Uncertain random time-varying system fault detection method considering output feedback control

Abstract

The invention relates to the technical field of state estimation and fault detection, solves the technical problem that a traditional closed-loop fault diagnosis method cannot effectively decouple a control signal and a residual signal, and particularly relates to an uncertain random time-varying system fault detection method considering output feedback control. Comprising the following steps: carrying out theoretical modeling by adopting a linear discrete system with model uncertainty to obtain a dynamic mathematical model of the linear discrete system; constructing a closed-loop residual generator used for generating a residual signal reflecting a system state based on the dynamic mathematical model; designing a fault detection threshold value; and analyzing the fault detectability, and verifying that the linear discrete system can be effectively identified and positioned when the fault occurs. According to the method, effective detection and diagnosis of the system fault are realized under the condition of uncertainty of the norm bounded model, and minimization of the upper bound of the estimation error variance of each step of recursion calculation is ensured, so that a reliable and efficient fault diagnosis technical solution is provided.

Description

Technical Field

[0001] The present invention relates to the technical field of state estimation and fault detection, and in particular to a method for detecting faults of an uncertain random time-varying system considering output feedback control. Background Art

[0002] Linear discrete systems are widely used in engineering and science, but they often face challenges brought by model uncertainty in practical applications. Traditional closed-loop fault diagnosis methods usually assume that the system model parameters are completely accurate, and use a complete matching method to construct the same control terms in the residual generator to achieve complete decoupling of the control law and the residual. However, in practical systems, model parameters are often not accurately known, such as linear systems and nonlinear systems with actuator coefficient matrix uncertainty. This makes traditional methods no longer applicable in the presence of model uncertainty, and feedback control will be explicitly reflected in the residual signal.

[0003] Existing studies on the treatment of model uncertainty systems mostly assume that there are upper and lower bounds on uncertainty, and design fault detectors under robust indicators. In addition, there are also studies on fault detection and separation of uncertain systems based on methods such as set element estimation and sliding mode observation. However, for closed-loop systems with uncertainty, relevant research is still relatively lacking. Existing studies such as Yang et al. designed an observer group based on a robust method for fuzzy systems with feedback, and used the upper and lower bounds of the uncertainty of the membership function to obtain the observer gain. Therefore, the problem of fault diagnosis of closed-loop uncertain systems still needs to be further explored. Summary of the invention

[0004] In view of the shortcomings of the prior art, the present invention provides a method for detecting faults in uncertain random time-varying systems taking into account output feedback control, which solves the technical problem that traditional closed-loop fault diagnosis methods cannot effectively decouple control signals and residual signals, while overcoming the difficulties caused by the unknown expectation and variance of model uncertainty.

[0005] To solve the above technical problems, the present invention provides the following technical solution: a method for detecting faults in an uncertain random time-varying system considering output feedback control, the method comprising the following process:

[0006] The linear discrete system with model uncertainty is used for theoretical modeling to obtain the dynamic mathematical model of the linear discrete system;

[0007] A closed-loop residual generator for generating a residual signal reflecting the system state is constructed based on a dynamic mathematical model;

[0008] Design a fault detection threshold that can detect faults in time when the residual signal exceeds the threshold;

[0009] Analyze fault detectability and verify that linear discrete systems can be effectively identified and located when faults occur.

[0010] Furthermore, the linear discrete system with model uncertainty modeled by the theoretical model is:

[0011]

[0012] Where k represents a discrete time moment; is the system status; To measure the output; is the control input; f k For failure; To measure noise; is the state noise; ΔA k and ΔB k Represents model uncertainty, all are unknown matrices, and have:

[0013] ||ΔA k ||≤δ A,k ,||ΔB k ||≤δ B,k .

[0014] And for the linear discrete system assumption:

[0015]

[0016]

[0017] Among them, δ A,k and δ B,k is a known scalar; the superscript T is the transposition symbol; represents mathematical expectation; The initial states x are 0 , state noise w k and measurement noise ν k The covariance matrix of ; ||·|| represents the vector norm; is the estimated value of the initial state of the system; is the initial state x 0 The covariance matrix of k is the measurement noise ν k The covariance matrix of k is the state noise w k The covariance matrix of is the initial state x 0 The upper bound of the norm of ; is the measurement noise ν k The upper bound of the norm of ; is the state noise w k The upper bound of the norm.

