Nonlinear discrete system actuator fault detection method based on set membership estimation index in finite frequency domain

Abstract

The invention relates to a nonlinear discrete system actuator fault detection method based on a set membership estimation index in a finite frequency domain, and the method comprises the steps: firstly building a nonlinear system model with actuator faults and unknown input interference; and then, carrying out augmentation operation on unknown input interference and original system state variables by proposing a matrix equation condition, and finally obtaining an augmented system. The method comprises the steps that firstly, an augmented system is obtained, then a corresponding observer is provided for the obtained augmented system, nonlinear terms in an error system generated by the observer and an original system are processed, so that the error system becomes a linear variable parameter system, then the error system is divided into two subsystems, and H-and L infinity parameter index design is conducted on the two subsystems respectively. And finally, generating a residual interval for the designed observer system by using a set membership estimation method to complete fault detection of system actuator faults. According to the method, the fault signal can be accurately detected, meanwhile, the unknown input interference of the system and the influence on fault detection are eliminated, the accuracy and speed of fault detection are improved, and the safety and reliability of the system are ensured.

Description

Technical Field

[0001] The present invention relates to robust fault detection of industrial nonlinear systems, and in particular to a method for detecting actuator faults of nonlinear discrete systems based on set membership estimation indices in a finite frequency domain. Background Art

[0002] With the advancement of modern science and technology, the level of automation in modern industry is increasing. Large-scale industrial scenarios, as well as control systems subject to national protection, place increasingly stringent requirements on system safety and reliability, such as nuclear, aerospace, and submarine systems. Consequently, fault diagnosis technology has garnered extensive attention from scholars. Fault diagnosis primarily encompasses fault detection, fault isolation, and fault estimation. For these systems, rapid and accurate fault detection is crucial, leading to in-depth and extensive research.

[0003] In the past decade, a large number of fault detection schemes have emerged, among which a relatively new scheme is the fault detection scheme based on interval observers. The so-called interval observer is to give a range of state changes by constructing upper and lower bound observers when uncertain factors appear in the system, which is the so-called interval. The interval observer is used as a residual generator to give the upper and lower bounds of the state at any time, thereby obtaining a natural threshold required for fault diagnosis. Therefore, the advantage of the fault diagnosis technology based on interval observers is that the interval observer can not only generate residuals but also give thresholds at the same time. Compared with the traditional observer-based fault diagnosis method, the two links of residual evaluation and threshold selection are omitted. The method is simple and concise. Based on H in the finite frequency domain _ / L ∞ The nonlinear discrete system actuator fault detection usually includes two technical indicators: H - and L ∞ The two indicators represent the observer's sensitivity to faults and robustness to disturbances. - Performance indicators usually require the design of several matrix inequalities to ensure the inequality relationship By designing a sufficiently large β f To obtain good enough fault sensitivity performance. ∞ The performance index also needs to design several matrix inequalities to make it tangible. The inequality relationship holds, where β d is the parameter to be designed, and its specific parameter form is determined by the actual design, ||d|| ∞ is the infinite norm of d(k). Compared with H ∞ Performance index, L ∞The advantage of the performance index is that the residual will obtain an initial residual value that is compatible with the initial error value, which ensures the rapidity of fault detection. ∞ Performance indicators require that the initial residual value converge to a designed residual interval before fault detection can be performed. In recent years, a new interval generation method has emerged in the field of fault diagnosis. This method uses set membership estimation and set theory to wrap the entire system state in a geometric shape. Currently, the mainstream geometric shapes are centrosymmetric polytopes and ellipsoids. Compared with traditional interval generation methods, intervals generated based on set membership estimation and set theory significantly reduce the conservatism in the interval estimation process and increase the observer's sensitivity to faults. Summary of the Invention

[0004] In view of the above-mentioned deficiencies in the prior art, the present invention proposes a method for detecting actuator faults of nonlinear discrete systems in a finite frequency domain based on set membership estimation technical indicators. First, the unknown input disturbance of the system is augmented with the system state to obtain a generalized system. Then, an observer is designed for the generalized system. The designed observer is subtracted from the generalized system to obtain an error system. The nonlinear terms in the error system are processed according to relevant theorems and merged with the system matrix of the error system to obtain a parameter matrix. Thus, the entire error system is transformed from the original nonlinear system into a linear variable parameter system. Then, according to the superposition principle of linear systems, the transformed error system is divided into two subsystems. The first subsystem only contains the fault component in the corresponding frequency domain; the second subsystem only contains noise; H is designed for the first subsystem. - performance index, making the system sensitive to the fault components in the corresponding frequency domain; design L for the second subsystem ∞ Performance indicators make the system robust to noise disturbances. Finally, the residual interval is obtained by applying set membership estimation and related set theory based on the obtained residual system to improve the speed and accuracy of fault detection.

