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CN113110182B - Fault-tolerant controller design method of leader following multi-agent system - Google Patents

Fault-tolerant controller design method of leader following multi-agent system Download PDF

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CN113110182B
CN113110182B CN202110418213.6A CN202110418213A CN113110182B CN 113110182 B CN113110182 B CN 113110182B CN 202110418213 A CN202110418213 A CN 202110418213A CN 113110182 B CN113110182 B CN 113110182B
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fault
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failure
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CN113110182A (en
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李建宁
缪坤忠
陈杨杰
王爱民
刘晓
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Hangzhou Dianzi University
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
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Abstract

本发明涉及一种领导跟随多智能体系统的容错控制器设计方法。该发明假定通信图是有向的和固定的，首先,基于故障的分布特征建立故障模型。而网络攻击模型由相互独立的随机伯努利变量描述，并采用拓扑结构分割的方法进行求解，为了避免零初始条件在传统的控制方法中，提出了适合于领导‑跟随的性能指标函数，接着，根据提供的故障模型和性能指标，设计了一种新型的容错控制器来确保领导‑跟随多智能体系统在发生网络攻击和执行器故障时仍可以实现容错均方一致性。最后，通过数值算例验证了该发明的有效性。

Figure 202110418213

The invention relates to a design method of a fault-tolerant controller for a leader-following multi-agent system. The invention assumes that the communication graph is directed and fixed. First, a fault model is established based on the distribution characteristics of faults. The network attack model is described by independent random Bernoulli variables and solved by the method of topological structure segmentation. In order to avoid zero initial conditions in the traditional control method, a performance index function suitable for leader-following is proposed, and then , according to the provided fault model and performance indicators, a novel fault-tolerant controller is designed to ensure that the leader-follower multi-agent system can still achieve fault-tolerant mean square consistency in the event of network attacks and actuator failures. Finally, the effectiveness of the invention is verified by a numerical example.

Figure 202110418213

Description

Fault-tolerant controller design method of leader following multi-agent system
Technical Field
The invention relates to a design method of a fault-tolerant controller of a leader follower multi-agent system, belonging to the field of fault-tolerant control.
Background
In recent decades, multi-agent systems have become widely used and are part of artificial intelligence. A multi-agent system is a computing system composed of a plurality of agents interacting in an environment, a complex large system is built into a plurality of small systems mutually communicating and coordinating, each small system is used as an agent easy to manage, and the cooperative relationship of coordination and interaction communication among the agents is emphasized, so that all the agents are converged to a state finally. Compared with a single intelligent agent system, the networked multi-intelligent agent system has higher efficiency, and more people develop research on the multi-intelligent agent system. Furthermore, multi-agent has wide application in many fields, such as cooperative control of Unmanned Aerial Vehicles (UAVs), formation control, clustering, etc., and the consistency problem is one of the key problems in multi-agent systems, and in order to achieve consistency, a distributed control law needs to be designed for each agent so that the final states of all agents approach the same.
With the complication of the multi-agent system, the possibility of failure is increased, the failure of one component can be evolved into the failure of the whole system, and in practical application, the actuator of the system can also be inevitably deviated, jammed and partially failed. Fault tolerant control is a control method that can automatically maintain the stability of a system and maintain a certain level of system performance when a system component fails. The introduction of fault tolerant control can prevent small faults from developing into a big problem. The method has profound practical significance for the research of fault-tolerant control.
In recent years, the rapid development of networks has made the control field more closely connected with the networks, and due to the introduction of networks, the efficiency of many aspects of control systems has been improved, for example, the processing speed of control systems has become faster. However, it also brings huge challenges such as data packet loss, network attack, network delay, etc., wherein network attack is a recent research hotspot. At present, there are some achievements about network attacks, and the existing literature researches the stability of a network control system under random network attacks. The above studies only consider one factor affecting the performance of the system, whereas in practical cases the system may be affected by multiple factors and the probability of a random cyber-attack occurring in the control input of each agent obeys a random bernoulli distribution.
Disclosure of Invention
Aiming at the defects of the prior art, the invention discloses a leader following multi-agent system fault-tolerant consistency method based on network attack and fault distribution. In order to improve the stability of the control system, H is provided when the system has actuator failure fault∞The invention designs a fault-tolerant controller leading to follow the multi-agent system, so that the system can keep the stable performance when an actuator fails and network attacks occur.
Aiming at the leader, selecting the following state system model:
Figure BDA0003026756710000021
wherein x is0(t)∈RnIs the state quantity of the system leader,
Figure BDA0003026756710000022
is the input of the system leader.
