affine spaces of strong linearizations for rational matrices and the recovery of eigenvectors and minimal bases abstract the main aim of this paper is two fold first to define and analyze two affine spaces of matrix pencils associated with a realization of an n n rational matrix g and show that almost all of these pencils are strong linearizations of g second to describe the recovery of eigenvectors minimal bases and minimal indices of g from those of the strong linearizations of g the affine spaces of pencils are constructed from the vector spaces of linearizations of matrix polynomials defined and analyzed by mackey et al in siam j matrix anal appl 28 2006 pp 971 1004 we show that eigenvectors minimal bases and minimal indices of g can be easily recovered from those of the strong linearizations of g in particular we construct a symmetric strong linearization of g when g is regular and symmetric we also describe the recovery of pole zero structure of g at infinity from the pole zero structure at infinity of a strong linearization of g

rational and polynomial matrix factorizations via recursive pole zero cancellation we develop a recursive algorithm for obtaining factorizations of the type r r1 r2 where all three matrices are rational and r1 is nonsingular moreover the factors r1 and r2 are such that either the poles of r1 1 and r2 are in a prescribed region of the complex plane or their zeros such factorizations cover the specific cases of coprime factorization inner outer factorization gcd extraction and many more the algorithm works on the state space or generalized state space realization of r and derives in a recursive fashion the corresponding realizations of the factors

on singular implicit linear dynamical systems we investigate properties of existence unicity representation of the causal solutions of implicit linear systems or generalized systems when the underlying matrix pencil is singular we relate the geometric and the algebraic approaches the main conclusion is that if the underlying matrix pencil is column singular i e has a nonempty set of column minimal indices the causal solutions when they exist can exactly be represented as the output of a classical two player dynamical system where the second player accounts for the nonunicity properties of the equivalent system are related to those of the singular matrix pencils made with the given matrices

rank revealing qr factorizations abstract an algorithm is presented for computing a column permutation pi and a qr factorization a pi qr of an m by n m or n matrix a such that a possible rank deficiency of a will be revealed in the triangular factor r having a small lower right block for low rank deficient matrices the algorithm is guaranteed to reveal the rank of a and the cost is only slightly more than the cost of one regular qr factorization a posteriori upper and lower bounds on the singular values of a are derived and can be used to infer the numerical rank of a keywords qr factorization rank deficient matrices least squares computation subset selection rank singular values

transformation of j spectral factorization of improper matrices to proper matrices we reduce the general problem of j spectral factorization with improper matrices given by a descriptor system realization to the problem of j spectral factorization with proper matrices

a comparison of solutions to the minimal design problem two of the recent algorithms for solving the minimal design problem in an efficient and apparently stable manner are presented in a common setting it is shown that the resulting algorithm maintains attractive features of both earlier algorithms including efficiency ease of use flexibility in selecting criteria of closeness as well as apparently numerical stability

comments on minimum gain minimum time deadbeat controllers abstract we point out that the problem formulated in the above paper by elabdalla and amin 1 was solved in an earlier paper 3 using a direct approach

an improved algorithm for the computation of structural invariants of a system pencil and related geometric aspects in this paper we propose a new recursive algorithm for computing the staircase form of a matrix pencil and implicitly its kronecker structure the algorithm compares favorably to existing ones in terms of elegance versatility and complexity in particular the algorithm without any modification yields the structural invariants associated with a generalized state space system and its system pencil two related geometric aspects are also discussed we show that an appropriate choice of a set of nested spaces related to the pencil leads directly to the staircase form we extend the notion of deflating subspace to the singular pencil case c 1997 elsevier science b v

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

an improved algorithm for the computation of kronecker s canonical form of a singular pencil we give an 0 t algorithm for computing the kronecker structure of an arbitrary m x n pencil xe a the algorithm is shown to he numerically stable because only unitary transformations are used the improved speed over earlier unitary methods is due to the efficient use of condensed forms which are maintained throughout the recursions of the algorithm

on the numerical computation of a structural decomposition in systems and control in this paper we develop a new numerical method for a special coordinate basis of a linear time invariant system such a special coordinate basis is essentially a structural decomposition which explicitly displays the finite and infinite zero structures as well as the invertibility structures of the given system the technique is playing important roles in numerous topics in system and control theory such as robust control h sub spl infin and h sub 2 optimal control almost disturbance decoupling and zero placement of linear systems just to name a few our method consists of three steps reduction by orthogonal transformations reduction by generalized sylvester equations and extraction of infinite zero structure the performance of our method is illustrated by some numerical examples

