a block pseudospectral method for maxwell s equations a block pseudospectral bps method is proposed as a new way to couple pseudospectral discretizations across interfaces in computations for a linear hyperbolic system the coupling is achieved via discretized derivative matching conditions obtained from the system compared to the standard technique of imposing compatibility conditions based on characteristics of the system the bps method offers better stability and accuracy especially in the case where equation coefficients are discontinuous computational examples for maxwell s equations in nonhomogeneous media demonstrate that bps retains high accuracy over times that are orders of magnitude larger than those for not only low order methods such as yee s but also high order methods such as characteristic based spectral elements

existence and comparison theorems for algebraic riccati equations for continuous and discrete time systems abstract we discuss two comparison theorems for algebraic riccati equations of the form xbr 1 b x x a br 1 c a br 1 c x q c r 1 c 0 simultaneously we give sufficient conditions to obtain the existence of the maximal hermitian solution of a riccati equation from the existence of a hermitian solution of a second riccati equation further similar results are given for discrete algebraic riccati equations

accuracy resolution and stability properties of a modified chebyshev method while the chebyshev pseudospectral method provides a spectrally accurate method integration of partial differential equations with spatial derivatives of order m requires time steps of approximately o n 2m for stable explicit solvers theoretically time steps may be increased to o n m with the use of a parameter alpha dependent mapped method introduced by kosloff and tal ezer em j comput phys 104 1993 pp 457 469 our analysis focuses on the utilization of this method for reasonable practical choices for n namely n lesssim 30 as may be needed for two or three dimensional modeling results presented confirm that spectral accuracy with increasing n is possible both for constant alpha hesthaven dinesen and lynov em j comput phys 155 1999 pp 287 306 and for alpha scaled with n alpha sufficiently different from 1 don and solomonoff em siam j sci comput 18 1997 pp 1040 1055 theoretical bounds however show that any realistic choice for alpha in which both resolution and accuracy considerations are imposed permits no more than a doubling of the time step for a stable explicit integrator in time much less than the o n improvement claimed by kosloff and tal ezer on the other hand by choosing alpha carefully it is possible to improve on the resolution of the chebyshev method in particular one may achieve satisfactory resolution with fewer than pi points per wavelength moreover this improvement is noted not only for waves with the minimal resolution but also for waves sampled up to about 8 points per wavelength our conclusions are verified by calculation of phase and amplitude errors for numerical solutions of first and second order one dimensional wave equations specifically while alpha can be chosen such that the mapped method improves the accuracy and resolution of the chebyshev method for practical choices of n it is not possible to achieve both single precision accuracy and gain the advantage of an o n m time step

the solution set of the algebraic riccati equation and the algebraic riccati inequality abstract we present a parametrization of the solution set of the algebraic riccati equation and the algebraic riccati inequality of optimal control where we assume only sign controllability of the underlying system furthermore we characterize the existence of maximal and minimal solutions of both the equation and the inequality by checkable conditions

brief suboptimal markovian smoothing estimates based on continuous curves of solutions of the algebraic riccati inequality based on a result on continuous dependence of solutions of an algebraic riccati equation on the data matrices we construct continuous curves of solutions of an algebraic riccati inequality and derive suboptimal markovian estimates for the steady state smoothing problem

multiobjective h sub 2 h sub spl infin control for a linear time invariant system with several disturbance inputs and controlled outputs we show how to minimize the nominal h sub 2 norm performance in one channel while keeping bounds on the h sub 2 norm or h sub spl infin norm performance implying robust stability in the other channels this multiobjective h sub 2 h sub spl infin problem in an infinite dimensional space is reduced to sequences of finite dimensional convex optimization problems we show how to compute the optimal value and how to numerically detect the existence of a rational optimal controller if it exists we reveal how the novel trick of optimizing the trace norm of the youla parameter over certain convex constraints allows one to design a nearly optimal controller whose youla parameter is of the same order as the optimal one

necessary and sufficient conditions for solvability of the riccati inequalities in the general case abstract this paper presents necessary and sufficient conditions for solvability of riccati inequalities in the general case in terms of purely imaginary eigenvalues of associated hamiltonian matrices

the eigenvalues of hermite and rational spectral differentiation matrices we derive expressions for the eigenvalues of spectral differentiation matrices for unbounded domains in particular we consider galerkin and collocation methods based on hermite functions as well as rational functions a fourier series combined with a cotangent mapping we show that i first derivative matrices have purely imaginary eigenvalues and second derivative matrices have real and negative eigenvalues ii for the hermite method the eigenvalues are determined by the roots of the hermite polynomials and for the rational method they are determined by the laguerre polynomials and iii the hermite method has attractive stability properties in the sense of small condition numbers and spectral radii