The equation of state for the ith follower agent:
Figure BDA0003026756710000023
wherein x isi(t)∈RnIs the state quantity of the ith follower,
Figure BDA0003026756710000024
representing the control input of the follower actuator,
Figure BDA0003026756710000025
is an external disturbance of the system. Matrix BwSatisfies BwBF A, B and BwIs the state matrix of the system with the appropriate dimensions and F is the known real matrix.
And (2) constructing a failure model of the executor with the leaders following the multi-agent. A general fault model for an actuator is now given as follows:
Figure BDA0003026756710000026
wherein: m isi＝diag{mi,1,mi,2,...mi,s}ui＝diag{ui,1,ui,2,...ui,s}i＝1,2,...,N,j＝1,2,....,s，
Redefining the model of the partial failure matrix:
Figure BDA0003026756710000027
wherein: m isi,jRepresenting the coefficient of the ith agent for the jth actuator failing,
Figure BDA0003026756710000028
and
Figure BDA0003026756710000029
represents the failure coefficient mi,jUpper and lower bounds.
According to the failure characteristics and upper and lower boundaries:
Figure BDA00030267567100000210
redefining the failure matrix may result in:
Figure BDA00030267567100000211
the form of the amplification matrix is as follows:
Figure BDA0003026756710000031
Figure BDA0003026756710000032
Figure BDA0003026756710000033
Figure BDA0003026756710000034
Figure BDA0003026756710000035
wherein:
Figure BDA0003026756710000036
augmented form of probability of failure of ith agent jth actuator of representative leadership multi-agent system, Γi0And
Figure BDA0003026756710000037
two failure coefficients selected from the segmented failure intervals are selected, based on the provided failure model,
Figure BDA0003026756710000038
can be rewritten as:
Figure BDA0003026756710000039
step (3) designs the network attack model of the invention
In the present invention, network attacks are considered, which are implemented by injecting misleading numbers into regular transmission data. To reduce system performance, the invention uses a non-linear function fj(x (t)) to represent a random network attack.
Figure BDA00030267567100000310
Wherein:
Figure BDA00030267567100000311
is represented by the xi(t) the signal received by the agent is from the xth agentj(t) the signals of the agents under network attack, alpha is more than or equal to 0j(t) 1 is the x-thi(t) the signal received by the agent is from the xth agentj(t) probability of network-attacked signal, (reduced to network-attacked signal)
Figure BDA00030267567100000312
The possibility of occurrence).
Step (4) is to establish a consistent control law equation of the whole system aiming at steps (1), (2) and (3):
first, a consistent control law is given for the whole leader-follower multi-agent system, i.e. for any initial conditions, if satisfied,
Figure BDA00030267567100000313
the entire system may implement a fault-tolerant mean square consistency protocol.
Designing a fault-tolerant controller:
ui(t)＝Kei(t),i＝1,...,N (9)
wherein:
Figure BDA00030267567100000314
the combination of equations (1), (2), (3), (7), (8), (10) yields the consistency equation for the entire system:
Figure BDA0003026756710000041
wherein: giIs representative of the strength of the communication link between the leader and the follower, aijThe information communication strength between follower agents is represented, and the follower agents and the information communication strength form a topological structure between the whole leader-following multi-agent agents.
Step (5) is to establish an error state equation of the whole system for steps (1), (2), (3) and (4):
Figure BDA0003026756710000042
the invention adopts a method of segmenting a topological structure to write an error state equation of the whole system into a form of an augmentation matrix:
Figure BDA0003026756710000043
Figure BDA0003026756710000044
wherein:
Figure BDA0003026756710000045
Figure BDA0003026756710000046
Figure BDA0003026756710000047
Figure BDA0003026756710000048
Figure BDA0003026756710000049
and (6) aiming at the state equation of the system described in the step four, selecting a proper Lyapunov function as follows, so that the system (11) can realize consistent stability of mean square and H-infinity performance index.
The designed Lyapunov function of the invention is as follows:
Figure BDA0003026756710000051
i.e. given the correct controller gain K>0, constant number
Figure BDA0003026756710000052
Variable cm>0, i-1, 2,3,4, and a matrix Q, T, F of the appropriate dimension, if given a positive definite matrix P of the appropriate dimension>0, N, satisfying the following linear matrix inequality holds, the system of step (5) can be implemented in the mean square sense with H∞And achieving the fault-tolerant consistency of leader-following multi-agent under the condition of the interference level gamma.
