CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 51, 2016 

A publication of 

 
The Italian Association 

of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Tichun Wang, Hongyang Zhang, Lei Tian 
Copyright © 2016, AIDIC Servizi S.r.l., 

ISBN 978-88-95608-43-3; ISSN 2283-9216 

A Temperature Compensation Algorithm Based on Curve 
Fitting and Spline Interpolation 

Shu-lan Xia*a, Ji-lin Wangb, Ru-gang Wangb, Li Zhaoc  
aCollege of Electrical Engineering, Yancheng Institute of Technology, Yancheng 224051, Jiangsu, PR China 
bCollege of Information Engineering, Yancheng Institute of Technology , Yancheng 224051, Jiangsu, PR China 
cCollege of Information Science and Engineering ,Southeast University, Nanjing 210096, Jiangsu, PR China 
xslnj@126.com 

The zero and sensitivity of piezo resistive pressure sensors will drift with temperature because of the 
temperature characteristics of the semiconductor, and it is the main factor causing the measurement error of 
pressure sensor. For high-precision pressure monitoring system, temperature drift has become an important 
obstacle to improve performance of the system, especially in the field of applications with large changes of 
ambient temperature. On the basis of analysing the advantages and disadvantages of a variety of temperature 
compensation methods, a temperature compensation method combined with polynomial curve fitting and 
three spline interpolations is proposed, and it can improve the performance of the system. 

1. Introduction 

The characteristic of piezo resistive pressure sensor is high sensitivity, dynamic response, high precision, and 
long working life; therefore it obtains more and more application in modern industry (Wang et al., 2007). It is a 
kind of pressure sensor using the piezo resistive effect of semiconductor materials. But because of the core 
component is piezo resistive diaphragm which  is sensitive when temperature changes, so it causes the 
sensor's zero point and sensitivity  changing when temperature drift. There are two kinds of common 
temperature compensation methods, hardware compensation method and software compensation method (Li 
et al., 2008; Palmer, 2006). The traditional compensate method according to the sensor's hardware 
characteristics has many faults, such as debug difficulties, poor universality and low accuracy and so on, it is 
detrimental to engineering(Ishigaki et al., 2007; Zhang, 2008; Wang et al., 2010). While the software 
compensation method can solve the above problems well, so it acquires more and more attention(Kowalski, 
1987; Pramanik, 2006; Guo, 2010). This paper uses software compensation method mainly, it adopt the 
appropriate sensor information fusion algorithm to compensate the influence of pressure sensor causing by 
the change from temperature, and puts forward a kind of temperature compensation method which combines 
polynomial curve fitting and cubic-spline interpolation, this method is flexible, and also can achieves high 
precision. 

2. High precision pressure detection system based on software compensation 

The method of software compensation combines the microprocessor and pressure sensor, it can make full 
use of microprocessor, and through compensation algorithms to correct the errors which caused by 
temperature (Handy et al., 2002; Li et al., 2009). Regardless using what type of software compensation 
method, the hardware structure is similar, the differences lies in algorithm (Zhao et al., 2005; Wu, 2007). The 
high precision pressure detecting system which we designed used the high performance micro-controller 
LPC2478 as its central processor, the processor based on the ARM architecture, its working frequency is 
72MHz, and it owns 512K flash memory and some serial interface. Meanwhile, the system using a AD which 
owns high sampling accuracy and also a piece of SDRAM. The System block diagram is shown in fig.1. 

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1651225

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Xia S.L., Wang J.L., Wang R.G., Zhao L., 2016, A temperature compensation algorithm based on curve fitting and 
spline interpolation, Chemical Engineering Transactions, 51, 1345-1350  DOI:10.3303/CET1651225   

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Figure 1: The System block diagram 

3. Algorithm analysis and the implementation  

At present, the software compensation method has interpolation method, curve/surface fitting method, 
querying table method and the method using BP neural network (Wu et al., 2012). In the interpolation method 
(Li et al., 2015), first it requires that the data is correct, and also requires knowing the relationship between 
dates. The curve fitting method tries to find a smooth curve, it can fit data’s well, but not necessary to pass any 
data. Querying table method is a method which must filling a series of parameters into a parameter list, after 
obtaining the measurements, process dates according to the relative parameters in the table. Querying table 
method requires large storage space, so it does not suit for microprocessor. Neural network method is a 
method that through establishing the artificial neural network model, and also through the sample to train the 
network parameters, but the   largest shortcoming is the instability of network, and the long training time (Ren 
et al., 2014). This paper puts forward a method combined with curve fitting and cubic spline interpolation 
compensation algorithm; this algorithm can increase the compensation performance significantly. The 
following content wants to explain the theory of polynomial fitting and cubic-spline interpolation firstly. 

