Use of squared magnitude function in approximation and hardware implementation of SISO fractional order system

This paper uses squared magnitude function and genetic algorithm (GA) to propose a generalized method for rational approximation of stable minimum phase fractional order transfer functions (FOTF). The unknown coefficients of an approximant are obtained by equating the squared magnitude of the approximant with that of the FOTF at frequency points (over a specified bandwidth) that are chosen optimally by using GA. Two FOTFs have been approximated by using the proposed method and one of the approximants has been realized in hardware using available discrete circuit components. The results of approximation and circuit realization strongly demonstrate the application of this work in approximation and/or realization of fractional order systems specifically, fractional order controller and filter of higher order.

[1]  A. Oustaloup,et al.  Fractional robust control with iso-damping property , 2008, 2008 American Control Conference.

[2]  Guido Maione,et al.  Continued fractions approximation of the impulse response of fractional-order dynamic systems , 2008 .

[3]  A. Charef.  Modeling and Analog Realization of the Fundamental Linear Fractional Order Differential Equation , 2006 .

[4]  D. Stewart,et al.  The missing memristor found , 2008, Nature.

[5]  Jan Terpak,et al.  Electronic realization of the fractional-order systems , 2007 .

[6]  Yih T. Tsay,et al.  A squared magnitude continued fraction expansion for stable reduced models , 1976 .

[7]  Y. Chen,et al.  A Modified Approximation Method of Fractional Order System , 2006, 2006 International Conference on Mechatronics and Automation.

[8]  B. Onaral,et al.  Linear approximation of transfer function with a pole of fractional power , 1984 .

[9]  D. Cafagna,et al.  Fractional calculus: A mathematical tool from the past for present engineers [Past and present] , 2007, IEEE Industrial Electronics Magazine.

[10]  G. Bohannan.  Analog Fractional Order Controller in Temperature and Motor Control Applications , 2008 .

[11]  S. Das,et al.  Functional Fractional Calculus for System Identification and Controls , 2007 .

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

[13]  B. Onaral,et al.  Fractal system as represented by singularity function , 1992 .

[14]  Chyi Hwang,et al.  Stable linear system reduction via a multipoint squared-magnitude continued-fraction expansion , 1988 .

[15]  Reza Ghaderi,et al.  Fractional order control of a coupled tank , 2010 .

[16]  Alain Oustaloup,et al.  Frequency-band complex noninteger differentiator: characterization and synthesis , 2000 .

[17]  Dingyu Xue,et al.  Time-constant robust analysis of a fractional order [proportional derivative] controller , 2011 .

[18]  Isabel S. Jesus,et al.  Development of fractional order capacitors based on electrolyte processes , 2009 .

[19]  Karabi Biswas,et al.  Realization of a Constant Phase Element and Its Performance Study in a Differentiator Circuit , 2006, IEEE Transactions on Circuits and Systems II: Express Briefs.

[20]  Vicente Feliu-Batlle,et al.  Robust fractional-order PI controller implemented on a laboratory hydraulic canal. , 2009 .

[21]  R Melicio,et al.  Fractional-order control and simulation of wind turbines with full-power converters , 2010, Melecon 2010 - 2010 15th IEEE Mediterranean Electrotechnical Conference.

[22]  Vishwesh A. Vyawahare,et al.  QFT LOOP SHAPING USING A FRACTIONAL INTEGRATOR - REAL-TIME APPLICATION TO A COUPLED-TANK SYSTEM , 2006 .

[23]  Guy Jumarie,et al.  Fractional multiple birth-death processes with birth probabilities λi(Δt)α+o((Δt)α) , 2010, J. Frankl. Inst..

[24]  A. Charef,et al.  Analogue realisation of fractional-order integrator, differentiator and fractional PI/spl lambda/D/spl mu/ controller , 2006 .

[25]  I. Podlubny,et al.  Analogue Realizations of Fractional-Order Controllers , 2002 .

[26]  AAM Arafa,et al.  Fractional modeling dynamics of HIV and CD4+ T-cells during primary infection , 2012, Nonlinear biomedical physics.

[27]  Igor Podlubny,et al.  Fractional-order systems and PI/sup /spl lambda//D/sup /spl mu//-controllers , 1999 .

[28]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[29]  Dominik Sierociuk,et al.  Experimental Evidence of Variable-Order Behavior of Ladders and Nested Ladders , 2013, IEEE Transactions on Control Systems Technology.

[30]  I. Podlubny.  Fractional-order systems and PIλDμ-controllers , 1999, IEEE Trans. Autom. Control..

[31]  Jie Li,et al.  Chaos in the fractional order unified system and its synchronization , 2008, J. Frankl. Inst..

[32]  Necati Özdemir,et al.  Fractional Order Control of Fractional Diffusion Systems Subject to Input Hysteresis , 2010 .

[33]  Rui Ding,et al.  Chaotic synchronization between different fractional-order chaotic systems , 2011, J. Frankl. Inst..

[34]  Hassen T. Dorrah,et al.  Design of aerospace control systems using fractional PID controller , 2012 .

[35]  A. Oustaloup,et al.  Numerical Simulations of Fractional Systems: An Overview of Existing Methods and Improvements , 2004 .

[36]  Luigi Fortuna,et al.  Fractional Order Systems: Modeling and Control Applications , 2010 .

[37]  Alina Voda,et al.  Optimal Approximation, Simulation and Analog Realization of the Fundamental Fractional Order Transfer Function , 2007, Int. J. Appl. Math. Comput. Sci..

[38]  Hironori A. Fujii,et al.  H(infinity) optimized wave-absorbing control - Analytical and experimental results , 1993 .

[39]  Alain Oustaloup,et al.  Numerical Simulations of Fractional Systems , 2003 .

[40]  Hyo-Sung Ahn,et al.  Fractional‐order iterative learning control for fractional‐order linear systems , 2011 .

[41]  Guido Maione,et al.  Laguerre approximation of fractional systems , 2002 .