9th

IEEE International Conference on Power Electronics & Drive Systems, Singapore, Dec 5-8, 2011.

Sliding Mode Control of a Feedback Linearized

Induction Motor using TS Fuzzy Based Adaptive

Iterative Learning Controller

Madhu Singh, Kanungo Barada Mohanty, Senior Member, IEEE, Bidyadhar Subudhi, Senior Member, IEEE

Department of Electrical Engineering, National Institute of Technology, Rourkela, INDIA

Email: madhu_nitjsr@rediffmail.com, kbmohanty@nitrkl.ac.in, bidyadhar@nitrkl.ac.in

Abstract—This paper presents an application of a Takagi Sugeno

(TS) Fuzzy logic based adaptive Iterative Learning Controller

(ILC) to reduce chattering, torque ripple and to improve

dynamic performance of a feedback linearized induction motor

drive with sliding mode controller for periodic speed tracking.

This ILC is connected to the forward path of sliding mode speed

control loop. At first, the state feedback linearization technique is

used for decoupling speed and flux control loop. It uses reference

frame transformation and control in a stationary (α-β) frame with

rotor flux and stator current components as the state variables.

Since the induction motor drive system is sensitive to parameter

variation, model uncertainties and load disturbances, a robust

control strategy based on sliding mode is designed. In the sliding

mode based scheme the chattering of state and control variables

and torque ripple are present. To reduce chattering and torque

ripple, a TS fuzzy logic based adaptive ILC is designed. Both

control schemes are simulated in SIMULINK environment.

Simulation results demonstrate that the performance of sliding

mode cum TS fuzzy logic based adaptive ILC is better than the

scheme with only sliding mode controller. These simulation

results are also verified with real time simulator, RT Lab.

Index Terms-- Feedback linearization, Decoupling control,

Sliding Mode Controller, Takagi Sugeno Fuzzy control, Adaptive

Iterative Learning Control, Real Time Simulator

I. INTRODUCTION

Induction motor drive is a multivariable, coupled and

nonlinear system. Nonlinear control theory [1]-[2] has been

applied continuously to improve performance of the drive

through nonlinear controllers or through linearization of the

system model. Many attempts have been made in past to

optimize the performance and simplify the control strategy of

the induction motor through field oriented control or vector

control [3]-[4], feedback linearization control [5]-[7], and

sliding mode control [8]-[12]. While vector control and

feedback linearization control schemes have successfully

eliminated coupling problem, leading to fast transient response

with decoupled flux and torque response, they are sensitive to

parameter variations, model uncertainties and load

disturbances, due to which the decoupling and transient

performance are affected. Sliding mode control combined with

vector control or feedback linearization control have tackled

this problem and give robust performance. But in sliding mode

control schemes chattering of state and control variables is the

main problem.

Iterative Learning Control is one of the recent emerging

control methodologies based on the combination of knowledge

and experience. Knowledge is concerned with information

about the system model, it’s environment and uncertainties,

while experience explores its repetitive behavior, previous

control efforts and some resulting error. This controller is

recommended at the places where system performs a task

repeatedly. This has been explored in [13], to design a long-

wall coal cutting process by using a method called multi-pass

process. It is investigated in [14], [15] that the performance of

repetitive tasks can be improved by using information taken in

the previous cycles. Thus learning was introduced in the

control of the repetitive system. During practical operation in

periodic speed tracking applications, induction motor exhibits

repetitive oscillations and chattering of state variables due to

PWM switching and limiter with conventional speed

controllers. It has adverse effect on the performance of whole

drive system. This problem is well tackled by ILC, because it

generates robust control command to check system dynamics

in every iterative step based on the previous systems dynamic

state.

This paper combines Takagi-Sugeno (TS) fuzzy logic [16],

iterative learning control [13]-[15] and sliding mode control

[8]-[12] in a novel way to retain robustness of sliding mode

control and to eliminate chattering of state and control

variables, and also to improve transient response. This novel

hybrid controller is applied to a feedback linearized induction

motor drive. At first, a feedback linearization scheme with

sliding mode speed and flux controllers is designed. In second

scheme, TS fuzzy logic based ILC is added in the forward path

of sliding mode speed control loop. Simulation results show

that the second scheme gives reduced chattering, reduced

torque ripple and improved transient response. The results are

verified with real time simulator.

