CN112987683B - Fault positioning method for dead zone non-smooth sandwich system - Google Patents

Fault positioning method for dead zone non-smooth sandwich system Download PDF

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CN112987683B
CN112987683B CN202110025922.8A CN202110025922A CN112987683B CN 112987683 B CN112987683 B CN 112987683B CN 202110025922 A CN202110025922 A CN 202110025922A CN 112987683 B CN112987683 B CN 112987683B
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CN112987683A (en
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周祖鹏
刘旭锋
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Guilin University of Electronic 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
    • 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
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Abstract

本发明公开了一种死区非光滑三明治系统的故障定位方法，该方法通过构建死区非光滑三明治系统的状态空间方程，然后根据构建的死区非光滑三明治系统的状态空间方程，构造能随系统工作区间变化而自动切换的非光滑观测器，并给出该观测器的存在条件；通过在三明治系统的输入端和输出端分别布置高精度光学数字编码器，通过高精度光学数字编码器获取系统的输入、输出数据，利用构造的观测器估计系统的状态，计算并绘制状态及状态估计误差；最后分析故障与状态估计误差对应关系，确定系统故障的发生位置及类型。该方法仅仅需要已知系统的输入和输出号，采样数据种类少，使用简单、定位可靠性高。

Figure 202110025922

The invention discloses a fault location method for a dead zone non-smooth sandwich system. The method constructs the state space equation of the dead zone non-smooth sandwich system, and then constructs the state space equation of the dead zone non-smooth sandwich system according to the constructed dead zone non-smooth sandwich system. A non-smooth observer that automatically switches when the working range of the system changes, and the existence conditions of the observer are given; by arranging high-precision optical digital encoders at the input and output ends of the sandwich system respectively, the high-precision optical digital encoders are used to obtain the Based on the input and output data of the system, the constructed observer is used to estimate the state of the system, and the state and state estimation error are calculated and plotted. Finally, the corresponding relationship between faults and state estimation errors is analyzed to determine the location and type of system faults. The method only needs to know the input and output numbers of the system, has few types of sampling data, is simple to use, and has high positioning reliability.

Figure 202110025922

Description

Fault positioning method for dead zone non-smooth sandwich system
Technical Field
The invention relates to the field of nonlinear system fault positioning, in particular to a fault positioning method for a dead zone non-smooth sandwich system.
Background
There is a class of system dead zone sandwich systems in the industry. By a linear subsystem L1Non-smooth link N, linear subsystem L2The three subsystems are connected in series. There is a possibility of failure of all three subsystems of the system. Mechanical devices with gear transmission or belt transmission can be regarded as sandwich systems, such as automobile gearboxes, transmission systems of lathes, production lines, steering systems of gun towers, angle adjusting systems of large astronomical telescopes and the like. When the system is in fault, the first condition for removing the fault is to find the position after the fault occurs. The current fault location technology is mainly applied to fault location in systems such as high-voltage transmission lines or power grids. For example, the invention patent with chinese patent publication No. CN110531222A discloses a fault location method for a high-voltage transmission line based on Matlab, which uses a symmetric component method to locate the distance of fault points on the high-voltage transmission line by analyzing the simulation results of Matlab and Simulink software for the possible short circuit and ground fault on the high-voltage transmission line. The invention patent with Chinese patent publication number CN111965490A discloses a simulation-based micro-grid fault positioning method in a micro-gridAnd each node is provided with a fault detection device for detecting the state signal of the node, the state characteristic of the node is provided through a microprocessor, and the fault type and the corresponding fault detection device are obtained through experimental simulation analysis, namely the fault node is determined. Very little research has been done on the localization of faults in sandwich systems. Aiming at the defects of the prior art, the invention discloses a fault of a state estimation error method positioning system.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a fault positioning method of a dead zone non-smooth sandwich system, which can accurately position three faults in the dead zone non-smooth sandwich system.
