time varying discrete riccati equation in terms of ben artzi gohberg dichotomy abstract necessary and sufficient conditions for the existence of the stabilizing solution to the time varying discrete riccati equation are derived in terms of the so called ben artzi gohberg dichotomy it is shown that the problem of such existence conditions is strongly related to that of unique solvability in vector valued l2 spaces of a system of singular difference equations which in this case is termed the extended hamiltonian system ehs connections between the existence of the stabilizing solution to the riccati equation and the bounded invertibility of an appropriate l2 operator are pointed out via the notion of disconjugacy

on computing the stabilizing solution of the discrete time riccati equation abstract a necessary and sufficient condition is given for the existence of the stabilizing solution of the discrete time riccati equation without any invertibility or positivity assumptions the result involves a special associated matrix pencil

on discrete time h problem with a strictly proper controller in this paper the standard discrete time h control problem is considered under the constraint imposed upon the controller to be strictly proper the necessary and eta sufficient conditions for the existence of a suboptimal controller are derived and the set of all linear time invariant controllers which solve the problem is parametrized in contrast with the unconstrained case the state space solution obtained is very transparent and actually is not more complex than the solution of the continuous time h problem

geometric insight into discrete time cheap and singular linear quadratic riccati lqr problems the hamiltonian system related to discrete time cheap linear quadratic riccati lqr problems is analyzed in a purely geometric context with the twofold purpose of getting a useful insight into its structural features and deriving a numerically implementable solution for the infinite horizon case by only using the standard geometric approach routines that are available

an extended symplectic pencil approach to the optimal h problem explicit state space formulas for the non canonical spectral factorization of a para hermitian rational matrix function which cannot be solved by using a riccati equation are constructed based on the esp method in discrete time the explicit indivisible criterion is given as means of the factorization using the esp method the derived centre factor satisfies the indivisible condition the results are also verified by a simple numerical example

probing convex polygons with x rays an x ray probe through a polygon measures the length of intersection between a line and the polygon this paper considers the properties of various classes of x ray probes and shows how they interact to give finite strategies for completely describing convex n gons it is shown that 3n 2 6 probes are sufficient to verify a specified n gon while for determining convex polygons 3n 1 2 x ray probes are necessary and 5n o 1 sufficient with 3n o 1 sufficient given that a lower bound on the size of the smallest edge of p is known

h output feedback control for linear discrete time varying systems via the bounded real lemma in this paper we develop a solution to the discrete time output feedback control problem for linear time varying ltv systems the solution is developed along the strategy set up by doyle et al in 1989 and the main ingredient in its derivation is the extension of the well known bounded real lemma to a discrete time varying context developed by van der veen and verhaegen in 1995 this approach contributes to the conceptual simplicity and hence to the accessibility of the solution apart from that we treat the infinite horizon case for ltv system of non uniform state dimension and varying input and output dimension both situations can easily occur in practice for example in multirate sampled data control systems the algorithm that can be derived from the solution presented in then applied to the output feedback of a dynamical system changing from one operation point in its operation envelope to another

time varying discrete riccati equation l sup 2 invertibility dichotomy and disconjugacy necessary and sufficient existence conditions for the stabilizing solution to the time varying discrete riccati equation are given the result is simply expressed in terms of the original input data via the l sup 2 uniformly bounded invertibility of an appropriate family of toeplitz like operators intimate connections with the notions of dichotomy and disconjugacy are also pointed out some aspects concerning the dependence of the existence conditions on the monotonicity of the input data are investigated as well

an island strategy for memetic discrete tomography reconstruction in this paper we present a parallel island model memetic algorithm for binary discrete tomography reconstruction that uses only four projections without any further a priori information the underlying combination strategy consists in separated populations of agents that evolve by means of different processes agents progress towards a possible solution by using genetic operators switch and a particular compactness operator a guided migration scheme is applied to select suitable migrants by considering both their own and their sub population fitness that is from time to time we allow some individuals to transfer to different subpopulations the benefits of this paradigm were tested in terms of correctness robustness and time of the reconstruction by considering publicly available datasets of images to tackle the so called stability problem we considered the case of noisy projections along four directions to simulate an instrumental error results show that the proposed method decreases the reconstruction error for all classes of images with respect to a serial implementation recently proposed by the authors and that such reconstruction error is almost invariant with respect to the number of demes moreover the computation time of the proposed parallel memetic algorithm scales in a quasi linear manner with respect to the demes number and is invariant with respect to the used number of migrations

reconstruction of convex lattice sets from tomographic projections in quartic time filling operations are procedures which are used in discrete tomography for the reconstruction of lattice sets having some convexity constraints many algorithms have been published giving fast implementations of these operations and the best running time s brunetti a daurat a kuba fast filling operations used in the reconstruction of convex lattice sets in proc of dgci 2006 in lecture notes in comp sci vol 4245 2006 pp 98 109 is o n 2logn time where n is the size of projections in this paper we improve this result by providing an implementation of the filling operations in o n 2 as a consequence we reduce the time complexity of the reconstruction algorithms for many classes of lattice sets having some convexity properties in particular the reconstruction of convex lattice sets satisfying the conditions of gardner gritzmann r j gardner p gritzmann discrete tomography determination of finite sets by x rays trans amer math soc 349 1997 2271 2295 can be performed in o n 4 time

a solution to hammer s x ray reconstruction problem we propose algorithms for reconstructing a planar convex body k from possibly noisy measurements of either its parallel x rays taken in a fixed finite set of directions or its point x rays taken at a fixed finite set of points in known situations that guarantee a unique solution when the data is exact the algorithms construct a convex polygon pk whose x rays approximate in the least squares sense k equally spaced noisy x ray measurements in each of the directions or at each of the points it is shown that these procedures are strongly consistent meaning that almost surely pk tends to k in the hausdorff metric as k this solves for the first time in the strongest sense hammer s x ray problem published in 1963

numerical solution to a descriptor discrete time algebraic riccati equation abstract we investigate the numerical solution of a special type of descriptor discrete time riccati equation involved in the solution of several key problems in robust control formulated under very general hypotheses necessary and sufficient existence conditions together with computable formulas are given for the stabilizing and antistabilizing solutions in terms of an associated matrix pencil two particular applications and numerical experiments of our results are included for squaring down with l norm preservation and j j spectral factorization both for a completely general discrete time system

a nested computational approach to the discrete time finite horizon lq control problem a new algorithmic setting is proposed for the discrete time finite horizon linear quadratic lq optimal control problem with constrained or unconstrained final state no matter whether the problem is cheap singular or regular the proposed solution based on matrix pseudoinversion is completed and made practically implementable by a nesting procedure for welding optimal subarcs that enables arbitrary enlargement of the control time interval

fast verified computation for stabilizing solutions of discrete time algebraic riccati equations fast iterative algorithms for computing interval matrices containing solutions of discrete time algebraic riccati equations are proposed these algorithms involve only cubic complexity per iteration the stabilizability and uniqueness of the contained solution can moreover be verified by these algorithms numerical results show the properties of the algorithms

an algorithm for discrete tomography abstract there are many algorithms in the literature for the approximating reconstruction of a binary matrix from its line sums in this paper we provide an algorithm which starts from the line sums of an unknown binary matrix f and outputs an integer matrix s with small entries in absolute values such that the line sums of f and s coincide we also give the results of some experiments with the algorithm

reconstruction of 4 and 8 connected convex discrete sets from row and column projections in this paper we examine the problem of reconstructing a discrete two dimensional set from its two orthogonal projection h v when the set satisfies some convexity conditions we show that the algorithm of the paper int j imaging systems and technol 9 1998 69 is a good heuristic algorithm but it does not solve the problem for all h v instances we propose a modification of this algorithm solving the problem for all h v instances by starting to build the spine the complexity of our reconstruction algorithm is o mn log mn min m2 n2 in the worst case however according to our experimental results in 99 of the studied cases the algorithm is able to reconstruct a solution without using the newly introduced operation in such cases the upper bound of the complexity of the algorithm is o mn log mn a systematic comparison of this algorithm was done and the results show that this algorithm has the better average complexity than other published algorithms the way of comparison and the results are given in a separate paper linear algebra appl submitted finally we prove that the problem can be solved in polynomial time also in a class of discrete sets which is larger than the class of convex polyominoes namely in the class of 8 connected convex sets

reconstructing convex polyominoes from horizontal and vertical projections in 1 we studied the problem of reconstructing a discrete set 5 from its horizontal and vertical projections we defined an algorithm that establishes the existence of a convex polyomino whose horizontal and vertical projections are equal to a pair of assigned vectors h v with h m and v n its computational cost is o n4m4 in this paper we introduce some operations for recontructing convex polyominoes by means of vectors h s and v s partial sums these operations allows us to define a new algorithm whose complexity is less than o n2m2

the class of suboptimal and optimal solutions to the discrete nehari problem characterization and computation simple and explicit formulae in a linear fractional transformation lft form for all suboptimal and optimal solutions to the discrete time version of the one block nehari problem are derived under no restrictive conditions imposed on the initial data using the so called signature condition a generalized popov theory type argument general solvability conditions in terms of a certain type of discrete time kalman szego popov yakubovich system are obtained for the class of optimal approximations derived via singular perturbation techniques the lft coefficients are all of mcmillan degree n r where n and r are the mcmillan degree and the multiplicity of the largest hankel singular value of the system to be approximated respectively the results also lead directly to a simple numerical algorithm

