riccati differential equation in optimal filtering of periodic non stabilizable systems this paper discusses the periodic solutions of the matrix riccati differential equation in the optimal filtering of periodic systems special emphasis is given to non stabilizable systems and the question addressed is the existence and uniqueness of a steady state periodic non negative definite solution of the periodic riccati differential equation which gives rise to an asymptotically stable steady state filter the results presented show that the stabilizability is not a necessary condition for the existence of such a periodic solution the convergence of the general solution of the periodic riccati differential equation to a periodic equilibrium solution is also investigated the results are extensions of existing time invariant systems results to the case of periodic systems

robust stabilization of discrete time linear systems with norm bounded time varying uncertainty abstract this paper presents an algorithm for the stabilization of a class of discrete time uncertain systems suffering from uncertainty of the norm bounded time varying type the stabilizing control gain matrix is obtained by solving a certain discrete riccati equation the solution of this riccati equation is also a lyapunov matrix which is used to establish the stability of closed loop system the results presented in this paper generalize those obtained in previous works for continuous time uncertain systems

on optimal pole assignment in a specified region in this paper the problem of pole assignment and linear quadratic design have been amalgamated a design method that has the merits of both approaches is presented and the problem of assigning the poles in a specified region is solved the characterizations for quadratic cost functions have been studied and these results used to assign poles in real schur form in a region so that the optimality is maintained designed procedures have been elaborated with examples

brief constrained linear state estimation a moving horizon approach this article considers moving horizon strategies for constrained linear state estimation additional information for estimating state variables from output measurements is often available in the form of inequality constraints on states noise and other variables formulating a linear state estimation problem with inequality constraints however prevents recursive solutions such as kalman filtering and consequently the estimation problem grows with time as more measurements become available to bound the problem size we explore moving horizon strategies for constrained linear state estimation in this work we discuss some practical and theoretical properties of moving horizon estimation we derive sufficient conditions for the stability of moving horizon state estimation with linear models subject to constraints on the estimate we also discuss smoothing strategies for moving horizon estimation our framework is solely deterministic

numerical j spectral factorization of general para hermitian matrices a numerical algorithm for j spectral factorization of para hermitian rational matrices is presented based on a generalization of the notion of separated realization of a para hermitian matrix the transformations used preserve the para hermitian matrix pencil structure

a successive optimal construction procedure for state feedback gains abstract this paper studies a successive optimal construction procedure of state feedback gains this topic was first studied in the context of the pole assignment problem for lq linear quadratic regulators but the selection of weighting matrices in each successive step was restricted in previous researches in this paper we consider more general selections of weighting matrices and give extended results in both the continuous time and the discrete time cases

brief extension of minimum variance estimation for systems with unknown inputs in this paper we address the problem of minimum variance estimation for discrete time time varying stochastic systems with unknown inputs the objective is to construct an optimal filter in the general case where the unknown inputs affect both the stochastic model and the outputs it extends the results of darouach and zasadzinski automatica 33 1997 717 where the unknown inputs are only present in the model the main difficulty in treating this problem lies in the fact that the estimation error is correlated with the systems noises this fact leads generally to suboptimal filters necessary and sufficient conditions for the unbiasedness of this filter are established then conditions under which the estimation error and the system noises are uncorrelated are presented and an optimal estimator and a predictor filters are derived sufficient conditions for the existence of these filters are given and sufficient conditions for their stability are obtained for the time invariant case a numerical example is given in order to illustrate the proposed method

abound rogue wave type solutions to the extended 3 1 dimensional jimbo miwa equation abstract under investigation in this article is an extended 3 1 dimensional jimbo miwa equation jm which can be used to describe many nonlinear phenomena in fluid dynamics by using the hirota bilinear form of the extended 3 1 dimensional jm equation thirty classes of rogue wave type solutions are found with the help of symbolic computations the rogue wave type solutions contain two important parameters a and b when a and b get different values we can present obvious rogue wave type solutions for example i taking a b 0 then we have algebraic solitary waves lump ii if one of a and b is fixed at 0 and the other is nonzero then we have an interaction solution between an algebraically decayed soliton and exponentially decayed soliton lumpoff iii let a 0 and b 0 then we have an interaction solution between lump and exponentially localized twin soliton wave instant or rogue wave the new rogue wave type solutions help us to know different physical worlds

