unified approach to h optimization hankel approximation and balanced realization problems a unified approach is proposed here to solve three problems simultaneously namely h optimization hankel approximation and minimal balanced realization for scalar systems it is shown that all these problems can be reduced to the same standard form wherein merely the singular value decomposition svd of an easily constructed square matrix is required to establish the final solutions the formulation is solely in terms of the coefficients of a transfer function and the resulting algorithm can be performed in a very computationally efficient way

one step extension approach to optimal hankel norm approximation and h sup infinity optimization problems a methodology is presented for hankel approximation and h sup infinity optimization problems that is based on a new formulation of a one step extension problem which is solved by the sarason interpolation theorem the parameterization of all optimal hankel approximants for multivariable systems is given in terms of the eigenvalue decomposition of an hermitian matrix composed directly from the coefficients of a given transfer function matrix phi rather than starting with the state space realization of phi the authors use polynomial coefficients of phi as input data in terms of these data a natural basis is given for the finite dimensional sarason model space and all computations involve only manipulations with finite matrices

the singular value decomposition of an infinite hankel matrix abstract let h be an infinite hankel matrix of known finite rank r a new algorithm for the numerical calculation of the singular values and vectors of h is presented the method proceeds by reduction to the singular value problem for an r r matrix this is achieved without solving for the poles of the symbol of h the resulting algorithm is of order r 3

robust stabilization of linear multivariable systems relations to approximation the stabilization of a linear system modelled as g a where g is a known rational transfer function and a is a perturbation satisfying w a2 g1 or equivalently there does not exist g with fewer poles in c than g such that w1 g g w2 a simple characterization of all robustly stabilizing controllers is then derived and state space formulae for maximally robust controllers are given finally reduced order controllers are considered

an efficient algorithm on optimal hankel norm approximation for multivariable systems this note provides a novel methodology for hankel approximation and h sup infinity optimization problems based on a new formulation of the one step extension problem which is solved by the sarason interpolation theorem the present method does not require an initial balanced realization and the parameterization of all optimal solutions is given in terms of the eigenvalue decomposition of a hermitian matrix composed directly from the coefficients of the given transfer function matrix

brief nonlinear observer design via passivation of error dynamics we present a new design scheme of nonlinear state observers global full order asymptotic observers through passivation of the error dynamics in order to consider passivity of the error dynamics for the observer problem we place a conceptual input and output on the generalized error dynamics which also includes the plant and the strictness of passivity is extended with respect to a set in which the estimation error becomes zero then output feedback passivation for the error dynamics will lead to the construction of a state observer it is also shown that a nonlinear observer is generally vulnerable to measurement disturbance in the sense that even an arbitrarily small measurement disturbance can lead to a blowup of the error state however due to the passivity of the error dynamics the proposed nonlinear injection gain can be easily modified for the observer to be robust to measurement disturbances

a mixed multiscale finite element method for elliptic problems with oscillating coefficients the recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large scale structure of the solutions without resolving all the fine scale structures motivated by the numerical simulation of flow transport in highly heterogeneous porous media we propose a mixed multiscale finite element method with an over sampling technique for solving second order elliptic equations with rapidly oscillating coefficients the multiscale finite element bases are constructed by locally solving neumann boundary value problems we provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain the asymptotic structure of the solutions numerical experiments are carried out for flow transport in a porous medium with a random log normal relative permeability to demonstrate the efficiency and accuracy of the proposed method

gramian matrices and balanced model of generalized systems in this work the generalization of reachability and observability gramian matrices of control generalized systems have been studied the properties of these matrices to be solutions of lyapunov equations have been analyzed in addition an algorithm to obtain balanced generalized systems is given

h sup infinity control for nonlinear systems with output feedback the basic question of nonlinear h sup infinity control theory is to decide for a given two port system when feedback that makes the full system dissipative and internally stable exists this problem can also be viewed as a question about circuits and after translation also has a game theoretic statement several necessary conditions for solutions to exist are presented and sufficient conditions for a certain construction to lead to a solution are given

