CN112995154B - Synchronous control method for complex network under aperiodic DoS attack - Google Patents

Synchronous control method for complex network under aperiodic DoS attack Download PDF

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CN112995154B
CN112995154B CN202110175213.8A CN202110175213A CN112995154B CN 112995154 B CN112995154 B CN 112995154B CN 202110175213 A CN202110175213 A CN 202110175213A CN 112995154 B CN112995154 B CN 112995154B
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dos attack
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CN112995154A (en
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曹杰
申冬琴
刘金良
李燕
赵慕阶
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Yunjing Business Intelligence Research Institute Nanjing Co ltd
Nanjing University of Science and Technology
Nanjing University of Finance and Economics
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Yunjing Business Intelligence Research Institute Nanjing Co ltd
Nanjing University of Science and Technology
Nanjing University of Finance and Economics
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L63/00—Network architectures or network communication protocols for network security
    • H04L63/14—Network architectures or network communication protocols for network security for detecting or protecting against malicious traffic
    • H04L63/1441—Countermeasures against malicious traffic
    • H04L63/1458—Denial of Service
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02D—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN INFORMATION AND COMMUNICATION TECHNOLOGIES [ICT], I.E. INFORMATION AND COMMUNICATION TECHNOLOGIES AIMING AT THE REDUCTION OF THEIR OWN ENERGY USE
    • Y02D30/00—Reducing energy consumption in communication networks
    • Y02D30/70—Reducing energy consumption in communication networks in wireless communication networks

Abstract

The invention discloses a complex network synchronous control method under aperiodic DoS attack, which comprises the steps of firstly, establishing a complex network system model and a synchronous error model, introducing an event triggering mechanism, establishing a network attack model based on the influence of the aperiodic DoS attack on network transmission data, and designing the synchronous error model of the complex network under the aperiodic DoS attack and the event triggering mechanism; obtaining sufficiency conditions for ensuring system index stability by utilizing Lyapunov stability theory; finally solving the inequality of the linear matrix to obtain an event trigger parameter and a feedback gain of the controller; the invention can effectively save network bandwidth resources, improve the communication capacity of a transmission channel, and simultaneously help a networked system to still keep running stably when coping with network attacks; by introducing the event triggering mechanism, the burden of network transmission can be effectively reduced.

Description

Synchronous control method for complex network under aperiodic DoS attack
Technical Field
The invention relates to the technical field of network control, in particular to a synchronous control method of a complex network system attacked by aperiodic DoS.
Background
In networked systems, the communication network may be controlled remotely, flexibly and economically, but as sampled data continues to surge, network resources become more and more limited, resulting in reduced system performance. In order to alleviate the dilemma of resource limitation, the event triggering mechanism is considered as an effective method for saving network resources due to the special advantages, and the main idea is that only when the corresponding triggering condition is met, the current data can be transmitted, so that the network resources are saved. Over the past several years, there have been many different event triggering schemes that are employed in corresponding engineering systems, such as sensor networks, multi-agent systems, and the like.
Despite the attractive advantages of event-triggered schemes, the performance of network systems is still challenged by various network attacks due to the openness of the communication network. DoS attacks are particularly common in the classification of existing network attacks because they tend to cause large transmission delays and severe packet losses, wasting network resources. DoS attacks are classified into periodic and aperiodic, and periodic DoS attacks have been widely studied. Thus, it is also a challenging problem to study event-triggered synchronization control of complex network systems under aperiodic DoS attacks.
Disclosure of Invention
The invention aims to: in order to overcome the defects in the prior art, the invention provides a complex network synchronous control method under aperiodic DoS attack, and a new state estimation system is established under the condition of considering an event triggering mechanism and the aperiodic DoS attack, so that the network load can be effectively reduced.
The technical scheme is as follows: in order to achieve the above purpose, the invention adopts the following technical scheme:
a complex network synchronous control method under aperiodic DoS attack includes building complex network system model and synchronous error model, introducing event trigger mechanism, building network attack model based on influence of aperiodic DoS attack on network transmission data, designing synchronous error model of complex network under aperiodic DoS attack and event trigger mechanism. And obtaining sufficiency conditions for ensuring system index stability by utilizing Lyapunov stability theory. And finally solving the inequality of the linear matrix to obtain the event triggering parameter and the feedback gain of the controller. The invention can effectively save network bandwidth resources, improve the communication capacity of the transmission channel, and simultaneously help the networked system to still keep running stably when coping with network attacks. The event triggering mechanism is introduced, so that the load of network transmission can be effectively reduced, and the method specifically comprises the following steps:
and S1, establishing a complex network system model and a synchronous error model according to the state variables of the nodes, the control inputs of the nodes, the coupling strength and the network matrix.
And S2, introducing an event triggering mechanism.
The measurement output is released into the network and transmitted to the state estimator when the following conditions are met:
Figure BDA0002939559850000021
wherein ,ρi E (0, 1) is a predefined parameter, Ω i Is a symmetric positive definite matrix, h is a constant sampling period,
Figure BDA0002939559850000022
representing the last sampling instant, indicating that the ith node has sampled within the kth sampling interval,/, is->
Figure BDA0002939559850000023
Is positive and has an initial value of 0, k denotes k-th sample interval, k=1, 2, …, < >>
Figure BDA0002939559850000024
For the current sampling time, the i node passes through u after the last data transmission time point i h time intervals are sampled again, u i Positive number, indicating the time interval, u i h represents the (u) i h time intervals, +.>
Figure BDA0002939559850000025
and />
Figure BDA0002939559850000026
Representing the last transmitted data and the current sampled data, respectively, T represents the transpose.