computation of transfer function matrices of generalized state space models a reliable computational algorithm is proposed for the evaluation of the transfer function matrices of linear multivariable systems described by generalized state space models the algorithm is based on a new efficient and numerically stable procedure for computing irreducible state space realizations of generalized state space models for each input output channel the proposed algorithm computes an irreducible realization using this procedure then the parameters of the corresponding transfer function gain poles zeros are determined from the resulting irreducible model

computing the zeros of periodic descriptor systems in this paper we give a numerically reliable algorithm to compute the zeros of a periodic descriptor system the algorithm is a variant of the staircase algorithm applied to the system pencil of an equivalent lifted time invariant state space system and extracts a low order pencil which contains the zeros both finite and infinite as well as the kronecker structure of the periodic descriptor system the proposed algorithm is efficient in terms of complexity by exploiting the structure of the pencil and is exclusively based on orthogonal transformations which ensures some form of numerical stability

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

robust stabilization of discrete generalized systems the robust stabilization problem for generalized discrete time systems described by polynomial or improper transfer function matrices subject to perturbations acting on the normalized coprime factors is solved the maximum achievable stability margin and the robust stabilizing controller are given in terms of realizations and solutions to appropriate riccati equations

an algorithm for pole assignment in high order multivariable systems an efficient computational method is presented for solving the problem of pole assignment by state feedback in linear multivariable systems with large dimensions a given multiinput system is first transformed to an upper block hessenberg form by means of orthogonal state coordinate transformations the state feedback problem is then reformulated in terms of the sylvester equation the transformed system matrices along with certain assumed block forms for unknown matrices enable the sylvester equation to be decomposed and solved effectively a distinct point of the proposed algorithm is that the solution procedure can be tailored to parallel implementation and is therefore fast computational aspects of the algorithm are discussed and numerical examples are provided

unfolding the zero structure of a linear control system abstract this paper is motivated by the problem of controller design for a parameter dependent linear system many compensation techniques depend critically on aspects of the zero structure of the system such as relative degree or the nonminimum phase property however the zero structure is structurally unstable meaning it may change discontinuously with small changes in the parameters thus it is crucial that the designer know what structures and structural transitions are possible this paper uses a miniversal deformation or unfolding of the kronecker form previously reported by the authors and applies it directly to pencils in a canonical form of the system matrix the unfolding is used to explore all zero structures in the neighborhood of a nominal system several examples are presented when possible the results are presented in the form of a bifurcation diagram

structural decomposition of linear singular systems the single input and single output case in this paper we present a structural decomposition for general single input and single output linear singular systems such a decomposition has a distinct feature of capturing and displaying all the structural properties such as the nite and in nite zero structures and redundant dynamics ofthe given system it is expected to be a powerful tool in solving control problems for singular systems such as h2 and h control model reduction and disturbance decoupling problems to name a few c 2002 elsevier science b v all rights reserved

a geometric approach to the synthesis of failure detection filters a geometric formulation of beard s failure detection filter problem is stated using the concepts of c a invariant and unobservability subspaces the notions of output separable and mutually detectable families of subspaces introduced by beard are also clarified it is shown that mutual detectability is a necessary and sufficient condition for the existence of a detection filter with arbitrarily assignable spectrum moreover it is shown that the failure detection falter problem has a computationally simple solution when the failure events satisfy some mild restrictions finally the complete duality between a generalization of beard s detection filter problem and the restricted control decoupling problem is illustrated

feedback design for regularizing descriptor systems this paper surveys numerical techniques for the regularization of descriptor generalized state space systems by proportional and derivative feedback we review generalizations of controllability and observability to descriptor systems along with definitions of regularity and index in terms of the weierstras canonical form three condensed forms display the controllability and observability properties of a descriptor system the condensed forms are obtained through orthogonal equivalence transformations and rank decisions so they may be computed by numerically stable algorithms in addition the condensed forms display whether a descriptor system is regularizable i e when the system pencil can be made to be regular by derivative and or proportional output feedback and if so what index can be achieved also included is a a new characterization of descriptor systems that can be made to be regular with index 1 by proportional and derivative output feedback

