passivity preserving model reduction by analytic interpolation abstract antoulas and sorensen have recently proposed a passivity preserving model reduction method of linear systems based on krylov projections the idea is to approximate a positive real rational transfer function with one of lower degree the method is based on an observation by antoulas in the single input single output case that if the approximant is preserving a subset of the spectral zeros and takes the same values as the original transfer function in the mirror points of the preserved spectral zeros then the approximant is also positive real however this turns out to be a special solution in the theory of analytic interpolation with degree constraint developed by byrnes georgiou and lindquist namely the maximum entropy central solution by tuning the interpolation points and the spectral zeros as prescribed by this theory one is able to obtain considerably better reduced order models we also show that in the multi input multi output case sorensen s algorithm actually amounts to tangential nevanlinna pick interpolation

eigenvalue based characterization and test for positive realness of scalar transfer functions an eigenvalue based characterization of positive realness of transfer functions of a single input single output time invariant linear system is derived based on this characterization we propose an efficient computational procedure to determine if a given transfer function is positive real the input for this eigenvalue based test is any given not necessarily minimal state space representation of the linear system the test only involves standard matrix computations such as computing eigenvalues of a matrix or a matrix pencil results of numerical experiments are reported

passivity preserving model reduction via interpolation of spectral zeros abstract an algorithm is developed for passivity preserving model reduction of linear time invariant systems implementation schemes are described for both medium scale dense and large scale sparse applications the algorithm is based upon interpolation at selected spectral zeros of the original transfer function to produce a reduced transfer function that has the specified roots as its spectral zeros these interpolation conditions are satisfied through the computation of a basis for a selected invariant subspace of a certain block matrix which has the spectral zeros as its spectrum explicit interpolation is avoided and passivity of the reduced model is established instead through satisfaction of the necessary conditions of the positive real lemma it is also shown that this procedure indirectly solves the associated controllability and observability riccati equations and how to select the interpolation points to give maximal or minimal solutions of these equations from these a balancing transformation may be constructed to give a reduced model that is balanced as well as passive and stable

balanced parametrization of classes of linear systems canonical forms and parametrizations are presented for several sets of minimal systems of given dimension asymptotically stable systems allpass systems bounded real systems positive real systems minimum phase systems and the class of all minimal systems the approach is based on balancing techniques for these classes of systems applications of these results to hankel operators and model reduction are discussed

complexity of critical functions for hamiltonian systems transition to predominantly chaotic motion in hamiltonian systems with two degrees of freedom is described by a complicated function of frequency which is called the critical function a graph of this function is a fractal set with the local structure which is believed to depend only on the arithmetic nature of the frequency we calculated numerically fractal dimensions of this function for a few typical systems using the method of modular smoothing and an efficient algorithm for computation of the fractal dimensions the dimensions which measure the complexity of the fractal are indeed the same within the error bounds and are equal to the dimension of the exponent of the brjuno function which is a purely arithmetic function

h_2 model reduction of linear network systems by moment matching and optimization in this article we compute the reduced order stable approximation of a linear network system preserving the topology and optimal w r t the h 2 norm of the approximation error our approach is based on time domain moment matching where we optimize over families of parameterized reduced order models matching moments at arbitrary interpolation points the low order models are parametrized in the free parameters i e the elements of the input matrix and the interpolation matrix we formulate an optimization based problem with the h 2 norm of the error as the objective function and with structural and physical properties as the constraints the problem is nonconvex and we write it in terms of the gramians of the error system we propose two solutions the first solution assumes that the error system admits a block diagonal observability gramian allowing for a simple convex reformulation as semidefinite programming but at the cost of some performance loss we also derive the sufficient conditions to guarantee the block diagonalization of the gramian the second solution employs a gradient projection method for a smooth reformulation yielding the locally optimal interpolation points and free parameters the potential of the methods is illustrated on a positive network and a power network