the algebraic riccati inequality parametrization of solutions tightest local frames and generalized feedback matrices abstract extending on previous work by faurre scherer and pavon a parametrization of the symmetric solutions of the algebraic riccati inequality is established this is then applied to derive new results on tightest local frames and on generalized feedback matrices that arise in stochastic realization theory

minimal representations of continuous time processes having spectral density with zeros in the extended imaginary axis given a process y with stationary increments and rational incremental spectral density s we parametrize minimal state space representations of dy we consider the case when s may have zeros on the imaginary axis including infinity and we analyze the zero structure of each state space representation

an exponentially fitted finite difference method for singular perturbation problems in this paper an exponentially fitted finite difference method is presented for solving singularly perturbed two point boundary value problems with the boundary layer at one end left or right point a fitting factor is introduced in a tridiagonal finite difference scheme and is obtained from the theory of singular perturbations thomas algorithm is used to solve the system the stability of the algorithm is investigated several linear and nonlinear problems are solved to demonstrate the applicability of the method it is observed that the present method approximates the exact solution very well

the regular indefinite linear quadratic optimal control problem stabilizable case this paper addresses an open problem in the area of linear quadratic optimal control we consider the regular infinite horizon stability modulo a subspace indefinite linear quadratic problem under the assumption that the dynamics are stabilizable our result generalizes previous works dealing with the same problem in the case of controllable dynamics we explicitly characterize the unique solution of the algebraic riccati equation that gives the optimal cost and optimal feedback control as well as necessary and sufficient conditions for the existence of optimal controls

numerical solution of the orr sommerfeld equation using the viscous green function and split gaussian quadrature we continue our study of the construction of numerical methods for solving two point boundary value problems using green functions building on the successful use of split gauss type quadrature schemes here we adapt the method for eigenvalue problems in particular the orr sommerfeld equation of hydrodynamic stability theory use of the green function for the viscous part of the problem reduces the fourth order ordinary differential equation to an integro differential equation which we then discretize using the split gaussian quadrature and product integration approach of our earlier work along with pseudospectral differentiation matrices for the remaining differential operators as the latter are only second order the resulting discrete equations are much more stable than those obtained from the original differential equation this permits us to obtain results for the standard test problem plane poiseuille flow at unit streamwise wavenumber and reynolds number 10000 that we believe are the most accurate to date

on hermitian solutions of the symmetric algebraic riccati equation the structure of the set of hermitian solutions of the matrix quadratic equation xdx xa a x c 0 is studied under the conditions that c c d is positive semidefinite and a d is stabilizable new features e g nonexistence of the minimal solution appear in contrast with the known case when a d is controllable

on the commutant algebras corresponding to the permutation representations of the full collineation groups of pg k s and eg k s s pr k 2 abstract in this paper the dimension t and a linear basis of the commutant algebra corresponding to the representation of the full collineation group as matrices permuting the flags incident point line or point hyperplane pairs have been determined for each one of the four geometries pg 2 s eg 2 s pg k s and eg k s s pr k 3 for the four geometries t 6 7 7 and 8 respectively and the corresponding linear bases are i g b t bt tb i g b t bt tb btb i g b t bt tb s and i g b t bt tb btb s i g b t are the relationship matrices of james ann math statist 28 1957 993 1082 and the matrix s was introduced by sysoev and shaikin avtomat i telemekh 5 1976 64 73

on the lyapunov equation coinvariant subspaces and some problems related to spectral factorizations a geometric approach to stochastic realization theory and hence to spectral factorization problems has been developed by lindquist and picci 1985 1991 and lindquist et al 1995 most of this work was done abstractly fuhrmann and gombani 1998 adopted an entirely hardy space approach to this set of problems studying the set of rectangular spectral factors of given size for a weakly coercive spectral function the parametrization of spectral factors in terms of factorizations of related inner functions as developed in fuhrmann 1995 had to be generalized this led to a further understanding of the partial order introduced by lindquist and picci in the set of stable spectral factors in the present paper we study the geometry of finite dimensional coinvariant subspaces of a vectorial hardy space h 2 via realization theory emphasizing the role of the lyapunov equation in lifting the hardy space metric to the statespace domain we follow this by deriving state space formulas for rectangular spect

a new robust line search technique based on chebyshev polynomials abstract newton s method is an important and basic method for solving nonlinear univariate and unconstrained optimization problems in this study a new line search technique based on chebyshev polynomials is presented the proposed method is adaptive where it determines a descent direction at each iteration and avoids convergence to a maximum point approximations to the first and the second derivatives of a function using high order pseudospectral differentiation matrices are derived the efficiency of the new method is analyzed in terms of the most popular and widely used criterion in comparison with newton s method using seven test functions

