CN109116736B - Fault-tolerant control method for actuator faults in linear multi-agent systems based on sliding mode - Google Patents

Abstract

The invention discloses a tracking fault-tolerant control method of a linear multi-agent system based on sliding mode control. Considering the actuator failure in the presence of external disturbances in a linear multi-agent tracking system, a fault-tolerant control method is proposed based on the sliding mode control method combined with adaptive control. Aiming at the possible failure of the actuator part of the agent, the state quantity error variable is defined according to the communication topology between multiple agents, and the integral sliding mode surface is designed based on the error variable, and the sufficient conditions for the asymptotic stability of the sliding mode are given. , and then use the adaptive method to estimate the unknown upper bound of the actuator fault, and propose a sliding mode fault-tolerant controller to ensure the tracking stability of the multi-agent system. The present invention designs an integral sliding mode surface according to the state error variable between multiple agents, enhances the robustness of the system, and proposes a fault-tolerant control law, which can have good performance when the system has partial actuator failures and external disturbances. The fault tolerance ability improves the tracking stability of the multi-agent system. The present invention is used for fault-tolerant control of linear multi-agent systems with actuator failure faults and external disturbances.

Description

Fault-tolerant control method for actuator faults in linear multi-agent systems based on sliding mode

technical field

The invention relates to a fault-tolerant control method of a linear multi-agent system based on sliding mode, and belongs to the field of multi-agent system control.

Background technique

With the rapid development of control theory, the control technology of a single agent is gradually becoming mature. In recent years, with the wide application of computer networks and the continuous development of artificial intelligence technology, multi-agent systems containing multiple individuals have aroused great research enthusiasm, which is largely due to the fact that multi-agent systems can The mutual coordination between them can complete relatively complex tasks, and at the same time save costs, more effectively complete tasks that cannot be completed by a single system. In view of the advantages of a wider range of tasks and higher efficiency, multi-agent systems have been used in many fields, such as robots, UAV formation control, automated traffic control, complex industrial process control, etc.

At present, the research results of multi-agent systems are mainly in the aspects of consistency, tracking and formation control, but these conclusions are based on the assumption that the agents do not have any faults. In a complex multi-agent system, the number of actuators is large and the distribution structure is complex, which is the focus of achieving the global goal. Once the actuators of some agents fail, the negative impact of the failure on a single agent may be through the multi-agent. The communication topology is enlarged to the whole world, which affects the performance of the entire system and even causes the task to fail. Therefore, the research on fault-tolerant control of multi-agent systems has important practical significance.

Sliding mode control is a special kind of nonlinear control, and has the advantages of fast response, insensitivity to uncertain parameters of the system, simple physical implementation, and good robustness. It has been widely used in theoretical research and practical engineering. Sliding mode control can force the system state to move along the pre-designed sliding mode surface to achieve asymptotic stability of the system, and the system reaching the sliding mode surface will no longer be affected by parameter changes and external disturbances, so it is very suitable for many Passive fault-tolerant control of agent systems.

In order to effectively deal with the possible existing actuator failures in multi-agent systems, there have been some research results in recent years, but they are still in their infancy. The scholar Zuo Zhiqiang proposed a control method of adaptive gain compensation for the partial failure of the actuators of a class of linear multi-agent systems. Deng Chao and others from Northeastern University designed an adaptive output feedback control method to solve the actuator failure problem of a class of nonlinear multi-agent systems. However, most of the existing solutions are fault-tolerant control based on adaptive control, and it is difficult to have good robustness to system parameter perturbation and external interference that cannot be avoided in practical engineering. Therefore, the present invention has certain innovation and practicability. .

SUMMARY OF THE INVENTION

Purpose of the invention: Aiming at the partial failure of the actuator of a class of linear multi-agent systems, combined with adaptive control, a sliding mode fault-tolerant control method is proposed to overcome the negative impact of faults on the system and ensure that the system can run stably.

Technical solution: A fault-tolerant control method for a linear multi-agent system based on sliding mode, characterized in that: considering the problems of actuator failure and external interference in the multi-agent system, according to the communication topology between the multi-agents, a The state error variable and an integral sliding mode surface are designed based on the error variable to ensure the asymptotic stability of the sliding mode of the system. An adaptive method is used to estimate the unknown upper bound of the fault information, and then a sliding mode fault-tolerant controller is proposed, so that the multi-agent system can continue to operate safely after the actuator fault occurs. It includes the following specific steps:

Step 1) Obtain the control model, actuator fault model and communication topology of the multi-agent system:

Step 1.1) The leader control model is shown in formula (1):

