CN108627841A - A kind of circle center locating method in shoal buried pipes supersonic sounding - Google Patents
A kind of circle center locating method in shoal buried pipes supersonic sounding Download PDFInfo
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- G—PHYSICS
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- G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
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Abstract
The invention discloses the circle center locating methods in a kind of shoal buried pipes supersonic sounding, are based on arrival time and least square method, include the following steps：Build the simplification highlight model of buried pipes；Using MATLAB design and simulation parameters, artificial echo signal is obtained；Build the brief single channel ultrasound emission and reception experiment porch under laboratory environment；Shoal complex environment is simulated, the Gaussian noise that signal-to-noise ratio is a certain threshold value is added in Initial experiments echo-signal, obtains echo-signal；Maximum amplitude method, characteristic parameter correlation detection, fast energy centre convergence method is respectively adopted, echo TOA estimations are carried out to echo-signal, therefrom selects best estimate method；According to the sound path simplified between highlight model, echo TOA estimations calculating energy converter and theoretical mirror image bright spot；It is fitted pipeline center of circle object function according to sound path and calculates numerical approximation solution, using numerical approximation solution as initial coordinate, the accurate solution in the pipeline center of circle is solved by interative computation.
Description
Technical Field
The invention relates to the field of ultrasonic detection of shallow buried pipelines, in particular to a pipeline circle center positioning method based on arrival time and a least square method.
Background
The long oil transportation pipelines buried in shoal zones (rivers, swamps, ponds, rice fields and the like) are generally laid below 1-2 meters in soil layers. After a leakage accident occurs, the conventional cofferdam operation for pipeline rush repair usually has the following three defects:
(1) the cofferdam building range is far larger than an actually required operation area;
(2) if the position of the cofferdam construction deviates from the leakage position, the cofferdam must be constructed again;
(3) the conventional cofferdam has safety risks of collapse and water seepage.
Therefore, if an automatic and intelligent mechanical device is provided in the process of the first-aid cofferdam operation of oil pipeline leakage, the position of the buried pipeline can be accurately positioned in the construction preparation stage, the edge position of the transverse circle of the pipeline can be monitored in real time in the construction process, and the pipeline is protected from being constructedMechanical or physical damage possibly caused in the industry can greatly shorten the period of emergency repair operation and prevent the error work or huge fund waste. At present, no special research equipment or system for monitoring the edge position of a buried pipeline target in real time in the construction and excavation operation process is available at home and abroad[1]。
The shoal environment is a heterogeneous medium similar to the mixing of sediment and water on the seabed. Numerous studies and mature applications have shown that ultrasound is an effective means of proximity detection and measurement in this type of environment[2]-[4]. However, in the application of the prior active sonar technology (such as a shallow layer profiler) in pipeline measurement, an acoustic wave signal is generally required to be transmitted at a long distance, so that the acoustic wave penetrates through two media, namely seawater and silt to obtain an echo.
In the process of implementing the invention, the inventor finds that at least the following disadvantages and shortcomings exist in the prior art:
the method can only qualitatively obtain the sonar image and the depth direction of the pipeline, cannot accurately position the radial cross section of the pipeline, and cannot accurately measure the edge position information of the pipeline.
Therefore, in the practical application of the ultrasonic detection engineering of the shoal pipeline, the accurate positioning and the real-time detection of the pipeline position have important significance.
Disclosure of Invention
The invention provides a circle center positioning method in ultrasonic detection of a shallow buried pipeline, which is based on a simplified bright spot model and provides a pipeline circle center positioning method based on arrival time and a least square method, so that the effect of efficiently and accurately positioning the circle center of a pipeline is realized, and the detailed description is as follows:
a circle center positioning method in ultrasonic detection of a shallow buried pipeline is based on time of arrival and a least square method and comprises the following steps:
constructing a simplified bright spot model of the buried pipeline; using MATLAB to design simulation parameters to obtain simulation echo signals;
a simple single-channel ultrasonic transmitting and receiving experiment platform under a laboratory environment is built;
simulating a complex shoal environment, and adding Gaussian noise with a signal-to-noise ratio of a certain threshold value into an original experiment echo signal to obtain an echo signal;
respectively adopting a maximum amplitude method, a characteristic parameter correlation detection method and a rapid energy center convergence method to carry out echo TOA estimation on echo signals, and selecting an optimal estimation method;
estimating and calculating a sound path between the transducer and a theoretical mirror image bright spot according to the simplified bright spot model and the echo TOA;
fitting a pipeline circle center objective function according to the sound path, calculating an approximate numerical solution, taking the approximate numerical solution as an initial coordinate, and solving an accurate solution of the pipeline circle center through iterative operation.
