Robust fault-tolerant control method for specified performance output feedback

Abstract

The invention discloses a specified performance output feedback robust fault-tolerant control method, and the method comprises the steps: firstly building a preliminary model of a control system according to the structure of the control system; secondly, giving out a partial failure coefficient of the actuator, and establishing an actuator fault model; and then considering transmission delay and an actuator fault model, designing a fault-tolerant controller, and introducing specified performance indexes into fault-tolerant control. And finally, establishing a closed-loop system with time lag and faults based on the preliminary model of the fault-tolerant controller and the control system with the introduced specified performance indexes, and completing fault-tolerant control. According to the method, the fault stability is ensured, the performance standard is met, and the controller can obtain good dynamic performance while having certain fault-tolerant capability.

Description

Technical Field

[0001] The present invention belongs to the technical field of fault-tolerant control, and relates to a prescribed performance output feedback robust fault-tolerant control method. Background Art

[0002] Robust control has received extensive attention in recent years. Its main goal is to ensure stability and performance (mainly focused on disturbance attenuation and state steady-state stability). Robust control is often used in the design of controllers, which can maintain stability and performance even when there are uncertainties in system modeling, external disturbances, and nonlinear systems. By implementing robust control methods, the ability of the system to resist various disturbances is significantly enhanced.

[0003] In the process of controller design, ensuring system robustness mainly involves identifying the bounds of uncertainties. To ensure stability under extreme conditions, a stronger control action is required compared to normal situations, which increases the conservatism of the controller and often impairs the dynamic performance. However, the prescribed performance control (PPC) that optimizes the dynamic performance provides a new perspective for this challenge.

[0004] Prescribed performance means ensuring that the tracking error converges to an arbitrarily small predefined region, while also ensuring that the convergence rate and overshoot satisfy the prescribed conditions. It requires satisfying both transient and steady-state performance, and its main goal is to improve system performance. In the application of prescribed performance, the most commonly used method is backstepping control, which is essentially a recursive design method. PPC integrates different innovative concepts. Some scholars combine prescribed performance with fuzzy control to stabilize a class of non-triangular structure nonlinear systems, and some scholars design an inverse neural network controller for stochastic nonlinear systems to achieve prescribed performance control.

[0005] In system performance analysis, the selection of the performance function is crucial. Common prescribed performance functions usually consider the exponential aspect. Generally, the determination of the performance function parameters is based on the control objective and the expected range of system response, mainly considering the overshoot and convergence rate under normal operating conditions. However, the dynamic response during system faults is often difficult to predict and is usually excluded from the design considerations. Therefore, when a fault occurs, it is difficult for the prescribed performance controller to maintain the optimal control performance. In this context, robust control provides a method to solve these problems.

[0006] Traditional robust control has the defect of ignoring the transient characteristics of the closed-loop system. On this basis, it is proposed to incorporate the prescribed performance index into the design of the robust fault-tolerant controller. This method combines the advantages of robust control and PPC, making the advantages and limitations of both complementary, so that the controller can obtain good dynamic performance while having a certain fault-tolerant ability. Summary of the Invention

[0007] The present invention provides a H with specified performance for the fault system with time delay. ∞ Output feedback control method. Aiming at the limitation that traditional robust control ignores the transient performance of closed-loop systems, a new method combining passive fault-tolerant control based on robust control theory and dynamic performance constraints is proposed.

[0008] The present invention provides a H having specified properties ∞ The fault-tolerant output feedback control method comprises the following steps:

[0009] Step 1: Establish a preliminary model of the control system based on the control system structure in general;

[0010] The preliminary model of the control system is expressed as:

[0011]

[0012] where x(t), u F (t), ω(t), and y(t) represent the selected state, actuator output signal, external disturbance, and system output with appropriate dimensions, respectively, while A, B, B ω and C is a parameter matrix with appropriate dimensions.

[0013] Step 2: Give the partial failure coefficient of the actuator and establish the actuator failure model;

[0014] ρ(t) is the actuator partial failure coefficient, which satisfies:

[0015] 0<ρ min ≤ρ(t)≤ρ max ≤1 (2)

[0016] where ρ max and ρ min denote the upper and lower bounds of ρ(t), respectively. In fact, partial actuator failure can be represented as a partial loss of actuator control effectiveness. Based on the above description, the model of partial actuator failure can be expressed as:

[0017] ρ(t)=τ(t)ρ 1 (t)+(1-τ(t))ρ 2 (t) (3)

[0018] in

[0019]

[0020] is a known scalar. represents the expectation of A, assuming that τ(t) follows the Bernoulli distribution, τ′ is a constant belonging to (0,1).

