Showing papers by "Shantanu Das published in 2011"

Functional Fractional Calculus

Abstract: When a new extraordinary and outstanding theory is stated, it has to face criticism and skeptism, because it is beyond the usual concept. The fractional calculus though not new, was not discussed or developed for a long time, particularly for lack of its application to real life problems. It is extraordinary because it does not deal with ordinary differential calculus. It is outstanding because it can now be applied to situations where existing theories fail to give satisfactory results. In this book not only mathematical abstractions are discussed in a lucid manner, with physical mathematical and geometrical explanations, but also several practical applications are given particularly for system identification, description and then efficient controls. The normal physical laws like, transport theory, electrodynamics, equation of motions, elasticity, viscosity, and several others of are based on ordinary calculus. In this book these physical laws are generalized in fractional calculus contexts; taking, heterogeneity effect in transport background, the space having traps or islands, irregular distribution of charges, non-ideal spring with mass connected to a pointless-mass ball, material behaving with viscous as well as elastic properties, system relaxation with and without memory, physics of random delay in computer network; and several others; mapping the reality of nature closely. The concept of fractional and complex order differentiation and integration are elaborated mathematically, physically and geometrically with examples. The practical utility of local fractional differentiation for enhancing the character of singularity at phase transition or characterizing the irregularity measure of response function is deliberated. Practical results of viscoelastic experiments, fractional order controls experiments, design of fractional controller and practical circuit synthesis for fractional order elements are elaborated in this book. The book also maps theory of classical integer order differential equations to fractional calculus contexts, and deals in details with conflicting and demanding initialization issues, required in classical techniques. The book presents a modern approach to solve the solvable system of fractional and other differential equations, linear, non-linear; without perturbation or transformations, but by applying physical principle of action-and-opposite-reaction, giving approximately exact series solutions.Historically, Sir Isaac Newton and Gottfried Wihelm Leibniz independently discovered calculus in the middle of the 17th century. In recognition to this remarkable discovery, J.von Neumann remarked, the calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more equivocally than anything else the inception of modern mathematical analysis which is logical development, still constitute the greatest technical advance in exact thinking.This XXI century has thus started to think-exactly for advancement in science & technology by growing application of fractional calculus, and this century has started speaking the language which nature understands the best.

On the selection of tuning methodology of FOPID controllers for the control of higher order processes.

Abstract: In this paper, a comparative study is done on the time and frequency domain tuning strategies for fractional order (FO) PID controllers to handle higher order processes. A new fractional order template for reduced parameter modelling of stable minimum/non-minimum phase higher order processes is introduced and its advantage in frequency domain tuning of FOPID controllers is also presented. The time domain optimal tuning of FOPID controllers have also been carried out to handle these higher order processes by performing optimization with various integral performance indices. The paper highlights on the practical control system implementation issues like flexibility of online autotuning, reduced control signal and actuator size, capability of measurement noise filtration, load disturbance suppression, robustness against parameter uncertainties etc. in light of the above tuning methodologies.

Introduction to Fractional Calculus

Abstract: Fractional calculus is three centuries old as the conventional calculus, but not very popular amongst science and or engineering community. The beauty of this subject is that fractional derivatives (and integrals) are not a local (or point) property (or quantity). Thereby this considers the history and non-local distributed effects. In other words perhaps this subject translates the reality of nature better! Therefore to make this subject available as popular subject to science and engineering community, adds another dimension to understand or describe basic nature in a better way. Perhaps fractional calculus is what nature understands and to talk with nature in this language is therefore efficient. For past three centuries this subject was with mathematicians and only in last few years, this is pulled to several (applied) fields of engineering and science and economics. However recent attempt is on to have definition of fractional derivative as local operator specifically to fractal science theory. Next decade will see several applications based on this three hundred years (old) new subject, which can be thought of as superset of fractional differintegral calculus, the conventional integer order calculus being a part of it. Differintegration is operator doing differentiation and sometimes integrations in a general sense.

