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EN
This paper focuses on the global practical Mittag-Leffler feedback stabilization problem for a class of uncertain fractional-order systems. This class of systems is a larger class of nonlinearities than the Lipschitz ones. Based on the quasi-one-sided Lipschitz condition, firstly, we provide sufficient conditions for the practical observer design. Then, we exhibit that practical Mittag-Leffler stability of the closed loop system with a linear, state feedback is attained. Finally, a separation principle is established and we prove that the closed loop system is practical Mittag-Leffler stable.
EN
Mathematical modeling with the original approach is used to study the dynamics of low-frequency electro-vibration machines while taking the technological load into account. Based on the mathematical analysis of differential equations describing the oscillatory motion of electromagnetic vibration exciters, the existing mathematical model has been corrected to describe the action of a semiconductor diode in the power circuits of single-cycle vibrators. Modeling nonlinear dynamic systems reveals the difficulties of obtaining the amplitude frequency characteristics of the associated discrete change in the frequency of the driving force, as well as ways to overcome them.
EN
In this paper, model reference output feedback tracking control of an aircraft subject to additive, uncertain, nonlinear disturbances is considered. In order to present the design steps in a clear fashion: first, the aircraft dynamics is temporarily assumed as known with all the states of the system available. Then a feedback linearizing controller minimizing a performance index while only requiring the output measurements of the system is proposed. As the aircraft dynamics is uncertain and only the output is available, the proposed controller makes use of a novel uncertainty estimator. The stability of the closed loop system and global asymptotic tracking of the proposed method are ensured via Lyapunov based arguments, asymptotic convergence of the controller to an optimal controller is also established. Numerical simulations are presented in order to demonstrate the feasibility and performance of the proposed control strategy.
4
Content available remote Study of forming process in memristive devices using rectangular waves
EN
This work focuses on the study of behavior of memristors during the process of forming, i.e., setting the resistance of the element and testing the stability of the resistance. Two types of BS-AF memristive elements are selected for laboratory experiments. The goal of the research is to investigate the conditions under which the resistance of the element is changed when voltages of different levels, duration and polarization are applied. Inparticular, weareinterestedinfindingthethresholdvoltagevaluewhichisneededtomodifyofstateoftheelement. Suchavaluedefinesthevoltage level below which one can safely measure instantaneous value of memristor’s resistance (data read operation) without changing this value. The results obtainedshowthatthethresholdvaluedependsonthepolarizationoftheappliedvoltage. Memristorsunderstudyaremuchmoresensitivetoapplying a negative voltage. This means, that a significantly smaller value of the voltage is needed to switch from low to high resistance level than to switch from high to low resistance level. It has also been observed that the final value of the memristor’s resistance depends on the number and level of applied voltage impulses. This study confirms that BS-AF memristors can operate in more than two states in a stable way, which can be beneficial when using memristors in memory structures or for neuromorphic applications.
PL
W pracy opisano wyniki badań nad procesem formowania memrystorów, tzn. programowaniem poziomu rezystancji chwilowej elementu oraz stabilnościa uzyskanej wartości. W części pomiarowej wykorzystano dwa rodzaje elementów memrystorowych z grupy BS-AF. Celem badań było określenie warunków niezbednych do zmiany rezystancji memrystora podczas wzbudzania napięciami o różnych poziomach, czasie trwania i polaryzacji. W szczególności poszukiwano wartości progowej napiecia zasilającego potrzebnej do modyfikacji stanu memrystora. Wartość taka definiuje poziom napięcia, poniżej którego możliwy jest pomiar wartości chwilowej memrystancji (operacja odczytu danych), bez zmiany opisywanej wartości. Uzyskane wyniki wskazują, że wartość ta zależy od polaryzacji przyłożonego napięcia. Badane memrystory są znacznie bardziej czułe na napięcie ujemne. Oznacza to, że znaczaco niższy poziom napiecia jest potrzebny aby przełączyć memrystor ze stanu niskiej do wysokiej rezystancji niż odwrotnie. Zaobserwowano również, że ostateczna wartość rezystancji chwilowej memrystora zależy od liczby i poziomu impulsów napięciowych. Eksperyment ten potwierdza,iz memrystory typu BS-AF mogą uzyskiwać więcej niż dwa poziomy rezystancji chwilowej w stabilny sposób, co może byc kluczowe dla wykorzystania ich w strukturach pamięci lub systemach neuromorficznych.
