separate joint optimization of error feedback and coordinate transformation for roundoff noise minimization in two dimensional state space digital filters this paper is concerned with the minimization of roundoff noise subject to l sub 2 norm dynamic range scaling constraints in two dimensional 2 d state space digital filters two methods are proposed with the first one using error feedback alone and the second one using joint error feedback and coordinate transformation optimization in the first method several techniques for the determination of optimal full scale block diagonal diagonal and scalar error feedback matrices for a given 2 d state space digital filter are proposed in the second method an iterative approach for minimizing the roundoff noise under l sub 2 norm dynamic range scaling constraints is developed by jointly optimizing a scalar error feedback matrix and a coordinate transformation matrix which may be regarded as an alternative approach to the conventional method for synthesizing the optimal 2 d filter structure with minimum roundoff noise an analytical method for the joint optimization of a general error feedback matrix and a coordinate transformation matrix under the scaling constraints is also proposed a numerical example is presented to illustrate the utility of the proposed techniques

jointly optimized error feedback and realization for roundoff noise minimization in state space digital filters roundoff noise rn is known to exist in digital filters and systems under finite precision operations and can become a critical factor for severe performance degradation in infinite impulse response iir filters and systems in the literature two classes of methods are available for rn reduction or minimization one uses state space coordinate transformation the other uses error feedback feed forward of state variables in this paper we propose a method for the joint optimization of error feedback feed forward and state space realization it is shown that the problem at hand can be solved in an unconstrained optimization setting with a closed form formula for gradient evaluation and an efficient quasi newton solver the unconstrained minimization problem can be solved efficiently with the infinite precision solution as a reference point we then move on to derive a semidefinite programming sdp relaxation method for an approximate solution of optimal error feedback matrix with sum of power of two entries under a given state space realization simulations are presented to illustrate the proposed algorithms and demonstrate the performance of optimized systems

identification of continuous time models the paper considers the problem of estimation of the transfer function of a continuous time dynamic system in the presence of colored noise the author introduces an operator transformation that allows for keeping a continuous time parametrization whereas the parameter estimation can be made by means of a discrete time maximum likelihood algorithm a comparison is made between the performance of the new method in comparison with a standard identification of an armax model the method is useful in cases where it is important to estimate the coefficients of a continuous time transfer function and where it is important to maintain a physical interpretation of the transfer function results

balanced forms canonicity and parameterization we show that balanced forms are canonical and continuously parameterize the set of systems of a given order which are asymptotically stable minimal and have distinct singular values the advantage of the balanced canonical form over other classical canonical forms lies precisely in the guaranteed stability and minimality of the parameterization we also derive formulas allowing the synthesis of systems using gradient methods in balanced form

frequency weighted model reduction via quasi kalman decomposition a frequency weighted model reduction method based on quasi kalman decomposition is proposed for discrete systems the method is essentially an extension of the technique presented by lu and lee 1985 to the frequency weighted case and can handle both single sided as well as double sided weights two illustrative examples show the effectiveness of the method and results are compared with the method developed by enns 1984

l sub 2 optimal filter reduction a closed loop approach this paper is concerned with the problem of order reduction of a full order kalman filter for a stable linear signal model so that the steady state filtering error variance associated with the reduced order filter is minimized by an orthogonal parameterization the above problem is formulated to minimize the filtering error variance over a set of orthogonal matrices both continuous and iterative algorithms are derived to compute an optimal reduced order filter the algorithms are shown to possess good properties including the desirable convergence property the proposed algorithms are simple and effective numerical examples are presented to demonstrate the effectiveness and the significant advantages of the proposed algorithms over the existing open loop methods such as the well known balanced truncation method

two dimensional geodesic flows having first integrals of higher degree we present a family of riemannian metrics on two sphere having the property that the geodesic flows admit first integrals which are fiberwise homogeneous polynomials of degree greater than 2 they also have the property that all geodesics are closed

