Syllabus

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), Various geometrical 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 fractional order 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 engineering systems, 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 motor and transmission lines, FO modeling of viscoelastic materials, concept of fractional damping, 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, non-minimum 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 lead lag 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.