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

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

time varying discrete hamiltonian systems abstract this paper is dealing with some relevant aspects which occur in the theory of time varying discrete hamiltonian systems the questions of dichotomy and disconjugacy are investigated in an enlarged context and some connections with the riccati equation theory are pointed out

regular switching components abstract we consider a problem of discrete tomography which consists of reconstructing a finite lattice set s z 2 from given horizontal and vertical x rays i e with prescribed numbers of points in each row and column without additional requirements the problem can be solved in polynomial time many variants require the solution to be in a chosen class a for instance the problem is np complete for the class a h v of hv convex lattice sets and it is polynomial for the class a h v p of hv convex polyominoes twenty years after these results the problem s complexity remains unknown for a c the class of c convex lattice sets i e two dimensional lattice polytopes the main difficulty to solve this problem comes from combinatorial structures called switching components switching components are closed path with horizontal and vertical edges such that the solutions s are composed of either the elements of even or of odd indices this binary choice can be encoded by a boolean variable associated with the switching component and the convexity constraints are simply encoded by sat clauses 2 clauses for hv convexity and 3 clauses for c convexity the purpose of the paper is to investigate the properties of the switching components and the consequences of the convexity requirements we divide the switching components in two classes regular if their turning angle is constant irregular otherwise we prove that adjacent regular switching components have the same boolean values this property allows us to merge them into extended switching components we prove that if all switching components are regular then the extended switching components are all independent then the number of solutions with the considered feet is 2 n where n is the number of extended switching components finally we prove that they are geometrically ordered

brief paper an efficient algorithm for discrete time h preview control an efficient numerical solution to the output feedback discrete time h preview control problem is presented the first step of the solution comes from embedding the preview control problem in a two degree of freedom generalised regulator framework in which the augmented system state includes a segment of the frequency weighted previewed signal an efficient method for solving the resulting inflated discrete algebraic riccati equations dares is then found it is demonstrated that one need only solve an estimation riccati equation with dimension that of the plant and a control riccati equation with dimension that of the plant and reference weighting taken together the reduced control riccati equation is the product of a recursive reduction enabled by the zero eigenvalues in the associated closed loop matrix an added feature of the presented algorithm is that no large matrix computations are required and no large matrices need be stored only the edges of the full information riccati equation solution are required when synthesising the controller

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

sensitivity analysis of the discrete time algebraic riccati equation abstract consider the discrete time algebraic riccati equation dare a t xa x a t xb s r b t xb 1 b t xa s t q 0 where a r n n b s r n m r rt r m m q qt r n n the available perturbation theory for the dare can only be applied to the case r 0 however in some control problems the matrix r can be singular in this paper we study perturbation properties of the dare without the restriction r 0 perturbation bounds and a relative condition number for the stabilizing solution of the dare are derived computable residual bounds for an approximate solution are also derived the theoretical results are illustrated by numerical examples

a strongly consistent information criterion for linear model selection based on m estimation in this paper a linear model selection procedure based on m estimation is proposed which includes many classical model selection criteria as its special cases it is shown that the proposed criterion is strongly consistent under certain mild conditions for instance without assuming normality of the distribution of the random errors the results from a simulation study are also presented

reconstruction of domino tiling from its two orthogonal projections we are interested in the reconstruction of a domino tiling of a rectangle from its two orthogonal projections we give polynomial algorithms for some subproblems when all the dominoes are of the same type and prove np completeness results when there are three types of dominoes when two types of dominoes are allowed we give a polynomial time transformation from a well known open problem

