Sensor Less Control For Im Via Fuzzy Observer Design

IEEE ISIE 2006, July 9-12, 2006, Montreal, Quebec, Canada

Sensorless

Control

for Induction Motors via Fuzzy Observer Design

Kuang-Yow Lian and Cheng-Yao Hung Department of Electrical Engineering Chung-Yuan Christian University Chung-Li 32023, Taiwan TEL: 886-3-2654815, FAX: 886-3-2654899 Email: lian dgdec.ee.cycu.edu.tw

Abstract-In this paper, sensorless control for induction motors is developed based on fuzzy observer design. First, the T-S fuzzy model is used to exactly represent the induction motor. Then, the fuzzy observer of the induction motor is straightforwardly constructed to estimate the immeasurable states of rotor speed and rotor flux, where the estimation gains are obtained by solving a set of linear matrix inequalities (LMIs). This observer scheme gives the estimated states converging to the real states exponentially. For the controller design, in light of the principles of vector control, a new concept, namely virtualdesired-variable synthesis is introduced to design the control law. Finally, numerical simulations and experiments are carried out to verify the theoretical results and show satisfactory performance.

I. INTRODUCTION

Induction motors (IMs) have been widely applied as electromechanical actuators because of their ruggedness, low maintenance, and low cost. Their usage in speed and torque tracking control application is expected to be quite popular in the near future [1], [2], [3], [4]. For advanced servo applications, the control technique mainly relies on high performance IM drive, where the need of precise rotor speed for feedback is essential. Optical encoders are usually used to detect the rotor speed. However, these speed sensors weaken the ruggedness, reliability and simplicity of an IM. It seems quite clear that the speed sensors cannot even be mounted in a hostile environment. In view of these points, many researches have focused on sensorless control. Due to the high order nonlinearity of IM's dynamics, the estimation of the rotor speed and flux becomes a challenging problem. To overcome these difficulties, various control algorithms have been proposed in the literature [5], [6], [7]. In this paper, we propose a new scheme for speed sensorless control for the full fifth-order model of IMs. The proposed scheme is mainly based on the fuzzy observer design. In recent years, fuzzy control has been widely applied to deal with nonlinear systems. Many researches on this issue are carried out based on Takagi-Sugeno (T-S) fuzzy models [8], [9]. The T-S fuzzy approach has been extensively used to model nonlinear systems. The basic idea for the approach is to decompose the model of a nonlinear system into a set of linear subsystems with associated nonlinear weighting functions. The stability analysis is carried out using Lyapunov direct method whereas the control problem is then fornulated into linear matrix inequalities (LMIs) [10], [11], [12]. This work pro1-4244-0497-5/06/$20.00 © 2006 IEEE

poses a fuzzy observer scheme to estimate the immeasurable variables of rotor speed and rotor flux of an IM. In traditional fuzzy observer design, the premise variables of fuzzy rules are assumed to be measurable. This is a strict constraint and limits its application [13], [14]. For many physical systems, including IMs, it is inevitable to use immeasurable states as the premise variables. The immeasurable premise variables often lead to non void disturbance of which the effect make the estimation error only remain in a residue set. Fortunately, we will point out that the membership functions of fuzzy sets for IMs satisfy a linear proportion type of property. The benefit of this type of membership functions is that the estimation error can converge to zero exponentially. Then, we obtain the observer gains by solving a set of LMIs using MATLAB LMI Toolbox. After finishing the speed and flux estimation design, the speed tracking control is investigated based on separation principle, i.e., the estimation error is supposed to be zero in the control design. At first, the speed tracking control is reformulated into the torque tracking problem. To achieve vector control, a set of virtual desired variables (VDVs) is introduced to synthesize the controller. The VDVs are determined in a straightforward manner based on the goal of achieving all of the desired torque, the constant desired flux, and the system stability. In the design, a skew-symmetric property pertaining to the dynamics of the IM is utilized to simplify the structure of the controller. To demonstrate the effectiveness of the proposed scheme, a voltage-fed model of drive system is set up to perform the task of speed tracking. The simulation and experimental results illustrate nice performance, even though in the case of low speed command is considered. II. DYNAMICAL MODEL OF INDUCTION MOTORS Let (isa, isb), (Aral Arb) and w denote the components of the stator current, rotor flux, and rotor speed, respectively. The induction motor is represented by a fifth-order model [15]: Zsa

