Complex State Variables Research Articles

Modelling discrete time fractional Rucklidge system with complex state variables and its synchronization

This article proposes a discrete time fractional order three dimensional Rucklidge system with complex state variables. The dynamical nature and chaotic behavior exhibited by the Rucklidge system with real state variables and higher dimensional system derived from complex state variables are compared. The stability of the proposed system is analyzed at their equilibrium states by obtaining eigenvalues numerically. Chaotic dynamics exhibited by the system with real and complex variables are investigated employing different methods like bifurcation analysis and maximum Lyapunov exponents via the Jacobian matrix method. Nonlinear controllers are introduced for chaos synchronization of the subsystems of the proposed discrete time Rucklidge system with fractional order. The article further discusses the coexisting behavior of the attractors for the Rucklidge system of real state variables with coexisting bifurcation diagrams. The impact of the parameters on the system dynamics is demonstrated with a sequence of bifurcation diagrams for the simultaneous variation of two parameters.

Complex State Variables as Analytical Tool for Control System Design of Medium-Voltage Drives

Complex state variables are used to analyze the dynamics of medium-voltage drives. These operate at low switching frequency to reduce the dynamic losses of the power semiconductor devices. It results in a low sampling rate of the digital signal processing system, causing signal delays that deteriorate the dynamic behavior. Undesired cross-coupling results between the current components id and iq. The effect is shown to be even more adverse than described by the conventional control theory. The solution is modeling the system by single-complex state variables. These permit designing a current controller of high dynamic performance, which exhibits zero cross-coupling. Improved performance at very low switching frequency is demonstrated by experimental results.

Modeling and Control of Low Switching Frequency High-Performance Induction Motor Drives

Complex state variables are used to study the dynamic behavior of induction motors considering the propagation in space of the distributed magnetic field inside the machine. The objective of this paper is to improve the dynamics of pulsewidth modulation inverters in medium-voltage drive systems. To keep the dynamic losses of the power devices at a tolerable level, the switching frequency must be below 1 kHz. The sampling rate of the digital signal processing system is then low which introduces considerable signal delay. The delay has an adverse influence on the dynamic behavior of the current control system. It introduces undesired cross coupling between the current components $i_{d}$ and $i_{q}$ . The degree of cross coupling is described by a cross-frequency transfer function. It is shown that the mechanism of cross coupling is different and more adverse than the conventional theory discloses. A current controller structure having poles and zeroes of the single-complex type is synthesized. Cross coupling is completely eliminated at any low switching frequency. Experimental results demonstrate that high dynamic performance and zero cross coupling is achieved even at very low switching frequency.

The Relationship Between Root Locus and Transient Field Components of AC Machines

The dynamic analysis of induction motors is supported by well-known theories: the two-axes transformation, and the space vector theory. Yet some inconsistencies with the theory of dynamic systems exist. The machine eigenvalues suggest the existence of two damped oscillators, physically not understandable. The respective eigenfrequencies change with the angular velocity of the reference frame. This contradicts the understanding that eigenfrequencies are inherent system properties. Physically, the dynamics depend on the continuous distribution of magnetic energy and its spatial displacement during transient processes. Information on the system dynamics is lost when dividing the continuum of magnetic energy into discrete portions. Complex state variables associate the dynamics to the propagation in space of distributed magnetic fields. The dynamic analysis reveals the existence single-complex eigenvalues. These define a novel class of system identifiers. They are characterized by having only one imaginary part instead of a conjugate complex pair. The use of complex state variables conveys insight and physical understanding of the dynamic processes within the machine. The approach constitutes an extension to the theory of dynamic systems.