on a connection between spectral factorization and geometric control theory we investigate here how the geometric control theory of basile marro and wonham can be obtained in a hilbert space context as the byproduct of the factorization of a spectral density with no zeros on the imaginary axis we show how controlled invariant subspaces can be obtained as images of orthogonal projections of co invariant subspaces onto a semi invariant markovian subspace of the hardy space of square integrable functions analytic in the right half plane output nulling subspaces are then related to a particular spectral factorization problem a similar construction is presented for controllability subspaces and a new algorithm for the computation of these subspaces is presented

matrix riccati inequality existence of solutions abstract we consider the various criteria including hamiltonian and frequency domain ones for solvability of the matrix riccati inequality arising from control problems some criteria are given in terms of symplectic algebra which is also used as an important technical tool

a fast spectrally accurate homotopy based numerical method for solving nonlinear differential equations abstract we present an algorithm for constructing numerical solutions to one dimensional nonlinear variable coefficient boundary value problems this scheme is based upon applying the homotopy analysis method ham to decompose a nonlinear differential equation into a series of linear differential equations that can be solved using a sparse spectrally accurate gegenbauer discretisation uniquely for nonlinear methods our scheme involves constructing a single sparse matrix operator that is repeatedly solved in order to solve the full nonlinear problem as such the resulting scheme scales quasi linearly with respect to the grid resolution we demonstrate the accuracy and computational scaling of this method by examining a fourth order nonlinear variable coefficient boundary value problem by comparing the scheme to newton iteration and the spectral homotopy analysis method which is the most commonly used implementation of the ham

acausal models and balanced realizations of stationary processes abstract we study acausal realizations of stationary or stationary increment processes in particular we characterize the family of models whose corresponding spectral factors have a fixed zero structure acausal models with a fixed zero structure are related to each other by a certain group of state feedback transformations which is naturally parametrized by the solution set of a homogeneous algebraic riccati equation each feedback transformation reflects some of the eigenvalues of the generator matrix a of the representation to a mirror image with respect to the imaginary axis dually acausal models with a fixed pole structure are parametrized by a dual riccati equation and by a corresponding family of output injection transformations from a general standpoint the results of this study clarify the role played by dual pairs of riccati equations in spectral factorization and may be relevant to other problem areas than stochastic modelling one natural application of the concepts discussed in the paper is to stochastic balancing balancing of models with an essentially arbitrary eigenvalue location can be accomodated very naturally in this framework a balancing algorithm involving the solution of a dual pair of riccati equations is discussed

efficient and stable generation of higher order pseudospectral integration matrices the main purpose of this work is to provide new higher order pseudospectral integration matrices hpims for the chebyshev type points and present an exact efficient and stable approach for computing the hpims the essential idea is to reduce the computation of hpims to that of higher order chebyshev integration matrices hcims and take a very simple and recursive way to compute the hcims efficiently and stably extensive numerical results show that the new approach for computing the hpims has better stability than that of the recently derived elgindy s approach for large number of chebyshev type points

the errors in calculating the pseudospectral differentiation matrices for cebysev gauss lobatto points abstract we discuss here the errors incurred using the standard formula for calculating the pseudospectral differentiation matrices for cebysev gauss lobatto points we propose explanations for these errors and suggest more precise methods for calculating the derivatives and their matrices

algebraic riccati equation and symplectic algebra questions of existence uniqueness and the parametric dependence of solutions of the algebraic riccati equation are considered the different criteria for the solubility of this equation are obtained with the help of symplectic algebra

hermitian solutions of the discrete time algebraic riccati equation this paper deals with the existence and uniqueness of unmixed solutions of the discrete time algebraic riccati equation dare sets of solutions of the dare are parametrized by invariant subspaces of the state transition matrix

asymmetricalgebraicriccatiequation ahomeomorphicparametrizationofthesetof solutions abstract in this paper asymmetric algebraic riccati equations are analyzed in particular we derive a new parametrization of the set of solutions generalizing on the symmetric case the proposed parametrization is obtained in terms of pairs of invariant subspaces of two related feedback matrices moreover the connection is clarified between the new parametrization and the classical homeomorphic one based on graph invariant subspaces of the pseudo hamiltonian matrix associated with the equation we finally show that also the newly introduced parametrization is given by a homeomorphic map

