numerical solution of projected algebraic riccati equations we consider the numerical solution of projected algebraic riccati equations using newton s method such equations arise for instance in model reduction of descriptor systems based on positive real and bounded real balanced truncation we also discuss the computation of low rank cholesky factors of the solutions of projected riccati equations numerical examples are given that demonstrate the properties of the proposed algorithms

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

tracking fast trajectories along a slow dynamics a singular perturbations approach controlled coupled slow and fast motions are examined in a singular perturbations setting the objective is to minimize a cost functional that takes into account both the fast motion supposing say tracking a fast target and the slow dynamics a method is offered to cope with the possibility that the fast flow has nonstationary limits invariant measures of the fast motion are then the controlled objects on the infinitesimal scale optimal amalgamation of them on the slow scale induces the variational limit whose solutions are near optimal solutions of the perturbed system

stability analysis and a priori error estimates of the third order explicit runge kutta discontinuous galerkin method for scalar conservation laws in this paper we present an analysis of the runge kutta discontinuous galerkin method for solving scalar conservation laws where the time discretization is the third order explicit total variation diminishing runge kutta method we use an energy technique to prove the mathrm l 2 norm stability for scalar linear conservation laws and to obtain a priori error estimates for smooth solutions of scalar nonlinear conservation laws quasi optimal order is obtained for general numerical fluxes and optimal order is given for upwind fluxes the theoretical results are obtained for piecewise polynomials with any degree k geq1 under the standard temporal spatial cfl condition tau leq gamma h where h and tau are the element length and time step respectively and the positive constant gamma is independent of h and tau

normal forms of symplectic pencils and the discrete time algebraic riccati equation abstract the solution of the discrete time algebraic riccati equation leads to symplectic pencils of matrices normal forms of such pencils under symplectic equivalence are determined special attention is given to characteristic roots of modulus 1 and their corresponding elementary divisors and inertial invariants

riccati equations in optimal filtering of nonstabilizable systems having singular state transition matrices until recently it was believed that a necessary and sufficient condition for convergence of the riccati difference equation of optimal filtering was that the system be both delectable and stabilizable recently it has been shown that the stabilizability condition can be removed but convergence has only established under restrictive assumptions including the requirement that the state transition matrix be nonsingular the present paper generalizes these results in several directions first properties of the algebraic riccati equation are established for the case of singular state transition matrix second several assumptions previously imposed in establishing convergence of the riccati difference equation for systems with unreachable modes on the unit circle are relaxed including replacing observability by detectability weakening the conditions on the initial covariance and allowing the state transition matrix to be singular third results on the convergence and properties of the riccati equations are expressed as both necessary and sufficient conditions whereas previous results were only sufficient these extensions mean that the results have wider applicability including fixed lag smoothing problems and filtering for systems with time delays the implications of the results in the dual problem of optimal control are also studied

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

covariance bounds for discrete time linear systems with time varying parameter uncertainty this paper deals with the computation of upper bounds for the state covariance matrix of discrete time linear systems subject to stochastic excitation and additive time varying uncertainty in the system dynamic matrix such upper bounds are obtained as the stabilizing solutions of suitable h type riccati equations a necessary and sufficient condition for the existence of such solutions is given in terms of the h norm of a suitable transfer function as for the computation of the optimal bound it is demonstrated that the bounds are a convex function of a scalar parameter so that efficient numerical schemes can be worked out

infinite horizon linear quadratic control with end point state penalty term the discrete time case abstract for a given discrete time system consider an infinite horizon linear quadratic control problem with positive semidefinite cost criterion and an extra penalty term for the state variable at infinity our central result shows that this problem is structurally equivalent to the associated problem where the state penalty term is required to vanish at infinity provided only that the latter problem has finite optimal cost everywhere for this case the optimal cost is represented by a unique solution of the possibly singular algebraic riccati equation and if in addition the underlying system is left invertible then optimal inputs for either problem are implementable as state feedback laws expressed in terms of the original system coefficients only even when the control weighting matrix in the cost criterion is not invertible

