Claudia A. Pérez-Pinacho

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.

Generalized multi-synchronization viewed as a multi-agent leader-following consensus problem

The work presents the generalized multi-synchronization of strictly different chaotic systems problem as a multi-agent leader-following consensus problem with fixed and not strongly connected topology from a differential algebra point of view. This problem is solved by using the differential primitive element as a linear combination of state measurements and control inputs from which it is possible to construct a family of transformations to carry out the multi-agent systems to a Generalized Observability Canonical Form Multi-output (GOCFM). Moreover, a dynamic consensus protocol is designed such that the states of the followers asymptotically converge to the state of the leader.

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.

Generalized multi-synchronization

In this paper, the problem of Generalized Multi-Synchronization (GMS) in master-slave topology is addressed. Within a differential algebraic framework this problem is interpreted as a leader-following consensus problem of Multi-Agent Systems (MAS). Here, a multi-agent system is treated as a network of interconnected systems with strictly different dynamics of same dimension, fixed and not strongly connected topology. Multi-agent system is carried out to a Multi-output Generalized observability Canonical Form (MGOCF) with a family of transformations obtained from an adequate selection of the differential primitive element as a linear combination of state measurements and control inputs. This allow us to explicitly give the synchronization algebraic manifold and design a dynamic consensus protocol able to asymptotically achieve consensus for all agents in the network. Finally a worked out example is provided to illustrate the methodology proposed.

Synchronization and Anti-synchronization of Fractional Order Chaotic Systems by Means of a Fractional Integral Observer

The problem of anti-synchronization is another phenomenon of interest that occurs in chaotic oscillators. This problem has appeared in modern repetitions of Huygens’ experiments (Bennett et al., Proc: Math Phys Eng Sci 458:563–579, 2002, [2]), lasers (Uchida et al., Phys Rev A 64:023801-1–023801-6, 2001, [2]), (Wedekind and Parlitz, Int J Bifurc Chaos 11(4):1141–1147, 2001, [3]), saltwater oscillators (Nakata et al., Phys D 115:313–320, 1998, [4]), and some biological systems where a nonchaotic signal is generated (Kim et al., Phys Lett A 320:39–46, 2003, [5]). Anti-synchronization has been treated as a direct modification of synchronization, simply with a sign change in the condition required for the error, and has been attacked with methods such as the active control (Emadzadeh and Haeri, Int J Electr Comput Energ Electron Commun Eng 1(6):898–901, 2007, [6]) (Guo-Hui, Chaos, Solitons Fractals 26:87–93, 2005, [7]) and the sliding mode control (Chen et al., Nonlinear Dyn 69:35–55, 2012, [8]). It can also be induced by noise (Kawamura, Phys D 270(1):20–29, 2014, [9]).

Generalized Multi-synchronization of Fractional Order Liouvillian Chaotic Systems Using Fractional Dynamical Controller

The attempt to understand the synchronize of a pair of systems have been extended to the study of a more complex problem involving multiple systems, of course, motivated by problems in the integer order case where synchronization is observed such as rendezvous, formation control, flocking and schooling, attitude alignment, sensor networks, distributed computing, consensus, and complex networks in general Florian Dorfler and Francesco Bullo, Automatica, 50(6):1539–1564, 2014, [1], Strogatz, Nature, 410(6825):268–276, 2001, [6], Martinez-Guerra et al., Applied Mathematics and Computation, 282:226–236, 2016, [4], Hale, J. Dyn. Diff. Eqns. 9(1):1–52, 1997, [2], Lin and Wang, Fuzzy Sets and Systems, 161(15):2066–2080, 2010, [3], Wang et al., IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 40(6):1468–1479, 2010, [7], Reza Olfati Saber et al., Proceedings of the IEEE, 95(1):215–233, 2007, [5], Wei Ren et al., IEEE Control Systems Magazine, 72–81, 2007, [8].