3.1 Polynomial curve fitting 
In the curve fitting method, polynomial fitting is the most appropriate method, because this method with easy 
calculation, and well-fitting effect (Tang et al., 2014). According to the given data (xi, yi), i=1, 2, …, m, 
polynomial fitting is the method which make a polynomial : 

 
k

n

k

k
xay 




0                                                                                                                                                

    (1)
 

 Let the squares error is minimum, i.e.  

min
1

2

0









  

 

m

i

n

k

k

iki
xayE                                                                                                                  (2) 

Therefore let: 

0




k
a

E
(k=1,2,…,m)                                                                                                                                       (3) 

Then: 

0 0 0

( )
n m m

j k j

i k i i

k i i

x a x y


  

            j=1, 2, … , n                                                                                     (4) 

Formula 1 can be unfolded as linear equations about coefficient ai, if this equation has a unique solution; the 
unique solution is the coefficient of the fitting polynomial. 

3.2 3-spline Interpolation 
Spline interpolation is a kind of improved subsection interpolation, it structures spline function in every 
adjacent area, meanwhile in order to guarantee the continuity at this node, it requires 2-order smoothness at 
the point, and namely it owns continuous 2-order derivatives.  
The function of 3-spline interpolation is defined as follows (Wang et al., 2007). 

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Assuming that the given interval [a, b] has n+1 nodes, and a=x0<x1<…<xn=b. If the function s(x) is a 
polynomial which order is no more than three in any interval [xk,xk+1],and s(x) owns continuous 2-order 
derivatives in the interval [a, b],So  can define s(x)  is cubic spline function of nodes x0,x1, …,xn. 
According to the above definition , s" is a linear function in every adjacent area [xk,xk+1].If making the 
s"(xk)=mk(k=1,2,…,n), s"(x) can be expressed as the next formula in area [xk,xk+1]. 

 
k

k

k

k

k

k

n

h

xx
m

h

xx
mxs











1

1                                                                                                  (5) 

In order to derivation conveniently, let hk=xk+1-xk. Then integral to formula (5) two times, so 

 
   

 
kkk

k

k
k

k

k
k

BxxA
h

xx
m

h

xx
mxs 










66

3

1

3

1                                (6) 

Take the value of xk, xk+1into the above formula, and then get the expression of Ak and Bk: 

 
kk

k

k

kk

k
mm

h

h

yy
A 








1

1

6
                                                                                   (7) 

6

2

k
kkk

h
myB 

                                                                                                                               (8) 

From formula (6) (7) (8) can find out that if the value of mk (k=1, 2, … , n) is known , the value of s(x)  can find 
out conveniently.  
The solution of  mk is  using the continuity characteristics of the nodes defined above, i.e. s´+(xk)=s´_(xk),And 
add the two boundary constraint conditions of endpoints, refer to the reference of Numerical Analysis(Yanfeng 
Zhang et al., 2007) ,can get the equations which can solve by chase-after method , it can reduce the 
computation greatly. 

3.3 Temperature compensation model and algorithm 
Pressure sensor is generally calibrated under standard temperature and standard pressure pj (j=1, 2, …, m) 
(Li et al., 2015), the number of marked point is n×m. Through the appropriate algorithm can obtain the relation 
among (ti, pj, uij) which comes from actual measurement processing, it is p=f (t,u).Here the parameter u is the 
output voltage of pressure sensor.  
If  the measurements Δpmax=max(Δp1, … , Δpj, …,Δpm) ,j=1,2, …,m ,according to p=f(t,u),while the true 
value is pj,so the measurement error is Δpj=∣p´j-pj∣, j=1,2, …,m. In the whole measurements, the maximum 
error of pressure sensor is Δpmax, so the composition error which relatives to the scale span pFS is 
ξ=(Δpmax/pFS)×100％.  
According to this temperature compensation model, and combining with curve fitting and cubic-spline 
interpolation, the procedure of the temperature compensation algorithm is as follows:  
Step 1 , fixed temperature t1 as a constant, doing a polynomial fitting with(u12,p1), (u22,p2) , …, 
(um2,pm),then can get the relation curve between p and u under temperature t1,p=f1(u).The order of  the 
polynomial is decided by measured data ,and usually using quadratic polynomial fitting ,it can also use linear 
fitting if the preference of the sensor is well, quadratic polynomial fitting can be realized through  Matlab .  
Step 2 similarly, fixed temperature t2 as a constant, then can get the relation curve between p and u under 
temperature t2, p= f2 (u). The rest can be done in the same manner, until fixed temperature tn as a constant, 
then can get the relation curve between p and u under temperature tn, p=fn (u). 
Step 3 take the actual measurement voltage u into p=f1 (u), p=f2 (u), ...  , p=fn(u), find out the pressure value 
under the standard temperature. 
Step 4 process the above pressure value and standard temperature using cubic spline interpolation, then 
obtain the relation curve between pressure p and temperature t, it is p=g(t), The program of cubic spline 
interpolation realizes on the ARM platform using C programming language. 
Step 5 Taking the real-time temperature into formula p=g (t), then will get the pressure value without 
temperature affects through the information fusion technology. 

