A dual-mode adaptive robust controller using the takagisugeno model applied to a three-phase induction motor

Copyright © IFAC Nonlinear Control Systems, Stuttgart, Gennany, 2004

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A DUAL-MODE ADAPTIVE ROBUST CONTROLLER USING THE TAKAGISUGENO MODEL APPLIED TO A THREE-PHASE INDUCTION MOTOR Aldayr D. Araujo*, Francisco Mota**, Renato E. F. Sousa*** *Department ofElectrical Engineering, Federal University ofRio Grande do Nor/e, Natal, Rio Grande do Norte, Brazil, 59072-970, e-mail: [email protected] **Department of Computer Engineering, Federal University ofRio Grande do Norte, Natal, Rio Grande do Norte, Brazil, 59072-970, e-mail: [email protected] ***Department ofElectrical Engineering, Federal University ofRio Grande do Norte, Natal, Rio Grande do Norte, Brazil, 59072-970, e-mail: [email protected]

Abstract: In this paper is presented the application of a control strategy that interpolates the Model Reference Adaptive Control (MRAC) and the Variable Structure Model Reference Adaptive Control (YS-MRAC) applied to an induction motor. The combination of these strategies is obtained with concepts from Fuzzy Logic, where is used the idea of error magnitude for activating each controller in correct time. We used the Takagi-Sugeno Inference Model to obtain an analytic expression for the adaptation law. The application to the induction motor shows that this new strategy adaptation incorporates the advantages of both MRAC and YS-MRAC. Copyright © 2004 IFAC Keywords: Model Reference Adaptive Control, Variable Structure Systems, Fuzzy Logic, Takagi-Sugeno Model, Three-Phase Induction Motor.

INTRODUCTION

minimized while the good transient properties are preserved. In this work we propose that the transition between the MRAC and YS-MRAC can be made on line, using the YS-MRAC during the transient and converging to MRAC when the system approaches the steady state. The goal is to have a robust system with fast response and small oscillations (characteristics of theYS-MRAC) and a smooth and low-level steady state control signal (characteristics of the MRAC). We propose to use Takagi-Sugeno fuzzy logic model (Takagi and Sugeno, 1989) to obtain an analytic expression for the control law that is, in fact, an interpolation of the MRAC and YS-MRAC controllers. We applied this new controller to the speed control of a three-phase induction motor and we verified that it incorporates the advantages of both MRAC and YS-MRAC strategies.

In the Model Reference Adaptive Control (MRAC) with integral adaptation laws, the plant output follows a specified reference model. Even with the modifications to increase the robustness of the conventional algorithm factor, nonnalization, etc.) (Ioannou and Sun, 1996), in general, the transient is slow and oscillatory. An alternative to improve the transient consists in using a variable structure controller (YSC), based on a switching function of the plant state variables. This switching function forces a trajectory to stay on a sliding surface, making the system insensitive to parametric uncertainties and disturbances. However, there is the necessity of using all plant state variables measurements (Utkin, 1992). This limitation is very restrictive, since the full state vector of the plant is often not available and the practical implementation can be not possible. Hsu and Costa (1989) (see also Hsu et aI., 1994) proposed a variable structure model reference adaptive controller (YS-MRAC), using the control structure of MRAC and switching control laws as in YSC. In spite of the good transient response, in general we have a control signal with high level and the occurrence of chattering phenomenon. The dual-mode controller, suggested by Hsu and Costa (1994) and Hsu et al. (1999) is based on the work of Emelyanov (1987) and proposes a connection between YS-MRAC and conventional MRAC. The idea is to have an algorithm between the conventional MRAC and YSMRAC in which the chattering problem can be

«j

2

MRAC AND YS-MRAC CONTROLLERS

The basic structure of the Model Reference Adaptive Control is shown in the Figure 1, where the control law is given by

where 8 = [ 8 1

82

r

=

(I)

is the parameter vector to be

[y

r

r is called regressor vector. The basic question in this strategy is the adequate choice of 8 in a such way that the plant adjusted and

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{jJ

output (y) follows the model reference output (Ym) or, in other words, to make the error (eo = Y -Ym) to converge to zero. A well known tuning strategy is the Model Reference Adaptive Control (MRAC) with 0'modification, where the parameter tuning is done by the differential equation

B= -(jO - reom

r

(2)

The aB factor is called forgetting factor and ]€oOJ is the learning factor. We have: I

B(t) =e-atB(O)- Jre-a(/-r) (eom)dr

(3)

o

Figure 2: VS-MRAC Controller

.r

3.1 Takagi-Sugeno Model The Takagi-Sugeno inference model has been discussed in the fuzzy systems literature in last years (Tanaka et aI., 1996; Chen and Pham, 2001; Driankov et aI., 1996). We consider a linguistic variable x that has values in the fuzzy sets Fk (with membership functions J.1", k

