Adaptive Fuzzy Control of Doubly-Fed Induction Machine

CEAI, Vol.16, No.2 pp. 98-110, 2014

Printed in Romania

Adaptive Fuzzy Control of Doubly-Fed Induction Machine Naâmane Bounar*, Abdesselem Boulkroune*, Farès Boudjema** *

**

Department of Automatic Control, University of Jijel, BP98 Ouled Aissa, Jijel, 18000, Algeria ( e-mail: [email protected], [email protected]) Department of Automatic Control, LCP, Ecole Nationale Polytechnique ENP, 10, Av. Hassen Badi, BP182, Algiers, Algeria (e-mail: [email protected])

Abstract: This paper presents a new adaptive fuzzy backstepping control (AFBC) scheme for doubly-fed induction machine (DFIM). This proposed controller guarantees speed tracking and reactive power regulation at stator side. The DFIM stator windings are directly connected to the line grid, while the rotor ones are controlled by means of an inverter. Using a state-all-flux model, the stator voltage vector oriented reference frame is adopted. The control design principle is particularly based on the decomposition of the motor model in two coupled subsystems, namely: the stator flux subsystem and the speed-rotor flux subsystem. The stator flux subsystem is stabilized independently of the speed behaviour. The DFIM unity power factor control and speed tracking problem is transferred to the rotor flux control problem. The unknown load torque is estimated on-line by a suitable adaptive law, the nonlinear functions appearing in the tracking errors dynamics and uncertainties are reasonably approximated by adaptive fuzzy systems. The proposed control scheme guarantees the tracking error exponential convergence to a small residual set. The performances of the proposed control system are evaluated in comparison with a non-adaptive backstepping control (NABC) scheme by simulation tests. Keywords: Doubly-fed induction motor, Backstepping approach, Fuzzy systems, Adaptive control, Load torque estimator, Lyapunov stability. INTRODUCTION In the last decade, significantly less attention has been paid to the control development of so called Double Fed Induction Machine (DFIM). The DFIM is a wound rotor asynchronous machine which can be controlled from the stator or rotor by various possible combinations. Connecting the stator windings of a DFIM directly to the line grid and the rotor windings to a controlled converter constitutes a typical connection scheme of this machine. This solution is very attractive where small speed variations around the synchronous velocity have to be imposed; the rotor power handled by the converter is a small fraction of the overall machine power. This allows the minimizing of converter size and therefore a decreased price of the whole system (Morel et al., 1998). However, DFIM usually presents the features of nonlinear and multivariable system with inevitable modeling uncertainties and hence can be considered as a challenging engineering problem. Different strategies were proposed in the literature to solve the DFIM control problem. One of the most significant developments in this area was the fieldoriented control that offers the decoupled control of the active and reactive powers (Hopfensperger et al., 1999; Peresada et al., 2003; Drid et al., 2005). Unfortunately, this control approach suffers from sensitivity to the machine parameter variations and inadequate rejection of external disturbances

and load changes (Wai, 2007). To partly overcome these insufficiencies of the field oriented control approach for DFIM, some advanced vector control techniques based on the nonlinear feedback linearization and sliding mode control principles have been addressed (Drid et al., 2005; Bekakra and Benattous, 2010; Ardjoun et al., 2011). The resulting controllers are not only robust to model uncertainty and to parameter variations; but also they having good disturbance rejection properties. Unfortunately, such performance is obtained at price of extremely high control activity. As consequence, the chattering phenomenon always occurs in the sliding and steady state modes, and may excite unmodeled high frequency dynamics. If system uncertainties are large, the sliding mode controller would require a high sliding gain causing higher chattering effect (Wai, 2007). So, the concept of “Intelligent Control” has been suggested as an alternative approach to conventional control techniques for complex control systems. The objective is to introduce new mechanisms permitting a more flexible control, but especially more robust one, able to deal with model uncertainties and parameter variations. One of these approaches is adaptive fuzzy logic control (Li and Lau, 1989). Using fuzzy systems for approximating of the nonlinear uncertain functions, adaptive fuzzy controllers for inductions motors (IM) have been developed in (Agamy et al., 2004; Youcef and Wahba, 2009). The aim of this paper is to present a new adaptive fuzzy controller for DFIM drives when the stator windings are directly connected to the line grid, while the rotor ones

CONTROL ENGINEERING AND APPLIED INFORMATICS

99

are controlled by means of a converter. Our control objectives are: · ·

Tracking of a smooth speed reference, in the presence of an unknown load torque. Reactive power regulation at stator side (unity power factor at stator).

For this, the machine model is decomposed in two coupled subsystems, namely: the stator flux subsystem and the speedrotor flux subsystem. First, the stator voltage vector oriented reference frame is adopted, and the control problem of a stator unity power factor is converted into a stator flux regulation problem. In fact, the time varying stator flux vector is required to be orthogonal to line voltage, and the daxis component of rotor flux appears as the control input for the stator flux subsystem. Then, with appropriate choice of the stator flux reference and the strict control of d-axis component of rotor flux to a suitable value, the stator flux error dynamics becomes linear and exponentially practically stable independently of the speed dynamics. Consequently, the DFIM stator unity power factor control and speed tracking problem is converted into a rotor flux control problem. The proposed adaptive fuzzy controller (AFC) is systematically constructed using backstepping technique. Fuzzy systems are used to reasonably approximate the unknown nonlinearities and uncertainties. While the adaptive laws, which are used to estimate on-line the load torque and the unknown fuzzy parameters, are derived in the sense of Lyapunov stability theorem. The main contributions of this paper lie in the following 1) A novel adaptive fuzzy controller for DFIM is proposed. To the authors’ best knowledge, there is no result reported in the literature on the adaptive fuzzy control design for doublyfed induction machine. Note that the design of the adaptive control, for a DFIM being controlled by acting on the rotor winding and with a stator which is directly connected to the grid, is very challenge. 2) An adaptive estimator is designed to approximate the unknown load torque. 3) A comparative study between our proposed adaptive controller and a non-adaptive controller has been addressed. 2. DFIM MODELLING By considering the classical simplifying assumptions, the dynamic model of the DFIM, in the synchronous d-q reference frame, can be described as (Drid et al., 2005). ì dj sd ï dt ï ï dj sq ïï dt í dj ï rd ï dt ï dj rq ï ïî dt

Rs j sd + Lss R = - s j sq + Lss R = - r j rd + Lrs R = - r j rq + Lrs =-

Rs M j rd + wsj sq + usd Ls Lrs Rs M jrq - wsj sd + u sq Ls Lrs Rr M j sd + wrj rq + urd Ls Lrs Rr M j sq - wrj rd + urq Ls Lrs

Stator and rotor flux equations are

ìj sd = Ls isd + M ird , j sq = Ls isq + M irq í îj rd = Lr ird + M isd , j rq = Lr irq + M isq

(2)

The mechanical equation is given by J

dW = Ge - Gl - k f W dt

(3)

