Sliding mode fault-tolerant consensus control algorithm for multi-agent systems under super-helix observer

Abstract

The present invention discloses a sliding mode fault-tolerant consistency control algorithm for a multi-agent system under a super-helical observer. Considering that the speed information of a second-order nonlinear multi-agent system is unmeasurable and an additive actuator fault occurs, a super-helical second-order sliding mode fault observer is first proposed, which can effectively estimate the speed state information of the agent and the actuator fault information. For the initial error problem and the peak problem, the observer estimation is made faster and more accurate by improving the observer structure and introducing the gain coefficient. Then, based on the integral sliding mode control, a fault-tolerant control method is proposed, the consistency error variable is defined according to the relative state information between the agents, and the sliding surface is designed based on the consistency error variable. Finally, a sliding mode fault-tolerant consistency control law is designed by combining the observer estimation value and the communication structure information between the agents. The control law can enable the multi-agent system to achieve the consistency target within a limited time under the influence of the additive actuator fault and the inherent nonlinear dynamics, thereby effectively avoiding the problem that the multi-agent system cannot complete the consistency task due to the actuator fault. The present invention can be used for the consistency control implementation problem of a class of multi-agent systems containing additive actuator faults and inherent nonlinear dynamics.

Description

Technical Field

[0001] The present invention relates to a sliding mode fault-tolerant consistency control algorithm for a multi-agent system under a super-helical observer, and belongs to the technical field of fault-tolerant control of multi-agent systems. Background Art

[0002] In recent years, with the rapid development of artificial intelligence (AI) technology, researchers have achieved numerous innovative advances and achievements in the field of intelligent algorithms. However, due to frequent and prolonged use, intelligent devices inevitably experience failures. Without pre-defined fault compensation measures, a failure can impact the entire task execution process and, in severe cases, even result in property damage or casualties. Consequently, fault-tolerant control algorithms have emerged and have become a focus of contemporary control research. Generally speaking, fault-tolerant control can be categorized as passive or active, depending on how control systems utilize redundancy to handle faults. Passive fault-tolerant control utilizes pre-defined system redundancy to achieve system insensitivity to faults. Drawing on the design principles of robust control, it does not modify controller parameters or structure, nor does it require the assistance of a fault diagnosis module; it simply suppresses specific faults. Active fault-tolerant control generally achieves fault tolerance by adjusting controller parameters or structure online after a fault occurs. It often relies on fault information obtained through fault diagnosis to adjust or restructure the control law.

[0003] Since the concept of multi-agent systems emerged, distributed collaborative control methods for multi-agent systems have attracted increasing attention due to their superior robustness, practicality, and flexibility compared to traditional centralized control. These systems have been widely applied in real-world applications, such as multi-robot systems and multi-aircraft systems. Consistency in multi-agent systems, as one of the most fundamental and important issues, has attracted a large number of researchers. Consistency refers to the ability of individual agents in a multi-agent system to converge toward a common desired state through information exchange under the influence of appropriate consistency control strategies. This principle is reflected in problems such as formation control and cluster control in multi-agent systems, demonstrating the importance of research on consistency control for multi-agent systems.

[0004] However, for a multi-agent system, a complex and large research subject, failures are more frequent and dangerous. For example, a multi-agent system composed of quadrotor unmanned aerial vehicles (UAVs) can be susceptible to various types of failures, including actuator failure, sensor failure, and communication failure, due to aging, wear, or unexpected events. These failures not only affect the faulty aircraft itself but also affect other aircraft in the system, thereby hindering the consistency control of the entire system. Therefore, it is crucial to introduce appropriate fault-tolerant controllers to compensate for the impact of failures when studying the consistency control of multi-agent systems.

