CN113110182A - 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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CN113110182A
CN113110182A CN202110418213.6A CN202110418213A CN113110182A CN 113110182 A CN113110182 A CN 113110182A CN 202110418213 A CN202110418213 A CN 202110418213A CN 113110182 A CN113110182 A CN 113110182A
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fault
leader
agent
failure
follower
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CN113110182B (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

The invention relates to a design method of a fault-tolerant controller of a leader follower multi-agent system. The invention assumes that the communication graph is directional and fixed, and first, a fault model is established based on the distribution characteristics of the fault. The network attack model is described by mutually independent random Bernoulli variables, a topological structure segmentation method is adopted for solving, a performance index function suitable for leader-following is provided in a traditional control method for avoiding zero initial conditions, and then a novel fault-tolerant controller is designed according to the provided fault model and performance index to ensure that the leader-following multi-agent system can still realize fault-tolerant mean square consistency when network attack and actuator faults occur. Finally, the effectiveness of the invention is verified through numerical calculation.

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 a matrix P of the correct dimension>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 leader-follower multi-agent system, a fault-tolerant controller is designed, so that the system can still keep stable operation when an actuator failure fault and a network attack exist.
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 a non-linear function fj(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∞Fault tolerance one for leading to follow multiple intelligent agents under interference level gamma conditionCausing sexual disorder.
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. A method of designing a fault-tolerant controller for a lead-follower multi-agent system, the method comprising the steps of:
designing state equations of a leader and a follower of a multi-agent system;
constructing a failure model of a leader-following multi-agent actuator;
according to the failure characteristics, the upper and lower bounds and the failure coefficients of the failure, the distribution rule of which meets Bernoulli distribution, a failure matrix model is defined:
Figure FDA0003026756700000011
miindicating the failure coefficient of the ith agent,
Figure FDA0003026756700000012
representing the probability of failure, Γ, of the ith agenti0And
Figure FDA0003026756700000013
representing two failure coefficients, G, selected from the partitioned failure intervalsiAnd
Figure FDA0003026756700000014
representing the failure coefficients reconstructed from the upper and lower bounds of the failure coefficients and the characteristics of the failure coefficients;
step (3) designing a model of network attack
By non-linear functions fj(x (t)) to represent random cyber attacks, wherein the random cyber attacks satisfy the Bernoulli distribution;
Figure FDA0003026756700000015
wherein:
Figure FDA0003026756700000016
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) signals received by the agentsFrom xj(t) probability of a signal being under network attack;
step (4) establishing a fault-tolerant consistency control law of the leader following multi-agent system;
giving a fault-tolerant consistency control law of the whole leader following multi-agent system:
Figure FDA0003026756700000017
wherein: giRepresenting the strength of communication connection between the leader and the follower, aijRepresenting strength of information communication, x, between follower agentsj(t) and xj(t) represents a follower state quantity, x0(t) represents the state quantity of the leader, ei(t) represents a consistency control law;
step (5) establishing an error state equation of the leader following multi-agent system
Figure FDA0003026756700000021
Writing an augmentation matrix form of an error state equation of the whole system by adopting a method of segmenting a topological structure:
Figure FDA0003026756700000022
Figure FDA0003026756700000023
wherein:
Figure FDA0003026756700000024
Figure FDA0003026756700000025
Figure FDA0003026756700000026
Figure FDA0003026756700000027
Figure FDA0003026756700000028
a, B, F are the state matrix of the system, K represents the gain of the controller, wi(t) is the external disturbance of the system, r0(t) is an input of the leader of the system, xi(t) is the state quantity of the ith follower,
Figure FDA0003026756700000029
is the control input of the follower actuator, and the error of the state quantities of the i-th leader and the follower is represented by deltai(t), L represents a topological structure matrix of the whole system, Q and H are two sub-matrixes of L respectively, and L is Q + H,
Figure FDA00030267567000000210
