on the existence of a solution to a spectral estimation problem a la byrnes georgiou lindquist a parametric spectral estimation problem in the style of byrnes georgiou and lindquist was posed in 1 but the existence of a solution was only proved in a special case based on their results we show that a solution indeed exists given an arbitrary matrix valued prior density the main tool in our proof is the topological degree theory

schur recursions error formulas and convergence of rational estimators for stationary stochastic sequences an exact and approximate realization theory for estimation and model filters of second order stationary stochastic sequences is presented the properties of j lossless matrices as a unifying framework are used necessary and sufficient conditions for the exact realization of an estimation filter and a model filter as a submatrix of a j lossless system are deduced an extension of the so called schur algorithm yields an approximate j lossless realization based on partial past information about the process the geometric properties of such partial realizations and their convergence are studied finally connections with the nevanlinna pick problem are made and how the techniques presented constitute a generalization of many aspects of the levinson szego theory of partial realizations is shown as a consequence generalized recursive formulas for reproducing kernels and christoffel darboux formulas are obtained in this paper the scalar case is considered the matrix case will be considered in a separate publication

passivity preserving model reduction by analytic interpolation abstract antoulas and sorensen have recently proposed a passivity preserving model reduction method of linear systems based on krylov projections the idea is to approximate a positive real rational transfer function with one of lower degree the method is based on an observation by antoulas in the single input single output case that if the approximant is preserving a subset of the spectral zeros and takes the same values as the original transfer function in the mirror points of the preserved spectral zeros then the approximant is also positive real however this turns out to be a special solution in the theory of analytic interpolation with degree constraint developed by byrnes georgiou and lindquist namely the maximum entropy central solution by tuning the interpolation points and the spectral zeros as prescribed by this theory one is able to obtain considerably better reduced order models we also show that in the multi input multi output case sorensen s algorithm actually amounts to tangential nevanlinna pick interpolation

optimization based computation of analytic interpolants of bounded complexity this paper provides a unifying algorithm for computing any analytic interpolant of bounded complexity such computation can be performed by solving an optimization problem due to a theorem by georgiou and lindquist this optimization problem is numerically solvable by a continuation method the proposed numerical algorithm is useful among other cases for designing a low degree controller for a benchmark problem in robust control the algorithm unifies previously developed algorithms for the caratheodory extension and the nevanlinna pick interpolation to one for more general interpolation problems

on the partial stochastic realization problem we describe a complete parameterization of the solutions to the partial stochastic realization problem in terms of a nonstandard matrix riccati equation our analysis of this covariance extension equation cee is based on a complete parameterization of all strictly positive real solutions to the rational covariance extension problem answering a conjecture due to georgiou 1987 in the affirmative we also compute the dimension of partial stochastic realizations in terms of the rank of the unique positive semidefinite solution to the cee yielding some insights into the structure of solutions to the minimal partial stochastic realization problem by combining this parameterization with some of the classical approaches in partial realization theory we are able to derive new existence and robustness results concerning the degrees of minimal stochastic partial realizations as a corollary to these results we note that in sharp contrast with the deterministic case there is no generic value of the degree of a minimal stochastic realization of partial covariance sequences of fixed length

equivalence between stability of partial realizations and feedback stabilization applications to reduced order stabilization abstract given a transfer function b z a z a classical feedback problem is to find a dynamic feedback d z c z such that the closed loop characteristic polynomial ca db z becomes stable antoulas has proved that this is equivalent to finding a stable partial realization of a certain sequence of real numbers the converse is also true every partial realization problem can be transformed into a feedback problem this paper gives a new formulation of the equivalence between feedback and realization of scalar systems as an application algebraic conditions for existence of reduced order stabilizing feedback are derived

analytic interpolation with a degree constraint for matrix valued functions we consider a nehari problem for matrix valued positive real functions and characterize the class of generically minimal degree solutions analytic interpolation problems such as the one studied herein for positive real functions arise in time series modeling and system identification the degree of positive real interpolants relates to the dimension of models and to the degree of matricial power spectra of vector valued time series the main result of the paper generalizes earlier results in scalar analytic interpolation with a degree constraint where the class of generically minimal degree solutions is characterized by an arbitrary choice of spectral zeros naturally in the current matricial setting there is freedom in assigning the jordan structure of the spectral zeros of the power spectrum i e the spectral zeros as well as their respective invariant subspaces the characterization utilizes rosenbrock s theorem on assignability of dynamics via linear state feedback

