Adaptive Fuzzy Modal Matching of Capacitive Micromachined Gyro Electrostatic Controlling

Abstract

A fuzzy PI controller was utilized to realize the modal matching between a driving and detecting model. A simulation model was built to study electrostatic decoupling controlling technology. The simulation results show that the modal matching can be gained by the fuzzy PI controller. The frequency difference between the driving mode and the detection mode is less than 1 Hz, and the offset of the input DC is smaller than 0.6 V. The optimal proportionality factor and integral coefficient are 1.5 and 20, respectively. The fuzzy PI controlling technology provides a good way for the parameter optimization to gain modal matching of micro gyro, via which the detecting accuracy and stability can be improved greatly.

Full text

Citation: Cheng, L.; Liu, R.; Guo, S.;
Zheng, G.; Liu, Y. Adaptive Fuzzy
Modal Matching of Capacitive
Micromachined Gyro Electrostatic
Controlling. Sensors 2023,23, 7422.
https://doi.org/10.3390/s23177422
Academic Editors: Olutunde Oyadiji
and Erwin Peiner
Received: 12 June 2023
Revised: 27 July 2023
Accepted: 23 August 2023
Published: 25 August 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
sensors
Article
Adaptive Fuzzy Modal Matching of Capacitive Micromachined
Gyro Electrostatic Controlling
Li Cheng 1, Ruimin Liu 2, Shumin Guo 3, Gaofeng Zheng 2and Yifang Liu 2,*
1Department of Aeronautical and Aviation Engineering, Hong Kong Polytechnic University, Hong Kong;
22043547g@connect.polyu.hk
2Pen-Tung Sah Institute of Micro-Nano Science and Technology, Xiamen University, Xiamen 361102, China;
wudideyueqiu@stu.xmu.edu.cn (R.L.); zheng_gf@xmu.edu.cn (G.Z.)
3School of Mathematical Sciences, Xiamen University, Xiamen 361002, China; shumin_guo@xmu.edu.cn
*Correspondence: yfliu@xmu.edu.cn; Tel.: +86-592-2185927
Abstract:
A fuzzy PI controller was utilized to realize the modal matching between a driving
and detecting model. A simulation model was built to study electrostatic decoupling controlling
technology. The simulation results show that the modal matching can be gained by the fuzzy PI
controller. The frequency difference between the driving mode and the detection mode is less than
1 Hz, and the offset of the input DC is smaller than 0.6 V. The optimal proportionality factor and
integral coefficient are 1.5 and 20, respectively. The fuzzy PI controlling technology provides a good
way for the parameter optimization to gain modal matching of micro gyro, via which the detecting
accuracy and stability can be improved greatly.
Keywords:
capacitive micro-machined gyro; electrostatic control; modal matching; numerical
simulation; feedback control
1. Introduction
Micro-electromechanical system technology (MEMS) is an important technology for
microsystem integration and has been applied in various fields, such as high-precision
sensing [
1
,
2
], automobile [
3
], electronic information [
2
,
4
], biomedical [
5
], aerospace [
6
],
and so on. There are also some novel methods that have been invented to improve the
sensing performance of gyros. Mao et al. [
7
] studied the mode-matching and Sagnac effect
in a millimeter-scale-wedged resonator gyroscope. Xue et al. [
8
] invented the all-polymer
monolithic resonant-integrated optical gyroscope. Wang et al. [
9
] utilized a resonator
with spun fiber to suppress the Kerr-effect-induced error in resonant fiber optic gyros.
Feng et al. [
10
] employed a Si
3
N
4
resonator to improve the long-term temperature bias
stability of integrated optical gyroscopes.
The capacitive micro gyro detecting angular velocity based on the Coriolis Effect has
become the most widely used angular velocity micro-sensor device with the advantages
of low manufacturing cost, good reliability, and easy integration [
11
–
13
]. However, the
resonant frequency of the gyro detection mode will drift because of process errors and
ambient temperature. So, the modal frequency between the driving mode and the detection
mode should be matched, which is the key that defines the sensitivity and stability of
micro-mechanical gyros [14,15].
Presently, there are several modal matching controlling methods, including the quality
adjustment method, material characteristic adjustment method, and stress adjustment
method [
16
]. The quality adjustment method that was designed based on theoretical anal-
ysis and process experience changes the resonant frequency by changing the resonator
quality or mass compensation [
17
]. The quality adjustment method is easily disturbed by
other factors, which limits the overall performance improvement of the gyro. The material
Sensors 2023,23, 7422. https://doi.org/10.3390/s23177422 https://www.mdpi.com/journal/sensors

Sensors 2023,23, 7422 2 of 12
property adjustment method changes the resonant frequency by depositing different mate-
rials on the surface of the resonator [
18
,
19
]. The materials deposition process was complex,
and it was difficult to gain the suitable thickness and quality of the material deposition
layer; therefore, the matching performance could not be realized easily. The methods of
quality adjustment and the material property adjustment are offline control methods and
are not suitable for long-term stable control. The above methods need to be implemented
during the design and manufacturing process; therefore, they are difficult to achieve and
expensive. The stress adjustment method is an online control method and has become the
most commonly used method, which does not require implementation during the design
and manufacturing process of micro gyros [20,21].
The stress adjustment methods are divided into the temperature adjustment method
and the external force adjustment method. The temperature adjustment method changes
the resonant frequency by heating the micro-vibration structure locally [
22
], which has the
disadvantages of poor control ability and high power. It is difficult to apply to different
material structures due to the different temperature-sensitive characteristics. The external
force control method changes the resonant frequency using residual internal stress [
23
],
electric field force, and magnetic field force [
24
]. Compared with other control technologies,
the external force adjustment method is an online control technology, which is beneficial to
the sensitivity and stability of the micro-mechanical gyro system with its high flexibility
and strong universality [25].
The static control method changes the resonant frequency by introducing an elec-
trostatic force on the harmonic oscillator to achieve modal matching control, and it has
become the method with the most development potential in the external force adjustment
method [
26
]. There is a difference between the resonant frequencies of the two directions be-
cause of the interference of the manufacturing process and the environment. The feedback
control system can be built based on the resonant frequency difference between the two
directions. The system adjusts the DC offset acting on the capacitor resonator to change the
resonant frequency to achieve modal matching to eliminate resonant errors. Static control
is an easily achieved online control method, which is helpful in speeding up the response
to noise interference and improving detection accuracy. Electrostatic control technology
has become the optimal choice for modal matching control of micro gyro systems [27].
In this paper, we analyze the theoretical derivation in the process of electrostatic
regulation of micro-capacitive gyros and build the modal matching system. We build the
numerical simulation model based on the Simulink module of MATLAB software (R2018a)
to achieve the modal matching of gyro and analyze the controller parameters. The influence
of the quality factor on the control process is researched. And the signal evolution law of
the modal matching process of the capacitive micro gyro is simulated.
2. Analysis of Electrostatic Control
It is shown in Figure 1a that the capacitive micro-mechanical gyro is mainly composed
of an inertial mass, internal and external frames, elastic elements, and a comb-tooth ca-
pacitor structure. The inertial mass has two degrees of freedom, and the Coriolis force
is transmitted to the comb capacitor structure through the elastic element when the gyro
vibrates. It is the particularity of the comb capacitor structure that makes both the driving
mode (vibration in the X direction) and detection mode (vibration in the Y direction) have
only one degree of freedom. The comb-tooth capacitor structure is shown in Figure 1b,
which includes a fixed plate and a movable plate. A sinusoidal signal is added to the fixed
plate, and a DC bias signal is added to the movable plate to drive the gyro. When the gyro
is in resonance, the DC signal and high-frequency signal are filtered out by the band-pass
filtering function of the structure of the internal phase-locked loop. The equations of driving
can be simplified to:
Fx=∂E
∂x=∂C
∂xVAsin(ωt)(1)

