homotopy solution of polynomial equations in h optimal control a method is described to solve the standard h optimal control problem polynomially the polynomial equations for equalizing solutions are of a homotopy type so that the equations may be transformed accordingly to a set of non linear ordinary differential equations

an information theoretic framework for deriving canonical decision feedback receivers in gaussian channels a framework is presented that allows a number of known results relating feedback equalization linear prediction and mutual information to be easily understood a lossless additive decomposition of mutual information in a general class of gaussian channels is introduced and shown to produce an information preserving canonical decision feedback receiver the approach is applied to intersymbol interference isi channels to derive the well known minimum mean square error mmse decision feedback equalizer dfe when applied to the synchronous code division multiple access cdma channel the result is the mmse or signal to interference ratio sir maximizing decision feedback detector which is shown to achieve the channel sum capacity at the vertices of the capacity region finally in the case of the asynchronous cdma channel we are able to give new connections between information theory decision feedback receivers and structured factorizations of multivariate spectra

optimal and self tuning deconvolution in time domain this paper is concerned with both the optimal minimum mean square error variance and self tuning deconvolution problems for discrete time systems when the signal model measurement model and noise statistics are known a novel approach for the design of the optimal deconvolution filter predictor and smoother is proposed based on projection theory and innovation analysis in the time domain the estimators are given in terms of an autoregressive moving average arma innovation model and one unilateral linear polynomial equation where the arma innovation model is obtained by performing one spectral factorization a self tuning scheme can be incorporated when the noise statistics the input model and or colored noise model are unknown the self tuning estimator is designed by identifying two arma innovation models

polynomial j spectral factorization several algorithms are presented for the j spectral factorization of a para hermitian polynomial matrix the four algorithms that are discussed are based on diagonalization successive factor extraction interpolation and the solution of an algebraic riccati equation respectively the paper includes a special algorithm for the factorization of unimodular para hermitian polynomial matrices and deals with canonical noncanonical and nearly noncanonical factorizations

brief paper identification of an additive nonlinear system and its applications in generalized hammerstein models an identification algorithm is developed for a class of nonlinear systems that are multi input and multi output in an additive form the convergence results are achieved and its applications to identification of a generalized hammerstein system is also discussed

feedforward control is dual to deconvolution a duality is demonstrated between optimal feedforward control and optimal deconvolution or input estimation these two problems are normally discussed separately in the literature but have close similarities duality between them can be demonstrated if and only if one uses general problem formulations with frequency shaped weighting in the criteria from one of the problems the dual problem can then be obtained immediately from the block diagram by reversing the directions of arrows interchanging summation points and node points and transposing all transfer function matrices this result applies for continuous and discrete time problems as well as for minimization of j g for any transfer function norms for which gt g a derivation of a polynomial solution to the frequency weighted discrete time mimo lqg feedforward control problem illustrates the use of the duality

non parametric nonlinear system identification a data driven orthogonal basis function approach in this paper a data driven orthogonal basis function approach is proposed for non parametric fir nonlinear system identification the basis functions are not fixed a priori and match the structure of the unknown system automatically this eliminates the problem of blindly choosing the basis functions without a priori structural information further based on the proposed basis functions approaches are proposed for model order determination and regressor selection along with their theoretical justifications

discrete j spectral factorization of possibly singular polynomial matrices we develop two state space algorithms for discrete time j spectral factorization of possibly singular possibly non column reduced and with possible zeros at the origin para hermitian matrices by assignment of matrices in optimal lq return difference equality the column degrees of the spectral factor equal the column degrees of the given matrix and the dimension of a discrete time algebraic riccati system equals the sum of the column degrees necessary and sufficient condition for the least order property of our state space realizations is column reduction of the given matrix

a novel derivation methodology for polynomial lq controller design a simple derivation method for optimization of linear quadratic lq controllers is presented in the discrete time polynomial systems framework a control law variation regarded as a potential feedforward from the innovations is used orthogonality is evaluated in the frequency domain by collectively cancelling unstable poles by zeros the suggested method summarized as a three step scheme is exemplified on a disturbance measurement feedforward and an output feedback problem it is a simple more direct alternative to the completing the squares approach

