International Electrical Engineering Journal (IEEJ)
Vol. 2 (2011) No. 3, pp. 571-580
ISSN 2078-2365
571
Abstract—In this paper, controller for a human brachial
artery system prototype has been devised with respect to a
dynamic model. Operational procedure of the designed
prototype has been pointed out. A linear model has been taken in
the consideration. Major structural components (pump and
shaker) have been treated autonomously during preliminary
analysis. Ziegler - Nichols tuning methods, Tyreus-Luyben
tuning technique and pole placement scheme have been utilized
to determine the suitable controller parameters. At the
concluding stage of analysis, the computed values are combined
to denote the overall response of the system. System has been
tested with typical test inputs: step, ramp and sine functions.
Obtained simulation results signify the correctness of the
completed tasks.
Index Terms—Arterial Stiffness, Dynamic Model, PID
Controller, Simulation Model
I. INTRODUCTION
Cardiovascular disease involves heart and/or blood vessels.
This is the leading cause of morbidity and mortality in many
developed and developing countries of the world [1]. Studies
have proved that cardiovascular diseases cause a heavy
burden on society in terms of disability, illness and financial
cost. By the time cardiovascular diseases are detected using
existing technologies such as magnetic resonance imaging,
angiography, ultrasound, X-ray computed tomography and
pulse wave velocity measurement, the underlying cause
(atherosclerosis) is usually in a complicated stage [2]. Major
constraints such as excessive cost, inaccuracy and necessity of
large equipments also exist for available detection
technologies. Thus, a large portion of cardiovascular patients
is currently undiagnosed, misdiagnosed or mistreated. A local
biomedical engineering company has recently developed an
inexpensive, precise and user-friendly device to detect arterial
S. Kabir graduated at AUT (Auckland University of Technology),
Auckland, New Zealand, with B.E (Honors), (e-mail:
N. Popovich, M.Sc. (Eng) is a Senior Lecturer at AUT (Auckland
University of Technology), Auckland, New Zealand (e-mail:
stiffness. Quality, consistency and performance are the three
most significant features of this device. This developed
device differs from currently available innovations as it
focuses on the complete engineering process in an innovative
manner. To verify the legitimacy of the formulated detection
process, a significance of fabrication of human brachial artery
system prototype is the paramount.
Section II of this paper presents information of designed
prototype [3]. Section III presents details of utilized controller
design and tuning technologies with verification strategies
with respect to a dynamic model. The conclusion fragment
summarizes key findings.
II. DESIGN OF PROTOTYPE
After performing comprehensive theoretical research,
numerical modeling and experimentations, prototype model
illustrated in Fig. 1 was devised in this paper.
Fig 1. Figure of the formulated prototype
In the designed prototype, a pump was incorporated to lift
fluid from the reservoir. The pump also supplied the required
base pressure. The flow rate and performance characteristics
of the pump were directed by voltage variations. Afterwards,
a closed tank was integrated to store fluid before directing to
the tested brachial artery. The tank removed fluctuations
generated by the pump. Subsequently, a shaker provided the
Calibration of a Human Brachial Artery System
Prototype Controller (Dynamic Model)
Shahariar Kabir, Nenad Popovich
Shahariar and Nenad , Calibration of a Human Brachial Artery System Prototype Controller (Dynamic Model)
572 | P a g e
obligatory fluctuations on the peak of the base pressure. The
characteristics of the shaker were also regulated by voltage
variations. The pressure sensor was incorporated in the
network to monitor accessible pressure level. The information
was sent to multiple controllers built in LabVIEW Software
where the difference between the required pressure and the
available pressure was analyzed [4-8]. Then based upon the
analysis results, feedback signals were directed to the pump
and the shaker via data acquisition cards and electronic
amplifiers [9-15].
The design was finalized after extensive experiments as
this was relatively uncomplicated, consistent and reasonably
priced with precise components. Also, there was no room for
human errors as the complete operational process was
automatic.
