FrA07.3
43rd IEEE Conference on Decision and Control December 14-17, 2004 Atlantis, Paradise Island, Bahamas
Bogdanov-Takens Bifurcation in Indirect Field Oriented Control of Induction Motor Drives Francisco Salas, Romeu Reginatto, Francisco Gordillo, Javier Aracil
Abstract— In this paper we explore further the occurrence of bifurcations in the indirect field oriented control of induction motors. This study is a continuation of previous publications [1], [2], [3]. New results reveal the occurrence of codimension-two bifurcation phenomena, such as a BogdanovTakens bifurcation.
I. I NTRODUCTION Due to its high reliability, field-oriented control (FOC) is the standard for high dynamic performance induction motor drives [4], [5], [6]. Historically, this remarkable controller was derived as a result of physical intuition and a deep understanding of the machine operation, with little concern about a rigorous analytical study of its stability and performance. Realizing the practical importance of IFOC, and motivated by the need to clarify its theoretical underpinnings, in the last few years a series of studies on indirect FOC for current-fed induction machines have been carried out [7], [8], [1], [2], [3] allowing a deeper understanding of its fundamental properties. The commissioning of an IFOC requires the knowledge of the rotor time constant, a parameter that can vary widely in practice and is known to cause performance and stability problems. This has motivated the research on the robustness of IFOC drives against rotor resistance uncertainty [8]. It has been shown that the speed control of induction motors through IFOC is globally asymptotically stable for any constant load torque if the rotor time constant is perfectly known or the error in its estimation is sufficiently small [1], [8]. In previous papers the occurrence of both saddlenode and Hopf bifurcations have been identified in IFOC as a result of rotor time constant mismatches [1], [2], [3]. The saddle-node bifurcation occurs for certain values of the mismatch in this estimation and certain load conditions. On the other hand, the occurrence of Hopf bifurcations also depends on the settings of the speed control loop. In this paper, we provide a more complete bifurcation analysis of the IFOC drive in order to understand the system behavior. First, we summarize the bifurcation analysis made
in previous papers and, then, a codimension-two BogdanovTakens bifurcation [9], [10] is presented and characterized for the case when the PI settings guarantee that no oscillations occur on the zero load case. The behavior for other PI settings is presented in [11]. The rest of the paper is organized as follows. In Section II, the system model and the control equations are given and a convenient framework is introduced. Local stability analysis is presented in section III comprising the occurrence of saddle-node, Hopf and Bogdanov-Takens bifurcations. Bifurcation diagrams and simulations that corroborate the theoretical predictions are given in section IV. We end the paper with Section V, which contains some concluding remarks. II. P ROBLEM FORMULATION We consider the indirect field oriented control (IFOC) with speed regulation for induction motor drives in its standard formulation (see [2], [12]), based on the currentfeed induction motor model expressed in the synchronously rotating reference frame. The implementation of IFOC requires the knowledge of the rotor time constant τr , which varies widely in practice [13], [14]. Let τˆr be the estimate of the rotor time constant employed in the IFOC commissioning, and define ∆
κ=
(1)
as the degree of tuning, i.e., if κ = 1 the system is said to be tuned, otherwise it is said to be detuned. We aim at studding the effect of this parameter mismatch on the dynamical phenomena of IFOC. Taking as state variables: x1 = λqr , x2 = λdr , the quadrature and direct axis components of the rotor flux; x3 = wref − w, the rotor speed error; and x4 = iqs the quadrature axis component of the stator current, this model can be stated as [12]
This work has been supported under MCyT-FEDER grants DPI200300429 and DPI2001-2424-C02-01. F. Salas, F. Gordillo and J. Aracil are with Departamento de Ingenier´ıa de Sistemas y Autom´atica, Universidad de Sevilla, Camino de los Descubrimientos s/n 4102 Sevilla, Spain
{fsalas,gordillo,aracil}@esi.us.es
R. Reginatto is with Department of Electrical Engineering, Universidade Federal do Rio Grande do Sul, Av. Osvaldo Aranha 103, 90035-190, Porto Alegre, RS, Brazil,
[email protected].