Figure BDA0003026756710000053
Figure BDA0003026756710000054
Wherein:
Figure BDA0003026756710000055
Figure BDA0003026756710000056
Figure BDA0003026756710000057
Figure BDA0003026756710000058
Figure BDA0003026756710000059
in classical H∞In theory, the zero initial condition must be satisfied, based on this build performance index J:
Figure BDA00030267567100000510
wherein:
Figure BDA00030267567100000511
step (7) is a further optimization for step (6), i.e. designing the gain of the controller.
Giving an appropriate constant
Figure BDA0003026756710000061
Variable cm>0, i-1, 2,3,4 and a matrix Q, T, F of the appropriate dimension, if there is one positive dimensionDefinite matrix P>0, N, and gain of the controller
Figure BDA0003026756710000062
The system of step (5) can be implemented in the mean square sense with H∞And achieving the fault-tolerant consistency of leading and following the multiple agents under the condition of the interference level gamma.
Figure BDA0003026756710000063
Figure BDA0003026756710000064
Wherein:
Figure BDA0003026756710000065
Figure BDA0003026756710000066
Figure BDA0003026756710000067
Figure BDA0003026756710000068
step (8) is further optimized for step (7), and the failure coefficients in step (6) and step (7) are unknown.
Giving an appropriate constant
Figure BDA0003026756710000069
Variable cm>0, i-1, 2,3,4 and moments Q, T, F of the appropriate dimension, if there is a matrix P of the appropriate dimension positive>0, N, gain of controller
Figure BDA00030267567100000610
The system of step (5) can be implemented in the mean square sense with H∞And achieving the fault-tolerant consistency of leading and following the multiple agents under the condition of the interference level gamma.
Figure BDA00030267567100000611
Figure BDA00030267567100000612
Wherein:
Figure BDA0003026756710000071
Figure BDA0003026756710000072
Figure BDA0003026756710000073
Figure BDA0003026756710000074
Figure BDA0003026756710000075
Figure BDA0003026756710000076
the invention has the beneficial effects that: the stability and dynamic performance index of the system formula (11) have H while considering the system stability∞And (4) performance. In order to improve the safety and reliability of the leading-following multi-agent system, the device is provided withA fault-tolerant controller is designed, so that the system can still keep stable operation when an actuator failure fault and a network attack exist in the system.
Drawings
FIG. 1: leader-follows the topology of the multi-agent;
FIG. 2: an attack signal of a first follower network attack;
FIG. 3: an attack signal of a second follower network attack;
FIG. 4: attack signals of a third follower network attack;
FIG. 5: an attack signal of a fourth follower network attack;
FIG. 6: leader-following the tracking trajectory in Multi agent State 1;
FIG. 7: leader-following the tracking trajectory in Multi agent State 2;
FIG. 8: leader-following the tracking trajectory error in multi-agent state 1;
FIG. 9: leader-follows the tracking trajectory error in multi-agent state 2.
Detailed Description
The invention will now be described in further detail with reference to examples shown in the accompanying drawings.
Aiming at the leader, selecting the following state system model:
Figure BDA0003026756710000081
wherein x is0(t)∈RnIs the state quantity of the leader and is,
Figure BDA0003026756710000082
an input that is a leader of the system.
The equation of state for the ith follower agent:
Figure BDA0003026756710000083
wherein: x is the number ofi(t)∈RnIs the state quantity of the ith follower,
Figure BDA0003026756710000084
representing the control input of the follower actuator,
Figure BDA0003026756710000085
is an external disturbance of the system. Matrix BwSatisfies BwBF A, B, F and BwIs a known real state matrix with the appropriate dimensions.
Step (2) constructs a failure model for the leader-follower multi-agent actuator of the invention. A general fault model for an actuator is now given as follows:
Figure BDA0003026756710000086
wherein: m isi＝diag{mi,1,mi,2,...mi,s}ui＝diag{ui,1,ui,2,...ui,s}i＝1,2,...,N,j＝1,2,....,s。
According to the characteristics of the failure fault of the actuator, redefining the model of a partial failure matrix:
Figure BDA0003026756710000087
wherein: m isi,jRepresenting the failure coefficient of the jth actuator of the ith agent,
Figure BDA0003026756710000088
and
Figure BDA0003026756710000089
represents the failure coefficient mi,jUpper and lower bounds.
Figure BDA00030267567100000810
According to the failure characteristics, the upper and lower bounds and the failure coefficients of the failure, the distribution rule of the failure satisfies Bernoulli distribution, and the failure matrix is redefined to obtain:
Figure BDA00030267567100000811
the form of the amplification matrix is as follows:
Figure BDA00030267567100000812
wherein:
Figure BDA0003026756710000091
Figure BDA0003026756710000092
probability of failure of the ith agent of the representative leader-follower multi-agent system for the jth actuator, Γi0And
Figure BDA0003026756710000093
two failure coefficients selected from the partitioned failure regions are used to follow the control input of the actuator based on the provided failure model
Figure BDA0003026756710000094
Can be rewritten as:
Figure BDA0003026756710000095
step (3) designs the network attack model of the invention
In the present invention, network attacks are considered, which are implemented by injecting misleading numbers into regular transmission data. To reduce system performance, the invention uses non-linear functionsfj(x (t)) to represent random cyber attacks, wherein the random cyber attacks satisfy the Bernoulli distribution.
