CN113625684B - Design method of tracking controller based on event trigger mechanism under hybrid network attack - Google Patents
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Abstract
The invention discloses a tracking controller and a method based on an event trigger mechanism under hybrid network attack. According to the current sampling data and the latest transmission data, establishing a trigger condition based on an event trigger mechanism; and respectively considering the influence of the deception attack and the DoS attack on the transmission data, and establishing a hybrid network attack model. And establishing an error system of the tracking controller and giving a tracking performance index. Sufficient conditions and controller gains to ensure tracking performance of the tracking system are obtained. The invention can effectively save network bandwidth resources and ensure the effectiveness of the system.
Description
Technical Field
The invention belongs to the field of network control, and particularly relates to a design method of a tracking controller with an event trigger mechanism and hybrid network attacks (including spoofing attacks and denial of service attacks).
Background
The goal of output tracking control is to ensure that the system output tracks the known reference model as closely as possible through a suitable controller. With the development of industry, the actual demand is gradually increased, the structure of a control system is increasingly complex, the position of a networked control system in the control system is more and more important, and tracking control is used as a basic problem in the research of control theory and application research and is widely applied in modern industry. Therefore, the research on the tracking control problem of the networked control system has certain theoretical and practical significance.
With the gradual expansion of the scale of the networked control system, the structure of the system is increasingly complex, and important problems such as network delay, data packet loss, transmission limitation and the like are inevitably brought, and the problems not only reduce the control performance of the system, but also influence the stability of the system. In addition, because the bandwidth of the communication channel in the networked control system is limited, the network load is increased, the network is blocked, and the like. The present invention therefore introduces an event triggering mechanism in order to reduce unnecessary waste of bandwidth resources in the network.
The wide introduction of the network system improves the performance of the control system in more aspects, such as resource sharing, convenient maintenance and the like. But at the same time, the system also generates a plurality of potential safety hazards, and the network system is easy to be attacked by the network. Generally, the cyber attack includes a replay attack, a spoofing attack, a denial of service (DoS attack), and the like. The basic principle of replay attacks is to transmit the previously intercepted data intact to the recipient. Spoofing attacks typically replace the actual data of the system with fake data to achieve the specific goals of the attacker. The DoS attack is to send a large number of requests to a server, occupy server resources, and make a user unable to respond in time, thereby losing normal network service.
However, it is understood that most of the existing research results only study one kind of network attack, but actually, these systems may suffer from various network attacks at the same time. To be closer to reality, two common cyber attacks are considered herein, including spoofing attacks and DoS attacks. To our knowledge, there is currently no relevant research effort to study the tracking control problem of networked control systems with event-triggered mechanisms and hybrid network attacks.
Disclosure of Invention
The purpose of the invention is as follows: in order to overcome the defects in the prior art, the invention provides a tracking controller and a method based on an event trigger mechanism under hybrid network attack, which can effectively save network bandwidth resources and ensure the effectiveness of a system by introducing an event trigger scheme while considering the influence of DoS attack and deception attack on network security.
The technical scheme is as follows: in order to achieve the purpose, the invention adopts the technical scheme that:
a tracking controller based on an event trigger mechanism under hybrid network attack is disclosed, wherein an error system of the tracking controller is as follows:
wherein,
denotes the derivative of e (t), t denotes the time, e (t) denotes the tracking error, e (t) x (t) -x r (t), x (t) denotes a systematic vector, x r (t) represents a reference model system vector, alpha (t) is a Bernoulli variable used for describing whether the spoofing attack occurs or not, wherein when the value is 1, the value does not occur, and when the value is 0, the value occurs; A. b, C, D, E, K is the required controller gain,
representing the actual input to the controller; h (e (t) k,n h) A non-linear function representing a spoofing attack; eta of 0 ≦ k,n (t)≤η m Representing a time delay, eta k,n (t) represents a time delay, η m An upper bound of the time lag is indicated,
definition of W e (t)＝(A-D)x r (t) + Cω (t) -Er (t), ω (t) representing system external disturbance, r (t) representing bounded reference input vector, ε k,n (t) represents the error threshold between the last transmitted signal and the current sampled signal, V 1,n-1 Indicating the moment at which DoS attack does not occur, V 2,n-1 Indicating the moment at which the DoS attack occurred.
The tracking performance indexes to be met are set as follows:
where U is a positive definite matrix. Gamma > 0 is a tracking performance indicator. t is t f Indicating the termination time and U represents a matrix of appropriate dimensions.
