CN111610779B - Method for identifying fault factors of data-driven nonlinear system actuator - Google Patents

Method for identifying fault factors of data-driven nonlinear system actuator Download PDF

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CN111610779B
CN111610779B CN202010626098.7A CN202010626098A CN111610779B CN 111610779 B CN111610779 B CN 111610779B CN 202010626098 A CN202010626098 A CN 202010626098A CN 111610779 B CN111610779 B CN 111610779B
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actuator
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estimate
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CN111610779A (en
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池荣虎
魏阳春
惠宇
姚文龙
林娜
张慧敏
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Qingdao University of Science and Technology
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B23/00—Testing or monitoring of control systems or parts thereof
    • G05B23/02—Electric testing or monitoring
    • G05B23/0205—Electric testing or monitoring by means of a monitoring system capable of detecting and responding to faults
    • G05B23/0218—Electric testing or monitoring by means of a monitoring system capable of detecting and responding to faults characterised by the fault detection method dealing with either existing or incipient faults
    • G05B23/0243—Electric testing or monitoring by means of a monitoring system capable of detecting and responding to faults characterised by the fault detection method dealing with either existing or incipient faults model based detection method, e.g. first-principles knowledge model
    • G05B23/0254—Electric testing or monitoring by means of a monitoring system capable of detecting and responding to faults characterised by the fault detection method dealing with either existing or incipient faults model based detection method, e.g. first-principles knowledge model based on a quantitative model, e.g. mathematical relationships between inputs and outputs; functions: observer, Kalman filter, residual calculation, Neural Networks

Abstract

The invention discloses a method for identifying fault factors of an actuator of a data-driven nonlinear system, and belongs to the field of intelligent control. In a data-driven framework, aiming at the problem of identifying fault failure factors of a nonlinear system with actuator faults, the identification method comprises the following steps: establishing a nonlinear system; dynamically linearizing the nonlinear system into an equivalent linear data model with an actuator fault failure factor; designing an updating algorithm to estimate a pseudo partial derivative in the linear data model; designing a state observer to estimate the state of the system; and designing an online estimation algorithm of the fault factor to realize the estimation of the fault factor. The method for identifying the fault factors of the actuator of the data-driven nonlinear system can estimate the fault failure factors of the actuator in the nonlinear system on line, and meets the requirement of estimating the fault of the actuator on the premise of good adaptability without depending on specific model information by using the data-driven method.

Description

Method for identifying fault factors of data-driven nonlinear system actuator
Technical Field
The invention belongs to the technical field of intelligent control, and particularly relates to an online identification method for a fault factor of a data-driven actuator of a nonlinear system with actuator faults.
Background
With the rapid development of modern economy, the scale and complexity of modern industrial systems are increasing, and such systems, once they fail, can cause significant economic losses. The fault identification method can estimate fault factors in the system, and further reduces the influence of faults on control.
For the research of the identification method of the actuator fault factor, the following two problems need to be considered: 1. the existing control system is more and more diversified and more complex, and the actuator fault factor identification method can be effectively applied to various systems; 2. the method can achieve the purpose of estimating the fault factors in the system while ensuring the adaptability.
The fault identification method is a new subject developed in the 60 s of the 20 th century, and provides guarantee for effective operation of an industrial system. The fault identification method realizes effective estimation of the size of the fault by monitoring the running state of the production process and judging the position and the degree of the fault. The task of fault identification is to select a convenient and effective method to find abnormal events in the process, and also to identify and diagnose the root cause of the abnormal events in the production process, thereby guiding an operator to correctly handle the process abnormality. The existing fault identification methods are divided into three categories: mathematical model-based methods, artificial intelligence-based methods, and data-driven-based methods. The mathematical model-based method firstly establishes a mathematical model of the process and then identifies the parameters of the system. The parameter estimation method, the state estimation method, the analytical redundancy method and the like are analysis-based methods, accurate mathematical models need to be obtained, and the problem of unmodeled dynamics exists. The artificial intelligence based method comprises an expert system, a neural network and the like, is suitable for a system which cannot obtain a detailed mathematical model, but needs a large amount of production experience and process knowledge and is poor in universality.
