Instrumentation & Measurement Magazine 26-1 - 42

Unconstrained Bezier Calibration
Method for Nonlinear Measurement
Calibration Applications:
A Comparison Study
Hongliang Hua, Jingbo Zhao, Zhenqiang Liao, and Yongjiang Chen
T
he primary motivation of this paper is to present
an accurate and universal method for the nonlinear
system modeling. An Unconstrained Bezier
Calibration Method (UBCM) is presented. By constraining
the boundary freedom of the UBCM, it could become a constrained
form called the Bezier Calibration Method (BCM),
which is suitable for situations with an accurate boundary
modeling requirement. A comparison study is performed
to compare the presented method with the most widely
utilized Polynomial Calibration Method (PCM) through
several nonlinear behavior modeling examples, including
sensor inherent nonlinearity calibration, sine wave, gauss
nonlinearity and nonlinear broken line. A comparison study
demonstrates that in the present verification examples the
UBCM has better calibration performance than that of the
BCM and PCM.
Calibration Modeling
It is well known that most sensors have an inherent nonlinearity
which could affect their measurement accuracy and
subsequently their control performance. Nonlinearity modeling
and calibration are an essential requirement for a nonlinear
measurement system [1], [2].
Nonlinear Modeling Method
Currently, several nonlinearity calibration methods have been
developed, such as Polynomial Calibration Method (PCM)
[3]-[5], neural network [6] and fuzzy logic calibration methods
[7]. Among these methods, the PCM is the most widely
utilized and has been applied in various fields such as sensor
nonlinearity compensation [3], [8], [9], information fusion
[10], and approximate modeling [11]-[15]. However, due to
the Runge phenomenon of the PCM [16]-[18], there often exists
a fluctuation error over the whole modeling range. The
Runge phenomenon is quite common in the modeling of a system
with strong nonlinearity, such as sinusoidal, Z type and
other nonlinear types [19], [20]. A variety of work has been
devoted to defeating the Runge phenomenon to improve the
modeling accuracy of the PCM [21], [22] in nonlinear system
42
applications. How to develop an accurate and universal calibration
method for the nonlinear system modeling is still a
challenging problem in the nonlinear measurement community
[23], [24].
Research Motivation
The primary motivation of this paper is to present an accurate
and universal calibration method for the nonlinear system
modeling. An Unconstrained Bezier Calibration Method
(UBCM) is presented, which is a more general form of our
previously presented BCM model [20]. In the UBCM, a series
of handle points and local weight coefficients are utilized
to control the local modeling features. As a result, the UBCM
could exhibit a strong nonlinearity modeling ability. A series
of nonlinear models are utilized to quantitatively evaluate the
calibration performance of the proposed UBCM and make a
comparison with the previous BCM and PCM. The main contributions
of this paper include:
◗ An Unconstrained Bezier Calibration Method (UBCM)
is presented for nonlinear system modeling. The UBCM
could achieve a nonlinear approximation of a set of
discrete data globally, which has a broad application prospects
such as sensor inherent nonlinearity calibration,
system modeling and dynamic control [25].
◗ A sensor inherent nonlinearity calibration example is
given to exhibit the effectiveness of the proposed UBCM
in reducing measurement error and enhancing the
measurement accuracy of a linear potentiometer.
◗ A comprehensive comparison of the UBCM, BCM and
PCM is performed to provide a reference for engineering
applications. In the comparison study, some typical
nonlinear functions including sine wave, gauss nonlinearity
and nonlinear broken line, are tested to evaluate
the nonlinear modeling performance of the UBCM, BCM
and PCM.
Calibration Method
In this case, we consider x and f as the calibrated data and reference
value, respectively.
IEEE Instrumentation & Measurement Magazine
1094-6969/23/$25.00©2023IEEE
February 2023
Instrumentation & Measurement Magazine 26-1

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