Preferably: the sufficient conditions of the tracking performance of the tracking controller are as follows:
Ξ 1 ＜0
Ξ 2 ＜0
the constraint conditions are as follows:
wherein: i is 1, 2; xi 1 Denotes an intermediate parameter I, xi 2 Representing the intermediate parameter two, P 1 、P 2 、Q i 、Q 3-i 、R i 、 R 3-i 、Z i 、Z 3-i All represent positive definite matrices; tau is 2 、τ 1 、β 1 、β 2 、τ 3-i 、
z min 、ι D Represents a given positive parameter; h denotes a sampling period.
Preferably: xi 1 The following were used:
Σ 11 ＝2β 1 P 1 +P 1 A+A T P 1 +Q 1 -g 1 R 1 -g 1 Z 1 n+U
Σ 32 ＝g 1 (R 1 +S 1 +Z 1 +M 1 )
Σ 21 ＝αK T B T P 1 +g 1 R 1 +g 1 S 1 +g 1 Z 1 +g 1 M 1
Σ 31 ＝-g 1 S 1 -g 1 M 1
Σ 32 ＝g 1 (R 1 +S 1 +Z 1 +M 1 )
Σ 33 ＝-g 1 Q 1 -g 1 R 1 -g 1 Z 1
Δ 22 ＝diag{-Ω,-γ 2 I,-I}
where α represents the expected value of the Bernoulli variable α (t), I represents the identity matrix of the appropriate dimension, Ω, L represent known matrices, η m 、ρ、β 1 Indicating a given positive parameter, U, R 1 、M 1 Representing a matrix of suitable dimensions, K representing the controller gain
It is preferable that：Ξ 2 The following were used:
Y 11 ＝-2β 2 P 2 +P 2 A+A T P 2 +Q 2 -g 2 Z 2 +U-g 2 R 2
Y 21 ＝g 2 (Z 2 +M 2 +R 2 +S 2 )
Y 31 ＝-g 2 M 2 -g 2 S 2
Y 32 ＝g 2 (Z 2 +M 2 +R 2 +S 2 )
Y 33 ＝-g 2 (Q 2 +Z 2 +R 2 )
Y 51 ＝η m P 2 A
Y 54 ＝η m P 2
Y 55 ＝-P 2 (R 2 +Z 2 ) -1 P 2
wherein, beta 2 Representing a known vector, Z 2 、M 2 Representing a matrix with appropriate dimensions.
Preferably: controller gain K of the tracking controller:
give DoS parameters
z min 、η m 、γ、ι D And positive scalar parameters alpha, h, c, beta 1 、β 2 、τ 1 、τ 2 Matrix L, if present
Matrix array
With the appropriate dimensions, we use the linear inequality to obtain:
Γ 1 ＜0
Γ 2 ＜0
the constraint conditions are as follows:
Λ 51 ＝η m AX 2
Λ 54 ＝η m
wherein i is 1, 2; y denotes a matrix, X 1 Representing a positive definite matrix, X 2 Denotes a positive definite matrix, τ 1 Denotes a positive real number, τ 2 Which represents a positive real number, is,
a positive definite matrix is represented and,
a positive definite matrix is represented and,
a positive definite matrix is represented and,
a positive definite matrix is represented and,
a matrix representing a suitable dimension is then formed,
a matrix representing a suitable dimension is then formed,
a positive definite matrix is represented and,
denotes positive real number, beta 2 Which represents a positive real number, is,
a matrix representing a suitable dimension is then formed,
a matrix representing a suitable dimension is then formed,
a positive definite matrix is represented and,
a positive definite matrix is represented and,
representing a positive definite matrix.
