Ashraf A. Zaher

Nonlinear control of permanent magnet stepper motors

Abstract The nonlinear dynamics of a permanent magnet stepper motor is studied by means of modern nonlinear theories such as bifurcation and chaos. A three-phase stepper motor is considered as a case study in this paper. The study shows that the system experiences a dynamic bifurcation (Hopf bifurcation) at high frequencies. Since this kind of motors is widely used in some important applications such as printers, disk drives, process control systems, X–Y records, and robotics, controlling such instabilities is the main concern of this paper. A nonlinear robust model-reference controller is introduced. The study shows how to stabilize the system, while having a satisfactory performance, even in the case when some of the motor parameters were uncertain.

Robust control of biped robots

Demonstrates a strategy for the design and implementation of robust controllers for robots. The design technique is based on constructing an error vector between the robot measurable states and the desired states, then forcing the gradient of this error vector to be negative via the use of a suitable Lyapunov function. The controller is robust in the sense that it accommodates unstructured uncertainties inherent in robotics. Different robots having different degrees of complexities are used to simulate the response of the proposed controller and compare it to existing ones. The simulation is made in a MATLAB environment. A few practical considerations are then addressed to investigate the causality of the proposed controller and its applicability to real-time situations. A conclusion is submitted with a few comments regarding both adaptivity and real-time compatibility of the proposed controller.

Nonlinear recursive chaos control

A nonlinear recursive chaos and bifurcation control approach, based on a backstepping technique, is presented to control bifurcation and chaos in engineering applications. Modern nonlinear system theory in bifurcation and chaos is used to study the dynamics of a 3rd order nonlinear system. The study considers the system when experiencing bifurcation and chaos state. In order to control it, a recursive backstepping controller is designed. We show how such a controller is effective in controlling the unstable bifurcation region.

Robust model-reference control for a class of non-linear and piece-wise linear systems

This paper demonstrates a strategy for designing model-reference controllers. The design technique is based on using Backstepping techniques as an extension to Lyapunov-based designs. The proposed controller has the capability of accomplishing both asymptotic stability and satisfactory transient performance. LMI techniques are used to investigate the stability of the proposed controller when the model could be expressed in an affine-based form. Two different approaches are used to deal with nonlinear systems. The first one is by using different affine models for the system with a scheduling parameter to switch between the models. The second one is by directly using the nonlinear system while adding pole-placement structure to the closed loop system. Full state feedback, output feedback and robustness problems are addressed in the existence of uncertainties. A simulated one-degree of freedom robot arm, in a MATLAB/SIMULINK environment, is used to exemplify the suggested techniques along with the LMI toolbox. Tradeoffs between stability and performance are carefully studied.

A nonlinear controller design for permanent magnet motors using a synchronization-based technique inspired from the Lorenz system

The dynamic behavior of a permanent magnet synchronous machine (PMSM) is analyzed. Nominal and special operating conditions are explored to show that the PMSM can experience chaos. A nonlinear controller is introduced to control these unwanted chaotic oscillations and to bring the PMSM to a stable steady state. The designed controller uses a pole-placement approach to force the closed-loop system to follow the performance of a simple first-order linear system with zero steady-state error to a desired set point. The similarity between the mathematical model of the PMSM and the famous chaotic Lorenz system is utilized to design a synchronization-based state observer using only the angular speed for feedback. Simulation results verify the effectiveness of the proposed controller in eliminating the chaotic oscillations while using a single feedback signal. The superiority of the proposed controller is further demonstrated by comparing it with a conventional PID controller. Finally, a laboratory-based experiment was conducted using the MCK2812 C Pro-MS(BL) motion control kit to confirm the theoretical results and to verify both the causality and versatility of the proposed controller.

