a global rational arnoldi method for model reduction abstract in this paper we propose two new approaches for model order reduction of large scale multi input multi output mimo linear time invariant dynamical systems lti these methods are based on a generalization of the global arnoldi algorithm which is used to generate projection subspaces an adaptive procedure for the selection of shift parameters is proposed and some simple arnoldi relations are established in order to compute error bounds on the transfer function numerical experiments showing the effectiveness of these approaches are provided

the estimation of the model order in exponential families estimators are sought that achieve high exponential rate of decrease in the underestimation probability while keeping the overestimation probability exponent at a certain prescribed level it is assumed that a given integer is known to upper bound the true order some examples are given of possible applications of the proposed method in specific order estimation and hypothesis testing problems

model reduction of nonstationary lpv systems this paper focuses on the model reduction of nonstationary linear parameter varying nslpv systems we provide a generalization of the balanced truncation procedure for the model reduction of stable nslpv systems along with a priori error bounds then for illustration purposes this method is applied to reduce the model of a two mass translational system furthermore we give an approach for the model reduction of stabilizable and detectable systems which requires the development and use of coprime factorizations for nslpv models for the general class of eventually periodic lpv systems which includes periodic and finite horizon systems as special cases our results can be explicitly computed using semidefinite programming

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

characterization of the solution to the optimal h model reduction problem abstract in this paper the h norm approximation of a given stable proper rational transfer function by a lower order stable proper rational transfer function is studied the solution method is based on the authors observation that the h model reduction problem can be converted into a hankel norm model reduction problem a simple characterization of the solution to the optimal h model reduction problem is developed

novel gramians for linear semistable systems in this paper the notions of pseudo gramians are introduced for linear time invariant semistable systems which allow multiple semisimple poles at the origin the proposed gramians are the generalizations of standard gramian matrices defined for asymptotically stable systems and they can be computed by a set of lyapunov equations furthermore it is shown that the controllability and observability of a semistable system are indicated by the ranks of the pseudo gramians and the controllability and observability energy functions are also characterized using the pseudo gramians additionally the h2 norm and h norm of a semistable system are analyzed and then the results are used for the model reduction of semistable systems finally the effectiveness of the methods is illustrated by an example of a gene regulation network

frequency weighted spl lscr sub spl infin norm and optimal hankel norm model reduction a new relative error model reduction method is proposed using frequency weighted balanced realization and explicit spl lscr sub spl infin norm error bounds are also derived for the relative error and multiplicative error the method only needs to solve two lyapunov equations it is further shown that this method is equivalent to the balanced stochastic truncation if the plant is square and minimum phase this paper also gives a complete solution to the frequency weighted hankel norm approximation with antistable weighting these results are then applied to spl lscr sub spl infin norm model reduction and several numerically effective algorithms are proposed it is shown through many numerical examples that these algorithms work very well and in many cases produce almost optimal solutions

a connection between state space and doubly coprime fractional representations explicit formulas for a doubly coprime fractional representation of the transfer function of a lumped lti system are given in terms of a stabilizable and detectable state space realization of the transfer function these formulas allow existing computational algorithms to be utilized for the purpose of computing doubly coprime factorizations several additional implications of this result are briefly discussed as are some immediate extensions and future research directions

universal detection of messages via finite state channels we propose a universal asymptotically optimum decision rule for deciding whether or not an observed sequence generated at the output of an unknown finite state channel corresponds to a given channel input message the hypothesized message is tested against a certain alternative message or several alternative messages under the neyman pearson criterion

an optimal controller reduction by covariance equivalent realizations based upon the theory of covariance equivalent realizations 1 this note presents a technique to obtain reduced order controllers via an optimal projection of full order lqg controllers this projection is optimal in the sense that it is a weighted least square approximation to a covariance equivalent projection of the full order optimal closed loop system the technique which is capable of extracting optimal minimal lqg controllers whenever they exist is also shown to yield controllers that are equivalent to those obtained by component cost analysis 12

