Control Theory: A Mathematical Perspective on Cyber-Physical Systems

Control theory is an interdisciplinary field that is located at the crossroads of pure and applied mathematics with systems engineering and the sciences. Recently the control field is facing new challenges motivated by application domains that involve networks of systems. Examples are interacting robots, networks of autonomous cars or the smart grid. In order to address the new challenges posed by these application disciplines, the special focus of this workshop has been on the currently very active field of Cyber-Physical Systems, which forms the underlying basis for many network control applications. A series of lectures in this workshop was devoted to give an overview on current theoretical developments in Cyber-Physical Systems, emphasizing in particular the mathematical aspects of the field. Special focus was on the dynamics and control of networks of systems, distributed optimization and formation control, fundamentals of nonlinear interconnected systems, as well as open problems in control

Invariance feedback entropy of uncertain nonlinear control systems

In der klassischen Kontrolltheorie geht man üblicherweise davon aus, dass Sensoren und Regler durch Punkt-zu-Punkt-Verkabelung miteinander verbunden sind. In vernetzten Kontrollsystemen (VKS) sind Sensoren und Regler oft räumlich verteilt und Daten werden mittels eines digitalen Kommunikationsnetzwerks übertragen. Im Vergleich zu klassischen Kontrollsystemen bieten VKS viele Vorteile wie z.B. reduzierte Verkabelung, geringe Installations- und Instandhaltungskosten, größere Systemflexibilität und einfache Modifizierbarkeit. VKS haben Anwendungen in vielen Bereichen, z.B. in der Fahrzeugtechnik, intelligenten Gebäuden und Transportnetzwerken. Jedoch macht die Verwendung von Kommunikationsnetzwerken in Regelschleifen die Analyse und den Entwurf von VKS wesentlich komplexer. Die Verwendung digitaler Kanäle in VKS beschränkt aufgrund der endlichen Bandbreite die Datenmenge, die pro Zeiteinheit von Sensoren zu Reglern übertragen werden kann. Dies führt zu Quantisierungsfehlern, welche die Regelungsperformance ungünstig beeinflussen können. Das Problem der Regelung und Zustandsschätzung über einen digitalen Kommunikationskanal mit beschränkter Bitrate hat in den letzten zwei Jahrzehnten viel Aufmerksamkeit erhalten. Eine scharfe untere Schranke der Datenrate eines digitalen Kanals zwischen dem Kodierer (in Sensornähe) und dem Regler, die zum Erreichen eines Regelungsziels wie z.B. Stabilisierung oder Invarianz benötigt wird, kann durch einen passenden Entropiebegriff als intrinsische Größe des Systems charakterisiert werden, und hängt nicht von der Wahl des Kodierers und Reglers ab. Im ersten Teil der Arbeit beschreiben wir die Invarianz-Feedback-Entropie (IFE), die den Begriff der Invarianz-Entropie für deterministische nichtlineare Kontrollsysteme auf unsichere Systeme erweitert. Die IFE charakterisiert die Zustandsinformation, die von einem Regler benötigt wird, um eine Teilmenge Q des Zustandsraums invariant zu machen. Wir diskutieren eine Anzahl von elementaren Eigenschaften der IFE, z.B. Bedingungen für ihre Endlichkeit und die im deterministischen Spezialfall vorliegende Äquivalenz zum wohlbekannten Begriff der Invarianz-Entropie (IED). Wir analysieren unsichere lineare Kontrollsysteme und leiten eine universelle Unterschranke der IFE her. Im zweiten Teil der Arbeit betrachten wir vernetzte Kontrollsysteme und streben eine obere Schranke der IFE eines Netzwerks in Termen der IFE der Teilsysteme an. Außerdem präsentieren wir drei technische Resultate. Zuerst zeigen wir, dass die IFE einer nichtleeren Teilmenge Q des Zustandsraums eines zeitdiskreten unsicheren Kontrollsystems nach oben durch die größte IFE der Mengen in einer beliebigen endlichen Partition von Q beschränkt ist. Im zweiten Resultat betrachten wir unsichere Kontrollsysteme S1 und S2 mit identischen Zustands- und Eingangsräumen. Die mengenwertigen Übergangsfunktionen F1 und F2 der beiden Systeme sind nach Annahme so beschaffen, dass das Bild eines beliebigen Zustands-Eingangs-Paars unter F1 in dem entsprechenden Bild unter F2 enthalten ist. Für eine gegebene nichtleere Teilmenge des Zustandsraums zeigen wir, dass die IFE von S2 größer oder gleich derjenigen von S1 ist. Das dritte Resultat zeigt, dass die IFE niemals kleiner wird, wenn man die Menge der Kontrolleingänge verkleinert. Um die Effektivität der Resultate zu illustrieren, berechnen wir eine Ober- und eine Unterschranke der IFE eines Netzwerks von unsicheren, linearen, zeitdiskreten Systemen, welche den zeitlichen Verlauf der Temperaturen in 100 Räumen eines zirkulären Gebäudes beschreiben. Im letzten Teil der Arbeit präsentieren wir Algorithmen für die numerische Abschätzung der IFE. Dazu betrachten wir zunächst eine Partition einer gegebenen Teilmenge Q des Zustandsraums. Dann wird ein Regler in Form einer Suchtabelle berechnet, die jedem Element der Partition eine Menge von Kontrollwerten zuordnet, welche die Invarianz von Q garantieren. Nach der Reduktion der Suchtabelle von einer mengenwertigen zu einer einwertigen Abbildung, wird ein gewichteter Graph konstruiert. Für deterministische Systeme liefert der Logarithmus des