CN115356929A - Proportional allowable tracking control method for actuator attack singularity multi-agent system - Google Patents

Proportional allowable tracking control method for actuator attack singularity multi-agent system Download PDF

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CN115356929A
CN115356929A CN202211012274.3A CN202211012274A CN115356929A CN 115356929 A CN115356929 A CN 115356929A CN 202211012274 A CN202211012274 A CN 202211012274A CN 115356929 A CN115356929 A CN 115356929A
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刘晓凡
吴宪祥
郭宝龙
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Xidian University
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Abstract

The invention discloses a proportional allowable tracking control method for a strange multi-agent system attacked by an executor, which comprises the following steps: step 1: constructing a follower singular multi-agent system, a leader singular multi-agent system and a nonlinear actuator attack model containing time lag; performing state estimation on the attack of the nonlinear actuator by adopting a radial neural network; step 2, designing a state observer for each follower agent; and step 3: constructing a self-adaptive distributed controller and an integral sliding-mode surface equation containing a singular matrix; and 4, step 4: analyzing the tolerance of a closed-loop system under the action of an adaptive controller through a singular system tolerance theory and a robust stability theory, and providing a singular multi-agent system proportion tolerance following control method; and 5: and (4) verifying the limited time accessibility of the designed integral sliding mode surface equation. The invention realizes the proportional allowable tracking control of the singular multi-agent system under the attack of an unknown nonlinear actuator, so that the system realizes the tracking and has good inhibiting effect on the attack of the actuator.

Description

Proportional allowable tracking control method for actuator attack singularity multi-agent system
Technical Field
The invention belongs to the technical field of multi-agent system control, and relates to a strange multi-agent system proportion allowable tracking control method with actuator attack.
Background
In recent years, the problem of distributed consistent control of multi-agent systems has received widespread attention and use. In general, consistent controls can be divided into leader-less consistent controls and leader-tracking controls. The tracking control is that all followers track one or a group of leaders under the action of the distributed controller, so that the method has the advantages of improving the communication efficiency and reducing the communication cost. In practical applications, different agents may perform different tasks. The proportional consistent control of a multi-agent system requires that agents converge according to different proportions, and the final convergence value of an agent is independent of the initial state of the system. Therefore, the proportional consistent control can effectively solve the problem of multi-scale coordination control among the intelligent agents. In some actual complex system modeling, a differential equation and an algebraic constraint equation are combined to establish an accurate system model. The dynamical model with algebraic constraints is called a singular system. A system formed by connecting a plurality of intelligent agents with singular dynamics models through a wired or wireless network is called a singular multi-intelligent-agent system.
The research of the fanciful multi-agent system has mostly focused on the problem of allowable consistent control without the leader, and the problem of allowable tracking control with the leader agent has not been fully studied. Meanwhile, the convergence targets of the singular multi-agent system depend on the initial state of the system, and the ratio-allowed coordination control of the singular multi-agent system with the convergence value independent of the initial state of the system is not involved. During the operation of a multi-agent system, attacks are one of the main threats to the security performance of the system. In order to ensure the security of a multi-agent system under attack conditions, the following two methods are mainly adopted at present. First, an attacked agent is detected, identified and deleted. Secondly, the external attack is restrained by combining the distributed controller under the condition that the attacked agent is not removed, and the system is guaranteed to have good resistance and recovery performance to the external attack. At present, the distributed consistent control research results with actuator attacks are concentrated on a normal multi-agent system, and the problem of allowable tracking control of the singular multi-agent system under the condition of an actuator is not considered yet. The sliding mode control is a nonlinear variable structure control, and when the system state passes through different areas, the sliding mode feedback control structure is controlled according to different rules. The sliding mode control has strong robustness on external interference of the system, can solve the problem of distributed coordination control of the singular multi-agent system with the external interference, and effectively reduces the damage of the external interference on the system performance.
At present, the multi-agent system tracking control research with actuator attack is mostly concentrated on a normal multi-agent system, and a singular multi-agent system with a kinetic model containing algebraic constraints is not fully considered. Further, the scale-tolerant tracking control can effectively solve the problem of multi-scale coordination control among singular multi-agents. Therefore, scale-tolerant tracking control of the singular multi-agent system with an actuator attack is an urgent problem to be solved.
Disclosure of Invention
The invention aims to provide a proportion-allowed tracking control method of a singular multi-agent system with actuator attack, which can provide a self-adaptive distributed controller based on a neural network and sliding mode control to drive the singular multi-agent system to realize the proportion-allowed tracking control.
The technical scheme adopted by the invention is as follows:
the proportional allowable tracking control method for the actuator attacking singular multi-agent system comprises the following steps:
step 1, constructing a follower singular multi-agent system, a leader singular multi-agent system and a nonlinear actuator attack model containing time lag; performing state estimation on the attack of the nonlinear actuator by adopting a radial neural network;
step 2, designing a state observer for each follower intelligent agent and constructing a state error and a proportion error;
step 3, constructing an integral sliding-mode surface equation containing a singular matrix and a self-adaptive distributed controller based on a neural network;
step 4, analyzing the tolerance of the closed-loop system under the action of the self-adaptive controller through a singular system tolerance theory and a robust stability theory, and providing a singular multi-agent system proportion tolerance following control method;
and 5, verifying the limited time accessibility of the designed integral sliding mode surface equation.
