Impact of susceptibility profiles of Gram-negative bacteria before and after the introduction of ertapenem at a medical center in northern Taiwan from 2004 to 2010

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TA8-

11540

Amarlscln Control Conkruncm Roc88dlnps 0 1 t h Balmom. Mayland June 1894

Nonlinear Observer-Based Tracking Control for Induction Motors

Jung

Hua Yang',

Wen Hab

Yu',

and

Li

Chen

Fu'v2

1. Department of Electrical Engineeging

2. Department of Computer Science

&

Information Engineering

National Taiwan University, Taipei, Taiwan,

R.O.C.

Abstract

In this paper, we propose a nonlinear observer-based controller for induction motors. Via the use of skew-symmetrical property of induction motors, a two- stage design and technique is applied to construct an observer-based controller for velocity and position tracking control. To demonstrate the effectiveness of the proposed scheme, the scheme is applied to the tracking control of robot manipulators. Simulation results have verified the effective performance of the controller. To implement the proposed scheme for experiment, a voltagecontrol type of drive system has been set up, which employs two personal computers, P C 486 and P C 286. One is adopted to perform the calculation of the control law and the other is to perform the function of pulse width modulation (PWM) and the generation of gating pulses. Satisfactory experimental studies are also shown in this paper.

Nomenclature

:

v a s , V I I , wcs(ua,, v b r , v c r ) : stator (rotor) phase voltages

I,,, i b " , ic,(iap, i b r , i c 7 ) : stator (rotor) phase currents

V d s ( V q . ) : d-axis (q-axis) stator voltage

i d . ( & ) : d-axis (q-axis) stator current

( P d r ( Q q r ) : d-axis (q-axis) rotor flux

9. : position of rotor flux vector

8 , : rotor angle

Bar : slip angle

w s : stator angular frequency ur : rotor angular speed

W ~ I : slip angular speed

Lrs(LiT) : leakage inductances of stator (rotor) windings

L m s ( L m , ) : magnetizing inductances of stator (rotor) windings

L., : mutual inductances between stator and rotor windings

R . ( R , ) : stator (rotor) resistance

t , ( L . , ) : stator (rotor) self-inductance

M : statorfrotor mutual inductance p : number of pole pairs J : rotor inertia D : damping coefficient

KT

: torque constant ( = 3 p M / 2 L , ) TL : disturbance torque a : leakage coefficient (= 1

- M ~ / L . L , )

c :1/uLs a1 :c(R.

+

M Z R , / L : ) a? : c ~ ~ r / ~ : a3 : p c M / L , a, : R , / L , as : M R , / L ,

1 Introduction

The induction motor has a wide variety of applications as the electromechanical actuator because of its ruggedness, low maintenance, and low cost. Advances in power electronics and microprocessor technology make it feasible to use the induction motors in place of dc and synchronous motors in a wide range of servo applications. However it is very difficult to achieve high performance with an induction motor, which is caused by the nonlinear coupling terms of the dynamics comparing the electrical part and the mechanical torque part.

In a separate-exdtation dc motor, the electromechanical arrangement provides decoupling between magnetic effects in the stator and in the rotor windings. This decoupling is reflected in the torque dynamical equation. The dc motor can be linearly operated simply by keeping the stator current constant.

However, for an induction motor, the stator windings produce an associated flux, and indirectly create a rotor flux as well. Angular separation between the fluxes results from the time delay inherent in the rotor circuit. As the stator current is the only mechanism for torque generation, the stator and rotor effects are tightly coupled. Thus, the motor machine cannot be operated as a linear device unless some kind of nonlinear feedback is applied.

To overcome the problems mentioned above in practical implementations, the field orientation control [12] is a widely used approach. This method uses a nonlinear feedback to obtain the approximate decoupling between the stator and rotor, and cause the dynamics of an induction motor to behave like those of a separate- excitation dc motor. When the full-state feedback is used, it is called direct field orientation control [8, 91. But. it requires accurate flux sensors which are expensive and suffer drawbacks such as temperature sensitivity and its induced changes to the motor structure. Another approach is to use observers for flux in place of the previous sensors. This method is called indirect field orientation control. Its stability, robustness, and related control properities are, however, not fully studied yet, although the field orientation control performs well in practice.

