International Electrical Engineering Journal (IEEJ)
Vol. 7 (2017) No.9, pp. 2377-2384
ISSN 2078-2365
http://www.ieejournal.com/
2377 Moustafa et. al., Control of nonlinear phenomena in a DC chopper-fed PMDC drive
Abstract— the effects of nonlinearity in a PMDC drive are the
main problems when apply a conventional control algorithm (p
or pi controller). As some system parameter such as the
controller gains or the supply voltage is being varied, the
nominal period-1 orbit in the drives may lose stability and lead
to nonlinear phenomena such as chaos and bifurcation. So that
we need to improve controllers that are match the parameter
variations. In this paper we use Simulink model to describe
fuzzy controller to control the nonlinear phenomena in a dc
chopper-fed PMDC drive and compare the results with p and pi
controller.
Index Terms—nonlinear phenomena, chaos, p controller, pi
controller, FLC, effects of nonlinearity, PMDC drive,
period-doubling bifurcation, Neimark-sacker bifurcation and
Simulink model
I. INTRODUCTION
The sources of nonlinearity in power electronics are power
switching devices (diode, SCR, BJT, power MOSFET and
IGBT), reactive components and electrical machine and
drives [1].
Nonlinear phenomena such as chaos and bifurcation can
lead system to harmful situations. So nonlinear phenomena
should be reduced as possible or totally suppressed [12].
In this paper we use fuzzy controller to control nonlinear
phenomena in a PMDC drive.
Lotfi Zadeh is the first one who propose fuzzy logic
controller in 1965. Fuzzy logic controller used in a lot of
intelligent applications [2, 7, 13]. The execution of fuzzy
rules depends on the operations done by human operators
does not need a mathematical model of the system [3]. The FLC steps is presented in section II, while in section III
present studying the stability of DC Chopper-Fed PMDC
Drives using Proportional integral (pi) Controller, section IV
present Designing of Fuzzy pi controller for the speed control
of nonlinear phenomena in a DC chopper-fed PMDC drive
and finally in section V present the conclusion for that
system.
II. FLC STEPS
Fuzzy logic controller (FLC) consists of fuzzification
interface, fuzzy control rules, inference engine and
defuzzification interface as shown in fig.1 [4].
Fig. 1 the basic structure of fuzzy logic controller
Where x1, x2 are the inputs of the FLC, uf is fuzzy control
action, u is the crisp control action and G1, G2, Gu are gains of
input and output.
A. FUZZIFICATION
The fuzzification strategy converts the crisp input data into
fuzzy sets (linguistic variables) and consists of membership
functions that describe the fuzzy rules. These functions can
be triangle, trapezoidal, quasi-linear and Gaussian shaped.
The triangular-shaped is usually used as membership.
B. FUZZY CONTROL RULES
We represent the fuzzy control rules by the form:
IF (Process state) THEN (actions can be inferred)
This describe what action should be taken from currently
information, which includes both input and feedback if a
closed-loop control system is applied [5, 6].
C. INFERENCE ENGINE
Converts the input fuzzy sets to the output fuzzy set. The
most important two types of fuzzy inference method are
Mamdani and Sugeno fuzzy inference methods [5, 6].
Fuzzy madmani inference system is shown in fig.2
Control of Nonlinear Phenomena in a
DC Chopper-Fed PMDC Drive
Eman Moustafa1, Belal Abou-Zalam2, Abdel-Azem Sobaih3
Industrial Electronics and Control Engineering Department, Faculty of Electronic Engineering
eman.osman88 @gmail.com
International Electrical Engineering Journal (IEEJ)
Vol. 7 (2017) No.9, pp. 2377-2384
ISSN 2078-2365
http://www.ieejournal.com/
2378 Moustafa et. al., Control of nonlinear phenomena in a DC chopper-fed PMDC drive
Figure (2) Fuzzy madmani inference system
D. DEFUZZIFICATION
The defuzzification is the reverse of fuzzification. [9]
Converts the output fuzzy set to crisp output. [5, 6] as the
requirements of the real world, the fuzzy output should
convert to a crisp output [9].
III. STUDY THE STABILITY OF THE SYSTEM WITH
PROPORTIONAL (P) CONTROLLER
A. SYSTEM OVERVIEW
This system consists of the power converter (dc chopper),
control electronic and PMDC motor. The shaft speed can be
controlled by control of the average voltage of the armature
via pulse width modulation (PWM). A Schematic diagram of
voltage mode controlled DC chopper-fed PMDC drive is
shown in Fig.3. And the Simulink model of DC chopper-fed
PMDC drive employing the P controller is shown in Fig.4.
Fig.3 a schematic diagram of the DC chopper-fed PMDC drive employing
the P controller.
Fig.4 the Simulink model of DC chopper-fed PMDC drive employing the PI
controller.
