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The development of intelligent sensors involves the design of reconfigurable systems capable of working with different input sensors signals. Reconfigurable systems should expend the least possible amount of time readjusting. A selfadjustment algorithm for intelligent sensors should be able to fix major problems such as offset, variation of gain and lack of linearity with good accuracy. This paper shows the performance of a progressive polynomial algorithm utilizing different grades of relative nonlinearity of an output sensor signal. It also presents an improvement to this algorithm which obtains an optimal response with minimum nonlinearity error, based on the number and selection sequence of the readjust points. In order to verify the potential of this proposed criterion, a temperature measurement system was designed. The system is based on a thermistor which presents one of the worst nonlinearity behaviors. The application of the proposed improved method in this system showed that an adequate sequence of the adjustment points yields to the minimum nonlinearity error. In realistic applications, by knowing the grade of relative nonlinearity of a sensor, the number of readjustment points can be determined using the proposed method in order to obtain the desired nonlinearity error. This will impact on readjustment methodologies and their associated factors like time and cost.
The development of intelligent sensors involves the use of selfadjustment algorithms that should be able to fix major problems such as offset, variation of gain and no linearity with good accuracy. Besides, it will be focused to simplify the calibration process.
The calibration stage is where many of the major problems begin because in the development and maintenance of measurement systems verification and readjustment processes are required. An accepted practice utilized in the past by measurement systems designers was to linearize the sensor output. The subject of linearization has been considered on different forms and stages, basically in the design of circuits with MOS and CMOS technologies [
Now, if the cost associated with the maintenance of the measurement systems is reviewed we can see that in a research done by Hutchins [
The development of intelligent sensors with readjustment capabilities is imperative in order to facilitate calibration while not increasing its costs. Today designers have different options for self readjustment sensors, some of which are artificial neural networks theory [
Several important works related to recursive algorithms that can be applied to the self readjustment of intelligent sensors exist, one of which is the progressive polynomial calibration method [
This paper presents the improvement of a progressive polynomial algorithm that facilitates the calibration process due to the selfadjustment of sensor. This proposal is based on the calibration method presented by Fouad [
One important point that needs clarification before proceeding is related to the meaning of the term “selfadjustment”. Adjustment is mainly concerned to the process of removing systematic errors in accordance with the definition in the Metrology and the International Vocabulary of Basic and General Terms in Metrology (VIM), ISO VIM [
The paper structure will be the following: the basic system design considerations are presented in Section 2. The improved polynomial progressive algorithm and its simulation results are described in Section 3. A practical implementation of an intelligent sensor with improved algorithm on small microcontroller (MCU) is showed in Section 4 and the tests and results are described in Section 5.
The term smart sensor was coined in the mid of 1980s [
This paper is focused on the compensation functionality. The compensation factor will be operating in three categories: Nonlinear compensation that linearizes the relationship between input and output, Crosssensitivity compensation due to ambient conditions and time based or long term drift compensation due to degradation of the sensor or its elements [
A previous step to the application of progressive polynomial calibration algorithm is the normalization of the input and output variables. For example, the output electrical signal
In most of the cases the input and output variables scales are different, and these variables need to be normalized. We suggest making the normalization of
After the normalization,
Once the input and output variables are normalized the four steps of the polynomial progressive polynomial method [
Based on the adjustment process of measuring systems [
With the first points
Using the points
For sensors with linear transfer function three foregoing steps are enough to selfreadjustment of the sensor, but if the transfer function is nonlinear, more coefficients
If the transfer function of the sensor is nonlinear, a new set of coefficients and functions are defined. The coefficients
Finally, the relative error metric,
Or another way to corroborate the method is by using the least mean square error, expressed by
Current literature qualifies this method just as a qualitative way; the criteria of how the sequence of readjustment points will be taken and its capability of quantitative means are not available. In the next section one proposal for effective algorithm evaluation with respect to the percent of nonlinearity of the input signal is presented, as well as the necessary modifications to improve the method to obtain the minimum relative error.
