a global rational arnoldi method for model reduction abstract in this paper we propose two new approaches for model order reduction of large scale multi input multi output mimo linear time invariant dynamical systems lti these methods are based on a generalization of the global arnoldi algorithm which is used to generate projection subspaces an adaptive procedure for the selection of shift parameters is proposed and some simple arnoldi relations are established in order to compute error bounds on the transfer function numerical experiments showing the effectiveness of these approaches are provided

slowly coupled oscillators phase dynamics and synchronization in this paper we extendthe results of frankel andkiemel siam j appl math 5 3 1993 pp 1436 1446 to a network of slowly coupledoscillators first we use malkin s theorem to derive a canonical phase model that describes synchronization properties of a slowly coupled network then we illustrate the result using slowly coupledoscillators 1 near and ronov hopf bifurcations 2 near saddle node on invariant circle bifurcations and 3 near relaxation oscillations we compare andcontrast synchronization properties of slowly andweakly coupledoscillators

identification of continuous time models the paper considers the problem of estimation of the transfer function of a continuous time dynamic system in the presence of colored noise the author introduces an operator transformation that allows for keeping a continuous time parametrization whereas the parameter estimation can be made by means of a discrete time maximum likelihood algorithm a comparison is made between the performance of the new method in comparison with a standard identification of an armax model the method is useful in cases where it is important to estimate the coefficients of a continuous time transfer function and where it is important to maintain a physical interpretation of the transfer function results

characterization of the solution to the optimal h model reduction problem abstract in this paper the h norm approximation of a given stable proper rational transfer function by a lower order stable proper rational transfer function is studied the solution method is based on the authors observation that the h model reduction problem can be converted into a hankel norm model reduction problem a simple characterization of the solution to the optimal h model reduction problem is developed

phaselocking in a reaction diffusion system with a linear frequency gradient a simple spatially inhomogeneous oscillatory reaction diffusion equation is studied existence of phaselocked waves is proven with a shooting argument stability is determined numerically applications to intestinal motility and a chemical experiment are described

balanced forms canonicity and parameterization we show that balanced forms are canonical and continuously parameterize the set of systems of a given order which are asymptotically stable minimal and have distinct singular values the advantage of the balanced canonical form over other classical canonical forms lies precisely in the guaranteed stability and minimality of the parameterization we also derive formulas allowing the synthesis of systems using gradient methods in balanced form

frequency locking in countable cellular systems localization of asymptotic quasi periodic solutions of autonomous differential systems we address the question of frequency locking in coupled differential systems and of the existence of some quasi periodic solutions of a certain kind of differential systems those systems are named cellular systems quite generally as they deal with countable numbers of coupled systems in some general banach spaces moreover the inner dynamics of each subsystem does not have to be specified we reach some general results about how the frequency locking phenomenon is related to the structure of the coupling map those results can be restated in terms of localization of a certain type of quasi periodic solution of differential systems that may be seen as cellular systems this paper gives some explanations about how and why synchronized behaviors naturally occur in a wide variety of complex systems

frequency weighted model reduction via quasi kalman decomposition a frequency weighted model reduction method based on quasi kalman decomposition is proposed for discrete systems the method is essentially an extension of the technique presented by lu and lee 1985 to the frequency weighted case and can handle both single sided as well as double sided weights two illustrative examples show the effectiveness of the method and results are compared with the method developed by enns 1984

reduced order compensators via balancing and central control design for a structural control problem real time control of a physical system necessitates controllers that are low order in this article we compare two balanced truncation methods as a means of designing low order controllers for a nonlinear cable mass system the first is the standard technique of balanced truncation the second linear quadratic gaussian lqg balanced truncation can be thought of as balancing based on the controller and states that are important from the perspective of control and filter design are retained the control design applied to each reduced order model is the central controller we provide an overview of the central controller and devote attention to the design of this controller in the presence of balancing also described in this article is a method for reducing computational time in solving algebraic riccati equations for the design of low order lqg balanced controllers

l sub 2 optimal filter reduction a closed loop approach this paper is concerned with the problem of order reduction of a full order kalman filter for a stable linear signal model so that the steady state filtering error variance associated with the reduced order filter is minimized by an orthogonal parameterization the above problem is formulated to minimize the filtering error variance over a set of orthogonal matrices both continuous and iterative algorithms are derived to compute an optimal reduced order filter the algorithms are shown to possess good properties including the desirable convergence property the proposed algorithms are simple and effective numerical examples are presented to demonstrate the effectiveness and the significant advantages of the proposed algorithms over the existing open loop methods such as the well known balanced truncation method