In the past few years, the advances in nonlinear control theroy have had a

notable impacts on the field of induction motor control. Input-output decoupling control was presented in [4] and [3] using geometric techniques. In [4]. a simplified model is used, i.e., only the electromagnetic part is considered and the speed is assumed to be a slowly varying parameter. Exact decoupling in the control of electric torque and flux amplitude using the amplitude and frequency of the voltage supply as inputs are achieved by a static state feedback. On the other hand, the results in [21, 221 are based on the adaptive control theories of linear systems so that linearized models of induction motors are used for controller design and stability analysis. Along the same line, the work in [18] proposed an adaptive tracking control scheme based on the nonlinear model. Moreover, Marino [5, 61 proposed a direct adaptive controller for speed regulation in which the motor model is input-output decoupled by a feedback-linearizing controller but with the load torque and rotor resistance being adapted in time. The results of that is the ensurance of asymptotic convergence of the system states and the estimated parameters.

In aforementioned schemes [4, 5, 6, 181, the full states must be available. So practical implementaions will require the flux obeservers, and hence a good deal of research effort has been directed towards the design of nonlinear controllers that employ flux observers. In [5, 61, the controller was combined

with the flux observer [lo]. However, no proof of stability wbs given for the resulting closed-loop system.

In [ 3 ] , the observer-based adaptive controller was designed and the closed-loop stability was proved under several restrictive assumptions. In [15], closed-loop stability is also proved by using design technique for nonlinear feedback control [16]. But the singularity of the control laws remains as in [3, 4, 5 , 6, 18, 151. The globally stable controller for regulating torque and flux amplitudes was presented in [17]. Using the skew-symmetric properties of the model of induction motors, a controller without singularity is designed by using a fourth-order observer, but only torque tracking and flux regulation are achieved.

In this paper, we propose a nonlinear observer-based controller for induction motors. Via the use of skew-symmetric properties of the model [li'], the tww stage design technique [la] is applied to construct an observer-based controller for velocity and position tracking control. To demonstrate the effectiveness of the proposed scheme, an application to the tracking control of robot manipulators is presented. Simulation results show that the tracking is satisfactory and both the input voltage signals as well as the current signals remain within the acceptable limits.

In order to implement the proposed scheme for experiment, a voltage-control type of drive system [13,14] has been set up, which employs two personal computers, namely, P C 486 and P C 286. One is adopted to perform the calculation of the control laws and the other is to perform the function of pulse width modulation (PWM) and generation of gating pulses.

The layout of this paper is as follows. The nonlinear model of the induction motors is shown in section 2. Such a model is then rearranged for the controller design, in which torque load is taken into more realistic considerations. In s e c t io n

3, the controller is proposed and the main results of this piper are derived as well. The simulation results are shown in section 4, and experimental setup as well as the results are presented in s e c tio n 5 . Finally, the conclusions are provided in section 6.

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2 Model

of

Induction Motors

With the notations in [l] [3], the dynamics of a balanced 3 - p h w Y-connected that has symmetric, and linear magnetic circuits can be described U a fifth-order stak-space dynamical system. Let the motor have p poles, then, after a coordinate transform onto the d-q coordinate frame rotating synchronously with an angular speed w,, the dynamical system can be represented as follows:

;d.

=

-mid.

+

w.iq.

+

azOdr

+

a3w,(~,,

+

Cvd. (1)

&,

=

-w.id.

-

ali,.

-

a w , ~ d ,

+

azo,,

+

CV,. (2)

bdr

=

-.addr

+

asid.

+

(w.

-

p w , ) ~ , , (3)

b,,

=

-or#,.

+

05i,.

-

(w.

-

pw+)@dt (4)

&

=

( - D w + + T e - T ~ ) / J (5)

Te

=

kT(@d& -@pride) ( 6 )

where Vd.,

V,,,

W. are control inputs and

T.

is the generated torque. The definitions of the notations and symbols are given in the N o m e n c l a t u r e

.

I t can be seen shortly that the dynamical model described by equations (1)-

( 6 ) possesses a skew-symmetric property, which will be used in the design of the controller. In order to manifest this property, we first rearrange the dynamical equations by using more compact notations as follows.

