Showing papers by "Shantanu Das published in 2017"

Modeling and control of a permanent-magnet brushless DC motor drive using a fractional order proportional-integral-derivative controller

Abstract: This paper deals with the speed control of a permanent-magnet brushless direct current (PMBLDC) motor. A fractional order PID (FOPID) controller is used in place of the conventional PID controller. The FOPID controller is a generalized form of the PID controller in which the order of integration and differentiation is any real number. It is shown that the proposed controller provides a powerful framework to control the PMBLDC motor. Parameters of the controller are found by using a novel dynamic particle swarm optimization (dPSO) method. The frequency domain pole-zero (p-z) interlacing method is used to approximate the fractional order operator. A three-phase inverter with four switches is used in place of the conventional six-switches inverter to suggest a cost-effective control scheme. The digital controller has been implemented using a field programmable gate array (FPGA). The control scheme is verified using the FPGA-in-the-loop (FIL) wizard of MATLAB/Simulink. Improvement in the overall performance of the system is observed using the proposed FOPID controller. The energy efficient nature of the FOPID controller is also demonstrated.

Charge-discharge energy efficiency analysis of ultracapacitor with fractional-order dynamics using hybrid optimization and its experimental validation

Abstract: Effective utilization of ultracapacitor for energy management and on-board power applications requires a thorough analysis of energy storage and delivery efficiency. Due to the presence of fractional-order dynamics, the power and energy behavior in ultracapacitors deviate significantly from standard dielectric capacitors, which follow the conventional integer-order calculus. In this regard, present work focuses on the energy efficiency analysis of ultracapacitor while considering fractional-order dynamics. A generalized approach based on the hybridization of an evolutionary seeker optimization algorithm and a local search technique has been proposed to investigate the effect of charging/discharging excitation by constant current and hence the fractional-order behavior for efficient power utilization. The proposed approach has been validated by suitable experiments on a commercially available ultracapacitor for different operating conditions.

Demonstrative fractional order - PID controller based DC motor drive on digital platform.

Abstract: In industrial drives applications, fractional order controllers can exhibit phenomenal impact due to realization through digital implementation. Digital fractional order controllers have created wide scope as it possess the inherent advantages like robustness against the plant parameter variation. This paper provides brief design procedure of fractional order proportional-integral-derivative (FO-PID) controller through the indirect approach of approximation using constant phase technique. The new modified dynamic particle swarm optimization (IdPSO) technique is proposed to find controller parameters. The FO-PID controller is implemented using floating point digital signal processor. The building blocks are designed and assembled with all peripheral components for the 1.5 kW industrial DC motor drive. The robust operation for parametric variation is ascertained by testing the controller with two separately excited DC motors with the same rating but different parameters.

Erratum to: Design of new practical phase shaping circuit using optimal pole---zero interlacing algorithm for fractional order PID controller

Abstract: This paper presents the implementation of fractional order PID (FO-PID) controller using hardwired modules of constant phase element (CPE). A new approach of phase shaping by slope cancellation of asymptotic phase plots for zeros and poles within the given bandwidth is realized. Analog circuits, which exhibit analog fractional-order integrator and fractional-order differentiator, are used for building the FO-PID controller. The design procedure is developed to obtain the optimal pole---zero pairs and respective "Fractance" components to realize for any value of fractional differ-integrator. These CPE elements give minimum error tolerance over the set phase value by using commercially available (R---C) components and Op-Amps. The pole---zero location in the root locus plot with constant asymptotic angle under various feed-forward gains is achieved with these analog integrodifferential circuits of the FO-PID. The iso-damping feature of the controller is practically demonstrated. A comparative performance is demonstrated under various settings of feed forward gains, which indicate the constant overshoot with FO-PID against the conventional PID. These circuits are developed and implemented with a DC motor emulator to confirm the designed performance of the controller.

