A method for evaluating the reliability of a reaction flywheel based on a multiplicative failure performance index and an estimation thereof

Abstract

The application relates to a reaction flywheel reliability evaluation method based on a multiplicative fault performance index and estimation thereof, aiming at the performance index selection and data acquisition problem of reaction flywheel reliability on-orbit monitoring. First, a small-angle linearization attitude dynamics model containing a multiplicative fault is established; second, the logical rationality of the multiplicative fault as a performance index for reaction flywheel reliability evaluation is analyzed; then, a multiplicative fault observer is designed to estimate the change value of the multiplicative fault in the working reaction flywheel; finally, based on the multiplicative fault estimation value, a three-parameter Weibull distribution fitting model is established, and a reaction flywheel reliability evaluation model is obtained in combination with a fault threshold. The application provides a multiplicative fault performance index for reaction flywheel reliability evaluation, which has good trend behavior under different working conditions, and the obtained on-orbit reliability evaluation result can provide powerful data support for fault diagnosis and health management of key components of a spacecraft.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of spacecraft reliability, and particularly relates to a reaction flywheel reliability evaluation method based on a multiplicative failure performance index and estimation thereof. BACKGROUND

[0002] The reaction flywheel is a commonly used actuator in a satellite attitude control system. Under the influence of interference and failure, on the one hand, the reaction flywheel needs to physically realize the precise adjustment of the satellite attitude by the control action, and on the other hand, the reaction flywheel needs to ensure the life index under the influence of a harsh environment and a continuous working load. Due to the influence of the harsh environment and the continuous working load, 44% of the actuator failures are the most serious and frequent spacecraft attitude control system failures. In 2013, the Kepler telescope developed by the United States National Aeronautics and Space Administration and worth 600 million US dollars was forced to stop serving due to the failure of two of the four flywheels. It can be seen that it is necessary to carry out dynamic evaluation and monitoring of the reliability of the reaction flywheel, which is beneficial to enabling engineers to grasp the state of the flywheel in real time, so that targeted maintenance measures can be implemented when the reaction flywheel fails.

[0003] Determining the performance index and the failure criterion based on the performance index is a prerequisite for carrying out reliability modeling and analysis based on degradation. Only in a few cases, a physical quantity that can directly represent the degradation process of a product can be obtained, for example, taking the crack width as the degradation feature of a structure or a material. In most cases, the degradation quantity of a product is difficult to directly observe, but a performance index that is related to the failure, reliability and life of the product can be observed. For modeling convenience, in one case, the performance of the product that is concerned in engineering is directly taken as the performance index, such as the capacity of a capacitor or a battery, the output intensity of a laser, etc. For some products, it is not appropriate to directly use the measured performance index as the degradation feature, and a more essential feature needs to be extracted through transformation. For example, for rotating parts, a frequency feature is often extracted from a vibration signal as a performance degradation index of the part. However, it must be noted that these physical quantities are generally related to the degradation failure process of the product, but not the direct features of the degradation failure process. For example, these physical quantities do not guarantee the regularity required by the degradation process, and the failure of the product cannot be determined without doubt based on these physical quantities. In addition, it is possible to monitor multiple physical quantities for a single product, which measure different aspects of the features of the product, and there is a complex correlation. At this time, it is necessary to simplify through appropriate data processing techniques, such as principal component analysis, selection based on expert knowledge, screening based on optimization algorithms, etc., to express the mode vector (coarse feature) of the original variable space of high dimension with the new mode vector (fine feature or secondary feature) of the feature space of low dimension, so as to find out the most representative and most effective feature as the performance index.

[0004] Similarly, in order to establish the reliability model of the reaction flywheel, a proper performance index must be determined first, which should be able to accurately capture the degradation process of the execution performance of the reaction flywheel. At present, the armature current, lubricant consumption and shaft temperature of the reaction flywheel are three commonly used performance indexes for describing the performance degradation. However, these performance indexes are only physical quantities that have a certain correlation with the execution performance degradation process, but not direct representations. At present, since the research attention is generally focused on the physical variables that can be directly measured, no performance index has been found that can directly represent the execution performance. Therefore, it is necessary to introduce a fault evolution variable to represent the real execution performance degradation of the reaction flywheel. Multiplicative fault is a representation of fault in the control field, which is usually modeled as a loss of partial execution performance, and it occurs with the aging of the reaction flywheel under continuous working load and environmental influence. Therefore, the change of the multiplicative fault can be used to simulate the real execution performance degradation of the reaction flywheel, and the logical rationality and evaluation accuracy of the current as a performance index can be analyzed. SUMMARY

