asymptotic formation and orbital stability of phase locked states for the kuramoto model abstract we discuss the asymptotic formation and nonlinear orbital stability of phase locked states arising from the ensemble of non identical kuramoto oscillators we provide an explicit lower bound for a coupling strength on the formation of phase locked states which only depends on the diameters of natural frequencies and initial phase configurations we show that when the phases of non identical oscillators are distributed over the half circle and the coupling strength is sufficiently large the dynamics of kuramoto oscillators exhibits two stages transition and relaxation stages in a transition stage initial configurations shrink to configurations whose diameters are strictly less than 2 in a finite time and then the configurations tend to phase locked states asymptotically this improves previous results on the formation of phase locked states by chopra spong 2009 26 and ha ha kim 2010 27 where their attention were focused only on the latter relaxation stage we also show that the kuramoto model is l 1 contractive in the sense that the l 1 distance along two smooth kuramoto flows is less than or equal to that of initial configurations in particular when two initial configurations have the same averaged phases the l 1 distance between them decays to zero exponentially fast for the configurations with different phase averages we use the method of average adjustment and translation invariant of the kuramoto model to show that one solution converges to the translation of the other solution exponentially fast this establishes the orbital stability of the phase locked states our stability analysis does not employ any standard linearization technique around the given phase locked states but instead we use a robust l 1 metric functional as a lyapunov functional in the formation process of phase locked states we estimate the number of collisions between oscillators and lower upper bounds of the transversal phase differences

emergent behaviors of a holonomic particle system on a sphere we study sufficient conditions for the asymptotic emergence of synchronous behaviors in a holonomic particle system on a sphere which was recently introduced by lohe non abelian kuramoto model and synchronization j phys a math theor 42 395101 395126 2009 these conditions depend only on the coupling strength and initial position diameter for identical particles we show that the position diameter approaches zero asymptotically under these sufficient conditions i e all particles approach to the same position for non identical particles the particle positions do not shrink to one point but can be squeezed into some small region whose diameter is inversely proportional to the coupling strength when the coupling strength is large we also provide several numerical results to confirm our analytical findings

synchronization of globally coupled phase oscillators singularities and scaling for general couplings abstract the onset of collective behavior in a population of globally coupled oscillators with randomly distributed frequencies is studied for phase dynamical models with arbitrary coupling the effect of a stochastic temporal variation in the frequencies is also included the fokker planck equation for the coupled langevin system is reduced to a kinetic equation for the oscillator distribution function instabilities of the phase incoherent state are studied by center manifold reduction to the amplitude dynamics of the unstable modes depending on the coupling the coefficients in the normal form can be singular in the limit of weak instability when the diffusive effect of the noise is neglected a detailed analysis of these singularities to all orders in the normal form expansion is presented physically the singularities are interpreted as predicting an altered scaling of the entrained component near the onset of synchronization these predictions are verified by numerically solving the kinetic equation for various couplings and frequency distributions

some new results in the theory of negative imaginary systems with symmetric transfer matrix function abstract this note represents a first attempt to provide a definition and characterisation of negative imaginary systems for not necessarily rational transfer functions via a sign condition expressed in the entire domain of analyticity along the same lines of the classic definition of positive real systems under the standing assumption of symmetric transfer functions we then derive a necessary and sufficient condition that characterises negative imaginary transfer functions in terms of a matrix sign condition restricted to the imaginary axis once again following the same line of argument of the standard positive real case using this definition even transfer functions with a pole at the origin with double multiplicity as well as with a possibly negative relative degree can be negative imaginary

tracking fast trajectories along a slow dynamics a singular perturbations approach controlled coupled slow and fast motions are examined in a singular perturbations setting the objective is to minimize a cost functional that takes into account both the fast motion supposing say tracking a fast target and the slow dynamics a method is offered to cope with the possibility that the fast flow has nonstationary limits invariant measures of the fast motion are then the controlled objects on the infinitesimal scale optimal amalgamation of them on the slow scale induces the variational limit whose solutions are near optimal solutions of the perturbed system

canonical form of symplectic matrix pencils abstract we obtain a real canonical form for real symplectic pencils it extends earlier results which were derived over the complex field so that the canonical form was complex and which were limited to the case where the elementary divisors of eigenvalues on the unit circle have an even degree