Among them, x ₀ (t)∈R ⁿ is the state quantity of the leader, and r ₀ (t)∈R ^m is the control input of the leader;

Step 1.2) The follower control model is shown in formula (2):

是已知的正常数；Among them, x _i (t)∈R ⁿ and _ui (t)∈R ^m represent the state quantity and control input of the i-th, i=1, 2,...,N followers, and are the given system matrix , and (A, B) is stable, f _i (t)∈R ^m represents the external disturbance received by the ith follower, and satisfies

is a known positive constant;

Step 1.3) The actuator fault model is shown in formula (3):

是带有失效故障的执行器输出，ρ_i(t)＝diag{ρ_i1(t)，ρ_i3(t)，...，ρ_im(t)}∈R^m×m是第i个跟随者的执行器失效矩阵，失效因子ρ_ip(t)，p＝1，2，...m是未知时变且有界的，当ρ_ip(t)＝0时，表示第i个跟随者的第p个执行器未发生故障，当0＜ρ_ip(t)＜1时，表示第i个跟随者的第p个执行器发生了部分失效故障，当ρ_ip(t)＝1时，表示第i个跟随者的第p个执行器完全失效，这里不考虑这种情况；where u _i (t) is the actuator input,

is the actuator output with failure fault, ρ _i (t)=diag{ρ _i1 (t),ρ _i3 (t),...,ρ _im (t)}∈R ^m×m is the ith follower The actuator failure matrix of the follower, the failure factor ρ _ip (t), p = 1, 2, ... m is unknown time-varying and bounded, when ρ _ip (t) = 0, it means the ith follower The p-th actuator of y does not fail. When 0<ρ _ip (t)<1, it means that the p-th actuator of the i-th follower has a partial failure. When ρ _ip (t)=1, Indicates that the p-th actuator of the i-th follower fails completely, which is not considered here;

Therefore, the agent model with partial actuator failure can be described as:

Step 1.4) Communication topology of the multi-agent system:

是跟随者之间的通讯拓扑图，其中

表示图G的邻接矩阵；记L为图G的Laplacian矩阵，定义

其中

是由各节点的度组成的对角矩阵，则l_ij的定义如式(5)所示：Consider a multi-agent system consisting of a leader labeled 0 and N followers labeled i = 1, 2, ..., N. The graph G = (V, E) represents that the leader and followers are included in the The communication topology diagram between all nodes within, where the node set V = {0, 1, 2, ..., N}, the communication link set between the nodes is E = V × V; the subgraph of G

is the communication topology between followers, where

Represents the adjacency matrix of graph G; denote L as the Laplacian matrix of graph G, define

in

is a diagonal matrix composed of the degrees of each node, then the definition of l _ij is shown in formula (5):

表示领导者与跟随者之间的邻接矩阵，如果领航者0与第i个跟随者之间有一条无向边，那么b_i＝1，否则，b_i＝0；定义

为第i个跟随者的邻集；make

Represents the adjacency matrix between the leader and the follower. If there is an undirected edge between the leader 0 and the i-th follower, then b _i = 1, otherwise, b _i = 0; definition

is the neighbor set of the i-th follower;

Step 2) According to the adjacent information obtained by the ith follower, define the state error variable shown in formula (6):

Among them, a _ij is the connection weight between the ith follower and the jth follower, b _i represents the connection weight between the _ith follower and the leader, and Ni is the neighbor set of the ith follower ;

则式(6)可以改写为如式(7)所示的全局误差变量：definition

Then Equation (6) can be rewritten as the global error variable shown in Equation (7):

in,

Step 3) For the multi-agent tracking control system with actuator fault shown in Equation (4), the sliding mode surface shown in Equation (8) is designed based on the state error variable Equation (6):

Among them, G=(B ^T B) ^-1 B ^T ∈R m ^×n is the sliding mode matrix, K∈R ^m×n is the controller gain to be designed; in order to ensure the asymptotic stability of the system sliding mode, satisfy Condition: Re[λ(A+BK)]<0, that is, all eigenvalues of matrix (A+BK) have negative real parts;

According to formula (7), formula (8) can be rewritten as

in,

Step 4) The adaptive method estimates the unknown upper bound of the fault information. First, define an unknown parameter θ _i =1/1-||ρ _i || to indirectly reflect the fault information, and design the adaptive law as shown in equation (10). to estimate θ _i :

是参数θ_i的估计值，且α是一个正常数，ω_i在步骤5)中给出；in,

is the estimated value of parameter θ _i , and α is a positive constant, ω _i is given in step 5);

Step 5) Synthesize step 3) and step 4), and design a complete fault-tolerant control law as shown in formula (11):

且η，k₁均为正常数；in,

And n, k ₁ are all positive numbers;

Combined with formula (7) and formula (9), formula (11) is rewritten as:

where, b=[b ₁ b ₂ ... b _N ] ^T ,

Step 5) According to the running state of the multi-agent system, select appropriate parameters to realize the tracking fault-tolerant control of the system.