[0018] Furthermore, the specific process of constructing the closed-loop residual generator includes:

[0019] Any two vectors of the same dimension Lemma 1 holds for the following inequality:

[0020] xy T +yx T ≤εxx T +ε -1 yy T ,

[0021] Among them, ε>0 is an arbitrary scalar;

[0022] Based on Lemma 1, we obtain Theorem 1 for designing the closed-loop residual generator, namely, γ i,k (i=1,2,3) is a positive real number. Consider the following matrix recursion equation:

[0023]

[0024] In the formula, Z k represents a part of the matrix used to calculate the observer gain, defined as Y k represents a matrix used to calculate the observer gain, defined as are known scalars δ A,k and δ B,k The square of ; I is the unit matrix;

[0025] Initial conditions At the same time, the following equations are satisfied:

[0026]

[0027] In the formula, γ 1,k for The maximum eigenvalue of 2,k represents the eigenvalues ​​of the measurement noise and control matrix; γ 3,k represents the maximum eigenvalue based on the measurement noise covariance matrix; λ max is the maximum eigenvalue of the matrix;

[0028] According to Theorem 1, the closed-loop residual generator and gain K are determined k ,as follows:

[0029]

[0030] In the formula, Represents the state estimate at the k+1th moment.

[0031] Furthermore, the specific process of designing the fault detection threshold includes:

[0032] Define the residual signal r k for:

[0033]

[0034] In the formula, is the measurement output; C k is a matrix of appropriate dimension; represents the estimated value of the state at the kth moment;

[0035] Theorem 2 is proposed based on the closed-loop residual generator, that is, in the absence of faults, the residual signal satisfies the following inequality:

[0036]

[0037] In the formula, ||·|| represents the vector norm; r k is the residual signal defined; δ e,k is the upper bound of the system state estimation error; is the upper bound of system disturbance and noise; δ r,k is the decision threshold for fault detection;

[0038] According to Theorem 2, the fault detection threshold is obtained:

[0039]

[0040] In the formula, ||r k || is the norm of the residual.

[0041] Furthermore, the analysis of fault detectability is specifically as follows:

[0042] If the fault f occurs at time k k , the following formula must be satisfied:

[0043]

[0044] Then at time k+1, the fault f k Can be detected.

[0045] Furthermore, the verification that the linear discrete system can be effectively identified and located when a fault occurs is to prove that:

[0046] ||r k+1 ||>δ r,k+1 ,

[0047] In the formula, ||·|| represents the vector norm; r k+1 is the residual signal at the k+1th moment; δ r,k+1is the fault detection threshold at the k+1th moment.

[0048] By means of the above technical solution, the present invention provides a method for detecting faults in an uncertain random time-varying system considering output feedback control, which has at least the following beneficial effects:

[0049] 1. The present invention focuses on solving the closed-loop fault diagnosis problem of linear discrete systems with model uncertainty, and has achieved remarkable innovative results by introducing a closed-loop residual generator of a robust least squares method. The design of the closed-loop residual generator achieves the greatest possible decoupling between the control signal and the residual signal, providing an efficient tool for dynamic system fault detection.

[0050] 2. This paper proposes a design scheme for the gain of the residual generator to minimize the error variance, and also ensures that the system can meet the condition of minimizing the upper bound of the estimated error variance through rigorous theoretical proof. This not only provides a solid theoretical basis for the research method, but also emphasizes the reliability and effectiveness of the proposed technology.

[0051] 3. The present invention successfully completes the fault detection of the closed-loop system based on the new robust least squares residual generator. This shows that the research has not only made important breakthroughs in theory, but also demonstrated wide applicability in practical applications. Therefore, this innovative research has made a significant contribution to promoting the development of the field of dynamic system fault detection and has provided beneficial patent effects for related technical fields. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] The drawings described herein are used to provide a further understanding of the present application and constitute a part of the present application. The illustrative embodiments of the present application and their descriptions are used to explain the present application and do not constitute an improper limitation on the present application. In the drawings:

[0053] Figure 1 It is a flow chart of the uncertain random time-varying system fault detection method in the present invention;

[0054] Figure 2 It is a curve diagram of the residual signal and the detection threshold verified by simulation in the present invention. DETAILED DESCRIPTION

[0055] In order to make the above-mentioned purposes, features and advantages of the present invention more obvious and easy to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific implementation methods, so that the implementation process of how the present application uses technical means to solve technical problems and achieve technical effects can be fully understood and implemented accordingly.