Under the event trigger mechanism, the next trigger time
Figure BDA0002939559850000027
Is described as：
Figure BDA0002939559850000028
Wherein min represents all u satisfying the condition i The smallest u in h i h, | represents a condition, u i h needs to satisfy the condition |following.
And S3, establishing a network attack model by considering the influence of the aperiodic DoS attack on the network transmission data.
And S4, considering the influence of the aperiodic DoS attack on the network transmission data, and adjusting an event triggering mechanism.
If a DoS attack in step S3 is introduced, the control signal will be blocked during the attack interval, which will result in the triggering condition in step S2 not being directly adopted, adjusted to:
Figure BDA0002939559850000029
where n represents the nth DoS attack interference time interval,
Figure BDA00029395598500000210
the time interval for indicating that the ith node satisfies the trigger condition is +.>
Figure BDA00029395598500000211
Personal (S)>
Figure BDA00029395598500000212
Indicate->
Figure BDA00029395598500000213
Time interval->
Figure BDA00029395598500000214
Namely k+u i Alternatively, the expression->
Figure BDA00029395598500000215
k i Representing the sum of the number of triggers of the ith node in the nth interference period, +.>
Figure BDA00029395598500000216
k i (n) represents the maximum number of times the ith node can be triggered, ψ, in the nth interference period i (n) represents the set of values of k that the ith node triggers in the nth cycle,/->
Figure BDA00029395598500000217
Representation definition->
Figure BDA00029395598500000218
Trigger time interval of ith node
Figure BDA00029395598500000219
Figure BDA00029395598500000220
Divided into->
Figure BDA0002939559850000031
wherein ：
Figure BDA0002939559850000032
wherein ,
Figure BDA0002939559850000033
represents the union of all time intervals, +.>
Figure BDA0002939559850000034
The mth time interval of the division is indicated,
Figure BDA0002939559850000035
the representation will->
Figure BDA0002939559850000036
Divided into a plurality of->
Figure BDA0002939559850000037
The maximum m, [ ] of the time interval represents the interval opening and closing.
Under event triggering and DoS attack, the actual time interval of data transmission of the ith node is:
Figure BDA0002939559850000038
will be
Figure BDA0002939559850000039
Divided into->
Figure BDA00029395598500000310
Figure BDA00029395598500000311
An intersection of an mth interval of data transmission representing an event of an ith node and an nth DoS attack interference dormancy interval, wherein:
Figure BDA00029395598500000312
Figure BDA00029395598500000313
representing the time interval during which data can be normally transmitted without DoS attack interference.
Two piecewise functions are defined:
Figure BDA00029395598500000314
and
Figure BDA0002939559850000041
the synchronization error is expressed as:
Figure BDA0002939559850000042
wherein ,
Figure BDA0002939559850000043
the method meets the following conditions:
Figure BDA0002939559850000044
wherein ,
Figure BDA0002939559850000045
representing defined piecewise functions, +.>
Figure BDA0002939559850000046
Representing defined piecewise functions, +.>
Figure BDA0002939559850000047
Data transmitted without DoS attack interference, < >>
Figure BDA0002939559850000048
Representation, Ω i Event-triggered parameter matrix, ρ, representing the ith node i Representing the event trigger parameters of the ith node, Γ represents the in-coupling matrix between complex network nodes.
And S5, designing a synchronous error model of the ith node of the complex network based on the DoS attack and the event trigger mechanism, and further obtaining the synchronous error model of the whole complex network system.
The synchronization error model of the whole complex network system is as follows:
Figure BDA0002939559850000049
/>
wherein ε (t) =col N {ε i (t)}，
Figure BDA00029395598500000410
G(t)＝col N {g(x i (t),s(t))}，/>
Figure BDA00029395598500000411
Figure BDA00029395598500000412
Figure BDA0002939559850000051
Wherein C represents a connection matrix between complex network nodes,
Figure BDA0002939559850000052
represents the Cronecker product, ψ (t) represents the initial state of the node, h represents the sampling time interval, col N {ε i (t) } represents a column vector of N consecutive columns, V k,n Representing event trigger time period irrespective of DoS attack, S n-1 Representing the period of time for which data is actually transmitted in consideration of the DoS attack.
And S6, obtaining sufficiency conditions for ensuring the index stability of the whole complex network system based on the Lyapunov stability theory.
And S7, solving a linear matrix inequality according to the sufficiency condition for ensuring the exponential stability of the whole complex network system obtained in the step 6, and obtaining an event triggering parameter and a controller feedback gain.
Preferably: the complex network system model established in the step S1 is as follows:
Figure BDA0002939559850000053
wherein ,xi (t) is the state variable of the ith node, x i (t)∈R n′ ，
Figure BDA0002939559850000054
Is x i Differentiation of (t), u i (t)∈R n′ ，u i (t) is the control input of the ith node, g: R n →R n Sigma > 0 is a given coupling strength as a continuous nonlinear vector function. C= [ C ] ij ] N×N Is a network matrix, wherein c ij When > 0, i is not equal to j, indicating that there is undirected connection between node i and node j, otherwise c ij =0. Diagonal element +.>
Figure BDA0002939559850000055
Γ＝diag{t 1 ,t 2 ,…,t N Is an in-coupling matrix and obeys +.>
Figure BDA0002939559850000056
wherein t i and />
Figure BDA0002939559850000057
Is a known constant, A is a constant matrix I, B is a constant matrix II, N is the number of complex network nodes, R n′ Representing the n' dimensional euclidean space.