consistent impulse response estimation and system realization from noisy data a novel method for the estimation of a finite part of the impulse response is presented the method is consistent in case the measured output data are prone to colored noise the combination of kung s realization algorithm yields a consistent system identification algorithm that is particularly useful for the identification of systems with uncoupled deterministic and stochastic dynamics this algorithm is called irear a detailed linear sensitivity analysis of the impulse response estimation and kung s realization algorithms yields novel error and covariance formulae for the estimated impulse response and system matrices and the poles and frequency response functions that result from the identified system description

identification and model reduction of time varying discrete time systems abstract linear discrete time variable systems are considered and an important class of uniform realizations is defined the necessary and sufficient conditions for a pulse response to be uniformly realizable are obtained two model reduction schemes one via balanced realizations and the other via hankel matrix are proposed the latter approach can be considered as an identification algorithm for the two dimensional pulse response sequence h k i it is shown that the two approaches yield an identical reduced model which is always asymptotically stable

parameter and state model reduction for large scale statistical inverse problems a greedy algorithm for the construction of a reduced model with reduction in both parameter and state is developed for an efficient solution of statistical inverse problems governed by partial differential equations with distributed parameters large scale models are too costly to evaluate repeatedly as is required in the statistical setting furthermore these models often have high dimensional parametric input spaces which compounds the difficulty of effectively exploring the uncertainty space we simultaneously address both challenges by constructing a projection based reduced model that accepts low dimensional parameter inputs and whose model evaluations are inexpensive the associated parameter and state bases are obtained through a greedy procedure that targets the governing equations model outputs and prior information the methodology and results are presented for groundwater inverse problems in one and two dimensions

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

probability one homotopy algorithms for solving the coupled lyapunov equations arising in reduced order h2 h modeling estimation and control optimal reduced order modeling estimation and control with respect to combined h 2 h criteria give rise to coupled lyapunov and riccati equations to develop reliable numerical algorithms for these problems this paper focuses on the coupled lyapunov equations which appear as a subset of the synthesis equations in particular this paper systematically examines the requirements of probability one homotopy algorithms to guarantee global convergence homotopy algorithms for nonlinear systems of equations construct a continuous family of systems and solve the given system by tracking the continuous curve of solutions to the family the main emphasis is on guaranteeing transversality for several homotopy maps based upon the pseudogramian formulation of the coupled lyapunov equations and variations based upon canonical forms these results are essential to the probability one homotopy approach by guaranteeing good numerical properties in the computational implementation of the homotopy algorithms

phase locking in chains of multiple coupled oscillators phase locking in chains of weakly coupled oscillators with coupling beyond nearest neighbors is studied starting with a piecewise linear coupling function a homotopy method is applied to prove the existence of phase locked solutions numerical examples are provided to illustrate the existence and the properties of the solutions differences between multiple coupling and nearest neighbor coupling are also discussed 2000 elsevier science b v all rights reserved

approximation of fir by iir digital filters an algorithm based on balanced model reduction an algorithm for the approximation of finite impulse response fir filters by infinite impulse response iir filters is presented the algorithm is based on a concept of balanced model reduction the matrix inversions normally associated with this procedure are notoriously error prone due to ill conditioning of the special matrix forms required this difficulty is circumvented here by directly formulating a reduced state space system description which is input output equivalent to the system that would more conventionally be obtained following the explicit step of constructing an interim balanced realization examples of fir by iir filter approximations are included

an implicit deflation method for ill conditioned sylvester and lyapunov equations this paper describes an effective algorithm for solving ill conditioned sylvester or lyapunov equations these equations can be solved by conventional methods for ill conditioned linear systems however such methods are not efficient since they require on the order of n 6 arithmetic operations on order n 2 data for these ill conditioned matrix equations an implicit deflation algorithm is proposed to implement a certain svd based minimum norm least squares approximate solution the method is practical since it relies only on the ability to solve a sylvester or lyapunov equation certain practical details are also discussed

subspace algorithms for the identification of multivarible dynamic errors in variables models abstract we consider the problem of identifying multivariable finite dimensional linear time invariant systems from noisy input output measurements apart from the fact that both the measured input and output are corrupted by additive white noise the output may also be contaminated by a term which is caused by a white input process noise furthermore all these noise processes are allowed to be correlated with each other we shall develop a solution to this problem in the framework of subspace identification and we shall show that our algorithms give consistent estimates when the system is operating in open or closed loop two realistic simulation studies are presented to demonstrate the practical applicability of the proposed algorithms 1997 elsevier science ltd

topology of the set of asymptotically stable minimal systems it is shown that the set of asymptotically stable and minimal single input single output systems of order n has n 1 connected components in the case of asymptotically stable and minimal multiple input multiple output systems however there is only one connected component