Figure BDA0003026756710000096
Wherein:
Figure BDA0003026756710000097
is represented by the xi(t) the signal received by the agent is from the xth agentj(t) a signal under network attack, 0. ltoreq. alphaj(t) 1 is the x-thi(t) the signal received by the agent is from the xth agentj(t) probability of network-attacked signal, (reduced to network-attacked signal)
Figure BDA00030267567100000910
The possibility of occurrence).
And (4) establishing a fault-tolerant consistency control law of the whole system aiming at the steps (1), (2) and (3).
First, the fault-tolerant consistent control law of the whole leader-follower multi-agent system is given, i.e. for any initial conditions, if satisfied,
Figure BDA0003026756710000098
the entire system may implement a fault-tolerant mean square consistency protocol.
Designing a fault-tolerant controller:
ui(t)＝Kei(t),i＝1,...,N (10)
wherein:
Figure BDA0003026756710000099
the combination of equations (1), (2), (3), (8), (9), (10) yields the consistency equation for the entire system:
Figure BDA0003026756710000101
wherein: giIs representative of the strength of the communication link between the leader and the follower, aijThe communication strength of the information between follower agents is represented, and the follower agents and the information form the communication strength of the topological structure between the whole leader-follower multi-agent agents.
Step (5) is to establish an error state equation of the whole leader-follower multi-agent system aiming at steps (1), (2), (3) and (4):
Figure BDA0003026756710000102
the invention adopts a method of segmenting a topological structure to write an augmentation matrix form of an error state equation of the whole system:
Figure BDA0003026756710000103
Figure BDA0003026756710000104
wherein:
Figure BDA0003026756710000105
Figure BDA0003026756710000106
Figure BDA0003026756710000107
Figure BDA0003026756710000108
Figure BDA0003026756710000109
and (6) aiming at the state equation of the system described in the step (4), selecting a proper Lyapunov function as follows, so that the system (11) can realize the consistent stability of fault-tolerant mean square and the performance index of H infinity.
The designed Lyapunov function of the invention is as follows:
Figure BDA0003026756710000111
i.e. if given the correct controller gain K>0, constant number
Figure BDA0003026756710000112
Variable cm>0, i-1, 2,3,4, and a matrix Q, T, F of appropriate dimension, and a positive matrix P of appropriate dimension>0, N, satisfying the following linear matrix inequality holds, the described system of step (5) can be implemented in the mean square sense with H∞And achieving the fault-tolerant consistency of leading and following the multiple agents under the condition of the interference level gamma.
Figure BDA0003026756710000113
Figure BDA0003026756710000114
Wherein:
Figure BDA0003026756710000115
Figure BDA0003026756710000116
Figure BDA0003026756710000117
Figure BDA0003026756710000118
Figure BDA0003026756710000119
in classical H∞In theory, the zero initial condition must be satisfied, based on this build performance index J:
Figure BDA00030267567100001110
wherein
Figure BDA00030267567100001111
Step (7) is a further optimization for step (6), i.e. designing the gain of the controller.
Giving an appropriate constant
Figure BDA0003026756710000121
Variable cm>0, i-1, 2,3,4 and moments Q, T, F of the appropriate dimension, if there is a matrix P of the appropriate dimension positive>0, N, and gain of the controller
Figure BDA0003026756710000122
The system of step (5) can be implemented in the mean square sense with H∞And achieving the fault-tolerant consistency of leading and following the multiple agents under the condition of the interference level gamma.
Figure BDA0003026756710000123
Figure BDA0003026756710000124
Wherein:
Figure BDA0003026756710000125
Figure BDA0003026756710000126
Figure BDA0003026756710000127
Figure BDA0003026756710000128
step (8) is further optimized for step (7), and the failure coefficients in step (6) and step (7) are known, but in most cases are unknown, and based on this, the following theorem is designed.
Giving an appropriate constant
Figure BDA0003026756710000129
Variable cm>0, i-1, 2,3,4 and moments Q, T, F of the appropriate dimension, if there is a matrix P of the appropriate dimension positive>0, N, the gain of the controller is full
Figure BDA00030267567100001210
The system of step (5) can be implemented in the mean square sense with H∞And achieving the fault-tolerant consistency of leading and following the multiple agents under the condition of the interference level gamma.
