solution regularity and smooth dependence for abstract equations and applications to hyperbolic pdes first we present a generalized implicit function theorem for abstract equations of the type f u 0 we suppose that u0 is a solution for 0 and that f is smooth for all but mainly we do not suppose that f u is smooth for all u we state conditions such that for all 0 there exists exactly one solution u u0 that u is smooth in a certain abstract sense and that the data to solution map u is smooth then we apply this to time periodic solutions of first order hyperbolic systems tuj aj x xuj bj t x u 0 and second order hyperbolic equations t2u a x 2 x2u b t x u tu xu 0 here we have to prevent small divisors from coming up moreover we need smooth dependence of bj and b on t to get smooth dependence of the solution on this is completely different to the case of parabolic pdes

boundary layer solutions to problems with infinite dimensional singular and regular perturbations abstract we prove existence local uniqueness and asymptotic estimates for boundary layer solutions to singularly perturbed equations of the type e x 2 u x f x u x g x u x e x u x 0 x 1 with dirichlet and neumann boundary conditions here the functions e and g are small and hence regarded as singular and regular functional perturbation parameters the main tool of the proofs is a generalization to banach space bundles of an implicit function theorem of r magnus

uniqueness of periodic solutions of a nonautonomous density dependent predator prey system abstract in this present paper we investigate the uniqueness of periodic solutions of a nonautonomous density dependent and ratio dependent predator prey system where not only the prey density dependence but also the predator density dependence are considered such that the studied predator prey system conforms to the realistically biological environment we start with a sufficient condition for the permanence of the system and then construct a weaker sufficient condition by introducing a specific set denoted as based on this and the brouwer fixed point theorem we obtain the existence condition of positive periodic solutions moreover since the uniqueness of positive periodic solutions can be ensured by global attractiveness we alternatively introduce a sufficient condition for global attractiveness similarly we also provide a sufficient condition for the uniqueness of non negative periodic solutions

analysis of the asymptotic properties of the moesp type of subspace algorithms the moesp type of subspace algorithms are used for the identification of linear discrete time finite dimensional state space systems they are based on the geometric structure of covariance matrices and exploit the properties of the state vector extensively in this paper the asymptotic properties of the algorithms are examined the main results include consistency and asymptotic normality for the estimates of the system matrices under suitable assumptions on the noise sequence the input process and the underlying true system

a numerical method for a nonlinear singularly perturbed interior layer problem using an approximate layer location a class of nonlinear singularly perturbed interior layer problems is examined in this paper solutions exhibit an interior layer at an a priori unknown location a numerical method is presented that uses a piecewise uniform mesh refined around numerical approximations to successive terms of the asymptotic expansion of the interior layer location the first term in the expansion is used exactly in the construction of the approximation which restricts the range of problem data considered denote the perturbation parameter as e and the number of mesh intervals to be used as n the method is shown to converge point wise to the true solution with a first order convergence rate overlooking a logarithmic factor for sufficiently small e n 1 a numerical experiment is presented to demonstrate the convergence rate established

the asymptotic variance of subspace estimates abstract we give new simple general expressions for the asymptotic covariance of the estimated system parameters a b c d in subspace identification the formulas can be applied to a whole class of subspace methods including n4sid moesp cva etc the asymptotic expressions highlight how the conditioning of the estimation problem influences the accuracy of the estimates

the role of vector autoregressive modeling in predictor based subspace identification subspace identification for closed loop systems has been recently studied by several authors a class of new and consistent closed loop subspace algorithms is based on identification of a predictor model in a way similar as prediction error methods pem do experimental evidence suggests that these methods have a behavior which is very close to pem in certain examples the asymptotical statistical properties of one of these methods have been studied recently allowing to show i its relation with cca and ii that cramer rao lower bound is not reached in general very little however is known concerning their relative performance in this paper we shall discuss the link between these predictor based methods to this purpose we exploit the role which vector auto regressive with exogenous input models play in all these algorithms the results of this paper provide a unifying framework under which all these algorithms can be viewed also the link with varx modeling have important implications as to computational complexity is concerned leading to very computationally attractive implementations we also hope that this framework and in particular the relation with varx modeling followed by model reduction will turn out to be useful in future developments of subspace identification such as the quest for efficient procedures and the statistical analysis with finite data

