System modeling method based on state and structure directed graph and application

Abstract

The invention relates to a state and structure directed graph-based system modeling method and application, belongs to the technical field of system digital simulation, and solves the technical problems that a system modeling method in the prior art is difficult to consider fault logic and function logic of a system at the same time and cannot perform quantitative evaluation on fault-tolerant capability of the system. The system modeling method comprises the following steps: step 1, determining a component structure, a component function and a component state of a system; obtaining a system state and state transition of the system state according to the component state; 2, constructing a component structure-oriented system directed graph according to a system state; and step 3, obtaining a system model based on the system directed graph. According to the system modeling method, based on the state and structure directed graph, the integrity of the function logic of the system can be accurately evaluated from the function logic and the system principle, and a complete quantitative evaluation method is provided.

Description

Technical Field

[0001] This invention belongs to the field of system digital simulation technology, specifically relating to a system modeling method and application based on state and structure directed graphs. Background Technology

[0002] With technological advancements and increasing system complexity, high reliability and safety have become critical requirements in aerospace, battery, energy, and transportation industries. In these fields, the failure of any single component can lead to the loss of entire system functionality, causing catastrophic consequences. Ensuring continuous system operation even when components fail is a key challenge in reliability engineering. The ability of a system to continue functioning despite component failures is called fault tolerance. To evaluate fault tolerance, a quantitative model of the system's state is needed, considering both failure and functional completion. Existing reliability research methods primarily focus on the logical connection structure between system components, extracting the system's fault logic topology from a graph theory perspective. However, these modeling methods, primarily logical, only consider the integrity of the system's functional logic and can only qualitatively describe the system's fault tolerance. However, complex systems often have multiple interrelationships between components. Hardware connections and functional associations between components influence the system's fault tolerance during operation. Therefore, combining functional and fault logic simultaneously and providing intuitive fault tolerance evaluation metrics plays a crucial role in system modeling. Summary of the Invention

[0003] In view of the above problems, the present invention provides a system modeling method and application based on state and structure directed graphs, which solves the technical problem that existing system modeling methods are unable to simultaneously consider the fault logic and functional logic of the system, and are unable to quantitatively evaluate the fault tolerance capability of the system.

[0004] This invention provides a system modeling method based on state and structure directed graphs, comprising the following steps: Step 1. Determine the system's component structure, component functions, and component states; obtain the system state and state transitions based on the component states; The system state transitions include the system state transition trajectories obtained from the component state transition trajectories; the system includes single-input single-output (SISO) systems and multiple-input multiple-output (MIMO) systems. The expression for the component state in the component structure is:

[0005] in, Representation Component c The state vector; Indicates the total number of components in the system; Representation Component c Total number of state categories; Considering the component state at different times, obtain the component state transition trajectory. The expression is:

[0006] in, Representation Component c The trajectory of state transition; Indicates the time of degradation The following components c The state vector; Indicates the degradation time; Represents the set of positive real numbers; The system state is obtained from the component state, expressed as:

[0007] in, Indicates the system status; This represents the mapping relationship between the component state vector space and the system state space; Represents the system state vector; State transitions from system state to system state; Step 2. Construct a component-oriented directed graph of the system based on the system state; Step 3. Obtain the system model based on the directed graph of the system.

[0008] Optionally, the degradation time The value range is set to the degradation time vector. Obtain the transition trajectory of the component's state. The state transition sequence is expressed as:

[0009] in, Indicates the time point of degradation The state vector of component c below; ; M This represents the total number of degradation time points.

[0010] Optionally, based on the state transition sequence of the component's state transition trajectory, the component's state transitions are merged into a component state transition matrix. The expression is:

[0011] in, Indicates the time point of degradation The following componentsc The state vector; ; Indicates the time point of degradation The following components c State vector.

[0012] Optionally, the system has a directed graph. The expression is:

[0013] in, V Represents the set of nodes in a directed graph; E This represents the set of edges in a directed graph.

[0014] Optionally, the node set of a directed graph V The expression is:

[0015] in, s The starting point of the directed graph represents the system's input. e The endpoint of the directed graph represents the system's output; For components Functional components.

