Design and implementation of Robust MultirateOutput Feedback based Sliding Mode Controller forInduction Motors using FPGA
Vini TutejaInstitute of Technology,Nirma UniversityAhmedabad, Gujarat, IndiaEmail: 10micc17@nirmauni.ac.inJignesh B. PatelInstitute of Technology,Nirma UniversityAhmedabad, Gujarat, IndiaEmail: jbpatel@nirmauni.ac.inAxay MehtaGuj. Power Engg.& Research Institute,Mehsana, Gujarat, IndiaEmail: draxaymehta@gmail.com
Abstract
—This paper demonstrates design of Multirate Output feedback(MROF) based Discretetime Sliding Mode Controller(DSMC) for sensorless control of Induction Motor(IM)implemented using System Generator (SysGen). A multirateoutput technique is used to estimate the state variables of Induction Motor. For designing the observer and controller, alinearised model is used. The efﬁcacy of the scheme is shown inthe simulation result.
Keywords
Multirate output feedback; Sliding Mode Controller(SMC); Field Programmable Gate Array (FPGA); Sensorless IMDrives; System Generator.
I. I
NTRODUCTION
Induction motors are the horsepower of industry andthey need to be controlled. There are various techniquesavailable in literature for the control of sensorless Inductiondrives[1],[2],[3].Variable speed drive for induction motor iswidely used drive in the industry. Generally, variable speeddrives (VSD) for induction motor require both wide range of operating speed and fast torque response, regardless of anydisturbances and uncertainties. This demands more advancedcontrol techniques for speed control of Induction Motor.Further, in the formulation of any control problem there willtypically be discrepancies between the actual plant and themathematical model developed for controller design. Thismismatch may be due to un modeled dynamics, variation inparameters or the approximation of complex plant behaviorby straightforward model. The designer has to ensure thatthe resulting controller has the ability to produce requiredperformance levels in practice despite such plant/modelmismatches. One particular approach to robust controllerdesign is Sliding Mode Control methodology [4],[5],[6],[7].A detailed survey on development and applications of SMCis presented in [8]. SMC is equally useful for linear as wellas certain class of nonlinear systems and that is why it is aleading contender for Induction Motor drives. It gives orderreduction, invariance to both torque variation and parameteruncertainties along with fast dynamic response. The realimplementation issues can be addressed with DSMC only.Discretetime counterpart of sliding mode theory is taken upby various research groups across the globe[7],[9].One of the popular and widely used control technique for theInduction Motor is Vector Control strategy which requiresﬂux position and speed measurement as input variables.Speed sensors reduce the reliability of the drive along withincreasing the price. Also, direct measurement of magneticﬁeld using hall effect sensors need the mounting of sensorsin the air gap of the machine increases the complexity.The ﬂux and speed estimator or another technique may beused to solve the said problems[2],[3]. Recently developedMROF based SMC is discussed in [10],[11], but the issues of implementation are not discussed yet.Also, most of the AC drives nowadays are implemented usingeither fully Digital Signal Processor(DSP) based controlstrategy or with both FPGA and DSP together, as DSP basedcircuits are simple and also ﬂexible in adapting variousapplications. But these circuits are sluggish and allow limitedcomputation resources due to the sequential computationfeature, complicated design process and long developmenttime cycle. Multiple DSPs can solve the problem but withincreased cost. FPGA provide an economic solution and fastcircuit response due to its simultaneous execution property.FPGA have high processing rate and consume less powercompared to DSPs [12],[13],[14],[15].In this paper, we propose an MROF based DSMC forsensorless IM using FPGA. The simulations have beencarried out using SysGenthe industry’s leading highlevelDSP design tool from XILINX. With SysGen, people withlittle knowledge of VHDL/Verilog, can create productionquality FPGA implementations using MATLABSimulink.SysGen’s Hardware Cosimulation feature allows one tovalidate working hardware as it supports JTAG communicationbetween FPGA hardware platform and Simulink [16]. It isa universally accepted simulation tool for modeling andanalysis of FPGA based concepts[17],[18],[19],[20].Thepaper is organized as follows: The basic concept of DSMCand MROF is discussed in section II. Section III discussesthe design. Section IV encapsulates the simulation results of MROF based SMC for sensorless Induction Motor followed
by the conclusion in section V.II. R
EVIEW OF
MROF
BASED
D
ISCRETE

TIME
SMC
TECHNIQUE
Consider the discretetime plant
x
(
k
+ 1) =
A
d
x
(
k
) +
B
d
u
(
k
)
(1)
y
(
k
) =
C
d
x
(
k
)
(2)where,
xǫR
n
,
uǫR
m
,
yǫR
p
,
A
d
ǫR
n
×
n
,
B
d
ǫR
n
×
m
and
C
d
ǫR
p
×
n
such that
C
d
B
d
is nonsingular. We also assumethat
(
A
dτ
,B
dτ
)
is completely controllable and
m < n
.The design of sliding mode control includes designing aswitching surface
s
(
x
(
k
)
,k
) =
{
x
(
k
)
/s
(
k
) =
Cx
(
k
) = 0
}
and design of a suitable control law
u
(
x
(
k
)
,k
)
such that anystate of the system outside the said switching surface is drivento reach the surface in ﬁnite time [21],[22].