a complete parameterization of all positive rational extensions of a covariance sequence in this paper we formalize the observation that filtering and interpolation induce complementary or dual decompositions of the space of positive real rational functions of degree less than or equal to n from this basic result about the geometry of the space of positive real functions we are able to deduce two complementary sets of conclusions about positive rational extensions of a given partial covariance sequence on the one hand by viewing a certain fast filtering algorithm as a nonlinear dynamical system defined on this space we are able to develop estimates on the asymptotic behavior of the schur parameters 1918 of positive rational extensions on the other hand we are also able to provide a characterization of all positive rational extensions of a given partial covariance sequence indeed motivated by its application to signal processing speech processing and stochastic realization theory this characterization is in terms of a complete parameterization using familiar objects from systems theory and proves a conjecture made by georgiou 1983 1987 our basic result however also enables us to analyze the robustness of this parameterization with respect to variations in the problem data the methodology employed is a combination of complex analysis geometry linear systems and nonlinear dynamics

on the solution of the minimal rational interpolation problem it is shown that the reachability and the observability indices of appropriately defined pairs of matrices provide the key for solving the minimal rational bitangen tial interpolation problem

on the partial realization problem abstract in this paper we take a unified approach to the partial realization problem in which we seek to incorporate ideas from numerical linear algebra most of which were originally developed in other contexts we approach the partial realization problem from several different angles and explore the connections to such topics as factoriza tion of hankel matrices block tridiagonalization generalizations of the lanczos process for biorthogonalization the euclidean algorithm and the principal part con tinued fractions of arne magnus the pade table and the berlekamp masseyalgo rithm in this way we are able to clarify some previous results by rissanen kalman and others and place them in a broader context this leads to several results and concepts which we think are new our analysis is restricted to the scalar case but some definitions and formulations have been rigged to facilitate an extension to the matrix case

the euclid algorithm and the fast computation of cross covariance and autocovariance sequences a simple linear procedure is given to compute the cross covariance sequence associated with the outputs of two rational digital transfer functions driven by the same white noise sequence such a computation often appears in the study of digital filters in wiener filtering in noise variance estimation in the study of low order approximations and in the study of multichannel systems a fast algorithm based on the euclid algorithm is introduced to solve the linear system of equations involved in the computation and a detailed analysis of the matrix is given the special case of the autocovariance computation is reviewed and the same study is performed alternate polynomial presentations are given and are shown to involve the same matrices and similar fast algorithms

on the nevanlinna pick interpolation problem analysis of the mcmillan degree of the solutions in this paper we investigate some aspect of the nevanlinna pick and schur interpolation problem formulated for schur functions considered on the right half plane of c we consider the well established parametrization of the solution q t s s 12 22 1 s 11 21 see e g j a ball i gohberg l rodman realization and interpolation of rational matrix functions operator theory adv appl 33 1988 1 72 h dym j contractive matrix functions reproducing kernel hilbert spaces and interpolation cbms regional conference series in mathematics no 71 amer math soc providence ri 1989 where the j inner function is completely determined by the interpolation data and s is an arbitrary schur function we then compare the relations between the realizations of q and s induced by we show in particular that s generates a solution with a low mcmillan degree if and only if s satisfies some interpolation conditions formulated on the left half plane of c this analysis can be considered to be partially complementary to the results of a lindquist c byrnes et al on caratheodory functions c byrnes a lindquist on the partial stochastic realization problem ieee trans automatic control 42 1997 1049 1070 c byrnes a lindquist s v gusev a s mateev a complete parametrization of all positive rational extension of a covariance sequence ieee trans automatic control 40 1995 1841 1857 c byrnes a lindquist s v gusev a convex optimization approach to the rational covariance extension problem tria mat 1997

geometry of the kimura georgiou parametrization of modelling filters we discuss questions concerning the geometry of the kimura georgiou para metrizalion of the set n of degree n positive real transfer functions with the first n coefficients in the laurent expansion about infinity prescribed for example one interesting question which has been raised is whether this set is star shaped about the maximum entropy solution this of course would be implied by convexity and would imply that plus n is diffeomorphic to euclidean n space all three of these geometric properties would be of interest for example when using geometric or optimization techniques to construct an n dimensional modelling filter with variations about the maximum entropy filter our first main result which also lends support to conjectures concerning convexity and star shapedness is that plus n is in fact diffeomorphic to euclidean n space our proof makes use of certain results from differential topology there are of course several intimate relations between positive reality and stabi

