A Fault Detection Method Based on Set Theory and Unknown Input Observer

Abstract

The present invention discloses a fault detection method based on set theory and an unknown input observer, comprising the following steps: S1: establishing a state-space model of a system to be detected with bounded random perturbations and measurement noise; S2: constructing an unknown input observer with unknown parameters based on the state-space model of the system to be detected; S3: establishing state estimation error and residual iterative equations based on the system model and the observer model; S4: deriving iterative formulas for the state estimation error set and the residual set based on the iterative equations; S5: establishing an optimization problem for minimizing the residual set size, and obtaining observer parameters that minimize the residual set size; S6: calculating the minimum residual set based on the observer parameters, and performing fault detection. The present invention provides a fault detection method framework that unifies set-theoretic unknown input observers and set-valued observers, achieving lower conservatism, requiring fewer conditions for the system to be detected, and having a wider range of applicability.

Description

Technical Field

[0001] The present invention relates to a fault detection technology, in particular to a fault detection technology based on set theory and an unknown input observer. Background Art

[0002] With the continuous innovation and development of modern industrial technology, industrial production systems are becoming increasingly automated, large-scale, and complex. At the same time, the potential for failures and risks within the system is also increasing. Therefore, highly reliable and real-time system model-based fault detection technology is of great significance for industrial applications with high safety and reliability requirements, such as military production, nuclear power facilities, chemical metallurgy, and aerospace. However, model-based fault detection technology relies on a system model of the object to be detected. However, due to the presence of disturbances, noise, and uncertainty, it is impossible to obtain an accurate model of the real system during the actual modeling process. Therefore, model-based fault detection solutions must be able to handle the impact of disturbances, noise, and modeling errors on fault detection. This technology is called robust fault detection.

[0003] From the 1980s to the early 21st century, most robust fault detection schemes employed what can be collectively referred to as active robust fault detection techniques. These schemes primarily employed methods to actively decouple the effects of noise, disturbances, and uncertainty on fault detection. Representative approaches during this period included unknown input observers, eigenstructure configuration, optimal parity relations, and frequency-domain design methods. However, active robust fault detection techniques aimed to completely decouple the effects of noise, disturbances, and uncertainty, resulting in relatively stringent application requirements and limited application scenarios. In recent years, advances in set theory, probability theory, and numerical optimization techniques have led to the development of a new paradigm in robust fault detection: passive fault detection techniques that are tolerant to noise, disturbances, and uncertainty. These techniques can derive reliable fault detection thresholds in real time while tolerating disturbances and noise, and therefore hold great promise for application in numerous industrial production processes.

[0004] Existing passive fault detection technologies are mainly divided into two categories: one is a random fault detection scheme based on the assumption that the disturbance and noise distribution is known. Representative methods include fault detection technology based on Kalman filters. The advantage of this type of scheme is that it can obtain less conservative fault detection results. However, the disadvantage is that it requires prior information on the disturbance and noise distribution. This requirement is difficult to achieve in some large and complex systems, and therefore its scope of application is limited.

[0005] Another type of passive fault detection scheme is a deterministic fault detection scheme based on unknown but bounded disturbance and noise distributions. Representative methods include set-theoretic unknown input observers and set-valued observers. The advantage of this type of scheme is that it does not require prior information about the disturbance and noise distribution and has a wider range of applications. The disadvantage is that the obtained fault detection results are more conservative. For example, the set-theoretic unknown input observer is more conservative when the system is mainly affected by measurement noise, while the set-valued observer is more conservative when the system is mainly affected by random disturbances. Summary of the Invention

[0006] The purpose of the present invention is to solve the technical problem that such fault detection schemes based on unknown but bounded disturbance and noise distributions are highly conservative, and to provide a fault detection method based on set theory and unknown input observer.

[0007] To achieve the above object, the present invention adopts the following technical solutions:

[0008] A fault detection method based on set theory and unknown input observer includes the following steps:

[0009] S1: Establish a state space model of the system to be detected with bounded random perturbations and measurement noise;

[0010] S2: Construct an unknown input observer with unknown parameters according to the state space model of the system to be detected;

[0011] S3: Establish the state estimation error and residual iterative equation based on the system model and observer model;

[0012] S4: Derive the iterative formula of state estimation error set and residual set according to the iterative equation;

[0013] S5: Establish an optimization problem to minimize the size of the residual set and find the observer parameters that minimize the size of the residual set;

[0014] S6: Calculate the minimum residual set based on the observer parameters and perform fault detection.

