CPS Modeling and Analysis Method Based on Aspect-Oriented Grey GSPN

Abstract

The present invention discloses a CPS modeling and analysis method based on aspect-oriented grey GSPN. First, according to the specific situation of the CPS to be modeled, an AGGSPN model is constructed; the grey system theory is introduced, and the greying processing method is used to solve the transition firing rate and the triggering probability of the stochastic switch in the AGGSPN model; the Markov method is used to solve the grey steady-state probability of the AGGSPN; a mathematical programming model is constructed based on the obtained grey steady-state probability to solve the final interval grey number representation; finally, the performance indicators of the industrial CPS are analyzed using the final interval grey number representation. The present invention adopts the analysis methods of grey theory, Markov chain and mathematical programming, effectively dealing with the uncertainties in the CPS, so that the results have high precision.

Description

Technical Field

[0001] The present invention relates to a method for modeling and analyzing a CPS, in particular to a method for modeling and analyzing a CPS based on aspect-oriented grey GSPN (Generalized Stochastic Petri Net). Background Art

[0002] A Cyber-Physical System (CPS) is a multi-dimensional complex system that integrates computing, communication, and control. It combines discrete computing processes and continuous physical processes, and realizes information fusion and behavior interaction between the physical world and the virtual world based on feedback loops.

[0003] During actual operation, a CPS uses sensors to sense the physical world in real time, processes the acquired information through a controller to generate instructions, and uses actuators to execute the generated instructions. The cooperation among the three ensures its effectiveness and real-time performance. The CPS has unique advantages and is widely used in fields such as intelligent transportation, aerospace, and intelligent healthcare. It is also at the core of Industry 4.0 and has become the focus of attention in the field of intelligent manufacturing.

[0004] Industrial CPS operates in a highly dynamic and complex environment and must have strong robustness and adaptability, which results in its relatively complex structure and poses additional challenges to its modeling and accurate analysis. A model with a clear structure, intuitive and easy to understand, and the corresponding performance analysis method not only helps with the early-stage research and development of the system, reliability assessment, etc., but also ensures the safe operation of the system and provides a theoretical basis for later maintenance and upgrade.

[0005] Currently, some methods and analysis tools have been proposed at home and abroad for the modeling, simulation, and various qualitative and quantitative analyses of CPS. For example, a stochastic modeling framework is proposed, and joint state and attack assessment problems are formulated and solved based on the framework; a method for co-modeling the structure and dynamic behavior of CPS is proposed. However, among these theoretical models and analysis tools, some models are relatively abstract and complex, difficult to understand, and cannot be directly compared with the actual production process; the modeling process of some methods is relatively cumbersome, and the reusability of the models is poor. If the subsequent production process is adjusted, the models need to be significantly changed; there are also some models that rarely consider the uncertainty of CPS operation in industrial scenarios, such as ignoring the activity duration, resulting in inaccurate analysis of the performance indicators of industrial CPS.

[0006] CN111722599A "CPS Modeling and Analysis Method Based on Object-Oriented Generalized Stochastic Petri Net" is a modeling, simulation, and various qualitative and quantitative analysis methods for CPS proposed by the inventor earlier. It uses an object-oriented method to abstract each module in CPS into different objects for modeling and introduces the method of fuzzy mathematics for performance analysis. However, this method will ignore some details in the CPS process, resulting in insufficient accuracy.

[0007] Based on the above analysis, in order to solve the problems of the existing models being relatively abstract, having high complexity, and insufficient description and analysis capabilities, a kind of aspect-oriented grey generalized stochastic Petri net is proposed for the structural process and time attributes of industrial CPS. This model is based on the generalized stochastic Petri net, combines aspect-oriented technology with the concept of color sets, and then introduces the grey system theory to characterize the uncertainty of industrial CPS, so as to be able to accurately analyze industrial CPS. Summary of the Invention

[0008] The purpose of the present invention is to propose a CPS modeling and analysis method based on aspect-oriented grey generalized stochastic Petri net for the deficiencies of the current technology.

[0009] The general idea of the method of the present invention is:

[0010] In the modeling stage of CPS, in order to illustrate the basic components and processes in CPS and consider the effectiveness and feasibility of the model in industrial CPS, aspect-oriented technology is combined with grey generalized stochastic Petri net, and aspect-oriented grey generalized stochastic Petri net (AGGSPN) is proposed as a new formal modeling method for CPS. Due to the introduction of aspect-oriented technology, an industrial CPS is modeled as a basic net and several aspect nets, rather than a whole, making it impossible to qualitatively and quantitatively analyze the complete process of industrial CPS. Therefore, the present invention proposes a weaving algorithm based on AGGSPN to correctly weave all aspect nets into the basic net to form a complete woven net, and then its performance can be analyzed.

[0011] In the performance analysis stage of CPS, considering that some production parameters in the working process of CPS have the characteristics of "small samples and poor information" and will change over time, it is impossible to simply represent them with a fixed statistical value. In order to improve the authenticity and accuracy of the established model, the grey system theory is introduced into the model, and interval grey numbers are used to represent the transition firing rate and the triggering probability of random switches in the model, and a mathematical programming method is used when solving the Markov chain corresponding to the model to obtain the final grey steady-state probability distribution.