Preferably: the outlier model established in step S1 is as follows:
Figure BDA0002939559850000058
let the synchronization error be epsilon i ＝x i (t) -s (t), the synchronization error system model is built as follows:
Figure BDA0002939559850000059
wherein ,g(xi (t),s(t))＝g(x i (t))-g(s(t))。
Wherein s (t) is an isolated point, t is time,
Figure BDA00029395598500000510
is s (t)) Is a derivative of ε i For synchronization error +.>
Figure BDA00029395598500000511
Is epsilon i Is a derivative of (a).
Preferably: in step S3, the method for establishing the network attack model considers the influence of the aperiodic DoS attack on the network transmission data:
considering the influence of non-periodic DoS attacks in a network channel, the attack signals are:
Figure BDA0002939559850000061
/>
wherein F (t) is a signal of a DoS attack,
Figure BDA0002939559850000062
and />
Figure BDA0002939559850000063
Is a real sequence, and d n+1 >d n +g n . In time interval +.>
Figure BDA0002939559850000064
wherein ,Dn ＝[d n +g n ,d n+1 ) Normal communications are blocked due to DoS attacks. In time interval +.>
Figure BDA0002939559850000065
wherein />
Figure BDA0002939559850000066
Normal communication is performed without interference from DoS attacks. The start time of the n+1th DoS attack period is d n +g n For a time d of n+1 -d n -g n ，/>
Figure BDA0002939559850000067
To attack the dormant set, d n For the period of active attack, g n For the period of time of attacking dormancy.
Setting the upper time limit of the duration of the DoS attack active period and the lower time limit of the sleep period as follows:
Figure BDA0002939559850000068
Figure BDA0002939559850000069
wherein ,
Figure BDA00029395598500000610
time upper bound representing duration of active period, +.>
Figure BDA00029395598500000611
A time lower bound representing sleep period, d max Represents the longest time of attack, g min Representing the shortest time to sleep.
Let n (t) be the number of attacks and sleep transitions made by DoS attacks in time interval [0, t), given two parameters
Figure BDA00029395598500000612
And eta is greater than or equal to 0, at->
Figure BDA00029395598500000613
The frequency of the internal DoS attack is:
Figure BDA00029395598500000614
wherein ,
Figure BDA00029395598500000615
represents the custom parameter I, eta represents the custom parameter II,>
Figure BDA00029395598500000616
representing a natural number set.
Preferably: in step S5, a synchronization error model of the ith node of the complex network is designed:
Figure BDA00029395598500000617
wherein sigma represents a coupling weight parameter, K i Representing a feedback gain matrix representing the ith node.
Preferably: in step S6, a method for obtaining a sufficiency condition for ensuring exponential stability of the whole complex network system based on Lyapunov stability theory is as follows:
setting a scalar alpha j' ＞0，μ j' ＞0，j'＝1,2，1＞ρ i > 0, i=1, 2..N, sampling interval h > 0, dos parameter η > 0,
Figure BDA00029395598500000618
feedback gain matrix->
Figure BDA00029395598500000619
When positive definite matrix P exists j' 、Q j' 、R j' 、Ω i' 、M j' and Nj' Is a matrix with proper dimension, so that when the following inequality and condition are satisfied, the system index is stable, and the attenuation rate is +.>
Figure BDA00029395598500000620
Figure BDA0002939559850000071
P 1 ≤μ 2 P 2 ,
Figure BDA0002939559850000072
Q 1 ≤μ 2 Q 2 ,Q 2 ≤μ 1 Q 1 ,
R 1 ≤μ 2 R 2 ,R 2 ≤μ 1 R 1 ,
Figure BDA0002939559850000073
wherein ,
Figure BDA0002939559850000074
Figure BDA0002939559850000075
Figure BDA0002939559850000076
ρ＝diag{ρ 1 ,...,ρ N },Ω＝diag{Ω 1 ,...,Ω N },
Figure BDA0002939559850000077
Figure BDA0002939559850000078
Figure BDA0002939559850000079
M 1 ＝[M 11 M 12 M 13 M 14 M 15 ],N 1 ＝[N 11 N 12 N 13 N 14 N 15 ],
M 2 ＝[M 21 M 22 M 23 M 24 ],N 2 ＝[N 21 N 22 N 23 N 24 ],
Γ 1 ＝[M 1 0 N 1 -M 1 -N 1 0],Γ 2 ＝[M 2 0 N 2 -M 2 -N 2 ],
Figure BDA00029395598500000710
wherein ,M11 、M 12 、M 13 、M 14 、M 15 Representing a free weight matrix M 1 Component N of (2) 11 、N 12 、N 13 、N 14 、N 15 Representing a free weight matrix N 1 Component of M 21 、M 22 、M 23 、M 24 Representing a free weight matrix M 2 Component N of (2) 21 、N 22 、N 23 、N 24 Representing a free weight matrix N 2 I represents the identity matrix.
