Design method for high-reliability fault observer of parameter stochastic system

Abstract

The invention provides a method for designing a high-reliability fault observer of a parameter stochastic system. The method comprises the following steps: firstly, constructing a state space model of a system containing uncertain random parameters according to dynamic characteristics of the system, constructing a state observer and a fault observer containing the random parameters, and forming a linear matrix inequality SLMI containing the random parameters; and then, in combination with a system limit state function and Taylor expansion mathematical approximation of random parameters, introducing a reliability index into the SLMI and converting the reliability index into a solvable probability linear matrix inequality PLMI, thereby ensuring that fault estimation under the influence of the random parameters has relatively high reliability. And finally, gains are solved according to the PLMI and substituted into the state observer and the fault observer. According to the method, the fault detection precision and reliability of a random parameter uncertainty system can be remarkably improved, the method can be used for guiding the design of fault observers of parameter random complex systems such as aerospace, electric power and intelligent manufacturing, and good social and economic benefits are created.

Description

Technical Field

[0001] The present invention belongs to the fields of control science and engineering, fault detection and diagnosis, etc. Specifically, it relates to a design method for a highly reliable fault observer for a parameter random system, and is applicable to dynamic systems that need to perform high-precision and high-reliability fault detection and diagnosis tasks. Background Art

[0002] Fault detection and diagnosis, as an important part of the safety guarantee of control systems, has long occupied an important position in theoretical research and engineering practice. Through effective fault estimation methods, anomalies in system operation can be detected in a timely manner, and corresponding measures can then be taken to ensure the safety and stability of the system. However, the widespread parameter uncertainties in complex systems pose severe challenges to fault detection and diagnosis technologies, especially in the face of random parameter uncertainties. Due to the random nature of such uncertainties, they bring many difficulties to the design and mathematical analysis of fault estimation methods, resulting in limited applicability of traditional methods in this regard. Therefore, researching new fault observer design methods for control systems with random parameter uncertainties has certain theoretical and engineering significance.

[0003] Most existing system fault estimation methods are based on the assumption that only exogenous uncertainties exist in the system, and rarely consider the possible randomness in the internal dynamic parameters of the system. This assumption limits the practical application ability of existing methods in complex systems to a certain extent. In fact, all mathematical models are essentially simplifications and approximations of real systems. During the modeling process, due to measurement errors, environmental disturbances, or limitations of the modeling method itself, system parameters are often affected by various uncertainty factors. This random parameter uncertainty significantly reduces the reliability of traditional fault estimation methods. Without modeling and dealing with these random characteristics, it will be difficult to meet the actual requirements for the accuracy and robustness of fault estimation.

[0004] In recent years, researchers have gradually realized the important impact of random parameter uncertainties on system performance and fault detection, and proposed some preliminary solutions. However, these methods usually have the following problems: 1) Ignoring the in-depth modeling of random parameter characteristics, resulting in the accuracy of fault estimation results being difficult to meet the requirements; 2) Traditional fault observers cannot converge in the fault estimation of systems with random parameters, with low reliability; 3) Lack of full utilization of the distribution characteristics of uncertainties, and the robustness of the method needs to be improved.

[0005] In view of the above problems, it is urgent to develop a design method for a system fault observer that can effectively cope with random parameter uncertainties. This method should be able to accurately characterize the impact of random parameter uncertainties on system performance theoretically and have efficient and reliable fault detection capabilities in practical applications. By combining advanced probability analysis methods and robust control theory, it is expected to provide an effective solution for the fault diagnosis of parameter random systems and further improve the safety and reliability levels of control systems. Summary of the Invention

[0006] To overcome the defects of existing methods, a design method for a highly reliable fault observer for parameter random systems is proposed. This method mainly consists of a state observer of the system, a state observer of the fault, and a probabilistic linear matrix inequality. The proposed method can effectively solve or improve the three problems described above. At the same time, the proposed method can ensure the stability requirements and H ∞ performance index of the fault observer with high reliability (specified probability) in the fault estimation of parameter random systems. To achieve the above objectives, the technical solution adopted in the present invention is as follows:

[0007] A design method for a highly reliable fault observer for parameter random systems includes the following steps:

[0008] First step, construct a state space model of a system with uncertain random parameters according to the dynamic characteristics of the target system. The so-called uncertain random parameters refer to the internal random parameters in the system matrix and the control matrix in the state space model of the system;

[0009] Second step, design a state observer and a fault observer with random parameters, and give a linear matrix inequality with random parameters based on the Lyapunov stability criterion;

[0010] Third step, based on the structural reliability theory, introduce a reliability index under the limit function of the system to quantify the reliability of the fault observer.

[0011] Fourth step, combine the Taylor expansion, linearly approximate the uncertainty parameters, and give a probabilistic linear matrix inequality with solvable gain according to the reliability index.

[0012] Fifth step, solve the probabilistic linear matrix inequality to obtain the gains of the state observer and the fault observer and other parameters to be determined.

[0013] Furthermore, the specific steps of the first step are as follows:

[0014] Construct a system state space model with random uncertain parameters according to the dynamic characteristics of the target system, which is expressed as follows:

[0015]

[0016] In Equation (1), represents the system state vector; represents the output vector of the system; represents the system input vector; is a slow-varying or constant fault that needs to be isolated and estimated, and it satisfies the L 2 norm bounded, that is A(ρ), B(ρ), F, and G represent matrices with appropriate dimensions, where A(ρ) and B(ρ) depend on the random parameter vector ρ = [ρ 1 , ρ 2 , …, ρ N T ; C y is the output matrix that maps x(t) to y(t); g(x(t)) is a non-linear function that satisfies the norm condition represented by the known matrix U as shown in Equation (2):

[0017]

[0018] The present invention focuses on slow-varying or constant faults because actual faults usually occur suddenly in the system and remain unchanged. The present invention follows the assumption This is because in most practical cases, f(t) usually lasts for a finite time until the fault is correctly diagnosed and the system is reconfigured.

[0019] The present invention focuses on the random parameter uncertainties in the system matrix A(ρ) and the control matrix B(ρ) because they usually contain uncertain physical parameters such as mass, damping, and stiffness. These uncertain physical parameters can be modeled by random distributions, and their statistical characteristics can be obtained through actual tests.

[0020] Furthermore, the specific steps of the second step are as follows:

[0021] First, construct the following state observer

[0022]

[0023] and the fault observer as

[0024]

[0025] In Equations (3) and (4), L represents the gain to be determined for the state observer, γ i > 0, i = 1, 2 represent the gains to be determined for the fault observer, represents the estimated value of the state vector x(t), represents the estimated value of the fault vector f(t), ​Represents the estimated value of the output vector y(t).

[0026] Define the state estimation error The dynamics of the estimation error can be expressed as:

[0027]

[0028] Since the fault f(t) considered in this method is a slowly varying or constant fault, it can be approximately considered that Then:

[0029]

[0030] To ensure that the designed fault diagnosis observer satisfies the H ∞ performance in a dynamic system with random parameters, this method introduces a generalized H ∞ filter. Define as the reference time-varying output and use it as the H ∞ performance variable for the above systems (5) and (6), with the initial condition being zero. The gain of the observer should be designed to minimize the upper bound of In the H ∞ optimization problem, the classical concept of disturbance attenuation can be extended to the related fault detection and diagnosis problem, which is defined as follows:

[0031]

[0032] will make thus achieving the generalized H ∞ performance.

[0033] Therefore, the Lyapunov function for the design of a fault observer for a dynamic system with random uncertain parameters can be designed as:

[0034]

[0035] When P in equation (8) is a positive definite matrix, Φ(t) is always greater than 0. At the same time, differentiating equation (8) gives:

[0036]

[0037] In equation (9), and there is

[0038]

[0039] To make the designed fault observer asymptotically stable, the matrix in equation (9) needs to be a negative definite matrix to ensure that equation (9) is less than zero.