model reduction of large scale systems by least squares abstract in this paper we introduce an approximation method for model reduction of large scale dynamical systems this is a projection which combines aspects of the svd and krylov based reduction methods this projection can be efficiently computed using tools from numerical analysis namely the rational krylov method for the krylov side of the projection and a low rank smith type iteration to solve a lyapunov equation for the svd side of the projection for discrete time systems the proposed approach is based on the least squares fit of the r 1 th column of a hankel matrix to the preceding r columns where r is the order of the reduced system the reduced system is asymptotically stable matches the first r markov parameters of the full order model and minimizes a weighted h 2 error the method is also generalized for moment matching at arbitrary interpolation points application to continuous time systems is achieved via the bilinear transformation numerical examples prove the effectiveness of the approach the proposed method is significant because it combines guaranteed stability and moment matching together with an optimization criterion

arma spectral estimation of narrow band processes via model reduction the problem of estimating autoregressive moving average arma models for narrowband processes is considered the following approach is proposed estimate a high order autoregressive ar approximation of the process by model reduction based on a truncated internally balanced realization or optimal hankel norm model reduction reduce the order of this high order ar estimate to find a lower order arma model this algorithm gives arma spectral estimates with excellent resolution properties without using iterative numerical minimization methods as for the maximum likelihood method how to take the narrowband assumption into account in the model reduction step is discussed in detail

the structure of characteristic values for classes of linear time invariant systems abstractin this paper a particular structure of the characteristic values found in some classes of systems is dealt with in particular for the linear time invariant systems in minimal form a b c d with transfer function matrix g s for which if i is a characteristic value then also 1 i is consequently when the system order is odd a singular value is equal to one the classes of systems for which this property holds are characterised by the existence of specific state space representations moreover the same structure of the characteristic values can be also retrieved for the class of even g s gt s and odd g s gt s systems several examples related to multi input multi output mimo systems are reported also referring to real physical systems

maximally differential prime ideals in a complete local ring let a be a noetherian local ring containing a field of characteristic zero it was proved recently by seibt 51 that if a is complete and p is a maximally differential prime ideal of a then p is permissible in 4 see section 1 for terminology in fact seibt has claimed this result for a more general class of rings but as we point out in remark 2 9 the proof given by him is valid only if a is complete or dim a p 1 seibt has also given 15 example 1 31 to show that the converse of the above result does not hold although this example is incorrect because the prime ideal considered in the example is not permissible it is true that the converse does not hold this is seen easily from our description of maximally differential prime ideals given in theorem 2 6 however see 4 for a specific example the main result of this paper is

a lyapunov theory for linear time delay systems some relationships between a lyapunov based stability test and linear control system models with time delays are explored the main result is a generalization of the result that if a b is a reachable pair of matrices then square a is a stability matrix if and only if there is a positive definite matrix k such that ak ka bb

balanced pod for model reduction of linear pde systems convergence theory we consider convergence analysis for a model reduction algorithm for a class of linear infinite dimensional systems the algorithm computes an approximate balanced truncation of the system using solution snapshots of specific linear infinite dimensional differential equations the algorithm is related to the proper orthogonal decomposition and it was first proposed for systems of ordinary differential equations by rowley int j bifurc chaos appl sci eng 15 3 997 1013 2005 for the convergence analysis we consider the algorithm in terms of the hankel operator of the system rather than the product of the system gramians as originally proposed by rowley for exponentially stable systems with bounded finite rank input and output operators we prove that the balanced realization can be expressed in terms of balancing modes which are related to the hankel operator the balancing modes are required to be smooth and this can cause computational difficulties for pde systems we show how this smoothness requirement can be lessened for parabolic systems and we also propose a variation of the algorithm that avoids the smoothness requirement for general systems we prove entrywise convergence of the matrices in the approximate reduced order models in both cases and present numerical results for two example pde systems

brief paper on the stability of a reduced order filter based on dominant singular value decomposition of the system dynamics the stability of a reduced order filter rof the gain of which is constructed on the basis of a subspace of dominant singular vectors of the system dynamics is examined a definition of s detectability is introduced it is found that the observability of all unstable and neutral singular vectors s detectability is a sufficient condition for the existence of a stable filter

higher derivations normal flatness and analytic products i z is maximally differential under a set of derivations of a ii there exist nonunits x1 x in a where r dim a z and a subring a of a such that x1 x are analytically independent over a a aoccx 1 x and z m a where m is the maximal ideal of a for a proof of 0 1 see 121 and for a proof of 0 2 see 11 theorem 2 61 noting that under the assumptions of 0 1 or 0 2 if z is maximally differential under a set of derivations of a then z is a prime ideal by 10 theorem 11 in the case of positive characteristic both 0 1 and 0 2 are clearly false