estimation in change point hazard rate models summary this paper discusses the estimation of parameters in hazard rate models with a changepoint due to the irregularity of the models the classical maximum likelihood method and the method of moments cannot be used a consistent estimator of the change point is obtained by examining the properties of the density represented as a mixture the performance of the estimator is checked via simulation

restricted range approximation by splines and variational inequalities error estimates are obtained for the approximation of nonnegative functions from below by nonnegative splines the maximum norm results are of the same order as the jackson type estimates for unconstrained approximation by splines the l 2 results are used to estimate the rate of convergence of the ritz method for certain variational inequalities the more general problem of approximating functions f in the set w g in c a b l x leqq g x leqq u x x in a b by splines in w is also discussed

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

algorithms for multidimensional spectral factorization and sum of squares in this paper algorithms are developed for the problems of spectral factorization and sum of squares of polynomial matrices with n indeterminates and a natural interpretation of the tools employed in the algorithms is given using ideas from the theory of lossless and dissipative systems these algorithms are based on the calculus of 2n variable polynomial matrices and their associated quadratic differential forms and share the common feature that the problems are lifted from the original n variable polynomial context to a 2n variable polynomial context this allows to reduce the spectral factorization problem and the sum of squares problem to linear matrix inequalities lmi s to the feasibility of a semialgebraic set or to a linear eigenvalue problem

semiparametric analysis of the additive risk model in contrast to the proportional hazards model the additive risk model specifies that the hazard function associated with a set of possibly time varying covariates is the sum of rather than the product of the baseline hazard function and the regression function of covariates this formulation describes a different aspect of the association between covariates and the failure time than the proportional hazards model and is more plausible than the latter for many applications in the present paper simple procedures with high efficiencies are developed for making inference about the regression parameters under the additive risk model with an unspecified baseline hazard function

on a schur algorithm approach to spectral factorization state space formulae abstract in some earlier publications 10 13 we indicated how the classical schur algorithm can be used as a method to solve a spectral factorization problem in the present paper indicate how to obtain a state space formulation of the theory and formulae for an iterative algorithm for spectral factorization using state space representations

a probabilistic approach to multivariable robust filtering and open loop control a new approach to robust filtering prediction and smoothing of discrete time signal vectors is presented linear time invariant filters are designed to be insensitive to spectral uncertainty in signal models the goal is to obtain a simple design method leading to filters which are not overly conservative modeling errors are described by sets of models parameterized by random variables with known covariances these covariances could either be estimated from data or be used as robustness tuning knobs a robust design is obtained by minimizing the spl hscrsub 2 norm or equivalently the mean square estimation error averaged with respect to the assumed model errors a polynomial solution based on an averaged spectral factorization and a unilateral diophantine equation is derived the robust estimator is referred to as a cautious wiener filter it turns out to be only slightly more complicated to design than an ordinary wiener filter the methodology can be applied to any open loop filtering or control problem in particular we illustrate this for the design of robust multivariable feedforward regulators decoupling and model matching filters

optimal multivariable regulation with disturbance measurement feedforward abstract the problem of optimal stochastic regulation based on output feedback and disturbance measurement feedforward for multivariable systems is solved using the polynomial equation approach two independent couples of polynomial matrix equations are derived which yield the feedback and the feedforward parts of the regulator

global convergence of the recursive kernel regression estimates with applications in classification and nonlinear system estimation an improved exponential bound on the l sub 1 error for the recursive kernel regression estimates is derived it is shown using the martingale device that weak strong and complete l sub 1 consistencies are equivalent consequently the conditions on a certain smoothing sequence are necessary and sufficient for strong l sub 1 consistency of the recursive kernel regression estimate the rates of global convergence are also given obtained results are applied to recursive classification rules and to nonlinear time series estimation