Denoting zT

=

[ZI zz z3 2 4 1 = [id. i,. @dr

@,.I,

XT

= [zT 251

,

UI

=

Vd.,

uz = v,,, u3

=

w., we can rewrite equations (1)-(6) as

21 = -a121

+

u32a

+

0123

+

a325z4

+

cu1 (7) i2 = -us21

-

a122

-

a32523

+

as24

+

cuz (8)

6 =

-a423

+

a5z1+ (u3 -P25)24 (9)

2.4 = - 0 4 2 4 + a m

-

(u3

-

P25)23 (10) z'5

=

( - D z 5 + T c - T ~ ) / J (11)

Te = k T ( 2 3 2 Z - 2 4 2 1 ) (12)

Note that the equations (7)-(10) can be further rewritten in a more compact symbolic form as

where

with

z

=

Az

+

Bu,

A

=

Ao + A I

+

A?, uT

=

[UI uz],

(13)

AO

g

diagr-al, -01, -a,, -a,], 0 u3 -a5 0 (U3 -PZ5)

'

-: 5

1

AI f -u3 0 a5 0 0 a5 4 1 1 3 -PZS) 0 0 0 (az + a s ) parza

1

0 and

It should be noted that AI is a skew-symmetric matrix, i.e., AI=

=

-Al. In the following, we assume that the torque load is a known function of the rotor speed. This assumption is more realistic than the one with constant load [15, 17, 181. It is well-known that the bearings and lots of viscous forces vary linearly with the sp+, while the large-scale fluid systems such as pumps and fans have loads that typically vary as the square of the speed [l9]. Hence, the torque load is assumed to be in the form of

TL =

W

+

C125

+

PzZ:, ₍₁₄₎ where

m,

p1, pz are some known constants.

3 Control

of

Induction Motors

Given the dynamic model of an induction motor described in the previonrscctiou, the control problem is stated subjected to some prior assumptions about the system in the following.

Assumptions :

(Al) The motor parameters are known in advance.

(A2) Rotor speed 2 5 and stator currents 21, zz are measurable.

(A3) The desired rotor speed Wd

E

C', i.e.,

w t ) ,

the i-th derivative of wd, i

=

1,2,

exist and are continuous.

Control O b j e c t i v e :

Given a d a i r e d rotor speed tracjectory wd(t), determine the control inputs

Y I , Y Z , U ~ such that the rotor speed w, can track the desired tracjectory exponen- t i d y in time t, i.e.,

2 5

-

Z5dv ast-oo.

assuming that the *ala zl, zz, and 25 are available.

hthermore, the rotor fluxes are also regulated according to the desired com- mands, i.e.,

ad?

-+

8s

a s t - 0 0

0,

-

0, ast-00.

In the following, the proposed controller is motivated by the two-stage analysis

fint praKnted in

[le].

The model of an induction motor can be viewed a cascade of two subsystem shown in Fig.1, or in other words a two-stage system.

3.1 Control

Strategy

3.1.1 The

First

Stage

Referring to equation ( l l ) , i.e.,

Jt5

+

Dz5

+

TL

=

T.,

the output signal in this stage is w, and the generated torque

T,

is viewed as the input. If we want the output signal 25 to track Z S ~ , a sufficient condition is to have the input signal

T,

be equal to the desired torque Td as follows :

Td

=

JS5d

+

kl25d

+

Bo

+

(PI

+

D

-

kl)25

+

PzZ:,

where

TL

is replaced by equation (14). Let e5

fi

25

-

ZSd, we then have

Ji5

+

k1e5

=

T.

-

Td.

Since

T. is

not the actual exogenous input, the right hand side of (16) can not be made identically aero.

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3.1.2

The Second

Stage

In thi stage, we just take the equations (7)-(10) into consideration. Here, the generated torque

T. is

viewed as the output, and the input signals are ut, uz, and

113. It is e u y to show that E,

=

24d

=

0, 23

=

23d

=

8,

zz

=

zzd

=

&,

the generated torque will coincide with Td from (12).

