CN103196481B - Calibrating device and calibrating method of mine fiber grating sensor - Google Patents

Abstract

The invention relates to a calibration device and a calibration method of a mine optical fiber grating sensor, belonging to the calibration device and the calibration method of a mine sensor. The device includes a protection cabinet, a fuel tank liquid level indicator placed outside the protection cabinet, a control box with a high-precision pressure gauge and function buttons, an operation console with a manual control valve and an electromagnetic control valve, a hydraulic station and an installation A test stand with a stretching cylinder, where the stretching cylinder is connected to the console through oil pipes, the piston rod of the cylinder connected to the stretching cylinder is placed in the test stand, and the clamp is set at one end of the test stand to fix the components to be calibrated. When calibrating the optical fiber grating sensor for mining, the oil pressure is applied to the sensor through the oil cylinder to complete the calibration, and the oil pressure loading process is stable, which ensures that the optical fiber grating sensor is evenly stressed; it can realize electromagnetic and manual dual control methods, and can switch seamlessly; the device Quick start, convenient control, safety and reliability, good versatility, and great practical application value.

Description

Calibration device and calibration method of mine fiber grating sensor

technical field

The invention relates to a calibration device and a calibration method of a mine sensor, in particular to a calibration device and a calibration method of a mine optical fiber grating sensor.

Background technique

Coal mine safety accidents often occur in our country, which makes the country and various departments pay more and more attention to coal mine safety. Although my country's coal mine safety monitoring has been well promoted and applied, the existing field of coal mine safety monitoring is still subject to unprecedented challenges. Since the advent of fiber gratings, they have been widely used in the field of fiber optic sensing, especially in recent years, the development and application of fiber grating sensors. Fiber Bragg grating sensing technology is a brand-new sensing technology rising with the development of fiber grating manufacturing technology.

At present, my country's coal mine safety monitoring system uses traditional mechanical or resistive sensors. Compared with traditional sensors, fiber grating sensors have the advantages of anti-electromagnetic interference, anti-corrosion, electrical insulation, and high sensitivity. Therefore, it has become a trend to use fiber grating sensors in petroleum, geotechnical, coal mines and other fields, and has been verified on site. Although fiber grating sensors are developing rapidly and are widely used, so far, there is no special calibration device and calibration method for mining fiber grating sensors in the market. The fiber grating sensors used in the society are based on the coefficients provided by suppliers. It is a general coefficient with low accuracy. When the working environment changes, it will greatly affect the measurement results. Especially in the special working environment of coal mines, the accuracy of fiber grating sensor measurement has a fatal impact on coal mines.

Contents of the invention

In order to overcome the deficiencies in the above-mentioned technologies, the object of the present invention is to provide a calibration device and calibration method for mining optical fiber grating sensors that realize electromagnetic control and manual control, work conveniently and reliably, load can be evenly loaded, and have high precision.

In order to achieve the purpose of the present invention, it is achieved by the following technical solutions: a calibration device of a fiber grating sensor for mines includes: a hydraulic system, a protection cabinet, a fuel tank liquid level indicator arranged outside the protection cabinet, a high-precision pressure gauge Control box with function buttons, console with manual control valve and electromagnetic control valve, hydraulic station, and experimental frame with stretching cylinder, where the stretching cylinder is connected to the console through oil pipes, and the oil cylinder connected to the stretching cylinder The piston rod is placed in the test frame, and the clamp is set at one end of the test frame to fix the components to be calibrated;

The hydraulic system includes: electromagnetic reversing valve, manual reversing valve, oil pump, motor, filter, relief valve, pressure converter and oil tank; the first inlet of the electromagnetic reversing valve and manual reversing valve are connected in parallel and simultaneously converted to pressure One end of the pressure converter is connected to the hydraulic station, and the other end of the pressure converter is connected to the pressure gauge; the second inlet of the electromagnetic reversing valve and the manual reversing valve are connected in parallel and connected with the other end of the hydraulic station at the same time, and the electromagnetic reversing valve and the manual reversing valve are connected in parallel. After the outlet of the valve is connected in series, one end is connected to the oil tank and one end of the overflow valve at the same time, and the other end is connected to the output end of the oil pump. The input end of the oil pump is connected to the oil tank through a filter, and the other end of the overflow valve is connected to the oil tank.