optimal and self tuning white noise estimators with applications to deconvolution and filtering problems using the innovation analysis method in the time domain based on the autoregressive moving average arma innovation model this paper presents a unified white noise estimation theory that includes both input and measurement white noise estimators and presents a new steady state optimal state estimation theory non recursive optimal state estimators are given whose recursive version gives a steady state kalman filter where a new algorithm of the kalman filter gain is proposed two new algorithms of the kalman predictor gain are also derived local asymptotic stability of the kalman filter is proved the classical kalman filtering theory is extended and modified the method used covers unstable systems with correlation noises and singular transition matrix and also covers the self tuning white noise signal and state estimators when the noise statistics is unknown to illustrate three simulation examples are given

nonparametric estimation and testing in a cure model nonparametric generalized maximum likelihood product limit point estimators and confidence intervals are given for a cure model with random censorship one two and k sample likelihood ratio tests for inference on the cure rates are developed in the two sample case its power is compared to the power of several alternatives including the log rank and gray and tsiatis 1989 biometrics 45 899 904 tests implications for the use of the likelihood ratio test in a clinical trial designed to compare cure rates are discussed

wiener design of adaptation algorithms with time invariant gains a design method is presented that extends least mean squared lms adaptation of time varying parameters by including general linear time invariant filters that operate on the instantaneous gradient vector the aim is to track time varying parameters of linear regression models in situations where the regressors are stationary or have slowly time varying properties the adaptation law is optimized with respect to the steady state parameter error covariance matrix for time variations modeled as vector arima processes the design method systematically uses prior information about time varying parameters to provide filtering prediction or fixed lag smoothing estimates for arbitrary lags the method is based on a transformation of the adaptation problem into a wiener filter design problem the filter works in open loop for slow parameter variations whereas a time varying closed loop has to be considered for fast variations in the latter case the filter design is performed iteratively the general form of the solution at each iteration is obtained by a bilateral diophantine polynomial matrix equation and a spectral factorization for white gradient noise the diophantine equation has a closed form solution further structural constraints result in very simple design equations under certain model assumptions the wiener designed adaptation laws reduce to lms adaptation compared with kalman estimators the channel tracking performance becomes nearly the same in mobile radio applications whereas the complexity is in general much lower

additive hazards models with latent treatment effectiveness lag time summary in many clinical trials for evaluating treatment efficacy it is believed that there may exist latent treatment effectiveness lag times after which medical treatment procedure or chemical compound would be in full effect in this paper semiparametric regression models are proposed and studied for estimating the treatment effect accounting for such latent lag times the new models take advantage of the invariant property of the additive hazards model in marginalising over an additive latent variable parameters in the models are thus easily estimated and interpreted while the flexibility of not having to specify the baseline hazard function is preserved monte carlo simulation studies demonstrate the appropriateness of the proposed semiparametric estimation procedure the methodology is applied to data collected in a randomised clinical trial which evaluates the efficacy of biodegradable carmustine polymers for treatment of recurrent brain tumours

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

testing if a nonlinear system is additive or not abstract additive nonlinear systems are one of the most widely used nonlinear and non parametric models to describe nonlinear behaviors a number of analysis and identification techniques have been developed in the literature for such systems to apply however one has to make sure that the system is additive or can be approximated well by an additive system this is a nontrivial problem and has eluded researchers for a long time to the best of our knowledge only scatter results are reported in the literature the difficulties lie in the fact that the function and its structure are unknown so estimating its derivatives under an unknown multidimensional density function could be subject to the curse of dimensionality in this paper we present two methods to check if a system is additive the first one estimates the squared derivative average via the fourier transform the other one directly estimates the squared derivative average in a reproducing kernel hilbert space rkhs setting for both methods convergence results are established and practical numerical algorithms are developed

discrete j spectral factorization abstract this paper deals with the j spectral factorization for general discrete rational matrices a simple approach based on the kalman filtering in krein space is proposed the main idea is to construct a stochastic state space filtering model in krein space such that the spectral matrix of the output is equal to the rational matrix to be factorized the spectral factor is then easily derived by using the generalized kalman filtering in krein space which is similar to the h 2 spectral factorization our approach unifies the treatment of the h 2 spectral factorization and the j spectral factorization the applications of the derived results in h and risk sensitive estimation for both nonsingular and singular systems are demonstrated