multiscale finite element methods for high contrast problems using local spectral basis functions in this paper we study multiscale finite element methods msfems using spectral multiscale basis functions that are designed for high contrast problems multiscale basis functions are constructed using eigenvectors of a carefully selected local spectral problem this local spectral problem strongly depends on the choice of initial partition of unity functions the resulting space enriches the initial multiscale space using eigenvectors of local spectral problem the eigenvectors corresponding to small asymptotically vanishing eigenvalues detect important features of the solutions that are not captured by initial multiscale basis functions multiscale basis functions are constructed such that they span these eigenfunctions that correspond to small asymptotically vanishing eigenvalues we present a convergence study that shows that the convergence rate in energy norm is proportional to h 1 2 where is proportional to the minimum of the eigenvalues that the corresponding eigenvectors are not included in the coarse space thus we would like to reach to a larger eigenvalue with a smaller coarse space this is accomplished with a careful choice of initial multiscale basis functions and the setup of the eigenvalue problems numerical results are presented to back up our theoretical results and to show higher accuracy of msfems with spectral multiscale basis functions we also present a hierarchical construction of the eigenvectors that provides cpu savings

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

a multiscale reduced basis method for parametrized elliptic partial differential equations with multiple scales we present a technique for solving parametrized elliptic partial differential equations with multiple scales the technique is based on the combination of the reduced basis method c prud homme d rovas k veroy y maday a t patera g turinici reliable real time solution of parametrized partial differential equations reduced basis output bound methods journal of fluids engineering 124 1 2002 70 80 and the multiscale finite element method t y hou x h wu a multiscale finite element method for elliptic problems in composite materials and porous media journal of computational physics 134 1 1997 169 189 to treat problems in which the differential coefficient is characterized by a large number of independent parameters for the multiscale finite element method a large number of cell problems has to be solved at the fine local mesh for each new configuration of the differential coefficient in order to improve the computational efficiency of this method we construct reduced basis spaces that are adapted to the local parameter dependence of the differential operator the approximate solutions of the cell problems are computed accurately and efficiently via performing galekin projection onto the reduced basis spaces and implementing the offline online computational procedure therefore a large number of similar computations at the fine local mesh can be carried out with lower computational cost for each new configuration of the differential coefficient numerical results are provided to demonstrate the accuracy and efficiency of the proposed approach

technical communique adaptive control for nonlinear nonnegative dynamical systems nonnegative and compartmental models are widespread in engineering systems and life sciences and play a key role in the understanding of these systems in this paper we develop a direct adaptive control framework for nonlinear uncertain nonnegative and compartmental dynamical systems the proposed framework is lyapunov based and guarantees partial asymptotic set point regulation that is asymptotic set point regulation with respect to part of the closed loop system states associated with the plant in addition the adaptive controller guarantees that the physical system states remain in the nonnegative orthant of the state space

h optimal observer based controller design for nonlinear multivariable systems abstract we propose a novel algorithm to synthesize an h optimal observer based controller for a nonlinear multivariable system based on the parametrization of the observer based controller a necessary and sufficient condition to achieve robust stability is developed moreover the nevanlinna pick interpolation theory is employed to test the solvability condition of the robust stabilization problem optimal robustness can be achieved via the technique of h optimization the exact solution of the model matching problem induced by the stability criterion is solved by the method of noniterative computation of the h norm

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

disturbance attenuation and h sub infinity control via measurement feedback in nonlinear systems a solution to the problem of disturbance attenuation via measurement feedback with internal stability is presented for an affine nonlinear system it is shown that the concept of disturbance attenuation in the sense of truncated l sub 2 norms can be given an interpretation in terms of the response to periodic inputs in the sense of rms amplitude even in the nonlinear setup in the case of state feedback a family of controllers is also provided the proofs of all these results are simple and provide deeper insight even in the analysis of the corresponding linear control problem

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

nonlinear h almost disturbance decoupling the l2 gain almost disturbance decoupling problem for siso nonlinear systems is formulated sufficient conditions are identified for the existence of a parametrized state feedback controller such that the l2 gain from disturbances to output can be made arbitrarily small by increasing its gain the controller is explicity constructed using a lyapunov based recursive scheme sufficient conditions for the solvability of the h almost disturbance decoupling problem and the explicit construction of teh controller are given for a more restrictive class of nonlinear systems