[0015] In some embodiments of the invention, the bounded random perturbation and measurement noise in step S1 are characterized by a set of centrally symmetric polyhedrons.

[0016] In some embodiments of the invention, the optimization problem in step S5 is an optimization problem of minimizing the Frobenius norm size of the residual set.

[0017] In some inventive embodiments, step S1 includes the following steps:

[0018] S1-1: Set up a linear discrete time-invariant system of the following form:

[0019]

[0020] in, and are all parameter matrices of the system, where n x , n y , n u , n ω and n η are the dimensions of the system state, system output, system input, random disturbance and measurement noise vector respectively, and the parameter matrix (A, C) is detectable. is the input of the system at the kth moment, is the output of the system at time k, is the state of the system at time k, is the random disturbance suffered by the system at the kth moment, is the measurement noise of the system at the kth moment.

[0021] S1-2: The initial state of the system, random perturbations and measurement noise are all considered to have bounded energy and are characterized by a set of centrally symmetric polyhedrons, denoted as and For any central symmetric polyhedron set Z=<p,G> , p is the center vector of the set, G is the generator matrix of the set, and the central symmetric polyhedron has the following two properties:

[0022] (1)

[0023] (2)

[0024] in, and are all centrally symmetric polyhedrons, M represents a linear transformation matrix with corresponding dimensions, and the operator is the Minkowski sum, and the Minkowski sum of sets X and Y is defined as

[0025] In some inventive embodiments, the method of step S2 includes:

[0026] Using the input vector u in the system (1) in step S1-1 k With the output vector y k , construct an unknown input observer of the following form:

[0027]

[0028] Among them, N, T, K and H are the observer parameters whose parameters are to be determined, z k is the state of the unknown input observer at the kth moment, is the state estimate of the system by the unknown input observer at the kth moment, is the output estimate of the system by the unknown input observer at the kth moment.

[0029] In some inventive embodiments, step S3 includes the following steps:

[0030] S3-1: Define the observer's estimation error and residual of the system state:

[0031]

[0032] Among them, e k is the state estimation error of the system by the observer, r k is the residual signal output by the observer to the system;

[0033] S3-2: Let the unknown input observer parameter K = K1 + K2. According to the definition of (3) in step S3-1, the state estimation error of the unknown input observer for the system (1) in step S1-1 can be expressed as:

[0034]

[0035] S3-3: Let the unknown input observer parameters satisfy the following constraints:

[0036]

[0037] Then the formula (4) in step S3-2 can be further transformed into:

[0038] e k+1 =(A-HCA-K1C)e k +(E-HCE)ω k -HFη k+1 -K1Fη k . (6)

[0039] S3-4: According to the definition of the residual signal in step S3-1 (3), the residual of the unknown input observer can be expressed as:

[0040] r k =Ce k +Fη k . (7)

[0041] In some inventive embodiments, step S4 includes the following steps:

[0042] S4-1: Using the state estimation error expression shown in formula (6) in step S3-3, the iterative calculation formula of the state estimation error set can be obtained:

[0043]

[0044] Among them, E k is the state estimation error set of the system (1) at step k, and it is assumed that the initial state estimation error set E0 is a centrally symmetric polyhedron set.

[0045] S4-2: Using the residual expression shown in formula (7), the iterative calculation formula of the residual set can be obtained:

[0046]

[0047] Among them, R k is the residual set of the k-th step system (1).

[0048] In some inventive embodiments, the following criteria are employed to achieve robust fault detection:

[0049]

[0050] This principle is Figure 2a and Figure 2b As shown, if the residual signal r is detected k Located in the residual set R k Outside, such as Figure 2b As shown, it can be confirmed that the system has failed. On the contrary, if the residual signal r k Falls into the residual set R k Inside, such as Figure 2a If the error is shown, it is considered that no system fault is detected.

