Andrzej Dzieliński

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.

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

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.

Fractional Order Model of Beam Heating Process and Its Experimental Verification

In the paper the application of fractional order calculus to the modelling of a beam heating process is discussed. The original process description in the form of the partial differential equation (Heat Transfer Equation) is transformed into a fractional order partial differential equation when the heat-flux is treated as the system input and the temperature is the system output. Using the Laplace transform, the transfer function of the beam heating system and its frequency response are obtained from the time-domain description. The theoretical results are verified with the experimental setup, using the thermoelectric (Peltier) module. The experimental results match the theoretical ones with high degree of accuracy.

Observer for discrete fractional order state-space systems

Abstract The paper is devoted to the design of an Luenberger-type observer for discrete fractional order systems. The discrete fractional order state space system is first recalled. Then its observability is discussed, and the observer equations are derived. Finally, some numerical examples are presented, and the applicability of the proposed observer is discussed. Also some numerical aspects of the observer algorithm are raised.

Modeling Heat Transfer in Heterogeneous Media Using Fractional Calculus

The paper presents the results of modeling 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 solid material (beam) can be described by integer order partial differential equation. However, in heterogeneous media it can be described by sub- or hyperdiffusion equation which results in fractional order partial differential equation. Taking into consideration that the 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 the new form. This leads to the transfer function which describes the dependency between the heat flux at the beginning of the beam and the temperature at the given distance. The article also presents the experimental results of modeling real plant in the frequency domain basing on the obtained transfer function.© 2011 ASME