CN107272660A - A kind of random fault detection method of the network control system with packet loss - Google Patents
A kind of random fault detection method of the network control system with packet loss Download PDFInfo
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- CN107272660A CN107272660A CN201710616552.9A CN201710616552A CN107272660A CN 107272660 A CN107272660 A CN 107272660A CN 201710616552 A CN201710616552 A CN 201710616552A CN 107272660 A CN107272660 A CN 107272660A
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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
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- G—PHYSICS
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- G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
- G05B2219/00—Program-control systems
- G05B2219/20—Pc systems
- G05B2219/24—Pc safety
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Abstract
The present invention discloses a kind of random fault detection method of the network control system with packet loss, consider that network control system has random loss, quantization error and situation about breaking down at random, initially set up the network control system model that there is random fault, resettle the model of fault Detection Filter, residual error evaluation mechanism is introduced to detect whether failure occurs, utilize Lyapunov Theory of Stability and LMI analysis method, obtain the adequate condition that augmented system meansquare exponential stability and fault Detection Filter are present, optimization problem is solved using Matlab LMI tool boxes, providing Optimal Fault Detection Filter parameter isThen whether occurred according to the residual error mechanism failure judgement set up.The present invention considers failure under actual conditions to be occurred at random, and the probability of happening of failure meets Bernoulli distributions, it is adaptable to general fault detection method, reduces conservative.
Description
Technical field
The present invention relates to the random fault inspection of network control system, the more particularly to network control system with packet loss
Survey method.
Background technology
Network control system (Networked Control System, NCS) is by shared communication channel (such as net
Network) each element in system is connected to the closed-loop feedback control system of composition, the feedback control with traditional point-to-point connection
System processed is compared, and network control system has the advantages that convenient for installation and maintenance, flexibility is high and is easy to reconstruct.But network
The problems such as introducing can bring the problem of some are new, such as data quantization, network-induced time delay, data-bag lost, influence system
Performance and stability, or even produce failure.In Practical Project, network control system is to performance, safety and reliability
It is required that it is very high, if some failures can not be excluded in time, greatly harm can be caused and lost, therefore fault detect is in recent years
The focus of research.
A crucial step for fault detect is exactly to be used as residual error generation mechanism by fault Detection Filter, obtains quick to failure
The residual signals of sense, recycle whether residual error evaluation mechanism failure judgement occurs.In recent years, increasing scholar's research is present
Exist in the fault detection problem of the network control system of chance phenomenon, such as system random delay, random loss or with
Machine nonlinear disturbance, very big harm will be produced to systematic function discounting for chance phenomenon.But most of document is being ground
When studying carefully NCS fault detection problem, all assume that what being to determine property of fault-signal occurred, but be due to the uncertain of network
In variation characteristic, actual conditions, failure occurs at random.
The content of the invention
For above-mentioned problems of the prior art, the invention provides the random of the network control system with packet loss
Fault detection method.In the case of considering that network control system has random loss, quantization error and broken down at random,
Devise full rank fault Detection Filter so that network control system remains to keep meansquare exponential stability simultaneously in these cases
And default H ∞ performance indications are met, effectively detection is out of order.
The technical solution adopted in the present invention is：A kind of random fault detection side of the network control system with packet loss
Method, comprises the following steps：
1) the controlled device mathematical modeling for the network control system that there is random fault is set up：
Wherein：For state vector,Preferably to measure output quantity,It is ω for finite energyk∈
l2The Unknown worm of [0, ∞],For fault-signal to be detected, A, E1,E2, C, D is the constant square with appropriate dimension
Battle array.αkThe probability that failure occurs in expression system, to meet the stochastic variable of Bernoulli 0-1 sequences distribution：
The measurement of system is output as after quantization：
Wherein：Δk=diag { Δs1,k,Δ2,k,…,Δm,k, | | Δk||2≤δ2, I is unit matrix.