on the newtonkleinman method for strongly stabilizable infinite dimensional systems we consider the newtonkleinman method for strongly stabilizable infinite dimensional systems under certain assumptions the maximal self adjoint solution to the associated control algebraic riccati equation is constructed the constructed solution is also the maximal solution to the corresponding control algebraic riccati inequality

some results on linear rational trigonometric interpolation abstract we present here formulae for calculating the p th derivative of a linear rational trigonometric interpolant written in barycentric form we give sets of interpolating points for which the interpolant converges exponentially towards the interpolated function

lattice properties of sets of semidefinite solutions of continuous time algebraic riccati equations the set of negative semidefinite solutions of the algebraic riccati equation a x xa xbb x c c 0 is studied under the most general hypothesis namely that a b is stabilizable modulo the undetectable subspace of a c if is a lattice then is isomorphic to 0 s where t0 is the direct sum of copies of the set of nonnegative real numbers and is a closure of which is order isomorphic to a system of well defined a invariant subspaces necessary and sufficient conditions are given such that or is a lattice

l sub 2 optimal filter reduction a closed loop approach this paper is concerned with the problem of order reduction of a full order kalman filter for a stable linear signal model so that the steady state filtering error variance associated with the reduced order filter is minimized by an orthogonal parameterization the above problem is formulated to minimize the filtering error variance over a set of orthogonal matrices both continuous and iterative algorithms are derived to compute an optimal reduced order filter the algorithms are shown to possess good properties including the desirable convergence property the proposed algorithms are simple and effective numerical examples are presented to demonstrate the effectiveness and the significant advantages of the proposed algorithms over the existing open loop methods such as the well known balanced truncation method

yet another characterization of solutions of the algebraic riccati equation abstract this paper deals with a characterization of the solution set of algebraic riccati equation are over reals for both controllable and uncontrollable systems we characterize all solutions using simple linear algebraic arguments it turns out that solutions of are of maximal rank have lower rank solutions encoded within it we demonstrate how these lower rank solutions are encoded within the full rank solution and how one can retrieve the lower rank solutions from the maximal rank solution we characterize situations where there are no full rank solutions to the are we also characterize situations when the number of solutions to the are is finite when they are infinite and when they are bounded we also explore the poset structure on the solution set of are which in some specific cases turns out to be a lattice which is isomorphic to lattice of invariant subspaces of a certain matrix we provide several examples that bring out the essence of these results

spectral factorization with imaginary axis zeros abstract we examine the problem of the existence and calculation of hermitian solutions p of a linear matrix inequality corresponding to the spectral factorization of a proper rational spectral density in particular zeros are permitted on the finite imaginary axis or at infinity as part of our construction of solutions we show explicitly that each such imaginary axis eigenvalue either finite or infinite defines a fixed part of all solutions p

the dual algebraic riccati equations and the set of all solutions of the discrete time riccati equation abstractin this paper two new pairs of dual continuous time algebraic riccati equations cares and dual discrete time algebraic riccati equations dares are proposed the dual dares are first studied with some nonsingularity assumptions on the system matrix and the parameter matrix then in the case of singular matrices a generalised inverse is introduced to deal with the dual dare problem these dual ares can easily lead us to an iterative procedure for finding the anti stabilising solutions especially to dare by means of that for the stabilising solutions furthermore we provide the counterpart results on the set of all solutions to dare inspired by the results for care two examples are presented to illustrate the theoretical results

a pseudospectral fictitious point method for high order initial boundary value problems when pseudospectral approximations are used for space derivatives one often encounters spurious eigenvalues these can lead to severe time stepping difficulties for pdes this is especially the case for equations with high order derivatives in space requiring multiple conditions at one or both boundaries we note here that a very simple to implement fictitious point approach circumvents most of these difficulties the new approach is tested on the kuramoto sivashinsky equation and on a dispersive linear pde featuring a time space corner singularity

on a hardy space approach to the analysis of spectral factors the paper presents a study of several problems related to spectral factorizations we assume only a very weak form of coercivity for the p x p spectral function phi and look at the set wm of all rectangular p x m spectral factors the main object is the arithmetization of the geometry of the set of minimal stable spectral factors by employing hardy space techniques and the arithmetic of inner functions particular attention is paid to the study of various partial orders associated with the set of spectral factors

continuous and discrete time riccati theory a popov function approach abstract necessary and sufficient conditions for the existence of the stabilizing solution of the algebraic riccati equation are derived for both the continuous and the discrete time case under the weakest possible assumptions imposed on the initial data both frequency domain conditions involving an associated popov function and time domain conditions involving an associated matrix pencil are investigated