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4. Experimental process and result analysis 

Table 1 gives the output voltage value of sensor when the marked point pressure is 0, 5, 10, 20, 30, 40, 60 
and the temperature is 0 ℃, 10 ℃, 20 ℃, 40 ℃and 60 ℃.  
According to the algorithm steps above , processing the experimental data , i.e. fix the temperature ,and then 
process p, u using 2-order fitting, On this basis, using cubic-spline interpolation to process the temperature 
and the pressure, fusion the information of the pressure and the temperature, so can get the correct pressure 
results. In the actual working temperature T = 27 ℃, the calibration pressure P=20.000 MPa, the system 
detects the voltage values u=62.335mv,take these results into the above curve equation, get the pressure 
value at different interpolation points .Then use cubic-spline interpolation to process the value of pressure and 
temperature, the cubic-spline interpolation curve is shown 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 
0 6.969 6.110 5.759 4.478 2.468 
5 23.215 22.363 21.879 20.811 19.147 
10 39.081 38.243 37.879 36.861 35.507 
15 54.874 54.040 53.815 53.041 51.745 
20 70.732 69.884 69.682 68.972 68.046 
25 86.514 85.672 85.496 84.836 83.966 
30 102.253 101.551 101.242 100.632 100.112 
35 118.046 117.214 116.930 116.233 115.772 
40 133.899 133.075 132.561 131.904 131.163 
45 149.298 148.624 148.123 147.745 147.046 
50 165.041 164.383 163.642 163.301 162.861 
55 180.857 180.019 179.095 178.96 178.440 
60 196.212 195.383 194.504 194.404 193.924 

 

Figure 2: Spline interpolation examples 

According to the aforementioned algorithm steps, the experimental data are processed. First, keeping the 
temperature constant, after curve fitting: 
t=10℃:p=0.0001u2+0.7412u-4.2437 
t=30℃:p=0.7314u-1.7433 
t=50℃:p=0.7891u-1.6154 
t=70℃:p=0.8432u-1.6014 
Take the actual temperature T = 27℃ into the function above, can compute the pressure value is 
P=19.994Mpa.The actual input pressure value 20MPa at this time, the measurement error relative to scale 
span is (Δp/pFS) × 100％= (0.006 Mpa /60 Mpa) ×100％=0.01％. 

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From the value can find out that the measurement error of system is small at this point, and it can eliminate 
interference caused by temperature well.  
The high pressure sensor used in this experimental is honeywell 13mm series which owns well linear 
characteristics (the scale span of it is 60MPa).This algorithm can develop the characteristic effectively. 
According to the calculation, the scale span integrated error compensated by temperature in the laboratory is 
(0.021／60)×100％=0.035％.The results show that after using the combining method of curve fitting and 
cubic-spline interpolation, the performance of the system improve significantly. Meanwhile, we can through 
increasing the marked point pressure and temperature to improve the performance further. 
After curve fitting keeping the temperature constant, it has a good linear characteristic. We tested at the 
temperature points: 8.2℃、4.6℃、22.3℃和 36.5℃. The pressure through temperature compensation is shown 
in Table.2 

Table 2: The pressure through temperature compensation 

P/MPa 
8.2℃ 4.6℃ 22.3℃ 36.5℃ 
U/mv U/mv U/mv U/mv 

0 9.99 9.994 0.02 9.992 
5 5.006 4.994 5.002 4.994 
20 9.993 9.993 20.02 9.996 
25 25.009 25.02 24.983 25.005 
30 30.006 29.994 29.004 30.006 
35 34.992 34.994 35.008 35.02 
40 40.005 39.994 39.993 39.996 

Table 3 gives the overall fusion effect of the algorithm performance test when the marked point pressure is 10, 
20, 30, 40, 50 and the temperature is 17.6 ℃, 23.3 ℃and 32.5 ℃(Test method is same as above). 

Table 3: The test result 

P/MPa 17.6℃ 23.3℃ 32.5℃ 
10 10.012 10.992 10.001 
20 20.004 19.986 19.998 
30 30.005 30.002 29.993 
40 40.008 39.992 40.021 
50 49.997 49.994 50.004 

5. Conclusion 

This paper puts forward a novel temperature compensation method which combines Quadratic curve fitting 
and cubic-spline interpolation, it can realize high precision temperature compensation when the marked points 
is less, so that it can reduce the sensor’s calibration time and workload effectively. For obtaining higher 

precision requirement of the pressure sensor, in full condition of cost, hardware’s compute power and running 
speed, through the marked point method can improve the system performance; this method has certain 
theoretical significance and engineering application value on solving the temperature compensation of high 
precision pressure sensor. 

Acknowledgements 

This work is supported by the Yancheng Institute of Technology Talents Project of China and the National 
Natural Science Foundation of China under Grant No. 60975017. 

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