Wojciech Mitkowski

The Comparison of Parameter Identification Methods for Fractional, Partial Differential Equation

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.

Simple Identification of Fractional Differential Equation

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.

Application of fractional order transfer functions to modeling of high — Order systems

The paper presents an application of elementary fractional order transfer funtions to efective modeling of high order control plants. An example of modeling experimental heat plant is also given.

An Estimation of Accuracy of Oustaloup Approximation

In the paper a new accuracy estimation method for Oustaloup approximation is presented. Oustaloup approximation is a fundamental tool to describe fractional-order systems with the use of integer-order, proper transfer function. The accuracy of approximation can be estimated via comparison of impulse responses for plant and Oustaloup approximation. The impulse response of the plant was calculated with the use of an accurate analytical formula and it can be interpreted as a standard. Approach presented in the paper can be applied to effective tuning of Oustaloup approximant for given application (for example in FO PID controller). The use of proposed method does not require us to know time response of a modeled controller. The proposed methodology can be easily generalized to another known approximations. Results of simulations show that the good performance of approximation is reached for low order and narrow angular frequency range.

Parameter identification for non integer order, state space models of heat plant

In the paper the parameter idetification problem for new, non integer order, state space models of heat transfer process is presented. The solution of all proposed state equations bases directly onto definitions of suitable non integer order differences. The parameters estimation is done with the use of MSE (Mean Square Error) cost function. The order of all proposed state-space models is also estimated. Results of experiments show, that all proposed non integer order models are more accurate in the sense of MSE cost function than known integer order model.

Tuning of the Half-Order Robust PID Controller Dedicated to Oriented PV System

In the paper tuning rules for half - order PID controller dedicated to control an oriented PV system were presented. The plant is described with the use of interval transfer function. Results were by simulations depicted.

Fractional-Order P2D β Controller for Uncertain Parameter DC Motor

In this paper an uncertain-parameter DC motor controlled with the use of non integer order P2D β controller with uncertain-parameters is considered. For this system an analysis of BIBO (Bounded Input Bounded Output) stability with respect to uncertainty of plant’s parameters was done. Results were with an example depicted.

Stabilization Results of Second-Order Systems with Delayed Positive Feedback

Oscillation and nonoscillation criteria are established for second-order systems with delayed positive feedback. We consider the stability conditions for the system without damping and with gyroscopic effect. A general algorithm for finding stability regions is proposed. Theoretical and numerical results are presented for single-input single-output case. These results improve some oscillation criteria of [1], [2] and [6].

An Estimation of Accuracy of Charef Approximation

In the paper a new accuracy estimation method for Charef approximation was presented. Charef approximation allows us to describe fractional-order systems with the use of integer-order, proper transfer function. The accuracy of approximation can be estimated with the use of comparison step responses of plant and Charef approximation. The step response of the plant was calculated with the use of an accurate analytical formula and it can be interpreted as a standard. Approach presented in the paper can be applied to effective tuning of Charef approximant for given plant. The use of proposed method does not require to know a step response of the modeled plant. The proposed methodology can be easily generalized to another known approximations. Results are by simulations illustrated.

Fractional-Order Models of the Ultracapacitors

In the paper, dynamic behavior of the ultracapacitors is investigated and analyzed. The ultracapacitors are represented by equivalent electrical circuit models and mathematically described by fractional-order differential equations. The identification procedure is proposed to identify the parameters of the models. The results of numerical simulations are compared with the results measured experimentally in the physical system.