The technical scheme for realizing the purpose of the invention is as follows:
a fault locating method for a dead zone non-smooth sandwich system comprises the following steps:
1) constructing a non-smooth state space equation capable of accurately describing a dead zone sandwich system with faults by using a key item separation principle and a switching function, wherein the dead zone nonlinear characteristic is characterized in that the size of output is only related to the size of input at the current moment and is not related to the input and the output at the previous moment, u (k) and y (k) are respectively measurable input and output variables in the dead zone sandwich system with faults, v (k) is a measurable variable, and v (k) is a measurable variable1(k) And v2(k) Is an unmeasurable intermediate variable, L1Representing the front-end linear subsystem, L2Representing the back-end linear subsystem, D1And D2Is the width of the dead zone, m1And m2Is the slope of the linear region, f (k) represents the fault of the system, and the state space equation of the system is established when the dead zone sandwich system containing the fault is provided, which is as follows:
1-1) establishing a state space equation of the linear subsystem: according to the linear system theory, the linear subsystem L1The state space equation of (a) is as follows:
Figure GDA0003342726260000021
according to the linear system theory, the linear subsystem L2The state space equation of (a) is as follows:
Figure GDA0003342726260000022
wherein
Figure GDA0003342726260000023
u∈R1 ×1，y∈R1×1，f∈R1×1，uf∈R1×1，af∈R1×1，bf∈R1×1，
Figure GDA0003342726260000024
Are respectively L1、L2The state transition matrix of (a) is,
Figure GDA0003342726260000025
are respectively L1、L2The input matrix of (a) is selected,
Figure GDA0003342726260000026
are respectively L1、L2The output matrix of (a) is obtained,
Figure GDA0003342726260000027
for the failure matrix, u ∈ R1×1As input, y ∈ R1×1For output, f ∈ R1×1Is a failure of the system, afAs a factor of a failed link, bfInput coefficients for unknown fault links; u. off∈R1×1The input for the failed link is unknown; a isfAnd bfObtained from a priori knowledge of system faults, known; suppose ufIs bounded, afIs less than 1, i.e. | af|<1, so that the fault system is stable according to the linear system stability condition, niFor the dimension of the i-th linear system, let
Figure GDA0003342726260000028
And is
Figure GDA0003342726260000029
1-2) establishing a state space equation of the dead zone subsystem: defining intermediate variables m (k), w1(k) Comprises the following steps:
m(k)＝m1+(m2-m1)h(k)
w1(k)＝m(k)(x(k)-D1h1(k)+D2h2(k))
wherein
Figure GDA00033427262600000210
In order to be a function of the switching,
obtaining the following according to the input-output relation of the dead zone:
v2(k)＝w1(k)-h3(k)w1(k)＝(1-h3(k))w1(k)，
wherein
Figure GDA0003342726260000031
Also as a switching function, when h3(k) When 0, the system operates in the linear region, and v2(k)＝w1(k) (ii) a When h is generated3(k) When 1, the system operates in the dead zone, and v2(k)＝w1(k)-w1(k) 0, according to the input-output characteristics of the dead zone:
Figure GDA00033427262600000311
due to the fact that
Figure GDA0003342726260000032
Substituting the formula (2) into the formula (3) to obtain:
Figure GDA0003342726260000033
1-3) establishing an integral state space equation of the dead zone sandwich system: according to formula (1), formula (2), formula (4) and
Figure GDA0003342726260000034
the state space equation of the linear motor drive system is as follows:
Figure GDA0003342726260000035
wherein
Figure GDA0003342726260000036
Figure GDA0003342726260000037
According to the characteristics of the sandwich system, only the output y (k) of the system can be directly measured, so that
Figure GDA0003342726260000038
Wherein
Figure GDA0003342726260000039
0 is a zero matrix of the corresponding order
Figure GDA00033427262600000310
Equation (5) is written as follows:
Figure GDA0003342726260000041
wherein etaiSwitching vectors introduced for considering the existence of dead zone non-linear characteristics of the system;
2) constructing a non-smooth observer capable of automatically switching along with the change of the working interval of the dead zone sandwich system with the fault according to the non-smooth state space equation of the dead zone sandwich system with the fault, which is constructed in the step 1), and giving the existence condition of the observer, wherein the method comprises the following steps:
Figure GDA0003342726260000042
wherein Kpj、kijRespectively the proportional gain and the integral gain of the ith working interval,
Figure GDA0003342726260000043
Figure GDA0003342726260000044
x (k), y (k), f (k), eta, respectivelyiWhen j is 1, 2, or 3, i is j, then
Figure GDA0003342726260000045
The convergence condition of the switching proportional-integral observer is Aj-KpjThe characteristic value of C is within the unit circle.
3) According to the characteristics of a practical second-order system, i.e. n1＝n2X is 21∈R2×1，x2∈R2×1Let x be1＝[X12，X12]、x2＝[X21，X22]X12 and X12 each represent L1X21 and X22 represent L, respectively2First and second state variables of; respectively arranging high-precision optical digital encoders at the input end and the output end of the sandwich system, acquiring input and output data of the system through the high-precision optical digital encoders, estimating the state of the system by adopting the observer constructed in the step 1), calculating and drawing a state and state estimation error diagram, and specifically, respectively aiming at the existence of L in the system by utilizing a Simulink module and an m editor in Matlab1Fault, or dead zone increasing fault, or L2When the fault occurs, estimating the state of the system, and calculating to obtain the state and a state estimation error;
the Simulink module comprises an ideal input module, a summation module, a controller and an L1Subsystem, dead zone link, L2Subsystem, feedback gain and connecting line, and also includes L1Fault signal, or L2A fault signal, or dead zone increasing module; when the system contains only L1In case of failure, will L2Fault signal f of2Setting the dead zone width to be zero, wherein the dead zone width is a normal width; when the system contains only L2In case of failure, will L1Fault signal f of1Setting the dead zone width to be zero, wherein the dead zone width is a normal width; when the system increases only the dead zone width, L is set1Fault signal f of1And L2Fault signal f of2Set to zero;
the m editor executes the following operations:
3-1) initializing the estimated values of the state variables:
order:
Figure GDA0003342726260000051
and calculate
Figure GDA0003342726260000052
Determining sampling frequency, simulation time and cycle number N according to the characteristics of an actual system, and collecting an input signal u and an output signal y;