[0005] The technical solution adopted by the present invention is: a method for detecting actuator faults in a nonlinear discrete system based on a set membership estimation index in a finite frequency domain, performing the following steps to obtain an observer for fault detection of an industrial application object; and determining whether an actuator to be detected in an industrial field is faulty. The method includes the following steps:

[0006] 1) Establish a nonlinear system model with actuator faults and unknown input disturbances;

[0007] 2) Propose matrix equality conditions and perform augmentation operations on the unknown input disturbance and the original system state variables to obtain the augmented system;

[0008] 3) Define an observer for the augmented system and process the nonlinear terms in the error system generated by the augmented system and the original system, so that the error system becomes a linear variable parameter system;

[0009] 4) Decompose the linear variable parameter error system into a fault vector subsystem and a noise vector subsystem, and perform H - and L ∞ Parameter index calculation;

[0010] 5) H - and L ∞ Substitute the parameter index into the matrix to obtain the observer for fault detection;

[0011] 6) For the fault detection observer, the set membership estimation method is used to generate the residual interval, which is used to determine whether there is a fault;

[0012] The industrial application objects include: actuators and controllers of electrical equipment at industrial sites, and sensor equipment for data acquisition.

[0013] The nonlinear system model is:

[0014]

[0015] Where: x(k) is the system state vector; y(k) is the system output signal; u(k) is the system input; d(k) is the unknown input interference; w(k) and v(k) are the measurement noise; f(k) is the system actuator fault;

[0016] Φ(x(k)) is the nonlinear term of the system and the nonlinear term satisfies the Lipschitz correlation property; A is the system state matrix; B is the system input gain matrix; C is the system output gain matrix; F is the fault distribution matrix; D is the unknown input interference gain matrix.

[0017] The augmented system is:

[0018]

[0019] Where: is the augmented state vector composed of the system state vector x(k) and the unknown input vector d(k);

[0020] Define the parameter matrices T, N, L so that holds true; therefore, the above system is further rewritten as:

[0021]

[0022] Define the observer form of the generalized system:

[0023]

[0024] Where: is the augmented state vector The estimated vector of , L is the gain matrix of the observer, and r(k) is the system residual;

[0025] Then the error system is:

[0026]

[0027] Where: e(k) is and The difference between is the error vector; ΔΦ(k) is the difference between the nonlinear terms of the generalized system and the observer system;

[0028] Since the nonlinear term satisfies the Lipschitz correlation property, we get:

[0029] Rewriting the nonlinear term of the error system, we have:

[0030]

[0031] Where:

[0032] The step of decomposing the linear variable parameter error system into a fault vector subsystem and a noise vector subsystem includes:

[0033] The fault vector subsystem is:

[0034]

[0035] The noise vector subsystem is:

[0036]

[0037] Among them, e f (k) is the error component containing only the fault component; r f (k) contains only e f The residual component of (k); e d (k) is the error component containing only interference and noise; r d (k) is only included in e d (k) the residual component;

[0038] Rewrite the above two subsystems as follows:

[0039]

[0040]

[0041] Where: D2=[0 I 0],ξ(k)=[w(k) T v(k) T v(k+1) T ] T .