A. DSMC design1) Switching surface design:
The switching function forDSMC is given as
s
(
k
) =
c
T
x
(
k
)
.
(3)Let the system in Eqn. 1 be transformed to a regular form by atransformation
¯
x
(
k
) =
T
r
x
(
k
)
, where
T
r
is the transformationmatrix, resulting in the following dynamics:
¯
x
(
k
+ 1) =
A
d
11
A
d
12
A
d
21
A
d
22
¯
x
(
k
) +
0
B
d
2
u.
(4)The sliding surface for the transformed system in Eqn. 4 isgiven by
¯
c
T
¯
x
= 0
, where
¯
c
T
=
K I
m
On the sliding surface, the system will follow the relation
¯
x
2
=
−
K
¯
x
1
, where
¯
x
2
comprises of last
(
m
)
states of
x
(
k
)
.Thus, the
¯
x
1
dynamics for the transformed equation becomes
¯
x
1
(
k
+ 1) = (
A
d
11
−
A
d
12
K
)¯
x
1
(
k
)
.
(5)From Eqn.5, it can be said that if
K
is chosen such that theeigenvalues of
(
A
d
11
−
A
d
12
K
)
are placed at the desiredlocations, then
¯
x
1
is stabilised and as
¯
x
2
=
−
K
¯
x
1
,
¯
x
2
alsobecomes stable on the switching surface.Thus,the sliding surface in terms of srcinal state coordinatesis given by
s
(
k
) = ¯
c
T
¯
x
(
k
) = ¯
c
T
T
r
x
(
k
)
.
(6)
2) Control law design:
The Gao’s reaching law [22] forDSMC is given by
s
(
k
+ 1)
−
s
(
k
) =
−
qτs
(
k
)
−
ǫτsgn
(
s
(
k
))
(7)where,
τ
is the sampling period
q,ǫ >
0
and
1
−
qτ >
0
should hold to guarantee the reaching phasestability. Using Eqn.1, Eqn.3 & Eqn.7, the control law canbe deﬁned as
u
=
Fx
(
k
) +
γsgn
(
s
(
k
))
(8)where,
F
=
−
(
c
T
B
d
)
−
1
c
T
[
A
d
−
I
+
qτ
]
γ
=
−
(
c
T
B
d
)
−
1
ǫτ.
In DSMC, the measurement and the control signal are updatedonly at regular intervals ,i.e. the sampling period and thecontrol input is considered to remain constant for a samplingperiod. In DSMC, unlike the continuous SMC, the states moveabout the sliding surface but are unable to stay on it. Thus,DSMC is said to exhibit Quasi Sliding mode.As discussed above, the DSMC design is based on statefeedback. However, in many systems it is not possible to getor measure all the state variables and so resort us to use outputmeasurements.It has been recently shown [23] that using multirate outputfeedback technique, it is always possible to obtain statevector for all controllable and observable systems within onesampling period. Moreover,it is also shown in [23],that MROFguarantees closed loop stability which is not the case in staticoutput feedback.MROF is the concept of sampling the control input andsensor output at different rates. Here the output is sampledfaster than the input[10],[11],[24],[25].It was found that the state feedback control law may berealised by the use of MROF, by representing the systemstates in terms of past control input and multirate sampledsystem outputs[23].Consider the continuous system
˙
x
=
Ax
+
Bu
(9)
y
=
Cx
(10)where,
xǫR
n
,
uǫR
m
,
yǫR
p
,
AǫR
n
×
n
,
BǫR
n
×
m
and
CǫR
p
×
n
.Let the above system be sampled at a sampling interval
τsec
and given as
x
(
k
+ 1)
τ
=
A
dτ
x
(
k
) +
B
dτ
u
(
k
)
(11)
y
(
kτ
) =
C
dτ
x
(
k
)
.
