new estimates for solutions of lyapunov equations results for estimating the solution of differential and algebraic lyapunov matrix equations are obtained and some of the well known results are generalized

solution bounds of the continuous riccati matrix equation a new approach is proposed for estimating the solution of the continuous algebraic riccati equation care upper and lower solution bounds of the care are presented comparisons show that the present bounds are more general and or tighter than existing results

new upper solution bounds of the continuous algebraic riccati matrix equation by utilizing some linear algebraic techniques new upper bounds of the solution of the continuous algebraic riccati equation care are derived according to the present bounds iterative procedures are also developed for obtaining more precise estimations comparing with existing results the obtained bounds are less restrictive

gradient based iterative algorithm for solving coupled matrix equations abstract this paper is concerned with iterative methods for solving a class of coupled matrix equations including the well known coupled markovian jump lyapunov matrix equations as special cases the proposed method is developed from an optimization point of view and contains the well known jacobi iteration gauss seidel iteration and some recently reported iterative algorithms by using the hierarchical identification principle as special cases we have provided analytically the necessary and sufficient condition for the convergence of the proposed iterative algorithm simultaneously the optimal step size such that the convergence rate of the algorithm is maximized is also established in explicit form the proposed approach requires less computation and is numerically reliable as only matrix manipulation is required some other existing results require either matrix inversion or special matrix products numerical examples show the effectiveness of the proposed algorithm

simultaneous eigenvalue lower bounds for the lyapunov matrix equation simultaneous eigenvalue lower bounds for the solution of the lyapunov equation are presented the n summation and the n product bounds generalize existing bounds such as those for the trace and the determinant these bounds are stronger than the majority of the relevant bounds shown in the literature

bounds in algebraic riccati and lyapunov equations a survey and some new results this paper summarizes and investigates the relations of a number of bounds for the solutions of the algebraic riccati and lyapunov equations that have been reported during the last two decades also presented are bounds for the unified riccati equation using the delta operator and it is shown that some bounds for the continuous and discrete riccati equations can be unified by them

lower summation bounds for the discrete riccati and lyapunov equations lower eigenvalue summation including trace bounds for the solution of the discrete algebraic riccati and lyapunov matrix equations are presented these are tighter than or supplement existing results

a matrix inequality abstract a new matrix inequality associated with bounds on solutions of algebraic riccati and lyapunov equations is presented the improvement of the obtained result over the previously reported results is shown

some bounds of the solution of the algebraic riccati equation bounds on extremal eigenvalues and lower and upper bounds of the trace for the solution of the algebraic riccati equation are presented through two examples it is shown that in some special cases the presented bounds can be better than the results recently published

upper matrix bound of the solution for the discrete riccati equation a new upper matrix bound of the solution for the discrete algebraic matrix riccati equation is developed this matrix bound is then used to derive bounds on the eigenvalues trace and determinant of the same solution it is shown that these eigenvalue bounds are less restrictive than previous results

upper summation and product bounds for solution eigenvalues of the lyapunov matrix equation upper bounds for summations including the trace and for products including the determinant of the eigenvalues of the solution of the continuous algebraic lyapunov matrix equation are presented the majority of the bounds are tighter than those in the literature and some are new

upper bounds for the eigenvalues of the solution of the discrete algebraic lyapunov equation upper bounds for summations including the trace and for products including the determinant of the eigenvalues of the solution of the discrete algebraic lyapunov equation dale are presented all bounds are derived from the matrix series solution of the dale the majority of the bounds are tighter than those in the literature

exponential condition number of solutions of the discrete lyapunov equation the condition number of the n spl times n matrix p is examined where p solves p apa sup bb sup and b is a n spl times d matrix lower bounds on the condition number spl kappa of p are given when a is normal a single jordan block or in frobenius form the bounds show that the ill conditioning of p grows as exp n d spl gt 1 these bounds are related to the condition number of the transformation that takes a to input normal in form a simulation shows that p is typically ill conditioned in the case of n spl gt 1 and d 1 when a sub ij has an independent gaussian distribution subject to restrictions we observe that spl kappa p sup 1 n spl sim 3 3 the effect of autocorrelated forcing on the conditioning on state space systems is examined

