A Fault-Tolerant Control Method for Multi-Agent Systems in Non-Cooperative Games

Abstract

A fault-tolerant control method for multi-agent systems in non-cooperative games. The present invention relates to a fault-tolerant control method for multi-agent systems in non-cooperative games. The purpose of the present invention is to solve the defects that the existing Nash equilibrium game control in non-cooperative games is not universal, and when any agent fails, the strategies of the remaining agents will be affected by the failure, resulting in the strategies converging to the wrong Nash equilibrium. The process is as follows: Step 1: Establish an overall state space model of the multi-agent system; Step 2: Establish a multi-agent communication topology model to obtain an adjacency matrix and a Laplacian matrix; Step 3: Set the payoff function of each agent; Step 4: Design a distributed controller for each agent; Step 5: Design an observer for each agent; Step 6: Design a fault-tolerant compensation controller for each agent. The present invention is used in the field of fault diagnosis and fault-tolerant control.

Description

Technical Field

[0001] The present invention relates to a fault-tolerant control method for multi-agent systems in non-cooperative games, belonging to the fields of fault diagnosis and fault-tolerant control. Background Art

[0002] In recent years, the field of multi-agent system games has developed rapidly, including formation control and rendezvous, coverage control, connectivity control, and network congestion control, etc., which have wide applications in practical systems. Multi-agent systems complete tasks through the mutual communication and interaction among agents, including many sensors, actuators, and communication networks. When performing tasks in a complex working environment, multi-agent systems will inevitably encounter faults, such as sensor faults, actuator faults, system faults, etc. Therefore, fault diagnosis and fault-tolerant control technologies are beneficial to ensuring that multi-agent systems complete tasks safely and effectively. Designing advanced fault-tolerant control methods to ensure the correct operation of multi-agent systems is very important.

[0003] The existing non-cooperative game Nash equilibrium game control only targets first-order, second-order, or higher-order integral-type agents, which is not universal in practical applications. And when any one agent actuator fails, the strategies of the remaining agents will be affected by the fault. In this case, the control strategy will cause the state to converge to the wrong Nash equilibrium. Therefore, when an agent fails, a new distributed control scheme needs to be designed to solve the distributed Nash equilibrium search problem of multi-agent systems in non-cooperative games. Summary of the Invention

[0004] The purpose of the present invention is to solve the defect that the existing non-cooperative game Nash equilibrium game control is not universal, and when any one agent fails, the strategies of the remaining agents will be affected by the fault, resulting in the strategy converging to the wrong Nash equilibrium, and to propose a fault-tolerant control method for multi-agent systems in non-cooperative games.

[0005] The specific process of a fault-tolerant control method for multi-agent systems in non-cooperative games is as follows:

[0006] Step 1: Establish an overall state space model of the multi-agent system;

[0007] Step 2: Establish a multi-agent communication topology model to obtain an adjacency matrix and a Laplacian matrix;

[0008] Step 3: Set the payoff function of each agent;

[0009] Step 4: Design a distributed controller for each agent;

[0010] Step 5: Design an observer for each agent;

[0011] Step 6: Design the fault-tolerant compensation controller for each agent.

[0012] The beneficial effects of the present invention are as follows:

[0013] The present invention proposes a distributed Nash equilibrium search strategy based on average consensus in a non-cooperative game framework and a fault-tolerant control method based on residual feedback. The present invention extends the distributed Nash equilibrium search problem of multi-agent systems to a more general multi-agent form and solves the multi-agent fault-tolerant control problem in a non-cooperative game framework.

[0014] In the existing non-cooperative games, the distributed Nash equilibrium search strategy only targets multi-agent systems of first-order, second-order, and higher-order integral types without state terms, which is not universal in practical applications. The present invention considers the distributed Nash equilibrium search strategy for more general forms of agents. Additionally, when any one agent fails, the control strategies of the remaining agents will be affected by the failure. In this case, the search strategy will cause the strategy to converge to the wrong Nash equilibrium. The present invention designs a fault-tolerant controller based on residual feedback, such that when an actuator failure occurs, the strategies of each agent can converge to the correct Nash equilibrium point, achieving fault-tolerant control.

