Observer-based fault-tolerant consensus sliding mode control algorithm for heterogeneous multi-agent systems

Abstract

The application discloses an observer-based fault-tolerant consensus sliding mode control algorithm for a heterogeneous multi-agent system, and considers a heterogeneous mixed-order nonlinear multi-agent system with unknown disturbances and actuator faults. In view of the characteristics that the disturbance and fault information of the system are difficult to obtain, a finite-time observer is first proposed to effectively estimate the disturbance and fault of the agent in real time. By introducing an adaptive gain, the dependence on unknown parameters is eliminated, and the tracking speed is improved. According to the relative state information between the multi-agents, a consensus error variable is defined. Then, based on terminal sliding mode control, an improved terminal sliding surface is proposed. By introducing a sine function link, the singularity and chattering problem is solved, and the fixed-time consensus tracking goal is achieved. The application can be used for fault-tolerant consensus control implementation of a kind of heterogeneous mixed-order nonlinear multi-agent system with unknown disturbances and actuator faults.

Description

Technical Field

[0001] This invention relates to an observer-based fault-tolerant and consistency sliding mode control algorithm for heterogeneous multi-agent systems, belonging to the field of fault-tolerant and consistency control technology for heterogeneous multi-agent systems. Background Technology

[0002] With the rapid development of society, communication, and artificial intelligence, multi-agent systems have become a research hotspot in the field of control engineering. Due to the collaboration and cooperation among individual agents, they can accomplish complex tasks that are difficult for individuals to complete alone. As research deepens and the application areas of multi-agent systems continue to expand, the collaborative tasks that multi-agent systems need to achieve are becoming more diverse, making heterogeneous hybrid multi-agent systems a hot research target. Since the concept of multi-agent systems emerged, their practicality, flexibility, and efficiency in engineering applications have attracted considerable attention. To ensure the stability and security of multi-agent systems in practical applications, it is essential to improve the system's robustness, and the consensus objective of multi-agent systems is one of the most fundamental issues. The consensus objective refers to the ability of each agent in the operation of a multi-agent system to achieve information interaction under certain control laws, ultimately reaching the same expected value. Consistency research is reflected in multiple engineering fields such as robot collaboration, drone swarming, and mobile sensor networks, and has significant research value.

[0003] However, in practical engineering applications, because multi-agent systems are significantly larger in hardware scale than single-agent systems, their susceptibility to unknown disturbances and failures also increases exponentially. Take a drone swarm system as an example: during mission execution, each drone experiences varying degrees of noise interference. These disturbances affect not only the individual drone's attitude but also its communication with other drones, ultimately impacting the overall system control. Therefore, achieving consistency quickly and effectively in multi-agent systems and designing more efficient controllers to improve system robustness are crucial. Summary of the Invention

[0004] Objective: To address the aforementioned research background, this invention proposes a fault-tolerant and consistent sliding mode control algorithm for heterogeneous multi-agent systems based on an observer. A finite-time observer capable of rapidly and accurately estimating unknown disturbances and fault information is designed. An improved terminal sliding surface is designed to enhance system robustness and significantly mitigate chattering issues caused by sliding mode. A sinusoidal function element is introduced to resolve the singularity problem of terminal sliding mode. Combining the relative state errors between agents, a non-singular terminal sliding mode fault-tolerant controller is designed, achieving consistent control of the multi-agent system within a fixed time.

