CN109116736B - Fault-tolerant control method for actuator fault of linear multi-agent system based on sliding mode - Google Patents

Fault-tolerant control method for actuator fault of linear multi-agent system based on sliding mode Download PDF

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CN109116736B
CN109116736B CN201811112494.7A CN201811112494A CN109116736B CN 109116736 B CN109116736 B CN 109116736B CN 201811112494 A CN201811112494 A CN 201811112494A CN 109116736 B CN109116736 B CN 109116736B
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杨蒲
王玉霞
疏琪堡
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Nanjing University of Aeronautics and Astronautics
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Abstract

The invention discloses a tracking fault-tolerant control method of a linear multi-agent system based on sliding mode control. Considering the fault of an actuator under the condition that the linear multi-agent tracking system has external interference, a fault-tolerant control method is provided based on a sliding mode control method and combined with self-adaptive control. Aiming at the possible failure fault of the actuator part of the intelligent agents, a state quantity error variable is defined according to the communication topology among the intelligent agents, an integral sliding mode surface is designed based on the error variable, sufficient conditions of gradual and stable sliding modes are given, then the unknown upper bound of the actuator fault is estimated by using a self-adaptive method, and the tracking stability of the intelligent agent system is ensured by providing a sliding mode fault-tolerant controller. According to the invention, an integral sliding mode surface is designed according to state error variables among the multi-agent systems, the robustness of the system is enhanced, a fault-tolerant control law is provided, and the system has good fault-tolerant capability when partial failure faults and external interference of an actuator exist, and the tracking stability of the multi-agent system is improved. The invention is used for fault-tolerant control of a linear multi-agent system with actuator failure faults and external interference.

Description

Fault-tolerant control method for actuator fault of linear multi-agent system based on sliding mode
Technical Field
The invention relates to a fault-tolerant control method of a sliding-mode-based linear multi-agent system, belonging to the field of multi-agent system control.
Background
With the rapid development of control theory, the control technology of a single intelligent agent gradually matures. In recent years, the wide application of computer networks and the continuous development of artificial intelligence technology have brought about great research enthusiasm of people due to the fact that a multi-agent system containing a plurality of individuals can complete relatively complex tasks through mutual coordination among agents, and the tasks which cannot be completed by a single system can be completed more effectively while the cost is saved. In view of the advantages of a multi-agent system, such as wider task field and higher efficiency, the multi-agent system is currently applied to a plurality of fields, such as robot and unmanned aerial vehicle formation control, automatic traffic control, complex industrial process control, and the like.
The current research efforts on multi-agent systems are mainly in terms of coherence, tracking and formation control, but these conclusions are based on the assumption that the agents do not fail. In a complex multi-agent system, the actuators are numerous and have complex distribution structures, which is the key point for realizing the global goal, once the actuators of some agents have faults, the negative influence of the faults on a single agent can be amplified to the global state through the communication topology among the multi-agents, the performance of the whole system is influenced, and even the task fails. Therefore, the method has important practical significance for the fault-tolerant control research of the multi-agent system.
The sliding mode control is a special nonlinear control, has the advantages of quick response, insensitivity to uncertain parameters of a system, simple physical implementation and good robustness, and is widely applied to theoretical research and actual engineering. The sliding mode control can force the state of the system to move along the pre-designed sliding mode surface, the gradual stability of the system is realized, and the system reaching the sliding mode surface is not influenced by parameter change and external disturbance any more, so the method is very suitable for passive fault-tolerant control of a multi-agent system.
In order to effectively handle existing actuator failures that may occur in multi-agent systems, there have been some research efforts in recent years, but still in the infancy. The learner's left concentration provides a self-adaptive gain compensation control method aiming at the failure fault of the actuator part of a linear multi-agent system. Duncao et al at university in northeast have designed a self-adaptive output feedback control method to solve the problem of actuator failure of a class of nonlinear multi-agent systems. However, most of the existing solutions are fault-tolerant control based on adaptive control, and the method hardly has good robustness to unavoidable perturbation of system parameters and external interference in actual engineering, so the method has certain innovativeness and practicability.
Disclosure of Invention
The purpose of the invention is as follows: aiming at the failure fault of an actuator part of a linear multi-agent system, a sliding mode fault-tolerant control method is provided by combining self-adaptive control, so that the negative influence of the fault on the system is overcome, and the stable operation of the system is ensured.