In order to solve the problems of unmodeled dynamics of a fault identification method based on a mathematical model method and poor universality of an artificial intelligence method, a data-driven fault identification method which does not depend on a system model and does not need a large amount of production experience and process knowledge is needed to be designed, and the purpose of estimating fault factors of an actuator in a system is achieved.
Disclosure of Invention
The invention discloses a method for identifying fault factors of an actuator of a data-driven nonlinear system, which aims to solve the problem that the designed identification method has good universality while ensuring the estimation of the fault factors of the actuator in the system.
The purpose of the invention is realized by the following technical scheme:
the invention discloses a method for identifying fault factors of an actuator of a data-driven nonlinear system, which mainly aims at the problem of online estimation of the fault factors of the actuator of the nonlinear system with dynamic actuator faults.
The invention discloses a method for identifying fault factors of a data-driven nonlinear system actuator, which comprises the following steps:
step 1, establishing a nonlinear system:
consider a discrete-time nonlinear system as follows:
y(k+1)＝f(y(k),…,y(k-ny),u(k),…,u(k-nu))；
wherein y (k) represents the output of the system at the k-th time; u (k) represents the control input of the system at the k-th moment; n isyAnd nuIs two unknown positive integers representing the order of the system; f (-) represents an unknown non-linear function;
the system meets the generalized Lipschitz condition, and when the delta u (k) is not equal to 0 and the delta y (k) is not equal to 0, the | delta y (k +1) | is not more than b1|△u(k)|+b2|△y(k)|；
Wherein, b1、b2Are all positive numbers; Δ u (k) — k (1), Δ y (k) ═ y (k) — y (k-1);
step 2, dynamically linearizing the nonlinear system into an equivalent linear data model with an actuator fault failure factor:
Figure GDA0002973270580000031
wherein w (k +1) represents the disturbance of the system at the k +1 th moment; v (k) represents the measurement noise of the system at time k; x (k) represents the state of the system at the k-th time; u (k) ═ u (k) u (k-1)]T；Bf(k)＝B(k)(1-δ(k))；
Figure GDA0002973270580000032
Figure GDA0002973270580000033
To define a pseudo partial derivative at time k; delta (k) represents a failure factor of the actuator failure at the kth moment, and the value range of the failure factor is delta (k) belonging to [0,1]]When δ (k) is equal to 1, it means that the actuator is finished at the k-th timeFull failure, δ (k) ═ 0, indicates that the actuator is working normally at the kth time, 0<δ(k)<1 represents the lost execution drive capability after an actuator failure;
step 3, designing an updating algorithm to estimate the pseudo partial derivatives in the linear data model:
Figure GDA0002973270580000034
Figure GDA0002973270580000035
if it is
Figure GDA0002973270580000036
Or | delta u (k-1) | is less than or equal to epsilon or
Figure GDA0002973270580000037
Wherein the content of the first and second substances,
Figure GDA0002973270580000038
representative pair
Figure GDA0002973270580000039
(ii) an estimate of (d); mu.s>0 is a weight factor; eta ∈ (0, 1)]Is a step size factor; ε is a positive number;
Figure GDA00029732705800000310
is that
Figure GDA00029732705800000311
An initial value of (1);
and 4, designing a state observer to estimate the system state:
Figure GDA00029732705800000312
wherein the content of the first and second substances,
Figure GDA00029732705800000313
represents an estimate of state x (k +1) at time k + 1;
Figure GDA00029732705800000314
represents an estimate of state x (k) at time k;
Figure GDA00029732705800000315
represents an estimate of state x (k +1) at time k; g (k +1) represents a gain operator at the k +1 th time; sigmaee(k +1) represents the variance of the state estimation bias at time k + 1; sigmaww(k +1) represents the variance of the system disturbance at time k + 1; sigmavv(k +1) represents the variance of the system measurement noise at time k + 1; the state estimation deviation at the k-th time is defined as
Figure GDA0002973270580000041
Step 5, designing an online estimation algorithm of the fault factors to realize the estimation of the fault factors:
Figure GDA0002973270580000042
wherein the content of the first and second substances,
Figure GDA0002973270580000043
