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EN
The paper presents a comparing integer order (IO) vs fractional order (FO) transfer function models of the forced air heating system applied in housing heating. The proposed FO models are simpler to identify than analogical IO models and their accuracy in the sense of fitting to experimental results is practically the same. Results are verified using experimental data from the real isothermal room, used in a pharmaceutical factory.
PL
W pracy zaprezentowano porównanie modeli transmitancyjnych całkowitego i niecałkowitego rz˛edu dla systemu nagrzewu powietrza stosowanego w ogrzewaniu budynków. Proponowane modele sa˛ tak samo dokładne jak modele całkowitego rze˛du, a ich identyfikacja jest prostsza ze wzgle˛du na mniejsza˛ liczbe˛ parametrów. Prezentowane wyniki zostały zweryfikowane z wykorzystaniem danych eksperymentalnych z pomieszczenia izotermicznego stosowanego w fabryce farmaceutycznej.
EN
In the paper two non-integer order, state space models of heat transfer process are compared. The first uses a known Caputo operator and the second – a new operator proposed by Caputo and Fabrizio in 2015. Both discussed models are modifications of a known, integer order, state space, semigroup model of heat transfer process. Parameters of both models were identified by means of optimization of MSE cost function with the use of simplex method, available in MATLAB. Both proposed models have been compared in the aspect of accuracy and convergence. Analytical and numerical results show that the Caputo-Fabrizio model is faster convergent and easier to implement than the Caputo model. However, its accuracy in the sense of MSE cost function is worse.
PL
W pracy omówiono podstawowe własności układów niecałkowitego rzędu. Podano wybrane przykłady zastosowań z następujących obszarów: modelowanie, identyfikacja, regulacja automatyczna. Przedstawiono również przykłady symulacji komputerowych.
EN
The paper discusses the basic properties of fractional systems. Given some examples of applications of the following areas: modeling, identification, automatic control. It also presents examples of computer simulations.
EN
The paper presents an approximation method for elementary fractional order transfer function containing both pole and zero. This class of transfer functions can be applied for example to build model - based special control algorithms. The proposed method bases on Charef approximation. The problem of cancelation pole by zero with useful conditions was considered, the accuracy discussion with the use of interval approach was done also. Results were depicted by examples.
EN
This paper considers a method of designing fractional-order observers for continuous-time linear fractional-order systems with unknown inputs. Conditions for the existence of these observers are given. Sufficient conditions for the asymptotical stability of fractional-order observer errors with the fractional order α satisfying 0 < α < 2 are derived in terms of linear matrix inequalities. Two numerical examples are given to demonstrate the applicability of the proposed approach, where the fractional order α belongs to 1 ≤ α < 2 and 0 < α ≤1, respectively. A stability analysis of the fractional-order error system is made and it is shown that the fractional-order observers are as stable as their integer order counterpart and guarantee better convergence of the estimation error.
EN
The optimization problem for fractional discrete-time systems with a quadratic performance index has been formulated and solved. The case of fixed final time and a free final state has been considered. A method for numerical computation of optimization problems has been presented. The presented method is a generalization of the well-known method for discrete-time systems of integer order. The efficiency of the method has been demonstrated on numerical examples and illustrated by graphs. Graphs also show the differences between the fractional and classical (standard) systems theory. Results for other cases of the fractional system order (coefficient ) and not illustrated with numerical examples have been obtained through a computer algorithm written for this purpose.
PL
W pracy rozpatrzono problem syntezy obserwatorów dodatnich liniowych układów dyskretnych ułamkowego rzędu. Wykorzystując podejście oparte na liniowych nierównościach macierzowych (LMI) sformułowano warunki istnienia dodatniego asymptotycznie stabilnego obserwatora. Podano warunki konieczne i wystarczające obserwowalności dodatnich dyskretnych układów ułamkowego rzędu. Rozważania zilustrowano przykładem symulacyjnym.
EN
The paper is devoted to observer synthesis for linear positive fractional discrete-time systems. The problem of finding a nonnegative gain matrix of the observer such that the observer is positive and asymptotically stable is formulated and solved by the use of LMI method. Necessary and sufficient conditions for the observability for linear positive fractional discrete-time systems are formulated and proved. The proposed approach to computation the nonnegative gain matrix is illustrated by numerical example.
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