=

=sb

( Uf+cL2

(

karsaA

2140

Arb

crL A,>b + )isa + c,2Ara +--

)isb

+L

Arb-LTA *L WAra +I U2

raWArb -

L

=T

Zsb

T-)

L

b +

_w

vUl

WAra

(1)

where (u1, u2) denotes the stator voltage; R5, Rr, L5, Lr and Lm are the stator resistance, rotor resistance, stator inductance, rotor inductance, and mutual inductance, respectively; a = Ls- L2/Lr; J, D, T1, are the mechanical inertia, natural damping, and loading torque, respectively; and T is the electromechanical coupling torque expressed as:

n.Lj (Ararsb -Arbsa) ' where np = (pole pair) x 3. Let us denote [X1 X2 X3 x41 = [ isa isb Ara Arb I T

(2)

=

x

The

model (1) can be rewritten as

M3+G(w)x+R(w) JA + Dw

=

T

=

T-T

(3) (4)

where M

Lr,oI2

[

I2 R(w) T

=

[I

[

J2

-

L

0

0

1

Rr Lr

LR

[ LrUi LrU2 0

-1

0

ImJ2

_Lm'2 r2

=[

0 -J2 J

G(w)- 00

0]

]', 7

I

'2 R,

=

+

c2

To express the induction motor (3) and (4) in terms of T-S fuzzy model, we rewrite the equations in the following form:

Plant Rule i: IF w is Fli and Ara is F2i and Arb iS F3i THEN (t) =AiC (t) + Bu (t) + b (6) y (t) = Cx(t), i =1, 2, .. 18 where W, Ara, and Arb are premise variables which are immeasurable. The fuzzy sets Fji (j = 1, 2, 3) and the system matrices Ai of subsystem i are set as follows:

F= l (z1)

d d+D1+ d

=

F21 (z2)

D1 i

F12 (Z1)

Di-dil

D2 D2 -d2

-

d2 +D2 d2 F22 (Z2) F31 (Z3) =D -3-d3 + Dd (z3) 3 (3 D3-d3, F32 -A A13 A12 O -A23 A21 A22

A31 A41

A32

D3 D3 -d3

A14'Oi

A24

A33 A43 O

X3

D1i-d X4

D2-d2 X5

D3-d3

A15 A25

-A34 O A35 A44 A42 A45 'S A54 A55 A53 L- A51 i A52 = = = where 01 = D3, V1 D2, O I= D1, 02 D3, 02 D2, Y02= di, 03 D3, 3 = d2, 503= D1; 04= D3, 4 = d2, ('4 = d, 05= d3, V5= D2,i50= D1, 06= d3, V6= D2, 9P6 = d1, 07= d3, V'7 d2, 07 = D108 = d3, V8 = d2, 508 = di. Using the singleton fuzzifier, product fuzzy inference and weighted average defuzzifier, the final output of the fuzzy system is inferred as follows: 8 ;z (t) E -ti(x(t)) {Aix (t) + Bu (t)} + b

Ai

=

y(t)

i=l

=

Cx(t),

(7) 8

(x (t)) with (x (t)) / E ti (x (t)) = (x(t)) = H=1 Fji (x (t)). Note that i=1 i ((t)) =1 for all t, where ,ui (x (t)) > 0 for all i = 1, 2, **, 8. Based on Cx(t) the setting of Fji and Ai, it can be checked that the inferred where the overall states x (t) = ( X X5 ), i.e., x (t) = output is exactly equivalent to the model of induction motor we note that all the membership functions [ isa isb Ara Arb wJ ] '; the measurable output y (t) = (5). (-)Moreover, the satisfy following property: Fij [isa isb]T; the control input u = [u1 u2]T, and the associated matrix and vector: Fij (x t) Fij (X-(t)) =Tlij (x() () A13 A14w A15 All A12 for some constant rij. Since Pi ( (0) A21 A22 -A23w A24 A25 Fli (x) F2i (x) F3i (x), it follows that A32 A(x) = -A34w A35 A31 A33 ;z (t) y (t)

A (x) x (t) + Bu + b

A41

A42

A43w

B=

Qoc o a

a

C

where AI1 = A51 = A52

,L A33

O T , b=

I

=

n,L, JL,

A44

=

[0

Pi( (t) - /-i('(t)) i (X - )F2i (X) F3i (X) + Fli () yi22(X -x)F3i (X) +Fli () F2i(x) Tyi3(X -) AT (X (t) x- (t))

