A Temperature Compensation Algorithm Based on Curve Fitting and Spline Interpolation

Transcription

1 1345 A publication of CHEMICAL ENGINEERING TRANSACTIONS VOL. 51, 2016 Guest Editors: Ticun Wang, Hongyang Zang, Lei Tian Copyrigt 2016, AIDIC Servizi S.r.l., ISBN ; ISSN Te Italian Association of Cemical Engineering Online at DOI: /CET A Temperature Compensation Algoritm Based on Curve Fitting and Spline Interpolation Su-lan Xia* a, Ji-lin Wang b, Ru-gang Wang b, Li Zao c a College of Electrical Engineering, Yanceng Institute of Tecnology, Yanceng , Jiangsu, PR Cina b College of Information Engineering, Yanceng Institute of Tecnology, Yanceng , Jiangsu, PR Cina c College of Information Science and Engineering,Souteast University, Nanjing , Jiangsu, PR Cina xslnj@126.com Te zero and sensitivity of piezo resistive pressure sensors will drift wit temperature because of te temperature caracteristics of te semiconductor, and it is te main factor causing te measurement error of pressure sensor. For ig-precision pressure monitoring system, temperature drift as become an important obstacle to improve performance of te system, especially in te field of applications wit large canges of ambient temperature. On te basis of analysing te advantages and disadvantages of a variety of temperature compensation metods, a temperature compensation metod combined wit polynomial curve fitting and tree spline interpolations is proposed, and it can improve te performance of te system. 1. Introduction Te caracteristic of piezo resistive pressure sensor is ig sensitivity, dynamic response, ig precision, and long woring life; terefore it obtains more and more application in modern industry (Wang et al., 2007). It is a ind of pressure sensor using te piezo resistive effect of semiconductor materials. But because of te core component is piezo resistive diapragm wic is sensitive wen temperature canges, so it causes te sensor's zero point and sensitivity canging wen temperature drift. Tere are two inds of common temperature compensation metods, ardware compensation metod and software compensation metod (Li et al., 2008; Palmer, 2006). Te traditional compensate metod according to te sensor's ardware caracteristics as many faults, suc as debug difficulties, poor universality and low accuracy and so on, it is detrimental to engineering(isigai et al., 2007; Zang, 2008; Wang et al., 2010). Wile te software compensation metod can solve te above problems well, so it acquires more and more attention(kowalsi, 1987; Pramani, 2006; Guo, 2010). Tis paper uses software compensation metod mainly, it adopt te appropriate sensor information fusion algoritm to compensate te influence of pressure sensor causing by te cange from temperature, and puts forward a ind of temperature compensation metod wic combines polynomial curve fitting and cubic-spline interpolation, tis metod is flexible, and also can acieves ig precision. 2. Hig precision pressure detection system based on software compensation Te metod of software compensation combines te microprocessor and pressure sensor, it can mae full use of microprocessor, and troug compensation algoritms to correct te errors wic caused by temperature (Handy et al., 2002; Li et al., 2009). Regardless using wat type of software compensation metod, te ardware structure is similar, te differences lies in algoritm (Zao et al., 2005; Wu, 2007). Te ig precision pressure detecting system wic we designed used te ig performance micro-controller LPC2478 as its central processor, te processor based on te ARM arcitecture, its woring frequency is 72MHz, and it owns 512K flas memory and some serial interface. Meanwile, te system using a AD wic owns ig sampling accuracy and also a piece of SDRAM. Te System bloc diagram is sown in fig.1. Please cite tis article as: Xia S.L., Wang J.L., Wang R.G., Zao L., 2016, A temperature compensation algoritm based on curve fitting and spline interpolation, Cemical Engineering Transactions, 51, DOI: /CET