the nevanlinna pick problem on the closed unit disk minimal norm rational solutions of low degree for the nevanlinna pick interpolation problem with n interpolation conditions interior and boundary we construct a family of rational solutions of degree at most n 1 we also establish necessary and sufficient conditions for the existence and the uniqueness of a solution with the minimally possible h norm and construct a family of minimal norm rational solutions of degree at most n 1 in the indeterminate case finally we supplement a result of ruscheweyh and jones showing that in case the interpolation nodes and the target values are all unimodular any rational solution of degree at most n 1 is necessarily a finite blaschke product

a maximum entropy principle for contractive interpolants abstract in this paper a maximum entropy principle is established for a class of continuous contractive m n matrix valued functions g with preassigned fourier coefficients gj j for j 0 1 the function g in this class which achieves the maximum entropy is characterized in a number of different ways and its entropy is computed in terms of a hankel operator based on the prescribed matrices 0 1 a pair of seemingly new szego type formulas are obtained enroute and some applications to nevanlinna pick interpolation are indicated

model reduction by moment matching for linear and nonlinear systems the model reduction problem for single input single output linear and nonlinear systems is addressed using the notion of moment a re visitation of the linear theory allows to obtain novel results for linear systems and to develop a nonlinear enhancement of the notion of moment this in turn is used to pose and solve the model reduction problem by moment matching for nonlinear systems to develop a notion of frequency response for nonlinear systems and to solve model reduction problems in the presence of constraints on the reduced model connections between the proposed results projection methods the covariance extension problem and interpolation theory are presented finally the theory is illustrated by means of simple worked out examples and case studies

rspop rough set based pseudo outer product fuzzy rule identification algorithm system modeling with neuro fuzzy systems involves two contradictory requirements interpretability verses accuracy the pseudo outer product pop rule identification algorithm used in the family of pseudo outer product based fuzzy neural networks popfnn suffered from an exponential increase in the number of identified fuzzy rules and computational complexity arising from high dimensional data this decreases the interpretability of the popfnn in linguistic fuzzy modeling this article proposes a novel rough set based pseudo outer product rspop algorithm that integrates the sound concept of knowledge reduction from rough set theory with the pop algorithm the proposed algorithm not only performs feature selection through the reduction of attributes but also extends the reduction to rules without redundant attributes as many possible reducts exist in a given rule set an objective measure is developed for popfnn to correctly identify the reducts that improve the inferred consequence experimental results are presented using published data sets and real world application involving highway traffic flow prediction to evaluate the effectiveness of using the proposed algorithm to identify fuzzy rules in the popfnn using compositional rule of inference and singleton fuzzifier popfnn cri s architecture results showed that the proposed rough set based pseudo outer product algorithm reduces computational complexity improves the interpretability of neuro fuzzy systems by identifying significantly fewer fuzzy rules and improves the accuracy of the popfnn

on the uniqueness result of theorem 6 in relative entropy and the multivariable multidimensional moment problem matrix valued covariance extension and multivariate spectral estimation are formulated as generalized moment problems in the three approach and its extensions under this context we discuss theorem 6 in the title paper concerning the bijectivity of a moment map defined over a parametric family of spectral densities in particular we provide a counterexample in which the moment map under consideration is shown to have a critical point namely a point at which the jacobian loses rank then with standard techniques in bifurcation theory we conclude further that the computed critical point is a bifurcation point which means that the moment map is not injective

the inverse problem of analytic interpolation with degree constraint and weight selection for control synthesis the minimizers of certain weighted entropy functionals are the solutions to an analytic interpolation problem with a degree constraint and all solutions to this interpolation problem arise in this way by a suitable choice of weights selecting appropriate weights is pertinent to feedback control synthesis where interpolants represent closed loop transfer functions in this paper we consider the correspondence between weights and interpolants in order to systematize feedback control synthesis with a constraint on the degree there are two basic issues that we address we first characterize admissible shapes of minimizers by studying the corresponding inverse problem and then we develop effective ways of shaping minimizers via suitable choices of weights this leads to a new procedure for feedback control synthesis