0-7803-8682-5/04/$20.00 ©2004 IEEE
τr τˆr
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x˙ 1 x˙ 2 x˙ 3 x˙ 4
κc1 x2 x4 u02 κc1 = −c1 x2 + c2 u02 + 0 x1 x4 u2 = −c3 x3 − c4 [c5 (x2 x4 − u02 x1 ) c3 −(Tm + wref )] c4 = (ki − kp c3 )x3 − kp c4 [c5 (x2 x4 − u02 x1 ) c3 −(Tm + wref )] c4 = −c1 x1 + c2 x4 −
(2) (3)
(4)
(5)
where Tm is the load torque; u02 is a constant reference for the rotor flux magnitude; c1 to c5 are machine parameters; kp and ki are the proportional and integral gains of the PI speed regulator, respectively. We assume that both wref , the speed reference, and Tm , the load torque, are constant. It is easy to show that for position regulation with a proportional-derivative controller the same model is obtained after a change of variables [15], so that all the results derived for speed regulation are also valid for position regulation.
Employing this parametrization, the equilibrium of the system (2)-(5) can be stated as [3], [12] ⎤ ⎡ e ⎤ ⎡ c2 u02 1−κ r x1 2 2 c1 1+κ r ⎥ ⎢ xe2 ⎥ ⎢ c2 u02 1+κr 2 ⎥ ⎢ e ⎥=⎢ (10) c1 1+κ2 r 2 ⎥ ⎢ ⎣ x3 ⎦ ⎣ ⎦ 0 xe4 u02 r where r is any real solution of the following 3rd order polynomial equation κr3 − r∗ κ2 r2 + κr − r∗ = 0
A. Parametrization of the tuned system In the tuned case, κ = 1, the model (2)-(5) simplifies considerably. By letting wref = 0 and Tm = 0, the standstill condition, the system reaches the equilibrium point x = xo = [0, cc21 u02 , 0, 0] (this is called the magnetization phase of IFOC). Starting from x(0) = xo , the fluxes x1 and x2 remain constant, regardless of the behavior of x4 . Thus, the model reduces to a 2nd order model given by −K x3 −c3 x˙ 3 = (ki − kp c3 ) −kp K x˙ 4 x4 c4 c3 wref + (6) kp c3 kp c4 Tm ∆ c c c u0
where we have defined K = 2 4c15 2 . We shall refer to the dynamic system (6) as the tuned system, which is a linear system and is usually taken as a base for setting the PI gains. The eigenvalues of system (6) can be arbitrarily assigned by the PI gains kp and ki . Instead of working directly with the PI gains kp and ki it is convenient for our study to introduce a normalized parametrization of the closed-loop eigenvalues in terms of a damping ξ and natural frequency ηc1 . For this choice of closed-loop eigenvalues for the tunned system, the PI gains are given by kp =
2ξηc1 − c3 , K
ki =
η 2 c21 K
(7)
B. Parametrization of the equilibria The equilibria of the system (2)-(5) has been studied previously (see [12], [16]) and parameterized in term of two dimensionless parameters, namely: ∆
r
=
r∗
=
∆
xe4 u02 (c4 Tm + c3 wref ) . K u02
(11)
It has been shown in [8] that this polynomial has a unique real solution provided that κ ≤ 3. For κ > 3, is has been shown in [1] that the number of equilibrium points change according to the following inequalities ra2 + 1 κ2 ra2 + 1 r2 + 1 ∆ |r∗ | ≤ rb∗ = κrb 2b 2 κ rb + 1 ∆
|r∗ | ≥ ra∗ = κra
(12) (13)
where we have defined √
2 ∆ k 2 − 3 + (k 2 − 9)(k 2 − 1) (14) ra = 2κ √
2 ∆ k 2 − 3 − (k 2 − 9)(k 2 − 1) rb = (15) 2κ If (12) and (13) are both satisfied strictly then the system (2)-(5) has three equilibrium points. If either (12) or (13) holds with equality, then the system has two equilibrium points. Otherwise the equilibrium point of this system is unique. This result is illustrated by Fig. 1, where the limits described by (12), ra∗ , and (13), rb∗ , are plotted. The point √ where the two curves intersect is κ = 3, r∗ = 33 . For large κ, ra∗ tends to zero and rb∗ tends to 0.5. The system has three equilibria if the degree of detuning and the loading condition lie in between the two curves and a unique equilibrium outside. At each one of the two curves two equilibria coincide. Given that the equilibria of the IFOC are now fully characterized and parameterized in terms of κ and r, we shall next study their stability properties employing a bifurcation analysis. III. B IFURCATION A NALYSIS
(8) (9)
The quantity r∗ represent the system loading, since it is proportional to the electrical torque developed in steady-state. On the other hand, r basically represents the quadrature axis component of the stator current. As it can be verified from the tuned system, the quantity r equals the value of r∗ in the tuned condition.
