Multi-agent sliding mode fault-tolerant control method under finite-time observer

Abstract

The present invention discloses a multi-agent sliding mode fault-tolerant control method under a finite-time observer. For a second-order nonlinear multi-agent system with unknown disturbances and actuator faults, a finite-time observer is proposed to achieve real-time observation of the disturbance and fault information suffered by the agent. According to the relative state information between the multi-agents, the consistency error variable is defined. Then a novel sliding surface is proposed, and a distributed adaptive sliding mode fault-tolerant method is designed by combining it with an artificial potential function. The present invention can eliminate the influence of faults and unknown disturbances on the system, and realize connectivity preservation when the multi-agent system reaches a consistent state.

Description

Technical Field

[0001] The invention relates to a multi-agent sliding mode fault-tolerant control method under a finite time observer, and belongs to the technical field of fault-tolerant consistency control of multi-agent systems. Background Art

[0002] With the rapid development of society, communication, artificial intelligence, etc., multi-agent systems have become a research hotspot in the field of control engineering. Due to the collaboration and cooperation between individuals, it can complete complex tasks that are difficult for individuals to complete. Since the concept of multi-agent systems was born, its practicality, flexibility, and efficiency in engineering applications have attracted enthusiastic attention. In order to ensure the stability and security of multi-agent systems in practical applications, the robustness of the system must be improved first, and the consistency goal of multi-agent systems is one of the most basic issues. The consistency goal means that during the operation of the multi-agent system, each agent can achieve information exchange under a certain control law and finally reach the same expected value. Consistency research is reflected in many engineering fields such as robot collaboration, drone formation, mobile sensor networks, etc., and has important research significance.

[0003] However, in actual engineering applications, since multi-agents are multiples of a single agent in terms of hardware scale, the possibility of being subject to unknown disturbances and failures is also doubled accordingly. Taking the drone swarm system as an example, when the drone formation performs a task, each drone will be subject to different degrees of noise interference. These disturbances will affect the individual's own posture on the one hand, and the information interaction between it and other individuals on the other hand, thus affecting the control of the entire system. Therefore, it is particularly important to quickly and effectively achieve the consistency of the multi-agent system and design a more efficient controller to improve the robustness of the system.

[0004] It is worth noting that in practical applications, due to the limitation of transmission power, the perception and communication range between adjacent agents is always limited. That is to say, the connectivity of the communication topology network can only be maintained within a limited distance. To address this problem, it is necessary to combine the artificial potential function with the control rate and further consider the realistic factors. Summary of the invention

[0005] Purpose of the invention: In view of the above research background, a multi-agent sliding mode fault-tolerant control method under a finite-time observer is proposed. A finite-time observer is designed to achieve fast and accurate estimation of unknown disturbances and fault information; an artificial potential function is designed to achieve connectivity preservation; a distributed adaptive sliding mode controller is designed in combination with the relative state error between agents, and the fault-tolerant consistency control of the multi-agent system is achieved in a finite time.

[0006] Technical solution:

[0007] The multi-agent sliding mode fault-tolerant control method under a finite time observer includes the following specific steps:

[0008] Step 1): Determine a dynamic model of a multi-agent system, wherein the multi-agent system includes a leader and n followers;

[0009] Step 2): Determine the communication topology of the multi-agent system;

[0010] Step 3): Construct a finite-time observer;

[0011] Step 4): Design a consistency control method, and verify that the multi-agent system achieves finite-time consistency under the designed consistency control method based on Lyapunov stability theory.