generalized multiscale finite element methods gmsfem in this paper we propose a general approach called generalized multiscale finite element method gmsfem for performing multiscale simulations for problems without scale separation over a complex input space as in multiscale finite element methods msfems the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space in the proposed approach we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage the enrichment in the online stage is performed based on a spectral decomposition of the offline space in the online stage for any input parameter a multiscale space is constructed to solve the global problem on a coarse grid the online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues the computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re used for solving the forward problem with any forcing and boundary condition compared with the other approaches where global snapshots are used the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse grid level we present various examples in the paper and some numerical results to demonstrate the effectiveness of our method

a new algorithm for the minimal balanced realization of a transfer function matrix abstract a new algorithm is proposed here to obtain a minimal balanced realization directly from the transfer function matrix tfm this method which employs the singular value decomposition svd of an infinite block hankel matrix requires neither an initial minimal realization nor the solution of a lyapunov matrix equation the formulation is solely in terms of the coefficients of the transfer function matrix

brief paper explicit construction of h control law for a class of nonminimum phase nonlinear systems this paper addresses a nonlinear h control problem for a class of nonminimum phase nonlinear systems the given system is first transformed into a special coordinate basis in which the system zero dynamics is divided into a stable part and an unstable part a sufficient solvability condition is then established for solving the nonlinear h control problem moreover based on the sufficient solvability condition an upper bound of the best achievable l 2 gain from the system disturbance to the system controlled output is estimated for the nonlinear h control problem the proofs of our results yield explicit algorithms for constructing required control laws for solving the nonlinear h control problem in particular the solution to the nonlinear h control problem does not require solving any hamilton jacobi equations finally the obtained results are utilized to solve a benchmark problem on a rotational translational actuator rtac system

minimal balanced realization of transfer function matrices using markov parameters a new relation between the controllability observability matrix and gramian for any stable realization is used to enable the development of an algorithm for obtaining a minimal balanced realization directly from the transfer function matrix the algorithm employs the markov parameters and a positive definite symmetric matrix which is composed solely from the coefficients of the least common denominator polynomial of the transfer function matrix

balanced realization of stable transfer function matrices simple formulas are presented to compute the internally balanced minimal realization and the singular decomposition of the hankel operator of a given continuous time p m stable transfer function matrix e s d s the proposed formulas involve the schwarz numbers of d s and the singular eigenvalues eigenmatrices of a suitable finite matrix similar results are also obtained for a given discrete time transfer function matrix

output feedback stabilization of uncertain dynamical systems given an uncertain dynamical system with a linear nominal part matched uncertainties and a measured output a sufficient condition is given for the existence of an output feedback stabilizing controller for the general multiinput multioutput case checking the sufficient condition involves the selection of a matrix f for the special class of single input single output systems f is a positive or negative scalar and it is shown that if the nominal system transfer function is strictly positive real then an output feedback stabilizing controller does indeed exist

generalized finite element methods their performance and their relation to mixed methods the notion of a generalized finite element method is introduced this class of methods is analyzed and their relation to mixed methods is discussed the class of generalized finite element methods offers a wide variety of computational procedures from which particular procedures can be selected for particular problems a particular generalized finite element method which is very effective for problems with rough coefficients is discussed in detail

multiscale finite element method for heat transfer problem during artificial ground freezing abstract in this work we consider the heat transfer problem during artificial ground freezing we present the mathematical model and the fine grid approximation for heterogeneous porous media where freezing pipes are considered as line source terms mathematical model is described by a two phase stefan problem fine grid approximation is performed using a finite element method the main goal of the work is the construction of the coarse grid approximation using the generalized multiscale finite element method gmsfem in gmsfem we solve local spectral problem in order to determine main heat flow characteristics in the heterogeneous porous media and construct special additional multiscale basis functions in order to perform an accurate approximation of the line source terms freezing pipes we present numerical results for two heat transfer model problems in the two and three dimensional formulations and investigate accuracy of the proposed multiscale methods for different numbers of the multiscale basis functions

brief paper full order observer design for a class of port hamiltonian systems we consider a special class of port hamiltonian systems for which we propose a design methodology for constructing globally exponentially stable full order observers using a passivity based approach the essential idea is to make the augmented system consisting of the plant and the observer dynamics to become strictly passive with respect to an invariant manifold defined on the extended state space on which the state estimation error is zero we first introduce the concept of passivity of a system with respect to a manifold by defining a new input and output on the extended state space and then perform a partial state feedback passivation which leads to the construction of the observer we then illustrate this observer design procedure on two physical examples the magnetic levitation system and the inverted pendulum on the cart system