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

representations of jointly stationary stochastic feedback processes abstract stable constant linear closed loop systems relating an input vector u to an output vector u and vice versa produce a jointly stationary y u procoss on the other hand it is often natural to split up a stationary vector random process z into component vectors yand u and to examine the closed loop relations between y and u this paper presents a number of new results on the spectral factorization and the closed loop representation of a jointly stationary vector y u proccss conditions are derived on the closed loop models for the joint process model to bo oe minimal degree stable and minimum phase relations between different joint process models producing the same spectrum yu z are established

on the relation between cca and predictor based subspace identification in this paper we investigate the relation between a recently proposed subspace method based on predictor identification pbsid known also as ldquowhitening filter algorithm rdquo and the classical cca algorithm the comparison is motivated by i the fact that cca is known to be asymptotically efficient for time series identification and optimal for white measured inputs and ii some recent results showing that a number of recently developed algorithms are very closely related to pbsid we show that pbsid is asymptotically equivalent to cca precisely in the situations in which cca is optimal while an ldquooptimizedrdquo version of pbsid behaves no worse than cca also for nonwhite inputs even though pbsid and its optimized version are consistent regardless of the presence of feedback in this paper we work under the assumption that there is no feedback to make the comparison with cca meaningful the results of this paper imply that the ldquooptimizedrdquo pbsid besides being able to handle feedback is to be preferred to cca also when there is no feedback only in very specific cases white or no inputs are the two algorithms asymptotically equivalent

on consistency of subspace methods for system identification subspace methods for identification of linear time invariant dynamical systems typically consist of two main steps first a so called subspace estimate is constructed this first step usually consists of estimating the range space of the extended observability matrix secondly an estimate of system parametersis obtained based on the subspace estimate in this paper the consistency of a large class of methods for estimating the extended observability matrix is analyzed persistence of excitation conditions on the input signal are given which guarantee consistent estimates for systems with only measurement noise for systems with process noise it is shown that a persistence of excitation condition on the input is not sufficient more precisely an example for which the subspace methods fail to give a consistent estimate of the transfer function is given this failure occurs even if the input is persistently exciting of any order it is also shown that this problem can be eliminated if stronger conditions on the input signal are imposed

subspace identification by data orthogonalization and model decoupling it has been observed that identification of state space models with inputs may lead to unreliable results in certain experimental conditions even when the input signal excites well within the bandwidth of the system this may be due to ill conditioning of the identification problem which occurs when the state space and the future input space are nearly parallel we have in particular shown in the companion papers automatica 40 4 2004 575 automatica 40 4 2004 677 that under these circumstances subspace methods operating on input output data may be ill conditioned quite independently of the particular algorithm which is used in this paper we indicate that the cause of ill conditioning can sometimes be cured by using orthogonalized data and by recasting the model into a certain natural block decoupled form consisting of a deterministic and a stochastic subsystem the natural subspace algorithm for the identification of the deterministic subsystem is then a weighted version of the pi moesp method of verhaegen and dewilde int j control 56 1993 1187 1211 the analysis shows that under certain conditions methods based on the block decoupled parametrization and orthogonal decomposition of the input output data perform better than traditional joint model based methods in the circumstance of nearly parallel regressors

diffusion in turbulence we prove long time diffusive behavior homogenization for convection diffusion in a turbulent flow that it incompressible and has a stationary and square integrable stream matrix simple shear flow examples show that this result is sharp for flows that have stationary stream matrices

identification of the deterministic part of mimo state space models given in innovations form from input output data the problem of linear multivariable state space model identification from input output data can under the presence of process and measurement noise be solved in a non iterative way when incorporating instrumental variables constructed from both input and output sequences in the recently developed class of multivariable output error state space model class of subspace model identification schemes abstract in this paper we describe two algorithms to identify a linear time invariant finite dimensional state space model from input output data the system to be identified is assumed to be excited by a measurable input and an unknown process tloise and the measurements are disturbed by unknown measurement noise both noise sequences are discrete zero mean white noise the first algorithm gives consistent estimates only for the case where the input also is zero mean white noise while the same result is obtained with the second algorithm without this constraint for the special case where the input signal is discrete zero mean white noise it is explicitly shown that this second algorithm is a special case of the recently developed multivariable output error state space moesp class of algorithms based on instrumental variables the usefulness of the presented schemes is highlighted in a realistic simulation study

