CN111310337B - Analysis method for on-orbit interruption frequency of navigation satellite signal - Google Patents

Analysis method for on-orbit interruption frequency of navigation satellite signal Download PDF

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CN111310337B
CN111310337B CN202010098951.2A CN202010098951A CN111310337B CN 111310337 B CN111310337 B CN 111310337B CN 202010098951 A CN202010098951 A CN 202010098951A CN 111310337 B CN111310337 B CN 111310337B
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interruption
satellite
frequency
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CN111310337A (en
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赵海涛
熊笑
谢军
杨慧
谷岩
郑晋军
张旭
张孝功
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Beijing Institute of Spacecraft System Engineering
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Abstract

The invention provides a navigation satellite signal on-orbit interruption frequency analysis method, which comprises the following specific processes: (1) Sequentially determining a subsystem, component equipment and a bottom event which possibly cause interruption of a navigation signal, wherein the bottom event comprises a fault switching event of a master/backup equipment and a soft fault, and the soft faults comprise a single-particle soft error generated by a single-particle sensitive device and a soft error caused by a software design defect; (2) Establishing an interrupt tree according to the analysis result in the step (1); (3) Establishing an interruption frequency analysis model according to the interruption tree, and calculating satellite interruption frequency indexes according to the occurrence frequency of various bottom events; (4) And according to the satellite outage frequency analysis result, the method is applied to satellite design initial index demonstration, distribution and index verification at the end of satellite development. The method can be applied to initial index demonstration of satellite design, distribution and index verification at the end of satellite development.

Description

Analysis method for on-orbit interruption frequency of navigation satellite signal
Technical Field
The invention relates to an analysis method of an on-orbit interruption frequency of a navigation satellite signal, which can be applied to allocation and verification of the interruption frequency of the navigation satellite signal and belongs to the technical field of navigation satellite design.
Background
The satellite navigation system is an important space infrastructure which is indispensable for national economy and national defense construction at present. The system is widely applied to various fields of national economy, and a huge satellite navigation industry is formed. In order to realize high-precision, all-day and all-weather navigation, positioning and time service of a satellite navigation system, a satellite navigation constellation must meet strict requirements on precision, availability, continuity and integrity. Wherein availability is the percentage of time the system can be used to navigate over a period of time based on reliable information provided to the user; continuity refers to the frequency at which the system maintains a prescribed performance throughout the duration of the operational phase, and may also be expressed as the frequency at which healthy spatial signals can continue to operate healthily without unintended interruption during prescribed time intervals. Availability and continuity determine the ability of a satellite navigation system to actually provide service, and if the navigation signal transmitted by the navigation system is frequently interrupted and in an unavailable state for various reasons, the accuracy of the system is not practical.
The continuity and availability of the spatial signal is related to the interruption of the navigation satellite. According to ECSS-Q-ST-30-09, availability analysis, interrupts (outages) are defined as states in which a product cannot function as desired. The interruption of the navigation satellite can be divided into four categories, including a long-term unplanned interruption caused by a permanent failure of the satellite, a long-term planned interruption caused by a satellite-to-life, a short-term unplanned interruption caused by a short-term failure, and a short-term planned interruption caused by a planned in-orbit maintenance operation. The short-term unplanned interruption has important importance in the design and analysis of navigation satellite interruption due to uncertainty of occurrence time and decisive influence on continuous broadcasting of navigation signals.
In order to ensure high continuity of navigation signals and high availability of navigation satellites, the design constraint of the interruption frequency as a satellite is put forward in the navigation satellite engineering of China. For a single star, the frequency of interruption generally corresponds to the short-term unplanned interruption, and is therefore equivalent to the average number of short-term unplanned interruptions. Short-term unplanned interruption refers to an interruption in which satellite signals are not available for only a certain period of time and are recoverable in a short period of time. Short-term unplanned interruptions to navigation satellites are typically caused by single event upset events, software errors, hard-failure post-cut-offs, and the like.
