1. Introduction

Applied mathematics is an interdisciplinary subject, which provides a broad qualitative and quantitative background for use in electrical engineering field. In this study we focus on aspects of theoretical and numerical analysis which is related to the bifurcation theory applied to the electrical motors.

The hallmark of the nonlinear behavior is the bifurcation, which is an abrupt qualitative change in the phase space motion which is refers to the small variations of one or more of system control parameters. The analysis of the behavior of dynamical systems can be carried out globally through bifurcation analysis, which describes the range of parameter values at which qualitative propriety changes occurs. Crossing a bifurcation point, existence and uniqueness of solutions is not guaranteed and a change in stability and/or order and/or the number of solutions occurs. In former studies ^{[1, 2]} three types of generic codimension one bifurcations of periodic solutions of nonlinear ODE were described: tangent, period doubling and Hopf bifurcation. If an equilibrium point satisfies two bifurcation conditions, we call the singularity as codimension two bifurcations. A typical case of codimension-2 bifurcation is the Fold-Hopf bifurcation (ZH) which is a transversal intersection of fold and hopf bifurcation curves at a bifurcation value. Aiming to study the existence of such particular bifurcation point in induction motor submitted to an Indirect Field-oriented control, analytical analysis are used to put into evidence the mathematical condition of their existence. Such parametric singularities can be caused by the parameter fluctuations namely the errors in the estimate of the rotor time constant which changes widely with temperature ^[3].

Field-oriented controllers FOC, frequently used as nonlinear controllers for induction machines, performs asymptotic linearization and decoupling ^[4]. Stability of FOC is generally investigated regarding errors in the estimate of the rotor resistance. It has been previously shown that the speed control of induction motors through under field-oriented control (IFOC) is globally asymptotically stable for any constant load torque.

An analysis of parameter plane singularities (saddle-node and Hopf bifurcations) in IFOC drives with respect to the rotor time constant variation provides a guideline for setting properly the motor parameters in order to avoid such bifurcations ^{[5, 6]}.

In the recent year, the qualitative approach has become increasingly useful tool in the analysis of nonlinear power systems. Aiming to understand the bifurcation mechanism and their associated responses, one should identify phase plane singularities (equilibrium, limit cycles, attraction basins) and parameter plane singularities (bifurcations, chaos) ^[7]. In former studies ^{[8, 9, 10]}, it has been shown the occurrence of either codimension one bifurcation such as saddle node bifurcation and Hopf bifurcation and codimension two such as Bogdanov-Takens or zero-Hopf bifurcation in IFOC induction motors. The cancelation of sustained oscillations which are, in general undesirable was the purpose of other studies which proposed the 'oscillation killer' in order to adjust the system and control parameters so that one can get rid of limit cycles ^[11]. Other results ^[12] permit to promote efficiency or improve dynamic characteristics of drives. An adequate combination between analytical and numerical tools may provide a deep comprehension of some nontrivial dynamical behavior related to bifurcation phenomena in a self-sustained oscillator ^[13].

The robustness margins for IFOC of induction motors can be deduced from the analysis of the bifurcation structures identified in parameter plane ^[14]. Since the self-sustained oscillations in IFOC for induction motors may be due to the appearance of a Hopf bifurcation ^[15], an exhaustive study of the bifurcation structures is mainly dedicated to preserve the local stability of the desired equilibrium point.

An outline of the paper is organized as follows. The governing equations of the IFOC induction motor and a general reminder will be described in section 2. The detailed analytical analysis of fold and zero-hopf bifurcations will be proposed in section 3, together with the numerical algorithm of ZH-bifurcation detection. Some numerical simulations will be given in section 4 and the conclusion in the section 5.

2. System Equation Formulations

Mathematically, the governing equations used for the modeling of indirect field-oriented control of induction motor can be described as ^[9] by the following set of four nonlinear autonomous differential equations:

(1)

(2)

(3)

(4)

For this set of equations,,, and denote the variable states. Preciously, and are the direct and the quadratic component of the rotor flux, respectively. being the difference between reference and the real mechanical speed. The last variable present’s the quadratic component of the stator current results from the outer loop PI. The parameter bifurcation is the ratio of the rotor time constant to its estimate and is a design parameter. and are the proportional and the integral PI controller gains, respectively. We can also define the following constants,, and , where:

is the rotor flux time constant.

, is the mutual inductance

, is the friction constant.

, where is the moment of inertia and is the pole pair number.

Finally, let be a constant where denotes the load torque.

