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SMALL SPEED ASYMPTOTIC STABILITY STUDY OF AN INDUCTION MOTOR SENSORLESS SPEED CONTROL SYSTEM WITH EXTENDED
GOPINATH OBSERVER
T. Pana and O. Stoicuta Electrical Drives and Robot Department, Technical University of Cluj-Napoca, Str. Constantin Daicoviciu No.15,
RO-400020 Cluj-Napoca, Romania, E-mail: [email protected] Control, Applied Informatics and Computers Department, University of Petrosani, Str. Universitatii No.20,
RO-332006 Petrosani, Romania, E-mail: [email protected]
KEYWORDS Extended Gopinath observer; induction motor, vector control system, direct rotor flux orientation method, asymptotic stability, sensorless. ABSTRACT
In this paper presents synthesis extended Gopinath observer (EGO) and analyzes the asymptotic stability of a vector control system for a squirrel-cage induction motor that contains in its loop an EGO. The studied control system is based on the direct rotor flux orientation method (DFCO) and the stability study is based upon the linearization theorem around the equilibrium points of the control system, emphasizing the estimated variation domain of the rotor resistance for which the control system remains asymptotically stable when the prescribed speed of the control system is close to zero. The stability study is made in continual case. The mathematical model of the vector regulating system is made using a value
e ed qλ λ− linked to stator current. INTRODUCTION
This paper presents a new flux and rotor speed observer called an Extended Gopinath observer (EGO). The design of the EGO observer is done based on an adaptive mechanism using the notion of Popov hyperstability (Popov 1973). Thus, this type of observer is included in the estimation methods based on an adaptation mechanism, along with the Extended Luenberger Observer (ELO) proposed by Kubota (Kubota et al. 1990) and the Model Adaptive System (MRAS) observer proposed by Schauder (Schauder 1992). In the second part of the paper a study is made about the asymptotic stability of the whole speed control system which contains in his loop an EGO estimator. THE GOPINATH EXTENDED OBSERVER
The EGO observer in composed of Gopinath rotor flux observer and adaptation mechanism used for the rotor speed estimation. The equations that define the rotor flux Gopinath observer are (Pana 2001):
* * *11 12 11
* *21 22
ˆ
ˆ ˆ
s s sr
s s sr r
d i a i a b udtd d da i a g i idt dt dt
ψ
ψ ψ
⎧ = ⋅ + ⋅ + ⋅⎪⎪⎨
⎡ ⎤⎪ = ⋅ + ⋅ + ⋅ −⎢ ⎥⎪ ⎣ ⎦⎩
(1)
where:
**11 * * * *
1 1
s r
aT T
σσ σ
⎛ ⎞−= − +⎜ ⎟⋅ ⋅⎝ ⎠
; * * *12 13 14= − ⋅ ⋅ ⋅ rpa a j a z ω ;
**13 * * * *
m
s r r
La
L L T σ=
⋅ ⋅ ⋅;
**14 * * *
m
s r
La
L L σ=
⋅ ⋅; * *
21 31a a= ;*
*31 *
m
r
La
T=
* *22 33= + ⋅ ⋅ rpa a j z ω ; *
33 *
1
r
aT
= − ; *11 * *
1
s
bL σ
=⋅
;*
**s
ss
LT
R= ;
**
*r
rr
LT
R= ;
( )2**
* *1 m
s r
L
L Lσ = −
⋅.
