PID Controller Response to Set-Point Change in DC-DC Converter Control

International Journal of Power Electronics and Drive System (IJPEDS)
Vol. 7, No. 2, June 2016, pp. 294~302
ISSN: 2088-8694  294
Journal homepage: http://iaesjournal.com/online/index.php/IJPEDS
PID Controller Response to Set-Point Change in DC-DC
Converter Control
Oladimeji Ibrahim, Nor Zaihar Yahaya, Nordin Saad
Departement of Electrical and Electronics Engineering, Universiti Teknologi PETRONAS, Malaysia
Article Info ABSTRACT
Article history:
Received Oct 26, 2015
Revised Mar 2, 2015
Accepted Apr 3, 2016
Power converter operations and efficiency is affected by variation in supply
voltage, loads current, circuit elements, ageing and temperature. To meet the
objective of tight voltage regulation, power converters circuit module and the
control unit must be robust to reject disturbances arising from supply, load
variation and changes in circuit elements. PID controller has been the most
widely used in power converter control. This paper presents studies of
robustness of PID controller tuning methods to step changes in the set point
and disturbance rejection in power converter control. A DC-DC boost
converter was modelled using averaged state-space mothod and PID
controllers were designed with five different tuning methods. The study
reveals the transient response and disturbance rejection capability of each
tuning methods for their suitability in power supply design applications.
Keyword:
Averaged state-space
Boost converter
Feedback
PID controller
Transient response Copyright © 2016 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Oladimeji Ibrahim,
Departement of Electrical and Electronics Engineering,
Universiti Teknologi PETRONAS,
32610 Bander Seri Iskander, Perak Malaysia.
Email: reacholaibrahim@gmail.com
1. INTRODUCTION
Power converters change electrical energy from one voltage, current or frequency level to another.
Distributed generations, communication systems, aviation and other electrical gadgets like computers,
electronic appliance require power supply for their operation. Some of these systems are sensitive to voltage
fluctuations while others tolerate certain level of variation that is within the international stipulated standard.
The high efficiency of switching mode power supply has made it the most widely used for power supply
design. Its output voltage is a function of the input line (supply) voltage, the duty cycle of the pulse width
modulation (PWM) control signal, the load current, and also the converter circuit element values [1, 2].
Power supply is design to produce output voltage that will remain within a specified range in the
face of load current change from no load to full load. To achieve constant output voltage, a negative feedback
control loop is incorporated to automatically adjust the control of PWM duty cycle regardless of supply input
variation, load change and variation of circuit elements. The error resulting from the difference between the
set point and output voltage is compensated by the controller which determines the PWM duty cycle for
ensuring that the output voltage is as close as possible to the reference value [3].
PID controller has widely been applied to compensate for error resulting from difference between
the set point and feedback in power converters design due to its simplicity and ruggedness [4-7]. Though in
recent time, research effort is towards investigating modern control theory like the fuzzy and state-space
control that could better account for modelling inadequacy and variation in circuit element values. The PID
control have shown high level of acceptance in terms the tight output voltage regulation in the face of
disturbance from line and load variations. This work focuses on PID tuning methods for power converter
applications by investigating their transient response to input supply variation and load disturbance. Five (5)
different PID tuning methods; namely Ziegler-Nichol Frequency Domain Method (ZN-FDM), Modified