The technical scheme is as follows: a fault-tolerant control method of a linear multi-agent system based on a sliding mode is characterized by comprising the following steps: considering the problems of actuator faults and external interference of a multi-agent system, state error variables are defined according to a communication topological structure among the multi-agents, an integral sliding mode surface is designed based on the error variables, and the gradual stability of the sliding mode of the system is ensured. And estimating the unknown upper bound of fault information by using a self-adaptive method, and further providing a sliding-mode fault-tolerant controller, so that the multi-agent system can continue to operate safely after an actuator fault occurs. The method comprises the following specific steps:
step 1) acquiring a control model, an actuator fault model and a communication topological structure of a multi-agent system:
step 1.1) the leader control model is as shown in formula (1):
Figure GSB0000190159010000021
wherein x is0(t)∈RnIs the state quantity of the leader, r0(t)∈RmIs a control input of the leader;
step 1.2) follower control model is shown as formula (2):
Figure GSB0000190159010000022
wherein x isi(t)∈RnAnd ui(t)∈RmState quantities and control inputs representing the i, i 1, 2, N followers are a given system matrix, and (a, B) is stable, fi(t)∈RmRepresents the external interference of the ith follower and meets the requirement
Figure GSB0000190159010000023
Is a known normal number;
step 1.3) the actuator fault model is shown as formula (3):
Figure GSB0000190159010000024
wherein u isi(t) is the actuator input and,
Figure GSB0000190159010000025
is the actuator output with failure, ρi(t)＝diag{ρi1(t)，ρi3(t)，...，ρim(t)}∈Rm×mIs the actuator failure matrix of the ith follower, failure factor ρip(t), p 1, 2,. m is unknown time-varying and bounded when ρipWhen (t) is 0, it indicates that the ith follower has not failed, and when 0 < ρ ≦ is setipWhen (t) < 1, it indicates that the ith follower has a partial failure in the pth actuator, and when ρipWhen (t) is 1, the pth actuator representing the ith follower is completely failed, and this case is not considered here;
thus, an agent model for the occurrence of an actuator partial failure fault may be described as:
Figure GSB0000190159010000026
step 1.4) communication topology of multi-agent system:
considering a multi-agent system comprising a leader, labeled 0, and N followers, labeled i 1, 2., N, a graph G (V, E) represents a communication topology graph between all nodes including the leader and followers, where the set of nodes V {0, 1, 2., N } is the set of communication links between the nodes E ═ V × V; subfigure of G
Figure GSB0000190159010000031
Is a communication topology between followers, wherein
Figure GSB0000190159010000032
An adjacency matrix representing diagram G; l is Laplacian matrix of graph G, and is defined
Figure GSB0000190159010000033
Wherein
Figure GSB0000190159010000034
Is a diagonal matrix composed of degrees of each node, thenijIs as defined in formula (5):
Figure GSB0000190159010000035
order to
Figure GSB0000190159010000036
Representing the adjacency matrix between the leader and the followers, if there is an undirected edge between leader 0 and the ith follower, then b i1, otherwise, b i0; definition of
Figure GSB0000190159010000037
A neighbor set for the ith follower;
step 2) defining a state error variable shown in a formula (6) according to adjacent information acquired by the ith follower:
Figure GSB0000190159010000038
wherein, aijIs the connection weight between the ith follower and the jth follower, biRepresenting the weight of the connection between the ith follower and the leader, NiIs the neighbor set of the ith follower;
definition of
Figure GSB0000190159010000039
Equation (6) can be rewritten as a global error variable as shown in equation (7):
Figure GSB00001901590100000310
wherein,
Figure GSB00001901590100000311
step 3) aiming at the multi-agent tracking control system with the actuator fault as shown in the formula (4), designing a sliding mode surface as shown in a formula (8) based on a state error variable formula (6):
Figure GSB00001901590100000312
wherein G ═ BTB)-1BT∈Rm×nIs a sliding mode matrix, K is an element of Rm×nIs the controller gain to be designed; in order to ensure the asymptotic stability of the system sliding mode, the following conditions are met: re [ lambda (A + BK)]< 0, i.e. all characteristic roots of the matrix (A + BK) have negative real parts;
according to the formula (7), the formula (8) can be rewritten as
Figure GSB0000190159010000041
Wherein,
Figure GSB0000190159010000042
step 4) estimating the unknown upper bound of the fault information by the self-adaptive method, firstly defining an unknown parameter thetai＝1/1-||ρiI indirectly reflects fault information, and an adaptive law shown as a formula (10) is designed to estimate thetai：
Figure GSB0000190159010000043
Wherein,
Figure GSB0000190159010000044
is the parameter thetaiAnd alpha is a normal number, omegaiGiven in step 5);
step 5) integrating the step 3) and the step 4), designing a complete fault-tolerant control law as shown in the formula (11):
Figure GSB0000190159010000045
wherein,
Figure GSB0000190159010000046
and η, k1Are all normal numbers;
by combining the formula (7) and the formula (9), the formula (11) is rewritten as:
Figure GSB0000190159010000047
wherein b ═ b1 b2 ... bN]T，
Figure GSB0000190159010000048
And 5) selecting proper parameters according to the running state of the multi-agent system to realize the tracking fault-tolerant control of the system.