complete aggregation of the lohe tensor model with the same free flow the lohe tensor model is a first order tensor valued continuous time dynamical system for the aggregation of tensors with the same rank and size it reduces to the well known aggregation models such as the kuramoto model the lohe sphere model and the lohe matrix model as special cases for low rank tensors we present conditions for the solution splitting property asymptotic convergence toward a set of equilibria and analyze two possible asymptotic states completely aggregate state and bi polar state which can emerge from a set of initial data moreover we provide a sufficient framework leading to the aforementioned two asymptotic states in terms of initial data and system parameters

efficient modal analysis of systems with local stiffness uncertainties the characterization of the uncertainty in modal quantities of an uncertain linear structural system is essential to the rapid determination of its response to arbitrary loadings although the size of many computational structural models used is extremely large i e thousands of equations the uncertainty to be analyzed is oftentimes localized to very small regions of the model this paper addresses the development of an efficient computational methodology for the modal analysis of linear structural systems with local stiffness uncertainties the newly developed methodology utilizes an enriched basis that consists of the sub spectrum of a nominal structural system augmented with additional basis vectors generated from a knowledge of the structure of the stiffness uncertainty in addition methods for determining bounds on the approximate modal frequencies and mode shapes are discussed numerical results demonstrate that the algorithm produces highly accurate results with greatly reduced computational effort copyright 2009 john wiley sons ltd

on polynomial matrix spectral factorization by symmetric extraction we revise the 1963 davis algorithm 2 for the spectral factorization of a para hermitian nonnegative polynomial matrix phi by symmetric factor extraction this algorithm is careless about zeros at infinity by introducing the notion of diagonal reducedness of phi we obtain an easy sufficient test for the absence of zeros at infinity we show then how to get phi diagonally reduced by diagonal excess reduction steps similar to the oono and yasuura steps removing all zeros at infinity and then how to remove synunetrically finite zeros while keeping el diagonally reduced hence free of zeros at infinity this results in a revised symmetric extraction spectral factorization algorithm with monotone degree control an example shows the didactical conceptual simplicity of the method appropriate symmetric extraction is discovered by revising and discovering important particular one sided factor extraction properties of polynomial matrices

a fast slow dynamical systems theory for the kuramoto type phase model abstract we present a fast slow dynamical systems theory for the kuramoto type phase model when the order parameters are frozen the fast system consists of independent oscillator equations whereas the slow system describes the evolution of order parameters we average out the slow system over the fast manifold to derive a weak form of an amplitude angle coupled system for the evolution of kuramoto s order parameters this yields the slow evolution of order parameters to be constant values which gives a rigorous proof to kuramoto s original assumption in his self consistent mean field theory

on the emergence and orbital stability of phase locked states for the lohe model we study the emergence and orbital stability of phase locked states of the lohe model which was proposed as a non abelian generalization of the kuramoto phase model for synchronization lohe introduced a first order system of matrix valued ordinary differential equations for quantum synchronization and numerically observed the asymptotic formation and orbital stability of phase locked states of the lohe model in this paper we provide an analytical framework to confirm lohe s observations of emergent phase locked states this extends earlier special results on lower dimensions to any finite dimension for the construction and orbital stability of phase locked states we introduce lyapunov functions to measure the ensemble diameter and dissimilarity between two lohe flows and using the time evolution estimates of these lyapunov functions we present an admissible set of initial states and show that an admissible initial state leads to a unique phase locked asymptotic state

algebraic riccati equation and j spectral factorization for h smoothing and deconvolution this paper deals with a general steady state estimation problem in the mathcal h infty setting the existence of the stabilizing solution of the related algebraic riccati equation are and of the solution of the associated j spectral factorization problem is investigated the existence of such solutions is well established if the prescribed attenuation level gamma is larger than gamma f the infimum of the values of gamma for which a em causal estimator with attenuation level gamma exists we consider the case when gamma leq gamma f and show that the stabilizing solution of the are still exists except for a finite number of values of gamma as long as a fixed lag acausal estimator smoother does the stabilizing solution of the are may be employed to derive a state space realization of a minimum phase j spectral factor of the j spectrum associated with the estimation problem this j spectral factor may be used in turn to compute the minimum lag smoothing estimator some of the aspects of the j spectral factorization problem and the properties of its solutions are discussed in correspondence to the finite number of values of gamma for which the stabilizing solution of the are does not exist