generalized multiscale finite element method symmetric interior penalty coupling motivated by applications to numerical simulations of flows in highly heterogeneous porous media we develop multiscale finite element methods for second order elliptic equations we discuss a multiscale model reduction technique in the framework of the discontinuous galerkin finite element method we propose two different finite element spaces on the coarse mesh the first space is based on a local eigenvalue problem that uses an interior weighted l 2 norm and a boundary weighted l 2 norm for computing the mass matrix the second choice is based on generation of a snapshot space and subsequent selection of a subspace of a reduced dimension the approximation with these multiscale spaces is based on the discontinuous galerkin finite element method framework we investigate the stability and derive error estimates for the methods and further experimentally study their performance on a representative number of numerical examples

a simple algorithm on minimal balanced realization for transfer function matrices given a transfer function matrix it is shown that its minimal balanced realization can be obtained directly from the singular values and the singular vectors of a constant matrix constructed from the coefficients of the least common denominator polynomial the algorithm is based on the close relationship of controllability and the observability gramians to the singular value decomposition of the associated infinite block hankel matrix

metric based upscaling we consider divergence form elliptic operators in dimension n ge 2 with l coefficients although solutions of these operators are only holder continuous we show that they are differentiable c1 with respect to harmonic coordinates it follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales this new numerical homogenization method is based on the transfer of a new metric in addition to traditional averaged homogenized quantities from subgrid scales into computational scales error bounds can be given and this method can also be used as a compression tool for differential operators 2006 wiley periodicals inc

multiscale finite volume method for compressible multiphase flow in porous media the multiscale finite volume msfv method has been recently developed and tested for multiphase flow problems with simplified physics i e incompressible flow without gravity and capillary effects and proved robust accurate and efficient however applications to practical problems necessitate extensions that enable the method to deal with more complex processes in this paper we present a modified version of the msfv algorithm that provides a suitable and natural framework to include additional physics the algorithm consists of four main steps computation of the local basis functions which are used to extract the coarse scale effective transmissibilities solution of the coarse scale pressure equation reconstruction of conservative fine scale fluxes and solution of the transport equations within this framework we develop a msfv method for compressible multiphase flow the basic idea is to compute the basis functions as in the case of incompressible flow such that they remain independent of the pressure the effects of compressibility are taken into account in the solution of the coarse scale pressure equation and if necessary in the reconstruction of the fine scale fluxes we consider three models with an increasing level of complexity in the flux reconstruction and test them for highly compressible flows tracer transport in gas flow imbibition and drainage of partially saturated reservoirs depletion of gas water reservoirs and flooding of oil gas reservoirs we demonstrate that the msfv method provides accurate solutions for compressible multiphase flow problems whereas slightly compressible flows can be treated with a very simple model a more sophisticate flux reconstruction is needed to obtain accurate fine scale saturation fields in highly compressible flows

feedback stabilization of general nonlinear control systems a passive system approach abstract this paper develops sufficient conditions for a general nonlinear control system 1 x f x u to be locally resp globally asymptotically stabilizable via smooth state feedback in particular it is shown that as in the case of affine systems this is possible if the unforced dynamic system of 1 is lyapunov stable and appropriate controllability like rank conditions are satisfied our results incorporate a series of well known stabilization theorems proposed in the literature for affine control systems and extend them to nonaffine nonlinear control systems

global stabilization of partially linear composite systems a linear stabilizable nonlinear asymptotically stable cascade system is globally stabilizable by smooth dynamic state feedback if a the linear subsystem is right invertible and weakly minimum phase and b the only linear variables entering the nonlinear subsystem are the output and the zero dynamics corresponding to this output both of these conditions are coordinate free and there is freedom of choice for the linear output variable this result generalizes several earlier sufficient conditions for stabilizability moreover the weak minimum phase condition for the linear subsystem cannot be relaxed unless a growth restriction is imposed on the nonlinear subsystem

time domain and frequency domain conditions for strict positive realness the author states various time domain and frequency domain conditions pertaining to multivariate strict positive real systems and establishes the relationship between these conditions their relationship is clarified by means of a three tier diagram each tier represents a set of equivalent conditions successively stronger than the lower tier the main theorem in the present study can be considered as the positive realness lemma for strictly positive real systems

