Discrete Time Systems

Discrete-Time Systems comprehend an important and broad research field. The consolidation of digital-based computational means in the present, pushes a technological tool into the field with a tremendous impact in areas like Control, Signal Processing, Communications, System Modelling and related Applications. This book attempts to give a scope in the wide area of Discrete-Time Systems. Their contents are grouped conveniently in sections according to significant areas, namely Filtering, Fixed and Adaptive Control Systems, Stability Problems and Miscellaneous Applications. We think that the contribution of the book enlarges the field of the Discrete-Time Systems with signification in the present state-of-the-art. Despite the vertiginous advance in the field, we also believe that the topics described here allow us also to look through some main tendencies in the next years in the research area

Distributed estimation techniques forcyber-physical systems

Nowadays, with the increasing use of wireless networks, embedded devices and agents with processing and sensing capabilities, the development of distributed estimation techniques has become vital to monitor important variables of the system that are not directly available. Numerous distributed estimation techniques have been proposed in the literature according to the model of the system, noises and disturbances. One of the main objectives of this thesis is to search all those works that deal with distributed estimation techniques applied to cyber-physical systems, system of systems and heterogeneous systems, through using systematic review methodology. Even though systematic reviews are not the common way to survey a topic in the control community, they provide a rigorous, robust and objective formula that should not be ignored. The presented systematic review incorporates and adapts the guidelines recommended in other disciplines to the field of automation and control and presents a brief description of the different phases that constitute a systematic review. Undertaking the systematic review many gaps were discovered: it deserves to be remarked that some estimators are not applied to cyber-physical systems, such as sliding mode observers or set-membership observers. Subsequently, one of these particular techniques was chosen, set-membership estimator, to develop new applications for cyber-physical systems. This introduces the other objectives of the thesis, i.e. to present two novel formulations of distributed set-membership estimators. Both estimators use a multi-hop decomposition, so the dynamics of the system is rewritten to present a cascaded implementation of the distributed set-membership observer, decoupling the influence of the non-observable modes to the observable ones. So each agent must find a different set for each sub-space, instead of a unique set for all the states. Two different approaches have been used to address the same problem, that is, to design a guaranteed distributed estimation method for linear full-coupled systems affected by bounded disturbances, to be implemented in a set of distributed agents that need to communicate and collaborate to achieve this goal

Set-based state estimation and fault diagnosis using constrained zonotopes and applications

This doctoral thesis develops new methods for set-based state estimation and active fault diagnosis (AFD) of (i) nonlinear discrete-time systems, (ii) discrete-time nonlinear systems whose trajectories satisfy nonlinear equality constraints (called invariants), (iii) linear descriptor systems, and (iv) joint state and parameter estimation of nonlinear descriptor systems. Set-based estimation aims to compute tight enclosures of the possible system states in each time step subject to unknown-but-bounded uncertainties. To address this issue, the present doctoral thesis proposes new methods for efficiently propagating constrained zonotopes (CZs) through nonlinear mappings. Besides, this thesis improves the standard prediction-update framework for systems with invariants using new algorithms for refining CZs based on nonlinear constraints. In addition, this thesis introduces a new approach for set-based AFD of a class of nonlinear discrete-time systems. An affine parametrization of the reachable sets is obtained for the design of an optimal input for set-based AFD. In addition, this thesis presents new methods based on CZs for set-valued state estimation and AFD of linear descriptor systems. Linear static constraints on the state variables can be directly incorporated into CZs. Moreover, this thesis proposes a new representation for unbounded sets based on zonotopes, which allows to develop methods for state estimation and AFD also of unstable linear descriptor systems, without the knowledge of an enclosure of all the trajectories of the system. This thesis also develops a new method for set-based joint state and parameter estimation of nonlinear descriptor systems using CZs in a unified framework. Lastly, this manuscript applies the proposed set-based state estimation and AFD methods using CZs to unmanned aerial vehicles, water distribution networks, and a lithium-ion cell.Comment: My PhD Thesis from Federal University of Minas Gerais, Brazil. Most of the research work has already been published in DOIs 10.1109/CDC.2018.8618678, 10.23919/ECC.2018.8550353, 10.1016/j.automatica.2019.108614, 10.1016/j.ifacol.2020.12.2484, 10.1016/j.ifacol.2021.08.308, 10.1016/j.automatica.2021.109638, 10.1109/TCST.2021.3130534, 10.1016/j.automatica.2022.11042

Model-based fault diagnosis for aerospace systems: a survey

http://pig.sagepub.com/content/early/2012/01/06/0954410011421717International audienceThis survey of model-based fault diagnosis focuses on those methods that are applicable to aerospace systems. To highlight the characteristics of aerospace models, generic nonlinear dynamical modeling from flight mechanics is recalled and a unifying representation of sensor and actuator faults is presented. An extensive bibliographical review supports a description of the key points of fault detection methods that rely on analytical redundancy. The approaches that best suit the constraints of the field are emphasized and recommendations for future developments in in-flight fault diagnosis are provided