In the above relations (1) marked with “*” the identified electrical sizes of the induction engine. The block diagram of the EGO is presented in figure 1
Figure 1: The Principle Schematic of the EGO Estimator
The essential element that determines the flux observer’s stability, and also his lack of sensibility to the engine’s parameters variation, is a g gate, which is a complex number of the form: a bg g j g= + ⋅ . In order to design this type of estimator we need to position the estimator’s poles in the left Nyquist plane so that the estimator’s stability is asured. The expresions
ag and bg after the pole positioning are (Pana 2001):
Proceedings 23rd European Conference on Modelling andSimulation ©ECMS Javier Otamendi, Andrzej Bargiela,José Luis Montes, Luis Miguel Doncel Pedrera (Editors)ISBN: 978-0-9553018-8-9 / ISBN: 978-0-9553018-9-6 (CD)
( ) ( )
( ) ( )
*33
* 2 2*1433
*33
2 *2* 1433
1 1
1
⎧ ⎡ ⎤⋅ ⋅ − ⋅⎪ ⎢ ⎥= ⋅ −⎪ ⎢ ⎥⋅ +⎪ ⎢ ⎥⎣ ⎦⎨
⎪ ⋅ ⋅ + ⋅⎪ = ⋅⎪ ⋅ +⎩
r pa
r p
r pb
r p
z ag
a z a
z ag
az a
ω β α
ω
ω α β
ω
(2)
The optimum position of the poles, the α and β values respectively, are obtained by the minimization of the induction motor’s rotor resistance variation over the stability of the flux observer. We get the following (Pana 2001):
( ) ( )2 2*
330; ; 0= = ⋅ ⋅ + >r pk z a kβ α ω (3)
In these conditions the Gopinath rotor flux observer is completely determined. Next, in order to determine the adaptation mechanism used to estimate the rotor speed, we will consider as a reference model the „stator curents - rotor fluxes” model of the induction engine and as an ajustable model, the model of the Gopinath rotor flux observer. The equations mentioned above written under the input-state-output canonic form are: • Reference model:
d x A x B udt
dy C xdt
⎧⎪⎪⎨⎪⎪⎩
= ⋅ + ⋅
= ⋅ (4)
• Ajustable model:
( )1ˆ ˆ ˆ
ˆ ˆ
xd x A x A B u G y ydt
dy C xdt
⎧ ⋅⎪⎪⎨⎪⎪⎩
= ⋅ + + ⋅ + ⋅ −
= ⋅ (5)
where:
11 12
21 22
a aA
a a⎡ ⎤
= ⎢ ⎥⎣ ⎦
;*12*22
00
aA
a⎡ ⎤
= ⎢ ⎥⎣ ⎦
;*11
1 *21
00
aA
a⎡ ⎤
= ⎢ ⎥⎣ ⎦
;0
Gg⎡ ⎤
= ⎢ ⎥⎣ ⎦
;
s
r
ix
ψ⎡ ⎤
= ⎢ ⎥⎣ ⎦
; s
r
ix
ψ
⎡ ⎤⎢ ⎥=⎢ ⎥⎣ ⎦
; su u= ; 11
0b
B⎡ ⎤
= ⎢ ⎥⎣ ⎦
; [ ]1 0C = .
In the above relations we marked with „~” the Gopinath estimator’s matrices which are dependent upon the rotor speed, which in turn needs to be estimated based on the adaptation mechanism. Next, in order to determine the expresion that defines the adaptation mechanism we will asume that the identified electric sizes are identical with the real electric sizes of the induction engine. In other words: * ; , 1, 2ij ija a i j= = and *
11 11b b=
In order to build the adaptive mechanism, for start we will calculate the estimation error given by the difference:
ˆe x xx = − (6) Derivation the relation (6) in relation with time and by using the relations (4) and (5) the relation (6) becomes:
( )1d dA A x A x G Cdt dt
e ex x= − ⋅ − ⋅ − ⋅ ⋅ (7)
If the determinant, ( )2det 0I G C+ ⋅ ≠ , then it exists a
unique inverse matrix ( ) 1
2M I G C−
= + ⋅ so that the
expression (7) can be written like this:
( ) ( )1 1d M A A M A A A xdt
e ex x= ⋅ − ⋅ + ⋅ − − ⋅ (8)
Equation (8) describes a linear system defined by the term ( )1M A A ex⋅ − ⋅ in inverse connection with a non
linear system defined by the term ( )yeΦ which receives
at input the error y xe C e= ⋅ between the models and has at the output the term:
( )1M A A A xρ = − ⋅ − − ⋅ (9)
The block diagram of the system that describes the dynamic evolution of the error between the state of the reference model and the state of the adjustable model is presented in figure 2:
Figure 2: The block diagram of the system that describes the dynamic evolution of the error between the state of
teh reference model and the state of the adjustable model As one may notice, this problem is frequently treated in the literature of the non-linear systems, being exactly the configuration of the Lure problem, and of one of the problems treated by Popov. Considering, according to the Popov terminology, the non-linear block described by ( )eyΦ the integral input-
output index associated to it is:
( ) ( ) ( )1
0 1 0, Re
t Tyt t e t t dtη ρ⎡ ⎤
⎢ ⎥⎣ ⎦= ⋅∫ (10)
In order for block to be hyper-stable a necessary condition is:
( ) ( ) ( ) ( )1 21 0
0, Re 0t T
yt e t t dtη ρ γ⎡ ⎤⎢ ⎥⎣ ⎦
= ⋅ ≥ −∫ (11)
for any input-output combination and where ( )0γ is a positive constant. In the above relation we marked with T
ye the following expression
0Tyye e⎡ ⎤= ⎣ ⎦ (12)
Obtained in order to keep the compatibility between the input and output dimensions, and ye represents the conjugate of the complex variable ye . Under these circumstances, using the relation (9) the expression (11) becomes:
( ) ( ) ( ) ( )1 21 10
00, Ret T
y M A A A xt e t dtη γ⎡ ⎤⋅ − − ⋅⎢ ⎥⎣ ⎦=− ⋅ ≥−∫ (13)
Next we assume that the error ( )1M A A A⋅ − − is
determined only by the rotor speed of the induction machine. In this case we may write:
( ) ( )1⋅ − − = − ⋅rr erM A A A Aω ω (14)
where: ( )14
14
0
0 1
− ⋅ ⋅⎡ ⎤= ⎢ ⎥
⋅ ⋅ + ⋅⎢ ⎥⎣ ⎦
p
erp
j a z
j z a gA .