[0040] Combined with the generalized H performance definition given by Equation (7), the matrix ∞ is corrected to to ensure that the system meets the generalized H performance while satisfying asymptotic stability, that is, the system needs to satisfy: ∞

[0041]

[0042] In Equations (10) and (11), L = P -1 Q, where sym(*) = * + * T .

[0043] Since Π 0 has a random parameter ρ, Equation (11) is a random linear matrix inequality. Random linear matrix inequalities cannot be solved directly. Steps 3 and 4 will be introduced later to transform the random linear matrix inequality into a deterministic linear matrix inequality and then solve for the observer gain that meets the specified reliability level.

[0044] Furthermore, the specific steps of the third step are as follows:

[0045] Based on structural reliability theory and Lyapunov stability criterion, assume that the limit state function of the system is where ρ is the random parameter to be processed. If G(ρ) < 0, the system will be asymptotically stable, and this region is reliable. If G(ρ) = 0, the system is critically stable, indicating that this region reaches the limit state surface.

[0046] This method proposes a reliability index α, which can be quantified and determined by solving the shortest distance from the origin to the limit state surface in the standard normal space, that is:

[0047]

[0048] where ρ = [ρ 1 , ρ 2 , …, ρ N T represents a vector of standard Gaussian random variables.

[0049] It should be noted that the random parameter problems solved by this method are not limited to the normal / Gaussian distribution case. Through the R-F or J-C theorem, various non-Gaussian random variables can be transformed into Gaussian random variables.

[0050] With the help of the reliability index α, the reliability R s and the failure probability F s can be obtained through Equations (13) and (14):

[0051] ​R s = Φ(α), (13)

[0052] F s = Φ(-α), (14)

[0053] where Φ(·) represents the quantile of the standard normal distribution.

[0054] Furthermore, the specific steps of the fourth step are as follows:

[0055] The linear matrix inequality (11) containing random uncertain parameters can be transformed into a probabilistic linear matrix inequality with a reliability index by this method. For this purpose, the following linear approximation needs to be performed on A(ρ):

[0056]

[0057] The result of Equation (15) is obtained through the following Taylor expansion:

[0058]

[0059] where f(ρ) represents any element of A(ρ). μ ρ represents the mean of ρ, and since ρ is a vector conforming to the standard Gaussian distribution, μ ρ = 0.

[0060] Equation (15) can be rewritten as

[0061]

[0062] where A N,j represents the j-th column of matrix A N ; U A,j is a row vector, where only the j th element is 1 and the rest of the elements are 0.

[0063] Since the necessary condition for Equation (11) to be negative definite is that the first term of the leading principal minor is negative definite, that is, the condition is satisfied. If the parameter uncertainty satisfies ρ T ρ ≤ α 2 , then if and only if there exist n positive constants ε j :

[0064]

[0065] To perform the calculation more clearly, Ψ 0 (P, Q) is defined here:

[0066]

[0067] By using the Schur complement, Equation (19) can be transformed into the following linear matrix inequality:

[0068]

[0069] where, Ψ α (P, Q) represents a matrix containing the reliability index α, and E = diag{ε 1 , ε 2 , …, ε n}.

[0070] Substitute Equation (20) into Equation (11) given in Step 2 to obtain the probabilistic linear matrix inequality with the reliability index:

[0071]

[0072] Given the system and the reliability index α, Equation (21) is solvable, and the solved state observer gain L, fault observer parameters Υ 1 and Υ 2 can ensure asymptotic stability, H ∞ performance, and high reliability requirements for fault detection under the condition that the system has random uncertain parameters.

[0073] Furthermore, the specific steps of the fifth step are as follows:

[0074] Given the reliability index α, solve the probabilistic linear matrix inequality (21) to obtain the gains and other parameters to be determined of the state observer and the fault observer, and substitute the parameters into Equations (3) and (4) to obtain the state observer and the fault observer given by the present invention.