hierarchical image segmentation based on iterative contraction and merging in this paper we propose a new framework for hierarchical image segmentation based on iterative contraction and merging in the proposed framework we treat the hierarchical image segmentation problem as a sequel of optimization problems with each optimization process being realized by a contraction and merging process to identify and merge the most similar data pairs at the current resolution at the beginning we perform pixel based contraction and merging to quickly combine image pixels into initial region elements with visually indistinguishable intra region color difference after that we iteratively perform region based contraction and merging to group adjacent regions into larger ones to progressively form a segmentation dendrogram for hierarchical segmentation comparing with the state of the art techniques the proposed algorithm can not only produce high quality segmentation results in a more efficient way but also keep a lot of boundary details in the segmentation results

disturbance decoupling for descriptor systems by state feedback we study the disturbance decoupling problem for linear time invariant descriptor systems we give necessary and sufficient conditions for the existence of a solution to the disturbance decoupling problem with or without stability via a proportional and or derivative feedback that also makes the resulting closed loop system regular and or of index at most one all results are proved constructively based on condensed forms that can be computed using orthogonal matrix transformations i e transformations that can be implemented in a numerically stable way

transmission zero assignment using semistate descriptions the problem of finite infinite transmission zero assignment is examined by squaring a system from the outputs to the inputs in particular this problem is studied in two cases state accessible systems and partially state accessible systems it is shown that the problem of transmission zero assignment for state accessible state variable systems is equivalent to a pole placement problem with state feedback for a generalized system which always has a solution in the case of partially state accessible systems it is shown that the transmission zero assignment problem is equivalent to a pole placement problem with output feedback for a generalized system in both cases the block hessenberg form of the system and the extended lower triangular hessenberg form are exploited to formulate the problem

determination of invariant zeros and zero directions of the system s a b c e mita 1977 presented a simple method for the invariant zeros of a linear multivariable system by constructing its maximal unobservable subspace mus however the results of this method have been restricted to a class of controllable and right invertible systems satisfying the diagonal decoupling condition and described by the triple a b c here a general method based on the theory of decoupling extends mita s results for all classes of systems described by the 4 tuples a b c e with no special requirements the mus of all classes of such systems is simply constructed and its order is given in terms of the system order n and the infinite zeros or the minimal observability indices of the system in addition the invariant zeros and zero directions are found from the eigenvalues and eigenvectors of a reduced order matrix having the dimension of the mus obtained directly from the state space parameters algorithms are presented for the invariant zeros and zero directions of invertible decouplable o

parametric state feedback control with model reduction a parametrization of the class of linear state feedback controllers that assign a set of desired self conjugate eigenvalues to the closed loop system is applied to a method of model reduction by making the maximum number of the closed loop modes unobservable while retaining arbitrary assignment of the remaining modes a lower order transfer function matrix is obtained the technique may be used to reduce a broad class of multivariable systems to first order multivariable systems the main result is a class of mode reducing fixed gain state feedback controllers explicitly specified by a set of free parameters which may be chosen to satisfy additional design requirements

computation of coprime factorizations of rational matrices abstract we propose numerically reliable state space algorithms for computing several coprime factorizations of rational matrices 1 factorizations with factors having poles in a given stability domain 2 factorizations with proper stable factors 3 factorizations with inner and j inner denominators the new algorithms are based on a recursive generalized schur algorithm for pole dislocation they are generally applicable whether the underlying descriptor state space representation is minimal or not and whether it is stabilizable detectable or not the proposed algorithms are useful in solving various computational problems for both standard and descriptor system representations

pole zero representation of descriptor systems we present an algorithm for computing the pole zero representation of descriptor systems whose generalized state space models are described by the 5 tuple e a b c d where e may be a singular matrix but del a e 0 the proposed algorithm uses only orthogonal transformations hence the computed results are numerically reliable numerical examples are included to illustrate the proposed results

fundamental reachability and observability matrices for discrete descriptor systems this paper uses the fundamental matrix of a regular discrete descriptor system to derive expressions for descriptor reachability and observability matrices reachable and unobservable subspaces are defined it is shown that the natural space for analyzing descriptor system properties is r 2n where n is the dimension of the system not rnas is the case for state space systems solutions are provided for the descriptor open loop control and estimation problems

numerical aspects of disturbance decoupling by measurement feedback a numerically stable algorithm is given to determine orthogonal similarity transformations such that the system attains a certain condensed form using this condensed form solvability of disturbance decoupling by measurement can be decided by inspection and controllers can be computed by solving linear equations

