## Fractional Order Systems: Modeling and Control ApplicationsThis book aims to propose implementations and applications of Fractional Order Systems (FOS). It is well known that FOS can be applied in control applications and systems modeling, and their effectiveness has been proven in many theoretical works and simulation routines. A further and mandatory step for FOS real world utilization is their hardware implementation and applications on real systems modeling. With this viewpoint, introductive chapters on FOS are included, on the definition of stability region of Fractional Order PID Controller and Chaotic FOS, followed by the practical implementation based on Microcontroller, Field Programmable Gate Array, Field Programmable Analog Array and Switched Capacitor. Another section is dedicated to FO modeling of Ionic Polymeric Metal Composite (IPMC). This new material may have applications in robotics, aerospace and biomedicine. |

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### Contents

1 Fractional Order Systems | 1 |

2 Fractional Order PID Controller and their Stability Regions Definition | 33 |

3 Fractional Order Chaotic Systems | 53 |

4 Field Programmable Gate Array Implementation | 77 |

5 Microprocessor Implementation and Applications | 91 |

6 Field Programmable Analog Array Implementation | 115 |

7 Switched Capacitor Integrated Circuit Design | 127 |

8 Fractional Order Model of IPMC | 145 |

167 | |

177 | |

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### Common terms and phrases

algorithm amplitude analog applied approximation attractor bilinear bilinear filter Bode diagrams calculated Caponetto characteristic equation Chua’s circuit closed-loop system coefficients comparison between predicted complex control system defined deflection differintegral discrete domain dynamical FPAA FPGA fractional calculus fractional derivatives fractional differential equation fractional integrator fractional order controller fractional order integrator fractional order system fractional system fractional-order system Frequency Hz hardware Hermite-Biehler theorem implementation initial conditions input signal integer order model integrator of order intentionally left blank loop Matignon method Nafion nonlinear obtained order q output parameters Kp Petráˇs phase margin Phase plane PID controllers Podlubny 1999a poles polynomial predicted and measured prefilter quasipolynomials rational realized resistor roots sampled data filter shown in Fig solution Stability region step response switched capacitor system where C(s Tavazoei tion transfer function Unit step responses values Vinagre Volta’s system voltage z-transformation zeros