The occurrence of saddle-node bifurcations in IFOC have been characterized in [1], [12]. As far as the Hopf bifurcation is concerned, partial characterizations have been provided in [8], [2], [3], [16]. Most of such results are summarized in this section, as a necessary background, and then a characterization of codimension-two bifurcations, namely a Bogdanov-Takens bifurcation, is provided. Once the closed-loop poles of the tuned system have been chosen for a given performance, the parameters kp and ki of the PI speed regulator can be assigned according to (7).
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v0
0.6
rb∗
0.5
r*
κ2 r4 + (3 − κ2 )r2 + 1 1 + κ2 r 2
(24)
A. Saddle-node bifurcation It is easy to verify that for κ = 1 (tuned condition) the eigenvalues of the Jacobian (16) are given by the roots of
0.4
(λ2 + 2ξηc1 λ + η 2 c21 )(λ + c1 )2
ra∗
0.3
0.2
0.1
0 0
5
κ
10
15
Fig. 1. Locus of the points in the parameter space where the number of equilibria changes.
Substituting (7) into (2)-(5) and calculating the Jacobian at the equilibria (10) we get ⎡ J
∆
=
−c1 c1 κr c1 c2 K c1 (2ξηc 1 − c3 ) c2
⎢ = ⎢ ⎣
−c1 κr −c1 − cc12 Kr − cc21 (2ξηc1 − c3 )r
c2 (1−κ) 1+κ2 r 2 c2 κ(1−κ) (1+κ2 r 2 )r 2 ) − K(1+κr (1+κ2 r 2 ) −c3 )(1+κr 2 ) − (2ξηc11+κ 2 r2
0 0 −c3 η 2 c21 −c3 (2ξηc1 −c3 ) K
⎤ ⎥ ⎥ ⎥ (16) ⎥ ⎦
The characteristic polynomial of J is P (λ) = λ4 + p3 λ3 + p2 λ2 + p1 λ + p0
(17)
The coefficients pi of the characteristic polynomial, and thus the eigenvalues of the Jacobian and the local stability properties of the closed-loop system, depend, for a given motor, only on κ, r∗ and the dynamics assigned for the tuned system. The occurrence of local bifurcations can then be characterized only in terms of these parameters. By neglecting friction, i.e. c3 = 0, the coefficients of the characteristic polynomial can be stated as p3 p2 p1 p0
= c1 (2ξηv2 + 2) = c21 η 2 v2 + 2ξηv1 + κ2 r2 + 1 = c31 η 2 v1 + ξηκv0 = c41 η 2 κv0 ∆
v2
=
v1
=
∆
which are in the open left-hand side of the complex plane, implying the expected local asymptotic stability of IFOC. As κ varies local stability can be lost due to either real or complex eigenvalues of the Jacobian crossing the jω axis toward the right-hand side of the complex plane. A saddlenode bifurcation can be detected by the Jacobian becoming singular or, equivalently, by the characteristic polynomial having a zero root. More specifically, we must have p0 = 0 in (17) which is equivalent of saying v0 = 0 in (21) or even κ2 r4 + (3 − κ2 )r2 + 1 = 0