The high-precision pressure gauge can display the force when the load is applied, and the range of the pressure gauge is 0MPa~60MPa.

The function buttons include a start and stop button, a start indicator light, a forward button and a backward button, and their functions are to control the start and stop of the system, indicate the working state of the device and control the telescopic action of the stretching cylinder.

The manual control valve and electromagnetic control valve arranged on the console are connected in series in the hydraulic system, and their functions are to independently operate and control the working mode of the stretching oil cylinder.

46# anti-wear hydraulic oil is used in the hydraulic station.

The bore of the stretching oil cylinder is 160mm.

The diameter of the piston rod of the oil cylinder is 70mm, and the maximum working stroke is 1000mm.

A kind of calibration device and calibration method of fiber grating sensor for mining, described calibration method can be calibrated according to the following steps according to the calibration idea of comparison method in static calibration:

(1) Installation: According to the invented device, according to the mine fiber grating sensors with different structures, connect and install the mine fiber grating sensors to be calibrated in the correct position, and the mine fiber grating sensors are connected to the fiber grating static demodulator through optical fibers On the top, the fiber grating static demodulator is connected to the computer terminal installed with monitoring and processing software through a network cable;

(2) Calibration:

a. Before calibration; after the mine fiber grating sensor is installed and checked correctly, press the start button on the operation box of the device to make the device enter the working state;

b. During calibration; press the forward and backward function buttons on the operation box to control the oil cylinder to act on the piston rod with oil pressure to make it stretch and retract. The action of the piston rod can force the fiber grating sensor and load it evenly Pressure, timely record the pressure value on the high-precision pressure gauge on the device; the fiber grating static demodulator demodulates the light wave signal of the optical fiber to the electrical signal, and the computer stores the demodulated electrical signal, and the monitoring and processing software records the calibration time of the fiber grating The wavelength change value of the sensor;

c. After calibration; post-processing the recorded pressure value and fiber wavelength change value to obtain the pressure-wavelength relationship curve, and perform curve fitting and regression analysis to obtain the fitting curve equation, thereby completing the linearity of the fiber grating sensor , Calibration of sensitivity;

(3) After the fiber grating sensor is calibrated, press the stop button on the operation box to stop the device;

The mine fiber Bragg grating sensor in the step (1) includes a mine fiber Bragg grating strain sensor, a mine fiber Bragg grating pressure sensor, and a mine fiber Bragg grating displacement sensor.

The curve fitting method adopted in c of the step (2) is the least squares method, so that the data points recorded in the calibration process ( , ) ( =0,1,2,..., ) represents the pressure value and wavelength value, where Indicates the number of pressure loading during the calibration process. The specific fitting process is as follows:

① Data points from known records ( , ) Draw a rough graph of the function - a scatter plot, and determine the degree of fitting polynomials ;

②According to the determined polynomial degree Write the equation of the fitted curve ,in , ,..., for each coefficient;

③According to the data ( , ) and times list calculation about pressure the expression of , ,..., The value of and about the wavelength the expression of , , ,..., value;

④ From linear algebra knowledge, we can know about polynomial coefficients , ,..., The system of linear equations can be expressed as a matrix:

It can be seen that this coefficient matrix is a symmetric positive definite matrix, so there is a unique solution; the coefficients can be solved through the matrix , ,..., The value of , and obtain the fitting curve equation.

