EPE Association: EPE-PEMC 2002: Special Session: Non Linear Dynamics in Power Electronics and Drives

EPE-PEMC 2002: Special Session: Non Linear Dynamics in Power Electronics and Drives
You are here: EPE Documents > 04 - EPE-PEMC Conference Proceedings > EPE-PEMC 2002 - Conference > EPE-PEMC 2002: Special Session: Non Linear Dynamics in Power Electronics and Drives

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   Bifurcation Analysis in PWM Regulated DC-DC Converters using Average Models
By V. Moreno; G. Olivar; E. Toribio; A. El Aroudi; B. Benader [View]
[Download]

Abstract: PWM Regulated Boost and Buck-Boost converters can show hysteresis phenomena in their bifurcation paths. Conditions for parameters that imply multiple attractors, i.e., multistability, are evaluated in the context of the average model approximation. The averaged system, without taking into account its physical limits (named border collisions), contains some symmetry, showing either one, always antisaddle (node or focus) or three, one saddle and the other two antisaddles, equilibrium points. Frequently, the antisaddles are spiral (foci) and show Hopf bifurcation. The global behaviour varying some parameter shows a pattern with two ending saddle-node (most frequently repellor) bifurcations. Then, the system changes from three to one equilibrium point (or vice-verse). The inclusion of border collisions bifurcations breaks the symmetry and modifies the mentioned scheme. Finally, limit cycles associated with Hopf bifurcation (which is subcritical) play an important role near stability transitions. Partitions in 2D parameter space are used as a tool in order to help to understand these phenomena.

   Bifurcation Phenomena in a Sliding-Mode Controlled Boost Converter
By Y. Zúniga; A. Miralles; C. Batlle; I. Massana; G. Olivar [View]
[Download]

Abstract: A Boost converter under sliding-mode control is numerically simulated with an ad-hoc C++ code. The trajectories can be analytically computed due to the existing linear topologies. Bifurcation diagrams (1D and 2D) show that a period-doubling route to chaos is common, although other types of non-smooth bifurcations are also present.

   Bifurcation Phenomena in Three-phase Space Vector Modulated Converters
By I. Nagy; Z. Suto [View]
[Download]

Abstract: The analysis and numerical results of a current controlled three-phase Space Vector Modulated (SVM) Voltage Source Converter (VSC) are presented. The aim of the current controller of VSC is to track the sinusoidal reference currents in each phase. Due to the saturation within the current controller the study leads to rather unexpected system states.

   Border Collision Bifurcations in a Chaotic PWM H-Bridge Single-phase Inverter
By C. Robert; B. Robert [View]
[Download]

Abstract: This paper deals with nonlinear dynamics of a PWM current-programmed H-Bridge. Fully chaotic behaviours appear and disappear under control tuning of the current loop. To explain how this strange dynamics evolve, we present a model that is a parametric one-dimensional piecewise linear map. We show how to apply a recent advance in chaos theory in order to determine the fixed points analytically, their domains of stability, and of the bifurcation points. Bifurcations which are nongeneric for smooth dynamical systems, also called Border Collision Bifurcations, allow a better understanding of the bifurcation diagram. With this example, we show that it is possible to predict the appearance of chaos in this converter in an entirely analytical way.

   Density aspects of chaotic DC-DC converters
By O. Woywode; H. Guldner [View]
[Download]

Abstract: The paper applies bifucation and statistical analysis to dc-dc converters. Both are able to analyze dc-dc converters regardsless whether the converters operate periodically or aperiodically. The dc component of the converter's inductor current is calculated with respect to ensemble (probablistic) averages of orbits. This will illustrate the density approach to dealing with arbitrary waveforms in power electronic systems. The orbits are generated by one-dimensional piecewise linear maps associated with samples of the inductor current taken in accordance with clock impulses. The benefits of the probablistic over the traditional approach are mentioned.

   Local Bifurcations in DC-DC Converters
By C.-C. Fang; E. H. Abed [View]
[Download]

Abstract: Three local bifurcations in DC-DC converters are reviewed They are period-doubling bifurcation, saddle-node bifurcation, and Neimark bifurcation. A general sampled-data model is employed to study types of loss of stability of the nominal (periodic) solution and their connection with local bifurcations. More accurate prediction of instability and bifurcation than using the averaging approach is obtained. Examples of bifurcations associated with instabilities in DC-DC converters are given.