B. Hopf bifurcations A Hopf bifurcation is characterized by the crossing of complex eigenvalues through the imaginary axis while the other eigenvalues remain the open left-hand side of the complex plane. This condition can be detected by applying the Routh-Hurwitz criterion to polynomial (17) which yields: λ4 λ3 λ2 λ1 λ0
(20) (21) ∆
2
1 + κr 1 + κ2 r 2 κ(3 − κ)r2 + κ + 1 1 + κ2 r 2
(22)
(26)
This equation is the same which describes the points where the number of equilibria changes [1], [12], whose solutions are given by r = ±ra and r = ±rb (see (14) (15)). Then the curves in Fig. 1 represent bifurcation surfaces in the parameter space, that is, they give the locus of all the turning points in this space [9], [10], [17]. The lower branch is given by the solution r = ra of (26) while the upper branch is give by the solution r = rb . The point where the two saddle-node bifurcation curves intersect at κ = 3 √ and r∗ = 33 is a codimension-two bifurcation named cusp or fold [10]. It is instructive to look at the branching diagram obtained varying r∗ for a given κ. Fig. 2 presents the branching diagram for κ = 4. Two turning points can be seen, associated to the crossing of each one of the two curves in Fig. 1. This branch topology is known to cause the occurrence of jumps and hysteresis [17].
(18) (19)
where we have defined
(25)
1 p3 q2 p3 q1 q2
p2 p1 p0
p0 (27)
p0 ∆
with q2 = p2 p3 − p1 and q1 = q2 p1 − p23 p0 . Since p3 is always positive, the condition for existence of Hopf bifurcation can be stated as follows:
(23)
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q1 = 0, and q2 > 0, p0 > 0
(28)
for the possible existence of double-zero eigenvalues of the Jacobian matrix (16), which is a necessary condition for the occurrence of such a bifurcation [18]. We carry out this analysis for the non-friction case, i.e., c3 = 0. The existence of a zero eigenvalue of the Jacobian has been characterized by (26) –saddle-node bifurcation– and is given by two curves in the (κ, r∗ ) plane given by ra∗ and rb∗ . Clearly, double-zero eigenvalues, if they exist, must necessarily occur on points lying on these two curves. A direct consequence of this fact is that it can only happen for κ > 3. From the characteristic polynomial of the Jacobian (17) we can obtain the condition for the existence of two zero eigenvalues as p0 = 0, p1 = 0 (31)
2.5
2
r
1.5
1
0.5
0 0
0.1
0.2
0.3
0.4
0.5
0.6
r*
Fig. 2.
From the structure of p0 and p1 (see (21) and (20)) with c3 = 0 it is easy to verify that (31) is equivalent to
Branching diagram for κ = 4.
