Observability of Speed in an Induction Motor from Stator Currents and ...

Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12-15, 2005

TuIB21.1

Observability of Speed in an Induction Motor from Stator Currents and Voltages Mengwei Li, John Chiasson, Marc Bodson, and Leon Tolbert Abstract— This paper describes a new approach to estimating the speed of an induction motor from the measured terminal voltages and currents without the use of a speed/position sensor. The new observer uses a purely algebraic speed estimator to stabilize a dynamic speed estimator and it is shown that it has the potential to provide low speed (including zero speed) control of an induction motor under full rated load. Index Terms— Sensorless Speed Observer, Induction Motor

II. M ATHEMATICAL M ODEL OF THE I NDUCTION M OTOR A (two-phase equivalent) state-space mathematical model of the induction motor (see [16][17]) written in space vector form [9], by defining lV , lVd + mlVe , # U , # Ud + m# Ue , and xV , xVd + mxVe > is g  1 (1  mqS $WU ) # U  lV + x l = gw V WU OV V 1 P g (1  mqS $WU ) # U + l = # = gw U WU WU V n o O qs P g$ Im lV # U  = gw MOU M

I. I NTRODUCTION Sensorless control of an induction motor refers the problem of controlling it without the use of a rotor position/speed sensor. Many different techniques have been proposed to estimate speed of an induction motor without a speed sensor. This area has a rather large literature and the reader is referred to [1][2][3][4][5][6][7][8][9][10] for an exposition of many of the existing approaches. The approach presented in this work is most closely related to the ideas described in [11][12][13][14][15]. In [11][12][13][14], observability is characterized as being able to reconstruct the unknown state variables as rational functions of the inputs, outputs, and their derivatives (See [12][13][14] for a more precise definition). We manage to obtain an algebraic expression for the rotor speed in terms of the machine inputs, machine outputs and their derivatives. In the systems theoretic approach considered in [15], the authors have shown that there are indistinguishable trajectories of the induction motor, i.e., pairs of different state trajectories with the same input/output behavior. That is, it is not possible to estimate the speed based on stator measurements for arbitrary trajectories [15]. A similar circumstance is shown here due to the fact that the "coefficients" of the algebraic expression for the speed all happen to be zero for some trajectories. We characterize a class of trajectories (or, modes of operation) from which the speed of the machine can be estimated from the stator currents and voltages. It is then shown how this speed estimate can be used in a field-oriented controller with the machine operating at low, or even zero, speed under full load. M. Li, J. Chiasson, and L. M. Tolbert are with the ECE Department, University of Tennessee, Knoxville, TN 37996, [email protected], [email protected], [email protected] M. Bodson is with the ECE Department, University of Utah, Salt Lake City, UT 84112, [email protected]. L. M. Tolbert is also with Oak Ridge National Laboratory, Oak Ridge TN. [email protected] Drs. Chiasson and Tolbert would like to thank Oak Ridge National Laboratory for partially supporting this work through the UT/Battelle contract no. 4000007596. Dr. Tolbert would also like to thank the National Science Foundation for partially supporting this work through contract NSF ECS-0093884.

0-7803-9568-9/05/$20.00 ©2005 IEEE

(1) (2) (3)

where,  is the position of the rotor, $ = g@gw> qs is the number of pole pairs, lVd > lVe are the (two phase equivalent) stator currents, and # Ud , # Ue are the (two phase equivalent) rotor flux linkages, UV and UU are the stator and rotor resistances, P is the mutual inductance, OV and OU are the stator and rotor inductances, M is the inertia of the rotor, and  O is the load torque. The symbols WU = =