minimal realization and dynamic properties of optimal smoothers smoothing algorithms of various kinds have been around for several decades however some basic issues regarding the dynamical structure and the minimal dimension of the steady state algorithm are still poorly understood in this paper we derive a realization of minimal dimension of the optimal smoother for a signal admitting a state space description of dimension n it is shown that the dimension of the smoothing algorithm can vary from n to 2n depending on the zero structure of the signal model the dynamics pole structure of the steady state smoother is also characterized explicitly and is related to the zero structure of the model we use several recent ideas from stochastic realization theory in particular a minimal markovian representation of the smoother is derived which requires solving a nonsymmetric wiener hopf factorization problem in this way the smoother is naturally expressed as the cascade of a whitening filter and a linear filter of least possible dimension whose state space is a minimal markovian subspace containing the smoothed estimate x spl i cap this among other aspects affords a very simple calculation of the error covariance matrix of the smoother a reduced order two filter implementation of the type due to mayne 1966 and fraser 1967 is obtained by solving a riccati equation of reduced dimension which is in general smaller than the dimension of the riccati equations considered in the literature

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

all optimal hankel norm approximations and their error bounds in discrete time this paper investigates a parallel problem to glover 1984 approximate a multivariable transfer function g z of mcmillan degree n by of mcmillan degree smaller than k in discrete time a state space solution is derived to the optimal hankel norm approximation problem together with characterization to all optimal hankel norm approximations it is shown that the error bound derived in glover 1984 holds for discrete time systems as well

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

characterization of solutions of non symmetric algebraic riccati equations abstract we give a new characterization of the solution set of non symmetric algebraic riccati equations involving real matrices our characterization involves the use of invariant subspaces of the coefficient matrices we also give a poset structure on the solutions of are and explore some properties of this poset partially ordered set

representation and factorization of discrete time rational all pass functions in this paper we obtain a general characterization of discrete time all pass rational matrix functions from state space representations we establish a general characterization of the solutions of lmi s and riccati equations finally we derive a complete factorization theory of all pass functions our results are obtained in the most general setting without introducing any ad hoc assumption and can be applied to a variety of problems such as the discrete time counterpart of the h infty model reduction problems solved by glover in continuous time

a step toward a unified treatment of continuous and discrete time control problems in this paper we introduce a new approach for a unified theory for continuous and discrete time optimal control problems based on the generalized cayley transformation we also relate the associated discrete and continuous generalized algebraic riccati equations we demonstrate the potential of this new approach by proving a new result for discrete algebraic riccati equations but we also discuss where this new approach as well as all other approaches still is nonsatisfactory we explain a discrepancy observed between the discrete and continuous case and show that this discrepancy is partly due to the consideration of the wrong analogues we also present an idea for an implication scheme that relates general theorems for discrete and continuous control problems

a parameterization of minimal stochastic realizations given a rational spectral density and a minimal spectral factor a parameterization of all minimal spectral factors is derived a minimal realization of each minimal spectral factor is also provided this parameterization is shown to hold under the condition of nonintersection between the spectra of two matrices this condition first introduced in previous work by ferrante et al 1993 is proved here to be also necessary finally these results are applied to the stochastic realization problem

a characterization of solutions of the discrete time algebraic riccati equation based on quadratic difference forms this paper is concerned with a characterization of all symmetric solutions to the discrete time algebraic riccati equation dare dissipation theory and quadratic difference forms from the behavioral approach play a central role in this paper along the line of the continuous time results due to trentelman and rapisarda h l trentelman p rapisarda pick matrix conditions for sign definite solutions of the algebraic riccati equation siam j contr optim 40 3 2001 969 991 we show that the solvability of the dare is equivalent to a certain dissipativity of the associated discrete time state space system as a main result we characterize all unmixed solutions of the dare using the pick matrix obtained from the quadratic difference forms this characterization leads to a necessary and sufficient condition for the existence of a non negative definite solution it should be noted that when we study the dare and the dissipativity of the discrete time system there exist two difficulties which are not seen in the continuous time case one is the existence of a storage function which is not a quadratic function of state another is the cancellation between the zero and infinite singularities of the dipolynomial spectral matrix associated with the dare due to the infinite generalized eigenvalues of the associated hamiltonian pencil one of the main contributions of this paper is to demonstrate how to resolve these difficulties

