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

on a class of functionals invariant under a zn action abstract this document considers a system of ordinary differential equations of the form q v sub q t q f t where f and v are periodic in t v is periodic in the components of q q sub 1 q sub m and the mean value of f vanishes by showing that a corresponding functional is invariant under a natural z sub n action a simple variational argument yields at least n 1 distinct periodic solutions of more general versions of are also treated as is a class of neumann problems for semilinear elliptic partial differential equations

exact operator solution of the calogero sutherland model the wave functions of the calogero sutherland model are known to be expressible in terms of jack polynomials a formula which allows to obtain the wave functions of the excited states by acting with a string of creation operators on the wave function of the ground state is presented and derived the creation operators that enter in this formula of rodrigues type for the jack polynomials involve dunkl operators

a two level atom coupled to a two dimensional supersymmetric and shape invariant system models a class of bound state problems which represents the coupling of a two level atom with a two dimensional supersymmetric system involving two shape invariant potentials is introduced we study two models with different coupling hamiltonians

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

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

balanced realization and model order reduction for nonlinear systems based on singular value analysis this paper discusses balanced realization and model order reduction for both continuous time and discrete time general nonlinear systems based on singular value analysis of the corresponding hankel operators singular value analysis clarifies the gain structure of a given nonlinear operator here it is proved that singular value analysis of smooth hankel operators defined on hilbert spaces can be characterized by simple equations in terms of their states a balanced realization and model order reduction procedure is derived based on it and several important properties such as stability balanced form hankel norm controllability and observability of the original system are preserved the work improves the earlier results of k fujimoto and j m a scherpen ieee trans automat control 50 2005 pp 2 18 and then continues with new balancing and model reduction results

model reduction for a class of convergent nonlinear systems in this technical note a model reduction procedure is presented for nonlinear systems that can be decomposed into a feedback interconnection of a linear and nonlinear subsystem conditions for stability of the reduced order model and an error bound are given herein the input to state convergence property is exploited which proves to be useful in the definition and derivation of the error bound the results are illustrated by application to a nonlinear mechanical system

an algebraic approach to shape invariance with reflections we consider a realization of supersymmetric shape invariant quantum systems where generalized ladder operators are constructed through the association of the supersymmetric partner operators the parameter potential translation operator and the reflection operator the basic algebraic relations as well as the system eigenstates and energy eigenvalues are obtained as an illustration we apply the generalized formalism for two important shape invariant potential systems the scarf hyperbolic and the morse potentials

existence and multiplicity for solutions of neumann problem for semilinear elliptic equations the existence and multiplicity results are obtained for solutions of neumann problem for semilinear elliptic equations by the least action principle and the minimax methods respectively

sign changing solutions on a kind of fourth order neumann boundary value problem abstract in this paper we consider the fourth order neumann boundary value problem u 4 t 2 u t u t f t u t for all t 0 1 and subject to u 0 u 1 u 0 u 1 0 using the fixed point index and the critical group we establish the existence theorem of solutions that guarantees the problem has at least one positive solution and two sign changing solutions under certain conditions

the hahn superalgebra and supersymmetric dunkl oscillator models a supersymmetric extension of the hahn algebra is introduced this quadratic superalgebra which we call the hahn superalgebra is constructed using the realization provided by the dunkl oscillator model in the plane whose hamiltonian involves reflection operators in this realization the reflections act as grading operators and the odd generators are part of the schwinger dunkl algebra which is a two parameter extension of the bosonic construction the even part of the algebra is built from bilinears in the odd generators and satisfies the hahn algebra supplemented with involutions a family of supersymmetric dunkl oscillator models in n dimensions is also considered the hamiltonians of these supersymmetric models differ from ordinary dunkl oscillators by pure reflection terms in two dimensions the supersymmetric dunkl oscillator is seen to have the even part of the hahn superalgebra as the invariance algebra

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

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

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

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

balanced truncation of linear time varying systems in this paper balanced truncation of linear time varying systems is studied in discrete and continuous time based on relatively basic calculations with time varying lyapunov equations inequalities we are able to derive both upper and lower error bounds for the truncated models these results generalize well known time invariant formulas the case of time varying state dimension is considered input output stability of all truncated balanced realizations is also proven the method is finally successfully applied to a high order model

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

multiple solutions of a class of neumann problem for semilinear elliptic equations abstract the multiplicity results are obtained for solutions of the neumann problem for nonlinear elliptic equations with unbounded nonlinearity by the implicit function theorem and the morse theory

