Revisiting four approximation methods for fractional order transfer function implementations: Stability preservation, time and frequency response matching analyses

Abstract Due to high computational load of ideal realization of fractional order elements, fractional order transfer functions are commonly implemented via integer-order, limited-band approximate models. An important side effect of such a non-ideal fractional order controller function realization for control applications is that the approximate fractional order models may deteriorate practical performance of optimal control tuning methods. Two major concerns come out for approximate realization in fractional-order control. These are stability preservation and model response matching properties. This study revisits four fundamental fractional order approximation methods, which are Oustaloup's method, CFE method, Matsuda's method and SBL fitting method, and considers stability preservation, time and frequency response matching performances. The study firstly presents a detailed review of Oustaloup's method, CFE method, Matsuda's method. Then, a modified version of SBL fitting method is presented. The stability preservation properties of approximation methods are investigated according to critical root placements of corresponding approximation method. Stability issue is highly significant for control applications. For this reason, a detailed analysis and comparision of stability preservation properties of these four approximation methods are investigated. Moreover, approximate implementations of an optimally tuned FOPID controller function are performed according to these four methods and compared for closed loop control of a large time delay system. Findings of this study indicate a fact that approximate models can considerably influence practical performance of optimally tuned FOPID control systems and ignorance of limitations of approximation methods in optimal tuning solutions can significantly affect real world performances.

[1]  Celaleddin Yeroglu,et al.  Limit Cycles in Nonlinear Systems with Fractional Order Plants , 2014 .

[2]  C. Yeroglu,et al.  Note on fractional-order proportional–integral–differential controller design , 2011 .

[3]  A. Kılıçman,et al.  Approximate solution of integro-differential equation of fractional (arbitrary) order , 2016 .

[4]  Baris Baykant Alagoz,et al.  FOPID Controllers and Their Industrial Applications: A Survey of Recent Results , 2018 .

[5]  Cemal Keles,et al.  Design of fractional-order PI controllers for disturbance rejection using RDR measure , 2014, ICFDA'14 International Conference on Fractional Differentiation and Its Applications 2014.

[6]  R. Iskander,et al.  Analog circuits sizing using the fixed point iteration algorithm with transistor compact models , 2012, Proceedings of the 19th International Conference Mixed Design of Integrated Circuits and Systems - MIXDES 2012.

[7]  Shantanu Das,et al.  Design of new practical phase shaping circuit using optimal pole–zero interlacing algorithm for fractional order PID controller , 2017 .

[8]  Sirish L. Shah,et al.  Continuous-time model identification of fractional-order models with time delays , 2011 .

[9]  Yong Wang,et al.  Rational approximation of fractional order systems by vector fitting method , 2017 .

[10]  Delfim F. M. Torres,et al.  Discrete direct methods in the fractional calculus of variations , 2012, Comput. Math. Appl..

[11]  Nusret Tan,et al.  An integer order approximation method based on stability boundary locus for fractional order derivative/integrator operators. , 2016, ISA transactions.

[12]  Antonio Visioli,et al.  Tuning rules for optimal PID and fractional-order PID controllers , 2011 .

[13]  Kevin L. Moore,et al.  Relay feedback tuning of robust PID controllers with iso-damping property , 2005, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[14]  J. Sabatier,et al.  From fractal robustness to the CRONE approach , 1998 .

[15]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

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

[17]  Djennoune Said,et al.  Comparison between two approximation methods of state space fractional systems , 2011 .

[18]  Baris Baykant Alagoz,et al.  Hurwitz stability analysis of fractional order LTI systems according to principal characteristic equations. , 2017, ISA transactions.

[19]  K. Moore,et al.  Discretization schemes for fractional-order differentiators and integrators , 2002 .

[20]  Alain Oustaloup,et al.  Fractional Order Differentiation and Robust Control Design: CRONE, H-infinity and Motion Control , 2015 .

[21]  Yangquan Chen,et al.  Two direct Tustin discretization methods for fractional-order differentiator/integrator , 2003, J. Frankl. Inst..

[22]  Baris Baykant Alagoz,et al.  A note on applications of time-domain solution of Cole permittivity models , 2017 .

[23]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[24]  Eduard Petlenkov,et al.  A Numerical Study for Plant-Independent Evaluation of Fractional-order PID Controller Performance 1 1This study is based upon works from COST Action CA15225, a network supported by COST (European Cooperation in Science and Technology). , 2018 .