II. SYSTEM DESCRIPTION

The schematic block diagram of the proposed system is

shown in Fig 1. This feedback linearizing control design

procedure is described in detail in [17], [18]. The brief

description of this control scheme follows. Two sliding mode

controllers are regulating flux and speed loop. Voltage model

[4] is used for flux estimation. Output of flux and speed

regulator and estimated flux are the inputs to the

IEEE PEDS 2011, Singapore, 5 - 8 December 2011

978-1-4577-0001-9/11/$26.00 ©2011 IEEE 625

9th

IEEE International Conference on Power Electronics & Drive Systems, Singapore, Dec 5-8, 2011.

Fig. 1 Block diagram of feedback linearized induction motor with sliding mode speed and flux controllers

feedback linearizing block and its output goes to the current

controller. Output of current controller is utilized to generate

gate drive signal for PWM voltage source inverter (VSI),

which forces reference current in the motor to develop

required torque. At first, the above control scheme is designed

and simulated. The mathematical development of sliding mode

speed and flux controllers is detailed in section-III. In the

second scheme, a TS fuzzy based iterative learning torque

compensator is added in series with sliding mode speed

controller. The design and development of the proposed

hybrid controller are detailed in section IV.

III. SLIDING MODE CONTROLLER

In sliding mode control entire system dynamics is governed

by sliding surface parameters. System response is insensitive

to parameter variation, model uncertainties and external

disturbances. The system response in the phase plane is forced

to follow a sliding line [1]. The dynamics of error e(t) and its

derivative ( )e t&

are driven to zero along the sliding

line ( )s t = ( )e t& + ( )e tλ [1]. In time domain, the corresponding

response is exponentially decaying. Its time constant (λ)

depends on the slope of the sliding line and control signal

forces the response to slide on slide-line and system state error

always remains on zero state. This process can be easily

implemented by switching process back and forth between

negative and positive controller gain. The system error can not

only be made zero, but its response can be made independent

of the plant parameters [8].

To design sliding mode speed and flux controllers, the

induction motor model is linearized using feedback

linearization technique as described in [17] Though

decoupling is obtained at steady state, it is sensitive to

disturbances, uncertainties and parameter variation. So, sliding

mode controller is designed as described below [1], [18].

The speed and flux error are:

1 ( ) r re t ω ω∗= − (1)

2( ) r re t ψ ψ∗= − (2)

Taking the derivative of error

1( )r r

e t ω ω∗= −& && (3)

2 ( )r r

e t ψ ψ∗= −& && (4)

Substituting the expressions of rω& and r

ψ& as obtained in [17]

1 1( ) 2 ( )T L

r r

K TBe t u d t

J J Jω ω∗= + − + +&&

(5)

2 2( ) 1 ( )mr

r r r

r r

LRe t R u d t

L Lψ ψ∗= + − +&& (6)

Where, extra-terms d1(t) and d2(t) are the external disturbances

and uncertainty appearing in speed and flux dynamics [17].

The sliding surface s(t) in integral form is defined as [1]:

1 1 1 1( ) ( ) ( )s t e t e t dtλ= + ∫ (7)

2 2 2 2( ) ( ) ( )s t e t e t dtλ= + ∫ (8)

So that we have

1 1 1 1( ) ( ) ( )s t e t e tλ= +& & (9)

2 2 2 2( ) ( ) ( )s t e t e tλ= +& & (10)

Substituting the value of 1 ( )e t& and 2 ( )e t&from equations

(5) and (6), we obtain variable surface equation as follows:

1 1 1 1( ) 2 ( )T L

r r

K TBs t u d t e

J J Jω ω λ∗= + − + + +&&

(11)

2 2 2 2( ) 1 ( )mr

r r r

r r

LRs t R u d t e

L Lψ ψ λ∗= + − + +&&

(12) The best value for 1& 2u u of a continuous control law that

would satisfy the precedent condition, i.e., 0s =&,

can be

considered in the following form [1]:

2 2 2eq nu u u= + (13)

u1 = u1eq

+ u1n (14)

Where, u1 and u2 are the control outputs, u1eq

and u2eq

are

the continuous control parts based on knowledge about system

dynamics, and u1n and u2

n are the switching control terms.

1( ) 0s t = 1( ) 0s t =&

2 0n

u = (15)

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9th

IEEE International Conference on Power Electronics & Drive Systems, Singapore, Dec 5-8, 2011.

2 ( ) 0s t = 2 ( ) 0s t =&

1 0n

u = (16)

Equivalent control for satisfying above condition is

1 12 ( )eq L

r r

TJ Bu e

K J Jω ω λ∗= + + +&

(17)

2 21 ( )eq r r

r r

m r r

L Ru e

L R Lψ ψ λ∗= + +&

(18) The switching control law is defined as

1 11 sgn( )nu sβ= −

(19)

2 22 sgn( )nu sβ= −

(20) where, sgn() is signum function. β1, β2 are the respective

switching gain and must be greater than total uncertainties

present in corresponding circuit guaranteed by the Lyapunov

stability criterion [1]. In order to reduce chattering J. J. E.