the generalised discrete algebraic riccati equation in linear quadratic optimal control this paper investigates the properties of the solutions of the generalised discrete algebraic riccati equation arising from the classic infinite horizon linear quadratic lq control problem in particular a geometric analysis is used to study the relationship existing between the solutions of the generalised riccati equation and the output nulling subspaces of the underlying system and the corresponding reachability subspaces this analysis reveals the presence of a subspace that plays an important role in the solution of the related optimal control problem which is reflected in the generalised eigenstructure of the corresponding extended symplectic pencil in establishing the main results of this paper several ancillary problems on the discrete lyapunov equation and spectral factorisation are also addressed and solved

on the structure of the solutions of discrete time algebraic riccati equation with singular closed loop matrix the classical discrete time algebraic riccati equation dare is considered in the case when the corresponding closed loop matrix is singular it is shown that in this case all the symmetric solutions of the dare coincide along some directions a parametrization of the set of solutions in terms of a reduced order dare is then obtained this parametrization provides an algorithm that appears to be computationally very attractive when the multiplicity of the eigenvalue spl lambda 0 of the closed loop matrix is large for the computation of the solutions of the dare the same issue for the generalized dare is also addressed

hermitian solutions of the equation x q nx 1n we consider the matrix equation x q nx 1n its hermitian solutions are parametrized in terms of the generalized lagrangian eigenspaces of a certain matrix pencil we show that the equation admits both a largest and a smallest solution the largest solution corresponds to the unique positive definite solution the smallest solution is the unique negative definite solution if and only if n is nonsingular if n is singular no negative definite solution exists an interesting relation between the given equation and a standard algebraic riccati equation of kalman filtering theory is also obtained finally we present an algorithm which converges to the positive definite solution for a wide range of initial conditions

the discrete algebraic riccati equation and linear matrix inequality we study the discrete time algebraic riccati equation in particular we show that even in the most general cases there exists a one one correspondence between solutions of the algebraic riccati equation and deflating subspaces of a matrix pencil we also study the relationship between algebraic riccati equation and the discrete time linear matrix inequality we show that in general only a subset of the set of rank minimizing solutions of the linear matrix inequality correspond to the solutions of the associated algebraic riccati equation and study under what conditions these sets are equal in this process we also derive very weak assumptions under which a riccati equation has a solution

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

existence uniqueness and stability of solutions to singular linear quadratic optimal control problems we consider general continuous and discrete time linear quadratic optimal control problems where the dynamics of the system is a linear differential or difference algebraic equation with constant coefficient matrices and the cost functionals are defined by infinite integrals or sums these problems occur in numerous applications in many different fields such as economics operations research electrical engineering model and system design biology and chemistry since they form two of the simplest optimal control problems and they are also used as linear approximations for many nonlinear problems which are much harder to solve we extend some of the known results to these general problems

modeling and estimation of discrete time gaussian reciprocal processes discrete time gaussian reciprocal processes are characterized in terms of a second order two point boundary value nearest neighbor model driven by a locally correlated noise whose correlation is specified by the model dynamics this second order model is the analog for reciprocal processes of the standard first order state space models for markov processes the model is used to obtain a solution to the smoothing problem for reciprocal processes the resulting smoother obeys second order equations whose structure is similar to that of the kalman filter for gauss markov processes it is shown that the smoothing error is itself a reciprocal process

error estimates to smooth solutions of runge kutta discontinuous galerkin methods for scalar conservation laws in this paper we study the error estimates to sufficiently smooth solutions of scalar conservation laws for runge kutta discontinuous galerkin rkdg methods where the time discretization is the second order explicit total variation diminishing tvd runge kutta method error estimates for the mathbb p 1 piecewise linear elements are obtained under the usual cfl condition tau leq gamma h for general nonlinear conservation laws in one dimension and for linear conservation laws in multiple space dimensions where h and tau are the maximum element lengths and time steps respectively and the positive constant gamma is independent of h and tau however error estimates for higher order mathbb p k k geq 2 elements need a more restrictive time step tau leq gamma h 4 3 we remark that this stronger condition is indeed necessary as the method is linearly unstable under the usual cfl condition tau leq gamma h for the mathbb p k elements of degree k geq 2 error estimates of o h k 1 2 tau 2 are obtained for general monotone numerical fluxes and optimal error estimates of o h k 1 tau 2 are obtained for upwind numerical fluxes