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

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

stochastic incremental h control for discrete time switched systems with disturbance dependent noise abstract in this paper the reverse mode dependent average dwell time reverse mdadt scheme incorporated with multiple incremental lyapunov functions are utilized to investigate the incremental h control for nonlinear discrete time stochastic switched systems sss the sufficient conditions in terms of reverse mdadt for the incremental h control are derived such that an incrementally globally asymptotically stable in the mean igasim system with a prescribed h disturbance attenuation level is obtained it is shown that an incremental h control could be attained when all of the subsystems are non igasim moreover when the discrete time sss is composed of some igasim and non igasim subsystems a reverse mdadt based relationship between these two kinds of subsystems is established such that an igasim discrete time sss with incremental l2 gain property is obtained two numerical examples are presented to demonstrate the effectiveness of the results

nonlinear input normal realizations based on the differential eigenstructure of hankel operators this paper investigates the differential eigenstructure of hankel operators for nonlinear systems first it is proven that the variational system and the hamiltonian extension with extended input and output spaces can be interpreted as the ga spl circ teaux differential and its adjoint of a dynamical input output system respectively second the ga spl circ teaux differential is utilized to clarify the main result the differential eigenstructure of the nonlinear hankel operator which is closely related to the hankel norm of the original system third a new characterization of the nonlinear extension of hankel singular values are given based on the differential eigenstructure finally a balancing procedure to obtain a new input normal output diagonal realization is derived the results in this paper thus provide new insights to the realization and balancing theory for nonlinear systems

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

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

data driven model reduction of monotone systems by nonlinear dc gains in this paper we develop data driven model reduction methods for monotone nonlinear control systems based on a nonlinear version of the dc gain the nonlinear dc gain is a function of the amplitude of the input and can be used to evaluate the importance of each state variable in fact the nonlinear dc gain is directly related to the infinity induced norm of the system as well as a notion of output reachability given the dc gain model reduction is performed by either truncating not so important state variables or aggregating state variables having similar importance under such truncation and clustering monotonicity and boundedness of the nonlinear dc gain are preserved moreover these two operations can be approximately performed based on simulation or experimental data alone this empirical model reduction approach is illustrated by an example of a gene regulatory network

nonlinear boundary value problems at resonance soit un espace de mesure fini et h l 2 l espace des fonctions a valeurs reelles de carre integrable mesurables de lebesgue sur on considere l equation lu r n u g u h ou h h l dom l h h est un operateur lineaire auto adjoint densement defini r n appartient au spectre l g r r est une fonction qui peut etre non lineaire on donne des resultats d existence generaux

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

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

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

successively iterative method of nonlinear neumann boundary value problems abstract two successively iterative sequences are constructed for computing solutions of the nonlinear neumann boundary value problems with time singularity the sequences start off with some constants main tool is the fixed point theorem of increasing operator on the order interval by considering convergence of the sequences we prove the existence of nontrivial sign changing solutions

the trigonometric rosen morse potential in the supersymmetric quantum mechanics and its exact solutions the analytic solutions of the one dimensional schrodinger equation for the trigonometric rosen morse potential reported in the literature rely upon the jacobi polynomials with complex indices and complex arguments we first draw attention to the fact that the complex jacobi polynomials have non trivial orthogonality properties which make them uncomfortable for physics applications instead we here solve the above equation in terms of real orthogonal polynomials the new solutions are used in the construction of the quantum mechanical superpotential

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

h sub infinity fixed lag smoothing filter for scalar systems the solution of the h sub infinity optimal linear fixed lag smoothing problem is considered using a polynomial matrix description for the discrete system the smoother is obtained from the solution of a linear equation and a spectral factorization calculation the pole zero properties of the optimal smoother are obvious in the polynomial representation and insights into the measurement noise rejection properties of the smoother are obtained allowance is made for both dynamic cost weighting and colored measurement noise the polynomial form of the smoothing filter may be incorporated into a self tuning algorithm and a simple adaptive smoother is discussed

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

model reduction for nonlinear systems with incremental gain or passivity properties in this paper model reduction techniques for a class of nonlinear systems are proposed specifically nonlinear systems are considered that can be decomposed as the feedback interconnection of a high order linear subsystem and a nonlinear subsystem of relatively low order allowing for the application of well developed reduction techniques for linear systems in this setting conditions are given under which internal stability as well as passivity or a bound on the l 2 gain are preserved for the reduced order nonlinear model additionally a priori error bounds are given in the derivation of the error bound an incremental gain or incremental passivity property of the nonlinear subsystem is shown to be instrumental additionally the techniques developed in this paper are applied in the scope of controller reduction as is illustrated by means of an industrial temperature control benchmark example

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

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

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

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

existence of critical point for abstract resonant problems with unbounded nonlinearities and applications to differential equations abstract in this paper using well known variational methods we prove a general abstract theorem on the existence of a critical point for resonant problems with unbounded nonlinearities and then apply it to ordinary and partial differential equations with generalized ahmad lazer paul conditions the abstract result contains several concrete results in the literature and can also be used to deal with some new cases for resonant differential equations

