Non-Euclidean Contraction Theory for Robust Nonlinear Stability

We study necessary and sufficient conditions for contraction and incremental stability of dynamical systems with respect to non-Euclidean norms. First, we introduce weak pairings as a framework to study contractivity with respect to arbitrary norms, and characterize their properties. We introduce and study the sign and max pairings for the $\ell_1$ and $\ell_\infty$ norms, respectively. Using weak pairings, we establish five equivalent characterizations for contraction, including the one-sided Lipschitz condition for the vector field as well as matrix measure and Demidovich conditions for the corresponding Jacobian. Third, we extend our contraction framework in two directions: we prove equivalences for contraction of continuous vector fields and we formalize the weaker notion of equilibrium contraction, which ensures exponential convergence to an equilibrium. Finally, as an application, we provide (i) incremental input-to-state stability and finite input-state gain properties for contracting systems, and (ii) a general theorem about the Lipschitz interconnection of contracting systems, whereby the Hurwitzness of a gain matrix implies the contractivity of the interconnected system

Relaxed ISS Small-Gain Theorems for Discrete-Time Systems

In this paper ISS small-gain theorems for discrete-time systems are stated, which do not require input-to-state stability (ISS) of each subsystem. This approach weakens conservatism in ISS small-gain theory, and for the class of exponentially ISS systems we are able to prove that the proposed relaxed small-gain theorems are non-conservative in a sense to be made precise. The proofs of the small-gain theorems rely on the construction of a dissipative finite-step ISS Lyapunov function which is introduced in this work. Furthermore, dissipative finite-step ISS Lyapunov functions, as relaxations of ISS Lyapunov functions, are shown to be sufficient and necessary to conclude ISS of the overall system.Comment: input-to-state stability, Lyapunov methods, small-gain conditions, discrete-time non-linear systems, large-scale interconnection

Adaptive control for time-varying systems: congelation and interconnection

This thesis investigates the adaptive control problem for systems with time-varying parameters. Two concepts are developed and exploited throughout the thesis: the congelation of variables, and the active nodes. The thesis first revisits the classical adaptive schemes and explains the challenges brought by the presence of time-varying parameters. Then, the concept of congelation of variables is introduced and its use in combinations with passivity-based, immersion-and-invariant, and identification-based adaptive schemes are discussed. As the congelation of variables method introduces additional interconnection in the closed-loop system, a framework for small-gain-like control synthesis for interconnected systems is needed.\vspace{2ex} To this end, the thesis proceeds by introducing the notion of active nodes. This is instrumental to show that as long as a class of node systems that possess adjustable damping parameters, that is the active nodes, satisfy certain graph-theoretic conditions, the desired small-gain-like property for the overall system can be enforced via tuning these adjustable parameters. Such conditions for interconnected systems with quadratic, nonlinear, and linearly parametrized supply rates, respectively, are elaborated from the analysis and control synthesis perspectives. The placement and the computation/adaptation of the damping parameters are also discussed. Following the introduction of these two fundamental tools, the thesis proceeds by discussing state-feedback designs for a class of lower-triangular nonlinear systems. The backstepping technique and the congelation of variables method are combined for passivity-based, immersion-and-invariance, and identification-based schemes. The notion of active nodes is exploited to yield simple and systematic proofs. Based on the results established for lower-triangular systems, the thesis continues to investigate output-feedback adaptive control problems. An immersion-and-invariance scheme for single-input single-output linear systems and a passivity-based scheme for nonlinear systems in observer form are proposed. The proof and interpretation of these results are also based on the notion of active nodes. The simulation results show that the adaptive control schemes proposed in the thesis have superior performance when compared with the classical schemes in the presence of time-varying parameters. Finally, the thesis studies two applications of the theoretical results proposed. The servo control problem for serial elastic actuators, and the disease control problem for interconnected settlements. The discussions show that these problems can be solved efficiently using the framework provided by the thesis.Open Acces