Preferably: the method comprises the following steps of:
the data transmitted under a spoofing attack is represented as:
wherein,
representing the transmitted data under the spoofing attack, alpha (t) is a Bernoulli variable used for representing whether the spoofing attack occurs or not, wherein 0 represents occurrence, 1 represents non-occurrence, and h (e (t) is k,n h) A non-linear function representing a spoofing attack, e (t) k,n h) Representing the transmitted data after passing through the transmission mechanism, e (t) k,n h)＝ε k,n (t)+e(t-η k,n (t))，ε k,n (t) represents the error threshold, η, between the last transmitted signal and the current sampled signal k,n (t) represents a time delay, and t represents a time;
the transmission data under DoS attack is represented as:
wherein,
showing the transmission data under the DoS attack, and zeta (t) represents the state of the DoS attack. ζ (t) ═ 1 denotes when t ∈ [ F ] n ,F n +z n ) Meanwhile, the DoS attack is in a sleep state. ζ (t) ═ 0 denotes when
t∈[F n +z n ,F n+1 ) Meanwhile, the DoS attack is in an active state, and the system is attacked by the DoS. F n Indicating the opening of the nth active periodFirst, F n+1 Indicating the end of the nth active period and the beginning of the (n + 1) th sleep period. z is a radical of n Indicating the length of the sleep period. Therefore, the start and end of DoS attack sleep cycle need to satisfy:
0≤F 0 ＜F 1 ＜F 1 +z 1 ＜F 2 ＜…＜F n ＜F n +z n ＜F n+1 。
a design method of a tracking controller based on an event trigger mechanism under hybrid network attack mainly aims at designing a tracking controller for a networked control system with an event trigger mechanism and hybrid network attack. An event-triggered mechanism is employed to alleviate network bandwidth load. By utilizing the Lyapunov stability theory, a sufficient condition for ensuring the good tracking performance of the tracking controller is obtained. In addition, the controller gain is obtained by solving a set of linear matrix inequalities. Finally, the effectiveness of the method is verified through a simulation example, and the method specifically comprises the following steps:
step 1: and establishing a system preliminary model and a preliminary reference model based on a networked control system, and designing a preliminary tracking controller.
Step 2: firstly, an event trigger mechanism is introduced, the problem of network resource limitation can be solved by reducing unnecessary data transmission by introducing the event trigger mechanism, and specifically, a trigger condition based on the event trigger mechanism is established according to current sampling data and latest transmission data;
and step 3: and (3) respectively considering the influence of the deception attack and the DoS attack on the transmission data, and establishing a hybrid network attack model.
And 4, step 4: and (3) establishing an error system of the tracking controller and giving a tracking performance index according to the system initial model, the initial reference model and the initial tracking controller established in the step (1), the trigger condition based on the event trigger mechanism established in the step (2) and the hybrid network attack model established in the step (3).
And 5: and obtaining sufficient conditions for ensuring the tracking performance of the tracking system by utilizing the Lyapunov stability theory.
Step 6: and solving the linear inequality to obtain a controller gain K of the tracking controller.
Preferably: the system preliminary model, the preliminary reference model and the preliminary tracking controller in the step 1 are as follows:
and (3) a system preliminary model:
a primary reference model:
definitions e (t) ═ x (t) — x r (t), designing the following preliminary tracking controller:
wherein,
denotes the derivative of x (t), x (t) denotes the system vector, u (t) denotes the controller output vector, ω (t) denotes the external input disturbance,
denotes x r Derivative of (t), x r (t) represents the reference model system vector, r (t) represents the bounded reference input vector, A, B, C, D, E all represent matrices of suitable dimensions, e (t) represents the tracking error, and K is the controller gain required;
representing the actual input to the controller.
Preferably: triggering conditions based on an event triggering mechanism in the step 2:
wherein epsilon k (t) represents the error threshold between the last transmitted signal and the current sampled signal, Ω represents a free weight matrix of suitable dimensions, c represents an event-triggered scalar parameter, e (t) k h + sh) represents the current sample data, e (t) k h) Indicating the latest transmitted data, t k h denotes the latest transmission time, sh denotes the current sampling time, s denotes a positive integer, and h denotes the sampling period
ε k (t)＝e(t k h+sh)-e(t k h)，Ω＞0。
The next transmission instant t k+1 h is expressed as:
wherein,
representing a positive integer.
Preferably: and 4, establishing an error system of the tracking controller and providing a tracking performance index:
due to the existence of DoS attacks, the transmission data will be affected, and the trigger condition based on the event trigger mechanism established in step 2 will no longer be applicable, so in consideration of the influence of DoS attacks, the transmission time is redefined as:
t k,n h＝{t k,a hsatisfying(1)|t k,a h∈V n-1,1 }∪{F n }
wherein, t k,n h denotes the trigger time, t, of the cycle in the nth DoS attack k,a h represents the trigger time of the a-th DoS period, n represents n DoS attack periods, a represents the a-th DoS period, k represents the number of trigger times in the n-th DoS period, and t k,a h. n, a are all non-negative integers;
definition of
Event interval X k,n The method is divided into the following cells:
the inter-cell representation of the event interval is as follows:
and is
V 1,n The method is divided into the following steps:
wherein,
two piecewise functions are defined:
and
then, the transmission data after the event triggering mechanism is represented as:
e(t k,n h)＝ε k,n (t)+e(t-η k,n (t)), wherein η k,n (t)∈[0,η M )。
The event triggering conditions at this time are:
wherein eta is k,n (t) represents the system time delay, ∈ k,n (t) represents the error threshold, η, between the last transmitted signal and the current sampled signal M Represents the time lag upper limit, and Ω represents a matrix of suitable dimensions;
and establishing an error system of the tracking controller according to the event triggering condition at the moment.
