A robust nonlinear controller for induction motor

A ROBUST NONLINEAR CONTROLLER FOR INDUCTION MOTOR...

14th World Congress of IFAC

G-2e-07-6

Copyright (() 1999 lfo'.qC

14th Triennial World Congress. Beijing P.R. China 7

A ROBUST NONLINEAR CONTROLLER FOR INDUCTION

MOTOR Guang Feng, Lipei Huang and Dongqi Zhu

Department of Elecfrical Engineering Tsinghua University Beijing. p~ R_ China, 100084 TeI: 86-10-62785520 Fax.-86-10-62783057 Email:guang.feng@mail. eic. tsinghua. edu. en

Abstract: In this paper. a ne\\.' method based on nonlinear auto-disturbance rejection

controller CADRe) has been developed for the induction motor drives. Via the use of ADRC, the stability is guaranteed for the speed servo control and insensitivity to uncertainties and disturbances are obtained as \veJI. The proposed control system has been successfully tested by means of sirnulations and experiments. The results sho\v that the controller perfonns quite robustly' under modeling uncertainty and measurement noise, and the dynamic performances are irnproved as \veH. Copyright © 199911r~4C' KeY'\'ord.s: disturbance, induction motors, stability.

1. INTRODUCTfON Motor control can be defined as a system integration technology- in the application of control theory, power electronics, and microcomputer control to achieve precision control of torque, velocity, and/or position of mechanical systems. At present, field oriented control (FOe) appears much more attractive due to the capability of torque/flux decoupling which gives high dynamic response and accurate motion profiles. Unfortunately, the high performance of such control strategy is often deteriorated due to significant plant uncertainties. The plant uncertainties, in general, include external disturbances, unpredictable parameter variations, and unmodeled plant nonlinear dynamics (Chang and Lee) 1996). Based on FOe, the

classic PID controllers are often used to regulate the static and dynamic perfonnance of control system. At some working area, the behavior of such controller could be satisfied but the overall optimal perfonnance couidnft be obtained by just one group of controller parameters. So the parameters of the PlO 1

insensitive~ non~inear control~

robustness,

controIJer should be modified according to the various operating condition of induction motor. This would add difficulty to the on-line debugging. To improve the perfonnance of field oriented control of induction motor, a new nonlinear auto-disturbance rejection controller (Han, 1998) is adopted. This system consists of three parts: tracking-differectiator (TD), extended state observer (ESO) and nonlinear state error feedback control law (NLSEF). By using ADRC, it is shown that the controller is robust against modeling uncertainty and measurement noise, so there is no need to estimate the rotor parameter precisely. Because the design of ADRC is independent of the controlled system model, this controller can manifest its adaptability and robustness not only in some operation area but also in the whole working area. To demonstrate the effectiveness of the proposed controllers, an application to the speed control of IM is presented. The hardware of the control system is based on the Dual-CPU flexible control system. Simulation and experimental results validate the controller's robustness and demonstrate

3301

Copyright 1999 IFAC

ISBN: 0 08 043248 4

A ROBUST NONLINEAR CONTROLLER FOR INDUCTION MOTOR...

14th World Congress of IFAC

3. CONTROL STRATEGY

that ADRC is effective in improving the stability and adaptability of system.

In order to control a class of uncertain system~ y (n) = J: y, y, .. , , y (n - I) , ( ) + '11 ( / ) + 11 ( t ) 2. THE INDUCTION MOTOR MODEL

jy, y,. -', y (n-l) ., t)

(Where Based on the reference-frame theory, the state space model of a squirrel case induction motor in a synchronous d-q reference frame can be described by fifth order nonlinear differential equation: :

= -

Idt

=-

i ql

k .

lldl

k

+

k

.

k [Iq! +

. 4Jdl

=

,

"4J q1 =

2 \V dl

2\V q 2,

Lm

Lm + -_- 'V q!OJ r + aL r Lm ~-"4JdzlDr ~ crL r I

-:r:-Jdl L T1ql - TI m

_

~'Vd2 -l.pq2U)r

•

\Vql

+ 'V,ll oo

•

Lq1W

j

.