[0012] Preferably, the implementation process of step 1) is:

[0013] Step 1.1): Determine the leader’s dynamic model as shown in (1):

[0014]

[0015] in, Respectively represent the position and velocity states of the leader; is the control input of the leader; m is the spatial dimension; t is the time variable;

[0016] Step 1.2): Determine the dynamic model of follower i (i = 1, 2, ..., n) with unknown disturbances and actuator failures as shown in (2):

[0017]

[0018] in, Respectively represent the position and velocity states of the i-th follower; is the control input of the ith follower; f i (x i (t),v i (t),t) represents the inherent nonlinear dynamic function of the ith follower; w i (t) represents the sum of the unknown disturbance and actuator fault of the ith follower, which is defined as the lumped fault;

[0019] Step 1.3): For the inherent nonlinear dynamic function f i (x i (t),v i (t),t) and the lumped fault w i (t) Make assumptions:

[0020]

[0021] in, and are all non-negative constants; ||·|| represents the bi-norm.

[0022] Preferably, the implementation process of step 2) is:

[0023] In a multi-agent system, the leader is marked as 0 and the follower is marked as i (i = 1, 2, ... n); the communication topology between followers is represented by a topological graph G = (V, E, A), where V = {1, 2, ..., n} represents a node set, represents the edge set, A=[a ij ] n×n represents the adjacency matrix, a ij Used to determine whether there is an edge between node i and node j; if (i, j) ∈ E, then there is a direct path between node i and node j, a ij =1; otherwise, a ij =0; define the degree of node i as The degree matrix of the topological graph G is expressed as D = diag{d1, d2, ..., d i ,…d n}; Laplace matrix is ​​expressed as L = [l ij ] n×n =DA,l ij is the i-th row and j-th column of the Laplace matrix, where when i=j, l ij =d i ; When i≠j, l ij =-a ij ; Define the adjacency matrix between the leader and the follower as B = diag{b1,b2,…,b i ,…b n}, b i Used to determine whether follower i can receive information from the leader; if follower i can obtain information directly from the leader, then b i =1; otherwise, b i = 0; the neighborhood of follower i is N i ={j|(j,i)∈E}.

[0024] Preferably, the implementation process of step 3) is:

[0025] Step 3.1): First, design a finite-time observer for each follower as shown in (4):

[0026]

[0027] in, represents the aggregate fault estimate of the ith follower; hi (t) is an auxiliary variable, a, c i is a normal number, satisfying The parameter α satisfies 0<α<1;

[0028] Step 3.2): Define the estimated error of the lumped fault As shown in (5):

[0029]

[0030] Step 3.3): Construct two auxiliary variables h i (t) and q i (t), in q i (t) with the help of the auxiliary variable h i (t) Designed as the estimated error of lumped fault Points:

[0031]

[0032] Adjust a, c i , the value of α makes the estimation error The multi-agent system is eventually uniformly bounded and finite-time stable, that is:

[0033]

[0034] Where T1 is the finite convergence time, V1(0) is the value of the selected Lyapunov function at t = 0; adjust a, c i , the value of α will estimate the error Adjust to a very small value of 0.

[0035] Preferably, the implementation process of step 4) is:

[0036] Step 4.1): Define the consistency tracking position error variable e xi (t) and the consistent tracking speed error variable e vi (t) is of the following form:

[0037]

[0038] Defining Position Error Speed ​​error Defining a vector collection Vector Collection Vector Collection Vector Collection

[0039] in, is the position error between the follower and the leader Vector collection of is the velocity error between the follower and the leader Vector collection of e x is the consistent tracking position error variable e xi (t) is a vector set, e v is the consistency tracking speed error variable e vi (t), then (8) can be rewritten into vector form as shown in (9):

[0040]

[0041] Among them, I m represents the m-dimensional identity matrix;

[0042] Step 4.2): Design sliding surface function s i (t) As shown in (10):

[0043]

[0044] Among them, k1, k2, k3 are positive constants, and the function The parameter ξ satisfies 0<ξ<0.2;

[0045] Step 4.3): Design the artificial potential function J i (t) As shown in (11):

[0046]

[0047] in, For any All have||x i (t)-x j (t)||≤R holds, N i is the neighborhood of follower i, R is the upper bound of the distance that can transmit information between the i-th follower and the j-th follower;

[0048] The artificial potential function J i (t) is combined with the sliding surface to obtain the sliding surface function s i (t) As shown in (12):

[0049]

[0050] Step 4.3): Design a consistency control method for the multi-agent system as shown in (13):

[0051]

[0052]

[0053] where k4, ι, ρ and k5 are positive constants, is an adaptive parameter; according to Lyapunov stability theory, it is verified that the multi-agent system achieves finite-time consistency under the action of the designed consistency control method.