Beneficial effects, due to the adoption of the above scheme, the present invention has the following positive effects and advantages: on the one hand, the calibration device can realize two working modes of electromagnetic control and manual control, so that the device can avoid the worry of being unable to work due to electromagnetic damage; On the one hand, the manual control valve and the electromagnetic control valve are connected in series in the hydraulic system, and the switching between the two working modes is simple and seamless switching can be realized; moreover, the hydraulic transmission system can apply a uniform and stable load during operation, which is easy to realize rapid start-up and control Convenient, safe and reliable, easy to operate. In terms of calibration method: On the one hand, advanced fiber grating sensors are used in calibration. Compared with traditional sensors, they have the advantages of intrinsic safety, strong anti-electromagnetic interference, electrical insulation, strong stability, and high sensitivity. Therefore, they can be calibrated in various environments. On the other hand, the mine-used fiber grating sensor is connected with the fiber grating static demodulator, and the measured data has high accuracy and good versatility. It is more accurate and has great practical application value.

Description of drawings

Fig. 1 is a structural diagram of the device of the present invention.

Fig. 2 is the working principle hydraulic system diagram of the device of the present invention.

Fig. 3 is a diagram of calibrating a fiber grating force-measuring anchor with the device of the present invention.

Fig. 4 is a diagram of calibrating the pressure gauge of the fiber grating support by the device of the present invention.

In the figure: 1. Protection cabinet; 2. Fuel tank liquid level indicator; 3. Control box; 4. High-precision pressure gauge; 5. Function buttons; 6. Manual control valve; 7. Electromagnetic control valve; 8. Operating console; 9. Hydraulic station; 10. Oil pipe; 11. Stretching oil cylinder; 12. Oil cylinder piston rod; 13. Experiment frame; 14. Fixture; 15. Electromagnetic reversing valve; 16. Manual reversing valve; 17. Oil pump; 18. Motor; 19, filter; 20, overflow valve; 21, pressure converter; 22, fuel tank; 23, fiber grating sensor; 24, anchor rod; 25, fiber grating support pressure gauge; 26, pressure display; Pressure hole; 28, optical fiber connection hole.

Detailed ways

The present invention will be further described below through specific embodiments and in conjunction with the accompanying drawings.

Embodiment 1: A kind of calibration device of fiber grating sensor for mining, described calibration device comprises: hydraulic system, protection cabinet 1, the oil tank liquid level indicator 2 that is arranged outside the protection cabinet, is equipped with high-precision pressure gauge 4 and Control box 3 with function button 5, console 8 with manual control valve 6 and electromagnetic control valve 7 installed, hydraulic station 9 and test frame 13 with stretching oil cylinder 11 installed, wherein the stretching oil cylinder is connected with the console through oil pipe 10 , the cylinder piston rod 12 connected with the stretching oil cylinder is placed in the test frame, and the clamp 14 is set at one end of the test frame for fixing the components to be calibrated.

The hydraulic system includes: electromagnetic reversing valve 15, manual reversing valve 16, oil pump 17, motor 18, filter 19, overflow valve 20, pressure converter 21 and oil tank 22; electromagnetic reversing valve 15 and manual reversing valve 16 The first inlet of the first inlet is connected in parallel with the pressure converter 21 and one end of the hydraulic station 9, and the other end of the pressure converter 21 is connected to the pressure gauge; the second inlet of the electromagnetic reversing valve 15 and the manual reversing valve 16 are connected in parallel and connected with the The other end of the hydraulic station 9 is connected. After the outlets of the electromagnetic reversing valve 15 and the manual reversing valve 16 are connected in series, one end is connected with the oil tank 22 and one end of the overflow valve 20 at the same time, and the other end is connected with the output end of the oil pump 17. The input end is connected to the oil tank 22 through the filter 19, and the other end of the overflow valve 20 is connected to the oil tank 22.

The high-precision pressure gauge 4 can display the force when the load is applied, and the range of the pressure gauge is 0MPa~60MPa.