The invention is also characterized in that:
the model of the follower singular multi-agent system in the step 1 is specifically as follows:
consider N follower singular multi-agent systems containing nonlinear effector attacks, where the model of the ith follower singular multi-agent system can be represented as:
Figure BDA0003811374120000031
x i (t)＝η i (t),t∈[-h,0] (1)
wherein x i (t)∈R n Is the state of the ith agent, and h (t) is 0-h (t) -h<Infinity and
Figure BDA0003811374120000032
time-varying time lag of u ir (t)∈R p Is a control input, w i (t)∈R q Is an external disturbance; function eta i (t) is an initial value of agent i. The matrix pair (E, A) is regular; wherein A ∈ R n×n ，B u ∈R n×p And B w ∈R n×q Is a known system parameter matrix, matrix B u Column full rank;
the model of the leader's singular multi-agent system is:
Figure BDA0003811374120000041
x 0 (t)＝η 0 (t),t∈[-h,0] (2)
wherein x 0 (t)∈R n Is the state of the leader's singular multi-agent system. Initial state of leader singular multi-agent system is eta 0 (t)；
The nonlinear actuator attack model containing time lag and the estimation method are as follows:
the actuator in the operation of the singular multi-agent system can be attacked, and the safety performance of the system is threatened: assuming that the actuators of each follower's singular multi-agent system are under attackIs different in strength, defines an attack coefficient alpha i ∈[0,1]. Then, the actuator attack function of the follower singular multi-agent system i can be expressed as:
u ir (t)＝u i (t)+α i φ i (x i (t),x i (t-h(t)),t) (3)
wherein the content of the first and second substances,
Figure BDA0003811374120000042
using radial neural networks
Figure BDA0003811374120000043
To phi i (x i (t),x i (t-h (t)), t) estimating with an estimation error of
Figure BDA0003811374120000044
Wherein the vector
Figure BDA0003811374120000045
Is the input variable of the neural network with an input deviation of-1, vector
Figure BDA0003811374120000046
Error vector
Figure BDA0003811374120000047
For any e i >0 satisfies
Figure BDA0003811374120000048
Matrix array
Figure BDA0003811374120000049
Representing the weight of the hidden layer to the output layer,
Figure BDA00038113741200000410
representing the weights of the input layer to the hidden layer. l i The number of nodes is hidden in the neural network, and p is the number of nodes in the output layer. Nonlinear equation σ i (. Is) an input-to-hidden-layer transfer function that can be expressed as a function vector of:
Figure BDA00038113741200000411
wherein
Figure BDA00038113741200000412
For i =1,2.., N, the hypothetical matrix
Figure BDA00038113741200000413
And
Figure BDA00038113741200000414
are respectively gamma i And Θ i The estimation matrix of (2); then, the estimation error of the weight matrix can be expressed as
Figure BDA00038113741200000415
And
Figure BDA00038113741200000416
assuming a non-linear function phi i (x i (t),x i (t-h (t)), t) an estimation function of
Figure BDA00038113741200000417
Then, the error function:
Figure BDA0003811374120000051
is expressed as
Figure BDA0003811374120000052
Residual error
Figure BDA0003811374120000053
Can be expressed as:
Figure BDA0003811374120000054
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003811374120000055
is a Jacobian matrix, and the infinitesimal term o (-) is expressed as
Figure BDA0003811374120000056
Norm of residual term
Figure BDA0003811374120000057
The following conditions are satisfied:
Figure BDA0003811374120000058
wherein beta is i ∈R 4 Is the unknown vector of the vector,
Figure BDA0003811374120000059
is beta i And the corresponding estimation error is
Figure BDA00038113741200000510
For an arbitrary matrix M ∈ R p×q ，
Figure BDA00038113741200000511
Wherein λ max (M T M) a representation matrix M T The maximum eigenvalue of M;
Figure BDA00038113741200000512
the step 2 specifically comprises the following steps:
step 2.1, designing a state observer for each follower singular multi-agent system:
Figure BDA00038113741200000513
Figure BDA00038113741200000514
wherein
Figure BDA00038113741200000515
Is state estimation, design u ai (t) for reducing non-linear actuator attacks a containing time lag i φ i (x i (t),x i (t-h (t)), t); and is provided with
Figure BDA00038113741200000516
Is an initial value of the observer system;
step 2.2. Defining state errors of follower singular multi-agent system and state observer
Figure BDA00038113741200000517
The expression for solving the state error equation by equation (8) is:
Figure BDA0003811374120000061
wherein the content of the first and second substances,
Figure BDA0003811374120000062
step 2.3. Construction of proportional error
Figure BDA0003811374120000063
And solving for the proportional error
Figure BDA0003811374120000064
The kinetic equation of (a):
Figure BDA0003811374120000065
wherein the content of the first and second substances,
Figure BDA0003811374120000066
v i and v j Is a proportional function allowing proportional tracking control, and v ij ＝v i /v j ，i＝1,2,...,N，j＝0,1,2,...,N。
The step 3 specifically comprises:
step 3.1, designing an integral sliding mode surface equation:
Figure BDA0003811374120000067
wherein X ∈ R n×n Is an unknown nonsingular matrix, K ∈ R p×n Is the feedback gain matrix to be designed, s is the integral variable, a ij For the connection weight between the follower singular multi-agent system j and the follower singular multi-agent system i, if the follower singular multi-agent system j is communicated with the follower singular multi-agent system i, a ij ＝a ji >0; otherwise, a ij ＝0；b i For the connection weights between the follower singular multi-agent and the leader singular multi-agent, if the follower singular multi-agent system i is in communication with the leader singular multi-agent system, b i >0; otherwise, b i ＝0；
Step 3.2, according to the sliding mode control theory, when the proportion error system reaches the sliding mode surface,
Figure BDA0003811374120000068
and
Figure BDA0003811374120000069
if true; obtaining an equivalent controller u by taking the integral sliding mode surface derivative as zero eqi (t) is:
Figure BDA0003811374120000071
substituting the equivalent controller into a proportional error equation to obtain a sliding-mode kinetic equation:
Figure BDA0003811374120000072
and 3.3, designing a self-adaptive distributed controller based on a neural network:
Figure BDA0003811374120000073
wherein alpha is i Is the attack coefficient, σ, of the actuator under attack i (. Is the input layer to hidden layer transfer function, p i (t)>0 is the neural network based adaptive distributed controller parameter to be solved,
Figure BDA0003811374120000074