Where Ge is the electromagnetic torque

(

pM j sqj rd - j sdj rq Ls Lrs

Ge =

where s,r d,q α,β R, L, M u, i,

)

Rotor and stator indices Synchronous reference frame Stationary reference frame Resistance, inductance and inductance Voltage, current and flux

j

mutual

qs ,qr

Stator and rotor electrical angles

q,W

Gl , Ge

Rotor mechanical position and speed Electrical frequencies of stator, rotor and shaft Load and electromagnetic torque

J,p σ =1-(M2/LsLr)

Inertia, number of pole pairs Leakage coefficient

w s = q&s , w r = q&r , w = q&

For all speed ranges the stator and the rotor angular frequencies are related to the shaft mechanical speed by ws = w r + w . Expressions of stator and rotor active and reactive powers are respectively given by ì Ps = u sd isd + u sq i sq , Q s = u sq i sd - u sd isq íP = u i + u i , Q = u i - u i rd rd rq rq r rq rd rd rq î r

(4)

3. DFIM CONTROL PROBLEM First, we suppose that the stator flux vector is aligned with daxis as shown in Fig.1. In the stationary frame abc, the component n of the stator voltage equation is given by (Hopfensperger et al., 1999). u sn = Rs isn +

dj sn dt

(5)

By neglecting the stator resistance, (5) can be rewritten as u sn »

(1)

dj sn dt

Then, the stator voltage vector is

(6) p in advance of the stator 2

flux. In the chosen reference frame, we can write

u sd = 0 , u sq = u s

(7)

Note that the stator electrical angle q s is calculated only with the grid voltage (Hopfensperger et al., 1999).

100

CONTROL ENGINEERING AND APPLIED INFORMATICS

p (8) 2 where q1 = arctan u sb u sa is the stator voltage vector angle q s = q1 -

(

)

in the stationary reference frame abc as shown in Fig.1. b -Rotor

q-axis

d-axis us

j rd =

b -Stator js q1

qr q

ws

* j sd =

a -Rotor a -Stator

The control objectives are the following

·

Tracking of a smooth speed reference, with unknown load torque. Reactive power regulation at stator side (unity power factor at stator).

It will be demonstrated that the stator-side reactive power regulation problem can be formalized as the requirement to guarantee that the line voltage vector and the stator flux vector are orthogonal. Considering the stator equations expressed in terms of stator fluxes and currents in the line voltage reference frame.

ìj&sd = -Rsisd + wsjsq í& îjsq = -Rsisq - wsjsd + us

(9)

side unity power factor control is reformulated as a stator flux orientation control objective, i.e. the stator flux vector is required to be orthogonal to line voltage vector. The stator flux subsystem control is designed in order to achieve asymptotic alignment of the stator flux vector with the d-axes of the line voltage vector reference frame, consequently, the stator voltage and flux vectors become orthogonal. Let us define stator flux tracking errors as (10)

Using (7) and (10), the stator flux dynamic equations in (1) can be written in error form as * * ìïj~& sd = -a1j~sd - a1j sd + a2j rd + wsj~sq - j&sd í ~& * ~ ~ ïî j sq = -a1j sq + a2j rq - wsj sd - wsj sd + us

where a1 = Rs / Lss , a2 = Rs M / Lr Lss

(12)

(

1 u s + a 2 j rq ws

)

(13)

ìïj~& sd = -a1j~sd + wsj~sq í ~& ~ ~ ïîj sq = -a1j sq - wsj sd

(14)

However, in a DFIM, The rotor flux is not available as control input and j rd in (12) can only represent the d-axis * rotor flux reference j rd for the real flux j rd . The rotor

voltages u rd and u rq are the only physical available control inputs of DFIM. From (14), one concludes that the dynamics * of the stator flux are exponentially stable (i.e. lim j sd = j sd t ®¥

and lim j sq = 0 ) provided that lim t ®¥

t ®¥

* j rd = j rd

.

Now, it is required to design a control law ( u rd and u rq ) * and lim W = W * . We which guarantee that lim j rd = j rd

From the second equation of (4), the unity power factor objective is equivalent to isd = 0 . In steady-state condition, all the derivatives are zero. According to the first equation of (9), j sq = 0 is necessary to ensure isd = 0 . Then, the stator-

* ~ j~sd = jsd - jsd , jsq = jsq

)

Using (12) and (13), (11) becomes

Fig. 1. Reference frames and angles for the oriented DFIM. ·

(

1 * * a1j sd + j& sd a2

with the d-axis stator flux reference computed from the second equation of (11).

w qs

To realize the required stator flux orientation, the d-axis component of rotor flux jrd can be considered as control input in (11), and should be

(11)

t ®¥

t ®¥

consider the reduced order DFIM model represented by the rotor flux and speed equations. ìx&1 = a5 ( x4 x3 - x5 x2 ) - a6 x1 - a7Gl ï í x&2 = -a3 x2 + a4 x4 - wr x3 + d1 ( x) + u1 ïx& = -a x + a x + w x + d ( x) + u 3 3 4 5 r 2 2 2 î 3

(15)

x1 = W

x , x 2 = j rq , x 3 = j rd , 4 = j sq , x5 = j sd , u = u rq , u 2 = u rd , a3 = Rr / Lr s , x = [x1, x2 , x3 , x4 , x5 ]T , 1 a =kf /J and a4 = Rr M / Lr Lss , a5 = pM / JLr Lss , 6 a7 = 1 / J .

with

where d 1 ( x ) and d 2 ( x) are the uncertainties and perturbations that can be naturally generated from the parameter variations. In the following, two nonlinear controllers are developed for the DFIM, namely · a non-adaptive backstepping controller (NABC) and · an adaptive fuzzy backstepping controller (AFBC). 3.1 Non-Adaptive Backstepping Control System The following realistic assumptions are used in the control design and the stability analysis. Assumption 1: We assume that the load torque satisfies the following relations

CONTROL ENGINEERING AND APPLIED INFORMATICS

Gl £ r 0 ,

101

G& l » 0

(16)

where r 0 is a known positive constant. Assumption 2 : The functions d 1 ( x ) and d 2 ( x) are uncertain. However, they are bounded by known positive nonlinear functions as follows: d 1 ( x ) £ r1 ( x ) (17) d 2 ( x) £ r 2 ( x)

(18)

Assumption 3: Assume that -the reference speed profile x1d = W* is bounded and sufficiently smooth, and 1 * * * -the reference signal of x3 i.e. x 3d = j rd = (a1j sd + j& sd ) a2 is assumed to be derivable and bounded. For the system (15), the backstepping design procedure (Krstic et al., 1995) is used for the construction of the control system which guarantees a practical exponential tracking of rotor speed and rotor flux reference signals. Step 1: For a continuous bounded reference signal x1d = W* , we define the tracking error e1 as follows

e1 = x1 - x1d

r2 X& 1 = -e1 e 2 - c1e12 - 0 e12 - a 7 Gl e1 4e 1 From Young’s inequality, one has - a 7 Gl e1 £ a 7 r 0 e1 £