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where Eis the electric field strength between the capacitor plates, Cis the capacitance value,
Vis the DC bias voltage value applied to the capacitor plates, and Ais the amplitude of the
sinusoidal signal.
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

󰇛󰇜
(1)
where E is the electric eld strength between the capacitor plates, C is the capacitance
value, V is the DC bias voltage value applied to the capacitor plates, and A is the amplitude
of the sinusoidal signal.
123
4 5 6
x
y
(a) (b)
Fixed Electrode
Dgap
Dantigap
Movable Electrode
Y
X
Figure 1. The structural features of capacitive micro gyro: (a) the structure of capacitive micro gyro;
(b) the comb-tooth capacitance structure.
While applying the voltage to the comb capacitor structure, there is a slight force in
the detection direction due to the particularity of its structure. We can obtain the detection
mode electrostatic negative stiness equation as follows:


 


󰇛󰇜
(2)
where N is the number of comb-tooth capacitor groups. Here, , H and L are the parame-
ters of the capacitor plate of dielectric constant, thickness, and length. Then, Dantigap and
Dgap are the structural parameters of the comb-tooth capacitor in Figure 1b, and V is the
DC bias voltage applied by the capacitor plate. Denote the product of the above parame-
ters and coecients with K. The design of the mechanical structure eliminates the modal
coupling between the driving direction and the detection direction. However, there are
orthogonal errors between the two resonance directions due to the dierences in the man-
ufacturing process [16]. In this paper, we build a modal matching closed-loop control sys-
tem based on the feedback of the phase between the quadrature error signal and the driv-
ing signal. The system eliminates static errors and improves detection accuracy by the PI
controller.
When the micro-mechanical gyro vibrates, both the driving mode and the detection
mode can be equivalent to a second-order spring damping system, from which the reso-
nant angle frequency of the detection mode can be calculated:



(3)
where ke is the eective static stiness of the detection mode, ms is the equivalent mass of
the detection mode, and km is the elastic coecient of the spring damping system equiv-
alent to the detection mode. A closed-loop control system of modal matching based on the
characteristic relationship between the input DC bias voltage and the detection terminal
signal is built in Equation (3). By adjusting the magnitude of the input DC oset, the res-
onant frequency between the driving mode and detection mode can be matched, realizing
the modal matching of the system stably.
Figure 1.
The structural features of capacitive micro gyro: (
a
) the structure of capacitive micro gyro;
(b) the comb-tooth capacitance structure.
While applying the voltage to the comb capacitor structure, there is a slight force in
the detection direction due to the particularity of its structure. We can obtain the detection
mode electrostatic negative stiffness equation as follows:
ke=∂F
∂y=−1
2NεHL D3
antiga p −D3
gap
Dantiga p ·Dga p)3V2=KV2(2)
where Nis the number of comb-tooth capacitor groups. Here,
ε
,Hand Lare the parameters
of the capacitor plate of dielectric constant, thickness, and length. Then, D
antigap
and D
gap
are the structural parameters of the comb-tooth capacitor in Figure 1b, and Vis the DC bias
voltage applied by the capacitor plate. Denote the product of the above parameters and
coefficients with K. The design of the mechanical structure eliminates the modal coupling
between the driving direction and the detection direction. However, there are orthogonal
errors between the two resonance directions due to the differences in the manufacturing
process [
16
]. In this paper, we build a modal matching closed-loop control system based on
the feedback of the phase between the quadrature error signal and the driving signal. The
system eliminates static errors and improves detection accuracy by the PI controller.
When the micro-mechanical gyro vibrates, both the driving mode and the detection
mode can be equivalent to a second-order spring damping system, from which the resonant
angle frequency of the detection mode can be calculated:
ωs=ske f f
ms
=skm+KV2
ms(3)
where k
eff
is the effective static stiffness of the detection mode, m
s
is the equivalent mass
of the detection mode, and km is the elastic coefficient of the spring damping system
equivalent to the detection mode. A closed-loop control system of modal matching based
on the characteristic relationship between the input DC bias voltage and the detection
terminal signal is built in Equation (3). By adjusting the magnitude of the input DC offset,
the resonant frequency between the driving mode and detection mode can be matched,
realizing the modal matching of the system stably.
3. Construction of the Modal Matching System
Then, a matching controlling system with voltage has been built to study the modal
matching controlling technology with a fuzzy PI controller. The block diagram of the modal
matching control system is shown in Figure 2.