[0054] Beneficial effects: For nonlinear multi-agent systems with unknown disturbances and actuator failures, a new multi-agent sliding mode fault-tolerant control method under a finite-time observer is designed. A finite-time observer is designed to achieve rapid estimation and compensation of disturbance and fault information. Based on the estimated value obtained by the observer and combined with the artificial potential function, a distributed adaptive sliding mode controller is designed to achieve the finite-time consistency control goal. In general, the present invention has the following advantages:

[0055] ①A finite-time observer is designed, which can realize rapid estimation and compensation of disturbance and fault information.

[0056] ② A novel sliding surface is designed to eliminate the need for fractional-order and sign functions, reduce the communication burden, and eliminate chattering.

[0057] ③ The distributed adaptive sliding mode control method designed by the present invention combines the artificial potential function with the proposed novel sliding mode surface, which can solve the consistency realization problem of a type of nonlinear multi-agent system with unknown disturbances and actuator failures. This method can ensure that the system completes convergence within a bounded time, improves the efficiency of the equipment to a certain extent, reduces the cost, and enhances the robustness and stability of the system, and realizes connectivity preservation;

[0058] The multi-agent sliding mode fault-tolerant control method under a finite-time observer proposed in the present invention has high accuracy, strong security, high efficiency, and not stringent hardware requirements. It has certain application significance and can be widely used in the consistency implementation problem of a class of nonlinear multi-agent systems with unknown disturbances and actuator failures. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 1 is a flow chart of the method of the present invention;

[0060] Figure 2 It is the communication topology network of multiple Qball-X4 quadrotor helicopter systems;

[0061] Figure 3 are the estimation curves of the two observers for the disturbance and failure of follower 2;

[0062] Figure 4 is the position tracking error curve under the control method designed by the present invention;

[0063] Figure 5 is a speed tracking error curve under the control method designed by the present invention;

[0064] Figure 6 is the position tracking error curve under the traditional sliding mode control method;

[0065] Figure 7 It is the speed tracking error curve under the traditional sliding mode control method;

[0066] Figure 8 is the response distance between agents under the control method designed by the present invention;

[0067] Fig. 9 is the response distance between agents under the traditional sliding mode control method. DETAILED DESCRIPTION

[0068] The present invention will be further explained below in conjunction with the accompanying drawings.

[0069] like Figure 1 As shown in the figure, a finite-time observer is designed for the case of unknown second-order system disturbances and actuator faults to achieve real-time observation of disturbances and fault information on the agent. Based on the design of a novel sliding surface, the connectivity preservation is achieved by combining it with an artificial potential function to solve the problem of bounded perception range between agents. Compared with conventional sliding mode control, this control law has a faster convergence speed, a limited convergence time, and higher robustness. A multi-agent sliding mode fault-tolerant control under a finite-time observer includes the following specific steps:

[0070] Step 1) Determine the multi-agent system dynamics model, including the following steps:

[0071] Step 1.1) Determine the leader’s dynamic model as shown in (1):

[0072]

[0073] in, Respectively represent the position and velocity states of the leader; Provide control input for leaders;

[0074] Step 1.2) Determine the dynamic model of follower i (i = 1, 2, ..., n) with unknown disturbances and actuator failures as shown in (2):

[0075]

[0076] in, Respectively represent the position and velocity states of the i-th follower; is the control input of the ith follower; f i (x i (t),v i (t),t) represents the inherent nonlinear dynamic function of the ith follower; wi (t) represents the sum of external disturbances and actuator-injected faults of the ith follower, which is defined as the lumped fault;

[0077] Step 1.3) For the inherent nonlinear dynamic function f i (x i (t),v i (t),t) and the lumped fault w i (t) Make reasonable assumptions:

[0078]

[0079] in, and are all non-negative constants;

[0080] Step 2) Determine the communication topology of the multi-agent system:

[0081] In a multi-agent system, the leader is marked as 0 and the follower is marked as i (i = 1, 2, ... n); the communication topology between followers is represented by a topological graph G = (V, E, A), where V = {1, 2, ..., n} represents a node set, represents the edge set, A=[a ij ] n×n represents the adjacency matrix, a ij Used to determine whether there is an edge between node i and node j; if (i, j) ∈ E, then there is a direct path between node i and node j, a ij =1; otherwise, a ij =0; define the degree of node i as The degree matrix of the topological graph G is expressed as D = diag{d1, d2, ..., d i ,…d n}; Laplace matrix is ​​expressed as L = [l ij ] n×n =DA,l ij is the i-th row and j-th column of the Laplace matrix, where when i=j, l ij =d i ; When i≠j, l ij =-a ij ; Define the adjacency matrix between the leader and the follower as B = diag{b1,b2,…,b i ,…b n}, b i Used to determine whether follower i can receive information from the leader; if follower i can obtain information directly from the leader, then b i =1; otherwise, b i = 0; the neighborhood of follower i is N i={j|(j,i)∈E}.

[0082] Step 3) Construct a finite-time fault observer, including the following steps:

[0083] Step 3.1) First, design a finite-time observer for each follower as shown in (4):

[0084]

[0085] in, represents the aggregate fault estimate of the ith agent; a, c i is a positive constant, 0<α<1,

[0086] Step 3.2) Define the estimated error of the lumped fault as shown in (5):

[0087]

[0088] Step 3.3) Construct two auxiliary variables, auxiliary variable h i (t) is designed as the integral of the lumped fault estimation error:

[0089]

[0090] By choosing appropriate parameters, the estimation error (5) is eventually uniformly bounded and the system is finite-time stable, that is:

[0091]

[0092] Where T1 is the finite convergence time, V1(0) is the value of the selected Lyapunov function at t = 0; i ,αselects a suitable value, the estimation error can be adjusted to a very small value;

[0093] Step 4) Design a consistency control method, including the following steps:

[0094] 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:

[0095]

[0096] definition Then (8) can be rewritten into vector form as shown in (9):

[0097]

[0098] Among them, I m represents the m-dimensional identity matrix;

[0099] Step 4.2): Design sliding surface function s i (t) As shown in (10):

[0100]

[0101] Among them, k1, k2, k3 are positive constants, and the function The parameter ξ satisfies 0<ξ<0.2;

[0102] Step 4.3): Design the artificial potential function J i (t) As shown in (11):

[0103]

[0104] in, For any All have||x i (t)-x j (t)||≤R holds; the artificial potential function J i (t) is combined with the sliding surface to obtain the sliding surface function s i (t) As shown in (12):

[0105]

[0106] Step 4.3): Design a consistency control method for the multi-agent system as shown in (13):

[0107]

[0108]

[0109] where k4, ι, ρ and k5 are positive constants, is an adaptive parameter; according to Lyapunov stability theory, it can be verified that the second-order nonlinear multi-agent system composed of (1) and (2) can achieve finite-time consistency under the control law (13).

[0110] The effectiveness of the implementation plan is illustrated by a real case simulation below.

[0111] The Qball-X4 quad-rotor helicopter flight control system developed by Canada's Quanser Company is used as the application research object. The multi-agent system consists of 5 Qball-X4 quad-rotor helicopters, including a leader marked as 0 and followers marked as i (i = 1, 2, 3, 4). Its communication topology is as follows: Figure 2As shown. Assuming that the weight of each edge in the communication topology graph is 1, the specific expressions of the Laplace matrix L and the adjacency matrix B can be calculated based on the communication topology structure:

[0112]

[0113] For the model of the multi-agent system, the leader's dynamic model is described as follows:

[0114]

