Robust control of pmsm using genetic algorithm analysis with pid

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 02 Issue: 11 | Nov-2013, Available @ http://www.ijret.org 287
ROBUST CONTROL OF PMSM USING GENETIC ALGORITHM
ANALYSIS WITH PID
T S Vishwanath1
, Subhash S K2
, Shivshankreppa B3
1
Professor, Dept. of ECE, BKIT Bhalki Karnataka, India
2
Principal, JPNCE Mahabob Nagar (AP), India
3
Associate Professor, GND Engg. College, Bidar India
tsvrec1@rediffmail.com, subhashsk@gmail.com
Abstract
In recent years, a remarkable evolution has been achieved by control systems in different application in Robot and many other Areas.
One of the significant applications of developing control systems is the Acceleration control in Permanent Magnet Synchronous
Motor. Control operations are performed statically even after the proposal of diverse techniques in the literature. In addition, H∞
controllers are hardly ever utilized to accomplish this. As a result of this, delayed stability problem occurs in Permanent Magnet
Synchronous Motor while controlling the acceleration or velocity. In this paper, a graphical-based acceleration stabilization
technique is proposed to accomplish effective stability in Permanent Magnet Synchronous Motor controlling operations. the proposed
technique is compared with PID Controller to obtain the best performance specifications such motors with large stability margins
with robust control can be effectly used in the field of robot application.
Keywords: H∞
Controller, PMSM, Stability, Acceleration, Optimisation.
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1. INTRODUCTION
Dc motors are generally controlled by conventional
proportional-integral-derivative (PID) controllers. Since they
can be designed easily and can be built with low cost in
expensive maintenance and effectiveness. It is necessary to
know system mathematical model or to make some
experiments for tuning PID parameters. However, it has been
known that conventional PID controller generally do not work
well for non-linear systems, and practically complex and vague
systems that have no precise mathematical models to overcome
these difficulties. Various types of modified conventional
controllers such as auto-tuning and adaptive PID controllers
can be used for this kind of problems. When compared to the
conventional controller.
In this paper the combined solution we have proposed and
designed a robust controller. We have used PID outer loop in
controller, the gains of the H∞
and PID are tuned on-line by use
of genetic algorithm. The paper investigates the design of H∞
controller for a permanent magnet synchronous motor drive
system, the robust PID and H∞
techniques have been applied to
design the controller. This paper proposes a technique to
develop a optimal robust PI and PID controller for PMSM to
achieve Robustness and performance.
1.1 Mathematical Modelling of a DC motor
DC motors are widely used in industrial and domestic
equipment, the control of speed/ position of a motor with high
accuracy in required the equivalent circuit of field and armature
circuit with its rotation mechanical models as shown in figure1.
Fig1: Separately Exicied DC Drive
Nomenclature:
The following are the physical parameters
Ea: Input/excitation voltage (v);
Eb: Back emf (v);
Ra: Resistance of armature winding (Ω);
Ia: Armature current (A);
La: Inductance of armature winding;
J: M.I of the motor rotor and load kg m2
/s2;
T: Motor torque (Nm);
W: The speed of shaft and the load (angular velocity);

__________________________________________________________________________________________
Φ: shaft position (rad);
ß: Damping ratio of mech.sys (Nm);
Tk: Torque factor constant (Nm/A);
Bk: The motor constant (V-S/rad );
A desired speed may be tracked when a desired shaft position is
also required. Single controller is enough to control both
position and speed of the reference signal in the form of voltage
determines the desired position and speed.
The controller is selected so that the error between the
synchronous and reference signal eventually tends to zero.
There are many DC motors depending upon the type of DC
motor may be controlled by varying the input voltage while for
some other motor speed can be controlled by controlling the
input current. In this work the DC motor is controlled by
varying input voltage. The control design for current is also
same.
For simplicity, a constant value as a reference signal is injected
to the synchronous to obtain a desired position. However the
method works successfully for any reference signal,
particularly for any stepwise continuous time function. This
signal may be a periodic signal or any signal to get a desired
shaft position.
Equations describing the dynamics of the input circuit are
expressed as follows
𝑉𝑡 = 𝐼𝑎 𝑅 𝑎 + 𝐿 𝑎 𝐼𝑎 + 𝐸𝑎
𝑉𝑡 = 𝐼𝑎 𝑅 𝑎 + 𝐿 𝑎
𝑑𝐼𝑎
𝑑𝑡
+ 𝐸𝑎
𝑇 = 𝐽
𝑑𝜔
𝑑𝑡
+ 𝐵𝜔 − 𝑇𝑙
𝑇 = 𝐽
𝑑
𝑑𝑡
𝑑𝜃
𝑑𝑡
+ 𝐵𝜔 − 𝑇𝑙
𝑇 = 𝐽
𝑑2
𝜃
𝑑𝑡2
+ 𝐵
𝑑𝜃
𝑑𝑡
− 𝑇𝑙
𝑇 = 𝐽𝜃" + 𝐵𝜃′ + 𝑇𝑙
𝑇 = 𝐾 𝑇 𝐼𝑎
𝐸𝑎 = 𝐾𝑎 𝜔
𝑑𝜔
𝑑𝑡
= ∅
The dynamic equation in the lap less domain and open loop
transfer function of DC motor
𝑠(𝐽𝑠 = 𝑏)𝜃(𝑠) = 𝑘𝐼 𝑠 (1)
(𝐿𝑠 + 𝑅)𝐼(𝑠) = 𝑉(𝑠) − 𝑘𝑠𝜃 𝑠 (2)
𝑃(𝑠) =
𝜃(𝑠)
𝑉(𝑠)
=
𝑘
𝐽𝑠 + 𝑏 𝐿𝑠 + 𝑅
+ 𝑡2
𝑟𝑎𝑑/𝑠𝑒𝑐
𝑣𝑜𝑙𝑡
The structure of the control system has the standard form.
Fig2: Standard Control Structure
The transfer function of the PID controller is
𝑛 𝑡 = 𝐾𝑝𝑒 𝑡 + 𝐾𝑖 𝑒(𝑡)𝑑𝑡 + 𝐾𝑝
𝑑𝑒
𝑑𝑡
𝑐 𝑠 = 𝑘𝑝 +
𝑘𝑖
𝑠
+ 𝑘𝑑𝑠 =
𝑘𝑑𝑠2
+ 𝑘𝑝𝑠 + 𝑘𝑖
𝑠
All the design criteria cannot be met well PID controller as
other shoot increases drastically with increase in proportional
gain constant. This proportional cannot meet all the design
requirements derivative or integral controller must be added to
the controller.
2. STANDARD H∞
DESIGN:
A standard H-infinity problem [5] which introduces a weight
function to output the error is depicted in Fig. 1. The closed
loop transfer function can be represented as,
Where, W(s) is the weight function and S(s)is the sensitivity
function. In order to make the system internally stable and to
     sSsWsTzw 

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minimize the H∞
norm of Tzw(s), an optimum feedback
controller K(s) is found out by formulating an optimal H∞
control problem, as follows,
0)(min 
sTzw
k (3)
To minimize the H∞
norm, the controller parameters has to be
optimized. Here, the optimization is done by determining the
system parameters through GA. The GA based system
parameters optimization is detailed below.
2.1 Parameters Optimization
The GA plays a major role in optimizing the system and
controller parameters and in obtaining an optimal H∞
controller.
Let NT be the number of system parameters to be optimized.
The parameters are considered to be multi-objective parameters.
Assuming the target parameters to be the gene of the
chromosomes, arbitrary chromosomes of length NT are
generated. The generated chromosomes can be represented as,
 )(
1
)(
1
)(
0
i
N
ii
i T
xxxX 
 
; ,
10  TNj (4)
Where, xj
(i)
is the jth
gene of ith
chromosome, NP is the
population pool and NT is the number of target parameters.
Here, the target parameters considered for optimization are Lq,
m,Rv, α and θ i.e. NT=5. Every gene of the chromosome is
generated arbitrarily within their corresponding minimum and
maximum intervals
i.e.
 minmin)(
0 , qq
i
LLx 
,
 maxmin)(
1 , vv
i
RRx 
,
 maxmin)(
2 ,mmx i

,
 maxmin)(
3 ,i
x
and
 maxmin)(
4 ,i
x
.
The generated gene values are subjected to check whether it
satisfies the controllability constraints or not. If any of the
chromosomes does not satisfy the controllability constraints, a
new chromosome is generated repeatedly until it is satisfied.
The controllability constraints are checked by generating a state
space model using the gene values of the generated
chromosome. For the state space model, the controllability
matrix H is then determined. The constraint can be expressed as
N1-Hrank≤HT, where, N1 is the row size of the matrix H, Hrank is
the rank of the matrix H and HT is the controllability threshold.
Thus, Np chromosomes are generated and the population pool is
filled up.
The fitness of the chromosomes that are in the population pool
is determined. To determine the fitness of the chromosomes,
initially, matrices of dimensions A and B are determined as
follows
























000
0
2
3
1
)(
0
)(
3
)(
2
)(
0
)(
3
)(
2
)(
3
)(
2
)(
1
)(
3
)(
2
)(
i
ii
i
p
ii
iipi
ii
i
x
xxR
x
xx
xxx
xx
A




(5)











0
1
0
1
)(
0
)(
i
i
x
B
(6)
With the matrices of dimension and the other system
parameters, the system and the process P(s) are developed as
follows









DC
BA
sP
ii
i
)()(
)(
)(
(7)
Where, D=0. Considering the system parameters and the plant
function, the fitness can be determined as
iNi
i
P
f

1
minarg
]1,0[ 

(8)
From the fitness function, the chromosomes that have
maximum fitness are placed in the selection pool and the
optimal solution for the H∞
control problem can be determined
as
||)(||minarg sTzw (9)
The fittest Np/2chromosomes that are present in the arbitrary
population pool are selected and they are subjected to the
genetic operations, crossover and mutation.
In the crossover operation, an exchange of genes is performed
between the two parent chromosomes. The crossover is
10  PNi

__________________________________________________________________________________________
performed with a crossover rate of Cr i.e.Cr.NT genes are
exchanged between two parent chromosomes. Hence, a child
chromosome Xchild is obtained for a pair of parent chromosomes.
In this manner, Np/2 parent chromosomes are selected in
sequence from the selection pool and the crossover operation is
performed on them. Hence, new Np/2 child chromosomes are
obtained from the crossover operation. After crossover
operation, the chromosomes are subjected to the next genetic
operation called mutation.
Mutation is an operation that mutates the genes of the
chromosomes to obtain new chromosomes. In our approach, an
adaptive mutation is performed for fast convergence of the
solution with a mutation rate of Mr. The mutation rate decides
the number of genes to be mutated. The mutation operation
performed over a child chromosome is described as follows
 A fittest chromosome, say Xfit
, is selected from the
selection pool i.e. the chromosome which has the
maximum fitness among all the chromosomes that are
present in the selection pool
Based on the gene values of Xfit
, the remaining chromosomes
that are
 present in the selection pool are modified as follows