parameter and state model reduction for large scale statistical inverse problems a greedy algorithm for the construction of a reduced model with reduction in both parameter and state is developed for an efficient solution of statistical inverse problems governed by partial differential equations with distributed parameters large scale models are too costly to evaluate repeatedly as is required in the statistical setting furthermore these models often have high dimensional parametric input spaces which compounds the difficulty of effectively exploring the uncertainty space we simultaneously address both challenges by constructing a projection based reduced model that accepts low dimensional parameter inputs and whose model evaluations are inexpensive the associated parameter and state bases are obtained through a greedy procedure that targets the governing equations model outputs and prior information the methodology and results are presented for groundwater inverse problems in one and two dimensions

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

probability one homotopy algorithms for solving the coupled lyapunov equations arising in reduced order h2 h modeling estimation and control optimal reduced order modeling estimation and control with respect to combined h 2 h criteria give rise to coupled lyapunov and riccati equations to develop reliable numerical algorithms for these problems this paper focuses on the coupled lyapunov equations which appear as a subset of the synthesis equations in particular this paper systematically examines the requirements of probability one homotopy algorithms to guarantee global convergence homotopy algorithms for nonlinear systems of equations construct a continuous family of systems and solve the given system by tracking the continuous curve of solutions to the family the main emphasis is on guaranteeing transversality for several homotopy maps based upon the pseudogramian formulation of the coupled lyapunov equations and variations based upon canonical forms these results are essential to the probability one homotopy approach by guaranteeing good numerical properties in the computational implementation of the homotopy algorithms

comparative study of finite wordlength effects in shift and delta operator parameterizations the sensitivity of the transfer function to finite wordlength effect errors in the implementation of the coefficients of both shift operator and delta operator parameterizations is analyzed the relationships between optimal realization sets in shift and delta operator are established it is shown that the optimal realization in delta operator parameterizations have better performance than those in their shift operator counterparts when the poles of the systems are clustered around z 1 a numerical example is given

subspace algorithms for the identification of multivarible dynamic errors in variables models abstract we consider the problem of identifying multivariable finite dimensional linear time invariant systems from noisy input output measurements apart from the fact that both the measured input and output are corrupted by additive white noise the output may also be contaminated by a term which is caused by a white input process noise furthermore all these noise processes are allowed to be correlated with each other we shall develop a solution to this problem in the framework of subspace identification and we shall show that our algorithms give consistent estimates when the system is operating in open or closed loop two realistic simulation studies are presented to demonstrate the practical applicability of the proposed algorithms 1997 elsevier science ltd

jointly optimal high order error feedback and realization for roundoff noise minimization in 1 d and 2 d state space digital filters joint optimization of high order error feedback and state space realization for minimizing roundoff noise at filter output subject to l2 scaling constraints is investigated for one and two dimensional state space digital filters linear algebraic techniques that convert the problems at hand into an unconstrained optimization problem are explored and an efficient quasi newton algorithm is then applied to solve the unconstrained optimization problem iteratively in this connection closed form formulas are derived for fast and accurate gradient evaluation finally case studies are presented to demonstrate that the high order error feedback does offer much improved performance and that the proposed joint optimization is superior relative to a sequentially optimized system where the state space coordinate transformation and high order error feedback matrices are optimized separately

on a class of integrable systems with a cubic first integral a few years ago selivanova gave an existence proof for some integrable models in fact geodesic flows on two dimensional manifolds with a cubic first integral however the explicit form of these models hinged on the solution of a nonlinear third order ordinary differential equation which could not be obtained we show that an appropriate choice of coordinates allows for integration and gives the explicit local form for the full family of integrable systems the relevant metrics are described by a finite number of parameters and lead to a large class of models mainly on the manifolds mathbb s 2 and mathbb h 2 many of these systems are globally defined and contain as special cases integrable systems due to goryachev chaplygin dullin matveev and tsiganov

fixed points of uniformly lipschitzian mappings in spaces with uniformly normal structure dans un espace de banach x on etudie quand reflexivite et structure normale sont suffisantes pour assurer pour de bon 1 la propriete de point fixe pour des auto applications uniformement lipschitziennes