[0015] Compared with the first-order, second-order, or higher-order integral agent models, the state space model considered by the present invention is more universal. The present invention can achieve the effect that when an actuator fails, it still converges to the Nash equilibrium point without failure through a residual-based fault-tolerant controller. Description of the Drawings

[0016] Figure 1 is the flowchart of the method of the present invention;

[0017] Figure 2 is the communication topology diagram between agents;

[0018] Figure 3 is an example diagram of the output trajectories of each agent without fault-tolerant control, y i1 is a 1D matrix of the output, y i2 is a 2D matrix of the output;

[0019] Figure 4 is an example diagram of the output trajectories of each agent with fault-tolerant control. Detailed Embodiments

[0020] Detailed Embodiment 1: The specific process of a fault-tolerant control method for a multi-agent system in a non-cooperative game in this embodiment is as follows:

[0021] The present invention aims to solve the problem of distributed Nash equilibrium search in a multi-agent system when actuator failures occur. The present invention estimates the strategies of non-neighbor agents through a distributed average consensus algorithm, and then updates the strategies of local agents. When an agent experiences an actuator failure, a fault-tolerant control method based on residual feedback is designed so that the strategies of all multi-agents can still converge to the Nash equilibrium point when actuator failures occur. The flowchart of the specific implementation is as Figure 1 shown.

[0022] Step 1: Establish an overall state space model for the multi-agent system;

[0023] Step 2: Establish a communication topology model for the multi-agent system to obtain an adjacency matrix and a Laplacian matrix;

[0024] Step 3: Set the payoff function of each agent based on the actual situation;

[0025] Step 4: Design a distributed controller for each agent;

[0026] Step 5: Design an observer for each agent;

[0027] Step 6: Design a fault-tolerant compensation controller for each agent.

[0028] Specific implementation method 2: The difference between this implementation method and the first specific implementation method is that in step 1, an overall state space model for the multi-agent system is established; the specific process is as follows:

[0029] Assume that N agents form a multi-agent system The state space model of agent i is expressed as follows:

[0030]

[0031] Among them, A i , B i , C i are the system parameter matrices of the i-th agent, i = 1, 2,..., N, x i ∈R n , u i ∈R l and y i ∈R m respectively represent the system state, control input, and output (i.e., the strategy of the agent) of the i-th agent; is the first derivative of x i , is the set of all agents, R represents real numbers, m, n, l are positive integers, R n is an n-dimensional real number, R l is an l-dimensional real number, and R m is an m-dimensional real number.

[0032] The agent is a mobile robot, a drone, etc.

[0033] Other steps and parameters are the same as those in the first specific implementation manner.

[0034] Specific implementation manner three: The difference between this implementation manner and the first or second specific implementation manner is that in step two, a multi-agent communication topology model is established to obtain an adjacency matrix and a Laplacian matrix; the specific process is as follows:

[0035] Establish a communication topology model according to the communication relationship between agents, and define the communication topology connection graph of the multi-agent system as

[0036] where is the edge set, and ε is the point set;

[0037] If (i,j) ∈ ε, it means that agent i and agent j are adjacent, and then communication and information interaction can be carried out;

[0038] If it means that agent i and agent j are not adjacent, and then communication cannot be carried out and information interaction is not allowed;

[0039] Obtain the adjacency matrix from the communication topology connection graph of the multi-agent system

[0040]

[0041] where, if (i,j) ∈ ε, then define a ij = 1, otherwise 0; a ij is an element in the adjacency matrix , i = 1, 2,..., N, j = 1, 2,..., N;

[0042] Obtain the Laplacian matrix from the adjacency matrix

[0043] where is the degree matrix, is the diagonal element in the degree matrix, diag() is the operation of putting the elements on the diagonal of the matrix, and R N×N is an N×N-dimensional real number.

[0044] Other steps and parameters are the same as those in the first or second specific implementation manner.