fit
j
k
jk
j
k
j
fit
j
k
j
k
j
fit
j
k
jk
j
k
j
new
j
xxif
x
x
xxifx
xxif
x
x
x k
)(
)(
)(
)()(
)(
)(
)(
;
1
;
;
1
(10)
 The genes of the obtained new chromosomes are
modified such that the following criterion is satisfied.
… (11)
In Eq. (10), xj
fit
, and xj
newk
are the genes of children
chromosomes that are obtained after performing crossover,
genes of the fittest chromosome and genes of the newly
obtained chromosome, respectively. Hence, all the genes
present in the chromosomes are modified as per the fitness
function. As this process relies on the fittest chromosome,
quick convergence can be accomplished. At the end of
mutation, Np/2 new chromosomes Xnew are obtained. For the
Np/2 new chromosomes, fitness is determined using the Eq.10.
The entire process is repeated for Imax iterations. Once it reaches
Imax iterations, the process is terminated. The chromosome with
maximum fitness present in the selection pool is chosen as the
best system parameters. On the basis of these parameters the
system and the controller are developed. The obtained optimal
H∞
controller can work satisfactorily for all the given system
parameters. Hence, the system can offer a good stabilization
over the velocity, which is considered as the system output.
From the obtained velocity, the acceleration can be determined
as
  bestvm mRfFdtdv //  (11)
Where, Fm is the output electromagnetic thrust, f is the total
friction, Rv is the damper coefficient, v is the velocity output
obtained from the system and mbest is the best mass value
obtained from the proposed technique.
3. PID
Integral term reduces the steady state error and adding
derivative term reduces the overshoot. PID controls which
small KD and Ki in this case the time required in large to go to
steady state. Design PID with appropriate Kp, kD & kI will
give satisfactory results. PID is used to get rid of the steady
state error due to disturbance. MATLAB provides tools for
automatically choosing optimal PID gains which makes to trial
and process.
The transfer function of the PI control
𝑢 = 𝑘𝑒 +
1
1 + 𝑠𝑡
𝑢 = 𝑘
1 + 𝑠𝑡
𝑠𝑡
𝑒 = (𝑘 +
𝑘
𝑠𝑡
)
The simplified dynamic model of the pm synchronous system is
as shown below.
The pm synchronous motor and the +inverter need to be
designed to obtain a mathematical model on which it design of
the robust controller is based the inverter output current closely
follow the reference current commands owning to the closed
current loop control.
The actual motor line currents are in d-q coordinates. zq and zd
are thus assumed to be equal to their reference commands z*
q
and z*
d respectively. The vector control loop places the stator
current vector is on the q-axis and the rotor flux on the d-axis
this implies that,
z*d = zd =0 and λq=0