[0115] Among them, the initial state values ​​of the leader are x0(0)=0.5, v0(0)=0.2;

[0116] The dynamic model of follower i (i = 1, 2, 3, 4) is described as follows:

[0117]

[0118] The inherent nonlinear dynamics f i (x i (t),v i (t),t)=-cos(x i (t))-cos(v i (t)), obviously ||f i (x i (t),v i (t), t)||≤2; the initial state values ​​of the followers are x1(0)=2.2, x2(0)=2.55, x3(0)=0.7, x4(0)=-2.4, v0(0)=0.2, v1(0)=3.2, v2(0)=1.98, v3(0)=-2.1, v4(0)=1.8.

[0119] In order to illustrate the effectiveness and superiority of the fault-tolerant control method designed by the present invention in solving the consistency control problem of a multi-agent system with unknown disturbances and actuator failures, it is assumed that follower 2 is faulty, the other followers are not faulty, but all followers are disturbed. The lumped fault equation of follower 2 is described as:

[0120] w2(t)=0.1sin(0.125πt)+0.1+0.2sin(0.5πt)

[0121] The lumped fault equation for the remaining followers is described as:

[0122] w i (t)=0.1+0.2sin(0.5πt)i=1,3,4

[0123] Select the observer parameters as: a = 1, ci =0.6, α=0.5. In order to better reflect the rapidity and accuracy of the observer designed by the present invention in disturbance and fault estimation, a traditional observer is selected for comparison, and the comparison curves of the observer and the traditional observer in disturbance and fault estimation are obtained as shown in the figure. Figure 3 By comparison, it can be seen that the observer designed by the present invention has a faster convergence speed, a smoother curve, and a lower peak value than the traditional observer. Therefore, the observer designed by the present invention has a better effect.

[0124] The controller parameters are selected as follows: k1 = 3, k2 = 0.1, k3 = 1, k4 = 0.5, k5 = 2, ξ = 0.1, ι = 1, ρ = 2, R = 5. In order to better reflect that the distributed adaptive sliding mode controller designed by the present invention has higher robustness, a traditional sliding mode controller is selected for comparison. Figure 4 and Figure 5 They are the position and velocity error curves of the follower under the action of the controller designed by the present invention, Figure 6 and Figure 7 It is the position and velocity error curve of the follower under the action of the traditional sliding mode controller. It can be seen from the curve that both methods can achieve the consistency convergence goal of the multi-agent system under disturbance and fault, but it is obvious that the controller designed by the present invention has a shorter convergence time and better robustness, and can better overcome the impact of faults and disturbances on the system. Figure 8 and Fig. 9 The response distances between agents under the two control methods are given respectively, and both methods can achieve connectivity preservation. In summary, this case can show that the control method is effective.

[0125] This method designs a sliding mode fault-tolerant control method for second-order nonlinear multi-agent systems with unknown disturbances and actuator failures. The characteristic of this method is that, for the case of unknown second-order system disturbances and actuator failures, a finite-time observer is designed to achieve real-time observation of the disturbances and fault information of the agent. On the basis of designing a novel sliding surface, the connectivity preservation is achieved by combining it with an artificial potential function to solve the problem of bounded perception range between agents. Compared with conventional sliding mode control, this control law has a faster convergence speed, a limited convergence time, and higher robustness.

[0126] 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 principle of the present invention. These improvements and modifications should also be regarded as the scope of protection of the present invention.