Preferably: in step S7, the method for acquiring the event trigger parameter and the feedback gain of the controller includes:
setting a scalar alpha j' ＞0，μ j' ＞0，κ j' ＞0，e j' >0，s j' ＞0，j'＝1,2，1＞ρ i > 0, i=1, 2..N, sampling interval h > 0, dos parameter η > 0,
Figure BDA0002939559850000081
matrix Λ, P j' ＞0，X j' ＞0，Y j' ＞0，/>
Figure BDA0002939559850000082
and />
Figure BDA0002939559850000083
For a matrix of appropriate dimensions, so that the following linear matrix inequality holds, the system state estimation model is exponentially stable and the decay rate is +.>
Figure BDA0002939559850000084
Figure BDA0002939559850000085
/>
Figure BDA0002939559850000086
Figure BDA0002939559850000087
Figure BDA0002939559850000088
Figure BDA0002939559850000089
wherein ：
Figure BDA00029395598500000810
Figure BDA00029395598500000811
Figure BDA00029395598500000812
Figure BDA00029395598500000813
Figure BDA00029395598500000814
Figure BDA0002939559850000091
the controller feedback gain is:
Figure BDA0002939559850000092
compared with the prior art, the invention has the following beneficial effects:
the synchronous control method of the complex network system under the aperiodic DoS attack effectively saves the bandwidth, reduces the network load, improves the communication capacity of the transmission channel, and can efficiently save the network bandwidth resource and reduce the network load. An event triggering mechanism is introduced, and the load of network transmission is reduced. The problem of synchronous control of a complex network system based on an event trigger mechanism is studied. The adeep stability theory and the linear matrix inequality technology are utilized to obtain the sufficiency condition of the index stability of the synchronous error system, and an ideal design method of the controller is provided.
Drawings
FIG. 1 is a flow chart of a state estimator design provided by the present invention.
Fig. 2 is a synchronization error diagram in an embodiment of the invention.
Fig. 3 is a diagram of control input signals under non-periodic DoS attack disturbance in an example of the present invention.
Fig. 4 is a diagram of event trigger timing in an example of the invention.
Detailed Description
The present invention is further illustrated in the accompanying drawings and detailed description which are to be understood as being merely illustrative of the invention and not limiting of its scope, and various equivalent modifications to the invention will fall within the scope of the appended claims to the skilled person after reading the invention.
The synchronous control method of the complex network system under the aperiodic DoS attack shown in fig. 1 comprises the following steps:
and S1, establishing a complex network system model and a synchronous error model according to the state variables of the nodes, the control inputs of the nodes, the coupling strength and the network matrix.
The complex network system model is built as follows:
Figure BDA0002939559850000093
wherein ,xi (t) is the state variable of the ith node, x i (t)∈R n ，
Figure BDA0002939559850000094
Is x i Differentiation of (t), u i (t)∈R n ，u i (t) is the control input of the ith node, g: R n →R n Sigma > 0 is a given coupling strength as a continuous nonlinear vector function. C= [ C ] ij ] N×N Is a network matrix, wherein c ij Indicating the existence of undirected connection between node i and node j, c ij When > 0, i is not equal to j, indicating whether there is a directional connection between nodes i and j, otherwise c ij =0. Diagonal line element
Figure BDA0002939559850000095
Γ＝diag{t 1 ,t 2 ,…,t N Is an in-coupling matrix and obeys
Figure BDA0002939559850000096
wherein t i and />
Figure BDA0002939559850000097
Is a known constant, A is a constant matrix one, and B is a constant matrix two.
The outlier model is as follows:
Figure BDA0002939559850000101
let the synchronization error be epsilon i ＝x i (t) -s (t), the synchronization error system model is built as follows:
Figure BDA0002939559850000102
wherein ,g(xi (t),s(t))＝g(x i (t))-g(s(t))。
And S2, introducing an event triggering mechanism.
The measurement output is released into the network and transmitted to the state estimator when the following conditions are met:
Figure BDA0002939559850000103
wherein ,ρi E (0, 1) is a predefined parameter, Ω i Is a symmetric positive definite matrix, h is a constant sampling period,
Figure BDA0002939559850000104
representing the last sampling instant, indicating that the ith node has sampled within the kth sampling interval,/, is->
Figure BDA0002939559850000105
Is positive and has an initial value of 0, k denotes k-th sample interval, k=1, 2, …, < >>
Figure BDA0002939559850000106
For the current sampling time, the i node passes through u after the last data transmission time point i h time intervals are sampled again, u i Positive number, indicating the time interval, u i h represents the (u) i h time intervals, +.>
Figure BDA0002939559850000107
and />
Figure BDA0002939559850000108
Representing the last transmitted data and the current sampled data, respectively, T represents the transpose.
Under the event trigger mechanism, the next trigger time
Figure BDA0002939559850000109
Is described as:
Figure BDA00029395598500001010
wherein min represents all u satisfying the condition i The smallest u in h i h, | represents a condition, u i h needs to satisfy the condition |following.
And step S3, a network attack model is established by considering the influence of the aperiodic DoS attack on the network transmission data.
Considering the influence of non-periodic DoS attacks in a network channel, the attack signals are:
Figure BDA00029395598500001011
wherein ,
Figure BDA00029395598500001012
and />
Figure BDA00029395598500001013
Is a real sequence, and d n+1 >d n +g n . In time interval +.>
Figure BDA00029395598500001014
wherein ,Dn ＝[d n +g n ,d n+1 ) Normal communications are blocked due to DoS attacks. In time interval +.>
Figure BDA00029395598500001015
wherein
Figure BDA00029395598500001016
Normal communication is performed without interference from DoS attacks. The start time of the n+1th DoS attack period is d n +g n For a time d of n+1 -d n -g n 。
Setting the upper time limit of the duration of the DoS attack active period and the lower time limit of the sleep period as follows:
Figure BDA0002939559850000111
Figure BDA0002939559850000112
wherein ,
Figure BDA0002939559850000113
time upper bound representing duration of active period, +.>
Figure BDA0002939559850000114
Representing the temporal lower bound of sleep periods.