In numerical methods or bifurcation problems, the phase plane analysis is an important technique for studying the behavior of nonlinear systems. Since, a two-parameter plane can be considered as made up of sheets as mentioned in previous paper ^[2], each one being associated with a well defined behaviour such as a fixed point, or an equilibrium or a periodic orbit.

With change in temperature or frequency there will be change in the values of the motor parameters. As the variation of these parameters increases, so will their effects on system dynamic behavior. In particular, the rotor time constant which may vary considerably over the operational range of the rotor resistance which changes widely with temperature. So, we choose in this study as the bifurcation parameter. This parameter is defined as the degree of tuning, i.e., if the system is considered to be tuned, otherwise it is said to be detuned ^[9].

The purpose of the present work is to give an analytical investigation of the influence of small variation of the parmeter on the dynamical behavior of motor model. A closed form analytical solution is developed for the tuned case characteristic of the motor (). In addition to this tuned phase, a considerable simplification of equation model (1)-(4) may be achieved in the magnetization phase of IFOC obtained by putting and . It is easy to verify that the equilibrium point is reached while the machine is in the standstill conditions.

In previous studies (see ^{[15, 16, 17]}) the equilibrium solutions of the system of equations (1)-(4) have been determined under general conditions. More importantly, it has been proven that an equilibrium point is parameterized in term of two dimensionless parameters (and) and it has the following analytical expression:

(5)

Analytically, presents any real solution of the following 3rd order polynomial equation:

(6)

In fact, is a mathematically term related to physical properties of the system loading. This term is defined by the following expression:

(7)

where is a constant denoted by .

On the other hand, the term r is related to the quadrature axis component of the stator current. This parameter is defined by the following simple expression:

(8)

In the previous paper ^[8], it has been proven that for any numerical parameter which verifies the following property, this polynomial has a unique real solution. When, it has been demonstrated also that the number of equilibrium points varies under the following two criteria:

(9)

(10)

The expresssions of and are defined by, respectively:

(11)

(12)

When , the IFOC equations (1)-(4) possess three, two or one equilibrium point according to the fact that criteria (9) and (10) are strictly verified or not. If either (9) and (10) are both verified with equality, then the system has 2 equilibrium points. In any other case the system has a unique equilibrium point.

A composite picture of the phase space behaviour can be gained by studying the 4th order autonomous ODEs (1-4) describing the IFOC induction motor (Equilibrium points, limit cycles, chaotic orbits). Each phasingularity involves four eigenvalues describing its stability. Generally, a local bifurcation at an equilibrium happens when some eigenvalues of the parametrized linear approximating differential equation cross some critical values such us the origin or the imaginary axis. Self-sustained oscillations in IFOC of induction motors can be originated by a codimension one bifurcation namely the Hopf bifurcation (H). Such kind of bifurcation can be computed from differential system (1)-(4), when a pair of complex conjugate eigenvalues among the eigenvalues set of the associate linearized system change from negative to positive real parts or vice versa.

Therefore the Hopf bifurcation results from the transversal crossing of the imaginary axis by a pair of complex conjugate eigenvalues. Such bifurcation is said to be supercritical if the periodic branch is initially stable and subcritical if the periodic branch is initially unstable.

The singular curves of the parameter plane corresponding to codimension-1 bifurcations may contain singular points of higher codimension ^{[18, 19]}. The simplest one located on a Hopf curve has the codimension-2 singular point, a Fold-Hopf (ZH). This bifurcation, also called the Zero-Hopf (ZH), is a codimension 2 bifurcation of an equilibrium point at which the critical equilibrium lies at a tangential intersection of curves of fold bifurcation curve and Andronov-Hopf bifurcation curve within the two parameter family.

Definition:

Consider an autonomous system of ordinary differential equations (ODEs),

(13)

where denotes the state vector, is a 2-dimensional parameters vector and is smooth.

Suppose that at the system has equilibrium. In the n-dimensional case with, the Jacobian matrix has:

1. a simple zero eigenvalue and a simple pair of purely imaginary eigenvalues, as well as

2. eigenvalues with , and

3. eigenvalues with , with .

3. Analytical Bifurcation Analysis

Local phase portrait near singularities can be very complicated. This phase-plane picture is developed by determining the form of the solution of the 4th order autonomous ODEs (1)-(4) describing the IFOC induction motor near each of its singularities (equilibrium points EP, limit cycles LC). Each one of these solutions involves four eigenvalues describing its stability. Evaluating the expression of Jacobian at the equilibrium point (5), we get the following form:

(14)

where and .