 ISSN: 2088-8694
IJPEDS Vol. 7, No. 2, June 2016 : 294 – 302
295
Ziegler-Nichol (MZN), Damped Oscillation Method (DOM), Tyreus-Luyben Tuning Method (TLM), and
Good Gain Method (GGM) are investigated. A DC-DC boost converter is modelled using averaged state-
space method and PID controllers are designed using the aforementioned tuning techniques. The Matlab-
Simulink is used to simulate the system in time-domain for analysing the transient response and also their
disturbance rejections capability.
2. BOOST CONVERTER MODELLING
A DC-DC boost converter is non-isolated power supply that produces output voltage greater than its
input. It comprises of at least a FET power switch (MOSFET/IGBT, Q1), a diode and an inductor (L) as
energy storage. Filter capacitor (C) is usually added to reduce the output voltage ripple. The operation can be
in continuous or discontinuous mode depending on the charging and discharging of the input inductor. The
steady and dynamic states model of continuous current mode (CCM) operation of non-ideal boost converter
is presented in subsections 2.1 and 2.2. The steady state model allows for design of circuit parameters while
the dynamic model is used for controller design and dynamic analysis.
2.1 Steady State Model
The PWM scheme is the mostly widely used for gate signal controlof field effect transistor (FET)
devicein power converters design. The power transfer stage of anon-isolated boost has the “on” and “off”
state as shown in Figure 1 and Figure 2 respectively. The total duration of the ON state (Ton) and OFF states
(Toff) in continuous conduction mode (CCM) is given by Eq. (1) and Eq. (2), where )(d is the duty cycle set
by the control circuit, and (Ts) is the switch period for complete one cycle.
son dTT  (1)
soff TdT )1(  (2)
Mode 1, when switch Q1 is in on state at time 0 < t < ton
ic(t)
Vc(t)
+
-
iout(t)
inV
LR
C
Lin ii t )(
L
)(toutVON
Q1
Figure 1. Equivalent circuit of boost converter at 0 < t < ton
The changes in inductor current ∆IL is given by:
L
tV
I onin
L  (3)
Mode 2 when switch Q1 is in off state at time ton < t  T

IJPEDS ISSN: 2088-8694 
PID Controller Response to Set-Point Change in DC-DC Converter Control (Oladimeji Ibrahim)
296
ic(t)
Vc(t)
+
-
iout(t)
inV
LR
C
Lin ii t )(
L
)(toutV
Q1
OFF
Figure 2. Equivalent circuit of boost converter at Ttton 
The inductor voltage changes by Vin – Vout and the current falls by ∆IL/Toff, therefore the inductor current
change is given by:
L
tVV
I
offinout
L
)( 
 (4)
Thus, the steady state input to output voltage conversion ratio is obtained as:
d
V
V in
out


1
(5)
The magnitude of peak-to-peak inductor current ripple LI is given by:
Lf
dV
I
s
in
L  (6)
And, also the output capacitor voltage ripple cV is:
Cf
dI
VV
s
o
outc  (7)
Table 1. Converter steady state parameters
Parameters Symbol Values
Input voltage inV 12 [V]
Output voltage outV 48 [V]
Output voltage ripple oV 500 [mV]
Rated power oP 200 [W]
Load resistance LR 11.5 [Ω]
Average output current oI 4.17 [A]
Duty cycle d 0.75
Inductor L 0.5 [m H]
Output capacitor C 125 [µF]
Switching frequency sf 50 [kHz]
MOSFET on resistance onR 40 [mΩ]
Inductor parasitic resistance Lr 10 [mΩ]
Capacitor ESR Cr 10 [mΩ]
2.2 Dynamic State Model
The state space averaged method is used to model a DC-DC boost converter considering parasitic
parameters of the system components. State space averaged technique gives complete model with both steady

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297
state (DC) and dynamic (AC) quantities making it easy to obtain the system transfer function for dynamic
state analysis [1, 8, 9].
The general state space averaged equation is given by Eq. (8) and Eq. (9)
)()()( tuBtxAtx 

(8)
)()( txCty  (9)
Where, the state averaged matrix )1(21 dAdAA  , )1(21 dBdBB  , and )1(21 dCdCC  .
The term )(tx is the converter DC state vector defined as inductors currents and capacitors voltages, u(t) is
converter DC input vector and y(t) is the converter DC output vector.
Considering the converter in mode 1 (Figure 1) when the switch Q1 is in on-state at time interval dTs,
applying KVL and KCL the differential state variables for the boost converter are obtained as follows:
)(
0
1
)(
)(
)(
1
0
0
)(
)(
)(
tVL
tV
ti
rRC
L
rR
dt
tdV
dt
tdi
in
c
L
cL
Lon
c
L

















































(10)
 


















)(
)(
)(
0)(
tV
ti
rR
R
tV
c
L
cL
L
out (11)
Similarly, mode 2 (Figure 2) when switch Q1 is in off-state at time interval (1-d)Ts, applying KVL
and KCL yields differential state variables:
)(
0
1
)(
)(
)(
1
)(
)(
//
)(
)(
tVL
tV
ti
rRCrRC
R
rRL
R
L
rRr
dt
tdVc
dt
tdi
in
c
L
cLcL
L
cL
LcLLL






















