a matrix valued kuramoto model the kuramoto model is an important model for studying the onset of phase locking in an ensemble of nonlinearly coupled phase oscillators each oscillator has a natural frequency often taken to be random and interacts with the other oscillators through the phase difference it is known that as the coupling strength is increased there is a bifurcation in which the incoherent state becomes unstable and a stable phase locked solution is born beginning with the work of lohe there have been a number of paper that have generalized the kuramoto model for phase locking to a non commuting setting here we propose and analyze another such model we consider a collection of matrix valued variables that evolve in such a way as to try to align their eigenvector frames the phase locked state is one where the eigenframes all align and thus the matrices all commute we analyze the stability of the equal frequency phase locked state and show that it is stable we also analyze for the case of symmetric matrices a dynamic analog of the twist states arising in the standard kuramoto model and show that these twist states are dynamically unstable

emergent behaviors of lohe tensor flocks we present a new aggregation model on the space of rank m tensors with the same size and study emergent dynamics of the proposed model our proposed new aggregation model is general enough to encompass lohe type models such as the kuramoto model the lohe sphere model and the lohe matrix models for the ensemble of real rank 0 rank 1 and rank 2 tensors respectively in this regard we call our proposed model as the lohe tensor model for rank m tensors with the same size for emergent dynamics of the proposed model we employ the ensemble diameter as a lyapunov functional to derive a riccati type differential inequality as a direct application of these differential inequalities we provide two sufficient frameworks leading to the emergent dynamics for homogeneous and heterogeneous ensembles in terms of system parameters and initial data

nonlinear stability of incoherence and collective synchronization in a population of coupled oscillators a mean field model of nonlinearly coupled oscillators with randomly distributed frequencies and subject to independent external white noises is analyzed in the thermodynamic limit when the frequency distribution isbimodal new results include subcritical spontaneous stationary synchronization of the oscillators supercritical time periodic synchronization bistability and hysteretic phenomena bifurcating synchronized states are asymptotically constructed near bifurcation values of the coupling strength and theirnonlinear stability properties ascertained

minimal realization and dynamic properties of optimal smoothers smoothing algorithms of various kinds have been around for several decades however some basic issues regarding the dynamical structure and the minimal dimension of the steady state algorithm are still poorly understood in this paper we derive a realization of minimal dimension of the optimal smoother for a signal admitting a state space description of dimension n it is shown that the dimension of the smoothing algorithm can vary from n to 2n depending on the zero structure of the signal model the dynamics pole structure of the steady state smoother is also characterized explicitly and is related to the zero structure of the model we use several recent ideas from stochastic realization theory in particular a minimal markovian representation of the smoother is derived which requires solving a nonsymmetric wiener hopf factorization problem in this way the smoother is naturally expressed as the cascade of a whitening filter and a linear filter of least possible dimension whose state space is a minimal markovian subspace containing the smoothed estimate x spl i cap this among other aspects affords a very simple calculation of the error covariance matrix of the smoother a reduced order two filter implementation of the type due to mayne 1966 and fraser 1967 is obtained by solving a riccati equation of reduced dimension which is in general smaller than the dimension of the riccati equations considered in the literature

complete synchronization of kuramoto oscillators with finite inertia abstract we present an approach based on gronwall s inequalities for the asymptotic complete phase frequency synchronization of kuramoto oscillators with finite inertia for given finite inertia and coupling strength we present admissible classes of initial configurations and natural frequency distributions which lead to the complete phase frequency synchronization asymptotically for this we explicitly identify invariant regions for the kuramoto flow and derive second order gronwall s inequalities for the evolution of phase and frequency diameters our detailed time decay estimates for phase and frequency diameters are independent of the number of oscillators we also compare our analytical results with numerical simulations

exponential stability of a class of nonlinear dynamical systems with uncertainties abstract exponential stability for a class of nonlinear dynamical systems with uncertainties is investigated based on the stability of the nominal systems a new approach to synthesizing a class of continuous state feedback controllers for uncertain nonlinear dynamical systems is proposed by such a class of feedback controllers we can guarantee exponential stability of uncertain nonlinear dynamical systems our approach can give a clear insight into system analysis finally an illustrative example is given to demonstrate the utilization of the approach developed in this paper

on the schrodinger lohe hierarchy for aggregation and its emergent dynamics the lohe hierarchy is a hierarchy of finite dimensional aggregation models consisting of the kuramoto model the complex lohe sphere model the lohe matrix model and the lohe tensor model in contrast the schrodinger lohe model is the only known infinite dimensional lohe aggregation model in literature in this paper we provide an explicit connection between the schrodinger lohe model and the complex lohe sphere model and then by exploiting this explicit relation we construct infinite dimensional liftings of the lohe matrix and the lohe tensor models in this way we establish the schrodinger lohe hierarchy which corresponds to the infinite dimensional extensions of the lohe hierarchy for the proposed hierarchy we provide sufficient frameworks leading to the complete aggregation in terms of coupling strengths and initial configurations