Compared with the prior art, the invention has the following beneficial effects:
1. the invention establishes a mathematical model of complex network attack aiming at a networked control system on the basis of considering deception attack and DoS attack.
2. And limited bandwidth is saved by adopting an event triggering scheme.
3. And (4) giving sufficient conditions of the tracking performance of the tracking controller by utilizing the Lyapunov theory.
4. The gain of the controller can be derived by solving a series of linear matrix inequalities.
Drawings
FIG. 1: the networked control system tracks the control plan.
FIG. 2: system x 1 (t) state trajectory and reference system x r1 (t) state trace.
FIG. 3: system x 2 (t) state trajectory and reference system x r2 (t) state trace.
FIG. 4: signal of DoS attack.
FIG. 5: the event triggers the release time and interval.
FIG. 6: and (4) spoofing the attack occurrence moment.
Detailed Description
The present invention is further illustrated in the accompanying drawings and described in the following detailed description, it is to be understood that such examples are included solely for the purposes of illustration and are not intended as a definition of the limits of the invention, since various equivalent modifications of the invention will become apparent to those skilled in the art after reading the present specification, and it is intended to cover all such modifications as fall within the scope of the invention as defined in the appended claims.
A method for designing a tracking controller based on an event trigger mechanism under a hybrid network attack, as shown in fig. 1, includes the following steps:
step 1: and establishing a system preliminary model and a preliminary reference model based on a networked control system, and designing a preliminary tracking controller.
And (3) a system preliminary model:
a primary reference model:
definitions e (t) ═ x (t) — x r (t), designing the following preliminary tracking controller:
in (1),
denotes the derivative of x (t), x (t) denotes the system vector, u (t) denotes the controller output vector, ω (t) denotes the external input disturbance,
denotes x r Derivative of (t), x r (t) represents the reference model system vector, r (t) represents the bounded reference input vector, A, B, C, D, E all represent matrices of suitable dimensions, e (t) represents the tracking error, and K is the controller gain required;
representing the actual input to the controller.
Step 2: in order to effectively save bandwidth, an event triggering scheme is introduced: according to the current sampling data and the latest transmission data, establishing a trigger condition based on an event trigger mechanism;
wherein epsilon k (t) represents the error threshold between the last transmitted signal and the current sampled signal, Ω represents a free weight matrix of suitable dimensions, c represents an event trigger scalar parameter, e (t) k h + sh) represents the current sample data, e (t) k h) Indicating the latest transmitted data, t k h represents the latest transmission time, sh represents the current sampling time, s represents a positive integer, and h represents the sampling period;
ε k (t)＝e(t k h+sh)-e(t k h)，Ω＞0。
the next transmission instant t k+1 h is expressed as:
wherein,
representing a positive integer.
And step 3: and respectively considering the influence of the deception attack and the DoS attack on the transmission data, and establishing a hybrid network attack model.
The data transmitted under a spoofing attack is represented as:
wherein,
representing the transmitted data under the spoofing attack, alpha (t) is a Bernoulli variable used for representing whether the spoofing attack occurs or not, wherein 0 represents occurrence, 1 represents non-occurrence, and h (e (t) is k,n h) A non-linear function representing a spoofing attack, e (t) k,n h) Representing the transmitted data after passing through the transmission mechanism, e (t) k,n h)＝ε k,n (t)+e(t-η k,n (t))，ε k,n (t) represents the error threshold, η, between the last transmitted signal and the current sampled signal k,n (t) represents a time delay, and t represents a time;
the transmission data under DoS attack is represented as:
wherein,
represents the transmission data under DoS attack, and ζ (t) represents the state of DoS attack. ζ (t) ═ 1 denotes when t ∈ [ F ] n ,F n +z n ) Meanwhile, the DoS attack is in a sleep state. ζ (t) ═ 0 denotes when t ∈ [ F ] n +z n ,F n+1 ) Meanwhile, the DoS attack is in an active state, and the system is attacked by the DoS. F n Indicating the start of the nth active period, F n+1 Indicating the end of the nth active period and the beginning of the (n + 1) th sleep period. z is a radical of n Indicating the length of the sleep period. Therefore, the beginning and end of DoS attack sleep cycles need to be satisfied:
0≤F 0 ＜F 1 ＜F 1 +z 1 ＜F 2 ＜…＜F n ＜F n +z n ＜F n+1 。
and 4, step 4: and (3) establishing an error system of the tracking controller and giving a tracking performance index according to the system initial model, the initial reference model and the initial tracking controller established in the step (1), the trigger condition based on the event trigger mechanism established in the step (2) and the hybrid network attack model established in the step (3).
