Dominik Sierociuk

Stability of Discrete Fractional Order State-space Systems

In this article, the stability problem for discrete-time fractional order systems is considered. The discrete-time fractional order state-space model introduced by the authors in earlier works is recalled in this context. The proposed stability definition is adopted from one used for infinite dimensional systems. Using this definition, the main stability result is presented in the form of a simple stability condition for the fractional order discrete state-space system. This is one of the first few attempts to give the stability conditions for this type of system. The condition presented is conservative1 the method gives only sufficient conditions, and the stability areas obtained when using it are smaller than those obtained from numerical solutions of the system. The relationship between the eigenvalues of the system matrix and the poles of the fractional-order system transfer function is also discussed. The main observation in this respect is that a set of L poles is related to every eigenvalue of the system matrix.

Modelling heat transfer in heterogeneous media using fractional calculus

This paper presents the results of modelling the heat transfer process in heterogeneous media with the assumption that part of the heat flux is dispersed in the air around the beam. The heat transfer process in a solid material (beam) can be described by an integer order partial differential equation. However, in heterogeneous media, it can be described by a sub- or hyperdiffusion equation which results in a fractional order partial differential equation. Taking into consideration that part of the heat flux is dispersed into the neighbouring environment we additionally modify the main relation between heat flux and the temperature, and we obtain in this case the heat transfer equation in a new form. This leads to the transfer function that describes the dependency between the heat flux at the beginning of the beam and the temperature at a given distance. This article also presents the experimental results of modelling real plant in the frequency domain based on the obtained transfer function.

Comparison and validation of integer and fractional order ultracapacitor models

In this article, the modeling of the ultracapacitor using different models of capacity part is shown. Two fractional order models are compared with the integer model of traditional capacitor. The identification was made using the diagram matching technique. Next, the derivation of time domain response of the ultracapacitor and system with the ultracapacitor are presented. The results of frequency domain identification were used to validate the response of the ultracapacitor in time domain. All theoretical results are compared with the response of the physical system with the ultracapacitor.

Experimental Evidence of Variable-Order Behavior of Ladders and Nested Ladders

The experimental study of two kinds of electrical circuits, a domino ladder and a nested ladder, is presented. While the domino ladder is known and already appeared in the theory of fractional-order systems, the nested ladder circuit is presented in this article for the first time. For fitting the measured data, a new approach is suggested, which is based on using the Mittag-Leffler function and which means that the data are fitted by a solution of an initial-value problem for a two-term fractional differential equation. The experiment showed that in the frequency domain the domino ladder behaves as a system of order 0.5 and the nested ladder as a system of order 0.25, which is in perfect agreement with the theory developed for their design. In the time domain, however, the order of the domino ladder is changing from roughly 0.5 to almost 1, and the order of the nested ladder is changing in a similar manner, from roughly 0.25 to almost 1; in both cases, the order 1 is never reached, and both systems remain the systems of non-integer order less than 1. Both studied types of electrical circuits provide the first known examples of circuits, which are made of passive elements only and which exhibit in the time domain the behavior of variable order.

Improved fractional Kalman filter and its application to estimation over lossy networks

This paper presents improvements on the fractional Kalman filter (FKF) based on the infinite dimensional form of a linear discrete fractional order state-space system. Furthermore, taking into account the considerable interest in estimation over networks with packet losses, the application and extension of the improved FKF are included. Some simulation cases are given in order to demonstrate the effectiveness of the proposed algorithms, with significant improvements in terms of estimation and smoothing results.

Adaptive Feedback Control of Fractional Order Discrete State-Space Systems

The paper is devoted to the application of fractional calculus concepts to modeling, identification and control of discrete-time systems. Fractional difference equations (FAE) models are presented and their use in identification, state estimation and control context is discussed. The fractional difference state-space model is proposed for that purpose. For such a model stability conditions are given. A fractional Kalman filter (FKF) for this model is recalled. The proposed state-space model and fractional order difference equation are used in an identification procedure which produces very accurate results. Finally, the state-space model is used in closed-loop state feedback control form together with FKF as a state estimator. The latter is also given in an adaptive form together with FKF and a modification of recursive least squares (RLS) algorithm as a parameters identification procedure. All the algorithms presented were tested in simulations and the example results are given in the paper

Identification of Parameters of a Half-Order System

This correspondence presents the half-order system behavior and its parameter identification. The identification is based on fitting the measured data using the Mittag-Leffler function. The data were collected for a discharge of a half-order system. The values of parameters obtained by a new identification method are in good agreement with the calculated interval for theoretical values, which takes into account the manufacturing tolerances of the used electrical elements.

Time domain validation of ultracapacitor fractional order model

In this paper, the modeling of the ultracapacitor using fractional order model is shown. The derivation of time domain response of the ultracapacitor and system with the ultracapacitor is presented. The results of frequency domain identification were used to validate the response of the ultracapacitor in time domain. All theoretical results are compared with the response of the physical system with the ultracapacitor. Then the issue of capacity for the ultracapacitors is shown and discussed.

New method of fractional order integrator analog modeling for orders 0.5 and 0.25

In the paper a novel method of analog modeling of fractional order integrators is presented. In particular, two special cases are discussed, i.e. integrators of order α = 0.25, and α = 0.5. The method proposed is based on the domino ladder approximation of irrational transfer function. It allows to obtain an analog model using only electric elements like resistors and capacitors of standard produced values.

Stability of discrete fractional order state-space systems

Abstract This paper presents simple stability condition for the fractional order discrete state-space system. This is one of very few attempts to give the condition of the stability of this type of systems. It is established for the state space model introduced by the Authors.