representation and identification of non parametric nonlinear systems of short term memory and low degree of interaction in this paper we propose a new representation which is particularly useful for a class of non parametric nonlinear systems that have short term memory and low degree of interaction advantages and disadvantages of this representation are discussed and compared to existing methods both theoretically and numerically furthermore results regarding structural estimation based on the analysis of variance and on full scale identification are also provided

design of reduced order optimal estimators directly in the frequency domain the frequency domain design of optimal reduced order estimators caiman filters for continuous time linear time invariant systems containing 0 k m noise free output measurements is considered in a direct frequency domain approach to state feedback and estimation the states of a system do not appear explicitly and therefore we here focus our attention mainly on the frequency domain characteristics of the optimal filter by suitably modifying known time domain results a simple polynomial matrix equation can be derived from which the optimal filter parameters are obtained by spectral factorization the equivalence between the time and the frequency domain results is established with the aid of a non minimal representation of reduced order observers

an efficient and versatile algorithm for computing the covariance function of an arma process computation of an arma covariance function is a common ingredient in analysis and synthesis of various problems in stochastic control estimation and signal processing several approaches can be used for this purpose in this paper we present an algorithm based on simple polynomial calculations compared with alternative strategies it has small computational load shows good numerical robustness and can be extended to handle multivariable arma processes even with complex valued coefficients

identification of nonlinear additive fir systems in this paper we consider identification of a nonlinear additive system an input signal is designed in such a way that the problem of identification of nonlinear additive systems is reduced to a problem of identification of static nonlinear functions then three approaches are established to estimate the order of the system the methods exploit the structure of the nonlinear additive model so that their implementations are easy

variable selection and identification of high dimensional nonparametric additive nonlinear systems this paper considers variable selection and identification of dynamic additive nonlinear systems via kernel based nonparametric approaches where the number of variables and additive functions may be large variable selection aims to find which additive functions contribute and which do not the proposed variable selection consists of two successive steps at the first step one estimates each additive function by kernel based nonparametric identification approaches without suffering from the curse of dimensionality at the second step a nonnegative garrote estimator is applied to identify which additive functions are nonzero by utilizing the obtained nonparametric estimates of each function under weak conditions the nonparametric estimates of each additive function can achieve the same asymptotic properties as for 1d nonparametric identification based on kernel functions it is also established that the nonnegative garrote estimator turns a consistent estimate for each additive function into a consistent variable selection with probability one as the number of samples tends to infinity two simulation examples are presented to verify the effectiveness of the variable selection and identification approaches proposed in the paper

a linear rank test for use when the main interest is in differences in cure rates for diseases with a positive probability of being cured a family of alternatives to the null hypothesis of equality of survival distributions is introduced which is designed to focus power against alternatives with differences in cure rates the optimal linear rank test for this alternative is derived and found to be substantially more efficient than the log rank test for this alternative when cure rates are less than 50 while there is little difference between the tests if the cure rates are 50 or greater the simple test based on the difference of kaplan meier estimates of the proportion cured is also examined and found to be fully efficient for this alternative with no censoring while its efficiency rapidly drops as censoring is increased the new test is not a pure test of equality of cure rates when the data are censored but rather is a test of equality of survival distributions that focuses power against late differences in the survival curves

a parametrisation for dissipative behaviours the matrix case we study linear time invariant dynamical systems that are dissipative with respect to a generalised power defined by a quadratic differential form we address several cases in an increasing order of complexity and show how dissipative systems can be parametrised in this process we also establish a number of results for polynomial matrices that are of independent interest the present article is a generalisation of our earlier work that dealt with single input single output dissipative systems pendharkar i and pillai h k 2004 a parametrization for dissipative behaviours systems and control letters 51 123 132

orthogonal least squares methods and their application to non linear system identification abstract identification algorithms based on the well known linear least squares methods of gaussian elimination cholesky decomposition classical gram schmidt modified gram schmidt householder transformation givens method and singular value decomposition are reviewed the classical gram schmidt modified gram schmidt and householder transformation algorithms are then extended to combine structure determination or which terms to include in the model and parameter estimation in a very simple and efficient manner for a class of multivariate discrete time non linear stochastic systems which are linear in the parameters

