A method and device for analyzing the availability of a satellite constellation considering satellite replenishment

Abstract

The present invention discloses a method and device for analyzing the availability of a satellite constellation considering satellite replacement, and the method includes: constructing the state space of the satellite system in the same orbit by using the k-out-of-n:G system; constructing the Markov chain state transition rate matrix of the satellite system in the same orbit; solving the state probabilities of the satellite system in the same orbit under different numbers of normally operating satellites; merging the state probabilities under the number of normally operating satellites that meet the set threshold, and calculating the state transition probability of the satellite system in the same orbit; constructing the state space of the satellite system between orbits; constructing the Markov chain state transition rate matrix of the satellite system between orbits; solving the state probabilities of the satellite system between orbits under different numbers of normally operating orbits; merging the state probabilities under the normally operating orbits that meet the threshold to obtain the normal operating probability of the system between orbits. Through the above method and the device for running the method, the present invention can reduce the impact of satellite node failures on the constellation mission capabilities.

Description

Technical Field

[0001] The present invention relates to the technical field of satellite constellation analysis, and more specifically, to a method and device for analyzing the availability of a satellite constellation considering satellite replacement. Background Art

[0002] At present, as satellite systems gradually develop into constellation systems, the number of artificial satellites is increasing. Satellite constellations need to consider not only cost issues but also the availability of the entire constellation system. The failure and degradation of satellites will become important factors affecting the availability of satellite systems or constellation systems. Currently, redundant design is commonly used in constellation reliability design. Backup satellites are placed in addition to the working satellites in the working orbit, or backup orbits are set up so that when some satellites fail, the backup satellites can quickly replace them, reducing the impact of satellite node failures on the constellation mission capabilities and improving the availability of the system.

[0003] Therefore, providing a method for calculating the availability of a constellation that can consider satellite replacement plays an important role in the design and analysis of constellation systems. Summary of the Invention

[0004] In view of this, the present invention provides a method and device for analyzing the availability of a satellite constellation considering satellite replacement.

[0005] To achieve the above object, the present invention adopts the following technical solutions:

[0006] First, the present invention discloses a method for analyzing the availability of a satellite constellation considering satellite replacement, including the following steps:

[0007] S1. Use a k-out-of-n:G system to construct the state space of the satellite system in the same orbit;

[0008] S2. Based on the state space of the satellite system in the same orbit, construct the Markov chain state transition rate matrix of the satellite system in the same orbit;

[0009] S3. Based on the Markov chain state transition rate matrix of the satellite system in the same orbit, solve the state probabilities of the satellite system in the same orbit under different numbers of normally operating satellites;

[0010] S4. According to the threshold that the number of normally operating satellites needs to meet when the satellite system in the same orbit is operating normally, merge the state probabilities under the number of normally operating satellites that meet the threshold, obtain the steady-state operating probability and steady-state failure probability of the satellite system in the same orbit, and calculate the state transition probability of the satellite system in the same orbit;

[0011] S5. Use a k-out-of-n:G system to construct the state space of the satellite system between orbits;

[0012] S6. Construct the Markov chain state transition rate matrix of the inter-orbit satellite system based on the state space of the inter-orbit satellite system and the state transition probability of the intra-orbit satellite system in the same orbit;

[0013] S7. Solve the state probabilities of the inter-orbit satellite system under different numbers of normally operating orbits based on the Markov chain state transition rate matrix of the inter-orbit satellite system;

[0014] S8. According to the threshold that the number of normally operating orbits needs to meet when the inter-orbit satellite system is operating normally, merge the state probabilities under the number of normally operating orbits that meet the threshold to obtain the normal operating probability of the inter-orbit satellite system.

[0015] Further, step S1 specifically includes:

[0016] Construct the state space of the intra-orbit satellite system: Y = {n, n - 1, …, k, k - 1, …, 1, 0};

[0017] where Y represents that there are n satellites in the same orbit and has n + 1 states; y ∈ Y represents the number of normally operating satellites in the same orbit; when the number of normally operating satellites in the same orbit is not less than k, the intra-orbit satellite system corresponding to the same orbit is in a normal operating state.