[0075] The advantages of the present invention compared with the prior art are as follows:

[0076] A design method for a highly reliable fault observer of a parameter random system involved in the present invention mainly models the possible random uncertainties in the internal dynamic parameters of the system, introduces reliability indexes, and thus realizes the design of a highly reliable fault observer. First, according to the dynamic characteristics of the system, a state space model of the system with uncertain random parameters is constructed, and then a state observer and a fault observer with random parameters are constructed to form a stochastic linear matrix inequality (SLMI). Then, combined with the system limit state function and the Taylor expansion mathematical approximation of the random parameters, the reliability index is introduced into the SLMI and transformed into a probabilistic linear matrix inequality (PLMI) with a reliability index to ensure that the fault estimation under the influence of random parameters has high reliability. Finally, the gain is solved according to the PLMI and substituted into the state observer and the fault observer. The present invention can significantly improve the fault detection accuracy and reliability of a system with random parameter uncertainties, and can be used to guide the design of fault observers for parameter random complex systems such as aerospace systems, power systems, and intelligent manufacturing systems, thus creating good social and economic benefits. Description of the Drawings

[0077] Figure 1 It is the implementation path of the technical solution of a design method for a highly reliable fault observer of a parameter random system of the present invention;

[0078] Figure 2 It is the comparison of the effect of constant fault estimation in the example verification;

[0079] Figure 3 It is the comparison of the effect of time-varying fault estimation in the example verification. Detailed Embodiment

[0080] In order to make the purpose, technical solution and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0081] The present invention takes a general class of dynamic systems with random parameters as an example to illustrate the specific implementation of the system and method.

[0082] As Figure 1 shown, the specific implementation steps of a design method for a highly reliable fault observer of a parameter random system of the present invention are as follows:

[0083] Step 1: Construct a state - space model of a system with uncertain random parameters according to the dynamic characteristics of the target system. The so - called uncertain random parameters refer to the internal random parameters in the system matrix and the control matrix in the state - space model of the system;

[0084] Given a second - order system with random parameters:

[0085]

[0086] where θ = [θ 1 , θ 2 , θ 3 , θ 4 contains random parameters following a normal distribution, with a mean of μ θ = [- 1.5, 0, - 2, 0] and a standard deviation of σ θ = [1.8, 0.5, 0.5, 2.3]. The upper bound of the non - linear term G(x(t)) is denoted as U = diag{0, 0.5}, where diag{·} represents a diagonal matrix. Referring to the generalized H ∞ The output performance is given by Equation (7), where C 1 = [1 1] T and C 2 = 0.1.

[0087] Step 2: Design a state observer and a fault observer with random parameters, and give a linear matrix inequality with random parameters based on the Lyapunov stability criterion;

[0088] The designed state observer and fault observer are shown in Equations (3) and (4).

[0089] Step 3: Based on the structural reliability theory, introduce a reliability index under the limit function of the system to quantify the reliability of the fault observer.

[0090] Set the reliability index as α = 2.0.

[0091] Step 4: Combine the Taylor expansion, linearly approximate the uncertainty parameters, and give a probability linear matrix inequality with a solvable gain according to the reliability index.

[0092] Through Equations (15 - 17) and the standardization of the normal distribution of θ, we can obtain:

[0093]

[0094] Step 5: Solve the probability linear matrix inequality to obtain the gains of the state observer and the fault observer and other parameters to be determined.

[0095] Under the specified reliability index α, the design method of a fault observer for a parametric uncertainty system with high reliability can be applied by solving the LMI of Equation (21). The obtained observer gain results in this section are all verified by the Monte Carlo sampling method, where the number of samples is set to N R = 100,000. Table 1 shows the true reliability R of the design method of a fault observer for a parametric uncertainty system with high reliability compared with two traditional methods s,re . The traditional method that does not consider random parameters uses the mean μ θ , but does not consider parametric uncertainty. Obviously, the method we proposed performs better in ensuring the reliability of the error system. When α = 2.0, the reliability has reached 0.99998. However, the reliability of other methods is lower. Generally speaking, the reliability design needs to meet the requirement of R s,re ≥ 0.99.

