Stability and Control in Complex Networks of Dynamical Systems

Stability analysis of networked dynamical systems has been of interest in many disciplines such as biology and physics and chemistry with applications such as LASER cooling and plasma stability. These large networks are often modeled to have a completely random (Erdös-Rényi) or semi-random (Small-World) topologies. The former model is often used due to mathematical tractability while the latter has been shown to be a better model for most real life networks. The recent emergence of cyber physical systems, and in particular the smart grid, has given rise to a number of engineering questions regarding the control and optimization of such networks. Some of the these questions are: How can the stability of a random network be characterized in probabilistic terms? Can the effects of network topology and system dynamics be separated? What does it take to control a large random network? Can decentralized (pinning) control be effective? If not, how large does the control network needs to be? How can decentralized or distributed controllers be designed? How the size of control network would scale with the size of networked system? Motivated by these questions, we began by studying the probability of stability of synchronization in random networks of oscillators. We developed a stability condition separating the effects of topology and node dynamics and evaluated bounds on the probability of stability for both Erdös-Rényi (ER) and Small-World (SW) network topology models. We then turned our attention to the more realistic scenario where the dynamics of the nodes and couplings are mismatched. Utilizing the concept of ε-synchronization, we have studied the probability of synchronization and showed that the synchronization error, ε, can be arbitrarily reduced using linear controllers. We have also considered the decentralized approach of pinning control to ensure stability in such complex networks. In the pinning method, decentralized controllers are used to control a fraction of the nodes in the network. This is different from traditional decentralized approaches where all the nodes have their own controllers. While the problem of selecting the minimum number of pinning nodes is known to be NP-hard and grows exponentially with the number of nodes in the network we have devised a suboptimal algorithm to select the pinning nodes which converges linearly with network size. We have also analyzed the effectiveness of the pinning approach for the synchronization of oscillators in the networks with fast switching, where the network links disconnect and reconnect quickly relative to the node dynamics. To address the scaling problem in the design of distributed control networks, we have employed a random control network to stabilize a random plant network. Our results show that for an ER plant network, the control network needs to grow linearly with the size of the plant network

Structural engineering of evolving complex dynamical networks

Networks are ubiquitous in nature and many natural and man-made systems can be modelled as networked systems. Complex networks, systems comprising a number of nodes that are connected through edges, have been frequently used to model large-scale systems from various disciplines such as biology, ecology, and engineering. Dynamical systems interacting through a network may exhibit collective behaviours such as synchronisation, consensus, opinion formation, flocking and unusual phase transitions. Evolution of such collective behaviours is highly dependent on the structure of the interaction network. Optimisation of network topology to improve collective behaviours and network robustness can be achieved by intelligently modifying the network structure. Here, it is referred to as "Engineering of the Network". Although coupled dynamical systems can develop spontaneous synchronous patterns if their coupling strength lies in an appropriate range, in some applications one needs to control a fraction of nodes, known as driver nodes, in order to facilitate the synchrony. This thesis addresses the problem of identifying the set of best drivers, leading to the best pinning control performance. The eigen-ratio of the augmented Laplacian matrix, that is the largest eigenvalue divided by the second smallest one, is chosen as the controllability metric. The approach introduced in this thesis is to obtain the set of optimal drivers based on sensitivity analysis of the eigen-ratio, which requires only a single computation of the eigenvector associated with the largest eigenvalue, and thus is applicable for large-scale networks. This leads to a new "controllability centrality" metric for each subset of nodes. Simulation results reveal the effectiveness of the proposed metric in predicting the most important driver(s) correctly.     Interactions in complex networks might also facilitate the propagation of undesired effects, such as node/edge failure, which may crucially affect the performance of collective behaviours. In order to study the effect of node failure on network synchronisation, an analytical metric is proposed that measures the effect of a node removal on any desired eigenvalue of the Laplacian matrix. Using this metric, which is based on the local multiplicity of each eigenvalue at each node, one can approximate the impact of any node removal on the spectrum of a graph. The metric is computationally efficient as it only needs a single eigen-decomposition of the Laplacian matrix. It also provides a reliable approximation for the "Laplacian energy" of a network. Simulation results verify the accuracy of this metric in networks with different topologies. This thesis also considers formation control as an application of network synchronisation and studies the "rigidity maintenance" problem, which is one of the major challenges in this field. This problem is to preserve the rigidity of the sensing graph in a formation during motion, taking into consideration constraints such as line-of-sight requirements, sensing ranges and power limitations. By introducing a "Lattice of Configurations" for each node, a distributed rigidity maintenance algorithm is proposed to preserve the rigidity of the sensing network when failure in a sensing link would result in loss of rigidity. The proposed algorithm recovers rigidity by activating, almost always, the minimum number of new sensing links and considers real-time constraints of practical formations. A sufficient condition for this problem is proved and tested via numerical simulations. Based on the above results, a number of other areas and applications of network dynamics are studied and expounded upon in this thesis