A45 A55

A54

T

_ J

0 0 0

[TIyIF2i (x) F3j (x) + Fli (x) y2F3i (x) + Fli () F2i(x) Ty31. Hence the grade functions ,ut have a nice

where AT

=1

0

0 01

0

0 1 0 0 0

A22 = _

i

A44

A53

-AslArb As2Ara

(L +

-

7

A13 -

=

L2_R A24

Lr A34

=

),

A31

=

A43

L2

=

,

A42 A14

=

=

L,R, Lr

7

A23 =

1, A55 A

D J

A15 = A21 = A25 = A32 = A35 = A41 = A45 =:::: A54 = 0. Then, according to [13], the T-S fuzzy model representation of (5) can be expressed by the following rules: A12 A53

=

where

property: Property: The error of grade functions _t (x(t)))-t (x(t)) is proportional to the error (x(t) -x(t)). Before designing the observer and controller, some assumptions are made as follows: A.1 The loading torque T1 is a known constant. A.2 The parameters J, D, Lm, L,r R5, Rr are known constants.

2141

A.3. The desired speed wd is a smooth and bounded signal. A.4. The stator current and voltage are measurable, while the rotor flux and speed are assumed to be unavailable. III. FuzzY OBSERVER OF INDUCTION MOTORS

In this section, we will design the fuzzy observer to estimate the immeasurable states under Assumptionsl-4. According to the fuzzy model (5), the fuzzy observer is given as follows: Observer Rule i: IF w is Fli and Ara is F2i and Arb is F3i THEN (8) (t) = Ai;(t) +Bu(t) +b +Li (y(t)-(t)) S (t) = C;z(t) I i = 1, 2,. ,8

where the premise variables W, Ara, and Arb are the estimated states of W, Ara, and Arb, respectively; x (t) and y (t) denote the estimations of x (t) and y (t), respectively; and Li is an observer gain to be determined. The inferred output of the observer is 8

:; (t)

E pi (x(t)) {Ai- (t) + Bu (t) + b i=l +Li(y (t)-y (t))} C:z (t) .

y (t)

Define the state estimation error e (t) ing (7) by (9), it leads to e (t)

8 =

=

x

(t)

-

(t). Subtract-

pi (x(t)) {(Ai -LiC) e} + 1 (t)

i=l

where

(t)

8 =

I i=l

(9)

(10)

According to (12), we obtain 8

V0 (e) < E

i=l

where Gi = (Ai- LiC)T P + P (Ai -LiC) + UTU + PP. Hence, the exponential convergence of e (t) is concluded if Gi < 0. Using Schur complement [16] and letting PLi = Zi, the inequality Gi < 0 can be converted to the following LMIs: [ATPP+PAi -CTZfT -Z'c+U U P < Vi=1, 2,

=

, 8.

(13) Design of the Fuzzy Observer: For the fuzzy observer (8), suppose that all states and control input are bounded. If there exists a common positive definite matrix P and Zi such that the LMIs (13) are feasible, then the estimation error converges to zero exponentially by letting observer gains Li = P-'Zi We can solve LMIs (13) using powerful packages like MATLAB Toolbox to obtain P and Zi. In turn, the observer gains are calculated from Li = P lZi. IV. CONTROLLER SYNTHESIS BY VIRTUAL DESIRED VARIABLES

The concept of control design for a sensorless induction motor is shown in Fig. 1. For simplicity, we assume the fuzzy observer provides a perfect estimation of x, i.e., we let A = A and w = w. The use of separation principle in the controller design is suggested by the exponential convergence of estimation error, which is endowed with the robustness for some amounts of uncertainty. _1

rl

{Aix (t)} i (ti (X)l-H(X:))

(1 1)

The term 1 (t) in (11) arises due to immeasurable premise variables W, Ara, and Arb. Recalling the property in Section II, we have

lTl

pi (x')e (t)' Gie

8

8

i=l

i=l

CT EAi {Aix (t) } E {Aix (t) } AT e-

Fuzzy Observer-Based

~0),IA MechanicalI IvvIV part

.I i

I

Electrical part

L -----__ Vsu, L/vn Vsw

v

Supposed that x (t) are bounded (this will be confirmed in controller design given later), the term 1 (t) has the following bounded fashion TIl < eTUTUe (12)

_ . . . .E

IM

s

er. . . . . . . . . . . . . . . . . . . . . . .