The simplified bright spot model specifically comprises:
taking the position of the transducer as an origin O, taking the radial cross section of the pipeline as a two-dimensional plane, and constructing a coordinate system of the position relation between each transducer and the pipeline when n transducers with the interval of d form a transducer array;
the working form of the transducer array is that a single transducer sequentially transmits and receives echoes, and n bright spot positions are correspondingly generated at the edge of a pipeline to obtain corresponding received echo signals.
Further, the bright spots are edge points of the cross-section circle of the pipeline, and the bright spots are located on a connecting line between the center of the transducer and the center of the cross-section circle of the pipeline.
Wherein, the position of pipeline cross section circle specifically is:
wherein (x)i,yi) Coordinates of the bright spots; (x)0,y0) As the coordinates of the circle center; (x'i，y′i) Are the coordinates of the transducer; liThe acoustic path between the transducer and the bright spot.
Further, the simple single-channel ultrasonic emission and reception experiment platform under the laboratory environment is specifically set up as follows:
selecting an initial coordinate right above the edge of the pipeline, wherein the vertical distance between the center of the pipeline and the transducer is a certain preset distance;
starting from the initial coordinate, setting a plurality of transducers from left to right at intervals of a certain preset value to receive the echo to be detected;
triggering once sound wave transmission and reception at each point to be measured, and performing once data acquisition; and obtaining echo signals obtained by a certain group of water tank experiments.
During specific implementation, the fitting of the pipeline circle center objective function according to the acoustic path and the calculation of the approximate numerical value solution are specifically as follows:
fitting the circle center and the radius according to a least square method; obtaining a corrected target function by using the coordinate of the transducer and the sound path distance relation between the transducer coordinate and the arc discrete point;
writing the modified objective function into an approximate form; and solving by utilizing a multivariate function extremum method to obtain an equation set, and further calculating to obtain a numerical solution.
Further, the step of solving an accurate solution of the center of the pipeline circle by iterative operation with the approximate numerical solution as the initial coordinate specifically comprises:
establishing a matrix P according to the initial coordinates and initial step length of the circle centeriWill matrix PiThe modified objective function is brought in to obtain another matrix FiFind another matrix FiThe smallest element F (x) inj,yj)；
If i is not equal to j, updating the coordinates of the circle center as xi＝xj，yi＝yj；
If i is j, updating the step length S to be S/2, and repeating the iteration step;
if the step length S is less than SminIteration stops, SminIs the set minimum iteration step size.
The technical scheme provided by the invention has the beneficial effects that:
1. the method is based on a simplified bright spot model, firstly, a correction target function is established based on the least square thought, then, a numerical solution of a circle center positioning point is deduced, then, the numerical solution is used as an initial coordinate, and a final circle center coordinate accurate solution is obtained through iterative calculation;
2. the method has important engineering significance for solving the problems of accurate positioning of the pipeline and real-time monitoring of the edge position in the leakage rush-repair operation of the oil pipeline buried in the shoal, and is a key step for realizing the automation and the intellectualization of detection;
3. the method verifies the calculation performance and accuracy of the positioning method through design simulation and experiments, and the result shows that the positioning method meets the precision requirement of actual engineering in water, thereby proving the feasibility of the positioning method in the application of the real environment.
Drawings
FIG. 1 is a flow chart of a circle center positioning method in ultrasonic detection of a shallow buried pipeline;
FIG. 2 is a schematic diagram of a single transducer tube bright spot model;
FIG. 3 is a bright spot model of a transducer array tube;
FIG. 4 is a diagram of a water tank experimental system;
FIG. 5 is a diagram of echo signals obtained from a water tank experiment;
FIG. 6 is a diagram of a simulated beach echo signal containing Gaussian noise;
FIG. 7 is a diagram illustrating the comparison of the results of three common TOA estimation methods.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, embodiments of the present invention are described in further detail below.