robust stabilization for a class of uncertain systems with multiple time delays via linear control in this paper robust stabilization for a class of uncertain systems with multiple time delays is considered a procedure of controller design for the global asymptotic stabilization of such systems via linear control is provided a numerical example is given to illustrate our main result

scaling of the discrete time algebraic riccati equation to enhance stability of the schur solution method a simple scaling procedure for discrete time riccati equations is introduced this procedure eliminates instabilities which can be associated with the linear equation solution step of the generalized schur method without changing the condition of the underlying problem a computable bound for the relative error of the solution of the riccati equation is also derived

on iterative solutions of general coupled matrix equations in this paper we study coupled matrix equations which are encountered in many systems and control applications first we extend the well known jacobi and gauss seidel iterations and present a large family of iterative methods which are then applied to develop iterative solutions to coupled sylvester matrix equations the basic idea is to regard the unknown matrices to be solved as parameters of a system to be identified and to obtain the iterative solutions by applying a hierarchical identification principle next we generalize the sylvester equations to general coupled matrix equations and propose a gradient based iterative algorithm for the solutions using a block matrix inner product the star star product we prove that the iterative algorithm always converges to the unique solutions for any initial values one advantage of the algorithms proposed is that they require less storage space in implementation than existing numerical methods finally we test the algorithms and show their effectiveness using numerical examples

matrix bounds of the discrete are solution this paper provides new lower and upper matrix bounds of the solution to the discrete algebraic riccati equation the lower bound always works if the solution exists the upper bounds are presented in terms of the solution of the discrete lyapunov equation and its upper matrix bound the upper bounds are always calculated if the solution of the lyapunov equation exists a numerical example shows that the new bounds are tighter than previous results in many cases

improved upper bounds for the solution of the continuous algebraic riccati matrix equation in this paper we present upper bounds for the solution matrix of the continuous algebraic riccati matrix equation care by utilizing some matrix inequalities and linear algebraic techniques also for the derived each bounds iterative algorithms are developed to obtain tighter solution estimates having compared some appearing results in the literature the obtained bounds are less restrictive and more efficient finally numerical examples are given to illustrate the effectiveness of the proposed results

diverse bounds for the eigenvalues of the continuous algebraic riccati equation summation bounds for the eigenvalues of a p pa are presented here matrices a p spl isin r sup n spl times n p is positive definite and a transpose of a these bounds are employed to derive new upper bounds for summations of eigenvalues of the solution of the continuous algebraic riccati equation

a generalization of the hermitian and skew hermitian splitting iteration method for solving sylvester equations this paper is concerned with a generalization of the hermitian and skew hermitian splitting hss iteration method for solving large sparse continuous sylvester equations the analysis shows that the ghss iteration method will converge under certain assumptions numerical results show that this new method is more efficient and robust than the existing ones

bounds for the solution of the lyapunov matrix equation a unified approach abstract in recent years several bounds have been reported for the solution of the continuous and the discrete lyapunov equations using the unified lyapunov equation we give in this paper bounds for the solution of this equation in the limiting cases the bounds reduce to existing bounds for both the continuous and discrete lyapunov equations

exploitation of symmetries for image reconstruction in linearized variable density diffraction tomography the problem of reconstructing an object s weakly varying compressibility and density distributions in three dimensional 3d acoustic diffraction tomography is studied based on the fourier diffraction projection theorem for acoustic media it is demonstrated that the 3d fourier components of an object s compressibility and density distributions can be decoupled algebraically thereby providing a method for separately reconstructing the distributions this is facilitated by the identification and exploitation of tomographic symmetries and the rotational invariance of the imaging model the developed reconstruction methods are investigated by use of computer simulation studies the application of the proposed image reconstruction strategy to other tomography problems is discussed