[0016] In another aspect, the present invention provides a method for obtaining the degree of system structural redundancy, wherein the system model obtained by the aforementioned system modeling method is used to obtain the degree of system structural redundancy, and the specific steps are as follows: Based on the system model, given the system state In the system directed graph corresponding to a single-input single-output (SISO) system or a multiple-input multiple-output (MIMO) system, obtain all functional paths that can realize the function from the starting point to the ending point of the directed graph. Based on the functional path, the degree of redundancy in the system structure is obtained.

[0017] Compared with the prior art, the present invention has at least the following beneficial effects: (1) The system modeling method of the present invention is based on the state and structure directed graph, which starts from the functional logic of the system. It can accurately evaluate the integrity of the system's functional logic and provide a complete quantitative evaluation method.

[0018] (2) The system state changes in the system modeling method of the present invention will change the structure diagram. Therefore, the impact of component failure on the system structure can be described.

[0019] (3) The system modeling method of the present invention finally obtains a formal expression of the system structure, and realizes the quantification of the redundancy of the system structure based on the directed graph of the system structure. Attached Figure Description

[0020] The accompanying drawings are for illustrative purposes only and are not intended to limit the scope of the invention.

[0021] Figure 1 This is a functional logic block diagram of the dual-redundant negative feedback control system in Embodiment 1 of the modeling method of the present invention; Figure 2 This is a structural diagram of the dual-redundant negative feedback control system in Embodiment 1 of the modeling method of the present invention; Figure 3 This is a structural diagram of a dual-redundant negative feedback control system in Embodiment 1 of the modeling method of the present invention when a component fails. Figure 4 This is a flowchart of the modeling method of the present invention.

[0022] Figure label: 1. Controller, 2. Sensor, 3. First redundant actuator unit, 4. First redundant actuator unit. Detailed Implementation

[0023] To better understand the above-described objectives, features, and advantages of the present invention, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that, unless otherwise specified, the embodiments of the present invention and the features thereof can be combined with each other. Furthermore, the present invention can be implemented in other ways different from those described herein; therefore, the scope of protection of the present invention is not limited to the specific embodiments disclosed below.

[0024] A specific embodiment of the present invention, such as Figure 1-4 A system modeling method based on state and structure directed graphs is disclosed, including the following steps: Step 1. Determine the system's component structure, component functions, and component states; obtain the system state and system state transitions based on the component states.

[0025] Furthermore, the component state of the component structure is obtained by the expression:

[0026] in, Representation Component c The state vector; Indicates the total number of components in the system; Representation Component c The total number of state categories.

[0027] Understandably, the component state is the component's failure mode.

[0028] Furthermore, the system state is obtained from the component state, expressed as:

[0029] in, Indicates the system status; This represents the mapping relationship between the component state vector space and the system state space; This represents the system state vector.

[0030] Furthermore, , Representation Component The state vector.

[0031] Furthermore, considering the component states at different times, the transition trajectory of the component states is obtained. Its expression is:

[0032] in, Indicates the time of degradation The following components c The state vector; Indicates the degradation time; It represents the set of positive real numbers.

[0033] It is understandable that the transition of component states occurs in a direction on the degradation timescale and is not reversible.

[0034] Furthermore, the system state transition trajectory The expression is:

[0035] in, Indicates the time of degradation The state vector of the system under the given conditions; Indicates the time of degradation The state of all components under; Represents the system state space. This represents the mapping relationship between the component state vector space and the system state space. It is generally believed that the system state is determined by the component state vector.

[0036] Furthermore, considering the discrete state, the degradation time... The range of values ​​is a set of degradation time vectors. Obtain components State transition trajectory The state transition sequence is expressed as:

[0037] in, Indicates the time point of degradation The state vector of component c below; .

[0038] Furthermore, the state transition sequences of all components are merged into a component state transition matrix. The expression is:

[0039] in, Indicates the time point of degradation The following components c The state vector; j =0,1,2,…, M , M This represents the total number of degradation time points.

[0040] Furthermore, ,in, Indicates the time point of degradation The following components c State vector.

[0041] It is understandable that the component state transition matrix The c Behavioral components State transition sequence The column vector represents the degradation time point. Below, the input to the system state function is The corresponding system status is , The state transition sequence of the system state transition trajectory is obtained by expression:

[0042] in, Indicates the time point of degradation The system status under the following conditions.