brief accuracy enhancing methods for balancing related frequency weighted model and controller reduction we consider the solution of the balancing related frequency weighted model and controller reduction problems using accuracy enhanced numerical algorithms we propose first new stability enforcing choices of the frequency weighted grammians which can guarantee the stability of reduced models for two sided frequency weights then we show that for the frequency weighted controller reduction problems with standard stability and performance enforcing frequency weights the computation of the frequency weighted grammians can be done by solving reduced order lyapunov equations for both frequency weighted model and controller reduction problems we indicate how to compute the grammians directly in terms of their cholesky factors this allows the extension of the square root and balancing free accuracy enhancing techniques to the frequency weighted case

coprime factorization controller reduction with bezout identity induced frequency weighting abstract this paper introduces a frequency weighted controller reduction scheme which offers good closed loop stability under use of the reduced order controller by combining the use of a stable fractional representation of the controller with a frequency weighting in the reduction procedure derived by consideration of the closed loop stability margin we formulate a frequency weighted controller reduction procedure with the weight simplified by the bezout identity some examples are used to illustrate the reduction procedure

asymptotically optimal classification for multiple tests with empirically observed statistics the decision problem of testing m hypotheses when the source is kth order markov and there are m or fewer training sequences of length n and a single test sequence of length n is considered k m n n are all given it is shown what the requirements are on m n n to achieve vanishing exponential error probabilities and how to determine or bound the exponent a likelihood ratio test that is allowed to produce a no match decision is shown to provide asymptotically optimal error probabilities and minimum no match decisions as an important serial case the binary hypotheses problem without rejection is discussed it is shown that for this configuration only one training sequence is needed to achieve an asymptotically optimal test

frequency weighted controller reduction methods and loop transfer recovery abstract this paper shows that if one designs an lqg controller using the conventional technique of loop transfer recovery ltr then two frequency weighted controller reduction methods the enns frequency weighted balanced truncation enns 1984a ph d thesis stanford university ca 1984b proc 23rd cdc las vegas nv 127 132 and the bezout identity induced frequency weighted reduction method anderson and liu 1987 proc amer control conf mn will be equivalent under the condition that the plant transfer function is square nonsingular and minimum phase we also show that enns method is equivalent to the bezout identity induced frequency weighted reduction method if the controller itself is stable and a particular representation for the controller is assumed

quasi steady state reduction for compartmental systems abstract we present a method to determine an asymptotic reduction in the sense of tikhonov and fenichel for singularly perturbed compartmental systems in the presence of slow transport it turns out that the reduction can be derived from the individual interaction terms alone we apply the result to spatially discretized reaction diffusion systems and obtain based on the reduced discretized systems a heuristic to reduce reaction diffusion systems in presence of slow diffusion

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

uniformly attracting solutions of nonautonomous differential equations abstract understanding the structure of attractors is fundamental in nonautonomous stability and bifurcation theory by means of clarifying theorems and carefully designed examples we highlight the potential complexity of attractors for nonautonomous differential equations that are as close to autonomous equations as possible we introduce and study bounded uniform attractors and repellors for nonautonomous scalar differential equations in particular for asymptotically autonomous polynomial and periodic equations our results suggest that uniformly attracting or repelling solutions are the true analogues of attracting or repelling fixed points of autonomous systems we provide sharp conditions for the autonomous structure to break up and give way to a bewildering diversity of nonautonomous bifurcations

reduced order compensators via balancing and central control design for a structural control problem real time control of a physical system necessitates controllers that are low order in this article we compare two balanced truncation methods as a means of designing low order controllers for a nonlinear cable mass system the first is the standard technique of balanced truncation the second linear quadratic gaussian lqg balanced truncation can be thought of as balancing based on the controller and states that are important from the perspective of control and filter design are retained the control design applied to each reduced order model is the central controller we provide an overview of the central controller and devote attention to the design of this controller in the presence of balancing also described in this article is a method for reducing computational time in solving algebraic riccati equations for the design of low order lqg balanced controllers