[0018] Further, step S2 specifically includes:

[0019] Determine the satellite states of the intra-orbit satellite system, and the satellite states include normal operating states and fault states;

[0020] Calculate the state transition probability: Calculate the failure rate λ of a single satellite in the same orbit transferring from the normal operating state to the fault state, and the repair rate μ of transferring from the fault state to the normal operating state through one satellite replenishment and repair;

[0021] Based on the failure rate λ and the repair rate μ, construct an (n + 1)×(n + 1) Markov chain state transition rate matrix of the intra-orbit satellite system:

[0022]

[0023] where Q represents the Markov chain state transition rate matrix of the intra-orbit satellite system.

[0024] Further, step S3 specifically includes:

[0025] Perform a diagonalization operation on the Markov chain state transition rate matrix Q of the intra-orbit satellite system to obtain the Jordan canonical form matrix D corresponding to Q;

[0026] The state transition probability matrix P of the in-orbit satellite system at any time t is obtained by the following expression using the Jordan canonical form matrix D corresponding to Q:

[0027] P = Me Dt M -1

[0028] where M is the Jordan canonical form matrix corresponding to the state transition probability matrix P;

[0029] When the in-orbit satellite system operates for a certain period of time and the fluctuation of any term in the state transition probability matrix P with respect to time is less than 1e-5, the state transition probability matrix P of the in-orbit satellite system converges stably to the vector P ST :

[0030]

[0031] According to the vector P ST Solve the state probability p of the in-orbit satellite system under different numbers of normally operating satellites i , i = 0, 1... n, where i represents the number of normally operating satellites.

[0032] Furthermore, step S4 specifically includes:

[0033] According to the threshold k that the number of normally operating satellites needs to satisfy when the in-orbit satellite system operates normally, sum up the state probabilities under the number of normally operating satellites that meet the threshold to obtain the steady-state working probability P1 and the steady-state failure probability P0 of the in-orbit satellite system:

[0034]

[0035] P0 = 1 - P1

[0036] From the relevant knowledge of graph theory, the probability of transitioning from the failure state to the working state after merging is the sum of the transition probabilities from the k states where the number of normally operating satellites does not meet the threshold to the state where all satellites are operating normally. It is easy to know that μ SI = μ. Calculate the probability λ of the in-orbit satellite system transitioning from the normal working state to the failure state based on the steady-state working probability P1 and the steady-state failure probability P0 SI , and the probability μ of transitioning from the failure state to the normal working state SI :

[0037]

[0038] The probability μ of the in-orbit satellite system transitioning from the failure state to the normal working state SIIt is numerically the same as the repair rate μ at which a single satellite in the corresponding orbit transfers from a faulty state to a normal operating state through one satellite replacement repair.

[0039] Furthermore, step S5 specifically includes:

[0040] Construct the state space of the inter-orbit satellite system composed of multiple different orbits: Y′ = {N, N - 1, …, K, K - 1, …, 1, 0};

[0041] where Y′ represents that there are N + 1 states in total for the inter-orbit satellite system composed of N different orbits; y' ∈ Y', representing the number of orbits in the inter-orbit satellite system that are operating normally; when the number of orbits operating normally is not less than K, the inter-orbit satellite system is in a normal operating state.

[0042] Furthermore, step S6 specifically includes:

[0043] According to the state space of the inter-orbit satellite system, and the probability λ that a satellite system in the same orbit transfers from a normal operating state to a faulty state SI , and the probability μ that it transfers from a faulty state to a normal operating state SI , construct the Markov chain transition rate matrix Q′ of the (N + 1) × (N + 1) inter-orbit satellite system:

[0044]

[0045] where Q′ represents the Markov chain transition rate matrix of the inter-orbit satellite system.

[0046] Furthermore, step S7 specifically includes:

[0047] Perform a diagonalization operation on the Markov chain transition rate matrix Q′ of the inter-orbit satellite system to obtain the Jordan canonical form matrix D′ corresponding to Q′;

[0048] Use the Jordan canonical form matrix D′ corresponding to Q′ to obtain the state transition probability matrix P′ of the inter-orbit satellite system at any time t through the following expression:

[0049] P' = M'e D't (M') -1 ;

[0050] where M' is the Jordan canonical form matrix corresponding to the state transition probability matrix P′;

[0051] When the inter-orbit satellite system runs for a certain period of time, and the fluctuation of any term in the state transition probability matrix P′ with respect to time is less than 1e-5, the state transition probability matrix P′ of the inter-orbit satellite system stably converges to the vector P ST ':

[0052]

[0053] According to the vector P ST 'Solve the state probability p' of the inter-orbit satellite system under different numbers of normally operating orbits j , j = 0, 1... N, where j represents the number of normally operating orbits.