For any positive derivable f function we can demonstrate the following inequality:
( )1 211 0
02
t KdfK f dt fdt
⎛ ⎞ ≥⎜ ⎟⎝ ⎠
⋅ ⋅ − ⋅∫ (15)
On the other hand, using the relation (14), the expression (13) becomes:
( ) ( ) ( ){ } ( )1 21 0
00, Re ⎡ ⎤⋅ ⋅⎣ ⎦=− ⋅ − ≥−∫t T
ry er rxt e t A dtη ω ω γ (16)
By combining the relations (15) and (16) we can write the following relations:
( ) 1ˆRe
⎧⎪⎨⎪⎩
= −
− ⋅ ⋅ = ⋅
rr
Ty er
fdfe A x Kdt
ω ω (17)
Because 1K is a constant and then, in case of a slower rω parameter variation related to the adaptive law, we can write:
( )Re ˆ⋅= ⋅ ⋅∫ Tr i y erk e A x dtω (18)
After replacing the variables that define the above expression (18) and taking into account the arbitrary nature of the iK positive constant we obtain:
( )⋅ ⋅ − ⋅= ∫r qr dri yd yqe ek dtψ ψω (19)
where dsyd dse i i= − and qsyq qse i i= − . Sometimes, insted of the adaptation law (19) we can use the following form:
( ) ( )⋅ − ⋅ + ⋅ ⋅ − ⋅= ∫r qr dr qr drR yd yq i yd yqe e e eK k dtψ ψ ψ ψω (20)
From the above relation we ca observe that a new proportional component apears from the desire to have 2 coefficients that can control the speed estimation dynamics. This fact isn’t always necesary because we can obtain very good results by using only expresion (19). Thus expresion (20) represents the general formula of the adaptation mechanism where RK represents the proportionality constant and =i R RK K T ; where RT represents the integration time of the proportional-integral regulator that defines the adaptation mechanism. THE MATHEMATICAL DESCRIPTION OF THE VECTOR CONTROL SYSTEM
The block diagram of the control system of the mechanical angular speed rω of the induction engine with a discreet orientation after the rotor flux (DFOC) is presented in figure 3.
Figure 3: The block diagram of the DFCO vectorial control system which contains an EGO loop.
In figure 3 we marked with B2 the control block of the speed control system with direct orientation after the rotor flux (DFCO) and with B1 the extended Gopinath estimator block (EGO). In order to mathematically describe the DFOC control system the following hypotheses have been considered: • The static frequency converter (CSF) is assumed to
contain a tension inverter. • The static frequency converter is considered ideal so
that the vector of the command measures is considered to be the entry vector of the induction motor.
• The dynamic measure transducers are considered ideal.
• The mathematical model of the vector control system will be written in an e ed qλ λ− axis reference bounded to the stator current.
Some of the equations that define the vector control system are given by the elements which compose the field orientation block and consist of: • Stator tensions decoupling block (C1Us):
