Chai Wah Wu

Localization of effective pinning control in complex networks of dynamical systems

This paper concerns pinning control in complex networks of dynamical systems, where an external forcing signal is applied to the network in order to align the state of all the systems to the forcing signal. By considering the control signal as the state of a virtual dynamical system, this problem can be studied in a synchronization framework. Prior studies have determined that a single controller can pin an entire network under certain conditions. This paper aims to further this study by looking at sufficient and necessary conditions for the possible locations where pinning control can be applied. We also study how the ease of control is influenced by the topology and in particular the algebraic connectivity of the network. In particular, we show that for systems with locally connected coupling it is harder to achieve pinning control than for systems with random or fully connected coupling. We also show that when pinning control is applied to a single system and the underlying topology of the network is a vertex-balanced graph, then the amount of control needed to effect pinning control grows at least as fast as the number of vertices. Furthermore, in order to achieve pinning control in systems coupled via locally connected graphs, as the number of systems grows, both the pinning control and the coupling among all systems need to increase.

Agreement and consensus problems in groups of autonomous agents with linear dynamics

We study two recent consensus problems in multi-agent coordination with linear dynamics. In Saber and Murray (2003) an agreement problem was studied which has linear continuous-time state equations and a sufficient condition was given for the given protocol to solve the agreement problem; namely that the underlying graph is strongly connected. We give sufficient and necessary conditions which include graphs that are not strongly connected. In addition, Saber and Murray show that the protocol solves the average consensus problem if and only if the graph is strongly connected and balanced. We show how multi-rate integrators can solve the average consensus problem even if the graph is not balanced. We give lower bounds on the rate of convergence of these systems which are related to the coupling topology. Saber and Murray also considered the case where the coupling topology changes with time but remain a balanced graph at all times. We relate this case of switching topology to synchronization of nonlinear dynamical systems with time-varying coupling and give conditions for solving the consensus problem even when the graphs are not balanced. Jadbabaie et al. (2003) study a model of leaderless and follow-the-leader coordination of autonomous agents using a discrete-time model with time-varying linear dynamics and show coordination if the underlying undirected graph is connected across intervals. Mureau (2003) extended this to directed graphs which are strongly connected across intervals. We prove that coordination is possible even if the graph is not strongly connected.

Global synchronization in coupled map lattices

This paper presents a global synchronization theorem for coupled map lattices. Roughly speaking the theorem states that the coupled map lattice x/sub n+1/=AF(x/sub n/) synchronizes if A has an eigenvalue 1 of multiplicity 1 corresponding to the synchronization manifold and the other eigenvalues of A are close to zero. Examples of coupled map lattices of logistic maps are used to illustrate the result. In particular, we give global results regarding synchronization in coupled map lattices for which previously only numerical evidence and local results were available. We show that (1) globally coupled maps synchronize if the coupling is large enough, (2) randomly coupled maps are synchronized if the number of couplings for each map is large enough and (3) coupled maps connected on a graph will synchronize if the ratio between the largest and the smallest nonzero eigenvalue of the Laplacian matrix of the graph is small.

More is more: The benefits of denser sensor deployment

Positioning disk-shaped sensors to optimize certain coverage parameters is a fundamental problem in ad hoc sensor networks. The hexagon lattice arrangement is known to be optimally efficient in the plane, even though 20.9% of the area is unnecessarily covered twice, however, the arrangement is very rigid—any movement of a sensor from its designated grid position (due to, e.g., placement error or obstacle avoidance) leaves some region uncovered, as would the failure of any one sensor. In this article, we consider how to arrange sensors in order to guarantee multiple coverage, that is, k-coverage for some value k > 1. A naive approach is to superimpose multiple hexagon lattices, but for robustness reasons, we may wish to space sensors evenly apart.n We present two arrangement methods for k-coverage: (1) optimizing a Riesz energy function in order to evenly distribute nodes, and (2) simply shrinking the hexagon lattice and making it denser. The first method often approximates the second, and so we focus on the latter. We show that a density increase tantamount to k copies of the lattice can yield k′-coverage, for k′ > k (e.g., k = 11, k′ = 12 and k = 21, k′ = 24), by exploiting the double-coverage regions. Our examples savings provably converge in the limit to the ≈ 20.9% maximum. We also provide analogous results for the square lattice and its ≈ 57% inefficiency (e.g., k = 3, k′ = 4 and k=5, k′ = 7) and show that for multi-coverage for some values of k′, the square lattice can actually be more efficient than the hexagon lattice.n We also explore other benefits of shrinking the lattice: Doing so allows all sensors to move about their intended positions independently while nonetheless guaranteeing full coverage and can also allow us to tolerate probabilistic sensor failure when providing 1-coverage or k-coverage. We conclude by construing the shrinking factor as a budget to be divided among these three benefits.