The function button 5 includes a start-stop button, a start indicator light, a forward button and a backward button, and its functions are to start and stop the control system, indicate the working state of the device and control the stretching action of the stretching cylinder.

The manual control valve 6 and electromagnetic control valve 7 arranged on the console 8 are connected in series in the hydraulic system, and their function is to independently operate and control the working mode of the stretching oil cylinder 11 .

Used in the described hydraulic station 9 is 46# anti-wear hydraulic oil.

The cylinder diameter of described stretching oil cylinder 11 is 160mm.

The diameter of the cylinder piston rod 12 is 70mm, and the maximum working stroke is 1000mm.

A kind of calibration device and calibration method of fiber grating sensor for mining, described calibration method can be calibrated according to the following steps according to the calibration idea of comparison method in static calibration:

(1) Installation: According to the inventive device, according to the mine fiber grating sensors with different structures, the mine fiber grating sensors to be calibrated are connected and installed in the correct position, and the mine fiber grating sensors are connected to the fiber grating static demodulator through optical fibers On the top, the fiber grating static demodulator is connected to the computer terminal installed with monitoring and processing software through a network cable;

(2) Calibration; a. Before calibration; after the mine fiber grating sensor is installed and checked correctly, press the start button on the device operation box to make the device enter the working state;

b. During calibration; press the forward and backward function buttons on the operation box to control the oil cylinder to act on the piston rod with oil pressure to make it stretch and retract. The action of the piston rod can force the fiber grating sensor, and Evenly load the pressure, and record the pressure value on the high-precision pressure gauge on the device in time; the fiber grating static demodulator demodulates the light wave signal of the optical fiber to the electrical signal, and the computer stores the demodulated electrical signal and records the calibration time by the monitoring and processing software The wavelength change value of the fiber grating sensor;

c. After calibration; post-processing the recorded pressure value and fiber wavelength change value to obtain the pressure-wavelength relationship curve, and perform curve fitting and regression analysis to obtain the fitting curve equation, thereby completing the linearity of the fiber grating sensor , Calibration of sensitivity;

(3) After the fiber grating sensor is calibrated, press the stop button on the operation box to stop the device;

In addition, the mine fiber Bragg grating sensor in the step (1) includes a mine fiber Bragg grating strain sensor, a mine fiber Bragg grating pressure sensor, and a mine fiber Bragg grating displacement sensor.

The curve fitting method adopted in c of the step (2) is the least squares method, so that the data points recorded in the calibration process ( , ) ( =0,1,2,..., ) represents the pressure value and wavelength value, where Indicates the number of pressure loading during the calibration process. The specific fitting process is as follows:

① Data points from known records ( , ) Draw a rough graph of the function - a scatter plot, and determine the degree of fitting polynomials ;

②According to the determined polynomial degree Write the equation of the fitted curve ,in , ,..., for each coefficient;

③According to the data ( , ) and times list calculation about pressure the expression of , ,..., The value of and about the wavelength the expression of , , ,..., value;

④ From linear algebra knowledge, we can know about polynomial coefficients , ,..., The system of linear equations can be expressed as a matrix:

It can be seen that this coefficient matrix is a symmetric positive definite matrix, so there is a unique solution. The coefficients can be solved through the matrix , ,..., The value of , and obtain the fitting curve equation.

Embodiment 2: FIG. 3 is a diagram of calibration of the fiber grating force measuring anchor by the device of the present invention. The anchor rod 24 with the fiber grating sensor 23 pasted on the surface is connected with the piston rod of the oil cylinder through the clamp 14, and the other parts of the device are the same as in the first embodiment.