2
11 13 3111
11 14 3111
1
1
r
ds ds rr r
r r
r r r
qsrr p qs
r
ds qsr rqs qs p r p ds
r
iu b v a a z i
b
i iu b v a z a z i
b
λ λ
λλ
λ λλ λ λ
ψ ωψ
ω ψ ωψ
∗ ∗ ∗ ∗ ∗∗
∗ ∗ ∗ ∗ ∗∗
⎧ ⎡ ⎤= ⋅ ⋅ − ⋅ − ⋅ − ⋅ ⋅⎪ ⎢ ⎥
⎪ ⎢ ⎥⎣ ⎦⎨
⋅⎡ ⎤⎪= ⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅ + ⋅ ⋅⎢ ⎥⎪
⎢ ⎥⎣ ⎦⎩
(21)
• PI flux regulator (PI_ψ) defined by the Kψ
proportionality constant and the Tψ integration time:
( )
6
6r
r r
ds r r
dxdt
Ki x K
Tψ
λ ψψ
ψ ψ
ψ ψ
∗
∗ ∗
⎧ = −⎪⎪⎨⎪ = ⋅ + ⋅ −⎪⎩
(22)
• couple PI regulator (PI_Me) defined by the MK
proportionality constant and the MT integration time:
( )
7
7r
e e
Mqs M e e
M
dxM M
dtK
i x K M MTλ
∗
∗ ∗
⎧ = −⎪⎪⎨⎪ = ⋅ + ⋅ −⎪⎩
(23)
• mechanical angular speed PI regulator (PI_W)
defined by the Kω proportionality constant and the Tω integration time:
( )
8
8
rr
re r
dxdt
KM x K
Tω
ωω
ω ω
ω ω
∗
∗ ∗
⎧ = −⎪⎪⎨⎪ = ⋅ + ⋅ −⎪⎩
(24)
• current PI regulator (PI_I) defined by the iK proportionality constant and the iT integration time:
( )
9
9
r r
r r r
ds ds
ids i ds ds
i
dxi i
dtK
v x K i iT
λ λ
λ λ λ
∗
∗ ∗
⎧ = −⎪⎪⎨⎪ = ⋅ + ⋅ −⎪⎩
(25)
( )
10
10
r r
r r r
qs qs
iqs i qs qs
i
dxi i
dtK
v x K i iT
λ λ
λ λ λ
∗
∗ ∗
⎧ = −⎪⎪⎨⎪ = ⋅ + ⋅ −⎪⎩
(26)
• Flux analyzer (AF):
2 2
sin ; cos
⎧ = +⎪⎪⎨⎪ = =⎪⎩
dr qrr
qr drr r
r r
ψ ψ ψ
ψ ψλ λ
ψ ψ
(27)
• The calculate of the couple block (C1Me):
= ⋅ ⋅
re a r qsM K i λψ (28) The other equations that define the mathematic model of the speed’s vector control system are: • The equations that define the stator currents – rotor
fluxes mathematical model of the induction engine; 4 equations defined based on the first relation in the canonic system (4) to which we can add the induction machine’s motion equation defined by the following expression:
1 2 3⎡ ⎤= ⋅ ⋅ − ⋅ − ⋅ − ⋅⎣ ⎦r m dr qs qr ds m r m r
d K i i K K Mdtω ψ ψ ω (29)
where: 1
32
= ⋅ ⋅p mm
r
z LK
J L;
2mFKJ
= ; 3
1=mK
J.
• The equations that define the extended Gopinath observer defined by the 4 relations that can be written based on system (1) with the equation that defines the speed adaptation mechanism (20). Expression (20) can also be written like below:
( ) ( )( ) ( )
15
15
⎧ = − ⋅ − − ⋅⎪⎪⎨
⎡ ⎤⎪ = ⋅ + ⋅ − ⋅ − − ⋅⎣ ⎦⎪⎩
ds qsqr drds qs
Rr ds qsqr drR ds qs
R
d x i i i idt
Kx K i i i i
T
ψ ψ
ω ψ ψ (30)
All these expressions form a 15 differential equations system with 15 unknown values. In order to offer a
coherent presentation of this differential equations system, we have used the following notations: • The state vector of the control system will be
[ ] 1,15i i
x x=
= (31)
where: 1 = edsx i λ ; 2 eqsx i λ= ; 3 edrx λψ= ; 4 eqrx λψ= ;
5 rx ω= ; 11 edsx i λ= ; 12 eqsx i λ= ; 13 edrx λψ= ; 14 eqrx λψ= .
• The input vector of the of the control system will be
[ ]1 2 3Tu u u u= (32)
where: *
1 ru ψ= ; *2 ru ω= ; 3 ru M= .