The parameters of the system are:
R=7.8 Ω, L=5mH, TL =0.087NM, Ke =0.0984Vs/rad,
Kt =0.09NM/A, ꙍref =100rad/s, B=0.000015Nm/rad/sec,
J=4.8400e-005Nm/rad/sec2, fs=20 kHz, T=0.05ms, VL =0
and VU=8V.
B. DYNAMICAL BEHAVIOR OF THE SYSTEM
The nominal behavior of a DC chopper-fed PMDC drive
employing the P controller is a period-1 orbit at Vin=24V and
KP=2200 (Figs. 5 and 6). But as the proportional gain (KP) or
the supply voltage (Vin) is varied, the Period-1 orbit loses
stability via period-doubling bifurcation [1, 11] and further
variation will lead to chaos.
When the proportional gain fixed at 2200 and varying the
supply voltage, the fast scale bifurcation occurred at Vin
=32V (figs.7 and 8). Further variation of the supply voltage
leads to chaos (figs.9 and 10).
(a)
(b)
Fig. 5 period-1(a) speed and (b) current trajectory in time
domain; KP=2200, Vin =24 V.
Fig.6 Period-1 phase portrait of speed against current
International Electrical Engineering Journal (IEEJ)
Vol. 7 (2017) No.9, pp. 2377-2384
ISSN 2078-2365
http://www.ieejournal.com/
2379 Moustafa et. al., Control of nonlinear phenomena in a DC chopper-fed PMDC drive
(a)
(b)
Fig. 7 period-2 (a) speed and (b) current trajectory in time
domain; KP=2200, Vin =32 V.
Fig.8 Period-2 phase portrait
(a)
(b)
Fig. 9 chaotic (a) speed and (b) current trajectory in time
domain; KP=2200, Vin =45 V.
Fig.10 chaotic phase portrait
IV. STUDY THE STABILITY OF THE SYSTEM WITH
PROPORTIONAL INTEGRAL (PI) CONTROLLER
A. SYSTEM OVERVIEW
A schematic diagram of the DC chopper-fed PMDC drive
employing the proportional integral (PI) controller is shown
in Fig.11.and the Simulink model of DC chopper-fed PMDC
drive employing the PI controller is shown in Fig.12. The
system is consists of three components are PMDC motor, DC
chopper and PI controller. The voltage produced from the
tacho-generator is proportional to the actual speed, this
actual speed compared with the reference speed to obtain an
error signal. This error signal is used by the controller to
produce a control voltage Vcon (t) that is compared with a
sawtooth signal Vramp (t) to produce the PWM signal.
When the PWM signal is high, the switch turns ON, and the
diode will be reverse biased (OFF). But when the PWM
signal is low, the switch turns OFF, and the diode will be
forward biased (ON), thus providing a return path for the
decaying armature current. The mathematical model of the
PI controlled PMDC drive is discussed in detail in [1, 10].
International Electrical Engineering Journal (IEEJ)
Vol. 7 (2017) No.9, pp. 2377-2384
ISSN 2078-2365
http://www.ieejournal.com/
2380 Moustafa et. al., Control of nonlinear phenomena in a DC chopper-fed PMDC drive
Fig.11 a schematic diagram of the DC chopper-fed PMDC drive employing
the PI controller.
Fig.12 the Simulink model of DC chopper-fed PMDC drive employing the
PI controller.
B. DYNAMICAL BEHAVIOR OF THE SYSTEM
The nominal behavior of a DC chopper-fed PMDC drive
employing the PI controller is a period-1 orbit (Figs. 13 and
14). But as the integral gain (KI) or the supply voltage (Vin) is
varied, the Period-1 orbit loses stability via Neimark-Sacker
bifurcation [1, 11] and a quasiperiodic orbit is born. Fixed KP
at 1 and KI at 1580 and varying the supply voltage, the
Neimark-Sacker bifurcation occurred at Vin =57V (Figs. 15
and 16). Further variation of the supply voltage (Vin = 65 V)
convert the system from CCM to DCM (Figs. 17 and 18).
(a)
(b)
Fig.13 period-1 (a) speed and (b) current trajectories in time domain at Vin
=24 V.
International Electrical Engineering Journal (IEEJ)
Vol. 7 (2017) No.9, pp. 2377-2384
ISSN 2078-2365
http://www.ieejournal.com/
2381 Moustafa et. al., Control of nonlinear phenomena in a DC chopper-fed PMDC drive
Fig.14 Period-1 phase portrait of speed against current and integrator
output.
(a)
(b)
Fig.15 (a) speed and (b) Quasi-periodic current trajectories at Vin =57 V.
Fig.16 Torus Phase portrait of speed against current and integrator output at
Vin =57V.
(a)
(b)
Fig.17 (a) speed and (b) current (transition from CCM to DCM) trajectories
at Vin =65 V.
Fig.18 DCM Phase portrait of speed against current and integrator output at
Vin =65V.
