CN107702908A - GIS mechanical oscillation signal Time-Frequency Analysis Methods based on VMD self adapting morphologies - Google Patents
GIS mechanical oscillation signal Time-Frequency Analysis Methods based on VMD self adapting morphologies Download PDFInfo
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- G—PHYSICS
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- G01M—TESTING STATIC OR DYNAMIC BALANCE OF MACHINES OR STRUCTURES; TESTING OF STRUCTURES OR APPARATUS, NOT OTHERWISE PROVIDED FOR
- G01M13/00—Testing of machine parts
Abstract
The invention discloses the GIS mechanical oscillation signal Time-Frequency Analysis Methods based on VMD self adapting morphologies, including：The different types of mechanical breakdown of Simulated GlS equipment；The vibration signal of GIS device of the repeated detection GIS device under nominal situation and simulation operating mode；Realize that vibration signal carries out time frequency analysis using variation mode decomposition, find out the vibration signal amplitude of GIS device with the change of frequency distribution by marginal spectrum；Comprehensive Hilbert analysis, obtains out of order characteristic criterion, by simulating different types of mechanical breakdown, establishes GIS mechanical fault diagnosis databases, realizes and carries out time frequency analysis to the vibration signal of above-mentioned GIS device.The present invention can effectively handle GIS vibration signals using through VMD algorithms to GIS mechanical oscillation signals progress time frequency analysis, so as to establish GIS mechanical fault diagnosis databases, to realize that live live detection GIS mechanical breakdowns provide theoretical foundation.
Description
Technical field
The present invention relates to signal processing technology field, more particularly to the GIS mechanical oscillation based on VMD self adapting morphologies
Signal Time-Frequency Analysis Method.
Background technology
Cubicle Gas-Insulated Switchgear (Gas Insulated Switchgear, GIS) can produce a variety of machineries
Defect, such as loosened screw, circuit breaker operation mechanism failure, transformer vibration etc..Therefore diagnosed by monitoring mechanical breakdown
Early defect can improve GIS operation stability inside GIS.Vibration signal passes through medium transmission caused by GIS internal faults
To GIS barrel body, for operating GIS device, sensor often is placed in drum surface, receives the vibration signal passed over,
So as to detect GIS whether operation exception, that is, break down.Due to vibration signal detection method strong antijamming capability, and and power network
In widely used ultra-high-frequency detection method can form complementation, it is convenient and reliable for low frequency signal in detecting.
But still focused mostly in shelf depreciation for GIS the Study on Fault at this stage[1]Direction, the vibration signal of collection
The mostly higher electromagnetic wave signal of frequency, it is less to mechanical fault signals research wider, that frequency is relatively low be present.Due to
The complexity of GIS structures, GIS operating to scene vibration characteristics are less studied, and document " is based on vibration signal HHT methods
GIS device fault diagnosis [J] China Powers, in 2013,39 (3) ", vibration letters of the Xu Tianle et al. to operating GIS
Number surveyed, but data acquisition is difficult and data volume is limited, fails to analyze the specific failure problems of GIS.Document is " extra-high
Press research [J] the power constructions of GIS/HGIS vibration equipments diagnostic methods, in 2009,30 (7) ", Cheng Lin et al. is to normal and different
Normal mixed type high-tension switch gear (Hybrid Gas Insulated Switchgear, HGIS) vibration signal carries out multiple
Detection process, and corresponding vibration amplitude and related figure of the number on frequency are drawn from statistics angle, but do not indicate failure
Problem.
The method of conventional processing vibration signal time-frequency is：Wavelet transformation, morphologic filtering, Hilbert-Huang conversion,
Empirical mode decomposition (Empirical Mode Decomposition, EMD), overall experience Mode Decomposition (Ensemble
Empirical Mode Decomposition, EEMD), local mean value decompose (Local Mean Decomposition, LMD)
Deng.The above method is obtained for good effect in GIS fault signal analysis, but is respectively provided with respective limitation.Small echo becomes
White noise can effectively be suppressed by changing, but suppressor pulse interference performance is not strong；Big iterative calculation amount, frequency aliasing, end be present in LMD
The problems such as point effect；Mathematical morphology has very strong suppressor pulse interference performance, algorithm simple possible, but intrinsic statistics be present
The select permeability of offset problem and optimum structure element.EMD is to analyze non-stationary signal and the strong instrument of nonlinear properties,
But modal overlap be present.
In summary, effective solve still is lacked for the time frequency analysis problem of GIS mechanical oscillation signals in the prior art
Scheme.
The content of the invention
In order to solve the deficiencies in the prior art, the invention provides the letter of the GIS mechanical oscillation based on VMD self adapting morphologies
Number Time-Frequency Analysis Method, by simulating different types of mechanical breakdown, can finally establish GIS mechanical fault diagnosis databases.
GIS mechanical oscillation signal Time-Frequency Analysis Methods based on VMD self adapting morphologies, including：
The different types of mechanical breakdown of Simulated GlS equipment；
The vibration signal of GIS device of the repeated detection GIS device under nominal situation and simulation operating mode；
Realize that vibration signal carries out time frequency analysis using variation mode decomposition：The vibration of the above-mentioned GIS device of detection is believed
Realize and constantly update number in a frequency domain, time domain is transformed into eventually through inverse Fourier transform, obtain nominal situation and simulation operating mode
The intrinsic mode functions group of the vibration signal of lower GIS device；
The vibration signal amplitude of GIS device is found out with the change of frequency distribution by marginal spectrum, with shaking under nominal situation
The HHT marginal spectrums of dynamic signal are compared, and oscillatory occurences and amplitude under simulated failure operating mode in the marginal spectrum of GIS vibration signals be present
Apparently higher than amplitude during nominal situation；
Comprehensive Hilbert analysis, obtains out of order characteristic criterion, by simulating different types of mechanical breakdown, establishes
GIS mechanical fault diagnosis databases, realize and time frequency analysis is carried out to the vibration signal of above-mentioned GIS device.
Further, it is main to construct loosened screw and based on winding in the different types of mechanical breakdown of Simulated GlS equipment
The transformer of deformation vibrates two kinds of common GIS mechanical breakdowns.
Above-mentioned fault type is relatively common and representative, and above-mentioned mechanical fault signals can show that the event of GIS device
Hinder defect.
Further, when constructing failure, set at GIS breaker operation mechanisms with disconnecting switch connecting screw and transformer
Put abnormal vibrations source, including loosened screw and the transformer vibration based on winding deformation.
Further, when detecting the vibration signal of GIS device of the GIS device under nominal situation and simulation operating mode outside
The piezoelectric acceleration transducer put detects on the downside of GIS peep-holes.
Further, also needed after the vibration signal of GIS device of the detection GIS device under nominal situation and simulation operating mode
Denoising is carried out, carries out variation mode decomposition again afterwards.
Further, the intrinsic mode functions group of vibration signal of GIS device is included respectively under nominal situation and simulation operating mode
Individual intrinsic mode function and residual components, intrinsic mode function accordingly contain the composition of different frequency sections from high to low, and
Change with the change of primary signal.
Further, the Hilbert spectrums are specially to do Hilbert transform to actual signal to ignore what discrepance obtained.
Further, realize that vibration signal carries out time frequency analysis using variation mode decomposition, key step is as follows：
One actual signal f is decomposed into K discrete mode uk, k ∈ 1,2 ..., K, ukBandwidth in a frequency domain all has
There is specific sparse attribute；
For each mode, by the Hilbert transformation calculations analytic signal related to each mode, it is unilateral to obtain its
Frequency spectrum；
The centre frequency each estimated is adjusted by adding exponential term in each mode, the frequency spectrum of each mode is adjusted
Make in corresponding base band；
Each mode is closely around in centerburst frequency wkNear, wkBandwidth by demodulated signal H1Gauss is put down
Slippery is estimated, obtains an affined variational problem；
On the basis of the above, make it be converted into no constraint variation using secondary penalty factor method and method of Lagrange multipliers to ask
Topic；
Variational problem is solved with alternating direction multiplier method, by alternately updating uk n+1、wk n+1、λn+1To seek augmentation
The saddle point of Lagrange expression formulas.
Further, above-mentioned actual signal is nominal situation and the vibration signal of the GIS device under simulation operating mode.
Further, realize that algorithm steps corresponding to vibration signal progress time frequency analysis are using variation mode decomposition：
(1) u is initializedk 1、wk 1、λ1It is 0 with n；
(2) n=n+1, the circulation of whole algorithm is started；
(3) u is constantly updatedkAnd wk, wherein k is recycled to K always since 1, and K is the mode decomposition number determined；
(4) λ is updated；
(5) judgement precision e is given>0, the iteration ends if predicated expressions are met, otherwise return to step (2).
