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

fuzzy geometry an updated overview abstract this paper briefly reviews work on the fuzzy topology and geometry of image subsets including adjacency separation and connectedness distance and relative position area perimeter and diameter convexity medial axes and thinning as well as some applications of these concepts in image processing and analysis

instability analysis of space trusses abstract reduction methods and computational procedures are presented for the instability analysis of space trusses with both geometric and material nonlinearities subjected to combined loads a mixed formulation is used with the fundamental unknowns consisting of both the member forces and the joint displacements the governing equations of the truss consist of the two sets of member constitutive relations and joint equilibrium equations the vector of fundamental unknowns is expressed as a linear combination of a small number of path derivatives derivatives of nodal displacements and member forces with respect to path parameters and a bubnov galerkin technique is used to approximate the governing equations by a small system of algebraic equations the reduced equations are then used to determine the stability boundary and trace the nonlinear equilibrium paths the potential of the proposed approach is discussed and its effectiveness is demonstrated by means of numerical examples of space trusses with combined geometric and material nonlinearities the constitutive relations in these examples are assumed for convenience to be represented by rambergsgood polynomials

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

connectedness of coherent fuzzy pictures abstract a fuzzy digital picture is called connected if it has only one connected component of constant fuzzy level that is maximal i e not adjacent to any component of higher fuzzy level in this paper we propose two acceptors defined over fuzzy digital pictures one is a one pebble acceptor and another is a deterministic one way nonwriting parallel sequential acceptor we also prove that both acceptors can determine whether a given coherent fuzzy digital picture is connected

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

model order reduction in hyperelasticity a proper generalized decomposition approach this paper deals with the extension of proper generalized decomposition methods to non linear problems in particular to hyperelasticity among the different approaches that can be considered for the linearization of the doubly weak form of the problem we have implemented a new one that uses asymptotic numerical methods in conjunction with proper generalized decomposition to avoid complex consistent linearization schemes necessary in newton strategies this approach results in an approximation of the problem solution in the form of a series expansion each term of the series is expressed as a finite sum of separated functions the advantage of this approach is the presence of only one tangent operator identical for every term in the series the resulting approach has proved to render very accurate results that can be stored in the form of a meta model in a very compact format this opens the possibility to use these results in real time reaching khz feedback rates or to be used in deployed handheld devices such as smartphones and tablets

structure preserving stability and accuracy properties of the energy conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models summary the computational efficiency of a typical projection based nonlinear model reduction method hinges on the efficient approximation for explicit computations of the scalar projections onto a subspace of a residual vector for implicit computations it also hinges on the additional efficient approximation of similar projections of the jacobian of this residual with respect to the solution the computation of both approximations is often referred to in the literature as hyper reduction to this effect this paper focuses on the analysis and comparative performance study for nonlinear finite element reduced order models of solids and structures of the recently developed energy conserving mesh sampling and weighting ecsw hyper reduction method unlike most alternative approaches this method approximates the scalar projections of residuals and or jacobians directly instead of approximating first these vectors and matrices then projecting the resulting approximations onto the subspaces of interest in this paper it is shown that ecsw distinguishes itself furthermore from other hyper reduction methods through its preservation of the lagrangian structure associated with hamilton s principle for second order dynamical systems this enables it to also preserve the numerical stability properties of the discrete system to which it is applied it is also shown that for a fixed set of parameter values the approximation error committed online by ecsw is bounded by its counterpart error committed off line during the training of this method therefore this error can be estimated in this case a priori and controlled the performance of ecsw is demonstrated first for two academic but nevertheless interesting nonlinear dynamic response problems for both of them ecsw is shown to preserve numerical stability and deliver the desired level of accuracy whereas the discrete empirical interpolation method and its recently introduced unassembled variant are shown to be susceptible to failure because of numerical instability the potential of ecsw for enabling the effective reduction of nonlinear finite element dynamic models of solids and structures is also highlighted with the realistic simulation of the nonlinear transient dynamic response of a complete car engine to thermal and combustion pressure loads using an implicit scheme for this simulation ecsw is reported to enable the reduction of the cpu time required by the high dimensional nonlinear finite element dynamic analysis by more than four orders of magnitude while achieving a very good level of accuracy copyright 2015 john wiley sons ltd

on the optimality of the discrete karhunen loeve expansion a short proof of the optimality of the discrete karhunen loeve expansion as the best linear approximation in the quadratic mean is presented