2 1346 Figure 1: Te System bloc diagram 3. Algoritm analysis and te implementation At present, te software compensation metod as interpolation metod, curve/surface fitting metod, querying table metod and te metod using BP neural networ (Wu et al., 2012). In te interpolation metod (Li et al., 2015), first it requires tat te data is correct, and also requires nowing te relationsip between dates. Te curve fitting metod tries to find a smoot curve, it can fit data s well, but not necessary to pass any data. Querying table metod is a metod wic must filling a series of parameters into a parameter list, after obtaining te measurements, process dates according to te relative parameters in te table. Querying table metod requires large storage space, so it does not suit for microprocessor. Neural networ metod is a metod tat troug establising te artificial neural networ model, and also troug te sample to train te networ parameters, but te largest sortcoming is te instability of networ, and te long training time (Ren et al., 2014). Tis paper puts forward a metod combined wit curve fitting and cubic spline interpolation compensation algoritm; tis algoritm can increase te compensation performance significantly. Te following content wants to explain te teory of polynomial fitting and cubic-spline interpolation firstly. 3.1 Polynomial curve fitting In te curve fitting metod, polynomial fitting is te most appropriate metod, because tis metod wit easy calculation, and well-fitting effect (Tang et al., 2014). According to te given data (xi, yi), i=1, 2,, m, polynomial fitting is te metod wic mae a polynomial : n y a x 0 (1) Let te squares error is minimum, i.e. 2 m n E yi a xi min (2) i1 0 Terefore let: E a 0 (=1,2,,m) (3) Ten: n m m j j ( xi ) a xi yi 0 i0 i0 j=1, 2,, n (4) Formula 1 can be unfolded as linear equations about coefficient ai, if tis equation as a unique solution; te unique solution is te coefficient of te fitting polynomial spline Interpolation Spline interpolation is a ind of improved subsection interpolation, it structures spline function in every adjacent area, meanwile in order to guarantee te continuity at tis node, it requires 2-order smootness at te point, and namely it owns continuous 2-order derivatives. Te function of 3-spline interpolation is defined as follows (Wang et al., 2007).

3 Assuming tat te given interval [a, b] as n+1 nodes, and a=x0<x1< <xn=b. If te function s(x) is a polynomial wic order is no more tan tree in any interval [x,x+1],and s(x) owns continuous 2-order derivatives in te interval [a, b],so can define s(x) is cubic spline function of nodes x0,x1,,xn. According to te above definition, s" is a linear function in every adjacent area [x,x+1].if maing te s"(x)=m(=1,2,,n), s"(x) can be expressed as te next formula in area [x,x+1] s n x m x x m 1 1 x x (5) In order to derivation conveniently, let =x+1-x. Ten integral to formula (5) two times, so s x m 3 x x x x 3 1 (6) m 1 A x x B 6 6 Tae te value of x, x+1into te above formula, and ten get te expression of A and B: A y 1 y 6 m 1 m (7) B y m 2 6 From formula (6) (7) (8) can find out tat if te value of m (=1, 2,, n) is nown, te value of s(x) can find out conveniently. Te solution of m is using te continuity caracteristics of te nodes defined above, i.e. s +(x)=s _(x),and add te two boundary constraint conditions of endpoints, refer to te reference of Numerical Analysis(Yanfeng Zang et al., 2007),can get te equations wic can solve by case-after metod, it can reduce te computation greatly. 3.3 Temperature compensation model and algoritm Pressure sensor is generally calibrated under standard temperature and standard pressure pj (j=1, 2,, m) (Li et al., 2015), te number of mared point is n m. Troug te appropriate algoritm can obtain te relation among (ti, pj, uij) wic comes from actual measurement processing, it is p=f (t,u).here te parameter u is te output voltage of pressure sensor. If te measurements Δpmax=max(Δp1,, Δpj,,Δpm),j=1,2,,m,according to p=f(t,u),wile te true value is pj,so te measurement error is Δpj= p j-pj, j=1,2,,m. In te wole measurements, te maximum error of pressure sensor is Δpmax, so te composition error wic relatives to te scale span pfs is ξ=(δpmax/pfs) 100%. According to tis temperature compensation model, and combining wit curve fitting and cubic-spline interpolation, te procedure of te temperature compensation algoritm is as follows: Step 1, fixed temperature t1 as a constant, doing a polynomial fitting wit(u12,p1), (u22,p2),, (um2,pm),ten can get te relation curve between p and u under temperature t1,p=f1(u).te order of te polynomial is decided by measured data,and usually using quadratic polynomial fitting,it can also use linear fitting if te preference of te sensor is well, quadratic polynomial fitting can be realized troug Matlab. Step 2 similarly, fixed temperature t2 as a constant, ten can get te relation curve between p and u under temperature t2, p= f2 (u). Te rest can be done in te same manner, until fixed temperature tn as a constant, ten can get te relation curve between p and u under temperature tn, p=fn (u). Step 3 tae te actual measurement voltage u into p=f1 (u), p=f2 (u),..., p=fn(u), find out te pressure value under te standard temperature. Step 4 process te above pressure value and standard temperature using cubic spline interpolation, ten obtain te relation curve between pressure p and temperature t, it is p=g(t), Te program of cubic spline interpolation realizes on te ARM platform using C programming language. Step 5 Taing te real-time temperature into formula p=g (t), ten will get te pressure value witout temperature affects troug te information fusion tecnology. (8)

