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:
according to the linear system theory, the linear subsystem L2The state space equation of (a) is as follows:
wherein
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,
Are respectively L
1、L
2The state transition matrix of (a) is,
are respectively L
1、L
2The input matrix of (a) is selected,
are respectively L
1、L
2The output matrix of (a) is obtained,
for the failure matrix, u ∈ R
1×1As input, y ∈ R
1×1For output, f ∈ R
1×1Is a failure of the system, a
fAs a factor of a failed link, b
fInput coefficients for unknown fault links; u. of
f∈R
1×1The input for the failed link is unknown; a is
fAnd b
fObtained from a priori knowledge of system faults, known; suppose u
fIs bounded, a
fIs less than 1, i.e. | a
f|<1, so that the fault system is stable according to the linear system stability condition, n
iFor the dimension of the i-th linear system, let
And is
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
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
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:
due to the fact that
Substituting the formula (2) into the formula (3) to obtain:
1-3) establishing an integral state space equation of the dead zone sandwich system: according to formula (1), formula (2), formula (4) and
the state space equation of the linear motor drive system is as follows:
wherein
According to the characteristics of the sandwich system, only the output y (k) of the system can be directly measured, so that
Wherein
0 is a zero matrix of the corresponding order
Equation (5) is written as follows:
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:
wherein K
pj、k
ijRespectively the proportional gain and the integral gain of the ith working interval,
x (k), y (k), f (k), eta, respectively
iWhen j is 1, 2, or 3, i is j, then
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:
and calculate
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
The following operations are performed:
if it is
The following operations are performed:
if it is
The following operations are performed:
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.
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:
according to the linear system theory, the linear subsystem L2The state space equation of (a) is as follows:
wherein
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,x
11And x
12Each represents L
1First and second state variables, x
21And x
22Each represents L
2The first and second state variables of (a),
are respectively L
1、L
2The state transition matrix of (a) is,
are respectively L
1、L
2The input matrix of (a) is selected,
are respectively L
1、L
2The output matrix of (a) is obtained,
for the failure matrix, u ∈ R
1×1As input, y ∈ R
1×1For output, f ∈ R
1×1Is a failure of the system, a
fAs a factor of a failed link, b
fInput coefficients for unknown fault links; u. of
f∈R
1×1The input for the failed link is unknown; a is
fAnd b
fObtained from a priori knowledge of system faults, known; suppose u
fIs bounded, a
fIs less than 1, i.e. | a
f|<1, so that the fault system is stable according to the linear system stability condition, n
iFor the dimension of the i-th linear system, let
And is
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
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
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:
due to the fact that
Substituting the formula (2) into the formula (3) to obtain:
1-3) establishing an integral state space equation of the dead zone sandwich system: according to formula (1), formula (2), formula (4) and
then:
wherein
According to the characteristics of the sandwich system, only the output y (k) of the system can be directly measured, so that
Wherein
0 is a zero matrix of the corresponding order
Equation (5) is written as follows:
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:
wherein K
pj、k
ijThe ith working interval is proportional gain and integral gain,
x (k), y (k), f (k), eta, respectively
iWhen j is 1, 2, or 3, i is j, then
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:
and calculate
Order:
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
The following operations are performed:
if it is
The following operations are performed:
if it is
The following operations are performed:
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.