= 1,2, ....M)

and the

set of M rules given by: Figure 1: MRAC Controller R(k):

The MRAC control with O'-modification is known to be robust to parameter uncertainties and external disturbances and, additionally, to have good steady state behavior (Ioannou and Sun, 1996). However its transient behavior is, in general, slow and oscillatory. Another control strategy (Figure 2), proposed by Hsu and Costa (1989), is obtained by the substitution of the integral laws of the MRAC by variable structure laws (switching laws), that is, the vector of parameters B is obtained in the following way:

O(t) =-8sgn(eom), 8=diag(BI,B2)

Bi,

where

i = 1,2 satisfies the inequality

=

1, ... ,M

(5)

According to the Takagi-Sugeno Model, given a numeric value for x the value for y is given by:

y=

L:lf'k (X)h (x) Lk=lf'k (X) M

(5)

The numeric value y obtained above can be thought as an interpolation of the values obtained using the functions

h (x ).

(4)

Bi > IOi *1

3.2 Interpolation ofthe MRAC and VS-MRAC Laws

and O· are the controller paranleters related to the

The basic idea to derive the control strategy is to use concepts from fuzzy logic in a way of combining the good behavior characteristics of the MRAC and VSMRAC controllers. In this sense we use the former (MRAC) when absolute value of the error is small and the last (YS-MRAC) when the absolute value of the error is big. The formalization of the "Small" and "Big" concepts will be made using fuzzy logic, where we consider the error as a linguistic variable whose fuzzy sets are S (Small) and B (Big). This idea was initially developed in Cunha et aI., 2002, where Mamdani inference rules were used. The basic diference in the approach presented here is the use of the Takagi-Sugeno Model, what allow us to obtain an analytic expression for the adaptive law. The plots of the membership functions of the value

I

matching condition (Ioannou and Sun, 1996). The variable structure controllers also present robustness to changes in the plant parameters and external disturbances and have a faster transient response than MRAC (Hsu and Costa, 1989). However, in general, this strategy presents the chattering phenomenom, what can be undesirable in some practical applications.

3

If x is Fk , then Y = fi(x), k

CONTROL STRATEGY

In this section we present the derivation of the adaptive law using concepts from fuzzy logic. In fact this derivation could also be obtained without these concepts. But the use of Takagi-Sugeno Model, as we will show, yields to a more intuitive mathematical derivation of the rules. In the following we present a simplified form of the Takagi-Sugeno in ference model (Takagi and Sugeno, 1989).

of the error in the fuzzy sets Sand B (J.1 s and J.1 B ' respectively) are shown in Figure 3, where, in this

830

Ps (eo) = e-eo

case,

we

P 8 (eo)

= 1- e-eo /0.1 = 1- Ps (eo) .

have

2 /

01 .

£ - angle between axis of stator phase 1 and rotor phase 1; is(t) - stator electrical current vector; lsd. i sq - stator electrical current vector components on direct and quadrature axis, respectively;

and

2

The parameter vector () of the controller will be adjusted according to the set of rules bellow:

R(l):

a(t) = dE(t) _ rotor angular mechanical speed. dt

If leol is S, then

Im

t

B(t) =e-atB(O)- Jre-lT(t-T) (eow)dr-

lS\! i

o

~'. ,I ' s

•.•••

(7) R(2):

aJ

If leol is B, then

!" ",

lSq

B(t) = -8sgn (eow), e = diag (81,8 2 )

/

(l)... 1f

"..

'-II

7

Re

•..• \'

ISd~\~

S--

\~---~s::~:r

1//////

1.0.-------=----~~-----=-----,

Figure 4: Yectorial Diagram to the Induction Motor 0.8

From Figure 4 we have 0.6

is (t ) = (iSd (t ) + iSq (t ). j ) eJP

0.4

\{J R(t)

= qJ Rd (t)eJ(P-E)

0.2

In order to obtain a simplified model of the induction motor it is used a Park Transform (Leonard, 1996). This transformation needs an estimation of the angle p, which can be calculated from 'l/rd, isq and 01

O+-----.--.~~....-_.___~~,L_~_____._--,.~.____~__j

-1.2

-0.8

0.4 o Membership Functions

-oA

0.8

Enor

Figure 3: Membership Functions for the Error Using the vectorial analysis with the rotor flux orientation (Cunha et aI., 2002; Leonhard, 1996), we obtain the following expression for the torque

According to the Takagi-Sugeno Model presented in . (6) ,and usmg Ps

+ P8 --

1 ,WIt .h

Ps (eo) -- e-eo

2 /

L

,

where L is a parameter to be adjusted, we obtain:

B(t) =fl s (eo)[e-mB( 0)- ye-m [em (eoW)dr] + (8)

where, rotor inductance by phase; magnetization inductance by phase; P number of poles pairs.

