Abstract. In the paper, mathematical models of the supercapacitors are investigated. The models are based on electrical circuits in the form of RC ladder networks. The elementary cell of the network may consist of resistances and capacitances that are connected in series or parallel. The dynamic behavior of the circuit is described using fractional-order differential equations and its properties are analyzed. The identification procedure with quadratic performance index is performed in time domain to identify the parameters of the supercapacitor models. The results of numerical simulations are compared with the results measured experimentally in the physical system. In addition, an example from the automotive industry is used for an experimental evaluation of the theoretical analysis and to present a perspective on the applicability of the approach for other industrial projects.
A new, state space, non-integer order model for the heat transfer process is presented. The proposed model is based on a Feller semigroup one, the derivative with respect to time is expressed by the non-integer order Caputo operator, and the derivative with respect to length is described by the non-integer order Riesz operator. Elementary properties of the state operator are proven and a formula for the step response of the system is also given. The proposed model is applied to the modeling of temperature distribution in a one dimensional plant. Results of experiments show that the proposed model is more accurate than the analogical integer order model in the sense of the MSE cost function.
In this paper the parameter identification methods for nonlinear models were compared for fractional, partial differential equation. The compared three methods are: the Levenberg-Marquardt algorithm, the Gauss-Newton algorithm and Nelder-Mead Simplex method. The series of numerical experiments were performed to test their robustness and calculation speed. The result of this tests were presented and described.
A new, state space, discrete-time, and memory-efficient model of a one-dimensional heat transfer process is proposed. The model is derived directly from a time-continuous, state-space semigroup one. Its discrete version is obtained via a continuous fraction expansion method applied to the solution of the state equation. Fundamental properties of the proposed model, such as decomposition, stability, accuracy and convergence, are also discussed. Results of experiments show that the model yields good accuracy in the sense of the mean square error, and its size is significantly smaller than that of the model employing the well-known power series expansion approximation.
In this paper the simple identification problem for fractional differential equation of Caputo type was considered. This is the problem of estimation parameters, for which the quadratic criterion is minimized. For solving this issue, the Non Linear Programing technique, based on Marquardt algorithm, was used. At the end of article the results for numerical experiments was presented.
In this paper, a new, state space, fractional order model of a heat transfer in two dimensional plate is addressed. The proposed model derives directly from a two dimensional heat transfer equation. It employes the Caputo operator to express the fractional order differences along time. The spectrum decomposition and stability of the model are analysed. The formulae of impluse and step responses of the model are proved. Theoretical results are verified using experimental data from thermal camera. Comparison model vs experiment shows that the proposed fractional model is more accurate in the sense of MSE cost function than integer order model.
Abstract. In the paper, a new method for solution of linear discrete-time fractional-order state equation is presented. The proposed method is simpler than other methods using directly discrete-time version of the Grünwald-Letnikov operator. The method is dedicated to use with any approximator to the operator expressed by a discrete transfer function, e.g. CFE-based Al-Alaoui approximation. A simulation example confirms the usefulness of the method. A new algorithm for a CFE-approximated solution of a discrete-time noninteger-order state equationbstract. In the paper, a new method for solution of linear discrete-time fractional-order state equation is presented. The proposed method simpler than other methods using directly discrete-time version of the Grünwald-Letnikov operator. The method is dedicated to use with ny approximator to the operator expressed by a discrete transfer function, e.g. CFE-based Al-Alaoui approximation. A simulation example onfirms the usefulness of the method. Many real applications, to mention model-based conrol, model-based fault detection, require to implement a oninteger-order model at a digital platform like PLC or PGA. Known discrete-time state space models of nonintegerrder systems are typically based on the Grünvald-Letnikov GL) definition. An accurate implementation of this model, in articular at the bounded resource platforms, requires a (very) ong-length approximation of the GL-based system [33].The purpose of this paper is to propose a new, discrete-time tate space model of a noninteger-order system, constructed ith the use of the continuous fraction expansion (CFE) imlemented for the Al-Alaoui operator. The use of such an aproximant enables to obtain a much more effective model in hat the memory length is quite low. This recommends its use t industrial digital platforms.The paper is organized as follows. Heaving recalled the ackground of the paper in Section 1, Section 2 outlines the undamentals of fractional-order calculus and introduces a operator is defined aswhere a and t denote time limits for calculation of the operator, α ∈ R denotes the noninteger order of the operation.Next, an idea of the Gamma (Euler) function (see for example [12]) can be given: DEFINITION 2. The Gamma function is defined asThe fractional-order, integro-differential operator (1) can be described by different definitions, given by Grünvald and Letnikov (GL definition), Riemann and Liouville (RL definition) and Caputo (C definition). All these definitions are given below. With respect to particular additional assumptions these definitions can be considered equivalent. A new algorithm for a CFE-approximated solution of a discrete-time noninteger-order state equationAbstract. In the paper, a new method for solution of linear discrete-time fractional-order state equation is presented. The proposed method is simpler than other methods using directly discrete-time version of the Grünwald-Letnikov operator. The method is dedicated to use with any approximator to the operator expressed by a discrete tra...
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