Wajdi Ahmad

Chaos in fractional-order autonomous nonlinear systems

We numerically investigate chaotic behavior in autonomous nonlinear models of fractional order. Linear transfer function approximations of the fractional integrator block are calculated for a set of fractional orders in ð0; 1� , based on frequency domain arguments, and the resulting equivalent models are studied. Two chaotic models are considered in this study; an electronic chaotic oscillator, and a mechanical chaotic ‘‘jerk’’ model. In both models, numerical simulations are used to demonstrate that for different types of model nonlinearities, and using the proper control parameters, chaotic attractors are obtained with system orders as low as 2.1. Consequently, we present a conjecture that third-order chaotic nonlinear systems can still produce chaotic behavior with a total system order of 2 þ e ,1 > e > 0, using the appropriate control parameters. The effect of fractional order on the chaotic range of the control parameters is studied. It is demonstrated that as the order is decreased, the chaotic range of the control parameter is affected by contraction and translation. Robustness against model order reduction is demonstrated. 2002 Elsevier Science Ltd. All rights reserved.

On nonlinear control design for autonomous chaotic systems of integer and fractional orders

Abstract In this paper, we address the problem of chaos control for autonomous nonlinear chaotic systems. We use the recursive “backstepping” method of nonlinear control design to derive the nonlinear controllers. The controller effect is to stabilize the output chaotic trajectory by driving it to the nearest equilibrium point in the basin of attraction. We study two nonlinear chaotic systems: an electronic chaotic oscillator model, and a mechanical chaotic “jerk” model. We demonstrate the robustness of the derived controllers against system order reduction arising from the use of fractional integrators in the system models. Our results are validated via numerical simulations.

Modeling of speech signals using fractional calculus

In this paper, we present a novel approach for speech signal modeling using fractional calculus. This approach is contrasted with the celebrated Linear Predictive Coding (LPC) approach which is based on integer order models. It is demonstrated via numerical simulations that by using a few integrals of fractional orders as basis functions, the speech signal can be modeled accurately. The new approach has the merit of requiring a smaller number of model parameters, and is demonstrated to be superior to the LPC approach in capturing the details of the modeled signal.

On the necessary and sufficient conditions for latch-up in sinusoidal oscillators

Two necessary and sufficient conditions for the occurrence of dynamical latch-up in sinusoidal oscillators are proposed. These conditions result from studying the stability of the equilibrium points associated with each region of operation of the fundamentally nonlinear driving-point characteristics that are necessary to model accurately the behaviour of sinusoidal oscillators. The fact that an equilibrium point can be either virtual or real is the key to predicting latch-up. We numerically validate our proposal on the derived models of some well-known oscillators by predicting and confirming either the possibility or the impossibility of latch-up.

SLIDING MODE CONTROL OF GENERALIZED FRACTIONAL CHAOTIC SYSTEMS

A sliding mode control technique is introduced for generalized fractional chaotic systems. These systems are governed by a set of fractional differential equations of incommensurate orders. The proposed design method relies on the fact that the stability region of a fractional system contains the stability region of its underlying integer-order model. A sliding mode controller designed for an equivalent integer-order chaotic system is used to stabilize all its corresponding fractional chaotic systems. The design technique is demonstrated using two generalized fractional chaotic models; a chaotic oscillator and the Chen system. The effect of the total fractional order is investigated with respect to the controller effort and the convergence rate of the system response to the origin. Numerical simulations validate the main results of this work.

PARAMETER IDENTIFICATION OF CHAOTIC SYSTEMS USING WAVELETS AND NEURAL NETWORKS

This paper addresses the problem of reconstructing a slowly-varying information-bearing signal from a parametrically modulated, nonstationary dynamical signal. A chaotic electronic oscillator model characterized by one control parameter and a double-scroll-like attractor is used throughout the study. Wavelet transforms are used to extract features of the chaotic signal resulting from parametric modulation of the control parameter by the useful signal. The vector of feature coefficients is fed into a feed-forward neural network that recovers the embedded information-bearing signal. The performance of the developed method is cross-validated through reconstruction of randomly-generated control parameter patterns. This method is applied to the reconstruction of speech signals, thus demonstrating its potential utility for secure communication applications. Our results are validated via numerical simulations.

Power factor correction using fractional capacitors

Power factor correction is investigated under the assumption of using a fractional-order capacitor. The effects of the fractional order of the capacitor on real and reactive source power, as well as on the value of the correcting capacitance are analyzed. Theoretical analysis supplemented by numerical simulations are presented, and an example is given.

Fractional-order phase-locked loop

In this paper, a fractional-order analog phase-locked loop (FAPLL) model is proposed. In the new model, the traditional analog phase locked loop (APLL) is generalized by allowing the loop filters as well as the voltage controlled oscillators to acquire fractional order dynamics. The new model offers superior performance compared to its integer order counterpart in terms of capture and locking time response. The improvement realized in the response of the FAPLL outweighs the slight increase in the loop bandwidth, typically seen in fractional order systems. The main points of this work are illustrated via numerical examples.

Noise Performance of Fractional-Order Phase-Locked Loop

An analog fractional-order phase-locked loop (FOPLL) is investigated under noisy conditions. The FOPLL using fractional voltage-controlled oscillator (FVCO) and a fractional loop filter (FLF) outperforms its integer-order counterpart for both noisy and noiseless signals. Reducing the loop fractional orders increases both the locking and capturing ranges, and the bandwidth of the FOPLL, which substantially improves the loop transient behavior. However, an increase in the FOPLL bandwidth makes it susceptible to noisy signals. Hence, a compromise has to be made between the loop gain and the loop fractional order to alleviate the effect of noisy signals. The main points of this work are illustrated via simple numerical example.

Modeling and simulation of a fractional order bioreactor system

Abstract In this paper, we propose a fractional order model for a well known bioreactor system. The model comprises a fractional differential equation for the ‘biomass’ and an integer order differential equation for the substrate’ dynamics. With an appropriate tuning of the ‘dilution rate’ control parameter, the proposed fractional order model behaves in a similar fashion to the parent integer order model, thus offering the advantage of model reduction while retaining the same qualitative steady state behavior. The dynamical responses are inherently slower than those obtained from the integer order model, and their speed increases with the system order. We also present a conjecture that for a fractional order bioreactor model of order 1 + α , biomass washout is obtained for dilution rates in excess of ‘ α ’ times the dilution rate for washout in the integer order model.