on computing the distance to stability for matrices using linear dissipative hamiltonian systems abstract in this paper we consider the problem of computing the nearest stable matrix to an unstable one we propose new algorithms to solve this problem based on a reformulation using linear dissipative hamiltonian systems we show that a matrix a is stable if and only if it can be written as a j r q where j j t r 0 and q 0 that is r is positive semidefinite and q is positive definite this reformulation results in an equivalent optimization problem with a simple convex feasible set we propose three strategies to solve the problem in variables j r q i a block coordinate descent method ii a projected gradient descent method and iii a fast gradient method inspired from smooth convex optimization these methods require o n 3 operations per iteration where n is the size of a we show the effectiveness of the fast gradient method compared to the other approaches and to several state of the art algorithms

agreeing in networks this paper considers a problem of output agreement in heterogeneous networks with dynamics on the nodes as well as on the edges the control and disturbance signals entering the nodal dynamics are unmatched meaning that some nodes are only subject to disturbances and not to the actuating signals to further enrich our model and motivated by synchronization problems in physical networks we accommodate solvable algebraic constraints resulting in a fairly general and heterogeneous network it is shown that appropriate dynamic feedback controllers achieve output agreement on a desired vector in the presence of physical coupling and despite the influence of constant as well as time varying disturbances furthermore we address the case of an optimal steady state deployment of the control effort over the network by suitable distributed controllers as a case study the proposed results are applied to a heterogeneous microgrid

the earth mover s distance as a metric for image retrieval we investigate the properties of a metric between two distributions the earth mover s distance emd for content based image retrieval the emd is based on the minimal cost that must be paid to transform one distribution into the other in a precise sense and was first proposed for certain vision problems by peleg werman and rom for image retrieval we combine this idea with a representation scheme for distributions that is based on vector quantization this combination leads to an image comparison framework that often accounts for perceptual similarity better than other previously proposed methods the emd is based on a solution to the transportation problem from linear optimization for which efficient algorithms are available and also allows naturally for partial matching it is more robust than histogram matching techniques in that it can operate on variable length representations of the distributions that avoid quantization and other binning problems typical of histograms when used to compare distributions with the same overall mass the emd is a true metric in this paper we focus on applications to color and texture and we compare the retrieval performance of the emd with that of other distances

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

weak form of stokes dirac structures and geometric discretization of port hamiltonian systems we present the mixed galerkin discretization of distributed parameter port hamiltonian systems due to the inherent definition of boundary interconnection ports this system representation is particularly useful for the modeling of interconnected multi physics systems and control at the prototypical example of a system of two conservation laws in arbitrary spatial dimension we derive the main contributions i a weak formulation of the underlying geometric stokes dirac structure ii its geometric approximation by a finite dimensional dirac structure using a mixed galerkin approach and power preserving maps on the space of discrete power variables and iii the approximation of the hamiltonian to obtain finite dimensional port hamiltonian state space models the power preserving maps on the discrete bond space offer design degrees of freedom for the discretization which is illustrated at the example whitney finite elements on a 2d simplicial triangulation the resulting schemes can be considered as trade offs between centered approximations and upwinding

deviation bounds in multi agent systems described by undirected graphs the theory of port hamiltonian systems is used to derive upper bounds for the state deviations in multi agent systems described by undirected graphs pinned to a reference signal the upper bounds for the deviations in networks of first or second order agents respectively depend on the minimal eigenvalue of the extended laplacian of the system in networks of first order agents the deviations decay exponentially with a rate depending on the same minimal eigenvalue in case networks of second order systems meet specific design properties it can be shown that the deviations also decay exponentially with half the rate compared to first order systems

approximating the nearest stable discrete time system abstract in this paper we consider the problem of stabilizing discrete time linear systems by computing a nearby stable matrix to an unstable one to do so we provide a new characterization for the set of stable matrices we show that a matrix a is stable if and only if it can be written as a s 1 u b s where s is positive definite u is orthogonal and b is a positive semidefinite contraction that is the singular values of b are less or equal to 1 this characterization results in an equivalent non convex optimization problem with a feasible set on which it is easy to project we propose a very efficient fast projected gradient method to tackle the problem in variables s u b and generate locally optimal solutions we show the effectiveness of the proposed method compared to other approaches