Due to the existence of DoS attacks, the transmission data will be affected, and the trigger condition based on the event trigger mechanism established in step 2 will no longer be applicable, so in consideration of the influence of DoS attacks, the transmission time is redefined as:
t k,n h＝{t k,a hsatisfying(1)|t k,a h∈V n-1,1 }∪{F n }
wherein, t k,n h denotes attack at the nth DoSTrigger time of stroke period, t k,a h represents the trigger time of the a-th DoS period, n represents n DoS attack periods, a represents the a-th DoS period, and t represents k,a h. n, a are all non-negative integers.
Definition of
Event interval X k,n The method is divided into the following cells:
the inter-cell representation of the event interval is as follows:
and is
V 1,n The method is divided into the following steps:
wherein,
two piecewise functions are defined:
and
then, the transmission data after the event triggering mechanism is represented as:
e(t k,n h)＝ε k,n (t)+e(t-η k,n (t)), wherein η k,n (t)∈[0,η M )。
The event triggering conditions at this time are:
wherein eta is k,n (t) represents the system time delay, ∈ k,n (t) represents the error threshold, η, between the last transmitted signal and the current sampled signal M Represents the time lag upper limit, and Ω represents a matrix of suitable dimensions;
the following tracking controller error system was set up:
wherein e (t) x (t) -x r (t), x (t) denotes a systematic vector, x r (t) denotes the reference model system vector, A, B, C, D, E denotes a matrix of appropriate dimensions, e (t) denotes the tracking error, and K is the required controller gain.
Representing the actual input to the controller. α (t) is a bernoulli variable used to describe whether a spoofing attack occurred, where a value of 1 indicates no occurrence and 0 indicates occurrence. Suppose h (e (t) k,n h) ) represents a non-linear function of a spoofing attack. Eta of 0 ≦ k,n (t)≤η m Representing a time delay. W e (t)＝(A-D)x r (t)+Cω(t)-Er(t)。
The tracking performance indexes to be met are set as follows:
where U is a positive definite matrix. Gamma > 0 is a tracking performance indicator. t is t f Indicating the termination time.
And 5: and a Lyapunov stability theory is utilized to obtain a sufficient condition for ensuring the tracking performance of the tracking system.
Give DoS parameters
z min 、η m 、γ、ι D And positive scalar parameters alpha, h, c, beta 1 、β 2 、τ 1 、τ 2 A matrix K, L, P if there is a matrix Ω > 0 1 >0、P 2 >0、 Q 1 ＞0,Q 2 ＞0,R 1 ＞0,R 2 ＞0,Z 1 ＞0,Z 2 > 0, matrix U, S 1 ,S 2 ,M 1 ,M 2 With the dimensions in place, the following inequality holds:
Ξ 1 ＜0
Ξ 2 ＜0
the constraint conditions are as follows:
wherein:
Σ 11 ＝2β 1 P 1 +P 1 A+A T P 1 +Q 1 -g 1 R 1 -g 1 Z 1 n+U,
Σ 32 ＝g 1 (R 1 +S 1 +Z 1 +M 1 )
Σ 21 ＝αK T B T P 1 +g 1 R 1 +g 1 S 1 +g 1 Z 1 +g 1 M 1
Δ 22 ＝diag{-Ω,-γ 2 I,-I}
Y 11 ＝-2β 2 P 2 +P 2 A+A T P 2 +Q 2 -g 2 Z 2 +U-g 2 R 2
Y 31 ＝-g 2 M 2 -g 2 S 2
Y 21 ＝g 2 (Z 2 +M 2 +R 2 +S 2 )
Y 32 ＝g 2 (Z 2 +M 2 +R 2 +S 2 )
Y 33 ＝-g 2 (Q 2 +Z 2 +R 2 )
Y 51 ＝η m P 2 A,Y 54 ＝η m P 2 ,Y 55 ＝-P 2 (R 2 +Z 2 ) -1 P 2
step 6: and solving the linear inequality to obtain a controller gain K of the tracking controller.