likelihood analysis of power spectra and generalized moment problems we develop an approach to the spectral estimation that has been advocated by a ferrante et al time and spectral domain relative entropy a new approach to multivariate spectral estimation ieee trans autom control vol 57 no 10 pp 2561 2575 oct 2012 and in the context of the scalar valued covariance extension problem by p enqvist and j karlsson minimal itakura saito distance and covariance interpolation in proc 47th ieee conf decision control 2008 pp 137 142 the aim is to determine the power spectrum that is consistent with given moments and minimizes the relative entropy between the probability law of the underlying gaussian stochastic process to that of a prior the approach is analogous to the framework of earlier work by byrnes georgiou and lindquist and can also be viewed as a generalization of the classical work by burg and jaynes on the maximum entropy method in this paper we present a new fast algorithm in the general case i e for general gaussian priors and show that for priors with a specific structure the solution can be given in closed form

the circulant rational covariance extension problem the complete solution the rational covariance extension problem to determine a rational spectral density given a finite number of covariance lags can be seen as a matrix completion problem to construct an infinite dimensional positive definite toeplitz matrix the northwest corner of which is given the circulant rational covariance extension problem considered in this paper is a modification of this problem to partial stochastic realization of periodic stationary process which are better represented on the discrete unit circle rather than on the discrete real line the corresponding matrix completion problem then amounts to completing a finite dimensional toeplitz matrix that is circulant another important motivation for this problem is that it provides a natural approximation involving only computations based on the fast fourier transform for the ordinary rational covariance extension problem potentially leading to an efficient numerical procedure for the latter the circulant rational covariance extension problem is an inverse problem with infinitely many solutions in general each corresponding to a bilateral arma representation of the underlying periodic process in this paper we present a complete smooth parameterization of all solutions and convex optimization procedures for determining them a procedure to determine which solution that best matches additional data in the form of logarithmic moments is also presented

the stability and instability of partial realizations a finite or infinite sequence of real numbers is said to be stable if it admits minimal realization by a stable linear system it is shown that the preservation of stability as such a sequence is truncated or extended is not a generic property even among stable sequences this is of interest for identification of linear systems from partial data and for partial realization of random systems in which cases it constitutes a negative result certain finite sequences have infinitely many minimal realizations each having different stability properties in this case a graphical criterion in the spirit of the nyquist criterion is derived to exploit this lack of uniqueness in order to determine whether one can achieve a stable pole placement by a judicious choice of partial realization

ar and arma identification algorithms of levinson type an innovations approach fast recursive in time identification procedures for both ar and arma processes e g chandrasekhar square root algorithms have been available for a few years these algorithms realize the desired transfer functions in the classical polynomial or rational form on the other hand the synthesis of polynomial and rational transfer functions in lattice and ladder form has fostered great interest in network theory by virtue of its pleasant properties this type of synthesis is strongly related to the theory of orthogonal polynomials on the unit circle an identification procedure with the realization of the desired whitening filter in lattice form was available for ar processes we give here a simple approach for obtaining such algorithms investigating furthermore the connections between the so obtained algorithms and the classical ones least squares procedure in the same way we obtain identification procedures with realization of the desired filter in lattice and ladder form for arma processes together with the connection with the classical extended least squares procedure the method is based upon a fairly general levinson orthogonalization lemma in hilbert space involving innovation techniques we extend the method to various other estimation problems the algorithms we obtain are fast even somewhat faster than the previous fast ones and seem to be well conditioned

a covariance extension approach to identification of time series in this paper we consider a three step procedure for identification of time series based on covariance extension and model reduction and we present a complete analysis of its statistical convergence properties a partial covariance sequence is estimated from statistical data then a high order maximum entropy model is determined which is finally approximated by a lower order model by stochastically balanced model reduction such procedures have been studied before in various combinations but an overall convergence analysis comprising all three steps has been lacking supposing the data is generated from a true finite dimensional system which is minimum phase it is shown that the transfer function of the estimated system tends in h to the true transfer function as the data length tends to infinity if the covariance extension and the model reduction is done properly the proposed identification procedure and some variations of it are evaluated by simulations

sensitivity shaping with degree constraint by nonlinear least squares optimization this paper presents a new approach to shaping of the frequency response of the sensitivity function in this approach a desired frequency response is assumed to be specified at a finite number of frequency points a sensitivity shaping problem is formulated as an approximation problem to the desired frequency response with a function in a class of sensitivity functions with a degree bound the sensitivity shaping problem is reduced to a finite dimensional constrained nonlinear least squares optimization problem to solve the optimization problem numerically standard algorithms for an unconstrained version of nonlinear least squares problems are modified to incorporate the constraint numerical examples illustrate how these design parameters are tuned in an intuitive manner as well as how the design proceeds in actual control problems