[0021] Step 3: Considering the transmission delay and actuator fault model, design a fault-tolerant controller;

[0022] The transmission delay needs to be considered during the controller design process. The fault-tolerant output feedback controller can be designed as:

[0023] u F (t)=-ρ(t)Ky(t-η(t)) (4)

[0024] where K represents the gain matrix to be designed, η(t) represents the transmission delay, and it satisfies:

[0025]

[0026] where η and μ are both known constants. Substituting (3) into (4), the fault-tolerant controller can be redesigned as:

[0027]

[0028] Step 4: Introduce the prescribed performance index into the fault-tolerant control to enable the controller to obtain good dynamic performance while having a certain fault-tolerant ability;

[0029] Define the tracking error as follows:

[0030]

[0031] represents the desired performance index, is the actual measured value related to the dynamic performance. Introduce the prescribed performance requirement into the design process of the fault-tolerant control, and give the definition of the prescribed performance as follows:

[0032]

[0033] where ζ min >0, ζ max >0, α>0, θ 1 >θ 0 >0. θ(t) is a performance function with an increasing property, θ(0)=θ 1 , The degree of constraint of min , ζ max and θ(t). For the above inequality constraints, it is usually difficult to directly apply them to the robust control scheme. For the convenience of subsequent research, it is necessary to equivalently transform the above conditions into an unconstrained type. Therefore, redefine:

[0034]

[0035] Among them, Γ(r(t)) is a strictly increasing smooth function, and r(t) represents the transformation error. Γ(r(t)) can be written as:

[0036]

[0037] Through the inverse function transformation, the transformation error can be expressed as:

[0038]

[0039] Taking the derivative of (11), we can obtain:

[0040]

[0041] Combining (9) - (11), obviously when r(t) approaches , Γ(r(t)) approaches 0, which means approaches 0. To facilitate the controller design, an auxiliary variable is introduced:

[0042]

[0043] Step Five, based on the above steps, a closed-loop system with time delay and faults is established;

[0044] Combining the fault-tolerant controller (6) introducing the specified performance index with the preliminary model (1) of the control system, the closed-loop system with time delay and faults can be summarized as:

[0045]

[0046] The tracking error is incorporated into the state as follows:

[0047]

[0048] The beneficial effects of the present invention: While maintaining the stability of the system, the system (14) has an H ∞ interference attention level γ. A method of H ∞ fault-tolerant control with specified performance is proposed. The concept of specified performance is introduced into robust control, an auxiliary variable is introduced, and the Lyapunov function of the auxiliary variable is redefined. The relationship between the stability condition of the system and the performance index is derived by using the Taylor expansion. Several assumptions are proposed to handle the time-varying terms, and the auxiliary matrix is used to handle the coupling terms.

[0049] By using the auxiliary matrix variable substitution technique and the Taylor expansion method, sufficient conditions for designing a fault-tolerant controller that meets the specified performance constraints are derived. Fault stability and compliance with performance standards are ensured. Computational examples show that the proposed method can effectively control a single-area power system. The present invention incorporates the specified performance index into the design of a robust fault-tolerant controller. This method combines the advantages of robust control and PPC, making the advantages and limitations of both complementary, so that the controller can obtain good dynamic performance while having a certain fault-tolerant ability. Description of the Drawings

[0050] Figure 1 It is a structural diagram of a fault closed-loop control system with specified performance constraints;

[0051] Figure 2 It is a control structure diagram of a single-area power system;

[0052] Figure 3 It is a frequency deviation trajectory diagram of a single-area power system with specified performance;

[0053] Figure 4(a) is a frequency deviation trajectory diagram of a single-area power system with new specified performance under the action of the original controller;

[0054] Figure 4(b) is a frequency deviation trajectory diagram of a single-area power system with new specified performance under the action of the reconstructed controller. Detailed Implementation Manner

[0055] The following further describes the present invention in detail with reference to the Figure 1 illustrated examples.

[0056] Step (1) Establish a preliminary model of the control system;

[0057] To maintain generality, according to the structure of the control system under general circumstances, the preliminary model of the control system is expressed as:

[0058]

[0059] where x(t), u F (t), ω(t) and y(t) respectively represent the selected state, actuator output signal, external disturbance and system output with appropriate dimensions, while A, B, B ω and C are parameter matrices with appropriate dimensions.

