CN115499153B - Optimal control method for electric CPS worm viruses based on benign worm interaction - Google Patents
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Abstract
The invention discloses an optimal control method of electric CPS worm viruses based on benign worm interaction, which comprises the following steps: s1, constructing a PLC-PC double-layer coupling network structure of benign worms; s2, constructing a PLC-PC node state transition diagram of benign worms; and S3, solving an optimal control strategy based on benign worm interaction. Modeling an electric CPS network into which benign worms are introduced according to infectious disease dynamics, and obtaining differential equations of all nodes with the benign worms; constructing a target cost function according to the introduction cost of benign worms, constructing a Hamiltonian function by utilizing an optimal control theory, solving a covariate differential equation set, a cross-sectional condition and an optimization condition by the Hamiltonian function, and finally obtaining an optimal control quantity according to the optimization condition.
Description
Technical Field
The invention relates to the technical field of power system networks, in particular to an optimal control method for electric CPS worm viruses based on benign worm interaction.
Background
With the wide application of the internet in various fields, the power system also becomes an interconnected network system, and the dependence on the information system is deeper and deeper, and the traditional power system gradually evolves into a power information physical fusion system (cyber PHYSICAL SYSTEM, CPS) of deep coupling of the power system and the information system. The safe and stable operation of the power CPS has close relation with the daily life and social stability of people, and a large amount of information technology and equipment are applied to the interconnection of the power grid in order to monitor the power CPS network and ensure the stable and safe operation of the power CPS network.
While the informatization degree of the electric power system is continuously improved, the information security problem becomes a non-negligible potential safety hazard, and the electric power CPS network may suffer various network attacks, such as denial of service attack, load redistribution attack, false data injection attack, worm virus propagation, and electric power information physical cooperative attack combining worm virus propagation and false data injection attack. Among the above network attacks, the most affecting power CPS networks is the worm virus. The virus is ingenious in design, can permeate and attack a network system, can avoid a security detection mechanism, can be transmitted between a programmable controller and a computer, and can cause extremely serious loss to an electric CPS network.
The main devices of the power CPS network for information exchange are computers and programmable controllers, which are always unidirectional when transmitting information. Hackers can use the internet to load computer worm viruses into computers to invade industrial computers, thereby causing the transmission of worm viruses among computer groups, benign worms can remove the worm viruses from being flooded in the network to a certain extent although the worm worms are worms, the benign worms can also be introduced into the network through being loaded into the computers, and the benign worms can interact with the worm viruses after being loaded into the computers, so that the transmission of the worm viruses is inhibited.
Therefore, how to design a method for introducing benign worms into a power CPS network, so that the worm viruses spread in the power CPS network is a technical problem that needs to be solved by those skilled in the art.
Disclosure of Invention
The invention provides an optimal control method of electric CPS worm virus based on benign worm interaction, which models an electric CPS network introduced with benign worm according to infectious disease dynamics to obtain differential equations of all nodes with benign worm; constructing a target cost function according to the introduction cost of benign worms, constructing a Hamiltonian function by utilizing an optimal control theory, solving a collaborative variable differential equation set, a cross-sectional condition and an optimization condition by the Hamiltonian function, and finally solving an optimal control quantity according to the optimization condition to solve the problems.
The invention provides the following technical scheme:
an optimal control method of electric CPS worm viruses based on benign worm interaction comprises the following steps:
S1, constructing a PLC-PC double-layer coupling network structure of benign worms;
s2, constructing a PLC-PC node state transition diagram of benign worms;
And S3, solving an optimal control strategy based on benign worm interaction.