Homogeneous Lp stability for homogeneous systems

The motivation of this paper comes from the fact that Lp−stability and Lp−gain, using the classical signal norms, is not well-defined for arbitrary continuous weighted homogeneous systems. However, using homogeneous signal norms it is possible to show that every internally stable homogeneous system has a globally defined finite homogeneous Lp−gain, for p sufficiently large. If the system has a homogeneous approximation, the homogeneous Lp−gain is inherited locally. Homogeneous Lp−stability can be characterized by a homogeneous dissipation inequality, which in the input affine case can be transformed to a homogeneous Hamilton-Jacobi inequality. An estimation of an upper bound for the homogeneous Lp−gain can be derived from these inequalities. Homogeneous L∞−stability is also considered and its strong relationship to Input-to-State stability is studied. These results are extensions to arbitrary homogeneous systems of the well-known situation for linear time-invariant systems, where the Hamilton-Jacobi inequality reduces to an algebraic Riccati inequality. A natural application of finite-gain homogeneous Lp−stability is in the study of stability for interconnected systems. An extension of the small-gain theorem for negative feedback systems and results for systems in cascade are derived for different homogeneous norms. Previous results in the literature use classical signal norms, hence, they can only be applied to a restricted class of homogeneous systems. The results are illustrated by several examples

Homogeneous finite-gain Lp-stability analysis on homogeneous systems

In dieser Arbeit wird gezeigt, dass die klassische Lp-Stabilität und Lp-Verstärkung für beliebige stetige, gewichtete homogene Systeme nicht wohldefiniert ist. Indem die klassische Lp-Norm von Signalen zu einer homogenen Lp-Norm so angepasst wird, dass diese bezüglich der Gewichtsvektoren homogen ist, ist es möglich zu zeigen, dass jedes intern stabile homogene System für hinreichend große p eine global definierte endliche homogene Lp-Verstärkung besitzt. Mit Hilfe einer homogenen Lyapunov-Funktion kann die homogene Lp-Stabilität durch eine homogene partielle Differentialungleichung charakterisiert werden, die sich im eingangsaffinen Fall in eine homogene Hamilton-Jacobi-Ungleichung transformieren lässt. Des Weiteren werden in dieser Arbeit detaillierte Methoden zur Abschätzung von oberen Schranken für homogene Lp-Verstärkungen aus diesen Ungleichungen abgeleitet. Dies schließt die homogene L∞-Verstärkung und die homogene Eingangs-Zustands-Verstärkung ebenfalls ein. Bei rückgekoppelten homogenen Systemen, bei denen die Gewichtsvektoren zwischen den Systemen zueinander passend sind, erlaubt die additive Ungleichung für die homogene Lp-Norm die Einführung des homogenen Small-Gain Theorems für beliebige p, wodurch eine Stabilitätsanalyse des geschlossenen Regelkreises ermöglicht wird. Weiterhin können homogene H∞-Regler entworfen werden, wenn das System eingangsaffin ist. Da die konventionellen Werkzeuge der linearen Systemtheorie nicht zur Verfügung stehen, können solche homogenen H∞-Regler nur garantieren, dass der geschlossene Regelkreis eine homogene Lp-Verstärkung hat, die kleiner als ein bestimmbarer Wert ist. Ihre Optimalität kann hingegen nicht garantiert werden. In jedem Kapitel werden mehrere kurze Beispiele vorgestellt, um zu veranschaulichen, wie eine solche homogene Lp-Verstärkung berechnet werden kann. Insbesondere ist eine detaillierte Analyse des “Continuous Super-Twisting Like Algorithm" mit tieferen Einblicken für interessierte Leser enthalten.In this thesis, it is shown that the classical Lp-stability and Lp-gain is not well-defined for arbitrary continuous weighted homogeneous systems. By modifying the classical Lp-norm of signals to be homogeneous w.r.t. some weight vectors, which is called homogeneous Lp-norm, it is possible to show that every internally stable homogeneous system has a globally defined finite homogeneous Lp-gain, for p sufficiently large. With the help of a homogeneous Lyapunov function, homogeneous Lp-stability can be characterized by a homogeneous partial differential inequality, which in the input affine case can be transformed to a homogeneous Hamilton-Jacobi inequality. Furthermore, in this thesis some detailed methods to calculate upper estimates for the homogeneous Lp-gain are provided from theses inequalities. This also includes the homogeneous L∞-gain and homogeneous Input-to-State gain. For feedback interconnected systems, if the weight vectors between plants are matched, the additive inequality for homogeneous Lp-norm allows the introduction of the homogeneous small gain theorem for each p, enabling stability analysis on the closed loop system. Finally, some homogeneous H∞-controllers can be designed, if the system is affine in the control input. Without the convenient tools for the linear systems, such homogeneous H∞-controllers can only guarantee that the closed loop system has homogeneous Lp-gain less than some derivable numbers, its optimality can not be guaranteed. Several short examples are presented within each chapter to illustrate how such homogeneous Lp-gain can be calculated. In particular a detailed analysis on the Continuous Super-Twisting Like Algorithm is included with deeper insight for interested readers