[0005] It is worth noting that most current research assumes that information such as the position and velocity of multi-agents is measurable. However, in actual engineering systems, given the cost of detection tools and the load on the agents, multi-agent systems may not be equipped with comprehensive sensor equipment, resulting in some states being unmeasurable. To address this practical issue, existing research often uses methods combined with observers to observe and estimate unmeasurable states. The field of multi-agent system research has made considerable progress in designing effective control schemes using the rational construction of state observers. Designing fault-tolerant control algorithms combined with observers to address the consistency issues of multi-agent systems in the presence of unmeasurable state information and actuator failures is both theoretically feasible and practically easy to implement. Summary of the Invention

[0006] Purpose of the invention: Based on the above research background, a novel sliding mode fault-tolerant consistency control algorithm for multi-agent systems under a super-helical observer is proposed. A super-helical second-order sliding mode observer is designed that can accurately estimate the additive faults of the actuators and the unmeasurable speed information. In order to better ensure the rapidity of observation, a gain coefficient is designed that can well handle the peak problem and enable the system to achieve rapid convergence when the initial error is too large. Combining the relative state information between the agents and the observer estimate, a sliding mode fault-tolerant consistency control law is designed. By designing an integral sliding surface instead of a linear sliding surface, the instability of the sliding mode approach process is avoided, so that the multi-agent system can still achieve the consistency control task within a limited time on the basis of fault tolerance.

[0007] Technical Solution: A novel sliding-mode fault-tolerant consistency control algorithm for multi-agent systems under a superhelical observer. To address the additive actuator faults and unmeasurable velocity information present in the system, a superhelical second-order sliding-mode observer with a gain coefficient is designed. This allows the observer and system to converge quickly and accurately obtain estimated values ​​even when initial values ​​differ significantly, effectively avoiding the impact of peaking problems. A consistency error variable is defined based on the relative state information between the agents, and an integral sliding surface is designed based on this to improve the robustness of the control system. A hyperbolic tangent term is introduced into the sliding-mode fault-tolerant consistency control law to mitigate the impact of chattering on the system, enabling the designed controller to achieve better control results.

[0008] A sliding mode fault-tolerant consistency control algorithm for a multi-agent system under a super-helical observer includes the following specific steps:

[0009] Step 1) Determining the multi-agent system model and its parameters applicable to the present invention includes the following steps:

[0010] Step 1.1) Determine the state space model of the leader agent as shown in formula (1):

[0011]

[0012] Among them, x0(t), v0(t)∈R m Represent the position and velocity state of the leader respectively; u0(t)∈R m For control input.

[0013] Step 1.2) Determine the state space model of the following agent with additive actuator failure and inherent nonlinear dynamics as shown in Equation (2):

[0014]

[0015] Among them, x i (t), v i (t)∈R m Represents the position and velocity state of the i-th agent respectively; u i (t)∈R m is the control input; f i (x i (t), v i (t), t)∈R m represents the inherent nonlinear dynamics of the ith agent; β i (t)∈R m represents the additive actuator failure caused by the time-varying behavior of the i-th agent, β i (t)=0 means no fault occurs, β i (t)≠0 means that the actuator of the i-th agent has an additive fault;

[0016] For the inherently nonlinear dynamic i (x i (t), v i (t), t) and actuator additive fault β i (t) and its first-order differential satisfy the following inequalities (3) and (4):

[0017]

[0018] in is a non-negative constant;

[0019]

[0020] in is a non-negative constant;

[0021] Step 2) Determine the communication topology of the multi-agent system to which the present invention is applicable:

[0022] Consider the communication topology of a multi-agent system consisting of a leader and n followers, where the leader is labeled 0 and the followers are labeled i=1, 2, ..., n. To describe the information interaction between all nodes including leaders and followers, represents the set of nodes in the graph, Represents the edge set; define a subgraph G = {V, E} to represent the communication topology network between followers, where V = {1, 2, ..., n} represents the set of nodes in the subgraph, Represents the edge set; define N i ={j|(i, j)∈E, i≠j} represents the set of nodes adjacent to node i, A=(a ij ) is the adjacency matrix of the subgraph G. If (i, j)∈E, i≠j, that means there is a direct path between nodes i and j, then a ij >0; otherwise, a ij =0; define the in-degree of node i as Then the in-degree matrix of subgraph G can be expressed as D=diag{d1,...,d n}. Thus, the Laplacian matrix of the subgraph can be expressed as L = [l ij ] n×n =DA;