[0043] It is understandable that the state of a system or component is its operating condition at a certain moment. As degradation time changes, the state of the component will change, which will further lead to a change in the state of the system. This invention refers to this change in component / system state with degradation time as state transition.

[0044] This invention describes the system reconfiguration caused by component failure through state transitions, thereby enabling the description of the dynamic changes in system reliability.

[0045] The system state at this time is determined through the above steps. Then, in this state, a directed graph model of the system is constructed.

[0046] Step 2. Construct a component-oriented directed graph of the system based on the system state. The expression is:

[0047] in, V Represents the set of nodes in a directed graph; E This represents the set of edges in a directed graph.

[0048] Furthermore, ,in, s The starting point of the directed graph represents the system's input. e The endpoint of the directed graph represents the system's output; For components Functional components.

[0049] It is understandable that the starting point of a directed graph is the first functional step that the system needs to go through to complete its function; the ending point of a directed graph is the system's output.

[0050] Furthermore, E The elements in the text represent the interaction relationships between components, for example, the flow of matter, energy, and information between various functional links.

[0051] It is understandable that a function requires multiple functional components, and the function can only be realized when all functional components are complete.

[0052] Step 3. Obtain the system model based on the directed graph of the system.

[0053] Another aspect of the present invention discloses a method for obtaining the degree of system structural redundancy using a system model based on a state and structure directed graph, the specific steps of which are as follows: Step 11. Based on the system model, determine the functional path; Specifically, in a given system state Below, in its corresponding directed graph system In the middle, obtain the starting point of the directed graph. s To the end of the directed graph e All functional paths that can achieve the desired functionality.

[0054] Understandably, a functional path represents the path that performs the desired function among all paths from system input to output. This applies to all paths starting from the finite graph. s To the end of the directed graph e The system identifies the paths that meet the system's functional requirements and calls them the system's functional paths.

[0055] Furthermore, for the directed graph of the system Starting from the point of the directed graph s To the end of the directed graph e The total number of all unique function paths .

[0056] Step 22. Based on the functional path, obtain the system redundancy level; Specifically, the system includes a single-input single-output (SISO) system and a multiple-input multiple-output (MIMO) system.

[0057] Next, for Single-Input Single-Output (SISO) and Multiple-Input Multi-Output (MIMO) systems, based on their respective characteristics, we will provide explanations based on the number of functional paths. The method of quantifying the redundancy of a system is one aspect of its fault tolerance capability.

[0058] Understandably, for the SISO system, there is only one start point and one end point in its directed graph, so it is sufficient to analyze only the total number of functional paths between this pair of start and end points.

[0059] For the system architecture diagram of the SISO system The degree of redundancy is measured from the node s To the node e Number of redundant functional paths After normalization, the structural tolerance is obtained, expressed as:

[0060] in, Indicates the system in state Structural tolerance; Indicates the system in state Lower system directed graph From the starting point s To the finish line e The total number of all unique function paths; Indicates the system in state The system is a directed graph; This indicates the system in its initial state. The system is a directed graph; Indicates the system's degradation time state of time The following is a directed graph of component structure.

[0061] Understandably, when This means that the system initially lacks redundancy and therefore structural fault tolerance. ;and This means that the system currently has no functional path and cannot perform the function; in this case, the system's structural fault tolerance is also defined as 0. It can be seen that... System status only The function.

[0062] For the directed graph of the MIMO system ,in, , Represents the set of starting points of a directed graph. Represents the set of endpoints in a directed graph. , , Indicates the first m A starting point of a directed graph, Indicates the first n The endpoint of a directed graph.

[0063] Furthermore, based on the directed graph of the MIMO system, the redundancy matrix between the starting and ending points of each directed graph is obtained. Redundancy matrix P The i The starting point and the first directed graph k The structural tolerance of the start and end nodes of a directed graph. The expression is:

[0064] in, , , Indicates the first i A starting point of a directed graph, Indicates the first k A directed graph endpoint; Indicating the initial state The system is a directed graph; m n represents the total number of starting points, and n represents the total number of ending points.