Figure BDA00030267567100001211
Figure BDA00030267567100001212
Wherein:
Figure BDA00030267567100001213
Figure BDA0003026756710000131
Figure BDA0003026756710000132
Figure BDA0003026756710000133
Figure BDA0003026756710000134
Figure BDA0003026756710000135
Figure BDA0003026756710000136
Figure BDA0003026756710000137
Figure BDA0003026756710000138
for ease of understanding, step (8) is now explained as follows: the fault-tolerant controller is designed to ensure that the system keeps the mean square consistency of the whole system under the fault condition and the network attack and has H∞The performance index γ.
(2) Firstly, a digital simulation example is used for verifying the effectiveness of the fault-tolerant control design method:
firstly, parameters of digital simulation are given:
A＝[-0.59 0.496；-5.513 -0.939]
B＝[0.06 0.06；1.879 2.328]
F＝[0.2 0；0.3 -0.5]
leader-Laplace array L following multiple agents satisfies:
Figure BDA0003026756710000139
follower actuator failure coefficient:
ρi1＝0.65 ρi2＝0.84 i＝1,2....4.
leader input: r is0(t)＝[sin(t)+14；cos(0.5×t-2.2)-12]T
External disturbance: w is ai(t)＝[sin(t)；1]T,i＝1,2,3,4
Probability of network attack occurrence: alpha is alphaj(t) ═ 0.3, performance index: gamma is 0.3
Through the step (8), the gains of the fault-tolerant controller of the invention can be obtained respectively:
K＝[2.2687 0.3449；26.6989 6.5273]
FIG. 1 illustrates a leader-follower multi-agent topology, and the present invention is directed to a system of five agents in a directed and directed topology.
Fig. 2, fig. 3, fig. 4, and fig. 5 are attack signals based on a case where the network attack occurrence probability is 30%.
FIGS. 6 and 7 show that the leaders-follow the tracks traced by the multi-agent in states 1,2, the tracks of the operations tend to be identical, i.e. the tracks of the operations tend to be identical
Figure BDA0003026756710000141
Thus, fault-tolerant consistency of five agents is achieved.
Fig. 8 and 9 show the tracking errors of five agents in two states, as shown in the figure, the tracking error will gradually converge to 0, so the controller gain obtained by theorem 3 can also realize the fault-tolerant consistency under the condition of actuator failure fault and network attack of the system.
The invention researches the consistency of the leader following the multi-agent system when the actuator fault and the network attack occur. First, a fault model suitable for a leader-follower multi-agent is established based on fault characteristics. The network attack model is described by mutually independent random Bernoulli variables, and a topological structure segmentation method is adopted for solving. Then, sufficient conditions for realizing the mean square consistency of the system are given, and finally, the effectiveness of the method is verified through a specific digital simulation example.

Claims (1)

1.领导跟随多智能体系统的容错控制器设计方法，该方法包括以下步骤：1. Lead a fault-tolerant controller design method for following multi-agent systems, which includes the following steps: 步骤(1)设计多智能体的领导者和跟随者的状态方程；Step (1) Design the state equations of the leaders and followers of the multi-agent; 步骤(2)构造领导-跟随多智能体的执行器的失效模型；Step (2) Construct the failure model of the leader-following multi-agent actuator; 根据故障失效的特征和上下界以及故障的失效系数，其分布规律满足伯努利分布，定义失效矩阵模型：According to the characteristics and upper and lower bounds of failure failure and the failure coefficient of failure, its distribution law satisfies Bernoulli distribution, and the failure matrix model is defined:
Figure FDA0003472385260000011
Figure FDA0003472385260000011
 mi表示第i个智能体的失效系数，
Figure FDA0003472385260000012
表示第i个智能体的发生失效故障的概率，Γi0和
Figure FDA0003472385260000013
表示从分割的失效区间中选取的两个失效系数，Gi和
Figure FDA0003472385260000014
表示根据失效系数的上界和下界和失效系数的特征后重新构造的失效系数；m i represents the failure coefficient of the ith agent,
Figure FDA0003472385260000012
represents the probability of failure of the i-th agent, Γ i0 and