In the previous section a recursive linearization method was described. However, it is necessary to define its performance under a set of important characteristics. Some of these characteristics include: number of adjustment points required, time expended in the adjustment process, the minimum nonlinearity error desired, and the applicability of the method to any sensor.
For example, a criterion to choose the number of calibration points based on a scale region where the sensor will be used is described in [
The first step is to select a nonlinear function as the input to the linearization process. In this document an increasing exponential function defined by
The next step consists of selecting a set of adjustment points, with cardinality
Two sets of calibration points were randomly generated in order to consider two different cases. These sets are shown bellow:
and
In the experiment, the range of
According to these results, if five points of adjustment are used, then the method can compensate sensors with a maximum nonlinearity relative error under 21%, yielding a maximum nonlinearity relative error output lower than 1% as shown in
According to these results, if seven points of adjustment are taken then the method can compensate sensors with a maximum nonlinearity relative error under 32%, yielding a maximum nonlinearity relative error output lower than 1% as shown in
Another important observation from
In order to establish a systematic method to avoid all the subjective aspects of the original algorithm an improvement to the algorithm was performed. This is proposed herein through the analysis of the response of the polynomial progressive algorithm. Then, considering the adjustment input vector for
So a set of new vectors
The numbers of
As it can be noticed from
The method can be generalized as follows: with
The elements
Now, the
The following experiment was done using simulation software in order to evaluate the improved algorithm: First, five sets of readjustment points were taken; the difference between each set was the number of adjustment points
After the PVA, the new vectors
Keep in mind that
By analyzing the results of
Temperature measurement systems are commonly used in almost any process. A thermistor as temperature sensor was selected in the construction of a measurement system. The thermistors, besides having a diversity of applications, can be found in a great variety of ways, sizes and characteristics. In this case, the major characteristic that will be analyzed is the
It is clear that in practical cases, the sensor will be on a circuit converting temperature to voltage capability, for example, a tension divider circuit or a Wheatstone bridge. The intelligent sensor was designed on a small MCU. Its topology is shown in
Five thermistors of different values of
The input signal
This signal was supplied to the improved algorithm. The results are presented in the next section.
The proposed improved algorithm was computed and the performance of the improved algorithm was compared against a Honeywell temperature meter, number UDC3000 with a thermocouple type K, span of −29 to 538°C (−20 to 1,000°F) and accuracy of ± 0.02%, taking 25 measures from a range of 0 to 120°C in 5°C steps using an oven system to change the temperature. The results are shown in
The output of the improved algorithm compared with the target straight line can be seen in
The same process as described above was done for the other thermistors and the obtained results are shown in
The results indicate that the maximum nonlinearity relative error, for each case, is less than 1%. These results were generated from a quantitative analysis of IPPAPVA, therefore, yielding the certainty of confidence that it is the optimal solution.
Moreover, it is important to comment on the sensor resolution. The ADC determinates the sensor resolution and this is defined by:
In this paper an improved progressive polynomial algorithm to perform compensation in intelligent sensors was presented. The improved algorithm needs to be executed only the first time the sensor is calibrated. Using data from
The proposed method assures the optimal solution using few adjustment points, reducing the time spent in the adjustment of the calibration process and reducing the calibration cost as a consequence. This method can be used for any sensor, as long as, the percentage of nonlinearity is under 40% and if the desired error is less than 1%.
The algorithm can be easily programmed into a microcontroller because it consists of simple operations and only the coefficients from
This work was supported by the: Instituto Tecnológico de Chihuahua, Dirección General de Educación Superior Tecnológica y La Universidad Autónoma de Querétaro, México (Project No. 83208P).
Intelligence Sensors classification base on its functionalities.
Input signal used for algorithm evaluation.
Algorithm response with five points of adjustment.
Algorithm response with seven points of adjustment.
Percentage of nonlinearity error of PVA.
Flow chart of the PVA Algorithm.
Results of the improved polinomial progressive algorithm for selfadjustment.
Topology of the intelligent sensor.
Response of thermistors used in the temperature measurement system (Ro is the resistance of the thermistor at 25°C).
Feature of nonlinearity of the thermistors used in the temperature measurement system (Ro is the resistance of the thermistor at 25°C).
Improved algorithm performance with six readjustment points
Improved algorithm relative error with six readjustment points
Improved Algorithm evaluation results.
Thermistors Features  % max. initial error ε_{r}  Order of the

% max ε_{r6} of


 
13.20% 

0.043 a 0.0192  
22.47% 

0.099 a 0.1383  
21.79% 

0.155 a 0.1931  
18.02% 

0.099 a 0.091  
19.17% 

0.0493 a 0.0718 