v0 = 0, In the important case of zero load operation, condition (28) can be given in a closed form as is stated in the next result [8], [1], [3]. Lemma 1: Let c3 ≡ 0 and Tm ≡ 0. Then, no Hopf bifurcation takes place for any κ > 0 provided that ξ, η satisfy the relation (4ξ 2 − 1)η + 2ξ > 0
∆
κ = κh =
η(1 + 2ξη) η − 2ξ(1 + 2ξη)
(30)
♦ For the general constant (non-zero) load case it is not possible to provide closed form solutions for the existence of Hopf bifurcations. However, it is always possible to provide numerical solutions for (28) which, despite being numerical, do not loose generality since the conditions hold for any IFOC drive. For the case c3 ≡ 0, the coefficients of P (λ) can be normalized to c1 , so that each element on the first column of the Routh-Hurwitz array, and consequently the condition for the existence of Hopf bifurcation (28), is a function of only four dimensionless parameters: κ, r∗ , ξ and η. Studies in this direction regarding the existence of Hopf bifurcations and its dependence on the PI settings (parameters ξ and η) have been provided in [2], [3]. In what follows we shall concentrate on codimension-two bifurcations. As mentioned in the introduction, we restrict our attention to the case the motor has no oscillations for zero load operation, so the PI settings must satisfy condition (29). C. Bogdanov-Takens bifurcations In order to provide a characterization for the occurrence of a Bogdanov-Takens bifurcation in IFOC, we first look
(32)
Since the denominators of both v0 and v1 are always positive quantities, condition (32) reduces to the following set of nonlinear equations κ(3 − κ)r2 + κ + 1 = κ2 r4 + (3 − κ2 )r2 + 1 =
0 0
(33) (34)
Solving (33) for r yields
(29)
If condition (29) is not satisfied, then a Hopf bifurcation takes place at
v1 = 0
r2 =
κ+1 κ(κ − 3)
(35)
which implies that real solutions can only exist for κ > 3. Substituting it into (34) yields the polynomial equation κ4 − 4κ3 − 2κ2 − 4κ + 9 = 0
(36)
whose unique real solution greater than 3 is given by √ √ ∆ 1 κ = κbt = 1+ 32 3 27+3 57+4 √ ≈ 4.5386 (37) √ 3 27+3 57
kbt + 1 ∆ r = rbt = ≈ 0.8906 (38) kbt (kbt − 3) The solution rbt above actually corresponds to r = ra for κ = κbt , as it can be verified by substituting κbt into (11). ∗ for which the double-zero The loading condition r∗ = rbt eigenvalue occurs can be obtained by substituting rbt and ∗ ≈ 0.418. κbt into (12), rbt We summarize the above development in the following result. Theorem 1: Let c3 = 0. There always exists a value of κ and r∗ for which the Jacobian of system (2)-(5) presents double-zero eigenvalues. This value is unique and is given ∗ . Moreover, this value does not by κ = κbt and r∗ = rbt depend on the choice of the PI setting. Remark 1: The point in the (κ, r∗ ) plane determined by ∗ κ = κbt and r∗ = rbt is on the lower curve in Fig. 3. At this point system (2)-(5) has 2 equilibrium points, exactly at the transition from a single equilibrium to 3 equilibrium
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points. The Jacobian has 2 zero eigenvalues exactly at the equilibrium that gives birth to the other 2 equilibrium of the system. Remark 2: For κ = κbt and r = rbt we have v2 > 0 implying that p2 > 0. As a result, the Jacobian of system (2)-(5) can never present 3 eigenvalues at the origin. A local view in the (κ, r∗ ) plane shows that from the point (κbt , rbt ) two branches of saddle-node bifurcations emerge, one of the characteristics of a Bogdanov-Takens bifurcation. To complete the local characterization of this bifurcation we look for a branch of Hopf bifurcations that emerge from the same point. Whereas the point (κbt , rbt ) does not depend on (ξ, η), the same cannot be expected for the existence of a Hopf bifurcation emerging from this point and so for the existence of the Bogdanov-Takens bifurcation itself. The above development can be summarized in the following theorem. Theorem 2: Let c3 = 0. Then, a branch of Hopf bifurcations emanate from the point (κbt , rbt ) in the (κ, r∗ ) plane provided that ξ and η are such that relation (28) can be satisfied for points arbitrarily close to (κbt , rbt ). Theorems 1 and 2 are an almost complete characterization of a Bogdanov-Takens bifurcation of IFOC