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

convergence properties of the riccati difference equation in optimal filtering of nonstabilizable systems this paper studies the convergence and properties of the solutions of the riccati difference equation special emphasis is given to systems which are not necessarily stabilizable in the filtering sense particularly those having uncontrollable roots on the unit circle besides generalizing and unifying previous work the results have application to a number of important problems including filtering and control of systems with purely deterministic disturbances such as sinusoids and drift components

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

criterion for the convergence of the solution of the riccati differential equation the optimal control problem for a linear system with a quadratic cost function leads to the matrix riccati differential equation the convergence of the solution of this equation for increasing time interval is investigated as a function of the final state penalty matrix a necessary and sufficient condition for convergence is derived for stabilizable systems even if the output in the cost function is not detectable an algorithm is developed to determine the limiting value of the solution which is one of the symmetric positive semidefinite solutions of the algebraic riccati equation examples for convergence and nonconvergence are given a discussion is also included of the convergence properties of the solution of the riccati differential equation to any real symmetric not necessarily positive semidefinite solution of the algebraic riccati equation

output feedback pole placement in the design of suboptimal linear quadratic regulators a design oriented methodology is presented for the construction of low order suboptimal output feedback compensators the design is suboptimal in the sense that it retains an l dimensional eigenspace from a reference optimal state feedback linear quadratic regulator problem if r is the number of outputs then for fixed l r it is known that the design requires a dynamic compensator of dimension l r the parameters of the compensator are determined by the solution of an associated output feedback pole placement problem using iterative dyadic pole placement a procedure is given which determines the solution of this problem without a priori assumptions on the dimension of the compensator the design is also extended to the class of stabilizable systems and the resulting compensator is shown to possess a separation property

new solution to the inverse regulator problem by the polynomial matrix method abstract this paper gives a new criterion for the inverse regulator problem in terms of polynomial matrix fractions the generalized kalman equation in polynomial matrix form leads to a necessary and sufficient condition for a stable feedback law to be optimal for some unknown quadratic performance index this criterion consists of i a symmetric factorizability of a polynomial matrix and ii a polynomial type criterion that guarantees detectability the relationship between the riccati and kalman equations is also established finally an example is given to illustrate the new result in contrast with the existing result

mean field effects for counterpropagating traveling wave solutions of reaction diffusion systems in many problems e g in combustion or solidification one observes traveling waves that propagate with constant velocity and shape in the x direction say are independent of y and z and describe transitions between two equilibrium states e g the burned and the unburned reactants as parameters of the system are varied these traveling waves can become unstable and give rise to waves having additional structure such as traveling waves in the y and z directions which can themselves be subject to instabilities as parameters are further varied to investigate this scenario we consider a system of reaction diffusion equations with a traveling wave solution as a basic state we determine solutions bifurcating from the basic state that describe counterpropagating traveling waves in directions orthogonal to the direction of propagation of the basic state and determine their stability specifically we derive long wave modulation equations for the amplitudes of the counterpropagating traveling waves that

parameterization of frequency weighting for a two stage linear quadratic regulator based design abstract the purpose of this paper is to parametrize a class of lq frequency weightings and the associated dynamic controllers and to utilize the available freedom in a controller design methodology a two stage methodology is described in the first stage an acceptable solution to the conventional lq problem is obtained utilizing established guidelines for the selection of constant cost matrices to achieve satisfactory closed loop performance and an acceptable reference spectrum for the design model in the second stage frequency weighting is introduced to produce controller parameterizations which retain a part or all of the reference spectrum to preserve the desired dynamics associated with the design model and make the controller more robust with respect to the differences between the design model and the actual system

new solutions of the 3 1 dimensional jimbo miwa equation we find new exact solutions of the 3 1 dimensional jimbo miwa equation by applying the extended homogeneous balance method we also get an iterative formula of finding exact solutions this paper shows that there exist a lot of solutions for the 3 1 dimensional jimbo miwa equation

discrete time optimal regulator with closed loop poles in a prescribed region this paper is concerned with the problem of designing discrete time optimal control systems with closed loop poles in a prescribed region of stability first by utilizing the property of the riccati equation with q being zero we develop a method for allocating poles in a disc with centre at the origin of the complex plane and with radius less than 1 secondly we deal with pole placement in a disc which is in the unit disc and also contacts the point 1 j0 of the complex plane to this end a bilinear transformation and continuous time regulator results are employed in each case the radius of the disc can be specified as a design parameter and the weighting matrices of the performance index are obtained to fulfil the desired pole allocations the design procedures are also illustrated by numerical examples