[0045] Specific implementation manner four: The difference between this implementation manner and one of the first to third specific implementation manners is that in step three, the payoff function of each agent is set based on the actual situation (according to the task); the specific process is as follows:

[0046] Each agent in the multi-agent system is a player in the non-cooperative game. According to the output (action) of the agent y = [y1, y2, …, y N T ∈R N×m , there will be a corresponding payoff function: J(y) = (J1(y),..., J N (y)) (the payoff function is determined according to the tasks that the agents need to complete. To complete a certain task, the agents need to rely on their outputs). When each player no longer changes its output (action) in order to obtain a greater payoff, the Nash equilibrium point is reached That is, it satisfies the following conditions:

[0047]

[0048] where, T is for transpose, R N×m is an N×m-dimensional real number, J(y) is the payoff function of the multi-agent system, J1(y) is the payoff function of the first agent, J N (y) is the payoff function of the Nth agent, y * is the output of the agent corresponding to the maximum payoff of the agent system (reaching the Nash equilibrium), is the output of agent i corresponding to the maximum payoff of the output (action) y i of agent i (reaching the Nash equilibrium), is the output when the payoffs of the other agents excluding agent i reach the maximum (reaching the Nash equilibrium), is the payoff of agent i when the payoffs of the other agents excluding agent i reach the maximum (reaching the Nash equilibrium), is the payoff of the ith agent when reaching the Nash equilibrium;

[0049] To reach the Nash equilibrium, each agent hopes to maximize its own payoff function J i (y i , y -i ) by changing its own output, that is:

[0050]

[0051] where, y -i = [y1, y2, …, y i-1 , y i+1 , …, y N T ∈R (N-1)×m is the vector excluding the output of the local agent.

[0052] Other steps and parameters are the same as those in any one of the specific embodiments one to three. ​​

[0053] Embodiment 5: The difference between this embodiment and any one of Embodiments 1 to 4 is that in step 4, the output of the distributed controller for each agent is set to (u c,i ); the specific process is as follows:

[0054] To achieve Nash equilibrium, a gradient-based control strategy is designed for agent i as follows:

[0055] u c,i = K c,i x i + K d,i (ω i - y i ) (4)

[0056] where K c,i and K d,i are controller parameters to be designed; ω i is an auxiliary variable, and u c,i is the controller output;

[0057] where:

[0058]

[0059] where β i is a small positive parameter to be designed, is the first derivative of ω i , is the gradient of the i-th profit function, and J i (z i ) is the profit of agent i. z i is the estimated value of the output of agent i to non-neighbor agents. Since an agent cannot obtain the output of non-adjacent agents, and an agent needs to know the output of other agents to change its actions to maximize the profit function.

[0060] Other steps and parameters are the same as any one of Embodiments 1 to 4.

[0061] Embodiment 6: The difference between this embodiment and any one of Embodiments 1 to 5 is that the solving process is as follows:

[0062]

[0063] is the gradient of the profit function with respect to the local output;

[0064] where y i is the output of agent i, and J i is the profit of agent i, and Ji (y) is the revenue of agent i, and y is the output of the agent system;

[0065] The estimator z of the output of non-neighbor agents by agent i i The solution process is as follows:

[0066] Here, the distributed leader-follower average consensus estimation algorithm is used to estimate the output of non-adjacent agents, and the expression is:

[0067]

[0068] In the formula, θ ij is a parameter to be designed, a ij is an element of the adjacency matrix in it, a ik is an element of the adjacency matrix in it, z ij is the estimation of the output of agent j by agent i, i≠j; z kj is the estimation of the output of agent j by agent k; ω j is an auxiliary variable, z i =[z i1 ,z i2 ,…,z iN T ∈R N is the estimation of the output of other agents by agent i.

[0069] Other steps and parameters are the same as those in any one of the specific embodiments one to five.

[0070] Specific embodiment seven: The difference between this embodiment and any one of the specific embodiments one to six is that the controller parameters K c,i and K d,i The solution process is as follows:

[0071] Combining the control strategy of the i-th agent with the state space model of the agent in step one, the closed-loop dynamic equation of agent i can be obtained:

[0072]

[0073] Define the augmented matrix and Furthermore, the closed-loop dynamic equation of the agent system can be obtained as:

[0074]

[0075] Among them, x is the system state of the agent, is the first derivative of x, ω is the auxiliary variable, and y is the output (action) of the agent system,​ is the augmented matrix, is the intermediate variable, is the first derivative of ω, β = diag{β1, β1, …, β N} are the parameters to be designed, Θ is the intermediate variable, Θ = diag{θ ij}; J i (z i ) is the payoff of agent i, z is the estimate of the output of the agent to other agents, is the first derivative of z, 1 N is a column vector with all N - dimensional elements being 1, is the Kronecker product symbol, vec is the vector operation, I N×N is the N×N identity matrix;

[0076] The matrices and variables in Equation (8) are as follows:

[0077] ω = [ω1, ω2,..., ω N T

[0078]

[0079]

[0080]

[0081] where diag is the operation of putting elements on the diagonal of the matrix;

[0082] To make Equation (8) converge, Equation (8) needs to satisfy the following conditions:

[0083] Design the parameters so that ψ is a stable matrix, ensuring that the strategy of each agent can reach the Nash equilibrium point;

[0084]

[0085] where m, l1 are positive constants, Φ is a positive definite matrix, β is the intermediate variable, β = diag{β1, β2,..., β N}.