III. FINE TUNING OF CONTROLLER PARAMETERS
The procedure of selecting the controller parameters to
analyze and control any physical process is known as
controller tuning [16-17]. A properly tuned controller offers
minimized energy cost, reduced process variability,
maximized efficiency and increased production rate.
In this paper, the following requirements were specified for
the prototype controller:
a. Minimum overshoot
b. Minimum settling time
c. Zero steady state error
P, I and D values for the PID
(Proportional-Integral-Derivative) controllers were identified
for a dynamic system. Thus, the investigation was performed
in a state where the relative positions of subsystems were
considered variant over time or where components and
structures were considered moving under the action of
external forces of equilibrium [18-20]. Since a linear system
was constructed, two prime mechanisms (pump and shaker)
were considered autonomously for the determinism of P, I and
D values. At the concluding stage, the calculated values were
united to identify the overall performance of the complete
system [21-24]. Thus, the following configurations were
analyzed:
(3.1) System with Pump and without Shaker
(3.2) System with Shaker and without Pump
(3.3) Complete System with Pump and Shaker
(3.1) System with Pump and without Shaker
A closed loop Simulink block diagram was created in this
section as shown in Fig. 2. In the designed Simulink diagram,
“Analog Filter” signified the low pass filter for pump.
Afterwards, “PID Controller” block showed the controller
used to control the attributes of the pump.
“Transfer Function for motor” represented the transfer
function of the motor section of the pump. Subsequently,
“Transfer Function for pump” signified the relationship that
was used to represent the rest of the pump. This segment
transferred the output speed from motor to the pressure output
from the pump. Next, “Transfer Function for tank”
corresponded to the tank that solved oscillation problem
created by the pump. “Gain for BA” characterized the tested
brachial artery and “Gain for sensor” symbolized the pressure
Fig 2. Simulink block diagram of the system with pump and without shaker
International Electrical Engineering Journal (IEEJ)
Vol. 2 (2011) No. 3, pp. 571-580
ISSN 2078-2365
573
sensor.
The following controller tuning methods were used in this
section:
(3.1.1) Ziegler – Nichols first tuning method
The Ziegler – Nichols tuning methods are heuristic
techniques for tuning PID controller [25]. The first tuning
method is an open loop technique, which means we needed to
define an open loop transfer function (OLTF) of the system in
the following form:
Afterwards, the open loop system was tested using a step
input and a tangent line was drawn at the inflection point of
the nearly “S-shaped” curve known as the reaction curve.
Then, a delay time L, as well as a time constant T, were
determined from the graph. This was also noted that the open
loop method or Ziegler – Nichols first tuning method
depended on the determination of the L and T constants in a
manual manner from the open loop transfer function graph of
the designed system. Thus, there were always some risks of a
human error.
Then the table depicted on Page 3 was utilized to find
parameters for P, PI and PID controllers:
TABLE I.
TABLE OF
ZIEGLER-N
ICHOLS
FIRST
TUNING
RULE [25]
Controller P
∞ 0
PI
0
PID
2L 0.5L
While using those calculated parameters, all responses
obtained from simulation analysis were unstable (note: for the
purpose of simplicity, responses are not shown in this paper).
So, this was concluded that satisfactory performance of the
system could not be reached using calculated parameter
values. This was also noted that the first tuning method was
only suitable for simple systems without any integrator or
dominant complex conjugate poles in open loop transfer
function [25]. Thus the cause of unsatisfactory behaviour
became evident.
(3.1.2) Tyreus-Luyben tuning method
To identify proportional, integral and derivative controller
values using Tyreus-Luyben tuning method, the closed loop
system depicted in Fig. 3 was designed incorporating open
loop transfer function (OLTF) obtained from previous
section.
Fig 3. Simulink block diagram with open loop transfer function
According to this method, only P controller had to be used
while setting derivative time constant as zero and integral
time constant as infinity. The gain in the PID controller block
was increased in steps of one-half the previous gain. The gain
Shahariar and Nenad , Calibration of a Human Brachial Artery System Prototype Controller (Dynamic Model)
574 | P a g e
was adjusted carefully to achieve sustain oscillation from the
system. The gain that caused sustained oscillation, as well as
the period of oscillation had to be recorded and used for
calculating controller’s parameters [25].