represents an estimate of the fault factor δ (k) at time k; delta sigma (k +1) ═ sigma'ee(k+1)-∑ee(k+1)；∑'ee(k +1) represents the variance of the estimated value of the post-fault system state at the time k + 1; sigmauu(k)＝E(u(k)u(k)T) Represents the variance of the system control input at time k;
Figure GDA0002973270580000044
further, the dynamic linearization process described in step 2 mainly includes the following steps:
step 2.1, for the nonlinear system described in step 1, using dynamic linearization, when Δ u (k) ≠ 0, there is a pseudo-partial derivative
Figure GDA0002973270580000045
Obtaining:
Figure GDA0002973270580000046
wherein the content of the first and second substances,
Figure GDA0002973270580000047
b is a normal number;
step 2.2, applying disturbance to the system, and making x (k) ═ y (k) obtain:
Figure GDA0002973270580000048
wherein u (k) ═ u (k) u (k-1)]T；
Figure GDA0002973270580000049
And 2.3, adding failure factors of actuator faults into the system, so that the original nonlinear system can be converted into:
Figure GDA00029732705800000410
wherein, Bf(k)＝B(k)(1-δ(k))；
Further, the step of designing the algorithm for updating the pseudo partial derivative is as follows:
step 3.1, consider the following objective function:
Figure GDA00029732705800000411
wherein μ >0 is a weighting factor;
to the parameter
Figure GDA0002973270580000051
Performing minimization to obtain pseudo partial derivativeAnd (3) estimation algorithm:
Figure GDA0002973270580000052
wherein the content of the first and second substances,
Figure GDA0002973270580000053
representative pair
Figure GDA0002973270580000054
(ii) an estimate of (d); eta ∈ (0, 1)]Is a step size factor;
step 3.2, in order to enable the algorithm to better estimate the time-varying parameters, the following reset algorithm is designed:
Figure GDA0002973270580000055
if it is
Figure GDA0002973270580000056
Or | delta u (k-1) | is less than or equal to epsilon or
Figure GDA0002973270580000057
Wherein the content of the first and second substances,
Figure GDA0002973270580000058
is that
Figure GDA0002973270580000059
An initial value of (1); ε is a positive number;
still further, considering that there is only one kind of fault at the same time, the design of the state observer proposed in step 4 is mainly implemented by the following method:
step 4.1, applying a Kalman filter to perform one-step budget:
Figure GDA00029732705800000510
wherein the content of the first and second substances,
Figure GDA00029732705800000511
represents an estimate of state x (k +1) at time k;
step 4.2, designing an estimation correction algorithm:
Figure GDA00029732705800000512
step 4.3, defining state estimation deviation
Figure GDA00029732705800000513
Obtaining:
Figure GDA00029732705800000514
step 4.4, taking variance of state estimation deviation:
Figure GDA00029732705800000515
and 4.5, obtaining a gain operator by optimizing the variance of the state estimation deviation:
g(k+1)＝(∑ee(k)+∑ww(k+1))(∑ee(k)+∑ww(k+1)+∑vv(k+1))-1；
furthermore, the design steps of the external fault failure factor online estimation algorithm in step 5 are as follows:
step 5.1, after the fault, the Kalman filter equation is converted into:
Figure GDA0002973270580000061
the state estimation bias equation becomes:
Figure GDA0002973270580000062
squaring the two sides of the equation and then taking the random average to obtain:
Figure GDA0002973270580000063
step 5.2, defining an expression delta sigma (k +1) ═ sigma'ee(k+1)-∑ee(k +1) to obtain:
Figure GDA0002973270580000064
the fault failure factor estimation expression is obtained as follows:
Figure GDA0002973270580000065
due to the parameters in the matrix B (k)
Figure GDA0002973270580000066
Is unknown, use
Figure GDA0002973270580000067
And (3) replacing:
the online estimation expression of the failure factor is obtained as follows:
Figure GDA0002973270580000068
wherein the content of the first and second substances,
Figure GDA0002973270580000069
has the advantages that:
1. the invention discloses a method for identifying fault factors of an actuator of a data-driven nonlinear system, which introduces a data-driven technology into the method for identifying the fault factors of the actuator, particularly obtains an equivalent linear data model with the fault failure factors of the actuator by dynamically linearizing the nonlinear system, designs a fault factor identification algorithm on the basis of the data model, has the advantage of not depending on specific model information, and improves the applicability of the algorithm.