numerical smoothness and error analysis for rkdg on the scalar nonlinear conservation laws the new concept of numerical smoothness is applied to the rkdg runge kutta discontinuous galerkin methods for scalar nonlinear conservations laws the main result is an a posteriori error estimate for the rkdg methods of arbitrary order in space and time with optimal convergence rate in this paper the case of smooth solutions is the focus point however the error propagation analysis framework is prepared to deal with discontinuous solutions in the future

numerical analysis of the acoustics of a diffusion flame the aim of this work is to obtain an analytical numerical solution for a jet diffusion flame in a closed domain and analyze its acoustic field the combustion chamber is a closed three dimensional region in which a jet of methane is injected separated from the jet of air to analyze the influence of the flame under the acoustic field two tests are performed in the first test occurs only the mixing and in the second test occurs the mixing and the chemical reactions the proposed model uses the navier stokes mass conservation and wave equations which are solved numerically the flow equations are coupled with the equations for the mass fraction of chemical species and temperature which are solved analytically based on the flamelet model the numerical scheme follows the finite difference method and the runge kutta multistage scheme is employed for the integration in time an estimate for the error is presented based on the solution of three grids using the error estimator of richardson the results show good agreement with data from the literature the proposed model is a quick and less expensive option to investigate the acoustic field of reactive flows in a closed region

long time error estimation for semi linear parabolic equations the long time error estimation approach of sun and ewing dyn contin discrete impuls systems ser b appl algorithms 9 2002 115 129 is applied here for the error analysis and estimation of linear and semilinear parabolic partial differential equations the analysis is carried out using the stability smoothing indicator the smoothing assumption the moving attractor the exact error propagation and the two level error propagation analysis introduced by sun and ewing dyn contin discrete impuls systems ser b appl algorithms 9 2002 115 129 moreover an inverse elliptic projection is employed here as a key technique in dealing with the spatial discretization error the error estimates obtained are uniform in time the results are substantiated by a complete mathematical analysis and numerical experiments

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

semiclassical asymptotics and gaps in the spectra of periodic schrodinger operators with magnetic wells we show that under some very weak assumption of effective variation for the magnetic field a periodic schrodinger operator with magnetic wells on a noncompact riemannian mani fold m such that h 1 m r 0 equipped with a properly discon nected cocompact action of a finitely generated discrete group of isometries has an arbitrarily large number of spectral gaps in the semi classical limit

periodic riccati difference equations approaching equilibria by implicit systems a parameterization of solutions to the periodic riccati difference equation rde is given by using behavioral tools the moving equilibria of the periodic rde are encompassed among the trajectories of the implicit ar model associated to the symplectic pencil of the corresponding hamiltonian equations this allows a parameterization of solutions of periodic rdes even in the case the dynamic matrix is singular

parametrization of all minimal square spectral factors abstract two different parametrizations of minimal square spectral factors are given these parametrizations extend recent results by picci and pinzoni since they characterize spectral factors without specification on the zero or pole structure

silverman algorithm and the structure of discrete time stochastic systems abstract it is a well known yet poorly understood fact that contrary to the continuous time case the same discrete time process y can be represented by minimal linear models see 1 1a 1 1b below which may either have a non singular or a singular d matrix in fact models with d 0 have been commonly used in the statistical literature on the other hand for models with a singular d matrix the riccati difference equation of kalman filtering involves in general the pseudo inversion of a singular matrix this cheap filtering problem dual to the better known cheap control problem has been studied for several decades in connection with the so called invariant directions of the riccati equation for a singular d a reduction of the order of the riccati equation is in general possible the reasons for such a reduction do not seem to be completely clear either in this paper we provide an explanation of this phenomenon from the classical point of view of zero flipping among minimal spectral factors changing d s occurs whenever zeros are flipped from z to their reciprocals at z 0 it is well known that for finite zeros the zero flipping process takes place by multiplication of the underlying spectral factor by a suitable rational all pass matrix function for infinite zeros zero flipping is implemented by a dual version of the silverman structure algorithm using this interpretation we derive a new algorithm for filtering of non regular processes based on a reduced order riccati equation we also obtain a precise characterization of the reduction of the order of the riccati equation which is afforded by zeros either at z or at the origin this order reduction has traditionally been associated with the study of invariant directions a point of view which as we show does not capture the essence of the phenomenon

