on the approximation power of convolution based least squares versus interpolation there are many signal processing tasks for which convolution based continuous signal representations such as splines and wavelets provide an interesting and practical alternative to the more traditional sine based methods the coefficients of the corresponding signal approximations are typically obtained by direct sampling interpolation or quasi interpolation or by using least squares techniques that apply a prefilter prior to sampling we compare the performance of these approaches and provide quantitative error estimates that can be used for the appropriate selection of the sampling step h specifically we review several results in approximation theory with a special emphasis on the strang fix 1971 conditions which relate the general o h sup l behavior of the error to the ability of the representation to reproduce polynomials of degree n l 1 we use this theory to derive pointwise error estimates for the various algorithms and to obtain the asymptotic limit of the l sub 2 error as h tends to zero we also propose a new improved l sub 2 error bound for the least squares case in the process we provide all the relevant bound constants for polynomial splines some of our results suggest the existence of an intermediate range of sampling steps where the least squares method is roughly equivalent to an interpolator with twice the order we present experimental examples that illustrate the theory and confirm the adequacy of our various bound and limit determinations

observability of structured linear systems in descriptor form a graph theoretic approach this paper deals with the right hand side observability of structured linear systems in descriptor form the method we propose is based on a graph theoretic approach it is applicable to a large class of descriptor systems including regular and nonregular systems even if they are square underdetermined or overdetermined after defining digraphs which represent such systems we express in graphic terms necessary and sufficient conditions to achieve the observability of the whole or of a given part of the state these conditions are easy to check because they are based on comparison of integers and on finding edges and subgraphs in a digraph therefore our approach is suited to study large scale and or uncertain systems

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

generalized proper inverse of polynomial matrices and the existence of infinite decoupling zeros a necessary and sufficient condition is presented for the existence of a generalized proper rational inverse for nonsquare polynomial matrices the condition is then proved to be equivalent to the absence of infinite decoupling zeros this condition provides a simple test for the absence of infinite decoupling zeros in the polynomial fraction form it can also be used to remove the infinite decoupling zeros in order to achieve a strongly irreducible system and to provide a new look of the unimodular matrix operation effect at infinity for the left and right polynomial fraction forms

pole placement for singular systems abstract in this paper we present a pole placement design for singular systems in polynomial fraction form the design can also be used for the role placement of singular systems in generalized state space form and requires no decomposition

the stochastic obstacle problem for the harmonic oscillator with damping abstract we study the kolmogorov equation associated with a second order stochastic variational inequality related to the harmonic oscillator

boundary value descriptor systems well posedness reachability and observability in this paper we introduce the class of two point boundary value descriptor systems tpbvds discrete time systems described by possibly linear dynamics and a set of boundary conditions constraining the values of the system state at the two endpoints of the system s interval of definition by introducing a standard form for regular pencils we obtain a new and simple generalized cayley hamilton theorem that simplifies our investigation of well posedness green s function solutions and reachability and observability for tpbvds there are two distinct notions of reachability and observability that one can define for tpbvds associated with processes that propagate inward from and outward toward the boundaries we investigate each of these in detail obtaining among other things far simpler forms for the reachability and observability results than found previously in the literature in addition we describe several methods for the efficient solution of tpbvds one involving recursions from each end of t

on conditions guaranteeing two polynomial matrices possess identical zero structures some sufficient conditions for two polynomial matrices to simultaneously possess identical finite and infinite zero structures are developed a special case of this problem that of obtaining conditions under which a polynomial matrix has no infinite zeros is also considered and certain connections with previous work are established

solvability controllability and observability of continuous descriptor systems in this paper we investigate the properties of the continuous descriptor system e dot x t ax t bu t 0 leqt leqb where e a and b are complex and possibly singular matrices and u t is a complex function differentiable sufficiently many times the traditional approach to such systems is to separate the state equations from the algebraic equations however such algorithms usually destroy the natural physically based sparsity and structure of the original system therefore we consider descriptor systems in their original form such systems possess numerous properties not shared by the well known state variable systems first we relate classical theories of matrix pencils to the solvability of descriptor systems then we extend the concepts of reachability controllability and observability of state variable systems to descriptor systems and describe the set of reachable states for descriptor systems

a nonlinear luenberger like observer for nonlinear singular systems this paper investigates observer design problem for a large class of nonlinear singular systems with multiple outputs we firstly regularize the singular system by injecting the derivative of outputs into the system then differential geometric method is applied to transform the regularized system into a simple normal form for which a luenberger like observer is proposed

analysis of descriptor systems using numerical algorithms in this paper we analyze numerical methods for the solution of the large scale dynamical system e dot y t ay t g t y t 0 y 0 where e and a are matrices possibly singular systems of this type have been referred to as implicit systems and more recently as descriptor systems since they arise from formulating system equations in physical variables special cases of such systems are algebraic differential systems we discuss the numerical advantages of this formulation and identify a class of numerical integration algorithms which have accuracy and stability properties appropriate to descriptor systems and which preserve structure detect nonsolvable systems resolve initial value consistency problems and are applicable to stiff descriptor systems we also present an algorithm for the control of the local truncation error on only the state variables

white noise driven quasilinear spdes with reflection we study reflected solutions of the heat equation on the spatial interval 0 1 with dirichlet boundary conditions driven by an additive space time white noise roughly speaking at any point x t where the solutionu x t is strictly positive it obeys the equation and at a point x t whereu x t is zero we add a force in order to prevent it from becoming negative this can be viewed as an extension both of one dimensional sdes reflected at 0 and of deterministic variational inequalities an existence and uniqueness result is proved which relies heavily on new results for a deterministic variational inequality