Spektralradius einer Übergangsmatrix, die aus dem Graphen ermittelt wird, eine obere Schranke der Entropie. Für unsichere Systeme stellt das maximale durchschnittliche Zyklusgewicht des Graphen eine Oberschranke der IFE dar. Im deterministischen Fall zeigen wir, dass der Wert der ersten Oberschranke nicht größer als derjenige der zweiten Oberschranke ist. Als nächstes präsentieren wir die Ergebnisse der Algorithmen angewandt auf drei deterministische Beispielsysteme, für welche der exakte Wert der IED bekannt ist oder durch andere Methoden abgeschätzt werden kann. Zusätzlich liefert unser Algorithmus ein statisches Kodierungs- und Regelungsprotokoll, das der Schranke an die Datenrate entspricht. Schließlich präsentieren wir die berechneten Oberschranken der IFE eines unsicheren linearen Kontrollsystems.In classical control theory, the sensors and controllers are usually connected through point-to-point wiring. In networked control systems (NCS), sensors and controllers are often spatially distributed and involve digital communication networks for data transfer. Compared to classical control systems, NCS provide many advantages such as reduced wiring, low installation and maintenance costs, greater system flexibility and ease of modification. NCS find applications in many areas such as automobiles, intelligent buildings, and transportation networks. However, the use of communication networks in feedback control loops makes the analysis and design of NCS much more complex. In NCS, the use of digital channels for data transfer from sensors to controllers limits the amount of data that can be transferred per unit of time, due to the finite bandwidth of the channel. This introduces quantization errors that can adversely affect the control performance. The problem of control and state estimation over a digital communication channel with a limited bit rate has attracted a lot of attention in the past two decades. A tight lower bound on the data rate of a digital channel between the coder (near the sensor) and the controller, to achieve some control task such as stabilization or invariance, can be characterized in terms of some appropriate notion of entropy which is described as an intrinsic property of the system and is independent of the choice of the coder-controller. In the first part of this thesis, we describe invariance feedback entropy (IFE) that extends the notion of invariance entropy of deterministic nonlinear control systems to those with uncertainty. The IFE characterizes the necessary state information required by any controller to render a subset Q of the state space invariant. We discuss a number of elementary properties of the IFE, e.g. conditions for its finiteness and its equivalence to the well-known notion of invariance entropy (IED) in the deterministic case. We analyze uncertain linear control systems and derive a universal lower bound of the IFE. In the second part of this thesis, we consider interconnected control systems and seek to upper bound the IFE of the network using the IFE of the subsystems. In addition, we present three technical results related to the IFE. First, we show that the IFE of a nonempty subset Q of the state space of a discrete-time uncertain control system is upper bounded by the largest possible IFE among the members of any finite partition of Q. Second, we consider two uncertain control systems, S1 and S2, that have identical state spaces and identical control input sets. The set valued transition functions, F1 and F2, of the two systems are such that the image of any state-input pair under F1 is a subset of that under F2. For a given nonempty subset of the state space, we show that the IFE of S2 is larger than or equal to the IFE of S1. Third, we show that the IFE will never decrease by reducing the set of control inputs. To illustrate the effectiveness of the results, we compute an upper bound and a lower bound of the IFE of a network of uncertain, linear, discrete-time subsystems describing the evolution of temperatures of 100 rooms in a circular building. In the last part of this thesis, we present algorithms for the numerical estimation of the IFE. In particular, given a subset Q of the state space, we first partition it. Then a controller, in the form of a lookup table that assigns a set of control values to each cell of the partition, is computed to enforce invariance of Q. After reduction of the lookup table to a single-valued map from a set-valued one, a weighted directed graph is constructed. For deterministic systems, the logarithm of the spectral radius of a transition matrix obtained from the graph gives an upper bound of the entropy. For uncertain systems, the maximum mean cycle weight of the graph upper bounds the IFE. For deterministic systems, the value of the first upper bound is shown to be lower than or equal to the value of the second upper bound. Next, we present the results of the algorithms applied to three deterministic examples for which the exact value of the IED is known or can be estimated by other techniques. Additionally, our algorithm provides a static coder-controller scheme corresponding to the obtained data-rate bound. Finally, we present the computed upper bounds of the IFE for an uncertain linear control system