2. The invention discloses a method for identifying fault factors of a data-driven nonlinear system actuator, which estimates the state of a system by designing a state observer, and designs an online fault failure factor estimation algorithm on the basis of the estimation of the state of the system, thereby achieving the purpose of estimating the fault failure factors in the system.
Other features and advantages of the present invention will become more apparent from the following detailed description of the invention when taken in conjunction with the accompanying drawings.
Drawings
FIG. 1 is a flow chart of a method for identifying fault factors of an actuator of a data-driven nonlinear system according to the present invention;
FIG. 2 is a graph of an update algorithm for pseudo-partial derivatives versus an estimate of the pseudo-partial derivatives;
FIG. 3 is a curve of the fault failure factor estimation by the actuator fault factor identification method of the present invention;
Detailed Description
For better illustrating the objects and advantages of the present invention, the present invention will be described in further detail with reference to the embodiments and the accompanying drawings.
Aiming at the problem of online estimation of actuator fault factors of a nonlinear system with dynamic actuator faults, the invention provides a method for identifying the actuator fault factors of the nonlinear system driven by data, wherein the method is based on a Kalman filter design state observer to estimate the system state under the framework of data driving and utilizes estimation deviation on the basis of a filter.
Referring to fig. 1, the method for identifying a fault factor of a data-driven nonlinear system actuator disclosed in the present embodiment includes the following steps:
step S1: a non-linear system is established.
Consider a discrete-time nonlinear system as follows:
y(k+1)＝f(y(k),…,y(k-ny),u(k),…,u(k-nu)) (1)
wherein y (k) represents the output of the system at the k-th time; u (k) represents the control input of the system at the k-th moment; n isyAnd nuIs two unknown positive integers representing the order of the system; f (-) represents the unknown non-linear function.
The system meets the generalized Lipschitz condition, and when the delta u (k) is not equal to 0 and the delta y (k) is not equal to 0, the | delta y (k +1) | is not more than b1|△u(k)|+b2|△y(k)|；
Wherein, b1、b2Are all positive numbers; Δ u (k) — u (k) -1, Δ y (k) ═ y (k) -1.
Step S2: and dynamically linearizing the nonlinear system into an equivalent dynamic linearized data model with an actuator fault failure factor.
Specifically, the method comprises the following steps:
step S21: for the nonlinear system (1) in step S1, using dynamic linearization, when Δ u (k) ≠ 0, there is a pseudo-partial derivative
Figure GDA0002973270580000081
Obtaining:
Figure GDA0002973270580000082
wherein the content of the first and second substances,
Figure GDA0002973270580000083
a pseudo partial derivative defined for time k;
Figure GDA0002973270580000084
b is a positive constant.
Step S22: applying a perturbation to the system (2) and letting x (k) y (k) obtain:
Figure GDA0002973270580000085
wherein u (k) ═ u (k) u (k-1)]T；
Figure GDA0002973270580000086
w (k +1) represents the disturbance of the system at the k +1 th moment; v (k) represents the measurement noise of the system at time k.
Step S23: adding failure factors of actuator faults into the system, the original nonlinear system can be converted into:
Figure GDA0002973270580000087
wherein, Bf(k) (1- δ (k)); delta (k) represents a failure factor of the actuator failure at the kth moment, and the value range of the failure factor is delta (k) belonging to [0,1]]When δ (k) is 1, it represents that the actuator completely fails at the kth time, and δ (k) is 0, which means that the actuator normally operates at the kth time, 0<δ(k)<1 denotes the execution drive capability lost after an actuator failure.
Step S3: the design update algorithm estimates the unknown pseudo-partial derivatives in the linear data model. Specifically, the method comprises the following steps:
step S31: consider the following objective function:
Figure GDA0002973270580000091
wherein μ >0 is a weighting factor.
To the parameter
Figure GDA0002973270580000092
And (3) carrying out minimization to obtain a pseudo partial derivative estimation algorithm:
Figure GDA0002973270580000093
wherein the content of the first and second substances,
Figure GDA0002973270580000094
representative pair
Figure GDA0002973270580000095
(ii) an estimate of (d); eta ∈ (0, 1)]Is the step size factor.
Step S32: in order to enable the algorithm to better estimate the time-varying parameters, the following reset algorithm is designed:
Figure GDA0002973270580000096
wherein the content of the first and second substances,
Figure GDA0002973270580000097
is that
Figure GDA0002973270580000098
An initial value of (1); ε is a positive number.