critical functions for complex analytic maps critical functions measure the width of the domain of stability around a given fixed point or an invariant circle for complex analytic and area preserving maps the author discusses their dependence on the rotation number of the invariant curves and proposes some new methods to determine them based on the existence of critical points and on some properties of quasiconformal maps by means of the majorant series method some rigorous estimates are given for complex area preserving maps like the semistandard map and the modulated singular map in particular the author makes use of the brjuno function to interpolate critical maps and proves that the convergence of the brjuno function is a necessary and sufficient condition for the existence of an analytic invariant curve of a given rotation number the author also discusses the optimality of the rigorous bounds obtained

on the solution of the minimal rational interpolation problem it is shown that the reachability and the observability indices of appropriately defined pairs of matrices provide the key for solving the minimal rational bitangen tial interpolation problem

on the partial realization problem abstract in this paper we take a unified approach to the partial realization problem in which we seek to incorporate ideas from numerical linear algebra most of which were originally developed in other contexts we approach the partial realization problem from several different angles and explore the connections to such topics as factoriza tion of hankel matrices block tridiagonalization generalizations of the lanczos process for biorthogonalization the euclidean algorithm and the principal part con tinued fractions of arne magnus the pade table and the berlekamp masseyalgo rithm in this way we are able to clarify some previous results by rissanen kalman and others and place them in a broader context this leads to several results and concepts which we think are new our analysis is restricted to the scalar case but some definitions and formulations have been rigged to facilitate an extension to the matrix case

modular smoothing and kam tori abstract transformation properties of kam tori under the action of a modular group on frequencies are analysed and used for calculating the kam tori of a hamiltonian system with one and a half degrees of freedom first we derive a relation between the kam torus with a frequency v and the kam torus with the frequency v 1 v 1 v where 1 v is the integer part of 1 v given by a continuous function l1 v in the double limit v m n and k 0 where m n is a rational frequency and k is a perturbation parameter the function l1 v was introduced by buric percival and vivaldi and also gives the transformation properties of the critical parameter as a function of the frequency this function provides a zeroth order modular smoothing approximation to the invariant tori which is better than a perturbation expansion the first order corrections to this approximation can be calculated from a function r1 v that appears to be smooth

model reduction by moment matching for linear and nonlinear systems the model reduction problem for single input single output linear and nonlinear systems is addressed using the notion of moment a re visitation of the linear theory allows to obtain novel results for linear systems and to develop a nonlinear enhancement of the notion of moment this in turn is used to pose and solve the model reduction problem by moment matching for nonlinear systems to develop a notion of frequency response for nonlinear systems and to solve model reduction problems in the presence of constraints on the reduced model connections between the proposed results projection methods the covariance extension problem and interpolation theory are presented finally the theory is illustrated by means of simple worked out examples and case studies

a framework for the solution of the generalized realization problem in this paper we present a novel way of constructing generalized state space representations e a b c d of interpolants matching tangential interpolation data the loewner and shifted loewner matrices are the key tools in this approach

finite frequency model reduction of two dimensional digital filters this technical note is concerned with the model reduction problem of two dimensional 2 d digital filters over finite frequency ranges the 2 d digital filter is described by the fornasini marchesini local state space fm lss model with the aid of the generalized kalman yakubovich popov gkyp lemma for 2 d systems sufficient conditions for the finite frequency model reduction problem are derived compared with full frequency methods the proposed finite frequency method can get a better approximation performance over finite frequency ranges an example is given to demonstrate the effectiveness of the proposed method

a note on potential diagonalizability of crystalline representations let k 0 mathbb q p be a finite unramified extension g k 0 the galois group text gal overline mathbb q p k 0 we show that all crystalline representations of g k 0 with hodge tate weights subseteq 0 ldots p 1 are potentially diagonalizable

stability preserving rational approximation subject to interpolation constraints a quite comprehensive theory of analytic interpolation with degree constraint dealing with rational analytic interpolants with an a priori bound has been developed in recent years in this paper we consider the limit case when this bound is removed and only stable interpolants with a prescribed maximum degree are sought this leads to weighted h 2 minimization where the interpolants are parameterized by the weights the inverse problem of determining the weight given a desired interpolant profile is considered and a rational approximation procedure based on the theory is proposed this provides a tool for tuning the solution to specifications the basic idea could also be applied to the case with bounded analytic interpolants