the kalman yakubovich popov inequality for differential algebraic systems abstract in this paper we revisit the kalman yakubovich popov lemma for differential algebraic control systems this lemma relates the positive semi definiteness of the popov function on the imaginary axis to the solvability of a linear matrix inequality on a certain subspace further emphasis is placed on the lur e equation whose solution set consists loosely speaking of the rank minimizing solutions of the kalman yakubovich popov inequality we show that there is a correspondence between the solution set of the lur e equation and the deflating subspaces of certain even matrix pencils finally we show that under certain conditions the lur e equation admits stabilizing anti stabilizing and extremal solutions we note that for our results we neither assume impulse controllability nor make any assumptions on the index of the system

zeros of spectral factors the geometry of splitting subspaces and the algebraic riccati inequality in this paper it is shown how the zero dynamics of not necessarily square spectral factors relate to the splitting subspace geometry of stationary stochastic models and to the corresponding algebraic riccati inequality the notion of output induced subspace of a minimal markovian splitting subspace which is the stochastic analogue of the supremal output nulling subspace in geometric control theory is introduced through this concept the analysis can be made coordinate free and straightforward geometric methods can be applied it is shown how the zero structure of the family of spectral factors relates to the geometric structure of the family of minimal markovian splitting subspaces in the sense that the relationship between the zeros of different spectral factors is reflected in the partial ordering of minimal splitting subspaces finally the well known characterization of the solutions of the algebraic riccati equation is generalized in terms of lagrangian subspaces invariant under the corresponding hamiltonian to the larger solution set of the algebraic riccati inequality

quadrature discretization method in tethered satellite control a pseudospectral method for solving the tethered satellite retrieval problem based on nonclassical orthogonal and weighted interpolating polynomials is presented traditional pseudospectral methods expand the state and control trajectories using global lagrange interpolating polynomials based on a specific class of orthogonal polynomials from the jacobi family such as legendre or chebyshev polynomials which are orthogonal with respect to a specific weight function over a fixed interval although these methods have many advantages the location of the collocation points are more or less fixed the method presented in this paper generalizes the existing methods and allows a much more flexible selection of grid points by the arbitrary selection of the orthogonal weight function and interval the trajectory optimization problem is converted to set of algebraic equations by discretization of the necessary conditions using a nonclassical pseudospectral method

generation of finite difference formulas on arbitrarily spaced grids on etablit des recurrences simples pour le calcul des poids dans les formules aux differences finies compactes pour les derivees de tous ordres avec une precision d ordre arbitraire sur des grilles a une dimension d espacement arbitraire

a parametrization of solutions of the discrete time algebraic riccati equation based on pairs of opposite unmixed solutions the paper describes the set of solutions of the discrete time algebraic riccati equation it is shown that each solution is a combination of a pair of opposite unmixed solutions there is a one to one correspondence between solutions and invariant subspaces of the closed loop matrix of an unmixed solution the results of the paper provide an extended counterpart of the parametrization theory of continuous time algebraic riccati equations by willems coppel and shayman

fast computation of the spectral differentiation by the fast multipole method abstract this paper is concerned with the application of the fast multipole method fmm to the pseudospectral methods the spectral differentiation in pseudospectral methods involves dense matrix vector products which if evaluated directly entail o n 2 operations the core of this work lies in the fact that the fmm leads to a significant speed up in cpu time with a major reduction in the amount of computer memory needed when performing matrix vector products the fmm is exploited to estimate the barycentric weights and the pseudospectral differentiation which reduces the complexity to o n furthermore we show the convergence analysis and present the applications to the second order boundary value problems numerical examples confirm the advantages of the method

series solution of nonlinear two point singularly perturbed boundary layer problems the present paper is concerned with the approximate analytic series solution of the nonlinear two point second order singularly perturbed boundary value problems in place of the traditional numerical perturbation or asymptotic methods a homotopy technique is employed it is shown that proper choices of an auxiliary linear operator and also an initial approximation during the implementation of the homotopy analysis method ham can yield uniformly valid and accurate solutions the fast convergence of the method is ensured by the optimal convergence control parameter obtained through the absolute residual error concept to demonstrate the favor of the ham over the traditional finite difference techniques several nonlinear problems have been solved and compared

on discrete algebraic riccati equations a rank characterization of solutions abstract we give a rank characterization of the solution set of discrete algebraic riccati equations dares involving real matrices our characterization involves the use of invariant subspaces of the system matrix we observe that a dare can be considered as a specific type of a non symmetric continuous time algebraic riccati equation we show how to extract the low rank solutions from the full rank solution of a homogeneous dare we also study the eigen structure of various feedback closed loop matrices associated with solutions of a dare we show as a consequence of our observations that a solution x of a dare for controllable systems and certain specific types of uncontrollable systems satisfies the inequality x m i n x x m a x we characterize conditions under which the solution set of dare is bounded unbounded we use the method of invariant subspaces to identify some invariant sets of a difference riccati equation