2d vortex methods and singular quadrature rules a new high order vortex method for the 2d euler equations is presented the method eliminates smoothing by constructing a singular quadrature rule for the biot savart law at each time step using quadtrees and orthogonal polynomials theory and numerical experiments show that the method is accurate and efficient yielding excellent long term accuracy in almost optimal cpu time

controllability and observability of second order descriptor systems we analyze controllability and observability conditions for second order descriptor systems and show how the classical conditions for first order systems can be generalized to this case we show that performing a classical transformation to first order form may destroy some controllability and observability properties as an example we demonstrate that the loss of impulse controllability in constrained multibody systems is due to the representation as a first order system to avoid this problem we will derive a canonical form and new first order formulations that do not destroy the controllability and observability properties

two polynomial matrix operations this note presents an algorithm to carry out the division of two polynomial matrices and an algorithm to transform a polynomial matrix into a row reduced one these algorithms are useful in the polynomial matrix fraction description of multivariable systems

equivalence transformations of rational matrices and applications the known theories of transformations between polynomial matrices are extended to the case of rational matrices specifically equivalence between rational matrices having possibly different dimensions is defined and this has the property of preserving the zero structure of rational matrices in the region c some implications of this new equivalence transformation for linear system theory are also provided

technical communique regularizability of linear time invariant descriptor systems under decentralized control regularizability conditions for linear time invariant descriptor systems under decentralized output feedback are presented in this paper the regularizability is characterized by rank conditions that involve the original system matrices only new rank conditions for regularity of the system are also derived

martingale solutions of a stochastic wave equation with reflection abstract we discuss an initial boundary value problem for a one dimensional stochastic wave equation with reflection for stochastic parabolic equations with reflection there are some well known results however there seems to be no existence result for a stochastic wave equation with reflection even for a deterministic wave equation the problem has not been completely resolved our goal is to establish the existence of a martingale solution for this problem

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

a tutorial on the geometric analysis of linear time invariant implicit systems abstract a background of recent work in nonregular and regular implicit systems of the form e x ax bu is provided then the output nulling unknown input and composite subspaces are defined for singular systems with recursions given for their computation inner subspaces that is those in the domain of e and a are distinguished from outer subspaces those in the codomain of e and a the notions of composite preimage and composite image are introduced the result is a framework that unifies some of the recent work in the geometric theory of implicit systems system properties are discussed for both regular and nonregular systems and duality is investigated

a forward trajectory global semi lagrangian transport scheme a forward trajectory semi lagrangian scheme for advection on the surface of the sphere is proposed the advection scheme utilizes the forward downstream trajectory originating at eulerian grid points and cascade interpolation a sequence of 1d interpolations to transfer data from the downstream lagrangian points to the eulerian points a new and more accurate algorithm determines pole values the resulting forward trajectory semi lagrangian scheme can easily incorporate high order trajectory integration methods this avoids the standard iterative process in a typical backward trajectory scheme two third order accurate schemes and a second order accurate scheme are presented a mass conservative version of the forward trajectory semi lagrangian scheme is also derived within the cascade interpolation framework mass from a lagrangian cell is transferred to the corresponding eulerian cell with two 1d remappings through an intermediate cell system mass in the polar region is redistributed by way of an efficient local approximation the resulting scheme is globally conservative but restricted to meridional courant number c 1

structure and smith macmillan form of a rational matrix at infinity the use of special row and column operations in the reduction of a rational matrix to its smith macmillan form at infinity is investigated the connections between this reduction procedure and the valuation approach are established a graphical method for finding the smith macmillan form of a rational matrix at infinity from its bode magnitude array and some new results on realization theory for polynomial matrices are presented

convergence of a variable blob vortex method for the euler and navier stokes equations convergence is proved of the variable blob vortex method proposed t hou d papageorgiou g papanicolaou and j zhu an adaptive vortex method for two dimensional viscous and incompressible flows preprint for the incompressible euler and navier stokes equations without boundaries this version of the variable blob method is designed to approximate directly the navier stokes equations without matching the interior flow to the boundary layer solution by choosing vortex blobs adaptively the numerical errors due to local shear are minimized