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

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

squaring down and the problems of almost zeros for continuous time systems abstract in this paper we construct precompensators which square down a system such that the resulting square system has the same unstable zeros and system gains we derive the minimal gain of the precompensator to achieve such a square system if the gain of this precompensator is too large then we derive an explicit trade off between the gain of the precompensator and the number of unstable zeros we allow the precompensator to introduce in the system

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

model reduction by differential balancing based on nonlinear hankel operators in this paper we construct balancing theory for nonlinear systems in the contraction framework first we define two novel controllability and observability functions via prolonged systems we analyze their properties in relation to controllability and observability and use them for so called differential balancing and its application to model order reduction one of the main contribution of this paper is showing that differential balancing has close relationships with the frechet derivative of the nonlinear hankel operator inspired by 3 we provide a generalization in order to have a computationally more feasible method moreover error bounds for model reduction by generalized balancing are provided

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

on existence and multiplicity of solutions of neumann boundary value problems for quasi linear elliptic equations abstract in the present paper the following neumann boundary value problem involving the p laplacian 1 1 p u f x u in u n 0 on is studied and some new existence and multiplicity results of solutions for the problem 1 1 are obtained

optimal hankel norm identification of dynamical systems abstract the problem of optimal approximate system identification is addressed with a newly defined measure of misfit between observed time series and linear time invariant models the behavioral framework is used as a suitable axiomatic setting for a non parametric introduction of system complexity and allows for a notion of misfit of dynamical systems that is independent of system representations the misfit function introduced here is characterized in terms of the induced norm of a hankel operator associated with the data and a co inner kernel representation of a model sets of pareto optimal models are defined as feasible trade offs between complexity and misfit of models and it is shown how all pareto optimal models are characterized as exact models of compressed data sets obtained from hankel norm approximations of data matrices this leads to new conceptual algorithms for optimal approximate identification of time series 1997 elsevier science ltd

model reduction for nonlinear systems by incremental balanced truncation in this paper the method of incremental balanced truncation is introduced as a tool for model reduction of nonlinear systems incremental balanced truncation provides an extension of balanced truncation for linear systems towards the nonlinear case and differs from existing nonlinear balancing techniques in the definition of two novel energy functions these incremental observability and incremental controllability functions form the basis for a model reduction procedure in which the preservation of stability properties is guaranteed in particular the property of incremental stability which provides a notion of stability for systems with nonzero inputs is preserved moreover a computable error bound is given next an extension towards so called generalized incremental balanced truncation is proposed which provides a reduction technique with increased computational feasibility at the cost of a potentially larger error bound the proposed reduction technique is illustrated by means of application to an example of an electronic circuit with nonlinear elements

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

coherent states for the harmonic oscillator representations of the orthosymplectic supergroup osp 1 2n r coherent states for the harmonic oscillator representations of the noncompact supergroup osp 1 2n r are introduced and the invariant integration measure is calculated by studying transformation properties of supercoset variables the generalized bogoliubov transformation for mixed systems of bosons and fermions is also obtained an example for the simple harmonic oscillator is given

a bosonized formulation of the supersymmetric quantum mechanics for shape invariant potential systems we consider the minimal bosonization realization of supersymmetric shape invariant systems where generalized supercharge operators are constructed using the partner supersymmetric operators the parameter potential translation formalism and the reflection operator we obtain the solution of the eigenvalue equation and study the quantum dynamics of the supersymmetric system including terms in the hamiltonian which are constructed using the combination of the bosonized supercharge operators the connections between the bosonized supersymmetric formalism the bose fermi transformation and the generalization of the deformed heisenberg algebra are discussed as an illustration we apply the generalized formalism for the case of the trigonometric rosen morse potential

generalized coherent squeezed and intelligent states for exactly solvable quantum systems and the analogue of the displacement and squeezing operators using an algebraic approach we construct purely coherent and squeezed states for a set of exactly solvable quantum confined systems assuming a generalization of both the ladder operator and the displacement operator methods showing the equivalence between the two procedures we obtain the analogue of the displacement and the squeeze operators and study their basic relations and their connections with the properties of the purely coherent and squeezed states that they produce in a second approach we construct generalized squeezed coherent and intelligent states for these systems considering the successive action of the displacement and the squeeze operators in the first case and the combined action of these operators in the second case we apply the generalized approach for shape invariant systems with potential parameters related by a translation and by a scaling and show that the presence of the generalizing functionals of the potential parameters in the formalism opens a large number of possibilities in the construction of these special quantum states

an algebraic q deformed form for shape invariant systems a quantum deformed theory applicable to all shape invariant bound state systems is introduced by defining q deformed ladder operators we show that these new ladder operators satisfy new q deformed commutation relations in this context we construct an alternative q deformed model that preserves the shape invariance property presented by the primary system q deformed generalizations of morse scarf and coulomb potentials are given as examples