Claims

1. A multi-agent sliding mode fault-tolerant control method under a finite time observer, characterized in that: The specific steps include: Step 1): Determine a dynamic model of a multi-agent system, wherein the multi-agent system includes a leader and n followers; Step 1.1): Determine the leader’s dynamic model as shown in (1): in, Respectively represent the position and velocity states of the leader; is the control input of the leader; m is the spatial dimension; t is the time variable; Step 1.2): Determine the dynamic model of followers i=1,2,…,n with unknown disturbances and actuator failures as shown in (2): in, Respectively represent the position and velocity states of the i-th follower; is the control input of the ith follower; f i (x i (t),v i (t),t) represents the inherent nonlinear dynamic function of the ith follower; w i (t) represents the sum of the unknown disturbance and actuator fault of the ith follower, which is defined as the lumped fault; Step 1.3): For the inherent nonlinear dynamic function f i (x i (t),v i (t),t) and the lumped fault w i (t) Make assumptions: in, and are all non-negative constants; ||·|| represents the bi-norm; Step 2): Determine the communication topology of the multi-agent system; In a multi-agent system, the leader is marked as 0 and the followers are marked as i=1,2,…n. The communication topology between followers is represented by a topological graph G=(V,E,A), where V={1,2,…,n} represents a node set. represents the edge set, A=[a ij ] n×n represents the adjacency matrix, a ij Used to determine whether there is an edge between node i and node j; if (i, j) ∈ E, then there is a direct path between node i and node j, a ij =1; otherwise, a ij =0; define the degree of node i as The degree matrix of the topological graph G is expressed as D = diag{d1, d2, ..., d i ,…d n }; Laplace matrix is ​​expressed as L = [l ij ] n×n =DA,l ij is the i-th row and j-th column of the Laplace matrix, where when i=j, l ij =d i ; When i≠j, l ij =-a ij ; Define the adjacency matrix between the leader and the follower as B = diag{b1,b2,…,b i ,…b n }, b i Used to determine whether follower i can receive information from the leader; if follower i can obtain information directly from the leader, then b i =1; otherwise, b i = 0; the neighborhood of follower i is N i ={j|(j,i)∈E} represents; Step 3): Construct a finite-time observer; Step 3.1): First, design a finite-time observer for each follower as shown in (4): in, represents the aggregate fault estimate of the ith follower; h i (t) is an auxiliary variable, a, c i is a normal number, satisfying The parameter α satisfies 0<α<1; Step 3.2): Define the estimated error of the lumped fault As shown in (5): Step 3.3): Construct two auxiliary variables h i (t) and q i (t), in q i (t) with the help of the auxiliary variable h i (t) Designed as the estimated error of lumped fault Points: Adjust a, c i , the value of α makes the estimation error The multi-agent system is eventually uniformly bounded and finite-time stable, that is: Where T1 is the finite convergence time, V1(0) is the value of the selected Lyapunov function at t = 0; adjust a, c i , the value of α will estimate the error Adjust to a very small value of 0; Step 4): Design a consistency control method, and verify that the multi-agent system achieves finite-time consistency under the action of the designed consistency control method according to Lyapunov stability theory; Step 4.1): Define the consistency tracking position error variable e xi (t) and the consistent tracking speed error variable e vi (t) is of the following form: Defining Position Error Speed ​​error Defining a vector collection Vector Collection Vector Collection Vector Collection in, is the position error between the follower and the leader Vector collection of is the velocity error between the follower and the leader Vector collection of e x is the consistent tracking position error variable e xi (t) is a vector set, e v is the consistency tracking speed error variable e vi (t), then (8) can be rewritten into vector form as shown in (9): Among them, I m represents the m-dimensional identity matrix, L is the Laplace matrix, and B is the adjacency matrix between the leader and the follower; Step 4.2): Design sliding surface function s i (t) As shown in (10): Among them, k1, k2, k3 are positive constants, and the function The parameter ξ satisfies 0<ξ<0.2; Step 4.3): Design the artificial potential function J i (t) As shown in (11): in, For any All have||x i (t)-x j (t)||≤R holds, N i is the neighborhood of follower i, R is the upper bound of the distance that can transmit information between the i-th follower and the j-th follower; The artificial potential function J i (t) is combined with the sliding surface to obtain the sliding surface function s i (t) As shown in (12): Step 4.3): Design a consistency control method for the multi-agent system as shown in (13): where k4, ι, ρ and k5 are positive constants, is an adaptive parameter; according to Lyapunov stability theory, it is verified that the multi-agent system achieves finite-time consistency under the action of the designed consistency control method.

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