canonical forms for hamiltonian and symplectic matrices and pencils abstract we study canonical forms for hamiltonian and symplectic matrices or pencils under equivalence transformations which keep the class invariant in contrast to other canonical forms our forms are as close as possible to a triangular structure in the same class we give necessary and sufficient conditions for the existence of hamiltonian and symplectic triangular jordan kronecker and schur forms the presented results generalize results of lin and ho w w lin t c ho on schur type decompositions for hamiltonian and symplectic pencils technical report institute of applied mathematics national tsing hua university taiwan 1990 and simplify the proofs presented there

adaptive finite element methods for conservation laws based on a posteriori error estimates we prove a posteriori error estimates for a finite element method for systems of strictly hyperbolic conservation laws in one space dimension and design corresponding adaptive methods the proof of the a posteriori error estimates is based on a strong stability estimate for an associated dual problem together with the galerkin orthogonality of the finite element method the strong stability estimate uses the entropy condition for the system in an essential way 1995 john wiley sons inc

existence of unmixed solutions of the discrete time algebraic riccati equation and a nonstrictly bounded real lemma abstract the existence of a solution of the discrete time algebraic riccati equation is established assuming modulus controllability and positive semidefiniteness on the unit circle of the popov function as an application a nonstrictly bounded real lemma is obtained

a survey of nonsymmetric riccati equations abstract we survey recent and also older results on nonsymmetric matrix riccati differential equations and in the time invariant case on the corresponding algebraic riccati equations in particular we cite various applications connected with matrix riccati equations

spectral gaps for periodic schrodinger operators with hypersurface magnetic wells analysis near the bottom we consider a periodic magnetic schrodinger operator hh depending on the semiclassical parameter h 0 on a noncompact riemannian manifold m such that h1 m r 0 endowed with a properly discontinuous cocompact isometric action of a discrete group we assume that there is no electric field and that the magnetic field has a periodic set of compact magnetic wells we suppose that the magnetic field vanishes regularly on a hypersurface s first we prove upper and lower estimates for the bottom 0 hh of the spectrum of the operator hh in l2 m then assuming the existence of non degenerate miniwells for the reduced spectral problem on s we prove the existence of an arbitrarily large number of spectral gaps for the operator hh in the region close to 0 hh as h 0 in this case we also obtain upper estimates for the eigenvalues of the one well problem

a discontinuous galerkin front tracking scheme and its optimal optimal error estimation an error estimate of optimal convergence rates and optimal error propagation optimal optimal was given for the numerical solutions produced by the runge kutta discontinuous galerkin rkdg method on the scalar nonlinear conservation laws in the case of smooth solutions in sun and rumsey 2013 this manuscript generalizes the problem to the case of a piecewise smooth solution containing one fully developed shock a front tracking technique is incorporated in the rkdg scheme to produce a numerical solution with a truly high order error the numerical smoothness approach of sun and rumsey 2013 is generalized to this particular case of a discontinuous solution

an exact line search method for solving generalized continuous time algebraic riccati equations we present a newton like method for solving algebraic riccati equations that uses an exact line search to improve the sometimes erratic convergence behavior of newton s method it avoids the problem of a disastrously large first step and accelerates convergence when newton steps are too small or too long the additional work to perform the line search is small relative to the work needed to calculate the newton step

unmixed solutions of the discrete time algebraic riccati equation the algebraic riccati equation of the optimal control problem associated with the discrete time system x k 1 fx k gu k is studied it is shown that in the case of a controllable system there exist solutions with prescribed unmixed characteristic polynomial of the corresponding closed loop matrix existence of solutions will also be proved under the weaker condition of modulus controllability maximal solutions are discussed