Has the advantages that: according to the fault-tolerant control method for the actuator fault of the sliding-mode-based linear multi-agent system, a state error variable is defined according to the communication topology among the multi-agents, an integral sliding mode surface is designed based on the state error variable, the asymptotic stability of the sliding mode of the system is guaranteed, the robustness of the system is enhanced, the unknown upper bound information of the fault is estimated by combining a self-adaptive method, a complete sliding-mode fault-tolerant controller is finally formed, and the system can be guaranteed to stably run and complete a tracking task after the actuator fault occurs.
Has the following advantages:
(1) aiming at a general linear multi-agent system, the problems of actuator partial failure fault and external interference are considered at the same time;
(2) according to a communication topological structure between intelligent agents, a global state error variable is constructed, an integral sliding mode surface is designed based on the variable, and the robustness of the system is enhanced;
(3) a sliding mode fault-tolerant controller is designed by combining a self-adaptive method, so that the multi-agent system can still complete the tracking task under the condition of failure of an actuator, and the fault-tolerant capability of the system is improved.
The method used by the invention is used as a fault-tolerant control method for faults of the linear multi-agent system actuator, has better fault-tolerant capability and robustness, strong operability and easy realization, has certain practical application value, and can be widely applied to the field of fault-tolerant control of faults of the multi-agent system actuator.
Drawings
FIG. 1 is a flow chart of the method of the present invention;
FIG. 2 is a schematic diagram of a Quanser quad-rotor aircraft Q-ball-X4 and its attitude motion;
FIG. 3 is a diagram of a communication topology for a multiple quad-rotor aircraft system;
FIG. 4 is a schematic block diagram of a single four-rotor aircraft fault-tolerant control;
FIG. 5 is an X-axis displacement tracking error plot;
FIG. 6 is an X-axis velocity tracking error curve;
FIG. 7 is an actuator dynamic error curve;
fig. 8 is a Sinmulink simulation.
Detailed Description
The invention is further explained below with reference to the drawings.
As shown in fig. 1, when an actuator failure fault and external disturbance occur in a linear multi-agent system, an error variable is defined according to communication topology between agents, an integral sliding mode surface is designed, and a sliding mode fault-tolerant controller is designed by combining acquired fault information. The method comprises the following specific steps:
step 1) acquiring a control model, an actuator fault model and a communication topological structure of a multi-agent system:
step 1.1) the leader control model is as shown in formula (1):
Figure GSB0000190159010000051
wherein x is0(t)∈RnIs the state quantity of the leader, r0(t)∈RmIs a control input of the leader;
step 1.2) follower control model is shown as formula (2):
Figure GSB0000190159010000052
wherein x isi(t)∈RnAnd ui(t)∈RmState quantities and control inputs representing the i, i 1, 2, N followers are a given system matrix, and (a, B) is stable, fi(t)∈RmRepresents the external interference of the ith follower and meets the requirement
Figure GSB0000190159010000061
Is a known normal number;
step 1.3) the actuator fault model is shown as formula (3):
Figure GSB0000190159010000062
wherein u isi(t) is the actuator input and,
Figure GSB0000190159010000063
is the actuator output with failure, ρi(t)＝diag{ρi1(t)，ρi3(t)，...，ρim(t)}∈Rm×mIs the actuator failure matrix of the ith follower, failure factor ρip(t), p 1, 2,. m is unknown time-varying and bounded when ρipWhen (t) is 0, it indicates that the ith follower has not failed, and when 0 < ρ ≦ is setipWhen (t) < 1, it means the ithPartial failure fault occurs to the p-th actuator of the follower when rhoipWhen (t) is 1, the pth actuator representing the ith follower is completely failed, and this case is not considered here;
thus, an agent model for the occurrence of an actuator partial failure fault may be described as:
Figure GSB0000190159010000064
step 1.4) communication topology of multi-agent system:
considering a multi-agent system comprising a leader, labeled 0, and N followers, labeled i 1, 2., N, a graph G (V, E) represents a communication topology graph between all nodes including the leader and followers, where the set of nodes V {0, 1, 2., N } is the set of communication links between the nodes E ═ V × V; subfigure of G
Figure GSB0000190159010000065
Is a communication topology between followers, wherein
Figure GSB0000190159010000066
An adjacency matrix representing diagram G; l is Laplacian matrix of graph G, and is defined
Figure GSB0000190159010000067
Wherein
Figure GSB0000190159010000068
Is a diagonal matrix composed of degrees of each node, thenijIs as defined in formula (5):
Figure GSB0000190159010000069
order to
Figure GSB00001901590100000610
Representing adjacency matrices between leader and followerIf there is an undirected edge between the navigator 0 and the ith follower, then b i1, otherwise, b i0; definition of
Figure GSB00001901590100000611
A neighbor set for the ith follower;
step 2) defining a state error variable shown in a formula (6) according to adjacent information acquired by the ith follower:
Figure GSB0000190159010000071
wherein, aijIs the connection weight between the ith follower and the jth follower, biRepresenting the weight of the connection between the ith follower and the leader, NiIs the neighbor set of the ith follower;
definition of
Figure GSB0000190159010000072
Equation (6) can be rewritten as a global error variable as shown in equation (7):
Figure GSB0000190159010000073
wherein,
Figure GSB0000190159010000074
step 3) aiming at the multi-agent tracking control system with the actuator fault as shown in the formula (4), designing a sliding mode surface as shown in a formula (8) based on a state error variable formula (6):
Figure GSB0000190159010000075
wherein G ═ BTB)-1BT∈Rm×nIs a sliding mode matrix, K is an element of Rm×nIs the controller gain to be designed; to ensure the gradual stability of the sliding mode of the systemQualitative, satisfying the condition: re [ lambda (A + BK)]< 0, i.e. all characteristic roots of the matrix (A + BK) have negative real parts;
according to the formula (7), the formula (8) can be rewritten as
Figure GSB0000190159010000076
Wherein,
Figure GSB0000190159010000077
step 4) estimating the unknown upper bound of the fault information by the self-adaptive method, firstly defining an unknown parameter thetai＝1/1-||ρiI indirectly reflects fault information, and an adaptive law shown as a formula (10) is designed to estimate thetai：
Figure GSB0000190159010000078
Wherein,
Figure GSB0000190159010000079
is the parameter thetaiAnd alpha is a normal number, omegaiGiven in step 5);
step 5) integrating the step 3) and the step 4), designing a complete fault-tolerant control law as shown in the formula (11):
Figure GSB00001901590100000710
wherein,
Figure GSB00001901590100000711
and η, k1Are all normal numbers;
by combining the formula (7) and the formula (9), the formula (11) is rewritten as:
Figure GSB0000190159010000081
wherein b ═ b1 b2 ... bN]T，
Figure GSB0000190159010000082
And 5) selecting proper parameters according to the running state of the multi-agent system to realize the tracking fault-tolerant control of the system.
The foregoing is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, various modifications and decorations can be made without departing from the principle of the present invention, and these modifications and decorations should also be regarded as the protection scope of the present invention.
The effectiveness of the implementation is illustrated in the following by a practical case simulation.
A four-rotor helicopter Qball-X4 manufactured by quanter company of Canada is used as a specific algorithm experiment simulation object. Fig. 2 is a diagram of Quanser's quad-rotor vehicle Qball-X4 and its attitude motion, and it can be seen from fig. 2 that there are six-dimensional variables (X, Y, Z, ψ, θ, φ) with respect to the ground for the quad-rotor vehicle, where the first three variables are position variables, i.e., the position with respect to the center of the inertial system. The last three variables are the attitude euler angles of the quadrotor helicopter: yaw ψ, pitch θ, roll φ. Without loss of generality, the displacement, the speed and the actuator dynamic in the X-axis direction are selected as state quantities to carry out simulation experiments on the state quantities.