[0012] The steps of the method adopted by the present invention to solve its technical problems are as follows:

[0013] A CPS modeling and analysis method based on an aspect-oriented grey generalized stochastic Petri net, including the following steps:

[0014] Step (1): According to the specific situation of the CPS to be modeled, abstract the industrial CPS into a basic net and several aspect nets; specifically, abstract the total operation process of the industrial CPS into a basic net, and then subdivide some sub-processes into several aspect nets;

[0015] The total operation process of the industrial CPS includes multiple sub-processes;

[0016] Step (2): Based on GSPN, introduce the concept of a color set into the basic net and aspect nets, that is, color the elements therein to obtain an AGGSPN model (Aspect-Oriented Grey Generalized Stochastic Petri Net);

[0017] 2-1 For the basic net, use the state of industrial equipment on the industrial CPS production line in the total operation process as the places of GSPN, use the resources or signals on the industrial CPS production line in the total operation process as the tokens of the places, and use the dynamic behavior of industrial equipment on the industrial CPS production line in the total operation process as the active transitions of GSPN; use the relationship between the places and active transitions in the total operation process as the arcs; and label different colors for different tokens for distinction; (The places, active transitions, tokens, and arcs in the above net are all elements of GSPN)

[0018] 2-2 For the aspect nets, use the specific state of industrial equipment on the industrial CPS production line in the sub-process as the places of GSPN, use the resources or signals on the industrial CPS production line in the sub-process as the tokens of the places, and use the specific dynamic behavior of industrial equipment on the industrial CPS production line in the sub-process as the active transitions of GSPN; use the relationship between the places and active transitions in the sub-process as the arcs; and label different colors for different tokens for distinction;

[0019] Step (3): On the premise of ensuring model consistency, weave the basic net and all aspect nets into a woven net for subsequent performance analysis of the model; specifically:

[0020] (3.1) For any aspect net, first connect it to the basic net according to the notification type of the aspect net, and delete the redundant auxiliary elements and the arcs associated with the redundant auxiliary elements;

[0021] The redundant auxiliary elements of the aspect net include a start element start, an end element end, and repeated places;

[0022] (3.2) If the aspect net and the basic net contain the same place P, the input arcs and output arcs of place P in the aspect net need to be converted into the input arcs and output arcs of place P in the basic net, and place P in the aspect net is deleted.

[0023] Step (4): Introduce the grey system theory, and use the greying processing method to solve the transition firing rate and the triggering probability of the random switch in the AGGSPN model woven in step (3); specifically, by introducing a perturbation grey element δ, the transition firing rate and the triggering probability in the AGGSPN model are converted into interval grey numbers, and a coefficient k is set for each transition, that is, the perturbation grey element coefficient of the time transition is set as k1, and the perturbation grey element coefficient of the random switch is set as k2. The specific process is as follows:

[0024] (4.1) If the active transition T1 in the AGGSPN is a time transition and the transition firing rate is λ1, its interval grey number is expressed as [λ1 - k1δ, λ1 + k1δ];

[0025] (4.2) Similarly, if the active transition T2 in the AGGSPN is a random switch and the triggering probability is Pr(T2), its interval grey number is expressed as [Pr(T2) - k2δ, Pr(T2) + k2δ].

[0026] Step (5): Use the Markov method to solve the grey steady-state probability of the AGGSPN; the specific process is as follows:

[0027] (5.1) Analyze the woven net in step (3) to obtain the reachable marking graph and the corresponding reachable marking set S. S is divided into the existing state set TS and the disappearing state set VS, that is, S = TS ∪ VS, The number of its elements is K T and K V , K S = K T + K V ;

[0028] (5.2) Obtain the transition probability grey matrix of the Embedded Markov Chain (EMC) according to the reachable marking graph Its calculation formula is as follows:

[0029]

[0030] From the transition probability matrix of the transition from the disappearing state to the disappearing state set The transition probability matrix for the transition from the disappearing state to the set of existing states constitutes;

[0031] It is composed of the transition probability matrix for the transition from the existing state to the set of disappearing states and the transition probability matrix for the transition from the existing state to the set of existing states constitutes;

[0032] The transition probability matrix of the EMC from the set of disappearing states to the set of disappearing states;

[0033] The transition probability matrix of the EMC from the set of disappearing states to the set of existing states;

[0034] The transition probability matrix of the EMC from the set of existing states to the set of disappearing states;

[0035] The transition probability matrix of the EMC from the set of existing states to the set of existing states;

[0036] (5.3) Remove the disappearing states from the EMC to obtain the REMC transition state matrix which is represented by Equation (2);

[0037]

[0038] Remove all the disappearing states in the EMC, leaving only the existing states, and a compressed EMC (Reduced EMC, REMC) can be obtained, which is the transition probability matrix of the REMC;

[0039] denotes the set of elements where represents the probability that the GSPN starts from the given disappearing state r and reaches the existing state k for the first time after any number of steps;

[0040] (5.4) Solve the grey steady-state probability of the existing states of the REMC through the linear equations of Equation (3);

[0041]

[0042] denotes a one-dimensional row vector, and the element represents the grey steady-state probability of the existing states of the REMC; (5.5) Calculate the average residence time of each existing state through Equation (4)