a derivative of the error representing the state quantities of the leader and follower,
Figure FDA00030267567000000211
respectively defined as the derivative of the state quantity of the ith follower and the state quantity of the leader; delta (t) is calculated from the sum of the values of delta (t),
Figure FDA0003026756700000031
the scaled versions of the corresponding variable or constant, respectively;
step (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 (6) can realize consistent stability of fault-tolerant mean square and has an H-infinity performance index;
designing a Lyapunov function:
Figure FDA0003026756700000032
i.e. if given the gain K of the appropriate controller>0, constant number
Figure FDA0003026756700000033
Variable cm>0, i-1, 2,3,4 and matrices Q, T, F of appropriate dimensions, and positive definite matrices P of appropriate dimensions>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∞The fault-tolerant consistency of leading and following the multiple agents is achieved under the condition of the interference level gamma;
Figure FDA0003026756700000034
Figure FDA0003026756700000035
wherein:
Figure FDA0003026756700000036
Figure FDA0003026756700000037
Figure FDA0003026756700000038
Figure FDA0003026756700000039
Figure FDA00030267567000000310
δT(t) represents the error-extended transposed form of the state quantities of the i-th leader and follower,
Figure FDA00030267567000000311
is the xi(t) the signal received by the agent comes from the first agent xj(t) the form of the probability expectation of the signal under network attack, P being a positive definite matrix of the appropriate dimension, cm>0, i-1, 2,3,4 is a constant satisfying the system requirements, γ is H in the mean square sense that the system can implement∞An indicator of the interference level, Γ11,Γ12,Γ13,Γ14,Γ15,
Figure FDA00030267567000000312
Are elements within the linear matrix inequality xi, respectively, which are formed by the already stated basic elements, xi representing a linear matrix inequality;
in classical H∞In theory, the zero initial condition must be satisfied, based on this build performance index J:
Figure FDA0003026756700000041
wherein
Figure FDA0003026756700000042
zT(t) and δ described aboveT(t) have the same meaning, e (t) is represented by
Figure FDA0003026756700000043
The new column vector is formed by the four elements, delta (0) is the value of delta (t) at the time t, and N is a positive definite matrix to be solved; j is a performance index meeting the zero initial condition of the system;
step (7) is further optimized aiming at step (6), namely the gain of the controller is designed;
giving an appropriate constant
Figure FDA0003026756700000044
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 FDA0003026756700000045
The system of step (5) can be implemented in the mean square sense with H∞The fault-tolerant consistency of leading and following the multiple agents is achieved under the condition of the interference level gamma;
Figure FDA0003026756700000046
Figure FDA0003026756700000047
wherein:
Figure FDA0003026756700000048
Figure FDA0003026756700000049
Figure FDA00030267567000000410
Figure FDA00030267567000000411
X1is the inverse of the matrix P and,
Figure FDA00030267567000000412
in its expanded form, the composition of the invention,
Figure FDA00030267567000000413
the form of gain amplification for the controller is the same as that for K,
Figure FDA0003026756700000051
is the linear matrix inequality of theorem 2,
Figure FDA0003026756700000052
is a linear matrix inequality
Figure FDA0003026756700000053
Consists of the elements already stated,
Figure FDA0003026756700000054
is an extended form of the matrix to be solved, the value of Y can be determined from
Figure FDA0003026756700000055
Calculating;
step (8) is further optimized for step (7), the failure coefficients in step (6) and step (7) are known, but in most cases the failure coefficients are unknown, and based on this, the following theorem is designed:
giving an appropriate constant
Figure FDA0003026756700000056
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 FDA0003026756700000057
The system of step (5) can be implemented in the mean square sense with H∞The fault-tolerant consistency of leading and following the multiple agents is achieved under the condition of the interference level gamma;
Figure FDA0003026756700000058
Figure FDA0003026756700000059
wherein:
Figure FDA00030267567000000510
Figure FDA00030267567000000511
Figure FDA00030267567000000512
Figure FDA00030267567000000513
Figure FDA00030267567000000514
Figure FDA00030267567000000515
Figure FDA00030267567000000516
Figure FDA00030267567000000517
e is a small integer positive, Ω is the linear matrix inequality of theorem 3,
Figure FDA00030267567000000518
II, Λ are basic elements within Ω, Ω11,Ω12,Ω13,Ω14,Ω15Is a matrix inequality
Figure FDA00030267567000000519
The inner elements, the already stated basic variables or constants, constitute the elements within the linear matrix inequality;
Figure FDA0003026756700000061
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.
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