with a constant matrix U. This undesired term 1 (t) will Fig. 1 The concept of control design for sensorless induction affect the estimation performance. Its effect, however, can be motor. exponentially attenuated to zero by suitably choosing observer gains Li. Now, we apply Lyapunov method to get the observer A. Speed Tracking Control gainsLi,i =1,2, ,8. First, denote the speed tracking error as w w- Wd. The Choose the Lyapunov function candidate V0 (e (t)) = tracking error dynamics can be rewritten as 4 (t) Pe (t). Taking the time derivative, we have 8 Jw + (D + kw) = T -Td + (Td- YO + kw), (14) Vo (e) < +Zi=l Hi P()e'K(Ai -LiC)P+P(Ai LiC)] e where Td denotes the desired torque which produces the desired speed; k, is an adjustable damping ratio; Tppe lTl. +C + 2142

Y= [1 CDd wdl is the regression vector; and the parameter vector 0= [Ti J D]T. For speed tracking control, the desired torque is selected as Td

This yields the following

=

For the last term, we rewrite it as (T- T) w = _X';, where L, [ X4d X3d X2 -x1 .Since R (w) depends on w, we further re-express (20) as =

YO -kw,

error

V7c

dynamics

Jw + (D + kw)

=

-

(15)

T -Td

Since the damping ratio D of induction motors is usually small, the damping term k,Cw plays a dominant role on improving the transient response for speed tracking. If T -Td is driven to zero, the rotor speed will converge to the desired value. In other words, the speed tracking control problem has been reformulated into the torque tracking problem, where the objective is T Td. The concept of virtual desired variables (VDVs) is introduced in the following to achieve the objective.

-LrJ2w ] ~+ f

-s X(+ kw[ 0

=

(D +kwA) 6J2 +

f;T

Lro7yI2 + L,R, Lr

K-'2

where

p=

After setting V

-

[

&LtR I2 R

r2

12x

-,

we

obtain

-(D + k,)2

xRx

(t),w)

1-

--

B. Concept of VDVs The VDVs consist with the virtual desired current (X1ld X2d) and virtual desired flux (X3d, X*4d). They will be specified by satisfying (i) the desired torque Td

=

nL. (Xx2dX3d-XldX4d);

(16)

(ii) the constant desired flux 2

2

2

c= X3cd + X4d;

It

can

be checked that the matrix R > 0 by choosing K >-LrRs.

Hence the exponential stability is shown if the VDVs defined.

well

D. Specijying VDVs In the remaining design procedure, control law u and VDVs -Xd are chosen such that (P = 0. The perturbed term (19) is explicitly rewritten as:

(17)

(iii) the stability when the true state variables (t) track the VDVs. Notice that the condition (17) for desired flux is to achieve the optimal torque [4]. Define the error signal for the electrical part as 5 = -d, where Xd = [ Xld X2d X3d X4d ] The control objective of steering T to track Td can be achieved if 0. To this end, the equation (3) is rewritten in terms

are

=

p

(p2

(p4

(p3

I

-XL(7yld +L

LrUi7;d + LmWX4d

LrU1

x

[ (pl

3d

LrU7;2d- LmWX3d- LrO77X2d + -X4d - JX4d + L Xld -RL X3d -R L 4d --'C4d + bJXf3d +LLR, X2d

Lr U2

X4d

1-(P

(21)

x

To satisfy (p3

(p4

=

=

0,

we can

obtain

-÷

of

X3d

as

Mx+G(w)x+R(w) where form

(P

is regarded

=

p

+

';4d

(18)

p

(_

+J L

perturbed term with the following

as a

(19) (p = T [Mid + G (w)Xd+ R (w) Xdl ~p. We will intend to set (P = 0 to specify the VDVs, which will

VC

(x (t), w) -

+

-Td)

LmRr

1-ld

Lr

X, 2d

X1 [

I

J

(22)

Xf2

PX3d

J

Substituting (23) into (22), Xld 0

=

PX4d

X3d

=

1

X3d

X4d

J

J

(23)

obtain

we

(r wp)J2+I2)[X3d]

-wnpJ2 C

Rr

(24)

1

L

2

Since the desired states also satisfy (16), substituting (24) into

(t)T R (w)j2 (t)(T+ (t)T -(

(D + kwg)

Lr PJ2

X4d

+

(17), ( X3d X4d ) = ( coCs (p (t)) csin (p (t)) ) for variable p (t) to be determined later. It follows that X4d

VC (t)i ) 2 (t)T M (t) + ijsJ2 The time derivative of Vc along (I15) and (I18) is

X3d

JJ2)

+

From

be addressed in Subsection D.