According to the foreign GPS system, galileo system and GLONASS system publications, the Galileo system and the GLONASS system are not known to provide interruption frequency indexes, and the GPS system provides average short-term unplanned interruption times at the constellation level (see Global Positioning System Standard Positioning Service Performance Standard (2008 edition)), but a detailed analysis method is not known. According to the construction condition of the domestic Beidou navigation system, china firstly puts forward the interruption frequency requirement in the construction stage of the global navigation system (see the Chinese Qingnian newspaper, 2017, 11, 6 and three-inquiry Chinese Beidou). Therefore, the interruption frequency index belongs to a new concept, and a specific analysis method is also lacking in navigation satellite engineering.
The analysis of the interrupt frequency needs to be focused on solving the following problems:
1) Analysis of range problems. Only faults related to navigation satellite service can cause interruption, one satellite has hundreds of devices and tens of thousands of components, and comprehensively analyzing the relationship between the devices and the components and the satellite interruption consumes a great deal of resources, and the analysis range needs to be determined in a simple and more efficient way so as to eliminate the devices and the components irrelevant to the satellite interruption.
2) Interrupt modeling problems. The analysis model of the interruption index is established by which method, various factors influencing interruption can be covered, and the analysis result can be conveniently obtained by a mathematical method.
3) Device function interruption rate. The function interruption rate of the device is basic input data of system interruption index analysis, and is the frequency of functional abnormality which can actually occur in orbit after the device is applied to a navigation satellite and single particle protection measures are taken. Because of the many relevant devices on the satellite, the index is difficult to obtain completely through a ground test or simulation method.
Disclosure of Invention
In view of this, the present invention provides a method for analyzing the frequency of in-orbit interruption of navigation satellite signals, which can be applied to the initial index demonstration of satellite design, the distribution and the index verification at the end of satellite development.
The technical scheme for realizing the invention is as follows:
the method for analyzing the frequency of the on-orbit interruption of the navigation satellite signal comprises the following steps:
(1) Sequentially determining a subsystem, a component device (single machine) and a bottom event which can cause the interruption of a navigation signal, wherein the bottom event comprises a fault switching event and a soft fault of a master/backup device, and the soft faults comprise two types, namely, a single particle soft error occurs in a single particle sensitive device, and the soft error is caused by a software design defect;
(2) Establishing an interrupt tree according to the analysis result in the step (1);
(3) Establishing an interruption frequency analysis model according to the interruption tree, and calculating satellite interruption frequency indexes according to the occurrence frequency of various bottom events;
(4) And according to the satellite outage frequency analysis result, the method is applied to satellite design initial index demonstration, distribution and index verification at the end of satellite development.
Further, under the condition of few bottom events and simple logic relationship of the interrupt tree, the top-level frequency of satellite signal interrupt can be decomposed into an OR gate, an AND gate and a combination thereof according to the interrupt tree;
the OR gate case is calculated as follows: let n be 1 Bottom event x i ,i=1,2,…,n 1 Any bottom event can cause satellite signal interruption, and the influence factor of the ith bottom event is beta i Frequency of satellite signal interruption
Wherein, MTBO i Representing the average occurrence interval time of the ith bottom event.
The AND gate condition is calculated as follows: let n be 2 Bottom event x j ,j=1,2,…,n 2 When all bottom events occur, satellite signal interruption can be caused, and the influence factor of the AND gate is beta j Frequency of satellite signal interruption
Wherein, MTBO j Representing the average occurrence interval time of the jth bottom event.
Substituting the frequency of occurrence of the bottom event into a calculation model or a simulation model according to the interruption frequency analysis, and analyzing to obtain an interruption frequency index.
Furthermore, the invention establishes a Monte Carlo simulation flow when the bottom events of the interrupt tree are more and the logic relationship is complex, thereby realizing the simulation calculation of the interrupt frequency.
Further, the index of the invention is demonstrated as follows:
a. applied to index demonstration
In the satellite feasibility demonstration stage, obtaining satellite interruption frequency indexes according to the prediction, and taking the satellite interruption frequency indexes as a basis for determining satellite development technical requirements;
b. application to index allocation
In the satellite scheme design stage, on the basis of interruption influence analysis, an interruption frequency index is preliminarily allocated to a related subsystem and equipment, the satellite interruption frequency index is obtained according to the estimation, whether the technical requirement value is met or not is judged, if not, the related subsystem and equipment index is adjusted, the interruption frequency index of the related subsystem and equipment is finally determined, and the whole satellite interruption frequency index is ensured to meet the requirement;
c. applied to index verification
In the satellite positive development stage, test data and simulation data of existing related equipment or devices are utilized, and according to the estimated interrupt frequency index of the final development realization state of the satellite is obtained, whether the technical index requirement is met is verified.