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

on the estimation of the order of a markov chain and universal data compression the authors estimate the order of a finite markov source based on empirically observed statistics the performance criterion adopted is to minimize the probability of underestimating the model order while keeping the overestimation probability exponent at a prescribed level a universal asymptotically optimal test in the sense just defined is proposed for the case where a given integer is known to be the upper bound of the true order for the case where such a bound is unavailable an alternative rule based on the lempel ziv data compression algorithm is shown to be asymptotically optimal also and computationally more efficient

model reduction by balanced truncation for systems with nuclear hankel operators we prove the h infinity error bounds for lyapunov balanced truncation and for optimal hankel norm approximation under the assumption that the hankel operator is nuclear this is an improvement of the result from glover curtain and partington siam j control optim 26 1998 pp 863 898 where additional assumptions were made the proof is based on convergence of the schmidt pairs of the hankel operator in a sobolev space we also give an application of this convergence theory to a numerical algorithm for model reduction by balanced truncation

consistent impulse response estimation and system realization from noisy data a novel method for the estimation of a finite part of the impulse response is presented the method is consistent in case the measured output data are prone to colored noise the combination of kung s realization algorithm yields a consistent system identification algorithm that is particularly useful for the identification of systems with uncoupled deterministic and stochastic dynamics this algorithm is called irear a detailed linear sensitivity analysis of the impulse response estimation and kung s realization algorithms yields novel error and covariance formulae for the estimated impulse response and system matrices and the poles and frequency response functions that result from the identified system description

order reduction of large scale second order systems using krylov subspace methods in order reduction of large scale linear time invariant systems krylov subspace methods based on moment matching are among the best choices today however in many technical fields models typically consist of sets of second order differential equations and krylov subspace methods cannot directly be applied two methods for solving this problem are presented in this paper 1 an approach by su and craig is generalized and the number of matching moments is increased 2 a new approach via first order models is presented resulting in an even higher number of matching moments both solutions preserve the specific structure of the second order type model

frequency dependent error bounds for uncertain linear models we study frequency dependent error bounds for approximation and truncation of linear dynamic models with uncertainty a possible application is in model simplification relevant for control design the uncertainty is described by quadratic constraints and the error bounds are calculated based on solutions to linear matrix inequalities

a subspace algorithm for balanced state space system identification in an earlier paper by moonen et at 1989 an algorithm has been introduced for identifying multivariable linear systems directly from input output data this new algorithm falls in the class of so called subspace methods and resembles impulse response based realization algorithms in this paper we extend this algorithm by incorporating a balancing step such that the identified model is always in balanced coordinates with this modification one obtains a data driven counterpart for kung s realization algorithm

identification and model reduction of time varying discrete time systems abstract linear discrete time variable systems are considered and an important class of uniform realizations is defined the necessary and sufficient conditions for a pulse response to be uniformly realizable are obtained two model reduction schemes one via balanced realizations and the other via hankel matrix are proposed the latter approach can be considered as an identification algorithm for the two dimensional pulse response sequence h k i it is shown that the two approaches yield an identical reduced model which is always asymptotically stable

estimating the number of states of a finite state source the problem of estimating the number of states of a finite alphabet finite state source is investigated an estimator is developed that asymptotically attains the minimum probability of understanding the number of states among all estimators with a prescribed exponential decay rate of overestimation probability the proposed estimator relies on the lempel ziv data compression algorithm in an intuitively appealing manner

additive watermark detection in the wavelet domain using 2d garch model watermark detection can be generally achieved by statistical decision methods so selection of an accurate statistical model is one of the major issues in watermark detection usually watermark detectors assume the wavelet coefficients to be independent and identically distributed therefore they cannot take into account important characteristics of the wavelet coefficients such as dependency and heteroscedasticity this paper presents a novel detector for wavelet domain additive image watermarking we use two dimensional generalized autoregressive conditional heteroscedasticity 2d garch model for the wavelet coefficients this model can capture the dependency heteroscedasticity and heavy tailed marginal distribution of these coefficients based on 2d garch model we design a new watermark detector and derive its receiver operating characteristics experimental results demonstrate the efficiency of proposed detector and its higher performance compared to alternative watermarking methods in the literature

