Analysis of fractional order systems using newton iteration-based approximation technique

Fractional differential equations play a major role in expressing mathematically the real-world problems as they help attain good fit to the experimental data. It is also known that fractional order controllers are more flexible than integer order controllers. But when it comes to the numerical approximation of fractional order functions inaccuracies arise if the conversion technique is not chosen properly. So, when a fractional order plant model is approximated to an integer order system, it is required that the approximated model be accurate, as the overall system performance is based on the estimated integer order model. Nitisha-Pragya-Carlson (NPC) is a recent approximation technique proposed in 2018 to derive the rational approximation of fractional order differ-integrators. In this paper, three fractional order plant models having fractional powers 3.1, 1.25 and 1.3 is analyzed in frequency domain in terms of magnitude and phase response. The performance of approximated third and second order NPC based integer model is studied and compared with the integer models developed using other existing technique. The approximation error is calculated by comparing the frequency response of the developed models with the ideal response. It has been found that in all the three examples NPC based models are very much close to the ideal values. Hence proving the efficacy of NPC technique in approximation of fractional order systems

Electronic realization of the fractional-order systems

This article is devoted to the electronic (analogue) realization of the fractional-order systems – controllers or controlled objects whose we earlier used, identified, and analyzed as a mathematical models only ��� namely a fractional-order differential equation, and solved numerically using a method based on the truncated version of the Grunwald - Letnikov formula for fractional derivative. The electronic realization of the fractional derivative is based on the continued fraction expansion of the rational approximation of the fractional differentiator from which we obtained the values of the resistors and capacitors of the electronic circuit. Along with the mathematical description are presented also simulation and measurement results

Exponential Integrator Methods for Nonlinear Fractional Reaction-diffusion Models

Nonlocality and spatial heterogeneity of many practical systems have made fractional differential equations very useful tools in Science and Engineering. However, solving these type of models is computationally demanding. In this work, we propose an exponential integrator method for nonlinear fractional reaction-diffusion equations. This scheme is based on using a real distinct poles discretization for the underlying matrix exponentials. Due to these real distinct poles, the algorithm could be easily implemented in parallel to take advantage of multiple processors for increased computational efficiency. The method is established to be second-order convergent; and proven to be robust for problems involving non-smooth/mismatched initial and boundary conditions and steep solution gradients. We examine the stability of the scheme through its amplification factor and plot the boundaries of the stability regions comparative to other second-order FETD schemes. This numerical scheme combined with fractional centered differencing is used for simulating many important nonlinear fractional models in applications. We demonstrate the superiority of our method over competing second order FETD schemes, BDF2 scheme, and IMEX schemes. Our experiments show that the proposed scheme is computationally more efficient (in terms of cpu time). Furthermore, we investigate the trade-off between using fractional centered differencing and matrix transfer technique in discretization of Riesz fractional derivatives. The generalized Mittag-Leffler function and its inverse is very useful in solving fractional differential equations and structural derivatives, respectively. However, their computational complexities have made them difficult to deal with numerically. We propose a real distinct pole rational approximation of the generalized Mittag-Leffler function. Under some mild conditions, this approximation is proven and empirically shown to be L-Acceptable. Due to the complete monotonicity property of the Mittag-Leffler function, we derive a rational approximation for the inverse generalized Mittag-Leffler function. These approximations are especially useful in developing efficient and accurate numerical schemes for partial differential equations of fractional order. Several applications are presented such as complementary error function, solution of fractional differential equations, and the ultraslow diffusion model using the structural derivative. Furthermore, we present a preliminary result of the application of the M-L RDP approximation to develop a generalized exponetial integrator scheme for time-fractional nonlinear reaction-diffusion equation

A novel ARX-based discretization method for linear non-rational systems

This paper presents a novel, simple, flexible and effective discretization method for linear non-rational systems including arbitrary linear fractional order systems (LFOS). The discretization algorithm relies on the direct integration in the complex domain and application of ARX (AutoRegressive eXogenous) model. Parameters of ARX-model are obtained by numerical inversion of Laplace transform from the set of input/output data from recorded step response to model of non-rational system. Numerical simulations of several representatives of LFOS (e.g. fractional order PID controller, fractional logarithmic filter, fractional oscillator etc.) are used to demonstrate the effectiveness of the proposed discretization method, both in the time and frequency domains. The obtained results indicate that the proposed ARX-based discretization method is adequate technique for obtaining digital approximation of LFOS