[0054] Furthermore, step S8 specifically includes:

[0055] According to the threshold K that the number of normally operating orbits needs to satisfy when the inter-orbit satellite system is operating normally, merge the state probabilities under the normally operating orbits that meet the threshold to obtain the steady-state operating probability P'1 of the inter-orbit satellite system:

[0056]

[0057] where p' j represents the state probability of the inter-orbit satellite system under j normally operating orbits.

[0058] On the other hand, the present invention also discloses a satellite constellation availability analysis device considering satellite replacement. The device includes a computer system, and when the computer system executes a computer program, it can implement the satellite constellation availability analysis method considering satellite replacement according to any one of the present invention.

[0059] Through the above technical solutions, it can be seen that compared with the prior art, the present invention discloses a satellite constellation availability analysis method and device considering satellite replacement, which has the following beneficial effects:

[0060] When analyzing the availability of the satellite constellation, the present invention fully considers the influence of satellite replacement, can reduce the influence of satellite node failures on the constellation mission capabilities, and improve the availability of the satellite constellation system. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the drawings in the following description are only the embodiments of the present invention, and for those of ordinary skill in the art, other drawings can be obtained according to the provided drawings without creative efforts.

[0062] Figure 1 It is a schematic diagram of the overall process of the satellite constellation availability analysis method considering satellite replacement provided by the present invention.

[0063] Figure 2 It is a schematic diagram of the state transition of the intra-orbit satellite system provided by the present invention.

[0064] Figure 3 The state transition diagram after combining the state probabilities of the in-orbit satellite system provided by the present invention.

[0065] Figure 4 The state transition diagram of the inter-orbit redundancy system provided by the present invention.

[0066] Figure 5 The state transition diagram of the in-orbit satellite system for a specific numerical example provided by the present invention.

[0067] Figure 6 The state transition diagram of the inter-orbit system for a specific numerical example provided by the present invention. Specific implementation manners

[0068] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0069] For a satellite constellation system with redundancy design, an embodiment of the present invention discloses a method for analyzing the availability of a satellite constellation considering satellite replacement, and the overall steps are as Figure 1 shown, including:

[0070] S1. Use the k-out-of-n:G system to construct the state space of the in-orbit satellite system;

[0071] S2. Based on the state space of the in-orbit satellite system, construct the Markov chain state transition rate matrix of the in-orbit satellite system;

[0072] S3. Based on the Markov chain state transition rate matrix of the in-orbit satellite system, solve the state probabilities of the in-orbit satellite system under different numbers of normally operating satellites;

[0073] S4. According to the threshold that the number of normally operating satellites needs to meet when the in-orbit satellite system is operating normally, merge the state probabilities under the number of normally operating satellites that meet the threshold, obtain the steady-state operating probability and steady-state failure probability of the in-orbit satellite system, and calculate the state transition probability of the in-orbit satellite system;

[0074] S5. Use the k-out-of-n:G system to construct the state space of the inter-orbit satellite system;

[0075] S6. Construct the Markov chain state transition rate matrix of the inter-orbital satellite system based on the state space of the inter-orbital satellite system and the state transition probability of the intra-orbital satellite system on the same orbit;

[0076] S7. Solve the state probabilities of the inter-orbital satellite system under different numbers of normally operating orbits based on the Markov chain state transition rate matrix of the inter-orbital satellite system;

[0077] S8. According to the threshold that the number of normally operating orbits needs to meet when the inter-orbital satellite system is operating normally, merge the state probabilities under the number of normally operating orbits that meet the threshold to obtain the normal operating probability of the inter-orbital satellite system.

[0078] In the embodiment of the present invention, within the same single orbit, a k-out-of-n:G system is used for redundant design, and in the design between orbits, a k-out-of-n:G system is also used for redundant design.