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

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

a note on balanced controller reduction the standard linear quadratic gaussian lqg optimal controller is not minimal in general even if the plant is minimal hence the order of the optimal controller can be less than that of the plant this note shows that the original balancing method of moore 1 can be used to generate these reduced order optimal controllers but the newer balancing methods recently proposed for controller reduction might not generate the reduced order optimal controllers even when they exist

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

approximation of fir by iir digital filters an algorithm based on balanced model reduction an algorithm for the approximation of finite impulse response fir filters by infinite impulse response iir filters is presented the algorithm is based on a concept of balanced model reduction the matrix inversions normally associated with this procedure are notoriously error prone due to ill conditioning of the special matrix forms required this difficulty is circumvented here by directly formulating a reduced state space system description which is input output equivalent to the system that would more conventionally be obtained following the explicit step of constructing an interim balanced realization examples of fir by iir filter approximations are included

slow manifold for a nonlocal stochastic evolutionary system with fast and slow components abstract this work is devoted to investigating invariant manifolds for a fast slow stochastic evolutionary system with nonlocal diffusion we establish the slow reduction via a random slow manifold which captures slow dynamics of the original stochastic fast slow system a simple example is shown to illustrate this slow reduction method

an implicit deflation method for ill conditioned sylvester and lyapunov equations this paper describes an effective algorithm for solving ill conditioned sylvester or lyapunov equations these equations can be solved by conventional methods for ill conditioned linear systems however such methods are not efficient since they require on the order of n 6 arithmetic operations on order n 2 data for these ill conditioned matrix equations an implicit deflation algorithm is proposed to implement a certain svd based minimum norm least squares approximate solution the method is practical since it relies only on the ability to solve a sylvester or lyapunov equation certain practical details are also discussed

topological decoupling linearization and perturbation on inhomogeneous time scales abstract we derive a linearization theorem in the framework of dynamic equations on time scales this extends a recent result from y xia j cao m han a new analytical method for the linearization of dynamic equation on measure chains j differential equations 235 2007 527 543 in various directions firstly in our setting the linear part need not to be hyperbolic and due to the existence of a center manifold this leads to a generalized global hartman grobman theorem for nonautonomous problems secondly we investigate the behavior of the topological conjugacy under parameter variation these perturbation results are tailor made for future applications in analytical discretization theory i e to study the relationship between odes and numerical schemes applied to them

controller reduction via stable factorization and balancing abstract a controller reduction procedure based on a representation of a controller as a matrix function defined using stable proper transfer functions and employing a balancing technique is studied in this paper for a certain right coprime factorization of an lqg designed controller k s n s d 1 s we approximate using a balancing technique the pair d s n s t by a low order pair d1 s n1 s t defining a factorization of the reduced order controller k 1 s n1 s d1 1 s we show that reducing the controller order in this way is motivated in a natural way which leads to the expectation of both good stability properties and good accuracy of approximation of closed loop behaviour this is also demonstrated in some examples

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

h optimal control part 1 model matching our aim in this paper is to develop a new approach for solving the h optimal control problem where the feedback arrangement takes the form of a linear fractional transformation the paper is in two parts in part 1 a basic kind of model matching problem is considered given rational matrices m s and n s the h norm of an error function defined as e s m s n s q s is minimized or bounded subject to e s and q s being stable closed form state space characterizations are obtained for both e s and q s the results established here will be used in part 2 of the paper hung 1989 to solve the h optimal control problem

an algorithmic method for the simplification of linear dynamic scalar systems a comprehensive method is proposed for obtaining reduced order models of a scalar linear time invariant system described by the transfer function g s the method is a viable alternative to the pade approximation technique for model order reduction it approximates the differential operations d i g s dsi proportional to the time moments by the divided difference equivalents and finally leads to linear equations in the unknown coefficients of the reduced model r s it is shown that a parameter governs the stability and the low frequency matching quality of the model heuristic criteria are given for selecting a proper the method is illustrated by some examples from the literature