matrix bounds of the solutions of the continuous and discrete riccati equations a unified approach in this paper a new scheme is introduced to measure the matrix bounds of the continuous and discrete riccati equations by estimating upper and lower matrix bounds of the solution of the unified algebraic riccati equation uare the same measurements for the solutions of the continuous and discrete riccati equations respectively can be obtained in limiting cases according to these obtained matrix bounds several eigenvalue bounds are also defined all the proposed results for the uare are new and more general than previous work some obtained results are compared with those of the literature via numerical examples it is shown that in some cases the presented results are tighter than the existing ones

new lower solution bounds of the continuous algebraic riccati matrix equation in optimal and robust control problems the so called continuous algebraic riccati equation care plays an important part by utilizing some matrix inequalities and linear algebraic techniques new lower matrix bounds for the solution of the care are derived following the derived bounds iterative algorithms are then developed to obtain sharper solution estimates in comparison to existing results the obtained bounds are less restrictive finally we give numerical examples to demonstrate the effectiveness of our results

on the trace bound of a matrix product this paper generalizes the main result given by fang et al ibid vol 39 p 2489 90 1994 to provide tighter bounds on the trace of the product of two real matrices

on the optimality of the gridding reconstruction algorithm gridding reconstruction is a method to reconstruct data onto a cartesian grid from a set of nonuniformly sampled measurements this method is appreciated for being robust and computationally fast however it lacks solid analysis and design tools to quantify or minimize the reconstruction error least squares reconstruction lsr on the other hand is another method which is optimal in the sense that it minimizes the reconstruction error this method is computationally intensive and in many cases sensitive to measurement noise hence it is rarely used in practice despite their seemingly different approaches the gridding and lsr methods are shown to be closely related the similarity between these two methods is accentuated when they are properly expressed in a common matrix form it is shown that the gridding algorithm can be considered an approximation to the least squares method the optimal gridding parameters are defined as the ones which yield the minimum approximation error these parameters are calculated by minimizing the norm of an approximation error matrix this problem is studied and solved in the general form of approximation using linearly structured matrices this method not only supports more general forms of the gridding algorithm it can also be used to accelerate the reconstruction techniques from incomplete data the application of this method to a case of two dimensional 2 d spiral magnetic resonance imaging shows a reduction of more than 4 db in the average reconstruction error

brief lower matrix bounds for the continuous algebraic riccati and lyapunov matrix equations in this paper we propose lower matrix bounds for the continuous algebraic riccati and lyapunov matrix equations we give comparisons between the parallel estimates finally we give examples showing that our bounds can be better than the previous results for some cases

a new approach for upper bound estimations of the solution of the continuous riccati equation by utilizing some linear algebraic techniques a new approach for measuring upper bounds of the solution of the continuous algebraic riccati equation care is developed two upper matrix bounds are developed comparing with those appeared in the literature the obtained bounds are less restrictive or more concise therefore they can be applied easily to deal with control problems

lower bounds for the trace of the solution of the discrete algebraic riccati equation two lower bounds for the trace of the solution of the discrete algebraic riccati equation dare are presented it is shown that in many cases these trace bounds are tighter than those in the literature and greater than the trace of the state weighting matrix even when the system matrix is singular the results are illustrated by an example

bounds on the solution of the algebraic riccati equation under perturbations in the coefficients abstract this paper is concerned with bounds on symmetric solutions of the continuous algebraic matrix riccati equation under perturbations on its coefficients special emphasis is given to the question of estimating the size of the perturbations on the stabilizing solution of the riccati equation when one or all its coefficients are subject to small perturbations upper bounds on the norm of the perturbations on the stabilizing solution are presented moreover it is shown that these perturbations can be determined as a single valued continuous function of the perturbations on the coefficient of the riccati equation

new upper solution bounds of the discrete algebraic riccati matrix equation in this note we present upper matrix bounds for the solution of the discrete algebraic riccati equation dare using the matrix bound of theorem 2 2 we then give several eigenvalue upper bounds for the solution of the dare and make comparisons with existing results the advantage of our results over existing upper bounds is that the new upper bounds of theorem 2 2 and corollary 2 1 are always calculated if the stabilizing solution of the dare exists whilst all existing upper matrix bounds might not be calculated because they have been derived under stronger conditions finally we give numerical examples to demonstrate the effectiveness of the derived results