(12)
 


















)(
)(
)(
//)(
tV
ti
rR
R
rRtV
c
L
cL
L
cLout (13)
The averaged state matrices for both “on”and “off”state with duty cycle (d) as weighting factor is
obtained as:






















)(
1
)(
)1(
)(
)1(1)//()(
cLcL
L
cL
LcLLLon
rRCrRC
dR
rRL
dR
L
drRrdrR
A (14)













0
1
L
B (15)

298








)(
)1(//
cL
L
cL
rR
R
drRC (16)
To carry out dynamic state analysis and controller design for the DC-DC boost converter, the
Matlab state-space (ss) to transfer function (tf) command is used to obtain the converter control-to-output
transfer function Gvd(s) using the steady state parameters presented in Table I.
062
06
057.1780
3.9979964
)(
ess
es.
sGvd


 (17)
3. PID CONTROLLER DESIGN
A proportional integral and derivative (PID) controller is considered as the most widely studied and
used in both academics and industries due to its simplicity and robustness. The PID controller is the
aggregates of the three sub-control units, the proportional, integral and derivative control modes. The
effective control signal u(t) by a PID controller in Laplace domain is given by Eq. (18) [10, 11].
dt
tde
kdtteKtektu d
t
ip
)(
)()()(
0
  (18)
The relationship of Eq. (19) is the continuous s-domain transfer function:
SK
S
K
KsC d
i
p )( (19)
Where, Kp, Ki and Kd are the proportional gain, integral gain and derivative gain respectively, Ti is
the integral time constant and Td derivative time constant.
PID controllers are designed using five (5) different tuning methods in order to improve the
converter transient response and eliminate steady state error. The procedures for each tuning methods
Ziegler-Nichol Frequency Domain Method (ZN-FDM), Modified Ziegler-Nichol (MZN), Damped
Oscillation Method (DOM), Tyreus-Luyben Tuning Method (TLM), and Good Gain Method (GGM) are
briefly explained in the subsections 3.1 to 3.5
3.1 Ziegler-Nichol Frequency Domain Method (ZN-FDM)
The Ziegler-Nichole frequency method was proposed by Ziegler and Nichole in the 1942 based on
sustained oscillation of the system response [10, 12]. The closed-loop system under proportional controller
(Kp) is driven to critically stable state by increasing the proportional gain with integral time constant (Ti) set
to infinity and derivative constant (Td) set to 0. The corresponding gain and period at this point are referred to
as ultimate gain uK and ultimate period uP obtained as 1 and 0.003s respectively for the converter. The
proportional gain (Kp), integral (Ti) and derivative constants (Td) for the PID controller are 0.6, 0.0015 and
0.000375 respectively using the Ziegler-Nichol tuning parameters in Table II.
Table 2. Ziegler-Nichol PID tuning parameters
Type of
Controller pK iT dT
P 0.5 uK  0
PI 0.45 uK 0.8 uP 0
PID 0.6 uK 0.5 uP 0.125 uP
3.2 Modified Ziegler-Nichol (MZN)
The Ziegler-Nichol frequency response method forces a system process to marginal stability leading
to unsafe operation for sensitive systems [10]. A more conservative method is proposed for 4/1 decay ratio
oscillation. The system is operated with low proportional gain control mode with integral time constant )( iT
set to (∞) and derivative time (Td) set to (0). The gain is gradually adjusted until a decay ratio of (1/4)th
is