blind identification of linear subsystems of lti zmnl lti models with cyclostationary inputs discrete time nonlinear models consisting of two linear time invariant lti filters separated by a finite order zero memory nonlinearity zmnl of the polynomial type the lti zmnl lti model are appropriate in a large number of practical applications we discuss some approaches to the problem of blind identification of such nonlinear models it is shown that for an nth order nonlinearity the possibly non minimum phase finite memory linear subsystems of lti zmnl and lti zmnl lti models can be identified using the n 1th order cyclic statistics of the output sequence alone provided the input is cyclostationary and satisfies certain conditions the coefficients of the zmnl are not needed for identification of the linear subsystems and are not estimated it is shown that the theory presented leads to analytically simple identification algorithms that possess several noise and interference suppression characteristics

least squares approximation of perfect reconstruction filter banks designing good causal filters for perfect reconstruction pr filter banks is a challenging task due to the unusual nature of the design constraints we present a new least squares ls design methodology for approximating prfbs that avoids most of these difficult constraints the designer first selects a set of subband analysis filters from an almost unrestricted class of rational filters then given some desired reconstruction delay this design procedure produces the causal and rational synthesis filters that result in the best least squares approximation to a prfb this technique is built on a multi input multi output mimo system model for filter banks derived from the filter bank polyphase representation using this model we frame the ls approximation problem for prfbs as a causal ls equalization problem for mimo systems we derive the causal ls solution to this design problem and present an algorithm for computing this solution the resulting algorithm includes a mimo spectral factorization that accounts for most of the complexity and computational cost for this design technique finally we consider some design examples and evaluate their performance

optimal parallel model solutions using the polynomial approach solutions to the optimal filtering and open loop control problems using a generalized parallel model scheme are presented the solutions are based on the algebra of polynomial matrices different c

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

the degree of copositive approximation and a computer algorithm the main results are as follows 1 let f in bf c 0 1 change its sign a finite number of times then the degree of copositive approximation of f by splines with n equally spaced knots is bounded by c omega 3 f 1 n for n large enough this rate is the best in the sense that omega 3 cannot be replaced by omega 4 2 an algorithm is developed basedon the proof 3 the first result above holds for a copositive polynomial approximation of f 4 if f in c 1 0 1 then the degree of approximation by copositive splines of order r is bounded by cn 1 omega r 1 f 1 n the results on f in bf c 0 1 fill a gap left by s p zhou israel j math 78 1992 pp 75 83 and y k hu d leviatan and x m yu j anal 1 1993 pp 85 90 j approx theory 80 1995 pp 204 218

weighted log rank type statistics for comparing survival curves when there is a time lag in the effectiveness of treatment summary in certain long term treatment trials one expects some lag period before the treatment is fully effective this paper presents two weighted log rank type statistics designed to have good efficiency over a wide range of lags a maximin efficiency robust statistic and a simplified version of this statistic both statistics can be computed as easily as the log rank statistic asymptotic efficiency calculations supported by small sample simulations show that the two proposed statistics are substantially more efficient than the conventional log rank statistic in certain lag situations with comparatively little efficiency loss relative to the log rank statistic when no lag exists we recommend the maximin statistic for situations where a lag is expected but cannot be specified precisely in advance

on the series expansion approach to the identification of hammerstein systems a polynomial identification algorithm for recovering a nonlinearity in the hammerstein system is proposed the estimate employs the legendre orthogonal system with adaptively selected number of terms the global consistency along with rates of convergence are established no assumptions concerning continuity of the nonlinearity or its functional form are made a data driven method using the cross validation technique for selecting the number of terms in the estimate is presented

detecting rigid convexity of bivariate polynomials abstract given a polynomial x r n p x in n 2 variables a symbolic numerical algorithm is first described for detecting whether the connected component of the plane sublevel set p x p x 0 containing the origin is rigidly convex or equivalently whether it has a linear matrix inequality lmi representation or equivalently if polynomial p x is hyperbolic with respect to the origin the problem boils down to checking whether a univariate polynomial matrix is positive semidefinite an optimization problem that can be solved with eigenvalue decomposition when the variety c x p x 0 is an algebraic curve of genus zero a second algorithm based on bezoutians is proposed to detect whether p has an lmi representation and to build such a representation from a rational parametrization of c finally some extensions to positive genus curves and to the case n 2 are mentioned

