A Novel Model Reference Adaptive Controller for Estimation of Speed and Stator Resistance for

Vector Controlled Induction Motor Drives A. V. Ravi Teja and Chandan Chakraborty, Senior Member IEEE

Department of Electrical Engineering, Indian Institute of Technology Kharagpur, WB-721302, India raviteja009@yahoo.co.in , chakraborty@ieee.org

Abstract- This paper presents a new adaptive speed estimation technique for the vector controlled induction motor drive. A model reference adaptive system (MRAS) is formed using instantaneous and steady state values of IV ×* in the reference and adjustable models respectively. Such formulation does not require flux estimation and is stable in all the 4-quadrants of operation. Proposed MRAS with stator resistance compensation is suitable for low and zero speed operation. The implementation does not require any additional sensors and therefore is very suitable for retrofit applications. Assuming that the speed is available from a speed encoder the controller may be slightly modified to estimate stator resistance. The proposed technique is simulated in MATLAB/SIMULINK and experimentally validated through a dSPACE-1104 based laboratory prototype. The stability of the proposed MRAS is verified through stability analysis and sensitivity of the controller to the variation of various machine parameters is also investigated.

I. INTRODUCTION

Vector controlled induction motor drive has become an industrial standard due to high dynamic performance and easy implementation of the scheme using simple analog/digital signal processing techniques. However, the necessity of the speed signal and involvement of crucial motor-parameters like Rs, Rr etc. make the system less robust and prone to deterioration in system dynamics [1-3]. The merits of sensorless controller are the removal of mechanical speed sensor thereby enhancing system reliability and reducing cost and size. Elimination of sensor also reduces the effect of sensor-noise and drift effects. On the other hand, a controller with on-line estimation and compensation of parameters is essential for satisfactory performance in a wider speed range. All these call for extensive investigation on the estimation of speed and parameters of vector controlled induction motor drive [1-10]. The available speed estimation techniques may be broadly classified into model based and signal injection based approaches. Of the model based methods, several observer based approaches are very attractive. In this area the use of extended Kalman filter dominated for some time [4]. However, the influence of noise characteristic, the absence of criteria for tuning and the computational burden of this stochastic observer are major limitations for wide acceptability. A series of works based on deterministic observer models (using Luenberger observer) is proposed [5-7]. However, these methods require the estimation of flux and in process become dependent on machine parameters. In this context a class of adaptive controllers based on model

reference adaptive system (MRAS) demands special mention. Flux, back-emf, reactive power based MRAS are proposed in literature [8-10]. Flux based MRAS is stable in all the four quadrants. However, these are not suitable for operation at low and very low speed due to the presence of a pure integrator [8]. Back-emf based MRAS also suffers from the deterioration in performance at low speed. Moreover, both flux and back-emf based MRAS depend on stator resistance [9]. A reactive power based MRAS overcomes all such problems but at the cost of loosing stability in the regenerative mode of operation [10]. This paper presents a

novel MRAS, which uses instantaneous value of IV ×* in the reference model and steady-state flux oriented value of

the same (i.e. IV ×* ) in the adjustable model. Selection of

IV ×* is a major success in the sense that the proposed system is now stable in all the four quadrants, operates very well at low speed including zero speed, does not require flux computation and easily implementable. It is important to note that the proposed controller is dependent on Rs. However, an active power based MRAS can easily be incorporated to make the system Rs independent. On the other hand, the proposed algorithm may be slightly modified to estimate Rs in case the speed signal is available. A proper estimation of Rs provides indirect measure of stator winding temperature and hence is extremely important for increased reliability, fault tolerant operation and system diagnostics. This paper has six sections. The following section presents the basic idea of the proposed MRAS. Section-III deals with the stability of the system and sensitivity to parameter variations. Simulation and experimental results are presented in Section-IV and V respectively. Section-VI concludes the work.

II. SYSTEM MODELING

A. A. Proposed MRAS Fig.1(a) shows the proposed MRAS. X (defined

as IV ×* ) is selected as the functional candidate to form the MRAS. Note that X is neither reactive power nor active power. The instantaneous value of the X (i.e. X1) is used in the reference model. On the other hand, the steady state value of X (i.e. X4) under flux oriented condition is considered for the adjustable model. The error of the two (i.e. ε=X1-X4) is fed to the adaptation mechanism, which yields the estimated

978-1-4244-6392-3/10/$26.00 ©2010 IEEE 1187

rotor speed (i.e. ωrest). Fig.1(b) shows the schematic diagram of the complete vector control of the induction motor with the proposed MRAS for speed estimation.