(20)

(21)

Choose a 5 x 5 x 2 as a virtual control to stabilize e1 and select

u given below as a desired reference signal for a 5 x 5 x 2 :

æ r2 ö u = a 5 x 4 x 3 + ç c1 + 0 ÷e1 - a 6 x1 - x&1d (22) ç 4e 1 ÷ø è where c1 > 0 is a free design constant and e 1 > 0 is a small design constant. This leads to the following dynamics ö ÷ e1 - a 7 Gl ÷ ø

The next step consists in stabilizing the tracking error e2 . Step 2: The time-derivative of (24) is given by

e& 2 = a 5 x 5 x& 2 + a 5 x& 5 x 2 - u& From (1), (15) and (22), we can write

e& 2 = h1 ( z1 ) + e1 + d 1 (x ) + a 5 x5 u1

1 2 e1 2

The time-derivative of (25) can be expressed as follows

(30)

with h1 ( z1 ) = -e1 + a 5 x 5 (- a3 x 2 + a 4 x 4 - w r x 3 ) + a 5 x 2 (- a1 x 5 + a 2 x 3 + w r x 4 ) a 5 x 3 (-a1 x 4 + a 2 x 2 - w s x 5 + u s ) a 5 x 4 (- a3 x 3 + a 4 x 5 + w r x 2 + u 2 ) + &x&1d + a 6 (a 5 ( x 4 x 3 - x 5 x 2 ) - a 6 x1 )

d 1 (x ) = a 5 x 5 d 1 ( x ) - a 5 x 4 d 2 ( x ) + (c1 + ( r 02 / 4e 1 ))a 7 Gl a 6 a 7 Gl where z 1 = [ x1 x 2 x 3 x 4 x 5 u 2 ] T , and e3 is the tracking error of x3 .

e3 = x3 - x 3d with x3d is the desired signal of x3 .

(31)

Define a Lyapunov function candidate for the (e1, e2)subsystem as

X 2 = X1 +

(23)

1 2 e2 2

(32)

(24)

(

)

X& 2 £ -c1e12 + e2 h1 ( z1 ) + d1 ( x) + a5 x5u1 + e 1a72 From Young’s inequality, one has

d1 (x )e2 £ d1 (x ) e2 £

(x r

2 2 2 2 2 5 1 ( x) + x4 r 2 ( x) + 2 r 0

)e

4e 2

2 æ ö æ r 02 ö ö 2 ç 2 æç 2 2÷ ÷ ç ÷ ç 2a5 + ç c1 + ç 4e ÷ ÷ a7 + a6 a7 ÷e 2 ç ÷ è 1 øø è è ø

The Lyapunov function candidate for the e1-subsystem is selected as

X1 =

(29)

Its time-derivative is given by

with e 2 is the tracking error of the variable a 5 x 5 x 2 , given by

e2 = a5 x5 x 2 - u

(28)

(c1 + ( r 02 / 4e 1 ))(e 2 + (c1 + ( r 02 / 4e 1 ))e1 ) +

From (15), we can write

æ r2 e&1 = - e 2 - ç c1 + 0 ç 4e 1 è

(27)

X& 1 £ -e1e 2 - c1e12 + e 1 a 72

Its time-derivative e&1 is given by

e&1 = a5 x 4 x3 - a 5 x5 x 2 - a 6 x1 - a 7 Gl - x&1d

r 02 2 e1 + e 1 a 72 4e 1

Using (27), (26) becomes

(19)

e&1 = x&1 - x&1d

(26)

(25) where

e2 > 0

is

a

small

design

(33)

2 2

+ (34)

constant.

102

CONTROL ENGINEERING AND APPLIED INFORMATICS

From (33), the control input u1 follows:

u1 =

can be chosen as

(

)

x2 r 2 (x) + x42 r22 ( x) + 2r02 ö÷ 1 æç - h1( z1) - c2e2 - 5 1 e2 ÷ a5 x5 çè 4e 2 ø (35)

where c 2 > 0 is a free design constant. Remark 1: The magnetising flux x5 must be non-zero (Due to the remanence flux). Using (34) and (35), (33) becomes X& 2 £ -c1e12 - c 2 e 22 + e 1a72 + æ ö æ r 02 ö ö 2 ç 2 æç ÷ a + a 2 a 2 ÷e ÷ ç a + c + 2 6 7÷ 2 ç 5 ç 1 ç 4e ÷ ÷ 7 ç ÷ è 1 øø è è ø 2

(2a

(36)

2 5

(37)

Then, let’s define a Lyapunov function candidate as follows

(42)

X& 3 £ - KX 3 + e

(43)

(

)

where e = e1a72 + 2a52 + (c1 + ( r02 / 4e1))2 a72 + a62 a72 e 2 + e 3 and K = min{2c1 , 2c 2 , 2c 3 } . Multiplying (43) by e Kt yields

)

d X 3 e Kt £ ee Kt dt

(44)

Integrating (44) over [0,t], it follows that 0 £ X 3 (t ) £

Step 3: At this step, we will construct the control law u 2 to stabilize the dynamics of x 3 . The time-derivative of (31) is given by

)

+ (c1 + ( r 02 / 4e 1 )) 2 a 72 + a 62 a 72 e 2

We can rewrite (42) as follows

(

The next step consists in stabilizing the tracking error e3 .

e& 3 = x& 3 - x& 3d = - x& 3d - a 3 x 3 + a 4 x 5 + w r x 2 + d 2 ( x) + u 2

X& 3 £ -c1e12 - c 2 e 22 - c 3 e 32 + e 1 a 72 + e 3 +

e æ e ö + ç X 3 (0) - ÷e - Kt K è Kø

(45)

This results in ultimately uniformly bounded (UUB) stabilization of the tracking errors ( e1 , e 2 , e3 ) . Since e can be chosen arbitrary and K only depends on the design parameters ( c1 , c 2 and c 3 ) , the ultimate error bounds can be made arbitrary small. 3.2 Adaptive Fuzzy Backstepping Control System

(38)

The control objective in this section is to design an adaptive fuzzy backstepping controller for DFIM guaranteeing global system stability with improved control robustness.

Using (35) and (36), the time-derivative of X 3 can be bounded by

The basic configuration of a fuzzy logic system consists of a fuzzifier, some fuzzy IF-THEN rules, a fuzzy inference engine and a defuzzifier, as shown in Fig. 2.