[0051] Since the residual set R k is the residual signal set r containing random disturbance and measurement noise k Therefore, criterion (10) is only a sufficient condition for the system to fail. In other words, even if The system may still be in a faulty state. Therefore, we should try to optimize the residual set R k In order to reduce the conservatism of fault detection, the size of the residual set R is used. k Size: For a set of centrally symmetric polyhedra Its Frobenius norm size is defined as where the operator ||·|| F represents the Frobenius norm.

[0052] In some inventive embodiments, step S5 includes the following steps:

[0053] S5-1: Establish the following optimization problem:

[0054]

[0055] Among them, H k and K1，k are the optimal Frobenius norm size values ​​of the residual set of the unknown input observer parameters H and K1 at time k, respectively. The other parameters of the observer are determined by the constraints described in formula (5);

[0056] S5-2: According to the state estimation error set (8) in step S4-1 and the residual set iterative formula (9) in step S4-2, and considering that the initial state estimation error set E0 of the system, the random disturbance set W and the measurement noise set V are all centrally symmetric polyhedrons, it is easy to obtain the state estimation error set E at each moment k and the residual set R k are all centrally symmetric polyhedrons, which are denoted as and Then the state estimation error set E at time k+1 is k and the residual set R k The generator matrices can be expressed as:

[0057]

[0058]

[0059] in,

[0060]

[0061]

[0062] S5-3: To solve the problem (11) in step S5-1, define J=G W (G W ) T and P=CC T , according to formula (13) in step S5-2, ψ(R k+1 ) can be further expressed as:

[0063]

[0064] in,

[0065]

[0066] Considering tr(FG V (G V ) T F T ) is a constant, so the problem (11) in step S5-1 can be equivalent to:

[0067]

[0068] Since problem (16) is an unconstrained convex optimization problem, the gain parameter of the unknown input observer at the kth moment can be obtained analytically by solving the following equation:

[0069]

[0070] The result is:

[0071]

[0072] in,

[0073] Among them, if and only if the matrix When the input is non-singular, the optimal unknown input observer parameters shown in formula (18) can be obtained: and are four positive semidefinite matrices ( ΛΛ T , ΓΓ T and ), is a singular matrix if and only if the null space intersection of these four positive semidefinite matrices is non-empty except for the origin. Therefore, in a practical system, the matrix It is often a non-singular matrix.

[0074] In some inventive embodiments, step S6 includes the following steps:

[0075] S6-1: Substitute the optimal unknown input observer parameter expression shown in equation (18) in step S5-3 into equation (15) in step S5-3 and eliminate Item, we can get:

[0076]

[0077] The quadratic matrix equation shown in formula (19) is the Riccati difference equation. When k→∞, S ∞ =S k+1 =S k , we can now get the following algebraic Riccati equation:

[0078]

[0079] S6-2: According to the theory of solving the algebraic Riccati equation, when the system (A, C) is detectable, there exists a certain S * ∈{S ∞} and the corresponding L * satisfy:

[0080]

[0081] So that the matrix The eigenvalue of is within the unit circle, so when the parameter of the unknown input observer is L * =[H ∞ K 1，∞ ], the unknown input observer is stable, and L * is the optimal steady-state parameter of the unknown input observer at infinity in the time domain;

[0082] S6-3: Solve the algebraic Riccati equation (20) in step S6-1 to obtain S ∞ Then, substitute it into formula (21) in step S6-2 to obtain L * =[H ∞ K 1，∞ ], where H ∞ and K 1，∞ That is, the steady-state optimal values ​​of the observer parameters H and K at infinity in the time domain;

[0083] S6-4: Given an initial state, estimate the error set E0 and obtain H ∞ and K 1，∞ After that, the initial state estimation error set E0 and H ∞ and K 1，∞ Substitute into equation (8) in step S4-1 and iteratively calculate the state estimation error set E at each moment k , further according to formula (9) in step S4-2, the residual set R under the minimum Frobenius norm at each moment can be obtained k .

[0084] In actual use, the Riccati equation shown in equation (20) in step S6-1 can be solved by a corresponding numerical method.

