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12
Olimpiu STOICUTA, Nadia STOICUTA
University of Petrosani, Romania, Teodor PANA
Technical University of Cluj Napoca, Romania
SMALL SPEED ASYMPTOTIC STABILITY STUDY OF A SPEED VECTOR CONTROL SYSTEM FOR AN INDUCTION MOTOR THAT CONTAINS IN ITS LOOP A GOPINATH OBSERVER
In this paper we analyze the asymptotic stability of a vector control system for an induction motor with short-circuited rotor that contains in its loop a Gopinath observer. The studied control system is based on the direct rotor flux orientation method (DFOC) 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 both the continual and discrete cases. The mathematical model of the vector regulating system is made using a value dXe-qXe linked to stator current. In order to mathematically describe the DFOC control system we will consider the following hypotheses: 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 system and axis transformation blocks are considered dynamically ideal; the mathematical model of the vector control system will be written in an dXe-qXe axis reference bounded to the stator current.
В статье анализируется асимптотическая стабильность векторной системы управления для асинхронного двигателя с короткозамкнутым ротором, которая содержит наблюдатель Гопина. Рассмотренная система управления базируется на прямом методе ориентации потока ротора (DFOC), а исследование стабильности основано на теореме линеаризации возле точек равновесия системы управления. При этом акцент сделан на предполагаемой области изменения сопротивления ротора, для которой система управления остается асимптотически устойчивой, когда предписанная скорость системы управления близка к нулю. Исследование стабильности проводилось в непрерывных и в дискретных случаях. Математическая модель векторной системы регулирования создана с использованием значений dke-qXe, связанных с потоком статора. Для математического описания системы управления DFOC, исходим из следующих предположений: статический преобразователь частоты (CSF), как предполагается, содержит инвертор напряженности; статический преобразователь частоты считают идеальным так, чтобы вектор управления был вектором входа асинхронного двигателя; динамические преобразователи измеренных значений считаются идеальными; система и блоки преобразования координат считаются динамически идеальными; математическая модель векторной системы управления будет создана в осях dke-qXe, связанных с потоком статора.
1. Introduction. This paper approaches a difficult problem within vector driving systems for induction motors namely the asymptotic stability study in a Lyapunov manner. The difficulty is brought by the mathematical model of the nonlinear analyzed system that makes the Lyapunov stability analysis methods difficult to apply. The novelty of the paper consists in obtaining the method and the form of the linerized mathematical model on which the analysis of the asymptotic stability is made. The shape and structure of the
mathematical model depends on the actual components within the analyzed control system and the way in which the state values are selected. In this paper we present the analysis of the asymptotic stability of a vector control system for an induction motor with a short-circuited rotor that contains in its loop a Gopinath observer. The analyzed control system is orientated according to the estimated rotor flux by the Gopinath observer.
We have realized the stability analysis for low speeds, emphasizing the influence of the iden-
tified rotor resistance within the stability of the control system. We have obtained the variation range of the identified rotor resistance for which the control system remains asymptotically stable.
2. The Mathematical Description of the Vector Control System. In order to mathematically describe the DFOC control system we will consider the following hypotheses:
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 system and axis transformation blocks are considered dynamically ideal.
The mathematical model of the vector control system will be written in an dA,e-qA,e axis reference bounded to the stator current.
Based on these hypotheses the block diagram of the direct vector control system that contains a Gopinath type rotor flux estimator in
its loop is presented within fig. 1 and the DFOC field orientation block within fig.2.
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 (Ci^s);
• couple PI regulator (PI_Me) defined by the KM proportionality constant and the TM integration time;
• PI flux regulator (PI_y) defined by the Kv proportionality constant and the T^ integration time;
• mechanical angular speed PI regulator (PI_W) defined by the Ka proportionality constant and the Ta integration time;
• current PI regulator (PI_I) defined by the K proportionality constant and the T integration time;
• Flux analyzer (AF).
The other equations that compose the mathematical model of the speed control system are those given by the relations that define the stator currents - rotor fluxes mathematical
Fig.1. Direct Vector Speed Control System
Fig.2. Internal structure of the field orientation bloc (DFOC)
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ISSN 0135-3500. Записки Горного института. Т.181
model of the induction motor as well as the equations that define the mathematical model of the observer for the Gopinath type rotor flux.
All these expressions can be put together as a 12 differential equations system with 12 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
x = [ xi]
where x1 = i
dsX e
7=1,12
X2 = iqsX e • X3 = VdrXe •
X4 - VqrXe ; x5 - ®r ; X11 " VdrXe ; X12 - VqrXe
The input vector of the of the control system will be
u - [u1u2u3]T
where u1 - y* ; u2 - ©*; u3 - Mr.