Dynamic Incentives for Optimal Control of Competitive Power Systems

This work presents a real-time dynamic pricing framework for future electricity markets. Deduced by first-principles analysis of physical, economic, and communication constraints within the power system, the proposed feedback control mechanism ensures both closed-loop system stability and economic efficiency at any given time. The resulting price signals are able to incentivize competitive market participants to eliminate spatio-temporal shortages in power supply quickly and purposively

Dynamic Incentives for Optimal Control of Competitive Power Systems

Technologisch herausfordernde Transformationsprozesse wie die Energiewende können durch passende Anreizsysteme entscheidend beschleunigt werden. Ziel solcher Anreize ist es hierbei, ein Umfeld idealerweise so zu schaffen, dass das Zusammenspiel aller aus Sicht der beteiligten Wettbewerber individuell optimalen Einzelhandlungen auch global optimal im Sinne eines übergeordneten Großziels ist. Die vorliegende Dissertation schafft einen regelungstechnischen Zugang zur Frage optimaler Anreizsysteme für heutige und zukünftige Stromnetze im Zieldreieck aus Systemstabilität, ökonomischer Effizienz und Netzdienlichkeit. Entscheidende Neuheit des entwickelten Ansatzes ist die Einführung zeitlich wie örtlich differenzierter Echtzeit-Preissignale, die sich aus der Lösung statischer und dynamischer Optimierungsprobleme ergeben. Der Miteinbezug lokal verfügbarer Messinformationen, die konsequente Mitmodellierung des unterlagerten physikalischen Netzes inklusive resistiver Verluste und die durchgängig zeitkontinuierliche Formulierung aller Teilsysteme ebnen den Weg von einer reinen Anreiz-Steuerung hin zu einer echten Anreiz-Regelung. Besonderes Augenmerk der Arbeit liegt in einer durch das allgemeine Unbundling-Gebot bedingten rigorosen Trennung zwischen Markt- und Netzakteuren. Nach umfangreicher Analyse des hierbei entstehenden geschlossenen Regelkreises erfolgt die beispielhafte Anwendung der Regelungsarchitektur für den Aufbau eines neuartigen Echtzeit-Engpassmanagementsystems. Weitere praktische Vorteile des entwickelten Ansatzes im Vergleich zu bestehenden Konzepten werden anhand zweier Fallstudien deutlich. Die port-basierte Systemmodellierung, der Verzicht auf zentralisierte Regeleingriffe und nicht zuletzt die Möglichkeit zur automatischen, dezentralen Selbstregulation aller Preise über das Gesamtnetz hinweg stellen schließlich die problemlose Erweiterbarkeit um zusätzliche optionale Anreizkomponenten sicher