inclusion principle for descriptor systems the purpose of this paper is to propose an expansion contraction framework for linear constant descriptor systems within the inclusion principle for dynamic systems our primary objective is to provide an explicit characterization of the expansion process whereby a given descriptor system is expanded into the larger space where all its solutions are reproducible by the expanded descriptor system if appropriate initial conditions are selected when a control law is formulated in the expanded space the proposed characterizations provide contractibility conditions for implementation of the control law in the original system a full freedom is provided for selecting appropriate matrices in the proposed expansion contraction control scheme in particular the derived theoretical framework serves as a flexible environment for expansion contraction control design of descriptor systems under overlapping information structure constraints

local heuristics and the emergence of spanning subgraphs in complex networks we study the use of local heuristics to determine spanning subgraphs for use in the dissemination of information in complex networks we introduce two different heuristics and analyze their behavior in giving rise to spanning subgraphs that perform well in terms of allowing every node of the network to be reached of requiting relatively few messages and small node bandwidth for information dissemination and also of stretching paths with respect to the underlying network only modestly we contribute a detailed mathematical analysis of one of the heuristics and provide extensive simulation results on random graphs for both of them these results indicate that within certain limits spanning subgraphs are indeed expected to emerge that perform well in respect to all requirements we also discuss the spanning subgraphs inherent resilience to failures and adaptability to topological changes

computing the radius of controllability for state space systems abstract in this paper we discuss the problem of computing the nearest uncontrollable system to a given control system represented by a matrix pair a b in order to do so we construct a sequence of structured matrices from given system matrices a and b controllability of the pair a b is equivalent to a condition on the null space dimension of an appropriate matrix in this sequence we show that the dimension of the reachability space is also related to the above condition further it is shown that the nearest structured low rank approximation slra of this structured matrix corresponds to a nearest uncontrollable system to the pair a b

an exact characterization of linear systems with zeros everywhere in the complex plane in this note an exact characterization of autonomous linear multivariable systems with zeros everywhere in the complex plane is given in terms of the system matrices such systems are called degenerate an important consequence of this characterization is that nondegeneracy is equivalent to the concept of functional reproducibility introduced by brockett and mesarovic based on this equivalence a new test for nondegeneracy in terms of the functional reproducibility matrix is derived an example is included to show that controllability observability and output controllability are not sufficient to guarantee the nondegeneracy of a linear system

the properties of reduced order minimum variance filters for systems with partially perfect measurements the problem of the finite time reduced order minimum variance full state estimation of linear continuous time invariant systems is considered in cases where the output measurement is partially free of corrupting white noise components the structure of the optimal filter is obtained and a link between this structure and the structure of the system invariant zeros is established using expressions that are derived in closed form for the invariant zeros of the system simple sufficient conditions are obtained for the existence of the optimal filter in the stationary case the structure and the transmission properties of the stationary filter for general left invertible systems are investigated a direct relation between the optimal filter and a particular minimum order left inverse of the system is obtained a simple explicit expression for the filter transfer function matrix is also derived the expression provides an insight into the mechanism of the optimal estimation

paper minimal representations of implicit linear systems definitions of external behaviour equivalence and minimality are given for a class of implicit non oriented linear dynamical systems of particular interest in computer aided analysis and design conditions for minimality are stated in terms of the rank properties of submatrices in the singular pencil realization and a geometric analysis is used in the proof a stable library quality numerical algorithm for minimization is included

design of l q regulators for state constrained continuous time systems a new methodology to the design of lq regulators for continuous time systems subject to linear state constraints is proposed consisting of two parts in the first one the positive invariance of the polyhedron defined by the constraints in the state space is imposed guaranteeing thereby that the constraints will not be violated furthermore the admissible constrained controllers are parameterized via the determination of the elements which are fixed in the state feedback matrix this parameterization enables in a second part the l q regulator to be obtained from the solution of a parameter optimization problem subject to linear constraints for which it is proposed a specialized feasible directions method

nearly singular filtering for uniform and non uniform rank linear continuous systems abstract the problem of the minimum variance stationary estimation of linear time invariant continuous systems is considered in the case where the norm of tho measurement white noise intensity approaches zero solutions in closed form are obtained for the kalman gain and the corresponding minimum error covariance matrix of the filtered estimate in terms of the noise intensity norm and the system transfer function matrix both uniform and non uniform rank systems are considered and for the first time a solution is obtained for the case where the infinite zeros of the system tond to infinity with multiple rates