3-2) let k be 3;
3-3) judging whether k is less than or equal to N, if k is less than or equal to N, executing the step 3-4), and if k is greater than N, finishing the operation;
3-4) if
Figure GDA0003342726260000053
The following operations are performed:
Figure GDA0003342726260000054
Figure GDA0003342726260000055
Figure GDA0003342726260000056
if it is
Figure GDA0003342726260000057
The following operations are performed:
Figure GDA0003342726260000058
Figure GDA0003342726260000059
Figure GDA00033427262600000510
if it is
Figure GDA00033427262600000511
The following operations are performed:
Figure GDA00033427262600000512
Figure GDA00033427262600000513
Figure GDA00033427262600000514
3-5) adding 1 to the value of k, and repeating the step 3-3);
4) the estimation errors of the 4 states of X12, X12, X21 and X22 are obtained through analysis and simulation, and the occurrence position and the type of the system fault are determined, specifically as follows:
4-1) if the estimation error of the state X11 converges to zero, the estimation error of the state X12 converges to zero, the estimation error of the state X21 converges to zero, and the estimation error of the state X22 converges to zero, no fault occurs;
4-2) if the state X11 estimation error converges to a fixed value, the state X12 estimation error converges to zero, the state X21 estimation error converges to zero, and the state X22 estimation error converges to zero, then a fault occurs at L1Subsystem, and step fault;
4-3) converging an interval if the estimation error of the state X11 changes according to a sine rule; the estimation error of the state X12 changes according to a sine rule and converges an interval; state X21 estimates error convergence to zero; state X22 estimates error convergence to zero; the fault occurs at L1Subsystem, and is sinusoidal fault with attenuated amplitude;
4-4) if the estimation error of the state X11 converges to zero, the estimation error of the state X12 converges to zero, the estimation error of the state X21 converges to a constant range, and the estimation error of the state X22 converges to zero, the fault is that the dead zone width is too large;
4-5) if the state X11 estimation error converges to a fixed value, the state X12 estimation error converges to a fixed value, the state X21 estimation error converges to a fixed value, and the state X22 estimation error converges to a fixed value, a fault occurs at L2Subsystem, and step fault;
4-6) converging an interval if the estimation error of the state X11 changes according to a sine rule; the estimation error of the state X12 changes according to a sine rule and converges an interval; the estimation error of the state X21 changes according to a sine rule and converges an interval; the estimation error of the state X22 changes according to a sine rule and converges an interval; the fault occurs at L2Subsystem, and is a sinusoidal fault with attenuated amplitude.
The method adopts a state estimation error method to position the fault of the dead zone non-smooth sandwich system, only needs the input and output numbers of the known system, has less types of sampling data, is simple to use and has high positioning reliability.
Drawings
FIG. 1 is a block diagram of a dead zone sandwich system with a fault;
FIG. 2 is a Simulink model diagram of three faults of a dead zone sandwich system;
FIG. 3 is the estimated error for 4 states of the system without failure;
FIG. 4 is an error of 4 states of the system with a step fault at L1;
FIG. 5 is an error of 4 states of the system with a sinusoidal fault at L1;
FIG. 6 is an error for 4 states of the system as the dead band width increases;
FIG. 7 is the error for 4 states of the system with a step fault at L2;
fig. 8 shows the error of 4 states of the system when L2 has a sinusoidal fault.
Detailed Description
The invention will be further elucidated with reference to the drawings and examples, without however being limited thereto.
Example (b):
a fault locating method for a dead zone non-smooth sandwich system comprises the following steps:
1) constructing a non-smooth state space equation capable of accurately describing a dead zone sandwich system with faults by using a key item separation principle and a switching function, wherein the dead zone nonlinear characteristic is characterized in that the size of output is only related to the size of input at the current moment and is not related to the input and the output at the previous moment, u (k) and y (k) are respectively measurable input and output variables in the dead zone sandwich system with faults, v (k) is a measurable variable, and v (k) is a measurable variable1(k) And v2(k) Is an unmeasurable intermediate variable, L1Representing the front-end linear subsystem, L2Representing the back-end linear subsystem, D1And D2Is the width of the dead zone, m1And m2Is the slope of the linear region, f (k) represents the fault of the system, and the state space equation of the system is established for the dead zone sandwich system with the fault as shown in fig. 1, which is as follows:
1-1) establishing a state space equation of the linear subsystem: according to the linear system theory, the linear subsystem L1The state space equation of (a) is as follows:
Figure GDA0003342726260000071
according to the linear system theory, the linear subsystem L2The state space equation of (a) is as follows:
Figure GDA0003342726260000072
wherein
Figure GDA0003342726260000073
u∈R1 ×1，y∈R1×1，f∈R1×1，uf∈R1×1，af∈R1×1，bf∈R1×1，x11And x12Each represents L1First and second state variables, x21And x22Each represents L2The first and second state variables of (a),
Figure GDA0003342726260000074
are respectively L1、L2The state transition matrix of (a) is,
Figure GDA0003342726260000075
are respectively L1、L2The input matrix of (a) is selected,
Figure GDA0003342726260000076
are respectively L1、L2The output matrix of (a) is obtained,
Figure GDA0003342726260000077
for the failure matrix, u ∈ R1×1As input, y ∈ R1×1For output, f ∈ R1×1Is a failure of the system, afAs a factor of a failed link, bfInput coefficients for unknown fault links; u. off∈R1×1The input for the failed link is unknown; a isfAnd bfObtained from a priori knowledge of system faults, known; suppose ufIs bounded, afIs less than 1, i.e. | af|<1, so that the fault system is stable according to the linear system stability condition, niFor the dimension of the i-th linear system, let
Figure GDA0003342726260000078
And is
Figure GDA0003342726260000079
1-2) establishing a state space equation of the dead zone subsystem: defining intermediate variables m (k), w1(k) Comprises the following steps:
m(k)＝m1+(m2-m1)h(k)
w1(k)＝m(k)(x(k)-D1h1(k)+D2h2(k))
wherein
Figure GDA0003342726260000081
In order to be a function of the switching,
obtaining the following according to the input-output relation of the dead zone:
v2(k)＝w1(k)-h3(k)w1(k)＝(1-h3(k))w1(k)，
wherein
Figure GDA0003342726260000082