[0043]

[0044] (5.6) The average residence time of each state and the corresponding gray stability probability can be used to obtain the cycle period of the system The calculation method is Equation (5);

[0045]

[0046] (5.7) The cycle period of the actual existence state can be obtained through and calculated;

[0047]

[0048] (5.8) The gray steady-state probability of the model Finally, it can be expressed by Equation (7), which is the ratio of the residence time of the actual existence state to its cycle period;

[0049]

[0050] After obtaining the transition state probability matrix, the GSPN steady-state probability calculation method of Equations (3)-(7) can be used to obtain represented by a composite gray number. Then, according to the interval gray number operation rules, the interval gray number representation about the perturbation gray element δ can be obtained. Since the operation between interval gray numbers will increase the uncertainty of the result, as δ continuously increases, the upper bound or lower bound of

[0051] Step (6). Based on the obtained gray steady-state probability, construct a mathematical programming model to solve the final interval gray number representation; finally, use the final interval gray number representation to analyze the performance indicators of this industrial CPS. The specific process is as follows:

[0052] (6.1) Under the conditions of meeting the actual production conditions (the transition firing rate is greater than 0, and the random switch trigger probability is between 0 and 1) and all are feasible solutions, find the maximum value of δ;

[0053]

[0054]

[0055] (6.2) Substitute the δ obtained from Equation (8) into to obtain the final interval gray number representation;

[0056] (6.3) Analyze the performance indicators of the industrial CPS using the obtained final interval grey number according to the specific CPS scenario.

[0057] Another object of the present invention is to provide an electronic device, which is characterized by comprising a processor and a memory, the memory stores machine-executable instructions that can be executed by the processor, and the processor executes the machine-executable instructions to implement the above method.

[0058] Yet another object of the present invention is to provide a machine-readable storage medium, which is characterized in that the machine-readable storage medium stores machine-executable instructions, and when the machine-executable instructions are called and executed by a processor, the machine-executable instructions cause the processor to implement the above method.

[0059] Advantages of the present invention:

[0060] The present invention is used for the modeling and analysis of CPS systems such as industrial intelligent manufacturing, aerospace, and intelligent healthcare. The invention proposes an aspect-oriented grey generalized stochastic Petri net model and uses it for the formal modeling of CPS, which can clearly express the logical structure and functional process of CPS. The present invention adopts the analysis methods of grey theory, Markov chain, and mathematical programming, effectively dealing with the uncertainties in CPS, so that the results have high accuracy. Description of the Drawings

[0061] Figure 1 Partial device diagrams for IPRPL-CPS, where (a) is a film laminating machine, (b) is a bending machine and sensors, (c) is a riveting machine, and (d) is a palletizing robot;

[0062] Figure 2 Core production flow chart of IPRPL-CPS;

[0063] Figure 3 Basic net BN of IPRPL-CPS

[0064] Figure 4 Two aspect nets AN1 and AN2 of IPRPL-CPS, where (a) is aspect net AN1 and (b) is aspect net AN2;

[0065] Figure 5 Conflict and resolution of time transitions, where (a) shows a conflict in time transitions, (b) shows an instantaneous transition occurring first, and (c) shows a random switch;

[0066] Figure 6 Weaving process of the aspect net with the notification type of after;

[0067] Figure 7The woven net WN for IPRPL-CPS;

[0068] Figure 8 is the reachability graph of WN;

[0069] Figure 9 is the graphical representation of the gray steady-state probability ; where (a) is (b) is (c) is (d) is (e) is (f) is (g) is (h) is Figure 10 Equipment utilization rate. Detailed implementation manners

[0070] The present invention will be further described below in conjunction with specific implementation examples.

[0071] The Intelligent Plate Reinforcement Production Line (IPRPL) is a typical industrial CPS, which includes a film laminating machine, a bending machine, a riveting machine, a palletizing robot, etc. Some of the equipment is as Figure 1 shown. Its specific working process is as follows:

[0072] The first step: The film laminating machine laminates the plate, and at the same time the bending machine bends the reinforcing rib raw material;

[0073] The second step: After waiting for the plate lamination to be completed and the reinforcing rib bending to be completed, the riveting machine rivets the plate. At the same time, a laser sensor and an industrial camera are used to detect the reinforcing rib to determine whether it meets the standard. If it meets the standard, the riveting machine is used to rivet the reinforcing rib; otherwise, the bending machine needs to correct it according to the deviation.

[0074] The third step: The palletizing robot palletizes the riveted plate, and then the production line starts to iterate from the first step again.

[0075] The above steps can be formally expressed as Figure 2 shown.

[0076] Before the specific implementation of the CPS modeling and analysis method based on the aspect-oriented gray generalized stochastic Petri net of the present invention, the relevant concept definitions and symbol explanations are as follows:

[0077] (1) The AGGSPN net system is defined as a binary tuple AGGSPN = (BN, ANS). Where: BN represents the basic net; ANS = {AN1, AN2,..., ANn} represents a finite set of aspect nets.

