optimal h model reduction via linear matrix inequalities continuous and discrete time cases necessary and sufficient conditions are derived for the existence of a solution to the continuous time and discrete time h model reduction problems these conditions are expressed in terms of linear matrix inequalities lmis and a coupling non convex rank constraint set in addition an explicit parametrization of all reduced order models that correspond to a feasible solution is provided in terms of a contractive matrix these results follow from the recent solution of the h control design problem using lmis particularly simple conditions and a simple parametrization of all solutions are obtained for the zeroth order h approximation problem and the convexity of this problem is demonstrated computational issues are discussed and an iterative procedure is proposed to solve the h model reduction problem using alternating projections although global convergence of the algorithm is not guaranteed

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

simultaneous eigenvalue lower bounds for the lyapunov matrix equation simultaneous eigenvalue lower bounds for the solution of the lyapunov equation are presented the n summation and the n product bounds generalize existing bounds such as those for the trace and the determinant these bounds are stronger than the majority of the relevant bounds shown in the literature

upper summation and product bounds for solution eigenvalues of the lyapunov matrix equation upper bounds for summations including the trace and for products including the determinant of the eigenvalues of the solution of the continuous algebraic lyapunov matrix equation are presented the majority of the bounds are tighter than those in the literature and some are new

h model reduction for discrete time singular systems this paper investigates the problem of h sub infinity model reduction for linear discrete time singular systems without decomposing the original system matrices necessary and sufficient conditions for the solvability of this problem are obtained in terms of linear matrix inequalities lmis and a coupling non convex rank constraint set when these conditions are feasible an explicit parametrization of the desired reduced order models is given particularly a simple lmi condition without rank constraint is derived for the zeroth order h sub infinity approximation problem finally an illustrative example is provided to demonstrate the applicability of the proposed approach

on response of nonlinear oscillators with random frequency of excitation abstract some types of nonlinear oscillators for which the frequency of excitation is stochastic are investigated the paper consists of two parts in the first part equations of motion are linearized with the aid of stochastic averaging differential equations for the mean and variance of the process are obtained and integrated numerically this approach is applicable for weakly nonlinear oscillators the case of strong nonlinearity is considered in the second part making use of computer simulation a number of stochastic realizations of the process are computed the stochastic process is characterized by the mean and standard deviation of these realizations calculations have been carried out for the duffing ueda and van der pol equations and for forced vibrations of a pendulum these calculations show that if attractors exist then the deterministic vibrations which may be chaotic turn regular by adding noise and the motion terminates in a stable fixed point or on a limit cycle

internal positivity preserved model reduction this article studies model reduction of continuous time stable positive linear systems under the hankel norm h norm and h 2 norm performance the reduced order systems preserve the stability as well as the positivity of the original systems this is achieved by developing new necessary and sufficient conditions of the model reduction performances in which the lyapunov matrices are decoupled with the system matrices in this way the positivity constraints in the reduced order model can be imposed in a natural way as the model reduction performances are expressed in linear matrix inequalities with equality constraints the desired reduced order positive models can be obtained by using the cone complementarity linearisation iterative algorithm a numerical example is presented to illustrate the effectiveness of the given methods

an extension to balanced truncation with application to structured model reduction balanced truncation is a standard method for model reduction possessing many desirable properties such as preservation of model stability and a priori error bounds balanced truncation is conducted using controllability and observability gramians generalized gramians can be found by solving a set of linear matrix inequalities in this paper we show that these linear matrix inequalities can be extended so that the number of decision variables are at least doubled leading to the concept of extended gramians herein it is shown that the desirable properties of balanced truncation also hold with extended gramians the extended gramians are especially useful for improving error bounds and for models possessing additional structural constraints as is demonstrated by means of examples herein

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

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

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

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

model reduction of a class of discrete time nonlinear systems abstract this paper considers the problem of model reduction of a class of discrete time systems subject to lipschitzian nonlinearities it is shown that under some conditions the nonlinear system can be either approximated by a discrete time linear time invariant system or a nonlinear system of reduced order the computation of the matrices of the reduced order system is carried out through the solutions of a set of linear matrix inequalities the proposed design is approved by the simulation of reduced order dynamics of a mass spring system subject to a nonlinear friction and a linear electric circuit with uncertain parameters

model reduction of multidimensional and uncertain systems model reduction methods are presented for systems represented by a linear fractional transformation on a repeated scalar uncertainty structure these methods involve a complete generalization of balanced realizations balanced gramians and balanced truncation model reduction with guaranteed error bounds based on solutions to a pair of linear matrix inequalities which generalize lyapunov equations the resulting reduction methods immediately apply to uncertainty simplification and state order reduction in the case of uncertain systems but also may be interpreted as state order reduction for multidimensional systems

analysis of bifurcations for non linear stochastic non smooth vibro impact system via top lyapunov exponent bifurcations are discussed by the criterion of top lyapunov exponent based on the local map and kaminski s algorithms a general formulation of the top lyapunov exponents is proposed for non linear vibro impact oscillators with gaussian white noise perturbation the analytical results are verified by phase portraits and bifurcation diagrams for a classical stochastic duffing vibro impact oscillator both results are consistent

model reduction in numerical heat transfer problems abstract four recent model reduction schemes namely balanced truncation singular perturbation balanced truncation hankel norm approximation and h norm approximation are applied to a two dimensional transient heat conduction problem the approximations of these four model reduction methods for step response in temperature variation were compared with the finite difference model it is found that a substantial model order reduction for heat transfer problem is possible the results are encouraging and attractive especially for the purpose of simulation and control problems