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

a characterization of multivariate quasi interpolation formulas and its applications let be a compactly supported function on s ands the linear space withgenerator that is s is the linear span of the multiinteger translates of it is well known that corresponding to a generator there are infinitely many quasi interpolation formulas a characterization of these formulas is presented which allows for their direct calculation in a variety of forms suitable to particular applications and in addition provides a clear formulation of the difficult problem of minimally supported quasi interpolants we introduce a generalization of interpolation called interpolation and a notion of higher order quasi interpolation called approximation a characterization of approximants similar to that of quasi interpolants is obtained with similar applications among these applications are estimating least squares approximants without matrix inversion surface fitting to incomplete or semi scattered discrete data and constructing generators with one point quasi interpolation formulas it will be seen that the exact values of the generator at the multi integers s facilitates the above study an algorithm to yield this information for box splines is discussed

vortex methods with spatially varying cores the accuracy of vortex methods employing smooth vortex particles blobs is determined by the blob size which can be viewed as a mollifier of the vorticity field for computational efficiency this core size needs to be spatially variable as particles are used to discretize different parts of the flow field such as the boundary layer and the wake in bluff body flows we derive here a consistent approximation for the viscous navier stokes equations using variable size vortex particles this derivation is based on the implementation of mappings that allow the consistent formulation of the diffusion and convection operators of the navier stokes equations in the context of vortex methods several local mappings can be combined giving the capability of mesh embedding to vortex methods it is shown that the proposed variable method offers a significant improvement on the computational efficiency of constant core size methods while maintaining the adaptive character of the method the method is ideally suited to flows such as wakes and shear layers and the validity of the approach is illustrated by showing results from cylinder flows and wall vortex interactions using this scheme previously unattainable simulations of cylinders undergoing rotary oscillations at high reynolds numbers reveal an interesting mechanism for drastic drag reduction

generalized functional observer a functional observer and state feedback are proposed for singular systems in the polynomial fraction form that requires no impulsive mode elimination the order of the compensator is determined by the newly defined generalized observability index that is associated with the mcmillan degree of the system a new generalized lyapunov equation is also proposed through a realization scheme that can be applied to both ordinary and singular systems the solution to the equation provides an algebraic approach to the observer of singular systems in the generalized state space form

backward differentiation approximations of nonlinear differential algebraic systems finite difference approximations of dynamical systems modeled by nonlinear semiexplicit differential algebraic equations are analyzed convergence for the backward differentiation method is proved for index two and index three problems when the numerical initial values obey certain constraints the appropriate asymptotic convergence rates and the leading error terms are determined

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

a geometric approach to eigenstructure assignment for singular systems moore s well known result on the feedback assignment of eigenstructure in linear state space systems 5 was generalized to descriptor systems in 9 in this note we make some useful extensions to fill out the theory to begin with we do not assume controllability of the plant it is shown that the maximum number of finite closed loop eigenvalues is equal to dim el where l is the supremal a e r b invariant subspace in order to relate the possible closed loop structure to the open loop plant structure we define a set of indexes for descriptor systems which are in many respects similar to the controllability indexes of state space systems in terms of these controllability indexes we give a systematic approach to selecting the closed loop eigenvectors a major contribution of this note is the fact that it is geometric in nature and hence avoids any decomposition of the system matrices into a special form e g weierstrass form 1 the geometric setting introduced here should constitute a basis for further research in generalizing the well known results in geometric system theory disturbance decoupling input output decoupling etc to singular systems

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

brief paper observability analysis and sensor location study for structured linear systems in descriptor form with unknown inputs this paper deals with the state and input right hand side observability and the sensor location problem to ensure this property for structured linear systems in the descriptor form the proposed method is based on a graph theoretic approach it is applicable to a large class of descriptor systems including regular and nonregular systems even if they are underdetermined overdetermined or square after defining a bipartite graph which represents such systems we deduce some graphic necessary and sufficient conditions to achieve the right hand side observability of the state and the input next we propose a sensor placement procedure to recover this property more precisely we provide the minimal number of the sensors required and either their pertinent location or a necessary and sufficient condition to check if a given sensor configuration is adequate or not to ensure state and input right hand side observability our approach uses algorithms with polynomial complexity orders and assumes only the knowledge of the system s structure and so is suited for large scale systems

uniform convergence of a linearly transformed particle method for the vlasov poisson system a particle method with linear transformation of the particle shape functions is studied for the 1d 1v vlasov poisson equation and a priori error estimates are proven which show that the approximated densities converge in the uniform norm when compared to standard fixed shape particle methods the present approach can be seen as a way to gain one order in the convergence rate of the particle trajectories at the cost of linearly transforming each particle shape it also allows to compute strongly convergent densities with particles that overlap in a bounded way

realizations in generalized state space form for polynomial system matrices and the definitions of poles zeros and decoupling zeros at infinity this paper contains three main results firstly a realization procedure is presented that brings a polynomial system matrix of a non proper multivariable system to generalized state space form such that all relevant properties including phenomena of redundancy associated with finite and infinite decoupling zeros are retained secondly new definitions are proposed for poles zeros and decoupling zeros at infinity of a general polynomial system matrix as a third result a theorem of rosenbrock 1974 e on the redundancy of lcr multiports is generalized to include the decoupling zeros at infinity