otherwisex
xxifx
xxifx
x
new
j
j
new
jj
j
new
jj
new
j
;
;
;
maxmax
minmin

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The mathematical model of pm synchronous motor can be
designed in the d-q synchronous rotating reference frame by the
following three-state non-linear differential equation.
4. ROBUST –CONTROL SCHEME
Fig.3
Thus the closed loop transfer function W to Z is given by
𝑇𝑧𝑤 𝑠 = 𝑐 𝑑 (𝑆𝐼 − 𝐴 𝑑)−1
𝐵𝑑 + 𝐷𝑑
The resulting closed-loop system is internally stable if and if
only the Eigen values of.
4.1 Mathematical Model Of PMSM
Research has indicated that the permanent magnet motor drive,
which include the PM synchronous Motor have became serious
competitors to the indication motor for servo applications the
PMSM has a sinusoidal back emf and require sinusoidal stator
currents to produce constant torque, PMSM is very similar to
the wound rotor synchronous M/c except that the PMSM that is
used for servo applications tends not to have any damper
windings and excitation is provided by the permanent magnet
instead of a field winding. Hence the d, q model of the PMSM
can be designed from the well known model of the synchronous
m/c with the equations of the damper windings and field
current-dynamics removed.
𝛹𝑎 = 𝐿 𝑎 𝑍 𝑎 + 𝐿 𝑏 + 𝑍 𝑏 + 𝐶𝑐 𝑍𝑐 + 𝛹𝑟𝑎
𝛹𝑏 = 𝐿 𝑏 𝑍 𝑎 + 𝐿 𝑏 𝑍 𝑏 + 𝐿 𝑐 𝑍𝑐 + 𝛹𝑏
𝛹𝑐 = 𝐿 𝑐 𝑍 𝑎 + 𝐿 𝑐 𝑍 𝑏 + 𝐿 𝑐 𝑍𝑐 +𝛹𝑐
𝐿 𝑎 , 𝐿 𝑏 , 𝐿 𝑐 are the self inductances of the stator a-b-c phases
respectively. a-b-c phases respectively,
𝐿 𝑎𝑏 , 𝐿 𝑏𝑐 , 𝐿 𝑎𝑐 Are mutual inductances, hence three phases, based
on the rotor angle position the flux linkages can be expressed as,
𝛹𝑎 = 𝜓𝑟 𝑐𝑜𝑠 𝜃
𝛹𝑏 = 𝜓𝑟 𝑐𝑜𝑠(𝜃 − 120°)
𝛹𝑐 = 𝜓𝑟 𝑐𝑜𝑠(𝜃 + 120°)
𝛹 = 𝛷𝑚 Or max value of flux.
The electrical dynamic equations in terms of phase variables
can be written as,
𝑉𝑠 = 𝑟𝑠 𝑧𝑠 +
𝑑
𝑑𝑡
[∆𝑠] ---- (12)
When the three phase system is symmetrical the voltage form a
balanced three phase set of sequences the sum of the set is zero
hence the phase the voltage can be expressed in state model on.
The voltage can be expressed in state model on
𝑉𝑎
𝑉𝑞
𝑉0
= 𝑟𝑠
𝑍 𝑑
𝑍 𝑞
𝑍0
+ 𝜔𝑟
0 −1 0
1 0 0
0 0 0
𝛹𝑑
𝛹𝑞
𝛹0
+ 𝑝
𝛹𝑑
𝛹𝑞
𝛹0
The d-q voltage model for the PMSM in matrix form,
𝑉𝑑
𝑉𝑞
=
(𝑟𝑠 + 𝑙 𝑑𝑝 ) 𝑙 𝑞 𝑤
𝑙 𝑑𝑤 𝑟𝑠 + 𝑙 𝑞 𝑝
𝑍 𝑑
𝑍 𝑞
+ 𝛹𝑟 𝜔
0
1
Where 𝑉𝑑&𝑉𝑞 are the d-q axis armature voltages r = R phase
resistance.
𝑑𝜃
𝑑𝑡
= 𝜔 In electrical analysis velocity =
𝑉𝜋
𝜁
Where, 𝜁 = 𝑝𝑜𝑙𝑒 𝑝𝑖𝑡𝑐ℎ
V= velocity of moving part
Now electromagnetic thrust is given by,
𝑃𝑜𝜔 = 𝐹𝑚 =
3
2
𝑃𝑛
𝜋
2
[𝜑 𝑑 𝐼𝑞 − 𝜑 𝑑 𝑧 𝑑]
If 𝜔𝑟 = (
𝑃
2
) 𝜔 𝑟𝑚 . 𝜔𝑟𝑚 is the mechanical rotor speed in rad/sec.
Now electromechanical torque
𝑃𝑧𝑑 =
𝑉𝑑 − 𝑅 𝑧𝑑 − 𝜔𝑠 𝐿 𝑞 𝑧 𝑞
𝐿 𝑑
𝑃𝑧 𝑑 =
𝑉𝑑 − 𝑅 𝑧𝑑 − 𝜔𝑠 𝐿 𝑞 𝑧 𝑞 − 𝜔𝑠 𝜑 𝑎 𝑓
𝐿 𝑞
𝑇𝑒𝑚 =
3
2
𝑃 𝜑 𝑎𝑓 𝐿 𝑞 + 𝐿 𝑑 − 𝐿 𝑞 𝑧 𝑑 𝑧 𝑞
= 𝐽𝑃𝜔𝑟 + 𝐵𝜔𝑟 + 𝑇𝑙
Where 𝑃 =
𝑑
𝑑𝑡
𝜔𝑟, 𝜔𝑠 are frequency rotor speed. P is number of pole pairs.