2) full rank fault Detection Filter is designed：
Wherein：For the state estimation of system,For the input of fault Detection Filter,For residual error
Signal, Af,Bf,Cf,DfIt is the parameter for the fault Detection Filter that needs are determined；
The input of fault Detection Filter is：
Wherein：βkThe random loss situation for occurring between controlled device and fault Detection Filter is represented, is met
The stochastic variable of Bernoulli 0-1 sequences distribution：
Introduce residual error evaluation mechanism to detect whether failure occurs, residual error evaluation function J (k) and threshold value J (th) are respectively：
Wherein：L is the maximum time span of evaluation function, and whether system, which breaks down, to be detected by formula (7)：
3) adequate condition that system meansquare exponential stability and fault Detection Filter are present is：
Wherein：
Wherein：* the transposition of symmetric position matrix is represented,U,X,W,It is the matrix with appropriate dimension
And be it is unknown,ε is known variables, and its dependent variable is all known.Utilize
Matlab LMI tool boxes are solved, and scalar γ ＞ 0 are given, if there is positive definite matrixU,X,
W and scalar ε ＞ 0 causes formula (8) to set up, then system is meansquare exponential stability, and meets H∞Performance indications, can obtain it is non-most
Excellent fault Detection Filter parameter, you can to proceed step 4)；If above-mentioned known variables are not solved, system is not equal
Side is exponentially stable and can not obtain non-optimal fault Detection Filter parameter, it is not possible to carry out step 4).
4) Optimal Fault Detection Filter parameter is calculated
By solving optimization problem formula (9)：
If there is solution, Optimal Fault Detection Filter parameter is obtainedOptimal H∞Performance indications are γmin,
Obtaining Optimal Fault Detection Filter parameter is：
Wherein：G3, V is nonsingular matrix.After obtaining Optimal Fault Detection Filter parameter, according to formula (3) and formula (4)
The residual signals r (k) of system can be obtained, then calculating formula (5) and formula (6), finally whether failure judgement occurs by formula (7).
If formula (9) can not obtain Optimal Fault Detection Filter without solution.
Compared with prior art, beneficial effects of the present invention：The present invention simultaneously consider imperfect measurement factor in system,
Packet loss, external disturbance and the failure occurred at random, are derived by a series of, give under the network environment and there is random event
The design method of fault Detection Filter in the case of barrier, only considered really during compared to traditional fault Detection Filter design setting model
The limitation of qualitative failure, this method, which has more, to be of practical significance, and reduces conservative.
Brief description of the drawings
Accompanying drawing 1 is the flow chart of the random fault detection method of the network control system with packet loss.
Accompanying drawing 2 is ω (k) ≠ 0,Residual signals figure during ρ=0.6.
Accompanying drawing 3 is ω (k) ≠ 0,Residual error evaluation function figure during ρ=0.6.
Accompanying drawing 4 is ω (k) ≠ 0,Residual error evaluation function figure during ρ=0.6.
Accompanying drawing 5 is ω (k) ≠ 0,Residual error evaluation function figure during ρ=0.6.
Accompanying drawing 6 is ω (k) ≠ 0,Residual error evaluation function figure during ρ=0.6.
Accompanying drawing 7 is ω (k) ≠ 0,Residual error evaluation function figure during ρ=0.6.
Embodiment
The embodiment to the present invention is described further below in conjunction with the accompanying drawings.
Referring to the drawings 1, a kind of random fault detection method of the network control system with packet loss comprises the following steps：
Step 1：Set up the controlled device mathematical modeling for the network control system that there is random fault
The controlled device that there is the network control system of random fault is formula (1), it is contemplated that in network control system,
The data of sampling will carry out the quantification treatment of data before network transmission is carried out, and the measurement of system is output as formula after quantization
(2)。
Step 2：Design full rank fault Detection Filter
Full rank fault Detection Filter formula (3) is designed, from αkRepresent the probability that failure occurs, αkTo meet Bernoulli
The stochastic variable of 0-1 sequences distribution, works as αkWhen=0, show that system does not break down, work as αkWhen=1, show that system determines hair
Raw failure,Bigger, the possibility broken down in expression system is bigger.