Beneficial effects: The fault-tolerant control method for actuator faults of a linear multi-agent system based on sliding mode proposed by the present invention defines a state error variable according to the communication topology between the multi-agents, and designs a state error variable based on the error variable. The integral sliding mode surface ensures the asymptotic stability of the sliding mode of the system and enhances the robustness of the system. Combined with the adaptive method to estimate the unknown upper bound information of the fault, a complete sliding mode fault-tolerant controller is finally formed to ensure that the system performs After the device fails, it can still run stably and complete the tracking task.

Has the following advantages:

(1) For the general linear multi-agent system, the problems of partial failure of the actuator and external disturbance are considered at the same time;

(2) According to the communication topology between agents, a global state error variable is constructed, and an integral sliding mode surface is designed based on this variable to enhance the robustness of the system;

(3) The sliding mode fault-tolerant controller is designed in combination with the adaptive method, so that the multi-agent system can still complete the tracking task in the event of actuator failure, which improves the fault-tolerant ability of the system.

As a fault-tolerant control method for actuator faults in a linear multi-agent system, the method used in the invention has good fault-tolerant ability and robustness, is highly operable and easy to implement, has certain practical application value, and can be widely used In the field of fault-tolerant control of actuator failures in multi-agent systems.

Description of drawings

Fig. 1 is the flow chart of the inventive method;

Figure 2 is a schematic diagram of the quadrotor aircraft Q-ball-X4 of Quanser and its attitude movement;

Fig. 3 is the communication topology structure diagram of the multi-quadcopter system;

Figure 4 is a block diagram of the fault-tolerant control principle of a single quadrotor aircraft;

Fig. 5 is the X-axis displacement tracking error curve;

Fig. 6 is the X-axis velocity tracking error curve;

Figure 7 is the actuator dynamic error curve;

Figure 8. Sinmulink simulation diagram.

Detailed ways

The present invention will be further explained below in conjunction with the accompanying drawings.

As shown in Fig. 1, considering a class of linear multi-agent systems when actuator failures and external disturbances occur, an error variable is defined according to the communication topology between the agents, and an integral sliding surface is designed, combined with the obtained faults Information Design Sliding Mode Fault Tolerant Controllers. Specific steps are as follows:

Step 1) Obtain the control model, actuator fault model and communication topology of the multi-agent system:

Step 1.1) The leader control model is shown in formula (1):

Among them, x ₀ (t)∈R ⁿ is the state quantity of the leader, and r ₀ (t)∈R ^m is the control input of the leader;

Step 1.2) The follower control model is shown in formula (2):

是已知的正常数；Among them, x _i (t)∈R ⁿ and _ui (t)∈R ^m represent the state quantity and control input of the i-th, i=1, 2,...,N followers, and are the given system matrix , and (A, B) is stable, f _i (t)∈R ^m represents the external disturbance received by the ith follower, and satisfies

is a known positive constant;

Step 1.3) The actuator fault model is shown in formula (3):

是带有失效故障的执行器输出，ρ_i(t)＝diag{ρ_i1(t)，ρ_i3(t)，...，ρ_im(t)}∈R^m×m是第i个跟随者的执行器失效矩阵，失效因子ρ_ip(t)，p＝1，2，...m是未知时变且有界的，当ρ_ip(t)＝0时，表示第i个跟随者的第p个执行器未发生故障，当0＜ρ_ip(t)＜1时，表示第i个跟随者的第p个执行器发生了部分失效故障，当ρ_ip(t)＝1时，表示第i个跟随者的第p个执行器完全失效，这里不考虑这种情况；where u _i (t) is the actuator input,

is the actuator output with failure fault, ρ _i (t)=diag{ρ _i1 (t),ρ _i3 (t),...,ρ _im (t)}∈R ^m×m is the ith follower The actuator failure matrix of the follower, the failure factor ρ _ip (t), p = 1, 2, ... m is unknown time-varying and bounded, when ρ _ip (t) = 0, it means the ith follower The p-th actuator of y does not fail. When 0<ρ _ip (t)<1, it means that the p-th actuator of the i-th follower has a partial failure. When ρ _ip (t)=1, Indicates that the p-th actuator of the i-th follower fails completely, which is not considered here;