[0060] Step (2) Establish an actuator fault model;

[0061] ρ(t) is the actuator partial failure coefficient, which satisfies:

[0062] 0 < ρ min ≤ ρ(t) ≤ ρ max≤1 (2)

[0063] where ρ max and ρ min represent the upper and lower bounds of ρ(t), respectively. In fact, the partial actuator fault can be represented as a partial loss of actuator control effectiveness. Based on the above description, the model of the partial actuator fault can be expressed as:

[0064] ρ(t) = τ(t)ρ 1 (t) + (1 - τ(t))ρ 2 (t) (3)

[0065] where

[0066]

[0067] is a known scalar. denotes the expectation of A. Assuming that τ(t) follows a Bernoulli distribution, τ′ is a constant belonging to (0, 1).

[0068] Design of the fault-tolerant controller in step (3);

[0069] The transmission delay needs to be considered during the controller design process. The fault-tolerant output feedback controller can be designed as:

[0070] u F (t) = -ρ(t)Ky(t - η(t)) (4)

[0071] where K represents the gain matrix to be designed, η(t) represents the transmission delay, and it satisfies:

[0072]

[0073] where η and μ are both known constants. Substituting (3) into (4), the fault-tolerant controller can be redesigned as:

[0074]

[0075] Step (4) Introduce the prescribed performance index into the fault-tolerant control;

[0076] First, define the tracking error as follows:

[0077]

[0078] represents the desired performance index, is the actual measurement value related to the dynamic performance. In order to introduce the specified performance requirements into the design process of fault-tolerant control, the definition of the specified performance is given as follows:

[0079]

[0080] where ζ min > 0, ζ max > 0, α > 0, θ 1 > θ 0 > 0. θ(t) is a performance function with an increasing characteristic, and θ(0) = θ 1 , The degree of constraint of min , ζ max and θ(t). For the above inequality constraints, it is usually difficult to directly apply them to the robust control scheme. For the convenience of subsequent research, it is necessary to equivalently transform the above conditions into an unconstrained type. Therefore, it is redefined as:

[0081]

[0082] where Γ(r(t)) is a strictly increasing smooth function, and r(t) represents the transformation error. Γ(r(t)) can be written as:

[0083]

[0084] Through the inverse function transformation, the transformation error can be expressed as:

[0085]

[0086] Taking the derivative of (11), we can obtain:

[0087]

[0088] Combining (9) - (11), obviously when r(t) approaches , Γ(r(t)) approaches 0, which means approaches 0. For the convenience of controller design, an auxiliary variable is introduced:

[0089]

[0090] Step (5) establishes a fault closed-loop control system with specified performance constraints;

[0091] Combining the designed fault-tolerant controller (6) with the preliminary model (1) of the control system, the closed-loop system with time delay and faults can be summarized as:

[0092]

[0093] Introduce the specified performance constraints into the fault-tolerant control, and the tracking error is incorporated into the state as follows:

[0094]

[0095] Let the dimensions of \(x(t)\), \(u_F(t)\), \(\omega(t)\), \(y(t)\), \(A\), \(B\), \(B\) ω , and \(C\) be \(n\times1\), \(j\times1\), \(s\times1\), \(m\times1\), \(n\times n\), \(n\times j(j\leq n)\), \(n\times s(s\leq n)\) and \(m\times n(m\leq n)\) respectively. The structure of the fault closed-loop control system with specified performance constraints is shown in Figure (1).

[0096] Step (6) gives the relevant symbol definitions;

[0097] \(He\) represents the sum of a matrix and its transpose. Given a square matrix \(X\), then \(He(X)=X + X^T\) T . \(diag\{...\}\) represents a diagonal matrix. For a symmetric matrix, \(*\) represents the symmetric terms. \(I\) is the identity matrix with appropriate dimensions. For a matrix \(M\), the symbol \(M>0\) indicates that the matrix \(M\) is symmetric positive definite, denotes the expectation of \(M\). For ease of representation, use \(\Gamma\) to represent \(\Gamma(r(t))\).