controller reduction concepts and approaches the problem of passing from a linear time invariant high order controller designed for a linear time invariant plant of presumably high order to a low order approximation of the controller is discussed the approximation problem is often best posed as a frequency weighted l sub infinity approximation problem many different controller representations are possible giving different performances of the various reduction algorithms

performance bounds for coprime factor controller reductions this paper investigates the performance of coprime factor based controller reduction methods we show that under some mild conditions closed loop performance bounds can be derived for observer based controllers the results in this paper lay a mathematical foundation for using coprime factor controller reduction methods in certain classes of controller reduction problems

topology of the set of asymptotically stable minimal systems it is shown that the set of asymptotically stable and minimal single input single output systems of order n has n 1 connected components in the case of asymptotically stable and minimal multiple input multiple output systems however there is only one connected component

improved spread spectrum a new modulation technique for robust watermarking this paper introduces a new watermarking modulation technique which we call improved spread spectrum iss when compared with traditional spread spectrum ss the signal does not act as a noise source leading to significant gains in some examples performance improvements over ss are 20 db in signal to noise ratio snr or ten or more orders of magnitude in the error probability the proposed method achieves roughly the same noise robustness gain as quantization index modulation qim but without the amplitude scale sensitivity of qim our proposed iss is as robust in practice as traditional ss

resolution of subgrid microscale interactions enhances the discretisation of nonautonomous partial differential equations we consider a general method for the macroscale modelling of nonlinear nonautomomous partial differential equations or difference equations our method is able to resolve the complex fluctuations which arise in nonliner nonautonomous systems these fluctuations are not accounted for by popular multiscale averaging procedures and homogenization our method is illustrated with the example of a forced burgers equation coarse grained macroscale spatial discretisations of nonlinear nonautonomous partial differential difference equations are given novel support by centre manifold theory dividing the physical domain into overlapping macroscale elements empowers the approach to resolve significant subgrid microscale structures and interactions between neighbouring elements the crucial aspect of this approach is that centre manifold theory organises the resolution of the detailed subgrid microscale structure interacting via the nonlinear dynamics within and between neighbouring elements the techniques and theory developed here may be applied to soundly discretise on a macroscale many dissipative nonautonomous partial differential difference equations such as the forced burgers equation adopted here as an illustrative example

balanced realizations for linear systems a variational approach this paper develops a new geometric approach to balanced realizations which enables balanced realizations for arbitrary linear systems to be defined and studied for an arbitrary unitarily invariant strictly plurisubharmonic function on the set of realizations of a given transfer function the class of realizations is considered that minimizes the function based on results from invariant theory and complex analysis a general theorem on the existence and uniqueness properties of such function minimizing realizations is derived if the function is the sum of the traces of the controllability and observability gramians the usual class of balanced realizations is obtained other choices of functions yield different new classes of function minimizing realizations

asymptotically optimum universal watermark embedding and detection in the high snr regime the problem of optimum watermark embedding and detection was addressed in a recent paper by merhav and sabbag where the optimality criterion was the maximum false negative error exponent subject to a guaranteed false positive error exponent in particular merhav and sabbag derived universal asymptotically optimum embedding and detection rules under the assumption that the detector relies solely on second order joint empirical statistics of the received signal and the watermark in the case of a gaussian host signal and a gaussian attack however closed form expressions for the optimum embedding strategy and the false negative error exponent were not obtained in that work in this paper we derive the false negative error exponent for any given embedding strategy and use such a result to show that in general the optimum embedding rule depends on the variance of the host sequence and the variance of the attack noise we then focus on high signal to noise ratio snr regime deriving the optimum embedding strategy for such a setup in this case a universally optimum embedding rule turns out to exist and to be very simple with an intuitively appealing geometrical interpretation the effectiveness of the newly proposed embedding strategy is evaluated numerically