[0005] Technical Solution: A fault-tolerant and consistent sliding mode control algorithm for heterogeneous multi-agent systems based on a finite-time observer. To address the unknown disturbances and actuator failures in heterogeneous hybrid-order nonlinear systems, a finite-time observer is designed, which can converge quickly and accurately obtain estimates, providing assistance for controller design. A consistency error variable is defined based on the relative state information between agents. Building upon terminal sliding mode control, an improved terminal sliding surface is designed to address chattering and singularity issues, and a sine function element is introduced, significantly mitigating chattering, resolving singularity problems, and improving system robustness. The specific steps of the observer-based fault-tolerant and consistent sliding mode control algorithm for heterogeneous multi-agent systems are as follows:

[0006] Step 1) Determine the dynamic model of the multi-agent system, including the following steps:

[0007] Step 1.1) Determine the leader's dynamic model as shown in (1):

[0008]

[0009] Where, x p，0 (t)∈R, x v，0 (t)∈R represent the leader's position and velocity state, respectively; u0(t)∈R is the control input;

[0010] Step 1.2) Determine the dynamic model of the follower i (i = 1, 2, ..., n) with unknown perturbations and actuator failures, which contains m second-order agents and nm first-order agents as shown in (2):

[0011]

[0012] Where, x p，i (t)∈R, x v，i (t)∈R represent the position and velocity states of the i-th follower agent, respectively; f i (x p，i (t), x v，i (t), t) and f i (x p，i (t) and t) represent the intrinsic nonlinear dynamic functions of the i-th second-order agent and the i-th first-order agent, respectively; d i (t) represents the unknown perturbation of the i-th agent; The actual control input for the i-th agent is expressed as shown in (3):

[0013]

[0014] Among them, u i(t)∈R represents the ideal control input for the i-th agent, 0<ρ i <1 indicates the actuator failure factor, ε i The increment of actuator faults is represented; the lumped fault of the i-th agent is defined as w. i (t), which includes unknown disturbances and actuator faults, is expressed as shown in (4):

[0015] w i (t)=d i (t)+ε i -ρ i u i (t) (4)

[0016] Step 1.3) For the inherent nonlinear dynamic function f i (x p，i (t), x v，i (t), t), f i (x p,i (t), t), Leader velocity state x v,0 and lumped failure w i (t) Make reasonable assumptions:

[0017]

[0018] in, and All are non-negative constants;

[0019] Step 2) Determine the communication topology of the multi-agent system:

[0020] Consider a multi-agent system consisting of one leader and n followers, where the leader is labeled 0 and the followers are labeled i (i = 1, 2, ..., n); the communication topology is represented by a topology graph. To describe it, in which, Represents a set of nodes. Denotes the set of edges. Let G represent the adjacency matrix; define the subsystem topology graph G = (V, E, A) to represent the communication network between followers, and correspondingly, V = {1, 2, ..., n} to represent the set of nodes. Let A represent the set of edges, and A represent the adjacency matrix. If there exists a directed edge from node j to node i, meaning node i can obtain information from node j, then (v j v i )∈E, a ij >0; otherwise, a ij =0; Define the in-degree of node i as Then the in-degree matrix of the topological graph G can be represented as D = diag{d1, d2, ..., d...} n}; The Laplace matrix is ​​represented as L = [l ij ] n×n =DA, where when i=j, l ij =d i When i ≠ j, l ij =-a ij Define the adjacency matrix between leaders and followers as B = diag{b1, b2, ..., b}. n If follower i can obtain information directly from the leader, then b i >0; otherwise, b i =0;

[0021] Step 3) Construct a finite-time observer, including the following steps:

[0022] Step 3.1) First, design a finite-time observer as shown in (6) for each follower i (i = 1, 2, ... n):

[0023]

[0024] in, Let represent the lumped fault estimate of the i-th agent; Indicates the estimation error. η1 and μ1 are both non-negative constants, 0 < γ1 < 1. sgn(·) is the sign function, sig γ The function (·) is defined as sig γ (x)=|x| γ sgn(x); Ξ is the adaptive gain to be designed;

[0025] Step 3.2) Define the lumped fault observer estimation error as shown in (7):

[0026]

[0027] Step 3.3) Design the adaptive gain as shown in (8):

[0028]

[0029] in

[0030]

[0031] Where, Δ1, Δ d ,ζ,κ1,κ2,χ and All are non-negative constants; by choosing appropriate parameters, the estimation error (7) will eventually be uniformly bounded and the error dynamics (7) of the system will be stable in finite time, that is:

[0032]

[0033] Among them, T 1i For finite convergence time, The initial estimation error for lumped faults is used; by selecting an appropriate value, the estimation error is adjusted to a minimum.