small roundoff noise realization of fixed point digital filters and controllers a method for finding digital realizations of discrete time transfer functions by fixed point arithmetic is presented the obtained realizations attain very small roundoff noise intensity this intensity is significantly smaller than the intensities that have been achieved so far the method applies many 0 1 coefficients in the realization it therefore requires a small number of multiplications and additions and it is significantly faster than many other realization methods a detailed comparison is made between this method and other existing methods for digital realization two different cases are investigated a butterworth filtering realization and the realization of an lqg linear quadratic gaussian optimal controller

comments on d stability of continuous time delay systems subjected to a class of highly structured perturbations in this paper we wish to point out that the proof of the main results in the above mentioned paper by lee ibid vol 40 1995 is incorrect furthermore we verify that there exists no continuous time delay system guaranteeing the d spl alpha r stability

a new approach to the design of cost optimal low noise digital filters an extension of the conventional state space description is proposed in this paper for the representation of digital filters in the proposed model the future values of the internal variables depend not only on the present values but also on that of k 1 previous time instants where k is the order of the model expressions for the roundoff noise and the dynamic range constraint have been obtained effects of structure transformation can be easily incorporated into these expressions the optimum digital filter structure developed using the proposed model in general is computationally more efficient than that developed using the conventional state space description this paper proposes three different noise cost performance measures which are indicative of the overall cost of the digital filter under a normed round off noise an illustrative example is included in the paper showing the improvement in the noise cost measure by using the new model under each of the three different measures simulation of this example using the dinap ii has confirmed the theoretical results

completely integrable 2d lagrangian systems and related integrable geodesic flows on various manifolds in this study we have formulated a theorem that generates deformations of the natural integrable conservative systems in the plane into integrable systems on riemannian and other manifolds by introducing additional parameters into their structures the relation of explicit solutions of the new and the original dynamics to the corresponding jacobi maupertuis geodesic flow is clarified for illustration we apply the result to three concrete examples of the many available integrable systems in the literature complementary integrals in those systems are polynomial in velocity with degrees 3 4 and 6 respectively as a special case of the first deformed system a new several parameter family of integrable mechanical systems and geodesic flows on s2 is constructed

k uniform rotundity is equivalent to k uniform convexity abstract r geremia and f sullivan proved that 2 uniform rotundity is equivalent to 2 uniform convexity this paper extends this result to k uniformly rotund spaces

j lossless and extended j lossless factorizations approach for domain h control the paper addresses problems of numerical conditioning of the discrete time h x control design based on j lossless factorizations of the chain scattering representation of a plant it is demonstrated that for a sufficiently small sampling period forward shift operator techniques may become ill conditioned and numerical robustness and reliability of design procedures can be significantly improved by utilizing a so called i operator form of the source problem state space models of all i domain controllers are given the solutions are obtained by considering two coupled i domain algebraic riccati equations that can be solved by applying a generalized eigenproblem formulation numerical conditioning of these solutions can effectively be evaluated by using a suitable relative condition number an extended j lossless factorization for generalized plants with zeros on the stability boundary circle is also given a so called zero compensator technique adapted to the i domain models is utilized to cancel suc

roundoff noise analysis of two efficient digital filter structures in this correspondence two efficient structures are analyzed for the implementation scheme of rounding before multiplication rbm the first one is the direct form ii transposed structure in spl rho operator spl rho dfiit recently proposed in the expression of roundoff noise gain is derived based on which a revised spl rho dfiit structure denoted as spl rho rdfiit is obtained it is shown that the spl rho rdfiit structure having the same implementation complexity yields a smaller roundoff noise gain than the spl rho dfiit the roundoff noise gain for the spl rho rdfiit structure with error feedbacks is also derived each structure can be optimized with regard to the free parameters it possesses with a genetic algorithm numerical examples are presented which show that the optimized structures are very competitive because they can even outperform the traditional optimal state space realization in terms of the roundoff noise performance as well as the implementation simplicity