C. Stability Analysis To analyze the stability of the system and design (P as well, we choose the following Lyapunov function candidate:

RrI2

-

(16) yields

(20) 2143

p

(t)

=

np

c2 Td

+

+

Lc 'A L

(X1X3d

+

X2X4d),

where p (t) is thus defined. On the other hand, to satisfy (P1 (p2= 0 in (21), the control law is formulated as follows: U2

U2

Xld [

X2d

+L'wJ L_ R, _2 L

T

lFXldlI

+

[X2dJ L

[

(('

XIi

Lr

+

[X2J L,

[X3d X4d

) J2 (2

Indeed, the implementation of the control law (25) is complicated due to the first term on the right-hand side, which includes the time derivative of Xld and X2d. But, we note that the exponential stability shown in Subsection C give a very nice robustness to the uncertainty. This property permits us to approximate the 'id by using aid Xid-Xid, where Xid + Xid = Xid. The resulting simplified control law will be adopted in our simulation and experiment. V. EXPERIMENTAL RESULTS

In this section, the performance of the control scheme will be verified by numerical simulations and experiments. For comparison, numerical simulation results are put next to associated experiments results. The specifications and parameters of the induction motor are listed in Table I. TABLE I THE SPECIFICATION AND PARAMETERS OF THE INDUCTION MOTOR

Rated Specification Pole Pair 3 Power 0.4 kW 120 V Voltage Current 3.4 A 1500 rpm Speed Parameters 2.85Q R, Rr 4.0Q 0.19667 H LI Lr 0.19667 H Lm 0.1886 H J 0.001 kg n2

the power inverter. Meanwhile, a set of I/0 modules are constructed in the card for the voltage/current measurement, encoder interface and some protections. For the purpose of comparison, the motor's instantaneous speed is measured by an optical incremental encoder with 1000 pulse/revolution. The software we adopt is Simulink 3.0 and Matlab 5.3. In addition, the system combines the motor control card with the Simulink/Real Time Workshop Toolbox such that the setup of the control law in the simulation can be directly applied to the experiment. The overall execution time interval is set as 6 sec for both simulations and experiments. Consider tacking of speed wd 30 + 20 sin(X) rad/sec. The control parameters are chosen as: k, = 0.47, c = 0.53. The simulation and experiment results of speed tracking for desired and actual speed are shown in Fig. 2(a). The maximal tracking error are 0.3 and 1 rad/sec for simulation and experiment, respectively. The desired and estimation rotor speed are shown in Fig. 2(b). The maximal tracking error are 0.4 and 0.5 rad/sec for simulation and experiment, respectively. The actual and estimation rotor speed are shown in Fig. 2(c). The speed estimation error (w -) is shown in Fig. 2(d). The experiment results of stator voltage and stator current for one phase are shown in Figs. 3. From these figures, we can find that the tracking error will tend to zero asymptotically when time goes to infinity. Furthermore, the stator current response, stator voltage, are satisfactorily. From the results, we can conclude the performance of the proposed control scheme. VI. CONCLUSION A fuzzy observer-based controller has been proposed to achieve speed tracking. To this end, some new concepts, such as virtual desired variables and Lipschitz-like condition are introduced to benefit the control design. Here, the general fuzzy model of induction motors is used to accomplish the design. The T-S fuzzy observer algorithm has been developed for the estimation of the rotor speed and the flux of an induction motor. The observer gains are obtained by solving a set of LMIs. The two-stage design technique is applied to construct a controller for speed tracking control. The numerical simulations and experimental results have illustrated the expected performance and indicate that the integration of the fuzzy observer and VDV-synthesis controller are very suitable in induction motor applications. ACKNOWLEDGMENT

This work was supported by the National Science Council, R.O.C., under Grant NSC 93-2213-E-033-008.

According to LMI (13), where we let U diag {0.9, 0.9, 0.9, 0.9, 0.9}, the observer gains obtained via LMI toolbox of Matlab. The induction motor is driven by a PC-based DSP controller. The overall system consists of the PC, DSP controller card, power inverter, encoder, and an induction motor. For implementing the control law, the DSP card provides not only a high speed floating-point computation but also a 100kHz PWM signal generation to 2144

REFERENCES [1] J. Li, L. Xu and Z. Zhang, "An adaptive sliding-mode observer for induction motor sensorless speed control," IEEE Trans. Ind. Appl., vol. 41, no. 4, pp. 1039-1046, Jul./Aug. 2005. [2] A. M. Lee, L. C. Fu, C. Y Tsai, and Y C. Lin, "Nonlinear adaptive speed and torque control of induction motors with unknown rotor resistance," IEEE Trans. Ind. Electron., vol. 48, no. 2, pp. 391-401, Apr. 2001. [3] K. K. Shyu, L. J. Shang, H. Z. Chen, and K. W. Jwo, "Flux compensated direct torque control of induction motor drives for low speed operation," IEEE Trans. Power Electron., vol. 19, no. 6, pp. 1608-1613, Nov. 2004.