[0085] The present invention has the following beneficial effects:

[0086] The present invention provides a fault detection method framework that unifies set-theoretic unknown input observers and set-valued observers, which can achieve lower conservatism, has lower requirements on the conditions that the detection system needs to meet, and has a wider range of applicability. BRIEF DESCRIPTION OF THE DRAWINGS

[0087] Figure 1 This is a flow chart of a fault detection method using set theory and an unknown input observer according to an embodiment of the present invention;

[0088] Figure 2a is a schematic diagram showing that no fault is detected in an embodiment of the present invention;

[0089] Figure 2b is a schematic diagram showing a fault detected in an embodiment of the present invention;

[0090] Figure 3ais a schematic diagram showing fault detection on the first component of the system output in an embodiment of the present invention;

[0091] Figure 3b is a schematic diagram showing fault detection on the second component of the system output in an embodiment of the present invention;

[0092] Figure 4 This is a diagram illustrating a visualization of residual sets of a system using set theory and an unknown input observer at times K=1 to k=25 according to an embodiment of the present invention;

[0093] Figure 5 1 is a schematic diagram comparing the conservatism of the embodiment of the present invention with that of the existing set-theoretic unknown input observer and set-valued observer under different disturbance and noise levels and Frobenius norm metrics;

[0094] Figure 6 3 is a schematic diagram comparing the conservatism of the embodiment of the present invention with that of the existing set-theoretic unknown input observer and set-valued observer at different times and Frobenius norm metrics. DETAILED DESCRIPTION

[0095] The following is a detailed description of the embodiments of the present invention. It should be emphasized that the following description is only exemplary and is not intended to limit the scope of the present invention and its application.

[0096] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be understood to indicate or imply relative importance or implicitly specify the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of such features. In the description of the embodiments of the present invention, "plurality" means two or more, unless otherwise specifically defined.

[0097] In order to overcome the defect of high conservatism of such fault detection schemes based on unknown but bounded disturbance and noise distribution in the above background technology, the present invention provides a fault detection method based on set theory and unknown input observer.

[0098] The following further describes the embodiments of the present invention in conjunction with the accompanying drawings. It should be emphasized that the embodiment described below is only an exemplary embodiment of the present invention, and the present invention is not limited to this embodiment.

[0099] Figure 1 FIG. 1 is a fault detection embodiment using set theory and an unknown input observer provided by the present invention, which includes the following steps:

[0100] S1: Establish a state space model of the system to be detected with bounded random perturbations and measurement noise, and the bounded random perturbations and measurement noise are characterized by a set of centrally symmetric polyhedrons;

[0101] S2: Construct an unknown input observer with unknown parameters according to the state space model of the system to be detected;

[0102] S3: Establish the state estimation error and residual iterative equation based on the system model and observer model;

[0103] S4: Derive the iterative formula of state estimation error set and residual set according to the iterative equation;

[0104] S5: Establish an optimization problem to minimize the Frobenius norm of the residual set and find the observer parameters that minimize the residual set size;

[0105] S6: Calculate the minimum Frobenius norm residual set based on the observer parameters and perform fault detection.

[0106] Step 5 is the core step in observer parameter design. Existing fault detection techniques based on set theory and observers differ primarily in the observer structure in step 2 and the observer parameter design method in step 5. The greatest contribution of this invention lies in proposing a framework that unifies existing methods and results in a less conservative approach. Therefore, the process flow differs little from existing techniques.

[0107] The following further describes embodiments of the present invention.

[0108] Consider a linear discrete time-invariant system of the following form:

[0109]

[0110] in, and are all parameter matrices of the system, where n x , n y , n u , n ω and n η are the dimensions of the system state, system output, system input, random disturbance and measurement noise vector respectively, and the parameter matrix (A, C) is detectable. is the input of the system at the kth moment, is the output of the system at time k, is the state of the system at time k, is the random disturbance suffered by the system at the kth moment, is the measurement noise of the system at the kth moment.