a modified chebyshev pseudospectral dd algorithm for the gbh equation in this paper a chebyshev spectral collocation domain decomposition dd semi discretization by using a grid mapping derived by kosloff and tal ezer in space is applied to the numerical solution of the generalized burger s huxley gbh equation to reduce roundoff error in computing derivatives we use the above mentioned grid mapping in this work we compose the chebyshev spectral collocation domain decomposition and kosloff and tal ezer grid mapping elaborately firstly the theory of application of the chebyshev spectral collocation method with grid mapping and dd on the gbh equation is presented this method yields a system of ordinary differential algebraic equations daes secondly we use a fourth order runge kutta formula for the numerical integration of the system of daes application of this modified method to the gbh equation show that this method m dd is faster and more accurate than the standard chebyshev spectral collocation dd s dd method

a boundary value technique for singular perturbation problems abstract a new way to solve singular perturbation problems is introduced it is designed for the practicing engineer or applied mathematician who needs a practical tool for these problems easy to use modest problem preparation and ready computer implementation as with other methods the original problem is partitioned into inner and outer solution differential equation systems however asymptotic solution techniques are not employed the method is distinguished by the following facts the inner solution problem is solved as a two point boundary problem where the terminal boundary conditions are supplied by the solution of the outer solution problem in turn the outer solution differential equations are solved as an initial value system where the initial conditions are supplied by the inner solution at its terminal point the method is iterative on the terminal point of the inner solution problem three numerical examples are included

lur e equations and even matrix pencils abstract in this work we consider the so called lur e matrix equations that arise e g in model reduction and linear quadratic infinite time horizon optimal control we characterize the set of solutions in terms of deflating subspaces of even matrix pencils in particular it is shown that there exist solutions which are extremal in terms of definiteness it is shown how these special solutions can be constructed via deflating subspaces of even matrix pencils

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

homogeneous riccati equations versus quadratic inequalities it is proved that under certain hypotheses the existence of hermitian solutions of a homogenized algebraic riccati equation of continuous type is guaranteed if the corresponding quadratic inequality admits a hermitian solution both finite dimensional matrices and infinite dimensional hilbert space operators riccati equations are considered

uncontrollable dissipative systems observability and embeddability the theory of dissipativity is well developed for controllable systems a more appropriate definition of dissipativity in the context of uncontrollable systems is in terms of the existence of a storage function namely a function such that along every system trajectory its rate of change at each time instant is at most the power supplied to the system at that time however even when the supplied power is expressible in terms of just the external variables the dissipativity property for uncontrollable systems crucially hinges on whether or not the storage function depends on variables unobservable hidden from the external variables this paper investigates the key aspects of both cases and also proposes another intuitive definition of dissipativity these three definitions are compared we show that drawbacks of one definition are addressed by another dealing first with observable storage functions under the conditions that no two uncontrollable poles add to zero and that dissipativity is strict as f

numerical treatment of singularly perturbed two point boundary value problems we propose a method for numerically solving linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval this is a practical method and can be easily implemented on a computer the original problem is divided into inner and outer region differential equation systems the reduced problem is solved to obtain the terminal boundary condition then a new inner region problem is created and solved as a two point boundary value problem tpbvp in turn the outer region problem is also solved as a tpbvp both these tpbvps are efficiently treated by employing a slightly modified classical finite difference scheme coupled with discrete invariant imbedding algorithm to obtain the numerical solutions the stability of some recurrence relations involved in the algorithm is investigated the proposed method is iterative on the terminal point some numerical examples are included and the computational results are compared with exact solutions it is observed that the accuracy predicted can always be achieved with very little computational effort

families of solutions of algebraic riccati equations abstract we consider homogeneous algebraic riccati equations in the general situation when the matrix of the dynamics can be mixed we show that in this case the equation may have infinitely many families of solutions an analysis of these families is carried over and an explicit formula is derived when the matrix of the dynamics is unmixed this formula provides an explicit parametrization of the whole set of the solutions we finally show that there are situations in which the equation has spurious solutions that do not belong to any of the families and derive sufficient conditions under which on the contrary the union of the families covers the whole set of solutions

