stochastic theory of continuous time state space identification this paper presents theory algorithms and validation results for system identification of continuous time state space models from finite input output sequences the algorithms developed are methods of subspace model identification and stochastic realization adapted to the continuous time context the resulting model can be decomposed into an input output model and a stochastic innovations model using the riccati equation we have designed a procedure to provide a reduced order stochastic model that is minimal with respect to system order as well as the number of stochastic inputs thereby avoiding several problems appearing in standard application of stochastic realization to the model validation problem

subspace based rational interpolation of analytic functions from phase data in this paper two simple subspace based identification algorithms to identify stable linear time invariant systems from corrupted phase samples of frequency response function are developed the first algorithm uses data sampled at nonuniformly spaced frequencies and is strongly consistent if corruptions are zero mean additive random variables with a known covariance function however this algorithm is biased when corruptions are multiplicative yet it exactly retrieves finite dimensional systems from noise free phase data using a finite amount of data the second algorithm uses phase data sampled at equidistantly spaced frequencies and also has the same interpolation and strong consistency properties if corruptions are zero mean additive random variables the latter property holds also for the multiplicative noise model provided that some noise statistics are known a priori promising results are obtained when the algorithms are applied to simulated data

a fast algorithm for subspace state space system identification via exploitation of the displacement structure two recent approaches van overschee de moor n4sid automatica 30 1 1994 75 verhaegen int j control 58 3 1993 555 in subspace identification problems require the computation of the r factor of the qr factorization of a block hankel matrix h which in general has a huge number of rows since the data are perturbed by noise the involved matrix h is in general full rank it is well known that from a theoretical point of view the r factor of the pr factorization of h is equivalent to the cholesky factor of the correlation matrix hth apart from a multiplication by a sign matrix in sima proceedings second niconet workshop paris versailles december 3 1999 p 75 a fast cholesky factorization of the correlation matrix exploiting the block hankel structure of h is described in this paper we consider a fast algorithm to compute the r factor based on the generalized schur algorithm the proposed algorithm allows to handle the rank deficient case c 2001 elsevier science b v all rights reserved

a p q oscillator realization of two parameter quantum algebras it is noted that the study of a quantum algebra sup q 2 with two independent deformation parameters p q leads to a p q oscillator realization for it the analysis is extended to the p q analogues of su 1 1 osp 2 1 and the centreless virasoro algebra the standard single parameter q deformations are obtained in the limit p q

a two parameter deformation of the jaynes cummings model path integral representation a two parameter p q deformation of the jaynes cummings model is obtained using a recently developed p q deformed oscillator in the rotating wave approximation rwa the dynamical symmetry of the model is the quantum superalgebra up q 1 mod 1 the partition function of the model is obtained as a path integral over generalized perelomov coherent states corresponding to the quantum algebra up q 1 mod 1 the complete spectrum of the model is extracted for both the case when the coupling constants are grassman valued and when they are c number valued it is noted that a relaxation of the rwa extends the dynamical symmetry to the quantum superalgebra ospp q 2 mod 2

application of a subspace model identification technique to identify lti systems operating in closed loop in this paper we reformulate the identification of linear time invariant lti systems operating in a closed loop with an lti compensator to an open loop multi input multi output mimo state space model identification problem followed by a model reduction step the open loop identification problem consists of the identification of a mimo state space model describing the deterministic transfer from freely excitable inputs to measurable constrained signals in the loop this problem is solved by the recently proposed moesp mimo output error state spac e model identification technique described in verhaegen and dewilde int j control 56 1187 1210 1992 the described state space approach does not constrain the delay structure of the controller plant combination it delivers an estimate of the controller and it allows an analytical reduction of the obtained state space model of the system or controller to half its original system order the open loop identification problem only requires the selection of the combined model and controller order the individual model and controller order selection is performed in the subsequent model reduction step furthermore specializing the presented approach to the identification of a siso system provided useful information regarding where to inject the signal in the loop in order to enhance numerical robustness the reliability of the outlined approach and the usefulness of the obtained insights are highlighted in two illustrative numerical examples