[0098] Step (7) gives Theorem 1 and constructs a suitable Lyapunov function according to the system model obtained in Step (5);

[0099] Theorem 1. For given positive constants \(\zeta\) min , \(\zeta_1\) max , \(c\), \(\tau'\), \(\mu\), \(\eta\) and the variables \(\rho_1(t)\), \(\rho_2(t)\) and a suitable controller gain matrix \(K\), the system (14) is mean-square stable and has an \(H\) 1 disturbance attenuation level \(\gamma\) if there exist positive definite matrices \(P>0\), \(Q>0\), \(W>0\), matrices \(E\) with appropriate dimensions and 2 and a positive scalar \(\gamma>0\) such that the following inequalities hold: ∞ where and positive scalar \(\gamma>0\) such that the following inequalities hold:

[0100]

[0101] where

[0102]

[0103]

[0104] Since it is necessary to incorporate the specified performance index into the controller design, the Lyapunov function is designed in two parts:

[0105] V(t) = V 1 (t) + V 2 (t) (17)

[0106] Where:

[0107]

[0108] V 1 (t) contains information related to specified performance, while V 2 (t) contains information about time delay and faults. Differentiate equation (17):

[0109]

[0110] First consider By combining it with equations (12) and (13) and using the rules of logarithmic operations, the following equation can be obtained:

[0111]

[0112] The specified performance constraint is:

[0113]

[0114] From the above constraints, the following conditions can be easily obtained:

[0115]

[0116] They strictly satisfy the conditions of the Taylor expansion of the typical logarithmic function ln(1 + x). Therefore, applying the Taylor expansion to gives:

[0117]

[0118] Where

[0119]

[0120] Since t approaches ∞ , if approaches 0, then can be regarded as an interference term According to it can be obtained:

[0121]

[0122] Combining (12), (21) and (22), the following result can be obtained:

[0123]

[0124] Reprocessing According to Lemma 1, it can be obtained that:

[0125]

[0126] By introducing the method of the freedom matrix and combining Any symmetric matrix E of appropriate dimension satisfies the following zero equality:

[0127]

[0128] Where:

[0129]

[0130] According to H ∞ Performance requirements, there are:

[0131]

[0132] Where:

[0133]

[0134] If Φ < 0 holds in Theorem 1, then holds, and further J < 0 holds. Considering the zero initial condition V(0) = 0, integrating both sides of (26) from 0 to t, when t approaches ∞ , the following inequality is obtained:

[0135]

[0136] Satisfies the H ∞ Performance requirements, and the proof is completed.

[0137] Step (8) is based on step (7), considering the existence of some time-varying terms I′Λ k I′ T , and Theorem 2 is given to handle this problem;

[0138] Theorem 2. For the given positive constants ζ min 、ζ max 、c、τ′、θ 0 、θ 1 、α、μ、 η and the variables ρ 1 (t), ρ 2 (t) and the appropriate controller gain matrix K, the system (14) is mean-square stable and has H ∞ interference attenuation level γ, if there exist positive definite matrices P > 0, Q > 0, W > 0, matrices E of appropriate dimension and positive scalar γ > 0, ε i> 0 (i = 1, 2) such that the following inequalities hold:

[0139]

[0140] Where:

[0141]

[0142]

[0143] Based on (10), there exists a sufficiently small scalar c to ensure the following condition holds:

[0144]

[0145] Based on this situation, an upper bound of Λ k and a lower bound of Λ and lower bound Λ k :

[0146]

[0147] It can be seen that the time-varying terms Λ 1 on the main diagonal, Λ 2 and Λ 5 can be scaled by their upper bounds. In contrast, for the off-diagonal terms Λ 3 and Λ 4 , their upper and lower bounds can be used for processing, and define:

[0148] Λ k = Λ k0 (1 + G k )(k = 3, 4) (31)

[0149] Where:

[0150]

[0151] Replace Λ 1 , Λ 2 and Λ 5 with their upper limits, and substitute (31) into (16). The matrix Φ can be divided into two parts, represents the constant part, ΔΦ represents the uncertain part, and

[0152]

[0153] Where:

[0154]

[0155] v k = [I′Λ k00 1×(4n+s+1) T ,β k =[0 1×n I′ T 0 1×(3n+s+1)

[0156] Combining Lemma 2 and Lemma 3, Theorem 2 is obtained.