(12)Consider the input to be sampled every
τsec
and the output besampled faster at a period
∆
sec
such that
∆ =
τ/N
, where Nis an integer greater than or equal to the observability indexof the system.[23]The system sampled at
∆
period be given by
x
(
k
+ 1)∆ =
A
d
∆
x
(
k
) +
B
d
∆
u
(
k
)
(13)
y
(
k
∆) =
C
d
∆
x
(
k
)
.
(14)If the past
N
sampled outputs are represented as
y
k
=
y
((
k
−
1)
τ
)
y
((
k
−
1)
τ
) + ∆
..y
(
kτ
−
∆)
,
(15)
and if
kτ
is replaced by
k
, then the multirate output sampledsystem can be written as
x
(
k
+ 1) =
A
dτ
x
(
k
) +
B
dτ
u
(
k
)
(16)
y
(
k
+ 1) =
C
0
x
(
k
) +
D
0
u
(
k
)
.
(17)where,
C
0
=
C CA
d
∆
CA
2
d
∆
..CA
N
−
1
d
∆
,D
0
=
0
CB
d
∆
C
(
B
d
∆
+
I
)
..C
Σ
N
−
2
j
=0
A
d
∆
j
B
d
∆
.
(18)Solving the MROF system equations, the estimated states
x
o
(
k
)
can be given by
x
o
(
k
) =
L
1
y
(
k
) +
L
2
u
(
k
−
1)
(19)where,
L
1
=
A
dτ
(
C
T o
C
o
)
−
1
C
T o
L
2
=
B
dτ
−
A
dτ
(
C
T o
C
o
)
−
1
C
T o
D
o
.
Substituting the value of the estimated states from Eqn.19into Eqn. 8, the control law for the MROF can be written as,
u
mrof
(
k
) =
Fx
o
(
k
) +
Gsgn
(
s
mrof
(
k
))
(20)where,
s
mrof
(
k
) =
c
T
x
o
(
k
)
F
=
−
(
c
T
B
dτ
)
−
1
c
T
(
qτ
−
I
+
B
dτ
)
G
=
−
(
c
T
B
dτ
)
−
1
ετ.
III. D
ESIGN OF
DSMC C
ONTROLLER
F
OR
I
NDUCTION
M
OTOR
The behaviour of a threephase, four pole, induction motorin the synchronous reference frame can be given by theequations given in [10].The parameters used for the induction motor modelare:Power: 3 HP/2.4 KWVoltage :460 Volts (LL,RMS)Frequency : 60 HzPhases:3Fullload current:4 AFullLoad efﬁciency: 80.0%Fullload speed:88.5%Power factor:80.0%No.of poles: 4
R
s
= 1.77
Ω
R
r
=1.34
Ω
X
ls
=5.25
Ω
(at 60 Hz)
X
m
=139
Ω
(at 60 Hz)
X
lr
=4.57
Ω
(at 60 Hz)The system given by the equations are linearised at a givenoperating point to give the state space model where
A
=
−
69 5359
−
39 51 5145
−
5359
−
69 12
−
5146 51
−
270
−
828 0
−
438
−
80367
−
5170 39
−
53
−
49635170 67
−
13 4963
−
53
B
=
38
.
96 00 38
.
960 0
−
37
.
72 00
−
37
.
72
C
=
1 0 0 0 00 1 0 0 0
where,
∆
x
(
k
) = [∆
i
ds
∆
i
qs
∆
ω
r
∆
i
dr
∆
i
qr
]
T
∆
u
(
k
) = [∆
V
ds
∆
V
qs
]
T
∆
y
(
k
) = [∆
i
ds
∆
i
qs
]
T
For N=3, this system is discretised at a sampling intervalof
τ
= 0
.
09
sec
to obtain the discretised state space model as
A
dτ
=
−
0
.
58
−
0
.
07
−
0
.
01
−
0
.
60
−
0
.
070
.
20
−
0
.
30
−
0
.
01 0
.
20
−
0
.
30
−
0
.
27 0
.
36 0
.
04
−
0
.
28 0
.
350
.
60 0
.
08 0
.
01 0
.
61 0
.
08
−
0
.
21 0
.
