Review on computational methods for Lyapunov functions

Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function

Inductive Certificate Synthesis for Control Design

The focus of this thesis is developing a framework for designing correct-by-construction controllers using control certificates. We use nonlinear dynamical systems to model the physical environment (plants). The goal is to synthesize controllers for these plants while guaranteeing formal correctness w.r.t. given specifications. We consider different fundamental specifications including stability, safety, and reach-while-stay. Stability specification states that the execution traces of the system remain close to an equilibrium state and approach it asymptotically. Safety specification requires the execution traces to stay in a safe region. Finally, for reach-while-stay specification, safety is needed until a target set is reached.The design task consists of two phases. In the first phase, the control design problem is reduced to the question of finding a control certificate. More precisely, the goal of the first phase is to define a class of control certificates with a specific structure. This definition should guarantee the following: ``Having a control certificate, one can systematically design a controller and prove its correctness at the same time."The goal in the second phase is to find such a control certificate. We define a potential control certificate space (hypothesis space) using parameterized functions. Next, we provide an inductive search framework to find proper parameters, which yield a control certificate. Finally, we evaluate our framework. We show that discovering control certificates is practically feasible and demonstrate the effectiveness of the automatically designed controllers through simulations and real physical systems experiments

On the synthesis of switched output feedback controllers for linear, time-invariant systems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2007.Includes bibliographical references (p. 189-193).The theory of switching systems has seen many advances in the past decade. Its beginnings were founded primarily due to the physical limitations in devices to implement control such as relays, but today there exists a strong interest in the development of switching systems where switching is introduced as a means of increasing performance. With the newer set of problems that arise from this viewpoint comes the need for many new tools for analysis and design. Analysis tools which include, for instance, the celebrated work on multiple Lyapunov functions are extensive. Tools for the design of switched systems also exist, but, in many cases, the method of designing stabilizing switching laws is often a separate process from the method which is used to determine the set of vector fields between which switching takes place. For instance, one typical method of designing switching controllers for linear, time-invariant (LTI) systems is to first design a set of stabilizing LTI controllers using standard LTI methods, and then design a switching law to increase performance. While such design algorithms can lead to increases in performance, they often impose restrictions that do not allow the designer to take full advantage of the switching architecture being considered.(cont.) For instance, if one switches between controllers that are individually stabilizing (without any switching), then, effectively, one is forced to switch only between stable systems and, hence, cannot take advantage of the potential benefits of switching between unstable systems in a stable way. It is, therefore, natural to wonder whether design algorithms can be developed which simultaneously design both the set of controllers to be switched and a stabilizing switching law. The work investigated here attempts to take a small step in the above direction. We consider a simple switching architecture that implements switched proportional gain control for second order LTI systems. Examination of this particular structure is motivated by its mathematical simplicity for ease of analysis (and, hence, as a means of gaining insight into the problem-at-large), but, as we will see, the design techniques investigated here can be extended to a larger class of (higher order, potentially non-linear and/or time-varying) systems using standard tools from robust control. The overall problem we investigate is the ability to create algorithms to simultaneously determine a set of switching gains and an associated switching law for a particular plant and performance objective.(cont.) After determining a set of necessary and sufficient conditions for a given second order plant to be stabilizable via the given switching architecture, we synthesize an algorithm for constructing controllers for which the corresponding closed-loop system dynamics are finite L, gain stable. Also, in an effort to demonstrate that the the given structure can, in fact. he used to increase performance. we consider a step-tracking design problem for a class of plants, where we use overshoot and settling time of the output step response to measure performance. We compare the results obtained using our switching architecture to the performance that can be obtained via two other LTI controller architectures to illustrate some of the performance benefits.by Keith Robert Santarelli.Ph.D

Data-driven Adaptive Stabilizer for Unknown Nonlinear Dynamic MIMO Systems Using a Cognition-based Framework