on the ill conditioning of subspace identification with inputs there is experimental evidence that the performance of standard subspace algorithms from the literature e g the n4sid method may be surprisingly poor in certain experimental conditions this happens typically when the past signals past inputs and outputs and future input spaces are nearly parallel in this paper we argue that the poor behavior may be attributed to a form of ill conditioning of the underlying multiple regression problem which may occur for nearly parallel regressors an elementary error analysis of the subspace identification problem shows that there are two main possible causes of ill conditioning the first has to do with near collinearity of the state and future input subspaces the second has to do with the dynamical structure of the input signal and may roughly be attributed to lack of excitation stochastic realization theory constitutes a natural setting for analyzing subspace identification methods in this setting we undertake a comparative study of three widely used subspace methods n4sid robust n4sid and po moesp the last two methods are proven to be essentially equivalent and the relative accuracy regarding the estimation of the a c parameters is shown to be the same

analysis of state space system identification methods based on instrumental variables and subspace fitting subspace based state space system identification 4sid methods have recently been proposed as an alternative to more traditional techniques for multivariable system identification the advantages are that the user has simple and few design variables and that the methods have robust numerical properties and relatively low computational complexities though subspace techniques have been demonstrated to perform well in a number of cases the performance of these methods is neither fully understood nor analyzed our principal objective is to undertake a statistical investigation of subspace based system identification techniques the studied methods consist of two steps the subspace spanned by the extended observability matrix is first estimated the asymptotic properties of this subspace estimate are derived herein in the second step the structure of the extended observability matrix is used to find a system model estimate two possible methods are considered the simplest one only uses a certain shift invariance property while in the other method a parametric representation of the null space of the observability matrix is exploited explicit expressions for the asymptotic estimation error variances of the corresponding pole estimates are given

n4sid subspace algorithms for the identification of combined deterministic stochastic systems abstract recently a great deal of attention has been given to numerical algorithms for subspace state space system identification n4sid in this paper we derive two new n4sid algorithms to identify mixed deterministic stochastic systems both algorithms determine state sequences through the projection of input and output data these state sequences are shown to be outputs of non steady state kalman filter banks from these it is easy to determine the state space system matrices the n4sid algorithms are always convergent non iterative and numerically stable since they only make use of qr and singular value decompositions both n4sid algorithms are similar but the second one trades off accuracy for simplicity these new algorithms are compared with existing subspace algorithms in theory and in practice

asymptotic properties of subspace estimators since the proposal of subspace methods in the 1980s and early 1990s substantial efforts have been made in the analysis of the statistical properties of the algorithms this paper surveys the literature on the asymptotic properties of particular subspace methods used for linear dynamic time invariant discrete time systems the goals of this paper are threefold first this survey tries to present the most relevant results on the asymptotic properties of estimators obtained using subspace methods secondly the main methods and tools that have been used in the derivation of these results are presented to make the literature more accessible thirdly main unsolved questions and rewarding research topics are identified some of which can be attacked using the toolbox discussed in the paper

some facts about the choice of the weighting matrices in larimore type of subspace algorithms in this paper the effect of some weighting matrices on the asymptotic variance of the estimates of linear discrete time state space systems estimated using subspace methods is investigated the analysis deals with systems with white or without observed inputs and refers to the larimore type of subspace procedures the main result expresses the asymptotic variance of the system matrix estimates in canonical form as a function of some of the user choices clarifying the question on how to choose them optimally it is shown that the cca weighting scheme leads to optimal accuracy the expressions for the asymptotic variance can be implemented more efficiently as compared to the ones previously published

unit canonical correlations between future and past on considere des processus arma stationnaires x t de rang complet et le cas ou il y a des fonctions lineaires du futur x t t 0 et du passe x t t 0 qui ont une correlation unite on montre que le nombre de paires de ce type lineairement independantes est le nombre de zeros du determinant des fonctions de transfert sur le cercle unite

periodic solutions of nonlinear boundary value problems one of the well known techniques in the theory of nonlinear boundary value problems bvp is the method of differential inequalities or the method of upper and lower solutions the method of alternative problems a global variant of the lyapunov schmidt method has been used in the study of problems at resonance the investigation of periodic bvp s forms an important subclass of problems at resonance our aim is to combine the two approaches to discuss nonlinear problems at resonance we restrict ourselves in this paper to the discussion of periodic boundary value problems in section 1 we shall indicate the method of upper and lower solutions the results for systems are given in a general way so as to include known results and also offer new directions section 2 deals with the abstract existence results at resonance in the desired framework in section 3 we consider the scalar periodic bvp s by combining the two techniques namely alternative methods and the method of differential inequalities