upper and lower matrix bounds of the solution for the discrete lyapunov equation new matrix bounds of the solution for the discrete algebraic matrix lyapunov equation are established the upper and lower eigenvalue bounds such as each eigenvalue including the extreme ones the trace and the determinant are also determined by these matrix bounds for the same solution the present schemes are tighter as compared with the majority of existing results

on bounds of the riccati and lyapunov matrix equations various bounds for the traces of the solutions of the algebraic riccati and lyapunov matrix equations are established in the continuous and discrete domain respectively the presented results give lower bounds for the sum of the eigenvalue of the matrix solutions

selection of a convolution function for fourier inversion using gridding computerised tomography application in the technique known as gridding the data samples are weighted for sampling density and convolved with a finite kernel then resampled on a grid preparatory to a fast fourier transform the authors compare the artifact introduced into the image for various convolving functions of different sizes including the kaiser bessel window and the zero order prolate spheroidal wave function pswf they also show a convolving function that improves upon the pswf in some circumstances

upper bounds for the solution of the discrete riccati equation upper bounds for individual eigenvalues and for summations of eigenvalues including the trace of the solution of the discrete algebraic riccati equation are presented some are new and some supplement bounds in the literature

higher order nonlinear diffraction tomography inversion of the rytov series nonlinear tomographic reconstruction algorithms are developed for inversion of data measured in scattering experiments in which the complex phase of the wavefields is modeled by an arbitrarily large possibly infinite number of terms in the rytov series the algorithms attain the form of a volterra series of nonlinear operators with the usual filtered backpropagation algorithm of diffraction tomography as the leading linear term a computer simulation study is included to illustrate the performance of the algorithms for the case of scattering objects with cylindrical symmetry

technical communique lower bounds on the solution of coupled algebraic riccati equation lower bounds for eigenvalues and matrix lower bound of a solution to continuous time coupled algebraic riccati equation and discrete time coupled algebraic riccati equation are developed as special cases of bounds for the unified coupled algebraic riccati equation they include bounds for the minimal eigenvalues the sums of the eigenvalues the trace and the determinant

simultaneous eigenvalue lower bounds for the riccati matrix equation the author derives n simultaneous lower bounds on the sums of the eigenvalues of the solution of the algebraic riccati equation and of the lyapunov matrix equation these bounds extend the single inequalities shown in the literature for the trace and for the maximum eigenvalue two inequality versions are derived and evaluated for application purposes and a comparison to relevant bounds in the literature is given

upper bounds for the eigenvalues of the solution of the lyapunov matrix equation eigenvalue upper bounds for the solution of the lyapunov matrix equation are presented for single eigenvalues summations of eigenvalues including the trace and products of eigenvalues including the determinant these bounds are compared to those for the trace and the extreme eigenvalues

upper matrix bounds for the discrete algebraic riccati matrix equation we propose upper matrix bounds for the discrete algebraic riccati matrix equation we show that our results are less restrictive than the previous bounds finally we give numerical examples in order to verify the effectiveness of our results

a brief summary of the bounds on the solution of the algebraic matrix equations in control theory abstract in recent years several bounds have been reported for different measures of the extent or size of the solution of the algebraic matrix equations arising in control theory such as the riccati equation and the lyapunov equation this paper collects the bounds that have been presented up to now and summarizes them in an unified form this will prove particularly convenient for those wishing to get a ready estimate of the solution while solving the equations numerically or to develop theoretical results that rely on these bounds

on solution bounds for three types of lyapunov matrix equations continuous discrete and unified equations an approach is proposed to obtain a series of solution bounds for the unified algebraic lyapunov matrix equation by making use of its feature i e a mediator between the continuous time and discrete time lyapunov equations due to the approach every existing solution bound for both the continuous and discrete equations becomes intact in the sense that it yields a corresponding bound for the unified equation the approach is endorsed by this simple fact the three types of lyapunov equations share the solution provided certain relationships are maintained among corresponding coefficient matrices of the equations some examples are presented to show that the proposed method actually produces new bounds and builds bridges over existing bounds