model reduction of large scale systems by least squares abstract in this paper we introduce an approximation method for model reduction of large scale dynamical systems this is a projection which combines aspects of the svd and krylov based reduction methods this projection can be efficiently computed using tools from numerical analysis namely the rational krylov method for the krylov side of the projection and a low rank smith type iteration to solve a lyapunov equation for the svd side of the projection for discrete time systems the proposed approach is based on the least squares fit of the r 1 th column of a hankel matrix to the preceding r columns where r is the order of the reduced system the reduced system is asymptotically stable matches the first r markov parameters of the full order model and minimizes a weighted h 2 error the method is also generalized for moment matching at arbitrary interpolation points application to continuous time systems is achieved via the bilinear transformation numerical examples prove the effectiveness of the approach the proposed method is significant because it combines guaranteed stability and moment matching together with an optimization criterion

arma spectral estimation of narrow band processes via model reduction the problem of estimating autoregressive moving average arma models for narrowband processes is considered the following approach is proposed estimate a high order autoregressive ar approximation of the process by model reduction based on a truncated internally balanced realization or optimal hankel norm model reduction reduce the order of this high order ar estimate to find a lower order arma model this algorithm gives arma spectral estimates with excellent resolution properties without using iterative numerical minimization methods as for the maximum likelihood method how to take the narrowband assumption into account in the model reduction step is discussed in detail

normal forms and invariant manifolds for nonlinear non autonomous pdes viewed as odes in infinite dimensions abstract we prove that a general class of nonlinear non autonomous ode s in frechet spaces are close to ode s in a specific normal form where closeness means that solutions of the normal form ode satisfy the original ode up to a residual that vanishes up to any desired order in this normal form the centre stable and unstable coordinates of the ode are clearly separated which allows us to define invariant manifolds of such equations in a robust way the main motivation is the case where the frechet space in question is a suitable function space and the maps involved in an ode in this space are defined in terms of derivatives of the functions so that the infinite dimensional ode is a finite dimensional pde we show that our methods apply to a relevant class of nonlinear non autonomous pde s in this way

robust controller design based on reduced order plants two dual controller design methods are proposed for linear time invariant multi input multi output systems where designs based on a reduced order plant robustly stabilizer higher order plants with additional poles or zeros in the stable region the additional poles or zeros are considered as multiplicative perturbations of the reduced plant the methods are tailored towards closed loop stability and performance and they yield estimates for the stability robustness and performance of the final design they can be considered as formalizations of two classical heuristic model reduction techniques one method neglects a plant pole sufficiently far to the left of dominant poles and the other cancels a sufficiently small stable plant zero with a pole at the origin

parametrization and optimization of reduced order filters this paper deals with the design of reduced order filters for linear signal models to ensure that the transfer function from the noise to the filtering error is not only stable but also has a minimum h sub 2 norm one of the major results in the paper is the discovery and parametrization of a set of filters of fixed order which lead to stable filtering error transfer functions such a parametrization is given in terms of an arbitrary orthogonal projection matrix and a given full order filter as a consequence the problem of minimizing the h sub 2 norm of the filtering error transfer function over the set of reduced order filters can be formulated as an equivalent unconstrained parametric optimization problem over a compact manifold a gradient flow based algorithm is proposed to find an optimal solution to the problem nice properties of the algorithm including the convergence property are established theoretically as well as demonstrated numerically with an example

structurally balanced controller order reduction with guaranteed closed loop performance abstract this paper considers controller order reduction with the consideration of closed loop stability and performance a closed loop structurally balancing method is proposed to reduce the controller orders for a class of systems and controllers the resulting reduced order controllers if they exist are guaranteed to stabilize the corresponding systems error bounds are also derived for the closed loop performance degradation