Preferably, in the step S1, the PLC-PC dual-layer coupling network structure includes a PC network and a PLC network, the PC network is denoted by a, the state a of the node in the PC network includes susceptibility, infection, quarantine, immunization and benign worm, the node in the susceptibility state is denoted by S A, the number at time t is denoted by S A (t), the node in the infection state is denoted by I A, The number at time t is denoted by I A (t), the isolated state node is denoted by Q A, the number at time t is denoted by Q A (t), the immune state node is denoted by R A, The number at time t is denoted by R A (t), the benign worm state node is denoted by a A, and the number at time t is denoted by a A (t); A PLC network is denoted by B, in which the state B of the node includes susceptible, infected, immunized and benign worms, the susceptible state node is denoted by S B, the number at time t is denoted by S B (t), the infected state node is denoted by I B, The number at time t is denoted by I B (t), the immune status node by R B, the number at time t by R B (t), the benign worm node by a B, the number at time t is denoted by a B (t).
More preferably, the total number of network nodes in the PC network is represented by N A, and the total number of nodes with the degree (i, j) in the PC network is represented byA representation; the total number of network nodes in the PLC network is represented by N B, and the total number of nodes with the degree of (k, l) in the PLC network is represented byAnd (3) representing.
More preferably, at any time t, the total number of the state nodes with the degree (i, j) in the PC network and the total number of the state nodes with the degree (k, l) in the PLC network are kept stable, the total number of the state nodes with the degree (i, j) is equal to the sum of the number of the state nodes, and the total number of the state nodes with the degree (k, l) is equal to the sum of the number of the state nodes, and the following formula is provided:
wherein, A PC network susceptibility status node with a representativeness of (i, j),A PC network infection status node of the representativeness (i, j),The PC network isolation status node represented as degree (i, j),A PC network immunity node with the representativeness of (i, j),A PC network benign worm state node of the degree of representativeness (i, j); a PLC network susceptibility status node with the representativeness of (k, l), A PLC network infection state node with the representativeness of (k, l),A PLC network immune status node with the representativeness of (k, l),A PLC network benign worm state node of the degree of representativeness (k, l).
Preferably, in the step S2, a state transition diagram of the PLC-PC node of the benign worm is constructed according to the PLC-PC double-layer coupling network structure into which the benign worm is introduced.
More preferably, in the PLC-PC node state transition diagram of the benign worm, the proportion of the susceptible state nodes born by the PC network in all the birth nodes is b, the proportion of the immune state nodes born by the PC network in all the birth nodes is 1-b-sigma 1, the proportion of the continuously introduced benign worm state nodes in all the birth nodes is sigma 1, the omega represents the benign worm introduction rate, and the numerical value is the same as that of sigma 1; in a PLC node network, only susceptible state nodes are born.
More preferably, in S3, the following objective cost function J (σ 1) is established:
Wherein c 1 represents a cost parameter for conducting benign worm state nodes introduced into the PC network; Representing the cost penalty spent introducing benign worm-state nodes of the PC network, the objective cost function J (σ 1) represents the cost penalty spent in controlling the benign worm-state node introduction rate σ 1 (t) of the PC network at time t, minimizing the number of worm-infected state nodes of the PC network and the PLC network when the optimal control reaches the terminal time t f, and at consecutive intervals [0, t f ].
More preferably, cross-sectional conditionsThe following are provided:
the beneficial effects of the invention are as follows:
According to the invention, the benign worm is introduced into the power CPS network, the change condition of the power CPS network after the benign worm is introduced is analyzed, and the benign worm is introduced into the power CPS network in a mode of loading the benign worm into a computer, so that the interaction of the worm virus can occur, and a worm virus infection node is converted into a benign worm node, thereby inhibiting the transmission of the worm virus in the power CPS network, and ensuring the information security and stable operation of the power CPS network.
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The invention will be further described with reference to the accompanying drawings, in which embodiments do not constitute any limitation of the invention, and other drawings can be obtained by one of ordinary skill in the art without inventive effort from the following drawings.
FIG. 1 is a flowchart of an optimal control method for a power CPS worm virus based on benign worm interactions of the present invention;
FIG. 2 is a block diagram of a PLC-PC dual-layer coupling network incorporating a benign worm of the present invention;
FIG. 3 is a state network diagram of a PLC-PC node incorporating a benign worm of the present invention;
fig. 4 is a network diagram of the state of another PLC-PC node of the present invention incorporating a benign worm.