wherein sgn (.) is a sign function, for any function x, when x>At 0, sgn (x) =1; sgn (x) =0 when x =0; when x is<At 0, sgn (x) = -1;
the updating rule of the parameters is as follows:
Figure BDA0003811374120000075
wherein the matrix
Figure BDA0003811374120000076
And
Figure BDA0003811374120000077
weight matrix gamma from hidden layer to output layer i And the weight Θ of the input layer to the hidden layer i Is determined by the estimation matrix of (a),
Figure BDA0003811374120000078
is an unknown vector beta i Is determined by the estimated value of (c),
Figure BDA0003811374120000079
is a Jacobian matrix, M i1 >0，M i2 >0，M i3 >0 is the gain matrix, ρ i (t)>0 is the adaptive distributed controller parameter, vector based on neural network to be solved
Figure BDA00038113741200000710
The step 4 specifically comprises the following steps:
step 4.1 when w i (t) =0, if matrix exists
Figure BDA00038113741200000711
And Q 2 ∈R n×n >0, satisfying:
Figure BDA0003811374120000081
Figure BDA0003811374120000082
wherein
Figure BDA0003811374120000083
The adaptive distributed controller (14) based on neural network can drive the state variables of the follower singular multi-agent system (1) to track the state of the leader singular multi-agent system (2) in a fixed proportion, i.e.,
Figure BDA0003811374120000084
establishing all the singular multi-agent systems;
step 4.2 when w i When (t) ≠ 0, given γ>0, if there is a matrix
Figure BDA0003811374120000085
H 1 ∈R n×n And positive definite moment of symmetry Q 1 ∈R n×n ，
Figure BDA0003811374120000086
H 1 ∈R n×n ，
Figure BDA0003811374120000087
H 2 ∈R q×q Satisfy the requirement of
Figure BDA0003811374120000088
Wherein
Figure BDA0003811374120000089
Figure BDA00038113741200000810
Figure BDA00038113741200000811
Figure BDA00038113741200000812
Matrix U satisfies
Figure BDA00038113741200000813
Is a diagonal matrix and feeds back a gain matrix
Figure BDA00038113741200000814
The adaptive distributed controller (14) based on the neural network can drive the state variables of the follower singular multi-agent system (1) to track the state of the leader singular multi-agent system (2) according to a fixed proportion, and has a restraining effect on the external interference of the follower singular multi-agent system, namely,
Figure BDA0003811374120000091
the external interference of the follower singular multi-agent system is satisfied when all the singular multi-agent systems are established:
Figure BDA0003811374120000092
wherein the content of the first and second substances,
Figure BDA0003811374120000093
e i (t)＝x i (t)-v i0 x 0 (t) and
Figure BDA0003811374120000094
Figure BDA0003811374120000095
the specific method of the step 4.1 is as follows:
step 4.1.1, adopting a singular system model decomposition method to carry out pair inequality A T X+X T A+Q 1 <0 is decomposed to obtain:
Figure BDA0003811374120000096
thus, det (A) 22 ) Not equal to 0,det (.) represents the determinant of the matrix, i.e. (E, A) no pulse;
step 4.1.2, constructing Lyapunov function of the formula (18)
Figure BDA0003811374120000097
Wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003811374120000098
Figure BDA0003811374120000099
Figure BDA00038113741200000910
Figure BDA00038113741200000911
the Lyapunov function is derived along an error system under the action of a self-adaptive controller based on a neural network, and the following can be obtained:
Figure BDA0003811374120000101
wherein the content of the first and second substances,
Figure BDA0003811374120000102
Figure BDA0003811374120000103
Figure BDA0003811374120000104
matrix of
Figure BDA0003811374120000105
Figure BDA0003811374120000106
l ij ＝-a ij All i is not equal to j; according to the formula (16), the
Figure BDA0003811374120000107
Thus, error function
Figure BDA0003811374120000108
And
Figure BDA0003811374120000109
the average index is stable; the proportional error tracking system has no pulse and satisfies
Figure BDA00038113741200001010
That is, the singular multi-agent system containing time-lapse nonlinear actuator attacks implements scale-tolerant tracking control.
The step 4.2 is specifically as follows:
when w (t) ≠ 0, the Lyapunov function constructed in the equation (18) is combined with the following energy function
Figure BDA00038113741200001011
The above equation is solved to obtain
Figure BDA00038113741200001012
Wherein
Figure BDA00038113741200001013
Figure BDA00038113741200001014
Figure BDA00038113741200001015
By means of matrices
Figure BDA0003811374120000111
And
Figure BDA0003811374120000112
right and left squaring respectively 5×5 Let us order
Figure BDA0003811374120000113
And
Figure BDA0003811374120000114
due to the matrix M L And M R If the matrix is a full rank matrix, J is less than or equal to 0; and feedback the gain matrix
Figure BDA0003811374120000115
The step 5 specifically comprises the following steps:
and 5.1, if the matrix K meets the condition in the step 4.2 and parameters in the adaptive controller (14) based on the neural network meet:
Figure BDA0003811374120000116
wherein, for all i =1,.., N,
Figure BDA0003811374120000117
is a normal number; the integral sliding mode surface equation (11) can be reached in limited time;
step 5.2, constructing a Lyapunov function of the formula (22):
Figure BDA0003811374120000118
the following are obtained by calculation:
Figure BDA0003811374120000119
wherein the content of the first and second substances,
Figure BDA00038113741200001110
thus, there is a moment
Figure BDA00038113741200001111
So that all T ≧ T satisfy V s (t) =0 and s (t) =0. Namely, the integral sliding mode surface equation can be reached within a limited time.
The invention has the beneficial effects that:
1. compared with the time-lag independent nonlinear actuator attack, the state and time-lag dependent actuator attack function designed by the invention can better simulate the actuator attack model in the actual system operation. And the invention adopts an estimation model based on a neural network method, thereby realizing the rapid estimation of the nonlinear actuator attack of the singular multi-agent system.
2. The invention designs a novel distributed self-adaptive controller based on sliding mode control and a neural network, the controller effectively reduces the influence of external interference on the tracking performance, and can drive the follower agent state to track the leader agent state according to a fixed proportion;
3. the proportion allowable tracking control of the singular multi-agent system under the condition of the actuator attack comprises the general conditions of allowable tracking control, proportion tracking control of a normal multi-agent system, tracking control of the normal multi-agent system and the like.