]diWl

1 cr

++

li d1

)

~ Uq~

a

+

tl-'q20)!

-

\f'dl(J)r

\J.IqZi dl ) -

TL

(1) where the variables and parameters are defined as follows: U d I' U q I

id

tagc

iq I

cl - axis(q

- axis) stalor current

d 2 ' Ij/ q2

d - axis(q

- axis) rotor flux

1 ,

If/

- axis) stator vol

d - axis(q

rotating

(J)]

speed

rotor angular

cv

rotor inertia p

0:::::

speed

Tracking-Differentiator (Han and Wang, 1994) is such a dynamic system that it can arrange the transition process according to the input reference signal and under controlled system (3). Assuming function x(n)::=: g(x, x(n-l), t) is stable. If the

x,· ..,

parameter r is properly chosen, the solution X(I) of

function .xc",)

R

inductance

q!

== 0

: . . I tal ~ -k.l d1 + k 2 \VdZ + l q1 ffi l + er

=-

4J d l

Lm

T1dl •

i J r,···

l'x(rr-

If

I

r(N-I))

IT Ix ( I) )

v (1)

ld l

=

o.

vT

> 0

Then as R -+ 00, x] (t) --* v( t). x( t) ---4' the first genera]ized derivative of v(t).

Extended ~state observer as an elementary unit for ADRC teclmology is improved so that it can track the system state variable and its derivatives successfully (Han~ 1995a). It is well known that the differential signal often can not be used because of the existing high frequency noises. But ESO could satisfactorily solve this problem.

the dynamical system can be re-expressed as:

.

v(t)~

3.2 Extended State Obsener

Lr-

By using the rotor flux orientation, W

g(x -

-~

R sL2r + R rL2m crL;

kI

n

(4) I i n1

S

r

= r

R- ~ x

L2

L

state

could satisfy the condition:

stator and rotor resislance

pole pairs

y is the measurable

variable), the structure of nonlinear auto-disturbance rejection controller is proposed (Shown in Fig.])ADRC is composed of three parts: trackingdifferentjator, extended state observer and nonlinear state error feedback control law.

of the coordinate

stator, rotor and mutual J

is an unknown disturbance, u ( t) is

law and

the control

load torque

n

)1' ( ( )

(3) uncertain

3. J Tracking-Differentialar T

r

cOr = k3('VdZiqJ -

function,

represents

(2 -1)

U dl

1 T 'V d2

(2 - 2)

We assume

yet) and u(t) can be rueasured.

Suppose the proposed system configuration is chosen

r

lD r

===

; ]

q

1 :::::

k3\V .d2 i ql -l~L k . Lm -- 11 nl - - - 'V.l2 lV • .... crL r

(2 - J) .

- ldlWI

I

+ -a

U .. l

"'\

(2 - 4)

(2) Previous equations show that the mathematical model of the induction motor is affected by the following drawback (Cecati, et aI., 1996): The system is nonlinear due to the coupling parts between state variables. motor parameters vary with operation condition. the load torque must be known.

Fig. 1. The Block Diagram of ADRC

3302

Copyright 1999 IFAC

ISBN: 0 08 043248 4

A ROBUST NONLINEAR CONTROLLER FOR INDUCTION MOTOR...

14th World Congress of IFAC

4- ADRC FOR

as follows: e == Zl

Zl

==

Given the dynamic model of an induction n1otor des.cribed in section 11, the control problem is subjected to some prior assulnptions about the system in the folJowing.