 ISSN: 2088-8694
299
obtained in the successive output response. The modified Ziegler-Nichol settings for response with overshoot
and no overshoot is presented in Table III. The proportional gain (Kp), integral (Ti) and derivative constants
(Td) for the PID controller are obtained as 0.194, 0.0015 and 0.00099 respectively.
Table 3. Modified Ziegler-Nichol tuning parameters
PID
Controller pK iT dT
Some
Overshoot
0.33 uK 0.5 uP 0.33 uP
No Overshoot 0.2 uK 0.5 uP 0.33 uP
3.3 Damped Oscillation Method (DOM)
Herriot proposed a slight modification to the closed loop Ziegler-Nichol tuning method [13]. This
methoddoes not allow the system to undergo oscillations so the process is not driven to marginal stability.
The closed loop system is operated initially with low gain proportional control mode with integral time
constant (Ti) = ∞ and derivative time constant Td = 0 as in the case of Ziegler Nichole ultimate gain method.
The gain is increased or decreased slowly until a (1/4)th
decay ratio of successive overshoot and undershoot is
obtained. At this point, proportional gain (Kp) and the period of damped oscillation Tud are used to calculate
the PID settings based on tuning parameters of Table IV. The optimum settings for the controller are obtained
as Kp = 0.09, (Ti) =0.002 and Td = 0.0005.
Table 4. Herriot tuning parameters for quarter-of-amplitude decay
Type of
Controller pK iT dT
P pdK - -
PID Adjusted 5.1/udT 6/udT
3.4 Tyreus-Luyben Tuning Method (TLM)
The Tyreus-Luyben method follows the same procedure as the Ziegler-Nichol method up till the
point of obtaining the ultimate gain and ultimate period. This method only propose settings for PI and PID
controller as presented in Table V [14]. The ultimate gain Ku and ultimate period Pu obtained under Ziegler-
Nichol method as 1 and 0.003s are used to calculate the PID tune parameters. The proportional gain (Kp),
integral (Ti) and derivative constants (Td) for the PID controller are obtained as 0.4545, 0.0066 and 0.000476
respectively
Table 5. Tyreus-Luyben tuning rules for PI and PID
Type of
Controller pK iT dT
P 2.3/uK uP2.2 -
PID 2.2/uK uP2.2 3.6/uP
3.5 Good Gain Method (GGM)
In the year 2010, Finn Haugen developed a new PID tuning method called the good gain method
[15]. The process is brought close to specified operating point under proportional controller (Kp) only set to
0 or 1, integral time Ti to ∞ and derivative time Td to 0. The proportional gain value Kp is adjusted until the
control loop has some overshoot and a barely observable undershoot as shown in Figure 3. The response is
assumed to represent good stability of the control system with the gain value KpGG. The integral time Ti is
computed from (20)
oui TT 5.1 (20)

300
Where (Tou) is the time between the overshoot and undershoot of the step response with only
proportional controller.
Figure 3. First overshoot and undershoot of the step response
The introduction of I-term in the loop having P-I controller in action might reduce the system stability. To
compensate for this, the can be reduced to 80% of the original value.
pGGp KK 8.0 (21)
The derivative-term can be added to make the controller a PID:
id TT 25.0 (22)
The proportional gain (Kp), integral constants (Ti) and derivative constants (Td) for the controller are obtained
as 0.04, 0.0045 and 0.001125 respectively.
4. RESULTS AND ANALYSIS
The performance analysis of each tuned PID controller using the five tuning methods on the
modelled DC-DC converter are carried out in the Matlab-Simulink environment. The unit step input transient
response in terms of the system rise time, settling time, steady state error and overshoot with unity feedback
without controller (NOC) is presented in Figure 4.
Figure 4. Tuned PID controller response to unit step input
The system time parametric response in Figure 4 shows that all the five controllers are able to
improve the system transient response and eliminate steady state error with no overshoot. The unity feedback
closed loop with no controller has a settling time of about 12 msec. while the good gain (GGM) PID
controller has the slowest response of 5sec. among the tuned controllers. The time response also shows that
0 0.005 0.01 0.015 0.02 0.025 0.03
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (Seconds)
Response
DOM
GGM
MZN
NOC
TLM
ZNFDM