the fixed point property in banach spaces with the nus property abstract in this paper we show that the weak nearly uniform smooth banach spaces have the fixed point property for nonexpansive mappings

roundoff noise minimization using delta operator realizations digital filters the authors examine the advantages of using delta operator state space realizations rather than shift operator realizations of transfer functions in terms of minimizing roundoff noise gain they give several conditions under which the optimal roundoff noise gain for delta operator realizations is smaller than the optimal gain for shift operator realizations they then illustrate that even sparse and hence nonoptimal delta operator realizations can have smaller roundoff noise gain than the optimal shift operator realizations

numerically robust delta domain solutions to discrete time lyapunov equations a problem of numerical conditioning of a special kind of discrete time lyapunov equations is considered it is assumed that a discretisation procedure equipped with the zero order holder mechanism is utilised that leads to the data matrices that are affinely related to the sampling period and matrices that are independent or linearly related to the squared sampling period it is shown that common forward shift operator techniques for solving these equations become ill conditioned for a sufficiently small sampling period and that numerical robustness and reliability of computations can be significantly improved via utilising the so called delta operator form of the origin equations

new examples of integrable conservative systems on s 2 and the case of goryachev chaplygin in this paper we propose new examples of conservative systems on s 2 possessing cubic integrals in momenta we obtain a classification of a large class of such systems we show that this class contains new families of integrable systems as well as the classical case of goryachev chaplygin

on a family of integrable systems on s2 with a cubic integral of motion we discuss a family of integrable systems on the sphere s2 with an additional third order integral in momenta this family contains the coryachev chaplygin top the goryachev system the system recently discovered by dullin and matveev and two new integrable systems on the non physical sphere with zero radius all these systems are isomorphic to each other

brief paper on the stability of a reduced order filter based on dominant singular value decomposition of the system dynamics the stability of a reduced order filter rof the gain of which is constructed on the basis of a subspace of dominant singular vectors of the system dynamics is examined a definition of s detectability is introduced it is found that the observability of all unstable and neutral singular vectors s detectability is a sufficient condition for the existence of a stable filter

a matrix factorization based structure for digital filters in this correspondence a novel digital filter structure is derived using matrix factorization this structure called the lcw structure is an improved version of the so called lgs structure proposed by li gevers and sun ldquoperformance analysis of a new structure for digital filter implementation rdquo ieee transactions on circuits and systems i vol 47 no 4 pp 474 482 april 2000 for a digital filter of order this new structure requires 4n 1 multiplications and 4n 1 additions for computing one filter output sample instead of 7n 3 and 6n 3 respectively which are needed for the lgs structure an expression of the roundoff noise gain is derived for the lcw structure design examples are presented to demonstrate the behavior of this structure and to compare with several well known digital filter structures in terms of minimizing roundoff noise and implementation efficiency

roundoff noise minimization and pole zero sensitivity in fixed point digital filters using residue feedback the problem of designing a finite wordlength fixed point realization of an nth order digital filter which uses residue feedback to minimize the output roundoff noise subject to l 2 scaling is considered the new structures require n extra additions but no more multiplications than the earlier low noise structures of mullis and roberts and have lower roundoff noise for sufficiently narrow bandwidth filters a new set of filter invariants called the residue modes are defined which characterize the new low noise structures and determine the output noise variance if the sum of the residue modes is less than the sum of the second order modes of mullis and roberts then lower roundoff noise is achieved every filter structure is shown to define a unique symmetrizing matrix q pole zero sensitivities and a new noise measure are defined in terms of q numerical results are included which compare the noise and pole zero sensitivity characteristics of different filter structures

optimal finite wordlength linear quadratic regulation the authors consider the design of an lqg regulator implemented in fixed point arithmetic which is optimized with respect to its state wordlength and structure the approach extends previous design methods so that arithmetic errors are directly included the optimal finite controller gains are computed as a function of the arithmetic wordlength subject to an l sub 2 scaling constant on the controller states the design method used permits the cost for the near optimal structure to be developed so that this can then be used as a basis for comparison to other less complex structures