~ g,(e)

Z2

gn_l(e)

Z n-I = Z n

in ::::

Zn+1

g" (e) + u(/)

-

Assumptions: (1) Rotor speed and stator currents are measurable. (2) Cl r ' "V d 2 , id I , i q I and their derivatives exist and

Z"+I = -gn+l(e)

(5) and

If the nonlinear functions gl (e),~o.· ,gn+l (e)

are continuous.

their related parameters are properly chosen, the state variables Z 1 (t ) i == 1,··· ,n will con verge to the filtered

state

variables

YI

i

( ( )

~

As it is shown in equation(2), the rotor flux is mainly controlled by equation(2-1),(2-2); and the speed is

I,··' ,n

affected mainly by equation (2-3),(2-4). From equation (2-2), the derivative of (2-2) is obtained as

successfully. In our case, we define a(t) =jy,y,_.o., y{n-I),t) + lV(t)

follows~

Then,

z..

a}

(t)

-?

-,

tj/d'~

a(t) ==fy,y,··· ~y("-I) ,t) + wet)

(t)

(r1+1)

~

], L... k L... ( k . , ) L =-yl!/d2+y 2'(J/dl+Y - ,1,J,+fql(()1 +rUd1 IH

r

(6) The observer's architecture(shown in equation (5) is not detennined by the actual expression of f (x, _i:, .... , X (n-l) , t), but oniy affected by the

variation range of y

INDUC1~ION MOl'OR

yet)

-

If Lm

-

so the robustness and

1;.

the (

-

r

coupling

part

is

k'I'd! +1. ja>1 ) q

r

r'

of

regarded

CT

(9) (9)---

equation as

mode ling

the

disturbance of system, it can be revvritten as: ..

adaptability of this observer are very good.

If'dI

= ~

Lm k

1.

V/ dl + ~-

T,

(Lm t) +

111/ dl --1 W 1

-r:;;.

lid I

( 10)

3,3. lVonlinear State Error Feedback Control Lavv'

where "\t'l

I~

A.. s sho\vn in Figure

Z12'

.•

o.

t

ZlH

generalized simplicity~

cO ;

1

J>(t),·-· ,y(n-l)(t).

derivatives

some

notations

are

defined

in

For

I

=Z

I J

Z

1J '

E2

::=:

Z 12 - Z 21 ' ••• , E: n = Z I n -

~

2

!!..L T 1.. J

(t)= -

w, ( 1)= - -L-rp

=

u dO

-lICI" -

-

iJI{i) I

+u

lV,(t)

L". I(T,.a)

III

(12)

sgn( c),

c/

/

02(1),.

T

Ifwe choose the control voltage reference as: (7)

fia I( E,a,uJ:')

U ql

(It)

U

bttfa/(c1 ,a ,o)+···+bt"fa/(£n,a )0)

Where

k:; \V d :: i Q 1 ---j \\' 1 ( t ) k . 1 = I1 q I + W ~ (t) + ~

LW!

Z 2tr

The control law u 0 (t) is selected as fa Hews: 1/ 0

=

the It-'

-

k I id I t i q I (j) l )

where

following: E

r

l ql

represent the output signal yet) and its

Z 211

-t;-(-

=

Continuing in this way, the coupling pan of equation (2-3) and (2-4) are regarded as modellng disturbance:

represent the

filtered reference signal and its generalized derivatives v(t), v(t),··' ,V·'1-II(t) respectively.~ Z In • ~.,

(t)

L

i51~u

,

(13)

The state feedback control law u(t) is detennined as : u(t) ::: uo(t) - zn+l(t) It is obvious that if

aCt)

(14)

(8) is precisely estimated by

The whole system can be re'Nfitten as: .. ¥/42

z "4-1 Ct) , we could realize modeling uncertainty and disturbance compensation (Han, 1995b). If the parameters and functions of the ADRC are properly chosen, the state variables have a desired behavior such as tracking, regulation and stability, and the control law u( t) is able to drive the state trajectory to the desired reference signa).

LiR k

1.

= -yl{/d2 + r r

cO,. == k3f.J.f

2¥/d2 +

r

TL

m UdO

r tJ

d2/lj'O

. i ql = -k,i q

I

}

+ - u qD a

Then~ the model of induction model could be decoupled into two linear subsystems: flux subsystem and speed subsystem ( shown in Fig.2 ) _

3303

Copyright 1999 IFAC

ISBN: 0 08 043248 4

A ROBUST NONLINEAR CONTROLLER FOR INDUCTION MOTOR...

u

-~

14th World Congress ofIFAC

Rs=2.92 Q Lr=O.285 H J = 0.1 kg

_AA_ ---

dl~~~--

Ca)

.