Under these circumstances the 15 differential equations system that define the mathematical model of the vector regulating system can be written as follows
( , )d x f x udt
= (33)
where: [ ] 1,15
( , ) ( , )i if x u f x u
== and the ( , )i if f x u=
functions are:
1 11 1 2 13 3 14 5 4 11( , ) = ⋅ + ⋅ + ⋅ + ⋅ ⋅ ⋅ + ⋅e p af x u a x x a x a z x x b uω (34)
2 1 11 2 14 5 3 13 4 11( , ) = − ⋅ + ⋅ − ⋅ ⋅ ⋅ + ⋅ + ⋅e p bf x u x a x a z x x a x b uω (35)
( )3 31 1 33 3 5 4( , ) = ⋅ + ⋅ + − ⋅ ⋅e pf x u a x a x z x xω (36)
( )4 31 2 5 3 33 4( , ) = ⋅ − − ⋅ ⋅ + ⋅e pf x u a x z x x a xω (37)
[ ]1 2 35 3 2 4 1 5 3( , ) = ⋅ ⋅ − ⋅ − ⋅ − ⋅m m mf x u K x x x x K x K u (38)
6 1 1( , ) = −f x u u g (39)
( )7 8 2 2 3( , ) = ⋅ + ⋅ − − ⋅aK
f x u x K u g K gTω
ωω
(40)
8 2 2( , ) = −f x u u g (41)
( ) 49 6 1 1
1
( , ) = ⋅ + ⋅ − −K gf x u x K u gT gψ
ψψ
(42)
310 7 7
1
( , ) ( , )= ⋅ + ⋅ −MM
M
gKf x u x K f x u
T g (43)
* * * *
11 11 1 2 13 13 14 2 14 11( , ) = ⋅ + ⋅ + ⋅ + ⋅ ⋅ ⋅ + ⋅e p af x u a x x a x a z g x b uω (44)
* * * *12 1 11 2 14 2 13 13 14 11( , ) = − ⋅ + ⋅ − ⋅ ⋅ ⋅ + ⋅ + ⋅e p bf x u x a x a z g x a x b uω (45)
* *13 1 13 2 14 11 11
3 1 4 2 1 2
( , )( , ) ( , )
= ⋅ + ⋅ − ⋅ ⋅ + ⋅ ⋅ +
+ ⋅ + ⋅ + ⋅ − ⋅a a b b
a b
f x u b x b x g b u g b ub x b x g f x u g f x u
(46)
* *
14 2 13 1 14 11 11
4 1 3 2 1 2
( , )( , ) ( , )
= − ⋅ + ⋅ − ⋅ ⋅ − ⋅ ⋅ −
− ⋅ + ⋅ + ⋅ + ⋅b a a b
b a
f x u b x b x g b u g b ub x b x g f x u g f x u
(47)
( ) ( )15 1 11 14 2 12 13( , ) = − ⋅ − − ⋅f x u x x x x x x (48)
where:
2 21 13 14= +g x x ;
( ) ( )2 15 1 11 14 2 12 13= ⋅ + ⋅ − ⋅ − − ⋅⎡ ⎤⎣ ⎦R
RR
Kg x K x x x x x x
T;
3 2 13 1 14= ⋅ − ⋅g x x x x ; 4 1 13 2 14= ⋅ + ⋅g x x x x ; * * *
1 33 13 14 2= − ⋅ − ⋅ ⋅ ⋅a b pb a g a g a z g ;
( )* *2 13 14 21= ⋅ − + ⋅ ⋅ ⋅ +b a p eb g a g a z g ω ;
* *3 31 11= − ⋅ − ⋅a b eb a g a g ω ; *
4 11= ⋅ − ⋅b a eb g a g ω ;
* 22 31
13e p
xz g a
xω = ⋅ + ⋅ ;
( ) ( )22 *2 33= ⋅ ⋅ +pk g z aα ;
( ) ( )*33
* 22 *14 2 33
1 1⎡ ⎤⋅− ⎢ ⎥= ⋅ +⎢ ⎥⋅ +⎣ ⎦
a
p
ag
a g z a
α;
( ) ( )2
2 *2 *142 33
1⋅ ⋅= ⋅
⋅ +
pb
p
g zg
ag z a
α;
9 9 ( , )= ⋅ + ⋅ia i
i
Kv x K f x u
T; 10 10 ( , )= ⋅ + ⋅i
b ii
Kv x K f x u
T;
23 3
1 13 1 31 2311
∗ ∗= ⋅ − ⋅ − ⋅ ⋅pg g
h a g a z ggg
;
3 4 42 14 2 1 31 23
11
∗ ∗ ⋅= ⋅ ⋅ ⋅ + ⋅ + ⋅ ⋅p p
g g gh a z g g a z g
gg;
( ) ( )13 11 1 14 11 211 1
1 ∗ ∗∗
⎡ ⎤= ⋅ ⋅ ⋅ − − ⋅ ⋅ +⎣ ⎦⋅a a bu x b v h x b v hb g
;
( ) ( )14 11 1 13 11 211 1
1 ∗ ∗∗
⎡ ⎤= ⋅ ⋅ ⋅ − + ⋅ ⋅ +⎣ ⎦⋅b a bu x b v h x b v hb g
.
111 1
s r
aT T
σσ σ
⎛ ⎞−= − +⎜ ⎟⋅ ⋅⎝ ⎠
; 13m
s r r
LaL L T σ
=⋅ ⋅ ⋅
;
14m
s r
LaL L σ
=⋅ ⋅
; 31m
r
LaT
= ; 331
r
aT
= − ; 111
s
bL σ
=⋅
;
ss
s
LTR
= ; rr
r
LTR
= ; 2
1 m
s r
LL L
σ = −⋅
; 32
ma p
r
LK z
L
∗
∗= ⋅ ⋅ .
Under these circumstances the mathematical model of the speed vector control system is fully determined as being defined by the non-linear differential equations system given by (33) whose initial condition is ( )0 0x = .