Let n (t) be the number of attacks and sleep transitions made by DoS attacks in time interval [0, t), given two parameters
Figure BDA0002939559850000115
And eta is greater than or equal to 0, at->
Figure BDA0002939559850000116
The frequency of the internal DoS attack is:
Figure BDA0002939559850000117
and S4, considering the influence of the aperiodic DoS attack on the network transmission data, and adjusting an event triggering mechanism.
If the DoS attack described above is introduced, the control signal will be blocked during the attack interval, which will result in the above trigger condition not being directly applicable, adjusted to:
Figure BDA0002939559850000118
wherein ,
Figure BDA0002939559850000119
k i indicating the sum of the number of triggers of the ith node in the nth interference period,
Figure BDA00029395598500001110
trigger time interval of ith node
Figure BDA00029395598500001111
Figure BDA00029395598500001112
Divided into->
Figure BDA00029395598500001113
wherein ：
Figure BDA00029395598500001114
wherein ,
Figure BDA00029395598500001115
represents the union of all time intervals, +.>
Figure BDA00029395598500001116
Represents the m-th time interval of the division, +.>
Figure BDA00029395598500001117
The representation will->
Figure BDA00029395598500001118
Divided into a plurality of->
Figure BDA00029395598500001119
The maximum m, [ ] of the time interval represents the interval opening and closing.
Under event triggering and DoS attack, the actual time interval of data transmission of the ith node is:
Figure BDA0002939559850000121
will be
Figure BDA0002939559850000122
Divided into->
Figure BDA0002939559850000123
Figure BDA0002939559850000124
An intersection of an mth interval of data transmission representing an event of an ith node and an nth DoS attack interference dormancy interval, wherein:
Figure BDA0002939559850000125
Figure BDA0002939559850000126
the time interval during which data can be normally transmitted without DoS attack interference.
Two piecewise functions are defined:
Figure BDA0002939559850000127
and
Figure BDA0002939559850000128
the synchronization error is expressed as:
Figure BDA0002939559850000129
wherein ,
Figure BDA00029395598500001210
the method meets the following conditions:
Figure BDA0002939559850000131
wherein ,
Figure BDA0002939559850000132
representing defined piecewise functions, +.>
Figure BDA0002939559850000133
Representing defined piecewise functions, +.>
Figure BDA0002939559850000134
Data transmitted without DoS attack interference, < >>
Figure BDA0002939559850000135
Representation, Ω i Event-triggered parameter matrix, ρ, representing the ith node i Representing the event trigger parameters of the ith node, Γ represents the in-coupling matrix between complex network nodes.
Step S5, designing a synchronous error model of the ith node of the complex network based on the DoS attack and the event trigger mechanism as follows:
Figure BDA0002939559850000136
the synchronization error model of the entire complex network system is as follows:
Figure BDA0002939559850000137
wherein ε (t) =col N {ε i (t)}，
Figure BDA0002939559850000138
G(t)＝col N {g(x i (t),s(t))}，/>
Figure BDA0002939559850000139
Figure BDA00029395598500001310
Figure BDA00029395598500001311
Wherein C represents a connection matrix between complex network nodes,
Figure BDA00029395598500001312
represents the Cronecker product, ψ (t) represents the initial state of the node, h represents the sampling time interval, col N {ε i (t) } represents a column vector of N consecutive columns, V k,n Representing event trigger time period irrespective of DoS attack, S n-1 Representing the period of time for which data is actually transmitted in consideration of the DoS attack.
And S6, obtaining sufficiency conditions for ensuring system index stability based on Lyapunov stability theory.
Setting a scalar alpha j' ＞0，μ j' ＞0，j'＝1,2，1＞ρ i > 0, i=1, 2..N, sampling interval h > 0, dos parameter η > 0,
Figure BDA00029395598500001313
feedback gain matrix->
Figure BDA00029395598500001314
When positive definite matrix P exists j' 、Q j' 、R j' 、Ω i' 、M j' and Nj' Is a matrix with proper dimension, so that when the following inequality and condition are satisfied, the system index is stable, and the attenuation rate is +.>
Figure BDA0002939559850000141
Figure BDA0002939559850000142
P 1 ≤μ 2 P 2 ,
Figure BDA0002939559850000143
Q 1 ≤μ 2 Q 2 ,Q 2 ≤μ 1 Q 1 ,
R 1 ≤μ 2 R 2 ,R 2 ≤μ 1 R 1 ,
Figure BDA0002939559850000144
wherein ,
Figure BDA0002939559850000145
/>
Figure BDA0002939559850000146
Figure BDA0002939559850000147
ρ＝diag{ρ 1 ,...,ρ N },Ω＝diag{Ω 1 ,...,Ω N },
Figure BDA0002939559850000148
Figure BDA0002939559850000149
Figure BDA00029395598500001410
M 1 ＝[M 11 M 12 M 13 M 14 M 15 ],N 1 ＝[N 11 N 12 N 13 N 14 N 15 ],
M 2 ＝[M 21 M 22 M 23 M 24 ],N 2 ＝[N 21 N 22 N 23 N 24 ],
Γ 1 ＝[M 1 0 N 1 -M 1 -N 1 0],Γ 2 ＝[M 2 0 N 2 -M 2 -N 2 ],
Figure BDA0002939559850000151
the proving process is as follows:
the Lyapunov function was constructed as follows:
Figure BDA0002939559850000152
wherein Pj' ＞0，Q j' ＞0，R j' ＞0，α j' ＞0，
Figure BDA0002939559850000153
Based on the above definition, consider the following two cases:
when j' =1, the derivative is calculated as follows:
Figure BDA0002939559850000154
wherein
Figure BDA0002939559850000155
Figure BDA0002939559850000156
Figure BDA0002939559850000157
G (·) satisfies G T (t)G(t)≤ε T (t)Λ T Λε(t).