[0078] (2) Since it is necessary to describe the time consumption of each activity and the flow of resources and signals in the industrial CPS, in the AGGSPN net system, each subnet of BN and ANS is based on GSPN and combines the concept of color sets. Among them, the basic net BN corresponds to the core production process in the industrial CPS, and its definition is as follows:

[0079] In the AGGSPN net system, the basic net BN is defined as a six-tuple BN = (P, T, F, C, M0, λ), where:

[0080] P = {P0, P1, …, P n} represents a finite set of places;

[0081] T = {T0, T1, …, T n} represents a finite set of activity transitions. The activity transition T is divided into a time transition set T T = {T0, T1, …, T k} and an instantaneous transition set T I = {T k+1 , T k+2 , …, T n} of two subsets, that is, T = T T ∪T I , The occurrence delay of time transitions follows an exponential random distribution, and the occurrence delay of instantaneous transitions is zero;

[0082] F = {F0, F1, …, F n} represents a finite set of arcs connecting places and activity transitions;

[0083] C = {C0, C1, …, C n} represents a finite set of colors;

[0084] M0: P → {0, 1, 2, …} represents the initial marking of the Petri net, which is a mapping from P to non-negative integers. For M0(P i ) represents the number of tokens in the place P i under the marking M0;

[0085] λ = {λ0, λ1, …, λ n} represents the set of transition firing rates associated with the time transition set T T .

[0086] Although the basic net BN describes the overall operation process of industrial CPS, some of its steps need to be further subdivided (for example, the palletizing process needs to be subdivided) or additional processing needs to be performed before and after them (for example, the bent stiffeners need to be inspected before clinching). Therefore, the aspect net AN is introduced. i Represents the specific details or additional processing of a certain step in the core functional process of industrial CPS, and its definition is as follows:

[0087] The aspect net AN in the aspect net set ANS of 2-2 i Is defined as a triple AN i =(Pc, Ad, VN), where:

[0088] Pc represents AN i The entry point corresponding to BN, which can be a place or an activity transition;

[0089] Ad = {before, after, around} represents AN i The notification type corresponding to BN, before means AN i Needs to be executed before the entry point, after means AN i Needs to be executed after the entry point, and around means using AN i To replace the entry point;

[0090] VN represents a six-tuple VN = (P, T, F, C, M0, λ), and its definition is the same as that of BN.

[0091] 2-3 Each aspect net AN i Corresponds to a certain step in the basic net. To facilitate its subsequent weaving into the basic net, two additional auxiliary elements are added to this aspect net: the start element start and the end element end, which represent the start and end points when this aspect net is woven into the basic net.

[0092] In industrial CPS, the flow of resources (such as stiffeners in IPRPL-CPS) and signals (such as the inspection results of stiffener bending in IPRPL-CPS) is involved. Therefore, in AGGSPN, tokens are colored to distinguish their types, and the dynamic behavior in industrial CPS is described by the occurrence rules of activity transitions. Before defining the occurrence rules of transitions, the pre-set and post-set of elements are defined as follows:

[0093] Let PN = (P, T, F) be a Petri net. For any element x ∈ P ∪ T, if there exists an element y ∈ P ∪ T such that the arc (y, x) ∈ F, then all elements y that satisfy the condition are called the pre-set of x, denoted as ·x = {y|(y, x) ∈ F}. Similarly, all elements y that satisfy the condition (x, y) ∈ F are called the post-set of x, denoted as x· = {y|(x, y) ∈ F}.

[0094] 2 - 5 associates an expression W: F → C with each arc, representing the color corresponding to the arc. The color of the arc is the same as the color of the connected place, that is, W(P i ,T i ) / W(T i ,P i ) = C(P i ). Then the occurrence rule of transition T i is as follows:

[0095] For if the place P i contains a token and its color is W(P i ,T i ), then the occurrence condition of T i is satisfied. The token can be removed from P i , and T i is enabled. After waiting for T i to complete, for the color of the token is changed to W(T i ,P i ) and it is placed into the place P i .

[0096] (3) To improve the authenticity and accuracy of the established model, the grey system theory is introduced into the model, and interval grey numbers are used to represent the transition firing rate and the triggering probability of the random switch in the model. Interval grey numbers are the most common grey numbers, and their usage is relatively simple. They can indicate the fluctuation range of a certain variable in actual production. Their definition is as follows:

[0097] Interval grey numbers refer to grey numbers with both an upper bound and a lower bound, denoted as a ≤ b. Where a is the lower bound of the interval grey number , and b is the upper bound of the interval grey number . If a = b, then the interval grey number is a white number, that is, a white number is a special interval grey number.