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3. Construction of the Modal Matching System
Then, a matching controlling system with voltage has been built to study the modal
matching controlling technology with a fuzzy PI controller. The block diagram of the
modal matching control system is shown in Figure 2.
Figure 2. Block diagram of the modal matching control system.
The gyro is driven by the constant amplitude and constant frequency signal and the
lower DC bias signal. And the signal of the drive module and the detection module of the
gyro are amplied through the front transimpedance amplier and trans-capacity ampli-
er, respectively. According to the principle of synchronous demodulation, the demodu-
lated signal is the strongest when the two signals are in the same phase as each other [17].
Therefore, we add a phase shift module before signal demodulation to shift the phase 90°
according to theoretical calculations. The phase dierence of 90° is a good way to make
sure the driving modal is perpendicular to the detecting modal and achieve an accurate
detection. The signals reach the phase detector, which consists of a multiplier and a low-
pass lter, and nally input the PI controller:
󰇛󰇜


󰇛󰇜
(4)
where x is the coecient of the driving mode, y is the coecient of the detection mode, Rf
is the value of the transimpedance amplier,  is the sinusoidal value of the quadra-
ture error angle, d is the parameters of the capacitive plate. We can further calculate:
 

󰇛󰇜



(5)
when two angular frequencies of both the driving modal and the detection modal are
equal, the modal matching is realized, and it is calculated according to the above formula:


(6)
The modal matching is realized when Verror(t) is zero. The PI controller outputs a DC
oset signal, which is superimposed on the initial input DC oset signal and then used as
the input of the DC oset signal of the gyro. When it is detected that the input of the PI
controller is zero, the system reaches modal matching, and the phase dierence between
the signal of the driving modal and the detected modal is 90°. The modal matching system
adjusts the amplitude of the input DC oset signal and uses PI control to adjust the DC
oset signal within 0.6 V accurately.
Figure 2. Block diagram of the modal matching control system.
The gyro is driven by the constant amplitude and constant frequency signal and the
lower DC bias signal. And the signal of the drive module and the detection module of
the gyro are amplified through the front transimpedance amplifier and trans-capacity
amplifier, respectively. According to the principle of synchronous demodulation, the
demodulated signal is the strongest when the two signals are in the same phase as each
other [
17
]. Therefore, we add a phase shift module before signal demodulation to shift the
phase 90
◦
according to theoretical calculations. The phase difference of 90
◦
is a good way
to make sure the driving modal is perpendicular to the detecting modal and achieve an
accurate detection. The signals reach the phase detector, which consists of a multiplier and
a low-pass filter, and finally input the PI controller:
Verror (t) = −
mdRfCxCyV2A2
xAyω3
xsin θ
Cmd2cos(δ2)(4)
where xis the coefficient of the driving mode, y is the coefficient of the detection mode, R
f
is the value of the transimpedance amplifier,
sin θ
is the sinusoidal value of the quadrature
error angle, dis the parameters of the capacitive plate. We can further calculate:
δ2=arctan cyωx
ky−msω2
x
=arctan2ζωx/ωs)2
1−ω2
x
ω2
s
(5)
when two angular frequencies of both the driving modal and the detection modal are equal,
the modal matching is realized, and it is calculated according to the above formula:
δ2=π
2(6)
The modal matching is realized when V
error
(t) is zero. The PI controller outputs a DC
offset signal, which is superimposed on the initial input DC offset signal and then used as
the input of the DC offset signal of the gyro. When it is detected that the input of the PI
controller is zero, the system reaches modal matching, and the phase difference between
the signal of the driving modal and the detected modal is 90
◦
. The modal matching system
adjusts the amplitude of the input DC offset signal and uses PI control to adjust the DC
offset signal within 0.6 V accurately.
4. Simulation and Result Analysis
As shown in Figure 3, the simulation model is built in MATLAB Simulink based on
the dynamic characteristics of micro gyro.

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4. Simulation and Result Analysis
As shown in Figure 3, the simulation model is built in MATLAB Simulink based on
the dynamic characteristics of micro gyro.
Figure 3. Simulation model of modal matching system.
The system response characteristics of modal matching and the action rules of vari-
ous parameters are analyzed via simulation. The system realizes the control without static
error through the PI adjustment method, which improves the control precision of the mi-
cro gyro.
4.1. Analysis of Modal Matching Signal
The system simulation model optimizes the characteristics of signal transmission and
realizes the signal matching between the two modes through closed-loop control. The out-
put signals of driving and detecting in modal matching are shown in Figure 4, respec-
tively.
The detection signal is weak and has large uctuations when the micro gyro is not
modal matching. When the system realizes the mode matching, the signal of the detection
mode of the system gradually increases and tends to be stable through the adjustment of
the closed-loop control. There is a phase shift circuit in the control system, which can per-
form phase shift calculation on the driving signal and improve the sensitivity and sensing
accuracy of the signal identication. When the system realizes modal matching, the phase
dierence of the two signals is 90°.
The signals of the two modes before and after the mode matching are shown in Fig-
ure 4. At the time of start-up, the system is in a state of modal mismatch, and the phase
dierence between the signal of the driving modal and the detection modal is not 90°. The
system gradually realizes modal matching via closed-loop adjustment, in which the phase
dierence of the two signals is equal to 90°.
The output of the PI controller is not an ideal DC signal but a signal with a weak
sinusoidal ripple. The driving mode and the detection mode of the gyro are not ideal for
mode matching. There will be a frequency dierence between them, so the designed sys-
tem controls the dierence within 1 Hz. The amplitude range of the control sine ripple can
be calculated as follows:
𝜔d−𝜔s<2𝜋Δ𝑓
(7)
Figure 3. Simulation model of modal matching system.
The system response characteristics of modal matching and the action rules of various
parameters are analyzed via simulation. The system realizes the control without static error
through the PI adjustment method, which improves the control precision of the micro gyro.
4.1. Analysis of Modal Matching Signal
The system simulation model optimizes the characteristics of signal transmission and
realizes the signal matching between the two modes through closed-loop control. The out-
put signals of driving and detecting in modal matching are shown in Figure 4, respectively.
Sensors 2023, 23, x FOR PEER REVIEW 6 of 12
𝜔d=√𝑘y−𝐾𝑉2
𝑚s
(8)
𝜔s=√𝑘y−𝐾(𝑉+Δ𝑉)2
𝑚s
(9)
The amplitude range of the sine ripple is ΔV<0.0041 V. We optimize the parameters
of the PI controller to meet the system design accuracy.
Figure 4. Signal evolution process of system modal matching.
4.2. Optimization and Analysis of PI Control
The inuence of the proportional and integral parameters of the PI controller on the
modal matching process is analyzed. Under the premise of satisfying the design accuracy,
the parameters of the PI controller, which decides the range of the amplitude adjustment
of the system, are determined. The impact of the initial input DC oset on the entire sys-
tem is analyzed nally.
It is shown in Figure 5 that Ki is 20, and Kp is 0.8, 1.5 and 2.0. Here, Ki is the integral
control coecient, and Kp is the proportional control coecient for the PI controller, re-
spectively. As Kp increases, the stability time of the system decreases, the response speed
increases, and the overshoot of the system decreases. However, when Kp is 2.0, the system
oscillates greatly, which means that the excessive Kp will cause the system to oscillate,
resulting in system instability.
Figure 4. Signal evolution process of system modal matching.
The detection signal is weak and has large fluctuations when the micro gyro is not
modal matching. When the system realizes the mode matching, the signal of the detection
mode of the system gradually increases and tends to be stable through the adjustment
of the closed-loop control. There is a phase shift circuit in the control system, which can
perform phase shift calculation on the driving signal and improve the sensitivity and