OU UU

P OV OU

 =1 =

P2 OV OU

UV 1 1 P2 + OV OV WU OU

have been used to simplify the expressions. WU is referred to as the rotor time constant while  is called the total leakage factor. III. A LGEBRAIC S PEED O BSERVER Differentiating (1) gives  g g2 g$ g l = (1  mqS $WU ) # U  mqS # U   lV gw2 V WU gw gw gw 1 g + (4) x = OV gw V Using the complex-valued equations (1) and (2), one can g eliminate # U and # U from (4) to obtain gw µ ¶ 1 g 1 g2 l = (1  mqS $WU ) x l + lV  gw2 V WU gw V OV V g 1 g P + 2 (1  mqS $WU ) lV   lV + x WU gw OV gw V µ ¶ g g$ mqS WU 1  xV lV + lV  = 1  mqS $WU gw OV gw (5)

3438

Solving (5) for g$@gw gives

At low speeds, defined by |t2 $| ¿ |t1 | > equation (9) reduces

(1  mqS $WU )2 1  mqS $WU g$ + × = gw mqS WU2 mqS WU g 1 g g2 P (1  mq $W ) l   +  l l x S U V V V WU2 gw OV gw gw2 V = 1 g xV lV + lV  gw OV (6) If the signals are measured exactly and the dynamic model is correct, the right-hand side must be real. From (1) it is seen that the denominator in the last term of (6) is equal to (@WU ) (1  mqS $WU ) # U and thus (6) is ¯singular (i.e., the ¯ ¯ ¯ denominator in (6) is zero), if and only if ¯# U ¯  0. Breaking down the right-hand side of (6) into its real and imaginary parts, the real part has the form g$ = d2 (xVd > xVe > lVd > lVe ) $ 2 + d1 (xVd > xVe > lVd > lVe ) $ gw (7) + d0 (xVd > xVe > lVd > lVe ) = The expressions for d2 (xVd > xVe > lVd > lVe ) > d1 (xVd > xVe > lVd > lVe ) > and d0 (xVd > xVe > lVd > lVe ) are lengthy and therefore not explicitly presented here (Appendix VII-B gives their steady-state expressions). It is shown in Appendix VII-C that (7) is never stable in steady state. On the other hand, the imaginary part of (6) has no derivatives in the speed leading to a 2qg degree polynomial equation in $ of the form t($) , t2 (xVd > xVe > lVd > lVe )$ 2 + t1 (xVd > xVe > lVd > lVe )$ + t0 (xVd > xVe > lVd > lVe )= (8) If $ is the speed of the motor, then t($) is zero. The expressions for t2 (xVd > xVe > lVd > lVe )> t1 (xVd > xVe > lVd > lVe ) > and t0 (xVd > xVe > lVd > lVe ) are lengthy and not explicitly presented here (Their steady-state expressions are given in Appendix VII-A.). There are two solutions to equation (8) and at least one of these two solutions must track the motor speed. This equation does not have any stability issue, but a procedure is required to determine which of the two solutions is correct. Further, there are situations when the speed cannot be determined by (8). For example, if xVd = constant and xVe = 0, it turns out that t2 (xVd > xVe > lVd > lVe ) = t1 (xVd > xVe > lVd > lVe ) = t0 (xVd > xVe > lVd > lVe )  0 and $ is not determinable from (8)1 . On the other hand, if the machine is operated at zero speed ($  0) with a load on it, then t2 (xVd > xVe > lVd > lVe )  0 and t1 (xVd > xVe > lVd > lVe ) 6= 0> and a unique solution is specified by (8) (see Appendix VIIA where this is proved in steady state). In fact, for low speed trajectories, consider equation (8) written in the form (t2 $ + t1 ) $ + t0 = 0=

(9)

1 An induction machine is not typically operated under these conditions. See [15] for more discussion of this issue.

t1 $ + t0 = 0 and $ is uniquely determined by $ = t0 @t1 = Appendix VII-B shows that, in steady state, |t2 $| ¿ |t1 | if 2 (WU qs $) ¿ 1= If t2 (xVd > xVe > lVd > lVe ) 6= 0> one determines the correct solution of (8) as follows: Differentiate equation (8) to obtain the new independent equation (2t2 $ + t1 )

g$ + t˙2 $ 2 + t˙1 $ + t˙0  0= gw

(10)