B. Formulation of the X-MRAS The induction motor stator voltages in the synchronously rotating reference frame may be expressed as:

)( rqrder

m

sqssdsesqssq

pLL

iLpiLiRV

ψψω

σσω

++

++= (1)

)( rdrqer

msds

sqsesdssd

pLL

iLp

iLiRV

ψψωσ

σω

−−+

−= (2)

Instantaneous value of IV ×* is defined as:

sqsdsdsq ivivX +=1 (3)

Substituting the values of sqv and sdv from (1) and (2) in (3), the instantaneous value of X becomes:

sqrdrqer

msdssqsesds

sdrqrder

msqssdsesqs

ipLLipLiLiR

ipLLiLpiLiRX

)]([

)]([2

ψψωσω

ψψωσω

−−+−

+++++= (4)

At steady state, the expression of X is

sqrqer

msqsesds

sdrder

msdsesqs

iLLiLiR

iLLiLiRX

)]([

)]([3

ψωσω

ψωω

−−+

++= (5)

For a filed oriented drive, substituting sdmrd iL=ψ , and

0=rqψ , the simplified expression of 1X becomes:

sqsdssqssdse iiRiLiLX 2][ 224 +−= σω (6)

The expression of X1 is independent of rotor speed. So, it is selected for the reference model. X2, X3 or X4 may be chosen as the adjustable model as they are dependent on the rotor speed ( rω ), however, X4 is selected in the adjustable model as this does not involve flux estimation and any derivative operations.

Fig 1(a) Proposed MRAS (i.e. X-MRAS) Structure for estimating rotor speed

Fig 1(b) Vector control block diagram with the proposed MRAS based speed

estimation technique Fig.1. Proposed X-MRAS based speed sensorless drive

III. STABILITY AND SENSITIVITY ANALYSIS

An investigation on stability of the system is reported in this section based on small signal analysis linearising machine equations around an operating point. The sensitivity of the proposed MRAS to the various machine parameters is also studied. The state-space representation of the machine using stator currents and rotor flux (in the synchronously rotating reference frame) as the state variables is given by,

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−

−⎥⎦

⎤⎢⎣

⎡+−−

⎥⎦

⎤⎢⎣

⎡+−

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

•

•

•

•

00

1

10

10

111

111

2

2

sq

sd

s

rq

rd

sq

sd

rsl

r

m

slrr

m

rr

m

sr

r

m

srr

ms

se

rr

m

srr

m

se

rr

ms

s

rq

rd

sq

sd

vv

Lii

L

LLL

LLL

LLL

RL

LL

LLL

LLL

RL

i

i

σψψ

τω

τ

ωττ

τσω

στσω

ωστσ

ωτσ

ψ

ψ

(7)

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

rq

rd

sq

sd

sq

sd ii

ii

ψψ0

000

10

01

(8)

This can be represented in short form as

BuAxx +=•

(9) Cxy = (10)

Linearising the state space equations around an operating point 0x , we get

0AxxAx Δ+Δ=Δ•

(11)

xCy Δ=Δ (12)

or, 01)( AxAsICy Δ−=Δ − (13)

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Where, [ ]Trqrdsqsd iix 00000 ψψ= (14) For checking the feasibility of the algorithm for speed estimation, we calculate AΔ in terms of rωΔ so that we can

get r

sdiωΔ

Δand

r

sqiωΔ

Δ

Now, from the small signal error equation of the X-MRAS, we can obtain

)()(1

)(

4

32

sG

sKKK

iKiK

imraspmras

r

sq

r

sd

r

=++

ΔΔ

+ΔΔ

=ΔΔ ωωωε

(15)

Where sLK =1 , sdessqssq iLiRvK ω222 −−=

sqessdssd iLiRvK ωσ223 +−= , 224 sqssds iLiLK σ−=

The closed loop transfer function of the X-MRAS based speed estimator is given by,

)()(

)(ˆ 1 sG

sK

K

sGimras

pmrasr

r

++=

−ωω (16)

Using equation (16), the stability study of the X-MRAS has been done in all the four quadrants of operation and the system has been found to be stable. Sensitivity to stator resistance variation: The sensitivity of the proposed speed estimation algorithm to the variation of stator resistance is studied here.

For this purpose, a plot of s

r

RΔΔω

versus rotor speed ( rω ) is

obtained.

rωΔ can be obtained from equation (15) as

41

32

)(

)(

Ks

KK

iKiKimras

pmras

sqsdr

++

Δ+Δ=Δ

−ω

For the sensitivity analysis, we assume that 0=Δ sdi as flux

remains unchanged in vector controlled operation and sdi is the flux component of current. Hence,

41

3

)(

)(

Ks

KK

iKimras

pmras

sqr

++

Δ=Δ

−ω (17)

Dividing by sRΔ on both sides, we get

41

3

)(

)(

Ks

KK

Ri

K

R imraspmras

s

sq

s

r

++

ΔΔ

=ΔΔ

−

ω (18)

Therefore, for getting the sensitivity to the stator resistance

variation, we need to compute s

sq

Ri

ΔΔ

which can be obtained

from the matrix ‘A’ when linearised with respect to Rs. The sensitivity plot is shown in Fig 2(a). Sensitivity to rotor time constant variation: The proposed speed estimation technique is also dependent on the rotor time constant. It is to be noted that in indirect field oriented control of induction motor drives, the slip is also dependent on rotor time constant. Therefore, the sensitivity to rotor time constant is split into two independent parts, one assuming slip is constant and other assuming rotor speed is constant.