1 X 3 = X 2 + e32 2

X& 3 £ -c1e12 - c2 e22 + e 1a72 + e3 (h2 ( z 2 ) + d 2 ( x ) + u 2 ) + 2 æ ö æ r 02 ö ö 2 ç 2 æç 2 2÷ ÷ ç ÷ 2 a + c + a + a a 6 7 ÷e 2 ç 5 ç 1 ç 4e ÷ ÷ 7 ç ÷ 1 øø è è è ø

Fuzzy Rule Base

(39) x

with h2 ( z 2 ) = - a3 x3 + a 4 x5 + w r x2 - x&3d , where

Fuzzy Inference Engine

z 2 = [ x1 x 2 x3 x5 ]T . To stabilize the dynamics (37), the control input u 2 can be chosen as follows

æ r 2 ( x) ö÷ u 2 = -h2 ( z 2 ) - ç c 3 + 2 e3 (40) ç 4e 3 ÷ø è where c 3 > 0 is a free design constant and e 3 > is a small design constant. From Young’s inequality, one has

d 2 ( x )e 3 £ r 2 ( x ) e 3 £

r 22 ( x)e32 4e 3

+e3

Using (40) and (41), (39) becomes

Defuzzifier

Fuzzifier

fˆ

Fig. 2. The basic configuration of a fuzzy logic system. The fuzzy inference engine uses the fuzzy IF-THEN rules to perform a mapping from an input vector T n ˆ x = [ x x ... x ] Î R to an output f Î R . The ith fuzzy 1

2

n

rule is written as R (i ) : if x1 is A1i and ... and xn is Ani then fˆ is f i

(41)

(46)

where A1i , A2i ,..., and Ani are fuzzy sets and f i is the fuzzy singleton for the output in the ith rule. By using the singleton fuzzifier, product inference, and center-average defuzzifier, the output of the fuzzy system can be expressed as follows:

CONTROL ENGINEERING AND APPLIED INFORMATICS

103

åi=1 f i æçè Õ j=1 m A ( x j ) ö÷ø fˆ ( x) = m n åi=1æçè Õ j=1 m A ( x j ) ö÷ø n

m

i j

i j

= q Ty (x )

(47)

where l1 > 0 is a design constant and Gˆl is the estimate of Gl . From (21) and (49), we can obtain the following dynamics: ~ (50) e&1 = a5 x 4 e3 - e2 - (l1 + a6 )e1 - a7 Gl

where m A i ( x j ) is the degree of membership of x j to Aij ,

~ where Gl = Gl - Gˆ l is the load torque estimation error, and

m is the number of fuzzy rules, q T = [ f 1 , f 2 , ... , f m ] is the

e2 is the tracking error of the variable a5 x5 x2 , given by

j

adjustable parameter vector (composed of consequent parameters), and y T = [y 1 y 2 ...y m ] with æç n m i ( x ) ö÷ Õ j=1 Aj j ø y i ( x) = è n m åi=1æçè Õ j =1 m Aij ( x j ) ö÷ø

~ 1 æç 2 Gl2 ö÷ V1 = e1 + 2 çè g l ÷ø

being the fuzzy basis function (FBF). Throughout the paper, it is assumed that the FBFs are selected so that there is always at least one active rule (Wang, 1994), n m i.e. å æç Õ m Ai ( x j ) ö÷ > 0 . i =1 j =1

è

j

ø

It is worth noting that the fuzzy system (47) is commonly used in control applications. Following the universal approximation results (Wang, 1994; Feriyonika and Dewantoro, 2013; Husek, and Cerman, 2013; Joelianto et al., 2013; Odior, 2013), the fuzzy system (47) is able to approximate any nonlinear smooth function f (x ) on a compact operating space to an arbitrary degree of accuracy. Of particular importance, it is assumed that the structure of the fuzzy system (i.e. the pertinent inputs, the number of membership functions for each input and the number of rules) and the membership function parameters are properly specified beforehand. The consequent parameters q are then determined by appropriate adaptation algorithms. Assumption 4: the functions d1 ( x ) and d 2 ( x) are smooth and completely unknown. Throughout the rest paper, we will exploit the following nice property with regard to function Tanh(.) 0 £ X - Tanh ( X / b i ) £ b i , where b i

is

a

small

for i = 1,2

positive

design

(48) constant

and

b i = 0.2785b i .

(52)

where g l > 0 is a design parameter. By assuming that the load torque is slowly time-varying ( G& l = 0 ), the time-derivative of (52) along (50) is given by & Gˆ ö ~æ V&1 = - e1 e 2 + a 5 x 4 e 3 e1 - (l1 + a 6 )e12 - Gl ç a 7 e1 + l ÷ (53) ç gl ÷ è ø If the load torque estimator is designed as ~ & Gˆ l = s l Gl - g l a7 e1 (54) where s l > 0 is a design parameter. Then, (53) can be written as

~ V&1 = -e1 e2 + a5 x4e3e1 - (l1 + a6 )e12 - (s l / g l )Gl2

(55)

The next step consists in stabilizing the tracking error e2 . Step 2: The time-derivative of (51) is given by

e& 2 = a 5 x 5 x& 2 + a 5 x& 5 x 2 - u&

(56) From the second subsystem of (1), (15) and (49), we can write

e& 2 = f 1 ( z1 ) + e1 + (a 5 a 2 x 2 - a5 x5w r - a 5 l1 x 4 )e3 - a 7 (s l + l1 )Gˆ l + a5 x5 u1

(57)

with f 1 ( z 1 ) = - e1 - a 5 a 3 x 5 x 2 + a 5 a 4 x 5 x 4 - a 5 x 5 w r x 3 d -

Now, we give the procedure of the backstepping design. Step 1: For a continuous bounded reference signal x1d , the tracking error e1 and its derivative e&1 are defined respectively by (19) and (21): e1 = x1 - x1d

e&1 = a5 x 4 x3 - a 5 x5 x 2 - a 6 x1 - a 7 Gl - x&1d Let Gˆ l be the estimate of Gl and select a new virtual control u as

u = a 5 x 4 x3 d + l1 e1 - a 6 x1d - x&1d - a 7 Gˆ l

e2 = a5 x5 x2 - u (51) Consider the following Lyapunov function candidate for the e1-subsystem

(49)

a 5 a1 x 5 x 2 + a 5 w s x 4 x 2 + a 5 a1 x 3 d x 4 + a 5 w s x 3 d x 5 - a 5 x 3 d u s - a 5 x 4 x& 3 d + a 6 x& 1d + &x&1d + l1 (e 2 + (l1 + a 6 )e1 ) - a 72 g l e1 +

a 7 (s l + l1 ) Gl + a 5 x 5 d 1 ( x )

where z1 = [ x1 x2 x4 x5 u Gl ]T and e3 is the tracking error of x3 . It is given by e3 = x3 - x3 d (58) * * * where x3d = j rd = (a1j sd + j& sd ) / a2 .