[0065] Furthermore, in MIMO systems, There are two scenarios: one is that the start and end node pairs of the directed graph are valid but there is no functional path, resulting in system failure; the other is that the start and end node pairs of the directed graph are invalid, and the lack of a functional path does not affect the system's functionality. For start and end node pairs that do not meet the requirements (i.e., the aforementioned system failure and invalid states), a Boolean matrix is ​​used. The redundancy matrix is ​​used as a filtering matrix to filter and correct it, and its first... i The starting point and the first directed graph k The start and end nodes of a directed graph. Satisfy: When the start and end node pairs When it is the start and end point of the system's functional path, A value of 1 indicates a valid start-end node pair; when the start-end node pair... When it is not the start or end point of the system's functional implementation path, A value of 0 indicates an invalid start-end node pair.

[0066] Based on the Boolean matrix, obtain the effective redundancy matrix. U , , No. i The starting point and the first directed graph k Redundancy of the start and end nodes of a directed graph satisfy:

[0067] in, This represents an invalid value, which is not included in the calculation. All addition and multiplication calculations return the original value, that is, for ,have , ; This indicates the values ​​involved in the calculation. This process distinguishes between invalid and valid start / endpoints, thus accurately quantifying the overall structural fault tolerance of the system.

[0068] After filtering and correcting the redundancy matrix of MIMO, the average structural fault tolerance of effective start-end node pairs is obtained based on the redundancy of the MIMO system. The expression is:

[0069] It is understandable that when the redundancy between any valid start-end pair is greater than 0, it means that there are redundant functional paths between all inputs and outputs of the system at this moment, and the average redundancy of each valid start-end pair is used as the system redundancy; when there is a pair of valid start-end pairs with no redundant functional paths, it is considered that the system no longer has structural fault tolerance, that is, the system no longer has redundancy.

[0070] This invention obtains the redundancy of system functional paths based on the directed graph of component structure; and obtains the system redundancy based on the redundancy of system functional paths.

[0071] A third aspect of the present invention also discloses a method for using a professional performance margin model of a system to obtain the degree of system redundancy, the specific steps of which are as follows: Step 101. Based on the interdisciplinary equation, degradation equation, and margin equation, construct the professional performance margin model of the system, expressed as:

[0072] in, For degradation time The professional performance margin of the timing system; Indicates the system in state The system's internal dependent variables; It is an external variable of the system.

[0073] For example, internal variables are variables that affect the system and are related to the system itself, such as the system's design parameters, material properties, or geometric dimensions; external variables are factors outside the system that affect the system, such as temperature, vibration, or voltage and current.

[0074] Furthermore, the interdisciplinary equation is:

[0075] in, P These are the system's performance parameters.

[0076] For example, for a mobile phone, the system's performance parameters include screen refresh rate, battery capacity, etc.

[0077] Furthermore, the degradation equation is:

[0078] in, P ( ) indicates the time of degradation System performance.

[0079] Furthermore, the margin equation is:

[0080] in, P th A threshold representing system performance.

[0081] Step 201. Based on the system's professional performance margin When retrieving the first pass, the expression is:

[0082] in, This indicates the moment when the system's performance margin first becomes less than 0; Indicates the infimum; Represent the set of positive real numbers; Indicates degradation time The system's professional performance margin.

[0083] Furthermore, based on a given discrete degradation time vector The transformation formula for obtaining the first penetration is:

[0084] in, Indicates the first j When a degeneration occurs The system's professional performance margin; Indicates the first j +1 degeneration time The system's professional performance margin; Understandably, First Passage Time refers to the time when a system or process first reaches a specific state or goal.

[0085] Step 301. Based on the reliable degradation time range of the system determined at the first crossing. To obtain the system's professional performance tolerance The expression is:

[0086] in, Indicates degradation time The professional performance tolerance of the time system; Indicates degradation time The differential; Indicates the initial degradation time.

[0087] Understandably, the professional performance tolerance of a system represents its fault tolerance capability in terms of professional performance, which is the integral of its professional performance margin over the remaining degradation time.

[0088] Step 401. Use the trapezoidal integral method to assess the system's performance tolerance. An approximate solution is performed to obtain the performance tolerance for degradation time in the discrete case, expressed as:

[0089] in, This indicates a focus on degradation time. Professional performance tolerance under discrete time conditions; q This represents the total number of elements in the degraded time series.

[0090] Furthermore, utilize specialized performance tolerances that focus on degradation time. Indicates system degradation time The approximate result of the professional performance tolerance of the system is expressed as:

[0091] in, Indicates degradation time The system is in state The system's internal dependent variables and external variables The approximate result of the professional performance tolerance under the given conditions.