Figure FDA0003472385260000013
represents two failure coefficients selected from the divided failure interval, G i and
Figure FDA0003472385260000014
Represents the reconstructed failure coefficient according to the upper and lower bounds of the failure coefficient and the characteristics of the failure coefficient; 步骤(3)设计网络攻击的模型Step (3) Design a network attack model 用非线性函数f(xj(t))来表示随机网络攻击，其中随机网络攻击满足伯努利分布；The random network attack is represented by a nonlinear function f(x j (t)), where the random network attack satisfies the Bernoulli distribution;
Figure FDA0003472385260000015
Figure FDA0003472385260000015
 其中：
Figure FDA0003472385260000016
代表的是第xi(t)个智能体接收到的信号来自于第xj(t)个智能体受到网络攻击的信号，0≤αj(t)≤1是第xi(t)个智能体接收到的信号来自于第xj(t)受到网络攻击的信号的概率；in:
Figure FDA0003472385260000016
Represents that the signal received by the xi (t) th agent comes from the signal that the x j (t) th agent is attacked by the network, and 0≤α j (t)≤1 is the xi (t) th The probability that the signal received by the agent comes from the signal that the x j (t)th is attacked by the network; 步骤(4)建立领导跟随多智能体系统容错一致性控制律；Step (4) establish a fault-tolerant consistency control law of the leader-following multi-agent system; 给出整个领导跟随多智能体系统的容错一致性控制律：The fault-tolerant consensus control law for the entire leader-following multi-agent system is given:
Figure FDA0003472385260000017
Figure FDA0003472385260000017
 其中：gi代表领导者和跟随者之间的通讯连通强度,aij代表跟随者智能体之间的信息连通强度,xj(t)和xj(t)代表跟随者状态量,x0(t)代表领导者的状态量,ei(t)代表一致性控制律；Among them: g i represents the communication connection strength between the leader and the follower, a ij represents the information connection strength between the follower agents, x j (t) and x j (t) represent the state quantity of the follower, x 0 (t) represents the state quantity of the leader, e i (t) represents the consistency control law; 步骤(5)建立领导跟随多智能体系统的误差状态方程Step (5) Establish the error state equation of the leader-following multi-agent system
Figure FDA0003472385260000021
Figure FDA0003472385260000021
 采用分割拓扑结构的方法，写出整个系统的误差状态方程的增广矩阵形式：Using the method of dividing the topology, write the augmented matrix form of the error state equation of the whole system:
Figure FDA0003472385260000022
Figure FDA0003472385260000022
Figure FDA0003472385260000023
Figure FDA0003472385260000023
 其中：in:
Figure FDA0003472385260000024
Figure FDA0003472385260000024
Figure FDA0003472385260000025
Figure FDA0003472385260000025
Figure FDA0003472385260000026
Figure FDA0003472385260000026
Figure FDA0003472385260000027
Figure FDA0003472385260000027
Figure FDA0003472385260000028
Figure FDA0003472385260000028
 A,B,F是系统的状态矩阵，K代表控制器的增益，wi(t)是系统的外部扰动，r0(t)是系统的领导者的输入,xi(t)是第i个跟随者的状态量，
Figure FDA0003472385260000029
是跟随者执行器的控制输入,第i领导者和跟随者的状态量的误差用di(t)来表示，L表示整个系统的拓扑结构矩阵，Q,H分别是L的两个子矩阵，其满足L＝Q+H,
Figure FDA00034723852600000210
代表领导者和跟随者的状态量的误差的导数，
Figure FDA00034723852600000211
分别定义为第i个跟随者的状态量和领导者状态量的导数；A, B, F are the state matrices of the system, K represents the gain of the controller, wi (t) is the external disturbance of the system, r 0 (t) is the input of the leader of the system, and xi (t) is the i-th the state quantity of a follower,
Figure FDA0003472385260000029
is the control input of the follower actuator, the error of the state quantity of the ith leader and the follower is represented by d i (t), L represents the topological structure matrix of the whole system, Q and H are the two sub-matrices of L respectively, It satisfies L=Q+H,
Figure FDA00034723852600000210
is the derivative of the error representing the state quantities of the leader and follower,
Figure FDA00034723852600000211
Defined as the derivative of the state quantity of the ith follower and the leader state quantity, respectively; d(t),
Figure FDA0003472385260000031
分别是对应的变量或者常量的扩维后的形式；d(t),
Figure FDA0003472385260000031
