Numerical simulation of fractional-order dynamical systems in noisy environments

In this paper, the fully discrete scheme is proposed based on the Simpson’s quadrature formula to approximate fractional-order integrals for noisy signals. This strategy is extended to simulate the response of fractional-order differential systems in noisy environments. The proposed technique is considered in determining statistical indicators for noisy signals in fractional electrical networks with white noise-influenced potential sources.

[1]  G. H. Erjaee,et al.  Derivation of a new Merton's optimal problem presented by fractional stochastic stock price and its applications , 2017, Comput. Math. Appl..

[2]  Dung Nguyen Tien.  Fractional stochastic differential equations with applications to finance , 2013 .

[3]  A. Friedman.  Stochastic Differential Equations and Applications , 1975 .

[5]  Esmat Sadat Alaviyan Shahri,et al.  Stabilization of Fractional-Order Systems Subject to Saturation Element Using Fractional Dynamic Output Feedback Sliding Mode Control , 2017 .

[6]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

[7]  E. Kolárová,et al.  Confidence intervals for RLCG cell influenced by coloured noise , 2017 .

[8]  Morad Nazari,et al.  Coefficient of restitution in fractional viscoelastic compliant impacts using fractional Chebyshev collocation , 2017 .

[9]  Fanhai Zeng,et al.  Numerical Methods for Fractional Calculus , 2015 .

[10]  José António Tenreiro Machado,et al.  Optimal variable-order fractional PID controllers for dynamical systems , 2018, J. Comput. Appl. Math..

[11]  E. Kolárová,et al.  Simulation of Higher-Order Electrical Circuits with Stochastic Parameters via SDEs , 2013 .

[12]  Michel Fogli,et al.  Reliability and Sensitivity Analysis of Transmission Lines in a Probabilistic EMC Context , 2016, IEEE Transactions on Electromagnetic Compatibility.

[13]  J. A. Tenreiro Machado,et al.  Stability analysis of a class of nonlinear fractional‐order systems under control input saturation , 2018 .

[14]  Eric A. Butcher,et al.  Numerical solution of multi-order fractional differential equations with multiple delays via spectral collocation methods , 2018 .

[15]  Xiaocui Li,et al.  Error Estimates of Finite Element Methods for Stochastic Fractional Differential Equations , 2017 .

[16]  R. Bagley,et al.  A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity , 1983 .

[17]  R. Bagley,et al.  On the Appearance of the Fractional Derivative in the Behavior of Real Materials , 1984 .

[18]  Behrouz Parsa Moghaddam,et al.  A numerical method for solving Linear Non-homogenous Fractional Ordinary Differential Equation , 2012 .

[19]  Pierre Bonnet,et al.  Stochastic EMC/EMI Experiments Optimization Using Resampling Techniques , 2016, IEEE Transactions on Electromagnetic Compatibility.

[20]  J. A. Tenreiro Machado,et al.  Chaos suppression in fractional systems using adaptive fractional state feedback control , 2017 .

[22]  Lubomír Brancík,et al.  Simulation of multiconductor transmission lines with random parameters via stochastic differential equations approach , 2016, Simul..

[23]  Chris Chatfield,et al.  Statistics for Technology (A Course in Applied Statistics) , 1984 .

[24]  Behrouz Parsa Moghaddam,et al.  A Robust Algorithm for Nonlinear Variable-Order Fractional Control Systems with Delay , 2018 .

[25]  Rahman Farnoosh,et al.  Parameters estimation for RL electrical circuits based on least square and Bayesian approach , 2012 .

[26]  J. Toivanen,et al.  ADI schemes for valuing European options under the Bates model , 2017, Applied Numerical Mathematics.

[27]  A. Luo,et al.  Fractional Dynamics and Control , 2011 .

[28]  E. Kolárová,et al.  Application of Stochastic Differential-Algebraic Equations in Hybrid MTL Systems Analysis , 2014 .

[29]  E. Kolafova.  Modelling RL Electrical Circuits by Stochastic Diferential Equations , 2005, EUROCON 2005 - The International Conference on "Computer as a Tool".

[30]  Rahman Farnoosh,et al.  Estimation of parameters in the state space model of stochastic RL electrical circuit , 2013 .

[31]  J. A. Tenreiro Machado,et al.  A NEW FRACTIONAL DERIVATIVE WITHOUT SINGULAR KERNEL Application to the Modelling of the Steady Heat Flow , 2015, 1601.01623.

[32]  E. Butcher,et al.  Stable fractional Chebyshev differentiation matrix for the numerical solution of multi-order fractional differential equations , 2017 .

[33]  Feng Gao,et al.  A new fractional derivative involving the normalized sinc function without singular kernel , 2017, 1701.05590.