Detailed Description
An optimal control method for power CPS worm viruses based on benign worm interactions is described in further detail below in connection with specific embodiments, which are for comparison and explanation purposes only, and the present invention is not limited to these embodiments.
Examples:
Referring to fig. 1-4, the power CPS worm virus optimal control method based on benign worm interaction provided by the embodiment of the invention comprises the following steps:
S1, constructing a PLC-PC double-layer coupling network structure of benign worms;
the individual devices of the computer and programmable controller can be considered as single nodes and classified into susceptible state nodes, infected state nodes, immune state nodes, benign worm state nodes, according to the degree of infection of the devices by worm viruses and benign worms. The infection state node refers to a node which is attacked by worm viruses, and the infection node randomly transmits information with worm viruses outwards; susceptible state nodes refer to nodes that have not installed the latest immune patch and are susceptible to worm virus infection; the immune state node is a node which is provided with the latest immune patch and completes virus searching and killing, the node is used for immunizing worm viruses within a certain time, and when the immune patch is invalid, the immune state node becomes a susceptible node; benign worm state nodes refer to nodes suffering from benign worm infection, the infection is benign and beneficial, the benign worm state nodes can carry out benign infection on susceptible state nodes and infected state nodes, and the susceptible state nodes and the infected state nodes suffering from benign worm infection are not infected by malignant worm viruses any more and only become immune state nodes after patching; the infected nodes isolated in the computer are called isolated state nodes, and the nodes do not send out worm virus attached information.
The relevant parameters of the network nodes are shown in table 1 below, the computer network and the programmable controller network being represented by a PC network a and a PLC network B, respectively.
TABLE 1 definition of parameters
Based on the definitions described in table 1, a two-layer coupled network structure as shown in fig. 2 can be constructed. (solid lines indicate normal transmission of nodes, dashed lines indicate limited transmission of nodes)
The degree of one PC node, denoted by (i, j), means that one PC node is connected to i other PC nodes and j PLC nodes; the degree of one PLC node, denoted by (k, l), means that one PLC node is connected to k other PLC nodes and to l PC nodes.
Defining the maximum value of the i node degree of the PC node as n 11; the maximum value of the j node degree of the PC node is n 12, the maximum value of the k node degree of the PLC node is n 21, and the maximum value of the l node degree of the PLC node is n 22.
The maximum value of the node degree is defined as follows:
Secondly, P A (i, j) and P B (i, j) are used for respectively representing vertex connectivity distribution of the PC network A and the PLC network B; the edge connectivity distributions of the PC network a and the PLC network B are defined as P A(i,·),PA (·, j) and P B(k,·),PB (·, l), respectively, and are specifically defined as follows:
meanwhile, the second moment defining the node's average degree and node degree is as follows:
It is assumed that at an arbitrary time t, the total number of PC nodes of degree (i, j) and the total number of PLC nodes of degree (k, l) remain stable, and the total number of nodes of degree (i, j) is equal to the sum of the number of nodes of each state, and the total number of nodes of degree (k, l) is equal to the sum of the number of nodes of each state. The formula is as follows:
wherein, A PC network susceptibility status node with a representativeness of (i, j),A PC network infection status node of the representativeness (i, j),The PC network isolation status node represented as degree (i, j),A PC network immunity node with the representativeness of (i, j),A PC network benign worm state node of the degree of representativeness (i, j); a PLC network susceptibility status node with the representativeness of (k, l), A PLC network infection state node with the representativeness of (k, l),A PLC network immune status node with the representativeness of (k, l),A PLC network benign worm state node of the degree of representativeness (k, l).