The following can be obtained:
Figure BDA0002939559850000158
using the Schur theorem, one can derive:
Figure BDA0002939559850000159
the finishing method can obtain:
Figure BDA00029395598500001510
thereby:
Figure BDA00029395598500001511
when j' =2, the derivative is calculated as follows:
Figure BDA0002939559850000161
using the same method, it is possible to obtain:
Figure BDA0002939559850000162
two cases are combined:
Figure BDA0002939559850000163
due to
Figure BDA0002939559850000164
Is available in the form of
Figure BDA0002939559850000165
wherein b1 ＝2η(α 1 +α 2 )h+ηln(μ 1 μ 2 )+2α 2 d max η-2α 1 g min η，
b 2 ＝(η+1)(2(α 1 +α 2 )h+ln(μ 1 μ 2 )+2α 2 d max η-2α 1 g min )。
Definition of the definition
Figure BDA0002939559850000166
The method can obtain:
Figure BDA0002939559850000167
thereby:
Figure BDA0002939559850000168
it can be derived that the system index is stable.
And S7, solving a linear matrix inequality, and acquiring an event triggering parameter and a controller feedback gain.
Setting a scalar alpha j' ＞0，μ j' ＞0，κ j' ＞0，e j' >0，s j' ＞0，j'＝1,2，1＞ρ i > 0, i=1, 2..N, sampling interval h > 0, dos parameter η > 0,
Figure BDA0002939559850000169
matrix Λ, P j' ＞0，X j' ＞0，Y j' ＞0，/>
Figure BDA00029395598500001610
and />
Figure BDA00029395598500001611
For a matrix of appropriate dimensions, so that the following linear matrix inequality holds, the system state estimation model is exponentially stable and the decay rate is +.>
Figure BDA00029395598500001612
/>
Figure BDA00029395598500001613
Figure BDA00029395598500001614
Figure BDA0002939559850000171
Figure BDA0002939559850000172
Figure BDA0002939559850000173
wherein ：
Figure BDA0002939559850000174
Figure BDA0002939559850000175
Figure BDA0002939559850000176
Figure BDA0002939559850000177
Figure BDA0002939559850000178
Figure BDA0002939559850000179
the controller feedback gain is:
Figure BDA00029395598500001710
the following was demonstrated:
using Schur's theorem, one can obtain:
Figure BDA00029395598500001711
wherein
Figure BDA00029395598500001712
/>
Figure BDA00029395598500001713
Figure BDA00029395598500001714
Figure BDA0002939559850000181
From the following components
Figure BDA0002939559850000182
The method can obtain:
Figure BDA0002939559850000183
definition X 1 ＝P 1 -1 And use
Figure BDA0002939559850000184
Replace->
Figure BDA0002939559850000185
A systematic exponential settling can be obtained.
The following method of simulation analysis is adopted to provide a specific embodiment, the gains of the estimator are solved by programming Matlab program to solve the inequality of the linear matrix, and a simulation curve is drawn, and the effectiveness of the invention is proved by a simulation example:
consider a complex network system with three nodes, the system parameters are:
Figure BDA0002939559850000186
the initial conditions of the system are as follows:
Figure BDA0002939559850000187
the nonlinear function and its upper bound Λ are expressed as:
Figure BDA0002939559850000188
let h=0.1, σ=0.8, α 1 ＝0.05，α 2 ＝0.5，μ 1 ＝μ 2 ＝1.01，d max ＝0.2，g min ＝1.78，ρ 1 ＝ρ 2 ＝ρ 3 ＝0.1，e j ＝10,s j =10 (j=1, 2). The initial state is
Figure BDA0002939559850000189
Figure BDA00029395598500001810
s T (0)＝[-0.5 0.5]The LMI toolbox using matlab derives the gain of the controller as:
K 1 ＝[-0.1039 -0.0823],K 2 ＝[-0.1042 -0.0838],K 3 ＝[-0.1057 -0.0875],
the obtained system synchronization error fluctuation is shown in fig. 2, the complex network system can be kept stable under the designed controller, the change track of the control input under the non-periodic DoS attack signal is shown in fig. 3, the release moment and the trigger interval of the event trigger of the designed controller are shown in fig. 4, and the complex network system can be kept stable under the control strategy of the designed controller and can effectively improve the data transmission efficiency, so that the designed controller is effective and has good performance.
The foregoing is only a preferred embodiment of the invention, it being noted that: it will be apparent to those skilled in the art that various modifications and adaptations can be made without departing from the principles of the present invention, and such modifications and adaptations are intended to be comprehended within the scope of the invention.

