A Fault Detection Method for Uncertain Stochastic Time-Varying Systems Considering Output Feedback Control
By designing a robust least squares closed-loop residual generator in an uncertain random time-varying system, the problem of fault detection under the difficulty of decoupling control signals and residual signals in traditional methods and model uncertainty is solved, and effective detection and diagnosis of system faults is achieved.
Patent Information
- Application Number
- CN202411516406.5
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2024-10-29
- Publication Date
- 2025-06-20
- Estimated Expiration
- 2044-10-29
AI Technical Summary
Traditional closed-loop fault diagnosis methods cannot effectively decouple control signals and residual signals, and cannot effectively detect system failures when model uncertainty exists.
The closed-loop residual generator is designed using a robust least squares method. By introducing output feedback control, the control signal and the residual signal are decoupled, and a fault detection threshold is designed to detect system failures.
It realizes effective detection and diagnosis of system faults under model uncertainty conditions, ensures the minimization of the upper bound of estimation error variance, and provides an efficient fault diagnosis tool.
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Figure CN119395998B_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the technical field of state estimation and fault detection, and particularly to a fault detection method for an uncertain stochastic time-varying system considering output feedback control. Background Art
[0002] Linear discrete systems are widely used in engineering and scientific fields, but in practical applications, they often face challenges brought by model uncertainties. Traditional closed-loop fault diagnosis methods usually assume that the system model parameters are completely accurate and use a completely 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 actual systems, the model parameters are often not precisely known, such as linear systems and nonlinear systems with uncertainties in the actuator coefficient matrix. This makes the traditional methods inapplicable in the presence of model uncertainties, and the feedback control will be explicitly reflected in the residual signal.
[0003] Existing research on the treatment of model uncertainty systems mostly assumes that the uncertainties have upper and lower bounds and designs fault detectors under robust metrics. In addition, there are also methods based on set membership estimation, sliding mode observation, etc. to study the fault detection and isolation of uncertainty systems. However, for closed-loop systems with uncertainties, relevant research is still relatively lacking. Existing research such as Yang et al. designed a set of observers for a fuzzy system with feedback based on a robust method and obtained the observer gains using the upper and lower bounds of the membership function uncertainties. Therefore, the fault diagnosis problem of closed-loop uncertainty systems still needs to be further explored. Summary of the Invention
[0004] Aiming at the deficiencies of the existing technology, the present invention provides a fault detection method for an uncertain stochastic time-varying system considering output feedback control, which solves the technical problem that the traditional closed-loop fault diagnosis method cannot effectively decouple the control signal and the residual signal, and at the same time overcomes the problem brought by the unknown expectation and variance of the model uncertainty.
[0005] To solve the above technical problems, the present invention provides the following technical solution: A fault detection method for an uncertain stochastic time-varying system considering output feedback control, the method includes the following processes:
[0006] Use a linear discrete system with model uncertainty for theoretical modeling to obtain the dynamic mathematical model of the linear discrete system;
[0007] Based on the dynamic mathematical model, construct a closed-loop residual generator for generating a residual signal reflecting the system state;
[0008] Design a fault detection threshold that can detect faults in time when the residual signal exceeds the threshold;
[0009] Analyze the fault detectability and verify that the linear discrete system can be effectively identified and located when a fault occurs.
[0010] Furthermore, the linear discrete system with model uncertainty in the theoretical modeling is as follows:
[0011]
[0012] where k represents the discrete time instant; is the system state; is the measurement output; is the control input; f k is the fault; is the measurement noise; is the state noise; ΔA k and ΔB k represent model uncertainties, both are unknown matrices, and there are:
[0013] ||ΔA k || ≤ δ A,k , ||ΔB k || ≤ δ B,k .
[0014] And for the linear discrete system, the assumptions are:
[0015]
[0016]
[0017] where δ A,k and δ B,k are known scalars; the superscript T is the transpose symbol; represents the mathematical expectation; are the covariance matrices of the initial state x0, the state noise w k and the measurement noise ν k respectively; ||·|| represents the vector norm; is the estimated value of the system initial state; is the covariance matrix of the initial state x0; V k is the covariance matrix of the measurement noise ν k ; W k is the covariance matrix of the state noise w k ; is the upper bound of the norm of the initial state x0; is the upper bound of the norm of the measurement noise ν k ; is the upper bound of the norm of the state noise w k .