Consider under packet drop, the input of fault Detection Filter is formula (4).β in formulakRepresent to occur in controlled device and
Random loss situation between fault Detection Filter, βkTo meet the stochastic variable of Bernoulli 0-1 sequences distribution.Work as βk
When=1,
Show that no data is lost, work as βkWhen=0, show data all loss.
Define residual error error signal：
ek=r (k)-fk (11)
Consider formula (1), (3), (4) and (11), following augmented system can be obtained by the method for state augmentation：
Wherein：
Construction residual error evaluation function J (k) and threshold value J (th) are respectively formula (5) and formula (6), and formula (7) can be for judgement event
Whether barrier occurs.Wherein：L is the maximum time span of evaluation function, when the value in residual error evaluation function is more than threshold value, is occurred
Failure and alarm, otherwise represent not break down.
Step 3：Construct Lyapunov functionsUtilize Lyapunov Theory of Stability and LMI
Analysis method, obtains the adequate condition that augmented system (12) meansquare exponential stability and fault Detection Filter are present.Step is as follows：
Step 3.1：The stability of augmented system is first determined whether, the adequate condition of system meansquare exponential stability is obtained.
Assuming that inequality (13) is set up, i.e.,
Wherein：* the transposition of symmetric position matrix, ψ are represented11=diag { Pl-G-GT,Pl-G-GT,Pl-G-GT,-I ,-I },
ψ22=diag {-Pl,-γ2I }, ψ33=diag {-ε-1I,-εδ-2I },
Wherein：
Φ1=Q+DFE+ETFTDT (15)
Wherein：F=Δskδ-1,
E=[0 0000 δ M δ N]
Lemma is amplified according to cross term, it is known that
Φ1≤Φ2=Q+ ε DDT+ε-1EET (16)
Φ2≤Φ3=Q+ ε DDT+ε-1(δ-1E)Tδ2(δ-1E) (17)
Φ3＜ 0 is of equal value with inequality (13), if inequality (13) is set up, Φ1＜ 0.
Due to Pl＞ 0, (Pl-G)TPl -1(Pl- G) >=0, therefore Pl-G-GT≥-GTPl -1G, so as to obtain
Wherein：Ξ11=diag {-GTPl -1G,-GTPl -1G,-GTPl -1G ,-I ,-I }, to (18) premultiplication diag { G-T,G-T,G-T,
I, I, I, I }, the right side, which multiplies its transposition, to be obtained
Mending lemma according to Schur can obtain
Υ1+Υ2+ Π ＜ 0 (20)
Wherein：
Work as θkWhen=0,
Wherein：
So as to
By non-zero initial condition andIt is available
Wherein：σ=(- λmax(Γ))-1λmax(Pl) ＞ 0, Asymptotic Stability on augmented system (12) mean square meaning can be obtained.
According to Lyapunov Theory of Stability, work as θkWhen=0, scalar γ ＞ 0 are given, there is positive definite matrix Pl＞ 0, scalar ε
＞ 0, nonsingular matrix G cause inequality (13) to set up.When the adequate condition of step 3.1 is set up, then perform step 3.2；Such as
The adequate condition of fruit step 3.1 is invalid, then augmented system (12) is not meansquare exponential stability, it is impossible to perform step 3.2.
Step 3.2：Work as θkWhen ≠ 0,
Wherein：
By (20), formula is obtained,
Consider zero initial condition, augmented system (12) exponential mean square stability further has
Meet H∞Performance indications.
Formula (13) can be write as the form of formula (8), can be obtained from formula (8),Represent Ω and W is nonsingular, can look for
To nonsingular matrix G3,G4So that
Make U=G1,
Define transposed matrix
In the case of without loss of generality, it is assumed that
It can obtain
So as to obtain
ZTΨ Z ＜ 0 (32)
Wherein：Z=diag { T, T, T, I, I, T, I, I, I }, formula (8) and formula (32) are of equal value.
