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Pozycja Open Access Data-Based Estimation of Parameters for Time-Inhomogeneous CTMC Birth-and-Death of Call Centers(EUROSIS-ETI, 2023-10-24) Burak, Maciej, Rafał; West Pomeranian University of Technology in Szczecin, PolandIn this paper, we present a novel approach to estimate parameters of a queuing model using real call center data. Our approach leverages the transient solution of an inhomogeneous continuous-time Markov chain (CTMC) queuing model emulating the behavior of the genuine system. Specifically, to assess the fidelity of our modeling approach, we use an example of approximating the real call center system with an inhomogeneous M(n)/M/c/K + M CTMC model. To do this, we utilize authentic call center data for replicating the behavior of the real system through our model. In particular, the model incorporates the true time-dependent rates for the arrival process, service rates, and the number of available servers. Our analysis focuses on assessing the accuracy of the Markovian assumptions made for modeling customer abandonment during waiting periods. Furthermore, we investigate the performance of our model under two distinct scenarios: overloaded systems and systems operating in a quality-driven mode. By examining these cases, we ascertain the effectiveness of our assumptions in accurately representing the behavior of the call center. Finally, we demonstrate the practical application of our findings by showcasing how a simple and computationally efficient M(n)/M/c/K+M Markovian approximation of a real call center can be used for accurate personnel planning while adhering to service-level constraints.Pozycja Open Access Convergence of the optimal non-linear GPC method with iterative state-dependent, linear time-varying approximation(2005) Orłowski, Przemysław; Technical University of SzczecinSolution of the optimal non-linear GPC method is mostly obtained in an iterative way with 3 following steps: initialisation, transformation from non-linear system given in general form into time-varying state-dependent form, and checking whether the convergence condition is satisfied. The main aim of this paper is to analyse how the transformation method from nonlinear model into time varying state-space dependent form have effect on the convergence of the algorithm. We try to answer following questions: If chosen transformation method is suitable ? If the convergence to the optimal solution is guaranteed ? If the number of iterations can be cut down ?Pozycja Open Access Periodic Linear Time-Varying System Norm Estimation Using Running Finite Time Horizon Transfer Operators(Department of Engineering, University of Ferrara, Ferrara, Italy, 2010-11) Orlowski, Przemyslaw; West Pomeranian University of Technology, Szczecin, Department of Control and MeasurementsA novel method for norm estimation for dynamical linear time-varying systems is developed. The method involves operators description of the system model i.e. transfer operator. The transfer operator defined for finite time horizon can be described by finite dimensional matrix whereas for infinite time horizon the operator is infinite dimensional. The norm estimate for infinite time horizon is based on analysis of a running series of the finite time horizon norm properties.Pozycja Open Access Robust control design for a class of non-linear systems(2003) Orłowski, Przemysław; Technical University of SzczecinThe paper develops design methods for a class of uncertain, non-linear control systems. It extends the robustness analysis techniques of linear time varying systems and in particular the associated computational methodology to a class of nonlinear systems. It can be divided into following four parts: theoretical background, identification procedure, structure of feedback control system and cost functional for control optimisation for uncertain non-linear systems, control optimisation algorithm and the method for estimate the worst case output uncertainty norm of the system.Pozycja Open Access An introduction to SVD-DFT frequency analysis for time-varying systems(Wydawnictwo Uczelniane Politechniki Szczecińskiej, 2003) Orłowski, Przemysław; Technical University of SzczecinThe paper develops tools and methods for linear time-varying, discrete-time sys-tems analysis. It consists of theoretical background, definitions and numerical algorithms for frequency characteristics approximation. The main method is based on Singular Value Decomposition SVD, Discrete Fourier Transform DFT and power density spectrum proper-ties. A few illustrative numerical examples are included. Three different models have been analysed: oscillatory element, low pass filter and variable structure system. For better evaluation, results for presented method are compared with classical Bode characteristics.Pozycja Open Access Selected Problems of Uncertainty Estimates for Non-linear Discrete-Time Systems(2000) Orłowski, Przemysław; Technical University of SzczecinThe paper develops a mathematical framework which helps to analyse the following class of finite horizon control problems for uncertain nonlinear discrete-time systems. [ WZORY] Uncertainty in the system description are modelled by unknown (norm bounded) mixed additive-multiplicative perturbations of the system matrix. It is assumed that the nominal control has feedback form. If the nominal control will be applied to the uncertain system the state and output will be (in general) different. Formulas for deriving estimates for the deviations of the output of perturbed system from the output of the nominal one has been presented. These estimates use norms of certain dynamical operators defined on a finite time interval.Pozycja Open Access Extension of SVD-DFT analysis for a class of non-linear systems(2006) Orłowski, Przemysław; Technical University of Szczecin, Control Engineering