Further, nominal situation system, for the spectrum distribution of GIS device vibration signal near 50Hz, frequency band is narrower；Screw
The frequency spectrum of looseness fault vibration signal is equally distributed near 50Hz, and frequency band is wider, and amplitude is apparently higher than normal vibration signal；
The spectrum distribution of transformer oscillation fault vibration signal is in 120~160Hz and 400~500Hz scopes.
Compared with prior art, the beneficial effects of the invention are as follows：
1st, VMD methods are introduced GIS mechanical oscillation signal process fields by the present invention, frequency division when being carried out to three kinds of vibration signals
Analysis, so that this kind can be applied to vibration signal processing deeper into ground more extensively for the processing method of nonlinear and non local boundary value problem
Field.
2nd, the present invention is effective realizes frequency division when the method based on VMD algorithms is carried out to the mechanical oscillation signal of GIS device
Analysis, by simulating different types of mechanical breakdown, can finally establish GIS mechanical fault diagnosis databases.
3rd, present invention use can effectively handle GIS to GIS mechanical oscillation signals progress time frequency analysis through VMD algorithms and shake
Dynamic signal, so as to establish GIS mechanical fault diagnosis databases, for realize live live detection GIS mechanical breakdowns provides theory according to
According to.
Brief description of the drawings
The Figure of description for forming the part of the application is used for providing further understanding of the present application, and the application's shows
Meaning property embodiment and its illustrate be used for explain the application, do not form the improper restriction to the application.
The vibration-testing platform that Fig. 1 embodiment of the present invention is built；
The normal GIS vibration signal time-domain diagrams of Fig. 2 (a)；
1 time GIS vibration signal time-domain diagram of Fig. 2 (b) failures；
2 times GIS vibration signal time-domain diagrams of Fig. 2 (c) failures；
Fig. 3 (a) nominal situations VMD is decomposed；
Fig. 3 (b) failures 1VMD is decomposed；
Fig. 3 (c) failures 2VMD is decomposed；
Fig. 4 (a) normal vibrations signal margin is composed；
The vibration signal marginal spectrum of Fig. 4 (b) failures 1；
The vibration signal marginal spectrum of Fig. 5 failures 2.
Embodiment
It is noted that described further below is all exemplary, it is intended to provides further instruction to the application.It is unless another
Indicate, all technologies used herein and scientific terminology are with usual with the application person of an ordinary skill in the technical field
The identical meanings of understanding.
It should be noted that term used herein above is merely to describe embodiment, and be not intended to restricted root
According to the illustrative embodiments of the application.As used herein, unless the context clearly indicates otherwise, otherwise singulative
It is also intended to include plural form, additionally, it should be understood that, when in this manual using term "comprising" and/or " bag
Include " when, it indicates existing characteristics, step, operation, device, component and/or combinations thereof.
As background technology is introduced, the time frequency analysis for GIS mechanical oscillation signals is present accurate in the prior art
Property it is not high the problem of, in order to solve technical problem as above, present applicant proposes based on VMD self adapting morphologies GIS machinery
Vibration signal Time-Frequency Analysis Method.
In order to study can live livewire work GIS vibration signal detection methods, grasp under different mechanical breakdown types
Vibration signal characteristic, establishes GIS mechanical fault diagnosis databases, and the application establishes one using the single-phase branch mailbox GIS devices of 110kV
Cover GIS mechanical fault detection platforms.Two kinds of common GIS mechanical breakdowns, and profit are vibrated by building loosened screw and transformer
GIS vibration signals under GIS normal operations vibration signal and two kinds of failures are measured with sensor, and are subject to variation mode point to signal
Solve (Variational Mode Decomposition, abbreviation VMD) processing.The analysis result explanation of measured signal, using warp
VMD algorithms carry out time frequency analysis to GIS mechanical oscillation signals can effectively handle GIS vibration signals, so as to establish GIS machineries
Fault Diagnosis Database, to realize that live live detection GIS mechanical breakdowns provide theoretical foundation.