104

The

CONTROL ENGINEERING AND APPLIED INFORMATICS

uncertain

continuous

function

f 1 ( z1 )

can

be

approximated by the fuzzy system (47) as follows fˆ1 ( z1 , q 1 ) = q 1Ty 1 ( z1 )

Replacing (63) into (57) and using (62) yield e&2 = e1 + ( a5a2 x2 - a5 x5wr - a5l1x4 )e3 æe ö ~ q1Ty 1 ( zˆ1 ) + J1( z1, zˆ1 ) - l2e2 - k1 Tanhçç 2 ÷÷ è b1 ø

(59)

(64)

~

where y 1 ( z1 ) is the FBF vector, which is fixed a priori by

where q1 = q1 - q1* is the parameter error vector.

the designer, and q1 is the adjustable parameter vector of the

Multiplying (64) by e2 and using the inequality (48), we get

fuzzy

e2e&2 = e1 e2 + (a5a2 x2 - a5 x5wr - a5l1x4 )e2e3

system.

Furthermore,

according

to

universal

approximation theorem (Wang, 1994), the functions f 1 ( z1 )

æe ö ~ - e2q1Ty1 ( zˆ1 ) + e2J1 ( z1, zˆ1 ) - l2e22 - k1 e2Tanhçç 2 ÷÷ è b1 ø

can be approximated optimally as follows f1 ( z1 ) = fˆ1 ( z1 ,q1* ) + v 1 ( z1 ) = q1*Ty 1 ( z1 ) + v 1 ( z1 )

£ e1 e 2 + (a 5 a 2 x 2 - a5 x 5 w r - a 5 l1 x 4 )e 2 e 3 -

(60)

(65) æe ö ~ e 2q1T y 1 ( zˆ1 ) - l 2 e 22 - k~1 e 2 Tanhçç 2 ÷÷ + k 1* b1 è b1 ø

where q1* is the optimal parameter vector and v 1 ( z1 ) is the unavoidable fuzzy approximation error which is generally assumed to be bounded as follows (Wang, 1994; Rusu, 2002; Boulkroune et al., 2008, 2009, 2010a, 2010b; Liu et al., 2011): v 1 ( z1 ) £ v 1 , "z1 Î W z1 , where v 1 is an unknown constant.

1

Define a Lyapunov function candidate for the (e1, e2)subsystem as follows V 2 = V1 +

Since the input vector available, it must zˆ = [ x , x , x , x , u , 1

where k~1 = k1 - k1* is the parameter error and b1 = 0.2785b1.

2

4

5

T

z1 = [ x1 , x 2 , x4 , x5 , u , Gl ] is not be replaced by its estimate ˆG ]T in (59). Thus, the fuzzy system l

(59) used to approximate f 1 ( z1 ) is replaced by the following fuzzy system fˆ1 ( zˆ1 , q 1 ) = q 1Ty 1 ( zˆ1 )

(61)

Taking the derivative of V2 with respect to time and using (55) and (65), one can obtain 1 ~T & 1 ~ V&2 = V&1 + e2 e& 2 + q1 q 1 + k 1k&1 g q1 g k1

£ ( a 5 a 2 x 2 - a 5 x5w r - a 5 l1 x 4 ) e2 e3 + a 5 x 4 e1e3 -

(l1 + a 6 )e12 - s l

f 1 ( z1 ) = f 1 ( z1 ) - fˆ1 ( z1 , q1* ) + fˆ1 ( z1 , q1* ) - fˆ1 ( zˆ1 , q1* ) + fˆ ( zˆ , q * ) 1

) + v 1 ( z1 ) + [q 1*Ty 1 ( z1

= q1*T y ( zˆ1

) + J1 ( z1 , zˆ1 )

) - q1*Ty 1 ( zˆ1

g q1

)]

upper bound, i.e. J1 ( z1 , zˆ1 ) £ k1* with k1* is an unknown positive constant (Wang, 1994; Boulkroune et al., 2010a). To stabilise the dynamics (57), the following fuzzy adaptive controller is proposed 1 u1 = a 7 (s l + l 1 ) Gˆ l - q 1T y 1 ( zˆ1 ) - l 2 e 2 a5 x5

(

)

~ Gl2 - l 2 e22 +

]

(63)

where l2 > 0 is a design constant, k 1 is the estimate of the unknown bound k1* and b1 > 0 is a small design constant.

(67)

æ e öù 1 ~é k 1 êk&1 - g k 1e2Tanh çç 2 ÷÷ ú + k 1* b 1 g k 1 ëê è b 1 ø ûú

(62)

where J1 ( z1 , zˆ1 ) = v 1 ( z1 ) + [q1*Ty 1 ( z1 ) - q1*Ty 1 ( zˆ1 )] is the fuzzy approximation error. Notice that J1 ( z1 , zˆ1 ) has an

æe ö k 1 Tanhçç 2 ÷÷ è b1 ø

[

gl

1 ~T & q1 q1 - g q 1e2y 1 ( zˆ1 ) +

1

= q1*T y ( zˆ1

(66)

where g q 1 and g k1 > 0 are design constants.

From (59)-(61), we have

1

1 2 1 ~T ~ 1 ~2 e2 + q1 q1 + k1 2 2g q 1 2g k 1

If the adaptation laws are designed as q& = -g s q + g e y ( zˆ ) 1

q1 q 1 1

q1 2

1

(68)

1

æe ö k&1 = -g k 1s k 1k 1 + g k 1e 2 Tanhçç 2 ÷÷ è b1 ø where s q 1 and s k 1 > 0 are small design constants.

(69)

Then, (67) can be expressed as V&2 £ ( a 5 a 2 x 2 - a 5 x 5 w r - a 5 l 1 x 4 ) e 2 e 3 + æs a 5 x 4 e 1 e 3 - (l 1 + a 6 )e 12 - çç l ègl ~T T ~ s q q - s k k + k *b q1 1

1

k1

1

1

1

ö~ 2 ÷ G l - l 2 e 22 - (70) ÷ ø 1

In the next step, we try to stabilize the tracking error e3 .

CONTROL ENGINEERING AND APPLIED INFORMATICS

105

Step 3: At this step, we will construct the control law u 2 . The time-derivative of (58) is given by (71) e&3 = - a3 x3 + a4 x5 + wr x2 + d 2 ( x) + u 2 - x&3d We can rewrite (71) as follows e&3 = -(a5a2 x2 - a5 x5wr - a5l1x4 )e2 - a5 x4e1 + f 2 ( z2 ) + u2 (72) with f 2 ( z2 ) = (a5a2 x2 - a5 x5wr - a5l1x4 )e2 + a5 x4e1 - a3 x3 + a4 x5 + wr x2 + d 2 ( x) - x&3d where z 2 = [ x1 x 2 x 3 x 4 x 5 ]T . The

uncertain

continuous

function

f 2 ( z2 )

can

where k~2 = k 2 - k 2* and b 2 = 0.2785b 2 . Define a Lyapunov function candidate as follows V3 = V 2 +

Taking the derivative of V3 with respect to time and using (78) and (70), one can obtain 1 ~T & 1 ~ V&3 = V&2 + e3 e&3 + q2 q2 + k 2k& 2 gq2 gk2 æ s ö~ £ -(l1 + a6 )e12 - çç l ÷÷Gl2 - l2e22 - l3e32 è gl ø ~T s q q -s k~T k + k *b + k * b +

be

q1 1

(73)

gq 2

follows

unavoidable fuzzy approximation error which is assumed to be bounded as follows, i.e.

v 2 ( z 2 ) £ k 2* , "z 2 Î W z 2

2 2

]

(80)

The adaptation laws are designed as q& = -g s q + g e y ( z ) 2

q2 q2 2

q2 3

2

(81)

2

æe ö k& 2 = -g k 2s k 2k 2 + g k 2 e3Tanhçç 3 ÷÷ è b2 ø where s q 2 and s k 2 > 0 are small design constants.