[0023] The adjacency matrix between the leader and the follower can be defined as B = diag (b1, b2, ..., b n ); When follower i can directly obtain information from the leader, b i > 0; otherwise, bi =0;

[0024] Step 3) Constructing a super-helical second-order sliding mode fault observer, including the following steps:

[0025] Step 3.1) First, construct a superhelix observer for each follower agent as shown in formula (5):

[0026]

[0027] in, and denote the speed estimation and actuator additive fault estimation of the i-th agent, α is an auxiliary variable, λ1 and λ2 are unknown parameters; the design gain coefficient η i (t), which can ensure that the system can quickly converge to the final state even when the initial error is large. Its expression is shown in formula (6):

[0028]

[0029] in, and σ i is a constant greater than 0. Since the gain coefficient decays rapidly over time, the peak problem caused by the difference between the initial value of the system and the estimated value is avoided.

[0030] Step 3.2) Define the estimated error of speed and fault state as shown in equation (7):

[0031]

[0032] By taking the derivative of Equation (7), we can obtain the error dynamics of the system as shown in Equation (8):

[0033]

[0034] Step 3.3) Under the condition that equation (4) holds, for the state space model (2) and the corresponding designed observer (5), if the parameters λ1 and λ2 satisfy

[0035]

[0036] Then the estimation error (7) is eventually uniformly bounded and the error dynamics (8) of the system is finite-time stable, that is:

[0037]

[0038]

[0039] Where T is the finite convergence time; V1(0) is the value of the selected Lyapunov function at t=0, and And P and Q are both positive definite matrices; λ min {·} and λ max {·} represents the minimum and maximum eigenvalues ​​of the matrix respectively; by selecting appropriate values ​​for λ1 and λ2, the estimation error can be adjusted to a very small value;

[0040] Step 4) Design a fault-tolerant control algorithm, including the following steps:

[0041] Step 4.1) Define the consistency tracking position error variable e based on the neighbor information obtained by the i-th agent xi (t) and the velocity error variable e vi (t), which has the following form:

[0042]

[0043] definition Formula (13) can be rewritten as a vector form as shown in Formula (14):

[0044]

[0045] Among them, I m is the m-dimensional identity matrix; define The obtained consistent tracking error system is shown in formula (15):

[0046]

[0047] Step 4.2) Design the integral sliding surface function as shown in formula (16)

[0048]

[0049] in s i (t), i=1,...,n are sliding mode variables; k1 and k2 are positive constants;

[0050] Step 4.3) Design a fault-tolerant consistency control algorithm for the follower agent as shown in Equation (17):

[0051]

[0052] where γ and ε are positive constants. Through theoretical derivation, it can be concluded that the multi-agent system composed of (1) and (2) can achieve finite-time consistency in the case of actuator failure under the fault-tolerant control law (17).

[0053] Beneficial Effects: For multi-agent systems with additive actuator faults, inherent nonlinear dynamics, and unpredictable speed, a novel sliding mode fault-tolerant consistency control algorithm for multi-agent systems under a super-helical observer is designed. A super-helical second-order sliding mode observer is designed to quickly and accurately estimate fault and speed information. Based on the estimated values, combined with the interaction information between the agents and the designed integral sliding surface, a sliding mode fault-tolerant consistency control algorithm is designed, which can achieve finite-time consistency of the multi-agent system under the influence of additive actuator faults and nonlinear dynamics. In general, it has the following specific advantages:

[0054] ① By designing a super-helical second-order sliding mode observer, the additive faults and unmeasurable speeds of the actuators in the system are accurately estimated within a limited time. By improving the observer structure, the rapidity of the estimation is further improved. This allows for accurate estimation of faults and states in situations where there is a large error between the initial estimated value and the actual value, while also saving equipment resources and improving hardware redundancy.

[0055] ② The designed sliding mode fault-tolerant consistency control law introduces the integral sliding surface and hyperbolic tangent term, which avoids instability during the approach process and weakens the chattering problem, which is conducive to ensuring the robustness of the system.

