Suppose there are N companion satellites numbered 1 to N orbiting a main spacecraft numbered 0 in low Earth orbit, where (x i,1 ,x i,2 ,x i,3 Let be the relative position coordinates of the i-th companion satellite and the main spacecraft in the Hill coordinate system. Only consider the in-plane heterogeneous spacecraft formation flight problem (i.e., x...). i,3 ≡0), therefore, the in-plane relative motion equation of the i-th companion satellite with respect to the main spacecraft is:

[0028]

[0029] Among them, R e is the Earth's orbital radius; μ is the gravitational parameter; It is the average orbital speed of the accompanying satellite; Definition (a) i,1 ,a i,2 ) is the additional acceleration of the i-th accompanying satellite due to aerodynamic drag; ι i It is the input time delay of the i-th companion satellite, ι i It can be arbitrarily large but bounded. In this case, the in-plane relative motion equation of the i-th companion satellite with respect to the main spacecraft is linearized as:

[0030]

[0031] in, This is the state of the i-th companion satellite; u i (t)=[a i,1 ,a i,2 ] T It is the input of the i-th companion satellite; y i (t) is the output of the i-th companion satellite; All are coefficient matrices, where I2 represents the 2nd order identity matrix, 0 2×2 This represents a zero matrix with a dimension of 2×2.

[0032] Steps 1 and 2: Establish the mathematical model and state-space equations of the signal from the master spacecraft to be tracked;

[0033] Considering that the mission of accompanying satellite formation flight is to achieve circular motion around the main spacecraft, let the output y of the i-th accompanying satellite be... i (t) needs to track a given signal of 50iv(t), where v(t) = [cos(w)] ref t),sin(w ref t)]T sin(·) is the sine function, cos(·) is the cosine function, and w ref =5ω is a given constant, and the given tracking signal v(t) can be described by a linear state-space equation:

[0034]

[0035] in, The output error of the i-th companion satellite is e i (t) can be described by the following equation:

[0036]

[0037] in The linearized equation of the in-plane relative motion of the i-th companion satellite with respect to the main spacecraft (2), the given tracking signal state equation (3), and the output error equation (4) of the i-th companion satellite are combined and discretized as follows:

[0038]

[0039] in T s With a sampling period, in this case, the mission of accompanying satellite formation flying around can be transformed into designing a control protocol so that the output error of each accompanying satellite tends to zero.

[0040] Specific Implementation Method Three: This implementation method differs from Specific Implementation Method One or Two in that the expression for constructing an explicit fully distributed observer for each accompanying satellite in step two is:

[0041]

[0042] Where, ε i (k) is the observer state, representing the estimate of the tracking signal given by the main spacecraft by the i-th companion satellite at time k, and ε0(k) = v(k); σ(k) is the piecewise constant switching signal; Let represent the communication weight between the j-th companion satellite or main spacecraft and the i-th companion satellite in the topology at time k. If the i-th companion satellite can obtain information from the j-th companion satellite or main spacecraft at time k, then... otherwise and τ ij (k) represents the time-varying communication delay between the j-th companion satellite or main spacecraft and the i-th companion satellite at time k; 0 < δ < 1. Furthermore, if the i-th companion satellite has no other companion satellite or main spacecraft communicating with it at time k, then let...

[0043] As can be seen, the observer constructed by formula (6) does not require global communication topology information between the accompanying satellite and the main spacecraft. Therefore, the observer of formula (6) is called a fully distributed observer, which makes the constructed control protocol more robust.

[0044] Specific Implementation Method Four: This implementation method differs from Specific Implementation Methods One, Two, or Three in that the process of designing an explicit linear state feedback gain based on the positive definite solution of the parametric Lyapunov equation in step three is as follows:

[0045] Establish the discrete parameter Lyapunov equations for the i-th companion satellite:

[0046] A T P i (γ i AP i (γ i )-A T P i (γ i )B(I2+B T P i (γ i B) -1 B T P i (γ i A = -γ i P i (γ i (7)

[0047] Where γ i It is the scalar to be designed; P i (γ i If ) is the unique positive definite solution of the discrete parameter Lyapunov equation (7); then the explicit linear state feedback gain K of the i-th companion satellite is... i (γ i It was designed as:

[0048] K i (γ i )=(I2+B T P i (γ i B) -1 B T P i (γ i A (8)

[0049] Furthermore, based on the solution of the time-delay regulation equation, an explicit feedforward gain process is designed as follows;

[0050] Construct the time-delay regulation equation for the i-th companion satellite:

[0051]

[0052] Where (X) i U i If ) is the solution to the time-delay regulation equation of the i-th companion satellite, then the feedforward gain L of the i-th companion satellite is... i Designed as:

[0053]

[0054] Furthermore, by utilizing state feedback gain, feedforward gain, and distributed observers of accompanying satellites, a corresponding fully distributed control protocol is established for accompanying satellite formation systems affected by time-varying topology and time-varying time delay, ensuring that each accompanying satellite completes its task of tracking the main spacecraft.