In a kind of typical embodiment of the application, as shown in figure 1, the application has built vibration-testing platform first,
Using the single-phase branch mailbox GIS devices of 110kV that certain switchgear plant is a whole set of, in GIS breaker operation mechanisms and disconnecting switch connecting screw
With setting abnormal vibrations source at transformer, including loosened screw and the transformer vibration based on winding deformation.Test real object and event
Hinder set location (failure 1 is loosened screw, and failure 2 is transformer winding deformation).Sensed using external piezoelectric type acceleration
Device detects three class vibration signals (normal, loosened screw, transformer vibration) on the downside of GIS peep-holes, and the information measured is inputted
Wave filter is filtered processing, transmits to amplifier being amplified processing again afterwards, finally transmits to computer and carries out algorithm
Processing and the analysis of waveform.
Analysis on vibration signal includes time-domain diagram analysis, variation mode decomposition (VMD), marginal spectrum are analyzed.
Wherein, time domain map analysis：Fig. 2 (a), Fig. 2 (b), Fig. 2 (c) are respectively oscillograph GIS is normal, failure 1, failure 2
When the vibration signal time-domain diagram that gathers, from the point of view of time-domain diagram, loosened screw and transformer vibrate the vibration signal under two kinds of operating modes
Amplitude is all higher than normal vibration signal, and under loosened screw failure, vibration signal amplitude amplification is obvious.Intuitively see, mutual inductance
Vibration signal dense degree highest under device vibration, thus it is speculated that have frequency-doubled signal superposition.
Variation mode decomposition (VMD)：Three groups of vibration signals decompose to obtain such as Fig. 3 after simple denoising, by VMD
(a), under three kinds of Fig. 3 (b), Fig. 3 (c) operating modes GIS vibration signals intrinsic mode functions group, each figure is followed successively by each solid from top to bottom
There are mode function (vimf) and residual components (res.).Vimf accordingly contains the composition of different frequency sections from high to low, and
Change with the change of primary signal.
Marginal spectrum is analyzed：Change of the amplitude with frequency distribution can more intuitively be found out by marginal spectrum.With normal operation
Under HHT marginal spectrums such as Fig. 4 (a) of vibration signal compare, the limit of 2 times GIS vibration signals of Fig. 4 (b) failures 1 and Fig. 5 failures
Can be very it can be clearly seen that oscillatory occurences, and amplitude at fundamental frequency 100Hz be present in fault signature, loosened screw failure in spectrum
Apparently higher than amplitude during normal operation, the energy of transformer oscillator signal focuses primarily upon 120~160Hz and 400~500Hz
Scope, its amplitude are far above the amplitude of fundamental frequency 50Hz or so under normal operation.Comprehensive Hilbert transform analysis, it can be deduced that two
The characteristic criterion of kind failure.
The application application VMD methods analyze and process GIS device vibration signal.Using VMD methods, in laboratory, pass through
Two kinds of common mechanical breakdowns (i.e. loosened screw and transformer vibration) are set to GIS device, with this method to GIS device three
Vibration signal is analyzed and researched under kind operating mode (normal, failure 1, failure 2).
Time-domain analysis based on VMD, natural mode group of functions is obtained, each the composition amplitude and phase of vibration signal can be obtained
Situation so that result is more accurate.
Analyzed based on VMD, can obtain the analysis of three kinds of operating mode vibration signal amplitudes and frequency, obtain vibrating under different faults
The characteristic criterion of signal.
For the spectrum distribution of normal vibration signal near 50Hz, frequency band is narrower；The frequency spectrum of loosened screw fault vibration signal
Equally it is distributed near 50Hz, frequency band is wider, and amplitude is apparently higher than normal vibration signal；Transformer oscillation fault vibration signal
Spectrum distribution in 120~160Hz and 400~500Hz scopes.
Described above, it is effective that the method based on VMD algorithms carries out time frequency analysis to the mechanical oscillation signal of GIS device
's.By simulating different types of mechanical breakdown, GIS mechanical fault diagnosis databases can be finally established.