a new approach to spectral estimation a tunable high resolution spectral estimator traditional maximum entropy spectral estimation determines a power spectrum from covariance estimates here we present a new approach to spectral estimation which is based on the use of filter banks as a means of obtaining spectral interpolation data such data replaces standard covariance estimates a computational procedure for obtaining suitable pole zero arma models from such data is presented the choice of the zeros ma part of the model is completely arbitrary by suitable choices of filter bank poles and spectral zeros the estimator can be tuned to exhibit high resolution in targeted regions of the spectrum

on the well posedness of a parametric spectral estimation problem and its numerical solution this paper concerns a spectral estimation problem in which we want to find a spectral density function that is consistent with estimated second order statistics it is an inverse problem admitting multiple solutions and selection of a solution can be based on prior functions we show that the problem is well posed when formulated in a parametric fashion and that the solution parameter depends continuously on the prior function in this way we are able to obtain a smooth parametrization of admissible spectral densities based on this result the problem is reparametrized via a bijective change of variables out of a numerical consideration and then a continuation method is used to compute the unique solution parameter numerical aspects such as convergence of the proposed algorithm and certain computational procedures are addressed a simple example is provided to show the effectiveness of the algorithm

model reduction of neutral linear and nonlinear time invariant time delay systems with discrete and distributed delays the problem of model reduction by moment matching for linear and nonlinear differential time delay systems is studied the class of models considered includes neutral differential time delay systems with discrete delays and distributed delays the description of moment is revisited by means of a sylvester like equation for linear time delay systems and by means of the center manifold theory for nonlinear time delay systems in addition the moments at infinity are characterized for both linear and nonlinear time delay systems parameterized families of models achieving moment matching are given the parameters can be exploited to derive delay free reduced order models or time delay reduced order models with additional properties e g interpolation at an arbitrary large number of points finally the problem of obtaining a reduced order model of an unstable system is discussed and solved

on the nonlinear dynamics of fast filtering algorithms the main purpose of this paper is to address a fundamental open problem in linear filtering and estimation namely what is the steady state or asymptotic behavior of the kalman filter or the kalman gain when the observed stationary stochastic process is not generated by a finite dimensional stochastic system or when it is generated by a stochastic system having higher dimensional unmodeled dynamics for example some time ago kalman pointed out that the usual positivity conditions assumed in the classical situation are not in fact necessary for the kalman filter to converge using a fast filtering algorithm which incorporates the statistics of the observation process as initial conditions for a dynamical system this question is analyzed in terms of the phase portrait of a universal nonlinear dynamical system this point of view has additional advantages as well since it enables one to use the theory of dynamical systems to study the sensitivity of the kalman filter to small changes in initial conditions e g to changes in the statistics of the underlying process this is especially important since these statistics are often either approximated or estimated in this paper for a scalar observation process necessary and sufficient conditions for the kalman filter to converge are derived using methods from stochastic systems and from nonlinear dynamics especially the use of stable unstable and center manifolds it is also shown that in nonconvergent cases there exist periodic points of every period p p ge 3 that are arbitrarily close to initial conditions having unbounded orbits rigorously demonstrating that the kalman filter can also be sensitive to initial conditions

new results on the rational covariance extension problem with degree constraint abstract a new theory for the rational covariance extension problem with degree constraint or simply the rcep has recently emerged with applications in high resolution spectral estimation speech synthesis and possibly new applications in time series analysis and system identification this paper establishes some new theoretical results on the rcep via an alternative analysis in one result we show the bijective correspondence between denominator polynomials of non strictly positive solutions of the rcep and the minimizers of a class of strictly convex functionals associated with non strictly positive pseudopolynomials defined on a subset of a finite dimensional space this result leads to an alternative constructive proof of a theorem of georgiou on complete parametrization of all solutions of the rcep and a new geometric proof with an extension to non real interpolators of a homeomorphism derived previously by blomqvist fanizza and nagamune we then generalize this homeomorphism to also allow for variation in the covariance data our contribution in the generalization is allowing for non strictly positive pseudopolynomials for the special case of strictly positive pseudopolynomials a stronger property of diffeomorphism in the context of the nevanlinna pick interpolation with degree constraint has been shown earlier by byrnes and lindquist the results of this paper have direct analogues to nevanlinna pick interpolation with degree constraint which is of interest to the robust control community