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

computational issues for linear periodic systems paradigms algorithms open problems linear periodic systems originate in various control fields involving periodic phenomena in the beginnings of algorithmic developments for periodic systems computational detours have been often employed to reduce the computational problems for periodic systems to those for standard systems well known techniques are the employment of lifted representations in discrete time or the use of periodic generators in continuous time new computational paradigms evolved later whose main ingredients are numerically reliable and efficient algorithms for manipulating matrix products without forming explicitly them reduction of large scale structured matrix pencils without building the underlying pencils computing with discrete time periodic system models with time varying state dimensions or solving periodic matrix differential equations using multiple shooting techniques the paper will give a succinct overview of all these new developments and will point out some of still existing open computational problems

a numerical design of interactors for general descriptor systems in this technical note we develop a numerically reliable method to design an interactor for a general descriptor system relative to a stability region cg and an offending zero set our main result is based on a condensed form this condensed form is independent of the offending zeros and is computed using only orthogonal transformations and hence is numerically backward stable

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

computation of the zeros and zero directions of linear multivariable systems a new geometric method of calculating multivariable system zeros and zero directions is presented by considering a particular choice of state feedback control law this control law has been motivated by the study of variable structure systems in the sliding mode the cases of singular and non singular cb have been treated separately

on the geometry of the inverse system the geometry of a particular reduced order state space realization of the inverse of a linear continuous time invariant strictly proper system is considered this geometry is characterized by the eigenstructure of the dynamic part of the inverse which is related to the invariant zero structure of the inverted system a method for deriving explicit expressions for the zeros and the zero directions is introduced this method is demonstrated in an example of a uniform rank system

computation of inner outer factorization of rational matrices we propose a new numerically reliable computational approach to determine the inner outer factorization of a rational transfer matrix g of a linear descriptor system in contrast to existing computationally involved one shot methods which require the solution of riccati or generalized riccati equations the new approach relies on an efficient recursive zeros dislocation technique the resulting inner and outer factors always have minimal order descriptor representations the approach proposed is completely general being applicable whenever g is proper strictly proper or not or of full column row rank or not

transfer function matrix of singular systems a formula is given which allows the determination of the transfer function matrix of linear multivariable singular systems i e systems of the form e dot x ax bu with e singular directly in terms of the matrices of the state space description and the characteristic polynomial without inverting a polynomial matrix an alternative closed form expression for the transfer function matrix in terms of the pencil a se is also given

the linear quadratic optimal regulator for descriptor systems in this paper we investigate the linear quadratic optimal regulator problem for the continuous time descriptor system e dot x ax bu where e is in general a singular matrix we solve first a general finite horizon problem by applying the calculus of variations to derive the optimal trajectory of the vector consisting of the concatenated descriptor codescriptor and control vectors from this trajectory the optimal feedback gain relating the control and descriptor variable can be computed by transforming to a coordinate system which can be computed by performing a singular value decomposition of e we derive several riccati differential equations all of which have the same solution this solution gives the optimal cost the steady state optimal feedback gain can be computed by solving an eigenvalue eigenvector problem formulated from the untransformed system parameters in general there does not exist a unique optimal feedback gain but rather the gain is constrained to lie in a linear variety whose dimension is equal to the number of inputs times the rank deficiency of e

an algorithm for the computation of 2 d eigenvalues in this paper a definition for the eigenvalues of two dimensional 2 d general discrete state space models is proposed this definition is consistent with that known for 2 d roesser models also a programmable algorithm is developed to compute the 2 d eigenvalues

the generalized eigenstructure problem in linear system theory the algebraic theory of linear time invariant systems has been studied in large detail during the past few decades and numerous computational algorithms have been developed to solve problems arising in this context in this paper the numerical aspects of a certain class of such algorithms dealing with what the author calls generalized eigenstructure problems are discussed some new and or modified algorithms are presented both the nmnerical stability of the algorithms and the conditioning of the problems they solve are analyzed using numerical criteria

computation of the structural invariants of linear multivariable systems with an extended version of the program zeros abstract the structural invariants like finite and infinite zeros play an important role in many problems in the analysis of linear systems in 1 emami naeini and van dooren presented the program zeros for computing the finite zeros of a linear system which in the author s opinion is at present the best commonly available program for this problem such a program does not exist for computing the other structural invariants like the infinite zeros and the kronecker indices this paper presents an extended version of the program zeros which computes the finite zeros as well as the other structural invariants

spectral matting we present spectral matting a new approach to natural image matting that automatically computes a basis set of fuzzy matting components from the smallest eigenvectors of a suitably defined laplacian matrix thus our approach extends spectral segmentation techniques whose goal is to extract hard segments to the extraction of soft matting components these components may then be used as building blocks to easily construct semantically meaningful foreground mattes either in an unsupervised fashion or based on a small amount of user input