4 Experimental process and result analysis Table 1 gives te output voltage value of sensor wen te mared point pressure is 0, 5, 10, 20, 30, 40, 60 and te temperature is 0, 10, 20, 40 and 60. According to te algoritm steps above, processing te experimental data, i.e. fix te temperature,and ten process p, u using 2-order fitting, On tis basis, using cubic-spline interpolation to process te temperature and te pressure, fusion te information of te pressure and te temperature, so can get te correct pressure results. In te actual woring temperature T = 27, te calibration pressure P= MPa, te system detects te voltage values u=62.335mv,tae tese results into te above curve equation, get te pressure value at different interpolation points.ten use cubic-spline interpolation to process te value of pressure and temperature, te cubic-spline interpolation curve is sown in fig.2. Table 1: Experimental data Press-ure P/MPa t=0 t=10 t=20 t=40 t=60 U/mv U/mv U/mv U/mv U/mv Figure 2: Spline interpolation examples According to te aforementioned algoritm steps, te experimental data are processed. First, eeping te temperature constant, after curve fitting: t=10 :p=0.0001u u t=30 :p=0.7314u t=50 :p=0.7891u t=70 :p=0.8432u Tae te actual temperature T = 27 into te function above, can compute te pressure value is P=19.994Mpa.Te actual input pressure value 20MPa at tis time, te measurement error relative to scale span is (Δp/pFS) 100%= (0.006 Mpa /60 Mpa) 100%=0.01%.

5 From te value can find out tat te measurement error of system is small at tis point, and it can eliminate interference caused by temperature well. Te ig pressure sensor used in tis experimental is oneywell 13mm series wic owns well linear caracteristics (te scale span of it is 60MPa).Tis algoritm can develop te caracteristic effectively. According to te calculation, te scale span integrated error compensated by temperature in te laboratory is (0.021/60) 100%=0.035%.Te results sow tat after using te combining metod of curve fitting and cubic-spline interpolation, te performance of te system improve significantly. Meanwile, we can troug increasing te mared point pressure and temperature to improve te performance furter. After curve fitting eeping te temperature constant, it as a good linear caracteristic. We tested at te temperature points: 和 Te pressure troug temperature compensation is sown in Table Table 2: Te pressure troug temperature compensation P/MPa U/mv U/mv U/mv U/mv Table 3 gives te overall fusion effect of te algoritm performance test wen te mared point pressure is 10, 20, 30, 40, 50 and te temperature is 17.6, 23.3 and 32.5 (Test metod is same as above). Table 3: Te test result 5. Conclusion P/MPa Tis paper puts forward a novel temperature compensation metod wic combines Quadratic curve fitting and cubic-spline interpolation, it can realize ig precision temperature compensation wen te mared points is less, so tat it can reduce te sensor s calibration time and worload effectively. For obtaining iger precision requirement of te pressure sensor, in full condition of cost, ardware s compute power and running speed, troug te mared point metod can improve te system performance; tis metod as certain teoretical significance and engineering application value on solving te temperature compensation of ig precision pressure sensor. Acnowledgements Tis wor is supported by te Yanceng Institute of Tecnology Talents Project of Cina and te National Natural Science Foundation of Cina under Grant No References Guo B., 2010, Information fusion for tactile sensor array signals, Journal of Congqing University, 33(6), Handy M.J., Haase M., Timmermann D., 2002, Low energy adaptive clustering ierarcy wit deterministic cluster -ead selection, Mobile and Wireless Communications Networ.

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