Example 1
A circle center positioning method in ultrasonic detection of a shallow buried pipeline, the circle center positioning method being based on time of arrival and a least square method, see fig. 1, the circle center positioning method comprising the steps of:
101: constructing a simplified bright spot model of the buried pipeline; using MATLAB to design simulation parameters to obtain simulation echo signals;
102: a simple single-channel ultrasonic transmitting and receiving experiment platform under a laboratory environment is built;
103: simulating a complex shoal environment, and adding Gaussian noise with a signal-to-noise ratio of a certain threshold value into an original experiment echo signal to obtain an echo signal;
104: respectively adopting a maximum amplitude method, a characteristic parameter correlation detection method and a rapid energy center convergence method to carry out echo TOA estimation on echo signals, and selecting an optimal estimation method;
105: estimating and calculating a sound path between the transducer and a theoretical mirror image bright spot according to the simplified bright spot model and the echo TOA;
106: fitting a pipeline circle center objective function according to the sound path, calculating an approximate numerical solution, taking the approximate numerical solution as an initial coordinate, and solving an accurate solution of the pipeline circle center through iterative operation.
The simplified bright spot model in step 101 is specifically:
taking the position of the transducer as an origin O, taking the radial cross section of the pipeline as a two-dimensional plane, and constructing a coordinate system of the position relation between each transducer and the pipeline when n transducers with the interval of d form a transducer array;
the working form of the transducer array is that a single transducer sequentially transmits and receives echoes, and n bright spot positions are correspondingly generated at the edge of a pipeline to obtain corresponding received echo signals.
Further, the bright spots are edge points of the cross-section circle of the pipeline, and the bright spots are located on a connecting line between the center of the transducer and the center of the cross-section circle of the pipeline.
Further, the step 102 of establishing a brief single-channel ultrasound transmitting and receiving experiment platform in a laboratory environment specifically includes:
selecting an initial coordinate right above the edge of the pipeline, wherein the vertical distance between the center of the pipeline and the transducer is a certain preset distance;
starting from the initial coordinate, setting a plurality of transducers from left to right at intervals of a certain preset value to receive the echo to be detected;
triggering once sound wave transmission and reception at each point to be measured, and performing once data acquisition; and obtaining echo signals obtained by a certain group of water tank experiments.
In the specific implementation, the fitting of the objective function of the center of the pipe according to the acoustic path and the calculation of the approximate numerical solution in step 106 are specifically:
fitting the circle center and the radius according to a least square method; obtaining a corrected target function by using the coordinate of the transducer and the sound path distance relation between the transducer coordinate and the arc discrete point;
writing the modified objective function into an approximate form; and solving by utilizing a multivariate function extremum method to obtain an equation set, and further calculating to obtain a numerical solution.
Further, taking the approximate numerical solution as the initial coordinate in step 106, and solving the precise solution of the center of the pipeline circle through iterative operation specifically includes:
establishing a matrix P according to the initial coordinates and initial step length of the circle centeriWill matrix PiThe modified objective function is brought in to obtain another matrix FiFind another matrix FiThe smallest element F (x) inj,yj)；
If i is not equal to j, updating the coordinates of the circle center as xi＝xj，yi＝yj；
If i is j, updating the step length S to be S/2, and repeating the iteration step;
if the step length S is less than SminIteration stops, SminIs the set minimum iteration step size.
In conclusion, the embodiment of the invention has important engineering significance for solving the problems of accurate positioning of the pipeline and real-time monitoring of the edge position in the leakage rush-repair operation of the oil pipeline buried in the shoal, and is a key step for realizing automation and intellectualization of detection.
Example 2
The scheme of example 1 is further described below with reference to specific calculation formulas and fig. 1 to 7, and is described in detail below:
201: constructing a simplified bright spot model of the buried pipeline;
the detailed operation of step 201 is:
1) as shown in fig. 2, a single underwater acoustic transducer and pipe position coordinate system is constructed with the transducer position as an origin O and the radial cross section of the pipe as a two-dimensional plane (XOY plane). Assuming that the cross section of the pipe, i.e. the circle O', is entirely within the-3 dB beam width angle of the transducer, the arc edge AB in fig. 2 is the illuminated area of the acoustic beam, and the other arc areas are the shadowgraph areas.
