CN114839946B - Network control system stabilizing method under replay attack based on switching system - Google Patents
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Abstract
The invention discloses a network control system stabilizing method under replay attack based on a switching system, which comprises the following steps: 1) Converting the network control system subjected to replay attack into a variable period sampling network control system, and establishing a continuous time state space model of the network control system under the replay attack; 2) Taking the time interval from the successful completion of the data transmission of the system to the successful completion of the data transmission of the next time as a sampling period, and converting the stability problem of the continuous time state space model of the network control system under replay attack into the stability problem of the discrete time switching system of the limited subsystem according to a variable period discrete method; 3) And determining the index stability and the system index decay rate of the switching system by utilizing the Lyapunov function method and the average residence time. The invention converts the random variable introduced by network attack into the system to switch among limited subsystems, thereby avoiding directly solving the system containing the random variable, reducing the calculated amount and widening the application of the idea of the switching system.
Description
Technical Field
The invention relates to the technical field of network attack and network control system stability, in particular to a network control system stability method under replay attack based on a switching system.
Background
In recent years, network security has received increasing attention, and data transmitted through a shared network is easily utilized by attackers. Because the sensors, the controllers and the executors in the network control system are connected through the shared network, the network attack can maliciously damage the stability of the network control system. Replay attack is used as a common attack mode of hackers, and the purpose of cheating the system and destroying the stability of the system is achieved by repeatedly sending a packet received by a host, so that the security of the network control system is seriously endangered.
Network attacks are often stochastic in nature, and in the approach of control system modeling, network attacks are often modeled as a random probability distribution or markov process. The main steps are as follows: 1. based on a discrete random control system stabilization method, the method constructs a random system Liapunov function, and the system stability condition is expected to be analyzed by solving the Liapunov function, but the method has complex model and large calculated amount; 2. based on a Markov process analysis method, the method researches a system Markov chain so as to judge the stability of the PDMP, but the obtained system stability condition has certain limitation.
Disclosure of Invention
The invention aims to overcome the defects and shortcomings of the prior art, and provides a network control system stabilizing method under replay attack based on a switching system, which converts random variables introduced by network attack into systems to be switched among limited subsystems, thereby avoiding directly solving the system containing the random variables, reducing the calculated amount and widening the application of the switching system idea.
In order to achieve the above purpose, the technical scheme provided by the invention is as follows: the network control system stabilizing method under replay attack based on the switching system comprises the following steps:
1) Because the sensor adopts a time driving mode, the controller and the executor adopt an event driving mode, according to the principle that the executor maintains the input of the last sampling period when the system suffers replay attack in one sensor period, the network control system suffering replay attack is converted into a variable period sampling network control system, and a continuous time state space model of the network control system under replay attack is established;
2) Taking the time interval from the successful completion of the data transmission of the system to the successful completion of the data transmission of the next time as a sampling period, and converting the stability problem of the continuous time state space model of the network control system under replay attack into the stability problem of the discrete time switching system of the limited subsystem according to a variable period discrete method;
3) And determining the index stability and the system index decay rate of the switching system by utilizing the Lyapunov function method and the average residence time.
Further, in step 1), the sensor adopts a time driving mode, the controller and the executor adopt an event driving mode, and the sampling period of the sensor is h; during a sensor sampling period h, whether a replay attack occurs on the "sensor-controller channel" or the "controller-actuator channel", the result is to have the actuator input not updated but to maintain the input of the previous sampling period; if at interval t i ,t j ) If n continuous replay attacks occur within i < j, then this is equivalent to the system at interval t i ,t j ) The sampling period in is (n+1) h, where t i 、t j N is any non-negative integer for the sampling moment of the sensor; converting the network control system suffered from replay attack into a variable period sampling network control system, and establishing a continuous time state space model of the network control system under the replay attack, wherein the model is expressed as:
wherein i is an integer of 0 or more, t is a time variable, t i+1 For the i+1th sampling time of the system, x (t) is the system state vector,is the differentiation of the system state vector, A p As a system matrix, B p For the system input matrix, u (t) is the system control input, K is the controller gain, x (t) i ) At t i Time of day system state vector.