deflation in eigenvalue assignment of descriptor systems using state feedback it is shown how deflation by unitary equivalence transformations can be used to solve the eigenvalue assignment problem for descriptor systems using state feedback the system is first transformed to a form that reveals its controllability the uncontrollable part of the system if any is then discarded and the assignments are continued with the completely controllable part of the system to give insight into the approach a mathematical method for the solution is derived without considering numerical issues using the idea of deflation it is shown how this method can be modified to derive a numerically efficient algorithm the numerical stability of the algorithm is proved and a numerical example is presented

on input allocation based regulation for linear over actuated systems results concerning the output regulation problem for over actuated linear systems are presented in this paper the focus is on the characterization of the solution of the full information regulator problem for systems which are right invertible but not left invertible and the input operator is injective the intrinsic redundancy in the plant model is exploited by parameterizing all solutions of the regulator equations and performing a static or dynamic optimization on the space of solutions this approach effectively shapes the non unique steady state of the system so that the long term behavior optimizes a given performance index in particular nonlinear cost functions that account for constraints on the inputs are considered within the general form of a hybrid system assumed for the allocation mechanism an example is given to illustrate the proposed methodology

spectral structures of the generalized companion form and applications in this note we investigate the relationship between the finite and infinite frequency structure of a regular polynomial matrix and that of a particular linearization called the generalized companion matrix a special resolvent decomposition of the regular polynomial matrix is proposed which is based on the weierstrass canonical form of this generalized companion matrix and the solution of a regular polynomial matrix description pmd is thus formulated from this resolvent decomposition both the initial conditions of the pseudostate and the input are considered

approximation algorithms for spanner problems and directed steiner forest we present an o nlogn approximation algorithm for the problem of finding the sparsest spanner of a given directed graph g on n vertices a spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph more precisely given a graph g v e with nonnegative edge lengths d e r 0 and a stretchk 1 a subgraph h v e h is a k spanner of g if for every edge s t e the graph h contains a path from s to t of length at most email protected d s t the previous best approximation ratio was email protected n 2 3 due to dinitz and krauthgamer stoc 11 we also improve the approximation ratio for the important special case of directed 3 spanners with unit edge lengths from email protected n to o n 1 3logn the best previously known algorithms for this problem are due to berman raskhodnikova and ruan fsttcs 10 and dinitz and krauthgamer the approximation ratio of our algorithm almost matches dinitz and email protected s lower bound for the integrality gap of a natural linear programming relaxation our algorithm directly implies an o n 1 3logn approximation for the 3 spanner problem on undirected graphs with unit lengths an easy o n approximation algorithm for this problem has been the best known for decades finally we consider the directed steiner forest problem given a directed graph with edge costs and a collection of ordered vertex pairs find a minimum cost subgraph that contains a path between every prescribed pair we obtain an approximation ratio of o n 2 3 e for any constant e 0 which improves the o n email protected min n 4 5 m 2 3 ratio due to feldman kortsarz and nutov email protected 12

singular systems a new approach in the time domain a new approach in the time domain is developed for the study of singular linear systems of the form e dot x ax bu y cx with e singular central to the approach is the fundamental triple alpha e a 1 e alphae a 1 b c with a real number satisfying det alpha e a neq 0 controllability and observability matrices are expressed in terms of the fundamental triple new tests for impulse controllability and impulse observability are also established feedback control problems including pole placement decoupling and disturbance localization are studied by use of a modified proportional and derivative feedback of the state in the form of u f alpha x dot x v

time delayed state and unknown input observation the paper deals with the use of output anticipations to relax restrictions of the existing unknown input observer uio design methods the anticipation signals are treated as additional outputs of the original system and a modified existence condition is presented the modified condition is shown equal to the condition of system inversion a computer simulation shows that the proposed uio using the anticipation signals extends the applicability of uio to a larger class of dynamic systems

computation of structural invariants of generalized state space systems in this paper we develop an algorithm for computing the zeros of a generalized state space model described by the matrix 5 tuple e a b c d where e may be a singular matrix but det a he 0 the characterization of these zeros is based on the system matrix of the corresponding 5 tuple both the characterization and the computational algorithm are extensions of equivalent results for state space models described by the 4 tuples a b c d we also extend these results to the computation of infinite zeros and left and right minimal indices of the system matrix several non trivial numerical examples are included to illustrate the proposed results