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

a new family of high resolution multivariate spectral estimators in this paper we extend the beta divergence family to multivariate power spectral densities similarly to the scalar case we show that it smoothly connects the multivariate kullback leibler divergence with the multivariate itakura saito distance we successively study a spectrum approximation problem based on the beta divergence family which is related to a multivariate extension of the three spectral estimation technique it is then possible to characterize a family of solutions to the problem an upper bound on the complexity of these solutions will also be provided finally we will show that the most suitable solution of this family depends on the specific features required from the estimation problem

a generalized entropy criterion for nevanlinna pick interpolation with degree constraint we present a generalized entropy criterion for solving the rational nevanlinna pick problem for n 1 interpolating conditions and the degree of interpolants bounded by n the primal problem of maximizing this entropy gain has a very well behaved dual problem this dual is a convex optimization problem in a finite dimensional space and gives rise to an algorithm for finding all interpolants which are positive real and rational of degree at most n the criterion requires a selection of a monic schur polynomial of degree n it follows that this class of monic polynomials completely parameterizes all such rational interpolants and it therefore provides a set of design parameters for specifying such interpolants the algorithm is implemented in a state space form and applied to several illustrative problems in systems and control namely sensitivity minimization maximal power transfer and spectral estimation

signal estimation via selective harmonic amplification music redux the technique known as multiple signal classification music is a semi empirical way to obtain pseudo spectra that highlight the spectral energy distribution of a time series it is based on a certain canonical decomposition of a toeplitz matrix formed out of an estimated autocorrelation sequence the purpose of this paper is to present an analogous canonical decomposition of the state covariance matrix of a stable linear filter driven by a given time series accordingly the paper concludes with a modification of music the new method starts with filtering the time series and then estimating the covariance of the state of the filter this step in essence improves the signal to noise ratio snr by amplifying the contribution to the actual value of the state covariance of a selected harmonic interval where spectral lines are expected to reside then the method capitalizes on the canonical decomposition of the filter state covariance to retrieve information on the location of possible spectral lines the framework requires uniformly spaced samples of the process

stability tests of interval time delay systems abstract this paper presents some sufficient conditions for robust stability independent of delay and decaying rate for uncertain time delay systems the obtained robustness results are extended to cover the class of interval time delay systems sufficient conditions are also derived for checking the instability of this class of systems the results are given in terms of norm and matrix measure easy to check a comparison between the derived conditions is given numerical computations are performed to illustrate the results

a convex optimization approach to arma modeling we formulate a convex optimization problem for approximating any given spectral density with a rational one having a prescribed number of poles and zeros n poles and m zeros inside the unit disc and their conjugates the approximation utilizes the kullback leibler divergence as a distance measure the stationarity condition for optimality requires that the approximant matches n 1 covariance moments of the given power spectrum and m cepstral moments of the corresponding logarithm although the latter with possible slack the solution coincides with one derived by byrnes enqvist and lindquist who addressed directly the question of covariance and cepstral matching thus the present paper provides an approximation theoretic justification of such a problem since the approximation requires only moments of spectral densities and of their logarithms it can also be used for system identification

brief paper spectral estimation by least squares optimization based on rational covariance extension this paper proposes a new spectral estimation technique based on rational covariance extension with degree constraint the technique finds a rational spectral density function that approximates given spectral density data under constraint on a covariance sequence spectral density approximation problems are formulated as nonconvex optimization problems with respect to a schur polynomial to formulate the approximation problems the least squares sum is considered as a distance properties of optimization problems and numerical algorithms to solve them are explained numerical examples illustrate how the methods discussed in this paper are useful in stochastic model reduction and stochastic process modeling

identifiability and well posedness of shaping filter parameterizations a global analysis approach in this paper we study the well posedness of the problems of determining shaping filters from combinations of finite windows of cepstral coefficients covariance lags or markov parameters for example we determine whether there exists a shaping filter with a prescribed window of markov parameters and a prescribed window of covariance lags we show that several such problems are well posed in the sense of hadamard that is one can prove existence uniqueness identifiability and continuous dependence of the model on the measurements our starting point is the global analysis of linear systems where one studies an entire class of systems or models as a whole and where one views measurements such as covariance lags and cepstral coefficients or markov parameters from data as functions on the entire class this enables one to pose such problems in a way that tools from calculus optimization geometry and modern nonlinear analysis can be used to give a rigorous answer to such problems in an algorithm independent fashion in this language we prove that a window of cepstral coefficients and a window of covariance coefficients yield a bona fide coordinate system on the space of shaping filters thereby establishing existence uniqueness and smooth dependence of the model parameters on the measurements from data

