Vishwesh A. Vyawahare

A new and simple method to construct root locus of general fractional-order systems

Recently fractional-order (FO) differential equations are widely used in the areas of modeling and control. They are multivalued in nature hence their stability is defined using Riemann surfaces. The stability analysis of FO linear systems using the technique of Root Locus is the main focus of this paper. Procedure to plot root locus of FO systems in s-plane has been proposed by many authors, which are complicated, and analysis using these methods is also difficult and incomplete. In this paper, we have proposed a simple method of plotting root locus of FO systems. In the proposed method, the FO system is transformed into its integer-order counterpart and then root locus of this transformed system is plotted. It is shown with the help of examples that the root locus of this transformed system (which is obviously very easy to plot) has exactly the same shape and structure as the root locus of the original FO system. So stability of the FO system can be directly deduced and analyzed from the root locus of the transformed IO system. This proposed procedure of developing and analyzing the root locus of FO systems is much easier and straightforward than the existing methods suggested in the literature. This root locus plot is used to comment about the stability of FO system. It also gives the range for the amplifier gain k required to maintain this stability. The reliability of the method is verified with analytical calculations.

Analysis of Fractional-order Point Reactor Kinetics Model with Adiabatic Temperature Feedback for Nuclear Reactor with Subdiffusive Neutron Transport

In this paper, a fractional-order nonlinear model is developed for the nuclear reactor with subdiffusive neutron transport. The proposed fractional-order point reactor kinetics model is a system of three coupled, nonlinear differential equations. The model represents subprompt critical condition. The nonlinearity in the model is due to the adiabatic temperature feedback of reactivity. This model originates from the fact that neutron transport inside the reactor core is subdiffusion and should be better modeled using fractional-order differential equations. The proposed fractional-order model is analyzed for step and sinusoidal reactivity inputs. The stiff system of differential equations is solved numerically with Adams-Bashforth-Moulton method. The proposed model is stable with self-limitting power excursions. The issue of convergence of this method for the proposed model for different values of fractional derivative order is also discussed.

Model predictive control for fractional-order system a modeling and approximation based analysis

A widely recognized advanced control methodology model predictive control is applied to solve a classical servo problem in the context of linear fractional-order (FO) system with the help of an approximation method. In model predictive control, a finite horizon optimal control problem is solved at each sampling instant to obtain the current control action. The optimization delivers an optimal control sequence and the first control thus obtained is applied to the plant. An important constituent of this type of control is the accuracy of the model. For a system with fractional dynamics, accurate model can be obtained using fractional calculus. One of the methods to implement such a model for control purpose is Oustaloups recursive approximation. This method delivers equivalent integer-order transfer function for a fractional-order system, which is then utilized as an internal model in model predictive control. Analytically calculated output equation for FO system has been utilized to represent process model to make simulations look more realistic by considering current and initial states in process model. The paper attempts to present the effect of modeling and approximations of fractional-order system on the performance of model predictive control strategy.

Design of Fractional-order PI controller for linear unstable systems

This paper presents a new strategy for designing of Fractional-order proportional integral (FOPI) controller for a linear unstable systems. The open loop transfer function is compared with the standard fractional-order (FO) transfer function whose closed loop response is always stable and comparison is done in frequency domain i.e. to match magnitude and phase response of the system with the standard FO transfer function. The slope of the Nyquist plot of the standard FO transfer function is also compared with the given open loop system with FOPI controller. The properly tuned FOPI controllers results are better than its integer order PI and PID counter-part.

A review of time domain, frequency domain and stability analysis of linear complex-order systems

The main objective of this paper is to present the time domain, frequency domain and stability analysis of linear systems represented by differential equations with complex-order derivatives. The impulse and step response of three different complex-order systems have been presented numerically with the help of MATLAB. For frequency domain analysis, Bode-plots of the same three complex-order systems have been sketched. Complex-order systems have infinite numbers of complex-conjugate poles. The stability analysis of the complex-order systems has been done in two ways. Firstly, for systems to be stable, the complex-conjugate poles in the principle Riemann sheet must be in the left half plane. Secondly, the complex-order q = u + iv of the complex-order systems must be interior to an open disk in the u-v plane, for systems to be stable.

Performance analysis of model predictive control for linear fractional-order system with bounded disturbance

Model predictive control is an advanced control methodology which has been utilized for many industrial processes. Model is the heart of this control strategy which is required for prediction of future states. In this paper, model predictive control is applied on a fractional-order system to test the control strategy against the disturbance by adding disturbance at the output of the process. Approximation method was used to deliver the rational function of fractional-order plant and to deliver state space model of a fractional-order system. It was observed that the model predictive control for fractional-order system works fine in the presence of unknown disturbance at the output.

Design of fractional-order controller for fractional-order systems using Bode's ideal loop transfer function method

Fractional-order models have been found to provide a more realistic and compact representation to real world and man-made systems. On the other side, fractional-order controllers have proved their efficacy over the conventional integer-order controllers by providing more flexibility in the design as well as by guaranteeing a more robust closed-loop configuration. In this paper, Bodes ideal loop transfer function method is used to design fractional-order controllers for fractional-order plants. To prove the efficacy of the designed method, the design methodology is tested for stable and unstable fractional-order plants. From the simulation results obtained, it is observed that with the designed controller the closed-loop response of the system is improved and is more robust.

Real time implementation of robust QFT controller and prefilter for magnetic levitation system

Controller design for magnetic levitation systems is considered to be difficult due to uncertainties and nonlinearities which existed in the system. The plant has inherently strong nonlinearities due to the natural properties of magnetic fields. Also the external disturbances will deteriorate the dynamic performance of the magnetic levitation system, and may give rise to system instability. These problem triggers enormous interests in designing various controllers for the non-linear dynamic system. Designing and implementing controllers for such systems is a meticulous work. In this paper, the robust QFT controllers and prefilters are designed using Interval Constraint Satisfaction Problem based method. The robust controller is designed by solving quadratic inequalities of robust stability and performance specifications. In QFT, the prefilter is designed to achieve the desired tracking specifications. The prefilter is designed by solving the quadratic inequalities of upper and lower tracking specifications. In this paper, the designed QFT controllers and prefilters are tested on the experimental setup designed by Educational Control Product Magnetic Levitation Setup ECP 730.

Analysis of Model Predictive Control for Fractional-Order System

This paper attempts to analyze the performance of model predictive control (MPC) strategy for a fractional-order system. MPC is a popular control technique that is extensively used for the control of industrial processes. Here MPC is applied to a system with fractional dynamics. The original fractional-order model of the system is considered as ‘plant’ and its integer-order approximation is considered as the ‘model’. The effect of approximation of fractional-order model on the MPC is studied. Also a study of the effect of uncertainties in the plant parameters and the non-integer derivative order is carried out.

Development and Analysis of Fractional-order Neutron Telegraph Equation

This chapter starts with the motivation to model neutron movement as anomalous diffusion, particularly subdiffusion, and then presents the derivation of the fractional-order neutron telegraph model using the stochastic framework of continuous-time random walk. Its longtime and short-time behaviors are analyzed. The mean-squared displacement of various transport models is calculated, and the comparative study is carried out. Finally, all the integer-order and fractional-order partial differential equation models are solved using separation of variables method to study the spatial distribution and time evolution of neutron flux in the slab reactor.