model order reduction of mimo bilinear systems by multi order arnoldi method abstract in this paper we present a time domain model order reduction method for multi input multi output mimo bilinear systems by general orthogonal polynomials the proposed method is based on a multi order arnoldi algorithm applied to construct the projection matrix the resulting reduced model can match a desired number of expansion coefficient terms of the original system the approximate error estimate of the reduced model is given and we also briefly discuss the stability preservation of the reduced model in some cases additionally in combination with krylov subspace methods we propose a two sided projection method to generate reduced models which capture properties of the original system in the time and frequency domain simultaneously the effectiveness of the proposed methods is demonstrated by two numerical examples

interpolatory projection methods for parameterized model reduction we provide a unifying projection based framework for structure preserving interpolatory model reduction of parameterized linear dynamical systems i e systems having a structured dependence on parameters that we wish to retain in the reduced order model the parameter dependence may be linear or nonlinear and is retained in the reduced order model moreover we are able to give conditions under which the gradient and hessian of the system response with respect to the system parameters is matched in the reduced order model we provide a systematic approach built on established interpolatory mathcal h 2 optimal model reduction methods that will produce parameterized reduced order models having high fidelity throughout a parameter range of interest for single input single output systems with parameters in the input output maps we provide reduced order models that are optimal with respect to an mathcal h 2 otimes mathcal l 2 joint error measure the capabilities of these approaches are illustrated by several numerical examples from technical applications

exact results for an approximate renormalisation scheme and some predictions for the breakup of invariant ori abstract an approximate renormalisation scheme is derived for the breakup of invariant tori of arbitrary winding ratio in hamiltonian systems of one and a half degrees of freedom similar to that of escande and doveil it is a free semi group with two generators this scheme is solved exactly for its orbits stable manifolds unstable manifolds and critical set various results are found including a cantor set of universal fractal diagrams the robustness of noble tori and a scaling law for areas near critical circles

a partial pade via lanczos method for reduced order modeling abstract the classical lanczos process can be used to efficiently generate pade approximants of the transfer function of a given single input single output time invariant linear dynamical system unfortunately in general the resulting reduced order models based on pade approximation do not preserve the stability and possibly passivity of the original linear dynamical system in this paper we describe the use of partial pade approximation for reduced order modeling partial pade approximants have a number of prescribed poles and zeros while the remaining degrees of freedom are used to match the taylor expansion of the original transfer function in as many leading coefficients as possible we present an algorithm for computing partial pade approximants via suitable rank 1 updates of the tridiagonal matrices generated by the lanczos process numerical results for two circuit examples are reported

finite time balanced truncation for linear systems via shifted legendre polynomials abstract in this paper we present a finite time model order reduction method for linear systems via shifted legendre polynomials the main idea of the approach is to use finite time empirical gramians which are constructed from impulse responses by solving block tridiagonal linear systems to generate approximate balanced system for the large scale system the balancing transformation is directly computed from the expansion coefficients of impulse responses in the space spanned by shifted legendre polynomials without individual reduction of the gramians and a separate eigenvector solve then the reduced order model is constructed by truncating the states corresponding to the small approximate hankel singular values hsvs the stability preservation of the reduced model is briefly discussed and in combination with the dominant subspace projection method we modify the reduction procedure to alleviate the shortcomings of the above method which may unexpectedly lead to unstable systems even though the original one is stable furthermore the properties of the resulting reduced models are considered finally numerical experiments are given to demonstrate the effectiveness of the proposed methods

multiplicites modulaires et representations de rm gl sb 2 mathbf z sb p et de rm gal overline mathbf q sb p mathbf q sb p en ell p appendice par guy henniart sur l unicite des types pour rm gl sb 2