From geometric acoustics theory, point L is the intersection of OO' and AB, which is also the theoretical specular reflection point whose reflection contributes most to the echo. Other illuminated areas on arc AB than point L will produce scattered echoes of equivalent bright spots. As is apparent from the positional relationship in fig. 1, the specular reflection echo at point L will reach point O before the other scattered echoes.
2) When n transducers at intervals d form a transducer array (T1, T2 … Tn), a coordinate system of individual transducer to pipe position is constructed, as shown in FIG. 3. Assuming that the working form of the transducer array is that a single transducer sequentially emits and receives echoes, n bright spot positions L1, L2, … … and Ln are correspondingly generated at the edge of the pipeline to obtain corresponding received echo signals yi(t)(i＝1,2,…,n)。
3) From the model fig. 3, it can be concluded that after the model is simplified, the bright spots are the edge points of the cross-sectional circle of the pipeline, and the bright spots are located on the connecting line between the center of the transducer and the center of the cross-sectional circle of the pipeline. If the coordinates of the transducer are (x'i，y′i) The coordinates of the bright spot are (x)i,yi)，liFor the acoustic path of the transducer and the bright spot, they have the following relationship:
wherein (x)0,y0) As the coordinates of the center of the circle.
202: using MATLAB to design simulation parameters to obtain simulation echo signals;
the method comprises the following specific steps: with transducer T1 as the origin of the coordinate system, the transducer spacing d is 0.01m, the transducer center frequency is 100kHz, -3dB beamwidth 17 °, 164dB per volt source level. The simulated oil pipeline is made of steel material, the O' coordinate of the center of the pipeline is (0.08, 0.53) (unit: m), and the outer radius of the pipeline is 0.08 m. Exciting single-frequency rectangular pulse signal frequency f0100KHz, 40us pulse width,the amplitude is 20V, the sound velocity in water is taken as c to be 1500m/s, and the sampling rate of the system is fs2MHz, single sample depth N2000.
203: a simple single-channel ultrasonic transmitting and receiving experiment platform under a laboratory environment is built;
the detailed operation of the step is as follows:
1) as shown in fig. 4, a signal generator is used as an excitation source and directly connected with a single transducer, and data is received by a data acquisition card, so as to form a simple single-channel acoustic wave receiving and transmitting experimental platform in a laboratory environment;
2) selecting the position right above the edge of the pipeline as an initial coordinate, wherein the vertical distance between the center of the pipeline and the transducer is 0.53m, and the parameters of the pipeline material, the pipeline radius, the excitation frequency and the like are consistent with the parameters under the simulation condition in the step 202;
3) starting from the initial coordinate, setting 8 transducers from left to right at intervals of 0.01m to receive the echo to be detected;
4) triggering once sound wave transmission and reception at each point to be measured, and performing once data acquisition;
wherein, the parameter setting of the data acquisition card is consistent with that in step 202.
5) The echo signals obtained by the experiment of obtaining a certain group of water tanks are shown in figure 5.
The embodiment of the invention does not limit the types of the signal generator, the transducer, the data acquisition card and the like, and only needs the device capable of completing the functions.
204: simulating a complex environment of a shoal, and adding Gaussian noise with a signal-to-noise ratio of 10dB into an original experimental echo signal to obtain an echo signal shown in FIG. 6;
205: respectively adopting a maximum amplitude method, a feature parameter correlation detection method and a rapid energy center convergence method to carry out echo TOA estimation on the echo signal obtained in the step 204, and selecting an optimal estimation method from the echo signal;
the above methods are all known to those skilled in the art, and are not described in detail in the embodiments of the present invention.
As can be seen from the comparison results shown in fig. 7, when gaussian noise is added, a better estimation effect can be obtained by using the fast energy-centric convergence method.
206: calculating the acoustic path l between the transducer and the theoretical mirror bright spot according to the echo TOA estimationi：
Wherein, tiFor echo arrival time, c is the speed of sound.