learning based hierarchical graph for unsupervised matting and foreground estimation automatically extracting foreground objects from a natural image remains a challenging task this paper presents a learning based hierarchical graph for unsupervised matting the proposed hierarchical framework progressively condenses image data from pixels into cells from cells into components and finally from components into matting layers first in the proposed framework a graph based contraction process is proposed to condense image pixels into cells in order to reduce the computational loads in the subsequent processes cells are further mapped into matting components using spectral clustering over a learning based graph the graph affinity is efficiently learnt from image patches of different resolutions and the inclusion of multiscale information can effectively improve the performance of spectral clustering in the final stage of the hierarchical scheme we propose a multilayer foreground estimation process to assemble matting components into a set of matting layers unlike conventional approaches which typically address binary foreground background partitioning the proposed method provides a set of multilayer interpretations for unsupervised matting experimental results show that the proposed approach can generate more consistent and accurate results as compared with state of the art techniques

hessenberg and hessenberg triangular forms in linear system thcory abstract in this paper we distinguish between free hessenberg forms where only the system matrix is transformed into a special structure and system hessenberg forms where either an input or output matrix is simultaneously simplified we show how the freedom in the choice of the transformation matrix for the free hessenberg form can be used to obtain a unique system hessenberg form we discuss numerical aspects and outline some basic applications for both types of hessenberg forms in control and system theory we show that the system hessenberg form can be extremely sensitive to perturbations a property not necessarily shared by the free hessenberg form several examples are explored and a first order perturbation analysis is given analogous aspects are also discussed for the hessenberg triangular forms resulting from reduction of a matrix pencil the main emphasis of this paper is on systems with one input

computation of matrix fraction descriptions of linear time invariant systems in this paper algorithms are presented for obtaining relatively prime matrix fraction descriptions mfd s of linear time invariant systems orthogonal coordinate transformations are used to reduce a given state space model to one whose state matrix is in block hessenberg form the observability controllability indexes of the system are also obtained in the process a recursive algorithm is then described for obtaining a relatively prime mfd from the block hessenberg representation it is then shown that some simplification can be made in the recursive algorithm by first reducing the block hessenberg form by means of a nonsingular triangular coordinate transformation to a more compact form such as the block frobenius form a permutation of the state variables of the block frobenius form yields a canonical representation similar to the luenberger canonical form some numerical and other properties of the algorithms are discussed and the use of the algorithms is illustrated by numerical examples the numerical performance of the algorithms is also compared with that of the structure algorithm of wolovich and falb

classification of proper bases of rational vector spaces minimal macmillian degree bases abstract the algebraic structure of the get of all proper rational vectors contained in a given rational vector space x1d4af s is shown to be that of a noothorian pr s motiulo m ropf pr a is the ring of proper rational functions the proper submodules mt of m form an ascending chain of submodules partially ordered by an invariant of mt defined as the valuation at s ofmt tho various bases of mt are examined and classified according to their property of column reduceness at s the concept of a prime column reduced at basis of mt is introduced it is shown that tho prime bases of mt can be further classified by their macmillan degrees and the existence of minimal macmillan degree bases for mt is established a prime and minimal macmillan degree basis ofmt extends forney s concept of a minimal polynomial basis of s for the rpr mt module the macmillan degrees of the columns of such bases form a set of invariants mt for which are defined as the x1d4af s generalized invariant dynamicol indices o

linear transformations which leave controllable multiinput descriptor systems controllable abstract a general linear descriptor system e itx ax bu where e a c n n b c n m x c n u c m e singular is called controllable if rank e a b n for all 0 0 c let f c n n m c n n m be a linear transformation form f x cxd we characterize all such linear transformations that leave the set c e a b rank e a b n 0 0 c invariant and show that only the well known transformations that leave controllability invariant can occur

generating low degree 2 spanners a k spanner of a connected undirected unweighted graph g v e is a subgraph g consisting of all the vertices of v and a subset of the edges with the additional property that the distance between any two vertices in g is larger than that distance in g by no more than a factor of k this paper is concerned with approximating the problem of finding a 2 spanner in a given graph with minimum maximum degree we first show that the problem is at least as hard to approximate as set cover then a randomized approximation algorithm is provided for this problem with approximation ratio of tilde o delta 1 4 we then present a probabilistic algorithm that is more efficient for sparse graphs our algorithms are converted into deterministic ones using derandomization