[0111] In this embodiment, the initial state of the system, random perturbations and measurement noise are all considered to have bounded energy and are characterized by a set of centrally symmetric polyhedrons, which are denoted as and For any central symmetric polyhedron set Z=<p,G> , p is the center vector of the set, and G is the generator matrix of the set. It should be noted that the central symmetric polyhedron has the following two properties:

[0112] (1)

[0113] (2)

[0114] in, and are all centrally symmetric polyhedrons, M represents a linear transformation matrix with corresponding dimensions, and the operator is the Minkowski sum, and the Minkowski sum of sets X and Y is defined as

[0115] Furthermore, using the input vector u in system (1) k With the output vector y k , construct an unknown input observer of the following form:

[0116]

[0117] Among them, N, T, K and H are the observer parameters whose parameters are to be determined, z k is the state of the unknown input observer at the kth moment, is the state estimate of the system by the unknown input observer at the kth moment, is the output estimate of the system by the unknown input observer at the kth moment.

[0118] Define the observer's estimation error and residual of the system state:

[0119]

[0120] Among them, e k is the state estimation error of the system by the observer, r k is the residual signal output by the observer to the system.

[0121] Furthermore, let the unknown input observer parameter K = K1 + K2. According to the definition in (3), the state estimation error of the unknown input observer for the system (1) can be expressed as:

[0122]

[0123] At this time, let the unknown input observer parameters satisfy the following constraints:

[0124]

[0125] Then the state estimation error formula in formula (4) can be further transformed into:

[0126] e k+1 =(A-HCA-K1C)e k +(E-HCE)ω k -HFη k+1 -K1Fη k . (6)

[0127] According to the definition of residual signal in (3), the residual of the unknown input observer can be expressed as:

[0128] r k =Ce k +Fη k . (7)

[0129] Using the state estimation error expression shown in formula (6), the iterative calculation formula of the state estimation error set can be obtained:

[0130]

[0131] Among them, E k is the state estimation error set of the system (1) at step k, and it is assumed that the initial state estimation error set E0 is a centrally symmetric polyhedron set. Further, using the residual expression shown in formula (7), the iterative calculation formula of the residual set can be obtained:

[0132]

[0133] Among them, R k is the residual set of the k-th step system (1).

[0134] In an embodiment of the present invention, the following criteria are adopted to implement robust fault detection:

[0135]

[0136] This principle is Figure 2a and Figure 2b As shown, if the residual signal r is detected k Located in the residual set R k Outside, such as Figure 2b As shown, it can be confirmed that the system has failed. On the contrary, if the residual signal r k Falls into the residual set R k Inside, such as Figure 2a If the error is shown, it is considered that no system fault is detected.

[0137] Since the residual set R k is the residual signal r containing random disturbance and measurement noise k Therefore, criterion (10) is only a sufficient condition for the system to fail. In other words, even if r k ∈R k The system may still be in a faulty state. Therefore, we should try to optimize the residual set R k In order to reduce the conservatism of fault detection, the size of the residual set R is used. k Size: For a set of centrally symmetric polyhedra Its Frobenius norm size is defined as where the operator ||·|| F represents the Frobenius norm.

[0138] Based on the above analysis, in order to reduce the conservatism of the fault detection method described in the embodiment of the present invention, the following optimization problem can be established:

[0139]

[0140] Among them, H k and K 1，k are the optimal Frobenius norm size values ​​of the residual set of the unknown input observer parameters H and K1 at time k, respectively. The other parameters of the observer are determined by the constraints described in formula (5).

[0141] According to the iterative formulas (8) and (9) of the state estimation error set and residual set, and considering that the initial state estimation error set E0, the random disturbance set W and the measurement noise set V of the system are all centrally symmetric polyhedrons, it is easy to obtain the state estimation error set E at each moment k and the residual set R k are all centrally symmetric polyhedrons, which are denoted as and Then the state estimation error set E at time k+1 is k and the residual set R k The generator matrices can be expressed as:

[0142]

[0143]

[0144] in:

[0145]

[0146]

[0147] To solve problem (11), we define J=G W (G W ) T and P=CC T According to formula (13), ψ(R k+1 ) can be further expressed as:

[0148]

[0149] in:

[0150]

[0151] Considering tr(FG V (G V ) T F T ) is a constant, so problem (11) can be equivalent to:

[0152]

[0153] Since problem (16) is an unconstrained convex optimization problem, the gain parameter of the unknown input observer at the kth moment can be obtained analytically by solving the following equation:

[0154]

[0155] The result is:

[0156]

[0157] in,

[0158] It should be noted that if and only if the matrix When the input is non-singular, the optimal unknown input observer parameters shown in formula (18) can be obtained: and are four positive semidefinite matrices ( ΛΛ T , ΓΓ T and ), is a singular matrix if and only if the null space intersection of these four positive semidefinite matrices is non-empty except for the origin. Therefore, in a practical system, the matrix It is often a non-singular matrix.