Further, the step 2) includes the steps of:
2.1 The time interval from the successful completion of the data transmission of the system to the successful completion of the data transmission of the next time is regarded as a sampling period, the size of the sampling period increases along with the increase of the number of times that the system continuously suffers from replay attack, and the continuous time state space model of the network control system under the replay attack is controlled according to the sampling period h according to a variable period discrete method i Performing discretization:
x(t i+1 )＝A(h i )x(t i )+B(h i )u(t i ),i＝0,1,2…
wherein i is an integer of 0 or more, t i 、t i+1 For the ith and (i+1) th sampling instants of the system, x (t i ) At t i Time system state vector, x (t i+1 ) At t i+1 Time system state vector, u (t i ) At t i Time of day system control input, A (h i ) Is h i Sampling period system matrix, B (h i ) Is h i A system input matrix of sampling periods, defined as:
wherein A is p As a system matrix, B p For the system input matrix, τ is the integral variable, sampling period h i For the time interval of successful completion of the system for two data transmissions, the sampling period h i Is time-varying and bounded, defined as:
h i ＝t i+1 -t i
h i ＝n i h
wherein h is the sampling period of the sensor, n i The number of sensor samples, n, between which the data transmission is successfully completed i E { 1..d+1 }, where d is the upper bound on the number of replay attacks that the network control system continuously takes place, and d is a positive integer;
2.2)A(h i ) And B (h) i ) The size of (2) is defined by the sampling period h i Each sampling period is determined to be regarded as a subsystem which is independent of each other, and the random variable h is contained by utilizing the idea of a switching system i The variable period sampling network control system is converted into a discrete time switching system with limited subsystems, and the whole network control system is switched among all subsystems according to a set rule:
in the method, in the process of the invention,for the ith subsystem>A state transition matrix for the ith subsystem, defined as:
in the method, in the process of the invention,for a segmented continuous switching signal, indicated at t i The system switches over to the ith subsystem at the moment +.>Indicating that subsystem not under replay attack is present, +.>Representing the occurrence of subsystems that are continuously subject to replay attacks q-1 times, where q.epsilon.2, d+1]Q is a positive integer, ">For the system matrix of the ith subsystem, +.>The system input matrix for the ith subsystem, K is the controller gain.
Further, in step 3), using the lyapunov function method means that, for a switching system having limited subsystems, lyapunov functions that make each subsystem exponentially stable are found separately, which are defined as:
wherein i is an integer of 0 or more, t i 、t i+1 For the ith and (i+1) th sampling instants of the system,for a segmented continuous switching signal, indicated at t i The system switches over to the ith subsystem at the moment +.>Lyapunov function at t for the ith subsystem i Function value of time->Lyapunov function at t for the ith subsystem i+1 Function value of time, x (t i ) At t i Time system state vector, x T (t i ) At t i Transpose of the time of day system state vector, +.>Lyapunov function parameters for the ith subsystem, +.>Is positive matrix, ++>Lyapunov function parameter for the (i+1) -th subsystem, ++>Is positive matrix lambda i Lambda is the attenuation index of the ith subsystem i Mu is the attenuation index of the switching system and is more than or equal to 1;
the attenuation track of the whole switching system is obtained through a recursion method, so that the Lyapunov function of exponential attenuation of the switching system needs to satisfy:
wherein t is 0 For the initial time of the system, m is the number of system switching times, t m For the time when the system switches to the mth system,for a segmented continuous switching signal, indicated at t m The time system switches to the mth subsystem, +.>Lyapunov function for the mth subsystem at t m Value of time of day->Lyapunov function to stay subsystem at initial time of switching system at t 0 Value of time of day->Is [ t ] 0 ,t m ) The switching times of the system in time, d is the upper bound of the continuous replay attack times of the network control system, d is a positive integer, s epsilon {1, …, d+1}, lambda s For the decay rate of subsystem for s times of continuous replay attack, r s The probability of subsystem occurrence for s consecutive occurrence of replay attacks;
average residence time refers to the average residence time of the whole switching system in each subsystem, and the minimum average residence time tau which can enable the system index to be stable is found by establishing the relation between the system attenuation rate and the average residence time of the system in each subsystem a The method comprises the following steps:
where lambda is the switching system attenuation index,as long as the whole switchThe average residence time of the system in each subsystem is greater than tau a The switching system index is stable, and the system index attenuation rate ρ is:
compared with the prior art, the invention has the following advantages and beneficial effects:
1. the invention applies the idea of switching the system to the stability analysis of the network control system under replay attack for the first time, and avoids the defects of complex system modeling and large calculated amount of a random system analysis method.