Rr= L 18 n Ls=O.285 H Lm=O.253 H Polepairs=2 III Rated Speed = 1430 r I min

All simulations arc done with a load torque disturbance applied to the induction motor. To see the intluence of the proposed controller, the perfonnances of ADRC and classic PID controller are compared. To verify the adaptability and robustness of ADRC; the same ADRC and PID controller are supplied to the high speed and Jow speed operation situation without changing their parameters.

flux subsystem

Fig. 4 shows respectively, the actual rotor angular speed aJ r' and its derivative. At t= I .15, load torque

TL

w;(tYI/sVl.1l)

(b)

of Induction Motor

It can be seen from Fig. 2 that the precise decoupljng of flux and speed control and exact linearization can be achieved if ESO can realize the state feedback and modeJ~ external disturbance compensatlon~ So~ it is convenience to use ADR.C to give the control signal an cl

I

Fig. 6 shows dynamic responses of ADRC \\o'ith step disturbance load (T,. = I 5 N' . '71 ). Compared to Fig. 6, Fig. 7 sho\vs dynamic responses of induction motor using classic PlO controllers. It is shown that the speed response of ADRC does not overshoot. Furthennore, the speed change of ADRC during load disturbance is less significant than that of PlO controller. So, it is obvious tbat ADRC can achieve higher dynamic perforn1ance (han PID controJler.

q0 , U q1 .

Fig.3 shows the proposed ADRL~ for induction motoL In order to achieve the rotor angular speed and the rotor flux regulation, the controller includes t"vo distinct control loops: the flux loop which employs one 2 nd ADRC to regulate the rotor flux; the speed loop, which employs two 1st order ADRC to control

the rotor speed

ro rand q-axis stator current i q I

In our case, the parameters of the ADRC and Pl D controller have been modified at rated speed (1430 rim in). Both controller function \vell at high speed range. Then, the same controllers are employed to regulate the IM at Iow speed (10 r/min) without changing their parameters. It can be seen from Fig. 8 to Fig. 9 that both the overall perfonnance and dynamic perfonnance of PlO controller deteriorate:l but ADRC can maintain its excellent perfonnance at low speed range. So there is no need for ADRC to adj ust its parameters according to the various operating conditions. The simulation results tested the affectivity of ADRC (shown in Fig. I0). These results verified that the adaptability and robustness of ADRC are all better than those of classic PlD controllers.

.

5. RESULTS In order to verify the perfonnance of the field oriented control based on the proposed ADRC, this system has been tested in several operating cond itions. A 2.2 kW induction motor with controller is simulated and tested by using the auto-disturbance rejection controller and the following motor parameters.

Fig. 3. The Control System of Induction Motor

15N . ID is added to the induction motor. Fig. 5

shows the rotor angular speed {j),., and its derivative that are estimated by ESQ. The simulation results demonstrate that the extended state observer can track the state variables of the induction motor and their derivatives successfully.

speed subsystem

Fig. 2. The Equivalent Dynamic Mathematical Model

U dI

=:

In order to verify the feasibility of ADRC, experiments are perfonned on the Dual-CPU flexible control system(shown in Fig.10). The core of the control system is a PC and a TMS320C30 DSP developing board. User interface, PWM pulse generation~ AID data acquisition and DSP control are all done by Cl Host PC. DSP deals with control algorithm, parameter estimation and state observation (Feng, et al.) 1997). Fig. 11 shows the starting curve of induction motor using ADRC. The speed and current signal is output directly from D/A. DSP

3304

Copyright 1999 IFAC

ISBN: 0 08 043248 4

A ROBUST NONLINEAR CONTROLLER FOR INDUCTION MOTOR...