[0096] Table 1 Comparison of the reliability performance between the proposed method and traditional methods

[0097]

[0098]

[0099] Under the solved gain γ 1 = 0.0070 and , given a constant fault signal , the fault estimation effect as shown in Figure 2 is obtained. It can be concluded from Figure 2 that the method proposed in this patent can still estimate the system fault well under the setting of multiple groups of uncertain parameters θ, while the method that does not consider random parameters estimates poorly or even diverges.

[0100] Given a time-varying fault signal , the fault estimation effect as shown in Figure 3 is obtained. It can be concluded from Figure 2 similarly that the method proposed in this invention can still estimate the time-varying fault of the system well under the setting of multiple groups of uncertain parameters θ, while the method that does not consider random parameters estimates poorly or even diverges.

[0101] The content not described in detail in the specification of this invention belongs to the prior art well-known to those skilled in the art. It is easy for those skilled in the art to understand that the above are only preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, and improvements made within the spirit and principles of the present invention shall be included within the protection scope of the present invention.

Claims

1. A method for designing a high-reliability fault observer for a parameter random system, characterized in that: The following steps are involved: The first step is to construct a state space model of the system with uncertain random parameters according to the dynamic characteristics of the target system; the so-called uncertain random parameters refer to the internal random parameters in the system matrix and the control matrix in the state space model of the system; The second step is to design the state observer and fault observer with random parameters, and give the linear matrix inequality with random parameters based on Lyapunov stability criterion; The third step is to introduce a reliability index under the limit function of the system based on the structural reliability theory to quantify the reliability of the fault observer; The fourth step is to combine Taylor expansion to linearly approximate the uncertainty parameters and give a probabilistic linear matrix inequality that can be solved based on the reliability index; Step 5: Solve the probability linear matrix inequality to obtain the gains and other parameters of the state observer and fault observer.

2. The method for designing a high-reliability fault observer for a parameter random system according to claim 1, characterized in that: The specific steps of the first step are as follows: According to the dynamic characteristics of the target system, a system state space model with random uncertain parameters is constructed, which is expressed as follows: In formula (1), represents the system state vector; represents the output vector of the system; represents the system input vector; is a slowly varying or constant fault that needs to be separated and estimated, which satisfies the bounded L2 norm, that is, A(ρ), B(ρ), F, and G represent matrices of appropriate dimensions, where A(ρ) and B(ρ) depend on the random parameter vector ρ = [ρ1, ρ2, …, ρ N ] T ; C y is the output matrix that maps x(t) to y(t); g(x(t)) is a nonlinear function that satisfies the norm condition represented by the known matrix U as shown in equation (2):