Drawings
FIG. 1 is a system control flow diagram of the method of the present invention;
fig. 2 is a communication topology diagram of an agent according to embodiment 1 and embodiment 2 of the present invention;
fig. 3 shows the locus diagrams of the proportional tracking error of the embodiment 1 when i =1,2,3,4 in (a), (b), (c) and (d), respectively;
fig. 4 is a schematic diagram of errors between the actuator attack and the neural network estimation function in the embodiment 1 when i =1,2,3,4 in (a), (b), (c), and (d), respectively;
in fig. 5, (a), (b), (c), and (d) are trajectory diagrams of sliding mode surface equations of example 1 when i =1,2,3,4, respectively;
fig. 6 is a trace diagram of proportional tracking errors of example 2 of the present invention when i =1,2,3,4 in (a), (b), (c), and (d), respectively;
fig. 7 is a schematic diagram of errors between the actuator attack and the neural network estimation function of embodiment 2 of the present invention when i =1,2,3,4 in (a) (b) (c) (d), respectively;
in fig. 8, (a), (b), (c), and (d) are trajectory diagrams of sliding mode surface equations for embodiment 2 of the present invention when i =1,2,3,4, respectively.
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.
The invention relates to a proportional allowable tracking control method for a singular multi-agent system under attack of an executor, as shown in figure 1, comprising the following steps of:
step 1, constructing an actuator attacking singular multi-agent system, wherein the actuator attacking singular multi-agent system comprises a follower singular multi-agent system, a leader singular multi-agent system and a nonlinear actuator attacking model containing time lag; performing state estimation on the attack of the nonlinear actuator by adopting a radial neural network;
step 2, designing a state observer for each follower intelligent agent and constructing a state error and a proportional error;
step 3, constructing an integral sliding-mode surface equation containing a singular matrix and a self-adaptive distributed controller based on a neural network;
step 4, analyzing the tolerance of the closed-loop system under the action of the self-adaptive controller through a singular system tolerance theory and a robust stability theory, and providing a singular multi-agent system proportion tolerance following control method;
and 5, verifying the limited time accessibility of the designed integral sliding mode surface equation.
The specific steps of the step 1 are as follows:
step 1.1. Model of follower singular multi-agent system:
consider N follower singular multi-agent systems containing nonlinear actuator attacks, where the model of the ith follower singular multi-agent system can be represented as:
Figure BDA0003811374120000131
x i (t)＝η i (t),t∈[-h,0] (1)
wherein x i (t)∈R n Is the state of the ith agent, and h (t) is 0-h (t) -h<Infinity and
Figure BDA0003811374120000141
time-varying time lag of u ir (t)∈R p Is a control input, w i (t)∈R q Is an external disturbance. Function eta i (t) is the initial value of agent i. The matrix pair (E, A) is regular. Wherein A ∈ R n×n ，B u ∈R n×p And B w ∈R n×q Is a known system parameter matrix, matrix B u Column full rank.
Step 1.2. The model of the leader's singular multi-agent system is:
Figure BDA0003811374120000142
x 0 (t)＝η 0 (t),t∈[-h,0] (2)
wherein x 0 (t)∈R n Is the state of the leader's singular multi-agent system. Initial state of leader singular multi-agent system is eta 0 (t)。
Step 1.3, a nonlinear actuator attack model containing time lag and estimation:
the actuators in the operation of the singular multi-agent system can be attacked, and the safety performance of the system is threatened. Assuming that the attack strength of actuators of each follower singular multi-agent system is different, defining attack coefficient alpha i ∈[0,1]. Then, the actuator attack function of the follower singular multi-agent system i can be expressed as:
u ir (t)＝u i (t)+α i φ i (x i (t),x i (t-h(t)),t) (3)
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003811374120000143
using radial neural networks
Figure BDA0003811374120000144
To phi i (x i (t),x i (t-h (t)), t) estimating with an estimation error of
Figure BDA0003811374120000145
Wherein the vector
Figure BDA0003811374120000146
Is an input variable of a neural network with an input offset of-1, vector
Figure BDA0003811374120000147
Error vector
Figure BDA0003811374120000148
For any e i >0 satisfies
Figure BDA0003811374120000149
Matrix of
Figure BDA00038113741200001410
Representing the weight of the hidden layer to the output layer,
Figure BDA00038113741200001411
representing the weights of the input layer to the hidden layer. l. the i The number of nodes is hidden in the neural network, and p is the number of nodes in the output layer. Nonlinear equation σ i (. Is) an input-to-hidden-layer transfer function that can be expressed as a function vector of:
Figure BDA0003811374120000151
wherein
Figure BDA0003811374120000152
For i =1,2.., N, the hypothetical matrix
Figure BDA0003811374120000153
And
Figure BDA0003811374120000154
are respectively gamma i And Θ i The estimation matrix of (2). Then, the estimation error of the weight matrix can be expressed as
Figure BDA0003811374120000155
And
Figure BDA0003811374120000156
assuming a non-linear function phi i (x i (t),x i (t-h(t)),t)Is an estimation function of
Figure BDA0003811374120000157
Then, the error function:
Figure BDA0003811374120000158
the expression of (a) is:
Figure BDA0003811374120000159
residual error
Figure BDA00038113741200001510
Can be expressed as:
Figure BDA00038113741200001511
wherein the content of the first and second substances,
Figure BDA00038113741200001512
is a Jacobian matrix with an infinitesimal term o (-) expressed as
Figure BDA00038113741200001513
Norm of residual term
Figure BDA00038113741200001514
The following conditions are satisfied:
Figure BDA00038113741200001515
wherein beta is i ∈R 4 Is the unknown vector of the vector,
Figure BDA00038113741200001516
is beta i And the corresponding estimation error is
Figure BDA00038113741200001517
For an arbitrary matrix M ∈ R p×q ，
Figure BDA00038113741200001518
Wherein λ max (M T M) a representation matrix M T The maximum eigenvalue of M;
Figure BDA00038113741200001519
wherein the step 2 specifically comprises:
step 2.1, designing a state observer for each follower singular multi-agent system:
Figure BDA00038113741200001520
wherein
Figure BDA00038113741200001521
Is state estimation, design u ai (t) for reducing non-linear actuator attacks a containing time lag i φ i (x i (t),x i (t-h (t)), t). And is
Figure BDA0003811374120000161
Is an initial value of the observer system.