Next, g$@gw is replaced by the right-hand side of equation (7) to obtain a new algebraic polynomial equation in $ given by j($) , 2t2 d2 $ 3 + (2t2 d1 + t1 d2 + t˙2 ) $ 2 + (2t2 d0 + t1 d1 + t˙1 ) $ + t1 d0 + t˙0 =

(11)

j($) is a third-order polynomial equation in $ for which the speed of the motor is one of its zeros. Dividing2 (11) by t($) from (8), the polynomial (11) has the form j($) = (2t2 d2 $ + 2t2 d1  t2 t1 d2 + t˙2 ) t($) + u1 (xVd > xVe > lVd > lVe ) $ + u0 (xVd > xVe > lVd > lVe ) = (12) where u1 (xVd > xVe > lVd > lVe ) , 2t22 d0  t2 t1 d1 + t2 t˙1  2t2 t0 d2 + t12 d2  t1 t˙2 (13) and u0 (xVd > xVe > lVd > lVe ) , t2 t1 d0 + t2 t˙0  2t2 t0 d1 + t0 t1 d2  t0 t˙2 = (14) If $ is equal to the speed of the motor, then both j($) = 0 and t($) = 0, and one obtains u($) , u1 (xVd > xVe > lVd > lVe ) $+u0 (xVd > xVe > lVd > lVe ) = 0= (15) This is now a first-order polynomial equation in $ with a unique solution as long as u1 (the coefficient of $) is nonzero (It is shown in Appendix VII-D that u1 6= 0 in steady state if t2 6= 0). The coefficients of u1 > u0 contain 3ug derivatives of the stator currents and 2qg derivatives of the stator voltages and, therefore, noise is a concern. Rather than use this purely algebraic estimator, it is now shown how to combine it with the dynamic model to obtain a smoother (yet stable) speed estimator. 2 Given the polynomials j($)> t($) in $ with deg{j($)} = qj > deg{t($)} = qt , the Euclidean division algorithm ensures that there are polynomials ($)> u($) such that j($) = ($)t($) + u($) and deg{u($)} $ deg{t($)} 3 1 = qt 3 1= Consequently, if $ 0 is a zero of both j($) and t($), then it must also be a zero of u($).

3439

IV. S TABLE DYNAMIC S PEED O BSERVER Dividing the right side of the differential equation model (7) by t($) (t2 (xVd > xVe > lVd > lVe ) 6= 0) > one obtains where

d2 $ 2 + d1 $ + d0 =  × t($> w) + $ + 

(16)

 , d1  d2 t1 @t2

(17)

 , d0  d2 t0 @t2 =

(18)

and

V. S IMULATION R ESULTS As a first look at the viability of the observer (22), simulations were carried out to test it. Here, a three-phase (two-phase equivalent) induction motor model was simulated using S IMULINK with parameter values chosen to be qs = 2> UV = 5=12 ohms> UU = 2=23 ohms, OV = OU = 0=2919 H> P = 0=2768 H, M = 0=0021 k-gm2 >  O_udwhg = 2=0337 N-m> Lmax = 2=77 A> Ymax = 230 V=

Then, as t($> w)  0, equation (7) may be rewritten as

Uw

(w> w0 ) , h

w0

( )g

the fundamental solution of (19), the full solution is given by Z w $(w) = (w> w0 )$(0) + (w>  )( )g = w0