Then the following table was utilized to find parameters for
PI and PID controllers:
TABLE II. TABLE OF TYREUS-LUYBEN TUNING RULE [25]
Controller
PI
2.2 Pu
PID
2.2 Pu
The calculated controller parameter values are stated below:
TABLE III. TABLE OF CALCULATED CONTROLLER VALUES
Controller PI 78.125 6.82
PID 115 6.82 0.496
So, for the PI controller, the controller parameters were
defined as following:
= 78.125
= 11.45
For the calculated parameters, the response demonstrated
in Fig. 4 was obtained due to unit step input:
Fig 4. Response of the system with PI controller due to unit step input
For the PID controller, the calculated parameters are stated
below:
= 115
= 11.45
=57.04
For the calculated parameters, the response demonstrated
in Fig. 5 was obtained due to unit step input:
Fig 5. Response of the system with PID controller due to unit step input
As stated, minimum settling time and zero steady state
error were defined as two significant parameters of the
designed system. It became evident from the response graphs
for both PI and PID controllers that there was always error
present in the system. The output did not settle down within
acceptable period of time as well. So, the controller values
were neglected.
(3.1.3) Ziegler – Nichols second tuning method
To identify P, PI and PID controller values using
Ziegler-Nichols second tuning method, the gain that caused
sustained oscillation for the open loop transfer function, as
well as the period of oscillation determined in the previous
section were used. This was noted during analysis that the
tuning method allowed fluctuations in the process variable as
long as each successive peak was not more than the size of
one-fourth of its predecessor. Then parameter values were
determined and placed in table below [25]:
TABLE IV. TABLE OF CALCULATED CONTROLLER VALUES
Controller P 125 ∞ 0
PI 112.5 2.588
( =43.47)
0
PID 150 1.55
( =96.738)
0.3875
( =58.125)
The following response was obtained for P controller:
International Electrical Engineering Journal (IEEJ)
Vol. 2 (2011) No. 3, pp. 571-580
ISSN 2078-2365
575
Fig 6. Response of the system with P controller due to unit step input
This was evident from the response graph that there was
always error present in the system because the proportional
controller could not completely remove the steady state error.
However, there was no overshoot in the system and the output
settled down within a reasonable period of time as well.
The following response was obtained for PI controller:
Fig 7. Response of the system with PI controller due to unit step input
It was concluded from analysis that the steady state error
was cancelled by the integral component of the controller.
However, integral component added oscillation in the system
which increased the overshoot. Thus, the output took longer
to settle down.
The response demonstrated in Fig. 8 was obtained for PID
controller.
Fig 8. Response of the system with PID controller due to unit step input
Therefore, it was clear from the response graph that the
PID controller offered the best response in terms of
overshoot, settling time and steady state error for designed
system. The method also incorporated dynamics of the
complete process which provided a clear picture of
operational procedure of the designed system.
(3.2) System with Shaker and without Pump
For shaker, the following Simulink block diagram was
developed:
Fig 9. Simulink block diagram of the system with shaker and without pump
In the above block diagram, “Analog Filter” symbolized
the analogue high pass filter for the shaker. “PID Controller”
block represented the controller that was used to control the
fluctuations produced by the shaker. “Transfer Fcn for
shaker” signified the physical structure of the shaker itself.