generalized multiscale discontinuous galerkin method for solving the heat problem with phase change abstract in this work we consider a numerical solution of a heat transfer problem with phase change in heterogeneous domains for simulation of heat transfer processes with phase transitions we use a classic stefan model computational implementation is based on generalized multiscale discontinuous galerkin method gmsdgm in this method the interior penalty discontinuous galerkin method is used for the global coupling on a coarse grid the main idea of these methods is to construct a small dimensional local solution space that can provide an efficient calculation on coarse grid level we present numerical results for different geometries to demonstrate an accuracy of the method

multi scale finite volume method for elliptic problems in subsurface flow simulation in this paper we present a multi scale finite volume msfv method to solve elliptic problems with many spatial scales arising from flow in porous media the method efficiently captures the effects of small scales on a coarse grid is conservative and treats tensor permeabilities correctly the underlying idea is to construct transmissibilities that capture the local properties of the differential operator this leads to a multi point discretization scheme for the finite volume solution algorithm transmissibilities for the msfv have to be constructed only once as a preprocessing step and can be computed locally therefore this step is perfectly suited for massively parallel computers furthermore a conservative fine scale velocity field can be constructed from the coarse scale pressure solution two sets of locally computed basis functions are employed the first set of basis functions captures the small scale heterogeneity of the underlying permeability field and it is computed in order to construct the effective coarse scale transmissibilities a second set of basis functions is required to construct a conservative fine scale velocity field the accuracy and efficiency of our method is demonstrated by various numerical experiments

exponential feedback passivity and stabilizability of nonlinear systems abstract motivated by n krasovskii s characterisation of exponential stability the concept of exponential passivity is introduced it is shown that to make a nonlinear system with factorisable high frequency gain matrix exponentially passive via either state or output feedback exponential minimum phaseness and invertibility conditions are necessary and sufficient these conditions also guarantee exponential output feedback stabilisability this result extends previous results concerning linear systems

passivity and disturbance attenuation via output feedback for uncertain nonlinear systems the authors address the problem of disturbance attenuation with internal stability via output feedback for a class of passive systems with uncertainties the problem is approached by means of adaptive output feedback control which does not require any state observer the results obtained extend an earlier result of steinberg and corless 1985 sufficient conditions are proposed under which a nonlinear system can be made locally or globally passive via output feedback

domain decomposition for multiscale pdes we consider additive schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic pdes with highly variable coefficients in contrast to standard analyses we do not assume that the coefficients can be resolved by a coarse mesh this situation arises often in practice for example in the computation of flows in heterogeneous porous media in both the deterministic and monte carlo simulated stochastic cases we consider preconditioners which combine local solves on general overlapping subdomains together with a global solve on a general coarse space of functions on a coarse grid we perform a new analysis of the preconditioned matrix which shows rather explicitly how its condition number depends on the variable coefficient in the pde as well as on the coarse mesh and overlap parameters the classical estimates for this preconditioner with linear coarsening guarantee good conditioning only when the coefficient varies mildly inside the coarse grid elements by contrast our new results show that with a good choice of subdomains and coarse space basis functions the preconditioner can still be robust even for large coefficient variation inside domains when the classical method fails to be robust in particular our estimates prove very precisely the previously made empirical observation that the use of low energy coarse spaces can lead to robust preconditioners we go on to consider coarse spaces constructed from multiscale finite elements and prove that preconditioners using this type of coarsening lead to robust preconditioners for a variety of binary i e two scale media model problems moreover numerical experiments show that the new preconditioner has greatly improved performance over standard preconditioners even in the random coefficient case we show also how the analysis extends in a straightforward way to multiplicative versions of the schwarz method

feedback passivation of discrete time systems under communication constraints passivity is a desirable property of a dynamical system because it implies stability and is invariant under negative feedback and parallel interconnections feedback passivation is the process of making a nonpassive system passive through feedback control in this technical note we study the problem of feedback passivation when the controller has limited information about the state of the plant nonlinear plants that are linear in the control inputs are considered the main result of the technical note is a certainty equivalence principle any state feedback controller that ensures closed loop input strict passivity with index mu using the exact state of the plant will also ensure closed loop stochastic quasi passivity using an estimate of the state provided that the infinity norm of the estimation error process is bounded by some function of mu a corollary is that for linear systems although passivity is more strict than stability feedback passivation does not place more constraints on the estimation error and hence does not demand more from the communication channel than mean square stabilization