[0056] ③ The fault-tolerant control algorithm designed by this invention can solve the problem of achieving finite-time consensus in a multi-agent system with unpredictable velocity states, subject to actuator failures and inherent nonlinear dynamics. This demonstrates that the algorithm design not only reduces the cost of state detection equipment but also enhances the robustness and fault tolerance of the entire system.

[0057] The sliding mode fault-tolerant consistency control algorithm for multi-agent systems under a superhelical observer proposed in the present invention has certain application significance, is easy to implement, has good real-time performance, high accuracy, can effectively improve the safety of the control system and has strong operability, saves time, and is more efficient. It can be widely used in the finite-time consistency implementation problem of a class of multi-agent systems with unmeasurable speed states under actuator failures and inherent nonlinear dynamics. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 is a flow chart of the method of the present invention;

[0059] Figure 2 This is the Qball-X4 quadcopter, an experimental device developed by Quanser, and its schematic diagram;

[0060] Figure 3 It is the communication topology network of multiple Qball-X4 quadrotor helicopter systems;

[0061] Figure 4is the failure of the second Qball-X4 quadrotor helicopter and its upper bound squared estimate curve;

[0062] Figure 5 The failure of the third Qball-X4 quadcopter and its upper bound squared estimate curve;

[0063] Figure 6 is the failure of the 4th Qball-X4 quadrotor helicopter and its upper bound squared estimate curve;

[0064] Figure 7 is the position tracking error of the multi-Qball-X4 quadrotor helicopter system.

[0065] Figure 8 is the velocity tracking error of the multi-Qball-X4 quadrotor helicopter system. DETAILED DESCRIPTION

[0066] The present invention will be further explained below with reference to the accompanying drawings.

[0067] like Figure 1 As shown in the figure, in response to the additive faults of the actuators and the unmeasurable speed information in the system, a super-helical second-order sliding mode observer with a gain coefficient is designed, so that the observer and the system can still converge quickly and accurately obtain the estimated value when the initial values ​​differ greatly, and effectively avoid the impact of the peak problem; the consistency error variable is defined according to the relative state information between the intelligent agents, and the integral sliding surface is designed based on this to improve the robustness of the control system; the hyperbolic tangent term is introduced in the sliding mode fault-tolerant consistency control law, thereby curbing the impact of vibration on the system, so that the designed controller can achieve better control effect.

[0068] A sliding mode fault-tolerant consistency control algorithm for a multi-agent system under a super-helical observer includes the following specific steps:

[0069] Step 1) Determining the multi-agent system model and its parameters applicable to the present invention includes the following steps:

[0070] Step 1.1) Determine the state space model of the leader agent as shown in formula (1):

[0071]

[0072] Among them, x0(t), v0(t)∈R m Represent the position and velocity state of the leader respectively; u0(t)∈R m For control input.

[0073] Step 1.2) Determine the state space model of the following agent with additive actuator failure and inherent nonlinear dynamics as shown in Equation (2):

[0074]

[0075] Among them, x i (t), v i (t)∈R m Represents the position and velocity state of the i-th agent respectively; u i (t)∈R m is the control input; f i (x i (t), v i (t), t)∈R m represents the inherent nonlinear dynamics of the ith agent; β i (t)∈R m represents the additive actuator failure caused by the time-varying behavior of the i-th agent, β i (t)=0 means no fault occurs, β i (t)≠0 means that the actuator of the i-th agent has an additive fault;

[0076] For the inherently nonlinear dynamic i (x i (t), v i (t), t) and actuator additive fault β i (t) and its first-order differential satisfy the following inequalities (3) and (4):

[0077]

[0078] in is a non-negative constant;

[0079]

[0080] in is a non-negative constant;

[0081] Step 2) Determine the communication topology of the multi-agent system to which the present invention is applicable:

[0082] Consider the communication topology of a multi-agent system consisting of a leader and n followers, where the leader is labeled 0 and the followers are labeled i=1, 2, ..., n. To describe the information interaction between all nodes including leaders and followers, represents the set of nodes in the graph, Represents the edge set; define a subgraph G = {V, E} to represent the communication topology network between followers, where V = {1, 2, ..., n} represents the set of nodes in the subgraph, Represents the edge set; define N i ={j|(i, j)∈E, i≠j} represents the set of nodes adjacent to node i, A=(a ij ) is the adjacency matrix of the subgraph G. If (i, j)∈E, i≠j, that means there is a direct path between nodes i and j, then a ij >0; otherwise, a ij =0; define the in-degree of node i as Then the in-degree matrix of subgraph G can be expressed as D=diag{d1,...,d n}. Thus, the Laplacian matrix of the subgraph can be expressed as L = [l ij ] n×n =DA;

[0083] The adjacency matrix between the leader and the follower can be defined as B = diag (b1, b2, ..., b n ); When follower i can directly obtain information from the leader, b i > 0; otherwise, b i =0;

[0084] Step 3) Constructing a super-helical second-order sliding mode fault observer, including the following steps:

[0085] Step 3.1) First, construct a superhelix observer for each follower agent as shown in formula (5):

[0086]

[0087] in, and denote the speed estimation and actuator additive fault estimation of the i-th agent, α is an auxiliary variable, λ1 and λ2 are unknown parameters; the design gain coefficient η i (t), which can ensure that the system can quickly converge to the final state even when the initial error is large. Its expression is shown in formula (6):

[0088]

[0089] in, and σ i is a constant greater than 0. Since the gain coefficient decays rapidly over time, the peak problem caused by the difference between the initial value of the system and the estimated value is avoided.

[0090] Step 3.2) Define the estimated error of speed and fault state as shown in equation (7):

[0091]

[0092] By taking the derivative of Equation (7), we can obtain the error dynamics of the system as shown in Equation (8):

[0093]

[0094] Step 3.3) Under the condition that equation (4) holds, for the state space model (2) and the corresponding designed observer (5), if the parameters λ1 and λ2 satisfy

[0095]

[0096] Then the estimation error (7) is eventually uniformly bounded and the error dynamics (8) of the system is finite-time stable, that is:

[0097]

[0098]

[0099] Where T is the finite convergence time; V1(0) is the value of the selected Lyapunov function at t=0, and And P and Q are both positive definite matrices; λ min {·} and λ max {·} represents the minimum and maximum eigenvalues ​​of the matrix respectively; by selecting appropriate values ​​for λ1 and λ2, the estimation error can be adjusted to a very small value;

[0100] Step 4) Design a fault-tolerant control algorithm, including the following steps:

[0101] Step 4.1) Define the consistency tracking position error variable e based on the neighbor information obtained by the i-th agent xi (t) and the velocity error variable e vi (t), which has the following form:

[0102]

[0103] definition Formula (13) can be rewritten as a vector form as shown in Formula (14):

[0104]

[0105] Among them, I m is the m-dimensional identity matrix; define The obtained consistent tracking error system is shown in formula (15):

[0106]

[0107] Step 4.2) Design the integral sliding surface function as shown in formula (16)

[0108]

[0109] in s i (t), i=1,...,n are sliding mode variables; k1 and k2 are positive constants;

[0110] Step 4.3) Design a fault-tolerant consistency control algorithm for the follower agent as shown in Equation (17):

[0111]

[0112] where γ and ε are positive constants. Through theoretical derivation, it can be concluded that the multi-agent system composed of (1) and (2) can achieve finite-time consistency in the case of actuator failure under the fault-tolerant control law (17).

[0113] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principles of the present invention. These improvements and modifications should also be regarded as within the scope of protection of the present invention.

[0114] The effectiveness of the implementation plan is illustrated by a real case simulation below.

