Mechanically actuated capacitor microphone control using mpc and narma l2 controllers

18
Mechanically Actuated Capacitor Microphone Control using MPC and NARMA-L2 Controllers
Mustefa Jibril 1
, Messay Tadese 2
, Eliyas Alemayehu Tadese 3
1.
School of Electrical & Computer Engineering, Dire Dawa Institute of Technology, Dire Dawa, Ethiopia
2.
School of Electrical & Computer Engineering, Dire Dawa Institute of Technology, Dire Dawa, Ethiopia
3.
Faculty of Electrical & Computer Engineering, Jimma Institute of Technology, Jimma, Ethiopia
mustefazinet1981@gmail.com
Abstract: In this paper, a capacitor microphone system is presented to improve the conversion of mechanical energy
to electrical energy using a nonlinear auto regressive moving average-L2 (NARMA-L2) and model predictive
control (MPC) controllers for the analysis of the open loop and closed loop system. The open loop system response
shows that the output voltage signal need to be improved. The comparison of the closed loop system with the
proposed controllers have been analyzed and a promising result have been obtained using Matlab/Simulink.
[Mustefa Jibril, Messay Tadese, Eliyas Alemayehu Tadese. Mechanically Actuated Capacitor Microphone
Control using MPC and NARMA-L2 Controllers. Researcher 2020;12(8):18-23]. ISSN 1553-9865 (print); ISSN
2163-8950 (online). http://www.sciencepub.net/researcher. 4. doi:10.7537/marsrsj120820.04.
Keywords: Microphone, Nonlinear auto regressive moving average-L2, Model predictive control
1. Introduction
A capacitor microphone is a device which makes
use of a capacitor to convert acoustical power into
electric energy. Capacitor microphones require
electricity from a battery or external supply. The
ensuing audio signal is stronger signal than that from
a dynamic. Capacitor microphone also have a
tendency to be more sensitive and responsive than
dynamics, making them properly-desirable to
capturing subtle nuances in a sound. They are not best
for excessive volume work, as their sensitivity makes
them susceptible to distort. A capacitor has two plates
with a voltage between them. In the capacitor mic,
this sort of plates is product of very light fabric and
acts as the diaphragm. The diaphragm vibrates when
struck through sound waves, changing the space
between the 2 plates and therefore changing the
capacitance. Specifically, whilst the plates are nearer
collectively, capacitance will increase and a charge
current occurs. When the plates are in addition apart,
capacitance decreases and a discharge current occurs.
A voltage is required across the capacitor for this to
paintings. This voltage is provided both by using a
battery inside the mic or via external phantom power.
2. Mathematical Modeling of a Capacitor
Microphone
Figure 1 shows a system in which that takes
place in a capacitor microphone. Plate a of the
capacitor is attached to the rigid body. Sound waves
impinge upon and exert a force on plate b of mass M,
which is suspended from the rigid body by a spring K
and which has damping B. The output voltage at the
resistor R is intended to reproduce electrically the
sound-wave patterns.
Figure 2 Capacitor microphone system
At equilibrium, with no external force extended
on plate b, there is a charge q0 on the capacitor. This
produces a force of attraction between the plates so
that the spring is stretched by an amount x1 and the
space between the plates is x0. When sound waves
exert a force on plate b there will be a resulting
motion x that is measured from the equilibrium
position. The distance between the plates will then be
x0-x, and the charge on the plates will be q0+q.

Researcher 2020;12(8) http://www.sciencepub.net/researcher RSJ
19
The capacitance is approximated by
 0
0 0
1
A A
C and C
x x x
 
 

where is the dielectric constant for air and A is
the area of the plate. We assume that the displacement
of 0x is big, so the circuit become open circuit and the
initial current becomes zero.
The energy expressions for this system are
 2 21 1
2
2 2
T Lq Mx  
 2 21 1
3
2 2
D Rq Bx  
      
2 2
0 0 1
1 1
4
2 2
V x x q q K x x
A
    
The two degrees of freedom are the displacement
x of plate b and the charge q on the capacitor.
Applying Lagrange’s equation twice gives
       2
0 1
1
5
2
Mx Bx q q K x x f t
A
      
