Abstract
Caputo definition of Fractional Diffintegration is numerically approximated for the real-time implementation in the digital/Computer/Embedded system. Fractional numerical methods and properties are discussed in this paper and the new method of software implementation is suggested for the fractional Proportional Integral-Derivative (PID) Controller. Comparison of the present “discretization method” with the exact known fractional diffintegration solution of algebraic functions has been discussed. Furthermore, Simulink implementation of the proposed algorithm with integer order solutions and compared with the solution of the available MATLAB tools. Performance of tuned integer order PID controller and manual tuned fractional PID controller using the present fractional numerical method is tabulated and showed that the present method is suitable for the real-time operation of the fractional-order PID controller. There is no need for conversion of the fractional-order controller to the higher order integer controller for practical implementation.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Caponetto R, Dongola G, Fortuna L, IPetras (2010) Fractional order systems modelling and control applications. World Scientific Series on Nonlinear Science, Series A, vol 72. World Scientific Publishing Co. Pvt. Ltd
Sabatier J, Agrawal OP, Tenreiro Machado JA (2007) Advances in fractional calculus: theoretical developments and applications in physics and engineering. Springer
Das S (2007) Functional fractional calculus for system identification and controls. Springer
Oldham KB, Spanier J (2006) The fractional calculus: theory and applications of differentiation and integration to arbitrary order (DoverBooks on Mathematics)
Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier
Ross B (ed) (1975) Fractional calculus and its applications. Springer, Berlin
Podlubny I (1999) Fractional differential equations. Academic Press, San Diego
Miller KS (1995) Derivatives of non-integer order. Math Mag 68(3):1–68
Vinagre BM et al (2006) On auto-tuning of fractional order PIλDμ controllers. In: Proceedings of IFAC workshop on fractional differentiation and its application (FDA’06) (Porto, Portugal)
Valerio et al (2006) Tuning of fractional PID controllers with Ziegler-Nichols-type rules. Signal Process 86:2771–2784
Dorcak L et al (2001) State-space controller design for the fractional order regulated system. International Carpathian Control Conference, Krynica, Poland
Nataraj PSV, Tharewal S (2007) On fractional-order QFT controllers. J Dyn Syst Meas Control 129:212–218
Boudjehem B, Boudjehem D (2012) Parameter tuning of a fractional-order PI controller using the ITAE criteria. Fractional Dynamics and Control, Springer Science Business Media
Padula F, Visioli A (2011) Tuning rules for optimal PID and fractional-order PID controllers. J Process Control 21:69–81
Cao J, Cao B-G (2006) Design of fractional order controllers based on particle swarm optimization. Industrial Electronics and Applications, IST IEEE Conference, pp 1–6
Cao J, Jin L, Cao B-G (2005) Optimization of fractional order PID controllers based on genetic algorithms. Machine Learning and Cybernetics, 2005. Proceedings of 2005 International Conference on, vol 9, pp 5686–5689
Khubalkar Swapnil, Junghare Anjali, Aware Mohan, Das Shantanu (2017) Modeling and control of a permanent-magnet brushless DC motor drive using a fractional order proportional-integral-derivative controller. Turk J Electr Eng Comput Sci 25:4223–4241
Das S, Saha S, Das S, Gupta A (2011) On the selection of tuning methodology of FO-PID controllers for the control of higher order processes. ISA Transactions 50(3):376–388
Jin Y, Branke J (2005) Evolutionary optimization in uncertain environments: a survey. IEEE Trans Evol Comput 9:303–317
Bhrawy AH et al (2014) A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations. J Comput Phys
Brunner Hermann et al (2010) Numerical simulations of two-dimensional fractional subdiffusion problems. J Comput Phys 229(18):6613–6622
Cao Jianxiong, Li Changpin, Chen YangQuan (2014) Compact difference method for solving the fractional reaction–subdiffusion equation with Neumann boundary value condition. Int J Comput Math. https://doi.org/10.1080/00207160.2014.887702
Langlands TAM, Henry BI (2005) The accuracy and stability of an implicit solution method for the fractional diffusion equation. J Comput Phys 205:719–736
Avinash KM, Bongulwar MR, Patre BM (2015) Tuning of fractional order PID controller for higher order process based on ITAE minimization. IEEE Indicon 1570186367
Center for Biotechnology Information. http://www.ncbi.nlm.nih.gov
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Gade, S., Kumbhar, M., Pardeshi, S. (2020). Numerical Approximation of Caputo Definition and Simulation of Fractional PID Controller. In: Gunjan, V., Suganthan, P., Haase, J., Kumar, A., Raman, B. (eds) Cybernetics, Cognition and Machine Learning Applications. Algorithms for Intelligent Systems. Springer, Singapore. https://doi.org/10.1007/978-981-15-1632-0_17
Download citation
DOI: https://doi.org/10.1007/978-981-15-1632-0_17
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-1631-3
Online ISBN: 978-981-15-1632-0
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)