Lr Lm

+J.18 (eo)[ -esgn (eo(tJ)] It is important to note that the adaptive law presented in Equation (8) is, in fact, a weighting of the MRAC and YS-MRAC laws, where the weights are given by the functions f..L s and f..L 8 related to

Equation (9) describes the induction motor torque in a similar way to the DC machine. The component of the rotor flux vector on the direct axis is equivalent to the field flux in a DC machine and the component of stator electrical current vector on the quadrature axis is equivalent to the armature current in a DC machine. Additionally, if the component of the rotor flux is kept constant, the torque can be controlled only by the component of the stator electrical current vector on the quadrature axis. Unfortunatelly, Equation (9) above is only valid if the motor parameters are known without uncertainties. In practice the decoupling between the direct and quadrature components isn't valid due changes mainly in the rotonc time constant. Therefore, to guarantee the control system robustness, these uncertainties are taken into account in the

eo'

4

THREE-PHASE INDUCTION MOTOR MODEL In this section we use a vectorial technique for modeling the induction motor, which is very important to study field orientation control (Cunha et aI., 2002; Leonhard, 1996). We define a system of complex orthogonal axis, d and q, where the rotor flux is the reference for the d axis. The motor vectorial diagram is presented in Figure 4, where

g - stator electrical current vector angle related to the rotor flux; p - rotor flux angle related to stator phase 1 axis; (tJs - stator electrical current vector angular speed;

parametersBi' i = 1,2.

'fIrd(t) - rotor flux related to the d axis;

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5

SIMULATIONS

Now we will show the results obtained with the interpolation of the MRAC and YS-MRAC laws, i.e., using the adaptation law presented in Equation (8). In Figure 7 we have the plant output using

In the following we will present the simulation results for the speed control of an induction motor, whose parameters were obtained from Cunha et aI., 2002. The structure of the loop control is presented in the Figures 1 and 2, where the reference r is a constant signal (step) of 180 rad/s and the plant has initial condition (speed) of 90 rad/s. The plant and the model transfer functions are, respectively:

3798

Y U

S + 11, 3

Y R

,

12

_

-m- - - -

Ps (eo) = e-e.'/o.s,

r = 0.01

the

a

values

= 1000,

8 == diag (0.0002, 0.004).

and

We

obtained, then, a system whose response is similar to the YS-MRAC, but without the chattering effect in the control signal u. We note that since the proposed control law is a weighted combination of the MRAC and YS-MRAC laws, we can obtain the behavior of each controller making L to vary between zero (YSMRAC) and infinity (MRAC).

(10)

s + 12

and

where y represents the angular speed of the rotor (00) and u (control signal) represents the stator electrical current on the quadrature axis (isq)' The stator electrical current on the direct axis (isd) is kept constant while isq is generated by the control algorithm. o

The system output, using the MRAC controller with a == 1.667 and r == 0.001 - see Equation (2) - is presented in Figure 5. In this case we can note an oscillatory behavior in the transient.

0.05

01

015

0.2

0.25

0.3

035

0.05

0.1

0.15

0.2

0.25

0.3

0 35

-

Plant Model

1-

11 4

[l 45

n 'i

-----

005

I1

:~~ o

0 05

01

U 1:'

n ~,

[125

nJ

IJ 35

01

0.15

[l

:2

0

~5

03

n 35

0 45

05

·----- Plant

I

Model

1 o L----'-------.J'----'-------:-'L.----'-------::-L.-~~c=::::c:=?

o

005

0.1

015

02

0.25

0.3

0 J5

(i

4

0.45

0.5

045

EXPERIMENTAL RESULT

In this section we present the application of the strategy to the speed control of a three-phase induction motor. The driver system used to implement the dual-mode adaptive robust controller is showed in Figure 8. It is composed by an fourpole, 60 Hz, 1725 rpm, squire cage-type, 0.25 HP induction motor fed by a three-phase YSIIPWM inverter with clUTent control by hysteresis window. In the ClUTent control, Hall effect sensors are used to measure the currents of two phases of the motor. A microcomputer receives the motor speed using a tachometer and, by a control software in C language, sends the necessary signal to the inverter. A mechanical load simulates disturbances. In this application we used a sampling period of 0.000194s, the program runs in DOS mode and the controller 0.0001, a == 1000, L == 1.5 parameters are:

Figure 5: MRAC Controller

-

Control

0.4

Figure 7: Dual-Mode Controller

11

CO",,"

04

1-

05

1

6

: : : : : : 1-

045

0.5

-0.5

1

04

05

r=

and 8 == diag (0.0002, 0.004). Figure 6: YS-MRAC Controller In Figure 6 we present the system output using the YS-MRAC controller, whose parameters were obtained from the law presented in Equation (4) where (3 == diag (0.0002, 0.004). In this case we have a transient without oscillations and a control signal u with chattering (high frequency switching).