[0034] Step 4) Design a consistency control algorithm, including the following steps:

[0035] Step 4.1) Based on the neighbor information obtained by the i-th agent, define the consistency tracking position error variable e. p，i and velocity error variable e v，i For the sake of brevity, the function variables are omitted, and its form is as shown in (11):

[0036]

[0037] The consistency error equation is shown in (12):

[0038]

[0039] Step 4.2) Design the terminal sliding surface function of the i-th agent as shown in (13):

[0040]

[0041] in, It is a scalar positive function; m1, n1, p1 and q1 are all positive odd numbers, satisfying m1 > n1, p1 < q1 < 2p1, m1 / n1 - p1 / q1 > 1; α1 and β1 are non-negative constants;

[0042] Step 4.3) Design a fault-tolerant and consistent control law for the i-th agent system as shown in (14):

[0043]

[0044] in The virtual control signal is defined as shown in (15):

[0045]

[0046] Where m2, n2, p2, and q2 are all positive odd numbers, satisfying m2 > n2 and p2 < q2; α2 and β2 are non-negative constants; μ τ (·) represents a continuous sine function element, and its expression is shown in (16):

[0047]

[0048] Where τ is a nonnegative constant;

[0049] Then the estimation error is eventually uniformly bounded and the system error dynamics are stable over a fixed time, that is:

[0050] T0<T1+T2+σ(τ) (17)

[0051] Where T0 is the convergence time. σ(τ) represents a very small time margin.

[0052] Beneficial Effects: For nonlinear multi-agent systems with unknown disturbances and actuator failures, a fault-tolerant and consistent sliding mode control algorithm based on a finite-time observer is designed for heterogeneous multi-agent systems. A finite-time observer is designed to achieve rapid estimation and compensation of disturbance and fault information. Based on the estimates obtained from the observer and combined with the relative state errors between agents, a non-singular terminal sliding mode controller is designed, achieving the fixed-time consistent control objective. Overall, this invention has the following advantages:

[0053] ① A finite-time disturbance observer was designed to achieve fast and accurate estimation of velocity and disturbance information. By designing an adaptive gain, the dependence on unknown parameters was eliminated, the difficulty of controller design was reduced, and the speed of estimation was improved. This not only saves equipment resources but also ensures fast and accurate estimation even in situations with large actual errors.

[0054] ② Based on the conventional terminal sliding surface, an improved terminal sliding surface was designed, which introduced a sinusoidal function element, greatly weakening the chattering effect caused by sliding, solving the terminal sliding singularity problem, and further improving the robustness of the system.

[0055] ③ The non-singular terminal sliding mode control algorithm designed in this invention can solve the fault-tolerant consistency control problem of a class of heterogeneous nonlinear multi-agent systems with unknown disturbances and actuator failures. This algorithm ensures that the system converges within a fixed time, improving equipment efficiency and reducing costs to a certain extent, while also enhancing the system's robustness and stability.

[0056] The observer-based fault-tolerant and consistent sliding mode control algorithm for heterogeneous multi-agent systems proposed in this invention has high accuracy, strong security, high efficiency, and relatively low hardware requirements. It has certain application significance and can be widely applied to the consistency realization problem of a class of heterogeneous nonlinear multi-agent systems with unknown disturbances and actuator failures. Attached Figure Description

[0057] Figure 1 This is a flowchart of the method of the present invention;

[0058] Figure 2 It is the communication topology network of the multi-Qball-X4 quadcopter system;

[0059] Figure 3 These are the estimation curves of the two observers for the lumped fault of follower 3;

[0060] Figure 4 It is a locally magnified curve of the estimated convergence time of follower 3 due to lumped fault;