(82)

(75)

where k 2* is an unknown constant. To stabilise the dynamics (71), the following fuzzy adaptive controller is proposed æe ö u 2 = -q 2Ty 2 ( z 2 ) - l 3 e3 - k 2 Tanh çç 3 ÷÷ (76) è b2 ø where l3 is a positive design constant, k 2 is the estimate of the unknown bound k 2* and b 2 is a small positive design constant. Replacing (76) into (71) and using (75) yield e& 3 = - ( a 5 a 2 x 2 - a 5 x 5 w r - a 5 l1 x 4 )e 2 - a 5 x 4 e1 æe ~ q 2T y 2 ( z 2 ) + v 2 ( z 2 ) - l 3 e 3 - k 2 Tanh çç 3 è b2 ~ * where q 2 = q 2 - q 2 is the parameter error vector.

ö ÷ ÷ ø

(77)

e 3 e&3 = -(a 5 a 2 x 2 - a 5 x 5w r - a 5 l1 x 4 )e 2 e 3 - a 5 x 4 e1 e3 æe ~ e 3q 2T y 2 ( z 2 ) + e 3v 2 ( z 2 ) - l 3 e32 - k 2 e3 Tanh çç 3 è b2

One can henceforth easily check that σ σ - σ κi κ~i κ i £ - κi κ~i2 + κi κ i*2 , for i = 1,2 2 2 σ θi ~ 2 σ θi * 2 ~T - σ θi θ i θ i £ θi + θ i , for i = 1,2 2 2 And using the previous inequalities and the adaptive laws (81)-(82), (80) becomes æ s ö~ σ ~ 2 V&3 £ -(l1 + a6 )e12 - çç l ÷÷Gl2 - l2e22 - l3e32 - θ1 θ1 2 è gl ø σ θ 2 ~ 2 σ κ1 ~ 2 σ κ 2 ~ 2 θ2 κ1 κ2 + h 2 2 2 where h = k1*b1 + k 2*b 2 +

Multiplying (77) by e3 and using the property (48), we get

ö ÷÷ ø

£ -(a 5 a 2 x 2 - a5 x 5 w r - a5 l1 x 4 )e 2 e3 - a5 x 4 e1 e3 æe ~ e3q 2Ty 2 ( z 2 ) - l3 e 32 - k~2 e3 Tanh çç 3 è b2

[

1 1

1

æ e öù 1 ~ é k 2 êk& 2 - g k 2e3 Tanh çç 3 ÷÷ú g k 2 êë è b 2 øúû

the functions f 2 ( z 2 ) can be optimally approximated as

where q 2* is the optimal parameter vector and v 2 ( z 2 ) is the

k1 1

1

1 ~T & q 2 q 2 - g q 2 e3y 2 ( z 2 ) +

According to universal approximaton theorem (Wang , 1994),

f 2 ( z 2 ) = fˆ2 ( z 2 , q 2* ) + v 2 ( z 2 ) = q 2*T y 2 ( z 2 ) + v 2 ( z 2 ) (74)

(79)

where g q 2 and g k 2 > 0 are design constants.

approximated by the adaptive fuzzy system (47) as follows fˆ2 ( z2 , q2 ) = q 2Ty 2 ( z2 )

1 2 1 ~T ~ 1 ~2 e3 + q2 q2 + k2 2 2g q 2 2g k 2

(78) ö ÷ + k 2* b 2 ÷ ø

σθ1 * θ1 2

2

+

σθ 2 * θ2 2

2

+

(83)

σ κ1 *2 σ κ 2 *2 κ1 + κ2 2 2

One can rewrite (83) as follows V&3 £ -z V3 + h

where

z = min{2(l1 + a6 ), 2sl , 2l2 , 2l3, σθ1 g θ1, σθ 2g θ 2, σk1gk1, σ k 2g k 2 }

Multiplying (84) by ezt yields

(84)

106

CONTROL ENGINEERING AND APPLIED INFORMATICS

(

)

d V3ez t £ hez t dt

(85)

Integrating (85) over [0 t], it follows that 0 £ V3 (t ) £

h æ h + ç V3 (0) z çè z

ö -z ÷÷e ø

t

(86)

This results in ultimately uniformly bounded (UUB) stabilization of the tracking errors ( e1 , e2 , e3 ) and the ~ ~ ~ parameter estimation errors ( q , q , k~ , k~ , G ) (Khalil, 1

2

1

2

l

Gˆ l , q1 , k1 , q 2 and k 2 ~ ~ ~ respectively established from that Gl , q1 , k~1 , q 2 and k~2 . 2001). The boundedness of

is

Remark 2: According to the definition of z and h , it can be seen that the size of z depends on the controller design parameters s l , γ θi , g ki , s qi , s ki , l1 , l 2 and l3 , and that of h depends on the controller design parameters b i , s qi and

s ki . It is clear that if we increase s l , γ θi , g ki , l1 , l2 , and l3 and decrease b i , s qi and s ki , it will help to reduce the term h / z . This implies that the tracking errors can be made arbitrary small by appropriately choosing those design parameters. Because e1, e3 Î L¥ and x1d , x3d Î L¥ , therefore x1 , x3 Î L¥ . From (11), one can write the dynamics of the tracking errors of the stator fluxes as follows

j~& sd = -a1j~sd + wsj~sq + a2e3

From those dynamics and since e3 Î L¥ , we can easily prove the boundedness of j~sd , j~sq and x4 . From ( x4 , x3d , e1 , x&1d , x1 , Gˆl ) Î L¥ , it can be concluded that u Î L¥ based on (49). Since x 2 = (e 2 + u ) / a5 x 5 , e 2 , u Î L¥ and * x5 > 0 , we can show that x2 Î L¥ . The boundedness of j sd and x follows that of j (or x ) and j~ . Due to the rq

2

sd

boundedness of x1 , x2 , x3 , x4 , x5 and Gˆ l and since q1, k1 ,q 2 , k 2 Î L¥ , we can conclude that the controls ( u1 and u 2 ) are also bounded. Now, taking a summary, under the above stator voltage vector orientation constraint, if the DFIM system given by (1) is directly connected to the grid by the stator winding and is controlled acting on the rotor winding by the proposed AFBC described by (49), (63), (68), (69), (76), (81) and (82) with the load torque estimator given by (54), then the practical stability of the closed-loop control system can be guaranteed. Let us consider the load torque adaptation law (54) that can be written in the following form & Gˆ l = s l Gl - s l Gˆ l - g l a7 e1