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

brief paper on the stability of a reduced order filter based on dominant singular value decomposition of the system dynamics the stability of a reduced order filter rof the gain of which is constructed on the basis of a subspace of dominant singular vectors of the system dynamics is examined a definition of s detectability is introduced it is found that the observability of all unstable and neutral singular vectors s detectability is a sufficient condition for the existence of a stable filter

hankel norm approximation for well posed linear systems the sub optimal hankel norm approximation problem is solved for a well posed linear system with generating operators a b c and transfer function g satisfying some mild assumptions in the special case of the sub optimal nehari problem an explicit parameterization of all solutions is obtained in terms of the system parameters a b c and g 0

the algebraic riccati equation conditions for the existence and uniqueness of solutions abstract unmixed solutions of the matrix equation xdx xa ax c 0 d 0 are studied

a spectral collocation method based on integrated chebyshev polynomials for two dimensional biharmonic boundary value problems this paper reports a new spectral collocation method for numerically solving two dimensional biharmonic boundary value problems the construction of the chebyshev approximations is based on integration rather than conventional differentiation this use of integration allows i the imposition of the governing equation at the whole set of grid points including the boundary points and ii the straightforward implementation of multiple boundary conditions the performance of the proposed method is investigated by considering several biharmonic problems of first and second kinds more accurate results and higher convergence rates are achieved than with conventional differential methods

the set of positive semidefinite solutions of the algebraic riccati equation of discrete time optimal control in this paper the algebraic riccati equation are of the discrete time linear quadratic lq optimal control problem and its set of positive semidefinite solutions is studied under the most general assumption which is output stabilizability with respect to an appropriate basis the discrete time algebraic riccati equation dare decomposes into a lyapunov equation and an irreducible riccati equation the focus is on the riccati part which amounts to studying a dare where all unimodular modes are controllable a bijection between positive semidefinite solutions and certain well defined sets of f invariant subspaces is established which together with its inverse is order reversing as an application issues concerning positive definite or strong solutions are clarified analogous results for negative semidefinite solutions are valid only under an additional assumption on the unobservable subspace

a numerical solution for non classical parabolic problem based on chebyshev spectral collocation method a numerical method based on chebyshev polynomials and local interpolating functions is proposed for solving the one dimensional parabolic partial differential equation subject to non classical conditions to assess its validity and accuracy the method is applied to solve several test problem

spectral differentiation matrices for the numerical solution of schrodinger s equation schrodinger s equation is solved numerically using the spectral collocation pseudospectral method based on hermite weighted polynomial interpolants several sets of numerical results from the literature are reproduced using this approach matlab codes that can serve as templates for further exploration are provided both in the paper and online a new proposal is made for solving schrodinger s equation in stokes wedges

designing parametric linear quadratic regulators for parametric lti systems via lmis this paper addresses the problem of determining parametric linear quadratic regulators lqrs for continuous time linear time invariant systems affected by parameters through rational functions th

a mixed binary non linear programming approach for the numerical solution of a family of singular optimal control problems abstractthis paper presents a new approach for the efficient solution of singular optimal control problems socps a novel feature of the proposed method is that it does not require a priori knowledge of the structure of solution at first the socp is converted into a binary optimal control problem then by utilising the pseudospectral method the resulting problem is transcribed to a mixed binary non linear programming problem this mixed binary non linear programming problem which can be solved by well known solvers allows us to detect the structure of the optimal control and to compute the approximating solution the main advantages of the present method are that 1 without a priori information the structure of optimal control is detected 2 it produces good results even using a small number of collocation points 3 the switching times can be captured accurately these advantages are illustrated through a numerical implementation of the method on four examples

on the solution of the riccati differential equation arising from the lq optimal control problem in this paper we consider the matrix riccati differential equation rde that arises from linear quadratic lq optimal control problems in particular we establish explicit closed formulae for the solution of the rde with a terminal condition using particular solutions of the associated algebraic riccati equation we discuss how these formulae change as assumptions are progressively weakened an application to lq optimal control is briefly analysed

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

brief paper positivity preserving h model reduction for positive systems this paper is concerned with the model reduction of positive systems for a given stable positive system our attention is focused on the construction of a reduced order model in such a way that the positivity of the original system is preserved and the error system is stable with a prescribed h performance based upon a system augmentation approach a novel characterization on the stability with h performance of the error system is first obtained in terms of linear matrix inequality lmi then a necessary and sufficient condition for the existence of a desired reduced order model is derived accordingly furthermore iterative lmi approaches with primal and dual forms are developed to solve the positivity preserving h model reduction problem finally a compartmental network is provided to show the effectiveness of the proposed techniques

global properties of pseudospectral methods abstract we apply polynomial interpolation methods both to the approximation of functions and to the numerical solutions of hyperbolic and elliptic partial differential equations the derivative matrix for a general sequence of collocation points is explicitly constructed we explore the effect of several factors on the performance of these methods we show that global methods cannot be interpreted in terms of local methods for example the accuracy of the approximation differs when large gradients of the function occur near the center of the region or in the vicinity of the boundary this difference does not depend on the density of the collocation points near the boundaries hence intuition based on finite difference methods can lead to false results