[0079] The specific content of the above steps will be further described below.

[0080] 1. Define and construct the state space of the satellite system within the same orbit using the k-out-of-n:G system

[0081] The k-out-of-n:G system contains n components. If the number of normally operating components is greater than or equal to k, the system G is regarded as operating normally, otherwise it is a failure. The present invention uses the k / n(G) system to construct the state space of the satellites within the orbit:

[0082] The state space of the satellite system within the same orbit is as follows: Y = {n, n - 1, …, k, k - 1, …, 1, 0};

[0083] Among them, Y represents that there are n satellites within the same orbit and has n + 1 states; y ∈ Y represents the number of normally operating satellites within the same orbit; when the number of normally operating satellites within the same orbit is not less than k, the corresponding intra-orbital satellite system is in a normal operating state.

[0084] Let the Markov chain {Y t} = {Y t |t = 0, 1…} be a random sequence. There are n satellites arranged on the orbit. Define the state space Y = {n, n - 1, …, k, k - 1, …, 1, 0}, indicating that the satellite system within the orbit has a total of n + 1 states. Among them, y ∈ Y represents that there are y satellites operating normally on the orbit. When the number of working satellites y is not less than k, the orbit can be regarded as being in a normal operating state.

[0085] 2. Construct the Markov chain state transition rate matrix of the satellite system within the same orbit

[0086] At \(t = 0\), all satellites in the system on the same orbit are in normal state, that is, there are \(n\) satellites in normal operation. And a satellite in normal operation state transfers to the failed state with a probability of \(\lambda\) (failure rate). Therefore, the state of having \(n\) satellites in normal operation will transfer to the state of having \(n - 1\) satellites in normal operation with a probability of \(n\lambda\). The transfer after failure between the remaining states follows the same pattern. Since the repair adopts a one-time satellite replenishment strategy, that is, one satellite replenishment repair can reach the full state, so the state of \(y\) satellites in normal operation transfers to the state of \(n\) satellites in normal operation with a probability of \(\mu\) (repair rate). In particular, at any moment if a satellite fails, the number of failed satellites is exactly one.

[0087] Based on the state space of the satellite system in the same orbit and the defined satellite failure and recovery probabilities above, establish the conversion relationship between each state quantity, and represent it in the form of a Markov chain. Let the Markov chain \(\{Y t \}=\{Y t |t = 0,1…\}\) be a random sequence. There are \(n\) satellites arranged on the orbit. Define the state space \(Y=\{n,n - 1,\ldots,k,k - 1,\ldots,1,0\}\), indicating that there are \(n + 1\) states in total for the satellite system in the orbit. Each \(Y t takes values in \(Y\) and has the property of no aftereffect (the situation at the next moment only depends on the current moment and has nothing to do with the past). A Markov chain is a random process in probability theory and mathematical statistics that has the Markov property (when a random process is given the current state and all past states, the conditional probability distribution of its future state depends only on the current state) and exists in a discrete exponential set and state space. The Markov chain has the characteristic of no aftereffect and can be defined by a transition matrix and a transition diagram.

[0088] The schematic diagram of the state transition of the satellite system in the same orbit is as Figure 2 shown. Among them, the head node is the intact state, and all satellites can operate normally. At this time, the failure rate of one satellite is \(n\lambda\); the next node after \(x = n\) is the state of one failed satellite. At this time, the failure probability of the next satellite is \((n - 1)\lambda\). At the same time, it can be restored to the head node state through repair; \(x = 0\) is the full-failure state, that is, the number of working satellites is 0. Repairing the state of this node can directly transfer to the situation of \(x = n\).

[0089] According to the above transfer process, the transition rate matrix \(Q\) of the Markov chain \(\{Y t \}\) of the satellite system in the same orbit can be constructed as:

[0090]

[0091] Matrix \(Q\) represents the state transition rate matrix of the Markov chain of the satellite system in the same orbit.