subspace based fault detection robust to changes in the noise covariances the detection of changes in the eigenstructure of a linear time invariant system by means of a subspace based residual function has been proposed previously while enjoying some success in its applicability in particular in the context of vibration monitoring the robustness of this framework against changes in the noise properties has not been properly addressed yet in this paper a new robust residual is proposed and the robustness of its statistics against changes in the noise covariances is shown the complete theory for hypothesis testing for fault detection is derived and a numerical illustration is provided

u 1 1 coherent states and a path integral for the jaynes cummings model the path integral over the coherent states of the u 1 1 superalgebra is constructed it is applied to the jaynes cummings model whose dynamical group is u 1 1

nuclear norm system identification with missing inputs and outputs we present a system identification method for problems with partially missing inputs and outputs the method is based on a subspace formulation and uses the nuclear norm heuristic for structured low rank matrix approximation with the missing input and output values as the optimization variables we also present a fast implementation of the alternating direction method of multipliers admm to solve regularized or non regularized nuclear norm optimization problems with hankel structure this makes it possible to solve quite large system identification problems experimental results show that the nuclear norm optimization approach to subspace identification is comparable to the standard subspace methods when no inputs and outputs are missing and that the performance degrades gracefully as the percentage of missing inputs and outputs increases

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

consistency and asymptotic normality of some subspace algorithms for systems without observed inputs asymptotic normality for a class of subspace algorithms which estimate the state in a first step is derived expressions for the asymptotic variance are given linear systems with unobserved white noise inputs are considered a class of subspace estimates for the system matrices obtained by estimating the state in the first step is analyzed the main result presented here states asymptotic normality of subspace estimates in addition a consistency result for the system matrix estimates is given an algorithm to compute the asymptotic variances of the estimates is presented in a final section the implications of the result are discussed

brief paper stochastic system transformation using generalized orthonormal basis functions with applications to continuous time system identification this paper studies the system transformation using generalized orthonormal basis functions that include the laguerre basis as a special case the transformation of the deterministic systems is studied in the literature which is called the hambo transform the aim of the paper is to develop a transformation theory for stochastic systems the paper establishes the equivalence of continuous and transformed discrete time stochastic systems in terms of solutions the method is applied to the continuous time system identification problem it is shown that using the transformed signals the po moesp subspace identification algorithm yields consistent estimates for system matrices an example is included to illustrate the efficacy of the proposed identification method and to make a comparison with the method using the laguerre filter

subspace identification of circulant systems this article concerns the identification of a class of large scale systems called circulant systems circulant systems have a special property that allows them to be decomposed into simpler subsystems through a state transformation this property has been used in literature for control design and here we show how it can be used for system identification the approach that is proposed here will both reduce the complexity of the problem as well as provide models which have a circulant structure that can be exploited for control design a novel identification algorithm for circulant systems based on subspace identification is presented the algorithm is then tested in simulation on an academic example of circulant system and on a realistic finite element model of a vibrating plate

variance analysis of identified linear miso models having spatially correlated inputs with application to parallel hammerstein models this contribution concerns variance analysis of linear multi input single output models when the inputs are temporally white but where different inputs may be correlated an expression is provided for the variance of a linearly parametrized estimate of the frequency response function from one block i e from one input to the output in particular this expression reveals that the variance increases in one block when the number of estimated parameters in another block is increased but levels off when the number of parameters in the other block reaches the number of parameters in the block in question it also quantifies exactly how correlation between inputs affects the resulting accuracy and a graphical representation is provided for this purpose the results are applicable to parallel miso hammerstein models when the nonlinearities are known and generalize an existing variance expression for this type of model

recursive subspace identification of linear and non linear wiener state space models the moesp class of identification algorithms are made recursive on the basis of various updating schemes for subspace tracking

the use of a bilinear transformation of the shift operator in subspace model identification proposes a mechanism which can improve the numerical robustness of a subspace based system identification method the pi scheme when the unknown system has poles situated close to z 1 a condition that often arises in applications where the sampling rate is too high the pi method is capable of solving a deterministic mimo identification problem in which the output can be corrupted by a very general perturbation including arbitrarily colored noise transients due to nonzero initial conditions and a deterministic zero bias by performing a bilinear transformation on the shift operator the authors are able to move the poles away from the point z 1 and a more robust identification results the implementation of this transformation gives rise to a series of anticausal filters applied to the input output data estimation accuracy is further improved by taking the unknown end conditions of the anticausal filters into account particularly when only short data records are available a numerical simulation highlights the improvements realized by the authors new algorithms