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

stable model order reduction for time domain exterior vibro acoustic finite element simulations abstract this paper presents a novel method that enables model order reduction of a fully coupled exterior vibro acoustic finite element model for time domain simulations the method preserves the stability of the full model and reduces the amount of degrees of freedom significantly with only a moderate amount of calculation complexity infinite elements are used on the finite element boundary to satisfy the sommerfeld radiation condition two different strategies to calculate the reduced order model are compared the first strategy works with a split reduced basis and can be applied on any fully stable model the second strategy starts from a modified everstine formulation and directly builds a reduced basis from the full model leading to more compact reduced order models furthermore a method is derived to perform explicit time integration on the reduced system while avoiding the inversion of the mass matrix which might not be possible due to the presence of the infinite elements also this method is shown to preserve the stability of the model and a computationally efficient way for implementation of the method is discussed the effectiveness of the novel methodology is demonstrated with two numerical models

optimal angle reduction a behavioral approach to linear system approximation we investigate the problem of optimal state reduction under minimization of the angle between system behaviors the angle is defined in a worst case sense as the largest angle that can occur between a system trajectory and its optimal approximation in the reduced order model this problem is analyzed for linear time invariant finite dimensional systems in a behavioral l2 setting without reference to input output decompositions and stability considerations the notion of a weakest past future link is introduced and it is shown how this concept is applied for the purpose of model reduction a method that reduces the state dimension by one is presented and shown to be optimal specific algorithms are provided for the numerical implementation of the approximation method the concepts and results are explicitly translated to an input output setting and related to balancing hankel norm reduction and normalized doubly coprime factorizations

a lyapunov theory for linear time delay systems some relationships between a lyapunov based stability test and linear control system models with time delays are explored the main result is a generalization of the result that if a b is a reachable pair of matrices then square a is a stability matrix if and only if there is a positive definite matrix k such that ak ka bb

balanced pod for model reduction of linear pde systems convergence theory we consider convergence analysis for a model reduction algorithm for a class of linear infinite dimensional systems the algorithm computes an approximate balanced truncation of the system using solution snapshots of specific linear infinite dimensional differential equations the algorithm is related to the proper orthogonal decomposition and it was first proposed for systems of ordinary differential equations by rowley int j bifurc chaos appl sci eng 15 3 997 1013 2005 for the convergence analysis we consider the algorithm in terms of the hankel operator of the system rather than the product of the system gramians as originally proposed by rowley for exponentially stable systems with bounded finite rank input and output operators we prove that the balanced realization can be expressed in terms of balancing modes which are related to the hankel operator the balancing modes are required to be smooth and this can cause computational difficulties for pde systems we show how this smoothness requirement can be lessened for parabolic systems and we also propose a variation of the algorithm that avoids the smoothness requirement for general systems we prove entrywise convergence of the matrices in the approximate reduced order models in both cases and present numerical results for two example pde systems

a new approach to the discretization of continuous time controllers a new method for discretizing continuous time controllers is derived using principles of controller approximation it focuses on the closed loop use of the discrete time controller the resulting approximation criterion is a measure for stability of the control system provides an upper bound on the sampling time for which stability can be guaranteed and because it is based on continuous time controller approximation it indicates the cost of discretization in terms of performance degradation the discrete time controller is obtained through minimization of this criterion which can be performed with standard software used in h sub infinity controller design

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

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

on the structure of balanced and other principal representations of siso systems a new matrix w co which can be considered as a cross gramian matrix which contains information about both controllability and observability is defined for single input single output linear systems using this matrix the structural properties of linear systems are studied in the context of principal component analysis the matrix w co can be used in obtaining balanced and other principal representations without computation of the controllability and the observability gramians the importance of this matrix in model order reduction is highlighted

goal oriented model constrained optimization for reduction of large scale systems optimization oriented reduced order models should target a particular output functional span an applicable range of dynamic and parametric inputs and respect the underlying governing equations of the system to achieve this goal we present an approach for determining a projection basis that uses a goal oriented model constrained optimization framework the mathematical framework permits consideration of general dynamical systems with general parametric variations and is applicable to both linear and nonlinear systems results for a simple linear model problem of the two dimensional heat equation demonstrate the ability of the goal oriented approach to target a particular output functional of interest application of the methodology to a more challenging example of a subsonic blade row governed by the unsteady euler flow equations shows a significant advantage of the new method over the proper orthogonal decomposition

computation of approximate null vectors of sylvester and lyapunov operators this paper describes an effective algorithm for computing approximate null vectors of certain matrix operators associated with sylvester or lyapunov equations the singular value decomposition and rank revealing qr methods are two widely used stable algorithms for numerical determination of the rank and nullity of a matrix a these methods however are not readily applicable to sylvester and lyapunov operators since they require on the order of n sup 6 arithmetic operations on order n sup 2 data for these problems a variant of inverse power iteration is employed to compute orthonormal bases for singular subspaces associated with the small singular values the method is practical since it relies only on the ability to solve a sylvester or lyapunov equation certain practical aspects are considered and a direct refinement technique is proposed to enhance the convergence of the algorithm