[0086] When Equation (8) converges, the controller parameters K c,i and K d,i are obtained.

[0087] Other steps and parameters are the same as those in any one of the specific embodiments one to six.

[0088] ​Embodiment 8: The difference between this embodiment and any one of Embodiments 1 to 7 is that in step 5, an observer for each agent is designed (formula (10)); the specific process is as follows:

[0089] When an actuator fault occurs in agent i, the state space model of agent i is described as:

[0090]

[0091] where f i is the actuator fault of agent i;

[0092] To implement residual-based output feedback fault-tolerant compensation control, an observer is designed for the i-th agent as follows:

[0093]

[0094] where r i is the residual signal, L i is the designed observer gain matrix, is the state estimation signal of the i-th agent, is the output estimation signal of the i-th agent.

[0095] Other steps and parameters are the same as any one of Embodiments 1 to 7.

[0096] Embodiment 9: The difference between this embodiment and any one of Embodiments 1 to 8 is that in step 6, a fault-tolerant compensation controller for each agent is designed (Equation (15)); the specific process is as follows:

[0097] When a fault occurs, the controller should implement fault compensation to reduce the impact of the fault on the entire system and make all multi-agent policies converge to the Nash equilibrium;

[0098] A fault-tolerant controller based on residual feedback is designed as:

[0099] u i = u c,i - u Q,i (11)

[0100] where u Q,i is the output of the fault-tolerant controller based on residual output feedback;

[0101] The state space model of the fault-tolerant controller Q i (S) is:

[0102]

[0103] where A Q,i , B Q,i , CQ,i , D Q,i is the parameter matrix of the fault-tolerant controller to be designed, and x Q,i is the state variable of the fault-tolerant controller, for x Q,i is the first derivative;

[0104] The fault-tolerant controller Q i (S) To achieve the effect of compensating for faults, that is, to minimize the difference between the output of the fault-tolerant controller and the fault:

[0105]

[0106] The other steps and parameters are the same as those in any one of the first to eighth specific embodiments.

[0107] Specific embodiment ten: The difference between this embodiment and any one of the first to ninth specific embodiments is that the observer parameter L i and the fault-tolerant controller parameters A Q,i , B Q,i , C Q,i , D Q,i are solved as follows:

[0108] From the above equations (9)-(11), the observed state error system is obtained as:

[0109]

[0110] The fault compensation error is as follows:

[0111] u Q,i - f i = C Q,i x Q,i + D Q,i r i - f i (14)

[0112] To make equations (13) and (14) converge, equations (13) and (14) need to satisfy the following conditions:

[0113] The design of the fault-tolerant controller and observer parameters needs to satisfy is a stable matrix;

[0114] is an intermediate variable,

[0115] is the element in the matrix ;

[0116]

[0117]

[0118]

[0119]

[0120]

[0121] where η 1,i , η 2,i is a positive number;

[0122] When equations (13) and (14) converge, the observer parameter L i and the fault-tolerant controller parameters A Q,i , B Q,i , C Q,i , D Q,i are obtained.

[0123] Other steps and parameters are the same as those in any one of the specific embodiments one to nine.

[0124] The following embodiments are used to verify the beneficial effects of the present invention:

[0125] The algorithm proposed by the invention is applied to a system composed of three agents, as Figure 2 shown, considering the connection control problem of three mobile sensor networks. The objective function of agent i is as follows:

[0126]

[0127] where p1 = [2, -2] T , p2 = [-2, -2] T , p3 = [-4, 2] T q1 = 3, q2 = 3, q3 = 6, m 12 = m 21 = m 23 = m 32 = 1, m 13 = m 31 = 0. According to the above profit function, the Nash equilibrium point before the actuator fault occurs can be obtained as:

[0128] y * = [-0.125, 0.75, 0.75, 0.5, 1.375, -0.25] T .