[0018] Furthermore, the specific process of constructing the closed-loop residual generator includes:
[0019] Propose any two vectors of the same dimension Lemma 1 for which the following inequality holds. The inequality is:
[0020] xy T + yx T ≤ εxx T + ε -1 yy T ,
[0021] where ε > 0 is an arbitrary scalar;
[0022] Based on Lemma 1, Theorem 1 for designing a closed-loop residual generator is obtained, i.e., γ i,k (i = 1, 2, 3) are positive real numbers. Consider the following matrix recurrence equation in the form:
[0023]
[0024] In the formula, Z k represents a part of the matrix for calculating the observer gain, defined as Y k represents a matrix for calculating the observer gain, defined as are the squares of the known scalars δ A,k and δ B,k respectively; I is the identity matrix;
[0025] Initial conditions simultaneously satisfy the following formula:
[0026]
[0027] In the formula, γ 1,k is the maximum eigenvalue of; γ 2,k represents the eigenvalue with respect to the measurement noise and the 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] Determine the closed-loop residual generator and the gain K k , as follows:
[0029]
[0030] In the formula, represents the state estimate value at the (k + 1)-th moment.
[0031] Furthermore, the specific process of designing the fault detection threshold includes:
[0032] Define the residual signal r k as:
[0033]
[0034] wherein, is the measurement output; C k is a matrix with appropriate dimensions; represents the state estimate value at the k-th moment;
[0035] Based on the closed-loop residual generator, Theorem 2 is proposed, that is, in the case of no fault, the residual signal satisfies the following inequality:
[0036]
[0037] wherein, ||·|| represents the vector norm; r k is the defined residual signal; δ e,k is the upper bound of the system state estimation error; is the upper bound of the 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, that is:
[0039]
[0040] wherein, ||r k || is the norm of the residual.
[0041] Furthermore, the analysis of fault detectability is specifically as follows:
[0042] If a fault f k occurs at the k-th moment, it needs to satisfy the following formula:
[0043]
[0044] Then at the (k + 1)-th moment, this 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:
[0046] ||r k+1 || > δ r,k+1 ,
[0047] wherein, ||·|| represents the vector norm; r k+1 is the residual signal at the (k + 1)-th moment; δ r,k+1 is the fault detection threshold at the (k + 1)-th moment.
[0048] With the above technical solution, the present invention provides a fault detection method for uncertain stochastic time-varying systems 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 uncertainties. By introducing a closed-loop residual generator based on a robust least squares method, significant innovative results have been achieved. The design of this closed-loop residual generator decouples the control signal and the residual signal as much as possible, providing an efficient tool for dynamic system fault detection.
[0050] 2. The present invention proposes a design scheme for the residual generator gain to minimize the error variance. At the same time, through strict theoretical proofs, it is ensured that the system can meet the condition of minimizing the upper bound of the estimation error variance. 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 a new robust least squares residual generator. This shows that this research has not only made important breakthroughs in theory, but also demonstrated wide applicability in practical applications. Therefore, this innovative research makes significant contributions to promoting the development of the dynamic system fault detection field and provides 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 form a part of the present application. The schematic embodiments of the present application and their descriptions are used to explain the present application and do not constitute an improper limitation of the present application. In the drawings:
[0053] Figure 1 is a flowchart of the fault detection method for the uncertain stochastic time-varying system in the present invention;
[0054] Figure 2 is a curve graph of the residual signal and the detection threshold for simulation verification in the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS
[0055] In order to make the above objects, features, and advantages of the present invention more obvious and understandable, the present invention will be further described in detail below with reference to the drawings and specific embodiments. Thereby, a full understanding of how the present application uses technical means to solve technical problems and achieve technical effects can be obtained and implemented accordingly.
[0056] This embodiment relates to the problems of state estimation and fault detection in the field of safety control, aiming to solve the problem that traditional closed-loop fault diagnosis methods cannot effectively decouple control signals and residual signals in a linear discrete system with model uncertainties, and at the same time overcome the difficulties brought by the unknown expectations and variances of model uncertainties. This embodiment proposes a fault detection method for uncertain stochastic time-varying systems considering output feedback control. By proposing a design scheme of a residual generator based on the robust least squares method, effective detection and diagnosis of system faults are achieved under the condition of norm-bounded model uncertainties, and the upper bound of the estimation error variance of each recursive calculation is minimized, thus providing a reliable and efficient fault diagnosis technical solution. Please refer to Figure 1 - Figure 2 , and the method includes the following steps:
[0057] S1. Use a linear discrete system with model uncertainties for theoretical modeling to obtain the dynamic mathematical model of the linear discrete system; in this embodiment, consider the following class of linear discrete systems with model uncertainties, that is:
[0058]
[0059] where k represents the discrete time instant; is the system state; is the measured output; is the control input; f k is the fault; is the measurement noise; is the state noise. The state noise w k and the measurement noise ν k are independent of each other and are both zero-mean white noises. The initial state x0 of the linear discrete system is random and independent of the noises.