Under these circumstances the 12 differential equations system that define the mathematical model of the vector regulating system can be written as follows
dx(t) = f (x(t), u (t)): dt
where f (x, u) =[f (x, u)].=—,
f = f (x, u) functions are:
( \
(1)
and the
fi( x, u) = anXj +
Z pX5 + a31
I
2,2 x11 + x12 j
X x2 + a13x3 + a14Zpx5x4 + bnudsXe ;
(
fi( x, u) = -
zpx5 + a31
x
2
I
22 v V x11 + x12 j
X x1 + anx2 - «14zpx5x3 + «13x4 + bnuqsXe
f3(X, u) = a31x1 + a33 x3 + a33
f 4 (X, u)
= a31x2 a31
x
2
I
■ x ;
x11 + x12
i
x 2 + x 2 11 12
x3 + a33x4 •
f5 (x, u) = Km1[x3x2 x4x1] Km2x5 Km3u3 •
f6 (X, u) = u1 - Vx121 + x122 •
K
f7(x, u) = TL x8 + K» (u 2 -x5)-
Kax2^x
-ax2y x11 + x12 •
f8( x, u) = u2- x5;
f9( x, u) = x6 + KJ\u1- Vxn+x122)- x1;
к
f10(x,u) = T^x7 + KM X TM
к»
t
v »
+ K» (u2 -x5) -KaW
x5> Kax2V x11 + x12
f11(X, u) = b1x11 + b2x12 - gab11udsXe +
+ gbb11uqsXe + b3 x1 + b4 x2 + + gaf1(X, u)- gbf2 (x u);
f12(X, u) = -b2x11 + b1 x12 - gabl1udsXe ~
+ gbb11uqsXe + b4 x1 + b3 x2 +
+ gbf1( x, u) + gaf2 (x, u ).
Where:
a11 =
a14 = '
( 1 1-a^
■ + ■
Lm
V Ts a Tr a j
Lm
a13 =-
LsLrTr a
LsLr a
31
Lm
T
33
1 T
Ls a
bu=Lr; Ts = ; Tr ; a =1-
L:
Rr
Lm
LsLr
K = • K = ^• K = A-
m1 2 J L/ m2 J ' m3 J'
3 I*
¡Г _ _ m .
K = — Z„~T ;
a11 =
a14 =
11
1 1-a
■ + -
2 p L
r
* \
VTs*a* Tr*a j
L*
a13 = * * * • LsLrTr a
L m • * * * 1 LsLr a * a31 = * a33
1 ^ * * ? Ls a * T ■■ s L* • * T -r _ L*
* a =1 2* L2 m • f f '' s r
X
x
2
X
X
1
x
2
a — Uzpx5 + (a*3)2 ;
S a
Sb —
1 1 + aa
33
* 2 2 / * \2 a14 ZpX5 + (a33)
1 aZpX5
* 2 2 / * \2 a*4 zpX5 + (a33)
* * * b1 — a33 - Saa13 - Sba14zpX5 +
+ zpx5 + a31
л/ХИ
Xi i X■
12
b2 — SbaU - (1 + Saa14)ZpX5;
b3 — a3l - sXi - Sb X
Л
zpx5 + a31
V2 2 X11 + x
b4 — Sba11 - Sa
zpx5 + a31
12 У
X
2
I 2 2 V xll + x
12 У
1
UdsXe — (b1 l^disXe h1) ; b11
UqsXe — ~7*~ (b1l^sXe - h2) ;
b11 K
VdsÀe — TLX9 + Kf9( X U );
и — * I 2 2"
hl — al 3 л / Xl l + Xu + a
* X2
31
- + zpx5x2 ;
Xi 1 + X
12
, * I 2 2"
h2 — a14 ZpX5\ X11 + X12 +
+a
* Xi X'
1Л2
31
Vx121
+ X
— + zpX5Xl . 12
Where Ls, Lr, Lm are the stator, rotor and mutual inductance, Rs, Rr are the stator and rotor resistance and a is the mutual coefficient of leakage.
In the relations above we noted with «*» the identified measures of the induction motor and k is the proportionality coefficient Gopinath.
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 (1) whose initial condition is x(0)=0.
3. The Asymptotic Stability Study of the Control System. In order to realize the analysis of the asymptotic stability we will consider an induction motor that has the following characteristics:
• electrical parameters
Rs = 0,371[Q]; Rr = 0,415[Q];
Ls = 0,08694[H]; Lr = 0,08762[H];
Lm = 0,08422[ H ].