Sensors 2023,23, 7422 6 of 12
sensing accuracy of the signal identification. When the system realizes modal matching,
the phase difference of the two signals is 90◦.
The signals of the two modes before and after the mode matching are shown in
Figure 4. At the time of start-up, the system is in a state of modal mismatch, and the phase
difference between the signal of the driving modal and the detection modal is not 90
◦
. The
system gradually realizes modal matching via closed-loop adjustment, in which the phase
difference of the two signals is equal to 90◦.
The output of the PI controller is not an ideal DC signal but a signal with a weak
sinusoidal ripple. The driving mode and the detection mode of the gyro are not ideal for
mode matching. There will be a frequency difference between them, so the designed system
controls the difference within 1 Hz. The amplitude range of the control sine ripple can be
calculated as follows:
ωd−ωs<2π∆f(7)
ωd=sky−KV2
ms(8)
ωs=sky−K(V+∆V)2
ms(9)
The amplitude range of the sine ripple is
∆V<
0.0041
V
. We optimize the parameters
of the PI controller to meet the system design accuracy.
4.2. Optimization and Analysis of PI Control
The influence of the proportional and integral parameters of the PI controller on the
modal matching process is analyzed. Under the premise of satisfying the design accuracy,
the parameters of the PI controller, which decides the range of the amplitude adjustment of
the system, are determined. The impact of the initial input DC offset on the entire system is
analyzed finally.
It is shown in Figure 5that K
i
is 20, and K
p
is 0.8, 1.5 and 2.0. Here, K
i
is the integral
control coefficient, and K
p
is the proportional control coefficient for the PI controller,
respectively. As K
p
increases, the stability time of the system decreases, the response speed
increases, and the overshoot of the system decreases. However, when K
p
is 2.0, the system
oscillates greatly, which means that the excessive K
p
will cause the system to oscillate,
resulting in system instability.
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𝜔d=√𝑘y−𝐾𝑉2
𝑚s
(8)
𝜔s=√𝑘y−𝐾(𝑉+Δ𝑉)2
𝑚s
(9)
The amplitude range of the sine ripple is ΔV<0.0041 V. We optimize the parameters
of the PI controller to meet the system design accuracy.
Figure 4. Signal evolution process of system modal matching.
4.2. Optimization and Analysis of PI Control
The inuence of the proportional and integral parameters of the PI controller on the
modal matching process is analyzed. Under the premise of satisfying the design accuracy,
the parameters of the PI controller, which decides the range of the amplitude adjustment
of the system, are determined. The impact of the initial input DC oset on the entire sys-
tem is analyzed nally.
It is shown in Figure 5 that Ki is 20, and Kp is 0.8, 1.5 and 2.0. Here, Ki is the integral
control coecient, and Kp is the proportional control coecient for the PI controller, re-
spectively. As Kp increases, the stability time of the system decreases, the response speed
increases, and the overshoot of the system decreases. However, when Kp is 2.0, the system
oscillates greatly, which means that the excessive Kp will cause the system to oscillate,
resulting in system instability.
Figure 5. The output signal of PI controller with different Kp.
As shown in Figure 6, as K
i
increases, the stability time of the system gradually
decreases, but the overshoot of the system increases too quickly. Increasing the value of K
i
as much as possible can reduce the stability time.

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Figure 5. The output signal of PI controller with dierent Kp.
As shown in Figure 6, as Ki increases, the stability time of the system gradually de-
creases, but the overshoot of the system increases too quickly. Increasing the value of Ki
as much as possible can reduce the stability time.
Figure 6. The output signal of the PI controller with dierent Ki parameters.
According to the simulation results, the optimal PI controller parameters are 1.5 for
Kp and 20 for Ki. At this point, the system can achieve a 0.6 V dierence adjustment.
Next, the impact of the input DC bias voltage on the system is analyzed. Take the
input DC oset voltage as 3.5 V, 3.7 V, and 3.9 V. As shown in Figure 7, the tunable voltage
of the system decreases, and the speed of system response is greatly accelerated with the
increase in DC oset voltage. However, the system oscillates strongly before the stability
stage with modal matching. The simulation results show that the system can achieve an
exact stable stage when the input DC oset voltage is 3.7 V.
Figure 7. The output signal of the PI controller with dierent DC bias voltages.
4.3. Analysis of the Performance of the Control System
The quality factor of capacitive micro-mechanical gyro will be reduced due to the
inuence of process and environment. According to Equation (10), we simulate the
changes in the parameters of the gyro to analyze the change in the quality factor.
󰇘󰇗
(10)