Using this calibration device to calibrate the fiber grating anchor dynamometer can be completed through the following steps:

Step 1: Prepare the fiber grating force measuring anchor rod to be calibrated, wherein the fiber grating strain sensor is attached to the anchor rod;

Step 2: Connect the FBG sensor to the FBG static demodulator;

Step 3: After the calibration anchor rod is installed and checked correctly, press the start button on the operation box to control the start of the calibration device, the stop button controls the device to stop, the forward button controls the cylinder piston rod to extend, and the back button controls the cylinder piston rod Retracted, the start indicator light can display the working status of the calibration device;

Step 4: After the calibration device works normally, control the oil cylinder to apply oil pressure to the piston rod, so as to realize the uniform load applied by the tension cylinder to the anchor rod;

Step 5: Use the fiber grating static demodulator to record the wavelength change of the fiber grating sensor, and at the same time record the load on the calibrated anchor shown by the high-precision pressure gauge, complete the relationship curve between the pressure value and the wavelength value, and perform fitting to obtain The fitting curve equation can complete the linearity and sensitivity calibration of the fiber grating force measuring anchor;

Step 6: After the anchor rod is calibrated, the stop button on the operation box can be used to control the device to stop.

Embodiment 3: FIG. 4 is a calibration diagram of the device of the present invention on the pressure gauge of the fiber grating support. The pressure measuring hole 27 is connected to the console 8 through 10 oil pipes. The pressure measuring hole 27 is located at the bottom of the pressure gauge 25 of the fiber grating bracket. There is an optical fiber connection hole 28 on one side of the pressure gauge 25 of the fiber grating bracket. There is a pressure indicator 26 in the middle part, and the other parts of the device are the same as the device of embodiment 1.

Using this calibration device to test the fiber grating support pressure gauge can be completed through the following steps:

Step 1: Prepare the fiber grating support pressure gauge to be calibrated;

Step 2: A fiber grating pressure sensor is installed inside the shell of the fiber grating support manometer, and a display is set outside the shell, which can display the current pressure of the support. The side end of the manometer is provided with an optical fiber connection hole, and the fiber grating static demodulator is connected to it connect.

Step 3: According to the structural schematic diagram of the invention, unplug the two oil pipes connected to the stretching oil cylinder, and connect them to the two pressure measuring holes of the fiber grating support pressure gauge;

Step 4: After the installation of the calibrated fiber grating support pressure gauge is completed, and after the inspection is correct, press the start button on the operation box to control the start of the calibration device, the stop button to control the device to stop, and the forward and backward function buttons to control the increase of hydraulic oil in the tank. Pressure and pressure relief, the start indicator can display the working status of the calibration device;

Step 5: After the calibration device works normally, control the oil cylinder to apply the oil pressure to the pressure gauge of the fiber grating support, and the oil cylinder applies a uniform load to make the fiber grating pressure gauge stable and uniform for calibration;

Step 6: Use the fiber grating static demodulator to record the wavelength change of the fiber grating sensor, and record the pressure value on the pressure display at the same time, complete the relationship curve between the pressure value and the wavelength value, and perform fitting to obtain the fitting curve equation, which can be completed Linearity and sensitivity calibration of fiber grating manometers;

Step 7: After the fiber grating support pressure gauge is calibrated, the control device can be stopped by the stop button on the operation box.

Through the above specific embodiments, the present invention provides a calibration device and a calibration method of a fiber grating sensor for mining. calibration device. In addition, the device can realize two working modes of electromagnetic control and manual control, so that the device can avoid the worry that the device cannot work due to electromagnetic damage; the manual control valve and the electromagnetic control valve are connected in series in the hydraulic system, and the switching between the two working modes is simple. To achieve seamless switching, the hydraulic transmission system can apply a uniform and stable load during operation, which is easy to achieve quick start, convenient control, safety and reliability, and simple operation; moreover, the hydraulic system of the inventive device acts on the sensor through the oil cylinder After the calibration is completed, the hydraulic system provides a uniform and stable load for the calibration device, which will make the fiber grating sensor bear uniform and stable force during calibration. At the same time, the fiber grating sensor to be calibrated has intrinsic safety, strong anti-electromagnetic interference ability, and Corrosion resistance, electrical insulation, strong stability, high sensitivity and other advantages, so it can be calibrated in a variety of environments without being affected; the fiber grating sensor is connected with the fiber grating static demodulator, and the measured data has high accuracy and good versatility At the same time, the least square method is used for post-processing the data, so the error is small, the calibration is more accurate, and it has great practical application value.