In addition, in order to better illustrate the algorithm of the application, VMD method general principles are described more fully below：
Variation mode decomposition (Variational Mode Decomposition, abbreviation VMD) be with classical Wiener filtering,
Solving Variational Problem method based on Hilbert transform and frequency compounding these three concepts.VMD can be by an actual letter
Number f is decomposed into K discrete mode uk(k∈1,2,…,K),ukBandwidth in a frequency domain all has specific sparse attribute.Tool
Body solution procedure is as follows：
(1) for each mode, by the Hilbert transformation calculations analytic signal related to each mode, can obtain
To its unilateral frequency spectrum：
(2) centre frequency each estimated is adjusted by adding exponential term in each mode, the frequency of each mode
Spectrum is modulated in corresponding base band：
(3) each mode can be closely around in centerburst frequency wkNear, wkBandwidth by more than reconcile signal
I.e. to the H of above-mentioned formula (1) (2) modulated signal1Gaussian smoothing degree is estimated, so can be obtained by an affined variation
Problem：
In formula：F is primary signal, uk(t) it is mode function, wkFor the centre frequency of each mode.
(4) on this basis, it is made to be converted into no constraint variation using secondary penalty factor method and method of Lagrange multipliers
Problem.Secondary penalty factor can ensure the reconstruction accuracy of signal in the presence of strong noise, and Lagrange multiplier causes about
Beam condition holding stringency, argument Lagrange expression formula are as follows：
(5) so variational problem can alternating direction multiplier method (Alternate Direction Method of
Mulatipliers, ADMM) to solve, pass through and replace renewal uk n+1、wk n+1、λn+1To seek the saddle of argument Lagrange expression formula
Point.Wherein uk n+1It can be expressed as：
Frequency domain is transformed into by Parseval/Plancherel Fourier's equilong transformations：
Pass through w=w-wkSubstitution of variable is carried out, non-negative frequency separation integrated form is then converted into, finally solves uk n +1More new-standard cement：
Above-mentioned algorithm corresponding algorithm flow in specific perform：
(1) u is initializedk 1、wk 1、λ1It is 0 with n；
(2) n=n+1, the circulation of whole algorithm is started；
(3) u is constantly updatedkAnd wk, wherein k is recycled to K always since 1, and K is the mode decomposition number determined；
(4) λ is updated
(5) judgement precision e is given>0, if meeting predicated expressions：
Then iteration ends, otherwise return to step (2).
As can be seen that VMD is to realize to constantly update in a frequency domain, time domain is transformed into eventually through inverse Fourier transform.
Composed on Hilbert：Hilbert transform is done to actual signal to obtain
In formula, s refers to actual signal.
Discrepance is now have ignored, above formula is referred to as Hilbert spectrums, is denoted as
Further define marginal spectrum
Marginal spectrum can characterize the accumulation amplitude distribution of each Frequency point of whole group data from the statistical significance.
Sum it up, vibration of the application to the single-phase branch mailbox GIS device surfaces of 110kV under a whole set of operation of certain switchgear plant
Signal measures, and constructs loosened screw and transformer two kinds of common GIS mechanical breakdowns of vibration based on winding deformation, more
Operation GIS vibration signals under secondary three kinds of operating modes of detection (normal signal, loosened screw, transformer vibration), VMD methods are introduced
GIS mechanical oscillation signal process fields, time frequency analysis is carried out to three kinds of vibration signals, so that this kind is believed for nonlinear and nonstationary
Number processing method can be applied to vibration signal processing field deeper into ground more extensively.
The preferred embodiment of the application is the foregoing is only, is not limited to the application, for the skill of this area
For art personnel, the application can have various modifications and variations.It is all within spirit herein and principle, made any repair
Change, equivalent substitution, improvement etc., should be included within the protection domain of the application.
Claims (10)
1. the GIS mechanical oscillation signal Time-Frequency Analysis Methods based on VMD self adapting morphologies, it is characterized in that, including：
The different types of mechanical breakdown of Simulated GlS equipment；
The vibration signal of GIS device of the repeated detection GIS device under nominal situation and simulation operating mode；
Realize that vibration signal carries out time frequency analysis using variation mode decomposition：The vibration signal of the above-mentioned GIS device of detection is existed
Realize and constantly update in frequency domain, time domain is transformed into eventually through inverse Fourier transform, obtain GIS under nominal situation and simulation operating mode
The intrinsic mode functions group of the vibration signal of equipment；
The vibration signal amplitude for finding out GIS device by marginal spectrum is believed with the change of frequency distribution with the vibration under nominal situation
Number HHT marginal spectrums compare, oscillatory occurences be present in the marginal spectrum of GIS vibration signals under simulated failure operating mode and amplitude be obvious
Amplitude during higher than nominal situation；
Comprehensive Hilbert analysis, obtains out of order characteristic criterion, by simulating different types of mechanical breakdown, establishes GIS machines
Tool Fault Diagnosis Database, realize and time frequency analysis is carried out to the vibration signal of above-mentioned GIS device.