asymptotic variance of closed loop subspace identification methods in this paper the asymptotic properties of a version of the innovation estimation algorithm by qin and ljung as well as of a version of the whitening filter based algorithm introduced by jansson are studied expressions for the asymptotic error as the sum of a bias term plus a variance term are given the analysis is performed under rather mild assumptions on the spectrum of the joint input output process however in order to avoid unnecessary complications the asymptotic variance formulas are computed explicitly only for finite memory systems i e of the arx type this assumption could be removed at the price of some technical complications the simulation results confirm that when the past horizon is large enough as compared to the predictor dynamics the asymptotic expressions provide a good approximation of the asymptotic variance also for armax systems

boundary value problem with unbounded fast oscillatory random flows abstract we use tartar s weak convergence method in conjunction with a variational principle to prove a sharp homogenization theorem for diffusion in steady random flows the flow has a stationary and square integrable stream matrix the key of our approach is introducing approximate correctors by means of a saddle point variational principle we also obtain the two term asymptotics

choice of state space basis in combined deterministic stochastic subspace identification this paper describes how the state space basis of models identified with subspace identification algorithms can be determined it is shown that this basis is determined by the input spectrum and by user defined input and output weightings through the connections between subspace identification and frequency weighted balancing the state space basis of the subspace identified models is shown to coincide with a frequency weighted balanced basis

parameterization and identification of multivariable state space systems a canonical approach in this paper the problem of determining a canonical state space representation for multivariable systems is revisited a method is derived to build a canonical state space representation directly from data generated by a linear time invariant system contrary to the classic construction methods of canonical parameterizations the technique developed in this paper does not assume the availability of any observability or controllability indices however it requires the a matrix of any minimal realization of the system to be non derogatory a subspace based identification algorithm is also introduced to estimate such a canonical state space parameterization directly from input output data

hopf bifurcation for semilinear dissipative hyperbolic systems abstract we consider boundary value problems for semilinear hyperbolic systems of the type t u j a j x x u j b j x u 0 x 0 1 j 1 n with smooth coefficient functions a j and b j such that b j x 0 0 for all x 0 1 r and j 1 n we state conditions for hopf bifurcation i e for existence local uniqueness up to phase shifts smoothness and smooth dependence on of time periodic solutions bifurcating from the zero stationary solution furthermore we derive a formula which determines the bifurcation direction the proof is done by means of a liapunov schmidt reduction procedure for this purpose fredholm properties of the linearized system and implicit function theorem techniques are used

quintic b spline method for solving second order linear and nonlinear singularly perturbed two point boundary value problems in this paper we have studied a numerical scheme to solve second order singularly perturbed two point linear and nonlinear boundary value problems the boundary layer of this type of problems exhibits at one end left or right point of the domain due to the presence of perturbation parameter the quintic b spline method and suitable piecewise uniform shishkin mesh have been used linear and nonlinear second order singularly perturbed boundary value problems have been solved by the present method the convergence analysis is also provided and the method is shown to have uniform convergence of fourth order numerical results have demonstrated the efficiency of the present method

an integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows precise necessary and sufficient conditions on the velocity statistics for mean field behavior in advection diffusion by a steady incompressible velocity field are developed here under these conditions a rigorous stieltjes integral representation for effective diffusivity in turbulent transport is derived this representation is valid for all peclet numbers and provides a rigorous resummation of the divergent perturbation expansion in powers of the peclet number one consequence of this representation is that convergent upper and lower bounds on effective diffusivity for all peclet numbers can be obtained utilizing a prescribed finite number of terms in the perturbation series explicit rigorous examples of steady incompressible velocity fields are constructed which have effective diffusivities realizing the simplest upper or lower bounds for all peclet numbers a nonlocal variational principle for effective diffusivity is developed along with applications to advection diffusion by random arrays of vortices a new class of rigorous examples is introduced these examples have an explicit stieltjes measure for the effective diffusivity furthermore the effective diffusivity behaves likek0 pe 1 2 in the limit of large peclet numbers wherek0 is the molecular diffusivity formal analogies with the theory of composite materials are exploited systematically

existence and asymptotic stability of periodic solutions with an interior layer of reaction advection diffusion equations abstract we consider a singularly perturbed parabolic periodic boundary value problem for a reaction advection diffusion equation we construct the interior layer type formal asymptotics and propose a modified procedure to get asymptotic lower and upper solutions by using sufficiently precise lower and upper solutions we prove the existence of a periodic solution with an interior layer and estimate the accuracy of its asymptotics moreover we are able to establish the asymptotic stability of this solution