Regelungstheorie

The workshop “Regelungstheorie” (control theory) covered a broad variety of topics that were either concerned with fundamental mathematical aspects of control or with its strong impact in various fields of engineering

Entwurf eines Beobachterbasierten Robusten Nichtlinearen Reglers

Due to observers ability in the estimation of internal system states, observers play an important role in the field of control and monitoring of dynamical systems. In reality, using sensors to measure the desired system states may be costly and/or affects the reliability of technical systems. Besides, some signals are impractical or inaccessible to be measured and using of sensors leads to significant errors such as stochastic noise. The solution of using observers is well-known since 1964. Besides the estimation of system states, some observers are able to estimate unknown inputs affecting the system dynamics such as disturbance forces or torques. These features are helpful for supervision and fault diagnosis tasks by monitoring the sensors and system components or for advanced control purposes by realizing observer-based control for practical systems. Among the state and disturbance observers, Proportional-Integral-Observer (PIO) is highly appreciated because of its simple structure and design procedure. Furthermore, using sufficiently high gain PIO, a robust estimation of system states and unknown inputs can be achieved. Besides taking the advantages of high gain design, the disadvantages of large overshoot and strong influence from measurement noise (as typical drawbacks of high gain utilization) in the control and estimation performance can not be neglected. Recently, some researches have been done to overcome the disadvantages of high gain observers and to adaptively adjust the gain of observer based on the resulting actual performance. Considering the advantages and disadvantages of high gain PIO besides the recent developments, it is evident that there are still open problems and questions to be solved in the area of optimal design of PIO and robust nonlinear control approaches based on PIO. On the other hand, the PI-Observer can be used in combination with linear/nonlinear control approaches (due to its simple structure and capability to estimate the system states and disturbances) to improve the performance and robustness of the closed-loop control results. Therefore, this thesis focuses on development and improvement of high gain Proportional-Integral-Observer as well as utilization of this observer in combination with well-known robust control approaches for possible general application in nonlinear systems. The Modified Advanced PIO (MAPIO) is introduced in this work as the extended version of Advanced PIO (APIO) to tune the gain of PIO according to the current situation. A cost function is defined so that the estimation performance and the related energy can be evaluated. Comparison between advanced observer design approaches has been done in the task of reconstructing the nonlinear characteristics and estimating the external inputs (contact forces) acting to elastic mechanical structures. Simulation results in open-loop and closed-loop cases verified that the performance of MAPIO in the task of unknown input estimation is more robust to different levels of measurement noise in comparison to previous methods e.g. APIO and standard high/low gain PIO. Furthermore, a new gain design approach of Proportional-Integral-Observer is proposed to overcome the disadvantages of high gain PIO and to realize the estimation of fast dynamical behaviors like unknown impact force. The dynamics of this force input is assumed as unknown. The idea of funnel control is taking into consideration to design the PIO gain. The important advantage of the proposed approach compared to previously published PIO gain design is the self-adjustment of observer gains according to the actual estimation situation inside the predefined funnel area. In this thesis it is shown that the proposed funnel PI-Observer algorithm allows adaptive PIO gain calculation, being able to be situatively adjusted even in the presence of measurement noise. Stability proof of funnel PI-Observer is investigated according to the switching observer condition and Lyapunov theory. The effectiveness of the proposed method is evaluated by simulation and experimental results using an elastic beam test rig. Furthermore, a nonlinear MIMO mechanical system is used to verify the effectiveness of the proposed method in the closed-loop context. Additionally, this thesis provides two new PI-Observer-based robust controllers as PIO-based sliding mode control and PIO-based backstepping control to improve the position tracking performance of a hydraulic differential cylinder system in the presence of uncertainties e.g. modeling errors, disturbances, and measurement noise. To use the linear PIO for estimation of system states and unknown inputs, the input-output feedback linearization approach is used to linearize the nonlinear model of hydraulic differential cylinder system. Thereupon the result of state and unknown input estimation is integrated into the structure of robust control design (here SMC and backstepping control) to eliminate the effects of uncertainties and disturbances. The introduced PIO-based robust controllers guarantee the ultimate boundness of the tracking error in the presence of uncertainties. The closed-loop stability is proved using Lyapunov theory in both cases. The proposed methods are experimentally