algebraic riccati equation and j spectral factorization for h estimation abstract in this paper we investigate on the existence of the stabilizing solution of the algebraic riccati equation are related to the h filtering problem with a prescribed attenuation level it is well known that such a solution exists and is positive definite for larger than a certain f and it does not exist for smaller than a certain 0 we consider the intermediate case 0 f and show that in this interval the stabilizing solution does exist except for a finite number of values of we show how the solution of the are may be employed to obtain a minimum phase j spectral factor of the j spectrum associated with the h filtering problem

paper efficient algorithm for matrix spectral factorization an algorithm is presented for the spectral factorization of polynomial or rational matrices arising in optimal control and filtering theory as well as in network theory there are two versions of the algorithm one applicable to continuous time problems the other to discrete time ones both versions are based on newton s method feature quadratic convergence and provide a significant improvement in efficiency over the existing methods

a parameterization of minimal stochastic realizations given a rational spectral density and a minimal spectral factor a parameterization of all minimal spectral factors is derived a minimal realization of each minimal spectral factor is also provided this parameterization is shown to hold under the condition of nonintersection between the spectra of two matrices this condition first introduced in previous work by ferrante et al 1993 is proved here to be also necessary finally these results are applied to the stochastic realization problem

the generalised discrete algebraic riccati equation in linear quadratic optimal control this paper investigates the properties of the solutions of the generalised discrete algebraic riccati equation arising from the classic infinite horizon linear quadratic lq control problem in particular a geometric analysis is used to study the relationship existing between the solutions of the generalised riccati equation and the output nulling subspaces of the underlying system and the corresponding reachability subspaces this analysis reveals the presence of a subspace that plays an important role in the solution of the related optimal control problem which is reflected in the generalised eigenstructure of the corresponding extended symplectic pencil in establishing the main results of this paper several ancillary problems on the discrete lyapunov equation and spectral factorisation are also addressed and solved

optimal lq feedback regulation of a nonisothermal plug flow reactor model by spectral factorization the linear quadratic lq optimal temperature and reactant concentration regulation problem is studied for a partial differential equation model of a nonisothermal plug flow tubular reactor by using a nonlinear infinite dimensional hilbert state space description first the dynamical properties of the linearized model around a constant temperature equilibrium profile along the reactor are studied it is shown that it is exponentially stable and approximately reachable next the general concept of lq feedback is introduced it turns out that any lq feedback is optimal from the input output viewpoint and stabilizing for the plug flow reactor linearized model a state lq feedback operator is computed via the solution of a matrix riccati differential equation mrde in the space variable thanks to the reachability property the computed lq feedback is actually the optimal one then the latter is applied to the nonlinear model and the resulting closed loop system dynamical performances are analyzed a criterion is given which guarantees that the constant temperature equilibrium profile is an asymptotically stable equilibrium of the closed loop system moreover under the same criterion it is shown that the control law designed previously is optimal along the nonlinear closed loop model with respect to some cost criterion the results are illustrated by some numerical simulations

generalized interpolation in h infty with a complexity constraint in a seminal paper sarason generalized some classical interpolation problems for h infinity functions on the unit disc to problems concerning lifting onto h 2 of an operator t that is defined on k

on the relaxation dynamics of lohe oscillators on some riemannian manifolds we study the collective relaxation dynamics appearing in weakly coupled lohe oscillators in a large coupling regime the lohe models on the unit sphere and unitary group were proposed as a nonabelian generalization of the kuramoto model on the unit circle and their emergent dynamics has been extensively studied in previous literature for some restricted class of initial data based on the lyapunov functional approach and order parameter approach in this paper we extend the previous partial results to cover a generic initial configuration via the detailed analysis on the order parameter measuring the modulus of the centroid in particular we present a detailed relaxation dynamics and structure of the resulting asymptotic states for the lohe sphere model we also present new gradient flow formulations for the lohe matrix models with the same one body hamiltonians on some group manifolds as a direct application of this new formulation we show that every bounded lohe flow which originated from any initial configuration converges asymptotically