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

frequency domain subspace based identification of discrete time power spectra from nonuniformly spaced measurements in this paper we present a new subspace based algorithm for the identification of multi input multi output square discrete time linear time invariant systems from nonuniformly spaced power spectrum measurements the algorithm is strongly consistent and it is illustrated with one practical example that solves a stochastic road modeling problem

guaranteed stability with subspace methods abstract we show how stability of models can be guaranteed when using the class of identification algorithms which have become known as subspace methods in many of these methods the a matrix is obtained or can be obtained as the product of a shifted matrix with a pseudo inverse we show that whenever the shifted matrix is formed by introducing one block of zeros in the appropriate position then a stable model results the cost of this is some possibly large distortion of the results but in some applications that is outweighed by the advantage of guaranteed stability

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

subspace identification for non linear systems with measured input non linearities this paper uses subspace methods to identify a class of multi input multi output discrete time non linear time varying systems specifically we identify systems that are non linear in measured data and linear in unmeasured states numerical examples are presented to demonstrate the efficacy of the method

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

subspace based prediction of linear time varying stochastic systems in this paper a new subspace method for predicting time varying stochastic systems is proposed using the concept of angle between past and present subspaces spanned by the extended observability matrices the future signal subspace is predicted by rotating the present subspace in the geometrical sense and time varying system matrices are derived from the resultant signal subspace proposed algorithm is improved for fast varying systems furthermore recursive implementation of both algorithms is developed

maximum entropy vector kernels for mimo system identification abstract recent contributions have framed linear system identification as a nonparametric regularized inverse problem relying on l 2 type regularization which accounts for the stability and smoothness of the impulse response to be estimated these approaches have been shown to be competitive w r t classical parametric methods in this paper adopting maximum entropy arguments we derive a new l 2 penalty to do so we exploit the structure of the hankel matrix thus controlling at the same time complexity measured by the mcmillan degree stability and smoothness of the identified models as a special case we recover the nuclear norm penalty on the squared block hankel matrix in contrast with the previous literature on reweighted nuclear norm penalties our kernel is described by a small number of hyper parameters which are iteratively updated through marginal likelihood maximization constraining the structure of the kernel acts as a hyper regularizer which helps controlling the effective degrees of freedom of our estimator to optimize the marginal likelihood we adapt a scaled gradient projection sgp algorithm which is proved to be significantly computationally cheaper than other first and second order off the shelf optimization methods the paper also contains an extensive comparison with many state of the art methods on several monte carlo studies which confirms the effectiveness of our procedure

subspace model identification part 2 analysis of the elementary output error state space model identification algorithm the elementary moesp algorithm presented in the first part of this series of papers is analysed in this paper this is done in three different ways first we study the asymptotic properties of the estimated state space model when only considering zero mean white noise perturbations on the output sequence it is shown that in this case the moespl implementation yields asymptotically unbiased estimates an important constraint to this result is that the underlying system must have a finite impulse response and subsequently the size of the hankel matrices constructed from the input and output data at the beginning of the computations depends on the number of non zero markov parameters this analysis however leads to a second implementation of the elementary moesp scheme namely moesp2 the latter implementation has the same asymptotic properties without the finite impulse response constraint secondly we compare the moesp2 algorithm with a classical state space model identification scheme the latter

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

order estimation for linear time invariant systems using frequency domain identification methods a two step coarse fine order estimation technique is proposed to determine the order of the numerator and the denominator polynomials of rational transfer function models for single input single output siso linear time invariant systems the coarse order estimation is based on rank detection by verification of the stochastic significance of the singular values of a linearized problem the fine order estimation is based on a statistical analysis of the maximum likelihood cost function the method is tested on measurements of low 4 zeros 6 poles and high 58 poles 58 zeros order systems