A sensor placement algorithm for redundant covering based on Riesz energy minimization

We present an algorithm for sensor placement with redundancy where each point in a 2-dimensional space is covered by at least k sensors under the constraint that all the sensors are located away from each other. We reduce the problem to distributing points evenly on the surface of a torus manifold and solve it computationally by minimizing the Riesz energy. We also study the case where the coverings are incrementally constructed. We illustrate our approach with numerical results and compare it to similar approaches in dispersed dither mask halftoning.

Topology design for fast convergence of network consensus algorithms

The quantities of coefficient of ergodicity and algebraic connectivity have been used to estimate the convergence rates of discrete-time and continuous-time network consensus algorithms respectively. Both of these two quantities are defined with respect to network topologies without the symmetry assumption, and they are applicable to the case when network topologies change with time. We present results identifying deterministic network topologies that optimize these quantities. We will also propose heuristics that can accelerate convergence in random networks by redirecting a small portion of the links assuming that the network topology is controllable.

Synchronization in systems coupled via complex networks

Recently, several models of complex networks such as small world networks and scale free networks have emerged in order to model real-life networks. This paper studies the synchronization that occurs in an array of identical chaotic systems coupled via such complex networks. We show that locally coupled arrays and randomly coupled arrays form two extremes in terms of their synchronization properties. Furthermore, we show that a locally coupled array can be made to synchronize by adding arbitrarily small couplings to an arbitrarily small fraction of coupling sites. We also present a criterion to classify such complex networks according to their synchronizability in the limit.

Direct multi-bit search (DMS) screen algorithm

Multi-bit screening is an extension of binary screening, in which every pixel in continuous-tone image can be rendered to one among multiple absorptance levels. Many multi-bit screen algorithms face the problem of contouring artifacts due to sudden changes in the majority absorptance level between gray levels. In this paper, we have extended the direct binary search to the multi-bit case where at every pixel the algorithm chooses the best drop absorptance level to create a visually pleasing halftone pattern without any user defined guidance. This is repeated throughout the entire range of gray levels to create a high quality multi-bit screen.

Synchronization in arrays of coupled nonlinear systems: passivity circle criterion and observer design

It has been shown that synchronization between two nonlinear systems can be studied as a control theory problem. We show that this relationship can be extended to synchronization in coupled arrays of nonlinear systems. In particular, we extend several stability conditions to synchronization criteria in arbitrarily coupled arrays: the passivity criterion, the circle criterion and a result on observer design of Lipschitz nonlinear systems.

More is More: The Benefits of Denser Sensor Deployment

Positioning disk-shaped sensors to optimize certain coverage parameters is a fundamental problem in ad-hoc sensor networks. The hexagon grid lattice is known to be optimally efficient, but the 20.9% of the area covered by two sensors may be considered a waste. Furthermore, any movement of a sensor from its designated grid position or sensor failure, due to placement error or obstacle avoidance, leaves some region uncovered, as would the failure of any one sensor. We explore how shrinking the grid can help to remedy these shortcomings. First, shrinking to obtain a denser hexagonal lattice allows all sensors to move about their intended positions independently while nonetheless guaranteeing full coverage. Second, sufficiently increasing the lattice density will naturally yield k-coverage for k > 1. Moreover, we show that a density increase tantamount to fc copies of the lattice can yield k -coverage, for kn jn > k (e.g. k = 11, kn jn = 12), through the exploitation of the double-coverage regions. Our examples savings provably converge in the limit to the ap 20.9% maximum. We also provide analogous results for the square lattice and its ap 57% inefficiency, including k = 3, kn jn = 4, k = 5,kn jn = 7, indicating that for multi-coverage, the square lattice can actually be more efficient than the hexagon lattice. All these efficiency gains can be used to provide 1-coverage or fc-coverage even in the face of probabilistic sensor failure. We conclude by construing the shrinking factor as a budget to be divided among these three benefits.