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

model approximation for discrete time state delay systems in the t s fuzzy framework this paper is concerned with the problem of h model approximation for discrete time takagi sugeno t s fuzzy time delay systems for a given stable t s fuzzy system our attention is focused on the construction of a reduced order model which not only approximates the original system well in an h performance but is also translated into a linear lower dimensional system by applying the delay partitioning approach a delay dependent sufficient condition is proposed for the asymptotic stability with an h error performance for the error system then the h model approximation problem is solved by using the projection approach which casts the model approximation into a sequential minimization problem subject to linear matrix inequality lmi constraints by employing the cone complementary linearization algorithm moreover by further extending the results h model approximation with special structures is obtained i e delay free model and zero order model finally two numerical examples are provided to illustrate the effectiveness of the proposed methods

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

hankel norm approximation of linear systems with time varying delay continuous and discrete cases this paper investigates the problem of hankel norm model reduction for linear systems with time varying delay in the state for a given stable system our attention is focused on the construction of reduced order model which guarantees the corresponding error system to be asymptotically stable and has a specified hankel norm error performance two different approaches are proposed to solve this problem one casts the model reduction into a convex optimization problem by using a linearization procedure and the other is based on the cone complementarity linearization idea which casts the model reduction into a sequential minimization problem subject to linear matrix inequality constraints both continuous and discrete time cases are considered a numerical example is provided to show the effectiveness of the proposed theory

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 of discrete time interval type 2 t s fuzzy systems this paper addresses the model reduction problem of discrete time interval type 2 it2 takagi sugeno t s fuzzy systems which represent the discrete time nonlinear systems subject to uncertainty with the use of it2 fuzzy sets the uncertainty of the discrete time nonlinear system can be captured by the lower and upper membership functions for a given high order discrete time it2 t s fuzzy system the purpose is to find a lower dimensional system to approximate the original system to achieve the approximation performance an mathcal h infty norm is used to suppress the error between the original system and its simplified system by introducing a membership functions dependent technique and applying a convex linearization method a membership functions dependent condition which takes the information of membership functions into account is obtained to reduce the dimensions of system matrices and the number of fuzzy rules of the system all the obtained theorems are represented as in the form of linear matrix inequalities finally simulation results are demonstrated to show the effectiveness of the derived results

improved results on h model reduction for markovian jump systems with partly known transition probabilities abstract this paper is concerned with the model reduction problem for markovian jump systems with partly known transition probabilities by making use of an existing result on the convex properties of the transition probabilities an improved condition is first derived for the h performance analysis of the error system two different approaches one in terms of linear matrix inequalities lmis and another in the form of an iterative algorithm subject to lmi constraints are then proposed to find a reduced order model such that the h performance of the error system is bounded by a specified level the first approach is obtained using finsler s lemma and involves fewer decision variables than the existing method it is shown that the proposed methods cover some existing results as special cases finally the advantages of our results are clearly illustrated by a numerical example

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

localization for a class of one dimensional quasi periodic schrodinger operators we prove for small and satisfying a certain diophantine condition the operator h varepsilon 2 delta frac 1 2 pi cos 2 pi j alpha theta j in mathbb z has pure point spectrum for almost all a similar result is established at low energy for h frac d 2 dx 2 k 2 cos 2 pi x cos 2 pi alpha x theta providedk is sufficiently large

model simplification for switched hybrid systems abstract this paper is concerned with the problem of h model reduction for switched system which is an important class of hybrid systems frequently encountered in practical situations two sharply different approaches are proposed to solve this problem the first approach casts the model reduction into a convex optimization problem which is the first attempt to solve the model reduction problem by using linearization procedure the second one based on the cone complementarity linearization idea casts the model reduction problem into a sequential minimization problem subject to linear matrix inequality constraints both approaches have their own advantages and disadvantages concerning conservatism and computational complexity a numerical example illustrates the effectiveness of the proposed theories

positive measure spectrum for schrodinger operators with periodic magnetic fields we study schrodinger operators with periodic magnetic field in r2 in the case of irrational magnetic flux positive measure cantor spectrum is generically expected in the presence of an electric potential we show that even without electric potential the spectrum has positive measure if the magnetic field is a perturbation of a constant one

all controllers for the general h control problem lmi existence conditions and state space formulas this paper presents all controllers for the general h control problem with no assumptions on the plant matrices necessary and sufficient conditions for the existence of an h controller of any order are given in terms of three linear matrix inequalities lmis our existence conditions are equivalent to scherer s results but with a more elementary derivation furthermore we provide the set of all h controllers explicitly parametrized in the state space using the positive definite solutions to the lmis even under standard assumptions full rank etc our controller parametrization has an advantage over the q parametrization the freedom q a real rational stable transfer matrix with the h norm bounded above by a specified number is replaced by a constant matrix l of fixed dimension with a norm bound and the solutions x y to the lmis the inequality formulation converts the existence conditions to a convex feasibility problem and also the free matrix l and the pair x y define a finite dimensional design space as opposed to the infinite dimensional space associated with the q parametrization

model reduction of delay systems using pade approximants this paper describes the rational approximation of a certain class of delay systems in the frequency domain using pad approximants of exp st three classes of approximants characterized by their relative degrees are considered easily computable a priori l and l l2error bounds are provided

delay dependent robust stabilization of uncertain state delayed systems this paper concerns a problem of robust stabilization of uncertain state delayed systems a new delay dependent stabilization condition using a memoryless controller is formulated in terms of matrix inequalities an algorithm involving convex optimization is proposed to design a controller guaranteeing a suboptimal maximal delay such that the system can be stabilized for all admissible uncertainties

modelling of weakly bonded laminated composite plates at large deflections abstract based on a general representation of displacement variation through the thickness of laminated plates a karman type nonlinear theory of laminated composite plates with weakened interfacial bonding is developed each weakly bonded interface is modelled by a spring layer model which has recently been used efficiently in the field of micromechanics of composites this spring layer model allows for a discontinuous distribution of displacements but requires the tractions to be continuous across each interface of adjacent layers the set of governing equations has variable coefficients in the most general form of bonding and includes conventional third order zigzag nonlinear theory of karman type for laminated composite plates as a special case when extreme values of interface parameters are used some simple numerical examples allowing for a closed form solution are presented to give an understanding of how a small amount of interfacial weakness affects the overall and local behaviour of laminated composite plates these include the important practical problem of reduced interface stresses due to weakened interfacial bonding which can be predicted by the theory presented herein