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

error bounds for dynamic responses in forced vibration problems when using mode superposition in large applications generally only relatively few approximate eigenmodes are linearly combined block lanczos iteration is an efficient method of determining such modes in this paper new a posteriori bounds are developed that estimate the error when approximating the exact result of mode superposition with a linear combination of the output vectors of block lanczos iteration mode superposition can be regarded as a way of computing g s f a function g of a selfadjoint matrix s applied to a vector one formula is developed that estimates the norm of the unknown error vector a second inequality gives a bound for the error when computing linear functionals v g s f of the response the error bounds require that f and possibly v are contained in the lanczos starting block and that all ritz vectors are used to compute the result no gaps in the spectrum of s need to be known the bounds can be evaluated at a small cost compared to the eigenpair extraction in large syste

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

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

zur theorie der linearen und nichtlinearen integralgleichungen fredholm hat eine auflosungsformel der inhomogenen linearen integralgleichung f left s right varphi left s right lambda int limits a b k left s t right varphi left t right dt entdeckt welche das resultat enthielt das diese gleichung nach s stets aufgelost werden kann wenn nicht eine der nullstellen einer gewissen ganzen transzendenten ist fur diese werte von in hilbertscher bezeichnung die sogenannten eigenwerte des kernes k s t und nur fur diese last wie fredholm ferner zeigte die homogene gleichung 0 varphi left s right int limits a b k left s t right varphi left t right dt losungen zu die nach hilbert zum betreffenden eigenwert des kernes k s t gehorige eigenfunktionen genannt werden hilbert hat die in der theorie der partiellen und gewohnlichen differentialgleichungen so wichtigen fragen nach der existenz sogenannter normalfunktionen und der entwickelbarkeit willkurlicher funktionen nach ihnen durch einfuhrung der greenschen funktion auf das viel allgemeinere problem zuruckgefuhrt die existenz von eigenfunktionen eines in s und t symmetrischen kernes k s t zu beweisen und die gesetze der entwickelbarkeit willkurlicher funktionen nach ihnen aufzustellen die greensche funktion selbst wird durch eine integralgleichung mit unsymmetrischem kerne bestimmt wobei die fredholmschen formeln zur anwendung kommen

a matrix deim technique for model reduction of nonlinear parametrized problems in cardiac mechanics abstract when relying on newton iterations to solve nonlinear problems in the context of reduced basis rb methods the assembling of the rb arrays during the online stage depends on the dimension of the underlying high fidelity approximation this is more of an issue when dealing with fully nonlinear problems for which the global jacobian matrix has to be entirely reassembled at each newton step in this paper the discrete empirical interpolation method deim and its matrix version mdeim are combined to evaluate both the residual vector and the jacobian matrix very efficiently in the case of complex parametrized nonlinear mechanical problems we compare this strategy with the classical deim approach and we derive a posteriori error estimates on the solution accounting for the contribution of deim mdeim errors the effectiveness of the proposed framework is assessed on quasi static nonlinear problems in particular we consider a nonlinear elasticity problem defined on a cube and a mechanical model describing heart contraction for an idealized left ventricle geometry the latter is a coupled problem in which the activation of the heart contraction is given by the solution of an electrophysiology model our numerical results show that mdeim is preferable to the classical deim both in terms of efficiency and accuracy

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

dimensional model reduction in non linear finite element dynamics of solids and structures a general approach to the dimensional reduction of non linear finite element models of solid dynamics is presented for the newmark implicit time discretization the computationally most expensive phase is the repeated solution of the system of linear equations for displacement increments to deal with this it is shown how the problem can be formulated in an approximation ritz basis of much smaller dimension similarly the explicit newmark algorithm can be also written in a reduced dimension basis and the computation time savings in that case follow from an increase in the stable time step length in addition the empirical eigenvectors are proposed as the basis in which to expand the incremental problem this basis achieves approximation optimality by using computational data for the response of the full model in time to construct a reduced basis which reproduces the full system in a statistical sense because of this global time viewpoint the basis need not be updated as with reduced bases computed from a linearization of the full finite element model if the dynamics of a finite element model is expressed in terms of a small number of basis vectors the asymptotic cost of the solution with the reduced model is lowered and optimal scalability of the computational algorithm with the size of the model is achieved at the same time numerical experiments indicate that by using reduced models substantial savings can be achieved even in the pre asymptotic range furthermore the algorithm parallelizes very efficiently the method we present is expected to become a useful tool in applications requiring a large number of repeated non linear solid dynamics simulations such as convergence studies design optimization and design of controllers of mechanical systems