[0056] This embodiment involves the state estimation and fault detection problems in the field of safety control, and aims to solve the problem that traditional closed-loop fault diagnosis methods cannot effectively decouple control signals and residual signals in linear discrete systems with model uncertainty, while overcoming the difficulties caused by the unknown expectation and variance of model uncertainty. This embodiment proposes a fault detection method for uncertain random time-varying systems considering output feedback control. By proposing a residual generator design scheme based on the robust least squares method, it realizes the effective detection and diagnosis of system faults under the condition of norm-bounded model uncertainty, and ensures that the upper bound of the estimated error variance of each recursive calculation step is minimized, thereby providing a reliable and efficient fault diagnosis technology solution. Please refer to Figure 1-Figure 2 , the method comprises the following steps:

[0057] S1. A linear discrete system with model uncertainty is used for theoretical modeling to obtain a dynamic mathematical model of the linear discrete system. In this embodiment, the following linear discrete system with model uncertainty is considered, namely:

[0058]

[0059] Where k represents a discrete time moment; is the system status; To measure the output; is the control input; f k For failure; To measure noise; is the state noise. k and measurement noise ν k are independent of each other and are all zero-mean white noise. The initial state x of the linear discrete system 0 is random and has nothing to do with noise.

[0060] For the linear discrete system corresponding to equation (1), assume that:

[0061]

[0062] Among them, the superscript T is the transposition symbol; represents mathematical expectation; The initial states x are 0 , state noise w k and measurement noise ν k The covariance matrix of ; ||·|| represents the vector norm; V k , W k , and All are known, among which, is the estimated value of the initial state of the system; is the initial state x0 The covariance matrix of k is the measurement noise ν k The covariance matrix of k is the state noise w k The covariance matrix of is the initial state x 0 The upper bound of the norm of ; is the measurement noise ν k The upper bound of the norm of ; is the state noise w k The upper bound of the norm.

[0063] In the linear discrete system adopted, A k , B k , C k , H k , ΔA k , ΔB k are matrices of appropriate dimensions, where A k , B k , C k , H k All are known matrices; ΔA k , ΔB k Represents model uncertainty, all are unknown matrices, and have:

[0064] ||ΔA k ||≤δ A,k ,||ΔB k ||≤δ B,k . (3)

[0065] Among them, δ A,k and δ B,k is a known scalar.

[0066] It is hoped that a residual generator of the following form can be designed for the linear discrete system corresponding to equation (1), namely:

[0067]

[0068] in, represents the estimated value of the state at the k+1th moment; represents the estimated value of the state at the kth moment, K k is the gain to be designed. The estimated error e k Denoted as:

[0069]

[0070] Estimated error variance P k Denoted as:

[0071]

[0072] Due to the existence of uncertainty, the control signal will directly affect the dynamic performance of the residual signal, and it is impossible to obtain an accurate estimate of the error variance. Therefore, the design goal is modified to construct a set of variables Guaranteed trouble-free At each step k, the gain K is designed. k , so that each step Minimum, that is, to achieve robust least squares.

[0073] S2. Constructing a closed-loop residual generator for generating a residual signal reflecting the system state based on a dynamic mathematical model; in this implementation method, the specific process includes:

[0074] Any two vectors of the same dimension Lemma 1 holds for the following inequality:

[0075] xy T +yx T ≤εxx T +ε -1 yy T , (7)

[0076] Among them, ε>0 is an arbitrary scalar;

[0077] Based on Lemma 1, we obtain Theorem 1 for designing the closed-loop residual generator, namely:

[0078] For the linear discrete system corresponding to equation (1), consider the residual generator of the form (4). If γ i,k (i=1,2,3) is a positive real number, γ i,k For the maximum eigenvalue of different parts of the system, consider the matrix recursion equation of the following form, namely:

[0079]

[0080] In the formula, Z k represents a part of the matrix used to calculate the observer gain, defined as Y k represents a matrix used to calculate the observer gain, defined as are known scalars δ A,k and δ B,k The square of ; I is the identity matrix.

[0081] Initial conditions Where P 0 is the initial estimation error variance, To correct the initial estimation error variance P 0 The designed initial variables satisfy the following formula:

[0082]

[0083]

[0084] in:

[0085]

[0086] In the formula, γ 1,k for The maximum eigenvalue of 2,k represents the eigenvalues ​​of the measurement noise and control matrix; γ 3,k represents the maximum eigenvalue based on the measurement noise covariance matrix; λ max is the largest eigenvalue of the matrix.

[0087] And there is a gain K k The design is as follows:

[0088]

[0089] but is the estimated error variance P when there is no fault k An upper bound of , and formula (15) makes Minimum.