EN
The non-linear stochastic dynamic behaviour of a high-rise vertical transportation system modelled as a concentrated mass and a cable with finite bending stiffness is considered. The slow time scale is defined and lateral cable displacements coupled with transverse motions are expanded in terms of approximating functions. The excitation of the high-rise building is assumed in the form of a narrow-band mean-square process equivalent to the harmonic process. The equivalent linearization technique is used to replace the original non-linear system with a linear approximation whose coefficients are determined from minimization of the mean-square equation difference between both systems.
EN
Although nonlinear mechanical systems have been the topic of numerous investigations during the last decades, the research on suitable analysis methods is still ongoing. One method that is commonly known and still sees a lot of interest is the Harmonic Balance method. In the basic version of this method, only one harmonic is used to approximate a periodic solution, which allows for fairly easy application. A drawback is that this approach may lead to solutions that are inaccurate or even artifacts which are solutions that possess no physical relevance. In this article, it is demonstrated how an error criterion can be used to access the accuraccy of solutions and how artifacts can be indentified based on this assessment. Subsequently, stability analysis is performed for solutions that possess small errors. The method is applied to an asymmetric Duffing oscillator as well as to a system that consists of two linearly coupled Duffing oscillators. The authors gave a corresponding presentation of their work at PCM-CCM Kraków 2019.
7
EN
Methods for solving non-linear control systems are still being developed. For many industrial devices and systems, quick and accurate regulators are investigated and required. The most effective and promising for nonlinear systems control is a State-Dependent Riccati Equation method (SDRE). In SDRE, the problem consists of finding the suboptimal solution for a given objective function considering nonlinear constraints. For this purpose, SDRE methods need improvement. In this paper, various numerical methods for solving the SDRE problem, i.e. algebraic Riccati equation, are discussed and tested. The time of computation and computational effort is presented and compared considering selected nonlinear control plants.
EN
Model predictive control (MPC) algorithms brought increase of the control system performance in many applications thanks to relatively easily solving issues that are hard to solve without these algorithms. The paper is focused on investigating how to further improve the control system performance using a trajectory of parameters weighting predicted control errors in the performance function of the optimization problem. Different shapes of trajectories are proposed and their influence on control systems is tested. Additionally, experiments checking the influence of disturbances and of modeling uncertainty on control system performance are conducted. The case studies were done in control systems of three control plants: a linear non-minimumphase plant, a nonlinear polymerization reactor and a nonlinear thin film evaporator. Three types of MPC algorithms were used during research: linear DMC, nonlinear DMC with successive linearization (NDMC–SL), nonlinear DMC with nonlinear prediction and linearization (NDMC–NPL). Results of conducted experiments are presented in greater detail for the control system of the polymerization reactor, whereas for the other two control systems only the most interesting results are presented, for the sake of brevity. The experiments in the control system of the linear plant were done as preliminary experiments with the modified optimization problem. In the case of control system of the thin film evaporator the researched mechanisms were used in the control system of a MIMO plant showing possibilities of improving the control system performance.
EN
This paper presents the design of digital controller for longitudinal aircraft model basedon the Dynamic Contraction Method. The control task is formulated as a tracking problem of velocity and flight path angle, where decoupled output transients are accomplished in spite of incomplete information about varying parameters of the system and external disturbances. The design of digital controller based on the pseudo-continuous approach is presented, where the digital controller is the result of continuous-time controller discretization. A resulting output feedback controller has a simple form of a combination of low-order linear dynamical systems and a matrix whose entries depend nonlinearly on certain known process variables. Simulation results for an aircraft model confirm theoretical expectations.