bifurcation and post buckling analysis of laminated composite plates via reduced basis technique abstract a reduced basis technique and a problem adaptive computational algorithm are presented for the bifurcation and post buckling analysis of laminated anisotropic plates the computational algorithm can be conveniently divided into three distinct stages the first stage is that of determining the bifurcation point the plate is discretized by using displacement finite element or finite difference models the special symmetries exhibited by the response of the anisotropic plate are used to reduce the size of the analysis region the vector of unknown nodal parameters is expressed as a linear combination of a small number of basis vectors and a rayleigh ritz technique is used to approximate the finite element equations by a small system of algebraic equations the reduced equations are used to determine the bifurcation point and the associated eigen mode of the panel in the second stage of the bifurcation buckling mode is used to obtain a nonlinear solution in the vicinity of the bifurcation point and new updated sets of basis vectors and reduced equations are generated in the third stage the reduced equations are used to trace the post buckling paths the effectiveness of the proposed technique for predicting the bifurcation and post buckling behavior of plates is demonstrated by means of numerical examples for plates loaded by means of prescribed edge displacements

variational integrators and the newmark algorithm for conservative and dissipative mechanical systems the purpose of this work is twofold first we demonstrate analytically that the classical newmark family as well as related integration algorithms are variational in the sense of the veselov formulation of discrete mechanics such variational algorithms are well known to be symplectic and momentum preserving and to often have excellent global energy behavior this analytical result is veried through numerical examples and is believed to be one of the primary reasons that this class of algorithms performs so well second we develop algorithms for mechanical systems with forcing and in particular for dissipative systems in this case we develop integrators that are based on a discretization of the lagrange d alembert principle as well as on a variational formulation of dissipation it is demonstrated that these types of structured integrators have good numerical behavior in terms of obtaining the correct amounts by which the energy changes over the integration run

the koiter newton approach using von karman kinematics for buckling analyses of imperfection sensitive structures the koiter newton approach is a reduced basis method for nonlinear structural analyses the method combines koiter s approach as a predictor step and the newton arc length technique as a corrector step to trace the entire load displacement equilibrium path of a structure in a step by step manner this computationally highly efficient and accurate solution method has recently demonstrated a superior performance in nonlinear analyses compared to standard techniques in this paper we propose an extension to buckling imperfection sensitivity analyses exploiting the method s stability and reliability we present two different modeling techniques for imperfection loads that both profit from the koiter newton approach reducing the initial computation steps to a small fraction thereof we introduce von karman kinematics which neglect some nonlinear terms of the green strain displacement relations to further increase the computational efficiency of the analysis by an essential reduction of the computation effort required to obtain higher order derivatives of the strain energy using various numerical examples and benchmark tests we demonstrate the overall high quality and performance of the proposed method

projection based reduction of fluid structure interaction systems using monolithic space time modes abstract the focus of this work is the development of reduced models for engineering applications in complex bidirectional fluid structure interaction in the simultaneous solution procedure velocity variables are used for both fluid and solid and the whole set of model equations is discretized by a stabilized time discontinuous space time finite element method flexible structures are modeled using a three dimensional continuum approach in a total lagrangian setting considering large displacements and rotations in the flow domain the incompressible navier stokes equations describe the newtonian fluid a continuous finite element mesh is applied to the entire spatial domain and the discretized model equations are assembled in a single set of algebraic equations considering the two field problem as a whole the continuous fluid structure mesh with identical orders of approximation for both solid and fluid in space and time automatically yields conservation of mass momentum and energy at the fluid structure interface a mesh moving scheme is used to adapt the nodal coordinates of the fluid space time finite element mesh to the structural deformation the computational approach for strongly coupled fluid structure interaction is used to create suitable reduced models of generic nonlinear problems reduction is performed with monolithic projection based space time modes ensuring strong coupling of fluid and structure in the reduced model the contribution discusses results using proper orthogonal decomposition pod for determination of monolithic space time modes in the reduction of fluid structure systems

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

on the recognition of properties of three dimensional pictures this correspondence proposes one pass algorithms that compute the numbers of objects cavities and holes in three dimensional 3 d digital pictures also it is shown that a one way 3 d parallel sequential acceptor cannot accept the class of 3 d binary arrays in which the set of 1 s is connected unlike the situation in the two dimensional 2 d case but it can do so if multipasses are allowed