model order reduction with preservation of passivity non expansivity and markov moments a new model order reduction technique is presented which preserves passivity and non expansivity it is a projection based method which exploits the solution of linear matrix inequalities to generate a descriptor state space format which preserves positive realness and bounded realness in the case of both non singular and singular systems solving the linear matrix inequality can be replaced by equivalently solving an algebraic riccati equation which is known to be a more efficient approach a new algebraic riccati equation and a frequency inversion technique are also presented to specifically deal with the important singular case the preservation of markov moments is also guaranteed by the judicious choice of a projection matrix three pertinent examples comparing the present approach with positive real balanced truncation show the strength and accuracy of the present approach

steady state matching and model reduction for systems of differential algebraic equations the problem of model reduction for nonlinear differential algebraic systems is addressed using the notions of moment and of steady state response these notions are formally introduced for this class of systems and families of nonlinear differential algebraic reduced order models achieving moment matching with additional properties are presented stronger results for the special class of linear singular systems are provided two simple examples illustrate the proposed technique

renormalization method for computing the threshold of the large scale stochastic instability in two degrees of freedom hamiltonian systems an approximate renormalization procedure is derived for the hamiltonianh v x t v2 2 m cosx p cosk x t it gives an estimate of the large scale stochastic instability threshold which agrees within 5 10 with the results obtained from direct numerical integration of the canonical equations it shows that this instability is related to the destruction of kam tori between the two resonances and makes the connection with kam theory possible improvements of the method are proposed the results obtained forh allow us to estimate the threshold for a large class of hamiltonian systems with two degrees of freedom

nonlinear moment matching based model order reduction in this paper we present a time domain notion of moments for a class of single input single output nonlinear systems in terms of the evolution of the output of a generalized signal generator driven by the nonlinear system we also define a new notion of moment matching and present a family of nonlinear parametrized reduced order models that achieve moment matching we establish relations with existing notions of moment for nonlinear systems showing that the newly derived and the existing families of reduced order models that achieve nonlinear moment matching respectively are equivalent furthermore we compute the reduced order model that matches the moments at two chosen signal generators one exciting the input of the system and another driven by the system simultaneously we also present a family of models computed on the basis of a nonlinear extension of the petrov galerkin projection that achieve moment matching finally we specialize the results to the case of nonlinear input affine systems

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

model reduction of neutral linear and nonlinear time invariant time delay systems with discrete and distributed delays the problem of model reduction by moment matching for linear and nonlinear differential time delay systems is studied the class of models considered includes neutral differential time delay systems with discrete delays and distributed delays the description of moment is revisited by means of a sylvester like equation for linear time delay systems and by means of the center manifold theory for nonlinear time delay systems in addition the moments at infinity are characterized for both linear and nonlinear time delay systems parameterized families of models achieving moment matching are given the parameters can be exploited to derive delay free reduced order models or time delay reduced order models with additional properties e g interpolation at an arbitrary large number of points finally the problem of obtaining a reduced order model of an unstable system is discussed and solved

structure preserving h 2 optimal model reduction based on the riemannian trust region method this paper studies stability and symmetry preserving h 2 optimal model reduction problems of linear systems which include linear gradient systems as a special case the problem is formulated as a nonlinear optimization problem on the product manifold of the manifold of symmetric positive definite matrices and two euclidean spaces to solve the problem by using the trust region method the gradient and hessian of the objective function are derived furthermore it is shown that if we restrict our systems to gradient systems the gradient and hessian can be obtained more efficiently more concretely by symmetry we can reduce linear matrix equations to be solved in addition by a simple example we show that the solutions to our problem and a similar problem in some literature works are not unique and the solution sets of both problems do not contain each other in general also it is revealed that the attained optimal values do not coincide numerical experiments show that the proposed method gives a reduced system with the same structure with the original system although the balanced truncation method does not

two sided time domain moment matching for linear systems in this technical note we study the problem of computing reduced order models of order nu that match 2 nu moments of the given to be reduced system at two sets of nu interpolation points called two sided moment matching in the time domain framework we also study the time domain moment matching problem of the state space representations of the first order derivative of a transfer function based on this we show that the solution of the two sided time domain moment matching problem at non distinct nu interpolation points yields a unique nu order model that satisfies the following properties the model matches the moments of the given system at the interpolation points and the same property holds for the first order derivatives of the corresponding transfer functions

potentially semi stable deformation rings let k qp be a finite extension and gk gal k k the galois group of an alge braic closure k let f be a finite field of characteristic p and vf a finite dimensional f vector space equipped with a continuous action of gkthe study of the defor mation theory of galois representations was initiated by mazur ma who showed that if vf has no non trivial endomorphisms then it admits a universal deformation ring ryw after the work of wiles wi and the conjectures of fontaine mazur fm it became clear that for arithmetic applications it was important to understand cer tain quotients of ry corresponding to deformations satisfying certain conditions for example wiles uses deformations which arise from finite flat group schemes and the corresponding quotient of ryw was constructed by ramakrishna ra suppose that l k is a finite extension and let a b be integers it seems to be a kind of folklore conjecture that there should be a quotient of ry whose points in finite extensions of w f l p correspond to deformations of vf which become semi stable over l and have hodge tate weights in the interval a b this is closely related to the final conjecture of fo 3 special cases of this when vf is 2 dimensional are conjectured in the papers of fontaine mazur fm p 191 breuil conrad diamond taylor bcdt conj 1 1 1 and breuil m zard bm 2 2 2 4 the purpose of this paper is to prove such a result

a survey of model reduction by balanced truncation and some new results balanced truncation is one of the most common model reduction schemes in this note we present a survey of balancing related model reduction methods and their corresponding error norms and also introduce some new results five balancing methods are studied 1 lyapunov balancing 2 stochastic balancing 3 bounded real balancing 4 positive real balancing and 5 frequency weighted balancing for positive real balancing we introduce a multiplicative type error bound moreover for a certain subclass of positive real systems a modified positive real balancing scheme with an absolute error bound is proposed we also develop a new frequency weighted balanced reduction method with a simple bound on the error system based on the frequency domain representations of the system gramians two numerical examples are illustrated to verify the efficiency of the proposed methods

matrix nearness based guaranteed passive system approximation abstract in this paper we present a new approach towards global passive approximation in order to find a passive real rational transfer function g s that is an arbitrarily close approximation of the passive transfer function nearest to a non passive square transfer function h s it is based on existing solutions to pertinent matrix nearness problems it is shown that the key point in constructing the passive real rational transfer function g s is to find a good rational approximation of the well known ramp function over an interval defined by the minimum and maximum dissipation of h s the proposed algorithms rely on the stable anti stable decomposition of a given transfer function pertinent examples are given to show the scope and accuracy of the proposed algorithms

converse kam theory and practice we unify extend reinterpret and apply criteria of birkhoff 1 herman 9 mather 2 3 aubry et al 4 5 and newman and percival 6 for the nonexistence of invariant circles for area preserving twist maps the criteria enable one to establish regions of phase space through which no rotational invariant circles pass for families of maps the same can be done for regions of the combined space of phase points and parameters the criteria can be implemented rigorously on a computer and give a practical method of proving quite strong results as an example we present a computer program which proved that the standard map has no rotational invariant circles for any parameter value k 63 64

modular smoothing of action the central quantity in the theory of transport for hamiltonian systems and in particular the area preserving twist maps is the action of rotational periodic orbits usually this is a complicated discontinuous function of two arguments some perturbation parameter k and a rational rotation number denoted by we applied the idea of modular smoothing to this complicated fractal our main result is that all the information contained in the fractal can be retrieved from a set of continuous or smooth functions of one variable

sylvester equations and projection based model reduction in this paper we establish a connection between krylov subspace techniques for multipoint pade interpolation and the use of sylvester equations for constructing reduced order models we also briefly point out that this connection partly extends to adi type techniques and to the smith iteration for computing approximate solutions of lyapunov equations

positive real and bounded real balancing for model reduction of descriptor systems we present an extension of the positive real and bounded real balanced truncation model reduction methods to large scale descriptor systems these methods are based on balancing the solutions of the projected lur e matrix equations important properties of these methods are that respectively passivity and contractivity are preserved in the reduced order models and that there exist approximation error bounds we also discuss the numerical solution of the projected lur e equations numerical examples are given