wiener filter design using polynomial equations a simplified way of deriving realizable and explicit wiener filters is presented discrete time problems are discussed in a polynomial equation framework optimal filters predictors and smoothers are calculated by means of spectral factorizations and linear polynomial equations a tool for obtaining these equations for a given problem structure is described it is based on the evaluation of orthogonality in the frequency domain by means of canceling stable poles with zeros comparisons are made to previously known derivation methodologies such as completing the squares for the polynomial systems approach and the classical wiener solution the simplicity of the proposed derivation method is particularly evident in multistage filtering problems to illustrate two examples are discussed a filtering and a generalized deconvolution problem a new solvability condition for linear polynomial equation appearing in scalar problems is also presented

brief risk sensitive filtering prediction and smoothing for discrete time singular systems this paper is concerned with steady state risk sensitive filtering prediction and smoothing problems for discrete time singular systems it is shown that a risk sensitive estimator can be obtained by ensuring the minimum of an indefinite quadratic form to be maximum minimum when the risk sensitivity parameter q is negative positive an auxiliary state space signal model and an innovation sequence in krein space are introduced to simplify the derivation of the estimator the estimator is calculated based on one j spectral factorization for risk seeking q 0 a numerical example is given to demonstrate the applicability of the result

discrete modulus of smoothness of splines with equally spaced knots we study the behavior of moduli of smoothness of splines s of order r with equally spaced knots x i x i 1 x i h the main results are as follows 1 for each 0 leqslant m r all quantities h j omega m j s j h p 0 leqslant j leqslant m are equivalent and can be measured by a discrete norm of the mth differences of the b spline coefficients of s which we call the mth discrete modulus of smoothness of s 2 all quantities h j omega m j s j h p m geqslant r 0 leqslant j leqslant r 1 are equivalent to omega r s h p which can be measured by the rth discrete modulus of s 3 when h is replaced by t 0 leq t leqslant h in the results above all the quantities can still be measured by the corresponding discrete modulus multiplied by a power of t h the results generalize the notion of a discrete norm of b spline series in the case of equal spacing as an application we use these results to prove that omega 3 is th

univariate splines equivalence of moduli of smoothness and applications several results on equivalence of moduli of smoothness of univariate splines are obtained for example it is shown that for any 1 k r 1 0 m r 1 and 1 p the inequality n v k v s v n 1 p k s n 1 p 1 v min k m 1 is satisfied where s c m 1 1 is a piecewise polynomial of degree r on a quasi uniform i e the ratio of lengths of the largest and the smallest intervals is bounded by a constant partition of an interval similar results for chebyshev partitions and weighted ditzian totik moduli of smoothness are also obtained these results yield simple new constructions and allow considerable simplification of various known proofs in the area of constrained approximation by polynomials and splines

a polynomial approach to minimax frequency domain optimization of multivariable feedback systems the properties of a feedback system where the plant has rational transfer matrix h and the compensator has transfer matrix g can be characterized through the system functions s i hg and t g i hg 1 good disturbance attenuation robustness limited bandwidth and compensator roll off may be obtained by minimizing a criterion of the form z where z v s t w1 wls t w2 w2t v with respect to the compensator transfer matrix g here v w1 and w2 are suitable rational weighting matrices the solution of the problem can be reduced to a pair of matrix polynomial equations since the optimal solution is highly non unique special solutions with additional optimality properties are considered as well the paper includes a discussion of the numerical solution of the polynomial equations and of the question how to choose the weighting matrices to ensure required feedback system properties an example illustrates the results

optimal reduced order estimators in the frequency domain the discrete time case the discrete time stationary kalman filter problem is solved in the z domain for nth order systems with m outputs where 0 k m measurements are noise free the design equations are very similar to the continuous time cage and they can be solved by spectral factorization giving the polynomial matrix d z which parametrizes the reduced order optimal estimator in the frequency domain by solving a single linear equation the equivalent time domain representation for the optimal filter can also be obtained a simple example demonstrates the filter design in the frequency domain