S2, constructing a PLC-PC node state transition diagram of benign worms;
Based on the PLC-PC double-layer coupling network structure constructed by S1 and introducing benign worm, the structure can be further simplified into a node state transition diagram, and as a computer (PC) and a programmable controller (PLC) always have unidirectional directions when transmitting information, the nodes have unidirectional directions when in transition, and the transmission of worm viruses is unidirectional. Worm viruses loaded on computers can not only spread among computers, but also among computers and programmable controllers. The process of spreading the worm virus of the programmable controller can also do the above-mentioned process.
Taking a PC node as an example, if a PC susceptible state node contacts a PC infected state node, the PC susceptible state node is converted into the PC infected state node in a proportion of beta 11; if one PC susceptible state node contacts one PLC infected state node, the PC susceptible state node is converted into the PC infected state node in a proportion of beta 12; if a PC susceptible state node touches a PC benign worm state node, the node will be converted to a PC benign worm state node in a proportion of alpha 11; if a PC infection state node touches a PC benign worm state node, the node will be converted to a PC benign worm state node in the proportion of c 11; if a PLC susceptibility status node touches a PC benign worm status node, the node will be converted into the PLC benign worm status node in the proportion of alpha 21; if a PLC infection state node contacts a PC benign worm state node, the node is converted into the PLC benign worm state node in the proportion of c 21; it should be noted here that there is a certain time delay τ when the PC benign worm state node interacts with the PLC infection state node. The PC infection state node can be immunized by installing a virus killing program to kill viruses and installing an immune patch, and is converted into an immunization state node, and the probability of successfully obtaining immunity is gamma 1; the PC benign worm state node can be converted into an immune state node by a patching mode, and the probability of successfully obtaining immunity is gamma 3. The conversion process of the PLC node is the same as that of the PC node.
Assuming that the total number of nodes of the model remains stable, μ 1 is the ratio of each node entering and leaving the network in the PC network, and μ 2 is the ratio of each node entering and leaving the network in the PLC network.
Assuming that the proportion of PC immune nodes with no immune function is η 1, the proportion of PLC immune state nodes with no immune function is η 2, and the proportion of PC infection state nodes isolated is δ 1. Meanwhile, the proportion of PC isolation nodes which are recovered to the immune state nodes after virus searching and killing and immune patch mounting is omega 1. In the PC node network, the proportion of the susceptible state nodes born by the PC network in all the birth nodes is b, the proportion of the immune state nodes born by the PC network in all the birth nodes is 1-b-sigma 1, the proportion of the benign worm state nodes continuously introduced in all the birth nodes is sigma 1, the benign worm introduction rate is represented by omega, and the numerical value of the benign worm introduction rate is the same as that of sigma 1; in a PLC node network, only susceptible state nodes are born.
The relevant conversion ratio definitions are shown in table 2.
Table 2 conversion ratio definition
Θ xy (t) is defined as the probability that a susceptible node has an adjacent infection state node, (x=1, 2; y=1, 2, where "1" represents PC network a and "2" represents PLC network B).
Wherein, the probability that PC susceptible state node and PC infected state node are adjacent is:
The probability that the PC susceptible state node is adjacent to the PLC infected state node is:
the probability that the PLC susceptibility state node is adjacent to the PC infection state node is:
The probability that the PLC susceptibility state node is adjacent to the PLC infection state node is:
Definition of the definition Probability of having neighboring benign worm nodes for susceptible/infected state nodes, (x=1, 2; y=1, 2, where "1" represents PC network a and "2" represents PLC network B).
Wherein the probability of a PC susceptible state node/infected state node being adjacent to a PC benign worm node is:
the probability of a PC susceptibility status node/infection status node being adjacent to a PLC benign worm node is:
the probability of a PLC susceptibility status node/infection status node being adjacent to a PC benign worm node is:
the probability that a PLC susceptible state node/infected state node is adjacent to a PLC benign worm node is:
After defining the above relevant definition, for the convenience of observation, the interaction process of the PC benign worm and the interaction process of the PLC benign worm are respectively represented by the following PLC-PC node state transition diagrams of fig. 3 and 4.