differentiation and regularity of semi discrete optimal transport with respect to the parameters of the discrete measure this paper aims at determining under which conditions the semi discrete optimal transport is twice dierentiable with respect to the parameters of the discrete measure and exhibits numerical applications the discussion focuses on minimal conditions on the background measure to ensure dierentiability we provide numerical illustrations in stippling and blue noise problems

load balancing of dynamical distribution networks with flow constraints and unknown in outflows we consider a basic model of a dynamical distribution network modeled as a directed graph with storage variables corresponding to every vertex and flow inputs corresponding to every edge subject to unknown but constant inflows and outflows as a preparatory result it is shown how a distributed proportional integral controller structure associating with every edge of the graph a controller state will regulate the state variables of the vertices irrespective of the unknown constant inflows and outflows in the sense that the storage variables converge to the same value load balancing or consensus this will be proved by identifying the closed loop system as a port hamiltonian system and modifying the hamiltonian function into a lyapunov function dependent on the value of the vector of constant inflows and outflows in the main part of the paper the same problem will be addressed for the case that the input flow variables are constrained to take value in an arbitrary interval we will derive sufficient and necessary conditions for load balancing which only depend on the structure of the network in relation with the flow constraints crown copyright c 2013 published by elsevier b v all rights reserved

a one dimensional reaction diffusion system with a fast reaction abstract we consider a system of second order ordinary differential equations describing steady state for a three component chemical system with diffusion in the case when one of the reactions is fast we discuss the existence of solutions and the existence uniqueness and characterization of a limit as the rate of the fast reaction approaches infinity

duality and network theory in passivity based cooperative control this paper presents a class of passivity based cooperative control problems that have an explicit connection to convex network optimization problems the new notion of maximal equilibrium independent passivity is introduced and it is shown that networks of systems possessing this property asymptotically approach the solutions of a dual pair of network optimization problems namely an optimal potential and an optimal flow problem this connection leads to an interpretation of the dynamic variables such as system inputs and outputs to variables in a network optimization framework such as divergences and potentials and reveals that several duality relations known in convex network optimization theory translate directly to passivity based cooperative control problems the presented results establish a strong and explicit connection between passivity based cooperative control theory on the one side and network optimization theory on the other and they provide a unifying framework for network analysis and optimal design the results are illustrated on a nonlinear traffic dynamics model that is shown to be asymptotically clustering

a discontinuous galerkin method for optimal control problems governed by a system of convection diffusion pdes with nonlinear reaction terms in this paper we study the numerical solution of optimal control problems governed by a system of convection diffusion pdes with nonlinear reaction terms arising from chemical processes the symmetric interior penalty galerkin sipg method with upwinding for the convection term is used as a discretization method we use a residual based error estimator for the state and the adjoint variables an adaptive mesh refinement indicated by a posteriori error estimates is applied the arising saddle point system is solved using a suitable preconditioner numerical results are presented to illustrate the performance of the proposed error estimator

singular perturbation analysis of a stationary diffusion reaction system whose solution exhibits a corner type behavior in the interior of the domain we consider a singularly perturbed system of second order differential equations describing steady state of a chemical process that involves three species two reactions one of which is fast and diffusion formal asymptotic expansion of the solution is constructed in the case when solution exhibits a corner type behavior in the interior of the domain of interest the theorem on estimation of the remainder is proved using a fixed point argument

a network dynamics approach to chemical reaction networks abstracta treatment of a chemical reaction network theory is given from the perspective of nonlinear network dynamics in particular of consensus dynamics by starting from the complex balanced assumption the reaction dynamics governed by mass action kinetics can be rewritten into a form which allows for a very simple derivation of a number of key results in the chemical reaction network theory and which directly relates to the thermodynamics and port hamiltonian formulation of the system central in this formulation is the definition of a balanced laplacian matrix on the graph of chemical complexes together with a resulting fundamental inequality this immediately leads to the characterisation of the set of equilibria and their stability furthermore the assumption of complex balancedness is revisited from the point of view of kirchhoff s matrix tree theorem both the form of the dynamics and the deduced behaviour are very similar to consensus dynamics and provide additional perspectives to the latte