Gl = -

1 (x&1 + a5 x5 x2 - a5 x4 x3 + a6 x1 ) a7

(87)

(88)

which leads to

s & Gˆ l = - l (x&1 + a5 x5 x 2 - a5 x4 x3 + a6 x1 ) - s l Gˆ l - g l a7 e1 a7

(89)

Because of the integral structure of the adaptation law (89), this updating law is implementable despite the presence of x&1 . In fact, it is can be rewritten as

s Gˆ l = Gˆ l (0) - l (x1 (t ) - x1 (0)) + ò h(t )dt a7 0 where t

æ ö s h = -çç s l Gˆ l + g l a 7 e1 + l (a 5 x5 x 2 - a5 x 4 x 3 + a 6 x1 )÷÷ a7 è ø

(90)

(91)

The overall scheme of the controlled DFIM is depicted in Fig.3 in which the stator is directly connected to the grid, and the DIFM is controlled acting on the rotor windings. ~& ~ Remark 3 : From (87), we can rewrite Gl = -s l Gl + g l a7 e1 , this equation can be seen as a standard disturbance observer. ~ In fact, if e1 converges to zero, then Gl also converges to zero. Consequently, Gˆ converges to G . l

j~& sq = - a1j~sq - w sj~sd

5

As the actual load torque Gl is unknown, the first equation in (15) will be used to compute its value. Consequently, Gl is given by

l

Remark 4 : Based on the universal approximation theorem and by incorporating fuzzy logic systems into adaptive control schemes, many adaptive controllers have been proposed in the literature, among them (Boulkroune et al., 2008, 2009) for single-input single-output (SISO) nonlinear systems, and (Boulkroune et al., 2010a, 2010b) for multipleinput multiple-output (MIMO) nonlinear systems. Generally, these adaptive fuzzy control approaches can have nice performance. However, they have been applied only to a relatively simple class of nonlinear systems. The key requirement is that the unknown nonlinearities appear on the same equation as the control input in a state space representation. Such restrictions on the location of the uncertain nonlinear functions are generally referred to as matching conditions. If practical systems are subject to some unknown nonlinear functions which do not satisfy the matching conditions (as in the case of the doubly-fed induction motor considered here), these adaptive fuzzy control approaches mentioned above cannot be implemented. In this paper, a fuzzy adaptive backstepping controller is developed for a DFIM. Compared to the above works (Boulkroune et al., 2008, 2009, 2010a, 2010b), the main contributions of this paper lie in the following ·

Because a part of the DFIM model is subject to some unknown nonlinear functions which do not satisfy the matching conditions, the backstepping approach has been used in the controller design.

CONTROL ENGINEERING AND APPLIED INFORMATICS · · ·

107

An adaptive estimator has been designed to approximate the unknown load torque. The comparative study between our proposed fuzzy adaptive backstepping controller and a non-adaptive backstepping controller has been addressed. The control design of the considered configuration (i.e. a DFIM being controlled by acting on the rotor winding and with a stator which is directly connected to the grid) is very challenge. To our best knowledge, in the literature, there are little works dealing with this control problem. p /2 -

Grid

q1

æu arctançç sb è u sa

+

ö ÷ ÷ ø

usa

u s, abc

3 to 2 usb

qs Speed and currents measurement, Flux , θ, θr , ωr calculation.

DFIM

u r ,abc u rd

dq to αβ 2 to 3

AFBC

Converter

u rq

= Grid

Fig. 3. The overall control scheme of the DFIM. 4. SIMULATION RESULTS The parameters of the tested DFIM are summarized in Table.1 (Drid et al., 2005). The controller parameters are set to the following values: For NABC system: c1 = 200, c 2 = 10000, c 3 = 50000 , e1 = e 2 = e 3 = 0.1 and r0 = 5 . For AFBC system: g l = 0.001 , s l = 200 , l1 = l2 = l3 = 200 ,

g q 1 = 100 ,

g k 1 = 0.05 ,

g q 2 = 1000 ,

g k 2 = 0.1 , s q 1 = s q 2 = 10 , s k 1 = s k 2 = 10 . The initial conditions are chosen as: k1 (0) = k 2 (0) = 0.2 , and q1i (0) = q 2i (0) = 0 . -3

The

fuzzy

system

[ x1 x2 x4 x5 u Gˆ l ]

T

-5

q1T y 1 ( zˆ1 )

has

the

vector

as input, while the fuzzy system

q 2T y 2 ( z 2 )

has the state vector [ x1 x 2 x3 x 4 x5 ]T as input. For each variable of the entries of these fuzzy systems, as in [19], we define three (one triangular and two trapezoidal) membership functions uniformly distributed on the intervals [- 0.5,1.5] for x 2 , x3 , x 4 and x5 , [- 150,200] for x1 , [- 50,50] for u , and [- 5,8] for Gˆ . l

Parameter Rated power voltage Current Synchronous speed Stator resistance Rotor resistance Stator inductance Rotor inductance Mutual inductance Inertia Friction coefficient Pole pairs

Value Pn =4 kW U =220/380 V I = 15/8.6 A w sn = 2p 50 Hz

Rs =1.2 W Rr =1.8 W Ls =1.1554H Lr =1.1568 H M =0.15 H J=0.2 kg.m2 kf=0.014 Nm.s/rad p=2

The simulation is carried out under the followings adverse conditions -Model uncertainties: d1 ( x ) = 3x2 and d 2 ( x ) = 4 x 2 + 2 x 3 and introduced at t = 0.8s . -Parameter variations: At t=0.4s and t=0.5 s, the stator and the rotor resistances are respectively increased at 50% of theirs ratted values. -External load torque: A load torque disturbance is applied as follows ì 0 Nm ï5 Nm ï Gl = í 0 Nm ï 5 Nm ï î 0 Nm

Reference signals

b1 = b 2 = 0.05,

Table1. Parameters of DFIM.

t £ 0.3s 0.3s £ t £ 0.7 s 0.7 s £ t £ 1s 1s £ t £ 1.4 s t ³ 1.4 s

Fig. 4, Fig. 5 and Fig. 6 show the simulation results of the NABC system. As it can be seen, the system control performances degrade in the presence of external disturbances and unstructured dynamical uncertainties. The simulation results of the proposed AFBC system are depicted in Fig.7, Fig.8 and Fig.9. From these simulation results, we can clearly see that a satisfactory behavior of the mechanical speed with regard to the imposed speed profile is achieved. Moreover, the load torque estimator gives a correct estimation for the actual load torque. The controller copes easily with the sudden external load disturbance, the parametric variations and the model uncertainties, and provides a fast tracking responses. We can observe clearly that the flux responses respect the imposed constraints. So, after transient, the stator and the rotor fluxes recover respectively their imposed values. Consequently, the flux orientation objective is guaranteed, and the stator reactive power converges to zero in steady-state operation. The AFBC approach is compared in similar operating conditions to the NABC approach. Fig. 10 shows the speed and flux tracking performances under the two control methods. It is evident that the proposed AFBC schema yields superior control performances than the NABC scheme. In fact, this new control scheme can achieve high accuracy in speed and flux tracking and shows very strong robustness to