Also as a switching function, when h3(k) When 0, the system operates in the linear region, and v2(k)＝w1(k) (ii) a When h is generated3(k) When 1, the system operates in the dead zone, and v2(k)＝w1(k)-w1(k) 0, according to the input-output characteristics of the dead zone:
Figure GDA00033427262600000811
due to the fact that
Figure GDA0003342726260000083
Substituting the formula (2) into the formula (3) to obtain:
Figure GDA0003342726260000084
1-3) establishing an integral state space equation of the dead zone sandwich system: according to formula (1), formula (2), formula (4) and
Figure GDA0003342726260000085
then:
Figure GDA0003342726260000086
wherein
Figure GDA0003342726260000087
Figure GDA0003342726260000088
According to the characteristics of the sandwich system, only the output y (k) of the system can be directly measured, so that
Figure GDA0003342726260000089
Wherein
Figure GDA00033427262600000810
0 is a zero matrix of the corresponding order
Figure GDA0003342726260000091
Equation (5) is written as follows:
Figure GDA0003342726260000092
wherein etaiSwitching vectors introduced for considering the existence of dead zone non-linear characteristics of the system;
2) constructing a non-smooth observer capable of automatically switching along with the change of the working interval of the dead zone sandwich system with the fault according to the non-smooth state space equation of the dead zone sandwich system with the fault, which is constructed in the step 1), and giving the existence condition of the observer, wherein the method comprises the following steps:
Figure GDA0003342726260000093
wherein Kpj、kijThe ith working interval is proportional gain and integral gain,
Figure GDA0003342726260000094
Figure GDA0003342726260000095
x (k), y (k), f (k), eta, respectivelyiWhen j is 1, 2, or 3, i is j, then
Figure GDA0003342726260000096
The non-smooth observer has a convergence condition of Aj-KpjThe characteristic value of C is within the unit circle.
3) According to the characteristics of the actual second-order system, then n1＝n2＝2，x1∈R2×1,x2∈R2×1Let x be1＝[X11，X12]T,x2＝[X21，X22]T. X11 and X12 each represent L1X21 and X22 represent L, respectively2First and second state variables. Arranging high-precision optical digital encoders at the input end and the output end of the sandwich system respectively, acquiring input and output data of the system through the high-precision optical digital encoders, estimating the state of the system by adopting the observer constructed in the step 1), calculating and drawing the state and the state estimation errorSpecifically, the method utilizes a Simulink module and an m editor in Matlab to respectively aim at the existence of L in the system1Fault, dead zone increasing fault, L2When the fault occurs, estimating the state of the system, and calculating to obtain the state and a state estimation error;
the Simulink module comprises an ideal input module, a summation module, a controller and an L1Subsystem, dead zone link, L2Subsystem, feedback gain and connecting line, and also includes L1Fault signal, or L2A fault signal, or dead zone increasing module; when the system contains only L1In case of failure, will L2Fault signal f of2Setting the dead zone width to be zero, wherein the dead zone width is 0.02 of the normal width; when the system contains only L2In case of failure, will L1Fault signal f of1Setting the dead zone width to be zero, wherein the dead zone width is 0.02 of the normal width; when the dead zone width of the system is increased, L is adjusted1Fault signal f of1And L2Fault signal f of2Set to zero;
the m editor executes the following operations:
3-1) initializing the estimated value of the state variable and the estimated value of the fault:
order:
Figure GDA0003342726260000101
and calculate
Figure GDA0003342726260000102
Order:
Figure GDA0003342726260000103
determining sampling frequency, state estimation time and defined cycle number N according to the characteristics of an actual system, and acquiring an input signal u and an output signal y;
3-2) let k be 3;
3-3) judging whether k is less than or equal to N, if k is less than or equal to N, executing the step 3-4), and if k is greater than N, finishing the operation;
3-4) if
Figure GDA0003342726260000104
The following operations are performed:
Figure GDA0003342726260000105
Figure GDA0003342726260000106
Figure GDA0003342726260000107
if it is
Figure GDA0003342726260000108
The following operations are performed:
Figure GDA0003342726260000109
Figure GDA00033427262600001010
Figure GDA00033427262600001011
if it is
Figure GDA00033427262600001012
The following operations are performed:
Figure GDA00033427262600001013
Figure GDA00033427262600001014
Figure GDA00033427262600001015
3-5) adding 1 to the value of k, and repeating the step 3-3);
4) the estimation errors of the 4 states of X12, X12, X21 and X22 are obtained through analysis and simulation, and the occurrence position and the type of the system fault are determined, specifically as follows:
4-1) if the estimation error of the state X11 converges to zero, the estimation error of the state X12 converges to zero, the estimation error of the state X21 converges to zero, and the estimation error of the state X22 converges to zero, no fault occurs;
4-2) if the state X11 estimation error converges to a fixed value, the state X12 estimation error converges to zero, the state X21 estimation error converges to zero, and the state X22 estimation error converges to zero, then a fault occurs at L1Subsystem, and step fault;
4-3) converging an interval if the estimation error of the state X11 changes according to a sine rule; the estimation error of the state X12 changes according to a sine rule and converges an interval; state X21 estimates error convergence to zero; state X22 estimates error convergence to zero; the fault occurs at L1Subsystem, and is sinusoidal fault with attenuated amplitude;
4-4) if the estimation error of the state X11 converges to zero, the estimation error of the state X12 converges to zero, the estimation error of the state X21 converges to a constant range, and the estimation error of the state X22 converges to zero, the fault is that the dead zone width is too large;
4-5) if the state X11 estimation error converges to a fixed value, the state X12 estimation error converges to a fixed value, the state X21 estimation error converges to a fixed value, and the state X22 estimation error converges to a fixed value, a fault occurs at L2Subsystem, and step fault;
4-6) converging an interval if the estimation error of the state X11 changes according to a sine rule; the estimation error of the state X12 changes according to a sine rule and converges an interval; the estimation error of the state X21 changes according to a sine rule and converges an interval; the estimation error of the state X22 changes according to a sine rule and converges an interval; the fault occurs at L2Subsystem, and is a sinusoidal fault with attenuated amplitude.
