**Rotor Speed Estimation Method Used in **

**Dynamic Control of the Induction Motor **

### CR CIUNA Gabriela

“Lucian Blaga” University of Sibiu, Romania,

Department of Computer and Electrical Engineering, Faculty of Engineering,

Postal address: 4 Emil Cioran Str., 550025 Sibiu, Romania, E-Mail:gabriela.craciunas@ulbsibiu.ro

**Abstract – In this paper it is proposed an algorithm for ****rotor speed estimation calculated directly from the ****rotor flux. The flux required for speed computation is ****estimated using Gopinath reduced order robust ****adaptive observer. In order to determine the structure ****of the observer we started from the state equations of ****the induction motor using spatial vectors written in ****fixed coordinates towards stator and considering the ****rotor speed constant. Quality of speed and rotor flux ****estimation was evaluated from the results obtained ****during different operation regimes. The proposed ****algorithm was then tested for its usability in the case of ****indirect field oriented control based on the rotor flux of ****the ****induction ****motor ****by ****the ****simulation ****in ****MATLAB/Simulink. **

**Keywords: three-phase induction motor; indirect field ****oriented control; Gopinath flux observer; rotor speed ****estimation. **

I. INTRODUCTION

Recent researches in high performance electrical drive were and are directed especially in solving the problems related to determine the rotor speed. Rotor speed is the main closed-loop value used in the field oriented control systems. So, it is used as feedback value in the speed loop and also as auxiliary feedback value in the flux observer loop.

The disappearance from the scheme of the position or speed sensor resulted in the appearance of the speed estimators and control systems without mechanical sensors.

In literature there are several methods proposed for speed estimation [1]-[4]. For example, speed estimation using adaptive methods “Model Reference Adaptive System” (MRAS), is based on two distinct rotor flux observers, derived from the induction motor model. This method offers good performances in steady-state and transient regime, except at low speed, caused by the presence in the algorithm of the pure integrator. This integrator raises problems related to the initial conditions and the drift. If the algorithm, where the

speed adaptation is implemented in the full-order observer, the drift problem is solved completely.

Another possible way to obtain an expression for the rotor speed is directly by using the dynamic model of the motor [2]. However, the accuracy of open-loop estimators depends greatly on the accuracy of the motor parameters used. At low rotor speed, the accuracy of the open-loop estimators is reduced, and in particular, parameter deviations from their actual values have great influence on the steady-state and transient performance of the drive system which uses an open-loop estimator.

In this paper it is proposed an algorithm that calculates the motor speed using the estimated rotor flux. The flux required for speed calculation can be estimated by an observer that is designed based on Gopinath’s reduced-order theory. Using MATLAB/Simuling computer simulation, the proposed algorithm has confirmed good performances even at low speed and during different operating conditions of the induction motor. The, the proposed algorithm was tested for usability in the case of indirect field oriented control (IFOC) based on the rotor flux of the induction motor [2], [4], [5].

II. MATHEMATICAL MODEL OF INDUCTION MOTOR

In order to determine the structure of the Gopinath reduced order flux observer, we start from the state equations of the induction motor expressed in the vector space. In the steady-state coordinate frame, these equations are [3],

*s*
*r*
*s*

*s* _{a}_{i}_{a}_{B}

*dt*
*i*
*d*

ν

+ Ψ +

= _{11} _{12} (1)

*r*
*s*

*r* _{a}_{i}_{a}

*dt*
*d*

Ψ + = Ψ

22

21 (2)

where,

σ

*s*

*L*
*D*

*a*_{11}=− = −

*r*
*m*
*r*
*r*
*r*

*m*

*s* *L*

*L*
*j*
*T*
*L*

*L*
*L*

*a* ω

σ

1

12

*r*
*m*

*T*
*L*
*a*_{21}= ;

*r*
*r*

*T*
*j*

*a*_{22}= ω − 1 (3)

_____________________________________________________________________________________________________________

σ

*s*

*L*
*B*= 1

2
2
*r*
*m*
*r*
*s*
*L*
*L*
*R*
*R*

*D*= +

*r*
*r*
*r*
*R*
*L*
*T* =
*r*
*sL*
*L*
*L*
*m*
2
1−
=
σ

The state variables of the motor are current *i _{s}* and
rotor flux linkage Ψ

*. The input variable is ν*

_{r}*. Motor leakage coefficient is σ and*

_{s}*T*is rotor time constant.