properties of a discrete time enhanced linear quadratic controller abstract this multi input multi output controller incorporates two features optimal gains associated with known exogenous inputs and penalties imposed both on squares of first backward differences in control actions and on control actions squared results are derived for both the time varying finite time problem and the time invariant infinite horizon problem the optimal control is expressed in terms of feedback gains that weight the most recent control vector in addition to the current system state and look ahead gains that weight known exogenous inputs for the infinite horizon problem results include 1 conditions under which the optimal gains exist and stabilize the system 2 time invariant properties of the look ahead gains and their characterization in terms of eigenvalues of the closed loop system 3 gains appropriate for constant exogenous inputs 4 steady state response conditions for constant exogenous inputs and 5 conditions under which both linear combinations of states and of control variables are reduced to zero in the steady state

the inverse problem of linear optimal control for constant disturbance this paper presents the solution to the inverse problem of linear optimal control systems to achieve pole placement and at the same time minimize the cost function and accommodate a constant but unknown disturbance the sufficient conditions for an optimal feedback law to have a positive r and positive semi definite q are found the relationships between the open loop poles and the closed loop poles that guarantee a positive r and q 0 are also obtained these relations offer a rule for selection of the appropriate closed loop poles which will guarantee the required feedback properties

analytic study on two nonlinear evolution equations by using the g g expansion method abstract the validity and reliability of the so called g g expansion method is tested by applying it to two nonlinear evolutionary equations solutions in more general forms are obtained when the parameters are taken as special values it is observed that the previously known solutions can be recovered new rational function solutions are also presented being concise and less restrictive the method can be applied to many nonlinear partial differential equations

optimal discrete systems with prescribed eigenvalues this paper deals with the design of optimal discrete control systems with prescribed eigenvalues solheim s method for such design is improved and extended to achieve the original design objective that is the minimization of a quadratic performance index with a prescribed control weighting matrix r and some state weighting matrix q a new recursive algorithm is also presented while this algorithm retains the simplicity provided by solheim s method it constructs the resultant state weighting matrix q as the sum of the recursive state weighting matrices the algorithm is general and applicable to any system structure for the sake of completeness the optimality of solheim s solution is tested by using a simple necessary condition which is developed in the paper it is shown that the optimality of solheim s solution is not in general guaranteed

technical communique reduced order kalman filter with unknown inputs this paper presents an optimal reduced order kalman filter for discrete time dynamic stochastic linear systems with unknown inputs the problem is to estimate a part of the state vector in the case where none of the observations are assumed to be noise free the proposed filter is obtained by minimizing the trace of the estimation error covariance matrix with respect to the remaining degrees of freedom after noninteresting state and unknown inputs decoupling the necessary and sufficient conditions for stability and convergence of the filter are established

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

pole placement by performance criterion modification a procedure is presented that modifies the state weighting matrix q and introduces a degree of relative stability into the original performance criterion to shape the resulting optimal dynamics by positioning the closed loop eigenvalues along the real axis of the optimal system in the linear quadratic regulator problem it is based on the results of the algebraic riccati equation are which establish the invariance of certain eigenspaces of the associated hamiltonian matrix with respect to certain perturbations of q and the degree of relative stability

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

all meromorphic solutions for two forms of odd order algebraic differential equations and its applications abstract in this article we employ the nevanlinna s value distribution theory to investigate the existence of meromorphic solutions of algebraic differential equations we obtain the representations of all meromorphic solutions for two classes of odd order algebraic differential equations with the weak p q and dominant conditions moreover we give the complex method to find all traveling wave exact solutions of corresponding partial differential equations as an example we obtain all meromorphic solutions of some generalized bretherton equations by using our complex method our results show that the complex method provides a powerful mathematical tool for solving great many nonlinear partial differential equations in mathematical physics and using the traveling wave nobody can find other new exact solutions for many nonlinear partial differential equations by any method

a rigorous justification of the matthews cox approximation for the nikolaevskiy equation abstract the nikolaevskiy equation is an example of a pattern forming system with marginally stable long modes it has the unusual property that the typical ginzburg landau scaling ansatz for the description of propagating patterns does not yield asymptotically consistent amplitude equations instead another scaling proposed by matthews and cox can be used to formally derive a consistent system of modulation equations we give a rigorous proof that this system makes correct predictions about the dynamics of the nikolaevskiy equation

robustness results in linear quadratic gaussian based multivariable control designs the robustness of control systems with respect to model uncertainty is considered using simple frequency domain criteria available and new results are derived under a common framework in which the minimum singular value of the return differences transfer matrix is the key quantity in particular robustness results associated with multivariable control systems designed on the basis of linear quadratic lq and the linear quadratic gaussian lqg design methodologies are presented