[0092] 3. Solve the state probabilities of the satellite system in the same orbit under different numbers of normally operating satellites

[0093] By using the Jordan diagonalization operation of Q, the standard matrix D can be obtained, and then the transition probability matrix P at any time t can be obtained:

[0094] P = Me Dt M -1

[0095] where M is the Jordan canonical form matrix corresponding to the transition probability matrix P; when the fluctuations of any term in the state transition probability matrix P with respect to time are less than 1e-5 after the system has run for a sufficiently long time, the state probability matrix will stably converge to the vector P ST , that is:

[0096] P ST Q = 0

[0097] From this, P can be solved ST ,

[0098] and then the state probabilities pi i , i = 0, 1... n; i represents the number of normally operating satellites in the same orbit.

[0099] 4. Combine the state probabilities of the satellite system in the same orbit under the number of normally operating satellites that meet the threshold

[0100] When the number of normally operating satellites is r (r ∈ Y), by summing the steady-state probabilities of the states with r > k - 1, the steady-state working probability P1 of the entire orbit and the corresponding steady-state failure probability P0 can be obtained

[0101]

[0102] P0 = 1 - P1

[0103] The state transition schematic diagram after combining the state probabilities is as Figure 3 shown. From this, the state space Y of the satellite system in the orbit can be split into two items: "normally operating" (Y1 = {n, n - 1,... k + 1, k}) and "not working" (Y0 = {k - 1,... 1, 0}). The "normally operating" state is denoted as "1", and the "not working" (fault state) is denoted as "0". The probability of transitioning from state "1" to "0" is λ SI , and vice versa is μ SI , λ SI is the probability of the satellite system in the orbit transitioning from the "not working" state (fault state) to the "normally operating" state, and μ SIThat is, the probability that the satellite system in orbit transfers from the "normal working" state to the "inoperable" state (fault state).

[0104] According to the state space transfer process in the orbit after the merged state, we can get λ SI With μ SI The relationship between:

[0105]

[0106] Since a single satellite in the same orbit is transferred from a faulty state to a normal working state through a satellite repair, the physical process is the same as that of all satellites in the same orbit after being merged. SI The physical process represented is also the probability of repair transition from the fault state (y = 0, 1 ... k-1) to the normal working state (y = n). Therefore, the probability μ of the satellite system in the same orbit transitioning from the fault state to the normal working state is SI The maintenance rate μ of a single satellite in the corresponding orbit transferring from a fault state to a normal working state through a satellite repair is numerically the same, and we can get:

[0107] μ=μ SI

[0108] By combining the two equations, we can respectively solve the probability λ that the satellite system in the same orbit changes from a normal working state to a fault state SI , and the probability μ of transition from fault state to normal working state SI :

[0109]

[0110] 5. Constructing the state space of inter-orbit satellite systems using k-out-of-n:G systems

[0111] Based on the above steps, each satellite system in each orbit is regarded as a subsystem, which together constitutes an inter-orbit system. The inter-orbit system is also a k / n(G) system. The subsystems that can work normally are counted, and "k" means that k out of n systems in the same orbit can work normally.

[0112] The state space of the inter-orbit satellite system can be expressed as: Y′ = {N, N-1, …, K, K-1, …, 1, 0};

[0113] Among them, Y′ indicates that the inter-orbital satellite system composed of N different orbits has a total of N+1 states; y'∈Y', represents the number of orbits in which the inter-orbital satellite system works normally; when the number of orbits in which the inter-orbital satellite system works normally is not less than K, the inter-orbital satellite system is in a normal working state.

[0114] Suppose the Markov chain {Y t'} = {Y t '|t = 0, 1…} is a random sequence. There are N subsystems in the system, each taking values in Y′ and having no aftereffect (the situation at the next moment only depends on the current moment and has nothing to do with the past). At t = 0, all subsystems of the system are in the normal state, that is, there are N normally operating subsystems, and a subsystem in the normal operating state transfers to the faulty state with a probability of λ SI . Therefore, the state of having N normally operating subsystems will transfer to the state of having N - 1 normally operating subsystems with a probability of nλ SI , and the subsequent transfers after failure between the remaining states follow the same pattern; because the repair adopts a one-time replenishment strategy, that is, one replenishment repair can achieve the full state situation, so the state of y′ normally operating subsystems transfers to the state of N normally operating subsystems with a probability of μ SI . In particular, at any moment, if there is a subsystem failure, the number of failures is exactly one. The schematic diagram of the state transition of the inter-orbital redundant system is as shown in Figure 4 , and the solved λ SI and μ SI are the transition probabilities of the constellation system. It can be solved according to the method of solving the state space of the constellation system.