“Gain for BA” symbolized the tested brachial artery and
“Gain for sensor” represented the pressure sensor. For this
system, the open loop transfer function (OLTF) was
calculated as:
The following block diagram was designed for simulation
incorporating the open loop transfer function:
Fig 10. Simulink block diagram with open loop transfer function
Shahariar and Nenad , Calibration of a Human Brachial Artery System Prototype Controller (Dynamic Model)
576 | P a g e
Subsequently, the designed closed loop system was tested
using various controller parameters as stated underneath:
Proportional gain constant = 50
Integral gain constant = 6
Derivative gain constant = 2
For the above controller parameters, the subsequent
response was obtained:
Fig 11. Response of the system with specified parameters due to step input
Proportional gain constant = 60
Integral gain constant = 10
Derivative gain constant = 2
For the stated controller parameters, the subsequent
response was obtained:
Fig 12. Response of the system with specified parameters due to step input
Proportional gain constant = 100
Integral gain constant = 20
Derivative gain constant = 5
For the stated controller parameters, the subsequent
response was obtained:
Fig 13. Response of the system with specified parameters due to step input
Thus, it became evident from simulation analysis that the
system would never become unstable or would never provide
the required “S” shape. Therefore, it was not possible to
utilize any of the Zeigler-Nichols methods or Tyreus-Luyben
tuning method, because it was impossible to achieve critical
gain, delay time and other necessary parameters for defining
P, I and D controllers. For this reason, a pole placement
method was utilized to place the closed loop poles in
pre-determined locations in s-plane. In the designed linear
system, poles influenced system response, band width,
transient response and stability [25].
3.2.1 Pole placement method
According to Fig. 10, two transfer functions were utilized
to find the overall transfer function of the complete system.
Thus, transfer function for the PID controller was defined as:
So, after rearranging, the closed loop transfer function
(CLTF) of the overall system was defined as:
The characteristic equation of the system was defined:
The major intention of determining the P, I and D values
was to accomplish the fastest non-oscillatory response with a
zero steady state error. To achieve this goal, it was required to
have double real poles, because this would provide the fastest
non-oscillatory response. From the open loop transfer
function (OLTF) of the designed system, the desired
characteristic equation of the system was defined:
0 (2)
By comparing (1) and (2), the controller’s parameters were
found as following:
Proportional gain constant = 161
Integral gain constant = 525
Derivative gain constant = 12.33
Note: Derivative gain constant value was much
smaller comparing to Proportional and Integral gain
constants, which means that Derivative gain constant had a
relatively insignificant influence to the system dynamic. So,
International Electrical Engineering Journal (IEEJ)
Vol. 2 (2011) No. 3, pp. 571-580
ISSN 2078-2365
577
only PI controller was used for the shaker. The following
satisfactory response was obtained from simulation analysis:
Fig 14. Response of the system with specified parameters due to step input
(3.3) Complete System with Pump and Shaker
At the concluding stage, the combined system
demonstrated in Fig. 15 was fabricated for simulation
purposes.
The following P, I and D controller values obtained from
previous sections were used for pump and shaker
respectively:
TABLE V. TABLE OF CALCULATED CONTROLLER VALUES FOR PUMP
Controller Value
P 150
I 96.738
D 58.125
TABLE VI. TABLE OF CALCULATED CONTROLLER VALUES FOR SHAKER
Controller Value
P 161
I 525
D 0
The responses demonstrated in Fig. 16, 17 and 18 were
obtained for unit step, ramp and sine functions respectively:
Fig 16. Response of the system due to unit step input
Fig 15. Simulink block diagram of the complete system with pump and shaker
Shahariar and Nenad , Calibration of a Human Brachial Artery System Prototype Controller (Dynamic Model)
578 | P a g e
Fig 17. Response of the system due to unit ramp input
Fig 18. Response of the system due to unit sine input
So, the simulation graphs proved the appropriateness of the
calculated controller values for both pump and shaker in
terms of overshoot, settling time and steady state error.
IV. CONCLUSION
The proper artistic response to the recently
developed arterial stiffness detection technology is to
embrace it as a new window of opportunity and to apply
it with passion, wisdom, fearlessness and delight. A
prototype has been designed to verify the authenticity
of the formulated technology. This paper presents
detailed information about design and tuning methods
with validation tactics that have been utilized for
prototype controller in consideration of a dynamic
model. Obtained simulation results signify the
correctness of the completed tasks.
ACKNOWLEDGMENT
The authors would like to acknowledge Professor
Adnan Al Anbuky, Engineer Shajib Khadem and Dr.
Tek Tjing Lie for their help and contribution in
developing this research paper.
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