Consequently, a sufficient condition for stability is that (w)   ? 0 for some  A 0= It is shown in Appendix VII-C that  A 0 in steady state> so the system is never stable in steady state. For the case that t2 (xVd > xVe > lVd > lVe ) 6= 0> consider (19) to be the induction motor “model” and the solution $ of algebraic estimator (15) to be the “measurement”. Then, let an observer be defined by gˆ $ = (w)ˆ $ + (w) + c ($  $ ˆ) = (20) gw If c  (w) A  A 0 for all w, then the estimator (20) is stable with a rate of decay of the error no less than . As this estimator is the result of integrating the signals (w)> (w)> and $ (from (15)), it is a smoother estimate than the purely algebraic estimate. In the case where t2 (xVd > xVe > lVd > lVe ) = 0> then the right side of equation (7) can be divided by t1 (xVd > xVe > lVd > lVe ) $ + t0 (xVd > xVe > lVd > lVe ) = 0 to obtain: gˆ $ = f(w) + c ($  $ ˆ) = (21) gw If c A  A 0 for all w, then the equation (21) is stable with a rate of decay of the error no less than = The estimate of speed proposed here is defined as the solution to the observer gˆ $ ˆ 2 + d1 (xVd > xVe > lVd > lVe ) $ ˆ , d2 (xVd > xVe > lVd > lVe ) $ gw ˆ) (22) + d0 (xVd > xVe > lVd > lVe ) + c ($  $ where $,

½

t0 @t1 u0 @u1

if |t2 $ ˆ |  0=05 |t1 | [See (8)] if |t2 $ ˆ | A 0=05 |t1 | [See (15)].

In Appendix VII-A it is shown that in steady state t1 6= 0 if t2 = 0 while in Appendix VII-D it is shown that u1 6= 0 if t2 6= 0=

The induction motor model for the simulation is based on equations (1), (2), and (3). In the control scheme, the estimated speed is fed back to a current command fieldoriented controller [9]. Figure 1 shows the simulation results of the motor speed and speed estimator with the motor under full load. From w = 0 to w = 0=4 seconds, a constant xVd is applied to the motor to build up the flux and the motor is considered to be held with a brake so that $  0= At w = 0=4 seconds, the brake is released and the machine is running on a low speed trajectory ($ max = 5 udg@v) with full load at the start. The estimated speed $ ˆ is used in the field-oriented controller. In this simulation, the observer gain c in equation (20) was chosen to be 1000=Figure 2 shows the simulation 6 4 2 Speed in rad/s

g$ = (w)$ + (w) (19) gw which is a linear first-order time-varying system. With

0

ωˆ ω

-2 -4 -6 -8

0

2

4 Time in seconds

6

8

Fig. 1. $ and $ ˆ with the motor tracking a low speed trajectory ($max = 5 udg@v) with full load at the start.

results of the motor speed and speed estimator with the motor under full load. From w = 0 to w = 0=4 seconds, a constant xVd is applied to the motor to build up the flux and the motor is considered to be held with a brake so that $  0= At w = 0=4 seconds, the brake is released and the machine is controlled to a zero speed trajectory ($  0) with full load at the start. $ ˆ is used in the field-oriented controller. The observer gain c in equation (20) was again chosen to be 1000=

3440

ωˆ

The explicit expression for t2 is 3 à ¡ ¢ !2 g l2Vd + l2Ve 1 2 2 t2 , qs × C OV WU 4 gw ¡2 ¢ g lVd + l2Ve  WU2 (xVd lVd + xVe lVe ) gw 2 ¡ ¢¡ ¢ W + U l2Vd + l2Ve x2Vd + x2Ve OV ¡2 ¢ ¶ µ 2 2 1 P 2 g lVd + lVe + 2 +  OV WU WU 4 gw µ ¶2 glVd glVe + OV WU2 lVe  lVd gw gw µ ¶ glVd glVe + 2WU2 lVe  lVd (xVe lVd  xVd lVe ) gw gw µ ¶ ¡ ¢2 P  +  OV WU2 l2Vd + l2Ve W µ U ¶ ¶ ¡2 ¢ P 2 2 +  2 WU lVd + lVe (xVd lVd + xVe lVe ) = WU

0

ω

Speed in rad/s

-1 -2 -3 -4 -5 -6 0

1

2 3 Time in seconds

4

Fig. 2. $ and $ ˆ with the motor tracking a zero speed trajectory ($ 0) with full load at the start.