(11)
In Equation (11), Q is the quality factor. We analyze the situations where Q is 200,
2000, 10,000, and 20,000. The output signals of the driving modal and detecting modal
before modal matching are shown in Figure 8. As the quality factor decreases by order of
Figure 6. The output signal of the PI controller with different Ki parameters.
According to the simulation results, the optimal PI controller parameters are 1.5 for K
p
and 20 for Ki. At this point, the system can achieve a 0.6 V difference adjustment.
Next, the impact of the input DC bias voltage on the system is analyzed. Take the
input DC offset voltage as 3.5 V, 3.7 V, and 3.9 V. As shown in Figure 7, the tunable voltage
of the system decreases, and the speed of system response is greatly accelerated with the
increase in DC offset voltage. However, the system oscillates strongly before the stability
stage with modal matching. The simulation results show that the system can achieve an
exact stable stage when the input DC offset voltage is 3.7 V.
Sensors 2023, 23, x FOR PEER REVIEW 7 of 12
Figure 5. The output signal of PI controller with dierent Kp.
As shown in Figure 6, as Ki increases, the stability time of the system gradually de-
creases, but the overshoot of the system increases too quickly. Increasing the value of Ki
as much as possible can reduce the stability time.
Figure 6. The output signal of the PI controller with dierent Ki parameters.
According to the simulation results, the optimal PI controller parameters are 1.5 for
Kp and 20 for Ki. At this point, the system can achieve a 0.6 V dierence adjustment.
Next, the impact of the input DC bias voltage on the system is analyzed. Take the
input DC oset voltage as 3.5 V, 3.7 V, and 3.9 V. As shown in Figure 7, the tunable voltage
of the system decreases, and the speed of system response is greatly accelerated with the
increase in DC oset voltage. However, the system oscillates strongly before the stability
stage with modal matching. The simulation results show that the system can achieve an
exact stable stage when the input DC oset voltage is 3.7 V.
Figure 7. The output signal of the PI controller with dierent DC bias voltages.
4.3. Analysis of the Performance of the Control System
The quality factor of capacitive micro-mechanical gyro will be reduced due to the
inuence of process and environment. According to Equation (10), we simulate the
changes in the parameters of the gyro to analyze the change in the quality factor.
󰇘󰇗
(10)


(11)
In Equation (11), Q is the quality factor. We analyze the situations where Q is 200,
2000, 10,000, and 20,000. The output signals of the driving modal and detecting modal
before modal matching are shown in Figure 8. As the quality factor decreases by order of
Figure 7. The output signal of the PI controller with different DC bias voltages.
4.3. Analysis of the Performance of the Control System
The quality factor of capacitive micro-mechanical gyro will be reduced due to the
influence of process and environment. According to Equation (10), we simulate the changes
in the parameters of the gyro to analyze the change in the quality factor.
m..
y=Fqcos ωt−c.
y−ky (10)
c=√ksms
Q(11)
In Equation (11), Qis the quality factor. We analyze the situations where Qis 200,
2000, 10,000, and 20,000. The output signals of the driving modal and detecting modal
before modal matching are shown in Figure 8. As the quality factor decreases by order
of magnitude, the detection signal decreases, although the phase difference between the
driving signal and the detection signal is still maintained at 90
◦
. When Qis 200, the
detection signal is hardly amplified. Therefore, we analyze the accuracy of the system when
Qis 2000, 10,000, and 20,000. From Figure 9, we find out that as Qincreases, the fluctuation
of the outputs of the PI controller increases, and the accuracy of the system reduces.

Sensors 2023,23, 7422 8 of 12
Sensors 2023, 23, x FOR PEER REVIEW 8 of 12
magnitude, the detection signal decreases, although the phase dierence between the
driving signal and the detection signal is still maintained at 90°. When Q is 200, the detec-
tion signal is hardly amplied. Therefore, we analyze the accuracy of the system when Q
is 2000, 10,000, and 20,000. From Figure 9, we nd out that as Q increases, the uctuation
of the outputs of the PI controller increases, and the accuracy of the system reduces.
Figure 8. Output signals of the drive mode and detection mode with dierent Q.
According to the simulation, it can be concluded that if the quality factor drops too
much, it will aect the results of modal matching, and it will even aect the realization of
modal matching. But when the magnitude is too high, the accuracy of the control system
will decrease, which increases the diculty of modal matching and causes a large error.
Figure 9. Output of the PI controller with dierent Q values.
As shown in Figure 10a, the random angle signal is input into the system to explore
the results of modal matching. The modal matching process after adding the angular ve-
locity is shown in Figure 10b. It can be seen that the system response speed is fast after
adding the angular velocity input. According to the stabilized enlarged signal, the gyro
has achieved modal matching. The increase in the amplitude of the detected modal indi-
cates that the detection mode is working.
Figure 8. Output signals of the drive mode and detection mode with different Q.
Sensors 2023, 23, x FOR PEER REVIEW 8 of 12
magnitude, the detection signal decreases, although the phase dierence between the
driving signal and the detection signal is still maintained at 90°. When Q is 200, the detec-
tion signal is hardly amplied. Therefore, we analyze the accuracy of the system when Q
is 2000, 10,000, and 20,000. From Figure 9, we nd out that as Q increases, the uctuation
of the outputs of the PI controller increases, and the accuracy of the system reduces.
Figure 8. Output signals of the drive mode and detection mode with dierent Q.
According to the simulation, it can be concluded that if the quality factor drops too
much, it will aect the results of modal matching, and it will even aect the realization of
modal matching. But when the magnitude is too high, the accuracy of the control system
will decrease, which increases the diculty of modal matching and causes a large error.
Figure 9. Output of the PI controller with dierent Q values.
As shown in Figure 10a, the random angle signal is input into the system to explore
the results of modal matching. The modal matching process after adding the angular ve-
locity is shown in Figure 10b. It can be seen that the system response speed is fast after
adding the angular velocity input. According to the stabilized enlarged signal, the gyro
has achieved modal matching. The increase in the amplitude of the detected modal indi-
cates that the detection mode is working.
Figure 9. Output of the PI controller with different Q values.
According to the simulation, it can be concluded that if the quality factor drops too
much, it will affect the results of modal matching, and it will even affect the realization of
modal matching. But when the magnitude is too high, the accuracy of the control system
will decrease, which increases the difficulty of modal matching and causes a large error.
As shown in Figure 10a, the random angle signal is input into the system to explore the
results of modal matching. The modal matching process after adding the angular velocity
is shown in Figure 10b. It can be seen that the system response speed is fast after adding the
angular velocity input. According to the stabilized enlarged signal, the gyro has achieved
modal matching. The increase in the amplitude of the detected modal indicates that the
detection mode is working.
Sensors 2023, 23, x FOR PEER REVIEW 8 of 12
magnitude, the detection signal decreases, although the phase dierence between the
driving signal and the detection signal is still maintained at 90°. When Q is 200, the detec-
tion signal is hardly amplied. Therefore, we analyze the accuracy of the system when Q
is 2000, 10,000, and 20,000. From Figure 9, we nd out that as Q increases, the uctuation
of the outputs of the PI controller increases, and the accuracy of the system reduces.
Figure 8. Output signals of the drive mode and detection mode with dierent Q.
According to the simulation, it can be concluded that if the quality factor drops too
much, it will aect the results of modal matching, and it will even aect the realization of
modal matching. But when the magnitude is too high, the accuracy of the control system
will decrease, which increases the diculty of modal matching and causes a large error.
Figure 9. Output of the PI controller with dierent Q values.
As shown in Figure 10a, the random angle signal is input into the system to explore
the results of modal matching. The modal matching process after adding the angular ve-
locity is shown in Figure 10b. It can be seen that the system response speed is fast after
adding the angular velocity input. According to the stabilized enlarged signal, the gyro
has achieved modal matching. The increase in the amplitude of the detected modal indi-
cates that the detection mode is working.
Figure 10.
The input and output signal:(
a
) Input of angular velocity; (
b
) output signal of the control
system with load when modal matching.
5. Simulation of Fuzzy Controller for Mode Matching
A fuzzy controller is utilized in the mode-matching system to improve the sensing
performance. Then, the Simulink simulation block diagram of the simulation system with a
fuzzy controller is shown in Figure 11. The fuzzy controller has the advantage of parameter