The state space model of the quad-rotor aircraft is as follows:
Figure GSB0000190159010000083
state space expressions written in standard form:
Figure GSB0000190159010000084
wherein,
Figure GSB0000190159010000085
u (t) is a control input, and y (t) is an X-axis displacement.
The airframe parameter values for this four-rotor aircraft are shown in table 1:
TABLE 1 numerical table of body parameters
Parameter(s) Value unit
K 120N
ω 15rad/see
M 1.4kg
Assuming θ is 0.035rad, the coefficient matrices in the nominal system can be obtained as follows:
Figure GSB0000190159010000091
here we consider a multi-quad rotor aircraft tracking control system consisting of one leader and four followers, where the leader is labeled 0 and the followers are labeled i (i ═ 1, 2, 3, 4). The system model of the leader Qball-X4 is:
Figure GSB0000190159010000092
considering the problem that the follower Qball-X4 has actuator failure fault and external interference, the system model is as follows:
Figure GSB0000190159010000093
assuming that the communication topology of the multi-quad rotor aircraft system is shown in fig. 3, we can obtain Laplacian matrix L and adjacency matrix as follows
Figure GSB0000190159010000094
Figure GSB0000190159010000095
Consider the actuator failure fault, ρ, of quadrotors 1 and 2 occurring during flight1＝0.4+0.1cos(t)，ρ20.3+0.2sin (2t), the remaining follower quadrotors were normal. Assume that the external disturbance experienced by four follower quad-rotor aircraft is set as: f. ofi(t) ═ 0.2sin (t) (i ═ 1, 2) and fi(t) 0.3cos (2t) (i 3, 4). Taking the state quantities of the leader control input and the follower of the system at the initial moment as follows: u. of0＝sin(t)，x0(0)＝[0.2 0.3 0.1]T，x1(0)＝[0 0.2 0.15]T，x2(0)＝[0.4 0.2 0.1]T，x3(0)＝[-0.6 0.4 0.25]T，x4(0)＝[-0.4 0.3 0.2]T. The values of the sliding mode surface matrix and the controller are as follows: g ═ 000.0667]，K＝[0.3611 -0.2078 -0.7733]，η＝30，α＝0.3，k1＝2，δ＝0.1，
Figure GSB0000190159010000096
According to the method, the system of the multi-four-rotor aircraft with actuator faults and external interference is subjected to fault-tolerant control, and the fault-tolerant control results are shown in figures 5 to 7. Fig. 5-7 are plots of X-axis displacement, velocity, and actuator dynamics tracking error, respectively.
As can be seen from fig. 5 to 7, when the system has an actuator failure, under the fault-tolerant control of the present invention, the tracking error of the X-axis displacement and the speed of each aircraft can approach zero in a short time, the tracking error curves of the four- rotor aircraft 3 and 4 that do not have a failure can approach zero in five seconds, and the four- rotor aircraft 1 and 2 that have a failure can also approach zero in about 7 seconds, that is, after the system has a failure, the aircraft can still successfully complete the tracking task, thereby avoiding the occurrence of accidents and the failure of the task.