Example (b):
(1) without failure
When there is no fault, the estimation errors for the 4 states (x11, x12, x21, x22) all eventually converge to zero, as shown in fig. 3.
(2) L1 step fault
Ideal input signal u is 0.5sin (5t), dead zone width is normal width 0.02, L2The fault signal f2 is zero and is given to L at 1s (100 th sampling point), respectively1And adding a step fault with the amplitude of 0v (no fault), a step fault with the amplitude of 1v and a step fault with the amplitude of 10v, and estimating the state of the system by adopting a non-smooth observer.
As shown in fig. 4, in the three cases, the estimation error of the state x11 converges to a fixed value after the occurrence of a failure (after 1 s) through 20 sampling points (0.2s), and the shape of the estimation error also coincides with the shape of the failure signal, but the estimation error convergence value of the state x11 also increases in proportion to the increase in the failure amplitude. The estimation errors for the remaining three states (states x12, x21, x22) eventually converge to "zero" (within 0.05% of the fault signal amplitude).
Further, as shown in fig. 5, when the fault frequency increases, the amplitude of the estimation error convergence value of the state x11 changes little, and the estimation errors of the remaining three states (states x12, x21, x22) eventually converge to "zero" (within 0.05% of the fault signal amplitude).
(3) Increase of dead zone width
Ideal input signal u is 0.5sin (5t), L1Fault signal f1 is zero, L2The fault signal f2 is zero. When the dead zone widths are 0.02 (normal width), 0.1, 0.2, 2, respectively, the state of the system is estimated using a non-smooth observer.
As shown in fig. 6, when the dead zone width is increased from 0.02 to 0.1, 0.2, 2, the estimation error of the state x21 converges to a certain range. As the dead band width increases to 0.2 and above, the estimated error curves for state x21 coincide, converging to a constant range. When the dead zone width is 0.2 or less, the estimation error of the state x21 converges to zero. The estimated error for the remaining 3 states (x11, x12, x22) converges to "zero" to within 0.1% of the input.
(4) L2 step fault
Ideal input signal u is 0.5sin (5t), dead zone width is normal width 0.02, L1The fault signal f1 is zero.
As shown in fig. 7, at 1s (100 th sampling point), a step fault (no fault) with an amplitude of 0v, a step fault with an amplitude of 1v, and a step fault with an amplitude of 2v are added, respectively. A non-smooth observer is used to estimate the state of the system. In the three cases, the estimation errors of 4 states (x11, x12, x21, and x22) converge to a fixed value after the occurrence of a fault (after 1 s) through 20 sampling points (0.2s), and the shape of the estimation error also matches the shape of the fault signal. As the magnitude of the fault increases, the convergence of the estimation error for state x11 also increases.
Further, the magnitude of the estimation error convergence value range of the 4 states (x11, x12, x21, x22) changes little when the failure frequency increases.