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

a unifying theorem for three subspace system identification algorithms the aim of this paper is to indicate and explore the similarities between three different subspace algorithms for the identification of combined deterministic stochastic systems the similarities between these algorithms have been obscured due to different notations and backgrounds it is shown that all three algorithms are special cases of one unifying theorem the comparison also reveals that the three algorithms use exactly the same subspace to determine the order and the extended observability matrix but that the weighting matrix used to calculate a basis for the column space of the observability matrix is different in the three cases

subspace identification for continuous time stochastic systems via distribution based approach this paper presents a novel approach for system identification of continuous time stochastic state space models from random input output continuous data the approach is based on the introduction of random distribution theory in describing higher time derivatives of stochastic processes and the input output algebraic relationship is derived which is treated in the time domain the efficacy of the approach is examined by comparing with other approaches employing the filters

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

criteria for informative experiments with subspace identification informative experiments are identification experiments which contain sufficient information for an identification algorithm to discriminate between different models in an intended model set in this paper a particular set of identification algorithms namely subspace based identification is considered criteria for experiments to be informative with these methods in the deterministic setup and the combined deterministic stochastic setup are presented it is pointed out that if these criteria are not satisfied interesting phenomena in which perfect cancellations of the deterministic components and the stochastic components occur in a subspace projection may occur it is further shown that such cancellations can indeed be avoided under mild conditions

a fourier spectral element algorithm for thermal convection in rotating axisymmetric containers we present a fourier spectral element approach for modeling thermal convection in a rotating axisymmetric container following the theory detailed in bernardi et al c bernardi m dauge y maday spectral methods for axisymmetric domains gauthier villars paris 1999 a fourier expansion of the field variables is performed in the periodic direction and the resulting collection of meridional problems is discretized by means of a parallel spectral element method a gauss lobatto jacobi 0 1 quadrature which incorporates the cylindrical radius in its weight is introduced to avoid a potential degeneracy of the discrete set of equations due to those nodes located on the axis of symmetry a second order timestepping scheme is presented which treats the coriolis and viscous forces implicitly numerical comparisons with analytical and published numerical solutions in spherical and cylindrical geometries are presented which highlight the accuracy of the model and demonstrate its spectral convergence properties

comment on the q deformed fermionic oscillator the q deformed fermionic oscillator is identified with the usual fermionic oscillator by virtue of a simple transformation so the realization of quantum algebra suq 2 can also be constructed from the usual fermionic oscillator

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

brief paper an insight into instrumental variable frequency domain subspace identification in this paper we study instrumental variable subspace identification of multi input multi output linear time invariant discrete time systems from non uniformly spaced frequency response measurements a particular algorithm which does not require noise covariance function to be known a priori is shown to be strongly consistent provided that disturbances have uniformly bounded second order moments and the frequencies satisfy a certain regularity condition interpolation properties of this algorithm and a related one are also studied a numerical example illustrating the properties of the studied algorithms is presented

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

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

a subspace algorithm for the identification of discrete time frequency domain power spectra in this paper we present a new subspace algorithm for the identification of multi input multi output linear discrete time systems from measured power spectrum data we show how the state space system matrices can be determined by taking the inverse discrete fourier transform of the given data and applying the result to a new realization algorithm special attention is paid to ensure the positive realness of the identified power spectrum the computational speed is improved by applying a lanczos algorithm the algorithm is illustrated with two practical examples

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

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

on line structured subspace identification with application to switched linear systems this article is concerned with the identification of switched linear multiple inputs multiple outputs state space systems in a recursive way first a structured subspace identification scheme for linear systems is presented which turns out to have many attractive features more precisely it does not require any singular value decomposition but is derived using orthogonal projection techniques it allows a computationally appealing implementation and it is closely related to input output models identification second it is shown that this method can be implemented on line to track both the range space of the extended observability matrix and its dimension and thereby the system matrices third by making use of an on line switching times detection strategy this method is applied to blindly identify switched systems and to label the obtained submodels simulation results on noisy data illustrate the abilities and the benefits of the proposed approach

subspace based identification of infinite dimensional multivariable systems from frequency response data a new identification algorithm which identifies low complexity models of infinite dimensional systems from equidistant frequency response data is presented the new algorithm is a combination of the fourier transform technique with the recent subspace techniques given noisefree data finite dimensional systems are exactly retrieved by the algorithm when noise is present it is shown that identified models strongly converge to the balanced truncation of the identified system if the measurement errors are covariance bounded several conditions are derived on consistency illustrating the trade offs in the selection of certain parameters of the algorithm two examples are presented which clearly illustrate the good performance of the algorithm