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

identification of arx systems with non stationary inputs asymptotic analysis with application to adaptive input design a key problem in optimal input design is that the solution depends on system parameters to be identified in this contribution we provide formal results for convergence and asymptotic optimality of an adaptive input design method based on the certainty equivalence principle i e for each time step an optimal input design problem is solved exactly using the present parameter estimate and one sample of this input is applied to the system the results apply to stable arx systems with the input restricted to be generated by white noise filtered through a finite impulse response filter or a binary signal obtained from the latter by a static nonlinearity

scanning integer matrices by means of two rectangular windows this paper deals with the reconstruction of integer matrices from rectangular scans in particular since the case of one rectangular scan has already been treated in a previous paper we consider two rectangular scans given as two integer matrices and we investigate the existence and the possibility of reconstruction of a third binary matrix which is compatible with them furthermore our inspection implies interesting side results about the number of these reconstructed matrices for different choices of the dimensions of two windows used in the input scans

convergence behavior of the schur recursion in the krein space for the j spectral factorization we present a krein space version of the schur recursion for the j spectral factorization which arises in h sup spl infin related problems the most notable difference of the proposed schur recursion from the ordinary one is that the proposed recursion can handle temporary changes of the inertia during the process we show that the schur recursion in the krein space converges to a j spectral factor exponentially under a suitable condition

a strongly consistent procedure for model selection in a regression problem summary we consider the multiple regression model yn xn en where yn and en are n vector random variables xn is an n x m matrix and 83 is an m vector of unknown regression parameters each component of 3 may be zero or nonzero which gives rise to 2 possible models for multiple regression we provide a decision rule for the choice of a model which is strongly consistent for the true model as n oo the result is proved under certain mild conditions for instance without assuming normality of the distribution of the components of en

discrete tomography determination of finite sets by x rays we study the determination of finite subsets of the integer lattice en n 2 by x rays in this context an x ray of a set in a direction u gives the number of points in the set on each line parallel to u for practical reasons only x rays in lattice directions that is directions parallel to a nonzero vector in the lattice are permitted by combining methods from algebraic number theory and convexity we prove that there are four prescribed lattice directions such that convex subsets of en i e finite subsets f with f en n conv f are determined among all such sets by their x rays in these directions we also show that three x rays do not suffice for this purpose this answers a question of larry shepp and yields a stability result related to hammer s x ray problem we further show that any set of seven prescribed mutually nonparallel lattice directions in 22 have the property that convex subsets of 22 are determined among all such sets by their x rays in these directions we also consider the use of orthogonal projections in the interactive technique of successive determination in which the information from previous projections can be used in deciding the direction for the next projection we obtain results for finite subsets of the integer lattice and also for arbitrary finite subsets of euclidean space which are the best possible with respect to the numbers of projections used

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

relationships between discrete time and continuous time algebraic riccati inequalities abstract the bilinear transformation is used to establish a direct relationship between a discrete time algebraic riccati inequality dari and an associated continuous time algebraic riccati inequality cari it is shown that under mild conditions the dari is solvable if and only if the corresponding cari is solvable the relationship between the dari and the cari is then used to translate the general solvability conditions for a cari given by scherer into analogous conditions for the dari it is shown how such conditions can be applied to determine the solvability of a discrete time h control problem whose solution set is characterized by two daris

general matrix pencil techniques for the solution of algebraic riccati equations a unified approach we present a unified theory of matrix pencil techniques for solving both continuous and discrete time algebraic riccati equations ares under fairly general conditions on the coefficient matrices the theory applies to a large class of ares and riccati like equations arising from the singular h sup spl infin and h sup 2 control problems singular linear quadratic control the 4 block nehari problem or from singular j spectral factorizations the underlying concept is the so called proper deflating subspace of a possibly singular matrix pencil in terms of which necessary and sufficient conditions for the solvability of riccati equations are given it is shown that these conditions can be checked and the solutions computed by a numerically sound algorithm

generalized continuous time riccati theory abstract a riccati like equation which incorporates as special cases the standard continuous time algebraic riccati equation and the constrained continuous time algebraic riccati equation is introduced and studied a characterization of the conditions under which an equation of general form has a stabilizing solution are investigated in terms of the so called proper deflating subspace of the extended hamiltonian pencil

reconstruction of convex polyominoes from orthogonal projections of their contours the problem of reconstructing a convex polyominoes from its horizontal and vertical projections when the projections are defined as the number of cells of the polyomino in the different lines and columns was studied by del lungo and m nivat in this paper we study the reconstruction of any convex polyomino when the orthogonal projections are defined as the contour length of the object intercepted by the ray we prove the np hardness of this problem for several classes of polyominoes general h convex v convex for hv convex polyominoes we give a polynomial time algorithm for the reconstruction problem

determination of q convex sets by x rays in this paper the problem of the determination of lattice sets from x rays is studied we define the class of q convex sets along a set d of directions which generalizes classical lattice convexity and we prove that for any d the x rays along d determine all the convex sets if and only if it determines all the q convex sets along d as a consequence any algorithm which reconstructs q convex sets from x rays can be used to reconstruct convex lattice sets from x rays along directions which provide uniqueness this gives a constructive answer to the discrete version of hammer s x ray problem

