A Dimensional Consistency Aware Time Domain Analysis of the Generic Fractional Order Biquadratic System
Keywords:fractional order biquadratic system, fractional differential equation, fractional time component parameter, dimensional consistency, time domain analysis
In this research, the time domain analysis of the fractional order biquadratic system with nonzero input and nonzero damping ratio has been performed. Unlike the previous works, the analysis has been generically done with dimensional consistency awareness without referring to any specific physical system where nonzero input and nonzero damping ratio have been allowed. The fractional differential equation of the system has been derived and analytically solved. The physical measurability of the dimensions of the fractional derivative terms which have been defined in Caputo sense, and response with significantly different dynamic from its dimensional consistency ignored counterpart have been obtained due to our dimensional consistency awareness. The resulting solution is applicable to the fractional biquadratic systems of any kind with any physical nature. Based on such solution and numerical simulations, the influence of the fractional order parameter to all major time domain parameters have been studied in detailed. The obtain results provide insight to the fractional order biquadratic system with dimensional consistency awareness in a generic point of view.
C.M.A. Pinto and A.R.M. Carvalho, ‘A latency fractional order model for HIV dynamics’, Journal of Computational and Applied Mathematics, vol. 312, pp. 240–256, 2017.
C.M.A. Pinto and A.R.M. Carvalho, ‘The HIV/TB coinfection severity in the presence of TB multi-drug resistant strains’, Ecological Complexity, vol. 32, pp. 1–20, 2017
P.D. Mandic’, T.B. Šcekara and M.P. Lazarevic’, “Dominant pole placement with fractional order PID controllers: D-decomposition approach’, ISA Transactions., vol. 67, pp. 76–86, 2017
P.D. Mandic’, M.P. Lazarevic’ and T.B. Šcekara, ‘D-Decomposition technique for stabilization of furuta pendulum: Fractional approach’, Bulletin of the Polish Academy of Science: Technical Science, vol. 64, pp. 189–196, 2016.
A. Allagui, T.J. Freeborn, A.S. Elwakil and B. Maundy, ‘Reevaluation of performance of electric double-layer capacitors from constant current charge/discharge and cyclic voltammetry’, Scientific Reports, vol. 6, 2016.
T.J. Freeborn, A. Allagui and A. Elwakil, ‘Modelling supercapacitors leakage behaviour using a fractional-order model’, Proceedings of the European Conference on Circuit Theory and Design 2017, pp. 1–4, 2017.
M. Cajic’, D. Karlicic’, and M. Lazarevic’, ‘Damped vibration of a nonlocal nanobeam resting on viscoelastic foundation: fractional derivative model with two retardation times and fractional parameters”, Meccanica, vol. 52, pp. 363–382, 2017.
X. Wang, H. Qi, B. Yu, Z. Xiong and H. Xu, ‘Analytical and numerical study of electroosmotic slip flows of fractional second grade fluids’, Communications in Nonlinear Science and Numerical Simulation, vol. 50, pp. 77–87, 2017.
Y. Jiang, H. Qi, H. Xu and X. Jiang, ‘Transient electroosmotic slip flow of fractional oldroyd-B fluids’, Microfluidics and Nanofluidics, vol. 21, 2017.
A. Ullah, W. Chen and M.A. Khan, “A new variational approach for restoring images with multiplicative noise’, Computers & Mathematics with Applications, vol. 71, pp. 2034–2050, 2016.
A. Ullah, W. Chen, S.G. Sun and M.A. Khan, “An efficient variational method for restoring images with combined additive and multiplicative noise,” International Journal of Applied and Computational Mathematics., vol. 3, pp. 1999–2019, 2017.
N. Shrivastava and P. Varshney, ‘Efficacy of order reduction techniques in the analysis of fractional order systems’, Proceedings of the IEEE region 10 conference 2017, pp. 2967–2972, 2017.
X. Yang, C. Li, T. Huang and Q. Song, ‘Mittag–Leffler stability analysis of nonlinear fractional-order systems with impulses’, Applied Mathematics and Computation., vol. 293, pp. 416–422, 2017.
Y. Tang, N. Li, M. Liu, Y. Lu and W. Wang, ‘Identification of fractional-order systems with time delays using block pulse functions’, Mechanical Systems and Signal Processing, vol. 91, pp. 382–394, 2017.
P.E. Jacob, S.M.M. Alavi, A. Mahdi, S.J. Payne and D.A. Howey, ‘Bayesian inference in non-Markovian state-space models with applications to battery fractional-order systems’, IEEE Transactions on Control Systems Technology, vol. 26, pp. 497–506, 2018.
K. Leyden and B. Goodwine, ‘Fractional-order system identification for health monitoring’, Nonlinear Dynamics, vol. 92, pp. 1317–1334, 2018.
S. Marir, M. Chadli and D. Bouagada, ‘New admissibility conditions for singular linear continuous-time fractional-order systems’, Journal of the Franklin Institute, vol. 354, pp. 752–766, 2017.
Y. Wei, B. Du, S. Cheng and Y. Wang, ‘Fractional order systems time-optimal control and its application’, Journal of Optimization Theory and Applications, vol. 174, pp. 122–138, 2017.
M. Guia, F. Gomez, and J. Rosales, ‘Analysis on the time and frequency domain for the RC electric circuit of fractional order’, Central European Journal of Physics, vol. 11, pp. 1366–1371, 2013.
P. V. Shah, A. D. Patel, I. A. Salehbhai, and A. K. Shukla, ‘Analytic solution for the electric circuit model in fractional order’, Abstract and Applied Analysis, vol. 2014, 2014.
F. Gomez, J. Rosales and M. Guia, ‘RLC electrical circuits of non-integer order’, Central European Journal of Physics, vol. 11, pp. 1361–1365, 2013.
G.-A. J. Francisco, R.-G. Juan, G.-C. Manuel and R.-H. J. Roberto, ‘Fractional order RC and LC circuits’, Ingeniería Investigación y Tecnología, vol. 15, pp. 311–319, 2014.
R. Banchuin and R. Chaisricharoen, ‘The FDE based time domain analysis of nonzero input/nonzero damping ratio fractional order biquadratic system’, Proceedings of the Joint International Conference on Digital Arts, Media and Technology with ECTI Northern Section Conference on Electrical, Electronics, Computer and Telecommunications Engineering, pp. 177–179, 2020.
I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering: Academic Press, 1999.
E. Kreyszig, Advanced Engineering Mathematics: John Wiley and Sons, 1999.
B.C. Kuo and F. Golnaraghi, Automatic Control Systems: John Wiley and Sons, 2003.
A. M. Mathai and H. J. Haubold, Special Functions for Applied Scientists: Springer, 2010.