Step S4: and designing a state observer to estimate the state of the system.
Specifically, the method comprises the following steps:
step S41: applying a Kalman filter to perform one-step budgeting:
Figure GDA0002973270580000099
wherein the content of the first and second substances,
Figure GDA00029732705800000910
represents an estimate of state x (k +1) at time k.
Step S42: designing an estimation correction algorithm:
Figure GDA00029732705800000911
where g (k +1) represents the gain operator at time k + 1.
Step S43: defining state estimation biases
Figure GDA00029732705800000912
Obtaining:
Figure GDA00029732705800000913
step S44: taking variance of state estimation bias:
Figure GDA0002973270580000101
therein, sigmaee(k +1) represents the variance of the state estimation bias at time k + 1; sigmaww(k +1) represents the variance of the system disturbance at time k + 1; sigmavv(k +1) represents the variance of the system measurement noise at time k + 1.
Step S45: and obtaining a gain operator by optimizing the variance of the state estimation deviation:
g(k+1)＝(∑ee(k)+∑ww(k+1))(∑ee(k)+∑ww(k+1)+∑vv(k+1))-1 (12)
step S5: designing an online estimation algorithm of the fault factor to realize the estimation of the fault factor:
specifically, the method comprises the following steps:
step S51: after a fault, the Kalman filter equation turns into:
Figure GDA0002973270580000102
the state estimation bias equation becomes:
Figure GDA0002973270580000103
squaring the two sides of the equation and then taking the random average to obtain:
∑'ee(k+1)＝(1-g(k+1))2∑'ee(k)+(1-g(k+1))2∑ww(k+1)+g(k+1)2∑vv(k+1)+(1-g(k+1))2(Bf(k)-B(k))(E(u(k)u(k)T))(Bf(k)-B(k))T
(15)
wherein, sigma'ee(k +1) represents the variance of the post-fault system state estimate value at time k + 1.
Step S52: expression ∑ (k +1) ═ Σ 'is defined'ee(k+1)-∑ee(k +1) to obtain:
Figure GDA0002973270580000104
the online estimation expression of the failure factor is obtained as follows:
Figure GDA0002973270580000111
wherein the content of the first and second substances,
Figure GDA0002973270580000112
represents an estimate of the fault factor δ (k); sigmauu(k)＝E(u(k)Tu (k)) represents the variance of the system control input at time k;
due to the parameters in the matrix B (k)
Figure GDA0002973270580000113
Is unknown, use
Figure GDA0002973270580000114
And (3) replacing:
the online estimation expression of the failure factor is obtained as follows:
Figure GDA0002973270580000115
wherein the content of the first and second substances,
Figure GDA0002973270580000116
the nonlinear system employed in the examples is as follows:
Figure GDA0002973270580000117
where a (k) is 1+ round (k/500) is a time-varying parameter, and the disturbance and measurement noise of the system are w (k +1) 0.1rand and v (k +1) 0.1rand, respectively.
The failure factor δ (k) for actuator failure is set to:
Figure GDA0002973270580000118
in the described embodiment, the controller parameters are set to η ═ 1, μ ═ 1, ρ ═ 1, and λ ═ 1. The initial value of the system is selected as
Figure GDA0002973270580000119
y(1)＝0.5，u(1)＝0，∑ee(1)＝0，∑'ee(1)＝0。
The estimation curve of the pseudo-partial derivative by the update algorithm of the pseudo-partial derivative provided by the invention is shown in FIG. 2; the estimation curve of the actuator fault factor identification method for the fault failure factor is shown in fig. 3.
As can be seen from fig. 2: the simulation time is 10-500, the estimated value of the pseudo partial derivative is 0.58, and the simulation is relatively stable; because the system changes when the simulation time is 500, the estimated value of the pseudo-partial derivative also changes, and when the simulation time is 500-1000, the estimated value of the pseudo-partial derivative is 0.64, and the change is relatively stable.
As can be seen from fig. 3: the simulation time is 0-800, and the estimated value of the fault failure factor is 1; in the simulation time of 801-900, the estimated value of the fault failure factor is 0.6; the simulation time is 901-1000, and the estimated value of the fault failure factor is 0.8. The fault factor identification method for the actuator can perfectly estimate the fault failure factor in the nonlinear system.