Further, the specific process of the step (1) of the invention is as follows:
a. and acquiring the composition, function and redundant design information of all subsystems of the navigation satellite, a satellite software matching table and a satellite FPGA matching table.
b. Analyzing the relation between each subsystem and navigation signal generation and broadcasting, and clearly influencing the subsystem of navigation signal interruption;
c. acquiring further information of subsystems possibly causing interruption of navigation signals, including equipment composition, functions, redundancy design, software and FPGA configuration conditions of each subsystem;
d. and determining the relation between each device and the generation and broadcasting of the navigation signal, and determining a bottom event which possibly causes the interruption of the navigation signal.
Further, the invention can be divided into three cases according to the influence degree of the bottom event on the satellite signal interruption, and the three cases are distinguished by the influence factor beta, and the value range and the description of beta are shown in a table 3;
TABLE 3 value ranges of influencing factors
Beta value Description of the invention
β=1 Failure necessarily results in the system being unusable
β=x,0<x<1 Failure causes the system to be unavailable at frequency x
β=0 Failure does not cause the system to be unusable
Compared with the prior art, the invention has the advantages that:
(1) The method provides a complete and feasible analysis flow of the interruption frequency of the navigation satellite, particularly an analysis method of the interruption frequency of the signal caused by single event, software error, backup failure and the like of the navigation satellite, and can be applied to initial index demonstration and distribution of satellite design and index verification at the end of satellite development.
(2) The method of the invention provides a simple and effective screening mode of a large amount of irrelevant information.
(3) The method provided by the invention provides interrupt tree modeling, can truly describe the influence and propagation characteristics of the bottom event, and objectively embody the effect of the influence factors.
(4) The method is particularly suitable for quantitative analysis of soft faults of spacecrafts with high requirements on-orbit continuity of navigation satellites and the like.
Drawings
FIG. 1 is a block flow diagram of the method of the present invention.
Fig. 2 is a functional tree example.
FIG. 3 is an interrupt tree example.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention more clear, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention.
The invention relates to an analysis method of the on-orbit interrupt frequency of a navigation satellite signal, which comprises the following specific processes:
(1) The method comprises the steps of sequentially determining a subsystem, a component device (single machine) and a bottom event which can cause the interruption of a navigation signal, wherein the bottom event comprises a fault switching event and a soft fault of a master/backup device, the soft fault comprises two types, namely, a single-particle soft error occurs in a single-particle sensitive device, and the soft error is caused by a software design defect.
a. Basic information is acquired. The system comprises the components, functions and redundant design information of all subsystems of the navigation satellite, a satellite software matching table and a satellite FPGA matching table.
b. And analyzing the relation between each subsystem and navigation signal generation and broadcasting, and definitely determining the influence of interruption.
c. Further information of the subsystems that may cause interruption of the navigation signal is obtained, including equipment composition, functionality, redundancy design, software and FPGA configuration of each subsystem.
d. And determining the relation between each device and the generation and broadcasting of the navigation signal, and determining a bottom event which possibly causes the interruption of the navigation signal.
The relation between the subsystem, the device and the navigation signal interruption can be rapidly determined according to the table 1 in the implementation process, and further analysis is performed on the basis.
Table 1 sub-system, apparatus and navigation signal interrupt relationship cue table
(2) Building interrupt trees
The interrupt tree takes the 'navigation satellite signal interrupt' as a top event, and the interrupt tree is built according to the analysis result in the step (1) by referring to the building method of the fault tree in the GJB/Z768A fault tree analysis guide. The interrupt tree is in the form shown in fig. 3.
The bottom event of the interrupt tree includes both soft faults caused by software errors, FPGA single-event soft errors, etc., and failover events of the primary/backup device (unit).
The interrupt tree is similar to the fault tree but also differs from the fault tree. The main differences between the interrupt tree and the fault tree proposed by the present invention are shown in table 2.
TABLE 2 major differentiation of interrupt trees from fault trees
The influence degree of the bottom event on the satellite signal interruption can be divided into three cases, and the three cases are distinguished by an influence factor beta, and the value range and the description of the beta are shown in a table 3.