Give DoS parameters
z min 、η m 、γ、ι D And positive scalar parameters alpha, h, c, beta 1 、β 2 、τ 1 、τ 2 Matrix L, if present
Matrix array
With the appropriate dimensions, this can be obtained using the linear inequality:
Γ 1 ＜0
Γ 2 ＜0
the constraint conditions are as follows:
the required controller gains are:
wherein:
simulation analysis:
the Matlab program is written to solve the linear matrix inequality to solve the gain of the tracking controller and draw a simulation curve, and the effectiveness of the method is demonstrated by using a simulation example.
Consider the parameters in the system model as:
B＝ [ 01 ] T ,C＝ [ 01 ] T
and the external disturbance input is: ω (t) ═ 8sin (t-0.5).
The reference model is:
wherein: r (t) sin (t + 0.5).
The nonlinear function under a spoofing attack is: h (e (t) [ -tanh [) T (0.15e 1 (t)) -tanh T (0.05e 2 (t))] T ；
The following scalar parameters are set: beta is a 1 ＝0.15,β 2 ＝2,τ 1 ＝1.02,τ 2 ＝1.02,η m ＝0.2,z min ＝1.3，
The event trigger parameter c is 0.4; the index parameter of the tracking performance is gamma which is 0.7; the bernoulli variable α is 0.6, indicating that the system is subject to spoofing and DoS attacks. Initial conditions were set to x (0) ═ 0.2-0.1] T ,x r (0)＝[0.5 0.1] T . By solving the linear inequality by Matlab, the following matrix parameters can be obtained:
Y＝[0.6030 -1.8109]，
by expression
The controller gain can be obtained as
K＝[0.0560 -0.0245]。
Fig. 2-5 were obtained from Matlab simulations. FIGS. 2 and 3 show the state traces of system x (t) and reference system x r (t), therefore, the designed method can ensure that the system state can track the state of the reference model and has good tracking performance; FIG. 4 shows signals of a DoS attack; FIG. 5 illustrates the event triggered release times and intervals; fig. 6 shows the occurrence time of a spoofing attack.
From the images obtained above, the following conclusions can be drawn: the controller of the networked control system based on the event trigger and the hybrid network attack can realize good tracking control.
The above description is only of the preferred embodiments of the present invention, and it should be 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 invention and these are intended to be within the scope of the invention.
Claims (8)
1. A design method of a tracking controller based on an event trigger mechanism under hybrid network attack is characterized by comprising the following steps:
step 1: establishing a system preliminary model and a preliminary reference model based on a networked control system, and designing a preliminary tracking controller;
step 2: according to the current sampling data and the latest transmission data, establishing a trigger condition based on an event trigger mechanism;
and step 3: respectively considering the influence of the deception attack and the DoS attack on the transmission data, and establishing a hybrid network attack model;
and 4, step 4: establishing a tracking controller error system and giving a tracking performance index according to the system initial model, the initial reference model and the initial tracking controller established in the step 1, the trigger condition based on the event trigger mechanism established in the step 2 and the hybrid network attack model established in the step 3;
the error system of the tracking controller is as follows:
wherein,
denotes the derivative of e (t), t denotes the time, e (t) denotes the tracking error, e (t) x (t) -x r (t), x (t) denotes a systematic vector, x r (t) represents a reference model system vector, alpha (t) is a Bernoulli variable used for describing whether the spoofing attack occurs or not, wherein the value is 1 to represent that the attack does not occur, and the value is 0 to represent that the attack occurs; A. b, C, D, E, K is the required controller gain,
representing the actual input to the controller; h (e (t) k,n h) A non-linear function representing a spoofing attack; eta of 0 ≦ k,n (t)≤η m Representing a time delay, η k,n (t) represents the time delay, η m Represents the upper bound of the time lag, defines W e (t)＝(A-D)x r (t) + Cω (t) -Er (t), ω (t) representing the system external disturbance, r (t) representing the bounded reference input vector, ε k,n (t) represents the error threshold between the last transmitted signal and the current sampled signal, V 1,n-1 Indicating the moment at which DoS attack does not occur, V 2,n-1 Representing the moment of occurrence of the DoS attack;
the tracking performance indexes to be met are set as follows:
wherein U is a positive definite matrix; gamma > 0 is a tracking performance index; t is t f Represents the termination time;
and 5: obtaining sufficient conditions for ensuring the tracking performance of the tracking system by utilizing the Lyapunov stability theory;
the sufficient conditions of the tracking performance of the tracking controller are as follows:
Ξ 1 ＜0
Ξ 2 ＜0
the constraint conditions are as follows:
wherein: i is 1, 2; xi 1 Denotes the intermediate parameter one, xi 2 Representing the intermediate parameter two, P 1 、P 2 、Q i 、Q 3-i 、R i 、R 3-i 、Z i 、Z 3-i All represent positive definite matrices; tau is 2 、τ 1 、β 1 、β 2 、τ 3-i 、
z min 、ι D Represents a given positive parameter; h represents a sampling period;
step 6: and solving the linear inequality to obtain a controller gain K of the tracking controller.