subspace based multivariable system identification from frequency response data two noniterative subspace based algorithms which identify linear time invariant mimo multi input multioutput systems from frequency response data are presented the algorithms are related to the recent time domain subspace identification techniques the first algorithm uses equidistantly in frequency spaced data and is strongly consistent under weak noise assumptions the second algorithm uses arbitrary frequency spacing and is strongly consistent under more restrictive noise assumptions promising results are obtained when the algorithms are applied to real frequency data originating from a large flexible structure

subspace identification with guaranteed stability using constrained optimization in system identification the true system is often known to be stable however due to finite sample constraints modeling errors plant disturbances and measurement noise the identified model may be unstable we present a constrained optimization method to ensure asymptotic stability of the identified model in the context of subspace identification methods in subspace identification we first obtain an estimate of the state sequence or extended observability matrix and then solve a least squares optimization problem to estimate the system parameters to ensure asymptotic stability of the identified model we write the least squares optimization problem as a convex linear programming problem with mixed equality quadratic and positive semidefinite constraints suitable for existing convex optimization codes such as sedumi we present examples to illustrate the method and compare to existing approaches

the singular value decomposition and long and short spaces of noisy matrices using geometrical algebraic and statistical arguments it is clarified why and when the singular value decomposition is successful in so called subspace methods first the concepts of long and short spaces are introduced and a fundamental asymmetry in the consistency properties of the estimates is discussed the model which is associated with the short space can be estimated consistently but the estimates of the original data which follow from the long space are always inconsistent an expression is found for the asymptotic bias in terms of canonical angles which can be estimated from the data this allows all equivalent reconstructions of the original signals to be described as a matrix ball the center of which is the minimum variance estimate remarkably the canonical angles also appear in the optimal weighting that is used in weighted subspace fitting approaches the results are illustrated with a numerical simulation a number of examples are discussed

on and off line identification of linear state space models abstract a geometrically inspired matrix algorithm is derived for the identification of statespace models for multivariable linear time invariant systems using possibly noisy input output i o measurements only as opposed to other mostly stochastic identification schemes no variance covariance information whatever is involved and only a limited number of i o data are required for the determination of the system matrices hence the algorithm can be best described and understood in the matrix formalism and consists of the following two steps first a state vector sequence is realized as the intersection of the row spaces of two block hankel matrices constructed with i o data then the system matrices are obtained at once from the least squares solution of a set of linear equations when dealing with noisy data this algorithm draws its excellent performance from repeated use of the numerically stable and accurate singular value decomposition also the algorithm is easily applied to slowly time

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

unitary representations of non compact supergroups we give a general theory for the construction of oscillator like unitary irreducible representations uirs of non compact supergroups in a super fock space this construction applies to all non compact supergroupsg whose coset spaceg k with respect to their maximal compact subsupergroupk is hermitean supersymmetric we illustrate our method with the example of su m p n q by giving its oscillator like uirs in a particle state basis as well as supercoherent state basis the same class of uirs can also be realized over the super hilbert spaces of holomorphic functions of az variable labelling the coherent states

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

hierarchical gradient based identification of multivariable discrete time systems in this paper we use a hierarchical identification principle to study identification problems for multivariable discrete time systems we propose a hierarchical gradient iterative algorithm and a hierarchical stochastic gradient algorithm and prove that the parameter estimation errors given by the algorithms converge to zero for any initial values under persistent excitation the proposed algorithms can be applied to identification of systems involving non stationary signals and have significant computational advantage over existing identification algorithms finally we test the proposed algorithms by simulation and show their effectiveness

generalized q fermion oscillators and q coherent states the algebra of q fermion operators developed earlier by two of the present authors is re examined it is shown that these operators represent particles that are distinct from usual spacetime fermions except in the limit q 1 it is shown that it is possible to introduce generalized q oscillators defined for infinity q or 1 in the range infinity q 0 these coincide with the q boson operators and for 0 q or 1 they coincide with q fermions the ordinary bosons and fermions may be identified with the limits q 1 and 1 respectively generalized q fermion coherent states are constructed by utilizing a nonlinear shift automorphism of the algebra of q fermion operators these are compared with the coherent states defined as eigenstates of annihilation operator matrix elements of the shift operator in the fock space basis are evaluated