Optimizing Continued Fraction Expansion Based IIR Realization of Fractional Order Differ-Integrators with Genetic Algorithm

Abstract: Rational approximation of fractional order (FO) differ-integrators via Continued Fraction Expansion (CFE) is a well known technique. In this paper, the nominal structures of various generating functions are optimized using Genetic Algorithm (GA) to minimize the deviation in magnitude and phase response between the original FO element and the rationalized discrete time filter in Infinite Impulse Response (IIR) structure. The optimized filter based realizations show better approximation of the FO elements in comparison with the existing methods and is demonstrated by the frequency response of the IIR filters.

Revisiting oustaloup's recursive filter for analog realization of fractional order differintegrators

Abstract: Rational approximation of fractional order (FO) elements is becoming increasing important in fractional order control design and signal processing. Among the available continuous time and discrete rational approximation techniques, the Oustaloup's recursive filter which gives a band-limited realization of FO elements is the focus of the present study. Its another variant, commonly known as modified Oustaloup's filter is also analyzed in the same light, so as to produce low order analog approximation of FO differ-integrators.

Identification of the core temperature in a fractional order noisy environment for thermal feedback in nuclear reactors

Abstract: In this paper, the core temperature of a 540 MWe Pressurized Heavy Water Reactor (PHWR) has been estimated from the measured temperature of the coolant using system identification techniques. The formulation is done considering fractional order (FO) dynamics for the sensor noise (i.e. 1/ƒα noise) and the conventional white noise in measurement. Performance study with different least square based linear estimators and nonlinear AutoRegressive eXogenous (ARX) and nonlinear Hammerstein-Wiener class of estimators for system identification has been done to show their relative merits to handle fractional order noise dynamics for estimating reactor core temperature.

Analog realization of fractional order hybrid differentiators via Carlson's approach

Abstract: Rational approximation of fractional order (FO) elements are now of prime importance for the need of its hardware implementation in control design and signal processing. Among the well known analog realizations methods, the Carlson's approach has been used in this paper due to its simplicity of calculation for designing a certain class of FO differentiators as hybrid filters. Impact of the independent parameters of the hybrid differentiator on the phase response has been depicted along with the achievable accuracies for increasing order of realization for filter design.

Control of nuclear reactor power with thermal-hydraulic effects via fuzzy PI γ D μ controllers

Abstract: A nonlinear nuclear reactor model is linearized at different initial powers for the purpose of controlling the reactor in load following mode A new fractional order (FO) fuzzy proportional integral derivative (PID) controller is tuned next using Genetic Algorithm (GA) to control the reactor at different operating conditions The effectiveness of using the fuzzy FOPID controller over conventional fuzzy PID controllers is shown with credible numerical simulations The controllers tuned with the highest and lowest power models are shown to work well at other operating conditions also and hence are robust with respect to the changes in nuclear reactor power level

System Order Identification and Control

Abstract: For unknown systems, ‘system identification’ has become the standard tool of the control engineer and scientists. Identifying a given system from data becomes more difficult, however when fractional orders are allowed. Here identification process is demonstrated using assumption that system order distribution is a continuous one. Frequency domain system identification can thus be performed using numerical methods demonstrated in this chapter. Here one concept of r-Laplace transforms is discussed (Laplace transform in log domain), to discuss the system order distribution. Also mentioned is variable order identification as further development where the system order also varies with ambient and time is highlighted. Here in this chapter, an identification method based on continuous order distribution, is discussed. This technique is suitable for both the standard integer order and fractional order systems. This is topic for further advance research as to qualify the procedure of system order identification and to have technique of tackling variable order. Extending this continuous order distribution discussion, the advance research of having a continuum order feedback and generalized PID control is elucidated. Also in this chapter some peculiarities of the pole property of fractional order system as ultra-damping, hyper-damping and fractional resonance is explained. Elaborate research in this direction is ongoing process; to crisply define the system identification, crisply define the variable order structure, along with generalized controller for future applications. The system identification in presence of disorder is what is challenging and some unification of disordered time-response that is relaxation is too discussed. This is general process of returning to equilibrium for say any stable system or properties of condense matter physics. The introduction to complex order calculus in system identification is too touched upon, in this chapter, along with identification of main parameters of ‘irregular’ stochastically behaving systems.

