There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

In this paper, we discuss consensus problems for networks of dynamic agents with fixed and switching topologies. We analyze three cases: 1) directed networks with fixed topology; 2) directed networks with switching topology; and 3) undirected networks with communication time-delays and fixed topology. We introduce two consensus protocols for networks with and without time-delays and provide a convergence analysis in all three cases. We establish a direct connection between the algebraic connectivity (or Fiedler eigenvalue) of the network and the performance (or negotiation speed) of a linear consensus protocol. This required the generalization of the notion of algebraic connectivity of undirected graphs to digraphs. It turns out that balanced digraphs play a key role in addressing average-consensus problems. We introduce disagreement functions for convergence analysis of consensus protocols. A disagreement function is a Lyapunov function for the disagreement network dynamics. We proposed a simple disagreement function that is a common Lyapunov function for the disagreement dynamics of a directed network with switching topology. A distinctive feature of this work is to address consensus problems for networks with directed information flow. We provide analytical tools that rely on algebraic graph theory, matrix theory, and control theory. Simulations are provided that demonstrate the effectiveness of our theoretical results.

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Consensus problems in networks of agents with switching topology and time-delays

In a recent Physical Review Letters article, Vicsek et al. propose a simple but compelling discrete-time model of n autonomous agents (i.e., points or particles) all moving in the plane with the same speed but with different headings. Each agent's heading is updated using a local rule based on the average of its own heading plus the headings of its "neighbors." In their paper, Vicsek et al. provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models. The Vicsek model proves to be a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function.

/pdf/coordination-of-groups-of-mobile-autonomous-agents-using-cqyatdq4iz.pdf

Coordination of groups of mobile autonomous agents using nearest neighbor rules

I have developed "tennis elbow" from lugging this book around the past four weeks, but it is worth the pain, the effort, and the aspirin. It is also worth the (relatively speaking) bargain price. Including appendixes, this book contains 894 pages of text. The entire panorama of the neural sciences is surveyed and examined, and it is comprehensive in its scope, from genomes to social behaviors. The editors explicitly state that the book is designed as "an introductory text for students of biology, behavior, and medicine," but it is hard to imagine any audience, interested in any fragment of neuroscience at any level of sophistication, that would not enjoy this book. The editors have done a masterful job of weaving together the biologic, the behavioral, and the clinical sciences into a single tapestry in which everyone from the molecular biologist to the practicing psychiatrist can find and appreciate his or

Principles of Neural Science

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

We explain the biology and physics underlying the chemotactic (foraging) behavior of E. coli bacteria. We explain a variety of bacterial swarming and social foraging behaviors and discuss the control system on the E. coli that dictates how foraging should proceed. Next, a computer program that emulates the distributed optimization process represented by the activity of social bacterial foraging is presented. To illustrate its operation, we apply it to a simple multiple-extremum function minimization problem and briefly discuss its relationship to some existing optimization algorithms. The article closes with a brief discussion on the potential uses of biomimicry of social foraging to develop adaptive controllers and cooperative control strategies for autonomous vehicles. For this, we provide some basic ideas and invite the reader to explore the concepts further.

Biomimicry of bacterial foraging for distributed optimization and control

From the Publisher:
Fuzzy control is emerging as a practical alternative to conventional methods of solving challenging control problems. Written by two authors who have been involved in creating theoretical foundations for the field and who have helped assess the value of this new technology relative to conventional approaches, Fuzzy Control is filled with a wealth of examples and case studies on design and implementation. Computer code and MATLAB files can be downloaded for solving the book's examples and problems and can be easily modified to implement the reader's own fuzzy controllers or estimators. Drawing on their extensive experience working with industry on implementations, Kevin Passino and Stephen Yurkovich have written an excellent hands-on introduction for professionals and educators interested in learning or teaching fuzzy control.

Fuzzy Control

In this note, we specify an "individual-based" continuous-time model for swarm aggregation in n-dimensional space and study its stability properties. We show that the individuals (autonomous agents or biological creatures) will form a cohesive swarm in a finite time. Moreover, we obtain an explicit bound on the swarm size, which depends only on the parameters of the swarm model.

/pdf/stability-analysis-of-swarms-56hd4m39zs.pdf

Stability analysis of swarms

In this article we specify an M-member "individual-based" continuous time swarm model with individuals that move in an n-dimensional space according to an attractant/repellent or a nutrient profile. The motion of each individual is determined by three factors: i) attraction to the other individuals on long distances; ii) repulsion from the other individuals on short distances; and iii) attraction to the more favorable regions (or repulsion from the unfavorable regions) of the attractant/repellent profile. The emergent behavior of the swarm motion is the result of a balance between inter-individual interactions and the simultaneous interactions of the swarm members with their environment. We study the stability properties of the collective behavior of the swarm for different profiles and provide conditions for collective convergence to more favorable regions of the profile.

/pdf/stability-analysis-of-social-foraging-swarms-3kyh0wptk6.pdf

Stability analysis of social foraging swarms

Stable direct and indirect adaptive controllers are presented, which use Takagi-Sugeno fuzzy systems, conventional fuzzy systems, or a class of neural networks to provide asymptotic tracking of a reference signal for a class of continuous-time nonlinear plants with poorly understood dynamics The indirect adaptive scheme allows for the inclusion of a priori knowledge about the plant dynamics in terms of exact mathematical equations or linguistics while the direct adaptive scheme allows for the incorporation of such a priori knowledge in specifying the controller We prove that with or without such knowledge both adaptive schemes can "learn" how to control the plant, provide for bounded internal signals, and achieve asymptotically stable tracking of a reference input In addition, for the direct adaptive scheme a technique is presented in which linguistic knowledge of the inverse dynamics of the plant may be used to accelerate adaptation The performance of the indirect and direct adaptive schemes is demonstrated through the longitudinal control of an automobile within an automated lane

Kevin M. Passino

Papers

Biomimicry of bacterial foraging for distributed optimization and control

Fuzzy Control

Stability analysis of swarms

Stability analysis of social foraging swarms

Stable adaptive control using fuzzy systems and neural networks