Hearing the clusters in a graph: A distributed algorithm

We propose a novel distributed algorithm to cluster graphs. The algorithm recovers the solution obtained from spectral clustering without the need for expensive eigenvalue/vector computations. We prove that, by propagating waves through the graph, a local fast Fourier transform yields the local component of every eigenvector of the Laplacian matrix, thus providing clustering information. For large graphs, the proposed algorithm is orders of magnitude faster than random walk based approaches. We prove the equivalence of the proposed algorithm to spectral clustering and derive convergence rates. We demonstrate the benefit of using this decentralized clustering algorithm for community detection in social graphs, accelerating distributed estimation in sensor networks and efficient computation of distributed multi-agent search strategies

QUALITY WATER BALANCE AS A BASE FOR WETLANDS RESTORATION IN THE UPPER BIEBRZA VALLEY

Main goal of presented research was the assessment of the influence of water damming in existing land reclamation systems on the surface water quality of the Upper Biebrza River catchment. Surface water quality was assessed on the concentration of BOD5, total phosphorus (TP) and total nitrogen (TN) recorded in 2014 at several monitoring points along Biebrza River and its tributaries. The upper Biebrza R. has a little (at the Sztabin gauging point even an insufficient) absorption capacity of organic pollutants and a high capacity for self-purifying and absorbing of TP and TN. The phosphorus binding capacity decreases along the river and in its upper reach it is necessary to reduce the load of P by 20% to maintain the river quality objectives. Water quality monitoring data and information about pollution sources showed high absorption capacities of TN in the monitored tributaries, which can receive an additional flux of this constituent in the amount exceeding the actual load up to several times. The absorption capacity of BOD5 and TP is lower by an order of magnitude. For Kropiwna R., it is required to reduce the load of organic components (measured as BOD5), which exceeds the requirements for the 1st quality class

A Homotopy Algorithm for Approximating Geometric Distributions by Integrable Systems

In the geometric theory of nonlinear control systems, the notion of a distribution and the dual notion of codistribution play a central role. Many results in nonlinear control theory require certain distributions to be integrable. Distributions (and codistributions) are not generically integrable and, moreover, the integrability property is not likely to persist under small perturbations of the system. Therefore, it is natural to consider the problem of approximating a given codistribution by an integrable codistribution, and to determine to what extent such an approximation may be used for obtaining approximate solutions to various problems in control theory. In this note, we concentrate on the purely mathematical problem of approximating a given codistribution by an integrable codistribution. We present an algorithm for approximating an m-dimensional nonintegrable codistribution by an integrable one using a homotopy approach. The method yields an approximating codistribution that agrees with the original codistribution on an m-dimensional submanifold E_0 of R^n

Scalable Approach to Uncertainty Quantification and Robust Design of Interconnected Dynamical Systems

Development of robust dynamical systems and networks such as autonomous aircraft systems capable of accomplishing complex missions faces challenges due to the dynamically evolving uncertainties coming from model uncertainties, necessity to operate in a hostile cluttered urban environment, and the distributed and dynamic nature of the communication and computation resources. Model-based robust design is difficult because of the complexity of the hybrid dynamic models including continuous vehicle dynamics, the discrete models of computations and communications, and the size of the problem. We will overview recent advances in methodology and tools to model, analyze, and design robust autonomous aerospace systems operating in uncertain environment, with stress on efficient uncertainty quantification and robust design using the case studies of the mission including model-based target tracking and search, and trajectory planning in uncertain urban environment. To show that the methodology is generally applicable to uncertain dynamical systems, we will also show examples of application of the new methods to efficient uncertainty quantification of energy usage in buildings, and stability assessment of interconnected power networks

Approximate feedback linearization of nonlinear control systems

Ph.D.Michael Los

Feedback Linearization of Transverse Dynamics for Periodic Orbits in R^3 with Points of Transverse Controllability Loss

In this paper we show how one can linearize the transverse dynamics of a nonlinear affine single-input system in R^3 in a neighborhood of a periodic orbit in the case when the transverse linear controllability fails in finite number of points along the periodic orbit. An autonomous feedback control providing stability of the periodic orbit is designed in the transverse coordinate system.Research supported in part by NSF under grant PYI ECS-9396296, by AFOSR under grant AFOSR 94NM006, and by a grant from Hughes Aircraft Company

Feedback Linearization of Transverse Dynamics for Periodic Orbits

This paper has been accepted for publication in Systems and Control Letters.In this paper we give necessary and sufficient conditions for feedback linearization of the transverse dynamics (TFL) of a nonlinear affine single-input system in a neighborhood of a periodic orbit. The TFL procedure provides a means of finding coordinates that are tuned to the structure of the control system with respect to the periodic orbit. An autonomous feedback control providing exponential stability of the periodic orbit is easily designed in the transverse coordinate system.NSF under grant PYI ECS-9396296 and a grant from Hughes Aircraft Compan

Least Squares Approximate Feedback Linearization

We study the least squares approximate feedback linearization problem: given a single input nonlinear system, find a linearizable nonlinear system that is close to the given system in a least squares (L_2) sense. A linearly controllable single input affine nonlinear system is feedback linearizable if and only if its characteristic distribution is involutive (hence integrable) or, equivalently, any characteristic one-form (a one-form that annihilates the characteristic distribution) is integrable. We study the problem of finding (least squares approximate) integrating factors that make a fixed characteristic one-form close to being exact in an L_2 sense. One can decompose a given one-form into exact and inexact parts using the Hodge decomposition. We derive an upper bound on the size of the inexact part of a scaled characteristic one-form and show that a least squares integrating factor provides the minimum value for this upper bound. We also consider higher order approximate integrating factors that scale a nonintegrable one-form in a way that the scaled form is closer to being integrable in L_2 together with some derivatives and derive similar bounds for the inexact part. One can use least squares approximate integrating factors in approximate feedback linearization of nonlinearizable single input affine systems. Moreover, least squares approximate integrating factors allow a unified approach to both least squares approximate and exact feedback linearization.NSF under grant PYI ECS-9396296, by AFOSR under grant AFOSR F49620-94-1-0183, and by a grant from Hughes Aircraft Company