In summary, the PLC-PC benign worm and worm virus cross-propagation model can be represented by the following differential equation set:
s3, solving an optimal control strategy based on benign worm interaction;
Loading the benign worms onto the computer network can inhibit the spread of worm viruses in the power CPS network, but the benign worms also have life cycles, and the PC benign worms are continuously introduced into the power CPS network, so that a certain cost is required. Therefore, taking the cost of introducing the PC benign worm into consideration, the PC benign worm introduction rate sigma 1 is selected as a control variable, so that the cost of introducing the PC benign worm is minimized.
The following objective cost function J (σ 1) is established:
Wherein c 1 represents a cost parameter for conducting benign worm state nodes introduced into the PC network; Representing the cost penalty spent introducing benign worm-state nodes of the PC network, the objective cost function J (σ 1) represents the cost penalty spent in controlling the benign worm-state node introduction rate σ 1 (t) of the PC network at time t, minimizing the number of worm-infected state nodes of the PC network and the PLC network when the optimal control reaches the terminal time t f, and at consecutive intervals [0, t f ].
Based on the principle of maximum value, a corresponding Hamiltonian H is constructed as follows:
Wherein λ1(t)、λ2(t)、λ3(t)、λ4(t)、λ5(t)、λ6(t)、λ7(t)、λ8(t) and lambda 9 (t) are covariates, and a differential equation set of covariates can be obtained according to the maximum principle, as follows:
The cross-sectional conditions are as follows:
according to the Pontrian maximum principle, the optimization conditions are calculated as follows:
from the optimization conditions, the method can be solved:
therefore, the optimal control solution is finally obtained as follows:
the power CPS worm virus optimal control method based on benign worm interaction provided by the embodiment of the invention is mainly characterized in that benign worms are introduced into the power CPS network, the change condition of the power CPS network after the benign worms are introduced is analyzed, and a worm virus optimal control strategy of the power CPS based on benign worm interaction is provided on the basis. Modeling an electric CPS network into which benign worms are introduced according to infectious disease dynamics, and obtaining differential equations of all nodes with the benign worms; constructing a target cost function according to the introduction cost of benign worms, constructing a Hamiltonian function by utilizing an optimal control theory, solving a covariate differential equation set, a cross-sectional condition and an optimization condition by the Hamiltonian function, and finally obtaining an optimal control quantity according to the optimization condition.
Finally, it should be noted that the above embodiments are only for illustrating the technical solution of the present invention, and not for limiting the scope of the present invention, and although the present invention has been described in detail with reference to the preferred embodiments, it should be understood by those skilled in the art that modifications or equivalent substitutions can be made to the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention.
Claims (7)
1. An optimal control method of electric CPS worm viruses based on benign worm interaction is characterized by comprising the following steps:
S1, constructing a PLC-PC double-layer coupling network structure of benign worms;
S2, constructing a PLC-PC node state transition diagram of the benign worm according to the PLC-PC double-layer coupling network structure;
S3, defining theta xy (t) as probability that the susceptible node has adjacent infection state nodes according to the PLC-PC node state transition diagram, and defining Probability of having adjacent benign worm nodes for susceptible state nodes/infected state nodes;
Wherein x epsilon {1,2}, y epsilon {1,2}, θ 11 (t) represents the probability that the PC susceptible state node is adjacent to the PC infected state node at time t, θ 12 (t) represents the probability that the PC susceptible state node is adjacent to the PLC infected state node at time t, θ 21 (t) represents the probability that the PLC susceptible state node is adjacent to the PC infected state node at time t, and θ 22 (t) represents the probability that the PLC susceptible state node is adjacent to the PLC infected state node at time t;
Representing the interaction process of the PC benign worm and the interaction process of the PLC benign worm as a differential equation set;
And solving an optimal control strategy based on benign worm interaction according to the differential equation set.