In steady state, let (see [9])

VI. C ONCLUSIONS AND F UTURE W ORK This paper introduced a new approach to speed sensorless control of an induction motor which entails using an algebraic estimate of the speed to stabilize a dynamic speed observer. The new observer does not require any sort of “slowly varying” speed assumption. The singularities of the observers were characterized under steady-state conditions. This sensorless speed controller shows potential for speed estimation at low speeds under full load. Future work will include the effect of parameter variation on the speed estimation as well as experimental results.

xVd + mxVe = X V hm$V w lVd + mlVe = L V hm$V w = The complex phasors UV and IV are related by ¯ ¯ XV  ¸= LV = 1+m VV UV + m$ V OV 1+m Vs Vs

Here Vs , LV =

VII. A PPENDIX : S TEADY-S TATE E XPRESSIONS In the following, $ V denotes the stator frequency and V denotes the normalized slip defined by V , ($ V  qs $) @$ V = With xVd + mxVe = X V hm$V w and lVd + mlVe = L V hm$V w , it is shown in [9] under steady-state conditions that the complex phasors X V and L V are related by (Vs , UU @ ($ V OU ) = 1@ ($ V WU )) LV

XV Ã

=

UV + m$ V OV =

³ UV +

1 + m VVs V 1 + m V s

!

XV = ´ 2 $ V OV (1+V 2 $2V WU ) +m 1+V 2 $ 2 W 2

(1)V$2V OV WU 2 1+V 2 $ 2V WU

V

U

A. Steady-state expressions for t2 > t1 > and t0 The steady-state expressions for t2 > t1 > and t0 are now derived. These expressions are then used to show that t2 A 0 for $ 6= 0> t2  0 for $ = 0> and t1 6= 0 if t2  0=

UU $ V OU

=

1 $V WU

XV h

UV + m$ V OV

=³ UV +

so that

1+mV$ V WU 1+mV$ V WU

i

XV ´ 2 $ V OV (1+V 2 $2V WU ) +m 1+V 2 $ 2 W 2

(1)V$ 2V OV WU 2 1+V 2 $ 2V WU

Further,

xVd lVd + xVe lVe xVe lVd  xVd lVe l2Vd + l2Ve x2Vd + x2Ve

= = = =

V

U

Re (X V L V ) Im (X V L V ) |L V |2 |X V |2 =

Thus, the 1vw > 2qg > and 4wk terms of t2 are all zero, i.e., Ã ¡ ¢ !2 g l2Vd + l2Ve 21 2 =0 qs OV WU 4 gw ¢ ¡2 2 2 2 g lVd + lVe qs WU (xVd lVd + xVe lVe ) = 0 gw ¢2 ¡ q2s (P@WU + 2) (1@4) OV WU2 g l2Vd + l2Ve @gw = 0=

3441

The 3ug term of t2 is given by W2 q2s U OV

¡2 ¢¡ ¢ lVd + l2Ve x2Vd + x2Ve =

W2 q2s U OV

The 5wk term of t2 is given by µ ¶2 glVd glVe 2 2 qs OV WU lVe  lVd gw gw 2

=

The 6

If $ = 0> then V = 1 and t1 6= 0= Finally, the steady-state expression for t0 is

|L V |2 |X V |2 =

t0 = µ³ UV +

2

OV WU2 $ 2V |L V | |X V | ´2 $2 O2 1+V 2 $2 W 2 2 = ( ) (1)V$ 2 O W UV + 1+V 2 $V2 WV2 U + V V 2 2 2V 2U V U (1+V $V WU )

×

q2s ³

The purpose of this appendix is to show that in steady state, |t2 $| ¿ |t1 | if (WU qs $)2 ¿ 1= In steady state, 2

|t2 $| = µ³ UV + ×

The 7 term of t2 is ¶ µ ¡ ¢2 P 2 qs  +  OV WU2 l2Vd + l2Ve WU ³ 2 ´ Uv (1)Uv + WU2 |L V |2 |X V |2 O W v U 2 = qs ³ ´2 $2 O2 1+V 2 $2 W 2 2 = ( ) (1)V$ 2 O W UV + 1+V 2 $V2 WV2 U + V V 2 2 2V 2U V U (1+V $V WU )