[0115] 6. Construct the Markov chain state transition rate matrix of the inter-orbital satellite system

[0116] According to the state space of the inter-orbital satellite system, and the probability λ SI of the satellite system within the same orbit transferring from the normal operating state to the faulty state, and the probability μ SI of transferring from the faulty state to the normal operating state, construct the (N + 1)×(N + 1) Markov chain transition rate matrix of the inter-orbital satellite system:

[0117]

[0118] Among them, the Q′ matrix represents the Markov chain transition rate matrix of the inter-orbital satellite system.

[0119] 7. Solve the state probabilities of the inter-orbital satellite system under different numbers of normally operating satellites

[0120] Perform a diagonalization operation on the Markov chain transition rate matrix Q′ of the inter-orbital satellite system to obtain the Jordan canonical form matrix D′;

[0121] According to the Jordan canonical form matrix D′

[0122] Obtain the state transition probability matrix P′ of the inter-orbital satellite system at any time t through the following expression:

[0123] P' = M'e D't (M') -1 ;

[0124] Wherein, M' is the Jordan canonical form matrix corresponding to the state transition probability matrix P'; when the satellite system between orbits operates for a certain period of time and the fluctuation of any term in the state transition probability matrix P' with respect to time is less than 1e-5, it can be considered that the state transition probability matrix P' of the satellite system between orbits converges stably to the vector P ST ':

[0125]

[0126] According to the vector P ST ' to solve the state probability p' of the satellite system between orbits under different numbers of normally operating orbits j , j = 0, 1... N, where j represents the number of normally operating orbits.

[0127] 8. Solve the normal operating probability of the system between orbits

[0128] According to the threshold K that the number of normally operating orbits needs to satisfy when the satellite system between orbits operates normally, the state probabilities under the normally operating orbits that meet the threshold are combined to obtain the steady-state operating probability P'1 of the satellite system between orbits:

[0129]

[0130] Wherein, p' j represents the state probability of the satellite system between orbits under j normally operating orbits.

[0131] In the present invention, the overall satellite constellation system can be a two-layer redundant system. Among them, the first-layer redundant system is the satellite system within the orbit, and the second-layer redundant system is the redundant system between orbits.

[0132] After the results are obtained by using this method for analysis, an analytical method can be used for scheme verification.

[0133] The following is a calculation example based on the above method:

[0134] 1) Define and construct the state space of the satellite system within the same orbit

[0135] Let the coefficients of the double-layer k / n(G) system (k-out-of-n:G system) be n = 5, k = 3, λ =

[0136] 0.1, μ = 0.9. Then it can be known that 5 satellites are arranged on the same orbit, and when the number of working satellites is not less than 3, the orbit can be regarded as being in a normal state.

[0137] 2) Construct the Markov chain state transition rate matrix of the satellite system in the same orbit

[0138] Figure 5 Figure 4 shows the schematic diagram of the state transition of the satellite system in the same orbit for a specific example. The numbers in the circles represent the number of satellites working properly in this orbit. As shown by Figure 5 it can be obtained that the transition rate matrix of the Markov chain of the system at the satellite level is:

[0139]

[0140] 3) Solve the state probabilities of the satellite system in the same orbit under different numbers of satellites working properly

[0141] From the transition rate matrix Q, the Jordan matrix D of Q can be further solved:

[0142]

[0143] When the satellite system in the orbit has been running for a period of time, the system state probability matrix will stabilize to the vector P ST , and from P ST the state matrix can be further solved to obtain the state probabilities p i :

[0144]

[0145] Table 1: Steady-state probabilities of each state of the inter-orbit satellite system

[0146] 4) Combine the state probabilities of the satellite system in the same orbit under the number of satellites working properly that meet the threshold

[0147] Since k = 3, only when the number of working satellites is greater than or equal to 3 can the orbit be considered to be working properly. Therefore, the steady-state working probability of the orbit can be calculated: P1 = 0.9726, P0 = 0.0274

[0148] Also because:

[0149]

[0150] The probability λ SI of the satellite system in the same orbit transferring from the normal working state to the failure state, and the probability μ SI of transferring from the failure state to the normal working state are respectively:

[0151] λ SI = 0.0254, μ SI = 0.9.