Generalized Frequency Domain Robust Tuning of a Family of Fractional Order PI/PID Controllers to Handle Higher Order Process Dynamics

Abstract: Generalization of the frequency domain robust tuning has been proposed in this paper for a family of fractional order (FO) PI/PID controllers. The controller tuning is enhanced with two new FO reduced parameter templates which are capable of capturing higher order process dynamics with much better accuracy. The paper validates the proposed methodology with a standard test-bench of higher order processes to show the relative merits of the family of FO controller structures.

Spreading of Non-Newtonian and Newtonian Fluids on a Solid Substrate under Pressure

Abstract: Strongly non-Newtonian fluids namely, aqueous gels of starch, are shown to exhibit visco-elastic behavior, when subjected to a load. We study arrowroot and potato starch gels. When a droplet of the fluid is sandwiched between two glass plates and compressed, the area of contact between the fluid and plates increases in an oscillatory manner. This is unlike Newtonian fluids, where the area increases monotonically in a similar situation. The periphery moreover, develops an instability, which looks similar to Saffman Taylor fingers. This is not normally seen under compression. The loading history is also found to affect the manner of spreading. We attempt to describe the non-Newtonian nature of the fluid through a visco-elastic model incorporating generalized calculus. This is shown to reproduce qualitatively the oscillatory variation in the surface strain.

Least square and Instrumental Variable system identification of ac servo position control system with fractional Gaussian noise

Abstract: In this paper, the classical Least Square Estimator (LSE) and its improved version the Instrumental Variable (IV) estimator have been used for the identification of an ac servo motor position control system The data for system identification has been collected from a practical test set-up for fixed command on the final angular position of the servo motor with varying level of velocity and acceleration The measured data is corrupted then with externally induced random noise having a Gaussian distribution, commonly known as white Gaussian noise (wGn) Performance of the LSE and IV estimators are also compared for fractional Gaussian noise (fGn) which have heavy tails in its statistical distribution and are capable of modeling real world signals having spiky nature

Estimation, Analysis and Smoothing of Self-Similar Network Induced Delays in Feedback Control of Nuclear Reactors

Abstract: This paper analyzes a nuclear reactor power signal that suffers from network induced random delays in the shared data network while being fed-back to the Reactor Regulating System (RRS). A detailed study is carried out to investigate the self similarity of random delay dynamics due to the network traffic in shared medium. The fractionality or self-similarity in the network induced delay that corrupts the measured power signal coming from Self Powered Neutron Detectors (SPND) is estimated and analyzed. As any fractional order randomness is intrinsically different from conventional Gaussian kind of randomness, these delay dynamics need to be handled efficiently, before reaching the controller within the RRS. An attempt has been made to minimize the effect of the randomness in the reactor power transient data with few classes of smoothing filters. The performance measure of the smoothers with fractional order noise consideration is also investigated into.

Genetic Algorithm Based Improved Sub-Optimal Model Reduction in Nyquist Plane for Optimal Tuning Rule Extraction of PID and PIlambdaDi Controllers via Genetic Programming

Abstract: Genetic Algorithm (GA) has been used in this paper for a new Nyquist based sub-optimal model reduction and optimal time domain tuning of PID and fractional order (FO) PIeDi controllers. Comparative studies show that the new model reduction technique outperforms the conventional H2-norm based reduced order modeling techniques. Optimum tuning rule has been developed next with a test-bench of higher order processes via Genetic Programming (GP) with minimum value of weighted integral error index and control signal. From the Pareto optimal front which is a trade-off between the complexity of the formulae and control performance, an efficient set of tuning rules has been generated for time domain optimal PID and PIeDi controllers.