InstituteThe paper develops computer algebra based method and tools for a class of non-linear time-varying (LTV), discrete-time systems. Proposed method base on two following transformations: from general NL system into state-space dependent piecewise linear (PWL) and from PWL into LTV. LTV system is then decomposed using method based on Singular Value Decomposition, Discrete Fourier Transform and Power Spectral Density into approximated Bode diagrams. Potential applications are simplified analysis and synthesis in frequency domain of weak and/or slow nonlinear systems. Especially the synthesis can be done for each time instant which pretend the method for applications in predictive control. The paper begins from short literature review, description of the model, detailed algorithm with description of the proposed method and two numerical examples.Pozycja Open Access Comparative boundary and sensitivity analysis for uncertain dynamical systems(Wydawnictwo Uczelniane Politechniki Szczecińskiej, 2002) Orłowski, Przemysław; Technical University of SzczecinThe paper develops analysis for models of uncertainty for dynamical, discretetime control systems on finite time horizon. Various models of uncertainty are analysed: additive, subtractive, and multiplicative. An analysis for electrical circuit with real, perturbed parameters is carried out. It is assumed, that the uncertain parameters are bounded, and could be described by rectangular distribution. A few cases of systems are considered. In the first case system is time-invariant. In the second case, system is time variant. Output errors are estimated using discrete evolution operators. As a comparison sensitivity analysis for time-invariant sis has been carried out. The results from estimations are compared to the set of the worst-case uncertain parameters related to output error.Pozycja Open Access Wprowadzanie i analiza zaburzeń parametrów w niepewnych układach dyskretnych(2002) Orłowski, Przemysław; Politechnika SzczecińskaW artykule przeprowadzono analizę niepewności występującej w układach dynamicznych, oraz przedstawiono sposoby wprowadzania jej do modelu matematycznego układu. Przeanalizowano sześć struktur wprowadzania niepewności do układu, m.in. postać addytywną, multiplikatywną, w sprzężeniu zwrotnym na wejściu i wyjściu układu. W dalszej części artykułu przeprowadzono analizę układu elektrycznego, z rzeczywistymi, niepewnymi parametrami. Przyjęto, że zakres zmian parametrów jest ograniczony i może być opisany rozkładem prostokątnym. Rozpatrzono dwa przykłady. Pierwszy, w którym występujące źródła zaburzeń poszczególnych parametrów układu są nieskorelowane oraz drugi, w którym wszystkie zaburzenia mają to samo źródło (w tym przypadku temperaturę). Błędy wyjścia zostały oszacowane bazując na metodzie wykorzystującej dyskretne operatory ewolucyjne. Rezultaty porównano z odpowiedziami układu wyznaczonymi dla ekstremalnych wartości parametrów.Pozycja Open Access Properties of the frequency SVD-DFT for discrete LTV systems based on first order examples(2006) Orlowski, Przemyslaw; Technical University of SzczecinThe paper develops frequency analysis tools for linear time-varying (LTV), discrete-time systems. The main method is based on the properties of the Singular Value De-composition (SVD), Discrete Fourier Transform (DFT) and Power Spectral Density (PSD). The analysis is carried out for a system with first order dynamics. The general objective of this paper is to examine how system frequency diagrams are depend on the variability of particular parameters. Especially, it is examined how the variability of three matrices of the state space model (scalars, in the 1st order case) influence the approximated Bode diagrams. A few cases of the variability of each matrix, in particular with different frequencies and phase shifting, are considered. Moreover, the analysis is carried out for different cases of the system parameters. The results of analysis for each case are shown in four diagrams: amplitude, phase, impulse and step responses. On the basis of these examples the most important features in each example are characterized.Pozycja Open Access Output uncertainty estimates for a class of non-linear discrete-time systems(Wydawnictwo Uczelniane Politechniki Szczecińskiej, 2004) Orłowski, Przemysław; Technical University of Szczecin, Control Engineering InstituteThe paper develops a mathematical framework which helps to analyse a class of non-linear, uncertain, discrete-time systems defined on finite horizon control. Uncertainty in the system is modelled by unknown (norm bounded) additive perturbations of the system matrices. The main purpose of the paper is to derive estimates for output deviations system from the output of the nominal (unperturbed) one. These estimates use norms of certain dynamical operators defined on a finite time interval.Pozycja Open Access Analiza częstotliwościowa układu zmiennego w czasie – algorytm i przykłady(Instytut Badań Systemowych Polskiej Akademii Nauk, 2005) Orłowski, Przemysław; Politechnika SzczecińskaW artykule zaproponowano nowe narzędzia oraz metodę do analizy układów niestacjonarnych dyskretnych w dziedzinie częstotliwości. W części teoretycznej zamieszczono najważniejsze twierdzenia i definicje wraz z dowodami oraz wynikający z nich algorytm do aproksymacji charakterystyk częstotliwościowych dla układów niestacjonarnych. Stosowane podejście bazuje na dekompozycji singularnej operatora układu, dyskretnej transformacie Fouriera i własnościach gęstości widmowej mocy. Wynikiem analizy jest funkcja przejścia dana w postaci zbioru częstotliwości wraz z odpowiadającym im zespolonym wektorem przejścia. Dla układów stacjonarnych funkcja ta pokrywa się z klasycznym odpowiednikiem – charakterystyką Bode’go i Nyquista. W końcowej części artykułu pokazano kilka przykładów charakterystyk dla różnych układów niestacjonarnych.