[0005] For the on-orbit reliability state dynamic evaluation and monitoring problem of the reaction flywheel as an actuator in micro-nano satellites, the present application proposes a reaction flywheel reliability evaluation method based on a multiplicative fault performance index and its estimation. Based on the qualitative multiplicative fault performance index rationality analysis and the quantitative fault observer design, the multiplicative fault performance index of the reaction flywheel is proposed and estimated, laying the foundation for the reliability modeling and evaluation of the reaction flywheel. First, a small-angle linearized attitude dynamics model containing a multiplicative fault is established. Second, the logical rationality of the multiplicative fault as a performance index for the reliability evaluation of the reaction flywheel is analyzed. Third, a multiplicative fault observer is designed to estimate the change value of the multiplicative fault in the working process of the reaction flywheel. Finally, based on the estimated value of the multiplicative fault, a three-parameter Weibull distribution fitting model is established, and the reliability evaluation model of the reaction flywheel is obtained in combination with the threshold value of the multiplicative fault.

[0006] To achieve the above object, the technical scheme adopted by the present application is as follows:

[0007] A reaction flywheel reliability evaluation method based on a multiplicative fault performance index and its estimation, comprising the following steps:

[0008] First, in the case of small Euler angles between the micro-nano satellite body coordinate system and the orbit coordinate system, a multiplicative fault diagonal matrix is introduced to the control input term based on the momentum theorem, wherein the element values on the diagonal line of the multiplicative fault diagonal matrix range from 0 to 1, and then a small-angle linearized attitude dynamics model containing a multiplicative fault is established;

[0009] The second step is to analyze the logical rationality of using multiplicative failure as a performance indicator for the reliability assessment of reaction flywheels, based on the definition and characteristics of performance indicators in degradation reliability modeling and evaluation.

[0010] The third step involves designing a multiplicative fault observer based on the micro-nano satellite attitude angle measurement, control input, and attitude dynamics model to estimate the multiplicative fault changes during the operation of the reaction flywheel.

[0011] The fourth step is to establish a three-parameter Weibull distribution model to fit the estimated multiplicative fault change value, considering that the degradation form of multiplicative faults is unclear and the value range is between 0 and 1.

[0012] The fifth step is to establish a reliability assessment model for the reaction flywheel based on the established three-parameter Weibull distribution model and the multiplicative fault threshold.

[0013] Furthermore, in the first step, a small-angle linearized attitude dynamics model incorporating multiplicative faults is established as follows:

[0014]

[0015] In the formula, This represents the roll angle γ(t) and angular rate ω of a micro / nano satellite. γ (t), pitch angle and its angular rate Yaw angle ψ(t) and its angular rate ω ψ The state vector consists of u(t), where t is time; u(t) = [u1(t), u2(t), u3(t)] T This represents the control vector composed of the three-axis control torques calculated by the attitude controller; d(t) = [d1(t), d2(t), d3(t)] T Let ρ(t) represent the disturbance vector composed of the lumped disturbance torques of the three axes; ρ(t) = diag(ρ1(t), ρ2(t), ρ3(t)) represents the diagonal matrix of multiplicative faults composed of the multiplicative faults of each flywheel on the three axes, where diag(·) is the sign of the diagonal matrix, and 0 ≤ ρ i (t)≤1, i=1,2,3; System matrix A and control matrix B are expressed as follows: Where I is the identity matrix of the corresponding dimension, M = diag{J1, J2, J3}, H = diag{4n} 2 (J2-J3),3n 2 (J1-J3),n 2 (J2-J1)}, J1, J2, and J3 represent the three-axis rotational inertia of the microsatellite, and n is the orbital angular rate.