Time independent fractional Schrodinger equation for generalized Mie-type potential in higher dimension framed with Jumarie type fractional derivative

Abstract: In this paper we obtain approximate bound state solutions of $N$-dimensional fractional time independent Schrodinger equation for generalised Mie-type potential, namely $V(r^{\alpha})=\frac{A}{r^{2\alpha}}+\frac{B}{r^{\alpha}}+C$. Here $\alpha(0<\alpha<1)$ acts like a fractional parameter for the space variable $r$. When $\alpha=1$ the potential converts into the original form of Mie-type of potential that is generally studied in molecular and chemical physics. The entire study is composed with Jumarie type fractional derivative approach. The solution is expressed via Mittag-Leffler function and fractionally defined confluent hypergeometric function. To ensure the validity of the present work, obtained results are verified with the previous works for different potential parameter configurations, specially for $\alpha=1$. At the end, few numerical calculations for energy eigenvalue and bound states eigenfunctions are furnished for a typical diatomic molecule.

Revisiting the Curie-Von Schweidler Law for Dielectric Relaxation and Derivation of Distribution Function for Relaxation Rates as Zipf's Power Law and Manifestation of Fractional Differential Equation for Capacitor

Abstract: The classical power law relaxation, i.e. relaxation of current with inverse of power of time for a step-voltage excitation to dielectric—as popularly known as Curie-von Schweidler law is empirically derived and is observed in several relaxation experiments on various dielectrics studies since late 19th Century. This relaxation law is also regarded as “universal-law” for dielectric relaxations; and is also termed as power law. This empirical Curie-von Schewidler relaxation law is then used to derive fractional differential equations describing constituent expression for capacitor. In this paper, we give simple mathematical treatment to derive the distribution of relaxation rates of this Curie-von Schweidler law, and show that the relaxation rate follows Zipf’s power law distribution. We also show the method developed here give Zipfian power law distribution for relaxing time constants. Then we will show however mathematically correct this may be, but physical interpretation from the obtained time constants distribution are contradictory to the Zipfian rate relaxation distribution. In this paper, we develop possible explanation that as to why Zipfian distribution of relaxation rates appears for Curie-von Schweidler Law, and relate this law to time variant rate of relaxation. In this paper, we derive appearance of fractional derivative while using Zipfian power law distribution that gives notion of scale dependent relaxation rate function for Curie-von Schweidler relaxation phenomena. This paper gives analytical approach to get insight of a non-Debye relaxation and gives a new treatment to especially much used empirical Curie-von Schweidler (universal) relaxation law.

A study of fractional Schrödinger equation composed of Jumarie fractional derivative

Abstract: In this paper we have derived the fractional-order Schrodinger equation composed of Jumarie fractional derivative. The solution of this fractional-order Schrodinger equation is obtained in terms of Mittag–Leffler function with complex arguments, and fractional trigonometric functions. A few important properties of the fractional Schrodinger equation are then described for the case of particles in one-dimensional infinite potential well. One of the motivations for using fractional calculus in physical systems is that the space and time variables, which we often deal with, exhibit coarse-grained phenomena. This means infinitesimal quantities cannot be arbitrarily taken to zero – rather they are non-zero with a minimum spread. This type of non-zero spread arises in the microscopic to mesoscopic levels of system dynamics, which means that, if we denote x as the point in space and t as the point in time, then limit of the differentials dx (and dt) cannot be taken as zero. To take the concept of coarse graining into account, use the infinitesimal quantities as (Δx) α (and (Δt) α ) with 0 < α < 1; called as ‘fractional differentials’. For arbitrarily small Δx and Δt (tending towards zero), these ‘fractional’ differentials are greater than Δx (and Δt), i.e. (Δx) α > Δx and (Δt) α > Δt. This way of defining the fractional differentials helps us to use fractional derivatives in the study of dynamic systems.