Claims (1)

1. A complex network synchronous control method under aperiodic DoS attack is characterized by comprising the following steps:
step S1, a complex network system model and a synchronous error model are established according to state variables of nodes, control inputs of the nodes, coupling weights and a connection matrix;
the complex network system model is built as follows:
Figure QLYQS_1
wherein ,xi (t) is the state variable of the ith node, x i (t)∈R n′ ，
Figure QLYQS_2
Is x i Differentiation of (t), u i (t)∈R n′ ，u i (t) is the control input of the ith node, g: R n →R n As a continuous nonlinear vector function, sigma > 0 is a given coupling weight parameter; c= [ C ] ij ] N×N Is a connection matrix between complex networks, where c ij When > 0, i is not equal to j, indicating that there is undirected connection between node i and node j, otherwise c ij =0; diagonal element +.>
Figure QLYQS_3
Γ＝diag{t 1 ,t 2 ,…,t N Is an in-coupling matrix between complex network nodes and obeys +.>
Figure QLYQS_4
wherein t i and />
Figure QLYQS_5
Is a known constant, A is a constant matrix I, B is a constant matrix II, N is the number of complex network nodes, R n′ Representing an n' dimensional Euclidean space;
the established outlier model is as follows:
Figure QLYQS_6
let the synchronization error be epsilon i (t)＝x i (t) -s (t), the synchronization error model is built as follows:
Figure QLYQS_7
wherein ,g(xi (t),s(t))＝g(x i (t))-g(s(t))；
Wherein s (t) is an isolated point, t is time,
Figure QLYQS_8
is the differentiation of s (t), ε i (t) is synchronization error,>
Figure QLYQS_9
is epsilon i Is a derivative of (2);
s2, introducing an event triggering mechanism;
the measurement output is released into the network and transmitted to the state estimator when the following conditions are met:
Figure QLYQS_10
wherein ,ρi E (0, 1) represents the event trigger parameter of the ith node, Ω i Is a symmetric positive definite matrix, h is a constant sampling period,
Figure QLYQS_11
representing the last sampling instant +.>
Figure QLYQS_12
Is positive and has an initial value of 0, k denotes the kth sampling interval, k=1, 2, …, < >>
Figure QLYQS_13
For the current sampling instant, u i h represents the (u) i h time intervals, +.>
Figure QLYQS_14
and />
Figure QLYQS_15
Respectively representing the last transmitted data and the current sampled data, wherein T represents transposition;
under the event trigger mechanism, the next trigger time
Figure QLYQS_16
Is described as:
Figure QLYQS_17
wherein min represents all u satisfying the condition i The smallest u in h i h, | represents a condition, u i h is required to satisfy the condition |following;
s3, establishing a network attack model by considering the influence of the aperiodic DoS attack on network transmission data;
the method for establishing the network attack model by considering the influence of the aperiodic DoS attack on the network transmission data comprises the following steps:
considering the influence of non-periodic DoS attacks in a network channel, the attack signals are:
Figure QLYQS_18
wherein F (t) is a signal of a DoS attack,
Figure QLYQS_19
and />
Figure QLYQS_20
Is a real sequence, and d n+1 >d n +g n The method comprises the steps of carrying out a first treatment on the surface of the At intervals of time
Figure QLYQS_21
wherein ,Dn ＝[d n +g n ,d n+1 ) Normal communications are blocked due to DoS attacks; in time interval +.>
Figure QLYQS_22
wherein />
Figure QLYQS_23
Normal communication is carried out, and interference of DoS attack is avoided; the start time of the n+1th DoS attack period is d n +g n For a time d of n+1 -d n -g n ，/>
Figure QLYQS_24
To attack the dormant set, d n For the period of active attack, g n A period of time for attacking dormancy;
setting the upper time limit of the duration of the DoS attack active period and the lower time limit of the sleep period as follows:
Figure QLYQS_25
Figure QLYQS_26
wherein ,
Figure QLYQS_27
time upper bound representing duration of active period, +.>
Figure QLYQS_28
A time lower bound representing sleep period, d max Represents the longest time of attack, g min Representing the shortest time to sleep;
let n (t) be the number of attacks and sleep transitions made by DoS attacks in time interval [0, t), given two parameters
Figure QLYQS_29
And eta is greater than or equal to 0, at->
Figure QLYQS_30
The frequency of the internal DoS attack is:
Figure QLYQS_31
wherein ,
Figure QLYQS_32
represents the custom parameter I, eta represents the custom parameter II,>
Figure QLYQS_33
representing a natural number set;
s4, considering the influence of the aperiodic DoS attack on the network transmission data, and adjusting an event triggering mechanism;
if a DoS attack in step S3 is introduced, the control signal will be blocked during the attack interval, which will result in the triggering condition in step S2 not being directly adopted, adjusted to:
Figure QLYQS_34
where n represents the nth DoS attack interference time interval,
Figure QLYQS_37
the time interval for indicating that the ith node satisfies the trigger condition is +.>
Figure QLYQS_39
Personal (S)>
Figure QLYQS_41
Indicate->
Figure QLYQS_36
Time interval->
Figure QLYQS_38
k i Representing the sum of the number of triggers of the ith node in the nth interference period, +.>
Figure QLYQS_40
k i (n) represents the maximum number of times the ith node can be triggered, ψ, in the nth interference period i (n) represents the set of values of k that the ith node triggers in the nth cycle,/->