Robust Estimation-Based Control of Chaotic Behavior in an Oscillator with Inertial and Elastic Symmetric Nonlinearities

In this paper, we study the dynamics of a forced nonlinear oscillator with inertial and elastic symmetric nonlinearities using modern nonlinear, bifurcation and chaos theories. The results for the response have shown that, for a certain combination of physical parameters, this oscillator exhibits a chaotic behavior or a transition to chaos through a sequence of period doubling bifurcations. The main objective of this paper is to control the chaotic behavior for this type of oscillator. A nonlinear estimation-based controller is proposed and the transient performance is investigated. The design of the parameter update mechanism is analyzed while discussing ways to extend its performance to further account for other types of uncertainties. We examine robustness problems as well as ways to tune the controller parameters. Simulation results are presented for the uncontrolled and controlled cases, verifying the effectiveness and the capability of the proposed controller. Finally, a discussion and conclusions are given with possible future extensions.

Tuning of Notch Filters for Controlling Chaos in a Chua's Circuit

This paper explores the use of notch filters for the purpose of damping out chaotic oscillations. The design of the filter and the way it is interfaced to the system are investigated from a signal-processing point of view. A Chuas circuit, that has typical applications in synchronization and secure communications, is used to exemplify the suggested methodology where both theoretical and experimental results are provided. The power spectrum of the original system is analyzed to selectively damp-out portions of the power spectrum, thus truncating period-doubling, the original cause of chaos. Both single and double notch filters are explored to examine their effect on the performance of the modified system. Practical implementations issues are addressed and advantages and limitations of the proposed method are highlighted.

Secure communication using Duffing oscillators

In this paper, a Duffing oscillator is used to construct a chaos-based secure communication system for transmitting digital signals. The synchronization of both the transmitter and the receiver is carried out using a Lyapunov-based control approach that observes the states of the transmitter. The parameters of the transmitter are assumed unknown, and are estimated at the receiver side to accomplish two tasks. The first one is to complete the synchronization process, while the second one is to implement a cryptography approach to secure the transmitted message. Different difficulty levels of constructing both the state observer and/or the parameters update laws are investigated, while highlighting the coupling effects between them. Tuning the proposed system, along with meeting hardware constraints when transmitting real-time signals are addressed, and recommendations for improving the performance are discussed.

Design of fast state observers using a backstepping-like approach with application to synchronization of chaotic systems.

A simple technique is introduced to build fast state observers for chaotic systems when only a scalar time series of the output is available. This technique relies on using a backstepping-like approach via introducing new virtual states that can be observed using the drive-response synchronization mechanism. The proposed dynamic structure of the virtual states allows for employing control parameters that can adjust the convergence rate of the observed states. In addition, these control parameters can be used to improve the transient performance of the response system to accommodate small and large variations of the initial conditions, thus achieving superior performance to conventional synchronization techniques. Simple Lyapunov functions are used to estimate the range of the control parameters that guarantees stable operation of the proposed technique. Three benchmark chaotic systems are considered for illustration; namely, the Lorenz, Chua, and Rossler systems. The conflict between stability and agility of the states observer is analyzed and a simple tuning mechanism is introduced. Implementation of the proposed technique in both analog and digital forms is also addressed and experimental results are reported ensuring feasibility and real-time applicability. Finally, advantages and limitations are discussed and a comparison with conventional synchronization methods is investigated.

A cryptography algorithm for transmitting multimedia data using quadruple-state CSK

A new technique for secure communication is introduced that aims at robustifying classical Chaotic Shift Keying (CSK) methods. The secret data are hidden within the chaotic transmitter states that can change among four different chaotic attractors such that binary information is effectively diffused. A novel cryptography algorithm is used to change the transmitter parameters such that they have a quadruple form; thus, breaking into the public communication channel using return map attacks will fail. At the receiver side, an adaptive control method is used to estimate the time-varying transmitter parameters via adopting a complete synchronization approach. Simulation results demonstrate the superior performance of the proposed technique in both time and frequency domains. A Duffing oscillator is used to build the proposed system using only the time series for the output. Different implementation issues are investigated for various digital multimedia data and an experimental investigation is carried out to verify the effectiveness of the proposed technique. Finally, generalizations to other chaotic systems as well as real-time compatibility of the design are discussed.