[0092] Furthermore, the aforementioned professional performance tolerance of this invention refers to a single professional performance characteristic. For a system with multiple professional performance characteristics, the professional performance margins are respectively... The corresponding professional performance tolerance is Then the overall professional performance tolerance of the system is: .

[0093] It is understandable that the tolerance for all professional performance characteristics is set to a smaller value.

[0094] Step 501. Based on the system structure fault tolerance and professional performance fault tolerance, obtain the overall fault tolerance capability of the system, expressed as:

[0095] in, Indicates degradation time The overall fault tolerance capability of the system; Indicates degradation time The corresponding system state at that time The system's structural fault tolerance.

[0096] To illustrate the effectiveness of the method proposed in this invention, the above technical solution of this invention will be described in detail below through two specific embodiments, as follows: Example 1: Taking a dual-redundant negative feedback control system as an example, its functional block diagram is as follows: Figure 1 As shown. The system components include a controller, dual-redundant actuator units, and sensors. The controller sends the input signal at the current time t to each of the two redundant actuator units (i.e., dual-redundant actuator units). x in ( t The sensor outputs the signal to the actuator at the current time t. x out ( t The measurement is performed and returned to the controller.

[0097] The system's components are decomposed into functional parts, revealing that the main functional components of the entire system are: an error module responsible for calculating errors based on the system's inputs and outputs, and a controller 1 that outputs control quantities. Sensor 2 is responsible for measuring the actual output and returning it to the controller. ), and the first redundant actuator unit 3, which is responsible for converting the control input into the actual output. ) and the first redundant actuator unit 4 ( The two are parallel processes, do not affect each other, and can each provide independent output to the system.

[0098] Based on this, the component-based directed graph of the system can be obtained, such as... Figure 2 As shown.

[0099] exist Figure 2 status Below, the directed graph of this system is: For example, the path from the starting point s to the ending point e in the system is denoted as: , , , , ... However, among these many paths, paths 1 and 2 do not pass through sensors, and these two paths cannot fully realize the system's function; therefore, they are not functional paths. Only paths 3 and 4 are functional paths. Therefore, the system redundancy in this state is equal to the total number of functional paths, so the total number of functional paths is 2.

[0100] When one of the actuator units fails, it will interact with the first redundant actuator unit 3 ( All related nodes and interactions disappear, resulting in the system structure diagram for this state as follows: Figure 3 Show.

[0101] exist Figure 3 status Below, the directed graph of this system is: At this point, the total number of paths from the starting point s to the ending point e in the system is 2, namely: , , However, of these two paths, path 2 does not pass through a sensor, and this path cannot fully realize the system's function; therefore, it is not a functional path. Only path 4 is a functional path. Therefore, the system redundancy in this state is equal to the total number of functional paths, resulting in a total path count of 1.

[0102] In addition, in real-world systems, the functional logic relationships between components cannot be constructed into a structure diagram solely through the methods described above. For example, some system components have... l In each component k The system will only malfunction when one of the components fails; functionally, this... k The relationships that occur when a failure happens cannot be described from a process perspective. Therefore, it is necessary to use some methods to simplify the modeling of this type of structure.

[0103] For example, for in lIn each component k When a failure occurs, define a pair of nodes. With one l Pick k Module These are redundant component nodes, connected in parallel in the graph. The expression for the total number of paths between them is:

[0104] in, This represents the total number of paths between these two nodes.

[0105] The number of paths here is the number of redundant components plus one. Without redundancy, The number of paths is 1.

[0106] For example, for Figure 4 The system shown (if l=4, k=2) has node pairs and The total number of function paths between them is equal to the number of their APs, which is: .

[0107] Example 2: Taking a solar cell array as an example, the components of a solar cell array consist of multiple photovoltaic units. For a single photovoltaic unit, there are two typical failure modes: short circuit and open circuit. (Note: The last sentence appears to be incomplete and possibly refers to a technical detail.) j The first in the string of batteries i The bypass diode connected to the first k Photovoltaic units c The state is Its state space is ,Right now

[0108] in, , Indicates the first j The total number of bypass diodes in the battery string; Indicates the total number of battery strings; Indicates the first j The first in the string of batteries i The total number of photovoltaic units connected to each bypass diode; ; The state vector of the solar cell array is .