Claims (3)

1.一种死区非光滑三明治系统的故障定位方法，其特征在于，包括如下步骤：1. a fault location method of dead zone non-smooth sandwich system, is characterized in that, comprises the steps: 1)利用关键项分离原则和切换函数，构建能准确描述含有故障的死区三明治系统的非光滑状态空间方程，由于死区非线性特性的特点是输出的大小只与当前时刻的输入的大小有关，与前一时刻的输入输出无关，其中在含有故障的死区三明治系统中，u(k)和y(k)分别是可测的输入、输出变量，v1(k)和v2(k)是不可测的中间变量，L1代表前端线性子系统，L2代表后端线性子系统，D1和D2是死区的宽度，m1和m2是线性区的斜率，f(k)代表系统的故障，针对含有故障的死区三明治系统时建立系统的状态空间方程，具体如下：1) Using the principle of separation of key terms and switching function, construct a non-smooth state space equation that can accurately describe the dead zone sandwich system with faults. Due to the nonlinear characteristics of the dead zone, the size of the output is only related to the size of the input at the current moment. , has nothing to do with the input and output at the previous moment, where in the dead-band sandwich system with faults, u(k) and y(k) are the measurable input and output variables, v 1 (k) and v 2 (k, respectively ) is an unmeasurable intermediate variable, L 1 represents the front-end linear subsystem, L 2 represents the back-end linear subsystem, D 1 and D 2 are the widths of the dead zone, m 1 and m 2 are the slopes of the linear zone, f(k ) represents the fault of the system, and the state space equation of the system is established for the dead zone sandwich system with faults, as follows: 1-1)建立线性子系统的状态空间方程：根据线性系统理论，线性子系统L1的状态空间方程如下所示：1-1) Establish the state space equation of the linear subsystem: According to the linear system theory, the state space equation of the linear subsystem L1 is as follows:
Figure FDA0003342726250000011
Figure FDA0003342726250000011
 根据线性系统理论，线性子系统L2的状态空间方程如下所示：According to linear system theory, the state - space equation of the linear subsystem L2 is as follows:
Figure FDA0003342726250000012
Figure FDA0003342726250000012
 其中
Figure FDA0003342726250000013
u∈R1×1，y∈R1×1，f∈R1×1，uf∈R1×1，af∈R1×1，bf∈R1×1，
Figure FDA0003342726250000014
分别为L1、L2的状态转移矩阵，
Figure FDA0003342726250000015
分别为L1、L2的输入矩阵，
Figure FDA0003342726250000016
分别为L1、L2的输出矩阵，
Figure FDA0003342726250000017
为故障矩阵，u∈R1×1为输入，y∈R1×1为输出，f∈R1×1为系统的故障，af为故障环节的系数，bf为未知的故障环节的输入系数；uf∈R1×1为故障环节的输入，是未知的；af和bf由系统故障的先验知识获得，是已知的；假设uf是有界的，af的范数小于1，即|af|<1，因此根据线性系统稳定性条件，故障系统是稳定的，ni为第i个线性系统的维数，设
Figure FDA0003342726250000018
且
Figure FDA0003342726250000019
in
Figure FDA0003342726250000013
u∈R 1×1 , y∈R 1×1 , f∈R 1×1 , u f ∈ R 1×1 , a f ∈ R 1×1 , b f ∈ R 1×1 ,
Figure FDA0003342726250000014
are the state transition matrices of L 1 and L 2 , respectively,
Figure FDA0003342726250000015
are the input matrices of L 1 and L 2 , respectively,
Figure FDA0003342726250000016
are the output matrices of L 1 and L 2 , respectively,
Figure FDA0003342726250000017
is the fault matrix, u∈R 1×1 is the input, y∈R 1×1 is the output, f∈R 1×1 is the fault of the system, a f is the coefficient of the fault link, and b f is the input of the unknown fault link coefficient; u f ∈ R 1×1 is the input of the fault link, which is unknown; a f and b f are obtained from the prior knowledge of the system fault and are known; assuming that u f is bounded, the range of a f The number is less than 1, that is |a f |<1, so according to the linear system stability condition, the fault system is stable, n i is the dimension of the ith linear system, let
Figure FDA0003342726250000018
and
Figure FDA0003342726250000019
 1-2)建立死区子系统的状态空间方程：定义中间变量m(k)、w1(k)为：1-2) Establish the state space equation of the dead zone subsystem: define the intermediate variables m(k) and w 1 (k) as: m(k)＝m1+(m2-m1)h(k)m(k)=m 1 +(m 2 -m 1 )h(k) w1(k)＝m(k)(x(k)-D1h1(k)+D2h2(k))w 1 (k)=m(k)(x(k)-D 1 h 1 (k)+D 2 h 2 (k)) 其中in
Figure FDA0003342726250000021
为切换函数，
Figure FDA0003342726250000021
is the switching function, 根据死区的输入输出关系得：According to the input and output relationship of the dead zone: v2(k)＝w1(k)-h3(k)w1(k)＝(1-h3(k))w1(k)，v 2 (k)=w 1 (k)-h 3 (k)w 1 (k)=(1-h 3 (k))w 1 (k), 其中in
Figure FDA0003342726250000022
Figure FDA0003342726250000022
 也为切换函数，当h3(k)＝0时，系统工作在线性区，且v2(k)＝w1(k)；当h3(k)＝1时，系统工作在死区，且v2(k)＝w1(k)-w1(k)＝0，根据死区的输入输出特性得：It is also a switching function. When h 3 (k)=0, the system works in the linear region, and v 2 (k)=w 1 (k); when h 3 (k)=1, the system works in the dead zone, And v 2 (k)=w 1 (k)-w 1 (k)=0, according to the input and output characteristics of the dead zone:
Figure FDA0003342726250000023
Figure FDA0003342726250000023
 由于
Figure FDA0003342726250000024
将公式(2)代入公式(3)中，得：because
Figure FDA0003342726250000024
Substituting formula (2) into formula (3), we get:
Figure FDA0003342726250000025
Figure FDA0003342726250000025