2. the GIS mechanical oscillation signal Time-Frequency Analysis Methods based on VMD self adapting morphologies as claimed in claim 1, it is special
Sign is, main to construct loosened screw and the transformer based on winding deformation in the different types of mechanical breakdown of Simulated GlS equipment
Vibrate two kinds of common GIS mechanical breakdowns.
3. the GIS mechanical oscillation signal Time-Frequency Analysis Methods based on VMD self adapting morphologies as claimed in claim 1 or 2, its
It is characterized in, when constructing failure, abnormal vibrations is set at GIS breaker operation mechanisms and disconnecting switch connecting screw and transformer
Source, including loosened screw and the transformer vibration based on winding deformation.
4. the GIS mechanical oscillation signal Time-Frequency Analysis Methods based on VMD self adapting morphologies as claimed in claim 1, it is special
Added when sign is the vibration signal for detecting GIS device of the GIS device under nominal situation and simulation operating mode using external piezoelectric type
Velocity sensor detects on the downside of GIS peep-holes.
5. the GIS mechanical oscillation signal Time-Frequency Analysis Methods based on VMD self adapting morphologies as claimed in claim 1, it is special
Sign is to also need to carry out at denoising after the vibration signal of GIS device of the detection GIS device under nominal situation and simulation operating mode
Reason, carries out variation mode decomposition again afterwards.
6. the GIS mechanical oscillation signal Time-Frequency Analysis Methods based on VMD self adapting morphologies as claimed in claim 1, it is special
Sign is that the intrinsic mode functions group of the vibration signal of GIS device includes each natural mode of vibration letter under nominal situation and simulation operating mode
Number and residual components, intrinsic mode function accordingly contain the composition of different frequency sections from high to low, and with primary signal
Change and change.
7. the GIS mechanical oscillation signal Time-Frequency Analysis Methods based on VMD self adapting morphologies as claimed in claim 1, it is special
Sign is that the Hilbert spectrums are specially to be HT to actual signal to ignore what discrepance obtained.
8. the GIS mechanical oscillation signal Time-Frequency Analysis Methods based on VMD self adapting morphologies as claimed in claim 1, it is special
Sign is to realize that vibration signal carries out time frequency analysis using variation mode decomposition, key step is as follows：
One actual signal f is decomposed into K discrete mode uk, k ∈ 1,2 ..., K, ukBandwidth in a frequency domain all has spy
Fixed sparse attribute；
For each mode, by the Hilbert transformation calculations analytic signal related to each mode, its unilateral frequency is obtained
Spectrum；
The centre frequency each estimated is adjusted by adding exponential term in each mode, the spectrum modulation of each mode is arrived
In corresponding base band；
Each mode is closely around in centerburst frequency wkNear, wkBandwidth by demodulated signal H1Gaussian smoothing degree
To estimate, an affined variational problem is obtained；
It is set to be converted into no constraint variation problem using secondary penalty factor method and method of Lagrange multipliers；
Variational problem is solved with alternating direction multiplier method, by alternately updating uk n+1、wk n+1、λn+1To seek argument Lagrange
The saddle point of expression formula.
9. the GIS mechanical oscillation signal Time-Frequency Analysis Methods based on VMD self adapting morphologies as claimed in claim 8, it is special
Sign is that above-mentioned actual signal is nominal situation and the vibration signal of the GIS device under simulation operating mode.
10. the GIS mechanical oscillation signal Time-Frequency Analysis Methods based on VMD self adapting morphologies as claimed in claim 8, it is special
Sign is to realize that algorithm steps corresponding to vibration signal progress time frequency analysis are using variation mode decomposition：
(1) u is initializedk 1、wk 1、λ1It is 0 with n；
(2) n=n+1, the circulation of whole algorithm is started；
(3) u is constantly updatedkAnd wk, wherein k is recycled to K always since 1, and K is the mode decomposition number determined；
(4) λ is updated；
(5) judgement precision e is given>0, the iteration ends if predicated expressions are met, otherwise return to step (2).
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