[0090] In this embodiment, the reliability of Theorem 1 is proved by mathematical induction, as follows:

[0091] From the initial value, we know that when k = 0 Established. Assumption Next we need to prove set up Where Ω k is the system state x k The covariance matrix of Indicates x k The matrix obtained after the expected operation. According to Lemma 1 and assumption We can get:

[0092]

[0093] When there is no fault, substituting equation (1) and equation (4) into equation (5) yields:

[0094] e k+1 =(A k -K k C k ) k +(ΔA k +ΔB k H k Ck )x k +w k +(ΔB k H k -K k )v k . (17)

[0095] From the above formula and formula (2), we can get that the accurate estimation error variance should be written as:

[0096]

[0097] In the formula, is the error e k and the system state x k relevance; is the system state x k and error e k of relevance.

[0098] Using Lemma 1 and taking ε as 1, the above formula can be written as:

[0099]

[0100] Substituting formula (16) into the above formula, we can further obtain:

[0101]

[0102] In order to deal with the influence of the uncertainty part in the above formula, using Lemma 1, we can get:

[0103]

[0104] From equations (9) to (11) and the assumption that the model uncertainty norm is bounded It can be seen that:

[0105]

[0106] According to the assumption And equation (12) and equation (13) can be obtained:

[0107]

[0108] Notice Apply the above formula to the gain K k The formula can be further organized to obtain:

[0109]

[0110] Obviously when the gain When , the right side of the above formula reaches a minimum, and

[0111]

[0112] The proof is finished, and Theorem 1 is established.

[0113] This embodiment proposes a design scheme for the gain of the residual generator to minimize the error variance, and also ensures that the system can meet the condition of minimizing the upper bound of the estimated error variance through rigorous theoretical proof. This not only provides a solid theoretical basis for the research method, but also emphasizes the reliability and effectiveness of the proposed technology.

[0114] S3. Design a fault detection threshold that can detect a fault in time when the residual signal exceeds the threshold. In this embodiment, the specific process includes:

[0115] Define the residual signal. In order to realize the fault detection of the linear discrete system corresponding to equation (1), this embodiment defines the following signal as the residual signal:

[0116]

[0117] In the formula, is the measurement output; C k is a matrix of appropriate dimension; Represents the state estimate at the kth moment; its threshold is determined by the following conclusion.

[0118] For the linear discrete system corresponding to equation (1), the closed-loop residual generator designed in Theorem 1 is considered. Based on the closed-loop residual generator, Theorem 2 is proposed, that is, in the absence of faults, the residual signal satisfies the following inequality:

[0119]

[0120]

[0121] In the formula, ||·|| represents the vector norm; r k is the residual signal defined; δ e,k is the upper bound of the system state estimation error; is the upper bound of system disturbance and noise; δ r,k is the decision threshold for fault detection.

[0122] The initial error value δ e,0 for:

[0123]

[0124] In the formula, x 0 is the initial state of the system; is the initial state x 0 The upper bound of the norm.

[0125] Therefore δ r,k Can be used as a fault detection threshold.

[0126] This embodiment proves Theorem 2 to determine the detection threshold so that the fault can be detected in time when the residual signal exceeds the threshold, as follows:

[0127] From the original system state corresponding to formula (1) and the residual generator dynamics (4), we can know:

[0128] r k =C k e k +ν k . (29)

[0129] In the formula, e k is the estimation error; C k is a matrix of appropriate dimension; To measure noise;

[0130] According to the triangle inequality:

[0131] ||r k ||≤||C k ||||e k ||+||ν k ||. (30)

[0132] According to the assumed formula (2), if we ensure that ||e k ||≤δ e,k , then we can get ||r k ||≤δ r,k . ||e k ||≤δ e,k It also needs to be proved by mathematical induction. According to the initial value formula (28), when k = 0, ||e 0 ||≤δ e,0 Established. Assume || e k ||≤δ e,k , we need to further prove that ||e k+1 ||≤δ e,k+1 According to formula (17), we can know that:

[0133] ||e k+1 ||≤||A k -K k C k ||||e k ||+||ΔA k +ΔB k H k C k ||||x k ||+||ΔBk H k -K k ||||ν k ||+||w k ||. (31)

[0134] Notice The above formula can be written as:

[0135]

[0136] Applying the triangle inequality we get:

[0137]

[0138] According to the assumption ||e k ||≤δ e,k And the Cauchy inequality gives:

[0139]

[0140] Substituting equation (2) and equation (3) into the above equation, we can get ||e k+1 ||≤δ e,k+1 , and then we can get ||r k ||≤δ r,k .