geometric aspects of the riccati difference equation in the nonsymmetric case abstract this paper provides a collection of geometric results concerning the nonsymmetric riccati difference equation it gives representation formulas which express all the solutions in terms of a suitable number of known ones further results include geometric properties of the difference of two solutions and the generalization of some of the author s earlier results to the nonsymmetric case

a necessary and sufficient condition for the existence of a tightest local frame abstract the local structure of the solution set of the algebraic riccati inequality can be described by the concept of tightest local frame introduced by lindquist and picci 1991 in a recent paper on stochastic modelling and estimation in this technical note we provide a necessary and sufficient condition for the existence of such a tightest local frame around a given solution of the riccati inequality in addition it is shown that this tightest local frame can actually be characterized if it exists

monotonicity of maximal solutions of algebraic riccati equations abstract a monotonicity result for the maximal solution of the equation xbb x a x xa q 0 q q a b stabilizable is proved

on the norms of inverses of pseudospectral differentiation matrices in this paper we give integral expressions for the elements of the inverses of second order pseudospectral differentiation matrices simple upper bounds are given for the maximum norms of these inverse matrices when chebyshev collocation points are used comment is made on the failure to obtain upper bounds that are uniform in the number of collocation points when the points are evenly spaced we also give integral expressions for inverses of first order chebyshev pseudospectral differentiation matrices

generalizing the algebraic riccati equation to higher order behavioral systems the well known kalman yakubovich popov lemma establishes an equivalence between dissipativity and the solvability of a linear matrix inequality see r a b van der geest and h l trentelman systems control lett 32 1997 pp 283 290 i masubuchi systems control lett 55 2006 pp 158 164 in this paper we strengthen this result by showing the equivalence of dissipativity to the solvability of the so called lur e equation t reis linear algebra appl 434 2011 pp 152 173 which mainly is a linear matrix inequality with a rank minimizing condition while the results about the solvability of the algebraic riccati equation j c willems ieee trans automat control 16 1971 pp 621 634 cannot be obtained from the kalman yakubovich popov lemma we will see that with the lur e equation this is possible

geometry of the algebraic riccati equation part ii we prove that the set of real symmetric solutions of the algebraic riccati equation is isomorphic to the algebraic variety of invariant subspaces of a related n times n matrix by characterizing the structure of this variety we obtain a detailed description of the geometric properties of the solution set of the algebraic riccati equation

generation of pseudospectral differentiation matrices i we present a simple method for computing n times n pseudospectral differentiation matrices of order p in cal o pn 2 operations for the case of quasi polynomial approximation the algorithm is based on fornberg s finite difference algorithm and is numerically stable a fortran implementation is included a necessary and sufficient condition for d p d p 1 is also given

directional interpolation approach to h optimization and robust stabilization it is shown that under a mild condition h infty optimization and robust stabilization are equivalent to a directional interpolation problem which is a matrix extension of the classical pick nevanlinna interpolation problem a classical iterative method which is an extension of the schur nevanlinna algorithm is given for solving the problem this method does not require the inner outer factorization nor the balanced realization of the original plant a circuit theoretical parameterization of all solutions is derived that is expected to enhance the physical insight to the h infty optimal control and robust stabilization this parameterization has the degree much less than the one obtained previously

a parametrization of the nonnegative definite solutions of the algebraic riccati equation in this paper the set of the hermitian nonnegative definite solutions of the algebraic riccati equation is studied the main result shows that any solution in this set is the product of a greatest solution and a projection the analysis only assumes the existence of a greatest solution and leads to generalize classification theorems via projections known in the stabilizable case

equations with infinite delay numerical bifurcation analysis via pseudospectral discretization abstract we address the problem of the numerical bifurcation analysis of general nonlinear delay equations including integral and integro differential equations for which no software is currently available pseudospectral discretization is applied to the abstract reformulation of equations with infinite delay to obtain a finite dimensional system of ordinary differential equations whose properties can be numerically studied with well developed software we explore the applicability of the method on some test problems and provide some numerical evidence of the convergence of the approximations

a fast algorithm for spectral differentiation a simple algorithm is presented for matrix multiplication in just over half the number of operations entailed by the conventional algorithm in cases where the matrix possesses the degree of symmetry widely exhibited by derivative matrices the algorithm is used to multiply the chebyshev derivative matrix by a vector for the larger values of n the ratio of execution times approached the expected value of 2