validated and the results are compared with the standard SMC and industrial standard approach P-Controller in the presence of measurement noise, model uncertainties, and external disturbances. A general comparison of SMC and backstepping control approaches is provided in the last part of this work.Die Regelung und Überwachung dynamischer Systeme kann voraussetzen, dass Informationen über interne Systemzustände bekannt sind. Die Verwendung von Sensoren zur Erfassung aller Systemzustände kann erhöhte Kosten zur Folge haben und die Systemzuverlässigkeit negativ beeinflussen. Weitere Probleme ergeben sich dadurch, dass ggf. nicht jeder Systemzustand sensorisch erfasst werden kann. Der Beobachter erlaubt die Rekonstruktion aller Systemzustände auf Grundlage weniger Messungen. Neben Systemzuständen können externe Eingangsgrößen wie Reibmomente und Störungen geschätzt werden. Als Konsequenz ermöglicht der Beobachter eine gegenüber Störungen robuste Regelung und Fehlerdiagnose technischer Systeme. Der Proportional-Integral-Observer (PIO) kann mittels bestehender Entwurfsverfahren einfach implementiert werden. Durch Anpassen der Rückkopplungsmatrix eignet sich der PIO zur kombinierten Schätzung von Zuständen und unbekannten Eingangsgrößen. In diesem Zusammenhang spielt die Wahl einer betragsmäßig großen Rückkopplungsverstärkungsmatrix, als sogenannter High Gain Ansatz, eine entscheidende Rolle. Weiterhin hängt die Performance des PIO von der unbekannten Charakteristik der zu schätzenden Eingangsgröße ab. Diese Arbeit befasst sich mit der Entwicklung optimierter Entwurfsverfahren für den Proportional-Integral-Observer und der Entwicklung und Anwendung beobachterbasierter Konzepte zur robusten Regelung nichtlinearer Systeme. In dieser Arbeit wird der modifizierte Advanced PIO (MAPIO) als erweiterte Version des Advanced PIO (APIO) eingeführt. Der Schätzfehler von MAPIO wird über ein Gütefunktional abgebildet. Das Gütefunktional wird durch Anpassung der Rückkopplungsverstärkungsmatrix an die Charakteristik der unbekannten Eingangsgröße minimiert. Die Performance der modifizierten Beobachterentwurfsansätze wird anhand eines praktischen Beispiels bewertet. Geschätzt wird eine unbekannte Kontaktkraft mit nichtlinearer Charakteristik, die auf ein mechanisches System wirkt. Anhand eines Simulationsbeispiels im offenen und geschlossenen Regelkreis wird die Performance von MAPIO gegenüber vorherigen Verfahren APIO und PIO verifiziert. Basierend auf der Idee des Funnel Reglers wird ein neuartiges Entwurfskonzept für den Proportional-Integral-Observer vorgestellt. Die Nachteile des PIO-Konzeptes mit hohem Verstärkungsfaktor können überwunden werden und Schätzungen schneller dynamischer Verhaltensweisen lassen sich realisieren. Der Vorteil der neuartigen Funnel PIO Methode ist, dass der Schätzfehler in einem definierten Bereich, der sogenannten Funnel-Area, verbleibt. In dieser Arbeit wird gezeigt, dass der vorgeschlagene Funnel PIO Algorithmus eine adaptive PIO Verstärkungsberechnung ermöglicht, die auch in Gegenwart von Messrauschen situativ eingestellt werden kann. Der Stabilitätsnachweis von Funnel PIO wird mittels der Lyapunov Theorie untersucht. Die Wirksamkeit der vorgeschlagenen Methode wird durch Simulation und experimentelle Ergebnisse validiert. Eine auf einen elastischen Balken wirkende äußere Kraft mit nichtlinearer Charakteristik wird geschätzt. Ein nichtlineares MIMO System wird verwendet, um die Wirksamkeit der vorgeschlagenen Methode im geschlossenen Regelkreis zu verifizieren. In dieser Arbeit werden zwei neue PI-Observer basierte robuste Regelungen (PIO-basierte Sliding Mode und PIO-basierte Backstepping Regelung) vorgestellt. Die Positionsregelung eines hydraulischen Differentialzylinders in Gegenwart von Modellunsicherheiten, Störungen und Messrauschen wird untersucht. Zur Anwendung der PIO-basierten Störgrößenschätzung wird eine Ein-/Ausgangs-Linearisierung des nichtlinearen Modells vorgenommen. Die Stabilität des geschlossenen Regelkreises wird in beiden Fällen mit der Lyapunov Theorie bewiesen. Die vorgeschlagenen Methoden werden experimentell validiert und die Ergebnisse werden mit dem Standard Sliding Mode Regler und einem P-Regler in Gegenwart von Messrauschen, Modellunsicherheiten und externen Störungen verglichen

Functional sets with typed symbols: Framework and mixed Polynotopes for hybrid nonlinear reachability and filtering

Verification and synthesis of Cyber-Physical Systems (CPS) are challenging and still raise numerous issues so far. In this paper, an original framework with mixed sets defined as function images of symbol type domains is first proposed. Syntax and semantics are explicitly distinguished. Then, both continuous (interval) and discrete (signed, boolean) symbol types are used to model dependencies through linear and polynomial functions, so leading to mixed zonotopic and polynotopic sets. Polynotopes extend sparse polynomial zonotopes with typed symbols. Polynotopes can both propagate a mixed encoding of intervals and describe the behavior of logic gates. A functional completeness result is given, as well as an inclusion method for elementary nonlinear and switching functions. A Polynotopic Kalman Filter (PKF) is then proposed as a hybrid nonlinear extension of Zonotopic Kalman Filters (ZKF). Bridges with a stochastic uncertainty paradigm are outlined. Finally, several discrete, continuous and hybrid numerical examples including comparisons illustrate the effectiveness of the theoretical results.Comment: 21 pages, 8 figure