[0042] The H _ Parameter indicator calculation includes:

[0043] For H in the finite frequency domain _ Indicator design, the generalized KYP lemma, Hermitian matrix, and Finsler lemma in the finite frequency domain are introduced to obtain the H _ Performance indicators:

[0044]

[0045] The L ∞ Parameter indicator calculation includes:

[0046] The Lyapunov function of the error system is defined as P d is a positive definite matrix;

[0047] Obtain the differential expression of the Lyapunov function of the noise vector subsystem:

[0048]

[0049] Rewrite the differential form of the above Lyapunov function and introduce Finsler's lemma to solve it, and finally get the system's L ∞ Performance indicators:

[0050] Let W = GL, Y = GS, and substitute them into H _ Performance indicators and L ∞ In the performance index design matrix, a method based on H in the finite frequency domain is obtained. _ / L ∞ Performance Indicators for Actuator Fault Detectors in Nonlinear Discrete Systems:

[0051] For a given matrix V, the scalars α1, α2, β d , if there exists a scalar β f >0 and the matrix P f >0, P d >0, Q>0, G, W, Y, so that for The following inequalities hold:

[0052]

[0053]

[0054]

[0055] Then the error system satisfies the designed H _ / L ∞ Performance indicators, observers

[0056]

[0057] To meet the designed H _ / L ∞ Fault detection observer based on performance indicators.

[0058] The residual interval is:

[0059]

[0060] The steps for real-time detection in industrial sites include:

[0061] 1) Collect real-time data of required detection variables;

[0062] 2) After modeling the observer, the residual interval is automatically calculated;

[0063] 3) If the residual signal between the actual output and the ideal output of the observer is within the residual interval, it is determined that there is no fault; otherwise, the current industrial object actuator is faulty and an alarm is issued.

[0064] The present invention also has the following beneficial effects and advantages:

[0065] 1. Compared with the traditional Lumberg-like observer, the method proposed in this invention has more design freedom.

[0066] 2. Since the present invention takes a nonlinear system as an example, many restrictions are reduced in terms of application.

[0067] 3. Compared with the traditional centrosymmetric polyhedral method, the use of ellipsoid analysis reduces a lot of calculation and conservatism. BRIEF DESCRIPTION OF THE DRAWINGS

[0068] Figure 1 Flow chart of the method of the present invention;

[0069] Figure 2 The simulation results of the nonlinear discrete system actuator fault detection method of the present invention are as follows; DETAILED DESCRIPTION

[0070] To make the above-mentioned objects, features, and advantages of the present invention more readily apparent, the specific implementation methods of the present invention are described in detail below with reference to the accompanying drawings. The following description sets forth many specific details to facilitate a full understanding of the present invention. However, the present invention can be implemented in many other ways than those described herein, and those skilled in the art may make similar modifications without departing from the scope of the invention. Therefore, the present invention is not limited to the specific implementation methods disclosed below.

[0071] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by those skilled in the art of the art to which the present invention pertains. The terms used in the specification of the invention herein are for the purpose of describing specific embodiments only and are not intended to limit the present invention.

[0072] Figure 1 The flowchart of the method of the present invention is as follows:

[0073] Step 1: First, a mathematical model of the nonlinear discrete system involved in the present invention is given. The model takes into account the nonlinear terms of the system, the noise interference, the unknown input interference and the actuator failure.

[0074] The model can be expressed as:

[0075]

[0076] Where: x(k) is the system state vector; y(k) is the system output signal; u(k) is the system control input; d(k) is the unknown input disturbance; w(k) and v(k) are measurement noises; f(k) is the system actuator fault; Φ(x(k)) is the nonlinear term of the system and the nonlinear term satisfies the Lipschitz correlation property; A is the system state matrix; B is the system input gain matrix; C is the system output gain matrix; F is the fault distribution matrix; and D is the unknown input disturbance gain matrix.

[0077] Step 2: Take the unknown input interference as the new state and form a new augmented state vector with the original system state vector So that we have the following system:

[0078]

[0079] Where: is the augmented state vector composed of the system state vector x(k) and the unknown input vector d(k). Then several parameter matrices T, N, and C to be designed are given so that Established.

[0080] Therefore, the above system can be further rewritten as:

[0081]

[0082] In this way, a new generalized system is obtained, formula (3). Then, such a new generalized system is subjected to H in the finite frequency domain. _ / L ∞ Design of actuator fault detection scheme for nonlinear discrete systems based on the index.