taking place at the point (κ, r∗ ) = (κbt , rbt ). What remains to be shown is the existence of a branch of homoclinic connection emerging from the same point. Due to the global characteristic of this type of phenomena, it is not possible to provide general condition for its existence. We can show however that such a phenomena does exist for particular IFOC drives by means of simulation and using continuation software AUTO 2000 [19]. IV. B IFURCATION D IAGRAMS FOR IFOC On the basis of the results presented in Section III we are able to provide rather complete branch diagrams for IFOC. A major property of such results is that they do not depend on particular induction motor parameters. With the exception of the homoclinic connection, all the results hold for any IFOC drive and are stated as functions of only 4 dimensionless parameters: κ, r∗ , ξ and η. To illustrate the bifurcation described in previous sections in Fig. 3 the bifurcation set in plane (κ, r∗ ) for PI settings that guarantee no oscillations in the zero load case and satisfy condition (28) is shown . Fig. 3 presents the Bogdanov-Takens bifurcation point (BT) at the values of κ and r∗ calculated in (37)-(38) where two branches of saddle-node bifurcations (SN a1) − (SN a2) corresponding to the saddle-node bifurcation at r∗ = rb∗ emerge. At this bifurcation point a supercritical Hopf bifurcation branch (HBa), analyzed in section III-B, and a Homoclinic Connection (HC) branch, obtained using the continuation software AUTO, appears. Fig. 4 shows the bifurcation diagram for κ = 3.8, which, according to Fig. 3, is a value of the bifurcation parameter for which no homoclinic connection occurs for any value
r∗ 2.0
1.6
1.2
HBb
0.8 HBa C
SNb SNa1
0.4
HC BT SNa2
0.0 0.0
2.0
4.0
6.0
8.0
10.0
12.0
κ
14.0
Fig. 3. Bifurcation set in the (κ, r ∗ ) plane (BT=Bogdanov-Takens bifurcation, HB=Hopf Bifurcation, SN=Saddle-Node bifurcation, C=cusp bifurcation, HC=Homoclinic Connection).
of the system load (r∗ ). In this figure we can see that when the parameter r∗ increases the system presents a saddlenode bifurcation (SN a1), a supercritical Hopf bifurcation (HBa), another saddle-node bifurcation (SN b) and another supercritical Hopf bifurcation (HBb) where the limit cycle born in the first Hopf bifurcation disappears. When κ = 5 the Bogdanov-Takens bifurcation has taken place and a homoclinic connection can be seen in Fig. 5. If we begin in this case from high enough load values, and decrease parameter r∗ , first a Hopf bifurcation occurs emerging a stable limit cycle (HBb), then there is a saddlenode bifurcation (SN b) and if we continue decreasing r∗ the amplitude of the limit cycle grows until a homoclinic connection (HC) occurs. Notice that the collision between the limit cycle and the saddle born in the saddle-node bifurcation can not be seen in Fig. 5, because the homoclinic connection is not plane due to the fact that it is far from its birth at the Bogdanov-Takens bifurcation. Fig. 6 is similar to Fig. 5 but, in this case, the saddle-node bifurcation (SN b) occurs before the limit cycle emerges from the Hopf bifurcation (HBb). In this figure we can better see the homoclinic connection. V. C ONCLUDING REMARKS In this paper, we have extended previous bifurcation analysis of indirect field oriented control of induction motors. While the previous results focused in codimensionone, local bifurcations (saddle-node and Hopf bifurcations), the present paper studies the existence of a codimensiontwo, namely Bogdanov-Takens (BT), bifurcation. Another novelty of the present study is to make evident the existence of homoclinic connections, which emerges of the BT bifurcation. The study, which does not complete the full
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6.
bifurcation diagram, shows the rich variety of behaviors of such standard industrial control of induction motors.
5.
R EFERENCES
4.
HBb x43. u02 2.
HBa 1.
SN a1
0. 0.00
0.20 0.10
0.40 0.30
SN b
0.60
0.80
0.50
0.70
1.00
r∗
0.90
x4 Fig. 4. Bifurcation diagram in the (r ∗ , u 0 ) plane for κ = 3.8. The 2 extreme values of stable limit cycles are represented by dots. Stable (unstable) equilibria are represented by solid (dashed) curves.
5.
4.
HC HBb
3.
x4 u02
2.
SN a1
1.