[0109] Furthermore, the system state vector of the solar cell array The expression is: .

[0110] For the j String of batteries, of which there are a total of Each substring has a bypass diode. Based on the fault-tolerance principle of bypass diodes, the remaining units... The relationship with component state is as follows:

[0111] but The expression is:

[0112] in, Representing the j The number of remaining cells in the battery string.

[0113] By combining the above equations, we can obtain the system state vector of the solar cell array. The various components are thus obtained. At this point, the system's structure function (state) is obtained. .

[0114] The input of a solar cell array is light energy, and the output is electrical energy. The nodes are individual photovoltaic units, and the structural connections represent the circuit topology. For battery strings, the interaction between units is complex due to the presence of bypass diodes. To simplify calculations, this invention integrates the fault-tolerant functional logic into the structural diagram, where the total number of functional paths in a battery string equals the number of remaining photovoltaic units.

[0115] in, This is the number of working battery cells in the battery string, which can be obtained from the system state function. For the first... j string batteries The number of remaining battery cells is the system state number. j element Therefore, we can obtain:

[0116] in, Indicates the first solar cell array system j The starting point of the directed graph of the series of batteries; Indicates the first solar cell array system j The endpoint of a directed graph of battery strings.

[0117] For the entire battery array system, since the individual battery strings are connected in parallel, the entire functional path of the system is the sum of the paths of each battery string, that is: .

[0118] For example, setting =6, and in battery strings 1-6, the number of working battery cells are 4, 4, 4, 5, 6, and 3 respectively. Therefore:

[0119]

[0120]

[0121]

[0122]

[0123]

[0124] For the entire battery array system, since the individual battery strings are connected in parallel, the entire functional path of the system is the sum of the paths of each battery string, that is: .

[0125] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention.

Claims

1. A system modeling method based on state and structure directed graphs, characterized in that, Includes the following steps: Step 1. Determine the system's component structure, component functions, and component states; The system state and its state transitions are obtained based on the component state. The system state transitions include the system state transition trajectories obtained from the component state transition trajectories; the system includes single-input single-output (SISO) systems and multiple-input multiple-output (MIMO) systems. The expression for the component state in the component structure is: in, Representation Component c The state vector; Indicates the total number of components in the system; Representation Component c Total number of state categories; Considering the component state at different times, obtain the component state transition trajectory. The expression is: in, Representation Component c The trajectory of state transition; Indicates the time of degradation The following components c The state vector; Indicates the degradation time; Represents the set of positive real numbers; The system state is obtained from the component state, expressed as: in, Indicates the system status; This represents the mapping relationship between the component state vector space and the system state space; Represents the system state vector; State transitions from system state to system state; Step 2. Construct a component-oriented directed graph of the system based on the system state; Step 3. Obtain the system model based on the directed graph of the system.

2. The system modeling method according to claim 1, characterized in that, Degradation time The value range is set to the degradation time vector. Obtain the transition trajectory of the component's state. The state transition sequence is expressed as: in, Indicates the time point of degradation The state vector of component c below; ; M This represents the total number of degradation time points.

3. The system modeling method according to claim 2, characterized in that, Based on the state transition sequence of the component state transition trajectory, the component state transitions are merged into a component state transition matrix. The expression is: in, Indicates the time point of degradation The following components c The state vector; ; Indicates the time point of degradation The following components c State vector.

4. The system modeling method according to claim 1, characterized in that, System directed graph The expression is: in, V Represents the set of nodes in a directed graph; E This represents the set of edges in a directed graph.

5. The system modeling method according to claim 4, characterized in that, Node set of a directed graph V The expression is: in, s The starting point of the directed graph represents the system's input. e The endpoint of the directed graph represents the system's output; For components Functional components.

6. A method for obtaining the degree of redundancy in system structure, wherein the system model obtained by the system modeling method according to any one of claims 1-5 is used to obtain the degree of redundancy in system structure, characterized in that, The specific steps are as follows: Based on the system model, given the system state In the system directed graph corresponding to a single-input single-output (SISO) system or a multiple-input multiple-output (MIMO) system, obtain all functional paths that can realize the function from the starting point to the ending point of the directed graph. Based on the functional path, the degree of redundancy in the system structure is obtained.

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