[0115] The flight control system of the Qball-X4 quadrotor helicopter developed by Quanser, Canada, is used as the research object. The multi-agent system consists of 6 Qball-X4 quadrotor helicopters, including a leader labeled 0 and followers labeled i, i = 1, ..., 5. The communication topology is as follows: Figure 2 Assume that the weight of each edge in the communication topology graph is 1. Then, according to the communication topology structure, the specific expressions of the Laplace matrix L and the adjacency matrix B can be calculated as follows:

[0116]

[0117] For the model of the multi-agent system, the state space model of the leader agent is described as follows:

[0118]

[0119] The initial state of the leader agent is x0(0) = 0, v0(0) = 1, and the control input is u0 = sin(t / 15). The state space model of follower agents i, i = 1, ..., 5 with additive actuator failure and inherent nonlinear dynamics is described as follows:

[0120]

[0121] The inherent nonlinear dynamic f i (x i (t), v i (t), t) = sin(x i (t))+sin(v i (t)), and satisfies ||f i (x i (t), v i (t), t)||≤2; the initial state values ​​of the following agent are x1(0)=1, x2(0)=1.2, x3(0)=0.5, x4(0)=-1, x5(0)=-1.5, v1(0)=1.5, v2(0)=1.2, v3(0)=2, v4(0)=0.5, v5(0)=0.

[0122] To illustrate the effectiveness and superiority of the fault-tolerant control algorithm designed in this invention in solving the consistent control problem of a multi-agent system with actuator failures, assume that additive actuator failures occur in follower agents 2 and 4, while the other aircraft are fault-free. The actuator failure of aircraft 2 is described as:

[0123] β2(t)=0.2+0.1sin(0.5πt)

[0124] The actuator fault description of aircraft 4 is:

[0125] β4(t)=0.1(1-e -0.2t )+0.2sin(0.5πt)cos(0.3t)

[0126] Obviously, there is an upper limit to the value of these actuator fault functions, which can be Therefore, according to step 3.3), the undetermined parameters in the superhelical second-order sliding mode observer can be set to λ1=5,λ2=13; the time-varying gain coefficient η i The parameters in (t) are set to σ=5. In order to better demonstrate the rapidity and accuracy of the observer designed in the present invention in fault estimation, we select the same fault situation and obtain the comparison curves of the present observer and the traditional observer in fault estimation (see Figure 3 and Figure 4As can be seen from the images, compared with the traditional observer, the fault observation curve of the observer designed in the present invention is smoother and has a lower peak value, which can effectively compensate for the impact of excessive initial error. At the same time, in terms of the time consumption of fault estimation, for two different actuator fault functions, the estimation time of the observer designed in the present invention is shorter than that of the traditional observer, indicating that the designed observer can achieve faster fault estimation when dealing with actuator additive faults.

[0127] The sliding surface coefficients are selected as k1=2, k2=1; the parameters in the fault-tolerant controller are set as γ=0.5, ε=0.15. Figure 5 and Figure 6 Figure 1 shows the position and velocity trajectory of each agent in the multi-agent system under the control of the controller. It can be seen that although some follower agents are affected by varying degrees of additive actuator failure, they can still quickly converge to the leader within a limited time through timely information exchange based on a fixed topology. Figure 7 is the control law curve of each intelligent agent. It can be seen that under the action of the sliding mode fault-tolerant consistency control law designed in the present invention, the control signal curve is smooth during the convergence process. Figure 8 This figure compares the control law curves of Agent 2 under a conventional fault-tolerant control algorithm and the fault-tolerant control algorithm designed in this invention. It can be seen that the fault-tolerant control algorithm designed in this invention better suppresses chattering, resulting in faster convergence and more stable fluctuations. In other words, the multi-agent system is more stable and performs better under the control of the controller designed in this invention. In summary, the control method simulated in this case is effective for multi-agent systems experiencing additive actuator failures.