M) that iT

=

z T - i T

=

[Ea, 3 4 1 = [23-& 2 4 -4.1, where 43, 54 are the estimates

kt e

=

2

-

Ed, where ZT

=

[Zld, Zzd, 23dl %'Id], ZT [23 241, iT

=

[?

441,

so that

i

=

(Ao + A I

-

K ) e

+

R ( f , I )

+

where

(a2 + a s ) &

+

asz&

-

( t l d

-

+d)

R(r, i )

=

-aazsZs

+

(az +a5134 0

-

(tzd

-

tzd)

[

0

K

=

diog[kz, k3, 0, 01

with

kz, ks

W i g feedback control gains, and vi, i

=

1,2,3,4, are extra control

tem nsed for cancelling the coupling terms in the subsequent

dosed

loop stability

UIrlYSiLl.

1407

(3)

3.1.3 Observer Design

In order to estimate the flux signals, we design a nonlinear observer UI follows :

$3

=

-4i.3 +a521

+

(.a

-

pz5)i.r

+

€11

+

€21 (25)

P 4

=

-a424

+

0522

-

(ua

-

pt5)i.s

+

€12

+

€22 (26) where €,,

,

1

I

i,

i

I

2, are some suitable functions to be specified in the following. Then, the estimation errors satisfy

i =

- a 4 I a i - ( u 3 - p z l ) J a i + E , (27)

where

Ia

2

diog[l, 11,

Here, €21, €22 are chosen for the purpose of canceling the coupling term in equ= tion (23), namely, R ( t , i), whereas €11, €12 are designed for canceling the coupling terms in the overall closed-loop stability analysis.

For simplicity, we first define some notations in the following:

Theorem 1 Consider an induction motor whose d w m i c s are gowmed by q u a -

tions (1)-(6) under the assumption8 (AI)-(AS). Then, the conhvl objective can be achieved provided the observer-based controller is designul according to (18)-(01)

with

where 53, i.4 are obtained from the following nonlinear observer

P3 = 4 4 i . 3

+

a521

+

(u3

-

pz5)i4

+

foe1

+

fie2 (31) i 4

=

-a454

+

a5za

-

(u3

-

p251i.3

+

fzei

+

f 3 e l (32)

Proof: Choose the Lyspounov function as

From equations (16)(22)(27), the derivative of

V

is obtained UI follows :

V

=

eT(Ao

+

K ) e

+

i T ( - a 4 Z a ) i

-

Lie:

+

e'R(t, i)

+

C I V I

+

e m

+ w 3

+

e 0 4

-

(61

+ € a i ) &

-

(€12

+

6 z ) h

+

es(Te

-

Td) First of all,

e5(T.

-

T d )

=

kreszaea

-

k~es24Gl

+

kre5(erzzd

-

G4Zld).

Next, let

which then yields that

€11

=

k ~ e s e a , Cia = - ~ T G J G I ,

V

=

-uollellz

-

a411illz

-

h e : +e3v3

+

e4v4

+

k ~ e ~ ( e a 2 a d

-

e421d)

(34)

+eTR(t, 3)

-

<21%3

-

<azi.r,

where

Furthermore, by substituting the definitions of v3 and v, into (34), we obtain

uo

=

manta1 + L a , aa

+

k31 a., a4)

V

I

-aollella

-

arllilla

-

tie:

+

e T R ( t , i ) - ( a i &

-

EZ2f4 (35)

It should be noted that rid which needs to be used for implementing 2 ) d

and

l t l d

indudes 2s.

This

is the reason why we design ; I d and

Lid

instead of z z d and zld.

Accordingly, more detailed expressions for i l d and ;ad are derived in the following.

and then substitute equations (11)-(12) into the above to yield

Note that ;id and &d are equal to & i d and & a d , respectively, except that 23, 2,

are replaced by i.3, i.4, then, we can easily find that

To cancel the term involving R(1, i) in equation (35), &I and &a are, hence, designed as

€a2

=

e1(a5z5

-

)

+

ez(aa

+

a5

+

-

as

P

Finally, we have

V

5

--Oolle(('

-

arllill'

-

k l e z

which thus concludes our proof by following a standard Lyapunov analysis (231.