    0 0
1
6Lq Rq x x q q E
A
     
These equations are nonlinear. However, a good
linear approximation can be obtained, because x and q
are very small quantities and, therefore, the x2, q2,
and xq terms can be neglected. This gives
 
  
2 2
0 0 0
0 0 0 0 0 0
2q q q q q
x x q q x q q x x q
  
    
With these approximations the system equations
become
   
2
0 0
1
2
7
2 2
q q q
Mx Kx Kx Bx f t
A A 
      
 0 0 0 0
8
x q q x x q
Lq Rq E
A A A  
     
From Equation (7), by setting f (t)=0 and taking
steady-state conditions, the result is
2
0
1 0
2
q
Kx
A
 
Similarly, in Equation (8) at equilibrium
0 0 0
0
x q q
E
A C
 
Therefore, the two system equations can be
written in linearized form as
   0
9
q
Mx Bx Kx q f t
A
    
 0
0
0 10
qq
Lq Rq x
C A
    
Let
 1 2 3 4, , ,x x x x v x q x q i and u f t       
The state space representation of the system
becomes
 
1 1
0
2 2
3 3
0 04 4
1
2
3
4
0 1 0 0
0
0
1
0 0 0 1 0
0
0
0 0 0 1
x xqK B
x xM M A
u
x x
q xx x
R
AL AL
x
x
y
x
x

 
 
      
       
             
      
        
 
 
 
 
 
 
 




The system parameters are shown in Table 1
below
Table 1 System parameters
No Parameter Symbol Value
1 Mass of the Plate M 0.1 Kg
2 Spring stiffness K 0.5 N/m
3 Damping coefficient B 0.12 N-s/m
4 Area of the plate A 0.00005 m^2
5 Inductance of the circuit L 5 H
6 Resistance of the circuit R 23 ohm
7 Capacitance dielectric constant  8.85x10^-12 F/m
8 Initial charge 0q 6.28x10^-15 C
9 Plates initial displacement 0x 7x10^-4 m

20
The transfer function between the input force
and the output charge become
 
 
 
 4 3 11 2 11 12
24.2 3.164 10 3.797 10 1
2.838s
.582
11
10s s
I s
F s s
G s
s     

 

The output voltage at the resistor R is
     12RV s RI s
Substituting Equation (12) in to Equation (11)
the transfer function between the input force and the
output resistance voltage become
 
 
  4 3 11 2 11 12
24.2 3.164 10 3.797 10 1.582 10
65.3R
s s s s
V s s
G s
F s       
 
3. Proposed Controllers Design
3.1 NARMA-L2 Controller Design
One of the primary capabilities of the NARMA-
L2 neuro-controller is to transform nonlinear system
dynamics into linear dynamics by canceling the
nonlinearities. We start off evolved by means of
describing how the identified neural community
model may be used to design a controller. The
advantage of the NARMA-L2 form is that you may
remedy for the control input that reasons the system
output to observe a reference signal:
Figure 2 NARMA-L2 Controller.
Table 2 illustrates the network architecture,
training data and training parameters of the proposed
controllers.
Table 2 Neural network Parameters
Network Architecture
Size of hidden layer 6 Delayed plant input 4
Sample interval (sec) 0.1 Delayed plant output 4
Training Data
Training sample 65 Maximum Plant output 2
Maximum Plant input 2 Minimum Plant output 1
Minimum Plant input 1 Max interval value (sec) 30
Min interval value (sec) 15
Training Parameters
Training Epochs 65
3.2 MPC Control
A block diagram of a model predictive control
system is shown in Figure 3. A system model is used
to count on the modern values of the output variables.
Figure 3 Block diagram for model predictive control.
The residuals, the variations a number of the
actual and predicted outputs, function the comments
sign to a Prediction block. The predictions are utilized
in two styles of MPC calculations which might be
achieved at each sampling right now: set-factor
calculations and manage calculations. Inequality
constraints at the input and output variables,
consisting of better and decrease limits, can be
included in both type of calculation.
4. Result and Discussion
4.1 Open Loop Response of the Capacitor
Microphone System
The response of the open loop capacitor
microphone for a 0.01 N impulse response is shown in
Figure 4 below.