[0098] Suppose there are two interval grey numbers After their mutual operation, the result is still an interval grey number. The operation rules of interval grey numbers are as follows, where during the operation, c ≠ 0, d ≠ 0;

[0099]

[0100] δ: The perturbed grey element in the ashing processing algorithm;

[0101] k j : The activity transition coefficient. The perturbed grey element coefficient of time transition is set as k1, and the perturbed grey element coefficient of the random switch is set as k2;

[0102] Pr = {Pr k+1 , Pr k+2 , …, Pr n} represents the triggering probability set of the instantaneous transition T i when it is a random switch;

[0103] τ = {τ0, τ1, …, τ k} represents the set of average implementation times corresponding to the time transition;

[0104] (7) Related concepts of Markov chain:

[0105] M i : A certain state in the reachable marking set S. If there is an instantaneous transition in the enabled state in this state, then M i is a disappearing state, otherwise it is an existing state;

[0106] S: The state space set of GSPN, with the number of elements K s , S can be divided into the existing state set TS and the disappearing state set VS, that is, S = TS ∪ VS, and the number of their elements are K T and K V , K S = K T + K V ;

[0107] The transition probability grey matrix of the isomorphic Markov chain (EMC) of GSPN;

[0108] It is composed of the transition probability matrix from the disappearing state to the disappearing state set and the transition probability matrix from the disappearing state to the existing state set;

[0109] It is composed of the transition probability matrix from the existing state to the disappearing state set and the transition probability matrix from the existing state to the existing state set;

[0110] The transition probability matrix of the EMC from the disappearing state set to the disappearing state set;

[0111] The transition probability matrix of the EMC from the disappearing state set to the existing state set;

[0112] The transition probability matrix of the EMC from the existing state set to the disappearing state set;

[0113] The transition probability matrix of the EMC from the existing state set to the existing state set;

[0114] After removing the disappearing states from the EMC, the EMC (Reduced EMC, REMC) that only contains existing states, is the transition probability matrix of the REMC;

[0115] The element therein represents the probability that the GSPN starts from a certain disappearing state r and reaches the existing state k for the first time after any number of steps;

[0116] represents a one-dimensional row vector, and the element represents the gray steady-state probability distribution of the existing state i in the REMC;

[0117] The average sojourn time of the existing states;

[0118] H i : The set of transitions that can be implemented in state i;

[0119] The cycle period of the existing states;

[0120] The cycle period of the system;

[0121] The gray steady-state probability of the model.

[0122] The specific implementation steps of the present invention are as follows:

[0123] Step (1) According to the specific situation of the modeled CPS, abstract the industrial CPS into a basic net and several aspect nets:

[0124] (1.1) Define a basic net and several aspect nets according to the production process of the industrial CPS. The basic net corresponds to the core production operation process;

[0125] (1.2) Since some of the steps need to be further subdivided or additional processing is required before and after them, define several aspect nets to represent the specific details or additional processing of a certain step in the core functional process of the industrial CPS;

[0126] Step (2) constructs an AGGSPN model, and the specific process is as follows:

[0127] (2.1) According to the structural process of IPRPL-CPS, it is divided into a basic network BN and two aspect networks AN1 and AN2; for the basic network, the states of industrial equipment on the industrial CPS production line in the overall operation process are used as the places of GSPN, the resources or signals on the industrial CPS production line in the overall operation process are used as the tokens of the places, and the dynamic behaviors of industrial equipment on the industrial CPS production line in the overall operation process are used as the activity transitions of GSPN; the relationship between the places and activity transitions in the overall operation process is used as the arcs; and different tokens are marked with different colors for distinction; (the places, activity transitions, tokens, and arcs in the above network are all elements of GSPN); for the aspect network, the specific states of industrial equipment on the industrial CPS production line in the subprocess are used as the places of GSPN, the resources or signals on the industrial CPS production line in the subprocess are used as the tokens of the places, and the specific dynamic behaviors of industrial equipment on the industrial CPS production line in the subprocess are used as the activity transitions of GSPN; the relationship between the places and activity transitions in the subprocess is used as the arcs; and different tokens are marked with different colors for distinction;

[0128] (2.2) In the AGGSPN model, each subnet is based on GSPN. The basic network BN is as Figure 3 shown, and the specific meanings of its places and transitions are shown in Table 1. The two aspect networks AN1 and AN2 are as Figure 4 shown, and the specific meanings of their places and transitions are shown in Table 2;

[0129] Table 1 Places and Transitions in BN and Their Meanings

[0130]

[0131]

[0132] Table 2 Places and Transitions in AN1 and AN2 and Their Meanings

[0133]

[0134] Step (3) proposes a weaving algorithm for performance analysis;

[0135] For any aspect network, its weaving process is mainly divided into two steps: first, connect the aspect network and the basic network together and delete the redundant auxiliary elements; then merge the same elements in the aspect network and the basic network. The redundant auxiliary elements of the aspect network include the start element start, the end element end, and the duplicate places. The specific steps are as follows:

[0136] (3.1) If the notification type of the aspect net is before, then insert the aspect net between the cut point and its predecessor set in the basic net, replacing its original input arc set; if the notification type of the aspect net is after, then insert the aspect net between the cut point and its successor set in the basic net, replacing its original output arc set; if the notification type of the aspect net is around, then replace the cut point in the basic net with the aspect net and connect it to its predecessor set and successor set. Then delete the auxiliary elements and the relevant arcs. For example Figure 6 Shown is the weaving process of an aspect net with a notification type of after and a cut point of T1.

[0137] (3.2) After initially connecting the aspect net and the basic net, if the aspect net and the basic net contain the same place P, then the input arcs and output arcs of place P in the aspect net need to be transformed into the input arcs and output arcs of place P in the basic net, and place P in the aspect net is deleted. For example Figure 3 the BN of Figure 4 (a) and the AN1 both contain place P1, then delete AN1.P1 and the arcs (AN1·T 10 , AN1.P1) and (AN1.P1, AN1·T9), and add new arcs (AN1·T 10 , BN.(BN1) and (BN1, AN1·T9).