Synchronization for Chaotic System Through an Observer Using the Immersion and Invariance (I&I) Approach

In this chapter, the named master–slave configuration, where nonlinear systems observers and chaos synchronization are used together. The key idea is to design observers to accomplish chaos synchronization, where the slave is actually an observer coupled to the master through its corresponding output. This chapter aims at the master–slave synchronization by applying the Immersion and Invariance (I&I) method to solve the chaos synchronization problem for a kind of simple chaotic systems. To this end, a class of feedback-linearized chaotic systems are characterized. Afterwards, the I&I method is applied to propose the corresponding observer, or slave system, for such systems. This observer, which has some robust properties, allows asymptotic estimation of the underlying dynamics of the master system. Notably, the I& I approach has been successfully applied to control, identify and observe a wide range of nonlinear systems. The seminal ideas of the I& I approach and its application can be found in Astolfi and Ortega (IEEE Trans. Autom. Control 48(4):590–606, 2003, [1], X. Liu (IEEE Trans. Autom. Control 55(9):2209–2214, 2010, [2]). The chapter is organized as follows. In Sect. 4.1, the problem statement is established. In Sect. 4.2, the observer is proposed to solve the synchronization problem, by applying the I& I method. Section 4.3 shows the results of some numerical comparisons with other well-known observers. The conclusions are given in Sect. 4.4.

Basic Concepts and Preliminaries

In this chapter, definitions and concepts about the fractional calculus are presented. Some of the concepts and definitions have been divided into two parts in order to better locate the topic of interest, such as the concepts of commensurate and incommensurate systems. Taking into account that there are tools that serve the same for both cases.

Fractional Generalized Quasi-synchronization of Incommensurate Fractional Order Oscillators

GS was introduced in Rulkov, Sushchik, Tsimring, Abarbanel (Phys Rev E 51, 980–994, 1995, [1]), but here definitions are extended and given in our own conception, for fractional order nonlinear systems, by using the fractional incommensurate differential primitive element. The problem of the Fractional Generalized Synchronization (FGS) was studied for a class of strictly different nonlinear commensurate fractional order systems in the master–slave configuration scheme (Martinez-Guerra, Mata-Machuca, Nonlinear Dyn Vol. 77, 1237–1244, 2014, [2]). Recently, numerous works have been reported on the problem of synchronization for incommensurate fractional order chaotic systems (Razminia, Majd, Baleanu, Adv Differ Equ 1–12, 2011, [3]), (Delshad, Asheghan, Beheshti, Commun Nonlinear Sci Numer Simul 3815–3824, 2011, [4]), (Wang, Zhang, Phys Lett A, 202–207, 2009, [5]), (Boulkroune, Bouzeriba, Bouden, Neurocomput Vol 173, 606–614, 2016, [6]). In general, study synchronization of strictly different systems is equivalent to study the asymptotic stability of the origin of the synchronization error or the stability of the synchronization manifold if possible. In many of these references, the stability of the incommensurate fractional order dynamics of the synchronization error is translated into a problem of stability of a commensurate fractional order or even an integer order system through a change of variable. In this chapter, we will show a convergence analysis directly from the incommensurate fractional order dynamics of the synchronization error. It is natural to present the incommensurate fractional order dynamics of the synchronization error in a modal decomposition due to each dynamics have different fractional order. Thus, we can obtain asymptotic convergence in a compact region near the origin in case of synchronization error for generalized synchronization of strictly different incommensurate fractional- order systems by using dynamical controllers obtained from differential algebraic techniques. In this chapter, the main contribution is a Fractional Generalized Synchronization constructive method for nonlinear incommensurate fractional-order chaotic systems in a master–slave topology, this phenomena is studied from an algebraic and differential point of view, that allows us to construct an Incommensurate Fractional Generalized Observability Canonical Form (IFGOCF) from an adequate selection of a fractional differential primitive element and moreover gives explicitly the form of the synchronization algebraic manifold for strictly different fractional order nonlinear systems. The former enables us to design an incommensurate fractional order dynamical controller able to achieve synchronization of strictly different incommensurate fractional order chaotic systems. Moreover, we introduce the concepts so-called Incommensurate Fractional Algebraic Observability and a fractional order Picard–Vessiot system. As far as we know, synchronization of strictly different incommensurate fractional order systems have not been tackled from this perspective.