gradient based iterative algorithms for solving a class of matrix equations in this note we apply a hierarchical identification principle to study solving the sylvester and lyapunov matrix equations in our approach we regard the unknown matrix to be solved as system parameters to be identified and present a gradient iterative algorithm for solving the equations by minimizing certain criterion functions we prove that the iterative solution consistently converges to the true solution for any initial value and illustrate that the rate of convergence of the iterative solution can be enhanced by choosing the convergence factor or step size appropriately furthermore the iterative method is extended to solve general linear matrix equations the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

iterative least squares solutions of coupled sylvester matrix equations abstract in this paper we present a general family of iterative methods to solve linear equations which includes the well known jacobi and gauss seidel iterations as its special cases the methods are extended to solve coupled sylvester matrix equations in our approach we regard the unknown matrices to be solved as the system parameters to be identified and propose a least squares iterative algorithm by applying a hierarchical identification principle and by introducing the block matrix inner product the star product for short we prove that the iterative solution consistently converges to the exact solution for any initial value the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

on upper bounds for the unified algebraic riccati equation in recent years several eigenvalues norms and determinants bounds have been investigated separately for the solutions of continuous and discrete riccati equations in this paper an upper bound for solution of the unified riccati equation is presented in the limiting cases the result reduces to a new upper bound for the solution of continuous and discrete riccati equation

on some bounds in the algebraic riccati and lyanupov equations some bounds for the trace and determinant of the solution to the algebraic riccati and lyapunov matrix equations are presented these bounds give information on mean size of the solution to the equations in a simple way the obtained determinant bound for the lyapunov equation shows the improvement over the previously reported one

on the discrete riccati equation several bounds have been reported recently for the trace of the solution to the discrete algebraic matrix riccati equation this note adds an alternative one to them

bounds on solutions to h algebraic riccati equations and h 2 properties of h central solutions abstract in this paper we give upper bounds on the stabilizing solutions of the algebraic riccati equations arising in standard h estimation and control problems for continuous time finite dimensional linear time invariant systems using these bounds we show that in h estimation or h feedback control via state feedback the so called central solution satisfies an interesting h 2 norm bound this bound may be readily used to find a central solution i e estimator or feedback controller that guarantees an equal comprise between h 2 and h objectives

reconstruction from projections under time frequency constraints low pass filtering computed tomography ct images to reduce noise may smooth or modify image features which are very important to the physician image features are often more easily identified and processed in the time frequency plane the authors use time frequency distributions for spatially varying filtering of noisy ct images constraining time frequency representation coefficients of the projection data or of the reconstructed image to be zero in certain regions of the time frequency plane the authors consider two different applications 1 filtering the projection data and then performing image reconstruction and 2 filtering the reconstructed image directly criteria minimized subject to constraints may be either a deterministic minimum weighted perturbation of the given projection data or a stochastic minimum mean square error in colored gaussian noise results show improvement over processing the image with a linear spatially invariant filter

a computational study of reconstruction algorithms for diffraction tomography interpolation versus filtered backpropagation from the standpoint of reporting a new contribution this paper shows that by using bilinear interpolation followed by direct two dimensional fourier inversion one can obtain reconstructions of quality which is comparable to that produced by the filtered backpropagation algorithm proposed recently by devaney for an n n image reconstructed from n diffracted projections the former approach requires approximately 4n fft s whereas the backpropagation technique requires approximately n2fft s we have also taken this opportunity to present the reader with a tutorial introduction to diffraction tomography an area that is becoming increasingly important not only in medical imaging but also in underwater and seismic mapping with microwaves and sound the main feature of the tutorial part is the statement of the fourier diffraction projection theorem which is an extension of the traditional fourier slice theorem to the case of image formation with diffracting illumination