__________________________________________________________________________________________
Final the motion equation can be written as
𝑑𝑣
𝑑𝑡
=
𝐹 𝑚 −𝑓−𝑅𝑣
𝑚
The inverter o/p currents are assumed to be very close to the
reference current commands. In the d-q co0ordinates, the motor
line currents Zq and Zd are approximated to the reference
commands Z*
q and Z*
d. Maximum torque can be obtained by
means of field oriented per ampere when the stator current is
properly placed on the q-axis and rotor flux on the d-axis.
When Zd ≈ Z*
d the maximum flux is constant. Hence torque is
directly proportional to q-axis current. The mathematical model
of PMSM is simuslated and simulation program is written in
MATLAB and used to verify the basic operation.
The mathematical model is capable of simulating the steady
state as well as dynamic respectively.
𝑑𝑧𝑞
𝑑𝑡
= (𝑉𝑞 − 𝑅 𝑧𝑞 − 𝜔𝑠𝛹𝑎𝑓)/𝐿 𝑞 (13)
𝑑𝜔𝑟
𝑑𝑡
=
𝑇𝑒 − 𝑇𝐿 − 𝐵 𝜔𝑟
𝐽
𝑇𝑒 = 𝑘𝑡 𝑧 𝑞 Where 𝐾𝑡 =
3𝑃𝛹𝑎
2
If the PMSM is driven by const-current-source inverter the zq is
known as acts as excitation current.
Here the internal motor characteristics and current feedback
control play an important role in establishing.
A simplified
To obtain the max torque/amp than the three phase current
command can be determined as follows.
𝑍 𝑎
∗
𝑍 𝑏
∗
𝑍𝑐
∗
=
𝑐𝑜𝑠 𝜃 − 𝑠𝑖𝑛 𝜃
𝑐𝑜𝑠(𝜃 − 120°) − 𝑠𝑖𝑛(𝜃 − 120°)
𝑐𝑜𝑠(𝜃 + 120°) − 𝑠𝑖𝑛(𝜃 + 120°)
𝑍 𝑑
∗
𝑍 𝑞
∗
(14)
𝑍 𝑎
∗
, 𝑍 𝑏
∗
, 𝑍𝑐
∗
are stationary frame a, b, c axis currents. 𝑍 𝑑
∗
, 𝑍 𝑞
∗
are
stationary frame d-q axis current commands.
5. DESIGN OF H∞
CONTROLLER
H∞
optimization has emerged on a unfair tool for Robust
control system design. The H∞
theory has sound back ground
for handling model uncertainties. And diff system operating
conditions, H∞
control design can early combine several
specifications such as disturbance attenuation, tracking,
Bandwidth, and Robust stabilizes into a single minimum
frequency domain probe using with mixed sensitivity
approaches loop shaping design procedure by mc far lane [7] in
this approach H∞
controller band on mix sensitivity approach is
synthesized for a plant transfer function designed for PMSM.
Using robust control tool box of MATLAB software the
optimization problem is solved by using this genetic algorithm
approach, selecting weights to obtain better performance and
robustness.
6. RESULTS AND DISCUSSIONS

__________________________________________________________________________________________
CONCLUSIONS
The performance of H∞
controller in stabilizing the
accelarattion and velocity of PMSM has considerabaly
improved as compared with PID controller. All the time
domain specificaitons have been met with a broader margin.
REFERENCES
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and P. Brot, "Challenges inBuilding Fault -Tolerant
Flight Control System for a Civil Aircraft",
IAENGInternational Journal of Computer Science,
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Wooseung Kim, and David J.Beebe, "Control
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[7]. V.Barbu and S.S.Sritharan, "H infinity-control theory
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[8]. Jayapal and J.K.Mendiratta, "H infinity Controller
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