Nonlinear analysis of dynamical complex networks

Copyright © 2013 Zidong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Complex networks are composed of a large number of highly interconnected dynamical units and therefore exhibit very complicated dynamics. Examples of such complex networks include the Internet, that is, a network of routers or domains, the World Wide Web (WWW), that is, a network of websites, the brain, that is, a network of neurons, and an organization, that is, a network of people. Since the introduction of the small-world network principle, a great deal of research has been focused on the dependence of the asymptotic behavior of interconnected oscillatory agents on the structural properties of complex networks. It has been found out that the general structure of the interaction network may play a crucial role in the emergence of synchronization phenomena in various fields such as physics, technology, and the life sciences

Static output-feedback stabilization of discrete-time Markovian jump linear systems: a system augmentation approach

This paper studies the static output-feedback (SOF) stabilization problem for discrete-time Markovian jump systems from a novel perspective. The closed-loop system is represented in a system augmentation form, in which input and gain-output matrices are separated. By virtue of the system augmentation, a novel necessary and sufficient condition for the existence of desired controllers is established in terms of a set of nonlinear matrix inequalities, which possess a monotonic structure for a linearized computation, and a convergent iteration algorithm is given to solve such inequalities. In addition, a special property of the feasible solutions enables one to further improve the solvability via a simple D-K type optimization on the initial values. An extension to mode-independent SOF stabilization is provided as well. Compared with some existing approaches to SOF synthesis, the proposed one has several advantages that make it specific for Markovian jump systems. The effectiveness and merit of the theoretical results are shown through some numerical example

Stabilizing Randomly Switched Systems

This article is concerned with stability analysis and stabilization of randomly switched systems under a class of switching signals. The switching signal is modeled as a jump stochastic (not necessarily Markovian) process independent of the system state; it selects, at each instant of time, the active subsystem from a family of systems. Sufficient conditions for stochastic stability (almost sure, in the mean, and in probability) of the switched system are established when the subsystems do not possess control inputs, and not every subsystem is required to be stable. These conditions are employed to design stabilizing feedback controllers when the subsystems are affine in control. The analysis is carried out with the aid of multiple Lyapunov-like functions, and the analysis results together with universal formulae for feedback stabilization of nonlinear systems constitute our primary tools for control designComment: 22 pages. Submitte

Mathematical problems for complex networks

Copyright @ 2012 Zidong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This article is made available through the Brunel Open Access Publishing Fund.Complex networks do exist in our lives. The brain is a neural network. The global economy is a network of national economies. Computer viruses routinely spread through the Internet. Food-webs, ecosystems, and metabolic pathways can be represented by networks. Energy is distributed through transportation networks in living organisms, man-made infrastructures, and other physical systems. Dynamic behaviors of complex networks, such as stability, periodic oscillation, bifurcation, or even chaos, are ubiquitous in the real world and often reconfigurable. Networks have been studied in the context of dynamical systems in a range of disciplines. However, until recently there has been relatively little work that treats dynamics as a function of network structure, where the states of both the nodes and the edges can change, and the topology of the network itself often evolves in time. Some major problems have not been fully investigated, such as the behavior of stability, synchronization and chaos control for complex networks, as well as their applications in, for example, communication and bioinformatics

Formal Controller Synthesis for Markov Jump Linear Systems with Uncertain Dynamics

Automated synthesis of provably correct controllers for cyber-physical systems is crucial for deployment in safety-critical scenarios. However, hybrid features and stochastic or unknown behaviours make this problem challenging. We propose a method for synthesising controllers for Markov jump linear systems (MJLSs), a class of discrete-time models for cyber-physical systems, so that they certifiably satisfy probabilistic computation tree logic (PCTL) formulae. An MJLS consists of a finite set of stochastic linear dynamics and discrete jumps between these dynamics that are governed by a Markov decision process (MDP). We consider the cases where the transition probabilities of this MDP are either known up to an interval or completely unknown. Our approach is based on a finite-state abstraction that captures both the discrete (mode-jumping) and continuous (stochastic linear) behaviour of the MJLS. We formalise this abstraction as an interval MDP (iMDP) for which we compute intervals of transition probabilities using sampling techniques from the so-called 'scenario approach', resulting in a probabilistically sound approximation. We apply our method to multiple realistic benchmark problems, in particular, a temperature control and an aerial vehicle delivery problem.Comment: 14 pages, 6 figures, under review at QES

Robust H∞ control of uncertain Markovian jump systems with time-delay

This correspondence is concerned with the robust stochastic stabilizability and robust H∞ disturbance attenuation for a class of uncertain linear systems with time delay and randomly jumping parameters The transition of the jumping parameters is governed by a finite-state Markov process. Sufficient conditionson the existence of a robust stochastic stabilizing and γ-suboptimal H∞ state-feedback controller are presented using the Lyapunov functional approach. It isshown that a robust stochastically stabilizing H∞ state-feedback controller can be constructed through the numerical solution of a set of coupled linear matrix inequalities. © 2000 IEEE.published_or_final_versio

Performance analysis with network-enhanced complexities: On fading measurements, event-triggered mechanisms, and cyber attacks

Copyright © 2014 Derui Ding et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Nowadays, the real-world systems are usually subject to various complexities such as parameter uncertainties, time-delays, and nonlinear disturbances. For networked systems, especially large-scale systems such as multiagent systems and systems over sensor networks, the complexities are inevitably enhanced in terms of their degrees or intensities because of the usage of the communication networks. Therefore, it would be interesting to (1) examine how this kind of network-enhanced complexities affects the control or filtering performance; and (2) develop some suitable approaches for controller/filter design problems. In this paper, we aim to survey some recent advances on the performance analysis and synthesis with three sorts of fashionable network-enhanced complexities, namely, fading measurements, event-triggered mechanisms, and attack behaviors of adversaries. First, these three kinds of complexities are introduced in detail according to their engineering backgrounds, dynamical characteristic, and modelling techniques. Then, the developments of the performance analysis and synthesis issues for various networked systems are systematically reviewed. Furthermore, some challenges are illustrated by using a thorough literature review and some possible future research directions are highlighted.This work was supported in part by the National Natural Science Foundation of China under Grants 61134009, 61329301, 61203139, 61374127, and 61374010, the Royal Society of the UK, and the Alexander von Humboldt Foundation of Germany