)0()0(

)1(

ˆ

)1(

ˆ

)1(

ˆ

=Δ=Δ

Δ

Δ+Δ

Δ=Δ

Δ∴

rslr

sl

r

r

r

r

ωω τ

ω

τ

ω

τ

ω (19)

A. Sensitivity to )0()1(

ˆ

=ΔΔ

Δ

slr

r

ωτω

:

Dividing equation (17) by )1(rτ

Δ , we get

41

3

)(

))1(

(

)1(ˆ

Ks

KK

iK

imraspmras

r

sq

r

r

++

ΔΔ

=Δ

Δ−

ττ

ω (20)

)1( r

sqiτΔ

Δ is calculated from ‘A’ matrix by linearising it with

respect to rτ

1

The sensitivity plot of 0)1(

ˆ=Δ

ΔΔ

slr

r ωτ

ω with respect to rotor

speed is shown in Fig 2(b).

Fig 2(a) Stator resistance sensitivity plot of X-MRAS

Fig 2(b) Rotor time constant sensitivity plot of X-MRAS assuming slip constant

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Fig 2(c) Rotor time constant sensitivity plot of X-MRAS assuming rotor speed constant

B. Sensitivity to)0(

)1(ˆ

=ΔΔΔ

rr

sl

ωτω :

In indirect rotor field oriented control of induction motor drives, the rotor slip, slω is given by,

sd

sq

rsl i

iτ

ω 1= (21)

Therefore, the sensitivity of slip with respect to rotor time constant is given by

)0(0

0

)1(1

)1(=Δ

ΔΔ

+=ΔΔ

rr

sq

sdrsd

sq

r

sl iii

i

ωτττω

(22)

Where )0(

)1(=Δ

ΔΔ

rr

sqi

ωτcan be obtained again by linearising ‘A’

matrix with respect to ⎟⎠⎞⎜

⎝⎛

rτ1 keeping 0=Δ rω .

The sensitivity plot of 0)1(

ˆ=Δ

ΔΔ

rr

sl ωτ

ω with respect to rotor

speed is shown in Fig. 2(c).

IV. SIMULATION RESULTS

The complete system is extensively simulated in MATLAB/ SIMULINK and this section presents some of the simulation results.

A Step change of rotor speed and zero speed operation The response of the induction motor for a step change in reference speed and zero speed operation can be seen in Fig 3. A step change in speed of 5 rad/sec is applied every 4 s

Fig 3(a) Reference speed and actual speed (in rad/sec) vs. time in seconds

Fig 3(b) Actual speed and estimated speed (in rad/sec) vs. time in seconds

Fig 3(c) d-axis and q-axis rotor flux (in Wb) vs. time in seconds from 0rad/sec (Fig 3(a)) and the actual speed is found to track the reference speed satisfactorily. The estimated speed is available in Fig 3(b), which shows that the estimated speed is very close to the actual rotor speed. Flux orientation is well maintained as depicted in Fig 3(c). The load applied to the motor is through a DC generator which offers 0.5pu torque. The proposed algorithm is also tested for forward and reverse motoring operation by applying a square wave speed-command as in Fig 4. The estimated speed is following the actual speed which in turn is matching with the reference speed as shown in Fig 4(a). In all these operations, the flux orientation is not disturbed as observed in Fig 4(b). Here also, the load torque is applied through a DC generator. This validates the proposed speed estimation algorithm experimentally.

B Ramp Response The performance of the algorithm at low speeds near zero is tested by applying a triangular wave input as in Fig 5. The estimated speed is following the actual speed which in turn is matching with the reference speed as shown in Fig 5(a). In all these operations, the flux orientation is not disturbed as observed in Fig 5(b). Here also, the load torque is applied through a DC generator. This validates the proposed speed estimation algorithm making it suitable for low speed applications.

C Operation at very low speed The performance of the algorithm at a low speed of 1 rad/sec is shown in Fig. 6. The estimated speed and the actual speed are shown in Fig 6(a). This shows that the proposed algorithm can estimate speed accurately even at very low speeds. The flux orientation is maintained which can be seen from Fig. 6(b).