0

3.2 Application to a Single-Link Manipulator

Previously, we have proposed a nonlinear observer-based tracking control scheme for induction motors with a known desired function of the rotor speed. However, the proposed scheme can be widely applied to the high performance control of systems using induction motors as actuctors. As an illustrative example, we will apply this scheme to the tracking control of robot manipulators actuated by induction motors. For simplicity, here we will only consider a Bingltlink manipulator

UI shown in Fig 2 which is driven by an induction motor. scribed as:

where

TL

= 18

+

Be

+

mgI sin 0,

so that

( J

+

1)6i

+

( B

+

D)8

+

mglsin 8 =

T,.

Note that l , m , l stand for the moment of inertia, the mass, and the length of the link, respectively, B , g represent the viscous frictional coefficient and gravity

constant, respectively, and 0 is the angle of rotation of the link. Similarly, we design the control input of the first stage as

Th e modeling of the mechanical part of the manipulator system can be de-

~ 6 i +

08 =

T.

-

T~~ (41)

T d

=

(J+I)6id+(B+D)8d+mgf sin 8 - ke(8-0d)

-

( k ,

-

( B + D ) ) ( 6 - 8 d ) , (42) Denoting eo

=

8

-

a d , e,

=

where ks, k, are some appropriate constants.

e

- i d , and e: = [eo e,], we then have

(43) i,

=

A,e,

+

B,(T.

-

T d ) , where 0 1 Am =

[-*

-a1

Bm

=

[ A ] .

1408

(4)

In the second stage, the design methodolow b the m e

h n

daeribed

earlier. Thus, for any Q

>

0, there exists P

>

0 such that PTA,

+

A,P P

-0.

so,

.fter some algebraic manipulation, we also have

V

5

-ao\\el\’

-

or\lillZ

-

eLQe,. In the same way, we define some notations in the following :

A

m =

tll

=

m =

I = TA

e

TB

e

A A A fB

e

TC

e

TD

4 PD

d

A A A go

=

81

=

92 = ;Id f t A ZZd

=

fo fl

e

Theorem 2 Consider a single link manipulator actuated bg a n induction motor whose dynamics are governed btr (1) and (6). Let the desind poaitionol tajectorg

be

ed(t), whoae @$’(t), i

=

1 , 2 , 3 are continuous. Based on the orrumption8 ( A l )

and (A2), the objective of position and velocity tmcking

,

i.e.,

~ ( t )

-.

ed(t), I ( t ) -+ id(:) ezponentioUg (U t

-

oo are achieved provided that the following nonlinear observer-bored controller U designed according to (18)-(21) and (42) with

v i = k~e:PB,ic

vz

=

-k~e:PB,*s

U3

=

-kTe:PBmZad

U4

=

kTefPBmzid where i 3 , i 4 are estimated bg the following nonlineor obrcntcr

P3

=

-aria

+

azzi

+

(us

-

PZ5)h +fa

i 4 = - 4 4 2 4 + a m

-

(us

-

pZ5)is

+

fi

with

PA:

+

AmP =

-9.

Proof: Choose a Lyaponuov function as

From equations (18)(21)(42), the derivative of V is obtained aa follows :

v

=

_eT(Ao

₊

_{K ) e}

₊

_{i T ( - a r I z ) i}

_-

_e:Qem

₊

_e:PB,(T.

-

_Td)

₊

_{R ( t , i )}

+CIVI + e a v i

+

e m

+

ervr

-

(€11

+

6 i ) &

-

(€12

+

bZ)o4 Following the results of Theorem 1, €11, & z , Ez1,md €m .re designed M

61 =

kTe:PBmez

€12

=

-kTe:PB,el

which then completes our proof. 0

4 Simulation Results

In this section, the preceding observer-based control scheme is applied to simulation studies of the velocity tracking of the induction motor and the tracking control of a single-link manipulator actuated by an induction motor. The numerical values of the six-pole squirrel-cage induction motor are

R. =

3.7450,

R,

=

3.58313, L .

=

163.3mH, L, = 163.3mH,

M

= 154.67mH, J = O.OSkgm’, and

D E

0.O65kgmzs-’.