[0138] After weaving the basic net of IPRPL-CPS and all aspect nets together according to this weaving algorithm, the woven net is as shown in Figure 7 Since the notification type of AN1 is before, it is woven before the cut point P6. The notification type of AN2 is aroind, so it serves as a replacement for the cut point T6. Therefore, the final woven net WN does not contain T6. Since BN and AN1 both contain P1, they are merged.

[0139] Figure 8 This is the reachable marking graph corresponding to the IPRPL-CPS woven net, where the dark color represents the existing state, the light color represents the disappearing state, the ellipse indicates that the state is an intermediate state, and the rectangle indicates that the state is an initial state.

[0140] Table 4 Reachable marking set S of WN

[0141]

[0142] In AGGSPN, if multiple timed transitions are enabled simultaneously, conflicts will occur, as shown in Figure 5 (a) where three timed transitions T1, T2, and T3 conflict. However, if there is an instantaneous transition present and enabled, the instantaneous transition will occur first, as shown in Figure 5As shown in (b) of [reference], T2 occurs prior to T1 and T3. Therefore, an instantaneous transition can be added before each time transition to eliminate conflicts. This structure is defined as a stochastic switch. As shown in Figure 5 (c) of [reference], T1, T2, and T3 form a stochastic switch with triggering probabilities Pr1, Pr2, and 1 - Pr1 - Pr2 respectively, which resolves the conflicts generated by T4, T5, and T6. This structure can be used in CPS modeling to describe the logical selection probabilities of some events. For example, in IPRPL-CPS, it is necessary to detect whether the bending of the reinforcing rib and the riveting of the sheet metal are qualified, and the qualification rate needs to be determined according to the actual production data statistics. Therefore, two sets of stochastic switches are defined in IPRPL-CPS, namely T8 and T9 in AN1 and T 12 and T 13 in AN2.

[0143] Step (4) introduces the grey system theory and uses the greying processing method to solve the transition firing rate and the triggering probability of the stochastic switch in the AGGSPN model woven in step (3). The specific approach is to transform the transition firing rate and the triggering probability of the model into interval functions by introducing a perturbation grey element δ, and a coefficient k is set for each transition. Specifically, the perturbation grey element coefficient of the time transition is set as k1, and the perturbation grey element coefficient of the stochastic switch is set as k2. The specific process is as follows:

[0144] (4.1) If the active transition T1 in AGGSPN is a time transition with a transition firing rate of λ1, its interval grey number representation is [λ1 - k1δ, λ1 + k1δ];

[0145] (4.2) Similarly, if the active transition T2 in AGGSPN is a stochastic switch with a triggering probability of [Pr(T 12 ), its interval grey number representation is [Pr(T 12 ) - k2δ, Pr(T 12 ) + k2δ];

[0146] By referring to the production data of IPRPL-CPS and measuring the time consumption of certain production steps on-site, the transition firing rate of the time transition and the triggering probability of the stochastic switch in the model can be obtained. Among them, the average implementation delay of the time transitions {T2, T3, T5, T 10 , T 14 , T 15} is: τ2 = 1, τ3 = 1 / 3, τ5 = 1 / 3, τ 10 = 1 / 4, τ 14 = 1 / 6, τ 15 = 1 / 6 (unit: min), that is, the corresponding transition firing rates are: λ2 = 1, λ3 = 3, λ5 = 3, λ 10 = 4, λ14 = 6, λ 15 = 6; The triggering probabilities of random switches T8 and T9 are Pr1 = 0.95 and Pr2 = 0.05 respectively. The triggering probabilities of random switches T 12 and T 13 are Pr3 = 0.98 and Pr4 = 0.02 respectively. Combining the actual production situation, the perturbation grey element coefficient k1 of the time transition firing rate is set to 10% of its own value, and the perturbation grey element coefficient k2 of the random switch triggering probability is set to 0.01. Then, these values are greyed out using this greying algorithm to obtain their interval grey number representations, as shown in Table 3.

[0147] Table 3 Interval Grey Number Representations of Time Transition Firing Rate and Random Switch Probability

[0148]

[0149]

[0150] Step (5) Solve the steady-state probability of the fuzzy GSPN using the Markov method. The specific process is as follows: Solve the grey steady-state probability of the AGGSPN using the Markov method; The specific process is as follows:

[0151] (5.1) Analyze and compile the net to obtain the reachable marking graph and the corresponding reachable marking set;

[0152] (5.2) According to the reachable marking graph, the transition probability grey matrix of the EMC can be obtained Its calculation formula is as follows:

[0153]

[0154] (5.3) Calculate the transition probability matrix U' of the REMC, which is represented by Equation (2);

[0155]

[0156] Taking IPRPL-CPS as an example, the transition probability matrix of the REMC isomorphic to this GSPN can be obtained

[0157]

[0158] (5.4) Solve the grey stable probability of the actual existence state of the REMC through the linear equation system of Equation (3);

[0159]

[0160] represents a one-dimensional row vector, and the element represents the grey steady-state probability of the actual existence state of the REMC;

[0161] (5.5) Calculate the average residence time of each actual existence state through formula (4).