14th World Congress of IFAC

calculates the rotor flux, and sends the signal to D/A port. Because of digitaJ interference, there exist some burrs in the curves. From Fig. 11, we could see the validity and the stability of the designed control system.

I

I

I

I

6. CONCLUSION

1(:0

This paper has presented a new nonlinear controller for induction motor. Extended state observers are used to implement the state estimation. The nonlinear state error feedback control law is adopted to realize state feedback and mode), external disturbance compensation. ADRC is robust against the modeling uncertainty and the external disturbance because it doesn't rely on the exact mathematical model of induction motor. In this paper, three auto-disturbance rejection controllers are used to regulate the speed, rotor flux and the stator current of the induction motor. Simulation and experimental results proved that ADRC has good robustness and adaptability. These results open new perspectives on utilization of nonlinear controllers on field orientation, and promote a new stream of studies on developing different structures of this class of controllers.

~."'~-----A_.

~ ~

... t

.

~ ...II_----",-._----L..--~

11 oil

l1li •

_. ....l... __

IIiI ..

......-----....~'--.-.---L---.--

~

ID

,

~

"T -

.2

1 ...

"I: •

(b) Fig. 5. (a) estimated rotor angular speed aJ r

~

and Cb)

it's derivative

I(S)

Fig. 6. Dynamic speed responses of ADRC due to load torque change (from 0 - 15 N . ID ) at rated speed

Fig. 7. Dynamic speed responses of classic PID controller due to load torque change (from 0 - 15 N . m ) at rated speed

(a)

I

i:

i-

~

~~ 5

i

-= "ii

i ~

~

(

Q

1

I Cs)

l'l~

(a)

(b)

Fig. 4. (a) actual rotor angular speed derivative

(j) r'

and (b) its -2 •

!

I

~

r-~-

h 1:1

~

I

r-

I I

a:=

J 1('11

1('1)

(a)

(b) Fig. 8. Global responses of speed regulation at low speed (10 r/min) (a) Using ADRC (b) Using PlO controller

3305

Copyright 1999 IFAC

ISBN: 0 08 043248 4

A ROBUST NONLINEAR CONTROLLER FOR INDUCTION MOTOR...

14th World Congress of IFAC

Ca)

Fig.ll. Dynamic responses of induction motor when starting

REFERENCES: (b) Fig. 9. Dynamic responses of speed regulation at low speed (10 rlmin) (a) Using ADRC (b) Using

Cecati, C. and G. Guidi (I 996). An adaptive nonlinear control algorithm for induction motor. In Proceedings of/he 1996 IEEE JECON, 1, pp.326331. Chang~ P~H_ and J. W. Lee (1996). A model reference observer for tirne-delay control and its applicatJon to robot trajectory control. IEEE Transactions on Control SystenlS Technology~ 4~ 2-10. Feng, G., H.l. Yu and L.P. Huang (J 997). Flexible control system of induction motor". In Proceedings ofIPEMC '97) 2, pp.96D-965~ Han, J.Q. and W. Wang (1994). NonlineartrackingDifferentiator. Trans. System Science and Mathematics (in CYhinese), 14~ ] 77-183. Han, J.Q. (1995a). The extended state observer of a class of uncertain systems. Trans. Control and Decision(in Chinese), 10,85-88. Han,J ~Q. (1995b). Nonlinear state error feedback

PlO controller

Operation Conunand Parameter Host Disp&set

TMS320C30 D&P Board

DJA Port

control Iaw-NLSEF. Trans. Control and Decision( in Chinese), 10,221-225. Han J .Q. (] 998)~ Auto-disturbances-rejection controller and it's applications. Trans. Control and Decision (in Chinese), 13, 19-23.

(Control Algurithm&Para. estimation)

T

Fig~

10. The structure of flexible control system

---.. X= O.2S/div

~..

. -.'..:'. ,,'" t _ ..

CHI: 6.0AJdiv

CH 1: stator current

CH2: 60.0rpm/div

CH2: speed

3306

Copyright 1999 IFAC

ISBN: 0 08 043248 4