The Stability and Control of Stochastically Switching Dynamical Systems

Inherent randomness and unpredictability is an underlying property in most realistic phenomena. In this work, we present a new framework for introducing stochasticity into dynamical systems via intermittently switching between deterministic regimes. Extending the work by Belykh, Belykh, and Hasler, we provide analytical insight into how randomly switching network topologies behave with respect to their averaged, static counterparts (obtained by replacing the stochastic variables with their expectation) when switching is fast. Beyond fast switching, we uncover a highly nontrivial phenomenon by which a network can switch between two asynchronous regimes and synchronize against all odds. Then, we establish rigorous theory for this framework in discrete-time systems for arbitrary switching periods (not limited to switching at each time step). Using stability and ergodic theories, we are able to provide analytical criteria for the stability of synchronization for two coupled maps and the ability of a single map to control an arbitrary network of maps. This work not only presents new phenomena in stochastically switching dynamical systems, but also provides the first rigorous analysis of switching dynamical systems with an arbitrary switching period

Cloud-efficient modelling and simulation of magnetic nano materials

Scientific simulations are rarely attempted in a cloud due to the substantial performance costs of virtualization. Considerable communication overheads, intolerable latencies, and inefficient hardware emulation are the main reasons why this emerging technology has not been fully exploited. On the other hand, the progress of computing infrastructure nowadays is strongly dependent on perspective storage medium development, where efficient micromagnetic simulations play a vital role in future memory design. This thesis addresses both these topics by merging micromagnetic simulations with the latest OpenStack cloud implementation while providing a time and costeffective alternative to expensive computing centers. However, many challenges have to be addressed before a high-performance cloud platform emerges as a solution for problems in micromagnetic research communities. First, the best solver candidate has to be selected and further improved, particularly in the parallelization and process communication domain. Second, a 3-level cloud communication hierarchy needs to be recognized and each segment adequately addressed. The required steps include breaking the VMisolation for the host’s shared memory activation, cloud network-stack tuning, optimization, and efficient communication hardware integration. The project work concludes with practical measurements and confirmation of successfully implemented simulation into an open-source cloud environment. It is achieved that the renewed Magpar solver runs for the first time in the OpenStack cloud by using ivshmem for shared memory communication. Also, extensive measurements proved the effectiveness of our solutions, yielding from sixty percent to over ten times better results than those achieved in the standard cloud.Aufgrund der erheblichen Leistungskosten der Virtualisierung werden wissenschaftliche Simulationen in einer Cloud selten versucht. Beträchtlicher Kommunikationsaufwand, erhebliche Latenzen und ineffiziente Hardwareemulation sind die Hauptgründe, warum diese aufkommende Technologie nicht vollständig genutzt wurde. Andererseits hängt der Fortschritt der Computertechnologie heutzutage stark von der Entwicklung perspektivischer Speichermedien ab, bei denen effiziente mikromagnetische Simulationen eine wichtige Rolle für die zukünftige Speichertechnologie spielen. Diese Arbeit befasst sich mit diesen beiden Themen, indem mikromagnetische Simulationen mit der neuesten OpenStack Cloud-Implementierung zusammengeführt werden, um eine zeit- und kostengünstige Alternative zu teuren Rechenzentren bereitzustellen. Viele Herausforderungen müssen jedoch angegangen werden, bevor eine leistungsstarke Cloud-Plattform als Lösung für Probleme in mikromagnetischen Forschungsgemeinschaften entsteht. Zunächst muss der beste Kandidat für die Lösung ausgewählt und weiter verbessert werden, insbesondere im Bereich der Parallelisierung und Prozesskommunikation. Zweitens muss eine 3-stufige CloudKommunikationshierarchie erkannt und jedes Segment angemessen adressiert werden. Die erforderlichen Schritte umfassen das Aufheben der VM-Isolation, um den gemeinsam genutzten Speicher zwischen Cloud-Instanzen zu aktivieren, die Optimierung des Cloud-Netzwerkstapels und die effiziente Integration von Kommunikationshardware. Die praktische Arbeit endet mit Messungen und der Bestätigung einer erfolgreich implementierten Simulation in einer Open-Source Cloud-Umgebung. Als Ergebnis haben wir erreicht, dass der neu erstellte Magpar-Solver zum ersten Mal in der OpenStack Cloud ausgeführt wird, indem ivshmem für die Shared-Memory Kommunikation verwendet wird. Umfangreiche Messungen haben auch die Wirksamkeit unserer Lösungen bewiesen und von sechzig Prozent bis zu zehnmal besseren Ergebnissen als in der Standard Cloud geführt

Data based identification and prediction of nonlinear and complex dynamical systems

We thank Dr. R. Yang (formerly at ASU), Dr. R.-Q. Su (formerly at ASU), and Mr. Zhesi Shen for their contributions to a number of original papers on which this Review is partly based. This work was supported by ARO under Grant No. W911NF-14-1-0504. W.-X. Wang was also supported by NSFC under Grants No. 61573064 and No. 61074116, as well as by the Fundamental Research Funds for the Central Universities, Beijing Nova Programme.Peer reviewedPostprin

An Application-Based Performance Characterization of the Columbia Supercluster

Columbia is a 10,240-processor supercluster consisting of 20 Altix nodes with 512 processors each, and currently ranked as the second-fastest computer in the world. In this paper, we present the performance characteristics of Columbia obtained on up to four computing nodes interconnected via the InfiniBand and/or NUMAlink4 communication fabrics. We evaluate floating-point performance, memory bandwidth, message passing communication speeds, and compilers using a subset of the HPC Challenge benchmarks, and some of the NAS Parallel Benchmarks including the multi-zone versions. We present detailed performance results for three scientific applications of interest to NASA, one from molecular dynamics, and two from computational fluid dynamics. Our results show that both the NUMAlink4 and the InfiniBand hold promise for application scaling to a large number of processors