In this embodiment, the estimation effect of the actuator fault factor is mainly used as a performance parameter, and fig. 3 shows an estimation curve of the fault failure factor by using the data-driven nonlinear system actuator fault factor identification method. As can be seen from fig. 2 and 3, the fault factor identification method for the actuator provided by the invention can achieve the purpose of estimating the fault factor, does not use any model information, and has good adaptability.
The above detailed description further illustrates the objects, technical solutions and advantages of the present invention, and it should be understood that the embodiments are only used for explaining the present invention and not for limiting the scope of the present invention, and modifications, equivalent substitutions, improvements and the like under the same principle and concept of the present invention should be included in the scope of the present invention.

Claims (5)

1. A method for identifying fault factors of a data-driven nonlinear system actuator is characterized by comprising the following steps: the method comprises the following steps:
step 1, establishing a nonlinear system:
consider a discrete-time nonlinear system as follows:
y(k+1)＝f(y(k),…,y(k-ny),u(k),…,u(k-nu))；
wherein the content of the first and second substances,
y (k) represents the output of the system at the k-th time;
u (k) represents the control input of the system at the k-th moment;
nyand nuIs two unknown positive integers representing the order of the system;
f (-) represents an unknown non-linear function;
the system meets the generalized Lipschitz condition, and when the delta u (k) is not equal to 0 and the delta y (k) is not equal to 0, the | delta y (k +1) | is not more than b1|△u(k)|+b2|△y(k)|；
Wherein the content of the first and second substances,
b1、b2are all positive numbers;
△u(k)＝u(k)-u(k-1)，△y(k)＝y(k)-y(k-1)；
step 2, dynamically linearizing the nonlinear system into an equivalent linear data model with an actuator fault failure factor:
Figure FDA0002973270570000011
wherein the content of the first and second substances,
w (k +1) represents the disturbance of the system at the k +1 th moment;
v (k) represents the measurement noise of the system at the k-th moment;
x (k) represents the state of the system at the k-th time;
u(k)＝[u(k) u(k-1)]T；
Bf(k)＝(1-δ(k))B(k)；
Figure FDA0002973270570000021
Figure FDA0002973270570000022
represents the pseudo-partial derivative defined at time k;
δ (k) is a failure factor of the actuator failure at the kth moment, and the value range of δ (k) is ∈ [0,1], when δ (k) is 1, the actuator completely fails at the kth moment, δ (k) is 0, the actuator works normally at the kth moment, and 0< δ (k) <1 represents the execution driving capacity lost after the actuator failure;
step 3, designing an updating algorithm to estimate the pseudo partial derivatives in the linear data model:
Figure FDA0002973270570000023
Figure FDA0002973270570000024
if it is
Figure FDA0002973270570000025
Or | delta u (k-1) | is less than or equal to epsilon or
Figure FDA0002973270570000026
Wherein the content of the first and second substances,
Figure FDA0002973270570000027
representing for time k
Figure FDA0002973270570000028
(ii) an estimate of (d);
μ >0 is a weighting factor;
eta ∈ (0, 1) is a step factor;
ε is a positive number;
Figure FDA0002973270570000029
is that
Figure FDA00029732705700000210
An initial value of (1);
and 4, designing a state observer to estimate the system state:
Figure FDA00029732705700000211
wherein the content of the first and second substances,
Figure FDA00029732705700000212
represents an estimate of state x (k +1) at time k + 1;
Figure FDA00029732705700000213
represents an estimate of state x (k) at time k;
Figure FDA00029732705700000214
represents the estimation of the state x (k +1) at the time kA value;
g (k +1) represents a gain operator at the k +1 th time;
∑ee(k +1) represents the variance of the state estimation bias at time k + 1;
∑ww(k +1) represents the variance of the system disturbance at time k + 1;
∑vv(k +1) represents the variance of the system measurement noise at time k + 1;
the state estimation deviation at the k-th time is defined as
Figure FDA0002973270570000031
Step 5, designing an online estimation algorithm of the fault factors to realize the estimation of the fault factors:
Figure FDA0002973270570000032