2. The optimal control method for a power CPS worm virus based on benign worm interactions as claimed in claim 1, wherein in S1, the PLC-PC dual-layer coupling network structure comprises a PC network and a PLC network, the PC network is denoted by a, the state a of the node in the PC network comprises susceptible, infected, quarantined, immunized and benign worms, the node in the susceptible state is denoted by S A, the number at time t is denoted by S A (t), The infection state node is denoted by I A, the number at time t is denoted by I A (t), the isolation state node is denoted by Q A, the number at time t is denoted by Q A (t), Immune status node is denoted by R A, number at time t is denoted by R A (t), benign worm status node is denoted by a A, number at time t is denoted by a A (t); A PLC network is denoted by B, in which the state B of the node includes susceptible, infected, immunized and benign worms, the susceptible state node is denoted by S B, the number at time t is denoted by S B (t), the infected state node is denoted by I B, The number at time t is denoted by I B (t), the immune status node by R B, the number at time t by R B (t), the benign worm node by a B, the number at time t is denoted by a B (t).
3. The optimal control method for a power CPS worm virus based on benign worm interactions as claimed in claim 2, wherein the total number of network nodes in said PC network is denoted by N A, and the total number of nodes of degree (i, j) in said PC network is denoted byA representation; the total number of network nodes in the PLC network is represented by N B, and the total number of nodes with the degree of (k, l) in the PLC network is represented byAnd (3) representing.
4. The optimal control method for the power CPS worm virus based on benign worm interaction according to claim 3, wherein at any time t, the total number of state nodes with the degree of (i, j) in the PC network and the total number of state nodes with the degree of (k, l) in the PLC network are kept stable, the total number of state nodes with the degree of (i, j) is equal to the sum of the number of state nodes, and the total number of state nodes with the degree of (k, l) is equal to the sum of the number of state nodes, and the following formula is adopted:
wherein, The number of PC network susceptibility status nodes of representativeness (i, j) at time t,The number of PC network infection state nodes of representativeness (i, j) at time t,The number of PC network isolated state nodes represented by the degree i, j at time t,The number of PC network immune nodes with the representativeness of i, j at the time t,The number of the PC network benign worm state nodes with the representativeness of i and j at the time t; The number of the PLC network susceptibility status nodes with the representativeness of k, l at the time t, The number of PLC network infection status nodes of representativeness k, l at time t,The number of the PLC network immune status nodes with the representativeness of k, l at the time t,The number of benign worm state nodes of the PLC network with a degree of representativeness k, l at time t.
5. The optimal control method for electric CPS worm virus based on benign worm interaction as claimed in claim 4, wherein in said PLC-PC node state transition diagram of benign worm, the proportion of PC network born susceptible state nodes in all birth nodes is b, the proportion of born immune state nodes in all birth nodes is 1-b- σ 1, the proportion of continuously introduced benign worm state nodes in all birth nodes is σ 1, and the benign worm introduction rate is represented by Ω, and the numerical value is the same as that of σ 1; in a PLC node network, only susceptible state nodes are born.
6. The optimal control method for a power CPS worm virus based on benign worm interactions as recited in claim 5, wherein in S3, the following objective cost function J (σ 1) is established:
Wherein c 1 represents a cost parameter for conducting benign worm state nodes introduced into the PC network; representing the cost penalty spent introducing benign worm-state nodes of the PC network;
σ 1 represents the proportion of continuously introduced benign worm state nodes in all birth nodes;
σ 1 (t) represents the benign worm state node introduction rate of the PC network at time t;
The objective cost function J (σ 1) represents the introduction rate σ 1 (t) of the benign worm state nodes of the PC network by controlling the time t, so that when the optimal control reaches the terminal time t f, the number of worm-infected state nodes of the PC network and the PLC network is at a minimum and the cost spent in the continuous interval [0, t f ] is at a minimum.
7. The optimal control method for a power CPS worm virus based on benign worm interactions as recited in claim 6, wherein the cross-sectional conditionThe following are provided:
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| CN109246106A (en) * | 2018-09-17 | 2019-01-18 | 平安科技（深圳）有限公司 | Worm-type virus program suppressing method, device and the equipment of wireless sensor network |
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