a novel reduced model for electrical networks with constant power loads we consider a network preserved model of power networks with proper algebraic constraints resulting from constant power loads both for the linear and the nonlinear differential algebraic model of the network we derive explicit reduced models which are fully expressed in terms of ordinary differential equations for deriving these reduced models we introduce the projected incidence matrix which yields a novel decomposition of the reduced laplacian matrix with the help of this new matrix we provide a complementary approach to kron reduction which is able to cope with constant power loads and nonlinear power flow equations

generalized port hamiltonian dae systems abstract motivated by recent work in this area we expand on a generalization of port hamiltonian systems that is obtained by replacing the hamiltonian function representing energy storage by a lagrangian subspace this leads to a new class of algebraic constraints and dae systems in physical systems modeling it is shown how dirac structures and lagrangian subspaces allow for similar representations and how this can be exploited to convert algebraic constraints originating from dirac structures into algebraic constraints corresponding to lagrangian subspaces and conversely

grid generation and optimization based on centroidal voronoi tessellations centroidal voronoi tessellations cvts are voronoi tessellations of a region such that the generating points of the tessellations are also the centroids of the corresponding voronoi regions such tessellations are of use in very diverse applications including data compression clustering analysis cell biology territorial behavior of animals and optimal allocation of resources in this paper we explore the use of cvts in grid generation in connection with finite element approximations of partial differential equations we being by describing these tessellations and methods for their determination we then discuss their application to mesh generation and finish with some examples of their use for the solution of partial differential equations

formation control using binary information in this paper we study the problem of formation keeping of a network of strictly passive systems when very coarse information is exchanged we assume that neighboring agents only know whether their relative position is larger or smaller than the prescribed one this assumption results in very simple control laws that direct the agents closer or away from each other and take values in finite sets we show that the task of formation keeping while tracking a desired velocity and rejecting matched disturbances is still achievable under the very coarse information scenario in contrast with other results of practical convergence with coarse or quantized information here the control task is achieved exactly

reduction of second order network systems with structure preservation this paper proposes a general framework for structure preserving model reduction of a second order network system based on graph clustering in this approach vertex dynamics are captured by the transfer functions from inputs to individual states and the dissimilarities of vertices are quantified by the mathcal h 2 norms of the transfer function discrepancies a greedy hierarchical clustering algorithm is proposed to place those vertices with similar dynamics into same clusters then the reduced order model is generated by the petrov galerkin method where the projection is formed by the characteristic matrix of the resulting network clustering it is shown that the simplified system preserves an interconnection structure i e it can be again interpreted as a second order system evolving over a reduced graph furthermore this paper generalizes the definition of network controllability gramian to second order network systems based on it we develop an efficient method to compute mathcal h 2 norms and derive the approximation error between the full order and reduced order models finally the approach is illustrated by the example of a small world network

convergence of the lloyd algorithm for computing centroidal voronoi tessellations centroidal voronoi tessellations cvts are voronoi tessellations of a bounded geometric domain such that the generating points of the tessellations are also the centroids mass centers of the corresponding voronoi regions with respect to a given density function centroidal voronoi tessellations may also be defined in more abstract and more general settings due to the natural optimization properties enjoyed by cvts they have many applications in diverse fields the lloyd algorithm is one of the most popular iterative schemes for computing the cvts but its theoretical analysis is far from complete in this paper some new analytical results on the local and global convergence of the lloyd algorithm are presented these results are derived through careful utilization of the optimization properties shared by cvts numerical experiments are also provided to substantiate the theoretical analysis

formation control of a multi agent system subject to coulomb friction this paper considers the formation control problem for a network of point masses which are subject to coulomb friction a dynamical model including the planar discontinuous friction force is presented in the port hamiltonian framework moreover continuous and discontinuous controllers are designed in order to achieve a desired prescribed formation the main results are derived using tools from nonsmooth lyapunov analysis it is shown that the continuous static feedback controller fails to achieve the exact formation while the discontinuous controller achieves the desired task exactly numerical simulations are provided to illustrate the effectiveness of the approach