[0055] Based on equations (6), (8), and (10), the control protocol for the i-th companion satellite is established as follows:

[0056]

[0057] Observing equation (5), it can be seen that the tracking problem of each accompanying satellite can be transformed into the output error e of each accompanying satellite. i The stabilization problem of (k). This is addressed by defining state transitions. From equations (5) and (9), we can derive:

[0058]

[0059] And we obtained:

[0060]

[0061] Based on the feedforward gain (10), the control protocol for the i-th companion satellite is rewritten as follows:

[0062]

[0063] in The control protocol for the i-th accompanying satellite affected by the input time delay is further described as follows:

[0064]

[0065] Substituting equation (15) into (12) yields:

[0066]

[0067] Based on the structure of the above equations, the exponential stability of equation (16) is equivalent to the exponential stability of the following system:

[0068]

[0069] Then there exists a scalar γ max ∈(0,1), such that equation (17) is valid for all γ i ∈(0,γ max All are exponentially stable. Therefore, the output error of the i-th companion satellite can be determined. The value approaches zero, which ensures that each accompanying satellite completes its task of tracking the main spacecraft's signals.

[0070] Example

[0071] Simulations are performed on a discretized heterogeneous spacecraft formation system with time-varying topology and time-varying time delay. The system consists of four companion satellites numbered 1, 2, 3, and 4, and a master spacecraft numbered 0 located in low Earth orbit.

[0072] The relevant technical parameters are: Earth orbital radius R e =6.3781×10 6 m; gravitational parameter μ = 3.9860 × 10 14 m 3 / s 2 The sampling period is T. s =1.

[0073] The time-varying communication topology of the companion satellite and the main spacecraft considered in this embodiment is as follows: Figure 1 and Figure 2 As shown, and assuming the communication topology is in Figure 1 and Figure 2 The system switches randomly between these time-varying time delays. Furthermore, the input time delays for the four companion satellites are set as follows: ι1 = 6, ι2 = 3, ι3 = 5, ι4 = 4; the time-varying communication time delays between companion satellites or the main spacecraft with communication capabilities are set as follows: τ 10 (k)=7+sign(sin(k)),τ 21 (k)=3+sign(sin(10k)),τ 32 (k)=5+sign(sin(20k)),τ 43 (k) = 4 + sign(sin(30k)), where sign(·) is the sign function.

[0074] Substituting the set parameters into the time delay regulation equation (9) of the accompanying satellite, the solution to the time delay regulation equation of the i-th accompanying satellite is obtained as follows:

[0075]

[0076]

[0077] According to control protocol (11), a fully distributed control protocol is obtained to ensure that the four companion satellites complete the tracking task. Its initial conditions are set as follows: Let the initial state of the linearized equation of the in-plane relative motion of the four companion satellites with respect to the main spacecraft be x1(0)=[340,340,3,2]. T x2(0) = [340, 300, 3, 1] T x3(0) = [260, 260, 2, 3] T x4(0) = 260, 220, 6, 5] T The initial conditions for the distributed observers of the four accompanying satellites are set to ε. i (t) = 0, the parameter γ of the i-th accompanying satellite i Let it be γ i =0.008.

[0078] The controlled trajectories of the four companion satellites relative to the main spacecraft and the control input signals of the four companion satellites are recorded respectively in Figure 3 and Figure 4 superior.

[0079] Figure 3 The trajectories of four companion satellites relative to the main spacecraft are shown at three different times (k=0, k=2000, k=5000), marked by *, ×, and This indicates that the four companion satellites reached their designated orbits at k=2000, and then each companion satellite revolved around the center of the main spacecraft with a different radius. Figure 4 This demonstrates that the amplitude of the control inputs to the four accompanying satellites does not exceed 0.12N, thus meeting the requirements of practical engineering. In summary, the designed fully distributed control method solves the formation problem of heterogeneous spacecraft with time-varying topology and time-varying time delay.

