Fractional Order Control of Fractional Diffusion Systems Subject to Input Hysteresis

This paper concerns the control of a time fractional diffusion system defined in the Riemann-Liouville sense. It is assumed that the system is subject to hysteresis nonlinearity at its input, where the hysteresis is mathematically modeled with the Duhem operator. To compensate the effects of hysteresis nonlinearity, a fractional order Proportional +Integral +Derivative (PID) controller is designed by minimizing integral square error. For numerical computation, the Riemann-Liouville fractional derivative is approximated by the Grunwald-Letnikov approach. A set of algebraic equations arises from this approximation, which can be solved numerically. Performance of the fractional order PID controllers are analyzed in comparison with integer order PID controllers by simulation results, and it is shown that the fractional order controllers are more advantageous than the integer ones.

[1]  A. Kilbas,et al.  A Cauchy-type problem for the diffusion-wave equation with Riemann-Liouville partial derivative , 2005 .

[2]  R. Bagley,et al.  On the Fractional Calculus Model of Viscoelastic Behavior , 1986 .

[3]  Serdar Ethem Hamamci,et al.  Fractional order PIλ control strategy for a Liquid Level System , 2010, 2010 Second World Congress on Nature and Biologically Inspired Computing (NaBIC).

[4]  I. Podlubny.  Fractional differential equations , 1998 .

[6]  Necati Özdemir,et al.  Fractional diffusion-wave problem in cylindrical coordinates , 2008 .

[7]  I. Schäfer,et al.  Modelling of coils using fractional derivatives , 2006 .

[8]  O. Agrawal.  A General Formulation and Solution Scheme for Fractional Optimal Control Problems , 2004 .

[9]  O. Agrawal.  Fractional Optimal Control of a Distributed System Using Eigenfunctions , 2007 .

[10]  B. D. Coleman,et al.  A constitutive relation for rate-independent hysteresis in ferromagnetically soft materials , 1986 .

[11]  Alexandra M. Galhano,et al.  Performance of Fractional PID Algorithms Controlling Nonlinear Systems with Saturation and Backlash Phenomena , 2007 .

[12]  F. Mainardi.  The fundamental solutions for the fractional diffusion-wave equation , 1996 .

[13]  A. Visintin.  Differential models of hysteresis , 1994 .

[14]  M. Krasnosel’skiǐ,et al.  Systems with Hysteresis , 1989 .

[15]  R. Hilfer.  Fractional Diffusion Based on Riemann-Liouville Fractional Derivatives † , 2000 .

[16]  O. Agrawal,et al.  AXIS-SYMMETRIC FRACTIONAL DIFFUSION-WAVE PROBLEM: PART I-ANALYSIS , 2008 .

[17]  J. Padovan,et al.  Diophantine type fractional derivative representation of structural hysteresis , 1997 .

[18]  Necati Özdemir,et al.  Fractional optimal control of a 2-dimensional distributed system using eigenfunctions , 2008 .

[19]  W. Wyss.  The fractional diffusion equation , 1986 .

[20]  I. Podlubny,et al.  Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives , 2005, math-ph/0512028.

[21]  Bernard D. Coleman,et al.  On a class of constitutive relations for ferromagnetic hysteresis , 1987 .

[22]  José António Tenreiro Machado,et al.  Application of Fractional Calculus in the Control of Heat Systems , 2007, J. Adv. Comput. Intell. Intell. Informatics.

[23]  Paolo Nistri,et al.  Mathematical Models for Hysteresis , 1993, SIAM Rev..

[24]  O. Agrawal.  Solution for a Fractional Diffusion-Wave Equation Defined in a Bounded Domain , 2002 .

[25]  Mohamed Abdalla Darwish,et al.  Existence of fractional integral equation with hysteresis , 2006, Appl. Math. Comput..

[26]  J. Padovan,et al.  Diophantine type fractional derivative representation of structural hysteresis , 1997 .

[27]  Yuriy Povstenko,et al.  Time-fractional radial diffusion in a sphere , 2008 .

[28]  Weihua Deng,et al.  Generating multi-directional multi-scroll chaotic attractors via a fractional differential hysteresis system , 2007 .

[29]  Mayergoyz,et al.  Mathematical models of hysteresis. , 1986, Physical review letters.

[30]  Necati Özdemir,et al.  Fractional optimal control problem of a distributed system in cylindrical coordinates , 2009 .

[31]  P. Lino,et al.  New tuning rules for fractional PIα controllers , 2007 .

[32]  C. Knospe,et al.  PID control , 2006, IEEE Control Systems.

[33]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .