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

comparison between h sub 2 and h sub infinity controllers the h sub infinity performances of a controller designed by h sub 2 methodology and a controller designed by h sub infinity methodology are quantitatively compared it is shown that the ratio of the h sub infinity performance of the two is no greater than 2k for the one block problem where k is the total number of right half plane transmission zeros for two subsystems namely the subsystem between the disturbance and the measurement and the subsystem between the control signal and the controlled output it is also shown that there are systems which can give a ratio arbitrarily close to the upper bound however those systems are of a pathological type some possible generalizations to two and four block problems are discussed

algorithms to evaluate multiple sums for loop computations we present algorithms to evaluate two types of multiple sums which appear in higher order loop computations we consider expansions of a generalized hyper geometric type sums n1 nn a1 n c1 a2 n c2 ap n cp b1 n d1 b2 n d2 bq n dq x1n1 xnnn with ai n j 1naijnj etc in a small parameter e around rational values of ci di s type i sum corresponds to the case where in the limit e 0 the summand reduces to a rational function of nj s times x1n1 xnnn ci di s can depend on an external integer index type ii sum is a double sum n 2 where ci di s are half integers or integers as e 0 and xi 1 we consider some specific cases where at most six functions remain in the limit e 0 the algorithms enable evaluations of arbitrary expansion coefficients in e in terms of z sums and multiple polylogarithms generalized multiple zeta values we also present applications of these algorithms in particular type i sums can be used to generate a new class of relations among generalized mult

an error bound for a discrete reduced order model of a linear multivariable system the design of feasible controllers for high dimension multivariable systems can be greatly aided by a method of model reduction in order for the design based on the order reduction to include a guarantee of stability it is sufficient to have a bound on the model error previous work has provided such a bound for continuous time systems for algorithms based on balancing in this note an linftybound is derived for model error for a method of order reduction of discrete linear multivariable systems based on balancing

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 state space formulation for optimal hankel norm approximations the optimal hankel norm approximation problem studied in 1 is reformulated in a state space setting the crucial extension theorem is reestablished in this framework the minimal degree optimal approximation is then derived in terms of state space parameters

phase transitions in the q voter model with noise on a duplex clique we study a nonlinear q voter model with stochastic noise interpreted in the social context as independence on a duplex network to study the role of the multilevelness in this model we propose three methods of transferring the model from a mono to a multiplex network they take into account two criteria one related to the status of independence local vs global and one related to peer pressure and vs or in order to examine the influence of the presence of more than one level in the social network we perform simulations on a particularly simple multiplex a duplex clique which consists of two fully overlapped complete graphs cliques solving numerically the rate equation and simultaneously conducting monte carlo simulations we provide evidence that even a simple rearrangement into a duplex topology may lead to significant changes in the observed behavior however qualitative changes in the phase transitions can be observed for only one of the considered rules local and for this rule the phase transition becomes discontinuous for q 5 whereas for a monoplex such behavior is observed for q 6 interestingly only this rule admits construction of realistic variants of the model in line with recent social experiments

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

testing validity of the kirkwood approximation using an extended sznajd model we revisit the deduction of the exit probability of the one dimensional sznajd model through the kirkwood approximation f slanina et al europhys lett 82 18006 2008 this approximation is peculiar in that in spite of the agreement with simulation results f slanina et al europhys lett 82 18006 2008 r lambiotte and s redner europhys lett 82 18007 2008 a m timpanaro and c p c prado phys rev e 89 052808 2014 the hypothesis about the correlation lengths behind it are inconsistent and fixing these inconsistencies leads to the same results as a simple mean field we use an extended version of the sznajd model to test the kirkwood approximation in a wider context this model includes the voter sznajd and united we stand divided we fall models r a holley and t m liggett ann prob 3 643 1975 k sznajd weron and j sznajd int j mod phys c 11 1157 2000 as different parameter combinations meaning that some analytical results from these models can be used to evaluate the performance of the kirkwood approximation we also compare the predicted exit probability with simulation results for networks with 10 3 sites the results show clearly the regions in parameter space where the approximation gives accurate predictions as well as where it starts failing leading to a better understanding of its reliability

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

a non iterative method for computing the infimum in h optimization this paper presents a simple and non iterative procedure for the computation of the exact value of the infimum in the singular h optimization problem and is an extension of our earlier work the problem formulation is general and does not place any restriction on the direct feedthrough terms between the control input and the controlled output variables and between the disturbance input and the measurement output variables our method is applicable to a class of singular h optimization problems for which the transfer functions from the control input to the controlled output and from the disturbance input to the measurement output have no invariant zeros on the jw axis and also satisfy certain geometric conditions the computation of the infimum in our method involves solving two well defined riccati and two lyapunov equations

technical communique the schur stability via the hurwitz stability analysis using a biquadratic transformation in this note it is shown that instead of a bilinear transformation a biquadratic transformation can be used to determine the schur stability of a given discrete time polynomial by determining the hurwitz stability of the corresponding continuous time polynomial however using a biquadratic transformation the analysis is not limited by any restriction on the root positions several numerical examples are provided to illustrate the procedure

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

approximate balanced truncation model reduction for large scale systems the complexity of a large scale system makes the computations of reduced order models based on balancing impractical however typically there is a weak coupling between the subsystems of a large scale system this is used here to derive approximate balanced truncation reduced order models for large scale systems with reliable and tractable computations

spectral analysis in linear process identification a new look at an old problem the problem of estimating the transfer function frequency response of a linear process by means of a modified spectral analysis method is treated a method of determining the maximum time lags of the correlation functions which makes use of the available a priori physical knowledge about the process and the input signal is proposed this leads in a natural way to the use of an asymmetrical window technique some simulation results are used to illustrate the method based on the asymptotic properties of the estimate the input design is studied finally it will be shown that this work can be used for the estimation of an upper bound of the model uncertainty in the frequency domain

mapping the q voter model from a single chain to complex networks abstract we propose and compare six different ways of mapping the modified q voter model to complex networks considering square lattices barabasi albert watts strogatz and real twitter networks we ask the question if always a particular choice of the group of influence of a fixed size q leads to different behavior at the macroscopic level using monte carlo simulations we show that the answer depends on the relative average path length of the network and for real life topologies the differences between the considered mappings may be negligible

exit probability of the one dimensional q voter model analytical results and simulations for large networks we discuss the exit probability of the one dimensional q voter model and present tools to obtain estimates about this probability both through simulations in large networks around 10 7 sites and analytically in the limit where the network is infinitely large we argue that the result e q q 1 q that was found in three previous works f slanina k sznajd weron and p przybyla europhys lett 82 18006 2008 r lambiotte and s redner europhys lett 82 18007 2008 for the case q 2 and p przybyla k sznajd weron and m tabiszewski phys rev e 84 031117 2011 for q 2 using small networks around 10 3 sites is a good approximation but there are noticeable deviations that appear even for small systems and that do not disappear when the system size is increased with the notable exception of the case q 2 we also show that under some simple and intuitive hypotheses the exit probability must obey the inequality q q 1 e 1 q in the infinite size limit we believe this settles in the negative the suggestion made s galam and a c r martins europhys lett 95 48005 2001 that this result would be a finite size effect with the exit probability actually being a step function we also show how the result that the exit probability cannot be a step function can be reconciled with the galam unified frame which was also a source of controversy

discrete system reduction via impulse response gramians and its relation to q markov covers a method for model reduction of linear discrete systems is proposed it is based on the impulse response gramian proposed by the authors 1989 for discrete systems this gramian is an extension of the one proposed for linear continuous systems and contains information on the input output behavior of the system the rth order reduced order models are made to retain the first r markov parameters and the first r r elements of the impulse response gramian of the original system the relation between this method and the q markov cover method is also discussed the method is illustrated by a numerical example

nonlinear q voter model we introduce a nonlinear variant of the voter model the q voter model in which q neighbors with possible repetition are consulted for a voter to change opinion if the q neighbors agree the voter takes their opinion if they do not have a unanimous opinion still a voter can flip its state with probability epsilon we solve the model on a fully connected network i e in mean field and compute the exit probability as well as the average time to reach consensus by employing the backward fokker planck formalism and scaling arguments we analyze the results in the perspective of a recently proposed langevin equation aimed at describing generic phase transitions in systems with two z2 symmetric absorbing states in particular by deriving explicitly the coefficients of such a langevin equation as a function of the microscopic flipping probabilities we find that in mean field the q voter model exhibits a disordered phase for high epsilon and an ordered one for low epsilon with three possible ways to go from one to the other i a unique generalized voter like transition ii a series of two consecutive transitions one ising like in which the z2 symmetry is broken and a separate one in the directed percolation class in which the system falls into an absorbing state and iii a series of two transitions including an intermediate regime in which the final state depends on initial conditions this third so far unexplored scenario in which a type of ordering dynamics emerges is rationalized and found to be specific of mean field i e fluctuations are explicitly shown to wash it out in spatially extended systems

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

brief paper positivity preserving h model reduction for positive systems this paper is concerned with the model reduction of positive systems for a given stable positive system our attention is focused on the construction of a reduced order model in such a way that the positivity of the original system is preserved and the error system is stable with a prescribed h performance based upon a system augmentation approach a novel characterization on the stability with h performance of the error system is first obtained in terms of linear matrix inequality lmi then a necessary and sufficient condition for the existence of a desired reduced order model is derived accordingly furthermore iterative lmi approaches with primal and dual forms are developed to solve the positivity preserving h model reduction problem finally a compartmental network is provided to show the effectiveness of the proposed techniques

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

a new frequency weighted balanced truncation method and an error bound in this paper a new frequency weighted model reduction method with an a priori error bound is proposed the method is a generalization of enns technique 1984 and yields stable models even when both input and output weightings are included the proposed method is compared with other existing methods using numerical examples

on the selection of the eigenvalues to be retained in the reduced order models this note points out that the model order selection criteria proposed earlier for davison s and marshall s reduced order models are not applicable in many of the practical cases a new criterion is proposed to overcome the deficiencies of the earlier methods

multiobjective h sub 2 h sub spl infin control for a linear time invariant system with several disturbance inputs and controlled outputs we show how to minimize the nominal h sub 2 norm performance in one channel while keeping bounds on the h sub 2 norm or h sub spl infin norm performance implying robust stability in the other channels this multiobjective h sub 2 h sub spl infin problem in an infinite dimensional space is reduced to sequences of finite dimensional convex optimization problems we show how to compute the optimal value and how to numerically detect the existence of a rational optimal controller if it exists we reveal how the novel trick of optimizing the trace norm of the youla parameter over certain convex constraints allows one to design a nearly optimal controller whose youla parameter is of the same order as the optimal one

an l convergent method to reduce frequency weighted infinite dimensional discrete time systems abstract in this paper we present a method based on balanced realizations truncations to reduce frequency weighted infinite dimensional discrete time systems the method is shown to be l convergent for a subclass of the nuclear systems class as its numerical computation is not easy we propose a practical method to reduce frequency weighted infinite dimensional systems and prove that it is l convergent for the class of systems whose transfer function has its derivative in the wiener algebra

discrete model reduction preserving bounded realness abstract in this paper we propose a discrete model reduction method that preserves the bounded realness we define the discrete bounded real br balancing and obtain the reduced order discrete models from the generalized singular perturbation approximation gspa and direct truncation dt of the br balanced system we show the properties of the reduced order systems and derive the h norm error bounds for the cases of the gspa at 1 of the complex plane and the dt

system identification using balanced parametrizations some general issues in the black box identification of multivariable systems are first discussed it is then suggested that balanced parametrizations can be used to give identifiable forms a particular advantage is that balanced parametrizations are known for several useful classes of linear dynamic models including stable minimal models minimum phase models positive real models and normalized coprime factor models before optimizing the parameters of balanced parametrizations an initial model must be found we use realization based methods and so called subspace methods for this purpose these methods are very effective at finding accurate initial models without preliminary estimation of various structural indexes the paper ends with two simulation examples which compare the use of balanced parametrizations with more traditional ones and three real examples based on practical problems a distillation column an industrial dryer and the irrational spectrum or sea waves

a generalised partial fraction expansion based frequency weighted balanced truncation technique in this paper we present some new results on a frequency weighted balanced truncation technique based on well known partial fraction expansion idea the reduced order models which are guaranteed to be stable in case of double sided weighting are obtained by direct truncation two sets of simple elegant and easily computable a priori error bounds are also derived relationships between the proposed method and the previous methods based on partial fraction idea are also derived the technique is illustrated using a numerical example of a practical application and then compared with other well known techniques to show the effectiveness of the method

a unified derivation and critical review of modal approaches to model reduction six of the most commonly used modal reduction techniques are represented by means of signal flow diagrams connecting inputs and state variables in this representation the derivation and the cross comparison of the various methods are greatly facilitated each method is then analysed to determine key characteristics i e i steady state agreement ii initial value agreement iii independence of the reduced model on the choice of retained state variables and inputs and iv eigen vector orientation realistic large scale systems are used to evaluate the static and dynamic characteristics of each model reduction method numerically these studies reveal some important and often unrecognized weaknesses of several traditional techniques and confirm the strength of the newly developed technique of litz

nested sums expansion of transcendental functions and multiscale multiloop integrals expansion of higher transcendental functions in a small parameter are needed in many areas of science for certain classes of functions this can be achieved by algebraic means these algebraic tools are based on nested sums and can be formulated as algorithms suitable for an implementation on a computer examples such as expansions of generalized hypergeometric functions or appell functions are discussed as a further application we give the general solution of a two loop integral the so called c topology in terms of multiple nested sums in addition we discuss some important properties of nested sums in particular we show that they satisfy a hopf algebra

subspace based multivariable system identification from frequency response data two noniterative subspace based algorithms which identify linear time invariant mimo multi input multioutput systems from frequency response data are presented the algorithms are related to the recent time domain subspace identification techniques the first algorithm uses equidistantly in frequency spaced data and is strongly consistent under weak noise assumptions the second algorithm uses arbitrary frequency spacing and is strongly consistent under more restrictive noise assumptions promising results are obtained when the algorithms are applied to real frequency data originating from a large flexible structure

exit probability in a one dimensional nonlinear q voter model we formulate and investigate the nonlinear q voter model which as a special case includes the linear voter and the sznajd model on a one dimensional lattice we derive an analytical formula for the exit probability and show that it agrees perfectly with monte carlo simulations the puzzle that we deal with here may be summarized by a simple question why does the mean field approach give the exact formula for the exit probability in the one dimensional nonlinear q voter model to answer this question we test several hypotheses proposed recently for the sznajd model including the finite size effects the influence of the range of interactions and the importance of the initial step of the evolution on the one hand our work is part of a trend of the current debate on the form of the exit probability in the one dimensional sznajd model but on the other hand it concerns the much broader problem of the nonlinear q voter model

frequency weighted model reduction with l error bounds abstract this paper considers frequency weighted model reduction the explicit lower and upper approximation error bounds are derived for certain classes of weighted model reduction problems the approximation is based on some unweighted approximations and the error bounds are given in terms of the hankel singular values of the weighted model

the use of model reduction techniques for designing iir filters with linear phase in the passband a technique to design iir filters with linear phase in the passband is presented this technique is based on model reduction of an fir prototype using frequency weighting to improve the approximation in the transition region filters designed using the proposed technique are compared with linear phase fir designs and to iir phase equalized designs with respect to computational complexity and group delay it is shown that for highly selective filters the proposed technique offers a good compromise

derivatives of horn hypergeometric functions with respect to their parameters the derivatives of eight horn hypergeometric functions four appell f1 f2 f3 and f4 and four degenerate confluent 1 2 1 and 1 with respect to their parameters are studied the first derivatives are expressed systematically as triple infinite summations or alternatively as single summations of two variable kampe de feriet functions taking advantage of previously established expressions for the derivative of the confluent or gaussian hypergeometric functions the generalization to the nth derivative of horn s functions with respect to their parameters is rather straightforward in most cases the results are expressed in terms of n 2 infinite summations following a similar procedure mixed derivatives are also treated an illustration of the usefulness of the derivatives of f1 with respect to the first and third parameters is given with the study of autoionization of atoms occurring as part of a post collisional process their evaluation setting the coulomb charge to zero provides the

formulation and solution of structured total least norm problems for parameter estimation the total least squares tls method is a generalization of the least squares ls method for solving overdetermined sets of linear equations ax spl ap b the tls method minimizes spl par e r spl par sub f where r b a e x so that b r spl isin range a e given a spl isin c sup m spl times n with m spl ges n and b spl isin c sup m spl times 1 the most common tls algorithm is based on the singular value decomposition svd of a b however the svd based methods may not be appropriate when the matrix a has a special structure since they do not preserve the structure previously a new problem formulation known as structured total least norm stln and the algorithm for computing the stln solution have been developed the stln method preserves the special structure of a or a b and can minimize the error in the discrete l sub p norm where p 1 2 or spl infin in this paper the stln problem formulation is generalized for computing the solution of stln problems with multiple right hand sides ax spl ap b it is shown that these problems can be converted to ordinary stln problems with one right hand side in addition the method is shown to converge to the optimal solution in certain model reduction problems furthermore the application of the stln method to various parameter estimation problems is studied in which the computed correction matrix applied to a or a b keeps the same toeplitz structure as the data matrix a of a b respectively in particular the l sub 2 norm stln method is compared with the ls and tls methods in deconvolution transfer function modeling and linear prediction problems

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

balanced performance preserving controller reduction two methods for controller reduction based on the closed loop system performance preserving criterion are proposed they are the closed loop block balanced and the closed loop structurally balanced truncating methods reduced order controllers are obtained directly by solving the closed loop system reduction problem with performance preserving weights

singular perturbational model reduction in the frequency domain singular perturbational approximations for linear continuous time and discrete time systems are developed in the frequency domain it is shown that the familiar singular perturbational result is an approximation at the origin in the complex plane however if the system has multiple time scale effects other approximations can be obtained at different locations on the negative real axis to emphasize such behavior the relationship between singular perturbational approximations and direct subsystem elimination is also investigated

model reduction via balanced realizations an extension and frequency weighting techniques two model reduction methods for discrete systems related to balanced realizations are described the first is a technique which utilizes the least controllable and observable subsystem in deriving a balanced discrete reduced order model for this technique as l sup infinity norm bound on the reduction error is given the second method is a frequency weighting technique for discrete and continuous time systems where the input normal or output normal realizations are modified to include a simple frequency weighting for this technique l sup infinity norm bounds on the weighted reduction errors are obtained

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

exit probability in generalised kinetic ising model in this paper we study generalised ising glauber models with inflow of information in one dimension and derive expressions for the exit probability using well established analytical methods the analytical expressions agree very well with the results obtained from numerical simulation only when the interaction is restricted to the nearest neighbor but as the range of interaction is increased the analytical results deviate from simulation results systematically the reasons for the deviation as well as some related open questions are discussed

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

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

universal features of exit probability in opinion dynamics models with domain size dependent dynamics we study the exit probability for several binary opinion dynamics models in one dimension in which the opinion state represented by of an agent is determined by dynamical rules dependent on the size of its neighbouring domains in all these models we find the exit probability behaves like a step function in the thermodynamic limit in a finite system of size l the exit probability e x as a function of the initial fraction x of one type of opinion is given by with a universal value of the form of the scaling function is also universal where is found to be dependent on the particular dynamics the variation of against the parameters of the models is studied c is non zero only when the dynamical rule distinguishes between states comparison with theoretical estimates in this case shows very good agreement

bilinear transformation and generalized singular perturbation model reduction a new general bilinear relationship is found between continuous and discrete generalized singular perturbation reduced order models this result is applied to the problem of deriving discrete analogs of continuous singular perturbation and direct truncation model reduction and leads to a new definition of discrete nyquist model reduction also unit circle bilinear transformations are used to relate several known facts about continuous and discrete balanced model reduction and incorporate them into a symmetrical unified framework

conjugation interpolation and model matching in h abstract the notion of conjugation is introduced to give a unified framework of algebraic operations connected to the analytic aspects of transfer functions two classes of conjugations are shown to be important one is the lossless conjugation that performs inner outer factorization and the other is the j lossless conjugation that solves the model matching problem related to h 8 control the solution of the model matching problem obtained by j lossless conjugation is regarded as the state space version of the directional interpolation theory and is much simpler than those obtained via the nehari approximation

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

properties of a low frequency approximation balancing method of model reduction this correspondence describes a balancing technique of model reduction which is characterized by exactness at zero frequency various interesting properties and error bounds of the reduced order model are derived in a rigorous manner a critical comparison is made to moore s reduced model 1981 obtained by direct truncation an illustrative numerical example is included

singular perturbational approximations for discrete time balanced systems a new zero order singular perturbational type approximation is developed for model reduction of discrete time linear systems this approximation is also suitable for model reduction of systems which have fast subsystems and which are represented in the internally balanced format subsystem elimination as suggested by moore for continuous time systems does not generalize for the discrete time case and the singular perturbational approximation gives a particular solution to the discrete time internally balanced model reduction problem

hankel norm approximation for well posed linear systems the sub optimal hankel norm approximation problem is solved for a well posed linear system with generating operators a b c and transfer function g satisfying some mild assumptions in the special case of the sub optimal nehari problem an explicit parameterization of all solutions is obtained in terms of the system parameters a b c and g 0

parametric generalized singular perturbation approximation for model order reduction a new model reduction technique for the approximation of balanced realization is introduced the method involves a further generalization of the generalized singular perturbation approximation by adding several parameters that can be tuned according to a specified performance criterion an a priori bound can be computed guaranteeing the quality of the approximated model in the whole range of parameter variations two numerical examples conclude this paper

a new parametrization of h suboptimal controllers elementary state space concepts are used to derive a transparent solution to the h control problem on an infinite horizon the main contribution of this solution is a novel representation of suboptimal controllers in terms of a pair of parametric riccati equations with a coupling constraint unlike the classical parametrization in terms of linear fractional transformation this state space representation has a homogeneous riccati formulation which should help to make the most of the suboptimal controller diversity potential applications include the design of reduced order controllers and more generally the selection of suboptimal controllers to meet or optimize additional constraints

a stable state space realization in the formulation of h norm computation in the two block hinftyoptimization problem usually we are given the state space realizations of the proper rational matrices r 1 s and r 2 s whose poles are all the open right half plane two problems are studied in the note the first is the evaluation of phi s r 1 s at s s k k 1 2 n where phi s is an inner function whose zeros s k k 1 2 n are the poles of r 1 s this evaluation is essential if chang and pearson s method is used for computing the optimal hinftynorm the problem is solved in state space via the solutions of lyapunov equations neither polynomial matrix manipulations nor numerical pole zero cancellations are involved in the evaluation the second problem is to find a stable state space realization of s s u s r 2 s where u s is an inner matrix this problem arises in the spectral factorization of gamma 2 r 2 ast r 2 doyle and chu had a method for constructing stable s s based on a minimal realization of r 2 s an alternate method is proposed the alternate method does not require a minimal realization of r 2 s and only a lyapunov equation is involved

a structured approximation problem with applications to frequency weighted model reduction the authors consider the problem of finding necessary and sufficient conditions for the existence of a q with no more than k poles in the closed left half plane such that the norm of a certain matrix

model reduction for state space symmetric systems in this paper the model reduction problem for state space symmetric systems is investigated first it is shown that several model reduction methods such as balanced truncation balanced truncation which preserves the dc gain optimal and suboptimal hankel norm approximations inherit the state space symmetric property furthermore for single input and single output siso state space symmetric systems we prove that the h norm of its transfer functions can be calculated via two simple formulas moreover the siso state space symmetric systems are equivalent to systems with zeros interlacing the poles zip under mild conditions

directional interpolation approach to h optimization and robust stabilization it is shown that under a mild condition h infty optimization and robust stabilization are equivalent to a directional interpolation problem which is a matrix extension of the classical pick nevanlinna interpolation problem a classical iterative method which is an extension of the schur nevanlinna algorithm is given for solving the problem this method does not require the inner outer factorization nor the balanced realization of the original plant a circuit theoretical parameterization of all solutions is derived that is expected to enhance the physical insight to the h infty optimal control and robust stabilization this parameterization has the degree much less than the one obtained previously

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

convolution and hankel operator norms for linear systems some norms are derived for convolution and hankel operators associated with linear time invariant systems in certain cases these norms are shown to be identical the tightest possible bound has been obtained for the absolute magnitude of the euclidean 2 or infinity norm of the time domain response of a multioutput system to certain classes of input disturbance

optimal sensitivity theory for multivariate distributed plants in this note we generalized the results of foias and tannenbaum 1986 a and foias et al 1987 on optimal sensitivity minimization to a much broader class of distributed plants we relate our results to certain kinds of hankel norm approximation problems that frequently appear in control engineering applications

expansion around half integer values binomial sums and inverse binomial sums i consider the expansion of transcendental functions in a small parameter around rational numbers this includes in particular the expansion around half integer values i present algorithms which are suitable for an implementation within a symbolic computer algebra system the method is an extension of the technique of nested sums the algorithms allow in addition the evaluation of binomial sums inverse binomial sums and generalizations thereof

harmonic sums and polylogarithms generated by cyclotomic polynomials the computation of feynman integrals in massive higher order perturbative calculations in renormalizable quantum field theories requires extensions of multiply nested harmonic sums which can be generated as real representations by mellin transforms of poincare iterated integrals including denominators of higher cyclotomic polynomials we derive the cyclotomic harmonic polylogarithms and harmonic sums and study their algebraic and structural relations the analytic continuation of cyclotomic harmonic sums to complex values of n is performed using analytic representations we also consider special values of the cyclotomic harmonic polylogarithms at argument x 1 resp for the cyclotomic harmonic sums at n which are related to colored multiple zeta values deriving various of their relations based on the stuffle and shuffle alge bras and three multiple ar gument relations we also consider infinite generalized nested harmonic sums at roots of unity which are related to the infinite cyclotomic harmonic sums basis representations are derived for weight w 1 2 sums up to cyclotomy l 20

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

robust reduced order modeling via the optimal projection equations with peterson hollot bounds an optimal reduced order modeling problem with parametric plant uncertainty is considered a model reduction bound suggested by recent work of i r petersen and c v hollot is utilized for guaranteeing robust reduced order modeling over a specified range of uncertain parameters necessary conditions which generalize the optimal projection equations for model reduction are used to characterize the reduced order model which minimizes the model reduction bound the optimality equations thus effectively serve as sufficient conditions for characterizing robust reduced order models