Solved using Matlab LMI tool boxes, work as θkWhen ≠ 0, scalar γ ＞ 0 are given, there is positive definite matrix Pl＞
0, scalar ε ＞ 0, nonsingular matrix G cause inequality (13) to set up.So augmented system (12) meansquare exponential stability, and full
Sufficient H∞Performance indications.When the adequate condition of step 3 is set up, i.e., when formula (8) is set up, then perform step 4；If step 3 is filled
Slitting part is invalid, then augmented system (12) is not meansquare exponential stability and non-optimal fault Detection Filter parameter can not be solved
, it is impossible to perform step 4.
Step 4：Calculate Optimal Fault Detection Filter parameter
For augmented system (12), obtaining Optimal Fault Detection Filter parameter by solving optimization problem formula (9) is
Formula (10), optimal H∞Performance indications are γmin, the residual signals r (k) of system can be obtained according to formula (3) and formula (4), then
Formula (5) and formula (6) are obtained, finally whether failure judgement occurs by formula (7)；If formula (9) can not obtain optimal failure without solution
Fault detection filter.
Embodiment：
Using a kind of random fault detection method of the network control system with packet loss proposed by the present invention, not outer
I.e. θ in the case that boundary is disturbed with failurekWhen=0, augmented system is meansquare exponential stability.Work as θkWhen ≠ 0, system is just to refer to
Number it is stable and meet H∞Performance indications.Concrete methods of realizing is as follows：
Controlled device is closed network networked control systems, and its controlled device mathematical modeling is formula (1), gives its systematic parameter
For
C=[0.6 0.8 0], D=0.1
Assuming that quantization resolution ρ=0.6 of system, using MATLAB LMI tool boxes, solves different packet loss probabilityWith it is random
Probability of malfunctionUnder optimal H∞Performance indications, as shown in table 1.As can be seen that with network channel probability of malfunction increase (lose
Bag probability reduces), corresponding performance indications γminAlso increase therewith, i.e. Disturbance Rejection degradation, illustrate that failure occurs general
The probability of rate and packet loss has important influence to systematic function.
Minimum γ in the case of the different faults of table 1 and drop probabilitiesmin
Assuming that the original state of system is x0=[0 0 0]T, ρ=0.6,With MATLAB LMI
Tool box, can be in the hope of γ for augmented system (11)min=1.1340, the optimized parameter of corresponding fault Detection Filter is
Cf=[- 0.00713 0.00441 0.00402], Df=-0.00369
Assuming that fault-signal and Unknown worm are respectively
ω (k)=e-0.02k sin(0.2k)
The residual error r (k) and residual error valuation functions J (k) of system are as shown in accompanying drawing 2 and accompanying drawing 3, according to of the present invention
Residual error evaluation mechanism takes assessment time span L=400, then the calculation formula of threshold value is
By 200 Monte-Carlo Simulations, the J that averages (th)=0.12588 is final threshold value,
0.11973=J (73) ＜ J (th) ＜ J (74)=0.12904
After illustrating that k=70 breaks down, fault-signal can be detected in 5 time steps, moreover it is possible to disturbance
Mutually distinguish.
Different faults probability in the case of ρ=0.6Residual error evaluation function and threshold value such as accompanying drawing 4, accompanying drawing 5, attached
Shown in Fig. 6 and accompanying drawing 7.
When, 0.02093=J (84) ＜ J (th)=0.02142 ＜ J (85)=0.02146
When, 0.07348=J (78) ＜ J (th)=0.07488 ＜ J (79)=0.07593
When, 0.10669=J (76) ＜ J (th)=0.10932 ＜ J (77)=0.11370
When, 0.12740=J (71) ＜ J (th)=0.13410 ＜ J (72)=0.14673
As can be seen that designed fault Detection Filter can effectively detect failure in out of order generation, system
The probability of generation is bigger, and the duration needed for detecting fault-signal is shorter, illustrates that the failure that research occurs at random is meaningful.