ADB–OECD Study on Enhancing Financial Accessibility for SMEs: Lessons from Recent Crises

During the era of global financial uncertainty, stable access to appropriate funding sources has been much harder for small and medium-sized enterprises (SMEs). The global financial crisis impacted SMEs and entrepreneurs disproportionately, exacerbating their traditional financing constraints. The financial conditions of many SMEs were weakened by the drop in demand for goods and services and the credit tightening. The sovereign debt crisis that hit several European countries contributed to further deterioration in bank lending activities, which negatively affected private sector development. The global regulatory response to financial crises, such as the Basel Capital Accord, while designed to reduce systemic risks may also constrain bank lending to SMEs. In particular, Basel III requires banks to have tighter risk management as well as greater capital and liquidity. Resulting asset preference and deleveraging of banks, particularly European banks with significant presence in Asia, could limit the availability of funding for SMEs in Asia and the Pacific. Lessons from the recent financial crises have motivated many countries to consider SME access to finance beyond conventional bank credit and to diversify their national financial system. Improving SME access to finance is a policy priority at the country and global level. Poor access to finance is a critical inhibiting factor to the survival and growth potential of SMEs. Financial inclusion is thus key to the development of the SME sector, which is a driver of job creation and social cohesion and takes a pivotal role in scaling up national economies. The Asian Development Bank (ADB) and the Organisation for Economic Co-operation and Development (OECD) have recognized that it is crucial to develop a comprehensive range of policy options on SME finance, including innovative financing models. With this in mind, sharing Asian and OECD experiences on SME financing would result in insightful discussions on improving SME access to finance at a time of global financial uncertainty. Based on intensive discussions in two workshops organized by ADB in Manila on 6–7 March 2013 and by OECD in Paris on 21 October 2013, the two organizations together compiled this study report on enhancing financial accessibility for SMEs, especially focusing on lessons from the past and recent crises in Asia and OECD countries. The report takes a comparative look at ADB and OECD experiences, and aims to identify promising policy solutions for creating an SME base that is resilient to crisis, from a viewpoint of access to finance, and which can help drive growth and development

Dynamic Resource Management for Cognitive Radios Using Limited-Rate Feedback

Abstract-Tailored for the emerging class of cognitive radio networks comprising primary and secondary wireless users, the present paper deals with dynamic allocation of subcarriers, rate and power resources based on channel state information (CSI) for orthogonal frequency-division multiple access (OFDMA). Users rely on adaptive modulation, coding and power modes that they select in accordance with the limited-rate feedback they receive from the access point. The access point uses CSI to maximize a generic concave utility of the average rates in the network while adhering to rate and power constraints imposed on the primary and secondary users to respect cognitive radio related hierarchies. When the channel distribution is available, optimum dual prices are found to optimally allocate resources across users dynamically per channel realization. In addition, a simple yet optimal online algorithm that does not require knowledge of the channel distribution and iteratively computes the dual prices per channel realization is developed using a stochastic dual approach. Analysis of the computational and feedback overhead along with simulations assessing the performance of the novel algorithms are also provided