numerical conditioning and asymptotic variance of subspace estimates new formulas for the asymptotic variance of the parameter estimates in subspace identification show that the accuracy of the parameter estimates depends on certain indices of near collinearity of the state and future input subspaces of the system to be identified this complements the numerical conditioning analysis of subspace methods presented in the companion paper on the ill conditioning of subspace identification with inputs automatica doi 10 1016 j automatica 2003 11 009

subspace model identification part 1 the output error state space model identification class of algorithms in this paper we present two novel algorithms to realize a finite dimensional linear time invariant state space model from input output data the algorithms have a number of common features they are classified as one of the subspace model identification schemes in that a major part of the identification problem consists of calculating specially structured subspaces of spaces defined by the input output data this structure is then exploited in the calculation of a realization another common feature is their algorithmic organization an rq factorization followed by a singular value decomposition and the solution of an overdetermined set or sets of equations the schemes assume that the underlying system has an output error structure and that a measurable input sequence is available the latter characteristic indicates that both schemes are versions of the mimo output error state space model identification moesp approach the first algorithm is denoted in particular as the elementary moesp scheme

asymptotic variance of subspace methods by data orthogonalization and model decoupling a comparative analysis this is a companion of the paper chiuso and picci 2004d where we do asymptotic error analysis of a weighted pi moesp type method and compare accuracy with respect to estimates obtained by customary joint subspace methods the analysis shows that under certain conditions methods based on orthogonal decomposition of the input output data and block decoupled parametrization perform better than traditional joint model based methods in the circumstance of nearly parallel regressors

a novel subspace identification approach with enforced causal models subspace identification methods sims for estimating state space models have been proven to be very useful and numerically efficient they exist in several variants but all have one feature in common as a first step a collection of high order arx models are estimated from vectorized input output data in order not to obtain biased estimates this step must include future outputs however all but one of the submodels include non causal input terms the coefficients of them will be correctly estimated to zero as more data become available they still include extra model parameters which give unnecessarily high variance and also cause bias for closed loop data in this paper a new model formulation is suggested that circumvents the problem within the framework the system matrices a b c d and markov parameters can be estimated separately it is demonstrated through analysis that the new methods generally give smaller variance in the estimate of the observability matrix and it is supported by simulation studies that this gives lower variance also of the system invariants such as the poles

time periodic boundary layer solutions to singularly perturbed parabolic problems abstract in this paper we present a result of implicit function theorem type which was designed for applications to singularly perturbed problems this result is based on fixed point iterations for contractive mappings in particular no monotonicity or sign preservation properties are needed then we apply our abstract result to time periodic boundary layer solutions which are allowed to be non monotone with respect to the space variable in semilinear parabolic problems with two independent singular perturbation parameters we prove existence and local uniqueness of those solutions and estimate their distance to certain approximate solutions

canonical correlation analysis approximate covariance extension and identification of stationary time series abstract in this paper we analyze a class of state space identification algorithms for time series based on canonical correlation analysis in the light of recent results on stochastic systems theory in principle these so called subspace methods can be described as covariance estimation followed by stochastic realization the methods offer the major advantage of converting the nonlinear parameter estimation phase in traditional arma models identification into the solution of a riccati equation but introduce at the same time some nontrivial mathematical problems related to positivity the reason for this is that an essential part of the problem is equivalent to the well known rational covariance extension problem therefore the usual deterministic arguments based on factorization of a hankel matrix are not valid for generic data something that is habitually overlooked in the literature we demonstrate that there is no guarantee that several popular identification procedures based on the same principle will not fail to produce a positive extension unless some rather stringent assumptions are made which in general are not explicitly reported in this paper the statistical problem of stochastic modeling from estimated covariances is phrased in the geometric language of stochastic realization theory we review the basic ideas of stochastic realization theory in the context of identification discuss the concept of stochastic balancing and of stochastic model reduction by principal subsystem truncation the model reduction method of desai and pal 1982 a realization approach to stochastic model reduction proc 1st decision and control conf pp 1105 1112 based on truncated balanced stochastic realizations is partially justified showing that the reduced system structure has a positive covariance sequence but is in general not balanced as a byproduct of this analysis we obtain a theorem prescribing conditions under which the subspace identification methods produce bona fide stochastic systems

realization of stochastic systems with exogenous inputs and subspace identification methods a stochastic realization theory for a discrete time stationary process with an exogenous input is developed by extending the classical cca technique some stochastic subspace identification methods are derived by adapting the realization procedure to finite input output data