[0157] Step (9) is based on step (8). Considering that ρ 1 (t) and ρ 2 (t) are unknown, in order to reduce the conservatism of the designed controller, Theorem 3 is given to solve this problem;

[0158] Theorem 3. For given positive constants ζ min 、ζ max 、c、τ′、θ 0 、θ 1 、α、μ、 η 、ρ 10 、ρ 20 and a suitable controller gain matrix K, the system (14) is mean-square stable and has an H ∞ disturbance attenuation level γ if there exist positive definite matrices P > 0, Q > 0, W > 0, matrices E with appropriate dimensions and positive scalars γ > 0, ε i > 0 (i = 1, 2), σ i > 0 (i = 1, 2) such that the following inequalities hold:

[0159]

[0160] where:

[0161]

[0162]

[0163] λ 1 = σ 1 (τ′ρ 10 ) 2 ,λ 2 = σ 2 [(1 - τ′)ρ 20 2 ,

[0164]

[0165] Definition:

[0166]

[0167] ​​​Therefore, the unknown variables ρ 1 (t) and ρ 2 (t) can be rewritten as:

[0168]

[0169] Substituting (34) into (28) and using Lemma 2 and Lemma 3, the corresponding result in (33) can be directly obtained, and the proof is completed.

[0170] Step (10) is based on step (9). Considering that there are still some coupling terms -ρ 0 EBKC, EBKC in (33), which makes it difficult to directly obtain the corresponding controller. Therefore, Theorem 4 is given to solve this problem;

[0171] Theorem 4. For given positive constants ζ min , ζ max , c, τ′, θ 0 , θ 1 , α, μ, η , ρ 10 , ρ 20 and S with a matrix of appropriate dimension, design the H ∞ fault-tolerant controller gain K = T -1 U. The system (15) is mean-square stable and has an H ∞ disturbance attenuation level γ. If there exist positive definite matrices P > 0, Q > 0, W > 0, T > 0, matrices E and U with appropriate dimensions and positive scalars γ > 0, ε i > 0, σ i > 0, such that the following inequalities hold:

[0172]

[0173] where:

[0174]

[0175] Ω 12 = [Π 1 Π 2 , Ω 22 = diag{-ε 1 , -ε 2 , -σ 1 I, -σ 2 I},

[0176]

[0177] By analyzing the structure of matrix (33), it is found that traditional decoupling methods, such as multiplying the left and right of column vectors or transforming into W or P problems, cannot solve the coupling terms. Therefore, a new decoupling method is proposed. Let:

[0178]

[0179] where S is an auxiliary matrix with appropriate dimensions. Therefore, the controller K is equal to T -1 U, and in (33), it has -ρ 0 The terms of EBKC and EBKC can be expressed as:

[0180]

[0181] where:

[0182]

[0183] By using Lemma 3, the following inequality holds:

[0184]

[0185] Combining Lemma 2 and Lemma 4, Theorem 4 is obtained, the proof is completed, and the controller gain K is designed as K = T -1 U.

[0186] Step (11) takes the load frequency control of a single-area power system as an example to illustrate that the designed fault-tolerant controller has the specified performance index.

[0187] The control structure of the single-area power system is as Figure 2 shown, and the single-area power system model can be described as:

[0188]

[0189] Δf, ΔP m , ΔP v and ΔP d respectively represent the frequency change, the change in turbine mechanical power output, the change in valve position, and the load change, while T g , T ch , M, D, and R these parameters respectively represent the governor time constant, the turbine time constant, the generator inertia, the generator damping coefficient, and the speed droop rate, and ACE is the input signal. The selected state x(t), system output y(t), and external disturbance ω(t) are as follows:

[0190] x(t) = [Δf ΔP m ΔP v ∫ACE] T ,

[0191] y(t) = [ACE ∫ACE] T , ω(t) = ΔP d . (40)

[0192] The considered power system can be described as:

[0193]

[0194] The control gain matrix is:

[0195] K = [K P K I (42)

[0196] K P and K I represent the proportional gain and integral gain respectively, and the performance constraint applies to the frequency deviation Δf. The parameters of the single-area power system are selected as M = 10, D = 1, R = 0.05, T ch = 0.3, T g = 0.1, β = 21, η = 0.1, η = 0.8, μ = 0.5, ρ min = 0.5, ρ max = 1, The parameters for specified performance are selected as ζ max = ζ min = 1, θ 1 = 0.15, α = 1, θ 1 = 0.005, c = 0.005 × ζ max . Under these parameter conditions, the variation of the frequency deviation over time in the single-area power system is as Figure 3 shown, and under the constraints of specified performance, the frequency deviation does not deviate from the expected range. Let the fault coefficient satisfy 0.5 ≤ ρ 1 (t) < 0.8 with a probability of Prob{ρ 1 (t) ∈ [0.5, 0.8)} = 0.3, 0.8 ≤ ρ 2 (t) < 1 with a probability of Prob{ρ 2 (t) ∈ [0.8, 1)} = 0.7. Taking as the disturbance signal, through Theorem 4, the fault-tolerant load frequency controller that satisfies the specified performance constraints is solved as:

[0197] K = [-0.0725 0.4093].