[0061] Figure 5 These are the estimation curves of the follower 4 being subjected to lumped faults by the two observers;

[0062] Figure 6 It is a locally magnified curve of the estimated convergence time of follower 4 due to lumped fault;

[0063] Figure 7 This is the position tracking error curve under the control algorithm designed in this invention;

[0064] Figure 8 This is the speed tracking error curve under the control algorithm designed in this invention;

[0065] Figure 9 This is the position tracking error curve under the traditional terminal sliding mode control algorithm;

[0066] Figure 10 This is the speed tracking error curve under the traditional terminal sliding mode control algorithm; Detailed Implementation

[0067] The invention will now be further explained with reference to the accompanying drawings.

[0068] like Figure 1 As shown, a finite-time observer is designed for heterogeneous hybrid-order systems with unknown disturbances and faults, enabling rapid estimation and compensation of disturbances. Based on terminal sliding mode control, an improved terminal sliding surface is designed to address the chattering problem inherent in sliding mode control, significantly reducing chattering; simultaneously, a sinusoidal function element is introduced to solve the singularity problem. Compared to conventional terminal sliding mode control, this control law has a faster convergence speed and a fixed convergence time, exhibiting higher robustness. A fault-tolerant and consistent sliding mode control for an observer-based heterogeneous multi-agent system includes the following specific steps:

[0069] Step 1) Determine the dynamic model of the multi-agent system, including the following steps:

[0070] Step 1.1) Determine the leader's dynamic model as shown in (1):

[0071]

[0072] Where, x p，0 (t)∈R, x v，0(t)∈R represent the leader's position and velocity state, respectively; u0(t)∈R is the control input;

[0073] Step 1.2) Determine the dynamic model of the follower i (i = 1, 2, ..., n) with unknown perturbations and actuator failures, which contains m second-order agents and nm first-order agents as shown in (2):

[0074]

[0075] Where, x p，i (t)∈R, x v，i (t)∈R represent the position and velocity states of the i-th follower agent, respectively; f i (x p，i (t), x v，i (t), t) and f i (x p,i (t) and t) represent the intrinsic nonlinear dynamic functions of the i-th second-order agent and the i-th first-order agent, respectively; d i (t) represents the unknown perturbation of the i-th agent; The actual control input for the i-th agent is expressed as shown in (3):

[0076]

[0077] Among them, u i (t)∈R represents the ideal control input for the i-th agent, 0<ρ i <1 indicates the actuator failure factor, ε i The increment of actuator faults is represented; the lumped fault of the i-th agent is defined as w. i (t), which includes unknown disturbances and actuator faults, is expressed as shown in (4):

[0078] w i (t)=d i (t)+ε i -ρ i u i (t) (4)

[0079] Step 1.3) For the inherent nonlinear dynamic function f i (x p,i (t), x v,i (t), t), f i (x p，i (t), t), Leader velocity state x v，0 and lumped failure w i (t) Make reasonable assumptions:

[0080]

[0081] in, and All are non-negative constants;

[0082] Step 2) Determine the communication topology of the multi-agent system:

[0083] Consider a multi-agent system consisting of one leader and n followers, where the leader is labeled 0 and the followers are labeled i (i = 1, 2, ..., n); the communication topology is represented by a topology graph. To describe it, in which, Represents a set of nodes. Denotes the set of edges. Let G represent the adjacency matrix; define the subsystem topology graph G = (V, E, A) to represent the communication network between followers, and correspondingly, V = {1, 2, ..., n} to represent the set of nodes. Let A represent the set of edges, and A represent the adjacency matrix. If there exists a directed edge from node j to node i, meaning node i can obtain information from node j, then (v j v i )∈E, a ij >0; otherwise, a ij =0; Define the in-degree of node i as Then the in-degree matrix of the topological graph G can be represented as D = diag{d1, d2, ..., d...} n}; The Laplace matrix is ​​represented as L = [l ij ] n×n =DA, where when i=j, l ij =d i When i ≠ j, l ij =-a ij Define the adjacency matrix between leaders and followers as B = diag{b1, b2, ..., b}. n If follower i can obtain information directly from the leader, then b i >0; otherwise, b i =0;