Modeling and control of four quadrant chopper fed DC series motor using two-degree of freedom digital fractional order PID controller

Abstract: The aim of the paper is to implement two degree of freedom digital fractional order proportional-integral-derivative (2-DOF FOPID) controller for speed control of DC series motor. Speed control is achieved by pulse width modulated control method through four quadrant chopper circuit. For realization of digital fractional order controller in integer order sense, pole-zero interlacing method is used. The controller parameters are tuned by using improved dynamic particle swarm optimization (IDPSO) technique. The effectiveness of proposed control scheme is simulated using FPGA (field programmable gate array)-in-loop wizard of MATLAB/Simulink. The performance comparison shows improvement with proposed controller as compared to integer counterpart.

Solution of space time fractional generalized KdV equation, KdV burger equation and Bona-Mahoney-Burgers equation with dual power-law nonlinearity using complex fractional transformation

Abstract: Fractional calculus is a rising subject in the current research field. The researchers of different disciplines are using fractional calculus models to investigate different practical problems. In this paper, we found the exact solutions of space-time fractional generalized KdV equation, KdV Burger equation and Benjamin-Bona-Mahoney-Burgers equation with dual power-law nonlinearity. The solutions are expressed in terms of hyperbolic, trigonometric and rational functions.

Application of Fractional Derivatives in Characterization of ECG graphs of Right Ventricular Hypertrophy Patients

Abstract: There are many functions which are continuous everywhere but non-differentiable at some or all points such functions are termed as unreachable functions. Graphs representing such unreachable functions are called unreachable graphs. For example, ECG is such an unreachable graph. Classical calculus fails in their characterization as derivatives do not exist at the unreachable points. Such unreachable functions can be characterized by fractional calculus as fractional derivatives exist at those unreachable points where classical derivatives do not exist. Definition of fractional derivatives has been proposed by several mathematicians like Grunwald-Letinikov, Riemann-Liouville, Caputo, and Jumarie to develop the theory of fractional calculus. In this paper, we have used Jumarie type fractional derivative and consequently the phase transition (P.T.) which is the difference between left fractional derivative and right fractional derivatives to characterize those points. A comparative study has been done between normal ECG sample and problematic ECG sample (Right Ventricular Hypertrophy) by the help of the above mentioned mathematical tool.

D Alemberts solution of fractional wave equations using complex fractional transformation

Abstract: Fractional wave equation arises in different type of physical problems such as the vibrating strings, propagation of electro-magnetic waves, and for many other systems. The exact analytical solution of the fractional differential equation is difficult to find. Usually Laplace-Fourier transformation method, along with methods where solutions are represented in series form is used to find the solution of the fractional wave equation. In this paper we describe the D Alembert s solution of the fractional wave equation with the help of complex fractional transform method. We demonstrate that using this fractional complex transformation method, we obtain the solutions easily as compared to fractional method of characteristics; and get the solution in analytical form. We show that the solution to the fractional wave equation manifests as travelling waves with scaled coordinates, depending on the considered fractional order value.

The new expansion method to solve Fractional KdV-Equations

Abstract: Fractional calculus of variation plays an important role to formulate the non-conservative physical problems. In this paper we use semi-inverse method and fractional variational principle to formulate the fractional order generalized Korteweg-deVries (KdV) equation with Jumarie type fractional derivative and proposed a new method to solve the non-linear fractional differential equation named as expansion method. Using this method we obtained the solutions of fractional order generalized KdV. The obtained solutions are more general compare to other method and the solutions are expressed in terms of the generalized hyperbolic, trigonometric functions and rational functions.

Characterization of Left Ventricular Hypertrophy via Fractional Derivatives

Abstract: In this paper, we have used the concepts of the fractional derivative of rough curves to characterize ECG of LVH patients and compared the results with normal ECGs. In mathematical language, an ECG is a rough curve having Q, R, S points as non-differentiable points where classical derivatives do not exist but fractional derivatives exist. We have calculated both left and right modified Riemann-Liouville fractional derivatives and their differences termed as phase transition at those non-differentiable points of V1, V2, V5, and V6 leads.Investigation shows that phase transition is higher for LVH patients than normal ones. This may be a method of determination of risk factor of LVH patients before doing Echocardiogram.