h model reduction of markovian jump linear systems abstract in this paper the h model reduction problem for linear systems that possess randomly jumping parameters is studied the development includes both the continuous and discrete cases it is shown that the reduced order models exist if a set of matrix inequalities is feasible an effective iterative algorithm involving linear matrix inequalities is suggested to solve the matrix inequalities characterizing the model reduction solutions using the numerical solutions of the matrix inequalities the reduced order models can be obtained an example is given to illustrate the proposed model reduction method

a computational scheme for h spl infin model reduction in this paper a computational scheme for the h spl infin model reduction problem is introduced the scheme is illustrated by three examples the algorithm is based on the recent characterization of the solution to the h spl infin model reduction problem a suboptimal computational procedure for the general multivariable continuous time rth order optimal h sub spl infin model reduction problem is developed

ltv model reduction with upper error bounds a new approach for computing upper error bounds for reduced order models of linear time varying systems is presented it is based on a transformation technique of the hankel singular values using positive real odd incremented functions by applying such time varying functions the singular values to be removed can be forced to become equal and constant so that they can be reduced two variations of this method are proposed one for finite time horizons and the other for infinite time problems including periodic systems

characterization and computation of the solution to the optimal l sub spl infin approximation problem the characterization of the solution to the problem of approximating a given stable proper rational transfer function in l sub spl infin norm sense by a rational function with prescribed number of stable and unstable modes is developed a simple state space suboptimal computational algorithm for the solution is presented

a gramian based approach to model reduction for uncertain systems the technical note considers a problem of model reduction for a class of uncertain systems with structured norm bounded uncertainty the technical note introduces controllability and observability gramians in terms of certain parameterized algebraic riccati inequalities based on these gramians three model reduction approaches are investigated for the underlying uncertain systems

h sub spl infin norm approximation of systems by constant matrices and related results in this work the h sub spl infin norm approximation of a given stable proper rational transfer function by a constant matrix is considered a new improved bound on h sub spl infin norm of a given stable transfer function is derived optimal approximation for relaxation systems is given

lyapunov exponent and chaos of duffing s equation perturbed by white noise the paper presents a method for computing the top lyapunov exponents of duffing s equation perturbed by white noise it is based on an extension of wedig s algorithm for linear stochastic systems the variation of the exponents with damping and harmonic forcing amplitudes is investigated it is found that for the range of forcing amplitude and damping considered noise tends to stabilize the system this effect is more pronounced for the case of varying harmonic forcing amplitude

lower bounds on the solution of lyapunov matrix and algebraic riccati equations in this note various lower bounds for all the eigenvalues of the solution matrix k of the lyapunov matrix equation are established a special case of this result is a generalization of that presented in 1 3 where lower bounds for the maximum and minimum eigenvalues of k are given moreover the approach used here enables one to establish various lower bounds for some of the largest eigenvalues of the solution matrix of the algebraic riccati equation

improved results on h model reduction for continuous time linear systems over finite frequency ranges this paper revisits the h model reduction problem for continuous time linear systems over finite frequency ranges given an asymptotically stable system our goal is to find a stable reduced order system in such a way that the error of the transfer functions between the original system and the reduced order one is bounded over a finite frequency range by virtue of the generalized kalman yakubovich popov gkyp lemma we first establish necessary and sufficient characterizations for this problem in terms of linear matrix inequalities lmis for the low and mid frequency cases through introducing a non conservative multiplier and resorting to the projection lemma the reduced order system matrices are decoupled with the matrix variables from the gkyp lemma then by introducing a new diagonal matrix variable and based on congruence transformation the reduced order system matrices are further decoupled with the matrix variable induced by the projection lemma and can be parameterized by a new matrix variable the results are extended to the high frequency case without the use of projection lemma to reduce the conservatism moreover an iterative convex optimization algorithm is developed to solve the conditions finally we demonstrate via numerical examples that our method can achieve much smaller approximation error than existing results

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

interpolatory h h model reduction abstract we introduce an approach to h model reduction that is founded on ideas originating in realization theory interpolatory h 2 optimal model reduction and complex chebyshev approximation within this new framework we are able to formulate a method that remains effective in large scale settings with the main cost dominated by sparse linear solves by employing loewner data driven partial realizations within each optimization cycle computationally demanding h norm calculations can be completely avoided several numerical examples illustrate that our approach will produce high fidelity reduced models consistently exhibiting better h performance than those produced by balanced truncation these models often are as good as and occasionally better than those models produced by optimal hankel norm approximation in all cases reduced models are produced at far lower cost than is possible either with balanced truncation or with optimal hankel norm approximation

induced l 2 norm model reduction of polytopic uncertain linear systems abstract in this paper we study an induced l 2 norm model reduction problem for polytopic uncertain linear systems the polytopic system has its state space data contained in a convex polytope a situation which often arises the problem we address is to approximate this type of system by a lower order polytopic uncertain linear system with a guaranteed induced l 2 norm error a sufficient solvability condition is provided in terms of lmis with one extra coupling rank constraint which generally leads to a nonconvex feasibility problem we use an improved computational scheme based on the alternating projection method to obtain solutions to the problem but only local convergence is guaranteed the reduced order uncertain system with its associated polytope is given explicitly by the solution of the feasibility problem

bounds for the eigenvalues of the solution of the discrete riccati and lyapunov equations and the continuous lyapunov equation we present some bounds for the eigenvalues and certain sums and products of the eigenvalues of the solution of the discrete riccati and lyapunov matrix equations and the continuous lyapunov matrix equation nearly all of our bounds for the discrete riccati equation are new the bounds for the discrete and continuous lyapunov equations give a completion of some known bounds for the extremal eigenvalues and the determinant and the trace of the solution of the respective equation