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

the discrete time generalized algebraic riccati equation order reduction and solutions structure abstract in this paper we discuss how to decompose the constrained generalized discrete time algebraic riccati equation arising in optimal control and optimal filtering problems into two parts corresponding to an additive decomposition x x 0 of each solution x the first part is trivial in the sense that it is an explicit expression of the addend x 0 which is common to all solutions so that it does not depend on the particular x the second part can be depending on the structure of the considered generalized riccati equation either a reduced order discrete time regular algebraic riccati equation whose associated closed loop matrix is non singular or a symmetric stein equation the proposed reduction is explicit so that it can be easily implemented in a software package that uses only standard linear algebra procedures

solving the algebraic riccati equation with the matrix sign function abstract this paper presents some improvements to the matrix sign function algorithm for the algebraic riccati equation a simple reorganization changes nonsymmetric matrix inversions into symmetric matrix inversions scaling accelerates convergence of the basic iteration and yields a new quadratic formula for certain 2 by 2 algebraic riccati equations numerical experience suggests the algorithm be supplemented with a refinement strategy similar to iterative refinement for systems of linear equations refinement also produces an error estimate the resulting procedure is numerically stable it compares favorably with current schur vector based algorithms

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

a birkhoff contraction formula with applications to riccati equations in this paper we show that the symplectic hamiltonian operators on a hilbert space give rise to linear fractional transformations on the open convex cone of positive definite operators that contract a natural invariant finsler metric the thompson or part metric on the convex cone more precisely the constants of contraction for the hamiltonian operators satisfy the classical birkhoff formula the lipschitz constant for the corresponding linear fractional transformations on the cone of positive definite operators is equal to the hyperbolic tangent of one fourth the diameter of the image by means of the close connections between hamilitonian operators and riccati equations this result and the associated machinery are applied to obtain convergence results for discrete algebraic riccati equations and riccati differential equations

defect correction method for the solution of algebraic riccati equations the solution of discrete and continuous algebraic riccati equations is considered it is shown that if an approximate solution is obtained then the defect for this solution again solves an algebraic riccati equation of the same form and that the system properties of detectability and stabilizability are inherited by this defect equation on the basis of these results a general defect correction method is proposed and numerical examples are given for the use of this method in combination with a bunse gerstner and v mehrmann s 1986 sr method

monotonicity and stabilizability properties of solutions of the riccati difference equation propositions lemmas theorems fallacious conjectures and counterexamples abstract the problem considered is that of selecting an initial covariance matrix for the kalman filter to ensure that the closed loop filter at every subsequent time instant is exponentially asymptotically stable as a time invariant filter sufficient conditions are derived based on monotonicity properties of the solution of the riccati difference equation the results have application in observer design and the cases of filtering for nonstabilizable systems and systems with singular system matrices are included

canonical correlation analysis approximate covariance extension and identification of stationary time series abstract in this paper we analyze a class of state space identification algorithms for time series based on canonical correlation analysis in the light of recent results on stochastic systems theory in principle these so called subspace methods can be described as covariance estimation followed by stochastic realization the methods offer the major advantage of converting the nonlinear parameter estimation phase in traditional arma models identification into the solution of a riccati equation but introduce at the same time some nontrivial mathematical problems related to positivity the reason for this is that an essential part of the problem is equivalent to the well known rational covariance extension problem therefore the usual deterministic arguments based on factorization of a hankel matrix are not valid for generic data something that is habitually overlooked in the literature we demonstrate that there is no guarantee that several popular identification procedures based on the same principle will not fail to produce a positive extension unless some rather stringent assumptions are made which in general are not explicitly reported in this paper the statistical problem of stochastic modeling from estimated covariances is phrased in the geometric language of stochastic realization theory we review the basic ideas of stochastic realization theory in the context of identification discuss the concept of stochastic balancing and of stochastic model reduction by principal subsystem truncation the model reduction method of desai and pal 1982 a realization approach to stochastic model reduction proc 1st decision and control conf pp 1105 1112 based on truncated balanced stochastic realizations is partially justified showing that the reduced system structure has a positive covariance sequence but is in general not balanced as a byproduct of this analysis we obtain a theorem prescribing conditions under which the subspace identification methods produce bona fide stochastic systems