Behavior Studies of Nonlinear Fractional-Order Dynamical Systems Using Bifurcation Diagram

Behavior Studies of Nonlinear Fractional-Order Dynamical Systems Using Bifurcation Diagram

Karima Rabah (20th August 1955 University of Skikda, Algeria)
DOI: 10.4018/978-1-5225-5418-9.ch006


Over the past decades, chaos has stimulated the interest of researchers due to its existence in different fields of science and engineering. The chaotic systems are characterized by their sensitivity to the initial conditions. This property makes the system unpredictable long term. Similar to the integer-order differential systems, fractional-order differential systems can exhibit chaotic behaviors. This type of system contains one or more elements of fractional order. The fractional calculus is recognized in the early seventeenth century but it has been widely applied in many fields and with intense growth just over the past decades. To avoid troubles arising from unusual behaviors of a chaotic system, chaos control has gained increasing attention in recent years. An important objective of a chaos controller is to suppress the chaotic oscillations completely or reduce them to the regular oscillations. The goal of this chapter is to present the evolution of chaotic systems in open and closed loop in function of their parameters and designing a controller using bifurcation diagrams.
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Fractional systems, or more non-integer order systems, can be considered as a generalization of integer order systems (Oldham & Spanier, 2006; Sabatier et al., 2007; Kilbas et al., 2006; Das, 2008).

Emergence of effective methods to solve differentiation and integration of non-integer order equations makes fractional-order systems more and more attractive for the systems control community (Rabah et al., 2015). In order to improve the performance of linear feedback systems, Podlubny (1999) proposed a generalization of the classical PID controller to PIλDμ form called the fractional PID, which has recently become very popular due to its additional flexibility to meet design specifications. Since, fractional order PID (FOPID) controllers have found application in several power systems as cited in the following. In Pan & Das (2012), a fractional order PID controller is designed to take care of various contradictory objective functions for an automatic voltage regulator (AVR) system. Bouafoura & Braiek (2010), deals with the design of fractional order PID controller for integer and fractional plants. In Chen et al. (2014), a FOPID is designed for the hydraulic turbine regulating system (HTRS) with the consideration of conflicting performance objectives. As well, the comparative study between the optimum of PID and FOPID controllers improve the superiority of the fractional order controllers over the integer controllers. In Faieghiet al. (2011), the author makes a design of Fractional-Order PID for ship roll motion control using chaos embedded PSO algorithm. In Tang et al. (2012), optimum design of fractional order PID controller for AVR system using chaotic ant swarm. Based on the fractional high gain adaptive control approach, Ladaci et al. (2009) introduced new tuning parameters for the performance behavior improvement of the controlled plant.

In this chapter, we are interested in the analysis of fractional order nonlinear systems and controls in temporal domain fractional PID controllers based on bifurcation diagram to avoid troubles arising from unusual and undesired behaviors.

The bifurcation diagram is one of the most important tools for determining the behavior of the dynamical systems. It shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the system. It rapidly evaluates all possible solutions of a system according to the variations of one of its parameters or its fractional order.

The aim of this work can be resumed in the following points:

  • 1.

    Study of the fractional-order chaotic system evolution in function of fractional-order using also the bifurcation diagram.

  • 2.

    Study of the newselected fractional-order nonlinear chaotic system behavior according to its own parameters using the bifurcation diagram.

  • 3.

    Designing a controller with a bifurcation diagram, particularly a fractional order PID controller, adjusted using this tool.

Simulation examples are given for the different categories of fractional-order chaotic systems, analysis and control design to illustrate the effectiveness of this mathematical tool.


Elements Of Fractional Calculus Theory

Fractional calculus is an old mathematical research topic, but it is retrieving popularity nowadays. Recent references Miller & Ross (1993) provide a good source of documentation on fractional systems and operators. However, the topics about application of fractional-order operator theory to dynamic systems control are just a recent focus of interest (Oustaloup, 1991, Ladaci & Charef, 2006, Rabah et al. 2017b).

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