A new Fractional Fourier transform based design of a band-pass FIR filter for power feedback in nuclear reactors under noisy environment

Abstract: A new discrete Fractional Fourier transform based Kaiser-window design technique has been adopted in this paper for the synthesis of a band-pass filter (BPF) in finite impulse response (FIR) form. The filter design is attempted with the aim of efficient power feedback with the measured sensor data, to the regulating system of nuclear reactors. The ill effects of integer and fractional order noise in the online power measurement and feedback have been reduced with the BPF which helps in efficient reactor control.

Application of Generalized Fractional Calculus in Other Science and Engineering Fields

Abstract: In this chapter a series of applications are described where fractional calculus is finding application. We start with diffusion model in electrochemistry, electrode electrolyte interface, capacitor theory, fractance circuits, and application in feed back control systems, viscoelasticity, and vibration damping system. This survey cannot cover complete applications like modern trends in electromagnetic theory like fractional multipole, hereditary prediction of gene behavior, fractional neural modeling in bio-sciences, communication channel traffic models, chaos theory, hence simple applications are provided for appreciation. However in the feedback control system section attempt is made to provide vector state feed back controller and observer available for multivariate control science, with explanation of fractional order feedback control and fractional phase shaper design to achieve robust iso-damped close loop performance.

Quantized Energy Momentum and Wave for an Electromagnetic Pulse—A Single Photon inside Negative Refractive Indexed Media

Abstract: An Electromagnetic (EM) radiation in dispersion less free space vacuum is represented by a photon, with corpuscular and wave nature. The discussions, for the past century aimed at the nature of photon inside a media having dispersion in the refraction property, other than free space. What about its nature if the space be of refractive index which is negative, is discussed in this paper. We call mechanical momentum, wave-momentum, and try to match our present theories with intriguing property of this ‘photon’ or pulse carrying EM energy packet, and more so we try to find its property energy, momentum inside a media a positive refractive media, and if the media show a negative refractive index behavior, then these queries are profound, and suitable explanations to these classical concepts of corpuscular-wave nature of photon inside these media are quest for the scientists dealing with these materials having negative index of refraction. Here some of this counterintuitive nature of corpuscular-wave nature of photon inside negative indexed material is brought out, with possible ‘new definition’ of its ‘wave-momentum’, the concept of ‘reactive energy’ inside negative indexed material, along with possible ‘new wave equation’. These definitions and expressions of ‘wave-momentum’ and ‘reactive energy’ pertaining to negative indexed material are new and discussed and derived by classical means.

Observation of Fractional Calculus in Physical System Description

Abstract: Fractional calculus allows a more compact representation and problem solution for some spatially distributed systems. Spatially distributed system representation allows a better understanding of the fractional calculus. The idea of fractional integrals and derivatives has been known since the development of regular calculus. Although not well known to most engineers, prominent mathematicians as well as scientists of the operational calculus have considered the fractional calculus. Unfortunately many of the results in the fractional calculus are given in language of advanced analysis and are not readily accessible to the general engineering and science community. Many systems are known to display fractional order dynamics. Probably the first physical system to be widely recognized as one demonstrating fractional behavior is the semi-infinite lossy (RC) transmission line. The current into the line is equal to the half-derivative of the applied voltage.

Denoising SPND signal by discrete wavelet analysis for efficient power feedback in regulating system of PHWRs under noisy environment

Abstract: This paper aims to denoise a neutron power sensor data which is the output of Self Powered Neutron Detector (SPND) in each of the Liquid Zone Compartments (LZC) in a nuclear reactor core. Noisy data measured over long period is analyzed with different types of wavelets by one dimensional Discrete Wavelet Transform (DWT) technique to investigate the local feature of the measured signal by selecting proper basis wavelet to feedback the processed signal for global reactor power control and safety purposes. Denoising has also been carried out, considering the SPND signal, corrupted with fractional order (FO) noise. A relative Maximum Absolute Error (MAE) based analysis, adopted in this paper is done to precisely compare the wavelet performances in integer and fractional order type noisy SPND signal.