[0159] Furthermore, the optimal unknown input observer parameters shown in Equation (18) are Substitute the expression into equation (15) and eliminate Item, we can get:

[0160]

[0161] The quadratic matrix equation shown in formula (19) is the Riccati difference equation. When k→∞, S ∞ =S k+1 =S k , we can now get the following algebraic Riccati equation:

[0162]

[0163] According to the relevant theories of solving the algebraic Riccati equation, when the system (A, C) is detectable, there exists a certain S * ∈{S ∞} and the corresponding L * satisfy:

[0164]

[0165] So that the matrix The eigenvalue of is within the unit circle, so when the parameter of the unknown input observer is L * =[H ∞ K 1，∞ ], the unknown input observer is stable, and L * is the optimal steady-state parameter of the unknown input observer at infinity in the time domain. It should be noted that the algebraic Riccati equation cannot usually be solved analytically. In practice, the Riccati equation shown in Equation (20) can be solved numerically.

[0166] Solve the algebraic Riccati equation (20) in step S6-1 to obtain S ∞ Then, substitute it into formula (21) in step S6-2 to obtain L * =[H ∞ K 1，∞ ], where H ∞ and K 1，∞ That is, the steady-state optimal values ​​of the observer parameters H and K at infinity in the time domain;

[0167] The initial state estimation error set E0 and H ∞ and K 1，∞ Substitute into equation (8) in step S4-1 and iteratively calculate the state estimation error set E at each moment k , further according to formula (9) in step S4-2, the residual set R under the minimum Frobenius norm at each moment can be obtained k .

[0168] Verification example:

[0169] To verify the effectiveness of the method described in the embodiment of the present invention, consider that system (1) has the following parameters:

[0170]

[0171]

[0172] And the random perturbation set W=<0,0.1I 4×4 >, measurement noise set V=<0,0.1I 2×2 >, the initial state estimate set E0 of the unknown input observer = <0, 0.01I 2×2 >.

[0173] According to the method described in the embodiment of the present invention, the optimal parameters of the unknown input observer at infinity in the time domain should be designed as:

[0174]

[0175]

[0176]

[0177] Assume that the initial state of the system is x0 = [0, 0] T , enter u k =[3sin(0.5k), 5sin(0.7k)] T , and the system does not fail during the period from k = 1 to k = 20. At k = 21, the system actuator fails, causing the system parameter matrix B to become

[0178]

[0179] For ease of explanation, define R k (1) and R k (2) is the residual set R k The projection onto the first and second dimensional components, and r k =[r k (1), r k (2)] T .from Figure 3a and Figure 3b It can be seen from FIG that when K=21, the system fails, and when k=22, the unknown input observer detects that the system fails on the second dimension component, thereby proving the effectiveness of the embodiment of the present invention. Figure 4 Then it is the residual set R of the system during k=1 to k=25 k A visual representation of .

[0180] Comparative Example:

[0181] Furthermore, to compare the conservatism of the proposed fault detection algorithm with the existing set-theoretic unknown input observer and set-valued observer methods, consider the system (1) with the following parameters:

[0182]

[0183]

[0184] And the measurement noise set V=<0,0.1I 2×2 >, the initial state estimate set E0 of the unknown input observer = <0, 0.01I 2×2 >. Assume that the initial state of the system is x0 = [0, 0] T , enter u k =[3sin(0.5k), 5sin(0.7k)] T Let the dimension of the first component of the random perturbation set be W1=<0, αI 1×1 >, and the noise sets of the other three component dimensions are W2=<0, αI 3×3 >. The size of parameter α can reflect the relative degree to which the system is affected by random disturbances and measurement noise: when parameter α is large, the system is mainly affected by random disturbances. When parameter α is small, the system is mainly affected by measurement noise.