2. The method for designing a tracking controller based on an event trigger mechanism under the hybrid network attack according to claim 1, wherein: the system preliminary model, the preliminary reference model and the preliminary tracking controller in the step 1 are as follows:
and (3) a system preliminary model:
a primary reference model:
definitions e (t) ═ x (t) — x r (t), designing the following preliminary tracking controller:
wherein,
denotes the derivative of x (t), x (t) denotes the system vector, u (t) denotes the controller output vector, ω (t) denotes the external input disturbance,
denotes x r Derivative of (t), x r (t) represents the reference model system vector, r (t) represents the bounded reference input vector, A, B, C, D, E all represent matrices of suitable dimensions, e (t) represents the tracking error, and K is the controller gain required;
representing the actual input to the controller.
3. The method for designing a tracking controller based on an event trigger mechanism under the hybrid network attack as claimed in claim 2, wherein: triggering conditions based on an event triggering mechanism in the step 2:
wherein epsilon k (t) represents the error threshold between the last transmitted signal and the current sampled signal, Ω represents a free weight matrix of suitable dimensions, c represents an event trigger scalar parameter, e (t) k h + sh) represents the current sample data, e (t) k h) Indicating the latest transmitted data, t k h represents the latest transmission time, sh represents the current sampling time, s represents a positive integer, and h represents the sampling period;
ε k (t)＝e(t k h+sh)-e(t k h)，Ω＞0；
the next transmission instant t k+1 h is expressed as:
wherein,
representing a positive integer.
4. The method for designing a tracking controller based on an event trigger mechanism under the hybrid network attack as claimed in claim 3, wherein: the method for establishing the hybrid network attack model in the step 3 comprises the following steps:
the data transmitted under a spoofing attack is represented as:
wherein,
representing the transmitted data under the spoofing attack, alpha (t) is a Bernoulli variable used for representing whether the spoofing attack occurs or not, wherein 0 represents occurrence, 1 represents non-occurrence, and h (e (t) is k,n h) A non-linear function representing a spoofing attack, e (t) k,n h) Representing the transmitted data after passing through the transmission mechanism, e (t) k,n h)＝ε k,n (t)+e(t-η k,n (t))，ε k,n (t) represents the error threshold, η, between the last transmitted signal and the current sampled signal k,n (t) represents a time delay, and t represents a time;
the transmission data under DoS attack is represented as:
wherein,
represents the transmission data under the DoS attack, and ζ (t) represents the state of the DoS attack; ζ (t) ═ 1 denotes when t ∈ [ F ] n ,F n +z n ) Meanwhile, the DoS attack is in a dormant state; ζ (t) ═ 0 indicates when t ∈ [ F ] n +z n ,F n+1 ) In the process, the DoS attack is in an active state, and the system is attacked by the DoS; f n Indicating the start of the nth active period, F n+1 Indicating the end of the nth active period and the beginning of the (n + 1) th sleep period; z is a radical of n Represents the length of the sleep period; therefore, the start and end of DoS attack sleep cycle need to satisfy:
0≤F 0 ＜F 1 ＜F 1 +z 1 ＜F 2 ＜…＜F n ＜F n +z n ＜F n+1 ；
for writing convenience, order
5. The method for designing a tracking controller based on an event trigger mechanism under the hybrid network attack as claimed in claim 4, wherein: and 4, establishing an error system of the tracking controller and giving a tracking performance index:
due to the existence of DoS attacks, the transmission data will be affected, and the trigger condition based on the event trigger mechanism established in step 2 will no longer be applicable, so in consideration of the influence of DoS attacks, the transmission time is redefined as:
t k,n h＝{t k,a h satisfies the formula (1) | t k,a h∈V 1,n-1 }∪{F n }
Wherein, t k,n h denotes the trigger time, t, of the cycle in the nth DoS attack k,a h represents the trigger time of the a-th DoS period, n represents n DoS attack periods, a represents the a-th DoS period, k represents the number of trigger times in the n-th DoS period, and t k,a h. n, a are all non-negative integers;
definition of
Event interval X k,n The method is divided into the following cells:
the inter-cell representation of the event interval is as follows:
and is
V 1,n The method is divided into the following steps:
wherein,
two piecewise functions are defined:
and
then, the transmission data after the event triggering mechanism is represented as:
e(t k,n h)＝ε k,n (t)+e(t-η k,n (t)), wherein η k,n (t)∈[0,η M )；
The event triggering conditions at this time are:
wherein eta is k,n (t) represents the system time delay, ε k,n (t) represents the error threshold, η, between the last transmitted signal and the current sampled signal M Represents the time lag upper limit, and Ω represents a matrix of suitable dimensions;
and establishing an error system of the tracking controller according to the event triggering condition at the moment.