[4] K. Y Lian, C. Y Hung, C. S. Chiu, and P. Liu, "Induction motor control with friction compensation: an approach of virtual-desired-varuable synthesis," IEEE Trans. Power Electron., vol. 20, no. 5, pp. 1066-1074, Sept. 2005. [5] H. T. Lee, L. C. Fu, and H. S. Huang, "Sensorless speed tracking of induction motor with unknown torque based on maximum power transfer," IEEE Trans. Ind. Electron., vol. 49, no. 4, pp. 911-924, Aug. 2002. [6] M. Feemster, P. Aquino, D. M. Dawson, and A. Behal, "Sensorless rotor velocity tracking control for induction motors," IEEE Trans. Contr Syst. Technol., vol. 9, no. 4, pp. 645-653, Jul. 2001. [7] J. Holtz and J. Quan, "Sensorless vector control of induction motors at very low speed using a nonlinear inverter model and parameter identification," IEEE Trans. Ind. Appl., vol. 38, no. 4, pp. 1087-1095, Jul./Aug. 2002. [8] T. Takagi and M. Sugeno, "Fuzzy identification of system and its applications to modeling and control," IEEE Trans. Syst., Man, Cybern., vol. SMC-15, no. 1, pp. 116-132, 1985. [9] Q. Gan and C. J. Harris, "Fuzzy local linearization and local basis function expansion in nonlinear system modeling," IEEE Trans. Syst., Man, Cybern. B, vol. 29, no. 4, pp. 559-565, Aug. 1999. [10] K. Tanaka and H. 0. Wang, Fuzzy Control Systems Analysis and Design: A Linear Matrix Inequality Approach. New York: Wiley, 2000. [11] X. J. Ma, Z. 0. Sun, and Y Y He, "Analysis and design of fuzzy controller and fuzzy observer," IEEE Trans. Fuzzy syst., vol. 6, no. 1, pp. 41-51, Feb. 1998. [12] A. Jadbabaie, M. Jamshidi, and A. Titli, "Guaranteed-cost design of continuous-time Takagi-Sugeno fuzzy controller via linear matrix inequalities," in Proc. FUZZ-IEEE, May 1998, pp. 268-273. [13] K. Y Lian, C. S. Chiu, T. S. Chiang, and P. Liu, "Secure communications of chaotic systems with robust performance via fuzzy observer-based design," IEEE Trans. Fuzzy syst., vol. 9, no. 1, pp. 212-220, Feb. 2001. [14] B. S. Chen, C. S. Tseng, and H. J. Uang, "Mixed H21Ho, fuzzy output feedback control design for nonlinear dynamic systems: an LMI approach," IEEE Trans. Fuzzy syst., vol. 8, no. 3, pp. 249-265, Jun. 2000. [15] B. K. Bose, Power Electronics and AC Drives. Englewood Cliffs, NJ: Prentice Hall, 1986. [16] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. cn

50

I~~ .

U)

o

-cn50

0v

0

.

cn II

II

3 1

cn

II

3 (c) cn

3

4

5

6

1

2

3

4

5

6

3

4

5

6

--1-

4

I1

3

Il cn

.w_

a

II

Times

l

4 3 (d) Fig. 2 The simulation and experimental results: (a) desired (-) an d actual (--) speed, (b) desired (-) and estimated (--) speed, (c) actual (- -) and estimated (--) speed, (d) speed estimation error.

0

I

2

5 6 Time(s)

4

-1-

1

100 0 -100

1

II

5 6 1~~~~~~~~~~~~~~~~~~~~~~~~~

4

2

-1-M---

-I1-

- - - -

--

I

- -

CZ

0

cn U)

(a)

Time(s)

,

50

0

5 6 Time(s) Fig. 3 The experimental results of stator voltage and stator current for one phase.

,

-cn

II

0

0

1

50

,. 0

2

. .~~~~~~~~I 1

2

3 (b)

4

5 6 Time(s) 2145

1

2

3

4