[0162]

[0163] (5.6) From the average residence time of each state and the corresponding grey stability probability the cycle period of the system can be obtained. The calculation method is formula (5);

[0164]

[0165] (5.7) The cycle period of the actual existence state can be obtained through and calculation;

[0166]

[0167] (5.8) The grey steady state probability of the model can finally be expressed by formula (7), which is the ratio of the residence time of the actual existence state to its cycle period;

[0168]

[0169] After obtaining the transition state probability matrix, the steady state probability of GSPN can be obtained through the calculation method of formulas (3)-(7). It is represented by a composite grey number, and then according to the operation rules of interval grey numbers, the interval grey number representation about the perturbation grey element δ can be obtained. Since the operation between interval grey numbers will increase the uncertainty of the result, as δ continuously increases, the upper bound or lower bound of

[0170] Step (6) constructs a mathematical programming model based on the obtained steady state probability and solves the final accurate numerical result. The specific process is as follows:

[0171] (6.1) Under the condition of meeting the actual production conditions (the transition firing rate is greater than 0, and the random switch triggering probability is between 0 and 1) and all are feasible solutions, find the maximum value of δ;

[0172]

[0173]

[0174] (6.2) Substitute the δ obtained from formula (8) into and then Final interval grey number representation;

[0175] Through the above steps, the interval grey number representation with respect to the perturbation grey element δ can be obtained, and its geometric representation is as Figure 9 shown.

[0176] Through Figure 10 it can be found that only the upper bound of then the rest of are all feasible solutions. Then, according to the mathematical programming model in step 4, the maximum value of δ can be obtained as 1.473. Substituting it into each to obtain its interval grey number representation. Substituting δ = 0 into each to obtain its whitenized value. The final interval grey number representation and its whitenized value of the grey steady-state probability are shown in Table 5.

[0177] Table 5 Grey steady-state probability Interval grey number representation and its whitenized value

[0178]

[0179] (6.3) According to the specific CPS scenario, use the obtained grey steady-state probability to analyze the performance indicators of this industrial CPS.

[0180] After obtaining the grey steady-state probability of AGGSPN, various indicators of the modeled system can be analyzed, such as equipment utilization rate, transition throughput rate, productivity, etc.

[0181] Taking the equipment utilization rate as an example. In AGGSPN, some places P i refer to whether a specific device is available. When there is a token in P i , it means the device is idle. If there is no token in P i , it means the device is working. Therefore, the utilization rate of this device can be expressed as the sum of the steady-state probabilities of all M(P i ) = 0.

[0182] In IPRPL-CPS, the utilization rate of the bending machine is The utilization rate of the film laminating machine is The utilization rate of the riveting machine is The utilization rate of the palletizing robot is The utilization rates of each device are as Figure 10As shown. It can be seen from the figure that the utilization rate of the palletizing robot is the lowest, the utilization rate of the bending machine is the highest, and the utilization rates of the film laminating machine and the riveting press are almost the same, but both are much lower than that of the bending machine. The utilization rate of the bending machine is the highest because it takes the most time to bend and calibrate the stiffeners. If the high utilization rate is maintained for a long time, machine failures may occur due to excessive load.

Claims

1. A CPS modeling and analysis method based on the oriented grey generalized stochastic Petri net, characterized in that It includes the following steps: Step (1): According to the specific situation of the industrial CPS to be modeled, abstract the industrial CPS into a basic net and several aspect nets; Step (2): The basic net and the aspect nets are based on GSPN, and the concept of color set is introduced, that is, colors are assigned to the elements therein, so as to obtain the AGGSPN model; Step (3): On the premise of ensuring model consistency, weave the basic net and all aspect nets into a woven net; Step (4): Introduce the grey system theory, and use the greying processing method to solve the transition firing rate and the triggering probability of the random switch in the AGGSPN model after weaving in Step (3); specifically, by introducing a perturbation grey element δ, the transition firing rate and the triggering probability in the AGGSPN model are transformed into interval grey numbers, and a coefficient k is set for each transition, that is, the perturbation grey element coefficient of the time transition is set as k1, and the perturbation grey element coefficient of the random switch is set as k2; Step (5): Use the Markov method to solve the grey steady-state probability of AGGSPN; the specific process is as follows: (5.1) Analysis step (3) Weave the net, obtain the reachability identification graph and the corresponding reachability identification set S. S is divided into the actual existence state set TS and the disappearance state set VS, that is, S = TS ∪ VS. The number of its elements is K T and K V , K S = K T + K V ; (5.2) Obtain the transition probability gray matrix of the embedded Markov chain EMC according to the reachability identification graph Its calculation formula is as follows: Among them It is composed of a transition probability matrix for transitioning from a disappearing state to a set of disappearing states and a transition probability matrix for transitioning from a disappearing state to a set of existent states constitute; A transition probability matrix for transitioning from an existing state to a set of disappearing states and a transition probability matrix for transitioning from an existing state to a set of existing states constitute; Denote the transition probability matrix of the EMC from the disappearing state set to the disappearing state set; Indicates the transition probability matrix of the EMC from the disappearing state set to the existing state set; Indicates the transition probability matrix of the EMC from the set of existent states to the set of disappearing states; It represents the transition probability matrix of the EMC from the set of actual existence states to the set of actual existence states; (5.3) Remove the disappearing state from the EMC, leaving only the existent state, to obtain the REMC transition state matrix It is expressed by Equation (2); wherein represents a set of elements , represents the probability that the GSPN starts from a given vanishing state r and reaches the existent state k for the first time after any number of steps (5.4) Solve the grey stable probability of the actual existence state of REMC through the linear equations in Equation (3); wherein represents a one-dimensional row vector, and the element represents the gray steady-state probability of the real storage state of REMC; (5.5) Calculate the average residence time of each actual existence state through formula (4). (5.6) The average residence time of each state and the corresponding gray stability probability are used to obtain the cycle period of the system The calculation method is shown in Equation (5); (5.7) Cycle period of the actual existence state Calculated by and ; (5.8) Grey steady-state probability of the model It is expressed by Equation (7) and is the ratio of the residence time of the actual existence state to its cycle period; Obtained by the GSPN steady-state probability calculation method of formulas (3)-(7) represented by the composite grey number, and then according to the interval grey number operation rules, we get the interval grey number representation of the perturbation grey element δ, which is the grey steady-state probability of AGGSPN; Step (6): Construct a mathematical programming model according to the grey steady-state probability of AGGSPN, and solve the final interval grey number representation; finally, analyze the performance indicators of this industrial CPS by using the final interval grey number representation.