frequency domain subspace system identification using non parametric noise models in the general case of non uniformly spaced frequency domain data and or arbitrarily coloured disturbing noise the frequency domain subspace identification algorithms described in mckelvey akcay and ljung ieee trans automatic control 41 7 1996 960 and van overschee and de moor signal processing 52 2 1996 179 are consistent only if the covariance matrix of the disturbing noise is known this paper studies the asymptotic properties strong convergence convergence rate asymptotic normality strong consistency and loss in efficiency of these algorithms when the true noise covariance matrix is replaced by the sample noise covariance matrix obtained from a small number of independent repeated experiments as an additional result the strong convergence in case of model errors the convergence rate and the asymptotic normality of the subspace algorithms with known noise covariance matrix follows

frequency domain subspace identification using nuclear norm minimization and hankel matrix realizations subspace identification techniques have gained widespread acceptance as a method of obtaining a low order model from data these are based on using the singular value decomposition as a means of estimating the underlying system order and extracting a basis for the extended observability space in the presence of noise rank determination becomes difficult and the low rank estimates lose the structure required for exact realizability furthermore the noise corrupts the singular values in a manner that is inconsistent with physical noise processes these problems are addressed by an optimization based approach using a nuclear norm minimization objective by using hankel matrices as the underlying data structure exact realizability of the low rank system models is maintained noise in the data enters the formulation linearly allowing for the inclusion of more realistic noise weightings a cumulative spectral weight is presented and shown to be useful in estimating models from data corrupted via noise a numerical example illustrates the characteristics of the problem

qsvd approach to on and off line state space identification moonen et al 1989 a presented an svd based identification scheme for computing state space models for multivariable linear time invariant systems in the present paper this identification procedure is reformulated making use of the quotient singular value decomposition qsvd here the input output error co variance matrix can be taken into account explicitly thus extending the applicability of the identification scheme to the case were the input and output data are corrupted by coloured noise it turns out that in practice due to the use of various pre filtering techniques anti aliasing etc this latter case is most often encountered the extended identification scheme explicitly compensates for the filter characteristics and the consistency of the identification results follows from the consistency results for the qsvd the usefulness of this generalization is demonstrated the development is largely inspired by recent progress in total least squares solution techniques van huffel 1989 for

brief recursive 4sid algorithms using gradient type subspace tracking sometimes we obtain some prior information about a system to be identified e g the order model structure etc in this paper we consider the case where the order of a mimo system to be identified is a priori known recursive subspace state space system identification algorithms presented here are based on the gradient type subspace tracking method used in the array signal processing the algorithms enable us to estimate directly the subspace spanned by the column vectors of the extended observability matrix of the system to be identified without performing the singular value decomposition also a new convergence proof of the gradient type subspace tracking is given in this paper under the condition of a step size between 0 and 1 we prove the convergence property of the recursive equation of the gradient type subspace tracking a numerical example illustrates that our algorithm is more robust with respect to the choice of the initial values than the corresponding past one

brief paper an approach to closed loop subspace identification by orthogonal decomposition a method of identifying closed loop systems is developed by using the orthogonal decomposition ort method the idea is to project the input and output data onto the space of exogenous inputs by using the lq decomposition to obtain their deterministic components the ort based method is then applied to deterministic components like the direct approach in order to derive state space models of the plant we also show that the present method is a subspace version of the two stage method for transfer function estimation from closed loop data some simulation results are included to show the applicability of the present method

brief consistency analysis of subspace identification methods based on a linear regression approach in the literature results can be found which claim consistency for the subspace method under certain quite weak assumptions unfortunately a new result gives a counter example showing inconsistency under these assumptions and then gives new more strict sufficient assumptions which however does not include important model structures such as e g box jenkins based on a simple least squares approach this paper shows the possible inconsistency under the weak assumptions and develops only slightly stricter assumptions sufficient for consistency and which includes any model structure