TABLE 3 value ranges of influencing factors
Beta value Description of the invention
β=1 Failure necessarily results in the system being unusable
β=x,0<x<1 Failure causes the system to be unavailable at frequency x
β=0 Failure does not cause the system to be unusable
(3) Determining occurrence frequency of various bottom events, and calculating satellite interruption frequency index
a. Calculating occurrence frequency of class bottom events of hard fault cutter
The device which is related to the broadcasting function and has cold backup or warm backup is generated by the navigation signal, after the main part of the device has hardware fault, the device needs to switch to backup work, and the navigation signal can be in an interrupt state during the period from the main part fault to the backup entering into normal work. The frequency of this bottom event, i.e., the frequency of device hardware failures. In general, the frequency of device hardware failures can be obtained using a method of reliability prediction. The standard basis of reliability prediction is GJB/Z299C reliability prediction Manual of electronic equipment and GJB 813 reliability model establishment and reliability prediction.
b. Calculating the occurrence frequency of soft fault bottom-like events
The soft fault bottom-like event generally has 2 cases, namely that single particle soft errors occur in single particle sensitive devices such as FPGA and the like, and the soft errors are caused by software design defects. The single particle soft error occurrence frequency of the sensitive devices such as the FPGA can be obtained by methods such as irradiation test, simulation analysis, on-orbit flight data statistics of similar devices and the like. The software error occurrence frequency can be obtained through a software reliability test.
c. Establishing an interrupt frequency analysis model according to the interrupt tree
Under the condition that the number of the bottom events of the interrupt tree is small and the logic relationship is simple, an analysis model can be established. The top level frequency of satellite signal interrupts can be broken down into or gates, and combinations thereof, depending on the interrupt tree.
The OR gate case is calculated as follows: let n be 1 Bottom event x i (i=1,2,…,n 1 ) Any occurrence of a bottom event may cause a satellite signal to be interrupted,the influence factor of the ith bottom event is beta i (i.e., the top event of satellite signal interruption at the occurrence of the ith bottom event has beta i Frequency occurrence of (a) of the satellite signal interruption
The AND gate condition is calculated as follows: let n be 2 Bottom event x j (j=1,2,…,n 2 ) When all bottom events occur, satellite signal interruption can be caused, and the influence factor of the AND gate is beta j (i.e., the top event of satellite signal interruption at the occurrence of the ith bottom event has beta j Frequency occurrence of (a) of the satellite signal interruption
When the number of the bottom events of the interrupt tree is large, the logic relationship is complex, the calculation amount of the analytic formula is large or the analytic formula is not suitable for calculation, a Monte Carlo simulation method can be adopted. According to the interrupt tree, a Monte Carlo simulation flow can be established, and the simulation calculation of the interrupt frequency is realized by utilizing a special reliability software tool or a programming simulation program.
d. Substituting the frequency of occurrence of the bottom event into a calculation model or a simulation model according to the interruption frequency analysis, and analyzing to obtain an interruption frequency index.
(4) And according to the satellite outage frequency analysis result, the method is applied to satellite design initial index demonstration, distribution and index verification at the end of satellite development.
a. Applied to index demonstration
In the satellite feasibility demonstration stage, the satellite interruption frequency index is obtained through preliminary prediction in the processes of (1), (2) and (3) and is used as a basis for determining the satellite development technical requirements.
b. Application to index allocation
In the design stage of the satellite scheme, on the basis of interruption influence analysis, an interruption frequency index is preliminarily allocated to related subsystems and equipment, the satellite interruption frequency index is obtained through the pre-estimation in the processes of (1), (2) and (3), whether the technical requirement value is met or not is judged, if the technical requirement value is not met, the related subsystems and equipment indexes are adjusted, iteration is repeated, the interruption frequency index of the related subsystems and equipment is finally determined, and the whole satellite interruption frequency index is ensured to meet the requirement.
c. Applied to index verification
In the positive development stage of the satellite, the interruption frequency index of the final development realization state of the satellite is obtained through the process analysis of (1), (2) and (3) by utilizing the test data and simulation data of the existing related equipment or devices, and whether the technical index requirement is met is verified.