[0016] Further, the logic rationality of the second step is as follows:

[0017] Without human maintenance in space, the multiplicative failure of the reaction flywheel will always show a gradual decline from 1 to 0 over time, that is, it has a good degradation trend behavior; Since the multiplicative failure is a direct representation of the execution performance of the reaction flywheel, the multiplicative failure is used as a performance index representing the degradation of the reaction flywheel, and is used for its reliability modeling and life prediction; The concept of multiplicative failure can be used for reliability modeling and prediction of reaction flywheels, and also for reliability modeling and prediction of typical actuators in general control systems.

[0018] Further, in the third step, the multiplicative failure observer is designed to estimate the change value of the multiplicative failure in the working process of the reaction flywheel as follows:

[0019]

[0020] In the formula, and are the estimated values of the corresponding state vector x(t) and multiplicative failure vector P(t) = [ρ1(t), ρ2(t), ρ3(t)] T ; e(t) represents the state estimation error vector, sign(·) is the sign function; ψ(t) = diag(u1(t), u2(t), u3(t)) is a diagonal matrix composed of control inputs; Λ = diag(Λ1, Λ2) and κ(t) = diag(κ1, κ2, κ3) are diagonal matrices of positive gain parameters, where Λ i = diag(Λ i1 , Λ i2 , Λ i3 ), i = 1, 2;

[0021] In addition, has the following adaptive update law:

[0022]

[0023] In the formula, is the angular rate estimation error vector, α i and β i are adjustable positive gains, and ζ is a positive value set artificially; |·| and ||·|| represent the absolute value symbol and the Euclidean norm, respectively;

[0024] The estimation of the multiplicative failure vector has the following adaptive estimation law:

[0025]

[0026] In the formula, the estimation of the multiplicative failure vector adjusted by the feedback of state error whose time-varying characteristics are determined by the control law ψ(t), and the positive gain Γ is used to adjust the convergence rate and error.

[0027] Further, in the fourth step, a three-parameter Weibull distribution model is established to fit the estimated multiplicative fault change value as follows:

[0028]

[0029] where Y(t) is the multiplicative fault fitting value, t is time, γ, θ, m are unknown parameters obtained by fitting the multiplicative fault estimation value, exp(·) is an exponential function, and ε is random noise with a cumulative distribution function G(ε) and a probability density function g(ε).

[0030] Further, in the fifth step, the reliability evaluation model is as follows:

[0031]

[0032] where R(t) represents reliability, Pr(·) represents a probability calculation symbol, represents the maximum value of Y(t) within 0 to t time; y th is a multiplicative fault threshold value.

[0033] Compared with the prior art, the present application has the following advantages: the present application proposes a multiplicative fault performance index of a reaction flywheel and an estimation method based on an observer design, aiming at the on-orbit reliability state dynamic evaluation and monitoring of the reaction flywheel as an actuator in a micro-nano satellite. The present application first conducts in-depth research on the reliability of the reaction flywheel from the perspective of attitude coupling dynamics and the concept of multiplicative fault: the multiplicative fault is a new and intrinsic performance index, which can be used in various different task scenarios. This discovery provides engineers with a new performance index for modeling and evaluating the reliability of the flywheel in more task scenarios, and can be extended to the reliability evaluation of actuators in other general control systems. BRIEF DESCRIPTION OF DRAWINGS

[0034] Figure 1 is a flow chart of a reaction flywheel reliability evaluation method based on a multiplicative fault performance index and estimation of the present application;

[0035] Figure 2 is a pitch angle tracking curve and a multiplicative fault estimation curve, where (a) is a pitch angle tracking curve, and (b) is a multiplicative fault estimation curve;

[0036] Figure 3 is a time-varying reliability curve based on multiplicative fault estimation. DETAILED DESCRIPTION

[0037] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0038] like Figure 1 As shown, this invention relates to a reliability assessment method for reaction flywheels based on multiplicative failure performance indicators and their estimation, comprising the following steps:

[0039] The first step is to establish a small-angle linearized attitude dynamics model that includes multiplicative faults, based on the angular momentum theorem and by introducing a multiplicative fault diagonal matrix into the control input term, where the values ​​of the elements on the diagonal of the multiplicative fault matrix are between 0 and 1.

[0040] The second step is to analyze the logical rationality of using multiplicative failure as a performance indicator for the reliability assessment of reaction flywheels, based on the definition and characteristics of performance indicators in degradation reliability modeling and evaluation.