21
Figure 4 Open loop response
The simulation result shows that an electrical
sound waves generated at the resistor and its value
decreased until becomes zero because of the capacitor
discharge the energy accumulated. The voltage output
need to be improved in gain value and minimize the
overshot and settling time that’s why we use a
controller to improve the system.
4.2 Comparison of the Capacitor Microphone
System with NARMA-L2 and MPC Controllers for
a Step Desired Sound Waves Voltage Input
The Simulink model of the capacitor microphone
system with NARMA-L2 and MPC controllers for a
step desired sound waves voltage input signal is
shown in Figure 5 below.
Figure 5 Simulink model of the capacitor microphone system with NARMA-L2 and MPC controllers for step
desired sound waves voltage input signal
The simulation result is shown in Figure 6.
Figure 6 Step response simulation result

22
The data of the rise time, percentage overshoot, settling time and peak value is shown in Table 3.
Table 3 Step response data
No Performance Data NARMA-L2 MPC
1 Rise time 1.08 sec 1.08 sec
2 Per. overshoot 12.5 % 32.5 %
3 Settling time 1.38 sec 1.38 sec
4 Peak value 4.5 Volt 5.3 Volt
As Table 3 shows that the capacitor microphone
system with NARMA-L2 controller improves the
performance of the system by minimizing the
percentage overshoot.
4.3 Comparison of the Capacitor Microphone
System with NARMA-L2 and MPC Controllers for
a Random Desired Sound Waves Voltage Input
The Simulink model of the capacitor microphone
system with NARMA-L2 and MPC controllers for a
random desired sound waves voltage input signal is
shown in Figure 7 below.
Figure 7 Simulink model of the capacitor microphone system with NARMA-L2 and MPC controllers for a random
desired sound waves voltage input signal
The simulation result is shown in Figure 8.
Figure 8 Random response simulation result

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The simulation result shows that the capacitor
microphone system with NARMA-L2 controller
improves the performance of the system by
minimizing the percentage overshoot with an exact
steady state value while the capacitor microphone
system with MPC controller losses its performance in
tracking the steady state value after 2 second.
5. Conclusion
In this paper, a capacitor microphone system is
designed to improve the performance of the
mechanical to electrical energy conversion with force
as a mechanical input and voltage as an electrical
output. The open loop response of the system shows
that the output signal need to be improved. NARMA-
L2 and MPC controllers are used to improve the
performance of the system. Comparisons of the
capacitor microphone system with NARMA-L2 and
MPC controllers has been done to track a step and
random desired voltage input signals. The simulation
result of the step input signal shows that the capacitor
microphone system with NARMA-L2 controller
improves the performance of the system by
minimizing the percentage overshoot while for the
random signal the capacitor microphone system with
NARMA-L2 controller improves the performance of
the system by minimizing the percentage overshoot
with an exact steady state value while the capacitor
microphone system with MPC controller losses its
performance in tracking the steady state value after 2
second. Finally, the comparative simulation proved
that the capacitor microphone system with NARMA-
L2 controller improved the performance of the
system.
Reference
1. Sedighe B. S et al. “Design and Modeling of a
Highly Sensitive Microelectromechanical
System Capacitive Microphone” J. of
Micro/Nanolithography, MEMS, and MOEMS,
Vol. 19, No. 2, 2020.
2. Lars U. et al. “A High Quality Digital Radio
Frequency Capacitor Microphone with Improved
Dynamic Range” The Journal of the Acoustical
Society of America, Vol. 147, No. 1953, 2020.
3. M. Ali Shah et al. “Design Approaches of
MEMS Microphones for Enhanced
Performance” Journal of Sensors, Vol. 2019,
Article ID 9294528, 26 pages, 2019.
4. Germano N. et al. “MEMS Capacitive
Microphones: Acoustical, Electrical and Hidden
Thermal-Related Issues” IEEE Sensors Journal,
Vol. 18, Issue 13, 2018.
5. Takanori U. et al. “System Identification of
Velocity Mechanomyogram Measured with a
Capacitor Microphone for Muscle Stiffness
Estimation” Journal of Electromyography and
Kinesiology, Vol. 33, pp. 57-63, 2017.
6. Shakiba Dowlati et al. “An Accurate Study on
Capacitive Microphone with Circular Diaphragm
using a Higher Order Elasticity Theory” Latin
American Journal of Solids and Structures, Vol.
13, Issue 4, pp. 590-609, 2016.
8/16/2020

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