stability of incoherence in a population of coupled oscillators we analyze a mean field model of coupled oscillators with randomly distributed frequencies this system is known to exhibit a transition to collective oscillations for small coupling the system is incoherent with all the oscillators running at their natural frequencies but when the coupling exceeds a certain threshold the system spontaneously synchronizes we obtain the first rigorous stability results for this model by linearizing the fokker planck equation about the incoherent state an unexpected result is that the system has pathological stability properties the incoherent state is unstable above threshold butneutrally stable below threshold we also show that the system is singular in the sense that its stability properties are radically altered by infinitesimal noise

the discrete algebraic riccati equation and linear matrix inequality we study the discrete time algebraic riccati equation in particular we show that even in the most general cases there exists a one one correspondence between solutions of the algebraic riccati equation and deflating subspaces of a matrix pencil we also study the relationship between algebraic riccati equation and the discrete time linear matrix inequality we show that in general only a subset of the set of rank minimizing solutions of the linear matrix inequality correspond to the solutions of the associated algebraic riccati equation and study under what conditions these sets are equal in this process we also derive very weak assumptions under which a riccati equation has a solution

a negative imaginary lemma and the stability of interconnections of linear negative imaginary systems the note is concerned with linear negative imaginary systems first a previously established negative imaginary lemma is shown to remain true even if the system transfer function matrix has poles on the imaginary axis this result is achieved by suitably extending the definition of negative imaginary transfer function matrices secondly a necessary and sufficient condition is established for the internal stability of the positive feedback interconnections of negative imaginary systems meanwhile some properties of linear negative imaginary systems are developed finally an undamped flexible structure example is presented to illustrate the theory

systems of matrix riccati equations linear fractional transformations partial integrability and synchronization we partially integrate a system of rectangular matrix riccati equations which describe the synchronization behavior of a nonlinear complex system of n globally connected oscillators the equations take a restricted form in which the time dependent matrix coefficients are independent of the node we use linear fractional transformations to perform the partial integration resulting in a system of reduced size which is independent of n generalizing the well known watanabe strogatz reduction for the kuramoto model for square matrices the resulting constants of motion are related to the eigenvalues of matrix cross ratios which we show satisfy various properties such as symmetry relations for square matrices the variables can be regarded as elements of a classical lie group not necessarily compact satisfying the matrix riccati equations trajectories lie either within or on the boundary of a classical domain and we show by numerical example that complete synchronization can occur even for the mixed case provided that certain unitarity conditions are satisfied we extend the definition of cross ratios to rectangular matrix systems and show that again the eigenvalues are conserved special cases are models with real vector unknowns for which trajectories lie on the unit sphere in higher dimensions with well known synchronization behavior and models with complex vector wavefunctions that describe synchronization in quantum systems possibly infinite dimensional we partially integrate a system of rectangular matrix riccati equations which describe the synchronization behavior of a nonlinear complex system of n globally connected oscillators the equations take a restricted form in which the time dependent matrix coefficients are independent of the node we use linear fractional transformations to perform the partial integration resulting in a system of reduced size which is independent of n generalizing the well known watanabe strogatz reduction for the kuramoto model for square matrices the resulting constants of motion are related to the eigenvalues of matrix cross ratios which we show satisfy various properties such as symmetry relations for square matrices the variables can be regarded as elements of a classical lie group not necessarily compact satisfying the matrix riccati equations trajectories lie either within or on the boundary of a classical domain and we show by numerical example that complete synchronization can occur even for the mixed ca

an improved computational technique for perturbations of the generalized symmetric linear algebraic eigenvalue problem a technique is developed and presented which improves the accuracy of the predictions for the eigenvalues of the usual first order perturbation scheme for the generalized symmetric linear algebraic eigenvalue problem the improved method very simply replaces the usual b norms appearing in the denominator of the expressions for the first order changes in the eigenvalues by bp norms where bp is the perturbed b matrix with increasing sizes for the perturbations the improved technique produces results whose accuracy approaches those of the usual second order predictions numerical studies are presented to support the analysis

a shocking display of synchrony abstract this article explores the kuramoto model describing the synchronization of a population of coupled oscillators two versions of this model are considered a discrete version suitable for a population with a finite number of oscillators and a continuum model found in the limit of an infinite population when the strength of the coupling between the oscillators exceeds a threshold the oscillators partially synchronize we explore the transition in the continuum model which takes the form of a bifurcation of a discrete mode from a continuous spectrum we use numerical methods and perturbation theory to study the patterns of synchronization that form beyond transition and compare with the synchronization predicted by the discrete model there are similarities with instabilities in ideal plasmas and inviscid fluids but these are superficial