Above is presently preferred embodiments of the present invention, not makees any formal limitation, every foundation to the present invention
The technical spirit of the present invention belongs to inventive technique to any simple modification made for any of the above embodiments, equivalent variations and modification
In the range of scheme.
Claims (1)
1. a kind of random fault detection method of the network control system with packet loss, it is characterised in that the failure in system is
Occur at random, specifically include following steps：
1) the controlled device mathematical modeling for the network control system that there is random fault is set up：
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2) full rank fault Detection Filter is designed：
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<mo>}</mo>
<mo>:</mo>
<mo>=</mo>
<mover>
<mi>&beta;</mi>
<mo>&OverBar;</mo>
</mover>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>p</mi>
<mi>r</mi>
<mi>o</mi>
<mi>b</mi>
<mo>{</mo>
<msub>
<mi>&beta;</mi>
<mi>k</mi>
</msub>
<mo>=</mo>
<mn>0</mn>
<mo>}</mo>
<mo>=</mo>
<mn>1</mn>
<mo>-</mo>
<mi>E</mi>
<mo>{</mo>
<msub>
<mi>&beta;</mi>
<mi>k</mi>
</msub>
<mo>}</mo>
<mo>:</mo>
<mo>=</mo>
<mn>1</mn>
<mo>-</mo>
<mover>
<mi>&beta;</mi>
<mo>&OverBar;</mo>
</mover>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>v</mi>
<mi>a</mi>
<mi>r</mi>
<mo>{</mo>
<msub>
<mi>&beta;</mi>
<mi>k</mi>
</msub>
<mo>}</mo>
<mo>=</mo>
<mi>E</mi>
<mo>{</mo>
<msup>
<mrow>
<mo>(</mo>
<msub>
<mi>&beta;</mi>
<mi>k</mi>
</msub>
<mo>-</mo>
<mover>
<mi>&beta;</mi>
<mo>&OverBar;</mo>
</mover>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>}</mo>
<mo>=</mo>
<mover>
<mi>&beta;</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>-</mo>
<mover>
<mi>&beta;</mi>
<mo>&OverBar;</mo>
</mover>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msubsup>
<mi>f</mi>
<mn>1</mn>
<mn>2</mn>
</msubsup>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
Introduce residual error evaluation mechanism to detect whether failure occurs, residual error evaluation function J (k) and threshold value J (th) are respectively：
<mrow>
<mi>J</mi>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>E</mi>
<mo>{</mo>
<msup>
<mrow>
<mo>&lsqb;</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mi>k</mi>
</munderover>
<msup>
<mi>r</mi>
<mi>T</mi>
</msup>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>)</mo>
</mrow>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
</mrow>
<mi>T</mi>
</msup>
<mo>}</mo>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>5</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>J</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mi>h</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<munder>
<mrow>
<mi>s</mi>
<mi>u</mi>
<mi>p</mi>
</mrow>
<mrow>
<mi>&omega;</mi>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>&Element;</mo>
<msub>
<mi>l</mi>
<mn>2</mn>
</msub>
<mo>,</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</mrow>
</munder>
<mi>J</mi>
<mrow>
<mo>(</mo>
<mi>L</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein：L is the maximum time span of evaluation function, and whether system, which breaks down, to be judged by formula (7)；
3) adequate condition that system meansquare exponential stability and fault Detection Filter are present is：
<mrow>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msub>
<mi>&Theta;</mi>
<mi>l</mi>
</msub>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<msub>
<mi>&Theta;</mi>
<mn>16</mn>
</msub>
</mtd>
<mtd>
<msub>
<mi>&Theta;</mi>
<mn>17</mn>
</msub>
</mtd>
<mtd>
<msub>
<mi>&Theta;</mi>
<mn>18</mn>
</msub>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<msub>
<mi>&Theta;</mi>
<mi>l</mi>