 ISSN: 2088-8694
301
the Ziegler-Nichol frequency domain, damped oscillation, and Tyreus-Luyben PID controllers has faster rise
time of 1.5 msec, 2msec and 2.2 msec respectively at start up.
Toinvestigate the robustness of the tuned PID controllers, the closed loop system comprising of the
modelled converter and the controllers are subjected to set point change. The parametric response in terms of
percentage overshoot, settling time and the controller ability to restore the system back to normal operation to
set point of 5V, 10V and 15V is presented in Figure 5.
Figure 5. Tuned PID controller transient response to set point change
Table 6. PID controller auto-tuning response at set point 5V, 10V and 15V
Set point
Maximum overshoot
(%)
Settling time
(msec.)
5 10 15 5 10 15
Ziegler-Nichol 0 65.5 70.7 1.5 1.5 1.5
Damped oscillation 0 58 34 2 2 1.5
Tyreus-Luyben 0 66 65.3 2.2 2.5 2
Modified Ziegler-Nichol 0 67 56 4 4 4.5
Good Gain 0 60 34.7 5 4 4.5
Table VI shows the summary of the controller’s response to set point change. At start up with 5V
input, the transient response show that Ziegler-Nichol PID controller has faster rise time and a settling time
of 1.5 msec. followed by damped oscillation and Tyreus-Luyben controllers with 2 msec. and 2.2 msec.
respectively. The Modified Ziegler-Nichol and Good gain controllers have the slowest response of 4 msec.
and 5 msec. respectively. It is observed that the five controller has no overshoot at start up but when set point
changed to 10V, the Modified Ziegler-Nichol has highest overshoot of 67% and damped oscillation has the
lowest of 58%. The set point is further increased to 15V, the Ziegler-Nichol has overshoot of 70.7 % and
damped oscillation still has the lowest overshoot of 34%. Among the tuned controllers, the Ziegler-Nichol
PID controller has the faster settling time but with high overshoot while the damped oscillation has the
second fastest settling time with advantage of lowest overshoot to set point change.
A further studies is carried out on the controller’s response to disturbance rejection capability. The
system is subjected to unit step disturbance after 15 msec. and the systems response is presented in Figure 6.
The Ziegler-Nichol PID controller is able to restore the system back to normal operating conditions in 2 msec
while it took 2.2 msec for bothTyreus-Luyben and damped oscillation controllers. The Modified Ziegler-
Nichol and Good gain PID controllers takes 4 msec. and 5 msec. respectively indicating their slow transient.
The controller’s autotunig parametric response to setpoint change and disturbance rejection shows that both
Ziegler-Nichol frequency domain and damped oscillation PID controller offer the best transient in terms of
rise and settling time. The damped oscillation has a distinct advantage of lower overshoot as compared to
Ziegler-Nichol frequency domain which will reduce voltage stress on field effect transitors (FET) device like
MOSFET during practical implementation.
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
0
10
20
30
40
Time (Seconds)
Steprespose
0.015 0.0151 0.0152 0.0153 0.0154
14.5
15
15.5
16
16.5
17
0.025 0.0251 0.0252 0.0253 0.0254
18
20
22
24
26
MZNstep
TLMstep
ZNFDMstep
GGMstep
DOMstep

302
Figure 6. Tuned PID controller response to unit step disturbance
5. CONCLUSION
PID controllers have been designed using different tuning methods and their performance have been
analysed on a DC-DC converter in Matlab-Simulink environment.The transient response in terms of rise time
and settling time of the controllers to step input change and disturbance rejection capability have been
investigated. The Ziegler-Nichol PID controller demonstrated fastest transient response followed by damped
oscillation and Tyreus-Luyben controller respectively. Contrary to fastest transient associated with Ziegler-
Nichol controller as compared to the second fastest damped oscillation PID controller, it has higher
overshoot. The lower overshoot with damped oscillation PID controller makes it a better choice in a sutuition
where there is need to reduce voltage stress on circuit elements like MOSFET during practical
implementation of power converters.
ACKNOWLEDGEMENTS
The authors would like to thank Universiti Teknologi PETRONAS and Malaysian Research
Assessment Instrument (MyRA) Grant No: 0153AB-J17 for supporting this work.
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0 0.005 0.01 0.015 0.02 0.025 0.03
0
0.5
1
1.5
2
2.5
3
Time (Seconds)
Stepresponse
0.015 0.016 0.017 0.018 0.019 0.02
1
1.2
1.4
1.6
1.8
2
DOMdist
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PID Controller Response to Set-Point Change in DC-DC Converter Control

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PID Controller Response to Set-Point Change in DC-DC Converter Control