Therefore, the agent model with partial actuator failure can be described as:

Step 1.4) Communication topology of the multi-agent system:

是跟随者之间的通讯拓扑图，其中

表示图G的邻接矩阵；记L为图G的Laplacian矩阵，定义

其中

是由各节点的度组成的对角矩阵，则l_ij的定义如式(5)所示：Consider a multi-agent system consisting of a leader labeled 0 and N followers labeled i = 1, 2, ..., N. The graph G = (V, E) represents that the leader and followers are included in the The communication topology diagram between all nodes within, where the node set V = {0, 1, 2, ..., N}, the communication link set between the nodes is E = V × V; the subgraph of G

is the communication topology between followers, where

Represents the adjacency matrix of graph G; denote L as the Laplacian matrix of graph G, define

in

is a diagonal matrix composed of the degrees of each node, then the definition of l _ij is shown in formula (5):

表示领导者与跟随者之间的邻接矩阵，如果领航者0与第i个跟随者之间有一条无向边，那么b_i＝1，否则，b_i＝0；定义

为第i个跟随者的邻集；make

Represents the adjacency matrix between the leader and the follower. If there is an undirected edge between the leader 0 and the i-th follower, then b _i = 1, otherwise, b _i = 0; define

is the neighbor set of the i-th follower;

Step 2) According to the adjacent information obtained by the ith follower, define the state error variable shown in formula (6):

Among them, a _ij is the connection weight between the ith follower and the jth follower, b _i represents the connection weight between the _ith follower and the leader, and Ni is the neighbor set of the ith follower ;

则式(6)可以改写为如式(7)所示的全局误差变量：definition

Then Equation (6) can be rewritten as the global error variable shown in Equation (7):

in,

Step 3) For the multi-agent tracking control system with actuator fault shown in Equation (4), the sliding mode surface shown in Equation (8) is designed based on the state error variable Equation (6):

Among them, G=(B ^T B) ^-1 B ^T ∈R m ^×n is the sliding mode matrix, K∈R ^m×n is the controller gain to be designed; in order to ensure the asymptotic stability of the system sliding mode, satisfy Condition: Re[λ(A+BK)]<0, that is, all eigenvalues of matrix (A+BK) have negative real parts;

According to formula (7), formula (8) can be rewritten as

in,

Step 4) The adaptive method estimates the unknown upper bound of the fault information. First, define an unknown parameter θ _i =1/1-||ρ _i || to indirectly reflect the fault information, and design the adaptive law as shown in equation (10). to estimate θ _i :

是参数θ_i的估计值，且α是一个正常数，ω_i在步骤5)中给出；in,

is the estimated value of parameter θ _i , and α is a positive constant, ω _i is given in step 5);

Step 5) Synthesize step 3) and step 4), and design a complete fault-tolerant control law as shown in formula (11):

且η，k₁均为正常数；in,

And n, k ₁ are all positive numbers;

Combined with formula (7) and formula (9), formula (11) is rewritten as:

where, b=[b ₁ b ₂ ... b _N ] ^T ,

Step 5) According to the running state of the multi-agent system, select appropriate parameters to realize the tracking fault-tolerant control of the system.

The above are only the preferred embodiments of the present invention. It should be pointed out that for those skilled in the art, without departing from the principles of the present invention, several improvements and modifications can be made. It should be regarded as the protection scope of the present invention.

The effectiveness of the implementation scheme is illustrated below with an actual case simulation.

The quadrotor helicopter Qball-X4 produced by Canadian quanser company is used as the specific algorithm experiment simulation object. Figure 2 is a schematic diagram of Quanser's quad-rotor aircraft Qball-X4 and its attitude motion. It can be seen from Figure 2 that there are six-dimensional variables (X, Y, Z, ψ, θ, φ) in the quad-rotor aircraft relative to the ground. The three variables are position variables, that is, the position relative to the center of the inertial frame. The last three variables are the Euler angles of the quadrotor helicopter attitude: yaw ψ, pitch θ, roll φ. Without loss of generality, the displacement, velocity and actuator dynamics in the X-axis direction are selected as the state quantities for simulation experiments.

The state space model of the quadrotor is as follows:

A state-space expression written in canonical form:

为状态量，u(t)为控制输入，输出y(t)为X轴位移量。in,

is the state quantity, u(t) is the control input, and the output y(t) is the X-axis displacement.