[0152] 5) Construct the state space of the inter-orbit satellite system

[0153] Regarding the satellite systems in each orbit as subsystems, they jointly form an inter-orbit system. The inter-orbit system is also a k / n(G) system. Count the subsystems that can work properly. The state space of the inter-orbit satellite system Y′ = {N, N - 1, …, K, K - 1, …, 1, 0}; "K" means that among the N satellite systems in the orbit that form the inter-orbit satellite system, K orbits can work properly. Similarly, taking N = 5 and K = 3, λ SI = 0.0254 and μ SI = 0.9 are used as the transition probabilities of the constellation system. The schematic diagram of the state transition of the formed inter-orbit system is as Figure 6 shown.

[0154] 6) Construct the Markov chain state transition rate matrix of the inter-orbit satellite system

[0155] The transition rate matrix of the Markov chain {Y t '} of the inter-orbit system is:

[0156]

[0157] 7) Solve the state probability of the inter-orbit satellite system

[0158] When the inter-orbit satellite system runs for a certain period of time and the fluctuation of any item in the state transition probability matrix P′ with respect to time is less than 1e-5, it can be considered that the state transition probability matrix P′ of the inter-orbit satellite system converges stably to the vector P ST ':

[0159]

[0160] From P ST 'Q' = 0, the stable vector P ST ' of the state probability matrix of the inter-orbit satellite system (constellation system) after running for a period of time can be obtained. From P ST ', the state probability p j under different numbers of normally working orbits of the inter-orbit system is further calculated:

[0161] Number j of tracks for normal operation of the inter-orbital system Probability 0 1.4359e-06 1 5.0879e-05 2 9.2683e-04 3 0.0116 4 0.1111 5 0.8763

[0162] Table 2: State probabilities of each state of the inter-orbit system

[0163] 8) Combine and solve the normal working probability of the inter-orbit system

[0164] Let P'1 be the steady-state working probability of the inter-orbit system, then:

[0165]

[0166] It can be calculated therefrom that the steady-state working probability of the inter-orbital system is 0.999.

[0167] After obtaining the results, an analytical method can be used for verification.

[0168] In specific applications, the present invention also discloses a satellite constellation availability analysis device considering satellite replacement, which device includes a computer system. When the computer system executes a computer program, it can implement the satellite constellation availability analysis method considering satellite replacement according to any one of the present invention.

[0169] In this specification, each embodiment is described in a progressive manner. The key point of each embodiment is to illustrate the differences from other embodiments. The same or similar parts among the embodiments can be referred to each other. For the device disclosed in the embodiments, since it corresponds to the method disclosed in the embodiments, the description is relatively simple, and the relevant parts can be referred to the description of the method part.

