From 2D to 3D Bifurcation Structures in Field Oriented Control of a PMSM

Copyright: © 2016 Souhail W, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. From 2D to 3D Bifurcation Structures in Field Oriented Control of a PMSM Wahid Souhail1*, Hedi Khamari2 and Mohamed Faouzi Mimouni1 1Electrical Engineering Department, National Engineering School of Monastir Monestir, Ibn al Jazar, Tunisia 2Computer Department, Faculty of Computer Science, Taief University Taief, Saudi Arabi


Introduction
The permanent magnet synchronous motors (PMSM) are among the main preferences for industrial control applications. It has high power density, fast dynamic response and high efficiency. Because of its high reliability, the Field -oriented based controller (FOC) is mainly used for high dynamic performance induction motor drives.
For several industrial implementations, the speed regulation of the PMSM is ensured with such type of controllers [1].
Sustained oscillations may arise near equilibrium in PMSM due to the nonlinearities inherent to it. Many undesirable behaviors of PMSM were well documented, but little understood.
Power systems including electromechanical machines present several challenging problems related to the nonlinear characteristics of its components. The resulting physical behaviors, including bifurcation, chaos, resonance, voltage collapse, may cause the loss of stability and the transition from normal to anomalous operating regimes [7][8][9]. Bifurcation theory not only renders some machines' defaults more comprehensible but uncovers new problems that need research.
In attempts to study the dynamics of electric machines, the phase space singularities as well as the parametric singularities were studied in order to define the stability domains and to avoid the undesirable behaviors [2,4,5,[10][11][12][13].
To overcome control draw backs and to solve the stabilization control problems in PMSM some control techniques were reported to be efficient enough to bring order to PMSM. The Lyapunov exponent based controller and the neural and back stepping techniques based nonlinear controllers were developed mainly to suppress chaos and to force the machine to a desired solution in an attraction basin [2][3][4][14][15][16]. In many nonlinear systems, Chaos can be originated from a succession of doubling period (Flip) bifurcations [17,18], these complex behaviors are usually twin. Thus not only chaos, but bifurcation control techniques are required for maintaining a system's behavior in a nominal operating state and to avoid loss of stability. The reference current qref i generated by the PI regulator of angular speed is given as: Ω Ω (6) The integral regulators of the system are defined as: The expression (6) becomes: The input voltages are expressed as: Differentiating the equations (7) with respect to t, gives the following differential system: Then, the machine control model is described by a system of six differential equations: (11) normal operating domains of PMSM. For that purpose, one can start by detecting a bifurcations in a 2D-parameteric plane, and then embed it in a 3D-parameteric space by varying a third parameter. The resulting structure is a bifurcation surface.
The numerical continuation methods are the main tool used to plot the bifurcation diagrams and to explore the dynamics of the PMSM submitted to a FOC. Section 2 is devoted to describe the system model and to define analytically the corresponding control model. Successive period doubling bifurcation leading to chaotic behavior is presented in section 3.
The section 4 is reserved to describe the existence conditions of three generic bifurcations, namely limit point (LP), Hopf (H) and Bogdanov-Takens (BT) bifurcations for particular sets of system and control parameters.
A different approach to characterize the PMSM dynamics, based on embedding a 2D-bifurcation structure into a 3D-parametric space is presented in section 5, and then the paper is ended by some concluding remarks.

Mathematical Model of PMSM Drive System and Preliminaries
Consider a PMSM submitted to a Field Oriented Control ( Figure 1). The outputs of the current controllers is transformed by an inverse Park transformation back from the d-q reference frame into the 2-phase system fixed with the stator which in turn is applied to the motor using the space vector modulation technique. Control of the motor speed is ensured by a reference current qref i generated by an outer loop.

The currents
The mathematical model of PMSM is given by: For investigation of the control problem of the PMSM with smooth air gap, the direct and the quadratic-axis winding inductances are equal The PI regulator is defined by the expression: Where, The classical control method includes three PI regulators to generate the voltages d q v , v having the following expressions: A doubling period bifurcation occurs when a branch of perioddoubled solutions is created or destroyed at the critical point. In the first case if the Hopf bifurcation is supercritical, a branch of stable perioddoubled solutions emerges and the original stable periodic solutions will be continued as a branch of unstable periodic solutions. In the second case, if the Hopf bifurcation is subcritical, an unstable perioddoubled solution is destroyed and then the stable periodic solutions evolve on a branch of unstable periodic solutions [8]. with the eigenvalues: Thus, the resulting limit cycle, with period T=9.33 s is stable and will undergo a cascade of period doubling bifurcation by decreasing p k Figure 2. As the parameter p k is varied, the machine enters into a complicated dynamics, through period doubling bifurcation, chaos intermittency and so on. Based on simulation results, the coordinates and the eigenvalues of the different PD bifurcation points are given in Table 1. It appears that in the PD critical points, two eigenvalues are equal to -1.
To make an overall inspection of the machine dynamics in different points of the bifurcation diagram, namely A, B, C, D, E, F, G, H and I, the phase portraits of the coordinate 1 x versus the variable 2 x are plotted in Table 2.