[25]  Y. Chen,et al.  A comparative introduction of four fractional order controllers , 2002, Proceedings of the 4th World Congress on Intelligent Control and Automation (Cat. No.02EX527).

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

[27]  Haakon Waadeland,et al.  Continued fractions with applications , 1994 .

[28]  Bilal Senol,et al.  A numerical investigation for robust stability of fractional-order uncertain systems. , 2014, ISA transactions.

[29]  D. Matignon.  Stability results for fractional differential equations with applications to control processing , 1996 .

[30]  A. Elwakil,et al.  On the stability of linear systems with fractional-order elements , 2009 .

[31]  Tianyu Liu,et al.  Fixed pole based modeling and simulation schemes for fractional order systems. , 2019, ISA transactions.

[32]  Antonio Visioli,et al.  On the fragility of fractional-order PID controllers for IPDT processes , 2017, MED.

[33]  Amit Konar,et al.  Approximation of a Fractional Order System by an Integer Order Model Using Particle Swarm Optimization Technique , 2008, ArXiv.

[34]  Marina Popolizio,et al.  Numerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag–Leffler Functions , 2018 .

[35]  S. Roy.  On the Realization of a Constant-Argument Immittance or Fractional Operator , 1967, IEEE Transactions on Circuit Theory.

[36]  Eduard Petlenkov,et al.  Model Reference Adaptive Control Scheme for Retuning Method-Based Fractional-Order PID Control with Disturbance Rejection Applied to Closed-Loop Control of a Magnetic Levitation System , 2018, J. Circuits Syst. Comput..

[37]  Eduard Petlenkov,et al.  Time-domain identification of One Noninteger Order Plus Time Delay models from step response measurements , 2019, Int. J. Model. Simul. Sci. Comput..

[38]  Robin De Keyser,et al.  An efficient algorithm for low-order direct discrete-time implementation of fractional order transfer functions. , 2018, ISA transactions.

[39]  Cemal Keles,et al.  Disturbance rejection performance analyses of closed loop control systems by reference to disturbance ratio. , 2015, ISA transactions.

[40]  Jun-Sheng Duan.  A generalization of the Mittag–Leffler function and solution of system of fractional differential equations , 2018, Advances in Difference Equations.

[41]  Jianfei Huang,et al.  The Grünwald-Letnikov method for fractional differential equations , 2011, Comput. Math. Appl..

[42]  D. Casagrande,et al.  The Integer–Order Approximation of Fractional–Order Systems in The Loewner Framework , 2019, IFAC-PapersOnLine.

[43]  Saptarshi Das,et al.  Fractional Order Signal Processing: Introductory Concepts and Applications , 2011 .

[44]  Yong Wang,et al.  State space approximation for general fractional order dynamic systems , 2014, Int. J. Syst. Sci..

[45]  Nusret Tan,et al.  A New Integer Order Approximation Table for Fractional Order Derivative Operators , 2017 .

[46]  Pragya Varshney,et al.  Rational approximation of fractional order systems using Carlson method , 2015, 2015 International Conference on Soft Computing Techniques and Implementations (ICSCTI).

[47]  YangQuan Chen,et al.  Tuning and auto-tuning of fractional order controllers for industry applications , 2008 .

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

[49]  Biswajit Basu,et al.  Fractional order models for system identification of thermal dynamics of buildings , 2016 .

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

[51]  B. Alagoz.  A note on robust stability analysis of fractional order interval systems by minimum argument vertex and edge polynomials , 2016, IEEE/CAA Journal of Automatica Sinica.

[52]  Antonio Visioli,et al.  On the fragility of fractional-order PID controllers for IPDT processes , 2017, 2017 25th Mediterranean Conference on Control and Automation (MED).

[53]  YangQuan Chen,et al.  Fractional order control - A tutorial , 2009, 2009 American Control Conference.

[54]  W. Glöckle,et al.  A fractional model for mechanical stress relaxation , 1991 .

[55]  Chi-Hsu Wang,et al.  Finding the integer order systems for fractional order systems via fractional operational matrices , 2012, Proceedings of 2012 9th IEEE International Conference on Networking, Sensing and Control.

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

[57]  B. T. Krishna.  Studies on fractional order differentiators and integrators: A survey , 2011, Signal Process..