First the control scheme is tested for load

Ts

=

po

+

pizs

+

pzz:, where

=

3.3, pl

=

0.25, pz = 0.004, and the response for sine wave tracking are shown in Fig.J(a)-(b).

Secondly, a single-link manipulator dynamics are used to demonstrate the proposed control law. Here, the numerical values of the manipulator are I =

0.0284kgm2, B

=

O.OO1kgmzs-l, in

=

0.3kg, g

=

9.8m/sZ, and I

=

0.5m. The desired trajectory is chosen as Bd = sin3t. Fig.l(a)-(b) show the performance for tracking control of this manipulator.

5 Experimental Results

The proposed controller is tested in a system with a squirrel-cage motor rated one house power (HP) with an optical encoder attached to its shaft, a PWM tr.nsister inverter, and two personal computers(PC486 with 16 bits A/D D/A card and PC286) communicating through 8255 cards. The configuration of the system is shown in Fig.5.

5.1 Rotor position and current detection

T h e rotor position is detected by a two pulse (with complementary outputs) o p tical encoder with two phases set apart by 90’. The resolution is 1024 pulses per revolution. The pulses go through a digital circuit, which accumulates the pulses with 12 bit counter. The corresponding speed is calculated by PC486.

Because of the balanced three phases, i.e., i.,

+

is,

+

ic. = 0 , two currents are adequate for the coordinate transformation. Currents are measured with Hall effect transducers (LB-IOGA) and low-pass filters.

5.2 Control Algorithm Processing

T h e control algorithm is performed by PC486. In each period of 2 ms, the pro- gram computes the following tasks: rotor position and current sampling, speed calculation, coordinate transformation, control algorithm execution, data fetching

to the common memory for PC286, and informing PC286.

5.3 Results

The squirrel-cage motor used in the experiment is manufactured by ELMA MO- TO& CO. with delta-connected stator. The parameters of the motor are shown below: Rated power

=

0.75 kW Rated current

=

4 A Rated voltage

=

220 V Rated frequency

=

60 Hz

Rated

speed

=

1120 rpm pol- = 6 R.

=

3.745

n

R

=

3.583

n

L,

=

163.3 mH

L. =

163.3 mH

L,

=

154.67 mH

Fig.6 shows the velocity response for 200 rpm

€21

=

eI(oz + a s )

+

s ( g o P B m T d

-

e:PBmw)

-

e a a d

+ kw

-

( B

+

D )

6 Conclusions

kTfl ’IZez

€22

=

4 5 e 3

+

z ( g 1 P B m T d

+

e r P B m m )

+

e& +os) From the skew-symmetric property of induction motors, the observer-based con-

troller is proposed. Without the measurement of fluxes, velocity and position tracking can

be

achieved under the assumptions that the system parameters are

Oh

B

k“

-

( B

+

D I m e z

(5)

The simulation and experiment results have shown the effectiveness of the proposed controller in combination with the flux observer. The future theoretical work is to enhance the robustness of the controller and/or to add parameter estimation with an adaptive observer.

References

[I] Paul C. Krause, Analysis of Electric Machinery, McGraw-Hill. 1986.

[2]

B.

K. Bow, Power Electronica and A C Drives, Prentice-Hall, 1986.

[3] Dong-I1 Kim, In-Jooug Ha and Myoung-Sam KO, “Control of Induction Mc- tors via Feedback Linearization with Input-Output Decoupling,” h t . J. Con- tro/, Vol. 51, No. 4, pp. 863-883, 1990.

(41 Alessandro

De

Luca and Giovanni Ulivi, “Design of an Exact Nonlinear

Con-

troller for Induction Motors,” IEEE Trans. on Automatic Control, Vol. 34, No. 12, December 1989.

[5] R. Marino, S. Pereada, P. Valii, ‘Adaptive Partial Feedbadc Linearization of Induction Motors,” Proc. 29th CDC, pp. 3313-3318, 1990.

161 R. Marino, S. Pereada, P. Valigi, *Adaptive Input-Output Linearizing Con- trol of Induction Motors,” IEEE Trans. Automat. Contr., Vol. 38, No. 2, pp.

208-221, Feb. 1993.

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