Slotine proposed an approach, by introducing boundary layer

of width Ф on either side of the switching surface [1].

Then switching control law redefines by

11

1

1 sgn( )n su β= −

Φ (21)

22

2

2 sgn( )n su β= −

Φ (22)

Finally the speed command and flux command variable u2

and u1 can be found as

11 1 1

1

2 sgn( )L

r r

T

TJ B su e

K J Jλ ω ω β∗ ∗

= + + + − Φ &

(23)

22 2 2

2

1 sgn( )r r

r r

m r r

L R su e

L R Lλ ψ ψ β∗ ∗

= + + − Φ &

(24)

With the above speed and flux command for the set value of

speed and flux are obtained.

IV. TS FUZZY ITERATIVE LEARNING CONTROLLER

In recent years, many research works have been reported

on application of ILC to servomechanism [13]-[15]. ILC is

basically an error correction algorithm and it has a memory

that stores previous controller output data. It removes periodic

error by using the previous information for the present trial

[14]. In this work P-I type ILC is designed for reducing the

chattering in state and control signals during periodic speed

tracking. It is connected to the forward path of speed control

loop. The error e(k) is obtained by subtracting previous speed

controller output from current one. Change of error ce(k) is

obtained by subtracting previous error from current one. Thus

( ) ( ) ( 1)

( ) ( ) ( 1)

s se k u k u k

ce k e k e k

= − −

= − − (25)

Where, e (k) and ce(k) are the speed error and the change in

speed error at the kth

sampling time. us(k) is sliding mode

speed controller output signal. The P-I type ILC control

algorithm is represented as:

2( ) ( ) ( )is

u k Ku k u k= + (26)

Where, u2(k) is the control output signal of ILC, K is the gain

constant, and ui(k) is the error compensator output, which

reduce chattering and compensate it. In this work Takagi-

Sugeno (TS) method is used to construct ILC. TS method is

utilized for designing adaptive error compensator ui(k), which

reduce chattering and ripple appearing in signals due to motor

limiter, relays and sliding mode control. SIMULINK model

for P-I type TS fuzzy adaptive iterative learning controller is

shown in Fig. 2.

For convenience of designing fuzzy control algorithm the

behavior of the dynamic chattering phenomenon is first

investigated. In general, waveform of chattering signal about

the set signal can be roughly illustrated as shown in Fig.3.

According to the sign of error and change of error the response

plane is roughly divided into four areas. X1, X2, X3 and X4 as

shown in Fig. 3.

Fig.3. The dynamic behavior chattering phenomena

The index used for identifying the response area is defined as

1 : ( ) 0X e k > and 1 : ( ) 0X ce k <

2: : ( ) 0X e k < and 2: : ( ) 0X ce k <

3 : ( ) 0X e k < and 1 : ( ) 0X ce k >

4 : ( ) 0X e k > and 1 : ( ) 0X ce k >

Generally signals have ripple within fixed boundary and

frequency is also fixed. Hence error (e(k)) and change of error

Fig . 2. Simulink model of TS fuzzy based ILC

N e

ce

P P

P P N N

N

X2 X3 X4 X1

Desirable

Signal Chattering

Signal

Chattering

plane Zone

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9th

IEEE International Conference on Power Electronics & Drive Systems, Singapore, Dec 5-8, 2011.

(ce (k)) have two states either positive (P) or negative (N).

Two trapezoidal membership functions namely P and N have

been selected for each, based on trial as shown in Fig. 4.

Fig. 4 Membership functions of e (k) and ce (k)

With the combination of two membership function four

possible inference rules are considered.

Rule 1: If e(k) is P and ce(k) is N then

ui1 (k)=K1 (a1e(k) +a2ce(k))

Rule2: If e (k) is P and ce(k) is N then ui2 (k) = K2 ui1

Rule 3: If e (k) is N and ce (k) is P then ui3 (k) = K3 ui1

Rule 4: If e (k) is N and ce (k) is N then ui4 (k) =K4 ui1

Here ui1, ui2, ui3, and ui4 represent the consequence of the TS

Fuzzy controller and K1, K2, K3 and K4 are the weighing

factors for corresponding rules. Output of TS fuzzy is

obtained using the centroid method of defuzzification.

4

1

4

1

( )

i j i j

j

i

i j

j

u

u k

µ

µ

=

=

=

∑

∑ (27)

Where, ij

µ represents the degree of fulfillment (DOF) of

corresponding rule (ij

u ).