_{r}III. MATHEMATICAL MODEL OF THE FLUX OBSERVER

In the proposed algorithm, the flux required for speed calculation will be estimated using a dynamic flux observer as follows. In the relations (1) and (2), two states of the induction motor are represented by two differential equations involving stator current and rotor flux. Stator current can be measured using current sensor, so it is regarded as a known state and rotor flux can be estimated by an observer that is designed based on Gopinath’s reduced order theory [2], [5]. The base structure of this adaptive observer is the existence of a simulator based on the current model and a corrector of the estimated value. This corrector is a feedback loop that amplifies the error between estimated and actual value. This difference leads the initial estimated value to the actual one, with controllable speed.

The observer equation is presented in the following form,

### (

### )

*s*

### (

### )

*r*

*s*

*s*

*r* _{gB}*dt*
*i*
*d*
*g*
*ga*
*a*
*i*
*ga*
*a*
*dt*
*d*
ν
−
+
Ψ
−
+
−
=

Ψˆ _{ˆ}

12 22 11

21 (4)

where it was noted with Ψˆ* _{r}*, estimated rotor flux space
vector.

To estimate the rotor flux it is preferred the motor
model written in the complex form. We chose this
solution because these types of observatories stability
study and determination of gate matrix parameters may
require fixing poles position in the complex plane. Thus,
the essential element is the gate *g*, a complex number
of form, (*g _{a}*+

*jg*). This element determines the flux observer stability, and insensitivity to variations in motor parameters. Observer poles are defined in the form of (−α+

_{b}*j*β). For each instantaneous rotor speed value is determined the gate matrix. Observer poles will be found permanently in fixed points or on certain paths in the complex plane, thus, the observer stability. The gate coefficients are [6],

*m*
*r*
*s*
*r*
*r*
*r*
*a*
*L*
*L*
*L*
*T*
*T*
*g* σ
ω
α
−
+
= 1
ˆ
1 2
2
;
*m*
*r*
*s*
*r*
*r*
*r*
*b*
*L*
*L*
*L*
*T*
*g* σ
ω
α
ω
2
2
ˆ
1
ˆ
+
= (5)
where,
2
2
ˆ
1
*r*
*r*
*T*
*k* ω

α = + ,

### (

*k*>0

### )

; β =0 (6) determine the optimum position of the poles on the negative real axis, thus minimizing the influence of induction motor parameter variation on the stability of flux observer. According to relations (5), gate coefficient values depend on the rotor speed ω*, so they must be adjusted in real time. To determine the estimated rotor flux, this speed will be replaced with the estimated rotor speed ωˆ*

_{r}*r*. This replacement is possible because the flux dynamics is much faster than the mechanical speed. Fig. 1 shows the configuration of the rotor flux observer.

Figure 1 Rotor flux observer configuration

If in equations (1) and (2), linking the stator voltage and current, rotor speed and rotor flux respectively, will be replaced coefficients from (3), we obtain the following form for the voltage space vector,

*r*
*r*
*r*
*m*
*r*
*r*
*r*
*m*
*s*
*s*
*s*
*r*
*m*
*r*
*s*
*s* *j*
*L*
*L*
*T*
*L*
*L*
*dt*
*i*
*d*
*L*
*i*
*L*
*L*
*R*

*R* + ⋅ + − Ψ + Ψ

= σ ω

ν

2 2

(6)

Writing relation (6) in complex form, the following expression results for estimated rotor speed,

## (

ˆ2 ˆ2## )

ˆ
ˆ
ˆ
*q*
*r*
*m*
*sd*
*s*
*sd*
*sd*
*rq*
*sq*
*s*
*sq*
*sq*
*rd*
*r*
*rd*
*L*
*L*
*dt*
*di*
*L*
*Di*
*dt*
*di*
*L*
*Di*
Ψ
+
Ψ
−
−
Ψ
−
−
−
Ψ
=
σ
ν
σ
ν
ω (7)

Fig. 2 shows the block diagram to calculate the rotor speed according to equation (7). Notice that for determining rotor speed were used as input in block the two orthogonal components of rotor flux determined by Gopinath flux observer of reduced order.