optimum quantization and its applications abstract minimum sums of moments or equivalently distortion of optimum quantizers play an important role in several branches of mathematics fejes toth s inequality for sums of moments in the plane and zador s asymptotic formula for minimum distortion in euclidean d space are the first precise pertinent results in dimension d 2 in this article these results are generalized in the form of asymptotic formulae for minimum sums of moments resp distortion of optimum quantizers on riemannian d manifolds and normed d spaces in addition we provide geometric and analytic information on the structure of optimum configurations our results are then used to obtain information on i the minimum distortion of high resolution vector quantization and optimum quantizers ii the error of best approximation of probability measures by discrete measures and support sets of best approximating discrete measures iii the minimum error of numerical integration formulae for classes of holder continuous functions and optimum sets of nodes iv best volume approximation of convex bodies by circumscribed convex polytopes and the form of best approximating polytopes and v the minimum isoperimetric quotient of convex polytopes in minkowski spaces and the form of the minimizing polytopes

port hamiltonian systems on graphs in this paper we present a unifying geometric and compositional framework for modeling complex physical network dynamics as port hamiltonian systems on open graphs the basic idea is to associate with the incidence matrix of any directed graph a dirac structure relating the flow and effort variables associated to the edges and vertices of the graph and to formulate energy storing or energy dissipating relations between the flow and effort variables of the edges and the internal vertices this allows for state variables associated to the edges and formalizes the interconnection of networks examples from different origins such as consensus algorithms that share the same structure are shown it is shown how the identified hamiltonian structure offers systematic tools for the analysis and control of the resulting dynamics

conditions on shifted passivity of port hamiltonian systems abstract in this paper we examine the shifted passivity property of port hamiltonian systems shifted passivity accounts for the fact that in many applications the desired steady state values of the input and output variables are nonzero and thus one is interested in passivity with respect to the shifted signals we consider port hamiltonian systems with strictly convex hamiltonian and derive conditions under which shifted passivity is guaranteed in case the hamiltonian is quadratic and state dependency appears in an affine manner in the dissipation and interconnection matrices our conditions reduce to negative semidefiniteness of an appropriately constructed constant matrix moreover we elaborate on how these conditions can be extended to the case when the shifted passivity property can be enforced via output feedback thus paving the path for controller design stability of forced equilibria of the system is analyzed invoking the proposed passivity conditions the utility and relevance of the results are illustrated with their application to a 6th order synchronous generator model as well as a controlled rigid body system

power diagrams properties algorithms and applications the power pow x s of a point x with respect to a sphere s in euclidean d space e d is given by d 2 x z r 2 where d denotes the euclidean distance function and z and r are the center and the radius of s the power diagram of a finite set s of spheres in e d is a cell complex that associates each s in s with the convex domain x in e d operatorname pow x s operatorname pow x t text for all t in s s the close relationship to convex hulls and arrangements of hyperplanes is investigated and exploited efficient algorithms that compute the power diagram and its order k modifications are obtained among the applications of these results are algorithms for detecting k sets for union and intersection problems for cones and paraboloids and for constructing weighted voronoi diagrams and voronoi diagrams for spheres upper space bounds for these geometric problems are derived

higher order time stepping methods for time dependent reaction diffusion equations arising in biology abstract this paper demonstrates the use of higher order methods to solve some time dependent stiff pdes in the past the most popular numerical methods for solving system of reaction diffusion equations was based on the combination of low order finite difference method with low order time stepping method we extend in this report the compatibility of fourth order finite difference scheme in space coupled with fourth order time stepping methods such as imexlm4 imexpc4 imexrk4 and etdrk4 b in time for direct integration of reaction diffusion equations in one space dimension some interesting numerical anomaly phenomenons associated with steady state solutions of the examples chosen from the literature are well presented to address the naturally arising points and queries our findings have led to the understanding of pattern formation such as spiral waves and patchy structures as well as some spatio temporal dynamical structures

centroidal voronoi tessellations applications and algorithms a centroidal voronoi tessellation is a voronoi tessellation whose generating points are the centroids centers of mass of the corresponding voronoi regions we give some applications of such tessellations to problems in image compression quadrature finite difference methods distribution of resources cellular biology statistics and the territorial behavior of animals we discuss methods for computing these tessellations provide some analyses concerning both the tessellations and the methods for their determination and finally present the results of some numerical experiments