[0141] The proof is finished, and Theorem 2 is established.

[0142] According to Theorem 2, the following fault detection strategy can be naturally proposed, namely:

[0143]

[0144] In the formula, ||r k || is the norm of the residual.

[0145] S4. Analyze the fault detectability and verify that the linear discrete system can be effectively identified and located when a fault occurs. In this embodiment, for the method for selecting the detection threshold of the closed-loop residual generator designed by Theorem 1 above, the fault detectability analysis is as follows:

[0146] Theorem 3: For equation (1), consider the closed-loop residual generator designed in Theorem 1 and the fault detection threshold in Theorem 2. If the fault f occurs at time k k Satisfy the following formula:

[0147]

[0148] Then the fault can be detected at time k+1.

[0149] This example proves Theorem 3 as follows:

[0150] When a fault occurs at time k, it can be known from formula (29):

[0151]

[0152] According to the triangle inequality:

[0153]

[0154] According to the Cauchy inequality:

[0155]

[0156] Further using the triangle inequality on the above formula we can get:

[0157]

[0158] Considering various model uncertainties and upper bounds of interference, we can obtain:

[0159]

[0160] In the formula, is the upper bound of system disturbance and noise; is the measurement noise ν k The upper bound of the norm.

[0161] Substituting formula (36) into the above formula, we can obtain:

[0162] ||r k+1 ||>δ r,k+1 ,

[0163] In the formula, ||·|| represents the vector norm; r k+1 is the residual signal at the k+1th moment; δ r,k+1 is the fault detection threshold at the k+1th moment;

[0164] The proof of Theorem 3 ends, i.e., the fault can be detected.

[0165] The present invention focuses on solving the closed-loop fault diagnosis problem of linear discrete systems with model uncertainty, and has achieved remarkable innovative results by introducing a closed-loop residual generator of a robust least squares method. The design of the closed-loop residual generator achieves the greatest possible decoupling between the control signal and the residual signal, providing an efficient tool for dynamic system fault detection.

[0166] This embodiment uses simulation to verify the above method to verify its effectiveness and reliability in practical applications, as follows:

[0167] In order to better understand the implementation of the present invention by researchers in this field, the present invention uses Matlab software for simulation. The specific information of the simulation software is as follows:

[0168] Software name: MATLAB;

[0169] Version information: 9.8.0.1380330 (R2020a) Update 2;

[0170] License number: 919961;

[0171] Operating system: Microsoft Windows 10 Home Chinese Edition Version 10.0 (Build 19042);

[0172] Java version: Java 1.8.0_202-b08 with Oracle Corporation Java HotSpot(TM)64-Bit Server VM mixed mode;

[0173] Special toolboxes: Statistics and Machine Learning Toolbox-11.7(R2020a), Aircraft Control Toolbox-1.0;

[0174] This embodiment is based on a three-tank system and considers a system with the following parameters:

[0175]

[0176] State noise w k and measurement noise ν k Independently and uniformly distributed in [-1×10 -6 ,1×10 -6 ].

[0177] The system model uncertainty is:

[0178]

[0179]

[0180] Among them, |δ A |<1.0×10 -4 ,|δ B |<1.0×10 -4 .

[0181] Consider a system with additive faults of the form k ,but:

[0182]

[0183] According to the simulation results, Figure 2 The residual obtained by the closed-loop residual generator proposed in Theorem 1 and the fault detection threshold calculated by Theorem 2 are shown. Figure 2 It can be seen that the proposed closed-loop fault diagnosis strategy can effectively detect additive faults that may occur in the system.

[0184] Therefore, this embodiment successfully completes the fault detection of the closed-loop system based on the new robust least squares residual generator. This shows that the research has not only made important breakthroughs in theory, but also demonstrated wide applicability in practical applications. Therefore, this innovative research has made significant contributions to promoting the development of the field of dynamic system fault detection and has provided beneficial patent effects for related technical fields.

[0185] Those skilled in the art can understand that all or part of the steps in the above-mentioned embodiment method can be completed by instructing the relevant hardware through a program, so the present application can take the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware. Moreover, the present application can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program codes.

[0186] Each embodiment in this specification is described in a progressive manner, and each embodiment focuses on the differences from other embodiments. The same or similar parts between the embodiments can be referred to each other. For the above embodiments, since they are basically similar to the method embodiments, the description is relatively simple, and the relevant parts can be referred to the partial description of the method embodiments.