10
Content available remote Optymalizacja procesów biologicznych w oczyszczalni ścieków typu SBR
PL
Biologiczna oczyszczalnia ścieków jest skomplikowanym systemem sterowania. Jednym z istotnych i kosztownych procesów tam zachodzących jest napowietrzanie ścieków. Prawidłowy jego przebieg ma decydujący wpływ na stopień oczyszczenia ścieków i koszty operacyjne pracy oczyszczalni. W związku z tym prowadzone są prace badawcze nad optymalizacją procesów biologicznych, pozwalającą redukować koszty, np. zużycie energii. W artykule opracowano model decyzyjny, a następnie rozwiązano zadanie optymalizacji - wyznaczenia optymalnej wartości zadanej stężenia tlenu. W badaniach symulacyjnych przedstawiono wyniki optymalizacji dla oczyszczalni ścieków typu SBR.
EN
A biological wastewater treatment plant is a complex control system. One of the important and costly processes is the aeration of wastewater. The correct of the process has a decisive influence on the degree of sewage treatment and the operating costs of the treatment plant. Therefore, research works are being carried out on optimisation of biological processes, allowing to reduce costs, e.g. energy consumption. In the paper, a decision model was developed and then the task of optimization - determining the optimal set point of dissolved oxygen was solved. In the simulation studies the optimization results for wastewater treatment plant type SBR were presented.
EN
In this paper, an operational method for solving linear and nonlinear systems described by ordinary differential equations is presented. The construction is based on the generalized derivative in the sense of distribution theory. The approach allows the response computation without needing the use of any integral transform as in the Mikusiński operational calculus. A general description of the algorithm is done and some illustrating examples are presented. The algorithm is recursive allowing to add and remove any pole or zero contribution. The extension to nonlinear systems is done by means of the Adomian polynomials.
PL
Biologiczna oczyszczalnia ścieków jest złożonym nieliniowym systemem sterowania. Jednym z istotnych i kosztownych procesów tam zachodzących jest napowietrzanie ścieków. Prawidłowy jego przebieg ma decydujący wpływ na stopień oczyszczenia ścieków i koszty operacyjne pracy oczyszczalni. W związku z tym prowadzone są prace badawcze nad nowymi strukturami i algorytmami sterowania. Powinny się one odznaczać wysoką efektywnością i niewielkim zapotrzebowaniem na energię elektryczną. W artykule zaprojektowano nowy adaptacyjny układ sterowania stężeniem tlenu uwzględniający instalację napowietrzającą jako system dostarczający powietrze do oczyszczalni ścieków. W badaniach symulacyjnych przedstawiono wyniki sterowania dla biologicznej oczyszczalni ścieków typu SBR.
EN
Biological wastewater treatment plant is a complex, nonlinear control system. One of the most important and expensive processes conducted is aeration. It’s valid flow has a deciding impact on the level of wastewater purification. Therefore a number of research are carried out to develop new control structures and algorithms. They should be characterized by high efficiency and low energy consumption. In the paper a novel adaptive control strategy of dissolved oxygen is employed. It considers aeration system as a source of the oxygen to the plant. Simulation tests presents control results for the biological wastewater treatment plant type SBR.
EN
This is a short tutorial on Volterra and Wiener series applications to modelling of nonlinear systems and phenomena, and also a survey of the recent achievements in this area. In particular, we show here how the philosophies standing behind each of the above theories differ from each other. On the other hand, we discuss also mathematical relationships between Volterra and Wiener kernels and operators. Also, the problem of a best approximation of large-scale nonlinear systems using Volterra operators in weighted Fock spaces is described. Examples of applications considered are the following: Volterra series use in description of nonlinear distortions in satellite systems and their equalization or compensation, exploiting Wiener kernels to modelling of biological systems, the use of both Volterra and Wiener theories in description of ocean waves and in magnetic resonance spectroscopy. Moreover, connections between Volterra series and neural network models, and also input-output descriptions of quantum systems by Volterra series are discussed. Finally, we consider application of Volterra series to solving some nonlinear problems occurring in hydrology, navigation, and transportation.
PL
W artykule zaproponowano różne modele łuku elektrycznego oraz analityczną metodę wyznaczania napięcia łuku elektrycznego na podstawie równania różniczkowego Cassiego i hybrydowego. Analizę przeprowadzono w dziedzinie czasu. W analizie zastosowano podejście deterministyczne. Wyniki porównano z pomiarami.