The airframe parameters of the quadrotor are shown in Table 1:

Table 1 Numerical table of airframe parameters

parameter value unit K 120N ω 15rad/see M 1.4kg

Assuming θ=0.035rad, the coefficient matrix in the nominal system can be obtained as follows:

Here we consider a multi-quadcopter tracking control system consisting of one leader and four followers, where the leader is marked as 0 and the follower is marked as i (i=1, 2, 3, 4). Then the system model of the leader Qball-X4 is:

Considering that the follower Qball-X4 has the problem of actuator failure and external disturbance, its system model is:

Assuming that the communication topology of the multi-quadcopter system is shown in Figure 3, we can obtain the following Laplacian matrix L and adjacency matrix

Considering that the actuator failures of quadrotors 1 and 2 during flight are ρ ₁ =0.4+0.1cos(t), ρ ₂ =0.3+0.2sin(2t), the rest of the follower quadrotors are normal. Suppose the external disturbances received by the four follower quadrotors are set as: f _i (t)=0.2sin(t)(i=1,2) and f _i (t)=0.3cos(2t)(i=3 , 4). Take the leader control input and the follower state quantities of the system at the initial moment as: u ₀ =sin(t), x ₀ (0)=[0.2 0.30.1] ^T , x ₁ (0)=[0 0.2 0.15] ^T , x ₂ (0)=[0.4 0.2 0.1] ^T , x ₃ (0)=[−0.6 0.4 0.25] ^T , x ₄ (0)=[−0.4 0.3 0.2] ^T . The values of the sliding mode surface matrix and the parameters of the controller are: G=[0 0 0.0667], K=[0.3611 -0.2078 -0.7733], η=30, α=0.3, k ₁ =2, δ=0.1,

According to the method of the present invention, fault-tolerant control is performed on a multi-quadrotor aircraft system with actuator faults and external disturbances. Figures 5 to 7 show the results of fault-tolerant control. Figures 5-7 are the tracking error curves of displacement, velocity and actuator dynamics in the X-axis direction, respectively.

It can be seen from Fig. 5-Fig. 7 that when an actuator failure occurs in the system, under the fault-tolerant control of the present invention, the tracking errors of the X-axis displacement and velocity of each aircraft can tend to zero in a relatively short period of time, and no failure occurs. The tracking error curves of quadrotors 3 and 4 tend to zero in five seconds, while quadrotors 1 and 2 in failure can also tend to zero in about 7 seconds, that is to say, after the system fails, the aircraft can still Successfully complete the tracking task to avoid accidents and task failures.

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):

wherein x is₀(t)∈RⁿIs the state quantity of the leader, r₀(t)∈R^mIs a control input of the leader;

step 1.2) follower control model is shown as formula (2):

wherein x is_i(t)∈RⁿAnd u_i(t)∈R^mState quantities and control inputs representing the i, i 1, 2, N followers are a given system matrix, and (a, B) is stable, f_i(t)∈R^mRepresents the external interference of the ith follower and meets the requirement

Is a known normal number;

step 1.3) the actuator fault model is shown as formula (3):

wherein u is_i(t) is the actuator input and,

is the actuator output with failure, ρ_i(t)＝diag{ρ_i1(t)，ρ_i3(t)，...，ρ_im(t)}∈R^m×mIs the actuator failure matrix of the ith follower, failure factor ρ_ip(t), p 1, 2,. m is unknown time-varying and bounded when ρ_ipWhen (t) is 0, it indicates that the ith follower has not failed, and when 0 < ρ ≦ is set_ipWhen (t) < 1, it indicates that the ith follower has a partial failure in the pth actuator, and when ρ_ipWhen (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:

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

Is a communication topology between followers, wherein

An adjacency matrix representing diagram G; l is Laplacian matrix of graph G, and is defined

Wherein

Is a diagonal matrix composed of degrees of each node, then_ijIs as defined in formula (5):

order to

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

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:

wherein, a_ijIs the connection weight between the ith follower and the jth follower, b_iRepresenting the weight of the connection between the ith follower and the leader, N_iIs the neighbor set of the ith follower;

definition of

Equation (6) can be rewritten as a global error variable as shown in equation (7):

wherein,

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):

wherein G ═ B^TB)^-1B^T∈R^m×nIs a sliding mode matrix, K is an element of R^m×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

Wherein,

step 4) estimating the unknown upper bound of the fault information by the self-adaptive method, firstly defining an unknown parameter theta_i＝1/1-||ρ_iI indirectly reflects fault information, and an adaptive law shown as a formula (10) is designed to estimate theta_i：

Wherein,

is the parameter theta_iAnd alpha is a normal number, omega_iGiven 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):

wherein,

and η, k₁Are all normal numbers;

by combining the formula (7) and the formula (9), the formula (11) is rewritten as:

wherein b ═ b₁ b₂... b_N]^T，

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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