[0185] For ease of explanation, and They are defined as the residual sets generated by the three algorithms described in this paper, the set theory unknown input observer and the set value observer at the kth step of the system. Figure 5 This is a comparison of the conservatism of the three methods under different perturbation and noise levels and Frobenius norm metrics. The time selected in the figure is k = 40 (without loss of generality, when k > 5, the residual sets generated by the three methods all converge). When α is small, it means that the system is mainly affected by measurement noise, and when α is large, it means that the system is mainly affected by random perturbations. It can be seen that the algorithm described in this article is the least conservative. Furthermore, Figure 6 The comparison of the conservatism of the three methods at different times and Frobenius norm is shown in the figure. α=0.2 is selected. Figure 6 It is also verified that the algorithm described in this article is the least conservative among the three types of algorithms.

[0186] The above is a further detailed description of the present invention in conjunction with specific preferred embodiments, and the specific implementation of the present invention should not be considered to be limited to these descriptions. Those skilled in the art will recognize that several equivalent substitutions or obvious variations can be made without departing from the scope of the present invention, and that any equivalent performance or application should be considered to fall within the scope of protection of the present invention.

Claims

1. A fault detection method based on set theory and unknown input observer, comprising the following steps: S1: Establish a state space model of the system to be detected with bounded random perturbations and measurement noise; the initial state, bounded random perturbations and measurement noise of the state space model are all characterized by a set of centrosymmetric polyhedra; S2: Construct an unknown input observer with unknown parameters according to the state space model of the system to be detected; S3: Establish the state estimation error and residual iterative equation based on the system model and observer model; S4: Derive the state estimation error set formula and the residual set iteration formula based on the iterative equation; S5: Establish an optimization problem to minimize the size of the residual set, and use numerical methods to obtain the observer parameters that minimize the size of the residual set; S6: Substitute the initial state error set and the observer parameters into the state estimation error set formula, iteratively calculate the state estimation error set, substitute the state estimation error set into the residual set iterative formula to calculate the minimum residual set, and perform fault detection; The initial state error set is characterized by a central polyhedron.

2. The method according to claim 1, characterized in that The optimization problem in step S5 is to minimize the Frobenius norm of the residual set.

3. The method according to claim 1, characterized in that Step S1 includes the following steps: S1-1: Set up a linear discrete time-invariant system of the following form: in, and are all parameter matrices of the system, where n x , n y ,n u ,n ω and n η are the dimensions of the system state, system output, system input, random disturbance and measurement noise vectors respectively, and the parameter matrix (A, C) is detectable; is the input of the system at the kth moment, is the output of the system at time k, is the state of the system at time k, is the random disturbance suffered by the system at the kth moment, is the measurement noise of the system at the kth moment; S1-2: The initial state of the system, random perturbations and measurement noise are all considered to have bounded energy and are characterized by a set of centrally symmetric polyhedrons, denoted as and For any central symmetric polyhedron set Z=<p,G> p is the center vector of the set, G is the generator matrix of the set, and the central symmetric polyhedron has the following two properties: (1) (2) in, and are all centrally symmetric polyhedrons, M represents a linear transformation matrix with corresponding dimensions, and the operator is the Minkowski sum, and the Minkowski sum of sets X and Y is defined as 4. The method according to claim 3, characterized in that The method of step S2 includes: Using the input vector u in the system (1) in step S1-1 k With the output vector y k , construct an unknown input observer of the following form: Among them, N, T, K and H are the observer parameters whose parameters are to be determined, z k is the state of the unknown input observer at the kth moment, is the state estimate of the system by the unknown input observer at the kth moment, is the output estimate of the system by the unknown input observer at the kth moment.