3. The method for designing a high-reliability fault observer for a parameter random system according to claim 2, characterized in that: The specific steps of the second step are as follows: First, construct the following state observer: And the fault observer is: In formula (3) and formula (4), L represents the desired gain of the state observer, i >0,i=1,2 represents the required gain of the fault observer, represents the estimated value of the state vector x(t), represents the estimated value of the fault vector f(t), Represents the estimated value of the output vector y(t); Define the state estimation error The dynamics of the estimation error can be expressed as: Since the fault f(t) considered in this method is a slowly changing or constant fault, then: In order to ensure that the designed fault diagnosis observer satisfies H ∞ Performance, this method introduces the generalized H ∞ Filter; Definition As the reference time-varying output, it is used for the H of the above systems (5) and (6) ∞ Performance variable, with initial condition equal to zero; the observer gain should be designed to minimize The upper bound of H ∞ In optimization problems, the classical concept of disturbance attenuation can be extended to the related fault detection and diagnosis problems, which is defined as follows: will make Thus, the generalized H ∞ performance; Therefore, the Lyapunov function designed for the fault observer of dynamic systems with random uncertain parameters can be designed as: When P in equation (8) is a positive definite matrix, Φ(t) is always greater than 0. At the same time, the derivative of equation (8) yields: In formula (9), And there is in, In order to make the designed fault observer satisfy asymptotic stability, the matrix in equation (9) It needs to be a negative definite matrix to ensure that equation (9) is less than zero; Combined with the generalized H given by formula (7), ∞ Performance Definition, Matrix Corrected to To ensure that the system is asymptotically stable and reaches the generalized H ∞ Performance, that is, the system needs to meet: Where sym(*)=*+* T , from equation (11), we can solve for the observer gain L = P -1 Q; Since Π0 has a random parameter ρ, equation (11) is a linear matrix inequality SLMI with random parameters, which cannot be solved directly. Steps 3 and 4 will be introduced later to transform SLMI into a solvable form and obtain the observer gain that meets the specified reliability level.

4. The method for designing a high-reliability fault observer for a parameter random system according to claim 3, characterized in that: The specific steps of the third step are as follows: Based on the structural reliability theory and Lyapunov stability criterion, it is assumed that the limit state function of the system is Where ρ is the random parameter that needs to be processed; if G(ρ)<0, the system will be asymptotically stable and the region is reliable; if G(ρ)=0, the system is critically stable, indicating that the region reaches the limit state surface; A reliability index α is introduced and proposed. α can be quantified and determined by solving the shortest distance from the origin to the limit state surface in the standard normal space, that is: where ρ=[ρ1,ρ2,…,ρ N ] T A vector representing a standard Gaussian random variable; With the help of reliability index α, reliability R s and the failure probability F s It can be obtained by equation (13) and equation (14): R s =Φ(a), (13) F s =Φ(-a), (14) Here, Φ(·) represents the quantile of the standard normal distribution.

5. The method for designing a high-reliability fault observer for a parameter random system according to claim 4, characterized in that: The specific steps of the fourth step are as follows: The linear matrix inequality (11) with random parameters can be transformed into the probabilistic linear matrix inequality PLMI with reliability index; for this purpose, the following linear approximation of A(ρ) is required: The result of formula (15) is obtained through the first-order Taylor expansion: where f(ρ) represents any element of A(ρ); μ ρ represents the mean of ρ, and since ρ is a vector that conforms to the standard Gaussian distribution, μ ρ =0; Formula (15) can be rewritten as: Where V A,j =[A 1,j ,A 2,j ,…,A N,j ], A N,j Represents the matrix A N The jth column of U A,j is a row vector, where only the jth th elements are 1, and the rest are 0; Since the necessary condition for the negative definiteness of formula (11) is that the first term of the sequential principal subformula is negative definite, that is, the condition is satisfied If the parameter uncertainty satisfies ρ T ρ≤α 2 , then if and only if there are n positive constants ε j hour: For a clearer use, let us define Ψ0(P,Q) as follows: Through Shur complementation, equation (19) is transformed into the following linear matrix inequality: Among them Ψ α (P,Q) represents the matrix containing the reliability index α, E = diag{ε1,ε2,…,ε n }; Substituting equation (20) into equation (11) given in step 2, we obtain the probability linear matrix inequality with reliability index: Under the premise of a given system and reliability index α, equation (21) is solvable, and the solved state observer gain L, fault observer parameters Υ1 and Υ2 can ensure the asymptotic stability of fault detection under the system with random uncertain parameters, H ∞ performance and high reliability requirements.

6. The method for designing a high-reliability fault observer for a parameter random system according to claim 5, characterized in that: The specific steps of the fifth step are as follows: Given the reliability index α, solve the probability linear matrix inequality (21) to obtain the gains and other parameters of the state observer and fault observer, and substitute the parameters into equations (3) and (4) to obtain the state observer and fault observer given by the present invention.

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