207: fitting a target function of the center of the pipeline and calculating an approximate numerical solution;
the detailed operation of the step is as follows:
1) fitting the circle center and the radius according to a least square method;
assuming that the radius of the oil pipeline is R, which is generally known in advance in practical engineering applications, the objective function of the least squares fitting circle can be expressed as:
2) obtaining a corrected target function by using the coordinate of the transducer and the sound path distance relation between the transducer coordinate and the arc discrete point;
due to the discrete points (x) of the circular arci,yi) Since it is an unknown parameter, the objective function cannot be directly calculated, and the objective function can be corrected by substituting equation (1) in step 201 for equation (3):
3) writing the modified objective function of equation (4) as an approximate form:
4) solving by using a multivariate function extremum method to obtain an equation set:
due to y'iWhen the value is 0, a numerical solution can be calculated after substitution:
wherein,each representsiThe same applies to other similar forms.
208: and taking the approximate numerical solution as an initial coordinate, and solving the precise solution of the center of the circle of the pipeline through iterative operation.
The detailed operation of the step is as follows:
1) assuming the initial coordinate of the center of a circle is (x)i,yi) Building 3 x 3 matrix Pi：
The elements being of equal step size, initial stepLength S, i.e. xi＝xi-1+S，yi＝yi-1+ S. Will matrix PiSubstituting the formula (4) in step 207 to obtain FiComprises the following steps:
2) matrix F is solvediThe smallest element F (x) inj,yj) If i is not equal to j, the circle center coordinate is updated to be xi＝xj，yi＝yj(ii) a If i equals j, update step S equals S/2, repeat iteration step 1).
3) If the step length S is less than SminIteration stops, SminIs the set minimum iteration step size.
In conclusion, the embodiment of the invention realizes the effect of efficiently and accurately positioning the circle center of the pipeline; the method has important engineering significance for solving the problems of accurate positioning of the pipeline and real-time monitoring of the edge position in the leakage rush-repair operation of the oil pipeline buried in the shoal, and is a key step for realizing automation and intellectualization of detection.
Example 3
The feasibility of the protocols of examples 1 and 2 is verified below in connection with the specific examples, tables 1-4, and described in detail below:
according to the circle center fitting method provided in the above embodiments 1 and 2, the accuracy of the proposed buried pipeline ultrasonic detection circle center positioning method is verified by performing calculation using experimental signals, and the detailed operation is as follows:
1) using the experiment platform and relevant experiment parameters in step 201 to perform 7 repeated experiments;
2) adopting a rapid energy center convergence method to carry out TOA estimation comparison on experimental echo signals of 7 groups of water tanks and simulated echo signals, wherein the result is shown in an attached table 1;
3) fitting a numerical solution of the circle center using the method of step 202;
by | δx|，|δyI respectively represents the error between the fitting value and the true value in the x direction and the y direction, and is used Representing the degree of deviation of the fitted circle center from the true circle center, and the obtained results are shown in table 2.
4) Solving the circle center by using the iterative algorithm in the step 203;
the initial step S is 0.1m, and the minimum iteration step Smin0.000001m, and the iteration initial coordinate is the numerical solution obtained in table 2. The results of the iterative calculations are shown in table 3.
5) And comparing the performance of the iterative algorithm under different initial coordinates.
The initial coordinates are respectively solved by the origin of coordinates and the circle center value obtained in table 2, and the iteration number ratio is shown in table 4.
TABLE 1 echo signal TOA estimation
TABLE 2 numerical solution of circle center coordinates
TABLE 3 results of iterative calculations
TABLE 4 comparison of iteration number
6) The experimental results are combined, so that the circle center coordinate error obtained through iterative calculation is small.
The requirement of positioning accuracy in practical engineering application is generally below 0.05m, and the measurement errors of 2-7 groups are below 0.01m except that the measurement error of a water tank experiment 1 group reaches 0.02m, so that the requirement of engineering accuracy is met. In addition, after the iterative computation is optimized, the iterative times are obviously reduced, and the computation speed is improved.
Reference to the literature
[1] Royal official article deep water submarine pipeline maintenance system engineering application research [ D ]. tianjin university: academy of construction engineering, 2010.
[2] Zhangxing, research on the propagation attenuation characteristics of ultrasonic non-uniform media [ D ], Shenyang university of industry, college of information science and engineering, 2015.