Figure 2 Speed calculation

_____________________________________________________________________________________________________________

IV. SIMULATION RESULTS

Algorithm proposed in this paper is verified in
indirect field oriented control with rotor flux orientation
of a three-phase induction machine, [2], [4]-[6] with
known parameters (*P _{N}* =5,5

*KW*;

*U*=230

_{N}*V*;

min
/
1500*rot*

*n _{s}* = ), Table 1.

*Table 1. The three-phase induction machine parameters *

*p*=2 *Rr*=2.408Ω

36

=
*s*

*Z* *Ls*=0.268*H*

28

=
*r*

*Z* *Lr*=0.274*H*

3

=

*m* *Lm*=0.265*H*

*Rs*=1.63Ω *J*=0.04 kgm2

Required speed in feedback loop so from indirect control block diagram Fig. 3 and from indirect field oriented control scheme Fig. 4 was obtained using the algorithm described in (7).

Figure 3 Indirect field oriented control using rotor flux and torque to produce command currents

Fig. 3 illustrates this concept and shows how the
rotor flux position can be obtained by adding the integral
of the slip frequency, calculated from the flux and
torque commands to produce an angular estimation of
the rotor flux position. Indirect field oriented control
command block, Fig.3 will calculate flux vector position
necessary for axes transformer blocks, Fig.4 from
two-phase mobile rotor reference to two-two-phase fixed stator
reference and vice versa. Thus, the relation for
determination of θ* _{e}* is [2], [5],

*dt*
*d*
*i*

*L*
*L*

*R* _{e}

*r*
*sq*

*r*
*m*
*r*
*r*
*e*

θ ω

ω =

Ψ +

=

* *

(8) The relation to calculate the imposed longitudinal stator current, written in rotor coordinates, is following,

*

* 1

*r*
*m*

*sd* _{L}

*i* = Ψ (9)

To calculate the imposed transversal current, written
in rotor coordinates, it was started from dynamic regime
equation. In both relations (9) and (10) it was considered
that Ψ* _{rd}* =Ψ

** and Ψ*

_{r}*=0 . Thus we have,*

_{rq}* 2

3 ** _{i}**

_{m}*L*

*Lm*
*p*
*dt*
*d*

*J* _{r}_{sq}

*r*

*r* _{=} _{Ψ} _{=}

ω

(10) resulting,

* 1 3

2

*

* _{m}

*L*
*L*
*p*
*i*

*r*
*m*
*r*
*sq*

Ψ

= (11)

In the indirect field oriented control case it is not
used a rotor flux control loop. Its value was imposed as
an input into the command block Ψ* _{r}**=0,9

*Wb*. Also, models used for PI controllers are those dealing with saturation of output value, limiting the controller output to the desired value.

In the control scheme it has not been introduced the PWM inverter, since the implementation of sensorless drive system allows replacement of reaction sizes (currents, voltage) with the corresponding command sizes.

Figure 4 Block diagram of sensorless vector control system with rotor speed estimation

Algorithm proposed in this paper was tested by real time simulation, using MATLAB/Simulink simulation environment.

0 0.05 0.1 0.15

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time (Seconds)
*r*
*r*
Ψ
Ψ
ˆ
...
____

0 0.05 0.1 0.15

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Time (Seconds)
*r*
*r*
Ψ
Ψ
ˆ
...
____

Figure 5 The real and estimated rotor flux for K=0.6 and K=2

Fig. 5 represent the actual rotor flux module
variation (Ψ* _{r}*), calculated from the motor equations in
two-phase axes system and estimated rotor flux module
variation (Ψˆ ), calculated with the equation (4).