the interpolation problem with a degree constraint the author previously 1983 1987 showed that there is a correspondence between nonnegative hermitian trigonometric polynomials of degree spl les n and solutions to the standard nevanlinna pick caratheodory interpolation problem with n 1 constraints which are rational and also of degree spl les n it was conjectured that the correspondence under suitable normalization is bijective and thereby that it results in a complete parametrization of rational solutions of degree spl les n the conjecture was proven by byrnes et al 1995 along with a detailed study of this parametrization however byrnes et al used a slightly restrictive assumption that the trigonometric polynomials are positive and accordingly the corresponding solutions have positive real part the purpose of the present note is to extend the result to the case of nonnegative trigonometric polynomials as well we present the arguments in the context of the general nevanlinna pick caratheodory fejer interpolation

a maximum entropy enhancement for a family of high resolution spectral estimators structured covariances occurring in spectral analysis filtering and identification need to be estimated from a finite observation record the corresponding sample covariance usually fails to possess the required structure this is the case for instance in the byrnes georgiou lindquist three like tunable high resolution spectral estimators there the output covariance of a linear filter is needed to initialize the spectral estimation technique the sample covariance estimate however is usually not compatible with the filter in this paper we present a new systematic way to overcome this difficulty the new estimate is obtained by solving an ancillary problem with an entropic type criterion extensive scalar and multivariate simulation shows that this new approach consistently leads to a significant improvement of the spectral estimators performances

cepstral coefficients covariance lags and pole zero models for finite data strings one of the most widely used methods of spectral estimation in signal and speech processing is linear predictive coding lpc lpc has some attractive features which account for its popularity including the properties that the resulting modeling filter i matches a finite window of n 1 covariance lags ii is rational of degree at most n and iii has stable zeros and poles the only limiting factor of this methodology is that the modeling filter is all pole i e an autoregressive ar model in this paper we present a systematic description of all autoregressive moving average arma models of processes that have properties i iii in the context of cepstral analysis and homomorphic filtering we show that each such an arma model determines and is completely determined by its finite windows of cepstral coefficients and covariance lags we show that these nth order windows form local coordinates for all arma models of degree n and that the pole zero model can be determined from the windows as the unique minimum of a convex objective function we refine this optimization method by first noting that the maximum entropy design of an lpc filter is obtained by maximizing the zeroth cepstral coefficient subject to the constraint i more generally we modify this scheme to a more well posed optimization problem where the covariance data enter as a constraint and the linear weights of the cepstral coefficients are positive in a sense that a certain pseudo polynomial is positive rather succinctly generalizing the maximum entropy method this new problem is a homomorphic filter generalization of the maximum entropy method providing a procedure for the design of any stable minimum phase modeling filter of degree less or equal to n that interpolates the given covariance window we present an algorithm for realizing these filters in a lattice ladder form given the covariance window and the moving average part of the model while we also show how to determine the moving average part using cepstral smoothing one can make use of any good a priori estimate for the system zeros to initialize the algorithm we conclude the paper with an example of this method incorporating an example from the literature on arma modeling

realization of power spectra from partial covariance sequences in this paper we address the problem of realization of a spectral density function from incomplete information about the underlying stochastic process the standing assumption is the availability of an incomplete partial sequence of covariance samples of the process we study the set of rational extensions of this finite sequence to an infinite covariance function that agrees with the available samples the classical theory of orthogonal polynomials with respect to the unit circle and the theory of moments have been utilized extensively in a variety of engineering problems including the one we are dealing with these have been known to provide a unifying framework for a variety of current spectral estimation techniques maximum entropy method pisarenko s harmonic decomposition etc in this work we consider and study the set of all covariance realizations of dimension lower than or equal to the length of the partial sequence and equal to the dimension of the maximum entropy realization the me solution is a point in this set other points correspond to pole zero models a general formula is obtained for recursively updated pole zero models of dimension increasing with the data record information about the zeros is obtained from the asymptotic behavior of the partial autocorrelation coefficients our approach combines techniques from the theory of orthogonal polynomials on the unit circle the theory of moments and also techniques from degree theory topology our objective is to develop a theory that will provide a frame for constructing recursively pole zero realizations of increasing dimension

further results on the convergence of the pavon ferrante algorithm for spectral estimation in this paper we provide a detailed analysis of the global convergence properties of an extensively studied and extremely effective fixed point algorithm for the kullback leibler approximation of spectral densities proposed by pavon and ferrante in their paper on the georgiou lindquist approach to constrained kullback leibler approximation of spectral densities our main result states that the algorithm globally converges to one of its fixed points