5. The method according to claim 4, characterized in that Step S3 includes the following steps: S3-1: Define the observer's estimation error and residual of the system state: Among them, e k is the state estimation error of the system by the observer, r k is the residual signal output by the observer to the system; S3-2: Let the unknown input observer parameter K = K1 + K2. According to the definition of (3) in step S3-1, the state estimation error of the unknown input observer for the system (1) in step S1-1 can be expressed as: S3-3: Let the unknown input observer parameters satisfy the following constraints: Then the formula (4) in step S3-2 can be further transformed into: by k+1 =(A-HCA-K1C)e k +(E-HCE)ω k -HFη k+1 -K1Fη k . (6) S3-4: According to the definition of the residual signal in (3) in step S3-1, the residual of the unknown input observer can be expressed as: R k =What k +Fη k . (7)。 6. The method according to claim 5, characterized in that Step S4 includes the following steps: S4-1: Using the state estimation error expression shown in formula (6) in step S3-3, the iterative calculation formula of the state estimation error set can be obtained: Among them, E k is the state estimation error set of the system (1) at step k, and it is assumed that the initial state estimation error set E0 is a centrally symmetric polyhedron set; S4-2: Using the residual expression shown in formula (7), the iterative calculation formula of the residual set can be obtained: Among them, R k is the residual set of the k-th step system (1).

7. The method according to claim 6, characterized in that Step S5 includes the following steps: S5-1: Establish the following optimization problem: Among them, H k and K 1，k are the optimal Frobenius norm size values ​​of the residual set of the unknown input observer parameters H and K1 at time k, respectively. The other parameters of the observer are determined by the constraints described in formula (5); S5-2: According to the state estimation error set (8) in step S4-1 and the residual set iterative formula (9) in step S4-2, and considering that the initial state estimation error set E0 of the system, the random disturbance set W and the measurement noise set V are all centrally symmetric polyhedrons, it is easy to obtain the state estimation error set E at each moment k and the residual set R k are all centrally symmetric polyhedrons, which are denoted as and Then the state estimation error set E at time k+1 is k and the residual set R k The generator matrices can be expressed as: in, S5-3: To solve the problem (11) in step S5-1, define J=G W (G W ) T and P=CC T , according to formula (13) in step S5-2, ψ(R k+1 ) can be further expressed as: in, Considering tr(FG V (G V ) T F T ) is a constant, so the problem (11) in step S5-1 can be equivalent to: Since problem (16) is an unconstrained convex optimization problem, the gain parameter of the unknown input observer at the kth moment can be obtained analytically by solving the following equation: The result is: in, Among them, if and only if the matrix When it is non-singular, the optimal unknown input observer parameters shown in formula (18) can be obtained; are four positive semidefinite matrices ( ΛΛ T , ΓΓ T and ), is a singular matrix if and only if the null space intersection of these four positive semidefinite matrices is nonempty except for the origin; therefore, in a practical system, the matrix It is often a non-singular matrix.

8. The method according to claim 7, characterized in that Step S6 includes the following steps: S6-1: Input the optimal unknown observer parameters shown in equation (18) in step S5-3 Substitute the expression into equation (15) in step S5-3 and eliminate Item, we can get: The quadratic matrix equation shown in formula (19) is the Riccati difference equation. When k→∞, S ∞ =S k+1 =S k , we can now get the following algebraic Riccati equation: S6-2: According to the theory of solving the algebraic Riccati equation, when the system (A, C) is detectable, there exists a certain S * ∈{S ∞ } and the corresponding L * satisfy: So that the matrix The eigenvalue of is within the unit circle, so when the parameter of the unknown input observer is L * =[H ∞ K 1，∞ ], the unknown input observer is stable, and L * is the optimal steady-state parameter of the unknown input observer at infinity in the time domain; S6-3: Solve the algebraic Riccati equation (20) in step S6-1 to obtain S ∞ Then, substitute it into formula (21) in step S6-2 to obtain L * =[H ∞ K 1,∞ ], where H ∞ and K 1，∞ That is, the steady-state optimal values ​​of the observer parameters H and K at infinity in the time domain; S6-4: Combine the initial state estimation error set E0 with H ∞ and K 1，∞ Substitute into equation (8) in step S4-1 and iteratively calculate the state estimation error set E at each moment k , further according to formula (9) in step S4-2, the residual set R under the minimum Frobenius norm at each moment can be obtained k .

9. The method according to claim 8, characterized in that In actual use, the Riccati equation shown in equation (20) in step S6-1 can be solved by a corresponding numerical method.

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