receding horizon recursive state estimation describes a receding horizon discrete time state observer using the deterministic least squares framework the state estimation horizon which determines the number of past measurement samples used to reconstruct the state vector is introduced as a tuning parameter for the proposed state observer a stability result concerning the choice of the state estimation horizon is established it is also shown that the fixed memory receding horizon state observer can be related to the standard dynamic observer by using an appropriate end point state weighting on the estimator cost function

anomalous scaling on a spatiotemporally chaotic attractor the nikolaevskiy model for pattern formation with continuous symmetry exhibits spatiotemporal chaos with strong scale separation extensive numerical investigations of the chaotic attractor reveal unexpected scaling behavior of the long wave modes surprisingly the computed amplitude and correlation time scalings are found to differ from the values obtained by asymptotically consistent multiple scale analysis however when higher order corrections are added to the leading order theory of matthews and cox the anomalous scaling is recovered

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

design of linear quadratic regulators with assigned eigenstructure a recursive design procedure is presented which assigns the closed loop eigenstructure in linear quadratic regulators at each stage the required solution for the steady state riccati matrix which shifts a pole or pole pair to specified values is obtained for pole pair placement a free parameter in the solution permits selection of closed loop eigenvectors the interactive design procedure is summarized and illustrated by an example

singular riccati equations stabilizing large scale systems in this paper we discuss the convergence of a stabilization algorithm based on a singular version of the discrete riccati difference equation this method is particularly appealing for large scale linear time invariant dynamical systems since one can nicely exploit the sparsity of such systems in order to reduce the complexity of the algorithm

difference and differential riccati equations a note on the convergence to the strong solution the convergence of the solutions of the differential difference riccati equations to the strong solution of the corresponding algebraic riccati equation are is addressed detectability only is required in the analysis and no assumption is made on the eigenvalues on the real imaginary axis on the unit circle in the discrete time case in particular from the results it follows that under the sole assumption of detectability a positive definite initial condition guarantees convergence to the strong solution even in the presence of unreachable eigenvalues on the imaginary axis or on the unit circle

meromorphic exact solutions of two extended 3 1 dimensional jimbo miwa equations abstract in this work abundant meromorphic exact solutions of two kinds of extended 3 1 dimensional jimbo miwa equations are obtained by making use of the complex method which contain rational solutions exponential function solutions and elliptic function solutions the properties of the results are also shown by some figures

a note on the maximal solution of the periodic riccati equation the periodic symmetric solutions of the periodic riccati differential equation associated with the filtering problem are considered by the authors it is proven that under the sole assumption of detectability there exists a maximal solution moreover such a solution turns out to be strong i e the characteristic multipliers of the associated closed loop system belong to the closed unit disk the proof relies on an iterative linearization technique which calls for a sequence of periodic lyapunov equations similar results are given for the minimal solution

on successive pole assignment by linear quadratic optimal feedbacks abstract this paper studies a method of shifting poles of linear constant systems via lq optimal feedback by making use of a solution to the so called inverse regulator problem this method enables us to shift poles successively by pairs while maintaining the well known advantages of lq regulators polynomial fractional representation is effectively used both to characterize the maximal pole assignable area which clarifies the theoretical bound of the lq optimal pole shifting and to derive an algorithm a numerical example is given to illustrate the results

deadbeat control and tracking of discrete time systems a new approach for forcing the state of a linear discrete time system to zero in a minimum number of steps is discussed the problem is formulated as the solution to a steady state optimal control problem with no cost on the control this problem in turn is set up as the solution to an associated eigenvalue problem no special assumption on the open loop system matrix and or the ratio of the number of states to controls is required stable numerical techniques are presented for solving for the feedback gain robust deadbeat tracking is also discussed

a complete optimality condition in the inverse problem of optimal control a complete optimality condition in the standard inverse problem of optimal control for the multi input case is established the optimality of a feedback control law is characterized completely in terms of a new geometric condition as well as the well known return difference condition the new condition not only provides a better insight into the well known sensitivity reduction property of optimal control for the multi input case but also indicates an essential difference between the solutions of the single and multi input inverse problem