• mechanical parameters
zD = 2; J = 0,15fe • m2];
F — 0,005
N ■ m ■ s rad
On the other hand, following the automated regulators tuning within the speed control system we obtain the following constants:
K
Kv = 501,3834; Tv =-;
v v 2374,7
Kt — 5,9881; T, —
K,
754,4176
Km —10,1988; TM —
K
M
1020
K» —10; Ta —
350
Under these circumstances by imposing the entry vector of the control system to be of the following type
U1 — — 0,69[Wb] ;
U2m — Wrm — nm
n 30
rad
(2)
u3 —Mr— MN — 93,269[N ■ m],
where nm = 100m[rpm] with m = 0,15 and the proportionality coefficient between the self values of the motor and those of the observer being k = 0,3 by solving the non-linear system f (x, u) = 0 using the Newton method having as the start point the vector:
X
2
X
2
X
s
-100
-200
-300
-400
-3-10 -2.5-1 Ü
-700
-600
-500
-400
-300
-200
-100
100
Real
Fig.3. The self values of the matrix AL; Mr=MN
x = [0 0 u 0 u2m 0 0 0 0 0 u 0] we obtain an equilibrium point xm = bm where:
bm = [bmi ]i=1,12 .
From those stated above by the linearization of the system (1) around the equilibrium
point xm = bm obtained for an u* = [u1u2mu3]r entry vector defined by (2) we obtain:
Ax (t ) = Al Ax(t ) + BL Au(t ) (3) where AL, BL matrixes are:
Al =
Bl =
dfi a *\
ir (bm , u )
dXj
'dfi
du
(bm,u )
k
i=1,12; j=1,12
i=1,12;k=1,3
Next, in order to study the asymptotic stability of the equilibrium points xm=bm we will analyze the self values of the AL matrix so that if they have a strictly negative real part the xm=bm equilibrium point is asymptotically stable for the linerized system (3). Under these circumstances according to the linearization theorem in a vicinity of the equilibrium point xm=bm the non-linear system (1) is asymptotically stable.
* Voicu M. Tehnici de analiza a stabilitatii sistemelor automate. Editura Tehnica. Bucuresti, Romania, 1986.
As the self values of the AL matrix are presented within fig.3, it results that the equilibrium points xm=bm of the linerized system (3) are asymptotically stable and according to the linearization theorem the equilibrium points Xm bm are asymptotically stable in certain vicinity for the non-linear system (1).
Also in fig.3 for a more in depth study of the asymptotic stability of the (1) system we emphasized the self values of the control system that does not contain a Gopinath observer.
From those presented above one may notice that the self values consist of the self values of the control system that does not contain a Gopinath observer as well as the self values of the Gopinath observer. The only self values that modify when a new entry vector is imposed are the self values of the Gopinath observer.
One may notice that inclusion of the Gopinath observer does not alter the self values of the control system that does not contain it.
Thus, one can conclude that we can study the asymptotic stability of a control system that contains in its loop a Gopinath observer by independently studying the asymptotic stability of the control system that does not contain a Gopinath observer and asymptotic stability of the Gopinath observer.
For a more detailed study of the asymptotic stability, next we will present the AL matrix self values under identical testing conditions
with those above with the difference that u3 =M= MN = 0[N • m].
As with the previous case from fig.4 one may notice that the xm=bm 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 (1).
Next, we will emphasize the influence of the estimated rotor resistance when studying asymptotic stability of the equilibrium point x0=b0 of the non-linear system (1) in a certain vicinity of this point.
The equilibrium point is obtained for an entry vector like u = [w1M20M3]r defined by (2), namely for a speed of n0 = 0[rpm].
Based on this study we can obtain information related to the range of variation of the estimated rotor resistance for which the nonlinear system (1) remains asymptotically stable around the x0 = b0 equilibrium point.
As the self values of the AL matrix are presented in fig.5, it results that the nonlinear system (1) around the equilibrium point x0=b0 is asymptotically stable for an es-
■ ,1 ш
-200
-300
-400
-3'I0 -2.5-IG
-700
-600
-500
-400
-300
-200
-100
100
Real
Fig.4. The self values of the matrix AL; Mr=0
Fig.5. Al self values by rotor resistance for n0=0[rpm] Mr=MN
150
100
50
E 0
-50
-100
-150
150
100
50
-50
-100
-150
■p* 100+1 .................•!»».................. 1 hill •K,
k = 10 m 1 ill = 0, 10 so
asJSl ill 40 50 60 70 80 80 7» : 60 »0 40 IS I! 10 21(10-100) 10'100 orf 2X00-100)
10" 1011 2X(10' H \ 10-100 100)
► TO
¡ noo 1 1 i i
-3-10 -2.510
Heal
Fig.6. Al self values by rotor resistance for n0=0[rpm] Mr=0
timated rotor resistance in range of D1 = = {R*;0,91Rr < R* < 1,48Rr}, becoming asymptotically unstable for an estimated rotor resistance in range of D2 = {R*;1,48Rr <R* < 0,91Rr}.