Let define the characteristic equation of :

(15)

being the eigenvalue parameter and denotes a unit matrix of order 4. It is easy to verify that is a 4th degree polynomial function:

(16)

The coefficients are defined as follows:

(17)

(18)

(19)

(20)

(21)

where and have the following expressions:

(22)

(23)

We next give the appropriate analytical conditions characterising the occurence of fold and fold-hopf bifurcations.

A Fold (F) bifurcation of equilibrium point, called also Saddle-Node or a limit point (LP) is a codimension one bifurcation which occurs when a single eigenvalue of the characteristic polynomial is equal to zero.

To check whether is a solution of, simply verify that the coefficient is equal to zero. Thus, announced the following theorem using the equations (16) and (21):

Theorem 1:

If there exists a real () and a real , solution of (6), for which the following condition is satisfied:

Then, the Jacobian of system (1)-(4) presents a single zero eigenvalue and fold bifurcation can occur.

In the general case, the necessary analytical condition for the existence of a Zero-Hopf bifurcation (ZH) for the system (1)-(4), at the equilibrium point (5), is that the equation has a simple zero eigenvalue and one pair of purely imaginary roots .

Let's solve the following four degrees equation:

(24)

Let , the condition of fold bifurcation occurence.

Then the equation (24) becomes:

(25)

Let's consider the following third degree polynomial function

(26)

Were, , and are the real coefficients.

Examing the zero of 3-dimensional polynomial.

Let the roots of be denoted ,and , then the factored form of the polynomial is given by:

(27)

Expanding the plynomial factors, we easily obtain the following relations:

(28)

The general form of the discriminant formula $\delta$ of a n-dimensional polynomial is given by:

(29)

Where are the roots of the polynomial.

For , the discriminant of is defined as follows:

(30)

The discriminant can be expressed in term of the coefficients as follows:

(31)

Finally,

(32)

When does a real polynomial have a pair of complex conjugate roots?

Computation of the purely imaginary eigenvalues based on elementary properties of cubic polynomials, we report on three cases of root types:

Case 1: , andare distinct real roots.

In which case, can be written as:

It is easy to verify that the discriminant is positive.

Case 2: Existence of two equal real roots.

We can find three possible double roots.

Since the discriminant is zero:

Case 3: Two complex conjugate roots.

By considering the symmetric properties, let denote the two complexe conjugate roots and denotes the real root of .

Roots of the polynomial can be written in the following forms:

(33)

Then the discriminant can be calculated as:

then, .

In this case, the discriminant is negative. We report all results on root types in the following Lemma:

Lemma 1:

If , then has three real roots.

If , then has a double root.

If , then has has a pair of complex conjugate roots.

In the case of , has one pair of complex conjugate roots. The Hopf bifurcation point can be detected when only one pair of the roots of have a zero real parts. This condition can be deduced generally from the generalized Routh-Hurwitz criterion applied to the characteristic polynomial (26) ^[9]:

if then one pair of purely imaginary roots can be detected.

Proof:

Let the purely imaginary roots of be denoted and , were is a non-zero real numbers. Each of the two roots satisfies the given relation:

(34)

As a result,

(35)

and

(36)

Subtraction of the equation (35) from the equation (36) gives:

(37)

Then

(38)

Addition of two equations (35) and (36) gives:

(39)

Then

(40)

The equality between two expressions (38) and (40) gives: , we then obtain .

The particular roots of are:

1. is real positive number.

2. Two purely imaginary roots

3. One real root.

Now, our main results, with respect to the above conditions, are announced in the following theorem.

Theorem:

If there exists a real (),and for which the following conditions are satisfied:

(ZH. 1) Occurrence test: ,

(ZH. 2) One zero root for: ,

(ZH. 3) One pair of purely imaginary roots for:

The Jacobian of system (1)-(4) presents one zero eigenvalue in addition of one pair of purely imaginary eigenvalues and a Fold-Hopf bifurcation can be detected.

Let’s now define the following numerical algorithm of ZH detection based on the announced theorem and in light of the obtained results discussed above. The proposed algorithm includes 7 steps detailed as follows:

Step 1: Set appropriate initial parameter values starting out from the tuned case and the equilibrium point.

Step 2: Identify the coefficients of the polynomials expressions.

Step 3: Computing the discriminant of, testing if.

Step 4: verify the condition =0.

Step 5: Verify the condition.

Step 6: The Zero-Hopf bifurcation is detected with eigenvalues , and is a non zero real number.