Claims (9)

1. the caliberating device of a mining optical fiber grating sensor, it is characterized in that: described caliberating device comprises: hydraulic system, protection cabinet (1), be placed in oil tank liquid level display meter (2) outside protection cabinet, be mounted with the control box (3) of high-precision pressure gauge (4) and function button (5), the operator's console (8) of manually operated control valve (6) and solenoid electric valve (7) is installed, Hydraulic Station (9) and be provided with the test stand (13) of stretching oil cylinder (11), wherein stretching oil cylinder is connected with operator's console by oil pipe (10), the cylinder piston rod (12) be connected with stretching oil cylinder is placed in experiment frame, fixture (14) is arranged on one end of experiment frame, for fixing component to be calibrated,

Hydraulic system comprises: solenoid directional control valve (15), hand-operated direction valve (16), oil pump (17), motor (18), filtrator (19), surplus valve (20), sensator (21) and fuel tank (22); Solenoid directional control valve (15) is connected with one end of sensator (21) and Hydraulic Station (9) afterwards with the first entrance parallel connection of hand-operated direction valve (16) simultaneously, another termination tensimeter of sensator (21); Solenoid directional control valve (15) is connected with the other end of Hydraulic Station (9) afterwards with the second entrance parallel connection of hand-operated direction valve (16) simultaneously, solenoid directional control valve (15) is connected with one end of fuel tank (22) and surplus valve (20) with one end after the outlet series connection of hand-operated direction valve (16) simultaneously, the other end is connected with the output terminal of oil pump (17), the input end of oil pump (17) passes through filtrator (19) connected tank (22), the other end connected tank (22) of surplus valve (20).

2. the caliberating device of a kind of mining optical fiber grating sensor according to claim 1, is characterized in that: described high-precision pressure gauge (4) can show power during loaded load, and pressure gage measuring range is 0MPa ~ 60MPa.

3. the caliberating device of a kind of mining optical fiber grating sensor according to claim 1, it is characterized in that: described function button (5) comprises start-stop button, starts pilot lamp, forwarding button and back, its function is the expanding-contracting action of the start and stop of control system, the duty of indicating device and restrained stretching oil cylinder.

4. the caliberating device of a kind of mining optical fiber grating sensor according to claim 1, it is characterized in that: described be located at manually operated control valve (6) on operator's console (8) and solenoid electric valve (7) is connected on hydraulic system, its function is the working method independently operating restrained stretching oil cylinder (11).

5. the caliberating device of a kind of mining optical fiber grating sensor according to claim 1, is characterized in that: the cylinder diameter of described stretching oil cylinder (11) is 160mm.

6. the caliberating device of a kind of mining optical fiber grating sensor according to claim 1, is characterized in that: the diameter of described cylinder piston rod (12) is 70mm, and maximum functional stroke is 1000mm.

7. use a scaling method for mining optical fiber grating sensor calibration apparatus, it is characterized in that: described scaling method can be demarcated as follows according to the demarcation thought of relative method in static demarcating:

(1) install; According to contrive equipment, according to the mining optical fiber grating sensor of different structure, mining optical fiber grating sensor to be calibrated is mounted on correct position, mining optical fiber grating sensor is received on the static (FBG) demodulator of fiber grating by optical fiber, the static (FBG) demodulator of fiber grating by netting twine with the computer end of monitoring process software be installed be connected;

(2) demarcate; Before a, demarcation; Mining optical fiber grating sensor installs, check errorless after, by the start button on lower device control box, make device enter duty;