[0083] Step 3: Give the observer form of the generalized system:

[0084]

[0085] Where: is the augmented state vector The estimated vector of , L is the gain matrix of the observer, and r(k) is the system residual. Through the given generalized system and its observer system, the error system of the system can be obtained:

[0086]

[0087] Where: e(k) is and The difference between the two is the error vector. ΔΦ(k) is the difference between the nonlinear terms of the generalized system and the observer system. According to the above, the nonlinear term satisfies the Lipschitz correlation property, giving Where: and They are φ ij The upper and lower bounds of φ ij Can be defined as in The specific calculation method is shown in the following formula: H ij As a matrix, it can be defined as Please note the e here n (i) is different from the error vector mentioned above and is defined as

[0088]

[0089] definition And ρ belongs to a convex set The vertex set of this set can be defined as: yes The vertices of the convex set, yes The (i,j)th element of .

[0090] According to the above properties and definitions, we can get So the nonlinear term of the error system is rewritten as:

[0091]

[0092] Where: Therefore, the original error system is transformed from a nonlinear system into a linear variable parameter system, which is convenient for the subsequent H _ / L ∞ Design indicators.

[0093] Step 4: For H - / L ∞ The rewritten error system is designed based on the indicators. The first subsystem only contains the fault vector in the finite frequency domain, and the second subsystem only contains the noise vector:

[0094]

[0095]

[0096] Among them, e f (k) is the error component containing only the fault component; r f (k) is only included in e f The residual component of (k); e d (k) is the error component containing only interference and noise; r d (k) is only included in e d (k) residual component; in order to simplify the expression in the design process, the above two subsystems are rewritten as

[0097]

[0098] Where: D2=[0 I 0],ξ(k)=[w(k) T v(k) T v(k+1) T ] T .

[0099] Design H for the first system - Indicators, for the second system design L ∞ index.

[0100] For H in the finite frequency domain - To design the indicators, it is necessary to introduce the relevant lemma in the finite frequency domain, namely the generalized KYP lemma: Consider the following discrete-time linear system

[0101]

[0102] There exist Hermitian matrices P and Q such that Q > 0 and the following inequality holds:

[0103]

[0104] Where: The selection of Ξ is shown in the table below, and for ∏, it is generally selected I is the identity matrix.

[0105] Note: If the matrix Q in the above inequality is set to zero, the finite frequency domain case will degenerate into the full frequency domain case.

[0106] Table 1 Values ​​of set Θ and matrix Ξ in different frequency domains

[0107]

[0108] In the table: β f is the parameter to be designed, θ l and θ h They are the upper frequency bound of the low frequency range and the lower frequency bound of the high frequency range respectively.

[0109] Replace the variables in the above inequality, 0→D, positive definite matrix P f →P. Then we get the following inequality by calculation:

[0110]

[0111] Where:

[0112]

[0113]

[0114]

[0115] Where * represents the conjugate symmetric term in the matrix;

[0116] Then rewrite the above inequality as:

[0117]

[0118] Where:

[0119]

[0120]

[0121]

[0122] Introducing Finsler's lemma: for vectors matrix and The following descriptions are equivalent: 1)

[0123] 2)

[0124] 3) Make in For any satisfaction The matrix of .

[0125] Therefore, according to the above Finsler lemma, the above inequality can be rewritten as:

[0126]

[0127] Then, applying the above lemma, the above inequality can be equivalent to:

[0128]

[0129] The sufficient condition for the above inequality to hold is:

[0130]

[0131] Where:

[0132]

[0133]

[0134]

[0135]

[0136]

[0137]

[0138]

[0139] The symbol He expresses He{A}=A+A T , the symbol * represents the symmetric term in the matrix. So we get the H - Performance indicators.

[0140] Next, we design the system L ∞ Performance indicators. First, define the Lyapunov function of the error system as P d is a positive definite matrix. To obtain the differential expression of the Lyapunov function V(k+1)-V(k)<0, according to the form of the subsystem described above, it is easy to obtain the specific form of the expression:

[0141]

[0142] Rewrite the difference form of the above Lyapunov function:

[0143]

[0144] From formula (12), we can get the following conclusions:

[0145]

[0146] From this conclusion, the following inequality can be established

[0147]

[0148] The above formula can be further simplified to:

[0149]

[0150] Multiply both sides of the above equation by and The following inequality can be obtained:

[0151]

[0152] Where: Symbols || || ∞ It can be seen from formula (16) that the system satisfies exponential stability, so the above matrix inequality can be combined with H - The same treatment of indicators:

[0153] make Then the above matrix inequality can be rewritten as:

[0154]

[0155] Introducing Finsler's lemma, we can get:

[0156]

[0157] Where: U d =[A d -I], So we can get:

[0158]

[0159] Where:

[0160]

[0161]

[0162]

[0163]

[0164]

[0165]

[0166] In order to obtain the residual L ∞ Performance indicators, design the following matrix:

[0167]

[0168] Multiply the left and right sides of the above inequality by and You can get:

[0169]

[0170] Finally, we can get: So we get the system's L ∞ Performance indicators.