EN
Different models of the electric arc and an analytical method of determining solution in close form for the arc voltage have been proposed in this article. This analysis has been conducted in the time domain. Deterministic approach has been used and results have been compared with measurements.
15
Content available Impact oscillators : fundamentals and applications
EN
The paper presents results of the analysis of nonlinear oscillators with nonsmooth elements and nonlinear systems with nonsmooth forcing components. Main attention has been focused on highlighting specific properties of nonsmooth systems compared to their smooth counterparts. Nonsmooth transformation of the time variable and the replacement of initial issues by boundary problems have been taken as the base for the analytical method. Results of numerical simulations and computing in the form of graphs of displacements and velocity waveforms and attractors are presented. To fully identify the system's behaviour and meet high performance specifications recourse to model all dynamics together with their interactions has been taken into account. Strong interactions among the parts of the system are considered and the phenomenon of the impact is exhibited. It has been found that non-smooth dynamical systems reveal significant wealth of nonlinear phenomena, including a chaotic, that are unique to this potentially important class of nonlinear systems. In non-smooth systems at small change of parameters, a sudden transition from a stable periodic oscillation to the full range of chaotic oscillations may often occur. The dynamics of nonsmooth oscillations with shock external forcing is analysed by using a relatively new mathematical tool, which appears to be hyperbolic algebra. The key idea of this tool is steeped in of non-smooth time transformations (NSTT) for strongly nonlinear, but still smooth models.
EN
This paper presents extensions of some results, obtained for the analysis of classical nonlinear control systems, to the nonlinear fractional order systems. It is shown that the results related to limit cycle prediction using describing function method can be applied to the fractional order plants. The frequency and the amplitude of the limit cycle are used for auto-tuning of the PID controller for nonlinear control systems with fractional order transfer functions. Fractional order control system with parametric uncertainty is also considered for the nonlinear case. On the other hand, a New method is provided for stability margin computation for fractional order nonlinear control system with parametric uncertainty structure using the Nyquist envelopes of the fractional order uncertain plant and the describing function that represents the nonlinearity of the system. Maximum perturbation bounds of the parameters of the fractional order plant are computed. Numerical examples are included to illustrate the methods presented.
17
EN
The paper deals with the problem of robust fault-tolerant control for non-linear discrete-time systems. The main part of this paper describes sensor a fault diagnosis scheme using virtual sensors, which recover the measurement of the fault sensor based on the fault-free ones. The virtual sensor is designed in such a way that a prescribed attenuation level is achieved with respect to the fault estimation error while guaranteeing the convergence of the robust observer underlying the virtual sensor. The subsequent part of the paper deals with the design of robust controller as well as the proposed fault-tolerant control scheme. The final part of the paper shows the experimental results regarding the multi-tank system, which confirm the effectiveness of proposed approach.
PL
Artykuł przedstawia problem odpornego sterowania tolerującego uszkodzenia układem nieliniowym w czasie dyskretnym. Głowna część artykułu opisuję diagnostykę czujników pomiarowych z wykorzystaniem wirtualnych czujników. Wirtualny czujnik został zaprojektowany w taki sposób, aby dla określonego poziomu tłumienia zakłóceń uzyskać możliwie mały błąd estymacji stanu oraz zapewnić dobra zbieżność obserwatora, na którym bazuje wirtualny czujnik. Kolejna część artykułu opisuję metodę projektowania odpornego regulatora oraz proponowaną metodę detekcji uszkodzeń. W końcowej części artykułu przedstawiono wyniki eksperymentalne dotyczące układu wielu zbiorników. Zastosowany układ wielu zbiorników charakteryzuję się nieliniowością, przez co może zostać wykorzystany do weryfikacji przedstawionych w artykule techniki sterowania tolerującego uszkodzenia. Przedstawione wyniki potwierdzają skuteczność proponowanego podejścia bazującego na detekcji uszkodzeń czujników pomiarowych z wykorzystaniem wirtualnych czujników pomiarowych. Wyniki obrazują działanie układu w przypadku wystąpienia uszkodzenia przy braku sterowania tolerującego uszkodzenia oraz działanie układu w przypadku zastosowania sterowania tolerującego uszkodzenia.