2

$ 2 OV (1  ) (1  V) × V =  (1 + V 2 $ 2V WU2 )

(WU qs $)2  (1 + V 2 $ 2V WU2 )

qs |$ V | OV (1  )2 |X V |4 |t1 | = µ³ ´2 $2 O2 1+V 2 $2 W 2 2 ¶2 ( ) (1)V$ 2 O W UV + 1+V 2 $V2 WV2 U + V V 2 2 2V 2U V U (1+V $V WU ) ¯³ ´¯ ¯ 2 ¯ ¯ 1  (WU qs $) ¯ × =  (1 + V 2 $ 2V WU2 )

Their ratio is then

¯ ¯ |t2 $| ¯¯ (WU qs $)2 ¯¯ =¯ ¯ ¯ 1  (WU qs $)2 ¯ |t1 | 2

which shows that (WU qs $) ¿ 1 =, |t2 $| ¿ |t1 | =

Finally, substituting these steady-state expressions into the expression for t2 > one obtains

(1)V$ 2V OV WU 2 1+V 2 $ 2V WU

4

|X V | ´2 $2 O2 1+V 2 $2 W 2 2 ¶2 ( ) + V V 2 2 2V 2U (1+V $V WU )

C. Steady-state expressions for d2 > d1 > d0 > and  The steady-state expressions for d2 > d1 > and d0 are now given and used to show that the steady-state value for  is always positive. The steady-state expressions for d2 > d1 > d0 are 4

(23)

With $ 6= 0> it is seen that t2 A 0 and t2 = 0 if and only if V = 1 (which is equivalent to $ = 0) = Similarly, it can be shown that the steady-state expression for t1 is

q2s |X V | d2 = µ³ ´2 $2 O2 1+V2 $2 W 2 2 ¶2 ( ) (1)V$ 2V OV WU + V V 2 2 2V 2U UV + 1+V 2 $2 W 2 V U (1+V $V WU ) 2

×

$ V (1  ) 1 ×  2 (1 + V 2 $ 2V WU2 ) ghq

4

qs $ V |X V | ´2 $2 O2 1+V 2 $2 W 2 2 ¶2 2 ( ) (1)V$ V OV WU + V V 2 2 2V 2U 2 1+V 2 $ 2V WU (1+V $V WU ) ³ ´ OV (1  )2 1  $ 2V WU2 (1  V)2 × = (24)  (1 + V 2 $ 2V WU2 )

t1 = µ³ UV +

4

qs |$ V | OV (1  ) |X V | ´2 $2 O2 1+V2 $2 W 2 2 ¶2 ( ) (1)V$ 2V OV WU + V V 2 2 2V 2U 2 1+V 2 $ 2V WU (1+V $V WU )

and

The 8wk term of t2 is ¶ µ ¡ ¢ P 2 qs  2 WU2 l2Vd + l2Ve (xVd lVd + xVe lVe ) WU ³ ´ 2 2 2Uv 1 2  O + WU WU |L V | |X V | v 2 =qs ³ ´2 $2 O2 1+V 2 $2 W 2 2 ( ) (1)V$ 2 O W UV + 1+V 2 $V2 WV2 U + V V 2 2 2V 2U V U (1+V $V WU ) µ ¶ (1  ) V$ 2V OV WU × UV + = 1 + V 2 $ 2V WU2

t2 = µ³ UV +

(25)

2

wk

q2s WU2

2

$ 2V OV (1  ) (1  V) =  (1 + V 2 $ 2V WU2 )