another balanced controller reduction algorithm the problem of controller reduction has been addressed by a number of authors 1 4 using the concept of balanced coordinates originally proposed by moore 5 this note establishes a more direct link between the riccati balancing methods in 2 4 and moore s work and proposes yet another candidate method as a logical extension of 2 4

a cyclic low rank smith method for large sparse lyapunov equations in this paper we present the cyclic low rank smith method which is an iterative method for the computation of low rank approximations to the solution of large sparse stable lyapunov equations it is based on a generalization of the classical smith method and profits by the usual low rank property of the right hand side matrix the requirements of the method are moderate with respect to both computational cost and memory furthermore we propose a heuristic for determining a set of suboptimal alternating direction implicit adi shift parameters this heuristic which is based on a pair of arnoldi processes does not require any a priori knowledge on the spectrum of the coefficient matrix of the lyapunov equation numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the adi parameters

on balanced realizations of bounded real and positive real functions based on normalized coprime factorizations with respect to indefinite metrics and the construction of suitable characteristic functions the ober balanced canonical forms for the classes of bounded real and positive real functions are derived this uses a matrix representation of the shift realization with respect to a basis related to sets of orthogonal polynomials

synchrony stability and firing patterns in pulse coupled oscillators we study non trivial firing patterns in small assemblies of pulse coupled oscillatory maps we find conditions for the existence of waves in rings of coupled maps that are coupled bi directionally we also find conditions for stable synchrony in general all to all coupled oscillators surprisingly we find that for maps that are derived from physiological data the stability of synchrony depends on the number of oscillators we describe rotating waves in two dimensional lattices of maps and reduce their existence to a reduced system of algebraic equations which are solved numerically

brief paper balanced truncation model reduction for systems with inhomogeneous initial conditions we present a rigorous approach to extend balanced truncation model reduction btmr to systems with inhomogeneous initial conditions we provide an estimate for the error between the input output maps of the original and of the reduced initial value system and we illustrate numerically the superiority of our approach over the naive application of btmr when btmr is applied to linear time invariant systems with inhomogeneous initial conditions it is crucial that the initial data are well represented by the subspaces generated by btmr this requirement is often ignored or it is avoided by making the restrictive assumption that the initial data are zero to ensure that the initial data are well represented by the btmr subspaces we add auxiliary inputs determined by the initial data

partial fraction expansion based frequency weighted model reduction technique with error bounds in this technical note we present a new frequency weighted balancing related technique which is based on a parametrized combination of the unweighted balanced truncation technique 9 8 and the partial fraction expansion technique 11 the reduced order models which are guaranteed to be stable in the case of double sided weighting are obtained by either direct truncation or singular perturbation approximation simple elegant and easily computable a priori error bounds are also derived numerical examples and comparison with other well known techniques show the effectiveness of the proposed technique

continuous time lti systems under lossless positive real transformations open loop balanced representation and truncated reduced order models abstractin this paper new results on the open loop balanced representation of continuous time linear time invariant systems are reported more particularly the effect of lossless positive real transformations on open loop balanced representations is investigated with specific attention to the problem of model order reduction the properties of systems where a lossless positive real transformation has been applied are discussed showing that if the original system is open loop balanced the resulting transformed system is still open loop balanced furthermore the singular values of the transformed system are related to those of the original one these results allow to derive a model order reduction strategy for this class of systems that leads to a consistent decrease of the numerical complexity the proposed approach reveals to be of particular interest for the design of reduced order systems with specific amplitude responses including analog multiband filters

a kind of structural frequency locking in generalized spatial automata abstract the classical concept of synchronization is usually related to the locking of the basic frequencies and instantaneous phases of regular oscillations and this question is addressed by studying specific kinds of coupled systems this work presents a different point of view we do not study the convergence of coupled systems to a synchronized behaviour but try to answer the following question in a population of coupled differential systems when each cell subsystem exhibits a periodic behaviour is the whole trajectory periodic we define generalized spatial automata with reference to continuous spatial automata by means of coupling maps and associated measures on the set of cells the main idea is the fact that a cell interprets its own environment via the states of the whole population and according to its own state a natural partition of periods is such that cells belong to the same class if their trajectories share a common period we demonstrate that in a general case where cells belong to an a priori unstructured set and their trajectories evolve in possibly distinct banach spaces the set of classes of periods is generally countable in particular when the set of cells is endowed with a borelian structure all the cells necessarily share a common period