Step 7: Update parameter, change by where is an appropriate small number. Go to step 2.

4. Numerical Simulations

The Zero-Hopf bifurcation mechanism’s generation from an equilibrium point, is illustrated by the graph of Figure 1. For the parameters,, and an equilibrium point (EP) is identified.

The EP vector is = (2.426, -1.76, 0.089, 0.495), solution of the differential system (2) for the initial conditions set =(0,0.5,0, 0).

Figure 1. Equilibrium Point: bifurcation process

Download as

PowerPoint Slide

Larger image(png format)

Figures index

Veiw figure

View current figure in a new window

View next figure

Starting from this located initial equilibrium, a continuation method permits to obtain the evolution of the direct compound flux versus the values of bifurcation parameter (see Figure 2). One Hopf bifurcation is obtained in such curve. This singularity has one pair of purely imaginary eigenvalues and the following coordinates in phase space: (3.694, -0.723, 0, and 0.211) for = 2.0135. The corresponding eigenvalues are (-1.61, -0.946, i2.936, - i2.936).

Figure 2. Bifurcation diagram

Download as

PowerPoint Slide

Larger image(png format)

Figures index

Veiw figure

View current figure in a new window

View previous figure

View next figure

Figure 3. Hopf bifurcation curve

Download as

PowerPoint Slide

Larger image(png format)

Figures index

Veiw figure

View current figure in a new window

View previous figure

View next figure

The continuation of the Hopf point (H) leads to trace the Hopf bifurcation curve shown in Figure 3. Such curve includes a 2-codimension Zero-Hopf bifurcation point in (2.2573, -1.5733, 0, 0.2318) for the parameters = 4.7792 and = 2.0965. The corresponding eigenvalues of this singularity are = (2.3386, i2.8593, - i2.8593, 0).

In fact, the ZH-bifurcation point in the parameters -plane is a critical point at which the critical equilibrium lies at a tangential intersection of fold bifurcation curve and Hopf bifurcation curve within the two parameter family (see Figure 4).

Figure 4. Fold-Hopf bifurcation detection

Download as

PowerPoint Slide

Larger image(png format)

Figures index

Veiw figure

View current figure in a new window

View previous figure

View next figure

For the parameters values = 3.2 and =2.409, the Fold curve includes two branches joining in a codimension two bifurcation points, namely cuspidal point (CP) having the following phase space coordinates = (2.01, -1.1547, 0, 0.5774).

Besides, such curve presents a Bogdanov-Taken bifurcation (BT) in = (1.062, -0.727, 0, 0.891) for the parameters values = 4.53852 and = 1.6721, with the associates eigenvalues = (-1.0796 + i4.2773, -1.0796- i4.2773, 0, 0).

Figure 5. Phase plane trajectory at Fold-Hopf bifurcation

Download as

PowerPoint Slide

Larger image(png format)

Figures index

Veiw figure

View current figure in a new window

View previous figure

The process of convergence of phase trajectory in the −plane shows a critical equilibrium at the Zero- Hopf bifurcation point (see Figure 5). The red points present the generated limit cycle.

5. Conclusion

This paper applies bifurcation and singularity analysis to study the complex dynamic behavior of IFOC induction motor. The computed steady state is shown to be adequat for low codimension bifurcation studies. Particularely, the paper clearly proposes an analytical analysis of fold and fold-hopf (ZH) bifurcations. This mathematical approach yields the theoretical test conditions under which the existence of such bifurcation singularities is verified and that can be solved analytically. The paper also introduces a computational algorithm for the detection of fold-hopf bifurcation using developed conditions. Finally, bifurcation curves that exhibit fold (F), hopf (H) and fold-hopf (ZH) bifurcations for various parameters are traced through numerical continuation methods.

Acknowledgement

This work was supported by ESIER-US, Industrial systems and Renewable Energy Unit of Search, Tunisia.