non abelian kuramoto models and synchronization we describe non abelian generalizations of the kuramoto model for any classical compact lie group and identify their main properties these models may be defined on any complex network where the variable at each node is an element of the unitary group u n or a subgroup of u n the nonlinear evolution equations maintain the unitarity of all variables which therefore evolve on the compact manifold of u n synchronization of trajectories with phase locking occurs as for the kuramoto model for values of the coupling constant larger than a critical value and may be measured by various order and disorder parameters limit cycles are characterized by a frequency matrix which is independent of the node and is determined by minimizing a function which is quadratic in the variables we perform numerical computations for n 2 for which the su 2 group manifold is s3 for a range of natural frequencies and all to all coupling in order to confirm synchronization properties we also describe a second generalization of the kuramoto model which is formulated in terms of real m vectors confined to the m 1 sphere for any positive integer m and investigate trajectories numerically for the s2 model this model displays a variety of synchronization phenomena in which trajectories generally synchronize spatially but are not necessarily phase locked even for large values of the coupling constant

extended kirsch combined method for eigenvalue reanalysis based on the kirsch combined method and the rayleigh quotient a new eigenvalue reanalysis method for modified structures is developed using this method the accuracy of the eigenvalues reanalysis of large modifications of structural parameters is improved as an application of the method a numerical example is given to illustrate the accuracy by comparing the second order perturbation method with the proposed method if the large modifications of structural parameters are made the calculated results show that the proposed method is effective

on the factorization of rational discrete time spectral densities in this paper we consider an arbitrary matrix valued rational spectral density phi z we show with a constructive proof that phi z admits a factorization of the form phi z w top z 1 w z where w z is stochastically minimal moreover w z and its right inverse are analytic in regions that may be selected with the only constraint that they satisfy some symplectic type conditions by suitably selecting the analyticity regions this extremely general result particularizes into a corollary that may be viewed as the discrete time counterpart of the matrix factorization method devised by youla in his celebrated work 48

conal distances between rational spectral densities this paper generalizes thompson and hilbert metrics to the space of spectral densities the resulting complete metric space has the differentiable structure of a finsler manifold with explicit geodesics the corresponding distances are filtering invariant can be computed efficiently and admit geodesic paths that preserve rationality these are properties of fundamental importance in many engineering applications

parametrization of all minimal square spectral factors abstract two different parametrizations of minimal square spectral factors are given these parametrizations extend recent results by picci and pinzoni since they characterize spectral factors without specification on the zero or pole structure

silverman algorithm and the structure of discrete time stochastic systems abstract it is a well known yet poorly understood fact that contrary to the continuous time case the same discrete time process y can be represented by minimal linear models see 1 1a 1 1b below which may either have a non singular or a singular d matrix in fact models with d 0 have been commonly used in the statistical literature on the other hand for models with a singular d matrix the riccati difference equation of kalman filtering involves in general the pseudo inversion of a singular matrix this cheap filtering problem dual to the better known cheap control problem has been studied for several decades in connection with the so called invariant directions of the riccati equation for a singular d a reduction of the order of the riccati equation is in general possible the reasons for such a reduction do not seem to be completely clear either in this paper we provide an explanation of this phenomenon from the classical point of view of zero flipping among minimal spectral factors changing d s occurs whenever zeros are flipped from z to their reciprocals at z 0 it is well known that for finite zeros the zero flipping process takes place by multiplication of the underlying spectral factor by a suitable rational all pass matrix function for infinite zeros zero flipping is implemented by a dual version of the silverman structure algorithm using this interpretation we derive a new algorithm for filtering of non regular processes based on a reduced order riccati equation we also obtain a precise characterization of the reduction of the order of the riccati equation which is afforded by zeros either at z or at the origin this order reduction has traditionally been associated with the study of invariant directions a point of view which as we show does not capture the essence of the phenomenon

on minimal spectral factors with zeroes and poles lying on prescribed regions in this note we consider a general discrete time spectral factorization problem for rational matrix valued functions we build on a recent result establishing existence of a spectral factor whose zeroes and poles lie in any pair of prescribed regions of the complex plane featuring a geometry compatible with symplectic symmetry in this general setting uniqueness of the spectral factor is not guaranteed it was however conjectured that if we further impose stochastic minimality uniqueness can be recovered the main result of his note is a proof of this conjecture