The method establishes a standardized flow of interrupt frequency index analysis, locates an analysis range through a functional analysis method, establishes an interrupt index model through an interrupt tree method, and obtains basic data such as FPGA functional interrupt rate through similar FPGA data and weighting factors. By using the method, the interruption frequency index can be demonstrated and distributed in the initial stage of satellite design, and the interruption frequency index can be verified in the delivery stage of the satellite.
Examples:
the following describes a specific embodiment by taking a certain navigation satellite outage frequency analysis as an example.
Step 1, acquiring subsystem basic information. The system comprises the components, functions and redundant design information of all subsystems of the navigation satellite, a satellite software matching table and a satellite FPGA matching table. In the example, the navigation satellite comprises 8 subsystems of a subsystem A, a subsystem B, a subsystem C and the like, and all the matched tables are complete.
And 2, analyzing the relation between each subsystem and navigation signal generation and broadcasting, and definitely interrupting the influence.
Firstly, analyzing whether each subsystem has short-term function interruption according to satellite hardware composition and redundancy design, a software matching table and an FPGA matching table. If some equipment executing subsystem functions has cold backup or warm backup, after the equipment main part fails and before switching to backup work, the subsystem is in a function interruption state; if the subsystem contains software and FPGA, the subsystem may be in a function interrupt state when faults such as software errors, FPGA single-particle soft errors, etc. occur.
Then, according to the satellite function tree, searching the function related to the broadcasting generated by the navigation signal, and if the short-term interruption possibility exists for the function, recording the related subsystem and the function name. The functional tree is derived from functional analysis results, and the method of functional analysis can be seen in ECSS-E-10-05A. A form of a navigation satellite function tree is shown in fig. 2.
The relationship between each subsystem analyzed and the navigation downlink signal interruption is shown in table 4.
TABLE 4 relation between subsystems and navigation downlink signal interruption
From the above analysis, subsystem a has the possibility of causing a downlink signal interruption.
And step 3, acquiring further information of the subsystem A possibly causing the interruption of the navigation signal, including the composition, the function, the redundant design information, the software and the FPGA configuration condition of each device.
And 4, analyzing the relation between each device and the generation and broadcasting of the navigation signal, and determining a bottom event possibly causing the interruption of the navigation signal.
Firstly, analyzing whether short-term function interruption of each device occurs according to device composition, redundancy design, software and FPGA configuration conditions. If the internal unit executing the function of the equipment has cold backup or warm backup, the equipment is in a function interruption state after the unit main part fails and before switching to backup work; if the device contains software and an FPGA, the device may be in a function interrupt state when faults such as software errors, single-particle soft errors of the FPGA and the like occur.
Then, according to the satellite and the subsystem function tree, the navigation signal generates information flow and function block diagram related to broadcasting, and the equipment with function interruption is analyzed and determined, and whether the function interruption affects the generation and broadcasting of the navigation signal or not is determined. If the function interruption of a device may cause the interruption of the navigation signal, the reason for the function interruption of the device is traced back, and the relevant reason is defined as a bottom event.
Through the step 3 and the step 4, the relation between each single machine in the subsystem A and the downlink signal interruption is obtained through analysis and is shown in a table 5.
TABLE 5 relationship between single machine and navigation downlink signal interruption in subsystem A
Step 5, according to the analysis conclusion of the table 5, an interrupt tree for navigating the downlink signal interrupt is established as shown in fig. 3. And determining the influence factors of each bottom event relative to the top event according to the past historical data.
And step 6, determining the occurrence frequency of various bottom events of the navigation satellite downlink signal interrupt tree according to the basic data of the bottom events, and calculating the satellite interrupt frequency index. Frequency of occurrence P of each bottom event xi The value data are shown in Table 6.
TABLE 6 navigation of the basic data (assumed values) for downlink signal interruption
According to the interrupt tree, the navigation downlink signal interrupt frequency can be calculated according to the following way
According to Table 6, the downlink signal interruption frequency is calculated as
And step 7, the interruption frequency index of the navigation satellite is required to be 0.5 times/year, and the interruption frequency index estimated by the current design is 0.173 times/year and is far greater than the task requirement, so that the current design can be confirmed to meet the technical requirement, and the next stage of work can be carried out.