THE ASYMPTOTIC STABILITY STUDY OF THE CONTROL SYSTEM
In order to realize the analysis of the asymptotic stability an induction with the following parameters has been considered: • Electrical parameters
[ ]0.371sR = Ω ; [ ]0.415rR = Ω ; [ ]0.08694 HsL = ;
[ ]0.08762 HrL = ; [ ]0.08422 HmL = . • Mechanical parameters
2pz = ; 20.15 kg mJ ⎡ ⎤= ⋅⎣ ⎦ ; N m s0.005rad
F ⋅ ⋅⎡ ⎤= ⎢ ⎥⎣ ⎦
On the other hand, following the automated regulators’ tuning within the speed control system the following constants have been obtained:
501.3834Kψ = ;2374.7
KT ψψ = ; 5.9881iK = ;
754.4176i
iKT = ;
10.1988MK = ; 1020
MM
KT = ; 10Kω = ; 350K
T ωω = ;
6RK = ; 4000
RR
KT = .
In the relations above, RK represents the proportionality constant and RT represents the integration time of the PI regulator from the speed estimator designed based on the Popov hyperstability. Under these circumstances by imposing the entry vector of the control system to be of the following type
[ ]*1 0.69 Wbru ψ= = ; *
2rad
30 sm rm mu n πω ⎡ ⎤= = ⋅ ⎢ ⎥⎣ ⎦;
[ ]3 93.269 N mr Nu M M= = = ⋅ (49) where [ ]100 rpmmn m= ⋅ with 0, 15m = and the proportionality coefficient 1.3=k by solving the non-linear system
( , ) 0f x u = (50)
Using the Newton method having as the start point the vector
[ ]*1 2 10 0 0 0 0 0 0 0 0 0 0 0 T
mx u u u= (51) The equilibrium point m mx b= results,
where [ ] 1,15m mi ib b
== .
From those stated above by the linearization of the system (33) around the equilibrium point m mx b=
obtained for an [ ]*1 2 3
Tmu u u u= entry vector
defined by (49) we get:
( ) ( ) ( )L Lx t A x t B u t•
Δ = ⋅Δ + ⋅Δ (52) where ,L LA B matrixes are
*
1,15; 1,15
( , ) ;iL m
j i j
fA b u
x= =
⎡ ⎤∂= ⎢ ⎥
∂⎢ ⎥⎣ ⎦
*
1,15; 1,3
( , ) .iL m
k i k
fB b u
u= =
⎡ ⎤∂= ⎢ ⎥∂⎣ ⎦
Next, in order to study the asymptotic stability of the equilibrium points m mx b= , the eigenvalues of the AL matrix will be analyzed, so that if they have a strictly negative real part the m mx b= equilibrium point is asymptotically stable for the linerized system (52). Under these circumstances according to the linearization theorem (Voicu 1986) in a vicinity of the equilibrium point m mx b= the non-linear system (33) is asymptotically stable. As the eigenvalues of the LA matrix are presented within Figure 4, it results that the equilibrium points m mx b= of the linerized system (52) are asymptotically stable and according to the linearization theorem the equilibrium points m mx b= are asymptotically stable in certain vicinity for the non-linear system (33).
Figure 4. The eigenvalues of the matrix ;L r NA M M= For a more detailed study of the asymptotic stability, next we will present the LA matrix eigenvalues under identical testing conditions with those above with the difference that [ ]3 0 N mr Nu M M= = = ⋅ . As with the previous case from Figure 5 one may notice that the m mx b= equilibrium points of the linerized system are asymptotically stable resulting that in certain vicinity these points are asymptotically stable for the non-linear system defined by (33).
Figure 5. The eigenvalues of the matrix ; 0L rA M = Next, the influence of the estimated rotor resistance will be emphasized, when studying asymptotic stability of the equilibrium point 0 0x b= of the non-linear system (33) in a certain vicinity of this point. The equilibrium point is obtained for an entry vector like
[ ]1 20 3Tu u u u= defined by (49); namely for a speed
of [ ]0 0 rpmn = . Based on this study we can obtain information related to the range of variation of the estimated rotor resistance for which the non-linear system (33) remains asymptotically stable around the 0 0x b= equilibrium point. As the eigenvalues of the AL matrix are presented in figure 6, it results that the non-linear system (33) around the equilibrium point 0 0x b= is asymptotically stable for an estimated rotor resistance in range of
{ }* *1 ;0.6 2.28r r r rD R R R R= ⋅ ≤ ≤ ⋅ ,becoming asymptotically
unstable for an estimated rotor resistance in range of { }* *
2 ;2.28 0.6r r r rD R R R R= ⋅ < < ⋅ .