Step 2.2. Defining state errors of follower singular multi-agent system and state observer
Figure BDA0003811374120000162
The expression for solving the state error equation by equation (8) is:
Figure BDA0003811374120000163
wherein the content of the first and second substances,
Figure BDA0003811374120000164
step 2.3. Construction of proportional error
Figure BDA0003811374120000165
And solving for the proportional error
Figure BDA0003811374120000166
The kinetic equation of (c):
Figure BDA0003811374120000167
wherein the content of the first and second substances,
Figure BDA0003811374120000168
v i and v j A proportional function for the allowable proportional tracking control, and v ij ＝v i /v j ，i＝1,2,...,N，j＝0,1,2,...,N.
Wherein the step 3 specifically comprises:
step 3.1, designing an integral sliding mode surface equation:
Figure BDA0003811374120000169
wherein X ∈ R n×n Is an unknown nonsingular matrix, K ∈ R p×n Is the feedback gain matrix to be designed, s is the integral variable, a ij For the connection weight between the follower singular multi-agent system j and the follower singular multi-agent system i, if the follower singular multi-agent system j is communicated with the follower singular multi-agent system i, then a ij ＝a ji >0; otherwise, a ij ＝0；b i For the connection weights between the follower singular multi-agent and the leader singular multi-agent, if the follower singular multi-agent system i is in communication with the leader singular multi-agent system, b i >0; otherwise, b i And =0. Step 3.2, according to the sliding mode control theory, when the proportion error system reaches the sliding mode surface,
Figure BDA0003811374120000171
and
Figure BDA0003811374120000172
this is true. Obtaining an equivalent controller u by taking the integral sliding mode surface derivative as zero eqi (t) is:
Figure BDA0003811374120000173
substituting the equivalent controller into a proportional error equation to obtain a sliding-mode kinetic equation:
Figure BDA0003811374120000174
step 3.3, designing the self-adaptive distributed controller based on the neural network
Figure BDA0003811374120000175
Wherein sgn (·) is a sign function, and for an arbitrary function x, sgn (x) =1 when x > 0; sgn (x) =0 when x =0; sgn (x) = -1 when x < 0.
The updating rule of the parameters is as follows:
Figure BDA0003811374120000176
wherein M is i1 >0，M i2 >0，M i3 >0 is the gain matrix, ρ i (t)>0 is the adaptive distributed controller parameter, vector based on neural network to be solved
Figure BDA0003811374120000177
Wherein the step 4 specifically comprises:
step 4.1 when w i (t) =0, if matrix exists
Figure BDA0003811374120000178
And Q 2 ∈R n ×n >0, satisfying:
Figure BDA0003811374120000181
Figure BDA0003811374120000182
wherein
Figure BDA0003811374120000183
The adaptive distributed controller (14) based on neural network can drive the state variables of the follower singular multi-agent system (1) to track the state of the leader singular multi-agent system (2) in a fixed proportion, i.e.,
Figure BDA0003811374120000184
the method is established for all the singular multi-agent systems, and specifically comprises the following steps:
step 4.1.1, adopting a singular system standard model decomposition method to carry out inequality A T X+X T A+Q 1 <0 is decomposed to obtain
Figure BDA0003811374120000185
Thus, det (A) 22 ) Not equal to 0,det (.) represents the determinant of the matrix, i.e. (E, A) no pulse;
step 4.1.2. Constructing the Lyapunov function of the formula (18):
Figure BDA0003811374120000186
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003811374120000187
Figure BDA0003811374120000188
Figure BDA0003811374120000189
Figure BDA00038113741200001810
the Lyapunov function is derived along an error system under the action of a self-adaptive controller based on a neural network, and the following can be obtained:
Figure BDA0003811374120000191
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003811374120000192
Figure BDA0003811374120000193
Figure BDA0003811374120000194
matrix array
Figure BDA0003811374120000195
Figure BDA0003811374120000196
l ij ＝-a ij For all i ≠ j. From the formula (16), it can be obtained
Figure BDA0003811374120000197
Thus, error function
Figure BDA0003811374120000198
And
Figure BDA0003811374120000199
the average index is stable. Error in proportionThe difference tracking system has no pulse and satisfies
Figure BDA00038113741200001910
That is, the singular multi-agent system containing time-lapse nonlinear actuator attacks implements scale-tolerant tracking control.
Step 4.2 when w i When (t) ≠ 0, given γ>0, if a matrix exists
Figure BDA00038113741200001911
H 1 ∈R n×n Positive definite moment of symmetry Q 1 ∈R n×n ，
Figure BDA00038113741200001912
H 1 ∈R n×n ，
Figure BDA00038113741200001913
H 2 ∈R q×q Satisfy the requirement of
Figure BDA00038113741200001914
Wherein
Figure BDA00038113741200001915
Figure BDA00038113741200001916
Figure BDA00038113741200001917
Figure BDA00038113741200001918
Matrix U satisfies
Figure BDA00038113741200001919
Is a diagonal matrix and feeds back a gain matrix
Figure BDA00038113741200001920
The adaptive distributed controller (14) based on the neural network can drive the state variables of the follower singular multi-agent system (1) to track the state of the leader singular multi-agent system (2) according to a fixed proportion, and has a restraining effect on the external interference of the follower singular multi-agent system, namely,
Figure BDA0003811374120000201
the external interference of the follower singular multi-agent system is satisfied when all the singular multi-agent systems are established:
Figure BDA0003811374120000202
wherein the content of the first and second substances,
Figure BDA0003811374120000203
e i (t)＝x i (t)-v i0 x 0 (t) and
Figure BDA0003811374120000204
Figure BDA0003811374120000205
when w (t) ≠ 0, the Lyapunov function constructed in equation (18) is combined with the following energy function:
Figure BDA0003811374120000206
solving the above equation can obtain:
Figure BDA0003811374120000207
wherein
Figure BDA0003811374120000208
Figure BDA0003811374120000209
Figure BDA00038113741200002010
By means of matrices
Figure BDA00038113741200002011
And
Figure BDA00038113741200002012
right and left multiplier xi, respectively 5×5 Let us order
Figure BDA00038113741200002013
And
Figure BDA00038113741200002014
due to the matrix M L And M R All are full rank matrices, then J ≦ 0 may be deduced. And feedback the gain matrix
Figure BDA00038113741200002015
And 5, verifying the limited time accessibility of the integral sliding mode surface:
step 5.1. If the matrix K satisfies the condition in step 4.2 and the parameters in the adaptive controller (14) based on the neural network satisfy
Figure BDA0003811374120000211
Wherein, for all i =1,.., N,
Figure BDA0003811374120000212
is a normal number. The integral sliding-mode surface equation (11) is reachable in a finite time.