This thesis focuses on a cognitive stabilizer concept which is an adaptive discrete control method based on a cognition-based framework. The aim of the cognitive stabilizer is to autonomously stabilize a specific class of unknown nonlinear multi-input-multi-output (MIMO) systems. The cognitive stabilizer is able to gain useful local knowledge of the unknown system and can autonomously define suitable control inputs to stabilize the system. The development of different kinds of adaptive, data-driven, and model-free controllers shows a clear tendency towards research on control methods with high autonomy. Here the term autonomy is used to describe the fact that the control approach/the related programming is organized such that the algorithm is able to handle the feedback design autonomously without instructions from outside the algorithm. Typical methods affected by this definition are adaptive control method, data-driven control method, and model-free control method. In this thesis, the state-of-the-art of them is reviewed. The main focus is given to the autonomy of the realized approaches. It can be concluded that the existing methods still show some open points achieving highly autonomous control. In order to address these open points, a framework similar to modeling approaches concerning the human cognition processes [Cac98] can be introduced in the engineering context, which is denoted as cognition-based framework. As stabilization control task is the most basic control task, the cognition-based framework for stabilization is established in this thesis. It is assumed, that the mathematical model of the system to be controlled is unknown and fully controllable, as well as the state vector can be measured. The cognitive stabilizer is realized based on the cognitive framework by its four main modules: (1) “perception and interpretation” using system identifier for the system local dynamic online identification and multi-step-ahead prediction; (2) “expert knowledge” relating to the stability criterion to guarantee the stability of the considered motion of the controlled system; (3) “planning” to generate a suitable control input sequence according to certain cost functions; (4) “execution” to generate the optimal control input in a corresponding feedback form. Each module can be realized using different methods. In this thesis, “perception and interpretation” is realized using neural networks, Gaussian process regression, or combined identifier. “Expert knowledge” consists of the data-driven quadratic stability criterion, the quadratic Lyapunov stability criterion with a certain Lyapunov function, and the uniform stability criterion. The modules “planning” and “execution” are realized together with exhaustive grid search method or direct input optimization using inverse model. The whole cognitive stabilizer is realized using the autonomous communication among each module. The cognitive stabilizer are tested using numerical examples or experimental results in this thesis. Pendulum system and Lorenz-system are considered as simulation examples. Both are benchmark examples for the nonlinear dynamic control design. The cognitive stabilizer is experimentally implemented and evaluated to a threetank-system. All the numerical examples and experimental results demonstrate the successful application of the proposed methods.Das Thema dieser Arbeit ist ein kognitives Stabilisierungsverfahren, das basierend auf einem kognitionsbasierten Framework ein adaptives diskretes Regelungsverfahren darstellt. Ziel des kognitiven Stabilisierungsverfahrens ist es, eine spezifische Klasse von unbekannten, nichtlinearen, Mehrgrößensystemen autonom zu stabilisieren. Das kognitive Stabilisierungsverfahren ist in der Lage, relevante lokale Informationen über das unbekannte System zu erlangen. Es kann autonom geeignete Steuergrößen definieren, um das System zu stabilisieren. Die Entwicklung von verschiedenen adaptiven, datenbasierten und modellfreien Reglern zeigte bereits die Tendenz der Erforschung von Regelungsmethoden mit hoher Autonomie. Der Begriff Autonomie wird hier verwendet, um die Tatsache zu beschreiben, dass das Regelungsverfahren bzw. die dazugehörige Programmierung so durchgeführt wird, dass der zugehörige Algorithmus den Rückführungsentwurf autonom ohne Einwirkungen von außerhalb des Algorithmus festlegen kann. Typische Methoden, die von dieser Definition beeinflusst werden sind die adaptive Regelungsmethode, die datenbasierte Regelungsmethode oder die modellfreie Regelungsmethode, deren Stand der Forschung in dieser Arbeit zusammengefasst wird. Der Hauptfokus liegt dabei auf der Autonomie der realisierten Verfahren. Es kann gezeigt werden, dass die existierenden Methoden immer noch einige offene Probleme aufweisen, um eine hohe autonome Regelung zu erreichen. Um diese offenen Probleme weiterzuentwickeln, kann ein Framework in den Ingenieurskontext eingeführt werden, das den Modellierungsverfahren bezüglich der menschlichen Kognitionsprozesse [Cac98] ähnelt und als kognitives Framework bezeichnet werden kann. Da Stabilisierungsaufgaben die elementarsten Regelungsaufgaben sind, wird in dieser Arbeit ein kognitionsbasiertes Framework zur Stabilisierung entwickelt. Zunächst wird angenommen, dass das mathematische Modell des zu regelnden Systems unbekannt, vollständig steuerbar und der Zustandsvektor messbar ist. Der kognitive Stabilisierungsregler wird basierend auf dem kognitiven System durch seine vier Hauptmodule realisiert: (1) ”Wahrnehmung und Interpretation“ durch einen Systemidentifikator zur Echtzeit-Identifikation der lokalen Systemdynamik und Mehr-Schritt-Vorhersage; (2) ”Expertenwissen“ bezogen auf das Stabilitätskriterium um die Stabilität der betrachteten Bewegung des geregelten Systems zu garantieren; (3) ”Planung“ um eine geeignete Eingangsgrößensequenz nach bestimmten Gütefunktionen zu erzeugen; (4) ”Ausführung“ um die optimalen Steuergrößen in eine entsprechende Rückführungsform zu generieren. Jedes Modul kann durch verschiedene Methoden realisiert werden. In dieser Arbeit wird das Modul ”Wahrnehmung und Interpretation“ durch neuronale Netzwerke, Gauß-Prozess-Regression oder einen kombinierten Identifikator umgesetzt. Das Modul ”Expertenwissen“ besteht aus dem datenbasierten quadratischen Stabilitätskriterium, dem quadratischen Lyapunov Stabilitätskriterium mit einer bestimmten Lyapunov-Funktion und dem gleichmäßigen Stabilitätskriterium. Die Module ”Planung“ und ”Ausführung“ werden zusammen durch das inverse Modell mit dem vollständigen ”Grid-Search“-Verfahren oder direkter Steuergrößenoptimierung realisiert. Die gesamte kognitive Stabilisierungsmethode wird durch die autonome Kommunikation zwischen jedem Modul realisiert. Die kognitive Stabilisierungsmethode wird in dieser Arbeit durch numerische Beispiele oder experimentelle Ergebnisse getestet. Zwei Simulationsbeispiele (Pendel-System sowie Lorenz-System) werden betrachtet. Beide sind Benchmarkbeispiele für den nichtlinearen dynamischen Regelungsentwurf. Die kognitive Stabilisierungsmethode wird experimentell auf das Drei-Tank-System angewendet und die entsprechenden Ergebnisse werden bewertet. Die numerischen Beispiele sowie die experimentelle Umsetzung zeigen die erfolgreiche Anwendung des dargestellten Verfahrens

Dynamical Systems

Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...