HC 0. 0.00
Fig. 5.
0.10
0.20
0.30
0.40
SN b 0.50
Bifurcation diagram in the (r ∗ ,
0.60
x4 ) u0 2
0.70
0.80
r∗
plane for κ = 5.
4.0 3.5
HC
[1] A. Bazanella and R. Reginatto, “Robustness margins for indirect field-oriented control of induction motors,” IEEE Transactions on Automatic Control, vol. 45, no. 6, pp. 1226–1231, 2000. [2] A. S. Bazanella and R. Reginatto, “Robust tuning of the speed loop in indirect field oriented control of induction motors,” Automatica, vol. 37, pp. 1811–1818, 2001. [3] F. Gordillo, F. Salas, R. Ortega, and J. Aracil, “Hopf bifurcation in indirect field-oriented control of induction motors,” Automatica, vol. 38, pp. 829–835, 2002. [4] W. Leonhard, Control of Electrical Drives. Springer-Verlag, 1985. [5] P. Vas, Vector Control of AC Machines. Clarendon Press, 1990. [6] D. W. Novotny and P. Sen, “Introduction to field orientation and high performance ac drives,” in IEEE Indus. Applicat. Soc. Annual Meeting, Sept 1986. [7] R. Ortega, A. Lor´ıa, P. Nicklasson, and H. Sira-Ram´ırez, Passivitybased Control of Euler-Lagrange Systems. Springer, 1998. [8] P. de Wit, R. Ortega, and I. Mareels, “Indirect field-oriented control of induction motors is robustly globally stable,” Automatica, vol. 32, no. 10, pp. 1393–1402, 1996. [9] J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer-Verlag, 1986, vol. 42. [10] J. Hale and H. Koc¸ak, Dynamics and bifurcation. Springer-Verlag, 1991, applied Mathematics Science Series, 3. [11] F. Salas, “Bifurcations in electromechanical system control (in Spanish),” Ph.D. dissertation, ESI. Sevilla, 2002. [12] A. Bazanella and R. Reginatto, “Robustness margins for indirect field-oriented control of induction motors,” in 37th CDC, 1998, pp. 1000–1006. [13] R. Krishnan and F. Doran, “Study of parameter sensitivity in high-performance inverter-fed induction motor drive systems,” IEEE Trans. Ind. Applic., vol. IA-23(4), pp. 623–635, Jul/Aug 1987. [14] R. Marino, S. Peresada, and P. Tomei, “Online stator and rotor resistance estimation for induction motors,” IEEE Transactions on Control Systems Technology, vol. 8(3), pp. 570–579, May 2000. [15] R. Reginatto and A. S. Bazanella, “Robustness of global asymptotic stability for indirect field-oriented control of induction motors,” IEEE Transactions on Automatic Control, vol. 48(7), pp. 1218–1222, July 2003. [16] F. Gordillo, F. Salas, R. Ortega, and J. Aracil, “Bifurcation analysis of indirect field-oriented control of induction motors,” in IFAC Workshop on Digital Control. PID’00, 2000. [17] R. Seydel, Practical bifurcation and stability analysis: from equilibrium to chaos. Springer-Verlag, New York, 1994, 2nd edition. [18] S. Wiggins, Introduction to applied nonlinear dynamical system and chaos. Springer-Verlag, 1990, vol. 2, applied Mathematics Science Series. [19] E. Doedel, R. Paffenroth, A. Champneys, T. Fairgrieve, Y. Kuznetsov, B. Oldeman, B. Sandstede, and X. Wang, AUTO 2000: continuation and bifurcation software for ordinary differential equations (with HomCont), June, 2002.
3.0
HBb
2.5
x2.0 4 u02 1.5
SN a1
1.0 0.5 0.0 0.00
Fig. 6.
HC 0.10
0.20
0.30
SN b 0.40
0.50
Bifurcation diagram in the (r ∗ ,
0.60
x4 ) u0 2
0.70
0.80
r∗
plane for κ = 6.
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