Instrumentation & Measurement Magazine 26-1 - 47

Fig. 8. Gauss nonlinearity modeling by PCM, UBCM and BCM, when a=1, b=0 and c=0.1. (a) and (b) Objective functions comparison. (c) Comparisons of the fifth
ordered f, fp
, fub and fb. (d)-(f) The distribution of ep
, eub and eb under different n.
Strong Nonlinearity (c=0.1): To further study the nonlinear
modeling ability of PCM, UBCM and BCM, parameter c is reduced
to 0.1 to enhance the nonlinearity of the gauss function.
As shown in Fig. 8c, the smaller c could lead to a narrower
gauss wave. This means the local nonlinearity of the present
gauss function will be stronger than that of c=2. Fig. 8a reveals
that the Op
and Ob produce little decrement as n increases from
2 to 10. The decrement of the Oub is relatively larger than that of
Op and Ob
/Op
and Ob
/Op
could achieve a minimum of 0.161 and
. This means that for the present gauss nonlinearity,
the UBCM could achieve a better calibration accuracy by increasing
the model order n. From Fig. 8b, it could be observed
that Oub
0.968, respectively.
Comparing Fig. 8d, Fig. 8e, and Fig. 8f, it could be observed
that the residual calibration errors mainly occur around x=0.
As n increases from 2 to 10, ep
and eb
around x=0 does not show
an obvious decreasing trend. This phenomenon could also be
observed from Fig. 8c. This indicates that the PCM and BCM
are unable to model the local strong nonlinearity of the gauss
function. Fig. 8d, Fig. 8e, and Fig. 8f reveal that the UBCM
could exhibit an ideal local nonlinearity modeling ability by increasing
the model order n.
Modeling of the Nonlinear Broken Line with
Inflection Points
In this section, a nonlinear broken line with two inflection
points, which is characterized by parameters r and 0, is utilized
to evaluate the calibration ability of the PCM, UBCM
and BCM, as depicted in Fig. 9. This kind of nonlinearity is
quite common in the micro potentiometer as described. Due
to the fabrication error, the micro potentiometer often has an
inherent saturation nonlinearity. To make a comprehensive
February 2023
evaluation, two parameters, r and 0, respectively, are utilized
to adjust the distortion degree and shape of the nonlinear broken
line.
Weak Nonlinearity (r=0.25): Generally, the larger r denotes
the stronger local nonlinearity of the nonlinear broken line.
In this case, r and 0 is set as 0.25 and 75°, respectively, to evaluate
the calibration performance of PCM, UBCM and BCM.
The calibration results of 0=15° are displayed in Fig. 10. From
Fig. 10a it could be observed that Oub
Op
and Ob
are smaller than
under different model ordered n. This means that both the
UBCM and BCM are optimal calibration approachs for the
present nonlinear broken line calibration. Fig. 10b reveals
Fig. 9. Illustration of the nonlinear broken line with inflection points.
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Instrumentation & Measurement Magazine 26-1

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