Sensors 2023,23, 7422 9 of 12
adaptation, which can overcome the shortcomings that stem from processing errors and
environmental drift.
Sensors 2023, 23, x FOR PEER REVIEW 9 of 12
Figure 10. The input and output signal:(a) Input of angular velocity; (b) output signal of the control
system with load when modal matching.
5. Simulation of Fuzzy Controller for Mode Matching
A fuzzy controller is utilized in the mode-matching system to improve the sensing
performance. Then, the Simulink simulation block diagram of the simulation system with
a fuzzy controller is shown in Figure 11. The fuzzy controller has the advantage of param-
eter adaptation, which can overcome the shortcomings that stem from processing errors
and environmental drift.
Figure 11. Simulink simulation block diagram of modal matching fuzzy control.
The fuzzy controller for modal frequency matching adopts dual input and single out-
put; the inputs are 𝑉𝑒𝑟𝑟𝑜𝑟 and 𝑉󰇗𝑒𝑟𝑟𝑜𝑟, and the output ∆V𝑓𝑢𝑧𝑧𝑦 of the controller is used to
change the resonance frequency of the gyro detection mode. When fuzzing the member-
ship function of 𝑉𝑒𝑟𝑟𝑜𝑟, we divide it into ve fuzzy sets {NB, NS, ZO, PS, PB}, and set the
scope of its universe to [−64, 64]. The membership function of the input 𝑉󰇗𝑒𝑟𝑟𝑜𝑟 is divided
into three fuzzy sets {N, Z, P}, and the scope of its universe is set to [−32, 32]. The mem-
bership function of the output ∆V𝑓𝑢𝑧𝑧𝑦 is divided into ve fuzzy sets {NB, NS, ZO, PS,
PB}, and the scope of the universe is set to [−64, 64]. In the fuzzy controlling set, NB (Neg-
ative Big), NS (Negative Small), ZO (Zero), PS (Positive Small), and PB (Positive Big) are
used to describe the ve levels of parameter deviation. And fuzzy sets {N, Z, P} are used
to describe the three levels of parameter deviation: N is Negative, Z is Zero, and P is Pos-
itive.
According to the two inputs, the fuzzy rule library is formulated as shown in Table
1. When the deviation change increases in the negative direction and the deviation is also
negative, we selected the maximum output in the positive direction, and vice versa. When
the deviation increases in the negative direction and the deviation is not positive, we
choose the maximum output in the positive direction and vice versa. When the deviation
changes to 0 and the deviation increases slightly in the negative direction, the positive
output is selected as a smaller output and vice versa.
Table 1. Fuzzy rule of modal matching controller.
𝑽󰇗𝒆𝒓𝒓𝒐𝒓\𝑽𝒆𝒓𝒓𝒐𝒓
NB
NS
ZO
PS
PB
N
PB
PB
ZO
NS
NS
Z
PB
PS
ZO
NS
NB
P
PS
PS
ZO
NB
NB
Figure 11. Simulink simulation block diagram of modal matching fuzzy control.
The fuzzy controller for modal frequency matching adopts dual input and single
output; the inputs are
Verror
and
.
Verror
, and the output
∆Vf uzzy
of the controller is used to
change the resonance frequency of the gyro detection mode. When fuzzing the membership
function of
Verror
, we divide it into five fuzzy sets {NB, NS, ZO, PS, PB}, and set the scope
of its universe to [
−
64, 64]. The membership function of the input
.
Verror
is divided into
three fuzzy sets {N, Z, P}, and the scope of its universe is set to [
−
32, 32]. The membership
function of the output
∆Vf uzzy
is divided into five fuzzy sets {NB, NS, ZO, PS, PB}, and the
scope of the universe is set to [
−
64, 64]. In the fuzzy controlling set, NB (Negative Big), NS
(Negative Small), ZO (Zero), PS (Positive Small), and PB (Positive Big) are used to describe
the five levels of parameter deviation. And fuzzy sets {N, Z, P} are used to describe the
three levels of parameter deviation: N is Negative, Z is Zero, and P is Positive.
According to the two inputs, the fuzzy rule library is formulated as shown in Table 1.
When the deviation change increases in the negative direction and the deviation is also
negative, we selected the maximum output in the positive direction, and vice versa. When
the deviation increases in the negative direction and the deviation is not positive, we choose
the maximum output in the positive direction and vice versa. When the deviation changes
to 0 and the deviation increases slightly in the negative direction, the positive output is
selected as a smaller output and vice versa.
Table 1. Fuzzy rule of modal matching controller.
.
Verror\Verror NB NS ZO PS PB
N PB PB ZO NS NS
Z PB PS ZO NS NB
P PS PS ZO NB NB
The quantification method of certainty is designed as a small method, and the de-
fuzzification method adopts the weight method. We calculated the results according to the
random point theory and used Table 2for statistics.