Figure 6. LA Eigenvalues by rotor resistance for
[ ]0 0 ; r Nn rpm M M= =
Next the influence of the estimated rotor resistance to the asymptotic stability of the non-linear system (33) will be presented, when the entry vector is [ ]1 20 3
Tu u u u= defined by (49) with the difference that [ ]3 0 N mr Nu M M= = = ⋅ . The eigenvalues of the AL
matrix are presented in Figure 7.
Figure 7. LA eigenvalues by rotor resistance for
[ ]0 0 ; 0rn rpm M= = As the eigenvalues of the AL matrix are presented in figure 7, it results that the non-linear system (33) around the equilibrium point 0 0x b= is asymptotically stable for an estimated rotor resistance in range of
{ }* *1 ;0.1 2.289r r r rD R R R R= ⋅ ≤ ≤ ⋅ ,becoming asymptotically
unstable for an estimated rotor resistance in range of { }* *
2 ;2.289 0.1r r r rD R R R R= ⋅ < < ⋅ . On the other hand in case we perform the digitization of the linear system (22) we get:
( ) ( ) ( )1 L Lx k F x k H u k+ = ⋅ + ⋅ (53) where: the ,L LF H matrixes are obtained from the
,L LA B matrixes using one of the two digitization types: • Simplified digitization:
15L LF I A T= + ⋅ ; L LH B T= ⋅ (54)
• Complete digitization:
2
215 2L L L
TF I A T A= + ⋅ + ⋅ ; 2
2L L L LTH B T A B= ⋅ + ⋅ ⋅ (55)
with the sampling time T. Proceeding in a similar manner the eigenvalues of the LF matrix in case the entry vector is defined by (49) and the
LF matrix is obtained by using simplified digitization using a [ ]53.3 secT μ= sampling time are graphically presented in Figure 8.
Figure 8: The eigenvalues of the ; 0L rF M = matrix when using simplified digitization
From the figure above one may notice that the m mx b= equilibrium points are asymptotically stable for the discrete system (53) in case the LF matrix is obtained through using the simplified digitization method. This conclusion also remains valid when the LF matrix is obtained through the complete digitization method both where the resistant couple is present or absent. Next we will present the influence of the estimated rotor resistance over the asymptotic stability of the discrete non-linear system when the entry vector is
[ ]1 20 3Tu u u u= and is defined by (49).
This study was conducted for a sampling time of [ ]53.3 secT μ= in case of simplified digitization.
The study helped us with the information that the 0 0x b= equilibrium point remains asymptotically stable for the linear (53) system for a variation of identified rotor resistance ranging from { }* *
1 ;0.71 2.272r r r rD R R R R= ⋅ ≤ ≤ ⋅ .
The 0 0x b= equilibrium point becomes asymptotically unstable when the variation of the identified rotor resistance varies from { }* *
2 ;2.272 0.71r r r rD R R R R= ⋅ < < ⋅
Figure 9: The eigenvalues of the ; 0L rF M = matrix when using complete digitization
When the study of the influence of identified rotor resistance variation is done for an entry vector of the
[ ]1 20 3Tu u u u= type defined by (49) with the
difference that [ ]3 0 N mr Nu M M= = = ⋅ then the stability domain for which the 0 0x b= equilibrium point
is asymptotically stable is { }* *1 ;0.18 2.278r r r rD R R R R= ⋅ ≤ ≤ ⋅
and the instability domain is { }* *2 ;2.278 0.18r r r rD R R R R= ⋅ < < ⋅ .
On the other hand the domains that define the upper and lower limits of variation of identified rotor resistance for which the 0 0x b= equilibrium point remains asymptotically stable for the (52) discrete linear system diminishes with the increase of the sampling time. The results presented above have been partially proved by experiment, due to the limitations introduced by the inverter. Next, the performances of the extended Gopinath estimator are presented in a variety of functional conditions. Thus the image below will present the graphics for the real and estimated rotors fluxes and also the graphics for the imposed speed, real speed and the estimated speed for small, medium and large imposed speeds.
Figure 10: drψ real flux compared to the drψ estimated
flux: * rad530 srπω ⎡ ⎤= ⋅ ⎢ ⎥⎣ ⎦
; 0rM = .
Figure 11: rω real speed compared to the rω estimated
speed and reference speed : * rad530 srπω ⎡ ⎤= ⋅ ⎢ ⎥⎣ ⎦
; 0rM = .
Figure 12: drψ real flux compared to the drψ estimated
flux: * rad150030 srπω ⎡ ⎤= ⋅ ⎢ ⎥⎣ ⎦
; 0rM = .
Figure 13: rω real speed compared to the rω estimated
speed and reference speed: * rad150030 srπω ⎡ ⎤= ⋅ ⎢ ⎥⎣ ⎦
; 0rM = .