PL
Biologiczna oczyszczalnia ścieków jest złożonym nieliniowym systemem sterowania. Jednym z istotnych i kosztownych procesów tam zachodzących jest napowietrzanie ścieków. Prawidłowy jego przebieg ma decydujący wpływ na stopień oczyszczenia ścieków i koszty operacyjne pracy oczyszczalni. W związku z tym prowadzone są prace badawcze nad nowymi strukturami i algo-rytmami sterowania. Powinny się one odznaczać wysoką efektywnością i niewielkim zapotrzebowaniem na energię elektryczną. W artykule zaprojektowano kaskadowy układ sterowania stężeniem tlenu uwzględniający instalację napowietrzającą jako system dostarczający powietrze do oczyszczalni ścieków. W badaniach symulacyjnych przedstawiono wyniki sterowania dla biologicznej oczyszczalni ścieków w Nowym Dworze Gdańskim.
EN
A biological wastewater treatment plant is a complex, nonlinear control system. Aeration is one of the most important and expensive processes conducted in it. Its valid flow has a deciding impact on the level of wastewater purification. Therefore a number of research are carried out to develop new control structures and algorithms. They should be characterized by high efficiency and low energy consumption. In the paper a novel cascade control strategy of dissolved oxygen is employed. It considers the aeration system as a source of the oxygen to the plant. Simulation tests present control results for the case study of the wastewater treatment plant located at Nowy Dwór Gdański. The paper is divided into 4 sections. Section 1 contains a short introduction to the main issues of research. The structure and models of the wastewater treatment plant at Nowy Dwor Gdanski and the aeration system are described in Section 2. Section 3 presents a nonlinear cascade control algorithm. The simulation results and analysis are described in Section 4. Section 5 concludes the paper.
PL
W artykule zaprezentowano adaptacyjny układ sterowania kątem kursowym i prędkością statku opisanego nieliniowym modelem dynamicznym MIMO (ang. Multi-input, Multi-Output) o czterech stopniach swobody ruchu (ang. Four-Degrees-of-Freedom, 4-DoF). Przedstawiony model statku jest silnie nieliniowy i uwzględnia nie tylko ograniczenia saturacyjne sygnałów sterujących ale także dynamikę maszyny sterowej i napędu głównego. Głównym zadaniem zaprojektowanego adaptacyjnego regulatora LQR jest sterowanie kątem kursowym statku i jego prędkością względem wody. Poszczególne zestawy parametrów regulatora użyte w procesie adaptacji zostały wyznaczone na podstawie linearyzacji modelu statku w wybranych nominalnych punktach pracy. W końcowej części artykułu przedstawiono wyniki badań symulacyjnych zaprojektowanego regulatora w układzie z pełnym nieliniowym modelem statku.
EN
In the paper an adaptive control system for the nonlinear 4-DoF model of a container vessel is presented. The considered model of the ship includes input signals saturations and dynamics of actuators. Main goal of the presented control system is control of the course angle and speed of the ship relative to water. The system synthesis is carried out by means of LQ-optimal control method. Sets of controller parameters are designed using linearization of the considered model in the nominal steady-state operating points of the ship. The final part of the paper includes simulation results of control system operation with full nonlinear MIMO model of the container vessel.
PL
Proces destylacji jest popularnym przemysłowym sposobem rozdziału ciekłych mieszanin wieloskładnikowych. W artykule rozważana jest mieszanina etanol-woda, będąca powszechnie destylowanym produktem. Celem artykułu jest omówienie procesu projektowania i symulacyjnej analizy działania trzech układów sterowania kolumną rektyfikacyjną z wykorzystaniem rozmytych regulatorów: P, PD i PI.
EN
Distillation is the most common unit operations in different branches of industry. In the paper a two product distillation process is considered (ethanol-water). The paper presents mathematical model of distillation column. Next, the three fuzzy controllers (P, PD, PI) are designed and implemented in MATLAB/Simulink environment. Finally, the results of simulation and a critical analysis are presented.
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