31 0
.
01
−
0
.
22 0
.
31
B
dτ
=
−
0
.
04 0
.
020
.
05 0
.
070
.
14 0
.
010
.
04
−
0
.
01
−
0
.
06
−
0
.
08
C
dτ
=
1 0 0 0 00 1 0 0 0
The system is also discretised at an interval of
∆ = 0
.
03
sec
to obtain the discretised state space model as
A
d
∆
=
5
.
11
−
2
.
70
−
0
.
15 5
.
24
−
2
.
63
−
0
.
60
−
1
.
86 0
.
16
−
0
.
82
−
2
.
00
−
13
.
34
−
4
.
10
−
0
.
01
−
13
.
74
−
4
.
11
−
5
.
25 2
.
68 0
.
16
−
5
.
40 2
.
610
.
83 2
.
00
−
0
.
17 1
.
05 2
.
06
B
d
∆
=
−
0
.
05
−
0
.
02
−
0
.
01 0
.
060
.
13 0
.
060
.
05 0
.
02
−
0
.
02
−
0
.
06
C
d
∆
=
1 0 0 0 00 1 0 0 0
The sliding surface coefﬁcient matrix
c
is obtained usingthe method given in [26]
c
=
379
.
9954
−
83
.
9690
−
144
.
9981 49
.
8898212
.
0087
−
46
.
0841
−
94
.
0999 19
.
5327
−
70
.
1751
−
3
.
2512
.
The states are estimated using Eqn.19 with
L
1
=
0
.
01
−
0
.
01
−
0
.
07
−
0
.
07
−
0
.
11 0
.
18
−
0
.
01 0
.
01 0
.
06 0
.
05
−
0
.
01 0
.
03
−
0
.
01
−
0
.
02
−
0
.
09 0
.
14 0
.
14
−
0
.
18
−
0
.
01 0
.
01 0
.
07 0
.
07 0
.
11
−
0
.
190
.
01
−
0
.
01
−
0
.
06
−
0
.
05 0
.
01
−
0
.
03
and
L
2
=
−
0
.
05 0
.
010
.
05 0
.
070
.
14
−
0
.
010
.
05 0
.
01
−
0
.
06
−
0
.
07
.
The control input is found using Eqn.20 with gains
F
=
−
97
.
1 36
.
13
−
24
.
74
−
33
.
50 21
.
0171
.
64
−
36
.
14 17
.
34 26
.
73
−
12
.
27
G
=
4
.
82
−
6
.
240
.
14 20
.
67
.
IV. S
IMULATION
R
ESULTS USING
S
YSTEM
G
ENERATOR
The linear system with error states has to track referenceof zero, for the nonlinear system to track a constant referencespeed. MROF based SMC is simulated using System Generator and the system model as shown in the Fig.1:
Fig.1 Simulink block diagram with System Generator for controllaw implementation
Fig.2 Output currents
∆
i
ds
and
∆
i
qs
.
Fig.3 Estimated states
ˆ
x
1
,
ˆ
x
2
,
ˆ
x
3
,
ˆ
x
4
,
ˆ
x
5
.
Fig.4 Control inputs
∆
V
ds
and
∆
V
qs
.
It is observed from Fig. 2 that the designed MROF basedSMC drives the dq axis current errors to zero. Fig.3 showsthat all the states estimated using Multirate output feedback converges quickly. The control inputs and the sliding surfacesare shown in Fig.4 and Fig.5 respectively.
Fig.5 Sliding surfacesFig.6 Observed and actual
∆
i
ds
Fig.7 Observed and actual
∆
i
qs
Fig.6 and Fig.7 show that the estimated and actual outputstates
∆
i
ds
and
∆
i
qs
track the reference zero as needed.Fig. 8 System response for external disturbanceFig.8 shows that the designed MROF based SMC lawbrings the speed error to zero, when external disturbanceis applied at 3 sec. and also it is inferred that the systemremains stable.V. C
ONCLUSION
In this paper, MROF based DSMC is designed for a sensorless induction motor. The speed of the motor is controlledby measuring the stator current only. The multirate outputfeedback approach drives the error states to zero swiftly. Thecontroller is simulated with System Generator for implementation on FPGA. The scheme also has the merits of robustnessof sliding mode control and the fast processing rate of FPGA.As seen from the plots, the MROF based DSMC simulatedusing System Generator is able to drive all the states to zerorapidly and rejects the disturbance also.R
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