verified computation for the hermitian positive definite solution of the conjugate discrete time algebraic riccati equation abstract a fast iterative algorithm is proposed for numerically computing an interval matrix containing the hermitian positive definite solution to the conjugate discrete time algebraic riccati equation this algorithm requires only cubic complexity per iteration and verifies the uniqueness of the contained solution numerical results show effectiveness and robustness of the algorithm

an algorithm reconstructing convex lattice sets in this paper we study the problem of reconstructing special lattice sets from x rays in a finite set of prescribed directions we present the class of q convex sets which is a new class of subsets of z2 having a certain kind of weak connectedness the main result of this paper is a polynomial time algorithm solving the reconstruction problem for the q convex sets these sets are uniquely determined by certain finite sets of directions as a result this algorithm can be used for reconstructing convex subsets of z2 from their x rays in some suitable sets of four lattice directions or in any set of seven mutually nonparallel lattice directions

an introduction to periodical discrete sets from a tomographical perspective in this paper we introduce a new class of binary matrices whose entries show periodical configurations and we furnish a first approach to their analysis from a tomographical point of view in particular we propose a polynomial time algorithm for reconstructing matrices with a special periodical behavior from their horizontal and vertical projections we succeeded in our aim by using a reduction involving polyominoes which can be characterized by means of 2 sat formulas

convex decomposition of u polygons when parallel x rays are considered in any finite set u of directions switching components with respect to u can be constructed this is true for u r n and also for any finite set of lattice directions r j gardner raised the problem of looking for a characterization of switching components in 2001 l hajdu and r tijdeman gave an answer by proving that a switching component is always the linear combination of switching elements though splendid this result fails to be a characterization theorem inside the class of convex bodies meaning that the switching element of the linear combination could be not convex even if the switching component is convex the purpose of this paper is to investigate the problem in the plane where a convex switching component with respect to u is a u polygon we prove that a u polygon can always be decomposed as a linear sum inside the class of u polygons

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

spectral and inner outer factorizations for discrete time systems the spectral and inner outer factorization problems are solved for discrete time systems with transfer matrices of arbitrary rank simple and explicit state space formulas expressed in terms of the stabilizing solution to the discrete time algebraic riccati system are derived the results are numerically effective since a reliable algorithm for computing such a stabilizing solution is available

x rays characterizing some classes of discrete sets in this paper we study the problem of determining discrete sets by means of their x rays an x ray of a discrete set f in a direction u counts the number of points in f on each line parallel to u a class f of discrete sets is characterized by the set u of directions if each element in f is determined by its x rays in the directions of u by using the concept of switching component introduced by chang and ryser comm acm 14 1971 21 combinatorial mathematics the carus mathematical monographs no 14 the mathematical association of america rahway 1963 and extended in discrete comput geom 5 1990 223 we prove that there are some classes of discrete sets that satisfy some connectivity and convexity conditions and that cannot be characterized by any set of directions gardner and gritzmann trans amer math soc 349 1997 2271 show that any set u of four directions having cross ratio that does not belong to 4 3 3 2 2 3 4 characterizes the class of convex sets we prove the converse that is if u s cross ratio is in 4 3 3 2 2 3 4 then the hv convex sets cannot be characterized by u we show that if the horizontal and vertical directions do not belong to u gardner and gritzmann s result cannot be extended to hv convex polyominoes if the horizontal and vertical directions belong to u and u s cross ratio is not in 4 3 3 2 2 3 4 we believe that u characterizes the class of hv convex polyominoes we give experimental evidence to support our conjecture moreover we prove that there is no number such that if u then u characterizes the hv convex polyominoes this number exists for convex sets and is equal to 7 see trans amer math soc 349 1997 2271

stabilizing solution to the reverse discrete time riccati equation a matrix pencil based approach abstract we derive necessary and sufficient conditions for the existence of the stabilizing solution to the reverse discrete time riccati equation a particular type of algebraic riccati equation which is related to the solution of the discrete time version of the extended two block nehari problem the conditions are expressed in terms of the left stable deflating subspace of an associated symplectic matrix pencil in particular a maximum phase spectral factorization of the popov function is obtained under very relaxed conditions imposed on the initial data it is also proved that under a restrictive additional assumption the stabilizing solution to the reverse riccati equation reduces to the antistabilizing solution to the usual discrete time riccati equation a reliable numerical algorithm for computing the stabilizing solution to the reverse riccati equation is also presented together with formulae for the solution to the extended nehari problem