[0198] Then set the specified performance parameters as and At this time, as shown in Fig. 4(a), the original controller K cannot meet the new prescribed performance constraints. Substituting the new prescribed performance constraints into Theorem 4, the controller gain can be solved as follows:

[0199] K 1 = [-0.0298 0.1455].

[0200] As shown in Fig. 4(b), the reconstructed controller can meet the new prescribed performance constraints. In summary, the fault-tolerant controller proposed in the present invention can achieve a stable goal while meeting the prescribed performance constraints.

[0201] The present invention aims at a faulty system with time delay and provides an H ∞ output feedback control method with prescribed performance. Aiming at the limitation that traditional robust control ignores the transient performance of the closed-loop system, a new method combining passive fault tolerance control based on robust control theory with dynamic performance constraints is proposed. By using the auxiliary matrix variable substitution technique and the Taylor expansion method, sufficient conditions for designing a fault-tolerant controller that meets the given performance constraints are derived. The fault stability and compliance with the performance criteria are ensured. The calculation example shows that the method can effectively control a single-area power system.

Claims

1. A method for robust fault-tolerant control with output feedback of specified performance, characterized in that: The following steps are involved: Step 1: Establish a preliminary model of the control system based on the control system structure; Step 2: Give the partial failure coefficient of the actuator and establish the actuator failure model; Step 3: Consider the transmission delay and actuator failure model and design a fault-tolerant controller; Step 4: Introduce the specified performance indicators into fault-tolerant control; Step 5: Based on the preliminary model of the fault-tolerant controller and control system obtained in step 4, a closed-loop system with time delay and fault is established to complete the fault-tolerant control.

2. A method for robust fault-tolerant control with output feedback of specified performance according to claim 1, characterized in that: The preliminary model of the control system described in step 1 is expressed as: where x(t), u F (t), ω(t), and y(t) represent the selected state, actuator output signal, external disturbance, and system output, respectively, while A, B, B ω and C is the parameter matrix.

3. A method for robust fault-tolerant control with output feedback of specified performance according to claim 2, characterized in that: The specific implementation process of step 2 is as follows: ρ(t) is the partial failure coefficient of the actuator, satisfying: 0<ρ min ≤ρ(t)≤ρ max ≤1; where ρ max and ρ min denote the upper and lower bounds of ρ(t), respectively. Actuator partial failure is represented as a partial loss of actuator control effectiveness. Therefore, the model of actuator partial failure is expressed as: ρ(t)=τ(t)ρ1(t)+(1-τ(t))ρ2(t) is a known scalar; represents the expectation of A, assuming that τ(t) follows the Bernoulli distribution, τ′ is a constant belonging to (0,1).

4. The method for robust fault-tolerant control with output feedback of specified performance according to claim 3, characterized in that: The specific design process of the fault-tolerant controller is as follows: Considering the transmission delay in the controller design process, the fault-tolerant output feedback controller is designed as: u F (t)=-ρ(t)Ky(t-η(t)) Where K represents the gain matrix to be designed, η(t) represents the transmission delay, and satisfies: in η and μ are both known constants, and the fault-tolerant controller is redesigned as:

5. A method for robust fault-tolerant control with output feedback of specified performance according to claim 4, characterized in that: The specified performance indicators are as follows: Define tracking error: in represents the expected performance indicator, It is the actual measured value related to dynamic performance; the specified performance requirements are introduced into the design process of fault-tolerant control, and the definition of specified performance is given as follows: where ζ min >0,ζ max >0, α>0, θ1>θ0>0; θ(t) is a performance function with increasing characteristics, θ(0)=θ1, The degree of constraint depends on ζ min , max and θ(t); Convert the above conditions to unconstrained types and redefine them as follows: Among them, Γ(r(t)) is an increasing smooth function, r(t) represents the transformation error, and Γ(r(t)) is written as: Through the inverse function transformation, the transformation error is expressed as: The derivative is:

6. A method for robust fault-tolerant control with output feedback of specified performance according to claim 5, characterized in that: The specific implementation process of step 5 is as follows: Combining the fault-tolerant controller with the specified performance index with the preliminary model of the control system, the closed-loop system with time delay and faults can be summarized as: Tracking Error Incorporated middle:

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