system zeros analysis via the moore penrose pseudoinverse and svd of the first nonzero markov parameter a new characterization of system zeros of an arbitrary linear system described by a state space model s a b c d is presented the transmission zeros are characterized as invariant zeros of an appropriate strictly proper system with a smaller number of inputs and outputs than the original system the approach is based on singular value decomposition svd of the first nonzero markov parameter this result together with characterization of invariant and decoupling zeros based on the moore penrose inverse of the first nonzero markov parameter and the kalman canonical decomposition theorem provided in the first part of the paper yield a complete characterization of system zeros of an arbitrary multi input multi output system

a numerical method for a generalized algebraic riccati equation in this paper we develop a numerical method for computing the semistabilizing solution of a generalized algebraic riccati equation gare the semistabilizing solution of such a gare has been used to characterize the solvability of the j j prime spectral factorization problem for general rational matrices which have poles and zeros on the extended imaginary axis the main difficulty for solving such a gare is that its associated skew hamiltonian hamiltonian pencil has eigenvalues on the extended imaginary axis consequently it is not clear which eigenspace of the associated skew hamiltonian hamiltonian pencil can characterize the desired semistabilizing solution i e it is not clear which eigenvectors and principal vectors corresponding to the eigenvalues on the extended imaginary axis should be contained in the eigenspace that we wish to compute and hence the well known generalized eigenvalue approach for the classical algebraic riccati equations cannot be directly employed for it our proposed method consists of computations of the eigendecomposition of the system pencil corresponding to the eigenvalues on the extended imaginary axis and the stable eigenspace of an augmented matrix pencil hence it is a generalization of the generalized eigenvalue approach for the classical algebraic riccati equations

the eigenstructure of an arbitrary polynomial matrix computational aspects we give a new numerical method to compute the eigenstructure i e the zero structure the polar structure and the left and right null space structure of a polynomial matrix p these structural elements are of fundamental importance in several systems and control problems involving polynomial matrices the approach is more general than previous numerical methods because it can be applied to an arbitrary m n polynomial matrix p with normal rank r smaller than m and or n the algorithm is then shown to compute the structure of the left and right null spaces of p as well the speed and accuracy of this new approach are also discussed

estimation of alpha mattes for multiple image layers image matting deals with the estimation of the alpha matte at each pixel i e the contribution of the foreground and background objects to the composition of the image at that pixel existing methods for image matting are typically limited to estimating the alpha mattes for two image layers only however in several applications one is interested in editing images with multiple objects in this work we consider the problem of estimating the alpha mattes of multiple n 2 image layers we show that this problem can be decomposed into n simpler subproblems of alpha matte estimation for two image layers moreover we show that by construction the estimated alpha mattes at each pixel are constrained to sum up to 1 across the multiple image layers a key feature of our framework is that the alpha mattes can be estimated in closed form we further show that due to the nature of spatial regularization used in the estimation the final estimated alpha mattes are not constrained to take values in 0 1 hence we study the optimization problem of estimating the alpha mattes for multiple image layers subject to the fact that the alpha mattes are nonnegative and sum up to 1 at each pixel we present experiments to show that our proposed method can be used to extract mattes of multiple image layers

a geometric approach to proportional plus derivative feedback using quotient and partitioned subspaces abstract in this paper a new characterization of invariant subspaces is presented using the notions of partitioned and quotient subspaces this classification is based on a feedback approach that exhibits the importance of these subspaces for the problem of proportional plus derivative feedback it also provides the ability to decompose the closed loop system into two subsystems that completely characterize the closed loop behavior a feedback free formulation is also considered which opens new horizons for the problem of general semistate feedback by utilizing nonsquare pencils for spectrum assignability the importance of this technique arises from the fact that we deal with reduced order pencils therefore the computational methods are more stable in order to accomplish this we use two generalized lyapunov equations and exploit the generalized hessenberg form

research on system zeros a survey this report provides a review of notable publications regarding the topic of zeros of multivariable systems in the period from 1970 into 1989 particular attention is paid to relationships between definitions of such zeros and their theoretical development transmission invariant system module infinite and finite zeros are addressed the latter subdivided into a hierarchical structure with respect to information considered in the definition process

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

singular perturbations and time scale methods in control theory survey 1976 1983 recent progress in the use of singular perturbation and two time scale methods of modeling and design for control systems is reviewed over 350 references are organized into major problem areas representative issues and results are discussed with a view to outlining research directions and indicating potential areas of application the survey is aimed at engineers and applied mathematicians interested in model order reduction separation of time scales and allied simplified methods of control system analysis and design the exposition does not assume prior knowledge of singular perturbation methods