computationally efficient optimal output decentralized estimation this paper presents decentralized computational architectures for the optimal state estimation in stochastic large scale linear systems the main feature of the proposed architectures lies on elaborating each subsystem s decision not only by processing its own local data but also by adjusting this decision with all other related subsystem s local data this adjustment procedure ensures the optimality of the decentralized filter it is emphasized that the kalman filter algorithm operates more efficiently when measurements are processed into low order subsets especially when they are processed one at a time thus using this feature in a decentralized scheme significantly increases computational savings and numerical stability architectures presented in this paper for the mechanization of decentralized estimators allow a high degree of parallelism and can be implemented on a wide range of computer networks

partial pole placement by lq regulators an inverse problem approach this paper gives a necessary and sufficient condition under which a state feedback control law places part of the closed loop poles exactly at specified points and at the same time is linear quadratic optimal for some quadratic weightings this is made possible by means of a solution to the inverse problem of optimal control a design example is given to illustrate the result

unbiased minimum variance estimation for systems with unknown exogenous inputs a new method is developed for the state estimation of linear discrete time stochastic systems in the presence of an unknown disturbance the filter obtained is optimal in the unbiased minimum variance sense the necessary and sufficient conditions for the existence and the stability of the filter are given

chemical turbulence equivalent to nikolavskii turbulence we find evidence that a certain class of reaction diffusion rd systems can exhibit chemical turbulence equivalent to nikolaevskii turbulence we study an extended complex ginzburg landau cgl equation derived from this class of rd systems first we show numerically that the power spectrum of this cgl equation in the neighborhood of a codimension two turing char21 benjamin feir point is qualitatively quite similar to that of the nikolaevskii equation then we demonstrate that the nikolaevskii equation can in fact be obtained from this cgl equation through a phase reduction procedure

h sub infinity optimal one step ahead output feedback control of discrete time systems the discrete time one step ahead standard four blocks h sub infinity optimal control problem is considered the output feedback problem is transformed into a one step ahead h sub infinity optimal state estimation problem and the existing solution is used to derive the controller it is assumed that only previous measurements are available for control two results are obtained the first requires the solutions of two coupled riccati equations where the necessary and sufficient condition for the existence of the h sub infinity optimal result requires only the positive semidefiniteness of these solutions the second result applies two decoupled riccati equations with a coupling constraint as in the continuous time case and two additional conditions the derivation procedure is most simple and may thus provide considerable insight into the solution of the discrete time standard problem a similar procedure can also be used to obtain a solution to problems where the current measurement is available

multiple soliton solutions for the calogero bogoyavlenskii schiff jimbo miwa and ytsf equations abstract in this work we employ the hirota s bilinear method to derive multiple front solutions for nonlinear evolution equations three models namely the 2 1 dimensional calogero bogoyavlenskii schiff equation the 3 1 dimensional jimbo miwa equation and the 3 1 dimensional potential ytsf equation are used as vehicles to conduct the analysis the work shows the power of the bilinear method and the variety of the obtained front solutions

on the solution of discrete time markovian jump linear quadratic control problems abstract a necessary and sufficient condition for the existence of a positive semidefinite solution of the coupled algebraic discrete time riccati like equation occurring in markovian jump control problems is derived by verifying a simple matrix inequality it is shown that such a solution exists and can be obtained as a limit of a monotonic sequence this leads to a straightforward numerical algorithm for the computation of the solution an example is given to illustrate the proposed method

critical exponents of nikolaevskii turbulence we study the spatial power spectra of nikolaevskii turbulence in one dimensional space first we show that the energy distribution in wave number space is extensive in nature then we demonstrate that when varying a particular parameter the spectrum becomes qualitatively indistinguishable from that of kuramoto sivashinsky turbulence next we derive the critical exponents of turbulent fluctuations finally we argue that in some previous studies parameter values for which this type of turbulence does not appear were mistakenly considered and we resolve inconsistencies obtained in previous studies

detectability and stabilizability of time varying discrete time linear systems the concepts of detectability and stabilizability are explored for time varying systems we study duality invariance under feedback an extended version of the lemma of lyapunov existence of stabilizing feedback laws linear quadratic filtering and control and the existence of approximate canonical forms

a new approach to the lq design from the viewpoint of the inverse regulator problem a new design method of lq regulators is developed by reverse application of pertinent results on the inverse regulator problem the lq regulator of interest here is unlike the usual one designed without specifying a quadratic cost but it proves to minimize some quadratic cost hence it possesses those well known robustness properties shared by all lq regulators a basic design procedure is developed first and followed by practical ones in which the design parameters are chosen based on asymptotic modal properties of the regulator this new method requires no riccati solutions thereby leading to simplification of the lq design as compared to the usual one

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