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Wykorzystanie estymatorów i obserwatorów stanu jako softwarowych czujników pomiarowych dla niedostępnych pomiarowo-zmiennych procesowych

Brylski Jędrzej

Elektronika : konstrukcje, technologie, zastosowania

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2023

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Vol. 64, Nr 4

35--42

PL

Artykuł przedstawia przegląd różniczkowych asymptotycznych obserwatorów (estymatorów) stanu typu Luenbergera lub Kalmana oraz całkowych obserwatorów odtwarzających stan dokładnie, stosowanych w liniowych układach dynamicznych. Estymatory i obserwatory mogą pracować jako softwarowe czujniki niedostępnych do pomiaru zmiennych wektora stanu układu. Takie soft-sensory, dostarczające wirtualnej informacji mogą być wykorzystane dla celów sterowania, do diagnostyki i wykrywania błędów w procesach technicznych oraz do monitoringu procesu, pod warunkiem, że znany jest model tego procesu. Obserwatory różniczkowe gwarantują jedynie asymptotyczne nadążanie za stanem rzeczywistym tzn. nieznany początkowy błąd odtwarzania z biegiem czasu maleje do zera, ale nie ma możliwości obliczenia jego bieżącej wartości, czyli rzeczywistego stanu. W drugiej części artykułu przedstawiono innego typu obserwator stanu, który ma strukturę opartą o operatory całkowe. Na podstawie pomiarów sygnałów wyjściowych i wejściowych układu na pewnym z góry określonym skończonym przedziale czasu T, zwanym oknem pomiarowym, może on po tym przedziale dokładnie odtworzyć stan z początku okna (stan początkowy) lub z końca okna (stan bieżący). W wersji on-line dokładna rekonstrukcja stanu x(t) jest wykonywana w sposób ciągły dla każdego t, na podstawie specjalnej procedury wykonywanej w dwóch równolegle i płynnie przesuwanych oknach o przyjętej szerokości T, na przedziale czasowym [t - T,t]. W jednym oknie przetwarzany jest sygnał sterowania, a w drugim sygnał z wyjścia systemu. Pokazano zasadnicze różnice tych obserwatorów.

EN

The paper presents an overview of differential, asymptotic state observers (estimators) like Luenberger or Kalman type as well as the integral exact state observers, both used in linear dynamical systems. Estimators and observers can work as soft-sensors for the measurement of the state vector variables which are not available for measurement. Such soft-sensors, providing virtual information, can be used for control purposes, for diagnostics and error detection in technical processes and for process monitoring, provided that the process model is known. Differential observers guarantee only asymptotic tracking of the real state, i.e. the unknown initial state reconstruction error decreases to zero over time, but it is not possible to calculate its current value, and thus the value of the real state. The second part of the paper presents a different type of state observer, which has a structure based on two integral operators. On the basis of measurements of the output and input signals of the system over a certain predetermined finite period of time T, called the measurement window, after this interval, the observer can reproduce the state exactly at the beginning of the window (initial state) or at the end of the window (current state). In the on-line version, the exact reconstruction of the state x(t) is performed continuously for each t, based on a special procedure performed in two parallel and smoothly sliding windows of the assumed width T, on a time interval [t - T,t]. In one window the control signal is processed, and in the second the signal from the system output. The main differences of these observers are shown.

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State estimators and observers for continuous and discrete linear systems. Part 2. Integral observers for exact state reconstruction

Byrski Jędrzej, Byrski Witold

Science, Technology and Innovation

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2019

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Vol. 5, no. 2

23--33

EN

In the paper, the exact state observers will be presented. The state estimators and observers can be used in technical processes for many purposes like the fault detection and diagnosis, the implementation of the state controllers, and soft reconstruction of inaccessible for measurements variables of the system. As the standard, for continuous systems the differential estimators of Kalman filter or Luenberger type observer are commonly used. However, if the initial conditions of the real state are unknown, both estimators guarantee only an asymptotic quality of the real state tracking. The paper presents another type of the state observers, which for continuous system have the structure given by two integral operators. Based on measurements of the system input and output signals on some predefined finite time interval T, they can reconstruct the initial state exactly. In on-line version, the exact state reconstruction is performed continuously for every t, based on special procedure executed within two moving windows of width T, on sliding time interval [t-T, t].

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An adaptive identification method based on the modulating functions technique and exact state observers for modeling and simulation of a nonlinear MISO glass melting process

Byrski Witold, Drapała Michał, Byrski Jędrzej

International Journal of Applied Mathematics and Computer Science

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2019

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Vol. 29, no. 4

739--757

EN

The paper presents new concepts of the identification method based on modulating functions and exact state observers with its application for identification of a real continuous-time industrial process. The method enables transformation of a system of differential equations into an algebraic one with the same parameters. Then, these parameters can be estimated using the least-squares approach. The main problem is the nonlinearity of the MISO process and its noticeable transport delays. It requires specific modifications to be introduced into the basic identification algorithm. The main goal of the method is to obtain on-line a temporary linear model of the process around the selected operating point, because fast methods for tuning PID controller parameters for such a model are well known. Hence, a special adaptive identification approach with a moving window is proposed, which involves using on-line registered input and output process data. An optimal identification method for a MISO model assuming decomposition to many inner SISO systems is presented. Additionally, a special version of the modulating functions method, in which both model parameters and unknown delays are identified, is tested on real data sets collected from a glass melting installation.