[0170] The above description of the disclosed embodiments enables those skilled in the art to implement or use the present invention. Various modifications to these embodiments will be obvious to those skilled in the art. The general principles defined herein can be implemented in other embodiments without departing from the spirit or scope of the present invention. Therefore, the present invention will not be limited to the embodiments shown herein, but will conform to the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for analyzing the availability of a satellite constellation considering satellite replacement, characterized in that It includes the following steps: S1. Construct the state space of the satellite system in the same orbit by using the k-out-of-n:G system; S2. Construct the Markov chain state transition rate matrix of the satellite system in the same orbit based on the state space of the satellite system in the same orbit; S3. Solve the state probabilities of the satellite system in the same orbit under different numbers of normally operating satellites based on the Markov chain state transition rate matrix of the satellite system in the same orbit; S4. According to the threshold that the number of normally operating satellites needs to meet when the satellite system in the same orbit is operating normally, merge the state probabilities under the number of normally operating satellites that meet the threshold, obtain the steady-state operating probability and steady-state failure probability of the satellite system in the same orbit, and calculate the state transition probability of the satellite system in the same orbit; S5. Construct the state space of the inter-orbit satellite system by using the k-out-of-n:G system; S6. Construct the Markov chain state transition rate matrix of the inter-orbit satellite system based on the state space of the inter-orbit satellite system and the state transition probability of the intra-orbit satellite system in the same orbit; S7. Solve the state probabilities of the inter-orbit satellite system under different numbers of normally operating orbits based on the Markov chain state transition rate matrix of the inter-orbit satellite system; S8. According to the threshold that the number of normally operating orbits needs to meet when the inter-orbit satellite system is operating normally, merge the state probabilities under the number of normally operating orbits that meet the threshold to obtain the normal operating probability of the inter-orbit satellite system; In S2 and S6, when constructing the Markov chain state transition rate matrix, it specifically includes: Determine the satellite states of the satellite system, including the normal operating state and the failure state; Calculate the failure rate of the satellite system from the normal operating state to the failure state, and the repair rate from the failure state to the normal operating state through one satellite replenishment repair; Construct the Markov chain state transition rate matrix based on the failure rate and the repair rate; In S3 and S7, when solving the state probabilities based on the Markov chain state transition rate matrix, it specifically includes: Perform a diagonalization operation on the Markov chain state transition rate matrix to obtain the corresponding Jordan canonical form matrix; Use the Jordan canonical form matrix corresponding to the Markov chain state transition rate matrix to obtain the state transition probability matrix of the satellite system at any time t; When the satellite system runs for a certain time and the fluctuation of any item in the state transition probability matrix with respect to time is less than 1e-5, the state transition probability matrix converges stably to a vector: Solve the state probabilities of the satellite system according to the vector after stable convergence.

2. The method for analyzing the availability of a satellite constellation considering satellite replenishment according to claim 1, wherein Step S1 specifically includes: Construct the state space of the satellite system in the same orbit: Y = {n, n - 1, …, k, k - 1, …, 1, 0}; Among them, Y represents that there are n satellites in the same orbit and has n + 1 states; y ∈ Y represents the number of normally operating satellites in the same orbit; when the number of normally operating satellites in the same orbit is not less than k, the intra-orbit satellite system corresponding to the same orbit is in the normal operating state.

3. The method for analyzing the availability of a satellite constellation considering satellite replenishment according to claim 1, wherein Step S4 specifically includes: According to the threshold k that the number of normally operating satellites needs to meet when the satellite system in the same orbit operates normally, the state probabilities under the number of normally operating satellites that meet the threshold are combined and summed to obtain the steady-state operating probability P1 and the steady-state failure probability P0 of the satellite system in the same orbit: P0 = 1 - P1 Calculate the probability λ that the satellite system in the same orbit transfers from the normal working state to the fault state based on the steady-state working probability P1 and the steady-state fault probability P0 SI , and the probability μ that it transfers from the fault state to the normal working state SI : Among them, the probability μ that the satellite system in the same orbit transfers from the fault state to the normal working state SI is numerically the same as the repair rate μ at which a single satellite in the corresponding orbit transfers from the fault state to the normal working state through one satellite replacement and repair.

4. The method for analyzing the availability of a satellite constellation considering satellite replacement according to claim 1, characterized in that Step S5 specifically includes: Construct the state space of the inter-orbit satellite system composed of multiple different orbits: Y′ = {N, N - 1, …, K, K - 1, …, 1, 0}; Among them, Y′ represents that there are N + 1 states in total for the inter-orbit satellite system composed of N different orbits; y' ∈ Y', which represents the number of orbits in the inter-orbit satellite system that are operating normally; when the number of normally operating orbits is not less than K, the inter-orbit satellite system is in a normal operating state.

5. The satellite constellation availability analysis method considering satellite replacement according to claim 1, characterized in that Step S8 specifically includes: According to the threshold K that the number of normal operating orbits needs to satisfy when the inter-orbital satellite system operates normally, the state probabilities under the normal operating orbits that satisfy the threshold are merged to obtain the steady-state operating probability P of the inter-orbital satellite system ' 1: where p ' j represents the state probability of the inter-orbital satellite system under the number j of normal operating orbits.

6. A satellite constellation availability analysis device considering satellite replacement, characterized in that It includes a computer system, and when the computer system executes a computer program, it can implement the method for analyzing the availability of a satellite constellation considering satellite replacement as described in any one of claims 1-5.

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