fast divergence conforming reduced basis methods for steady navier stokes flow abstract reduced basis methods rb methods or rbms form one of the most promising techniques to deliver numerical solutions of parametrized pdes in real time with reasonable accuracy for incompressible flow problems rbms based on lbb stable velocity pressure spaces do not generally inherit the stability of the underlying high fidelity model and instead additional stabilization techniques must be introduced one way of bypassing the loss of lbb stability in the rbm is to inflate the velocity space with supremizer modes this however deteriorates the performance of the rbm in the performance critical online stage as additional dofs must be introduced to retain stability while these dofs do not effectively contribute to accuracy of the rb approximation in this work we consider a velocity only rb approximation exploiting a solenoidal velocity basis the solenoidal reduced basis emerges directly from the high fidelity velocity solutions in the offline stage by means of piola transforms the solenoidality of the velocity space is retained under geometric transformations making the proposed rb method suitable also for the investigation of geometric parameters to ensure exact solenoidality of the high fidelity velocity solutions that constitute the rb we consider approximations based on divergence conforming compatible b splines we show that the velocity only rb method leads to a significant improvement in computational efficiency in the online stage and that the pressure solution can be recovered a posteriori at negligible extra cost we illustrate the solenoidal rb approach by modeling steady two dimensional navier stokes flow around a naca0015 airfoil at various angles of attack

surrogate modeling of multiscale models using kernel methods this work investigates the possibilities of acceleration and approximation of multiscale systems using kernel methods the key element is to learn the interface between the different scales using a fast surrogate for the microscale model which is given by multivariate kernel expansions the expansions are computed using statistically representative samples of in and output of the microscale model as learning methods we apply both support vector machines and a vectorial kernel greedy algorithm we demonstrate the applicability of the resulting surrogate models using two multiscale models from different engineering disciplines first we consider a human spine model coupling a macroscale multibody system with a microscale intervertebral spine disc model and second a model for simulation of saturation overshoots in porous media involving nonclassical shock waves

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

tracing post limit point paths with reduced basis technique abstract a reduced basis technique and a problem adaptive computational algorithm are presented for predicting the post limit point paths of structures in the proposed approach the structure is discretized by using displacement finite element models the nodal displacement vector is expressed as a linear combination of a small number of vectors and a rayleigh ritz technique is used to approximate the finite element equations by a small system of nonlinear algebraic equations to circumvent the difficulties associated with the singularity of the stiffness matrix at limit points a constraint equation defining a generalized arc length in the solution space is added to the system of nonlinear algebraic equations and the rayleigh ritz approximation functions or basis vectors are chosen to consist of a nonlinear solution of the discretized structure and its various order derivatives with respect to the generalized arc length the potential of the proposed approach and its advantages over the reduced basis load control technique are outlined the effectiveness of the proposed approach is demonstrated by means of numerical examples of structural problems with snap through and snap back phenomena

model order reduction for dynamic simulation of a flexible multibody system via absolute nodal coordinate formulation abstract the absolute nodal coordinate formulation ancf can be used to model a flexible multibody system subject to both overall motions and large deformations with an increasing number of finite elements of ancf to mesh flexible bodies however the computation cost will become extremely high to improve the computational efficiency of ancf for a large scaled flexible multibody system a systematic method is proposed in this study for the model order reduction based on the proper orthogonal decomposition and the galerkin projection at first an approach for the selection of reduced constraint equations is proposed to deal with the singularity of the coefficient matrix of the reduced order model rom then the computation of the reduced stiffness matrix and generalized force vector of the rom are parallelized via the openmp directives afterwards two parametric approaches are presented to make the rom be adaptive to the change of system parameters one is the interpolation approach of reduced order basis vectors on a manifold to obtain the parametric rom and the other is the fast to slow rom approach via the reduced order basis vectors extracted from the fast dynamic responses to simulate slow dynamic responses finally four numerical examples are given to validate the efficacy of the proposed method for the dynamic simulations of both rigid and flexible multibody systems

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

dynamic analysis using a reduced basis of exact modes and ritz vectors the problem of obtaining an approximate solution for the response of a structure under dynamic forces is addressed the structure is modeled using mass and stiffness matrices and has n degrees of freedom an dimensional approximation is sought in a subspace defined by an arbitrary m orthogonal array a residual vector which is smaller in magnitude the more accurate the approximation is derived the residual contains two terms one is the error involved representing the force vector in the subspace the other is smaller the better the array leads to m approximate solutions of the exact eigenproblem numerical experiments are performed using arrays that are combinations of exact eigenvectors and of ritz vectors in one case when 11 different 10 dimensional bases are used to approximate the solution of a 60 degree of freedom problem it is found that a root mean square measure of relative error is smallest for the basis formed using the 6 lowest exact eigenvectors plus 4 ritz vectors generated from the spatial distribution of the loading

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

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

on connectivity properties of grayscale pictures abstract a grayscale digital picture is called connected if it has only one connected component of constant gray level that is maximal i e not adjacent to any component of higher gray level this note establishes some equivalent conditions for connectedness and also defines a grayscale generalization of the genus in terms of sums of local property values