predictability and unpredictability in kalman filtering the authors study the dynamical behavior of the kalman filter when the given parameters are allowed to vary in a way which does not necessarily correspond to an underlying stochastic system this may correspond to situations in which the basic parameters are chosen incorrectly through estimates the authors show that as has been suggested by kalman the filter equations converge to a limit corresponding to a steady state filter for a subset of the parameter space which is much larger than that corresponding to bona fide stochastic systems more surprisingly in the complement of this subset the filtering equations behave in both a regular and an unpredictable manner representative of some of the basic aspects of chaotic dynamics this interesting dynamical behavior occurs already for one dimensional filters and a complete phase portrait in this case is given

time and spectral domain relative entropy a new approach to multivariate spectral estimation the concept of spectral relative entropy rate is introduced for jointly stationary gaussian processes using classical information theoretic results we establish a remarkable connection between time and spectral domain relative entropy rates this naturally leads to a new spectral estimation technique where a multivariate version of the itakura saito distance is employed it may be viewed as an extension of the approach called three introduced by byrnes georgiou and lindquist in 2000 which in turn followed in the footsteps of the burg jaynes maximum entropy method spectral estimation is here recast in the form of a constrained spectrum approximation problem where the distance is equal to the processes relative entropy rate the corresponding solution entails a complexity upper bound which improves on the one so far available in the multichannel framework indeed it is equal to the one featured by three in the scalar case the solution is computed via a globally convergent matricial newton type algorithm simulations suggest the effectiveness of the new technique in tackling multivariate spectral estimation tasks especially in the case of short data records

on control system design using random samples of contractive block toeplitz matrices contractive lower triangular block toeplitz ltbt matrices have recently been parametrized in a closed form by a sequence of unstructured contractive matrices a problem naturally arises between uniform samplings over contractive ltbt matrices and their matrix parameters that is which one is more appropriate in control system analysis and synthesis this paper makes it clear that uniform sampling over matrix parameters results in a contractive ltbt matrix with a more rapid concentration speed towards the boundary of the uncertainty set this is not an attractive property in control related problems numerical comparisons also show that uniform sampling directly over contractive ltbt matrices outperforms that over the matrix parameter set in robustness analysis for a closed loop system as well as in strongly stable closed loop system design with complexity constraints on the controller as a sacrifice uniform sampling over contractive ltbt matrices is more mathematically complicated and more time cons

important moments in systems and control the moment problem matured from its various special forms in the late 19th and early 20th centuries to a general class of problems that continues to exert profound influence on the development of analysis and its applications to a wide variety of fields in particular the theory of systems and control is no exception where the applications have historically been to circuit theory optimal control robust control signal processing spectral estimation stochastic realization theory and the use of the moments of a probability density many of these applications are also still works in progress in this paper we consider the generalized moment problem expressed in terms of a basis of a finite dimensional subspace p of the banach space c a b and a positive sequence c but with a new wrinkle inspired by the applications to systems and control we seek to parameterize solutions which are positive rational measures in a suitably generalized sense our parameterization is given in terms of smooth objects in particular the desired solution space arises naturally as a manifold which can be shown to be diffeomorphic to a euclidean space and which is the domain of some canonically defined functions the analysis of these functions and related maps yields interesting corollaries for the moment problem and its applications which we compare to those in the recent literature and which play a crucial role in part of our proof

arma identification of graphical models consider a gaussian stationary stochastic vector process with the property that designated pairs of components are conditionally independent given the rest of the components such processes can be represented on a graph where the components are nodes and the lack of a connecting link between two nodes signifies conditional independence this leads to a sparsity pattern in the inverse of the matrix valued spectral density such graphical models find applications in speech bioinformatics image processing econometrics and many other fields where the problem to fit an autoregressive ar model to such a process has been considered in this paper we take this problem one step further namely to fit an autoregressive moving average arma model to the same data we develop a theoretical framework and an optimization procedure which also spreads further light on previous approaches and results this procedure is then applied to the identification problem of estimating the arma parameters as well as the topology of the graph from statistical data

on the problem of stable rational interpolation abstract aspects of the stability issue in connection with rational interpolation are investigated in particular it is shown that unconstrained interpolation of a given set of points together with an associated mirror image set of points yields a one parameter family of stable interpolating functions as an application it is also shown that if a certain number of markov parameters are given by appropriate choice of the moments stable realizations are obtained