discrete time average positivity and spectral factorization in a behavioral framework abstract quite recently willems and trentelman siam j control opt 36 1998 1703 1749 have proposed quadratic differential forms for system analysis and synthesis from the behavioral point of view in this paper we develop their discrete time version called quadratic difference forms and some related fundamental concepts with respect to dissipativeness for a dynamical system which interacts with its environment along square summable trajectories moreover we provide an algorithm for spectral factorization of polynomial matrices as one example of quadratic difference forms a numerical example will be given to show the validity of our result

symmetric matrix polynomial equation interpolation results new numerical procedures are proposed to solve the symmetric matrix polynomial equation a t s x s x t s a s 2b s that is frequently encountered in control and signal processing an interpolation approach is presented that takes full advantage of symmetry properties and leads to an equivalent reduced size linear system of equations it results in a simple and general characterization of all solutions of expected column degrees several new theoretical results concerning stability theory and reduced sylvester resultant matrices are also developed and used to conclude a priori on the existence of a solution by means of numerical experiments it is shown that our algorithms are more efficient than older methods and namely appear to be numerically reliable

a variational approach to the polynomial optimization of multivariable control with coloured noise at the controller output a method is derived for the design of an optimal continuous time controller for a multivariable plant in the presence of stochastic inputs which in this case includes coloured noise at the controller output the design processes are polynomial and the derivation is based on the calculus of variations an example illustrates the design process

constrained spline smoothing several results on constrained spline smoothing are obtained in particular we establish a general result showing how one can constructively smooth any monotone or convex piecewise polynomial function ppf or any q monotone ppf q geq 3 with one additional degree of smoothness to be of minimal defect while keeping it close to the original function in the mathbb l p quasi norm it is well known that approximating a function by ppfs of minimal defect splines avoids introduction of artifacts which may be unrelated to the original function thus it is always preferable on the other hand it is usually easier to construct constrained ppfs with as few requirements on smoothness as possible our results allow us to obtain shape preserving splines of minimal defect with equidistant or chebyshev knots the validity of the corresponding jackson type estimates for shape preserving spline approximation is summarized in particular we show that the mathbb l p estimates p geq 1 can be immediately derived from the mathbb l infty estimates

h sub 2 optimal linear robust sampled data filtering design using polynomial approach a new frequency domain approach to robust multi input multi output mimo linear filter design for sampled data systems is presented the system and noise models are assumed to be represented by polynomial forms that are not perfectly known except that they belong to a certain set the optimal design guarantees that the error variance is kept below an upper bound that is minimized for all admissible uncertainties the design problem is cast in the context of h sub 2 via the polynomial matrix representation of systems with norm bounded unstructured uncertainties the sampled data mix of continuous and discrete time systems is handled by means of a lifting technique however it does not increase the dimensionality or alter the computational cost of the solution the setup adopted allows dealing with several filtering problems a simple deconvolution example illustrates the procedure

semiparametric analysis of general additive multiplicative hazard models for counting processes the additive multiplicative hazard model specifies that the hazard function for the counting process associated with a multidimensional covariate process z w t x t t takes the form of t z g 0 t w t 0 t h 0 t x t where 0 0 t 0 t t is a vector of unknown regression parameters g and h are known link functions and 0 is an unspecified baseline hazard function in this paper we develop a class of simple estimating functions for 0 which contains the partial likelihood score function in the special case of proportional hazards models the resulting estimators are shown to be consistent and asymptotically normal under appropriate regularity conditions weak convergence of the aalen breslow type estimators for the cumulative baseline hazard function 0 t 0 t 0 u du is also established furthermore we construct adaptive estimators for 0 and o that achieve the semiparametric information bounds finally a real example is provided along with some simulation results

b spline interpolation compared with finite difference finite element and finite volume methods which applied to two point boundary value problems abstract this paper considers the b spline interpolation and compares this method with finite difference finite element and finite volume methods which applied to the two point boundary value problem d d x p x d u d x f x a x b u a u b 0