B. (WU qs $) ¿ 1 =, |t2 $| ¿ |t1 |

term of t2 is µ ¶ glVd glVe 2 2 qs 2WU lVe  lVd (xVe lVd  xVd lVe ) gw gw 2 $ V OV (1+V 2 $ 2V WU ) 2WU2 $ V |L V |2 |X V |2 2 $2 W 2 1+V 2 V U = qs ³ ´2 $2 O2 1+V 2 $2 W 2 2 = ( ) (1)V$ 2 O W UV + 1+V 2 $V2 WV2 U + V V 2 2 2V 2U V U (1+V $V WU ) wk

 |X V |4 ´2 $2 O2 1+V 2 $2 W 2 2 ¶2 ( ) (1)V$ 2V OV WU + V V 2 2 2V 2U 2 1+V 2 $ 2V WU (1+V $V WU )

(26)

4

d1 = µ³ UV +

3442

×

qs |X V | ´2 $2 O2 1+V 2 $2 W 2 2 ¶2 ( ) (1)V$ 2V OV WU + V V 2 2 2V 2U 2 1+V 2 $ 2V WU (1+V $V WU )

2$ 2V (1  )2 (1  V) 1 × 2 2 2 2  (1 + V $ V WU ) ghq

(27)

and 4

d0 = µ³ UV + ×

 |X V | ´2 (1)V$ 2V OV WU + 2 2 2 1+V $ W V

2 $ 2V O2V (1+V 2 $ 2V WU )

U

2

(1+V 2 $2V WU2 )

$ 3V (1  )2 (1  V)2 1 × = 2 2 2 2  (1 + V $ V WU ) ghq

2

¶2 (28)

where 4

³

2 (1) 1+(V1)V$ 2V WU 2 WU 1+V 2 $ 2V WU

´2

qs WU |X V | ghq , µ³ ´2 $2 O2 1+V 2 $2 W 2 2 ¶2 ( ) (1)V$ 2V OV WU + V V 2 2 2V 2U UV + 1+V 2 $2 W 2 V U (1+V $V WU ) µ ¶2 2 $ V OV (1+V 2 $ 2V WU ) 4  $V qs WU |X V | 2 1+V 2 $ 2V WU + µ³ ´2 $2 O2 1+V 2 $2 W 2 2 ¶2 = ( ) (1)V$2V OV WU + V V 2 2 2V 2U UV + 1+V 2 $2 W 2 V U (1+V $V WU ) Remark Recall that it was pointed out in Section ¯ ¯ III ¯ ¯ (following equation (6)) that ghq = 0 if and only if ¯# U ¯  0= To compute the steady-state value of > note that by (17),  = d1  d2 t1 @t2 = It is then easily seen that d1 A 0> d2 t1 ? 0> and t2 A 0> so that in the steady state  A 0= That is, the system (19) is never stable in steady state. D. Steady-state expression for u1 It is now shown that the steady-state value of u1 (xVd > xVe > lVd > lVe ) in (13) is nonzero. Substituting the steady-state values of t2 > t1 > t0 > d2 > d1 > and d0 (noting that t˙1  0 and t˙2  0 in steady state) into (13) gives  |X V |12 ´2 $2 O2 1+V 2 $2 W 2 2 ¶6 ( ) (1)V$ 2V OV WU + V V 2 2 2V 2U 2 1+V 2 $ 2V WU (1+V $V WU ) µ ¶3 q4s (1  )6 $ 3V O2V 1 × × 1 + V 2 $ 2V WU2 4 ³ ´2 1 2 × 1 + WU2 $ 2V (1  V) × ghq where ghq is given in Appendix VII-C. It is then seen that u1 6= 0 in steady state. u1 = µ³ UV +

E. Steady-state speed Substituting in the steady-state values of d2 > d1 > and d0 , it is seen that d21  4d2 d0  0> so that, interestingly, the steady-state value of the right-hand side of (7) may be rewritten as µ ¶2 d1 2 d2 $ + d1 $ + d0 = d2 $ + 2d2

where d2 is nonzero by (26) = On the other hand, the steady-state solutions of (8) are p t1 + t12  4t2 t0 =$ $1 , 2t2 and p t1  t12  4t2 t0 1 = 2 2 = $2 , 2t2 WU qs $