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 error exponent and capacity games of private watermarking systems watermarking systems are analyzed as a game between an information hider a decoder and an attacker the information hider is allowed to cause some tolerable level of distortion to the original data within which the message is hidden and the resulting distorted data can suffer some additional amount of distortion caused by an attacker who aims at erasing the message two games are investigated the error exponent game and the coding capacity game motivated by a worst case approach we assume that the attacker is informed of the hiding strategy taken by the information hider and the decoder which are uninformed of the attacking scheme this approach leads to the maximin error exponent and maximin coding capacity as objective functions it is assumed that the host data is drawn from a finite alphabet memoryless stationary source and its realization side information is available at the encoder and the decoder a single letter expression for the maximin error exponent is found under large deviations distortion constraints moreover we find an asymptotically optimal random coding distribution a universal decoder and a worst case attack channel it is proved that there is a saddle point in the asymptotic exponent and that the minimax and the maximin error exponents are equal finally a single letter expression for the coding capacity i e the maximin reliable information rate is found

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

partial fraction expansion based frequency weighted model reduction technique with error bounds in this technical note we present a new frequency weighted balancing related technique which is based on a parametrized combination of the unweighted balanced truncation technique 9 8 and the partial fraction expansion technique 11 the reduced order models which are guaranteed to be stable in the case of double sided weighting are obtained by either direct truncation or singular perturbation approximation simple elegant and easily computable a priori error bounds are also derived numerical examples and comparison with other well known techniques show the effectiveness of the proposed technique

optimal watermark embedding and detection strategies under limited detection resources an information theoretic approach is proposed to watermark embedding and detection under limited detector resources first the attack free scenario is considered under which asymptotically optimal decision regions in the neyman pearson sense are proposed along with the optimal embedding rule later the case of zero mean independent and identically distributed i i d gaussian covertext distribution is explored with unknown variance under the attack free scenario for this case a lower bound on the exponential decay rate of the false negative probability is proposed it is proven that the optimal embedding and detecting strategy is superior to the customary linear additive embedding strategy in the exponential sense finally these results are extended to the case of memoryless attacks and general worst case attacks optimal decision regions and embedding rules are offered and the worst attack channel is identified

balanced gains and their significance for l 2 model reduction the singular values do not give enough information for model reduction in the l2sense other invariants are introduced which help describe the contribution of each state to the l2magnitude of the impulse response this leads to another suboptimal balanced truncation criterion

controller reduction by component cost analysis component cost analysis 1 is used to develop a method for controller reduction the reduction of the controller is based upon the participation of the controller states in the value of a quadratic performance metric the controller states which have the smallest contribution to the performance metric are truncated to produce the reduced controllers an error index is defined to evaluate the reduced controller compared to the optimal lqg controller and bounds on this index are derived a numerical example is included to illustrate the procedure

controller reduction using canonical interactions a conceptual approach to controller reduction is proposed that is based on closed loop considerations the method assesses the interactions between the plant and a high order controller as described by canonical correlation coefficients a free parameter in the algorithm of the reduced controller makes it possible to trade stability robustness versus performance several examples demonstrate the efficacy of the method

convergence rate for an approximation approach to h norm optimization problems with an application to controller order reduction in a recent paper by fisher et al 1989 a constrained h smooth optimization technique proc 28th cdc florida u s a a smooth approximation technique is proposed to solve a general class of constrained h norm optimization problems the aim of this paper is to show that such approximation has a convergence rate of o in p2p furthermore this technique is applied to handle a general class of frequency weighted controller order reduction problems