Claims (1)

1. A fault-tolerant control method of a linear multi-agent system based on a sliding mode is characterized by comprising the following steps: considering a communication topological structure among multiple intelligent agents and possible actuator partial failure faults of a single intelligent agent, combining adaptive control and sliding mode control, providing an adaptive sliding mode tracking fault-tolerant control method, so that a multi-intelligent-agent system can still realize tracking stability after a fault occurs, and keep good dynamic quality, according to the communication topological structure among the multiple intelligent agents, firstly defining a state error variable, designing an integral sliding mode surface based on the state error variable, increasing the robustness of the system, estimating the unknown upper bound of fault information through an adaptive method, and further designing a corresponding sliding mode fault-tolerant controller, wherein the method comprises the following specific steps:
step 1) acquiring a control model, an actuator fault model and a communication topological structure of a multi-agent system:
step 1.1) the leader control model is as shown in formula (1):
Figure FSB0000190157000000011
wherein x is0(t)∈RnIs the state quantity of the leader, r0(t)∈RmIs a control input of the leader;
step 1.2) follower control model is shown as formula (2):
Figure FSB0000190157000000012
wherein x isi(t)∈RnAnd ui(t)∈RmState quantities and control inputs representing the i, i 1, 2, N followers are a given system matrix, and (a, B) is stable, fi(t)∈RmRepresents the external interference of the ith follower and meets the requirement
Figure FSB0000190157000000013
Figure FSB0000190157000000014
Is a known normal number;
step 1.3) the actuator fault model is shown as formula (3):
Figure FSB0000190157000000015
wherein u isi(t) is the actuator input and,
Figure FSB0000190157000000016
is the actuator output with failure, ρi(t)＝diag{ρi1(t)，ρi3(t)，...，ρim(t)}∈Rm×mIs the actuator failure matrix of the ith follower, failure factor ρip(t), p 1, 2,. m is unknown time-varying and bounded when ρipWhen (t) is 0, it indicates that the ith follower has not failed, and when 0 < ρ ≦ is setipWhen (t) < 1, it indicates that the ith follower has a partial failure in the pth actuator, and when ρipWhen (t) is 1, the pth actuator representing the ith follower is completely failed, and this case is not considered here;
thus, an agent model for the occurrence of an actuator partial failure fault may be described as:
Figure FSB0000190157000000021
step 1.4) communication topology of multi-agent system:
considering a multi-agent system comprising a leader, labeled 0, and N followers, labeled i 1, 2., N, a graph G (V, E) represents a communication topology graph between all nodes including the leader and followers, where the set of nodes V {0, 1, 2., N } is the set of communication links between the nodes E ═ V × V; subfigure of G
Figure FSB0000190157000000022
Is a communication topology between followers, wherein
Figure FSB0000190157000000023
An adjacency matrix representing diagram G; l is Laplacian matrix of graph G, and is defined
Figure FSB0000190157000000024
Wherein
Figure FSB0000190157000000025
Is a diagonal matrix composed of degrees of each node, thenijIs as defined in formula (5):
Figure FSB0000190157000000026
order to
Figure FSB0000190157000000027
Representing the adjacency matrix between the leader and the followers, if there is an undirected edge between leader 0 and the ith follower, then bi1, otherwise, bi0; definition of
Figure FSB0000190157000000028
As a neighbor set for the ith follower；
Step 2) defining a state error variable shown in a formula (6) according to adjacent information acquired by the ith follower:
Figure FSB0000190157000000029
wherein, aijIs the connection weight between the ith follower and the jth follower, biRepresenting the weight of the connection between the ith follower and the leader, NiIs the neighbor set of the ith follower;
definition of
Figure FSB00001901570000000210
Equation (6) can be rewritten as a global error variable as shown in equation (7):
Figure FSB00001901570000000211
wherein,
Figure FSB00001901570000000212
step 3) aiming at the multi-agent tracking control system with the actuator fault as shown in the formula (4), designing a sliding mode surface as shown in a formula (8) based on a state error variable formula (6):
Figure FSB00001901570000000213
wherein G ═ BTB)-1BT∈Rm×nIs a sliding mode matrix, K is an element of Rm×nIs the controller gain to be designed; in order to ensure the asymptotic stability of the system sliding mode, the following conditions are met: re [ lambda (A + BK)]< 0, i.e. all characteristic roots of the matrix (A + BK) have negative real parts;
according to the formula (7), the formula (8) can be rewritten as
Figure FSB0000190157000000031
Wherein,
Figure FSB0000190157000000032
step 4) estimating the unknown upper bound of the fault information by the self-adaptive method, firstly defining an unknown parameter thetai＝1/1-||ρiI indirectly reflects fault information, and an adaptive law shown as a formula (10) is designed to estimate thetai：
Figure FSB0000190157000000033
Wherein,
Figure FSB0000190157000000034
is the parameter thetaiAnd alpha is a normal number, omegaiGiven in step 5);
step 5) integrating the step 3) and the step 4), designing a complete fault-tolerant control law as shown in the formula (11):
Figure FSB0000190157000000035
wherein,
Figure FSB0000190157000000036
and η, k1Are all normal numbers;
by combining the formula (7) and the formula (9), the formula (11) is rewritten as:
Figure FSB0000190157000000037
wherein b ═ b1 b2... bN]T，
Figure FSB0000190157000000038
And 5) selecting proper parameters according to the running state of the multi-agent system to realize the tracking fault-tolerant control of the system.
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