108

CONTROL ENGINEERING AND APPLIED INFORMATICS

-4

0,4 x 10 0,7 x 10-4

3,6 x 10 5,4 x 10-4

The superiority of the AFBC over NABC is clearly shown in these results. Based on the simulation results, numerical comparison of the two control methods developed in this work is shown in Table. 2. 5. CONCLUSION A new adaptive fuzzy backstepping control scheme has been proposed for high-performance DFIM drive. The backstepping technique has been applied to systematically construct our controller which guarantees uniform ultimate boundedness of all signals in the closed-loop system. Lyapunov approach has been adopted to derive the parameter adaptation laws. The tracking error dynamics have been proved to exponentially converge to a residual adjustable set. The obtained results confirmed the effectiveness of the proposed AFBC scheme for control of a DFIM. It has been shown that the proposed controller allows good tracking performances and stator reactive regulation to zero in steady state, and can deal with the unavoidable parameters variations, external disturbance and model uncertainties. Comparison results against a NABC system show better robustness to parameter variations and system uncertainty. It is worth noting that the control methodology proposed here can be easily extended to other electric drives. Our future work will address the experimental implementation of this proposed AFBC scheme and the design of a speed sensorless controller.

tracking of phisq (Wb)

1

Flux φrd

-3

4,1 x 10 8,4 x 10-5

1.1

0

0.5 1 time (s)

1

0

0.5 1 time (s)

0.5

1

1.5

0.5

electromagnetic torque (Nm)

time (s) 20 10 0 -10 -20

0

0.5

1 time (s)

1.5 (c)

0.5

1 time (s)

3

x 10

0

-1

0

0.5

1 time (s)

1.5 (a)

500

0

-500

2 1 0 -1

0

0.5

1 time (s)

1.5 (d)

Fig.4. NABC scheme: (a) Rotor speed ( W solid line, W * dotted line). (b) actual load torque. (c) Torque. (d) Stator reactive power.

382

381

380.5

0

0.5

1 time (s)

1.5 (c)

380

0

0.5

1 time (s)

1.5 (b)

0

0.5

1 time (s)

1.5 (d)

600 400 200 0 -200

Fig.6. NABC scheme: (a) Stator voltage u sd . (b) Stator voltage u sq . (c) Rotor voltage urd . (d) Rotor voltage u rq . load torque es tim ation (Nm )

150 100 50

0

0.5

1 time (s)

1.5 (b)

4

1.5 (d)

381.5

-0.5

0

0

0.5 1 time (s)

382.5

1

0 -2

1.5 (a)

-0.2

1.5 (c)

1.5

10 0 -10

0

0.5

1 time (s)

4 2 0 -2

0

0.5

1.5 (c)

1 tim e (s)

20

-20

6

(a)

s tator reac tiv e power (V A R)

0

0

0

* (dotted Fig. 5. NABC scheme: (a) j sd (solid line) and j sd * line). (b) j sq (solid line) and j sq (dotted line). (c) j rd * (solid line) and j rd (dotted line). (d) j rq .

elec trom agnetic torque (Nm )

0

2

1.5 (b)

-0.1

0.5

rotor s peed (rd/s )

load torque (Nm)

50

stator reactive power (VAR)

rotor speed (rd/s)

100

4

0.5 1 time (s)

0.1

1.5

6

150

0

0.2

2

200

200

-0.2

1.5 (a)

2.5

0

0

-0.1

rotor flux phirq (Wb)

Flux φsq

-4

0.1

1.2

tracking of phird (Wb)

40 x 10 9,3 x 10-2

Flux φsd

d-axis stator voltage (V)

-2

d-axis rotor voltage (V)

NABC AFBC

Speed Ω

1.3

q-axis stator voltage (V)

MSE (%) Control Strategy

0.2

q-axis rotor voltage (V)

Table 2. Numerical comparison between the NABC and AFBC systems.

1.4 tracking of phisd (Wb)

external load disturbance and the system uncertainty. Moreover, the control effort of the AFBC is smaller than that of the NABC.

3

x 10

1.5 (b)

4

2 1 0 -1

0

0.5

1 time (s)

1.5 (d)

Fig.7. AFBC scheme: (a) Rotor speed: W (solid line) and W* (dotted line). (b) Load torque: Ge (solid line) and Gˆ e (dotted line). (c) Torque. (d) Stator reactive power.

CONTROL ENGINEERING AND APPLIED INFORMATICS

0.2 tracking of phisq (Wb)

tracking of phisd (Wb)

1.4 1.3

1.1 0

0.5

1

0

-0.1

1.5 (a)

2

0.2

1.5

0.1

1 0.5

0

0.5

1

1.5 (b)

1

1.5 (d)

time (s)

rotor flux phirq

tracking of phird (Wb)

time (s)

0

REFERENCES

0.1

1.2

1

109

0

-0.1

0

0.5

1 time (s)

-0.2

1.5 (c)

0

0.5 time (s)

* Fig.8. AFBC scheme: (a) j sd (solid line) and j sd (dotted * line). (b) j sq (solid line) and j sq (dotted line). (c) j rd * (solid line) and j rd (dotted line). (d) j rq .

382.5

q-axis stator voltage (V)

d-axis stator voltage (V)

1.5 1 0.5 0 -0.5 -1

0

0.5

1

1.5 (a)

600 400 200 0 -200 0

0.5

1 time (s)

1.5 (c)

380

0

0.5

1

1.5 (b)

1

1.5 (d)

time (s) q-axis rotor voltage (V)

d-axis rotor voltage (V)

381

380.5

time (s)

-400

382

381.5

600 400 200 0 -200

0

0.5 time (s)

0.2

NABC AFBC

1

0

-1

tracking error of phisq (Wb)

NABC AFBC

0.1 0

-0.1

0

0.5

1 time (s)

1.5 (a)

0.2 NABC AFBC

0.1 0 -0.1 -0.2

tracking error of phisd (Wb)

2

-0.2

0

0.5

1

1.5 (b)

time (s) tracking error of phird (Wb)

speed tracking error (rd/s)

Fig.9. AFBC scheme: (a) Stator voltage u sd . (b) Stator voltage u sq . (c) Rotor voltage urd . (d) Rotor voltage u rq .

1 NAFC AFBC

0.5 0

-0.5

0

0.5

1 time (s)

1.5 (c)

-1

0

0.5

1 time (s)

Fig.10. Tracking performances for both controllers

1.5 (d)

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