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State estimators and observers for continuous and discrete linear systems. Part 1. Differential asymptotic state estimators

Byrski J., Byrski W.

Science, Technology and Innovation

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2018

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Vol. 3, no. 2

62--68

EN

In the paper an overview of state estimators and state observers used in linear systems, will be presented. The state estimators and observers can be used in many applications like the state reconstruction for the control purposes or for the diagnosis and fault detection in technical processes or for the virtual measurements of inaccessible variables of the system as well as for the best filtration of the differential equation solution. As the standard most commonly the Kalman filter and Luenberger type observers are used. Although the Kalman filter guarantees optimal filtering quality of the state, reconstructed from the noisy measurements, both Kalman filter and the Luenberger observer guarantee only asymptotic quality of the real state changes and tracking, basing on the current measurements of the system output and input signals. Unfortunately, the value of the estimation error at any moment of time cannot be calculated. The discussion on differences between continuous and two types of discrete Kalman Filter will be presented. This paper is planned to be the introduction to presentation of another type of the state observers which have the structure given by the integral operators. Based on measurements of the system output and input signals on some predefined finite time interval, they can reconstruct, after this interval, the observed state exactly.

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A double window state observer for detection and isolation of abrupt changes in parameters

Byrski J., Byrski W.

International Journal of Applied Mathematics and Computer Science

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2016

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Vol. 26, no. 3

585--602

EN

The paper presents a new method for diagnosis of a process fault which takes the form of an abrupt change in some real parameter of a time-continuous linear system. The abrupt fault in the process real parameter is reflected in step changes in many parameters of the input/output model as well as in step changes in canonical state variables of the system. Detection of these state changes will enable localization of the faulty parameter in the system. For detecting state changes, a special type of exact state observer will be used. The canonical state will be represented by the derivatives of the measured output signal. Hence the exact state observer will play the role of virtual sensors for reconstruction of the derivatives of the output signal. For designing the exact state observer, the model parameters before and after the moment of fault occurrence must be known. To this end, a special identification method with modulating functions will be used. A novel concept presented in this paper concerns the structure of the observer. It will take the form of a double moving window observer which consists of two signal processing windows, each of width T . These windows are coupled to each other with a common edge. The right-hand side edge of the left-side moving window in the interval [t − 2T, t − T ] is connected to the left-hand side edge of the right-side window which operates in the interval [t − T, t]. The double observer uses different measurements of input/output signals in both the windows, and for each current time t simultaneously reconstructs two values of the state—the final value of the state in the left-side window zT (t − T ) and the initial value of the state z0(t − T ) in the right-side window. If the process parameters are constant, the values of both the states on the common joint edge are the same. If an abrupt change (fault) in some parameter at the moment tA = t − T occurs in the system, then step changes in some variables of the canonical state vector will also occur and the difference between the states will be detected. This will enable localization of the faulty parameter in the system.

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The Application of the Exact State Estimation Method in Electric Power Systems

Dziewiński T., Włodarczyk P.

Przegląd Elektrotechniczny

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2013

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R. 89, nr 7

164--168

EN

A large share of distributed, renewable, intermittent energy resources resulted in the increased dynamics of the electricity grid and made it less predictable. Electric Power Systems (EPS), which used to be considered quasi-static systems of high order, with the presence of Distributed Energy Resources (DERs) at the distribution level, are constantly changing into active dynamic systems. This implies the application of the Dynamic State Estimation (DSE) which can estimate voltage and phase in real-time as well as can be used for the diagnostic purposes, hardware maintenance and control. In this paper Exact State Estimator for the EPS, sometimes referred also as Integral State Observer (ISO) is designed. Furthermore the comparison study with the classical Kalman-Bucy filter is made showing the advantages of the ISO over classical methods.

PL

Znaczący udział rozproszonych oraz odnawialnych źródeł enegii o dostawach nieciągłych spowodował zmianę dynamiki sieci elektroenergetycznej, utrudniając przewidywalność dostaw energii. System elektroenergetyczny traktowany wcześniej jako quasi-statyczny system wysokiego ręedu, ewoluuje obecnie w stronę systemu dynamicznego ze względu na obecność rozproszonych źrodeł enegii na poziomie dystrybucji. Implikuje to koniecznosc dynamicznej obserwacji stanu systemu do celów określenia wartosci napięcia i fazy w czasie rzeczywistym oraz na potrzeby systemów diagnostyki oraz systemów sterowania. Poniższy artykuł prezentuje metodę Dokładnego Odtwarzania Stanu, znaną również pod nazwą metody Obserwatorów Całkowych. W artykule dokonano porównania metody dokładnego odtwarzania stanu z klasyczną metodą opartą o Filtr Kalmana-Bucy, wskazując zalety tej pierwszej.