families of moment matching based low order approximations for linear systems the problem of finding a framework for linear moment matching based approximation of linear systems is discussed it is shown that it is possible to define five equivalent notions of moments i e one from a complex s domain point of view two using krylov projections and two from a time domain perspective based on these notions classes of parameterized reduced order models that achieve moment matching are obtained we analyze the controllability and observability properties of the models belonging to each of the classes of reduced order models we find the subclasses of models of orders lower than the number of matched moments i e we find the sets of parameters that allow for pole zero cancellations to occur furthermore we compute the subclass of minimal order model s the minimal order is equal to half of the number of matched moments i e generically the largest number of possible pole zero cancellations is half of the number of matched moments finally we present classes of reduced order models that match larger numbers of moments based on interconnections between reduced order models

transformation properties of perturbation expansions around elliptic fixed points transformation properties of perturbation expansions of vibrational quasi periodic orbits upon the modular transformations of frequencies are studied using simple but nontrivial examples the siegel complex quadratic map and special solutions of the real area preserving quadratic map it is shown that the transformation properties are similar to those in the previously studied case of the rotational invariant tori except for some special features of the real map related to the atypical nature of 1 3 resonance in this case

chaotic boundary of a hamiltonial map abstract the half plane maps are a class of complex hamiltonial maps whose invariant curves at most fixed irrational frequencies can be obtained as a convergent taylor series expansion for these maps the boundary between regular ordered motion on invariant curves and irregular chaotic motion is given by the radius of convergence of the series the successive terms of the series oscillate wildly due to the presence of small divisors methods are presented for taming the series based on the conversion of the convergenceexponentc ln c into the integral of a continuous but nondifferentiable lambda function whose graph whows a similarity structure on small scales self similarity properties are illustrated for the chaotic boundary function c where v is the frequency

potential automorphy and change of weight we prove an automorphy lifting theorem for l adic representations where we impose a new condition at l which we call potentential diagonalizability this result allows for change of weight and seems to be substantially more exible than previous theorems along the same lines we derive several applications for instance we show that any irreducible totally odd essentially self dual regular weakly compatible system of l adic representations of the absolute galois group of a totally real eld is potentially automorphic and hence is pure and its l function has meromorphic continuation to the whole complex plane and satises the expected functional equation

controllability observability and the solution of ax xb c a closed form finite series representation of the unique solution x of the matrix equation ax xb c is developed using this representation the image kernel and rank of x are related to the controllable and unobservable subspaces of the a c and c b pairs respectively bounds on the rank of x are obtainedin terms of the dimensions of these subspaces in the case that c has unitary rank an exact calculation of rank x is made the generic rank of x with a b fixed and c generic is evaluated

solving the algebraic riccati equation with the matrix sign function abstract this paper presents some improvements to the matrix sign function algorithm for the algebraic riccati equation a simple reorganization changes nonsymmetric matrix inversions into symmetric matrix inversions scaling accelerates convergence of the basic iteration and yields a new quadratic formula for certain 2 by 2 algebraic riccati equations numerical experience suggests the algorithm be supplemented with a refinement strategy similar to iterative refinement for systems of linear equations refinement also produces an error estimate the resulting procedure is numerically stable it compares favorably with current schur vector based algorithms

a new result on passivity preserving model reduction abstract the problem of model reduction with preservation of passivity is investigated the approach is based on positive real interpolation and is inspired by the similarity between lowner and pick matrices the former are important in problems of general rational interpolation while the latter in problems of interpolation by positive real functions it follows that interpolation of the original set of data together with an appropriately defined mirror image set of data yields automatically positive real interpolants subsequently we show how this result can be implemented using a krylov projection procedure the ensuing model reduction method preserves stability and passivity and can be implemented efficiently for the large scale systems

universal small scale structure near the boundary of siegel disks of arbitrary rotation number abstract we present numerical evidence that for analytic maps of the complex plane with a siegel disk of any irrational rotation number and a quadratic critical point on its boundary the small scale structure near the critical point can be described asymptotically by a universal two parameter family we propose an explanation in terms of an ergodic attractor for a renormalisation operator