almost global consensus on the n sphere this paper establishes novel results regarding the global convergence properties of a large class of consensus protocols for multiagent systems that evolve in continuous time on the n dimensional unit sphere or n sphere for any connected undirected graph and all n in mathbb n backslash lbrace 1 rbrace each protocol in said class is shown to yield almost global consensus the feedback laws are negative gradients of lyapunov functions and one instance generates the canonical intrinsic gradient descent protocol this convergence result sheds new light on the general problem of consensus on riemannian manifolds the n sphere for n in mathbb n backslash lbrace 1 rbrace differs from the circle and mathsf so 3 where the corresponding protocols fail to generate almost global consensus moreover we derive a novel consensus protocol on mathsf so 3 by combining two almost globally convergent protocols on the n sphere for n in lbrace 1 2 rbrace theoretical and simulation results suggest that the combined protocol yields almost global consensus on mathsf so 3

realization theory for multivariate stationary gaussian processes this paper collects in one place a comprehensive theory of stochastic realization for continuous time stationary gaussian vector processes which in various pieces has appeared in a number of our earlier papers it begins with an abstract state space theory based on the concept of splitting subspace these results are then carried over to the spectral domain and described in terms of hardy functions finally differential equations type stochastic realizations are constructed the theory is coordinate free and it accommodates infinite dimensional representations minimality and other systems theoretical concepts being defined by subspace inclusion rather than by dimension we have strived for conceptual completeness rather than generality and the same framework can be used for other types of stochastic realization problems

formation of phase locked states in a population of locally interacting kuramoto oscillators abstract in this paper we present an asymptotic formation of phase locked states from the ensemble of kuramoto oscillators with a symmetric and connected interaction topology for a limited interaction topology that does not have an all to all interaction lyapunov type approaches based on phase and frequency diameters do not work due to the lack of completeness thus we employ an energy method together with the connectedness of underlying interaction topologies to determine the complete synchronization estimates our synchronization estimation method consists of two parts first we establish that the uniform boundedness of fluctuations yields the asymptotic formation of phase locked states using ojasiewicz gradient inequality second we show that for the initial configurations lying in the half circle the uniform boundedness of fluctuations can be derived by a comparison with solutions to the linear gronwall s differential inequality for the total phase variance

asymptotic numerical methods and pade approximants for non linear elastic structures in this paper we apply asymptotic numerical methods for computing non linear equilibrium paths of elastic beam plate and shell structures the non linear branches are sought in the form of asymptotic expansions and they are determined by solving numerically fem several linear problems with a single stiffness matrix a large number of terms of the series can be easily computed by using recurrence formulas in comparison with a more classical step by step procedure the method is rapid and automatic we show with some examples that the choice of the expansion s parameter and the use of pade approximants play an important role in the determination of the size of the domain of convergence

the spectrum of the locked state for the kuramoto model of coupled oscillators we analyze the linear stability of the phase locked state in the kuramoto model of coupled oscillators the main result is the first rigorous characterization of the spectrum and its associated eigenvectors for any finite number of oscillators all but two of the eigenvalues are negative and merge into a continuous spectrum as the number of oscillators tends to infinity one eigenvalue is always zero by rotational invariance the final eigenvalue corresponding to a collective mode determines the stability of the locked state

emergent behavior of a second order lohe matrix model on the unitary group we study a second order extension to the first order lohe matrix model on the unitary group which can be reduced to the second order kuramoto model with inertia as a special case for the proposed second order model we present several sufficient frameworks leading to the emergence of the complete and practical synchronizations in terms of the initial data and the system parameters for the identical hamiltonians we show that the complete synchronization emerges asymptotically in contrast for the non identical hamiltonians the practical synchronization occurs for some class of initial data when the product of the coupling strength and inertia is sufficiently small

acausal models and balanced realizations of stationary processes abstract we study acausal realizations of stationary or stationary increment processes in particular we characterize the family of models whose corresponding spectral factors have a fixed zero structure acausal models with a fixed zero structure are related to each other by a certain group of state feedback transformations which is naturally parametrized by the solution set of a homogeneous algebraic riccati equation each feedback transformation reflects some of the eigenvalues of the generator matrix a of the representation to a mirror image with respect to the imaginary axis dually acausal models with a fixed pole structure are parametrized by a dual riccati equation and by a corresponding family of output injection transformations from a general standpoint the results of this study clarify the role played by dual pairs of riccati equations in spectral factorization and may be relevant to other problem areas than stochastic modelling one natural application of the concepts discussed in the paper is to stochastic balancing balancing of models with an essentially arbitrary eigenvalue location can be accomodated very naturally in this framework a balancing algorithm involving the solution of a dual pair of riccati equations is discussed