Sensors 2023,23, 7422 10 of 12
Table 2. The theoretical calculation results of the input and output of fuzzy controller.
Verror\.
Verror −35 −15 10 36
−66 64 48 47 32
−35 58 38 32 32
−20 32 22 18 20
10 −10 −12 −14 −13
38 −32 −32 −36 −54
68 −32 −41 −50 −64
The initial value of the center frequency of the band-pass filter is 3381 Hz, and the
sinusoidal signal frequency generated by the direct digital synthesizer (DDS) module is
3051 Hz. The address increments output by the fuzzy controller are sequentially searched
to the center frequency of the band-pass filter corresponding to each address, which is
shown in Figure 12. According to the partially amplified waveform, when reaching stability,
the frequency of the band-pass filter changes in the range of 3051 Hz~3056 Hz, and the
frequency error of the analog modal matching is within 5 Hz. And then, the frequency
difference between driving and detecting is lower than the 1 Hz that was the base for the
modal matching of micro-mechanical gyro.
Sensors 2023, 23, x FOR PEER REVIEW 10 of 12
The quantication method of certainty is designed as a small method, and the de-
fuzzication method adopts the weight method. We calculated the results according to
the random point theory and used Table 2 for statistics.
Table 2. The theoretical calculation results of the input and output of fuzzy controller.
𝑽𝒆𝒓𝒓𝒐𝒓\𝑽󰇗𝒆𝒓𝒓𝒐𝒓
−35
−15
10
36
−66
64
48
47
32
−35
58
38
32
32
−20
32
22
18
20
10
−10
−12
−14
−13
38
−32
−32
−36
−54
68
−32
−41
−50
−64
The initial value of the center frequency of the band-pass lter is 3381 Hz, and the
sinusoidal signal frequency generated by the direct digital synthesizer (DDS) module is
3051 Hz. The address increments output by the fuzzy controller are sequentially searched
to the center frequency of the band-pass lter corresponding to each address, which is
shown in Figure 12. According to the partially amplied waveform, when reaching stabil-
ity, the frequency of the band-pass lter changes in the range of 3051 Hz~3056 Hz, and the
frequency error of the analog modal matching is within 5 Hz. And then, the frequency
dierence between driving and detecting is lower than the 1 Hz that was the base for the
modal matching of micro-mechanical gyro.
Figure 12. Center frequency and DDS frequency.
The simulation results show that the fuzzy PI controller can provide an excellent
method for detection. The comparison of control performance with dierent controllers is
shown in Table 3. The fuzzy PI displays a good method to decrease the response time,
seling errors, and overshoots, which is the base for the stable detection of capacitive mi-
cromachined gyros.
Table 3. Comparison of control performance with dierence controllers.
Method
Response Time
Settling Errors
Overshoot
PI controller
~1.5 s
~5%
~10%
Fuzzy PI controller
<0.4 s
<1%
<5%
6. Conclusions
There are orthogonal errors in the detection mode when driving the gyro because of
the process errors in the manufacturing of micro-mechanical gyro. In this paper, a PI
closed-loop control system based on the phase relationship between the signals of driving
mode and output orthogonal error is designed. The modal matching of the micro-mechan-
ical gyro is realized through the changing of the input DC oset voltage. The frequency
Figure 12. Center frequency and DDS frequency.
The simulation results show that the fuzzy PI controller can provide an excellent
method for detection. The comparison of control performance with different controllers
is shown in Table 3. The fuzzy PI displays a good method to decrease the response time,
settling errors, and overshoots, which is the base for the stable detection of capacitive
micromachined gyros.
Table 3. Comparison of control performance with difference controllers.
Method Response Time Settling Errors Overshoot
PI controller ~1.5 s ~5% ~10%
Fuzzy PI controller <0.4 s <1% <5%
6. Conclusions
There are orthogonal errors in the detection mode when driving the gyro because
of the process errors in the manufacturing of micro-mechanical gyro. In this paper, a
PI closed-loop control system based on the phase relationship between the signals of
driving mode and output orthogonal error is designed. The modal matching of the micro-
mechanical gyro is realized through the changing of the input DC offset voltage. The
frequency difference between the driving and detecting of the micro-mechanical gyro is
controlled within
1 Hz
using electrostatic control technology. The simulation model is
built in Simulink to optimize the parameters of PI control, realizing the system to adjust
the DC offset in 0.6 V and improving the response speed and stability of the system to