[0084] Step 3) Construct a finite-time observer, including the following steps:

[0085] Step 3.1) First, design a finite-time observer as shown in (6) for each follower i (i = 1, 2, ... n):

[0086]

[0087] in, Let represent the lumped fault estimate of the i-th agent; Indicates the estimation error. η1 and μ1 are both non-negative constants, 0 < γ1 < 1. sgn(·) is the sign function, sig γ The function (·) is defined as sig γ (x)=|x| γ sgn(x); Ξ is the adaptive gain to be designed;

[0088] Step 3.2) Define the lumped fault observer estimation error as shown in (7):

[0089]

[0090] Step 3.3) Design the adaptive gain as shown in (8):

[0091]

[0092] in

[0093]

[0094] Where, Δ1, Δ d ξ, κ1, κ2, x and All are non-negative constants; by choosing appropriate parameters, the estimation error (7) will eventually be uniformly bounded and the error dynamics (7) of the system will be stable in finite time, that is:

[0095]

[0096] Among them, T 1i For finite convergence time, The initial estimation error for lumped faults is used; by selecting an appropriate value, the estimation error is adjusted to a minimum.

[0097] Step 4) Design a consistency control algorithm, including the following steps:

[0098] Step 4.1) Based on the neighbor information obtained by the i-th agent, define the consistency tracking position error variable e. p，i and velocity error variable e v,i For the sake of brevity, the function variables are omitted, and its form is as shown in (11):

[0099]

[0100] The consistency error equation is shown in (12):

[0101]

[0102] Step 4.2) Design the terminal sliding surface function of the i-th agent as shown in (13):

[0103]

[0104] in, It is a scalar positive function; m1, n1, p1 and q1 are all positive odd numbers, satisfying m1 > n1, p1 < q1 < 2p1, m1 / n1 - p1 / q1 > 1; α1 and β1 are non-negative constants;

[0105] Step 4.3) Design a fault-tolerant and consistent control law for the i-th agent system as shown in (14):

[0106]

[0107] in The virtual control signal is defined as shown in (15):

[0108]

[0109] Where m2, n2, p2, and q2 are all positive odd numbers, satisfying m2 > n2 and p2 < q2; α2 and β2 are non-negative constants; μ τ (·) represents a continuous sine function element, and its expression is shown in (16):

[0110]

[0111] Where τ is a nonnegative constant;

[0112] Then the estimation error is eventually uniformly bounded and the system error dynamics are stable over a fixed time, that is:

[0113] T0<T1+T2+σ(τ) (17)

[0114] Where T0 is the convergence time. σ(τ) represents a very small time margin.

[0115] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

[0116] The effectiveness of the implementation plan is illustrated below using a real-world case simulation.

[0117] The Qball-X4 quadcopter flight control system, developed by Quanser Industries, Canada, was used as the research object. A multi-agent system consisting of six Qball-X4 quadcopters, including a leader (labeled 0) and followers (labeled i = 1, 2, 3, 4, 5), was constructed. Its communication topology is as follows: Figure 2As shown. The leader is a second-order agent, agents labeled 1, 2, and 3 are second-order agents, and agents labeled 4 and 5 are first-order agents. Assuming the weight of each edge in the communication topology graph is 1, the specific expressions for the Laplace matrix L and the adjacency matrix B can be calculated based on the communication topology structure as follows:

[0118]

[0119] For this multi-agent system model, the dynamics of the leader are described as follows:

[0120]

[0121] The initial state value of the leader is [x] p，0 (0), x v，0 [0] = [0, 1]; obviously |x v，0 |≤1.