</msub>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<msub>
<mi>&Theta;</mi>
<mn>26</mn>
</msub>
</mtd>
<mtd>
<msub>
<mi>&Theta;</mi>
<mn>27</mn>
</msub>
</mtd>
<mtd>
<msub>
<mi>&Theta;</mi>
<mn>28</mn>
</msub>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<msub>
<mi>&Theta;</mi>
<mi>l</mi>
</msub>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<msub>
<mi>&Theta;</mi>
<mn>37</mn>
</msub>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mrow>
<mo>-</mo>
<mi>I</mi>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<msub>
<mi>&Theta;</mi>
<mn>46</mn>
</msub>
</mtd>
<mtd>
<msub>
<mi>&Theta;</mi>
<mn>47</mn>
</msub>
</mtd>
<mtd>
<msub>
<mi>&Theta;</mi>
<mn>48</mn>
</msub>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mrow>
<mo>-</mo>
<mi>I</mi>
</mrow>
</mtd>
<mtd>
<msub>
<mi>&Theta;</mi>
<mn>56</mn>
</msub>
</mtd>
<mtd>
<msub>
<mi>&Theta;</mi>
<mn>57</mn>
</msub>
</mtd>
<mtd>
<msub>
<mi>&Theta;</mi>
<mn>58</mn>
</msub>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<msub>
<mi>&Theta;</mi>
<mn>66</mn>
</msub>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<msup>
<mi>M</mi>
<mi>T</mi>
</msup>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mrow>
<mo>-</mo>
<msup>
<mi>&gamma;</mi>
<mn>2</mn>
</msup>
<mi>I</mi>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<msup>
<mi>N</mi>
<mi>T</mi>
</msup>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mrow>
<mo>-</mo>
<msup>
<mi>&epsiv;</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mi>I</mi>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mo>*</mo>
</mtd>
<mtd>
<mrow>
<mo>-</mo>
<msup>
<mi>&epsiv;&delta;</mi>
<mrow>
<mo>-</mo>
<mn>2</mn>
</mrow>
</msup>
<mi>I</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo><</mo>
<mn>0</mn>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>8</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein：
Wherein：* the transposition of symmetric position matrix is represented,U,X,W,It is the matrix with appropriate dimension and is
Unknown,ε is known variables, and its dependent variable is all known；Utilize Matlab
LMI tool boxes are solved, and scalar γ ＞ 0 are given, if there is positive definite matrixU, X, W and mark
Measuring ε ＞ 0 causes formula (8) to set up, then system is meansquare exponential stability, and meets H∞Performance indications, can obtain non-optimal failure
Fault detection filter parameter, you can to proceed step 4)；If above-mentioned known variables are not solved, system is not square index
It is stable and non-optimal fault Detection Filter parameter can not be obtained, it is not possible to carry out step 4)；
4) Optimal Fault Detection Filter parameter is calculated
By solving optimization problem formula (9)：
If there is solution, Optimal Fault Detection Filter parameter is obtainedOptimal H∞Performance indications are γmin, obtain most
Excellent fault Detection Filter parameter is：
Wherein：G3, V is nonsingular matrix；The residual signals r (k) of system is obtained further according to formula (3) and formula (4), then calculating formula
(5) and formula (6), finally by formula (7), whether failure judgement occurs；
If formula (9) can not obtain Optimal Fault Detection Filter without solution.
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Effective date of registration: 20210802 Address after: 710000 No. x376, axis 11-21, shop on the third floor, No. 1, Changle slope, Changle East Road, Chanba Ecological District, Xi'an City, Shaanxi Province Patentee after: Xi'an Sixiang Internet Technology Co.,Ltd. Address before: 313000 1-C, building 1, No. 656 Qixing Road, high tech Zone, Wuxing District, Huzhou City, Zhejiang Province Patentee before: Huzhou xinbeilian Network Technology Co.,Ltd. |