2. The CPS modeling and analysis method based on the oriented grey generalized stochastic Petri net according to claim 1, characterized in that Specifically, in Step (1), the total operation process of the industrial CPS is abstracted into a basic net, and then some sub-processes are subdivided into several aspect nets.

3. The CPS modeling and analysis method based on the oriented grey generalized stochastic Petri net according to claim 1, characterized in that Specifically, Step (2) is as follows: For the basic net, take the state of industrial equipment on the industrial CPS production line in the total operation process as the places of GSPN, take the resources or signals on the industrial CPS production line in the total operation process as the tokens of the places, and take the dynamic behavior of industrial equipment on the industrial CPS production line in the total operation process as the active transitions of GSPN; take the relationship between the places and the active transitions in the total operation process as the arcs; and different tokens are marked with different colors for distinction; For the aspect nets, take the specific state of industrial equipment on the industrial CPS production line in the sub-process as the places of GSPN, take the resources or signals on the industrial CPS production line in the sub-process as the tokens of the places, and take the specific dynamic behavior of industrial equipment on the industrial CPS production line in the sub-process as the active transitions of GSPN; take the relationship between the places and the active transitions in the sub-process as the arcs; and different tokens are marked with different colors for distinction.

4. The CPS modeling and analysis method based on the oriented grey generalized stochastic Petri net according to claim 3, characterized in that Specifically, Step (3) is as follows: (3.1) For any aspect net, first connect it to the basic net according to the notification type of this aspect net, and delete the redundant auxiliary elements and the arcs associated with the redundant auxiliary elements; The redundant auxiliary elements of the aspect net include the start element start, the end element end and the repeated places; (3.2) If the aspect net and the basic net contain the same place P, the input arcs and output arcs of the place P in the aspect net need to be transformed into the input arcs and output arcs of the place P in the basic net, and the place P in the aspect net is deleted.

5. The CPS modeling and analysis method based on the oriented grey generalized stochastic Petri net according to claim 1, characterized in that Specifically, Step (4) is as follows: (4.1) If the active transition R1 in AGGSPN is a time transition and the transition firing rate is λ1, its interval grey number is expressed as [λ1 - k1δ, λ1 + k1δ]; (4.2) Similarly, if the active transition R2 in AGGSPN is a random switch and the triggering probability is Pr(T2), its interval grey number is expressed as [Pr(T2) - k2δ, Pr(T2) + k2δ].

6. The CPS modeling and analysis method based on the oriented grey generalized stochastic Petri net as claimed in claim 5, wherein (6.3) According to the specific CPS scenario, analyze the performance indicators of the industrial CPS by using the obtained final interval grey number representation. (6.1) Under the condition of meeting the actual production conditions and all being feasible solutions, find the maximum value of δ; (6.2) Substitute the δ obtained from Equation (8) into to obtain the final interval grey number representation; (6.3) According to the specific CPS scenario, use the obtained final interval grey number representation to analyze the performance indicators of the industrial CPS.

7. The CPS modeling and analysis method based on the direction-oriented grey generalized stochastic Petri net according to claim 6, characterized in that The actual production condition is that the transition firing rate is greater than 0, and the triggering probability of the random switch is between 0 and 1.

8. An electronic device, characterized in that, It includes a processor and a memory. The memory stores machine-executable instructions that can be executed by the processor, and the processor executes the machine-executable instructions to implement the method according to any one of claims 1-7.

9. A machine-readable storage medium, characterized in that, The machine-readable storage medium stores machine-executable instructions. When the machine-executable instructions are called and executed by the processor, the machine-executable instructions cause the processor to implement the method according to any one of claims 1-7.

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