Stabilization control of inverted pendulum systems by fractional order PD controller based on D-decomposition technique

Many systems in nature are inherently under-actuated, with fewer actuators than degrees of freedom. However, even with reduced number of actuators, these systems are able to produce complex movements. To be capable of performing such motions, complex control algorithms must be implemented. Classical benchmark examples for studying problems of this kind include inverted pendulum systems. This paper deals with stability problem of two types of inverted pendulum controlled by a fractional order PD controller. Rotational and cart inverted pendulum are highly nonlinear mechanical systems with one control input and two degrees of freedom. Detailed mathematical model of both pendulums are derived using the Rodriguez method. Stabilization of pendulum around its unstable equilibrium point is achieved by using the fractional order PDα controller, in combination with partial feedback linearization technique. There are several methods for determining stability region of a closed loop system, and D-decomposition is one of them. Herein, D-decomposition method is applied to the inverted pendulum case, and determining its stability regions in parameters space of a fractional order PD controller is presented. D-decomposition for linear fractional systems is investigated, and for the case of linear parameters dependence. Fractional order control laws are represented by a transfer functions which are not rational, which gives rise to a problem of practical implementation of the corresponding control algorithms. A method for rational approximation of linear fractional order systems used in this paper is computationally efficient, accurate, and relies on the interpolation of the frequency characteristics of the system on a predefined set of target frequencies. The performance of the proposed method is demonstrated with experimental verification of the stabilization control of the cart pendulum system

PARAMETERS ESTIMATION OF FRACTIONAL ORDER SYSTEM WITH DOMINANT POLE USING CO-EVOLUTIONARY PARTICLE SWARM OPTIMIZZATION (CPSO) ALGORITHM

ABSTRACT This paper deals with fractional order systems parameters estimation by use of Co-evolutionary Particle Swarm Optimization (CPSO) method. in some cases such as fractional order systems identification in spite of existing different methods, it is difficult to obtain estimation of model structure parameters and generally it leads to solving the with constrained complex non-linear optimization problems and this topic is one of the identification challenges of these systems. Since some of systems are inherently fractional order and because of having special behavior in these systems which in its similar integer order systems are not found. There for necessity of fractional modeling is double for such systems. In this paper, at first, we assume that the measured out-input data exists and for approximation to reality is considered that these data has been corrupted with noise. Then considering model structure as the linear combination of fractional orthogonal basis functions by use of CPSO suitable algorithm leads to estimation of fractional order system parameters and related to the complexity level of master system, suitable or acceptable approximation is obtained. In finally, by simulating of physical-typical sample system in noisy conditions leads to system identification which gained results shows the effectiveness of presented method. KEYWORDS: Fractional Order Systems, Parameter Estimation, System Identification, Co-Evolutionary Particle Swarm Optimization (Cpso) Algorithms Although the mathematics of fractional calculations has a few hundred years old, but in the two decades ago, it has been attracted in research and applicable fields of various sciences. Also, it was seen that some of the real systems have inherent fractional order behaviour and for example we can refer to real systems such as: viscoelastic materials, cell diffusion processes, transmission of signals via strong magnetic fields and some systems with disturbance characteristics that they have inherent fractional order behaviour One of the features of behaviour of fractional order systems is presence of non-periodic modes that they are decay in polynomial form and also a behaviour that it is called long memory that we can't find its similarity in integer order rational systems . So, if modelling, identification, controlling and other studies on these systems want to be accurate and close to reality, it should be based on fractional order model of these systems. Even in integer order systems, modelling in the form of fractional order mode or controller design with fractional model is also more effective, because of its more degrees of freedom and also the systems with integer order are special state of fractional order systems. This topic has been shown in several researches, therefore, the importance of fractional models and their synthesis is clear in practic

An efficient implementation of an implicit FEM scheme for fractional-in-space reaction-diffusion equations

Fractional differential equations are becoming increasingly used as a modelling tool for processes with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues, which impose a number of computational constraints. In this paper we develop efficient, scalable techniques for solving fractional-in-space reaction diffusion equations using the finite element method on both structured and unstructured grids, and robust techniques for computing the fractional power of a matrix times a vector. Our approach is show-cased by solving the fractional Fisher and fractional Allen-Cahn reaction-diffusion equations in two and three spatial dimensions, and analysing the speed of the travelling wave and size of the interface in terms of the fractional power of the underlying Laplacian operator