Interval Quadratic Connected Control Systems in Industrial Robotics and Mechatronics

Industrial robotic and mechatronic systems in real machinery are quadratically connected dynamical systems with parametric uncertainties, whose mathematical models can be extremely complex. Program motion stabilization of such systems, especially those described in quasi-coordinates, is a topical scientific problem, which does not yet have an acceptable for engineering developments comprehensive solution. The article proposes a possible concept of such a solution, based on the Lyapunov’s direct method, extended to the interval dynamic systems domain with time derivatives of state variables written in the interval cubic forms. The authors conceptual approach is based on new concepts introduced in the article: covering form, form compensator, modular variables form. According to this approach, first of all, the transition to new complex system state variables formed with signatures is carried out. The mathematical apparatus is based on special interval matrix algebra transformations, which could not be included in the article because of the volume, and therefore are only stated. As a result, the authors obtain a new class of quasi-relay type controls with system state dependent modulated impact jumps, and therefore combining the properties of both soft smooth and hard forcing types.

Module-phase synchronization in complex dynamic system

The concept of module-phase synchronization was proposed. The chaos synchronization between drive system and response system was achieved in module space and phase space respectively (module-phase synchronization). Different from the evolutions in real space, there is no pseudorandom behavior in phase space when module-phase synchronization achieve. All the phases of complex state variables switched between two fixed values which are determined by initial values of drive system. And the modules varied within a bounded field. The theoretical analysis and the simulations were also given.

Mapping Model of Chaotic Phase Synchronization

A coupled map model for the chaotic phase synchronization and its desynchronization phenomenon is proposed. The model is constructed by integrating the coupled kicked oscillator system, kicking strength depending on the complex state variables. It is shown that the proposed model clearly exhibits the chaotic phase synchronization phenomenon. Furthermore, we numerically prove that in the region where the phase synchronization is weakly broken, the anomalous scaling of the phase difference rotation number is observed. This proves that the present model belongs to the same universality class found by Pikovsky et al.. Furthermore, the phase diffusion coefficient in the de-synchronization state is analyzed.

Design of Fast and Robust Current Regulators for High-Power Drives Based on Complex State Variables

High-power pulsewidth-modulated inverters for medium-voltage applications operate at switching frequencies below 1 kHz to keep the dynamic losses of the power devices at a permitted level. Also, the sampling rate of the digital signal processing system is then low, which introduces considerable signal delays. These have adverse effects on the dynamics of the current control system and introduce undesired cross coupling between the current components i/sub d/ and i/sub q/. To overcome this problem, complex state variables are used to derive more accurate models of the machine and the inverter. From these, a novel current controller structure employing single-complex zeros is synthesized. Experimental results demonstrate that high dynamic performance and zero cross coupling is achieved even at very low switching frequency.

On the spatial propagation of transient magnetic fields in AC machines

AC motors have proliferated as the most important machine type used in speed variable drive systems. The dynamic analysis and description of revolving field machines is supported by well-established theories: Park's transformation (1929), and the space vector theory by Kovacs and Racz (1959). Yet some inconsistencies with the theory of dynamic systems exist; the machine eigenvalues suggest the existence of two damped oscillators. A transient condition is described by time-dependent oscillations of terminal quantities. It appears unsatisfactory that the respective eigenfrequencies change with the velocity of the reference frame. This contradicts the common understanding according to which the eigenfrequency is an inherent system property. A clarification is reached using a novel approach for the dynamic analysis. The approach is based on complex state variables. These characterize a transient condition by the spatial propagation of distributed magnetic fields. The formal analysis is an extension to the space vector theory and the theory of dynamic systems.

The representation of AC machine dynamics by complex signal flow graphs

Induction motors are modeled by nonlinear higher-order dynamic systems of considerable complexity. The dynamic analysis based on the complex notation exhibits a formal correspondence to the description using matrices of axes-oriented components; yet differences exist. The complex notation appears superior in that it allows the distinguishing between the system eigenfrequencies and the angular velocity of a reference frame which serves as the observation platform. The approach leads to the definition of single complex eigenvalues that do not have conjugate values associated with them. The use of complex state variables further permits the visualization of AC machine dynamics by complex signal flow graphs. These simple structures assist to form an understanding of the internal dynamic processes of a machine and their interactions with external controls. >