Gian Carlo Gómez-Cortés

Synchronization of integral and fractional order chaotic systems : a differential algebraic and differential geometric approach with selected applications in real-time

A New Model-free Based Proportional Reduced-order Observer Design for the synchronization of Lorenz System.- A Model-free Sliding Observer to Synchronization Problem Using Geometric Techniques.- Experimental Synchronization by Means of Observers.- Synchronization of an Uncertain Rikitake System With Parametric Estimation.- Secure Communications and Synchronization via Sliding-mode Observer.- Synchronization and Anti-Synchronization of Chaotic Systems: A Differential and Algebraic Approach.- Synchronization of Chaotic Liouvillian Systems. An Application to Chuas Oscillator.- Synchronization of Partially Unknown Nonlinear Fractional Order Systems.- Generalized Synchronization via the Differential Primitive Element.- Generalized Synchronization for a Class of Non-Differentially Flat and Liouvilion Chaotic Systems.- Generalized Multi-synchronization by Means of a Family of Dynamical Feedbacks.- Fractional Generalized Synchronization in Nonlinear Fractional Order Systems Via a Dynamical Feedback.- An Observer for a Class of Incommensurate Fractional Order Systems.- Index.

Synchronization of nonlinear fractional order systems by means of PIrα reduced order observer

This paper deals with the master-slave synchronization scheme for partially known nonlinear fractional order systems, where the unknown dynamics are considered as the master system and we propose the slave system structure which estimates the unknown state variables. Besides it is introduced a new fractional model free reduced order observer inspired on the new concept of Fractional algebraic observability (FAO); we applied the results to a Rossler hyperchaotic fractional order system and Lorenz fractional order system, and by means of some simulations we show the effectiveness of the suggested approach.

Synchronization of incommensurate fractional order system

In this paper we present a new observer model free type for synchronization of incommensurate fractional order systems. We propose an observer structure that estimates the unknown state variables (master system), the estimates and the output are the slave system. For solving this problem, we introduce a new incommensurate fractional algebraic observability (IFAO) property which is used as the main ingredient in the design of the slave system. Some numerical results show the effectiveness of the suggested approach.

Control Theory and Synchronization

This chapter focuses on the main concepts necessary to a proper understanding of the topics developed in this book. In reading this chapter, the emphasis should be on definitions, and in addition, some examples will be given in order to clarify the concepts of control theory and synchronization. The most important issues are topics on differential algebra, differential geometry, and their applications.

Generalized Synchronization for a Class of Nondifferentially Flat and Liouvillian Chaotic Systems

In this chapter, we study the problem of generalized synchronization (GS) for a class of nondifferentially flat and Liouvillian systems. In nonlinear systems, the GS of a state exists if it is possible to define a mapping such that the states of one system are equal to those of another. From the point of view of differential algebra, we propose a very special transformation in order to achieve GS for this class of systems that have a nonflat output. The transformation is generated via the differential primitive element, which is a linear combination of states and inputs of the system with coefficients in a differential field. Finally, we construct a dynamical control obtained through a chain of integrators to reach the GS. This is illustrated by means of numerical simulations to show the effectiveness of the methodology proposed.

Generalized Multisynchronization by Means of a Family of Dynamical Feedbacks

In this chapter, we study the problem of generalized multisynchronization (GMS) of multiple decoupled chaotic systems under the master–slave configuration. From the viewpoint of differential algebra, we propose a family of transformations in order to achieve GMS for the whole slave family. The family of transformations is generated via the family of differential primitive elements, which are linear combinations of states and inputs of the families of master systems and slave systems with coefficients in a differential field. Finally, we construct multiple dynamical controls to reach the GMS. The effectiveness of the methodology proposed is illustrated by means of numerical simulations.

An Observer for a Class of Incommensurate Fractional-Order Systems

In this chapter, we present a new observer-model-free type of synchronization for a certain class of incommensurate fractional-order systems. We apply our proposals to the master–slave synchronization scheme, where the unknown dynamics are considered the master system, and we propose an observer structure as a slave system that estimates the unknown state variables. For solving this problem, we introduce a new incommensurate fractional algebraic observability (IFAO) property, which is used as the main ingredient in the design of the slave system. Some numerical results show the effectiveness of the suggested approach.

A Model-free Model-Free-Based Proportional Reduced-order observer Reduced-Order Observer Design for the Synchronization of Lorenz system Lorenz Systems

In this chapter, we deal with the synchronization problem of a Lorenz system using a proportional reduced-order observer design in the algebraic and differential settings. We prove the asymptotic stability of the resulting error system, and by means of algebraic manipulations, we obtain estimates of the current states (master system). In this chapter, the construction of a proportional reduced-order observer is the main ingredient in our approach. Finally, we present simulations to illustrate the effectiveness of the suggested approach.

Synchronization of an Uncertain Rikitake System with Parametric Estimation

In this chapter, we deal with the synchronization and parameter estimations of an uncertain Rikitake system. The strategy consists in proposing a slave system that has to follow asymptotically the unknown Rikitake system, referred to as the master system. The gains of the slave system are adjusted continually according to a convenient adaptation control law until the measurable output errors converge to zero. The convergence analysis is carried out using Barbalat’s lemma.

Synchronization of Partially Unknown Nonlinear Fractional-Order Systems

In this chapter, a synchronization scheme for partially known nonlinear fractional order systems is proposed. In this approach, the unknown dynamics is considered as the master system, and the slave system estimates the unknown variables. For solving this problem, we introduce a fractional algebraic observability property, which is used as the main tool in the design of the master system. As a numerical example, we consider a fractional-order Lorenz chaotic system, and by means of some simulations, we demonstrate the effectiveness of the suggested approach.