another balanced controller reduction algorithm the problem of controller reduction has been addressed by a number of authors 1 4 using the concept of balanced coordinates originally proposed by moore 5 this note establishes a more direct link between the riccati balancing methods in 2 4 and moore s work and proposes yet another candidate method as a logical extension of 2 4

a cyclic low rank smith method for large sparse lyapunov equations in this paper we present the cyclic low rank smith method which is an iterative method for the computation of low rank approximations to the solution of large sparse stable lyapunov equations it is based on a generalization of the classical smith method and profits by the usual low rank property of the right hand side matrix the requirements of the method are moderate with respect to both computational cost and memory furthermore we propose a heuristic for determining a set of suboptimal alternating direction implicit adi shift parameters this heuristic which is based on a pair of arnoldi processes does not require any a priori knowledge on the spectrum of the coefficient matrix of the lyapunov equation numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the adi parameters

on the structure of digital controllers with finite word length consideration it is well known that a stable discrete time control system may become unstable when the digital controller is actually implemented with a digital control processor due to the finite word length effects and that the stability behavior of the system depends on the structure of the digital controller due to the difficulties in adopting the existing stability robustness measures a new measure is derived which is tractable and hence enables us to handle the digital controller structure problem easily the optimal controller structure problem is formulated by maximizing this measure an analytical solution is presented an algorithm is developed for finding sparser structures

minimization of roundoff errors in digital realizations of kalman filters a comparison between d williamson s method ieee trans autom contr vol ac 30 p 930 9 1985 and the 0 1 method for the design of the digital kalman filter is made the 0 1 method is shown to be much better than williamson s method and the other methods from the estimation error pint of view whereas williamson s method is slightly better than the 0 1 method from the computation time point of view comparison with all the other transfer function based realizations shows that the designer of the digital kalman filter has to choose only between the 0 1 and williamson methods

on balanced realizations of bounded real and positive real functions based on normalized coprime factorizations with respect to indefinite metrics and the construction of suitable characteristic functions the ober balanced canonical forms for the classes of bounded real and positive real functions are derived this uses a matrix representation of the shift realization with respect to a basis related to sets of orthogonal polynomials

brief paper balanced truncation model reduction for systems with inhomogeneous initial conditions we present a rigorous approach to extend balanced truncation model reduction btmr to systems with inhomogeneous initial conditions we provide an estimate for the error between the input output maps of the original and of the reduced initial value system and we illustrate numerically the superiority of our approach over the naive application of btmr when btmr is applied to linear time invariant systems with inhomogeneous initial conditions it is crucial that the initial data are well represented by the subspaces generated by btmr this requirement is often ignored or it is avoided by making the restrictive assumption that the initial data are zero to ensure that the initial data are well represented by the btmr subspaces we add auxiliary inputs determined by the initial data

delay replacement in direct form structures a class of high order low sensitivity digital filter structures is generated by means of a direct transformation of a high order direct form df structure the proposed structures have low coefficient dynamic range and exhibit low coefficient frequency response sensitivity and roundoff noise characteristics yet require only 3n multiplications per output sample as compared to 2n in the direct form case

roundoff noise minimization for 2 d state space digital filters using joint optimization of error feedback and realization the joint optimization problem of error feedback and realization for two dimensional 2 d state space digital filters to minimize the effects of roundoff noise at the filter output subject to l2 norm dynamic range scaling constraints is investigated it is shown that the problem can be converted into an unconstrained optimization problem by using linear algebraic techniques the unconstrained optimization problem at hand is then solved iteratively by applying an efficient quasi newton algorithm with closed form formulas for key gradient evaluation analytical details are given as to how the proposed technique can be applied to the cases where the error feedback matrix is a general block diagonal diagonal or block scalar matrix a case study is presented to illustrate the utility of the proposed technique