Next we will present the influence of the estimated rotor resistance to the asymptotic stability of the non-linear system (1) when the entry vector
is u = [u1u20u3]r defined by (2) with the difference that u3 =Mr= MN = 0[N ■ m]. The self values of the Al matrix are presented in fig.6.
As the self values of the AL matrix are presented in fig.6, it results that the non-linear system (1) around the equilibrium point x0=b0 is asymptotically stable for an estimated rotor resistance in range
of D1 = {R*;0,1Rr <R* <2,61-106Rr}, becoming asymptotically unstable for an estimated rotor resistance in range of D2 =
= {R*;2,61Rr < R* < 0,1Rr}.
On the other hand in case we realize the digitization of the (3) linear system we obtain:
x(k +1) = FLx(k) + HLu (k), (4)
where the FL, HL matrixes are obtained from the Al, Bl matrixes using one of the two digitization types:
• simplified digitization:
(5)
Fl = I12 + AlT ; complete digitization:
Hl = BlT ;
rp 2
Fl = I12 + AlT + A¡ —;
2
Hl = BLT + ALBL—,
where: T is the sampling time. Proceeding in a similar manner the self values of the Fl matrix in case the entry vector is defined by (2) and the FL matrix is obtained by using simplified digitization using a T = 53,3[^ sec]sampling time were considered.
From this consideration we can see, that the xm=bm equilibrium points are asymptotically stable for the discrete system (4) in case the Fl matrix is obtained through using the simplified digitization method.
This conclusion also remains valid when the Fl 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 u = [u1u20u3]T and is defined by (2). This study was conducted for a sampling time of T = 53,3[^sec] in case of simplified digitization. After the study we obtained the information that the x0=b0 equilibrium point remains asymptotically stable for the (5) linear system for a variation of identi-
fied rotor resistance ranging from D1 =
* *
= {Rr;0,91Rr < Rr < 1,48Rr}.
The Xo=bo equilibrium point becomes asymptotically unstable when the variation of the identified rotor resistance varies from D2 = = {R*;1,48Rr < R*r < 0,91Rr}.
When the study of the influence of identified rotor resistance variation is done for an entry vector of the u = [u1u2u3]T type defined by (2) with the difference that u3 =M r= MN = 0[N ■ m] then the stability domain for which the x0=b0 equilibrium point is asymptotically stable is D1 = {R*;0,1Rr < R < 306,39Rr} and the instability domain is D2 = {R*;306,39Rr < R < 0,1Rr}.
On the other hand the domains that define the upper and lower limits of variation of identified rotor resistance for which the x0=b0 equilibrium point remains asymptotically stable for the (5) discrete linear system diminishes with the increase of the sampling time.
4. Conclusions. In this paper is presented under a unitary entity the mathematical model of the speed vector control system that has in its loop a Gopinath observer. This mathematical model allows for external stability study as well as internal stability study of the control system.
After the study we observed that the analyzed regulating system is asymptotically stable for the imposed speed range (0...1500)[rpm] both, when a resistant couple is lacking or present as Mr=MN. This conclusion is valid in both the continual and discrete cases where the sampling time is T = 53,3[^ sec].
From those presented above we can say that the Gopinath observer adds self values to the control system that does not contain it in its loop without modifying the other self values. This conclusion is only valid when the matrixes that are the basis of projecting the Gopinath observer are the same size with those of the induction motor.
In this paper we determined the upper and lower variation limits for the identified rotor resistance for which the x0=b0 equilibrium point remains asymptotically in both the discrete and continual cases.
After the study we obtained the following information emphasized within Table 1 for the continual case ant Table 2 for the discrete case.
Table 1
Continual case
M=MN Mr=0
LI 0,9lRr 0,1 Rr
LS 1,48Rr 2,61-106R,.
Table 2
Discrete case
M=Mn Mr=0
LI 0,91Rr 0,1 Rr
LS 1,48Rr 306,39Rr
Where, through LI we noted the lower limit and through LS we noted 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 of the continual case when the resistant couple is null. The upper limit stays the same in both the discrete and continual cases when Mr=MN.
The lower limit for the discrete case is the same as the lower limit of the continual case in both the lack or presence of the resistant couple.
Also, from the tables above one may notice that the upper limit in case of lacking resistant couple increases compared to the upper limit in case of having resistant couple and the upper limit in case of lacking the resistant couple decreases compared to the lower limit in case of having resistant couple. This conclusion remains valid for both, the continual and discrete cases.
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ISSN 0135-3500. Записки Горного института. Т.181