[0060] For the linear discrete system corresponding to Equation (1), assume:
[0061]
[0062] where the superscript T is the transpose symbol; represents the mathematical expectation; are the covariance matrices of the initial state x0, the state noise w k and the measurement noise ν k respectively; ||·|| represents the vector norm; V k , W k , and are all known, where is the estimated value of the system initial state; is the covariance matrix of the initial state x0; V kTo measure the covariance matrix of the noise ν k ; W k is the covariance matrix of the state noise w k ; is the upper bound of the norm of the initial state x0; is the upper bound of the norm of the measurement noise ν k ; is the upper bound of the norm of the state noise w k ;
[0063] In the adopted linear discrete system, A k , B k , C k , H k , ΔA k , ΔB k are all matrices with appropriate dimensions, where A k , B k , C k , H k are all known matrices; ΔA k , ΔB k represent model uncertainties and are all unknown matrices, and there are:
[0064] ||ΔA k || ≤ δ A,k , ||ΔB k || ≤ δ B,k . (3)
[0065] where, δ A,k and δ B,k are known scalars;
[0066] It is desired to design a residual generator in the following form for the linear discrete system corresponding to Equation (1), that is:
[0067]
[0068] where, represents the state estimate at the (k + 1)-th time step; represents the state estimate at the k-th time step, and K k is the gain to be designed. Denote the estimation error e k as:
[0069]
[0070] Denote the estimation error variance P k as:
[0071]
[0072] Due to the existence of uncertainties, the control signal will directly affect the dynamic performance of the residual signal, and it is impossible to obtain the accurate estimated error variance. Therefore, the design objective is modified to construct a set of variables to ensure that when there is no fault it holds at each time instant k, and design the gain K k such that at each step is minimized, that is, robust least squares is achieved.
[0073] S2. Construct a closed-loop residual generator based on the dynamic mathematical model to generate a residual signal reflecting the system state; in this implementation method, the specific process includes:
[0074] Propose Lemma 1 for any two vectors of the same dimension such that the following inequality holds. The inequality is:
[0075] xy T +yx T ≤εxx T +ε -1 yy T , (7)
[0076] where ε > 0 is an arbitrary scalar;
[0077] Based on Lemma 1, obtain Theorem 1 for designing the closed-loop residual generator, that is:
[0078] For the linear discrete system corresponding to Equation (1), consider a residual generator in the form of Equation (4). If γ i,k (i = 1, 2, 3) are positive real numbers, and γ i,k is the largest eigenvalue of different parts of the system, consider the following form of matrix recurrence equation, that is:
[0079]
[0080] In the formula, Z k represents a part of the matrix for calculating the observer gain, defined as Y k represents a matrix for calculating the observer gain, defined as are respectively the squares of the known scalars δ A,k and δ B,k ; I is the identity matrix.
[0081] Initial conditions where P0 is the initial estimated error variance, is the initial variable designed to correct the initial estimated error variance P0, and at the same time satisfies the following formula:
[0082]
[0083]
[0084] Wherein:
[0085]
[0086] In the formula, γ 1,k is the largest eigenvalue; γ 2,k represents the eigenvalue with respect to the measurement noise and the control matrix; γ 3,k represents the largest 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 designed as shown in the following formula:
[0088]
[0089] Then is an upper bound of the estimation error variance P k when there is no fault, and Equation (15) makes the smallest at each step.