[0171] Step 5: Let W = GL, Y = GS, and substitute them into H _ Performance indicators and L ∞ The performance index design matrix can be expressed as Equations (11), (19) and (20).

[0172] In summary, we can get a method based on H in finite frequency domain. _ / L ∞ Design scheme for actuator fault detection of nonlinear discrete systems based on performance index: For a given matrix V, scalars α1, α2, β d , if there exists a scalar β f > 0 and a matrix P of appropriate dimensions f >0, P d >0, Q>0, G, W, Y, so that for The following inequalities hold:

[0173]

[0174]

[0175]

[0176] Then the error system satisfies the designed H _ / L ∞ Performance indicators, observers

[0177]

[0178] To meet the designed H - / L ∞ Fault detection observer based on performance indicators.

[0179] Where:

[0180]

[0181] Step 6: After completing the corresponding performance indicator design, the residual interval of the system can be expressed. In this part, the aforementioned set membership estimation theory is used for design.

[0182] First, the relevant knowledge content that will be used next is explained.

[0183] Definition 1: The Minkowski sum of two sets can be defined as follows:

[0184]

[0185] symbol Represents the Minkowski sum operation between sets.

[0186] Definition 2: An ellipsoid It is composed of a unit ball The ellipsoid can be defined as:

[0187]

[0188] In the formula is the center of the ellipsoid ε(c,M), is the generating matrix or shape matrix of the ellipsoid ε(c,M).

[0189] Property 1: Applying a linear transformation of the form y = Kx + b to an ellipsoid x∈εc,M) yields an ellipsoid of the form:

[0190] Kε(c,M)+b=ε(Kc+b,KM)

[0191] Where: and are known vectors and matrices.

[0192] Property 2: Given N ellipsoids ε(c i ,M i ), i = 1, ..., N, then the Minkowski sum of these N ellipsoids satisfies the following conclusions:

[0193]

[0194] Where: Vector σ=[σ1,...,σN ] satisfies σ i >0 and

[0195] Note: Different parameters α will produce ellipsoids of different shapes and sizes ε(c m ,M m (σ)). The trace of the generator matrix It is related to the sum of the squares of the semi-axes of the ellipsoid, so the trace of the generating matrix is ​​selected to optimize the parameters so that the optimized parameters The trace of the generator matrix Satisfy the following forms:

[0196]

[0197] Property 3: Given an ellipsoid ε(c,M), ​​the minimum box BOX(ε(c,M)) that can enclose the ellipsoid ε(c,M) has the following form:

[0198]

[0199] Where: X = MM T , B n =[-1,1] n is an n-dimensional unit hypercube. We can get an interval estimate of x in the following form:

[0200]

[0201] Where: and are the upper and lower bounds of the ellipsoid variable x∈ε(c,M) respectively.

[0202] According to the above theoretical basis, the residual interval is evaluated and the fault diagnosis scheme can be obtained.

[0203] According to the above system and observer, we can give x(0)∈ε(c0,M0), Then we can know that x(k) is wrapped in the ellipsoid ε(x(k),M k ). Similarly, we can also obtain the ellipsoid that contains noise and disturbance:

[0204] w(k)∈ε(0,W), v(k)∈ε(0,V)

[0205] Where: W, V are both known matrices of suitable dimension. In addition, it is also easy to find two normal numbers Satisfy the following formula:

[0206]

[0207] According to the definition of error You can get the ellipsoid of the initial value of the wrapping error:

[0208]

[0209] By applying Property 3, we have:

[0210]

[0211] Where: n x is the dimension of the system state vector. According to the properties of Lipschitz nonlinear system, we can get:

[0212]

[0213] The above formula also shows that ΔΦ0∈ε(0,Φ0), Therefore, based on the error system of the above content, under the premise of no fault, let The ellipsoid of the wrapping error can be obtained by continuous iteration:

[0214]

[0215] According to the above properties, the above formula can be rewritten as:

[0216] According to the residual equation, the following expression for the residual can be obtained:

[0217]

[0218] Where: σ=[σ1,...,σ N ], and X(σ)=M m (σ)M m (σ) T , according to property 2, the optimized X(σ * ), where X(σ * ) meets the following conditions:

[0219]

[0220] Where:

[0221]

[0222] Let X k+1 =X(σ * ), we can get:

[0223]

[0224] In summary, the residual threshold of the system based on the set membership estimation technology in the absence of faults can be obtained, which is expressed as:

[0225]

[0226] The above content is based on L ∞ The indicator also obtains the residual threshold as:

[0227]

[0228] This threshold can be used to generate an ellipsoid ε(0,||r d ||I), the threshold value of the ellipsoid can be expressed as:

[0229]

[0230] Comparing the residual threshold obtained using this ellipsoid with the threshold generated by the previous ellipsoid, a less conservative residual interval can be obtained:

[0231]

[0232] So the following fault detection scheme can be given:

[0233]

[0234] Example:

[0235] Here, a DC servo motor is used as an example to further demonstrate the accuracy and speed of the proposed method for detecting actuator faults in nonlinear systems. The mathematical model of this system can be expressed as:

[0236]

[0237]

[0238] According to this model and the above method of dealing with nonlinear terms, the value range of the nonlinear term after transformation can be obtained:

[0239]

[0240] After selecting relevant parameters and solving the optimization problem, we can get β d =0.5, μ1=0.1277, μ2=0.1284, and the corresponding parameter matrix is:

[0241]

[0242] Given the initial state value is x0 = [0.5 0.5 0.5] T , the input is uk =0.2sin(0.1k);

[0243] Measurement noise v k =[-0.1 0.1]×[sin(k) cos(k)] T ;

[0244] Unknown disturbance w k =[-0.1 0.1]×[sin(0.1k) cos(0.1k)] T ;

[0245] The initial value of the observer can be chosen to be

[0246] Taking the motor speed as an example, a method based on H in the finite frequency domain _ / L ∞ The simulation results of the actuator fault detection scheme for nonlinear discrete systems with the following indicators are shown in Figure 2. Figure 2 When a fault occurs, the residual value will exceed the residual interval, thereby triggering the fault detection condition and realizing the fault detection.

[0247] The present invention provides a method for detecting actuator faults of nonlinear discrete systems based on set membership estimation in a finite frequency domain. The method comprises the following steps for real-time detection at an industrial site:

[0248] 1) Collect real-time data of required detection variables, such as current and voltage;

[0249] 2) Modeling based on industrial field equipment, designing observers, and automatically calculating residual intervals;

[0250] 3) If the difference between the actual output and the ideal output of the observer, that is, the residual signal, is included in the calculated residual interval, it is determined that there is no fault; otherwise, the current industrial object actuator is faulty and an alarm is issued.

[0251] The above is a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principles of the present invention. These improvements and modifications should be regarded as within the scope of protection of the present invention.

Claims

1. A method for actuator fault detection of nonlinear discrete systems based on set membership estimation index in finite frequency domain, characterized in that: The following steps are performed to obtain an observer for fault detection of an industrial application object; the observer is used to determine whether an actuator to be detected at an industrial site is faulty. The method includes the following steps: 1) Establish a nonlinear system model with actuator faults and unknown input disturbances; 2) Propose matrix equality conditions and perform augmentation operations on the unknown input disturbance and the original system state variables to obtain the augmented system; 3) Define an observer for the augmented system and process the nonlinear terms in the error system generated by the augmented system and the original system, so that the error system becomes a linear variable parameter system; 4) Decompose the linear variable parameter error system into a fault vector subsystem and a noise vector subsystem, and perform H - and L ∞ Parameter index calculation; 5) H - and L ∞ Substitute the parameter index into the matrix to obtain the observer for fault detection; 6) For the fault detection observer, the set membership estimation method is used to generate the residual interval, which is used to determine whether there is a fault; The industrial application objects include: actuators and controllers of electrical equipment at industrial sites, and sensor equipment for data acquisition.