[0187] The above implementation methods have been described in detail. Specific examples are used herein to illustrate the principles and implementation methods of the present invention. The description of the above embodiments is only used to help understand the method of the present invention and its core idea. At the same time, for those skilled in the art, according to the idea of ​​the present invention, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as a limitation on the present invention.

Claims

1. A method for detecting faults in uncertain random time-varying systems considering output feedback control, characterized in that: The Method The process includes: The linear discrete system with model uncertainty is used for theoretical modeling to obtain the dynamic mathematical model of the linear discrete system; A closed-loop residual generator for generating a residual signal reflecting the system state is constructed based on a dynamic mathematical model; Design a fault detection threshold that can detect faults in time when the residual signal exceeds the threshold; Analyze fault detectability and verify that linear discrete systems can be effectively identified and located when faults occur.

2. The method for detecting faults in an uncertain random time-varying system according to claim 1, characterized in that: The linear discrete system with model uncertainty modeled by the theory is: Where k represents a discrete time moment; is the system status; To measure output; is the control input; f k For failure; To measure noise; is the state noise; ΔA k and ΔB k Represents model uncertainty, all are unknown matrices, and have: ||ΔA k ||≤δ A,k ,||ΔB k ||≤δ B,k . And for the linear discrete system assumption: Among them, δ A,k and δ B,k is a known scalar; the superscript T is the transposition symbol; represents mathematical expectation; They are the initial state x0, state noise w k and measurement noise ν k The covariance matrix of ; ||·|| represents the vector norm; is the estimated value of the initial state of the system; is the covariance matrix of the initial state x0; V k is the measurement noise ν k The covariance matrix of k is the state noise w k The covariance matrix of is the upper bound of the norm of the initial state x0; is the measurement noise ν k The upper bound of the norm of ; is the state noise w k The upper bound of the norm.

3. The method for detecting faults in an uncertain random time-varying system according to claim 2, characterized in that: The specific process of constructing the closed-loop residual generator includes: Any two vectors of the same dimension Lemma 1 holds for the following inequality: xy T +yx T ≤εxx T +ε -1 yy T , Among them, ε>0 is an arbitrary scalar; Based on Lemma 1, we obtain Theorem 1 for designing the closed-loop residual generator, namely, γ i,k (i=1,2,3) is a positive real number. Consider the following matrix recursion equation: In the formula, Z k represents a part of the matrix used to calculate the observer gain, defined as Y k represents a matrix used to calculate the observer gain, defined as are known scalars δ A,k and δ B,k The square of ; I is the unit matrix; Initial conditions At the same time, the following equations are satisfied: In the formula, γ 1,k for The maximum eigenvalue of γ 2,k represents the eigenvalues ​​of the measurement noise and control matrix; γ 3,k represents the maximum eigenvalue based on the measurement noise covariance matrix; λ max is the maximum eigenvalue of the matrix; According to Theorem 1, the closed-loop residual generator and gain K are determined k ,as follows: In the formula, Represents the state estimate at the k+1th moment.

4. The method for detecting faults in an uncertain random time-varying system according to claim 3, characterized in that: The specific process of designing the fault detection threshold comprises: Define the residual signal r k for: In the formula, is the measurement output; C k is a matrix of appropriate dimension; represents the estimated value of the state at the kth moment; Theorem 2 is proposed based on the closed-loop residual generator, that is, in the absence of faults, the residual signal satisfies the following inequality: In the formula, ||·|| represents the vector norm; r k is the residual signal defined; δ e,k is the upper bound of the system state estimation error; is the upper bound of system disturbance and noise; δ r,k is the decision threshold for fault detection; According to Theorem 2, the fault detection threshold is obtained: In the formula, ||r k || is the norm of the residual.

5. The method for detecting faults in an uncertain random time-varying system according to claim 4, characterized in that: The analysis of fault detectability is specifically as follows: If the fault f occurs at time k k , the following formula must be satisfied: Then at time k+1, the fault f k Can be detected.

6. The method for detecting faults in an uncertain random time-varying system according to claim 1, characterized in that: The verification that the linear discrete system can be effectively identified and located when a fault occurs is to prove that: ||r k+1 ||>d r,k+1 , In the formula, ||·|| represents the vector norm; r k+1 is the residual signal at the k+1th moment; δ r,k+1 is the fault detection threshold at the k+1th moment.

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