Interestingly the correct steady-state speed is found by choosing the + sign in the quadratic formula when computing the roots of (8). However, this is not necessarily true if the system is not in steady state. R EFERENCES [1] K. Rajashekara, A. Kawamura, and K. Matsuse, eds., Sensorless Control of AC Motor Drives - Speed and Position Sensorless Operation. IEEE Press, 1996. [2] P. Vas, Sensorless Vector Control and Direct Torque Control. Oxford University Press, 1998. [3] M. Vélez-Reyes, K. Minami, and G. Verghese, “Recursive speed and parameter estimation for induction machines,” in Proceedings of the IEEE Industry Applications Conference, pp. 607–611, 1989. San Diego, California. [4] M. Vélez-Reyes, W. L. Fung, and J. E. Ramos-Torres, “Developing robust algorithms for speed and parameter estimation in induction machines,” in Proceedings of the IEEE Conference on Decision and Control, pp. 2223–2228, 2001. Orlando, Florida. [5] M. Vélez-Reyes, Decomposed algorithms for parameter estimation. PhD thesis, Massachusetts Institute of Technology, 1992. [6] M. Vélez-Reyes and G. Verghese, “Decomposed algorithms for speed and parameter estimation in induction machines,” in Proceedings of the IFAC Nonlinear Control Systems Design Symposium, pp. 156–161, 1992. Bordeaux, France. [7] M. Bodson and J. Chiasson, “A comparison of sensorless speed estimation methods for induction motor control,” in Proceedings of the 2002 American Control Conference, pp. 3076–3081, May 2002. Anchorage, AK. [8] E. G. Strangas, H. K. Khalil, B. A. Oliwi, L. Laubnger, and J. M. Miller, “A robust torque controller for induction motors without rotor position sensor: Analysis and experimental results,” IEEE Transactions on Energy Conversion, vol. 14, pp. 1448–1458, December 1999. [9] W. Leonhard, Control of Electrical Drives. 3rd Edition, SpringerVerlag, Berlin, 2001. [10] M. Ghanes, J. Deleon, and A. Glumineau, “Validation of an interconnected high gain observer for sensorless induction motor on low frequencies benchmark: Application to an experimental set-up,” IEE Proceedings Control Theory and Applications, vol. 152, pp. 371–378, July 2005. [11] D. Neši´c, I. M. Y. Mareels, S. T. Glad, and M. Jirstrand, “Software for control system analysis and design: symbol manipulation,” in Encyclopedia of Electrical Engineering, John Wiley & Sons, J. Webster, Editor, 2001. available online at http://www.interscience.wiley.com:83/eeee/. [12] M. Diop and M. Fliess, “On nonlinear observability,” in Proceedings of the 1st European Control Conference, pp. 152–157, Hermès, Paris, 1991. [13] M. Diop and M. Fliess, “Nonlinear observability, identifiability and persistent trajectories,” in Proceedings of the 36th Conference on Decision and Control, pp. 714–719, Brighton England, 1991. [14] M. Fliess and H. Síra-Ramirez, “Control via state estimation of some nonlinear systems,” in Symposium on Nonlinear Control Systems (NOLCOS-2004), September 2004. Stuttgart, Germany. [15] S. Ibarra-Rojas, J. Moreno, and G. Espinosa-Pérez, “Global observability analysis of sensorless induction motors,” Automatica, vol. 40, pp. 1079–1085, 2004. [16] M. Bodson, J. Chiasson, and R. Novotnak, “High performance induction motor control via input-output linearization,” IEEE Control Systems Magazine, vol. 14, pp. 25–33, August 1994. [17] R. Marino, S. Peresada, and P. Valigi, “Adaptive input-output linearizing control of induction motors,” IEEE Transactions on Automatic Control, vol. 38, pp. 208–221, February 1993.

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