[0041] The third step involves designing a multiplicative fault observer based on the micro-nano satellite attitude angle measurement, control input, and attitude dynamics model to estimate the multiplicative fault changes during the operation of the reaction flywheel.

[0042] The fourth step is to establish a three-parameter Weibull distribution model to fit the estimated multiplicative fault change value, considering that the degradation form of multiplicative faults is unclear and the value range is between 0 and 1.

[0043] The fifth step is to establish a reliability assessment model for the reaction flywheel based on the established three-parameter Weibull distribution model and the multiplicative fault threshold.

[0044] Specifically, in the first step, the small-angle linearized attitude dynamics model containing multiplicative faults is:

[0045]

[0046] In the formula, This represents the roll angle γ(t) and angular rate ω of a micro / nano satellite. γ (t), pitch angle and its angular rate Yaw angle ψ(t) and its angular rate ω ψ The state vector consists of u(t) and u(t); u(t) = [u1(t), u2(t), u3(t)] T This represents the control vector composed of the three-axis control torques calculated by the attitude controller; d(t) = [d1(t), d2(t), d3(t)] Tdenotes the disturbance vector composed of three-axis lumped disturbance torques; p(t) = diag(p1(t), p2(t), p3(t)) denotes the multiplicative fault diagonal matrix composed of three-axis flywheel multiplicative faults, diag(·) is the diagonal matrix symbol, and 0 < p i (t) < 1, i = 1, 2, 3; the system matrix A and the control matrix B can be expressed as where I is the unit matrix of the corresponding dimension, M = diag{J1, J2, J3}, H = diag{4n 2 (J2-J3), 3n 2 (J1-J3), n 2 (J2-J1)}, J1, J2, J3 represent the three-axis rotational inertia of the micro-nano satellite, and n is the orbital angular rate.

[0047] In the second step, without human maintenance in space, the flywheel multiplicative fault will always show a gradual decrease from 1 to 0 over time, that is, it has a good degradation trend behavior. In addition, since the multiplicative fault is a direct representation of the execution efficiency of the flywheel, engineers do not have to worry about choosing how many external performance indicators and analyzing their relationship to represent the actual degradation process of the execution efficiency. Therefore, using the multiplicative fault as a performance indicator to indicate the degradation of the flywheel and using it for reliability modeling and life prediction has a unique advantage. In addition, due to the universality of the multiplicative fault concept, it can not only be used for reliability modeling and prediction of flywheels, but also be used for reliability modeling and prediction of other typical actuators in general control systems.

[0048] In the third step, based on equation (1), the following multiplicative fault observer can be designed:

[0049]

[0050] wherein and are the estimated values of the corresponding state vector x(t) and multiplicative fault vector P(t) = [p1(t), p2(t), p3(t)] T ; denotes the state estimation error vector, sign(·) is the sign function; ψ(t) = diag(u1(t), u2(t), u3(t)) is the diagonal matrix composed of control inputs; Λ = diag(Λ1, Λ2) and κ(t) = diag(κ1, κ2, κ3) are positive gain parameter diagonal matrices, where Λ i = diag(Λ i1 , Λ i2 , Λ i3 ), i = 1, 2. In addition, It has the following adaptive update law:

[0051]

[0052] In the formula, Let α be the angular rate estimation error vector. i and β i ζ is an adjustable positive gain, which is a positive value set manually; |·| and ||·|| represent the absolute value sign and the Euclidean norm, respectively.

[0053] The dynamics of the state estimation error can be obtained from equations (1) and (2), that is:

[0054]

[0055] In the formula,

[0056] To design an adaptive law for multiplicative fault estimation, consider the following Lyapunov function:

[0057]

[0058] In the formula, Γ is the positive gain parameter. For simplicity, the time symbol t is ignored in equations (5) and (6). Taking the time derivative of equation (5) along the trajectory of equation (2) yields equation (6):

[0059]

[0060] In the formula, h is the derivative of the multiplicative fault P. The upper bound of the norm; λ max (Ξ) represents the largest eigenvalue of matrix Ξ. It is an unknown positive value.