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

subspace based methods for the identification of linear time invariant systems subspace based methods for system identification have attracted much attention during the past few years this interest is due to the ability of providing accurate state space models for multivariable linear systems directly from input output data the methods have their origin in classical state space realization theory as developed in the 1960s the main computational tools are the qr and the singular value decompositions here an overview of existing subspace based techniques for system identification is given the methods are grouped into the classes of realization based and direct techniques similarities between different algorithms are pointed out and their applicability is commented upon we also discuss some recent ideas for improving and extending the methods a simulation example is included for comparing different algorithms the subspace based approach is found to perform competitive with respect to prediction error methods provided the system is properly excited

identification of linear systems via spectral analysis given time domain data consistency reduced order approximation and performance analysis estimation of a parametric input output i o infinite impulse response transfer function given time domain i o data is considered some of the desirable properties of any approach to this problem are unimodality of the performance surface consistent identification in the sufficient order case and stability of the fitted model under undermodeling we first consider a frequency domain solution to the least squares equation error identification problem using the power spectrum and the cross spectrum of the i o data to estimate the i o parametric transfer function the proposed approach is shown to yield a unimodal performance surface consistent identification in colored noise and sufficient order case and stable fitted models under undermodeling for arbitrary stationary inputs so long as they are persistently exciting of sufficiently high order asymptotic performance analysis is carried out for both sufficient order and reduced order cases these asymptotic results are then used to derive statistics on the corresponding estimated transfer function we also investigate an iterative pseudomaximum likelihood approach and analyze its performance under sufficient order modeling finally computer simulation examples are provided to illustrate the two approaches

a class of subspace model identification algorithms to identify periodically and arbitrarily time varying systems abstract subspace model identification algorithms that allow the identification of a linear time varying ltv state space model from an ensemble set of input output measurements are presented in this paper each pair of input and output sequences in this ensemble is recorded when the underlying system to be identified undergoes the same time varying behavior the algorithms operate directly on the available ensemble of input output data and are a generalization of the recently proposed multivariable output error state space moesp class of algorithms to this ensemble type of identification problems a special case is considered in this paper where the repetition of this time varying behavior is intrinsic namely in periodically time varying systems an example of identifying a multirate sampled data system from a recorded input and output sequence demonstrates some of the capabilities of the presented subspace model identification algorithms

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

nonstationary consistency of subspace methods in this paper we study ldquononstationary consistencyrdquo of subspace methods for eigenstructure identification i e the ability of subspace algorithms to converge to the true eigenstructure despite nonstationarities in the excitation and measurement noises note that such nonstationarities may result in having time varying zeros for the underlying system so the problem is nontrivial in particular likelihood and prediction error related methods do not ensure consistency under such situation because estimation of poles and estimation of zeros are tightly coupled we show in turn that subspace methods ensure such consistency our study carefully separates statistical from nonstatistical arguments therefore enlightening the role of statistical assumptions in this story

high performance numerical algorithms and software for subspace based linear multivariable system identification basic algorithmic and numerical issues involved in subspace based linear multivariable discrete time system identification are described a new identification toolbox slident has been developed and incorporated in the freely available subroutine library in control theory slicot reliability efficiency and ability to solve industrial identification problems received a special consideration two algorithmic subspace based approaches moesp and n4sid and their combination and both standard and fast techniques for data compression are provided structure exploiting algorithms and dedicated linear algebra tools enhance the computational efficiency and reliability extensive comparisons with the available computational tools based on subspace techniques show the better efficiency of the slident toolbox at comparable numerical accuracy and its capabilities to solve identification problems with many thousands of samples and hundreds of parameters

on gradient based search for multivariable system estimates this paper addresses the design of gradient based search algorithms for multivariable system estimation in particular the paper here considers so called ldquofull parametrizationrdquo approaches and establishes that the recently developed ldquodata driven local coordinaterdquo methods can be seen as a special case within a broader class of techniques that are designed to deal with rank deficient jacobians this informs the design of a new algorithm that via a strategy of dynamic jacobian rank determination is illustrated to offer enhanced performance

computing real roots of a polynomial in chebyshev series form through subdivision with linear testing and cubic solves abstract an arbitrary polynomial of degree n f n x can always be represented as a truncated chebyshev polynomial series chebyshev form this representation is much better conditioned than the usual power form of a polynomial we describe a new method for finding the real roots of f n x in chebyshev form the canonical interval x 1 1 is subdivided into n s subintervals each is tested for zeros using the error bound for linear interpolation on zero possible intervals f n is approximated by a cubic polynomial whose roots are then found by the usual sixteenth century formulas