Computation of equilibrium points
The equation (11) can be written in the form: 2  3  4  5  6 , , , , , ( , , , , , ) The equilibrium points are obtained by equating the righthand side of the equation (12) to zero as follows: x i With . . The global stability and dynamic characteristics of the equilibrium point are profoundly affected by the parametric singularities namely the bifurcation phenomena that will be discussed in the next sections.

Period doubling bifurcation of limit cycles
Analysis of the dynamical behavior of the PMSM led to identify a sequence of particular set of control bifurcation points. The transition from an equilibrium point to a limit cycle through a Hopf bifurcation is followed by a cascade of doubling period bifurcations (PD) which constitutes a veritable route to chaos.

Existence Conditions of Certain Parametric Singularities
Some control approaches reset the Integral action of the PI when the Saturation is reached and particularly the anti wind up methods based on removing the integral part from the input [19].
The integral correctors of the machine drive have an important effect on the system dynamics. The case of a machine driver without integral correctors is considered, so the mathematical model of system given in (11) can be transformed as: With: The characteristic polynomial of Jacobian matrix is:

Limit point bifurcation
Investigation was conducted to establish the conditions leading to limit point bifurcation. The LP bifurcation, which results from interaction between stable and unstable equilibrium points, has three eigenvalues, one of which is 0 and two are nonzero. Therefore, the necessary existence condition is derived from the equation ( ) 0  The Routh-Hurwitz stability criterion is applied to polynomial ( ) p λ , in order to derive the condition for existence of Hopf bifurcation [13] (Table 3)

Bogdanov-Takens bifurcation
The Bogdanov-Takens bifurcation (BT) is a local codimension 2 bifurcation of an equilibrium point. In the parameter plane, the critical equilibrium has a zero eigenvalue of multiplicity two. For near parameter values, the system has a saddle and a non-saddle points which collide and disappear via a saddle node bifurcation. The non-saddle equilibrium turns into a limit cycle when it crosses an Andronov-Hopf bifurcation H. This cycle changes into an orbit homoclinic to the saddle and vanishes via a saddle homoclinic bifurcation [6].
Looking for the possible existence of double-zero eigenvalues of the Jacobian matrix (18), the necessary condition for the occurrence of such a bifurcation is: The existence of Bagdanov-Takens bifurcation BT is controlled by the condition: λ is in the right half of the complex plane. The corresponding behavior is anunstable equilibrium point which appears also in areas 2' and 2". The areas 2 and 2' are separated by a limit point curve Lp + .  (Table 4).

Two-parameter bifurcation plane: The continuation of the limit points
In this point the motor tends to reverse rotation and continues to spin at low For generator operating mode, the same types of bifurcations are shown in an apparent symmetry on the LP bifurcation curve.
Effect of the variation of a third parameter on the bifurcation sets: More features of the bifurcation structure displayed in a parameter plane could be revealed by varying a third parameter. In the parameter plane  (Tables 5 and 6).
Embedding the bifurcation structure shown in Figure 8   x β − plane. bifurcation point in parameter plane, maps to a bifurcation curve in 3D parameter space. Based on the results obtained in Figure 8 and under realistic assumptions, the 3D parameter space can be considered as a set of codimension one bifurcation surfaces connected through codimension two bifurcation curves. This study paves the way to further researches aiming to explore more generic structures of bifurcation surfaces in 3D parametric space and to study the effect of varying a fourth parameter on their shapes and sizes. Also the study aims to strengthening the knowledge and practice on the combined effect of a broader set of parameters on the PMSM dynamics ( Figures  9 and 10).

Conclusion
The simulation results not only can reveal the dependence of PMSM behavior on the control parameters, but also it can be used for control and design purposes. Methods from bifurcation theory are applied to identify and characterize complex bifurcation sets of PMSM behavior in both motor and generator operating modes.
After identifying a period doubling bifurcation cascade under the variation of a proportional control parameter, the analytical necessary conditions for existence of Hopf and Bogdanov-Takens and limit point bifurcations are given in this paper. These parametric space singularities permit to understand the mechanism of transition from equilibrium dot to limit cycle and from one limit cycle to another limit cycle with different order and stability.
Embedding 2D bifurcation structures into 3D bifurcation ones is mainly introduced to study the combined effect of a larger set of parameters on the PMSM dynamics and to widen the understanding of certain types of parametric and phase space singularities.
The bifurcation surfaces established correspond to limit point bifurcation for the motor and generator operating mode, and for period doubling period bifurcation.