Sensors 2023,23, 7422 11 of 12
achieve modal matching. The influence of system quality factor on the modal matching
process is analyzed. A fuzzy controller is utilized in the system to ensure modal matching
and improve sensing performance. The frequency difference during modal matching is
controlled within 5 Hz, which has high control accuracy and stability. The feasibility of the
system through the simulation input of angular velocity is verified. The results are helpful
in improving the adaptability and accuracy of micro-mechanical gyros.
Author Contributions:
Methodology, modelling, writing—original draft preparation, L.C. and R.L.;
writing—review and editing, S.G.; conceptualization and funding acquisition, G.Z.; supervision and
funding acquisition, Y.L. All authors have read and agreed to the published version of the manuscript.
Funding:
This research was funded by the Natural Science Foundation of Guangdong Province
(grant numbers 2022A1515010923 and 2022A1515010949) and the Natural Science Foundation of
Fujian Province (grant number 2020H6003).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
References
1.
Kuang, Y.; Hou, Z.; Jian, D.; Liu, G.; Ma, C.; Xiao, D.; Wu, X. Reducing the Sensitivity of Fabrication Error in the MEMS Butterfly
Gyroscope by Adopting an Improved Vibration Beam. IEEE Sens. J. 2023,23, 5609–5617. [CrossRef]
2.
Elnemr, Y.E.; Abu-Libdeh, A.; Raj, G.C.A.; Birjis, Y.; Nazemi, H.; Munirathinam, P.; Emadi, A. Multi-Transduction-Mechanism
Technology, an Emerging Approach to Enhance Sensor Performance. Sensors 2023,23, 4457. [CrossRef] [PubMed]
3.
Guo, K.; Zhou, X. Research Report on the Application of MEMS Sensors Based on Copper Oxide Nanofibers in the Braking of
Autonomous Vehicles. J. Environ. Public Health 2022,2022, 5852729. [CrossRef] [PubMed]
4.
Luo, Z.; Zhang, A.; Huang, W.; Shao, S.; Liu, Y.; Wu, T.; Zou, Y. Aluminum Nitride Thin Film Based Reconfigurable Integrated
Photonic Devices. Ieee J. Sel. Top. Quantum Electron. 2023,29, 9300119. [CrossRef]
5.
Liao, Z.; Zoumhani, O.; Boutry, C.M. Recent Advances in Magnetic Polymer Composites for BioMEMS: A Review. Materials
2023
,
16, 3802. [CrossRef]
6.
Wang, Z.; Zhou, W.; Xiao, Z.; Yao, Q.; Xia, X.; Mei, J.; Zhang, D.; Chen, P.; Li, S.; Wang, Y.; et al. A High-Temperature
Accelerometer with Excellent Performance Based on the Improved Graphene Aerogel. ACS Appl. Mater. Interfaces
2023
,15,
19337–19348. [CrossRef]
7.
Mao, X.; Yang, H.; Long, D.; Wang, M.; Wen, P.Y.; Hu, Y.Q.; Wang, B.Y.; Li, G.Q.; Gao, J.C.; Long, G.L. Experimental demonstration
of mode-matching and Sagnac effect in a millimeter-scale wedged resonator gyroscope. Photonics Res.
2022
,10, 2115–2121.
[CrossRef]
8.
Xue, X.M.; Tang, J.; Zhou, H.L.; Qin, Y.Y.; Song, Y.J.; Li, R.Z.; Zhang, X.Y.; Zhang, T. All-polymer monolithic resonant integrated
optical gyroscope. Opt. Express 2022,30, 42728–42737. [CrossRef]
9.
Wang, Z.; Wang, G.C.; Gao, W.; Cheng, Y. Suppression of Kerr-effect induced error in resonant fiber optic gyro by a resonator
with spun fiber. Opt. Express 2021,29, 19631–19642. [CrossRef]
10.
Feng, C.K.; Zhang, Y.G.; Ma, H.H.; Li, H.; Feng, L.S. Improving long-term temperature bias stability of an integrated optical
gyroscope employing a Si3N4 resonator. Photonics Res. 2022,10, 1661–1668. [CrossRef]
11.
Kock, G.; Combette, P.; Tedjini, M.; Schneider, M.; Gauthier-Blum, C.; Giani, A. Experimental and numerical study of a thermal
expansion gyroscope for different gases. Sensors 2019,19, 360. [CrossRef] [PubMed]
12.
Crescenzi, R.; Castellito, G.V.; Quaranta, S.; Balucani, M. Design of a Tri-Axial Surface Micromachined MEMS Vibrating Gyroscope.
Sensors 2020,20, 2822. [CrossRef] [PubMed]
13.
Rahmani, M.; Rahman, M.H.; Nosonovsky, M. A new hybrid robust control of MEMS gyroscope. Microsyst. Technol.
2020
,26,
853–860. [CrossRef]
14.
Jeong, J.; Suh, J.; Choi, H. Angular momentum sensitivity of a MEMS gyroscope for condensed matter systems. Phys. Scr.
2023
,
98, 065925. [CrossRef]
15.
Chang, C.-Y.; Wang, S.-W.; Li, S.-S. Performance enhancement of piezoelectric bulk mode MEMS mode-matched gyroscopes
based on a secondary phase feedback loop. J. Micromechanics Microeng. 2023,33, 044001. [CrossRef]
16.
Zhang, H.; Zhang, C.; Chen, J.; Li, A. A Review of Symmetric Silicon MEMS Gyroscope Mode-Matching Technologies. Microma-
chines 2022,13, 1255. [CrossRef]
17.
Liu, Y.; Cai, Y.; Zhang, Y.; Tovstopyat, A.; Liu, S.; Sun, C. Materials, Design, and Characteristics of Bulk Acoustic Wave Resonator:
A Review. Micromachines 2020,11, 630. [CrossRef]

Sensors 2023,23, 7422 12 of 12
18.
Mei, Y.; Bi, X.; Cao, Z.; Weng, J.; Guo, X.; Xu, Q. Automatic Power Control for a Frequency Tunable Optoelectronic Oscillator. Ieee
Trans. Circuits Syst. Ii-Express Briefs 2023,70, 1385–1389. [CrossRef]
19.
Zhang, X.; Xie, F.; Chen, M.; Yang, Y. A highly stable frequency single-mode optical fiber laser based on a Fabry-Perot etalon. Opt.
Fiber Technol. 2022,73, 103047. [CrossRef]
20.
Wang, X.; Zheng, X.; Shen, Y.; Xia, C.; Liu, G.; Jin, Z.; Ma, Z. A Digital Control Structure for Lissajous Frequency-Modulated Mode
MEMS Gyroscope. IEEE Sens. J. 2022,22, 19207–19219. [CrossRef]
21.
Uzunoglu, B.E.; Erkan, D.; Tatar, E. A Ring Gyroscope With On-Chip Capacitive Stress Compensation. J. Microelectromechanical
Syst. 2022,31, 741–752. [CrossRef]
22.
Yang, L.; Su, Y.; Qiu, A.-P.; Xia, G.-M. Effect of pulse density feedback on force balance micro-machined gyroscope. Guangxue
Jingmi Gongcheng/Opt. Precis. Eng. 2013,21, 2087–2094.
23.
Wei, L.; You, Z.; Kuai, X.; Zhang, M.; Yang, F.; Wang, X. MEMS thermal-piezoresistive resonators, thermal-piezoresistive oscillators,
and sensors. Microsyst. Technol. 2023,29, 1–17. [CrossRef]
24.
Sonmezoglu, S.; Alper, S.E.; Akin, T. An automatically mode-matched MEMS gyroscope with wide and tunable bandwidth.
J. Microelectromechanical Syst. 2014,23, 284–297. [CrossRef]
25.
Zhang, H.; Chen, W.; Yin, L.; Fu, Q. An Interface ASIC Design of MEMS Gyroscope with Analog Closed Loop Driving. Sensors
2023,23, 2615. [CrossRef] [PubMed]
26.
Wagner, A.; Putnik, M.; Hiller, T.; Degenfeld-Schonburg, P.; Zimmermann, A. Influence of Parametric Nonlinearities on Closed-
Loop Operated MEMS Gyroscopes. J. Microelectromechanical Syst. 2022,31, 348–357. [CrossRef]
27.
Li, Q.; Ding, L.; Liu, X.; Zhang, Q. Research on a Silicon Gyroscope Interface Circuit Based on Closed-Loop Controlled Drive
Loop. Sensors 2022,22, 834. [CrossRef]
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