 1-3)建立死区三明治系统的整体状态空间方程：根据公式(1)、公式(2)、公式(4)和
Figure FDA0003342726250000026
则有1-3) Establish the overall state space equation of the dead zone sandwich system: according to formula (1), formula (2), formula (4) and
Figure FDA0003342726250000026
then there are
Figure FDA0003342726250000027
Figure FDA0003342726250000027
 其中in
Figure FDA0003342726250000028
Figure FDA0003342726250000028
Figure FDA0003342726250000031
Figure FDA0003342726250000031
 根据三明治系统的特性可知，只有系统的输出y(k)能够被直接测量，则令
Figure FDA0003342726250000032
其中
Figure FDA0003342726250000033
0是相应阶数的零矩阵，设According to the characteristics of the sandwich system, only the output y(k) of the system can be directly measured, then let
Figure FDA0003342726250000032
in
Figure FDA0003342726250000033
0 is the zero matrix of the corresponding order, let
Figure FDA0003342726250000034
则公式(5)写成如下形式：
Figure FDA0003342726250000034
Then formula (5) can be written in the following form:
Figure FDA0003342726250000035
Figure FDA0003342726250000035
 其中ηi为考虑系统存在死区非线性特性而引入的切换向量；where η i is the switching vector introduced considering the existence of dead zone nonlinearity in the system; 2)根据步骤1)构建的含有故障的死区三明治系统的非光滑状态空间方程，构造能随含有故障的死区三明治系统工作区间变化而自动切换的非光滑观测器，并给出该观测器的存在条件，包括如下步骤：2) According to the non-smooth state space equation of the dead-band sandwich system with faults constructed in step 1), construct a non-smooth observer that can automatically switch with the working interval of the dead-zone sandwich system with faults, and give the observer Existence conditions, including the following steps:
Figure FDA0003342726250000036
Figure FDA0003342726250000036
 其中Kpj、kij分别为第i个工作区间的比例增益和积分增益，
Figure FDA0003342726250000037
Figure FDA0003342726250000038
分别为x(k)、y(k)、f(k)、ηi的估计值，若j＝1、2、3时，i＝j，则
Figure FDA0003342726250000039
非光滑观测器的收敛条件为Aj-KpjC的特征值在单位圆内；where K pj and k ij are the proportional gain and integral gain of the ith working interval, respectively,
Figure FDA0003342726250000037
Figure FDA0003342726250000038
are the estimated values of x(k), y(k), f(k), and η i , respectively. If j=1, 2, and 3, i=j, then
Figure FDA0003342726250000039
The convergence condition of the non-smooth observer is that the eigenvalues of A j -K pj C are in the unit circle; 3)根据实际二阶系统的特性，即n1＝n2＝2，则x1∈R2×1，x2∈R2×1，假设x1＝[X12，X12]T、x2＝[X21，X22]T，X12和X12分别代表L1的第一个和第二个状态变量，X21和X22分别代表L2的第一个和第二个状态变量；在三明治系统的输入端和输出端分别布置高精度光学数字编码器，通过高精度光学数字编码器获取死区三明治系统的输入、输出数据，采用步骤1)构造的观测器估计系统的状态，计算并绘制状态及状态估计误差图，具体是利用Matlab中的Simulink模块及m编辑器，分别针对系统存在L1故障、死区增大故障、L2故障时，估计系统的状态，并计算得到状态计误差；3) According to the characteristics of the actual second-order system, that is, n 1 =n 2 =2, then x 1 ∈ R 2×1 , x 2 ∈ R 2×1 , assuming x 1 =[X12, X12] T , x 2 = [X21, X22] T , X12 and X12 represent the first and second state variables of L1, respectively, and X21 and X22 represent the first and second state variables of L2, respectively; at the input of the sandwich system and The high-precision optical digital encoders are arranged at the output ends, and the input and output data of the dead-zone sandwich system are obtained through the high-precision optical digital encoders. The observer constructed in step 1) is used to estimate the state of the system, and the state and state estimation error are calculated and plotted. Figure, specifically using the Simulink module in Matlab and the m editor, respectively, when the system has L1 fault, dead zone increase fault, and L2 fault, the state of the system is estimated, and the state meter error is calculated ; 4)通过分析仿真得到X12、X12、X21和X22这4个状态的估计误差，确定系统故障的发生位置及类型，具体如下：4) Obtain the estimated errors of the four states of X12, X12, X21 and X22 through analysis and simulation, and determine the location and type of system faults, as follows: 4-1)若状态X11估计误差收敛到零，状态X12估计误差收敛到零，状态X21估计误差收敛到零，状态X22估计误差收敛到零，则无故障发生；4-1) If the estimated error of state X11 converges to zero, the estimated error of state X12 converges to zero, the estimated error of state X21 converges to zero, and the estimated error of state X22 converges to zero, then no fault occurs; 4-2)若状态X11估计误差收敛于固定值，状态X12估计误差收敛到零，状态X21估计误差收敛到零，状态X22估计误差收敛到零，则故障发生在L1子系统，且为阶跃故障；4-2) If the estimated error of state X11 converges to a fixed value, the estimated error of state X12 converges to zero, the estimated error of state X21 converges to zero, and the estimated error of state X22 converges to zero, then the fault occurs in the L1 subsystem and is of order jump fault; 4-3)若状态X11估计误差按正弦规律变化，收敛一个区间；状态X12估计误差按正弦规律变化，收敛一个区间；状态X21估计误差收敛到零；状态X22估计误差收敛到零；则故障发生在L1子系统，且为幅值衰减的正弦故障；4-3) If the estimated error of state X11 changes according to the