In b, demarcation; Push case moves forward and backward function button and oil cylinder with controllable by oil pressure effect on the piston rod, it is made to extend and indentation, the action of piston rod can make fiber-optic grating sensor stressed, and uniform load pressure, the pressure value on timely pen recorder on high-precision pressure gauge; The static (FBG) demodulator of fiber grating is by the lightwave signal of demodulation optical fiber to electric signal, and computer stores the electric signal after demodulation and by the wavelength variations numerical value of monitoring process software record timing signal fiber-optic grating sensor;

After c, demarcation; The pressure value of record and fiber optic wavelength change numerical value are carried out post-processed, obtains the relation curve of pressure-wavelength, and carry out curve fitting and regretional analysis, obtain fit curve equation, thus complete the demarcation to the fiber-optic grating sensor linearity, sensitivity;

(3), after fiber-optic grating sensor is demarcated and terminated, the stop button on push case makes device quit work.

8. the scaling method of a kind of mining optical fiber grating sensor according to claim 7, is characterized in that the mining optical fiber grating sensor in described step (1) comprises mining optical fiber grating strain transducer, mining optical fiber grating pressure transducer, mining optical fiber grating displacement transducer.

9. the scaling method of a kind of mining optical fiber grating sensor according to claim 7, is characterized in that: the method for the curve adopted in the c of described step (2) is least square method, makes the data point (x recorded in calibration process _i, y _i) (i=0,1,2 ..., m) represent pressure value and wavelength values, wherein m represents pressure-loaded number of times in calibration process, concrete fit procedure following steps:

1. by the data point (x of known record _i, y _i) draw the rough figure of function---scatter diagram, determine the frequency n of polynomial fitting;

2. fit curve equation y=a is write out according to the degree of polynomial n determined ₀x ⁰+ a ₁x ¹+ ... + a _nx ⁿ, wherein a ₀, a ₁..., a _nfor each term coefficient;

3. according to data (x _i, y _i) and number of times m list calculate about pressure x _iexpression formula value and about wavelength y _iexpression formula value;

4. known about multinomial coefficient a by mathematical statistics knowledge ₀, a ₁..., a _nsystem of linear equations, can be expressed in matrix as:

$\left[\begin{array}{cccc}m+1& \underset{i=0}{\overset{m}{\mathrm{\Σ}}}{x}_i& ...& \underset{i=0}{\overset{m}{\mathrm{\Σ}}}{x}_i^n\\ \underset{i=0}{\overset{m}{\mathrm{\Σ}}}{x}_i& \underset{i=0}{\overset{m}{\mathrm{\Σ}}}{x}_i^2& ...& \underset{i=0}{\overset{m}{\mathrm{\Σ}}}{x}_i^{n+1}\\ .& .& .& .\\ .& .& .& .\\ .& .& .& .\\ \underset{i=0}{\overset{m}{\mathrm{\Σ}}}{x}_i^n& \underset{i=0}{\overset{m}{\mathrm{\Σ}}}{x}_i^{n+1}& ...& \underset{i=0}{\overset{m}{\mathrm{\Σ}}}{x}_i^{2n}\end{array}\right]\left[\begin{array}{c}{a}_0\\ a_1\\ .\\ .\\ .\\ a_n\end{array}\right]\left[\begin{array}{c}\underset{i=0}{\overset{m}{\mathrm{\Σ}}}{y}_i\\ \underset{i=0}{\overset{m}{\mathrm{\Σ}}}{x}_i{y}_i\\ .\\ .\\ .\\ \underset{i=0}{\overset{m}{\mathrm{\Σ}}}{x}_i^n{y}_i\end{array}\right]$

This matrix of coefficients known is-individual symmetric positive definite matrix, therefore existence and unique solution; Each term coefficient a can be solved by matrix ₀, a ₁..., a _nvalue, near and draw fit curve equation.