on the complete synchronization of the kuramoto phase model abstract we discuss the asymptotic complete phase frequency synchronization for the kuramoto phase model with a finite size n we present sufficient conditions for initial configurations leading to the exponential decay toward the completely synchronized states our new sufficient conditions and decay rate depend only on the coupling strength and the diameter of initial phase and natural frequency configurations but they are independent of the system size n hence they can be used for the mean field limit for the complete synchronization estimates we estimate the time evolution of the phase and frequency diameters for configurations the initial phase configurations for identical oscillators located on the half circle will converge to the complete synchronized states exponentially fast in contrast for the non identical oscillators the complete frequency synchronization will occur exponentially fast for some restricted class of initial phase configurations our estimates are based on the monotonicity arguments of extremal phase and frequencies which do not employ any linearization procedure of nonlinear coupling terms and detailed information on the eigenvalue of the linearized system around the complete synchronized states we compare our analytical results with numerical simulations

averaging of globally coupled oscillators abstract we study a specific system of symmetrically coupled oscillators using the method of averaging the equations describe a series array of josephson junctions we concentrate on the dynamics near the splay phase state also known as the antiphase state ponies on a merry go round or rotating wave we calculate the floquet exponents of the splay phase periodic orbit in the weak coupling limit and find that all of the floquet exponents are purely imaginary in fact all the floquet exponents are zero except for a single complex conjugate pair thus nested two tori of doubly periodic solutions surround the splay phase state in the linearized averaged equations we numerically integrate the original system and find startling agreement with the averaging results on two counts the observed ratio of frequencies is very close to the prediction and the solutions of the full equations appear to be either periodic or doubly periodic as they are in the averaged equations such behavior is quite surprising from the point of view of generic dynamical systems theory one expects higher dimensional tori and chaotic solutions we show that the functional form of the equations and not just their symmetry is responsible for this nongeneric behavior

on exponential synchronization of kuramoto oscillators in this technical note we study the problem of exponential synchronization for one of the most popular models of coupled phase oscillators the kuramoto model we consider the special case of finite oscillators with distinct bounded natural frequencies our first result derives a lower bound on the coupling gain which is necessary for the onset of synchronization this bound improves the one derived by jadbabaie we then calculate a lower bound on the coupling gain that is sufficient to guarantee oscillator synchronization and derive further sufficient conditions to ensure exponential synchronization of the angular frequencies of all oscillators to the mean natural frequency of the group we also characterize the coupling gain that is sufficient for the oscillator phase differences to approach any desired compact set in finite time

on the solvability of the positive real lemma equations abstract in this paper we consider the classical equations of the positive real lemma under the sole assumption that the state matrix a has unmixed spectrum a a without any other system theoretic assumption observability reachability stability etc we derive a necessary and sufficient condition for the solvability of the positive real lemma equations

rapid reanalysis of the generalized eigenvalue problem of locally modified linear dynamical systems abstract a computationally efficient method for the reanalysis of the damped generalized eigenvalue problems associated with linear dynamical systems with local damping and stiffness modifications is developed no restrictions are imposed on the damping of the system and the modifications in the damping and the stiffness can be independent of each other the application of the developed method to the undamped generalized eigenvalue problem with mass and stiffness modifications is also discussed the proposed method achieves up to three orders of magnitude gains in computational efficiency compared to the conventional solution procedures

eigenvalue error analysis of viscously damped structures using a ritz reduction method the efficient solution of the eigenvalue problem that results from inserting passive dampers with variable stiffness and damping coefficients into a structure is addressed eigenanalysis of reduced models obtained by retaining a number of normal modes augmented with ritz vectors corresponding to the static solutions resulting from the load patterns introduced by the dampers has been empirically shown to yield excellent approximations to the full eigenvalue problem an analysis of this technique in the case of a single damper is presented a priori and a posteriori error estimates are generated and tested on numerical examples comparison theorems with modally truncated models and a markov parameter matching reduced order model are derived these theorems corroborate the heuristic that residual flexibility methods improve low frequency approximation of the system the analysis leads to other techniques for eigenvalue approximation approximate closed form solutions are derived that include a refinement to eigenvalue derivative methods for approximation an efficient newton scheme is also developed a numerical example is presented demonstrating the effectiveness of each of these methods