References

[1]	H. Kawakami. Bifurcations of periodic responses in forced dynamic non linear circuits, computation of bifurcation values of the system parameter, IEEE trans. circuits and systems, cas-31, pp 248-260, 1984.
	In article
[2]	H. Khammari, C. Mira and J.P. Carcass` es. Behaviour of harmonics generated by a Duffing type equation with a nonlinear damping, Part I. International Journal of Bifurcation and Chaos, Vol. 15, No. 10, 3181-3221, 2005.
	In article		CrossRef
[3]	K. Yahyia, S. zouzou, F. Benchabane. Indirect vector control of induction motor with on line rotor resistance identification, Asian Journal of Information and Technology Volume (5) 12, 2006, pp. 1410-1415.
	In article
[4]	Seok Ho Jeon, Dane Baang, and Jin Young Choi. Adaptive Feedback Linearization Control Based on Airgap Flux Model for Induction Motors, International Journal of Control, Automation, and Systems, vol. 4, no. 4, pp. 414-427, August 2006.
	In article
[5]	A. S. Bazanella R. Reginatto. Instability Mechanisms in Indirect Field Oriented Control Drives: Theory and Experimental Results, IFAC 15th Triennial World Congress, Barcelona, Spain, 2002.
	In article
[6]	A.S. Bazanella, R. Reginatto and R. Valiatil. On Hopf bifurcations in indirect field oriented control of induction motors: Designing a robust PI controller, 1, Proceedings of the 38th Conference on Decision and Control, Phoenix, Arizona USA, December, 1999.
	In article
[7]	C. Mira, J. P. Carcass` es, C. Simo, J. C. Tatjer. Crossroad area-spring area transition. (II) Foliated parametric representation, Int. J. Bifurcation and Chaos 1 (2), 339-348, 1991.
	In article		CrossRef
[8]	R. Reginatto, F. Salas, F. Gordillo and J. Aracil. Zero-Hopf Bifurcation in Indirect Field Oriented Control of Induction Motors, First IFAC Conference on Analysis and Control of Chaotic Systems: CHAOS’ 06, 2006.
	In article
[9]	F. Salas, R. Reginatto, F. Gordillo and J. Aracil. Bogdanov-Takens Bifurcation in Indirect Field Oriented Control of Induction Motor Drives, 43rd IEEE Conference on Decision and Control, Bahamas, 2004.
	In article
[10]	F. Salas, F. Gordillo, J. Aracil and R. Reginatto. Codimension-two Bifurcations In Indirect Field Oriented Control of Induction Motor Drives, International Journal of Bifurcation and Chaos, 18 (3), 779-792, 2008.
	In article		CrossRef
[11]	C. C. de-Wit, J. Aracil, F. Gordillo and F. Salas. The Oscillations Killer: a Mechanism to Eliminate Undesired Limit Cycles in Nonlinear Systems, CDC-ECC’ 05, Seville, Spain, 2005.
	In article
[12]	B. Zhang, Y. Lu and Z. Mao. Bifurcations and chaos in indirect field oriented control of induction motors, Journal of Control Theory and Applications, 2, 353-357, 2004.
	In article		CrossRef
[13]	A. Algaba, E. Freire, E. Gamero and A. J. Rodriguez-Luis. Analysis of Hopf and Takens Bogdanov Bifurcations in a Modified van der PolDuffing Oscillator, Nonlinear Dynamics 16: 369404, 1998.
	In article		CrossRef
[14]	A. S. Bazanella and R. Reginatto. Robustness Margins for Indirect Field- Oriented Control of Induction Motors, IEEE Transaction on Automatic Control, 45, 1226-1231, 2000.
	In article		CrossRef
[15]	F. Gordillo, F. Salas, R. Ortega and J. Aracil. Hopf Bifurcation in indirect field-oriented control of induction motors, Automatica, 38, 829- 835, 2002.
	In article		CrossRef
[16]	N. Jabli, H. Khammari, M.F. Mimouni, An Analytical Study of Low- Codimension Bifurcations of Indirect Field-Oriented Control of Induction Motor. ‘International Journal of Mathematical Models and Methods in Applied Sciences’, Vol. 4, Issue 2.
	In article
[17]	N. Jabli, H. Khammari, M.F. Mimouni. An analytical study of bifurcation and nonlinear behavior of indirect filed-oriented control induction motor. 6th WSEAS International Conference on dynamical systems & control, CONTROL’ 10, Sousse, May 3-6, 2010.
	In article
[18]	N. Jabli, H. Khammari, M.F. Mimouni. Bifurcation scenario and chaos detection under variation of PI controller and induction motor parameters. 9th WSEAS International Conference on Non-Linear Analysis, Non-Linear System and Chaos, NOLASC’ 10, Sousse, Tunisia, 3-6 Mai 2010.
	In article
[19]	N. Jabli, H. Khammari, M.F. Mimouni, Multi-Parameter Bifurcation Analysis of a Three-Phases Induction Motor.’International Review of Electrical Engineering’, IREE, volume 5, Issue 2, 2010.
	In article