[0061] If equation (6) holds, we can obtain the adaptive law for estimating the multiplicative fault value:

[0062]

[0063] And two assumptions must be met:

[0064] Assumption 1: Lumped disturbance d(t) = [d1(t), d2(t), d3(t)] T It is bounded and always exists. Make in It is an unknown positive value.

[0065] Assumption 2: The derivative of the multiplicative fault P It is norm-bounded, that is

[0066] In fact, the adaptive law (7) has physical meaning. The estimation of P(t) is regulated by the feedback of state error , while its time-varying characteristic is determined by the control law ψ(t), and the positive gain Γ can be used to regulate the convergence rate and error.

[0067] Before giving the final theorem, the following lemma is introduced:

[0068] Lemma 1: For the system where f(·) is locally Lipschitz with respect to x(t) and piecewise continuous with respect to time t, there exists a continuously differentiable function V(x(t), t) such that along any trajectory of the system, the following holds:

[0069]

[0070] where η1 and η2 are class functions, and η3 is a class function. Then, the solution x(t) of the system is uniformly ultimately bounded.

[0071] Based on Lemma 1, the following theorem is given:

[0072] Theorem 1: For the attitude control system (1) with flywheel faults, if Assumption 1 and Assumption 2 are satisfied, and λ max (Ξ) < 0 holds, then the unknown multiplicative fault estimation can be achieved by the multiplicative fault observer (2) and the adaptive law (7), and the estimation error (4) can converge to a small neighborhood of the origin. Therefore, the adaptive law (7) will converge to a small neighborhood of the true value of the unknown multiplicative fault vector P(t).

[0073] The fourth step includes:

[0074]

[0075] where Y(t) is the multiplicative fault fitting value, t is time; γ, θ, m are unknown parameters obtained by fitting the multiplicative fault estimation value; exp(·) is the exponential function; ε is random noise, and its cumulative distribution function is G(ε), and its probability density function is g(ε).

[0076] The fifth step includes:

[0077]

[0078] where R(t) represents reliability; Pr(·) represents the probability calculation symbol; represents the maximum value of Y(t) within the time from 0 to t; yth is a multiplicative fault threshold.

[0079] The application is described in detail below with specific examples.

[0080] Firstly, a small-angle linearized attitude dynamics model containing multiplicative fault is established, taking the pitch channel as an example:

[0081]

[0082] where the expected pitch attitude angle is assumed to be θ d (t) = 3 x sin(0.6 / π x t + 0.2) to generate a persistent excitation.

[0083] Secondly, the definition and characteristics of the performance index based on the degradation reliability modeling evaluation are evaluated, and the logical rationality of the multiplicative fault as a performance index for the reliability evaluation of the reaction flywheel is analyzed in depth.

[0084] Thirdly, based on the attitude angle measurement value of the micro-nano satellite, the control input and the attitude dynamics model, a multiplicative fault observer is designed to estimate the change value of the multiplicative fault in the working process of the reaction flywheel.

[0085] The given formula (2) of the multiplicative fault observer and the formula (7) of the corresponding adaptive law are used to estimate the assumed time-varying multiplicative fault. In the simulation, the parameters in formula (2) and formula (7) are set to Λ = diag{1, 5}, α2 = 3, β2 = 1, and Γ = 400. Figure 2 Figure (a) of the above shows that the attitude tracking performance decreases as the flywheel execution performance decreases. From Figure 2 Figure (b) of the above, it can be seen that the multiplicative fault observer can well estimate the assumed real multiplicative fault, and the estimation error is small.

[0086] Fourthly, based on the estimated change value of the multiplicative fault, and considering that the form of the multiplicative fault degradation is not clear and the value range is between 0 and 1, a three-parameter Weibull distribution model is established to fit the estimated change value of the multiplicative fault.

[0087] Using the estimated value of the multiplicative fault to fit formula (9) gives the estimated value of the parameters as γ = 0, θ = 159.7, m = 6.7, and ε ~ N(0, 0.0115).

[0088] Fifthly, based on the established three-parameter Weibull distribution model, the reliability evaluation model of the reaction flywheel is established in combination with the multiplicative fault threshold.

[0089] If the failure threshold of the multiplicative fault loss is set to y th = 0.815, the change trend graph of the flywheel reliability is obtained based on formula (10), as shown in Figure 3 .