[34]  José António Tenreiro Machado,et al.  SM-Algorithms for Approximating the Variable-Order Fractional Derivative of High Order , 2017, Fundam. Informaticae.

[35]  G. S. Ladde,et al.  Development of modified Geometric Brownian Motion models by using stock price data and basic statistics , 2009 .

[36]  E. Kolárová,et al.  Application of Stochastic Differential Equations in Second-Order Electrical Circuits Analysis , 2012 .

[37]  Juan J. Nieto,et al.  Analytical Solutions for Multi-Time Scale Fractional Stochastic Differential Equations Driven by Fractional Brownian Motion and Their Applications , 2018, Entropy.

[38]  Morad Nazari,et al.  The spectral parameter estimation method for parameter identification of linear fractional order systems , 2016, 2016 American Control Conference (ACC).

[39]  Alireza Alfi,et al.  Bilateral control of uncertain telerobotic systems using Iterative Learning Control: Design and stability analysis , 2019 .

[40]  M. Crow,et al.  Numerical simulation of Stochastic Differential Algebraic Equations for power system transient stability with random loads , 2011, 2011 IEEE Power and Energy Society General Meeting.

[41]  Abdon Atangana,et al.  Electrical circuits RC, LC, and RL described by Atangana–Baleanu fractional derivatives , 2017, Int. J. Circuit Theory Appl..

[42]  Isabel S. Jesus,et al.  Fractional control of heat diffusion systems , 2008 .

[43]  Qin Tang,et al.  Stochastic Analysis of Deep-Submicrometer CMOS Process for Reliable Circuits Designs , 2011, IEEE Transactions on Circuits and Systems I: Regular Papers.

[44]  P. Balasubramaniam,et al.  Moment stability via resolvent operators of fractional stochastic differential inclusions driven by fractional Brownian motion , 2017, Appl. Math. Comput..

[45]  Kai Strunz,et al.  Stochastic circuit modelling with Hermite polynomial chaos , 2005 .

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

[47]  Sandip Banerjee,et al.  Mathematical Modeling , 2014 .

[48]  Farshid Mirzaee,et al.  Application of hat basis functions for solving two-dimensional stochastic fractional integral equations , 2018 .

[49]  Philippe Besnier,et al.  The Adaptive Controlled Stratification Method Applied to the Determination of Extreme Interference Levels in EMC Modeling With Uncertain Input Variables , 2016, IEEE Transactions on Electromagnetic Compatibility.

[50]  J. Machado,et al.  EXACT TRAVELING-WAVE SOLUTION FOR LOCAL FRACTIONAL BOUSSINESQ EQUATION IN FRACTAL DOMAIN , 2017 .

[51]  Athanasios Papoulis,et al.  Probability, Random Variables and Stochastic Processes , 1965 .

[52]  Morad Nazari,et al.  Optimal Periodic-Gain Fractional Delayed State Feedback Control for Linear Fractional Periodic Time-Delayed Systems , 2018, IEEE Transactions on Automatic Control.

[53]  Zu-Guo Yu,et al.  Modeling and simulation of the horizontal component of the geomagnetic field by fractional stochastic differential equations in conjunction with empirical mode decomposition , 2010 .

[54]  E. Kolarova.  Statistical Estimates of Stochastic Solutions of RL Electrical Circuits , 2006, 2006 IEEE International Conference on Industrial Technology.

[55]  Hari M. Srivastava,et al.  A new computational approach for solving nonlinear local fractional PDEs , 2017, J. Comput. Appl. Math..

[56]  E. Kolárová.  Applications of second order stochastic integral equations to electrical networks , 2015 .

[57]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[58]  H. T. Tuan,et al.  Asymptotic separation between solutions of Caputo fractional stochastic differential equations , 2017, 1711.08622.

[59]  JF Gómez-Aguilar,et al.  Electrical circuits RC and RL involving fractional operators with bi-order , 2017 .

[60]  J. F. Aguilar.  Behavior characteristics of a cap-resistor, memcapacitor, and a memristor from the response obtained of RC and RL electrical circuits described by fractional differential equations , 2016 .

[61]  B. Øksendal.  Stochastic Differential Equations , 1985 .

[62]  Intermittency fronts for space-time fractional stochastic partial differential equations in $(d+1)$ dimensions , 2016, 1602.07262.

[63]  Morad Nazari,et al.  Stability and Control of Fractional Periodic Time-Delayed Systems , 2017 .

[64]  Muhammad Aslam Uqaili,et al.  A comparative mathematical analysis of RL and RC electrical circuits via Atangana-Baleanu and Caputo-Fabrizio fractional derivatives , 2018 .