[0122] The dynamic model of follower i (i = 1, 2, 3) is described as follows:

[0123]

[0124] The inherent nonlinear dynamic f i (x p，i (t), x v，i (t), t)=cos(x) p，i (t))+cos(x v，i (t)), obviously |f i (x p，i (t), x v，i (t), t)|≤2; the perturbation equation is described as: d i =0.1(1-e -0.1t )+0.2sin(2t)i=1,2,3.

[0125] The dynamic model of follower i (i = 4, 5) is described as follows:

[0126]

[0127] The inherent nonlinear dynamic f i (x p，i (t), t)=cos(x) p，i (t)), obviously |f i (x p，i (t), t)|≤1; the perturbation equation is described as: d i =0.1+0.2sin(0.5πt)i=4,5.

[0128] The initial state values ​​of the followers are x p，1(0) = 5, x p,2 (0) = 1.5, x p，3 (0) = -2, x p，4 (0) = -4.2, x p，5 (0) = 2.5, x v，1 (0) = 1.2, x v，2 (0) = 0.5, x v，3 (0) = 0. To illustrate the effectiveness and superiority of the fault-tolerant control algorithm designed in this invention in solving the consistency control problem of a multi-agent system with actuator failures, it is assumed that followers 3 and 4 suffer actuator failures, while the other followers are fault-free. Where ρ3 = 0.2, ρ4 = 0.3, ε3 = 0.2, ε4 = 0.1.

[0129] The observer parameters are selected as follows: η1=0.9, μ1=1.2, γ1=0.7, Δ1=5, Δ d =20, Ψ0=0.7, ζ=4, κ1=0.4, κ2=3, χ=0.9, To better demonstrate the speed and accuracy of the observer designed in this invention during estimation, a traditional observer was selected for comparison. The comparison curves of the observer and the traditional observer during disturbance estimation are shown below. Figure 3 and Figure 5 , Figure 4 and Figure 6 The local magnified curves of the convergence time are presented. The comparison shows that the observer designed in this invention converges faster, produces a smoother curve, and has a lower peak value compared to traditional observers. Therefore, the observer designed in this invention performs better.

[0130] The controller parameters are selected as follows: α1 = β1 = α2 = β2 = 2, m1 = 9, n1 = 5, p1 = 7, q1 = 9, m2 = 11, n2 = 9, p2 = 5, q2 = 7, τ = 0.1. To better demonstrate the higher robustness of the integral nonsingular terminal sliding mode controller designed in this invention, a traditional terminal sliding mode controller is selected for comparison. Figure 7 and Figure 8 These are the position and velocity error curves of the follower under the action of the controller designed in this invention. Figure 9 and Figure 10 These are the position and velocity error curves of the follower under the action of a traditional terminal sliding mode controller. It can be seen that both methods can achieve the goal of consistent convergence in a multi-agent system under uncertainty and disturbances. However, the controller designed in this invention clearly has a shorter convergence time and significantly solves the chattering problem. Therefore, this case demonstrates that the control method is effective.