Step 5.2, constructing a Lyapunov function of the formula (22):
Figure BDA0003811374120000213
is obtained by calculation
Figure BDA0003811374120000214
Wherein the content of the first and second substances,
Figure BDA0003811374120000215
thus, there is a moment
Figure BDA0003811374120000216
So that all T ≧ T satisfy V s (t) =0 and s (t) =0. Namely, the integral sliding mode surface equation can be reached within a limited time.
Example 1
In order to verify the proposed ratio allowable tracking control effect, the method provided by the invention is adopted to carry out simulation verification by using matlab. In this embodiment, FIG. 2 is an experimental topology diagram, which includes a leader node 0 and 4 follower agent nodes 1,2,3,4. The matrix parameters of the agent are
E＝[1 0 0；0 1 0；0 0 0],A h ＝[0.4 -0.3 0.2；0.2 0.4 0.4；0 0 0.1],
A＝[1 2 1；3 2 4；1 2 1],B u ＝[1 2 0] T ,B w ＝[0 1 -1] T
The time-lag nonlinear actuator attack function is defined as
Figure BDA0003811374120000217
And let h (t) =0.2sin (t). Assuming that the attack coefficient of each follower singular multi-agent system i is alpha i =1, i.e. all follower singular multi agent systems are exposedAnd (4) attack of the section. The scale function of the allowable tracking is v 0 ＝1，v 1 ＝1.5，v 2 ＝1，v 3 ＝1.5，v 4 And =1. The simulation results of FIG. 3 show the proportional allowable tracking error e i (t)＝x i (t)-v i0 x 0 (t) state trace. From FIG. 3, the state trace e of the scale-tolerant tracking error can be seen i (t)＝x i (t)-v i0 x 0 (t) converges to zero after 6s, i.e. proportional tolerant tracking control can be achieved. Suppose the number of hidden layers of the neural network is 10, i.e. node =10, and the initial value of the weight function is
Figure BDA0003811374120000221
And
Figure BDA0003811374120000222
and matrix
Figure BDA0003811374120000223
M i2 ＝4diag(1,1,1,1)，M i3 =5diag (1,1,1,1). FIG. 4 is a trace of error between time-lapse nonlinear actuator attacks and neural network estimated parameters
Figure BDA0003811374120000224
As can be seen from fig. 4, the designed neural network function can realize fast estimation of the time-lag nonlinear actuator attack of the follower singular multi-agent system i =1,2,3,4. Fig. 5 is a track of a sliding mode surface equation, and it can be known from the figure that the sliding mode surface equation is asymptotically stable.
Example 2
Considering the singular multi-agent system scale tolerant tracking control without time lag, i.e. the time lag function h (t) =0. Wherein the matrix parameters of the singular multi-agent system are the same as those in embodiment 1, and the actuator attack model is
Figure BDA0003811374120000225
The scale-tolerant tracking function and neural network are the same as in example 1, and the singular agent system i assumes an attack coefficient α for each follower i =1, i.e. all follower singular multi-agent systems are subject to external attacks. The scale function of the allowable tracking is v 0 ＝1，v 1 ＝1.5，v 2 ＝1，v 3 ＝1.5，v 4 =1, FIG. 6 is the singular multi-agent system proportional tracking error e without time lag i (t)＝x i (t)-v i0 x 0 The state trajectory of (t) is faster in the rate of convergence of the proportional tracking error function of the singular multi-agent system containing no time lag than in example 1. Further, fig. 7 is a difference value between the neural network estimation function and the actuator attack function without time lag, and it can be known from the figure that the designed neural network function can realize fast estimation of the actuator attack without time lag. Fig. 8 is a track of a sliding mode surface equation without time lag, and it can be known from the figure that the designed integral sliding mode surface equation is asymptotically stable.

Claims (6)

1. The proportional allowable tracking control method for the actuator attack singularity multi-agent system is characterized by comprising the following steps of:
step 1, constructing a follower singular multi-agent system, a leader singular multi-agent system and a nonlinear actuator attack model containing time lag; performing state estimation on the attack of the nonlinear actuator by adopting a radial neural network;
step 2, designing a state observer for each follower intelligent agent and constructing a state error and a proportion error;
step 3, constructing an integral sliding mode surface equation containing a singular matrix and a self-adaptive distributed controller based on a neural network;
step 4, analyzing the tolerance of the closed-loop system under the action of the self-adaptive controller through a singular system tolerance theory and a robust stability theory, and providing a singular multi-agent system proportion tolerance following control method;
step 5, verifying the limited time accessibility of the designed integral sliding mode surface equation;
the step 2 specifically comprises:
step 2.1, designing a state observer for each follower singular multi-agent system:
Figure FDA0003811374110000011
wherein E, A h ，B u For the parameter matrix of the known singular multi-agent system, h (t) is equal to or more than 0 and equal to or less than h (t)<Infinity and
Figure FDA0003811374110000012
time-varying time lag of u i (t) is the distributed controller to be designed,
Figure FDA0003811374110000013
Figure FDA0003811374110000014
is state estimation, design u ai (t) for reducing non-linear actuator attacks a containing time lag i φ i (x i (t),x i (t-h (t)), t); and is provided with
Figure FDA0003811374110000015
Is an initial value of the observer system;
step 2.2, state errors of follower singular multi-agent system and state observer are defined
Figure FDA0003811374110000021
Wherein x is i (t) is the state of the ith agent, and the expression for solving the state error equation by equation (8) is:
Figure FDA0003811374110000022
wherein the content of the first and second substances,
Figure FDA0003811374110000023
step 2.3, construction of proportional error
Figure FDA0003811374110000024
And solving for the proportional error
Figure FDA0003811374110000025
The kinetic equation of (a):
Figure FDA0003811374110000026
wherein the content of the first and second substances,