In summary, the above embodiments are only preferred embodiments of the present invention, and are not intended to limit the scope of the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (5)

1. The method for analyzing the frequency of the on-orbit interruption of the navigation satellite signal is characterized by comprising the following specific processes:
(1) Sequentially determining a subsystem, component equipment and a bottom event which possibly cause interruption of a navigation signal, wherein the bottom event comprises a fault switching event of a master/backup equipment and a soft fault, and the soft faults comprise a single-particle soft error generated by a single-particle sensitive device and a soft error caused by a software design defect;
(2) Establishing an interrupt tree according to the analysis result in the step (1);
(3) Establishing an interruption frequency analysis model according to the interruption tree, and calculating satellite interruption frequency indexes according to the occurrence frequency of various bottom events;
under the conditions of few bottom events and simple logic relationship of the interrupt tree, the top-level frequency of satellite signal interrupt can be decomposed into an OR gate, an AND gate and a combination thereof according to the interrupt tree;
the OR gate case is calculated as follows: let n be 1 Bottom event x i ,i=1,2,…,n 1 Any bottom event can cause satellite signal interruption, and the influence factor of the ith bottom event is beta i Frequency of satellite signal interruption
Wherein, MTBO i Representing the average occurrence interval time of the ith bottom event;
the AND gate condition is calculated as follows: let n be 2 Bottom event x j ,j=1,2,…,n 2 When all bottom events occur, satellite signal interruption can be caused, and the influence factor of the AND gate is beta j Frequency of satellite signal interruption
Wherein, MTBO j Representing the average occurrence interval time of the jth bottom event;
substituting the frequency of occurrence of the bottom event into a calculation model or a simulation model according to the interruption frequency analysis, and analyzing to obtain an interruption frequency index;
(4) And according to the satellite outage frequency analysis result, the method is applied to satellite design initial index demonstration, distribution and index verification at the end of satellite development.
2. The method for analyzing the frequency of the on-orbit interruption of the navigation satellite signal according to claim 1, wherein the simulation calculation of the interruption frequency is realized by establishing a Monte Carlo simulation flow in the condition that the number of the bottom events of the interruption tree is large and the logic relationship is complex.
3. The method for analyzing the frequency of the in-orbit interrupt of a navigation satellite signal according to claim 1, wherein the application scene of the index is:
a. applied to index demonstration
In the satellite feasibility demonstration stage, obtaining satellite interruption frequency indexes according to the prediction, and taking the satellite interruption frequency indexes as a basis for determining satellite development technical requirements;
b. application to index allocation
In the satellite scheme design stage, on the basis of interruption influence analysis, an interruption frequency index is preliminarily allocated to a related subsystem and equipment, the satellite interruption frequency index is obtained according to the estimation, whether the technical requirement value is met or not is judged, if not, the related subsystem and equipment index is adjusted, the interruption frequency index of the related subsystem and equipment is finally determined, and the whole satellite interruption frequency index is ensured to meet the requirement;
c. applied to index verification
In the satellite positive development stage, test data and simulation data of existing related equipment or devices are utilized, and according to the estimated interrupt frequency index of the final development realization state of the satellite is obtained, whether the technical index requirement is met is verified.
4. The method for analyzing the frequency of an on-orbit interrupt of a navigation satellite signal according to claim 1, wherein the specific process of step (1) is as follows:
a. acquiring composition, function and redundant design information of all subsystems of a navigation satellite, and a satellite software matching table and a satellite FPGA matching table;
b. analyzing the relation between each subsystem and navigation signal generation and broadcasting, and clearly influencing the subsystem of navigation signal interruption;
c. acquiring further information of subsystems possibly causing interruption of navigation signals, including equipment composition, functions, redundancy design, software and FPGA configuration conditions of each subsystem;
d. and determining the relation between each device and the generation and broadcasting of the navigation signal, and determining a bottom event which possibly causes the interruption of the navigation signal.
5. The method for analyzing the frequency of the on-orbit interruption of the navigation satellite signal according to claim 1, wherein the degree of influence of the bottom event on the interruption of the satellite signal can be divided into three cases and distinguished by an influence factor beta, and the value range and description of beta are shown in table 3;
TABLE 3 value ranges of influencing factors
Beta value Description of the invention β=1 Failure necessarily results in the system being unusable β=x,0<x<1 Failure causes the system to be unavailable at frequency x β=0 Failure does not cause the system to be unusable
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