wherein the content of the first and second substances,
Figure FDA0002973270570000033
represents an estimate of the fault factor δ (k) at time k;
△∑(k+1)＝∑'ee(k+1)-∑ee(k+1)；
∑'ee(k +1) represents the variance of the estimated value of the system state after the fault at the k +1 th moment;
∑uu(k)＝E(u(k)u(k)T) Represents the variance of the system control input at time k;
Figure FDA0002973270570000034
2. the method of claim 1, further comprising: the dynamic linearization process of the step 2 mainly comprises the following steps:
step 2.1, for the nonlinear system described in step 1, use is made ofDynamic linearization, when Δ u (k) ≠ 0, there is a pseudo-partial derivative
Figure FDA0002973270570000035
Obtaining:
Figure FDA0002973270570000036
wherein the content of the first and second substances,
Figure FDA0002973270570000037
b is a normal number;
step 2.2, applying disturbance to the system, and making x (k) ═ y (k) obtain:
Figure FDA0002973270570000038
wherein u (k) ═ u (k) u (k-1)]T；
Figure FDA0002973270570000039
And 2.3, adding failure factors of actuator faults into the system, so that the original nonlinear system can be converted into:
Figure FDA0002973270570000041
wherein, Bf(k)＝B(k)(1-δ(k))。
3. The method of claim 1, further comprising: the updating algorithm design for estimating the pseudo partial derivative in the linear data model in the step 3 mainly comprises the following steps:
step 3.1, consider the following objective function:
Figure FDA0002973270570000042
wherein μ >0 is a weighting factor;
to the parameter
Figure FDA0002973270570000043
And (3) carrying out minimization to obtain a pseudo partial derivative estimation algorithm:
Figure FDA0002973270570000044
wherein the content of the first and second substances,
Figure FDA0002973270570000045
representative pair
Figure FDA0002973270570000046
(ii) an estimate of (d); eta ∈ (0, 1)]Is a step size factor;
step 3.2, in order to enable the algorithm to better estimate the time-varying parameters, the following reset algorithm is designed:
Figure FDA0002973270570000047
if it is
Figure FDA0002973270570000048
Or | delta u (k-1) | is less than or equal to epsilon or
Figure FDA0002973270570000049
Wherein the content of the first and second substances,
Figure FDA00029732705700000410
is that
Figure FDA00029732705700000411
An initial value of (1); ε is a positive number.
4. The method of claim 1, further comprising: considering the case where only one type of fault is present at the same time, the design of the state observer proposed in step 4 mainly comprises the following steps:
step 4.1, applying a Kalman filter to perform one-step budget:
Figure FDA00029732705700000412
wherein the content of the first and second substances,
Figure FDA00029732705700000413
represents an estimate of state x (k +1) at time k;
step 4.2, designing an estimation correction algorithm:
Figure FDA00029732705700000414
step 4.3, defining state estimation deviation
Figure FDA00029732705700000415
Obtaining:
Figure FDA00029732705700000416
step 4.4, taking variance of state estimation deviation:
Figure FDA0002973270570000051
and 4.5, obtaining a gain operator by optimizing the variance of the state estimation deviation:
g(k+1)＝(∑ee(k)+∑ww(k+1))(∑ee(k)+∑ww(k+1)+∑vv(k+1))-1。
5. the method of claim 1, further comprising: the design of the fault failure factor online estimation algorithm in the step 5 mainly comprises the following steps:
step 5.1, after the fault, the Kalman filter equation is converted into:
Figure FDA0002973270570000052
the state estimation bias equation becomes:
Figure FDA0002973270570000053
squaring the two sides of the equation and then taking the random average to obtain:
Figure FDA0002973270570000054
step 5.2, defining an expression delta sigma (k +1) ═ sigma'ee(k+1)-∑ee(k +1) to obtain:
Figure FDA0002973270570000055
therein, sigmauu(k)＝E(u(k)Tu (k)) represents the variance of the system control input at time k;
the estimated expression for the fault failure factor is obtained as follows:
Figure FDA0002973270570000056
due to the parameters in the matrix B (k)
Figure FDA0002973270570000057
Is unknown, use
Figure FDA0002973270570000058
And (3) replacing:
the online estimation expression of the failure factor is obtained as follows:
Figure FDA0002973270570000061
wherein the content of the first and second substances,
Figure FDA0002973270570000062
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