are the expanded form of the corresponding variable or constant, respectively; 步骤(6)针对步骤(4)所描述的系统的状态方程，通过选择如下合适的李雅普诺夫函数，使得系统(6)可以实现容错均方一致稳定和具有H∞性能指标；Step (6) is aimed at the state equation of the system described in step (4), by selecting the following suitable Lyapunov function, so that the system (6) can achieve fault-tolerant mean square consistent stability and have H∞ performance index; 设计李雅普诺夫函数：Design the Lyapunov function:
Figure FDA0003472385260000032
Figure FDA0003472385260000032
 即如果给定恰当的控制器的增益K>0，常数
Figure FDA0003472385260000033
变量cm>0,i＝1,2,3,4和具有恰当维数的矩阵Q,T,F，以及恰当维数正定矩阵P>0,N，满足下面的线性矩阵不等式成立，则步骤(5)的所描述的系统可以实现在均方意义下中以H∞干扰水平γ条件下达到领导跟随多智能体的容错一致性；That is, if given an appropriate controller gain K>0, the constant
Figure FDA0003472385260000033
Variables c m >0, i=1, 2, 3, 4 and matrices Q, T, F with appropriate dimensions, and positive definite matrices P > 0, N with appropriate dimensions, satisfy the following linear matrix inequality, then step The system described in (5) can achieve fault-tolerant consistency of leader-follower multi-agents with H ∞ interference level γ in the mean square sense;
Figure FDA0003472385260000034
Figure FDA0003472385260000034
Figure FDA0003472385260000035
Figure FDA0003472385260000035
 其中：in:
Figure FDA0003472385260000036
Figure FDA0003472385260000036
Figure FDA0003472385260000037
Figure FDA0003472385260000037
Figure FDA0003472385260000038
Figure FDA0003472385260000038
Figure FDA0003472385260000039
Figure FDA0003472385260000039
Figure FDA00034723852600000310
Figure FDA00034723852600000310
 dT(t)代表了第i领导者和跟随者的状态量的误差扩维后的转置形式，
Figure FDA00034723852600000311
是第xi(t)个智能体接收到的信号来自于第个智能体xj(t)受到网络攻击的信号的概率期望的形式，P为恰当维数正定矩阵，cm>0,i＝1,2,3,4是一个满足系统需求的常数，γ是系统可以实现均方意义下中以H∞干扰水平的指标,Γ11,Γ12,Γ13,Γ14,Γ15,
Figure FDA00034723852600000312
分别是线性矩阵不等式Ξ内的元素，其由已经陈述的基本元素所构成，Ξ代表了一个线性矩阵不等式；d T (t) represents the transposed form of the error expansion of the state quantities of the ith leader and follower,
Figure FDA00034723852600000311
is the form of the probability expectation that the signal received by the xi (t) th agent comes from the signal that the th agent x j (t) is attacked by the network, P is a positive definite matrix of appropriate dimension, c m >0,i =1, 2, 3, 4 is a constant that satisfies the system requirements, γ is the indicator that the system can achieve the H ∞ interference level in the mean square sense, Γ 11 , Γ 12 , Γ 13 , Γ 14 , Γ 15 ,
Figure FDA00034723852600000312
are the elements in the linear matrix inequality Ξ, respectively, which are composed of the basic elements that have been stated, and Ξ represents a linear matrix inequality; 在经典的H∞理论中，零初始条件必须被满足，基于此构造性能指标J：In the classical H ∞ theory, the zero initial condition must be satisfied, and the performance index J is constructed based on this:
Figure FDA0003472385260000041
Figure FDA0003472385260000041
 其中in
Figure FDA0003472385260000042
Figure FDA0003472385260000042
 zT(t)和上述的dT(t)的含义一致,e(t)是由
Figure FDA0003472385260000043
这四个元素构成的新的列向量，d(0)是d(t)在t时刻的值，N是待求的正定矩阵；J为满足系统零初始条件的性能指标；z T (t) has the same meaning as the above d T (t), e(t) is defined by
Figure FDA0003472385260000043
The new column vector formed by these four elements, d(0) is the value of d(t) at time t, N is the positive definite matrix to be found; J is the performance index that satisfies the zero initial condition of the system; 步骤(7)是针对步骤(6)做的进一步优化，即设计控制器的增益；Step (7) is a further optimization done for step (6), namely designing the gain of the controller; 给一个恰当的常数
Figure FDA0003472385260000044
变量cm>0,i＝1,2,3,4和具有恰当维数的矩Q,T,F，如果存在一个具有恰当维数正定的矩阵P>0,N，且控制器的增益
Figure FDA0003472385260000045
give an appropriate constant
Figure FDA0003472385260000044
Variables c m > 0, i=1, 2, 3, 4 and moments Q, T, F of appropriate dimensions, if there exists a positive definite matrix P > 0, N of appropriate dimensions, and the gain of the controller
Figure FDA0003472385260000045
 则步骤(5)的系统可以实现在均方意义下中以H∞干扰水平γ条件下达到领导跟随多智能体的容错一致性；Then the system in step (5) can achieve the fault-tolerant consistency of the leader-follower multi-agent under the condition of H ∞ interference level γ in the mean square sense;