Claims

1. A fully distributed control method for heterogeneous spacecraft formation considering time-varying topology and time-varying time delay, characterized in that: Includes the following steps: Step 1: Establish an orbital dynamics model of the accompanying satellite formation with mutually different input time delays and obtain its state-space equations; establish a signal model of the master spacecraft to be tracked and obtain its state-space equations. The specific process of step one is as follows: Step 11: Establish the mathematical model and state-space equations of the accompanying satellite formation flight system: Suppose there are N companion satellites numbered 1 to N orbiting a main spacecraft numbered 0 in low Earth orbit, where (x i,1 ,x i,2 ,x i,3 ) represents the relative position coordinates of the i-th companion satellite and the main spacecraft in the Hill coordinate system. Establish based on x i,3 The in-plane relative motion equation of the i-th companion satellite with respect to the main spacecraft in the case of ≡0 is: Among them, R e is the Earth's orbital radius; μ is the gravitational parameter; It is the average orbital speed of the accompanying satellite; Definition (a) i,1 ,a i,2 ) is the additional acceleration of the i-th accompanying satellite due to aerodynamic drag; ι i The input time delay of the i-th companion satellite is given, and the in-plane relative motion equation of the i-th companion satellite with respect to the main spacecraft is linearized as follows: in, This is the state of the i-th companion satellite; u i (t)=[a i,1 ,a i,2 ] T It is the input of the i-th companion satellite; y i (t) is the output of the i-th companion satellite; All are coefficient matrices, where I2 represents the 2nd order identity matrix, 0 2×2 This represents a zero matrix with dimension 2×2; Steps 1 and 2: Establish the mathematical model and state-space equations for the signal from the master spacecraft to be tracked: The mission of accompanying satellites in formation flight is to achieve circular motion around the main spacecraft. Let the output y of the i-th accompanying satellite be... i (t) needs to track a given signal of 50iv(t), where v(t) = [cos(w)] ref t),sin(w ref t)] T sin(·) is the sine function, cos(·) is the cosine function, and w ref =5ω is a given constant, and the given tracking signal v(t) is described by a linear state-space equation: in, The output error of the i-th companion satellite is e i (t) can be described by the following equation: in The linearized equation of the in-plane relative motion of the i-th companion satellite with respect to the main spacecraft (2), the given tracking signal state equation (3), and the output error equation (4) of the i-th companion satellite are combined and discretized as follows: in, T s The sampling period; Step 2: Construct explicit, fully distributed observers for each accompanying satellite to obtain the status information of the master spacecraft, which is constrained by time-varying communication delays; The expression for constructing an explicit, fully distributed observer for each accompanying satellite in step two is as follows: Where, ε i (k) is the observer state, representing the estimate of the tracking signal given by the main spacecraft by the i-th companion satellite at time k, and ε0(k) = v(k); σ(k) is the piecewise constant switching signal; Let $\mathbf{i}$ represent the communication weight between the $j$-th companion satellite or master spacecraft and the $i$-th companion satellite in the topology at time $k$. If companion satellite $i$ can obtain information from companion satellite or master spacecraft $j$ at time $k$, then... otherwise and τ ij (k) represents the time-varying communication delay between the j-th companion satellite or main spacecraft and the i-th companion satellite at time k, 0 < δ < 1; Step 3: Based on the positive definite solution of the parametric Lyapunov equation and the solution of the time-delay regulation equation, design explicit linear state feedback gain and feedforward gain respectively. Using the state feedback gain, the feedforward gain and the fully distributed observer, establish a corresponding fully distributed control protocol for the accompanying satellite formation system affected by time-varying topology and time-varying time delay, so as to ensure that each accompanying satellite completes the task of tracking the main spacecraft.

2. The fully distributed control method for heterogeneous spacecraft formation considering time-varying topology and time-varying time delay as described in claim 1, characterized in that: The process of designing the explicit linear state feedback gain based on the positive definite solution of the parametric Lyapunov equation in step three is as follows: Establish the discrete parameter Lyapunov equations for the i-th companion satellite: A T P i (c i )AP i (c i )-A T P i (c i )B(I2+B T P i (c i )B) -1 B T P i (c i )A=-γ i P i (c i ) (7) Where, γ i It is the scalar to be designed; P i (γ i If ) is the unique positive definite solution of the discrete parameter Lyapunov equation (7); then the explicit linear state feedback gain K of the i-th companion satellite is... i (γ i It was designed as: K i (c i )=(I2+B T P i (c i )B) -1 B T P i (c i )A (8).

3. The fully distributed control method for heterogeneous spacecraft formation considering time-varying topology and time-varying time delay as described in claim 2, characterized in that: The process of designing an explicit feedforward gain based on the solution of the time-delay adjustment equation in step three is as follows: Construct the time-delay regulation equation for the i-th companion satellite: Where (X) i U i If ) is the solution to the time-delay regulation equation of the i-th companion satellite, then the feedforward gain L of the i-th companion satellite is... i Designed as:

4. The fully distributed control method for heterogeneous spacecraft formation considering time-varying topology and time-varying time delay as described in claim 3, characterized in that: In step three, the process of establishing a corresponding fully distributed control protocol for a companion satellite formation system affected by time-varying topology and time-varying time delay, using state feedback gain, feedforward gain, and the fully distributed observer, is as follows: Based on equations (6), (8), and (10), the control protocol for the i-th companion satellite is established as follows: Observing equation (5), it can be seen that the tracking problem of each accompanying satellite can be transformed into the output error e of each accompanying satellite. i The stabilization problem of (k) can be addressed by defining state transitions. From equations (5) and (9), we can derive: And we obtained: Based on the feedforward gain (10), the control protocol for the i-th companion satellite is rewritten as follows: in The control protocol for the i-th accompanying satellite affected by the input time delay is further described as follows: Substituting equation (15) into (12) yields: According to equation (16) above, the exponential stability of this system is equivalent to the exponential stability of the following system: There exists a scalar γ max ∈(0,1), such that the system corresponding to equation (17) is for all γ i ∈(0,γ max All of them are exponentially stable.

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