[0090] The content described in the specification of the present application is the prior art known to those skilled in the art. Those skilled in the art can easily understand that the above description is only the preferred embodiment of the present application and is not intended to limit the present application. Any modification, equivalent replacement and improvement made within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims

1. A method for reliability assessment of a flywheel with reaction based on multiplicative failure performance indicators and their estimates, characterized by, Comprising the following steps: In the first step, in the case of small Euler angles between the micro-nano satellite body coordinate system and the orbit coordinate system, based on the momentum theorem and introducing a multiplicative fault diagonal matrix multiplied by the control input term, the multiplicative fault diagonal matrix has a value range of 0 to 1 on the diagonal line, and then a small-angle linearized attitude dynamics model containing multiplicative fault is established as follows: （1） In the formula, denotes a state vector composed of the roll angle and its angular rate , the pitch angle and its angular rate , the yaw angle and its angular rate , is time; denotes a control vector composed of three-axis control moments calculated by the attitude controller; denotes an interference vector composed of three-axis collective interference moments; denotes a multiplicative fault diagonal matrix composed of three-axis individual flywheel multiplicative faults, is a diagonal matrix symbol, and ; the system matrix , the control matrix is expressed as , wherein is a unit matrix of the corresponding dimension, , , , , , denotes the three-axis rotational inertia of the micro-nano satellite, is the orbital angular rate; In the second step, based on the definition and characteristics of performance index in the degradation reliability modeling evaluation, the logical rationality of using multiplicative fault as performance index for the reliability evaluation of reaction flywheel is analyzed in depth; In the third step, based on the attitude angle measurement value of micro-nano satellite, control input and attitude dynamics model, a multiplicative fault observer is designed to estimate the change value of multiplicative fault in the working process of reaction flywheel as follows: （2） where and are the respective state vectors and multiplicative fault vectors are the estimates of the respective state vectors denotes the state estimation error vector, is the sign function; is a diagonal matrix of control input components; and are positive gain parameter diagonal matrices, where ; Furthermore, has an adaptive update law of the form （3） wherein is the angular rate estimation error vector, and is a positive gain that is adjustable, is a positive value that is set artificially; denotes the absolute value symbol; Estimation of multiplicative fault vectors There is an adaptive estimation law as follows: (7) where the estimate of the multiplicative fault vector is regulated by the feedback of the state error whose time-varying characteristics are determined by the control law , the positive gain is used to regulate the convergence rate and the error; In the fourth step, based on the estimated change value of multiplicative fault, and considering that the degradation form of multiplicative fault is not clear and the value range is between 0 and 1, a three-parameter Weibull distribution model is established to fit the estimated change value of multiplicative fault; In the fifth step, based on the established three-parameter Weibull distribution model, combined with the threshold value of multiplicative fault, a reliability evaluation model of reaction flywheel is established.

2. The flywheel reliability evaluation method based on multiplicative failure performance index and its estimation according to claim 1, characterized in that: The logical rationality of the second step is as follows: In the absence of human maintenance in space, the multiplicative fault of reaction flywheel will always show a gradual decline from 1 to 0 over time, that is, it has a good degradation trend behavior; Since multiplicative fault is a direct representation of the execution efficiency of reaction flywheel, multiplicative fault is used as a performance index to represent the degradation of reaction flywheel, and is used for its reliability modeling and life prediction; The concept of multiplicative fault is used for reliability modeling and prediction of reaction flywheel or for reliability modeling and prediction of typical actuators in general control systems.

3. The method for flywheel reliability evaluation based on multiplicative failure performance index and its estimation according to claim 1, characterized in that: In the fourth step, the three-parameter Weibull distribution model is established to fit the estimated change value of multiplicative fault as follows: (9) wherein is the multiplicative fault fit value, is time; , , is an unknown parameter fitted from the multiplicative fault estimate; is an exponential function; is random noise with a cumulative distribution function and a probability density function .

4. The flywheel reliability evaluation method based on multiplicative failure performance index and its estimation according to claim 3, characterized in that: In the fifth step, the reliability evaluation model is as follows: (10) wherein represents a reliability; represents a probability calculation symbol; represents is the maximum value in 0 to time; is a multiplicative failure threshold.

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