[0129] Assume that the model of each agent is:

[0130]

[0131]

[0132] where x i = [x i1 , x i2 T ∈R 2 is the position signal of the position sensor of agent i, v i = [v i1 , v i2 T ∈R 2 is the velocity signal of the velocity sensor of agent i, u i = [u i1 , u i2 T ∈R 2 as the control input. The parameter matrices of the system are as follows:

[0133]

[0134] In the simulation, take β i = 0.03, θ ij = 1, and the observer gain matrix is:

[0135]

[0136] The controller parameters in formula (4) are:

[0137]

[0138] The parameters of the compensation fault-tolerant controller in formula (12) are set as:

[0139]

[0140] The initial output coordinates of the agents are given as y0 = [3 0 1 1 2 1] T , assuming that the actuator fault occurs in the second agent at the 100th second, Figure 3 Compared with Figure 4 it can be seen that the outputs of all agents can converge to the Nash equilibrium point under the condition of having a fault-tolerant compensation controller.

[0141] The present invention may also have many other embodiments. Without departing from the spirit and essence of the present invention, those skilled in the art can make various corresponding changes and deformations according to the present invention, but these corresponding changes and deformations should all fall within the protection scope of the appended claims of the present invention.​​​

Claims

1. A fault-tolerant control method for multi-agent systems in non-cooperative games, characterized in that: The specific process of the method is as follows: Step 1: Establish an overall state space model of the multi-agent system; Step 2: Establish a multi-agent communication topology model to obtain the adjacency matrix and Laplacian matrix; Step 3: Set the payoff function for each agent; Step 4: Design a distributed controller for each agent; Step 5: Design an observer for each agent; Step 6: Design a fault-tolerant compensation controller for each agent; In Step 1, the overall state space model of the multi-agent system is established; the specific process is as follows: Suppose that N agents form a multi-agent system The state space model of agent i is represented as follows: Among them, A i , B i , C i is the system parameter matrix of the i-th agent, i = 1, 2, …, N, x i ∈R n , u i ∈R l and y i ∈R m respectively represent the system state, control input, and output of the i-th agent; is the first derivative of x i , is the set of all agents, R represents the real numbers, m, n, l are positive integers, R n is the n-dimensional real numbers, R l is the l-dimensional real numbers, R m is the m-dimensional real numbers; In Step 2, the multi-agent communication topology model is established to obtain the adjacency matrix and Laplacian matrix; the specific process is as follows: Establish a communication topology model according to the communication relationship between agents, and define the communication topology connection graph of the multi-agent system as Among them, is the edge set, is the vertex set; If it indicates that agent i and agent j are adjacent, and communication and information interaction can be carried out; If it indicates that agent i and agent j are not adjacent, and thus cannot communicate or interact with each other; The adjacency matrix is obtained from the communication topology connection graph of the multi-agent system Among them, if there is then define a ij = 1, otherwise 0; a ij is an element in the adjacency matrix where i = 1, 2, …, N, j = 1, 2, …, N; Obtaining the Laplacian matrix from the adjacency matrix Among them, is the degree matrix, is the diagonal element in the degree matrix, diag() is the operation of placing elements on the diagonal of the matrix, and R N×N is a real number of N×N dimension; In Step 3, the payoff function for each agent is set; the specific process is as follows: Each agent in the multi-agent system is a player in the non-cooperative game. According to the output of the agent \(y = [y_1, y_2, \ldots, y N T \in\mathbb{R} N×m , there will be a corresponding payoff function: \(J(y)=(J_1(y),\ldots,J N (y))\);​ To achieve Nash equilibrium, each agent hopes to maximize its own profit function J by changing its output i (y i ,y -i ) That is: where y -i = [y1, y2,..., y i-1 , y i+1 ,..., y N T ∈ R (N-1)×m is the vector excluding the output of the local agent;​ In Step 4, the output of the distributed controller designed for each agent is; the specific process is as follows: To achieve Nash equilibrium, a gradient-based control strategy is designed for agent i as follows: u c,i = K c,i x i + K d,i (ω i - y i ) (4) where K c,i and K d,i are controller parameters; ω i is an auxiliary variable, and u c,i is the controller output. Where: where β i is a positive parameter, is the first derivative of ω i with respect to, is the gradient of the i-th payoff function J i (z i ) is the payoff of agent i, and z i is the estimate of the output of agent i for non-neighbor agents; The solution process is as follows: where y i is the output of agent i, and J i is the payoff of agent i, and J i (y) is the payoff of agent i, where y is the output of the agent system; The estimator z output by the agent i to non-neighbor agents i is solved as follows: The average consensus estimation algorithm based on distributed leader-follower is used to estimate the output of non-adjacent agents, and the expression is: where θ ij is a parameter, a ij is an element in the adjacency matrix a ik is an element in the adjacency matrix z ij is the estimate output by agent i to agent j, i≠j; z kj is the estimate output by agent k to agent j; ω j is an auxiliary variable, z i =[z i1 ,z i2 ,...,z iN T ∈R N is the estimate output by agent i to other agents.​ 2. The fault-tolerant control method for a multi-agent system in a non-cooperative game according to claim 1, characterized in that: The controller parameters K c,i and K d,i are solved as follows: Combining the control strategy of the i-th agent with the state space model of the agent in Step 1, the closed-loop dynamic equation of agent i can be obtained: Define the augmented matrix and Furthermore, the closed-loop dynamic equation of the agent system can be obtained as follows: where x is the system state of the agent, is the first derivative of x, ω is the auxiliary variable, and y is the output of the agent system, is the augmented matrix, is the intermediate variable, is the first derivative of ω, β = diag{β1, β1, …, β N} is the parameter, Θ is the intermediate variable, Θ = diag{θ ij}; J i (z i ) is the payoff of agent i, z is the estimate of the output of other agents by the agent, is the first derivative of z, 1 N is the column vector with all N - dimensional elements being 1, is the Kronecker product symbol, vec is the vector operation, I N×N is the N×N identity matrix; The matrices and variables in Equation (8) are as follows: Where, diag is the operation of placing elements on the diagonal of the matrix; To make Equation (8) converge, Equation (8) needs to satisfy the following conditions: Design parameters such that ψ is a stable matrix to ensure that the strategy of each agent can reach the Nash equilibrium point; where m and l1 are positive constants, Φ is a positive definite matrix, β is an intermediate variable, β = diag{β1, β2,..., β N}; The controller parameters K are obtained when Equation (8) converges c,i and K d,i .