Figure QLYQS_42
The definition of the representation is given by,
Figure QLYQS_35
trigger time interval of ith node
Figure QLYQS_43
Divided into->
Figure QLYQS_44
Wherein: />
Figure QLYQS_45
wherein ,
Figure QLYQS_46
represents the union of all time intervals, +.>
Figure QLYQS_47
Represents the m-th time interval of the division, +.>
Figure QLYQS_48
The representation will->
Figure QLYQS_49
Divided into a plurality of->
Figure QLYQS_50
The maximum m of the time interval, [) represents the interval opening and closing;
under event triggering and DoS attack, the actual time interval of data transmission of the ith node is:
Figure QLYQS_51
will be
Figure QLYQS_52
Divided into->
Figure QLYQS_53
Figure QLYQS_54
An intersection of an mth interval of data transmission representing an event of an ith node and an nth DoS attack interference dormancy interval, wherein:
Figure QLYQS_55
Figure QLYQS_56
a time interval which indicates that data can be normally transmitted without DoS attack interference;
two piecewise functions are defined:
Figure QLYQS_57
and
Figure QLYQS_58
the synchronization error is expressed as:
Figure QLYQS_59
/>
wherein ,
Figure QLYQS_60
the method meets the following conditions:
Figure QLYQS_61
wherein ,
Figure QLYQS_62
representing defined piecewise functions, +.>
Figure QLYQS_63
Representing defined piecewise functions, +.>
Figure QLYQS_64
Representing data transmitted without DoS attack interference, Ω i Event-triggered parameter matrix, ρ, representing the ith node i An event trigger parameter representing an ith node, Γ representing an in-coupling matrix between complex network nodes;
s5, designing a synchronous error model of an ith node of the complex network based on the DoS attack and the event trigger mechanism, and further obtaining the synchronous error model of the whole complex network system;
the synchronization error model of the whole complex network system is as follows:
Figure QLYQS_65
wherein ε (t) =col N {ε i (t)}，
Figure QLYQS_66
G(t)＝col N {g(x i (t),s(t))}，/>
Figure QLYQS_67
Figure QLYQS_68
Wherein C represents a connection matrix between complex network nodes,
Figure QLYQS_69
represents the Cronecker product, ψ (t) represents the initial state of the node, h represents the sampling time interval, col N {ε i (t) } represents a column vector of N consecutive columns, V k,n Representing event trigger time period irrespective of DoS attack, S n-1 Representing a period of time for actually transmitting data in consideration of DoS attack;
designing a synchronous error model of an ith node of the complex network:
Figure QLYQS_70
wherein sigma represents a coupling weight parameter, K i A feedback gain matrix representing the ith node;
s6, obtaining sufficiency conditions for ensuring the index stability of the whole complex network system based on Lyapunov stability theory;
the method for obtaining the sufficiency condition for ensuring the exponential stability of the whole complex network system based on Lyapunov stability theory comprises the following steps:
setting a scalar alpha j' ＞0，μ j' ＞0，j'＝1,2，1＞ρ i > 0, i=1, 2..N, sampling time interval h > 0, dos parameter η > 0,
Figure QLYQS_71
feedback gain matrix->
Figure QLYQS_72
When positive definite matrix P exists j' 、Q j' 、R j' 、Ω i' 、M j' and Nj' Is a matrix with proper dimension, so that when the following inequality and condition are satisfied, the system index is stable, and the attenuation rate is +.>
Figure QLYQS_73
/>
Figure QLYQS_74
P 1 ≤μ 2 P 2 ,
Figure QLYQS_75
Q 1 ≤μ 2 Q 2 ,Q 2 ≤μ 1 Q 1 ,
R 1 ≤μ 2 R 2 ,R 2 ≤μ 1 R 1 ,
Figure QLYQS_76
wherein ,
Figure QLYQS_77
Figure QLYQS_78
Figure QLYQS_79
ρ＝diag{ρ 1 ,...,ρ N },Ω＝diag{Ω 1 ,...,Ω N },
Figure QLYQS_80
Figure QLYQS_81
Figure QLYQS_82
M 1 ＝[M 11 M 12 M 13 M 14 M 15 ],N 1 ＝[N 11 N 12 N 13 N 14 N 15 ],
M 2 ＝[M 21 M 22 M 23 M 24 ],N 2 ＝[N 21 N 22 N 23 N 24 ],
Γ 1 ＝[M 1 0 N 1 -M 1 -N 1 0],Γ 2 ＝[M 2 0 N 2 -M 2 -N 2 ],
Figure QLYQS_83
wherein ,M11 、M 12 、M 13 、M 14 、M 15 Representing a free weight matrix M 1 Component N of (2) 11 、N 12 、N 13 、N 14 、N 15 Representing a free weight matrix N 1 Component of M 21 、M 22 、M 23 、M 24 Representing a free weight matrix M 2 Component N of (2) 21 、N 22 、N 23 、N 24 Representing a free weight matrix N 2 I represents an identity matrix;
s7, solving a linear matrix inequality according to the sufficiency condition for ensuring the exponential stability of the whole complex network system obtained in the step 6, and obtaining an event triggering parameter and a controller feedback gain;
the method for acquiring the event triggering parameters and the feedback gain of the controller comprises the following steps:
setting a scalar alpha j' ＞0，μ j' ＞0，κ j' ＞0，e j' >0，s j' ＞0，j'＝1,2，1＞ρ i > 0, i=1, 2..N, sampling time interval h > 0, dos parameter η > 0,
Figure QLYQS_84
matrix Λ, P j' ＞0，X j' ＞0，Y j' ＞0，/>
Figure QLYQS_85
Figure QLYQS_86
and />
Figure QLYQS_87
For a matrix of appropriate dimensions, so that the following linear matrix inequality holds, the system state estimation model is exponentially stable and the decay rate is +.>
Figure QLYQS_88
Figure QLYQS_89
Figure QLYQS_90
Figure QLYQS_91
Figure QLYQS_92
Figure QLYQS_93
wherein ：
Figure QLYQS_94
Figure QLYQS_95
Figure QLYQS_96
Figure QLYQS_97
Figure QLYQS_98
Figure QLYQS_99
the controller feedback gain is:
Figure QLYQS_100
/>
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