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

reduced basis technique for collapse analysis of shells a hybrid finite element rayleigh ritz technique is used to predict the collapse behavior of shells in this hybrid technique the modeling versatility of the finite element method is preserved and a significant reduction in the number of degrees of freedom is achieved by expressing the nodal displacement vector as a linear combination of a small number of basis vectors a rayleigh ritz technique is used to approximate the finite element equations of the discretized shell by a reduced system of nonlinear algebraic equations a scalar function is introduced to measure the degree of nonlinearity of the structure for the case of loading applied by means of axial end shortening also a quantitative measure for the error of the reduced system of equations is proposed some insight is given as to why and when the reduced basis technique works and the effectiveness of the technique for predicting the collapse behavior of shells is demonstrated by means of a numerical example of elastic collapse of an axially compressed pear shaped cylinder

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

structure preserving model reduction for mechanical systems this paper focuses on methods of constructing of reduced order models of mechanical systems which preserve the lagrangian structure of the original system these methods may be used in combination with standard spatial decomposition methods such as the karhunen loeve expansion balancing and wavelet decompositions the model reduction procedure is implemented for three dimensional finite element models of elasticity and we show that using the standard newmark implicit integrator significant savings are obtained in the computational costs of simulation in particular simulation of the reduced model scales linearly in the number of degrees of freedom and parallelizes well

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

a reduction method for nonlinear structural dynamic analysis abstract a computational algorithm for predicting the nonlinear dynamic response of a structure is presented the nonlinear system of ordinary differential equations resulting from the finite element discretization is highly reduced by means of a rayleigh ritz analysis the basis vectors are chosen to be the current tangent eigenmodes together with some modal derivatives that indicate the way in which the spectrum is changing only a few basis updatings are required during the whole time integration the truncation error introduced at every change of basis is pointed out as the cause for a divergence type behaviour and some means for eliminating it are discussed results for examples involving large displacements are shown and compared to the results obtained by integrating the complete system of equations

characteristics of subspace in reduced basis technique in the reduced basis technique the coefficients of global shape functions are taken as generalized degrees of freedom the non linear system is solved iteratively in a reduced subspace using considerably less unknowns this paper investigates the characteristics of the reduced system the relationship between the reduced system and the main system the criterion to judge the critical point with reduced quantities and the criterion of convergence in the reduced subspace

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

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

an object oriented framework for reduced order models using proper orthogonal decomposition pod this work presents an object oriented design for reduced order modeling using the proper orthogonal decomposition pod technique object oriented design attributes such as data encapsulation inheritance and polymorphism are shown to produce efficient implementations that optimize code reuse and maintainability in reduced order models this paper demonstrates how existing finite element libraries can be used to efficiently carry out integration of field quantities and their gradients over complex domains for producing reduced order models that are based on the weak form galerkin method and pod furthermore numerical examples are used to show how reduced order models are computationally feasible for reducing cpu time as long as the computational expense incurred in solving the associated systems of linear equations is significantly higher than the computational expense related to the assembly of the system of 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

nonlinear dynamical response of impulsively loaded structures a reduced basis approach the order of a dynamical system is reduced by a rayleigh ritz technique using selective basis vectors numerical results are obtained using ritz vectors and their derivatives as basis vectors both use of such derivatives and use of updating basis vectors provide excellent results

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

the perimeter of a fuzzy set abstract in pattern recognition one often wants to measure the perimeters of regions in images this is straightforward if the region is crisply defined but if it is fuzzy it is not obvious how its perimeter can be measured this paper proposes a definition of perimeter for fuzzy subsets of the plane and shows that it reduces to the standard definition if the fuzzy subset is an ordinary subset the isoperimetric inequality does not generalize to fuzzy subsets but certain properties of the perimeters of convex sets do generalize to fuzzy perimeters of convex fuzzy subsets

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

improving the efficiency of large scale topology optimization through on the fly reduced order model construction summarytopology optimization of large scale structures is computationally expensive notably because of the cost of solving the equilibrium equations at each iteration reduced order models by projection also known as reduced basis models have been proposed in the past for alleviating this cost we propose here a new method for coupling reduced basis models with topology optimization to improve the efficiency of topology optimization of large scale structures the novel approach is based on constructing the reduced basis on the fly using previously calculated solutions of the equilibrium equations the reduced basis is thus adaptively constructed and enriched based on the convergence behavior of the topology optimization a direct approach and an approach with adjusted sensitivities are described and their algorithms provided the approaches are tested and compared on various 2d and 3d minimum compliance topology optimization benchmark problems computational cost savings by up to a factor of 12 are demonstrated using the proposed methods copyright textcopyright 2014 john wiley sons ltd