[3] Leonie Ex. research on transmission rule of ultrasonic waves in slurry [ D ]. China university of Petroleum: oil and gas well engineering, 2007.
[4] Ultrasonic ranging error analysis and correction in slurry, Caopenong, Zhang-Yi Fang, metering technology, 1996 (10):23-24.
Those skilled in the art will appreciate that the drawings are only schematic illustrations of preferred embodiments, and the above-described embodiments of the present invention are merely provided for description and do not represent the merits of the embodiments.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.
Claims (7)
1. A circle center positioning method in ultrasonic detection of a shallow buried pipeline is characterized in that the circle center positioning method is based on arrival time and a least square method, and comprises the following steps:
constructing a simplified bright spot model of the buried pipeline; using MATLAB to design simulation parameters to obtain simulation echo signals;
a simple single-channel ultrasonic transmitting and receiving experiment platform under a laboratory environment is built;
simulating a complex shoal environment, and adding Gaussian noise with a signal-to-noise ratio of a certain threshold value into an original experiment echo signal to obtain an echo signal;
respectively adopting a maximum amplitude method, a characteristic parameter correlation detection method and a rapid energy center convergence method to carry out echo TOA estimation on echo signals, and selecting an optimal estimation method;
estimating and calculating a sound path between the transducer and a theoretical mirror image bright spot according to the simplified bright spot model and the echo TOA;
fitting a pipeline circle center objective function according to the sound path, calculating an approximate numerical solution, taking the approximate numerical solution as an initial coordinate, and solving an accurate solution of the pipeline circle center through iterative operation.
2. The method of claim 1, wherein the simplified bright spot model is specifically:
taking the position of the transducer as an origin O, taking the radial cross section of the pipeline as a two-dimensional plane, and constructing a coordinate system of the position relation between each transducer and the pipeline when n transducers with the interval of d form a transducer array;
the working form of the transducer array is that a single transducer sequentially transmits and receives echoes, and n bright spot positions are correspondingly generated at the edge of a pipeline to obtain corresponding received echo signals.
3. The method as claimed in claim 2, wherein the bright spots are the edge points of the pipe cross-section circle, and the bright spots are located on the line connecting the center of the transducer and the center of the pipe cross-section circle.
4. The method as claimed in claim 3, wherein the location of the circle center in the ultrasonic detection of the shallow buried pipeline is specifically as follows:
wherein (x)i，yi) Coordinates of the bright spots; (x)0，y0) As the coordinates of the circle center; (x'i，y′i) Are the coordinates of the transducer; liThe acoustic path between the transducer and the bright spot.
5. The method as claimed in claim 2, wherein the simple single-channel ultrasonic transmitting and receiving experiment platform under the laboratory environment is specifically:
selecting an initial coordinate right above the edge of the pipeline, wherein the vertical distance between the center of the pipeline and the transducer is a certain preset distance;
starting from the initial coordinate, setting a plurality of transducers from left to right at intervals of a certain preset value to receive the echo to be detected;
triggering once sound wave transmission and reception at each point to be measured, and performing once data acquisition; and obtaining echo signals obtained by a certain group of water tank experiments.
6. The method of claim 1, wherein the fitting of the pipe circle center objective function according to the acoustic path and the calculation of the approximate numerical solution are specifically as follows:
fitting the circle center and the radius according to a least square method; obtaining a corrected target function by using the coordinate of the transducer and the sound path distance relation between the transducer coordinate and the arc discrete point;
writing the modified objective function into an approximate form; and solving by utilizing a multivariate function extremum method to obtain an equation set, and further calculating to obtain a numerical solution.
7. The method as claimed in claim 6, wherein the approximate numerical solution is used as the initial coordinate, and the exact solution for solving the center of the pipeline by iterative operation is specifically:
establishing a matrix P according to the initial coordinates and initial step length of the circle centeriWill matrix PiAfter correctionAn objective function, obtaining another matrix FiFind another matrix FiThe smallest element F (x) inj，yj)；
If i is not equal to j, updating the coordinates of the circle center as xi＝xj，yi＝yj；
If i is j, updating the step length S to be S/2, and repeating the iteration step;
if the step length S is less than SminIteration stops, SminIs the set minimum iteration step size.
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