Claims

1. A sliding mode fault-tolerant consistency control algorithm for a multi-agent system under a super-helical observer, characterized in that: The specific steps include: Step 1) Determine the multi-agent system model and its parameters, including the following steps: Step 1.1) Determine the state space model of the leader agent as shown in formula (1): Among them, x0(t), v0(t)∈R m Respectively represent the position and velocity state of the leader; u0(t)∈R m is the control input; Step 1.2) Determine the state space model of the follower agent with actuator additive faults and inherent nonlinear dynamics as shown in equation (2): Among them, x i (t), v i (t)∈R m Respectively represent the position and velocity state of the i-th agent; u i (t)∈R m is the control input; f i (x i (t), v i (t), t)∈R m represents the inherent nonlinear dynamics of the ith agent; β i (t)∈R m represents the additive actuator failure caused by the time-varying behavior of the i-th agent, β i (t) = 0 means no fault occurs, β i (t)≠0 means that the actuator of the i-th agent has an additive fault; For the inherently nonlinear dynamic i (x i (t), v i (t), t) and actuator additive fault β i (t) and its first-order differential satisfy the following inequalities (3) and (4): in is a non-negative constant; in is a non-negative constant; Step 2) Determine the communication topology of the multi-agent system: Consider the communication topology of a multi-agent system consisting of a leader and n followers, where the leader is labeled 0 and the followers are labeled i=1, 2, ..., n. To describe the information interaction between all nodes including leaders and followers, represents the set of nodes in the graph, represents the edge set; defines a subgraph G = {V, E} to represent the communication topology network between followers, where V = {1, 2, ..., n} represents the set of nodes in the subgraph, Represents the edge set; define N i ={j|(i, j)∈E, i≠j} represents the set of nodes adjacent to node i, A=(a ij ) is the adjacency matrix of the subgraph G. If (i, j)∈E, i≠j, which means there is a direct path between nodes i and j, then a ij >0; otherwise, a ij =0; define the in-degree of node i as Then the in-degree matrix of the subgraph G can be expressed as D = diag{d1, ..., d n }; From this, the Laplacian matrix of the subgraph can be expressed as L = [l ij ] n×n =DA; The adjacency matrix between the leader and the follower can be defined as B = diag (b1, b2, ..., b n ); When follower i can directly obtain information from the leader, b i > 0; otherwise, b i =0; Step 3) constructing a super-helical second-order sliding mode fault observer, comprising the following steps: Step 3.1) First, construct a superhelix observer for each follower agent as shown in formula (5): in, and denote the speed estimation and actuator additive fault estimation of the ith agent, respectively; α is an auxiliary variable; λ1 and λ2 are unknown parameters; the design gain coefficient η i (t), which can ensure that the system can quickly converge to the final state even when the initial error is large. Its expression is shown in formula (6): in, and σ i is a constant greater than 0; since the gain coefficient decays rapidly over time, the peak problem caused by the difference between the initial value of the system and the estimated value is also avoided; Step 3.2) Define the estimated error of speed and fault state as shown in equation (7): By taking the derivative of equation (7), we can get the error dynamics of the system as shown in equation (8): Step 3.3) Under the condition that equation (4) holds, for the state space model (2) and the corresponding designed observer (5), if the parameters λ1 and λ2 satisfy Then the estimation error (7) is ultimately uniformly bounded and the system error dynamics (8) is finite-time stable, that is: Where T is the finite convergence time; V1(0) is the value of the selected Lyapunov function at t=0, and And P and Q are both positive definite matrices; λ min {·} and λ max {·} respectively represent the minimum and maximum eigenvalues ​​of the matrix; by selecting appropriate values ​​for λ1 and λ2, the estimation error can be adjusted to a very small value; Step 4) Design a fault-tolerant control algorithm, including the following steps: Step 4.1) Define the consistent tracking position error variable e according to the neighbor information obtained by the i-th agent xi (t) and the speed error variable e vi (t) is of the following form: definition Formula (13) can be rewritten as a vector form as shown in Formula (14): Among them, I m is the m-dimensional identity matrix; definition The obtained consistent tracking error system is shown in formula (15): Step 4.2) Design the integral sliding surface function as shown in formula (16): in s i (t), i=1,...,n are sliding mode variables; k1 and k2 are positive constants; Step 4.3) Design a fault-tolerant consistency control algorithm for the follower agent as shown in formula (17): where γ and ε are positive constants. Through theoretical derivation, it can be concluded that the multi-agent system composed of (1) and (2) can achieve finite-time consistency in the case of actuator failure under the fault-tolerant control law (17).

Images