2. Compared with a public Lyapunov function method, the method has the advantage that the system stability judging condition is more universal.
3. The method has wide use space in the stability analysis of the network control system under the network attack, simple modeling, strong adaptability and wide application prospect.
Drawings
Fig. 1 is a schematic diagram of an example of a replay attack next-stage linear inverted pendulum system used in the present invention.
Detailed Description
The present invention will be described in further detail with reference to examples and drawings, but embodiments of the present invention are not limited thereto.
The embodiment provides a network control system stabilizing method under replay attack based on a switching system, which uses a multi-Lyapunov function method and average residence time, and comprises the following steps:
1) The sensor adopts a time driving mode, the controller and the executor adopt an event driving mode, and the sampling period of the sensor is h; during a sensor sampling period h, whether a replay attack occurs on the "sensor-controller channel" or the "controller-actuator channel", the result is to have the actuator input not updated but to maintain the input of the previous sampling period; if at interval t i ,t j ) If n continuous replay attacks occur within i < j, then this is equivalent to the system at interval t i ,t j ) The sampling period in is (n+1) h, where t i 、t j N is any non-negative integer for the sampling moment of the sensor; converting the network control system suffered from replay attack into a variable period sampling network control system, and establishing a continuous time state space model of the network control system under the replay attack, wherein the model is expressed as:
wherein i is an integer greater than or equal to 0, t is a time variable, t i+1 For the i+1th sampling time of the system, x (t) is the system state vector,is the differentiation of the system state vector, A p As a system matrix, B p For the system input matrix, u (t) is the system control input, K is the controller gain, x (t) i ) At t i Time of day system state vector. The first-order linear inverted pendulum system shown in fig. 1, wherein the mass M of the trolley is=1.32 kg, the mass M of the pendulum rod is=0.109 kg, the friction coefficient b of the trolley and the guide rail is=0.1N/M/s, and the gravity acceleration g is=9.8M/s 2 The length L=0.25m from the rotation axis of the swing rod to the mass center of the swing rod, and the mass I=0.0023kg.m of the swing rod 2 The state space model of the primary inverted pendulum system can be obtained by analyzing the stress of the trolley and the swinging rod:
wherein the system matrixSystem input matrix->Controller gain k= [134.8981 74.2585 206.5828 38.0107 ]]。
2) The time interval from the successful completion of the data transmission of the system to the successful completion of the data transmission of the next time is regarded as a sampling period, and the stability problem of the continuous time state space model of the network control system under replay attack is converted into the stability problem of the discrete time switching system of the limited subsystem according to a variable period discrete method, and the method is as follows:
the continuous time state space model of the network control system under replay attack is discretized according to different periods, and the idea of a switching system is utilized to contain a random variable h i The variable period sampling network control system of (2) is converted into a discrete time switching system with a limited subsystem, and the method comprises the following steps of:
2.1 The time interval from the successful completion of the data transmission by the system to the successful completion of the data transmission by the next time is regarded as one sampling period, and the size of the sampling period increases with the increase of the number of times the system continuously suffers from replay attack. According to the variable period discrete method, the continuous time state space model of the network control system under replay attack is according to the sampling period h i Performing discretization:
x(t i+1 )＝A(h i )x(t i )+B(h i )u(t i ),i＝0,1,2…
wherein i is an integer of 0 or more, t i 、t i+1 For the ith and (i+1) th sampling instants of the system, x (t i ) At t i Time system state vector, x (t i+1 ) At t i+1 Time system state vector, u (t i ) At t i Time of day system control input, A (h i ) Is h i Sampling period system matrix, B (h i ) Is h i A system input matrix of sampling periods, defined as:
wherein A is p As a system matrix, B p For the system input matrix, τ is the integral variable, sampling period h i For the time interval of successful completion of the system for two data transmissions, the sampling period h i Is time-varying and bounded, defined as:
h i ＝t i+1 -t i
h i ＝n i h
wherein h is the sampling period of the sensor, n i The number of sensor samples, n, between which the data transmission is successfully completed i E { 1..d+1 }, where d is the upper bound on the number of replay attacks that the network control system continuously takes place, and d is a positive integer; as shown in fig. 1, the sensor sampling period h=10ms of the one-stage linear inverted pendulum system, and the upper bound of the number of replay attacks continuously occurs in the system.
2.2)A(h i ) And B (h) i ) The size of (2) is defined by the sampling period h i Each sampling period is determined to be regarded as a subsystem which is independent of each other, and the random variable h is contained by utilizing the idea of a switching system i The variable period sampling network control system is converted into a discrete time switching system with limited subsystems, and the whole network control system is switched among all subsystems according to a set rule:
wherein,for the ith subsystem>A state transition matrix for the ith subsystem, defined as:
wherein,for a segmented continuous switching signal, indicated at t i The system switches over to the ith subsystem at the moment +.>Indicating that subsystem not under replay attack is present, +.>Representing the occurrence of subsystems that are continuously subject to replay attacks q-1 times, where q.epsilon.2, d+1]Q is a positive integer, ">For the system matrix of the ith subsystem, +.>The system input matrix for the ith subsystem, K is the controller gain. The closed loop networked inversion system shown in fig. 1 can be described by a discrete time switching system with four subsystems:
wherein,A p for the system matrix, i.e. [1,4 ]]I is a positive integer, ">j is a sum variable, B p A matrix is input for the system.
3) The stable switching system index and the system index attenuation rate are determined by utilizing the Lyapunov function method and the average residence time, and the method is specifically as follows:
the use of the lyapunov function method refers to finding a lyapunov function that stabilizes the index of each subsystem for a switching system having limited subsystems, respectively, which is defined as:
wherein i is an integer of 0 or more, t i 、t i+1 For the ith and (i+1) th sampling instants of the system,for a segmented continuous switching signal, indicated at t i The system switches over to the ith subsystem at the moment +.>Lyapunov function at t for the ith subsystem i Function value of time->Lyapunov function at t for the ith subsystem i+1 Function value of time, x (t i ) At t i Time system state vector, x T (t i ) At t i Transpose of the time of day system state vector, +.>Lyapunov function parameters for the ith subsystem, +.>Is positive matrix, ++>Lyapunov function parameters for the (i+1) -th subsystem，/>Is positive matrix lambda i Lambda is the attenuation index of the ith subsystem i Mu is the attenuation index of the switching system and mu is more than 1;
the attenuation track of the whole switching system is obtained through a recursion method, so that the Lyapunov function of exponential attenuation of the switching system needs to satisfy:
wherein t is 0 For the initial time of the system, m is the number of system switching times, t m For the time when the system switches to the mth system,for a segmented continuous switching signal, indicated at t m The time system switches to the mth subsystem, +.>Lyapunov function for the mth subsystem at t m Value of time of day->Lyapunov function to stay subsystem at initial time of switching system at t 0 Value of time of day->Is [ t ] 0 ,t m ) The switching times of the system in time, d is the upper bound of the continuous replay attack times of the network control system, d is a positive integer, s epsilon {1, …, d+1}, lambda s For the decay rate of subsystem for s times of continuous replay attack, r s The probability of subsystem occurrence for s consecutive occurrence of replay attacks;
average residence time refers to the average residence time of the entire switching system in each subsystem by establishing the systemRelationship between decay rate and average residence time of system in each subsystem, find minimum average residence time tau for stabilizing system index a The method comprises the following steps:
wherein lambda is the attenuation index of the switching system,as long as the average residence time of the entire switching system in each subsystem is greater than τ a The switching system index is stable, and the system index attenuation rate ρ is:
the primary linear inverted pendulum system shown in fig. 1 takes m=10, and utilizes a one-dimensional search method to obtain lambda by taking the subsystem attenuation rate as large as possible 1 ＝1.035，λ 2 ＝λ 3 ＝1.030，λ 4 =1.020, μ=1.05, λ=1.030, the minimum average residence time τ is calculated a As long as the average residence time of the whole switching system in each subsystem is greater than or equal to 0.8ms, =0.8 ms, the system is exponentially stable and the system attenuation rate ρ=1.01.
The above examples are preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the above examples, and any other changes, modifications, substitutions, combinations, and simplifications that do not depart from the spirit and principle of the present invention should be made in the equivalent manner, and the embodiments are included in the protection scope of the present invention.
Claims (3)
1. The network control system stabilizing method under replay attack based on the switching system is characterized by comprising the following steps:
1) Because the sensor adopts a time driving mode, the controller and the executor adopt an event driving mode, according to the principle that the executor maintains the input of the last sampling period when the system suffers replay attack in one sensor period, the network control system suffering replay attack is converted into a variable period sampling network control system, and a continuous time state space model of the network control system under replay attack is established;
2) Taking the time interval from the successful completion of the data transmission of the system to the successful completion of the data transmission of the next time as a sampling period, and converting the stability problem of the continuous time state space model of the network control system under replay attack into the stability problem of the discrete time switching system of the limited subsystem according to a variable period discrete method;
3) Determining the index stability and the system index attenuation rate of the switching system by utilizing the Lyapunov function method and the average residence time;
the use of the lyapunov function method refers to finding a lyapunov function that stabilizes the index of each subsystem for a switching system having limited subsystems, respectively, which is defined as:
wherein i is an integer of 0 or more, t i 、t i+1 For the ith and (i+1) th sampling instants of the system,for a segmented continuous switching signal, indicated at t i The system switches over to the ith subsystem at the moment +.>Lyapunov function at t for the ith subsystem i Function value of time->Lyapunov function at t for the ith subsystem i+1 Function value of time, x (t i ) At t i Time system state vector, x T (t i ) At t i Transpose of the time of day system state vector, +.>Lyapunov function parameters for the ith subsystem, +.>Is positive matrix, ++>Is the lyapunov function parameter of the (i+1) th subsystem,is positive matrix lambda i Lambda is the attenuation index of the ith subsystem i Mu is the attenuation index of the switching system and is more than or equal to 1;
the attenuation track of the whole switching system is obtained through a recursion method, so that the Lyapunov function of exponential attenuation of the switching system needs to satisfy:
wherein t is 0 For the initial time of the system, m is the number of system switching times, t m For the time when the system switches to the mth system,for sectionally continuously switching signals, a tableShown at t m The time system switches to the mth subsystem, +.>Lyapunov function for the mth subsystem at t m Value of time of day->Lyapunov function to stay subsystem at initial time of switching system at t 0 Value of time of day->Is [ t ] 0 ,t m ) The switching times of the system in time, d is the upper bound of the continuous replay attack times of the network control system, d is a positive integer, s epsilon {1, …, d+1}, lambda s For the decay rate of subsystem for s times of continuous replay attack, r s The probability of subsystem occurrence for s consecutive occurrence of replay attacks;
average residence time refers to the average residence time of the whole switching system in each subsystem, and the minimum average residence time tau which can enable the system index to be stable is found by establishing the relation between the system attenuation rate and the average residence time of the system in each subsystem a The method comprises the following steps:
where lambda is the switching system attenuation index,as long as the average residence time of the entire switching system in each subsystem is greater than τ a The switching system index is stable, and the system index attenuation rate ρ is:
2. the method for stabilizing a network control system under replay attack based on a switching system according to claim 1, wherein in step 1), a sensor adopts a time driving mode, a controller and an actuator adopt an event driving mode, and a sensor sampling period is h; during a sensor sampling period h, whether a replay attack occurs on the "sensor-controller channel" or the "controller-actuator channel", the result is to have the actuator input not updated but to maintain the input of the previous sampling period; if at interval t i ,t j ) If n continuous replay attacks occur within i < j, then this is equivalent to the system at interval t i ,t j ) The sampling period in is (n+1) h, where t i 、t j N is any non-negative integer for the sampling moment of the sensor; converting the network control system suffered from replay attack into a variable period sampling network control system, and establishing a continuous time state space model of the network control system under the replay attack, wherein the model is expressed as:
wherein i is an integer of 0 or more, t is a time variable, t i+1 For the i+1th sampling time of the system, x (t) is the system state vector,is the differentiation of the system state vector, A p As a system matrix, B p For the system input matrix, u (t) is the system control input, K is the controller gain, x (t) i ) At t i Time of day system state vector.
3. The method for stabilizing a network control system under a replay attack based on a handover system according to claim 1, wherein the step 2) includes the steps of:
2.1 Successfully completing one data transmission to the systemThe next time interval for successfully completing data transmission is regarded as a sampling period, the size of the sampling period is increased along with the increase of the number of times that the system continuously suffers from replay attack, and the continuous time state space model of the network control system under the replay attack is according to the sampling period h according to a variable period discrete method i Performing discretization:
x(t i+1 )＝A(h i )x(t i )+B(h i )u(t i ),i＝0,1,2…
wherein i is an integer of 0 or more, t i 、t i+1 For the ith and (i+1) th sampling instants of the system, x (t i ) At t i Time system state vector, x (t i+1 ) At t i+1 Time system state vector, u (t i ) At t i Time of day system control input, A (h i ) Is h i Sampling period system matrix, B (h i ) Is h i A system input matrix of sampling periods, defined as:
wherein A is p As a system matrix, B p For the system input matrix, τ is the integral variable, sampling period h i For the time interval of successful completion of the system for two data transmissions, the sampling period h i Is time-varying and bounded, defined as:
h i ＝t i+1 -t i
h i ＝n i h
wherein h is the sampling period of the sensor, n i The number of sensor samples, n, between which the data transmission is successfully completed i E { 1..d+1 }, where d is the upper bound on the number of replay attacks that the network control system continuously takes place, and d is a positive integer;
2.2)A(h i ) And B (h) i ) The size of (2) is defined by the sampling period h i Each sampling period is determined to be regarded as a subsystem which is independent of each other, and the random variable h is contained by utilizing the idea of a switching system i The variable period sampling network control system is converted into a discrete time switching system with limited subsystems, and the whole network control system is switched among all subsystems according to a set rule:
in the method, in the process of the invention,for the ith subsystem>A state transition matrix for the ith subsystem, defined as:
in the method, in the process of the invention,for a segmented continuous switching signal, indicated at t i The system switches over to the ith subsystem at the moment +.>Indicating that subsystem not under replay attack is present, +.>Representing the occurrence of subsystems that are continuously subject to replay attacks q-1 times, where q.epsilon.2, d+1]Q is a positive integer, ">For the system matrix of the ith subsystem, +.>The system input matrix for the ith subsystem, K is the controller gain.
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