decay rate estimations for linear quadratic optimal regulators let u t fx t be the optimal control of the open loop system x t ax t bu t in a linear quadratic optimization problem by using different complex variable arguments we give several lower and upper estimates of the exponential decay rate of the closed loop system x t a bf x t main attention is given to the case of a skew hermitian matrix a given an operator a for a class of cases we find a matrix b that provides an almost optimal decay rate we show how our results can be applied to the problem of optimizing the decay rate for a large finite collection of control systems a bj j 1 n and illustrate this on an example of a concrete mechanical system at the end of the article we pose several questions concerning the decay rates in the context of linear quadratic optimization and in a more general context of the pole placement problem

spectral norm and trace bounds of algebraic matrix riccati equations upper bounds on spectral norm and trace for the solution of continuous algebraic matrix riccati equations are proposed the proposed bounds are based on a form of the general analytic solution it is shown that in many cases the spectral norm and the trace bounds are better than other results

bounds for the eigenvalues of the solution of the unified algebraic riccati matrix equation abstract in recent years several eigenvalues bounds have been investigated separately for the solutions of the continuous and the discrete riccati and lyapunov matrix equations in this paper lower bounds for the eigenvalues of the solution of the unified riccati equation relatively to continuous and discrete cases are presented in the limiting cases the results reduce to some new bounds for both the continuous and discrete riccati equation

eigenvalue upper bounds of the solution of the continuous riccati equation new upper bounds for the solution eigenvalues of the continuous algebraic matrix riccati equation are developed they include bounds of the extreme eigenvalues the summation and product of eigenvalues the trace and the determinant it is shown that the majority of the present eigenvalue bounds expressed in concise forms are less restrictive and sharper than existing results

direct fourier reconstruction in computer tomography direct fourier transform ft reconstruction of images in computerized tomography ct is not widely used because of the difficulty of precisely interpolating from polar to cartesian samples in this paper an exact interpolation scheme is proposed which in practice can be approached with arbitrary accuracy using well conditioned algorithms several features of the direct ft method are discussed a method that allows angular band limiting of the data before processing to avoid angular aliasing artifacts in the reconstructed image is discussed and experimentally verified the experimental results demonstrate the feasibility of direct ft reconstruction of ct data

lower bounds on the solution of lyapunov matrix and algebraic riccati equations in this note various lower bounds for all the eigenvalues of the solution matrix k of the lyapunov matrix equation are established a special case of this result is a generalization of that presented in 1 3 where lower bounds for the maximum and minimum eigenvalues of k are given moreover the approach used here enables one to establish various lower bounds for some of the largest eigenvalues of the solution matrix of the algebraic riccati equation

the least squares anti bisymmetric solution and the optimal approximation solution for sylvester equation in this paper a modified conjugate gradient iterative method for solving sylvester equation is presented by using this iterative method the least squares anti bisymmetric solution and the optimal approximation solution can be obtained here we present the derivation and theoretical analysis of our iterative method numerical results illustrate the feasibility and effectiveness of the proposed iterative method

universal spectral parameter dependent lax operators for the drinfeld double of the dihedral group d3 two universal spectral parameter dependent lax operators are presented in terms of the elements of the drinfeld double d d3 of the dihedral group d3 applying representations of d d3 to these yields matrix solutions of the yang baxter equation with a spectral parameter

a stability enhancing scaling procedure for schur riccati solvers abstract scaling riccati equations so that the solution has unit norm eliminates instabilities associated with the inversion step of invariant subspace based schur methods without changing the relative condition number of the underlying problem such optimal scaling makes it easy to estimate this condition number and to bound the relative error of the computed solution moreover the determination of whether the first step of newton s method will improve this solution is also simplified

bounds for the eigenvalues of the solution of the discrete riccati and lyapunov equations and the continuous lyapunov equation we present some bounds for the eigenvalues and certain sums and products of the eigenvalues of the solution of the discrete riccati and lyapunov matrix equations and the continuous lyapunov matrix equation nearly all of our bounds for the discrete riccati equation are new the bounds for the discrete and continuous lyapunov equations give a completion of some known bounds for the extremal eigenvalues and the determinant and the trace of the solution of the respective equation