sine law and converges to an interval; the estimated error of state X12 changes according to the sine law and converges to an interval; the estimated error of state X21 converges to zero; the estimated error of state X22 converges to zero; then the fault occurs In the L 1 subsystem, and it is a sinusoidal fault with amplitude decay; 4-4)若状态X11估计误差收敛到零，状态X12估计误差收敛到零，状态X21估计误差收敛到一个恒定的范围，状态X22估计误差收敛到零，则故障为死区宽度过大；4-4) If the estimated error of state X11 converges to zero, the estimated error of state X12 converges to zero, the estimated error of state X21 converges to a constant range, and the estimated error of state X22 converges to zero, the fault is that the dead zone width is too large; 4-5)若状态X11估计误差收敛于固定值，状态X12估计误差收敛于固定值，状态X21估计误差收敛于固定值，状态X22估计误差收敛于固定值，则故障发生在L2子系统，且为阶跃故障；4-5) If the estimated error of state X11 converges to a fixed value, the estimated error of state X12 converges to a fixed value, the estimated error of state X21 converges to a fixed value, and the estimated error of state X22 converges to a fixed value, then the fault occurs in the L2 subsystem, And it is a step fault; 4-6)若状态X11估计误差按正弦规律变化，收敛一个区间；状态X12估计误差按正弦规律变化，收敛一个区间；状态X21估计误差按正弦规律变化，收敛一个区间；状态X22估计误差按正弦规律变化，收敛一个区间；则故障发生在L2子系统，且为幅值衰减的正弦故障。4-6) If the estimated error of state X11 changes according to the sine law, and converges an interval; the estimated error of state X12 changes according to the sine law, and converges an interval; the estimated error of state X21 changes according to the sine law, and converges to an interval; the estimated error of state X22 changes according to the sine law It changes regularly and converges to an interval; then the fault occurs in the L 2 subsystem, and it is a sinusoidal fault with attenuation of amplitude. 2.根据权利要求1所述的一种死区非光滑三明治系统的故障定位方法，其特征在于，所述的Simulink模块包括理想输入、求和模块、控制器、L1子系统、死区环节、L2子系统、反馈增益和连接线，还包括L1故障信号，或L2故障信号，或死区增大模块；当仅有L1故障时，将L2的故障信号f2设定为零，死区宽度为正常宽度；当系统仅含L2故障时，将L1的故障信号f1设定为零，死区宽度为正常宽度；当系统仅死区宽度增大时，将L1的故障信号f1和L2的故障信号f2设定为零。2. the fault location method of a kind of dead zone non-smooth sandwich system according to claim 1, is characterized in that, described Simulink module comprises ideal input, summation module, controller, L 1 subsystem, dead zone link , L2 subsystem, feedback gain and connecting line, also include L1 fault signal, or L2 fault signal, or dead zone increase module ; when only L1 fault, set the L2 fault signal f2 is zero, the dead - band width is the normal width; when the system only contains L2 faults, set the fault signal f1 of L1 to zero, and the dead - band width is the normal width; when the system only increases the dead-band width, set the The fault signal f 1 of L 1 and the fault signal f 2 of L 2 are set to zero. 3.根据权利要求1所述的一种死区非光滑三明治系统的故障定位方法，其特征在于，所述的m编辑器执行如下操作：3. the fault location method of a kind of dead zone non-smooth sandwich system according to claim 1, is characterized in that, described m editor performs the following operations: 3-1)初始化状态变量的估计值：3-1) Initialize the estimated value of the state variable: 令：
Figure FDA0003342726250000051
并计算
Figure FDA0003342726250000052
根据实际系统特征确定采样频率、采样时间及循环次数N，并采集系统输入信号u和输出信号y；make:
Figure FDA0003342726250000051
and calculate
Figure FDA0003342726250000052
Determine the sampling frequency, sampling time and cycle number N according to the actual system characteristics, and collect the system input signal u and output signal y; 3-2)令k＝3；3-2) Let k=3; 3-3)判断k是否小于等于N，若k小于等于N，则执行步骤3-4)，若k>N,则结束运行；3-3) Determine whether k is less than or equal to N, if k is less than or equal to N, then execute step 3-4), if k>N, then end the operation; 3-4)若
Figure FDA0003342726250000053
则进行如下操作：3-4) If
Figure FDA0003342726250000053
Then do the following:
Figure FDA0003342726250000054
Figure FDA0003342726250000054
Figure FDA0003342726250000055
Figure FDA0003342726250000055
Figure FDA0003342726250000056
Figure FDA0003342726250000056
 若
Figure FDA0003342726250000057
则进行如下操作：like
Figure FDA0003342726250000057
Then do the following:
Figure FDA0003342726250000058
Figure FDA0003342726250000058
Figure FDA0003342726250000059
Figure FDA0003342726250000059
Figure FDA00033427262500000510
Figure FDA00033427262500000510
 若
Figure FDA00033427262500000511
则进行如下操作：like
Figure FDA00033427262500000511
Then do the following:
Figure FDA00033427262500000512
Figure FDA00033427262500000512
Figure FDA00033427262500000513
Figure FDA00033427262500000513
Figure FDA00033427262500000514
Figure FDA00033427262500000514
 3-5)k的值加1，重复步骤3-3)。3-5) Add 1 to the value of k, and repeat steps 3-3).
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