Stability Analysis and Control of Several Classes of Logical Dynamic Systems and the Applications in Game Theory

With the rapid development of complex networks, logical dynamic systems have been commonly used mathematical models for simulating Genetic Regulatory Networks (GRNs) and Networked Evolutionary Games (NEGs), which have attracted considerable attention from biology, economy and many other fields. By resorting to the Semi-Tensor Product (STP) of matrices, logical dynamic systems can be equivalently converted into discrete time linear systems with algebraic forms. Based on that, this thesis analyzes the stability and studies the control design problems of several classes of logical dynamic systems. Moreover, the obtained results are applied to investigate the control and optimization problems of NEGs. The main results of this thesis are the following. • The stability and event-triggered control for a class of k-Valued Logical Networks (KVLNs) with time delays are studied. First, some necessary and sufficient con- ditions are obtained to detect the stability of Delayed k-Valued Logical Networks (DKVLNs). Second, the global stabilization problem under event-triggered control is considered, and some necessary and sufficient conditions are presented for the sta- bilization of Delayed k-Valued Logical Control Networks (DKVLCNs). Moreover, an algorithm is proposed to construct all the event-triggered state feedback controllers via antecedence solution technique. • The robust control invariance and robust set stabilization problems for a class of Mix- Valued Logical Control Networks (MVLCNs) with disturbances are studied. First, a calculation method for the Largest Robust Control Invariant Set (LRCIS) contained in a given set is introduced. Second, based on the Robust Control Invariant Subset (RCIS) obtained, the robust set stabilization of MVLCNs is discussed, and some new results are presented. Furthermore, the design algorithm of time-optimal state feedback stabilizers via antecedence solution technique is derived. • The robust set stability and robust set stabilization problems for a class of Probabilis- tic Boolean Control Networks (PBCNs) with disturbances are studied. An algorithm to determine the Largest Robust Invariant Set (LRIS) with probability 1 of a given set for a Probabilistic Boolean Network (PBN) is proposed, and the necessary and sufficient conditions to detect whether the PBN is globally finite-time stable to this invariant set with probability 1 are established. Then, the PBNs with control inputs are considered, and an algorithm for LRCIS with probability 1 is provided, based on which, some necessary and sufficient conditions for finite-time robust set stabiliza- tion with probability 1 of PBCNs are presented. Furthermore, the design scheme of time-optimal state feedback stabilizers via antecedence solution technique is derived. • The stabilization and set stabilization problems for a class of Switched Boolean Con- trol Networks (SBCNs) with periodic switching signal are studied. First, algebraic forms are constructed for SBCNs with periodic switching signal. Second, based on the algebraic formulations, the stabilization and set stabilization of SBCNs with peri- odic switching signal are discussed, and some new results are presented. Furthermore, constructive procedure of open loop controllers is given, and the design algorithms of switching-signal-dependent state feedback controllers via antecedence solution tech- nique are derived. • The dynamics and control problems for a class of NEGs with time-invariant delay in strategies are studied. First, algebraic forms are constructed for Delayed Networked Evolutionary Games (DNEGs). Second, based on the algebraic formulations, some necessary and sufficient conditions for the global convergence of desired strategy pro- file under a state feedback event-triggered controller are presented. Furthermore, the constructive procedure and the number of all valid event-triggered state feedback controllers are derived, which can make the game converge globally. • The evolutionary dynamics and optimization problems of the networked evolutionary boxed pig games with the mechanism of passive reward and punishment are studied. First, an algorithm is provided to construct the algebraic formulation for the dynamics of this kind of games. Then, the impact of reward and punishment parameters on the final cooperation level of the whole network is discussed

Information flow and cooperative control of vehicle formations

We consider the problem of cooperation among a collection of vehicles performing a shared task using intervehicle communication to coordinate their actions. Tools from algebraic graph theory prove useful in modeling the communication network and relating its topology to formation stability. We prove a Nyquist criterion that uses the eigenvalues of the graph Laplacian matrix to determine the effect of the communication topology on formation stability. We also propose a method for decentralized information exchange between vehicles. This approach realizes a dynamical system that supplies each vehicle with a common reference to be used for cooperative motion. We prove a separation principle that decomposes formation stability into two components: Stability of this is achieved information flow for the given graph and stability of an individual vehicle for the given controller. The information flow can thus be rendered highly robust to changes in the graph, enabling tight formation control despite limitations in intervehicle communication capability

34th Midwest Symposium on Circuits and Systems-Final Program

Organized by the Naval Postgraduate School Monterey California. Cosponsored by the IEEE Circuits and Systems Society. Symposium Organizing Committee: General Chairman-Sherif Michael, Technical Program-Roberto Cristi, Publications-Michael Soderstrand, Special Sessions- Charles W. Therrien, Publicity: Jeffrey Burl, Finance: Ralph Hippenstiel, and Local Arrangements: Barbara Cristi

