Fractional Order Systems

The following sections are included:

• Preface

• Acknowledgments

• Contents

• List of Figures

• List of Tables

In this chapter a brief introduction to Fractional Calculus will be given. Starting from the historical roots of the Science of Fractional Calculus, this chapter aims to define fundamental notions and observable behaviours of FOS.

A new procedure that makes it possible to define the parameters of fractional order PI^λD^μ controller, designed to stabilize a first-order plant with delay-time, is proposed in this chapter. The complete set of the stabilizing PI^λD^μ parameters is determined using a version of the Hermite-Biehler theorem applicable to quasipolynomials. The widespread industrial use of PID controllers and the potentiality of their non integer order representation justifies a timely interest in PI^λD^μ tuning techniques.

A brief introduction on the fractional order chaotic systems will be given in this chapter. In such a system the order of derivatives in its state-space representation is a fractional one. A survey of several well-known integer and fractional order chaotic systems will be presented.

In this chapter the fundamental operator s^m, where m is a real number, is approximated via the binomial expansion of the backward difference. A hardware implementation of the differintegral operator is then proposed using the Field Programmable Gate Array (FPGA). Digital hardware implementation of a fractional-order differintegral operator requires a careful consideration of certain issues including system performance, hardware cost, and hardware speed. FPGA-based implementations are up to one hundred times faster than implementations based on microprocessors; this extra speed can be exploited to allow a higher performance in terms of digital approximations of fractional-order systems.

In this chapter the examples for digital implementation of fractional order controllers (FOC) will be given. A survey of several implementations and applications based on processor devices is presented. For illustration we present fractional order integrator, fractional order PD controller and fractional order ID controller.

The aim of this chapter is to propose an analog implementation of PI^λD^μ controller based on Field Programmable Analog Arrays (FPAAs). The proposed approach guarantees a good degree of approximation and the possibility to easily fix the integrative and derivative order. The frequency analysis of the reported examples shows the feasibility and reliability of the proposed approach.

A Switched Capacitors (SC) implementation of fractional differintegral operator is proposed in this chapter. Time and frequency domain result tests validate the feasibility and reliability of the SC circuit implementation. A detailed analysis of the influence of the non-idealities is proposed in order to obtain a design ready for Integrated Circuit (IC) implementation. The proposed approach may guarantee a good degree of approximation of the fractional differintegral operator and therefore may allow the possibility to realize a PI^λD^μ controller IC based on switched capacitors technology.

In this chapter a new model for Ionic Polymer Metal Composites (IPMC) actuators based on non-integer order models is proposed. IPMCs are very interesting polymers because of their capability to transform electrical energy into mechanical energy and vice-versa, making them particularly attractive for possible applications in different fields, such as robotics, aerospace, biomedicine, etc. An experimental setup has been realized to study the IPMCs behavior and an algorithm has been developed in Matlab environment in order to identify a fractional order model of IPMC actuators.

The following sections are included:

• Bibliography

• Index