Figure FDA0003472385260000046
Figure FDA0003472385260000046
Figure FDA0003472385260000047
Figure FDA0003472385260000047
 其中：in:
Figure FDA0003472385260000048
Figure FDA0003472385260000048
Figure FDA0003472385260000049
Figure FDA0003472385260000049
Figure FDA00034723852600000410
Figure FDA00034723852600000410
Figure FDA00034723852600000411
Figure FDA00034723852600000411
 X1是矩阵P的逆矩阵，
Figure FDA00034723852600000412
为其增广形式，
Figure FDA00034723852600000413
为控制器的增益的增广形式和K的含义相同，
Figure FDA0003472385260000051
是定理2的线性矩阵不等式，
Figure FDA0003472385260000052
是线性矩阵不等式
Figure FDA0003472385260000053
中的元素,由已经陈述的元素构成，
Figure FDA0003472385260000054
是待求矩阵的增广形式，Y的值可由
Figure FDA0003472385260000055
求出；X 1 is the inverse of matrix P,
Figure FDA00034723852600000412
in its augmented form,
Figure FDA00034723852600000413
is the augmented form of the gain of the controller and has the same meaning as K,
Figure FDA0003472385260000051
is the linear matrix inequality of Theorem 2,
Figure FDA0003472385260000052
is a linear matrix inequality
Figure FDA0003472385260000053
The elements in , consisting of the elements already stated,
Figure FDA0003472385260000054
is the augmented form of the matrix to be solved, and the value of Y can be determined by
Figure FDA0003472385260000055
ask for; 步骤(8)是针对步骤(7)做了进一步的优化，步骤(6)和步骤(7)中的失效系数是已知的，但是在大部分的情况下失效系数都是未知的，基于此，设计了如下的定理：Step (8) is further optimized for step (7). The failure coefficients in steps (6) and (7) are known, but in most cases the failure coefficients are unknown. Based on this , devised the following theorem: 给一个恰当的常数
Figure FDA0003472385260000056
变量cm>0,i＝1,2,3,4和具有恰当维数的矩Q,T,F，如果存在一个具有恰当维数正定的矩阵P>0,N，控制器的增益满
Figure FDA0003472385260000057
则步骤(5)的系统可以实现在均方意义下中以H∞干扰水平γ条件下达到领导跟随多智能体的容错一致性；give an appropriate constant
Figure FDA0003472385260000056
Variables c m > 0, i = 1, 2, 3, 4 and moments Q, T, F with appropriate dimensions, if there is a positive definite matrix P > 0, N with appropriate dimensions, the gain of the controller is full
Figure FDA0003472385260000057
Then the system in step (5) can achieve the fault-tolerant consistency of the leader-follower multi-agent under the condition of H ∞ interference level γ in the mean square sense;
Figure FDA0003472385260000058
Figure FDA0003472385260000058
Figure FDA0003472385260000059
Figure FDA0003472385260000059
 其中:in:
Figure FDA00034723852600000510
Figure FDA00034723852600000510
Figure FDA00034723852600000511
Figure FDA00034723852600000511
Figure FDA00034723852600000512
Figure FDA00034723852600000512
Figure FDA00034723852600000513
Figure FDA00034723852600000513
Figure FDA00034723852600000514
Figure FDA00034723852600000514
Figure FDA00034723852600000515
Figure FDA00034723852600000515
Figure FDA00034723852600000516
Figure FDA00034723852600000516
Figure FDA00034723852600000517
Figure FDA00034723852600000517
 ε为一个很小的整正数，Ω是定理3的线性矩阵不等式，
Figure FDA00034723852600000518
Π,Λ是Ω内的基本元素,Ω11,Ω12,Ω13,Ω14,Ω15是矩阵不等式
Figure FDA00034723852600000519
里面的元素，已经陈述的基本变量或者常量构成了线性矩阵不等式里面的元素；ε is a small positive integer, Ω is the linear matrix inequality of Theorem 3,
Figure FDA00034723852600000518
Π,Λ are the basic elements in Ω, Ω 11 ,Ω 12 ,Ω 13 ,Ω 14 ,Ω 15 are matrix inequalities
Figure FDA00034723852600000519
The elements inside, the basic variables or constants that have been stated constitute the elements inside the linear matrix inequality;
Figure FDA0003472385260000061
Figure FDA0003472385260000061
 为了方便理解，现在对步骤(8)进行以下解释：设计容错控制器，使系统在故障情况和网络攻击下使整个系统保持均方一致性，并且具有H∞性能指标γ。In order to facilitate understanding, step (8) is now explained as follows: design a fault-tolerant controller, which makes the system maintain mean square consistency for the whole system under fault conditions and network attacks, and has H ∞ performance index γ.
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