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  • 7:06 PM, Monday, 18 Oct 2021

Course Postgraduate
Semester Electives
Subject Code AVC866
Subject Title Fractional Calculus and Control


Fractional Calculus: Review of basic definitions of integer‐order (IO) derivatives and integrals and their geometric and physical interpretations, Definition of Riemann‐Liouville (RL) integration, Definitions of RL, Caputo and Grunwald‐Letnikov (GL) fractional derivatives (FDs), Variousgeometrical and physical interpretations of these FDs, Computation of these FDs for some basic functions like constant, ramp, exponential, sine, cosine, etc., Laplace and Fourier
transforms of FDs.

Fractional‐order Differential Equations: Study of basic functions like Gamma function, Mittag‐Leffler function, Dawson’s function, Hypergeometric function, etc, Analysis of linear fractionalorder differential equations (FDEs): formulation, Solution with different FDs, Initial conditions, Problem of initialization and the remedies.

Fractional‐order Modeling: Concepts of ‘memory’ and ‘non‐locality’ in real‐world and engineeringsystems, non‐exponential relaxation , ‘Mittag‐Leffler’ type decay and rise, Detailed analysis of fractional‐order (FO) modeling of: electrical circuit elements like inductor, capacitor, electrical machines like transformer, induction motorand transmission lines, FO modeling of viscoelastic materials, concept of fractionaldamping, Models of basic circuits and mechanical systems using FO elements, Concept of anomalous diffusion, non‐ Gaussian probability density function and the development of corresponding FO model, FO models of heat transfer, A brief overview of FO models of biological systems.

Linear Fractional‐order Systems: Review of basic concepts of complex analysis, Concepts of multivalued functions, branch points, branch cuts, Riemann surface and sheets, Fractional‐order transfer function (FOTF) representation, Concepts like commensurate and non‐commensurate TFs, stability, impulse, step and ramp response, Frequency response, nonminimum phase systems, Root locus, FO pseudo state‐space (PSS) representation and the associated concepts like solution of PSS model, controllability, observability, etc.

Fractional‐order Control: Detailed discussion and analysis of superiority of FO control over the conventional IO control in terms of closed‐loop performance, robustness, stability, etc., FO leadlag compensators, FO PID control, design of FO state‐feedback, Realization and implementation issues for FO controllers, survey of various realization methods and the comparative study.

Text Books

Same as Reference

  1. K. B. Oldham and J. Spanier. The Fractional Calculus . Dover Publications, USA, 2006.
  2. Kilbas, H. M. Srivastava, and J. J. Trujillo. Theory and Applications of Fractional Differential Equations, Elsevier, Netherlands, 2006.
  3. Podlubny. Fractional Differential Equations . Academic Press, USA, 1999.
  4. C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue, and V. Feliu. Fractional‐order Systems and Control: Fundamentals and Applications Springer‐Verlag London Limited, UK, 2010.
  5. R. L. Magin. Fractional Calculus in Bioengineering. Begell House Publishers, USA, 2006.
  6. R. Caponetto, G. Dongola, L. Fortuna, and I. Petras. Fractional Order Systems: Modeling and Control Applications. World Scientific, Singapore, 2010.
  7. K. S. Miller and B. Ross. An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons, USA, 1993.
  8. S. Das. Functional Fractional Calculus for System Identification and Controls, Springer, Germany, 2011.
  9. M. D. Ortigueira. Fractional Calculus for Scientists and Engineers. Springer, Germany, 2011.
  10. Petras. Fractional‐Order Nonlinear Systems: Modeling, Analysis and Simulation Springer, USA, 2011.