2. The method for actuator fault detection of nonlinear discrete systems based on set membership estimation index in finite frequency domain according to claim 1, characterized in that: The nonlinear system model is: Where: x(k) is the system state vector; y(k) is the system output signal; u(k) is the system input; d(k) is the unknown input interference; w(k) and v(k) are measurement noises; f(k) is the system actuator fault; Φ(x(k)) is the nonlinear term of the system and the nonlinear term satisfies the Lipschitz correlation property; A is the system state matrix; B is the system input gain matrix; C is the system output gain matrix; F is the fault distribution matrix; D is the unknown input interference gain matrix.

3. The method for actuator fault detection of nonlinear discrete systems based on set membership estimation index in finite frequency domain according to claim 1, characterized in that: The augmented system is: Where: is the augmented state vector composed of the system state vector x(k) and the unknown input vector d(k); Define the parameter matrices T, N, L so that holds true; therefore, the above system is further rewritten as:

4. The method for actuator fault detection of nonlinear discrete systems based on set membership estimation index in finite frequency domain according to claim 1, characterized in that: include: Define the observer form of the generalized system: Where: is the augmented state vector The estimated vector of , L is the gain matrix of the observer, and r(k) is the system residual; Then the error system is: Where: e(k) is and The difference between is the error vector; ΔΦ(k) is the difference between the nonlinear terms of the generalized system and the observer system; Since the nonlinear term satisfies the Lipschitz correlation property, we get: Rewriting the nonlinear term of the error system, we have: Where:

5. The method for actuator fault detection of nonlinear discrete systems based on set membership estimation index in finite frequency domain according to claim 1, characterized in that: The step of decomposing the linear variable parameter error system into a fault vector subsystem and a noise vector subsystem includes: The fault vector subsystem is: The noise vector subsystem is: Among them, e f (k) is the error component containing only the fault component; r f (k) contains only e f The residual component of (k); e d (k) is the error component containing only interference and noise; r d (k) contains only e d (k) the residual component; Rewrite the above two subsystems as follows: Where: D2=[0 I 0],ξ(k)=[w(k) T v(k) T v(k-1) T ] T .

6. The method for actuator fault detection of nonlinear discrete systems based on set membership estimation index in finite frequency domain according to claim 1, characterized in that: The H - Parameter indicator calculation includes: For H in the finite frequency domain - Indicator design, the generalized KYP lemma, Hermitian matrix, and Finsler lemma in the finite frequency domain are introduced to obtain the H _ Performance indicators:

7. The method for actuator fault detection of nonlinear discrete systems based on set membership estimation index in finite frequency domain according to claim 1, characterized in that: The L ∞ Parameter indicator calculation includes: The Lyapunov function of the error system is defined as P d is a positive definite matrix; Obtain the differential expression of the Lyapunov function of the noise vector subsystem: Rewrite the differential form of the above Lyapunov function and introduce Finsler's lemma to solve it, and finally get the system's L ∞ Performance indicators:

8. The method for actuator fault detection of nonlinear discrete systems based on set membership estimation index in finite frequency domain according to claim 1, characterized in that: Let W = GL, Y = GS, and substitute them into H - Performance indicators and L ∞ In the performance index design matrix, a method based on H in the finite frequency domain is obtained. - / L ∞ Performance Indicators for Actuator Fault Detectors in Nonlinear Discrete Systems: For a given matrix V, the scalars α1, α2, β d , if there exists a scalar β f >0 and the matrix P f >0, P d >0, Q>0, G, W, Y, so that for The following inequalities hold: Then the error system satisfies the designed H _ / L ∞ Performance indicators, observers To meet the designed H _ / L ∞ Fault detection observer based on performance indicators.

9. The method for actuator fault detection of nonlinear discrete systems based on set membership estimation index in finite frequency domain according to claim 1, characterized in that: The residual interval is:

10. The method for actuator fault detection of nonlinear discrete systems based on set membership estimation index in finite frequency domain according to claim 1, characterized in that: The steps for real-time detection in industrial sites include: 1) Collect real-time data of required detection variables; 2) After modeling the observer, the residual interval is automatically calculated; 3) If the residual signal between the actual output and the ideal output of the observer is within the residual interval, it is determined that there is no fault; otherwise, the current industrial object actuator is faulty and an alarm is issued.

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