   Low speed sensorless control of a class of electrical machines
By R. Ortega; A. Astolfi; M. B. Becherif [View]
[Download]

Abstract: The problem of sensorless (local) speed regulation of a class of electrical machines is addressed and solved using a simple linear-time varying controller. The class, which contains permanent magnet synchronous and stepper motors, consists of all fully actuated machines whose magneto motive force can be approximated by a first harmonic Fourier expansion. The controller—which contains the internal model of the steadystate solution—is able to asymptotically reconstruct the control signal necessary to achieve speed regulation, even in the presence of unknown but constant load torque. To prove global stability and boundedness of the unforced system we exploit the by now well–known passivity property of electro-mechanical systems. We work out in detail the problem of speed regulation for a permanent magnet synchronous motor, for which normalized simulations that illustrate the properties of the design are provided.

   Modeling and Instability of Average Current Control
By C.-C. Fang [View]
[Download]

Abstract: Dynamics and stability of average current control of DC-DC converters are analyzed by sampled- data modeling. Orbital stability is studied and it is found unrelated to the ripple size of the orbit. Compared with the averaged modeling, the sampled-data modeling is more accurate and systematic. An unstable range of compensator pole is found by simulations, and is predicted by sampled-data modeling and harmonic balance modeling.

   Stability Analysis and Bifurcations of Switching Regulators with PI and Sliding Mode Control
By L. Martínez-Salamero; J. Calvente; A. El Aroudi; R. Giral; M. Debbat; G. Olivar [View]
[Download]

Abstract: This paper deals with the study of nonlinear behaviour of hysteresis DC-DC switching regulators. General space state switched model is presented for the three elementary converters, buck, boost and buck-boost. Proportional and Integral (PI) control are taken into account in the model. It is shown that the introduction of the integral term increases the order of the system and may produce subharmonic oscillations and even chaos. The Poincaré map 'P' as function of the circuit parameters is obtained using closed form expressions. The Jacobian matrix 'DP' evaluated at the fixed points of the map is also obtained. The loci of the eigenvalues of 'DP', when some parameters are varied, show the occurence of flip bifurcation. Time domain simulations and bifurcation diagrams for some examples of DC-DC switching regulators show a period doubling route to chaos.

EPE-PEMC 2002: Special Session: Non Linear Dynamics in Power Electronics and Drives
You are here: EPE Documents > 04 - EPE-PEMC Conference Proceedings > EPE-PEMC 2002 - Conference > EPE-PEMC 2002: Special Session: Non Linear Dynamics in Power Electronics and Drives
	[return to parent folder]

	Bifurcation Analysis in PWM Regulated DC-DC Converters using Average Models By V. Moreno; G. Olivar; E. Toribio; A. El Aroudi; B. Benader	[View] [Download]
Abstract: PWM Regulated Boost and Buck-Boost converters can show hysteresis phenomena in their bifurcation paths. Conditions for parameters that imply multiple attractors, i.e., multistability, are evaluated in the context of the average model approximation. The averaged system, without taking into account its physical limits (named border collisions), contains some symmetry, showing either one, always antisaddle (node or focus) or three, one saddle and the other two antisaddles, equilibrium points. Frequently, the antisaddles are spiral (foci) and show Hopf bifurcation. The global behaviour varying some parameter shows a pattern with two ending saddle-node (most frequently repellor) bifurcations. Then, the system changes from three to one equilibrium point (or vice-verse). The inclusion of border collisions bifurcations breaks the symmetry and modifies the mentioned scheme. Finally, limit cycles associated with Hopf bifurcation (which is subcritical) play an important role near stability transitions. Partitions in 2D parameter space are used as a tool in order to help to understand these phenomena.

	Bifurcation Phenomena in a Sliding-Mode Controlled Boost Converter By Y. Zúniga; A. Miralles; C. Batlle; I. Massana; G. Olivar	[View] [Download]
Abstract: A Boost converter under sliding-mode control is numerically simulated with an ad-hoc C++ code. The trajectories can be analytically computed due to the existing linear topologies. Bifurcation diagrams (1D and 2D) show that a period-doubling route to chaos is common, although other types of non-smooth bifurcations are also present.