6. The method for designing a tracking controller based on an event trigger mechanism under the hybrid network attack as claimed in claim 5, wherein: intermediate parameter xi 1 The following were used:
Σ 11 ＝2β 1 P 1 +P 1 A+A T P 1 +Q 1 -g 1 R 1 -g 1 Z 1 +U
Σ 32 ＝g 1 (R 1 +S 1 +Z 1 +M 1 )
Σ 21 ＝αK T B T P 1 +g 1 R 1 +g 1 S 1 +g 1 Z 1 +g 1 M 1
Σ 31 ＝-g 1 S 1 -g 1 M 1
Σ 32 ＝g 1 (R 1 +S 1 +Z 1 +M 1 )
Σ 33 ＝-g 1 Q 1 -g 1 R 1 -g 1 Z 1
Δ 22 ＝diag{-Ω,-γ 2 I,-I}
Δ 33 ＝diag{-I,-P 1 (R 1 +Z 1 ) -1 P 1 ,-P 1 (R 1 +Z 1 ) -1 P 1 },
where α represents the expected value of the Bernoulli variable α (t), I represents the identity matrix of the appropriate dimension, Ω, L represent known matrices, η m 、ρ、β 1 Indicating a given positive parameter, U, R 1 、M 1 A matrix with appropriate dimensions is shown and K represents the controller gain.
7. The method for designing a tracking controller based on an event trigger mechanism under the hybrid network attack as claimed in claim 6, wherein: intermediate parameter two xi 2 The following were used:
Y 11 ＝-2β 2 P 2 +P 2 A+A T P 2 +Q 2 -g 2 Z 2 +U-g 2 R 2
Y 21 ＝g 2 (Z 2 +M 2 +R 2 +S 2 )
Y 31 ＝-g 2 M 2 -g 2 S 2
Y 32 ＝g 2 (Z 2 +M 2 +R 2 +S 2 )
Y 33 ＝-g 2 (Q 2 +Z 2 +R 2 )
Y 51 ＝η m P 2 A
Y 54 ＝η m P 2
Y 55 ＝-P 2 (R 2 +Z 2 ) -1 P 2
wherein, beta 2 Representing a known vector, Z 2 、M 2 Representing a matrix with appropriate dimensions.
8. The method for designing a tracking controller based on an event trigger mechanism under the hybrid network attack as claimed in claim 7, wherein: controller gain K of the tracking controller:
give DoS parameters
z min 、η m 、γ、ι D And positive scalar parameters alpha, h, c, beta 1 、β 2 、τ 1 、τ 2 Matrix L, if present
Matrix array
Y has a suitable dimension and is obtained using the linear inequality:
Γ 1 ＜0
Γ 2 ＜0
the constraint conditions are as follows:
Λ 51 ＝η m AX 2
Λ 54 ＝η m
where i ═ 1,2, Y denotes a matrix, X 1 Representing a positive definite matrix, X 2 Denotes a positive definite matrix, τ 1 Denotes a positive real number, τ 2 Which represents a positive real number, is,
a positive definite matrix is represented and,
a positive definite matrix is represented and,
a positive definite matrix is represented and,
a positive definite matrix is represented and,
a matrix representing a suitable dimension is then formed,
a matrix representing a suitable dimension is then formed,
a positive definite matrix is represented and,
denotes positive real number, beta 2 Which represents a positive real number, is,
a matrix representing a suitable dimension is then formed,
a matrix representing a suitable dimension is then formed,
a positive definite matrix is represented and,
a positive definite matrix is represented and,
representing a positive definite matrix.
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