832

~

'..'." .." .-"""'J' InV~f1.~r Pcntiulll I_'<,I_~)I·.... ·~·m·=·····"·1

without oscillations and a smooth and low-level steady-state control signal. Additionally, it presented robustness to uncertain parameters and disturbances. Experimental tests confirmed the simulation results.

1 - :_ _

700

[=

~IHz l

t

if;

)1I -l L~3 '.: i !

;:;i;Ti,h

..

i_V-""-"J{:~7~.J················~r~~·rt[,--.·~-~.-·_·---,rl hchomder

~'.:::••••:••••""')"

REFERENCES

[:~

Indll~1jon

;\,j~chanical

~·Ia(.:hitlt:

Load

Chen, G. and Pham, T. T. (2001). Introduction to Fuzzy Sets, Fuzzy Logic and Fuzzy Control Systems, CRC Press LLC, Boca Raton. Cunha, C. D., Araujo, A.D., Barbalho, D.S. and Mota, F. (2002). A Dual -Mode Adaptive Robust Controller Applied to the Speed Control th of a Three-Phase Induction Motor. 7 International Workshop on Variable Structure Systems. Sarajevo, 2002. Driankov, D., Hellendoorn, H. and Reinfrank, M. (1996). An Introduction to Fuzzy Control, Springer-Verlag, Berlin Heidelberg. Emelyanov, S. V. (1987), Binary Automatic Control Systems, MlR Publishers, Moscow (english translation). Hsu, L. and Costa, R.R. (1994), B-MRAC Global Exponential Stability with a new Model Reference Adaptive Controller Based on Binary Control Theory, Control-Theory and Advanced Technology, Vol.] 0, N.4, Part 1, pp. 649-668. Hsu, L., Real, 1.A. and Costa, R.R., (1999). DualMode Adaptive Control using Gaussian Networks: Stability Analysis, Proceedings of the IFAC'99 World Congress Beijing, July 1999. Hsu, L., Araujo, A. D. and Costa, R. R. (1994). Analysis and Design of I/O Based Variable Structure Adaptive Control, IEEE Trans. Automatic Control 39(1): 4-21. Hsu, L. and Costa, R. R. (1989). Variable Structure Model Reference Adaptive Control Using only Input and Output Measurements, Inl. 1. Control 49(2): 399-416. loannou, P. A. and Sun, 1. (1996). Robust Adaptive Control, Prentice Hall, Inc., Englewood Cliffs, NJ. Leonhard, W. (1996). Control of Electrical Drivers, Springer-Verlag, second edition. Takagi, T. and Sugeno, M. (1989). Fuzzy Identification of Systems and its Applications to Modeling and Control, IEEE Trans. Syst., Man, Cybern. 15( 1): 116-132. Tanaka, K., Ikeda, T. and Wang, H. O. (1996). Robust Stabilization of a Class of Uncertain Non-Linear Systems via Fuzzy Control: Quadratic Stabilizability, Hoo Control Theory, and Linear Matrix Inequalities, IEEE Trans. Fuzzy Systems 4(1): 1-13. Utkin, V.I. (1992). Sliding Modes in Control and Optimization, Springer-Verlag.

Figure 8: Driver System The experimental result is showed in the Figure 9, where the plant and reference model speeds are given in rpn1 and the control signal u =

isq is given in mA.

In the practical test the motor and the reference model outputs start with zero initial speed. The reference is initially assumed as 900 rpm, after a certain time it is increased to 1200 rpm, and finally it is returned to 1000 rpm. It is important to note that changes in the reference imply parametric variations due to rotoric time constant changes. After that, it is introduced a 30% nominal load disturbance by some seconds. The plant follows the reference model with the speed converging to the specified reference in each case (the model gain is one). The disturbance effect is felt only by increasing the control signal during the time interval in which the load disturbance is applied (Figure 9). The transient behavior is fast and with no oscillations as in the simulation results. The total time of the experiment was 26s.

..__~__.

_

_.__.__.

..

]]·o:oo:.~~

Figure 9: Experimental Result

7

CONCLUSIONS

A dual-mode controller to the induction motor speed control was proposed. The control strategy can be seen as an interpolation of the MRAC and VSMRAC laws using concepts from fuzzy logic. The Takagi-Sugeno model allowed us to obtain an analytic expression form to the adaptation law. According to simulations, it can be verified that the dual-mode algorithm provided a fast transient

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