Distributed control of deregulated electrical power networks

A prerequisite for reliable operation of electrical power networks is that supply and demand are balanced at all time, as efficient ways for storing large amounts of electrical energy are scarce. Balancing is challenging, however, due to the power system's dimensions and complexity, the low controllability and predictability of demand, and due to strict physical and security limitations, such as finitely fast generator dynamics and finite transmission-line capacities. The need for efficient and secure balancing arrangements is growing stronger with the increasing integration of distributed generation (DG), the ongoing deregulation of production and consumption of electrical energy, and thus, also the provision of many of the ancillary services that are essential for network stability. DG is mostly based on renewable, intermittent sources such as wind and sun, and consequently, it is associated with a much larger uncertainty in supply than conventional, centralized generation. Moreover, with the emergence of deregulated energy markets as core operational mechanism, the prime goal of power system operation is shifted from centralized minimization of costs to the maximization of individual profit by a large number of competing, autonomous market agents. The main objective of this thesis is to investigate the control-technical possibilities for ensuring efficient, reliable and stable operation of deregulated and badly predictable electrical power networks. Its contributions cover aspects of power system operation on a time scale ranging from day-ahead trading of electrical energy to second-based load-frequency control. As a first contribution, we identify the maximization of security of supply and market efficiency as the two main, yet conflicting objectives of power system operation. Special attention is paid to congestion management, which is an aspect of power system operation where the tension between reliability and efficiency is particularly apparent. More specifically, the differences between locational pricing and cost-based congestion redispatch are analyzed, followed by an assessment of their effects on grid operation. Next, we demonstrate that the current synchronous, energy-based market and incentive system does not necessarily motivate producers to exchange power profiles with the electricity grid that contribute to network stability and security of supply. The thesis provides an alternative production scheduling concept as a means to overcome this issue, which relies on standard market arrangements, but settles energy transactions in an asynchronous way. Theoretical analysis and simulation results illustrate that by adopting this method, scheduling efficiency is improved and the strain on balancing reserves can be reduced considerably. A major part of this thesis is dedicated to real-time, i.e., closed-loop, balancing or load-frequency control. With the increasing share of badly predictable DG, there is a growing need for efficient balancing mechanisms that can account for generator and transmission constraints during the operational day. A promising candidate solution is model predictive control (MPC). Because the large dimensions and complexity of electrical power networks hamper a standard, centralized implementation of MPC, we evaluate a number of scalable alternatives, in which the overall control action is computed by a set of local predictive control laws, instead. The extent of inter-controller communication is shown to be positively correlated with prediction accuracy and, thus, attainable closed-loop performance. Iterative, system-wide communication/coordination is usually not feasible for large networks, however, and consequently, Pareto-optimal performance and coupled-constraint handling are currently out of reach. This also hampers the application of standard cost-based stabilization schemes, in which closed-loop stability is attained via monotonic convergence of a single, optimal system-wide performance cost. Motivated by the observations regarding non-centralized MPC, the focus is then shifted to distributed control methods for networks of interconnected dynamical systems, with power systems as particular field of application, that can ensure stability based on local model and state information only. First, we propose a non-centralized, constraint-based stabilization scheme, in which the set of stabilizing control actions is specified via separable convergence conditions for a collection of a-priori synthesized structured max-control Lyapunov functions (max-CLFs). The method is shown to be non-conservative, in the sense that non-monotonic convergence of the structured functions along closed-loop trajectories is allowed, whereas their construction establishes the existence of a control Lyapunov function, and thus, stability, for the full, interconnected dynamics. Then, an alternative method is provided in which also the demand for a monotonically converging full-system CLF is relaxed while retaining the stability certificate. The conditions are embedded in an almost-decentralized Lyapunov-based MPC scheme, in which the local control laws rely on neighbor-to-neighbor communication only. Secondly, a generalized theorem and example system are provided to show that stabilization methods that rely on the off-line synthesis of fixed quadratic storage functions (SFs) fail for even the simplest of linear, time-invariant networks, if they contain one or more subsystems that are not stable under decoupled operation. This may also impede the application of max-CLF control. As key contribution of this thesis, to solve this issue, we endow the storage functions with a finite set of state-dependent parameters. Max-type convergence conditions are employed to construct a Lyapunov function for the full network, whereas monotonic convergence of the individual SFs is not required. The merit of the provided approach is that the storage functions can be constructed during operation, i.e., along a closed-loop trajectory, thus removing the impediment of centralized, off-line LF synthesis associated with fixed-parameter SFs. It is shown that parameterized-SF synthesis conditions can be efficiently exploited to obtain a scalable, trajectory-dependent control scheme that relies on non-iterative neighbor-to-neighbor communication only. For input-affine network dynamics and quadratic storage functions, the procedure can be implemented by solving a single semi-definite program per node and sampling instant, in a receding horizon fashion. Moreover, by interpolating a collection of so-obtained input trajectories, a low-complexity explicit control law for linear, time-invariant systems is obtained that extends the trajectory-specific convergence property to a much stronger guarantee of closed-loop asymptotic stability for a particular set of initial conditions. Finally, we consider the application of max-CLF and parameterized SFs for real-time balancing in multimachine electrical power networks. Given that generators are operated by competitive, profit-driven market agents, the stabilization scheme is extended with the competitive optimization of a set of arbitrarily chosen, local performance cost functions over a finite, receding prediction horizon. The suitability of the distributed Lyapunov-based predictive control and parameterized storage function algorithms is evaluated by simulating them in closed-loop with the 7-machine CIGRÉ benchmark system. The thesis concludes by summarizing the main contributions, followed by ideas for future research

Control Theory: On the Way to New Application Fields

Control theory is an interdisciplinary field that is located at the crossroads of pure and applied mathematics with systems engineering and the sciences. Recently, deep interactions are emerging with new application areas, such as systems biology, quantum control and information technology. In order to address the new challenges posed by the new application disciplines, a special focus of this workshop has been on the interaction between control theory and mathematical systems biology. To complement these more biology oriented focus, a series of lectures in this workshop was devoted to the control of networks of systems, fundamentals of nonlinear control systems, model reduction and identification, algorithmic aspects in control, 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