	Bifurcation Phenomena in Three-phase Space Vector Modulated Converters By I. Nagy; Z. Suto	[View] [Download]
Abstract: The analysis and numerical results of a current controlled three-phase Space Vector Modulated (SVM) Voltage Source Converter (VSC) are presented. The aim of the current controller of VSC is to track the sinusoidal reference currents in each phase. Due to the saturation within the current controller the study leads to rather unexpected system states.

	Border Collision Bifurcations in a Chaotic PWM H-Bridge Single-phase Inverter By C. Robert; B. Robert	[View] [Download]
Abstract: This paper deals with nonlinear dynamics of a PWM current-programmed H-Bridge. Fully chaotic behaviours appear and disappear under control tuning of the current loop. To explain how this strange dynamics evolve, we present a model that is a parametric one-dimensional piecewise linear map. We show how to apply a recent advance in chaos theory in order to determine the fixed points analytically, their domains of stability, and of the bifurcation points. Bifurcations which are nongeneric for smooth dynamical systems, also called Border Collision Bifurcations, allow a better understanding of the bifurcation diagram. With this example, we show that it is possible to predict the appearance of chaos in this converter in an entirely analytical way.

	Density aspects of chaotic DC-DC converters By O. Woywode; H. Guldner	[View] [Download]
Abstract: The paper applies bifucation and statistical analysis to dc-dc converters. Both are able to analyze dc-dc converters regardsless whether the converters operate periodically or aperiodically. The dc component of the converter's inductor current is calculated with respect to ensemble (probablistic) averages of orbits. This will illustrate the density approach to dealing with arbitrary waveforms in power electronic systems. The orbits are generated by one-dimensional piecewise linear maps associated with samples of the inductor current taken in accordance with clock impulses. The benefits of the probablistic over the traditional approach are mentioned.

	Local Bifurcations in DC-DC Converters By C.-C. Fang; E. H. Abed	[View] [Download]
Abstract: Three local bifurcations in DC-DC converters are reviewed They are period-doubling bifurcation, saddle-node bifurcation, and Neimark bifurcation. A general sampled-data model is employed to study types of loss of stability of the nominal (periodic) solution and their connection with local bifurcations. More accurate prediction of instability and bifurcation than using the averaging approach is obtained. Examples of bifurcations associated with instabilities in DC-DC converters are given.

	Low speed sensorless control of a class of electrical machines By R. Ortega; A. Astolfi; M. B. Becherif	[View] [Download]
Abstract: The problem of sensorless (local) speed regulation of a class of electrical machines is addressed and solved using a simple linear-time varying controller. The class, which contains permanent magnet synchronous and stepper motors, consists of all fully actuated machines whose magneto motive force can be approximated by a first harmonic Fourier expansion. The controller—which contains the internal model of the steadystate solution—is able to asymptotically reconstruct the control signal necessary to achieve speed regulation, even in the presence of unknown but constant load torque. To prove global stability and boundedness of the unforced system we exploit the by now well–known passivity property of electro-mechanical systems. We work out in detail the problem of speed regulation for a permanent magnet synchronous motor, for which normalized simulations that illustrate the properties of the design are provided.

	Modeling and Instability of Average Current Control By C.-C. Fang	[View] [Download]
Abstract: Dynamics and stability of average current control of DC-DC converters are analyzed by sampled- data modeling. Orbital stability is studied and it is found unrelated to the ripple size of the orbit. Compared with the averaged modeling, the sampled-data modeling is more accurate and systematic. An unstable range of compensator pole is found by simulations, and is predicted by sampled-data modeling and harmonic balance modeling.

	Stability Analysis and Bifurcations of Switching Regulators with PI and Sliding Mode Control By L. Martínez-Salamero; J. Calvente; A. El Aroudi; R. Giral; M. Debbat; G. Olivar	[View] [Download]
Abstract: This paper deals with the study of nonlinear behaviour of hysteresis DC-DC switching regulators. General space state switched model is presented for the three elementary converters, buck, boost and buck-boost. Proportional and Integral (PI) control are taken into account in the model. It is shown that the introduction of the integral term increases the order of the system and may produce subharmonic oscillations and even chaos. The Poincaré map 'P' as function of the circuit parameters is obtained using closed form expressions. The Jacobian matrix 'DP' evaluated at the fixed points of the map is also obtained. The loci of the eigenvalues of 'DP', when some parameters are varied, show the occurence of flip bifurcation. Time domain simulations and bifurcation diagrams for some examples of DC-DC switching regulators show a period doubling route to chaos.