Claims

1. A fault-tolerant and consistent sliding mode control method for heterogeneous multi-agent systems based on observers, comprising the following specific steps: Step 1) Determine the dynamic model of the multi-agent system, including the following steps: Step 1.1) Determine the leader's dynamic model as shown in (1): in, x p，0 (t)∈R, x v，0 (t)∈R represent the leader's position and velocity state, respectively; u0(t)∈R is the control input; Step 1.2) Determine the dynamic model of the follower i (i = 1, 2, ..., n) with unknown perturbations and actuator failures, which contains m second-order agents and nm first-order agents as shown in (2): Where, x p，i (t)∈R, x v，i (t)∈R represent the position and velocity states of the i-th follower agent, respectively; f i (x p，j (t), x v，i (t), t) and f i (x p，i (t) and t) represent the intrinsic nonlinear dynamic functions of the i-th second-order agent and the i-th first-order agent, respectively; d i (t) represents the unknown perturbation of the i-th agent; The actual control input for the i-th agent is expressed as shown in (3): Among them, u i (t)∈R represents the ideal control input of the i-th agent, 0<ρ i <1 indicates the actuator failure factor, ε i The increment of actuator faults is represented; the lumped fault of the i-th agent is defined as w. i (t), which includes unknown disturbances and actuator faults, is expressed as shown in (4): w i (t)=d i (t)+ε i -ρ i u i (t) (4) Step 1.3) For the inherent nonlinear dynamic function f i (x p，i (t), x v，i (t), t), f i (x p，i (t), t), Leader velocity state x v，0 and lumped failure w i (t) Make reasonable assumptions: in, and All are non-negative constants; Step 2) Determine the communication topology of the multi-agent system: Consider a multi-agent system consisting of one leader and n followers, where the leader is labeled 0 and the followers are labeled i (i = 1, 2, ..., n); the communication topology is represented by a topology graph. To describe it, in which, Represents a set of nodes. Denotes the set of edges. Let G represent the adjacency matrix; define the subsystem topology graph G = (V, E, A) to represent the communication network between followers, and correspondingly, V = {1, 2, ..., n} to represent the set of nodes. Let A represent the set of edges, and A represent the adjacency matrix. If there exists a directed edge from node j to node i, meaning node i can obtain information from node j, then (v j v i )∈E, a ij >0; otherwise, a ij =0; Define the in-degree of node i as Then the in-degree matrix of the topological graph G can be represented as D = diag{d1, d2, ..., d...} n }; The Laplace matrix is ​​represented as L = [l ij ] n×n =DA, where when i=j, l ij =d i When i ≠ j, l ij =-a ij Define the adjacency matrix between leaders and followers as B = diag{b1, b2, ..., b}. n If follower i can obtain information directly from the leader, then b i >0; otherwise, b i =0; Step 3) Construct a finite-time observer, including the following steps: Step 3.1) First, design a finite-time observer as shown in (6) for each follower i (i = 1, 2, ... n): in, Let represent the lumped fault estimate of the i-th agent; Indicates the estimation error. η1 and μ1 are both non-negative constants. sgn(·) is the sign function, sig γ The function (·) is defined as sig γ (x)=|x| γ sgn(x); Ξ is the adaptive gain to be designed; Step 3.2) Define the lumped fault observer estimation error as shown in (7): Step 3.3) Design the adaptive gain as shown in (8): in Where, Δ1, Δ d ,ζ,κ1,κ2,χ and All are non-negative constants; by choosing appropriate parameters, the estimation error (7) will eventually be uniformly bounded and the error dynamics (7) of the system will be stable in finite time, that is: Among them, T 1i For finite convergence time, The initial estimation error for lumped faults is used; by selecting an appropriate value, the estimation error is adjusted to a minimum. Step 4) Design a consistency control algorithm, including the following steps: Step 4.1) Based on the neighbor information obtained by the i-th agent, define the consistency tracking position error variable e. p，i and velocity error variable e v,i For the sake of brevity, the function variables are omitted, and its form is as shown in (11): The consistency error equation is shown in (12): Step 4.2) Design the terminal sliding surface function of the i-th agent as shown in (13): in, It is a scalar positive function; m1, n1, p1 and q1 are all positive odd numbers, satisfying m1 > n1, p1 < q1 < 2p1, m1 / n1 - p1 / q1 > 1; α1 and β1 are non-negative constants; Step 4.3) Design a fault-tolerant and consistent control law for the i-th agent system as shown in (14): in The virtual control signal is defined as shown in (15): Where m2, n2, p2, and q2 are all positive odd numbers, satisfying m2 > n2 and p2 < q2; α2 and β2 are non-negative constants; μ τ (·) represents a continuous sine function element, and its expression is shown in (16): Where τ is a nonnegative constant; Then the estimation error is eventually uniformly bounded and the system error dynamics are stable over a fixed time, that is: T0<T1+T2+σ(τ) (17) Where T0 is the convergence time. σ(τ) represents a very small time margin.

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