_{r}_____________________________________________________________________________________________________________

Simulation results confirm that the initial estimation
error reduces rapidly and the estimated value converges
to the actual one, for a coefficient *K* small, *K=0.6*. Same
can not tell if the coefficient *K* is greater, *K=2*, where
large errors exist between actual and estimated rotor
flux. In this case, the value of the estimated rotor flux is
strongly influenced by the rotor resistance.

Results that the coefficient *K* can control the
correction weight generated by the corrector and applied
to the simulator, from the structure of the reduced order
Gopinath observer.

In Fig. 6 we can see that the rotor speed is well
estimated for *K=0.6* through the proposed algorithm,
without error in this regime. Thus, for a well chosen
coefficient *K* is confirmed that the initial estimation
error reduces rapidly and the estimated value converges
to the actual one. Not the same can be said for *K=2*

when between the estimated and the reference value there is an error.

0 0.05 0.1 0.15

0 50 100 150 200 250 300 350

Time (Seconds)
*r*
*r*

ω ω ˆ ... ____

0 0.05 0.1 0.15

0 50 100 150 200 250 300 350

Time (Seconds)
*r*
*r*
ω
ω
ˆ
...
____

Figure 6 The real and estimated speed for K=0.6 and K=2

Simulation results were obtained for a no load start
of the motor, followed then at *t=0.2 s* by applying a load
torque *T _{L}*=5

*Nm*.

Also by simulation, it will be tested the
performances of the indirect field oriented drive
schemes, with estimation robust adaptive of the rotor
flux in which rotor speed is estimated with the proposed
algorithm. The proposed algorithm was tested both for
high and low rotor speed values. Thus to the control
system was applied a reference of type step speed with
value ω** _{r}* =157

*rad*/sec, corresponding nominal speed of the induction motor. In Fig, 7 is presented the system response to the application of this step speed. Note that after a slower progress due to increased inertia of the

drive system, estimated rotor speed reaches the imposed reference value.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0

50 100 150

Time (Seconds)

*r*
*r*
ω
ω
ˆ
...
____ *

Figure 7 The reference and estimated speed for speed value 157rad/s and 20rad/s

Also, a very good progress towards the prescribed
rotor speed is shown by the graph in Fig. 7, were it was
applied a step speed with the value ω** _{r}* =20

*rad*/sec.

V. CONCLUSION

From this paper emerges that the algorithm proposed
by me where the rotor speed is calculated directly by the
robust adaptive reduced order Gopinath flux observer,
proved to be efficient both in transient regime and in
indirect field oriented control system with three-phase
induction motor. For a more stable flux observer poles
were placed on the negative real axis, thus obtaining
relative simple calculations in real time. Also, the
coefficient *K* can control the correction weight generated
by the corrector and applied to the simulator, from the
flux observer structure. Thus, for a well chosen
coefficient *K*, the simulation results confirmed that the
initial estimation error reduces rapidly and the estimated
value converges to the actual one.

The speed required by the indirect field oriented control block was obtained using the proposed algorithm, which confirmed the good performance both at high and at low rotor speed levels.

REFERENCES

[1] M. Ahmad, “High Performance AC Drives – Modelling Analysis and Control”, Springer, 2010.

[2] P. Vas, “Sensorless Vector and Direct Torque Control”, Oxford University Press, Oxford, 1998.

[3] T. Pana, O. Stoicuta, "Small Speed Asymptotic Stability
*Study of an Induction Motor Sensorless Speed Control *
*System with Extended Gopinath Observer", Advances in *
Electrical and Computer Engineering, vol. 11, no. 2, pp.
15-22, 2011.

[4] H. Kubota, K. Matsuse, “Speed Sensorless Field Oriented
*Control of Induction Machines Using Flux Observer”, *
IEEE Trans. Ind. Appl. 30, pp. 1219–1224, 1994.

[5] P. Wach, “Dynamics and Control of Electrical Drives”, Springer, 2011.

[6] G. Cr ciuna , "Performances of Gopinath Flux Observer
*Used in Direct Field Oriented Control of Induction *
*Machines", Advances in Electrical and Computer *
Engineering, vol. 11, no. 1, pp. 73-76, 2011.