V. DESIGNING OF FUZZY PI CONTROLLER FOR THE SPEED
CONTROL OF NONLINEAR PHENOMENA IN A DC
CHOPPER-FED PMDC DRIVE
Fuzzy logic is suitable for a model that is difficult to control
and non-linear ones [2, 7].Fuzzy controller provide better
results than other controllers. Fuzzy logic controller (FLC)
can decrease the nonlinearity effectiveness in a DC motor
and progress the execution of a controller [8].
A. SYSTEM OVERVIEW
The basic structure of fuzzy pi controller is shown in fig. 19
International Electrical Engineering Journal (IEEJ)
Vol. 7 (2017) No.9, pp. 2377-2384
ISSN 2078-2365
http://www.ieejournal.com/
2382 Moustafa et. al., Control of nonlinear phenomena in a DC chopper-fed PMDC drive
Fig. 19 The basic structure of fuzzy pi controller
As shown in Fig.19 there are two inputs ,One is the control
error e(k), which is the difference between the reference
signal r(k) and the output signal y(k), the other one is the
change in this error ∆e(k).and one output is change of control
output ∆u(k). The equations of inputs and output is expressed
as:
e (k) = r(k)- y(k) (1)
∆e (k) = e (k) – e (k-1) (2)
u (k)= u(k-1)+ ∆u(k) (3)
Where r (k) is reference speed, y (k) is actual speed, e (k-1)
is previous error, u (k) is control output and u (k-1) is
previous control output.
Simulink model of the system using Fuzzy pi controller is
shown in fig.20 and the Simulink model of fuzzy pi controller
block is shown in fig.21
Fig.20 Simulink model of the system using Fuzzy pi controller
Fig.21 Simulink model of fuzzy pi controller block
Fuzzy mamdani inference developed for the FLC of the
system is shown in fig.22.
In this design 5 membership functions for each input (error
& change in error) and output (change in control signal) are
used: NB (Negative Big), NM (Negative Medium), NS
(Negative Small), ZE (Zero), PS (Positive Small), PM
(Positive Medium) and PB (Positive Big) and are shown in
Fig.23
Fig.22 Fuzzy mamdani inference developed for the FLC of the system
(a) Membership functions for error
(b) Membership functions for change of error
(c) Membership functions for change of control signal.
Fig.23 Membership functions for inputs and output
International Electrical Engineering Journal (IEEJ)
Vol. 7 (2017) No.9, pp. 2377-2384
ISSN 2078-2365
http://www.ieejournal.com/
2383 Moustafa et. al., Control of nonlinear phenomena in a DC chopper-fed PMDC drive
In order to model the actions that a human operator would
decide the change in the controller output (∆u) according to
the error e and its change ∆e, it is necessary to observe the
behaviors of the error signal e and its change ∆e on different
operating regions. Therefore, the signs of e and ∆e are used to
determine the signs of ∆u. The sign of ∆u should be positive if
u is required to be increased and it should be negative
otherwise. The fuzzy rule base (fuzzy decision table) is shown
in Table I.
Table I fuzzy rule base for FLC
∆e
e
NB NS ZZ PS PB
PB ZZ PS PS PB PB
PS NS ZZ PS PS PB
ZZ NS NS ZZ PS PS
NS NB NS NS ZZ PS
NB NB NB NS NS ZZ
B. DYNAMIC BEHAVIOR OF DC CHOPPER FED PMDC
DRIVES EMPLOYING THE FUZZY PI CONTROLLER.
Fuzzy pi controller make the PMDC motor speed control
smooth, FLC Performs fast tracking speed and zero or very
small steady state error is observed as shown in fig.24. FLC
leads to a stable system.
When Vin =24 volt the period-1 speed and current are
shown in fig.24 and period-1 phase portrait is shown in
fig.25. By varying the supply voltage to 57 the system still
stable as shown in figs. (26 and 27).and further variation of
the supply voltage doesn’t lead to change the stability of the
system and still in period-1 as shown in figs. (28 and 29).
(a)
(b)
Fig.24 period-1 (a) speed and (b) current trajectories in time domain at
Vin =24 V.
Fig.25 Period-1 phase portrait of speed against current at Vin =24 V.
(a)
International Electrical Engineering Journal (IEEJ)
Vol. 7 (2017) No.9, pp. 2377-2384
ISSN 2078-2365
http://www.ieejournal.com/
2384 Moustafa et. al., Control of nonlinear phenomena in a DC chopper-fed PMDC drive
(b)
Fig.26 period-1 (a) speed and (b) current trajectories in time domain at
Vin =57 V.
Fig.27 Period-1 phase portrait of speed against current at Vin =57 V.
(a)
(b)
Fig.28 period-1 (a) speed and (b) current trajectories in time domain at
Vin =65 V.
Fig.29 Period-1 phase portrait of speed against current at Vin =65 V.
VI. CONCLUSIONS
In this paper we use waveforms and phase portrait to study
the occurrence of nonlinear phenomena.
Fuzzy pi controller make the PMDC motor speed control
smooth, FLC Performs fast tracking speed and zero or very
small steady state error is observed. FLC leads to a stable
system as shown in V.
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