A new look at the nature of linear momentum and the energy of electromagnetic radiation inside negative refractive media

Abstract: More than a century has passed since the concept of wave–particle duality was introduced, where electromagnetic (EM) radiation in a dispersionless free space vacuum is represented by a photon that exhibits both corpuscular and wave natures. Historically, discussions have concentrated on the nature of a photon inside a medium that exhibits dispersion in the refraction property, other than free space. In this paper we call mechanical momentum, wave momentum, and try to match our 'thought experiments' with the properties of the photon, or pulse, carrying the EM energy packet. We also try to find the photon's property energy and momentum inside a positive refractive medium. If the medium shows a negative refractive index behavior, then this has profound consequences, and suitable explanations to the classical concept of the corpuscular–wave nature of a photon inside this medium are needed. In this paper, some of the counterintuitive details of the corpuscular–wave nature of a photon inside a negative indexed material are discussed, producing a possible new definition of wave momentum: the concept of 'reactive energy' inside a negative indexed material. These definitions and expressions of 'wave-momentum' and 'reactive energy' pertaining to negative indexed material are new and are discussed and derived by classical means.

A new hardwire circuit for metering reactivity

Abstract: Sub-criticality, criticality, and reactivity are the terms which are not annunciated in Indian PHWRs (Pressurised Heavy Water Reactors) and are thus not very clear to the reactor operators. These parameters are prime movers defining nuclear reactor and should be annunciated in control room by simple means to aid operator to view reactivity excursions while in power operation or during shutdown maintenance. A very novel circuit of a hardwire scheme to solve the coupled neutron kinetics is presented and its output is compared against analytical solutions. This scheme does not require any estimation technique and is robust as the instrument is based on basic reactor governing laws. This instrument is able to show the transients in reactivity as well as steady state values and is aimed for all Indian PHWRs for multi-decade operation. This technique is tunable for any research reactor or fast reactor too.

Spreading of Non-Newtonian and Newtonian Fluids on a Solid Substrate under Pressure

Abstract: Strongly non-Newtonian fluids namely, aqueous gels of starch, are shown to exhibit visco-elastic behavior, when subjected to a load. We study arrowroot and potato starch gels. When a droplet of the fluid is sandwiched between two glass plates and compressed, the area of contact between the fluid and plates increases in an oscillatory manner. This is unlike Newtonian fluids, where the area increases monotonically in a similar situation. The periphery moreover, develops an instability, which looks similar to Saffman Taylor fingers. This is not normally seen under compression. The loading history is also found to affect the manner of spreading. We attempt to describe the non-Newtonian nature of the fluid through a visco-elastic model incorporating generalized calculus. This is shown to reproduce qualitatively the oscillatory variation in the surface strain.

Functions Used in Fractional Calculus

Abstract: This chapter presents a number of functions that have been found useful in the solution of the problems of fractional calculus. The base function is the Gamma function, which generalizes the factorial expression, used in multiple differentiation and repeated integrations, in integer order calculus. The Mittag-Leffler function is the basis function of fractional calculus, as exponential function is to integer order calculus. Several modifications of the Mittag-Leffler functions, along with other variants are introduced which are developed since 1903, for study of the fractional calculus. These functions are called Higher Transcendental Functions and its use in solving Fractional Differential Equations is as similar to use of transcendental functions for solving Integer Order Differential Equations. Use of these functions is demonstrated for solving Fractional differential equations with Laplace Transform Technique. Here, some interesting physical interpretation is given, for memory integrals for relaxation laws for generalized system dynamics (with memory); along with basic definition and physical interpretation of rough functions, and its fractal dimension. Several examples are solved to get fractional integration and fractional differentiation of standard function and use of introduced higher transcendental functions is demonstrated especially for solving Fractional Differential Equations.

A new mechanics of corpuscular-wave transport of momentum and energy inside negative indexed material

Abstract: A century has passed regarding wave-particle duality, well an electromagnetic (EM) radiation in dispersionless free space vacuum is represented by a photon, with corpuscular and wave nature. The discussions from the past century aimed at the nature of photon inside a media having dispersion in the refraction property, other than free space.