Figure FDA0003811374110000027
v i and v j A proportional function for the allowable proportional tracking control, and v ij ＝v i /v j ，i＝1,2,...,N，j＝0,1,2,...,N；
The step 3 specifically comprises:
step 3.1, designing an integral sliding mode surface equation:
Figure FDA0003811374110000028
where X is an unknown nonsingular matrix, K is a feedback gain matrix to be designed, s is an integral variable, a ij For the connection weight between the follower singular multi-agent system j and the follower singular multi-agent system i, if the follower singular multi-agent system j is communicated with the follower singular multi-agent system i, then a ij ＝a ji >0; otherwise, a ij ＝0；b i For the connection weight between the follower singular multi-agent and the leader singular multi-agent, if the follower singular multi-agent system i is communicated with the leader singular multi-agent system, b i >0; otherwise, b i ＝0；
Step 3.2, according to the sliding mode control theory, when the proportion error system reaches the sliding mode surface,
Figure FDA0003811374110000031
and
Figure FDA0003811374110000032
establishing; obtaining an equivalent controller u by taking the integral sliding mode surface derivative as zero eqi (t) is:
Figure FDA0003811374110000033
substituting the equivalent controller into a proportional error equation to obtain a sliding-mode kinetic equation:
Figure FDA0003811374110000034
step 3.3, designing the self-adaptive distributed controller based on the neural network
Figure FDA0003811374110000035
Wherein alpha is i Is the attack coefficient, σ, of the actuator under attack i (. Is) the input layer to hidden layer transfer function, ρ i (t)>0 is the neural network based adaptive distributed controller parameter to be solved,
Figure FDA0003811374110000036
sgn (.) is a sign function, for an arbitrary function x, when x>At 0, sgn (x) =1; sgn (x) =0 when x =0; when x is<At 0, sgn (x) = -1;
the updating rule of the parameters is as follows:
Figure FDA0003811374110000037
wherein the matrix
Figure FDA0003811374110000038
And
Figure FDA0003811374110000039
weight matrix gamma from hidden layer to output layer i And the weight Θ of the input layer to the hidden layer i Is determined by the estimation matrix of (a),
Figure FDA00038113741100000310
is an unknown vector beta i Is determined by the estimated value of (c),
Figure FDA00038113741100000311
is a Jacobian matrix, M i1 >0，M i2 >0，M i3 >0 is a gain matrix, vector
Figure FDA00038113741100000312
2. The method for proportional-tolerant tracking control of actuator attack singular multi-agent system as claimed in claim 1, wherein the model of the follower singular multi-agent system in step 1 is specifically:
consider N follower singular multi-agent systems containing nonlinear actuator attacks, where the model of the ith follower singular multi-agent system can be represented as
Figure FDA0003811374110000041
x i (t)＝η i (t),t∈[-h,0] (1)
Wherein x i (t)∈R n Is the state of the ith agent, and h (t) is 0-h (t) -h<Infinity and
Figure FDA0003811374110000044
time-varying time lag of u ir (t)∈R p Is to control the transmissionIn, w i (t)∈R q Is an external disturbance; function eta i (t) is the initial value of agent i; the matrix pair (E, A) is regular; wherein A ∈ R n×n ，B u ∈R n×p And B w ∈R n×q Is a known system parameter matrix, matrix B u Column full rank;
the leader's singular multi-agent system model is:
Figure FDA0003811374110000042
x 0 (t)＝η 0 (t),t∈[-h,0] (2)
wherein x 0 (t)∈R n Is the state of the leader's singular multi-agent system; initial state of leader singular multi-agent system is eta 0 (t)；
The nonlinear actuator attack model containing time lag and the estimation method are as follows:
the actuator in the operation of the singular multi-agent system can be attacked, and the safety performance of the system is threatened: assuming that the attacking strength of actuators of each follower singular multi-agent system is different, defining an attack coefficient alpha i ∈[0,1](ii) a Then, the actuator attack function of the follower singular multi-agent system i can be expressed as:
u ir (t)＝u i (t)+α i φ i (x i (t),x i (t-h(t)),t) (3)
wherein the content of the first and second substances,
Figure FDA0003811374110000043
using radial neural networks
Figure FDA0003811374110000051
To phi i (x i (t),x i (t-h (t)), t) estimation is performed with an estimation error of
Figure FDA0003811374110000052
Wherein the vector
Figure FDA0003811374110000053
Is the input variable of the neural network with an input deviation of-1, vector
Figure FDA0003811374110000054
Error vector
Figure FDA0003811374110000055
For any e i >0 satisfies
Figure FDA0003811374110000056
Matrix of
Figure FDA0003811374110000057
Representing the weight of the hidden layer to the output layer,
Figure FDA0003811374110000058
representing the weight from the input layer to the hidden layer; l i The number of nodes is hidden in the neural network, and p is the number of nodes in an output layer; nonlinear equation sigma i (. Is) an input-to-hidden-layer transfer function that can be expressed as a function vector of:
Figure FDA0003811374110000059
wherein
Figure FDA00038113741100000510
For i =1,2.., N, the hypothetical matrix
Figure FDA00038113741100000511
And
Figure FDA00038113741100000512
are respectivelyΓ i And Θ i The estimation matrix of (2); then, the estimation error of the weight matrix can be expressed as
Figure FDA00038113741100000513
And
Figure FDA00038113741100000514
assuming a non-linear function phi i (x i (t),x i (t-h (t)), t) an estimation function of
Figure FDA00038113741100000515
Then, the error function:
Figure FDA00038113741100000516
is expressed as
Figure FDA00038113741100000517
Residual error
Figure FDA00038113741100000518
Can be expressed as
Figure FDA00038113741100000519
Wherein, the first and the second end of the pipe are connected with each other,
Figure FDA00038113741100000520
is a Jacobian matrix, and the infinitesimal term o (-) is expressed as
Figure FDA00038113741100000521
Norm of residual term
Figure FDA00038113741100000522
The following conditions are satisfied:
Figure FDA00038113741100000523
wherein beta is i ∈R 4 Is a vector that is not known and is,