[0090] In this embodiment, the reliability of Theorem 1 is proved by mathematical induction as follows:
[0091] From the initial value, it can be seen that when k = 0 holds. Assume Next, it is necessary to prove Let where Ω k is the covariance matrix of the system state x k , represents the matrix obtained after taking the expectation operation on x k . It is known that From Lemma 1 and the assumption it can be obtained that:
[0092]
[0093] When there is no fault, substituting Equation (1) and Equation (4) into Equation (5) gives:
[0094] e k+1 =(A k -K k C k )e k +(ΔA k +ΔB k H k C k )x k +wk +(ΔB k H k -K k )v k . (17)
[0095] From the above equation and Equation (2), the accurate estimated error variance should be written as:
[0096]
[0097] where is the correlation between the error e k and the system state x k . is the correlation between the system state x k and the error e k .
[0098] Using Lemma 1 and taking ε as 1 in it, the above equation can be written as:
[0099]
[0100] Substituting Equation (16) into the above equation, we can further obtain:
[0101]
[0102] To handle the influence of the uncertainty part in the above equation, using Lemma 1, we can get:
[0103]
[0104] From Equation (9) - Equation (11) and the assumption that the model uncertainty norm is bounded we know that:
[0105]
[0106] According to the assumption and Equation (12) and Equation (13), we can get:
[0107]
[0108] Note that Completing the square for the above equation with respect to the gain K k , we can further organize it to get:
[0109]
[0110] Obviously, when the gain , the right side of the above equation reaches the minimum, and
[0111]
[0112] The proof is completed, and thus the content of Theorem 1 holds.
[0113] This embodiment proposes a design scheme for the gain of the residual generator to minimize the error variance. At the same time, through strict theoretical proofs, it is ensured that the system can meet the condition of minimizing the upper bound of the estimation error variance. 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 faults in a timely manner when the residual signal exceeds the threshold; in this embodiment, the specific process includes:
[0115] Define the residual signal. To implement fault detection for the linear discrete system corresponding to Equation (1), this embodiment defines the following signal as the residual signal:
[0116]
[0117] where is the measured output; C k is a matrix with appropriate dimensions; represents the state estimate value at the k-th moment; its threshold is determined by the following conclusion.
[0118] For the linear discrete system corresponding to Equation (1), consider the closed-loop residual generator designed in Theorem 1. Based on the closed-loop residual generator, Theorem 2 is proposed, that is, in the case of no fault, the residual signal satisfies the following inequality:
[0119]
[0120]
[0121] where ||·|| represents the vector norm; r k is the defined residual signal; δ e,k is the upper bound of the system state estimation error; is the upper bound of the system disturbance and noise; δ r,k is the decision threshold for fault detection.
[0122] The initial value of the error magnitude at the initial moment δ e,0 is:
[0123]
[0124] where x0 is the initial state of the system; is the upper bound of the norm of the initial state x0.
[0125] Therefore, δ r,k can be used as the fault detection threshold.
[0126] In this embodiment, Theorem 2 is proved to determine the detection threshold so that a fault can be detected in time when the residual signal exceeds the threshold, as follows:
[0127] From the original system state corresponding to Equation (1) and the residual generator dynamics (4), it can be seen that:
[0128] r k = C k e k + ν k . (29)
[0129] Where e k is the estimation error; C k is a matrix with appropriate dimensions; is the measurement noise;
[0130] According to the triangle inequality, it can be known that:
[0131] ||r k || ≤ ||C k ||||e k || + ||ν k ||. (30)
[0132] According to the assumed Equation (2), it can be known that if ||e k || ≤ δ e,k is ensured at each moment, then ||r k || ≤ δ r,k . ||e k || ≤ δ e,k also needs to be proved by mathematical induction. According to the initial value Equation (28), it can be seen that ||e0|| ≤ δ e,0 holds when k = 0. Assume ||e k || ≤ δ e,k , and further it is necessary to prove ||e k+1 || ≤ δ e,k+1 . According to Equation (17), it can be known that:
[0133] ||e k+1 || ≤ ||A k - K k C k ||||e k || + ||ΔA k + ΔB k H k C k ||||x k || + ||ΔB k H k - K k ||||ν k || + ||wk ||. (31)
[0134] It is noted that The above equation can be written as:
[0135]
[0136] Applying the triangle inequality, we can obtain:
[0137]
[0138] According to the assumption ||e k || ≤ δ e,k and the Cauchy inequality, we can obtain:
[0139]
[0140] Substituting equations (2) and (3) into the above equation, we know that ||e k+1 || ≤ δ e,k+1 , and then we can obtain ||r k || ≤ δ r,k .
[0141] The proof is completed, that is, the content of Theorem 2 holds.