3. A fault-tolerant control method for a multi-agent system in a non-cooperative game according to claim 2, characterized in that: In Step 5, an observer for each agent is designed; the specific process is as follows: When a fault occurs in the actuator of agent i, the state space model of agent i is described as: where, f i is the actuator fault of agent i; An observer is designed for the i-th agent as follows: where, r i is the residual signal, L i is the designed observer gain matrix, is the state estimation signal of the i-th agent, is the output estimation signal of the i-th agent.

4. A fault-tolerant control method for multi-agent systems in non-cooperative games according to claim 3, characterized in that: In Step 6, a fault-tolerant compensation controller for each agent is designed; the specific process is as follows: A fault-tolerant controller based on residual feedback is designed as: u i = u c,i - u Q,i (11) where, u Q,i is the output of the residual output feedback fault-tolerant controller; Fault-tolerant controller Q i (S)'s state-space model is as follows: Among them, A Q,i , B Q,i , C Q,i , D Q,i are the parameter matrices of the fault-tolerant controller, and x Q,i is the state variable of the fault-tolerant controller, is the first derivative of x Q,i ; Fault-tolerant controller Q i (S) To achieve the effect of compensating for faults, that is, to minimize the difference between the output of the fault-tolerant controller and the fault:

5. A fault-tolerant control method for a multi-agent system in a non-cooperative game according to claim 4, characterized in that: The observer parameter L i and the fault-tolerant controller parameters A Q,i , B Q,i , C Q,i , D Q,i The solution process is as follows: From the above equations (9)-(11), the observed state error system is obtained: The fault compensation error is as follows: u Q,i -f i = C Q,i x Q,i + D Q,i r i -f i (14) To make Equations (13) and (14) converge, Equations (13) and (14) need to satisfy the following conditions: The design of the fault-tolerant controller and observer parameters needs to satisfy is a stable matrix; is an intermediate variable, is a matrix in the element; where η 1,i , η 2,i is a positive number; When the equations (13) and (14) converge, the observer parameter L is obtained. i and the fault-tolerant controller parameters A Q,i , B Q,i , C Q,i , D Q,i .

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