Fig 4(a) Actual speed and estimated speed (in rad/sec) vs. time in seconds

Fig 4(b) d-axis and q-axis rotor flux (in Wb) vs. time in seconds

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D Stator Resistance Estimation From Fig 1(a), it can be seen that the adjustable model is dependent on rotor speed as well as stator resistance. So far, we have estimated rotor speed assuming stator resistance and other parameters are known. But, we can also estimate the stator resistance if the rotor speed is known as shown in figure 7(a). The estimated stator resistance and the stator resistance varied in the machine are shown in fig 7(b).

Fig 5(a) Actual speed and estimated speed (in rad/sec) vs. time in seconds

Fig 5(b) d-axis and q-axis rotor flux (in Wb) vs. time in seconds

Fig 6(a) Actual speed and estimated speed (in rad/sec) vs. time in

seconds

Fig 6(b) d-axis and q-axis rotor flux (in Wb) vs. time in seconds

Fig 7(a) Stator Resistance Estimation Algorithm using X-MRAS

Fig 7(b) Actual stator resistance and stator resistance (in ohms) estimated using X-MRAS

V. EXPERIMENTAL RESULTS

The proposed MRAS based speed estimation algorithm is extensively tested in hardware using a dSPACE-1104 based laboratory prototype. Some typical results are presented here.

A Step change of rotor speed and zero speed operation The response of the induction motor for a step change in reference speed and zero speed operation can be seen in Fig 8. A step change in speed of 5 rad/sec is applied every 4 secs from 0rad/sec and the actual speed is found to track the reference speed (Fig 8(a)) satisfactorily. The estimated speed is very close to the actual rotor speed (Fig 8(b)). Flux orientation is well maintained as depicted in Fig 8(c). The load applied to the motor is through a DC generator as detailed in simulation. The proposed algorithm is also tested in hardware for a step positive and negative speed command as in Fig 9. The estimated speed is following the actual speed as shown in Fig 9(a). The flux orientation is not disturbed as observed in Fig 9(b)

B Triangular wave response The performance of the algorithm for a triangular wave input is shown in Fig 10. The estimated speed is matching with the actual speed as shown in Fig 10(a). The flux orientation is not disturbed as observed in Fig 10(b). This validates the proposed speed estimation algorithm experimentally making it suitable for low speed applications.

C Very low speed operation The hardware result corresponding to the operation of the proposed controller at 1 rad/sec reference can be seen from Fig. 11(a) which shows that the estimated speed matches with the actual speed satisfactorily. The flux is oriented as seen from Fig. 11(b).

Fig 8(a) Reference speed and actual speed (in rad/sec) vs. time in seconds

Fig 8(b) Actual speed and estimated speed (in rad/sec) vs. time in seconds

Fig 8(c) d-axis and q-axis rotor flux (in Wb) vs. time in seconds

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Fig 9(a) Actual speed and estimated speed (in rad/sec) V/s time in seconds

Fig 9(b) d-axis and q-axis rotor flux (in Wb) V/s time in seconds

Fig 10(a) Actual speed and estimated speed (in rad/sec) vs. time in seconds

Fig 10(b) d-axis and q-axis rotor flux (in Wb) V/s time in seconds

D Stator Resistance Estimation Stator Resistance estimation using the controller of Fig 7(a) is tested in hardware. An external resistance is suddenly connected in series with the stator at 5s and removed at 15s. The estimated stator resistance is correct as shown in Fig 12.

Fig 11(a) Actual speed and estimated speed (in rad/sec) vs. time in seconds

Fig 11(b) d-axis and q-axis rotor flux (in Wb) vs. time in seconds

Fig 12 Estimated stator resistance (in ohms) estimated using X-MRAS

VI. CONCLUSIONS

This paper has presented a new MRAS based speed estimation technique for the vector controlled induction

motor drive. The proposed method used IV ×* as the functional candidate to formulate the MRAS. Such MRAS is found to be stable in all the four quadrants of operation. Moreover there is no need for flux estimation. This made the method very suitable for low speed operation. The usefulness of the proposed algorithm has been confirmed by stability studies, MATLAB/SIMULINK based simulation and corresponding experimental results from dSPACE1104 based laboratory prototype. The sensitivity of the proposed algorithm to various machine parameters is also reported.

TABLE I INDUCTION MACHINE PARAMETERS

Symbol Meaning Value P Pole pair 2 Ls

Lr Lm

Rs

Rr J B

Stator self-inductance Rotor self-inductance

Magnetizing Inductance Stator Resistance Rotor Resistance Machine Inertia

Viscous coefficient

0.6848 H 0.6848 H 0.6705 H 5.71 Ω

4.0859 Ω 0.011 Kg-m2

0.0015

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