[0142] According to Theorem 2, the following fault detection strategy can be naturally proposed, that is:
[0143]
[0144] where ||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 above method of selecting the detection threshold of the closed-loop residual generator designed by Theorem 1, 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 occurring at time k k satisfies the following equation:
[0147]
[0148] then this fault can be detected at time k + 1.
[0149] This embodiment proves Theorem 3 as follows:
[0150] In the case of a fault occurring at time k, from equation (29), we know that:
[0151]
[0152] According to the triangle inequality, it can be known that:
[0153]
[0154] According to the Cauchy inequality, it can be known that:
[0155]
[0156] Further using the triangle inequality for the above formula, we can get:
[0157]
[0158] Considering the upper bounds of various model uncertainties and disturbances, we can get:
[0159]
[0160] In the formula, is the upper bound of system perturbation and noise; is the measurement noise ν k the upper bound of the norm.
[0161] Substituting Equation (36) into the above formula, we can get:
[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 + 1)-th moment; δ r,k+1 is the fault detection threshold at the (k + 1)-th moment;
[0164] The proof of Theorem 3 is completed, that is, the fault can be detected.
[0165] The present invention focuses on solving the closed-loop fault diagnosis problem of linear discrete systems with model uncertainties. By introducing a closed-loop residual generator of a robust least squares method, remarkable innovative achievements have been made. The design of the closed-loop residual generator realizes as much decoupling as possible 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 and verify its effectiveness and reliability in practical applications, as follows:
[0167] For researchers in the field to better understand the implementation of the present invention, 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 class of three - tank systems, considering a system with the following parameters:
[0175]
[0176] State noise w k and measurement noise ν k are independent of each other and uniformly distributed over [-1×10 -6 , 1×10 -6 .
[0177] The system model uncertainty is:
[0178]
[0179]
[0180] where, |δ A | < 1.0×10 -4 , |δ B | < 1.0×10 -4 .
[0181] Considering the system additive fault f in the following form k , then:
[0182]
[0183] According to the simulation results, it can be seen that Figure 2Shows the residuals obtained by the closed-loop residual generator proposed in Theorem 1 and the fault detection threshold calculated by Theorem 2. From Figure 2 it can be seen that the proposed closed-loop fault diagnosis strategy can effectively detect the possible additive faults in the system.
[0184] Therefore, this embodiment has successfully completed the fault detection of the closed-loop system based on the novel robust least squares residual generator. This shows that this research has not only made important breakthroughs theoretically, 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 provided beneficial patent effects for related technical fields.
[0185] Those of ordinary skill in the art can understand that all or part of the steps in implementing the methods of the above embodiments can be completed by instructing relevant hardware through a program. Therefore, this application can adopt the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware aspects. Moreover, this application can adopt 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 code.
[0186] Each embodiment in this specification is described in a progressive manner. The key point of each embodiment is to illustrate the differences from other embodiments. The same or similar parts among 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 embodiments have introduced the present invention in detail. Specific examples are used in this article to elaborate on the principles and implementation methods of the present invention. The descriptions of the above embodiments are only used to help understand the method and its core idea of the present invention; at the same time, for those of ordinary skill in the art, according to the idea of the present invention, there will be changes in the specific implementation methods and application scopes. In summary, the content of this specification should not be construed as a limitation to 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. The specific process 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, is the estimated error variance P when there is no fault k An upper bound of 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; A k , C k are all known matrices with appropriate dimensions in linear discrete systems; V k is the measurement noise ν k The covariance matrix of k is the state noise w k The covariance matrix of 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; H k is a known matrix with appropriate dimension in the linear discrete system; Ω k is the system state x k The covariance matrix of According to Theorem 1, the closed-loop residual generator and gain K are determined k ,as follows: In the formula, represents the estimated value of the state at the k+1th moment; B k is a known matrix with appropriate dimension in the linear discrete system; To measure the output; is the control input; represents the estimated value of the state at the kth moment; 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; 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, the measurement noise ν k and state noise w 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; 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 1, 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 known matrix with appropriate dimension in the linear discrete system; 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.
4. The method for detecting faults in an uncertain random time-varying system according to claim 3, characterized in that: The analysis of fault detectability is specifically as follows: If the fault f occurs at time k k Satisfy the following formula: Then at time k+1, the fault f k Can be detected; In the formula, is the measurement noise ν k The upper bound of the norm.
5. 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.