Figure FDA00038113741100000524
is beta i And the corresponding estimation error is
Figure FDA00038113741100000525
For an arbitrary matrix M ∈ R p×q ，
Figure FDA0003811374110000061
Wherein λ is max (M T M) representation matrix M T The maximum eigenvalue of M;
Figure FDA0003811374110000062
3. the method for proportional-tolerant tracking control of an actuator attack singular multi-agent system as claimed in claim 2, wherein said step 4 is specifically:
step 4.1 when w i (t) =0, if matrix exists
Figure FDA0003811374110000063
Q 1 ∈R n×n >0 and Q 2 ∈R n×n >0, satisfying:
Figure FDA0003811374110000064
Figure FDA0003811374110000065
wherein
Figure FDA0003811374110000066
The adaptive distributed controller (14) based on neural network can drive the state variables of the follower singular multi-agent system (1) to track the state of the leader singular multi-agent system (2) in a fixed proportion, i.e.,
Figure FDA0003811374110000067
establishing all the singular multi-agent systems;
step 4.2 when w i When (t) ≠ 0, given gamma>0, if a matrix exists
Figure FDA0003811374110000068
H 1 ∈R n×n Positive definite moment of symmetry Q 1 ∈R n×n ，
Figure FDA00038113741100000611
H 1 ∈R n×n ，
Figure FDA0003811374110000069
H 2 ∈R q×q Satisfy the requirements of
Figure FDA00038113741100000610
Wherein:
Figure FDA0003811374110000071
Figure FDA0003811374110000072
Figure FDA0003811374110000073
Figure FDA0003811374110000074
matrix U satisfies
Figure FDA0003811374110000075
Is a diagonal matrix and feeds back a gain matrix
Figure FDA0003811374110000076
The adaptive distributed controller (14) based on the neural network can drive the state variables of the follower singular multi-agent system (1) to track the state of the leader singular multi-agent system (2) according to a fixed proportion, and has a restraining effect on the external interference of the follower singular multi-agent system, namely,
Figure FDA0003811374110000077
the external interference of the follower singular multi-agent system is satisfied when the external interference of all the singular multi-agent systems is satisfied:
Figure FDA0003811374110000078
wherein the content of the first and second substances,
Figure FDA0003811374110000079
e i (t)＝x i (t)-v i0 x 0 (t) and
Figure FDA00038113741100000710
Figure FDA00038113741100000711
4. the actuator attack singular multi-agent system scale tolerance tracking control method as claimed in claim 3, wherein said step 4.1 is embodied by:
step 4.1.1, adopting a singular system model decomposition method to carry out pair inequality A T X+X T A+Q 1 <0 is decomposed to obtain:
Figure FDA00038113741100000712
thus, det (A) 22 ) Not equal to 0,det (.) represents the determinant of the matrix, i.e. (E, A) no pulse;
step 4.1.2. Constructing the Lyapunov function of the formula (18):
Figure FDA00038113741100000713
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003811374110000081
Figure FDA0003811374110000082
Figure FDA0003811374110000083
Figure FDA0003811374110000084
the Lyapunov function is derived along an error system under the action of a self-adaptive controller based on a neural network, and the following can be obtained:
Figure FDA0003811374110000085
wherein the content of the first and second substances,
Figure FDA0003811374110000086
Figure FDA0003811374110000087
Figure FDA0003811374110000088
Figure FDA0003811374110000089
matrix array
Figure FDA00038113741100000810
Figure FDA00038113741100000811
l ij ＝-a ij All i is not equal to j; from the formula (16), it can be obtained
Figure FDA00038113741100000812
Thus, error function
Figure FDA00038113741100000813
And
Figure FDA00038113741100000814
the average index is stable; the proportional error tracking system has no pulse and satisfies
Figure FDA00038113741100000815
Namely, bagsA singular multi-agent system with time-lag nonlinear actuator attack realizes proportion-tolerant tracking control.
5. The method for proportional-tolerant tracking control of an actuator attack singular multi-agent system as claimed in claim 3, wherein said step 4.2 is embodied as:
when w (t) ≠ 0, combining the Lyapunov function constructed in the equation (18) with the following energy function:
Figure FDA0003811374110000091
solving the above equation can obtain:
Figure FDA0003811374110000092
wherein:
Ξ 11 ＝Π 11 ,Ξ 12 ＝Π 12 ,
Figure FDA0003811374110000093
Ξ 15 ＝Π 15 ,Ξ 22 ＝Π 22 ,
Figure FDA0003811374110000094
Ξ 44 ＝Π 44 ,Ξ 55 ＝Π 55 ,
Figure FDA0003811374110000095
by means of matrices
Figure FDA0003811374110000096
And
Figure FDA0003811374110000097
right and left multiplier xi, respectively 5×5 Let us order
Figure FDA0003811374110000098
And
Figure FDA0003811374110000099
due to the matrix M L And M R If the matrix is a full rank matrix, J is less than or equal to 0; and feedback the gain matrix
Figure FDA00038113741100000910
6. The method for proportional-tolerant tracking control of an actuator attack singular multi-agent system as claimed in claim 3, wherein said step 5 is specifically:
step 5.1. If the matrix K satisfies the condition in step 4.2 and the parameters in the adaptive controller (14) based on the neural network satisfy
Figure FDA00038113741100000911
Wherein, for all i =1, N,
Figure FDA00038113741100000912
is a normal number; the integral sliding mode surface equation (11) can be reached in a limited time;
step 5.2, constructing a Lyapunov function of the formula (22):
Figure FDA0003811374110000101
is obtained by calculation
Figure FDA0003811374110000102
Wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003811374110000103
thus, there is a moment
Figure FDA0003811374110000104
So that all T ≧ T satisfy V s (t) =0 and
Figure FDA0003811374110000105
namely, the finite time of the integral sliding mode surface equation can be reached.
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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116627042A (en) * 2023-07-20 2023-08-22 南京邮电大学 Distributed collaborative tracking method for asymmetric saturated multi-self-body system of actuator
CN116627042B (en) * 2023-07-20 2023-09-29 南京邮电大学 Distributed collaborative tracking method for asymmetric saturated multi-self-body system of actuator

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