V. RESULTS AND DISCUSSIONS

The proposed control scheme is simulated in SIMULINK as

well as in the real-time simulator (RT-Lab). First scheme uses

two sliding mode controllers for controlling speed and flux

and second scheme uses one more TS fuzzy based ILC

connected in the forward path of speed control loop. The

specifications of motor and controllers are as detailed below.

Motor Specifications

Three Phase Squirrel Cage Induction Motor– 5 HP (3.7 kW),

4 pole, ∆-connected, 415 V, 1445 rpm, Rs = 7.34 Ω, Lls

=0.021 H, Lm = 0.5H, Rr = 5.64 Ω, Llr = 0.021 H, J=0.16 kg-

m2, B=0.035 kg-m

2/s.

Sliding Mode Controller Specifications

Flux Controller: λ1 = 120, β1 = 110

Speed Controller: λ2 = 200, β2 = 0.5

Iterative Learning Controller Specification

a1= 0.1, a2=10; K1=1,K2=0.25, K3= −0.25 and K4= −1

The periodic reference speed and load torque are taken as:

30sin10r

tω∗ = , 10sin10l

T t=

Simulation results of speed response, speed error, torque

response and reference torque, rotor flux linkage, α-β components of rotor flux and stator current are presented.

A. Scheme-I : With Sliding Mode Controller

The simulation results are shown in Fig. 5. This shows the

motor tracks the set speed. It has been observed that the α-β

axis current and rotor flux components are completely

decoupled even during transient condition. The robustness

over external disturbance is also obtained. Chattering of

torque, rotor flux and stator current is also significant even

with the gain reduction within boundary layer.

B. Scheme-II : With Sliding Mode cum ILC

The same set-up is tested with sliding mode controller with

ILC. The proposed iterative learning controller gives reduced

torque ripple and reduced chattering in stator current and rotor

flux as these are evident from simulation results shown in Fig.

6. The inclusion of ILC in the forward path of speed control

loop checks instantaneous difference between present speed

error and previous instant speed error and generates equivalent

control signal for reducing the error. For this reason very

simple TS fuzzy iterative learning controller with merely four

rules is proposed and number of iteration used is one. This

proposed controller is extremely simple and gives better result

in all responses such as speed error, torque, flux and winding

current as clearly evident from Fig.6. The objective alleviation

of chattering is almost fulfilled.

C. Real Time Simulation of Schemes- I and II

The schemes-I and II are implemented and tested in RT-

Lab environment. Fig. 7 shows the real-time simulation results

of rotor speed, torque, α-β components of stator current and

rotor flux, and speed error for scheme-I (with sliding mode

controller). Fig. 8 shows the corresponding results for scheme-

II (with ILC).

VI. CONCLUSION

The feedback linearized induction motor drive with two

different control schemes are developed and compared. First

one uses only sliding mode controller. The second is a hybrid

of sliding mode controller along with TS fuzzy based iterative

learning controller (ILC). Both the schemes are simulated in

SIMULINK environment and real time simulator, RT Lab.

The performances of the system with both controllers are

compared in terms of speed, speed error, torque, motor current

and flux response. It is clearly established from simulation

results that sliding mode controller cum TS fuzzy adaptive

ILC gives decoupling at all stages, reduces torque ripple and

better rotor flux and stator current response than the sliding

mode controller. The proposed controller is also extremely

simple and easy to implement. The results are tested and

verified by real-time simulation using RT-Lab simulator.

P 1

N

0 +1 0.5

0.5

-0.5 -1

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9th

IEEE International Conference on Power Electronics & Drive Systems, Singapore, Dec 5-8, 2011.

Fig. 5. Simulation results of the motor drive system with sliding mode

controller Fig. 6. Simulation results of the motor drive system with sliding mode

cum TS fuzzy adaptive ILC

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9th

IEEE International Conference on Power Electronics & Drive Systems, Singapore, Dec 5-8, 2011.

Fig. 7 Real time simulation results of the sliding mode controlled induction

motor drive system

Fig. 8 Real time simulation results of the sliding mode cum TS fuzzy

adaptive ILC controlled induction motor drive system

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Reference Speed

Actual Speed

Load Torque

Actual Torque

Beta current component

Alpha current

component

Beta rotor flux component

Alpha rotor flux

component

Actual, reference speed and speed error

Actual, reference speed and speed error

Beta rotor flux component

Alpha rotor flux component

Beta current component

Alpha current component

Reference Speed

Actual Speed

Load Torque

Actual Torque

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