Figure 14: drψ real flux compared to the drψ estimated
flux: * rad300030 srπω ⎡ ⎤= ⋅ ⎢ ⎥⎣ ⎦
; 0rM = .
Figure 15: rω real speed compared to the rω estimated
speed and reference speed: * rad300030 srπω ⎡ ⎤= ⋅ ⎢ ⎥⎣ ⎦
; 0rM = .
On the other hand we will present in the following image the effect of the rotors resistance emphasizing the dynamic performances of the EGO estimator. Thus we will present the graphic between the real and estimated rotor fluxes and also the graphic between the imposed speed, real speed and estimated speed for low, medium and high imposed speeds when the rotor resistance in at the lower limit and at the upper limit respectively, limits that separate the stability from the asymptotic instability.
Figure 16: drψ real flux compared to the drψ estimated
flux: * rad530 srπω ⎡ ⎤= ⋅ ⎢ ⎥⎣ ⎦
; 0rM = ; * 1.7r rR R= ⋅ .
Figure 17: rω real speed compared to the rω estimated
speed and reference speed: 0rM = . * 1.7r rR R= ⋅ .
Figure 18: drψ real flux compared to the drψ estimated
flux: * rad150030 srπω ⎡ ⎤= ⋅ ⎢ ⎥⎣ ⎦
; 0rM = . * 1.7r rR R= ⋅ .
Figure 19: rω real speed compared to the rω estimated
speed and reference speed: 0rM = ; * 1.7r rR R= ⋅ .
Figure 20: drψ real flux compared to the drψ estimated
flux: 0rM = ; * 1.7r rR R= ⋅ .
Figure 21: rω real speed compared to the rω estimated
speed and reference speed: * rad300030 srπω ⎡ ⎤= ⋅ ⎢ ⎥⎣ ⎦
; * 1.7r rR R= ⋅ .
CONCLUSIONS
This paper presents analytically in a single form the mathematical model of the speed vector-controlled system with an extended Gopinath estimator, suitable for stability analysis. The mathematical model is written in a e ed qλ λ− orientation value linked to the stator current vector. This mathematical model allows for both the internal asymptotic study as well as the external stability of the control system. The study helped us to conclude that the analyzed control system is asymptotically stable for the imposed speed range ( ) [ ]0 1500 rpm… both, when a resistant couple is lacking or present as r NM M= . This conclusion is valid in both the continual and discrete cases where the sampling time is [ ]53.3 secT μ= . When the estimator from within the control system has not the same order with the motor, the estimator adds eigenvalues and affects the eigenvalues of the regulating system that does not contain in its loop an extended Gopinath estimator. The Gopinath flux estimator, whose gate matrix is calculated with the (2) relations, ensures the adjustment system very good dynamic performances that gives us the possibility to assert, that such an estimator can be successfully used in industry. In this paper we determined the upper and lower variation limits for the identified rotor resistance for which the
0 0x b= equilibrium point remains asymptotically in both the discrete and continual cases. The following information emphasized within Table 1 for the continual case ant Table 2 for the discrete case have been obtained.
Table 1
r NM M= 0rM = LI 0.6 rR⋅ 0.1 rR⋅ LS 2.28 rR⋅ 2.289 rR⋅
Table 2
r NM M= 0rM =
LI 0.71 rR⋅ 0.18 rR⋅ LS 2.272 rR⋅ 2.278 rR⋅
where, LI is the lower limit and LS the upper limit of variation of identified rotor resistance. From the tables above one may notice that the upper limit for the discrete case decreases compared to the upper limit for the continual case while the lower limit for the discrete case increases compared to the lower limit of the continual case. This conclusion remains valid in both cases whet the resistant couple is either lacking or present.
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AUTHOR BIOGRAPHIES
TEODOR PANA was born in Cluj – Napoca (05.09.1954), Romania, where he graduated the Faculty of Electrical Engineering, Technical University of Cluj – Napoca. He is full professor at the Department of Electrical Drives and Robots of the
Technical University of Cluj – Napoca. He got his PhD degree in Electrical Engineering in 1995. His major fields of interest are: Sensorless Vector Control of Electrical Drives, Real-Time Programming Applications”. Contact address: [email protected]
OLIMPIU STOICUTA was born in Petrosani, (11.10.1977), Romania, where he attended Faculty of Mechanical and Electrical Engineering in University of Petrosani, specialization Automation and Applied Informatics. Since 2001
is member of the Department of Automation, Applied Informatics and Computer of the University of Petrosani, Romania. Since 2003 is PhD student in Electrical Engineering on Technical University of Cluj – Napoca being coordinated by Mr. PhD engineering professor Teodor Pana. Is interested of the following areas: “Nonlinear control systems stability” and “Systems identification”. Can be contacted at the following e-mail address: [email protected]