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

stabilization of nonlinear systems under variable sampling a fuzzy control approach this paper investigates the problem of stabilization for a takagi sugeno t s fuzzy system with nonuniform uncertain sampling the sampling is not required to be periodic and the only assumption is that the distance between any two consecutive sampling instants is less than a given bound by using the input delay approach the t s fuzzy system with variable uncertain sampling is transformed into a continuous time t s fuzzy system with a delay in the state though the resulting closed loop state delayed t s fuzzy system takes a standard form the existing results on delay t s fuzzy systems cannot be used for our purpose due to their restrictive assumptions on the derivative of state delay a new condition guaranteeing asymptotic stability of the closed loop sampled data system is derived by a lyapunov approach plus the free weighting matrix technique based on this stability condition two procedures for designing state feedback control laws are given one casts the controller design into a convex optimization by introducing some over design and the other utilizes the cone complementarity linearization idea to cast the controller design into a sequential minimization problem subject to linear matrix inequality constraints which can be readily solved using standard numerical software an illustrative example is provided to show the applicability and effectiveness of the proposed controller design methodology

an overview of periodic elliptic operators the article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas close attention is paid to studying analytic properties of bloch and fermi varieties which influence significantly most properties of such operators the approaches described are applicable not only to the standard model example of schrodinger operator with periodic electric potential v x but to a wide variety of elliptic periodic equations and systems equations on graphs operator and other operators on abelian coverings of compact bases many important applications are mentioned however due to the size restrictions they are not dealt with in details

memoryless h controllers for discrete time systems with time delay this paper considers the problem of h stabilization of discrete time systems with state delays and exogenous bounded energy l 2 disturbances specifically an h state feedback control design problem for discrete time systems with time delay is addressed the principal result involves sufficient conditions in terms of a modified riccati equation for characterizing state feedback controllers that enforce a bound on h performance and guarantee closed loop stability in the face of system state delay

h model reduction for discrete time delay systems delay independent and dependent approaches this paper investigates the problem of h model reduction for linear discrete time state delay systems for a given stable system our attention is focused on the construction of reduced order models which guarantee the corresponding error system to be asymptotically stable and have a prescribed h error performance both delay independent and dependent approaches are developed with sufficient conditions obtained for the existence of admissible reduced order solutions since these obtained conditions are not expressed as strict linear matrix inequalities lmis the cone complementary linearization method is exploited to cast them into sequential minimization problems subject to lmi constraints which can be readily solved in standard numerical software in addition the development of reduced order models with special structures such as delay free models and zeroth order models is also addressed the approximation methods presented in this paper can be further extended to cope with systems with unc

an asymptotic method based composite plate model considering imperfect interfaces abstract this paper presents an asymptotic method based analysis of composite laminates having interfacial imperfections in general imperfect interfaces are simply modeled by introducing a linear spring layer model which empirically assumes that the displacement jumps that occur at the weakened interface are proportional to the transverse shear stresses of interface positions in this study we propose a composite plate model derived by using asymptotic expansion that does not make any assumptions other than the scaling of coordinate systems within the framework of the asymptotic analysis the spring layer model is introduced to describe the effect of weakened interfaces which is realized by the separation of domains in the through the thickness direction and the integration of piecewise continuous warping functions as a result we newly define a spring parameter that is exactly the same as the stiffness of a spring therefore a set of spring elements are added to the through the thickness modeling of the microscopic analysis and the plate equations derived in the macroscopic problem are the same as those of the perfectly bonded laminates as a consequence we also derive the proposed plate model with the mathematical rigor that the previous asymptotic models contain we provide some numerical results verifying that the proposed method shows good agreement with the elasticity and 3d fem solutions

ℋ model reduction for linear parameter varying systems with distributed delay this paper is concerned with the model reduction for linear parameter varying lpv systems with both discrete and distributed delays for a given stable system our attention is focused on the construction of reduced order models which approximate the original system well in an norm sense first a sufficient condition is proposed for the asymptotic stability with an performance of the error system by using the parameter dependent lyapunov functional method then the decoupling technique is applied such that there does not exist any product term between the lyapunov matrices and the system matrices in the parametrised linear matrix inequality plmi constraints thus a new sufficient condition is obtained based on the new condition two different approaches are developed to solve the model reduction problem one is the convex linearisation approach and the other is the projection approach finally a numerical example is provided to demonstrate the effectiveness of the proposed design method

reduced order aggregated models for bilinear time invariant dynamical systems a necessary and sufficient condition for a bilinear time invariant dynamical system to admit an aggregated reduced order bilinear model is given

brief paper weighted h model reduction for linear switched systems with time varying delay this paper is concerned with model reduction for continuous time linear switched systems with time varying delay for a given stable switched system our attention is focused on construction of a reduced order model such that the error system is exponentially stable with a prescribed weighted performance by applying the average dwell time approach and the piecewise lyapunov function technique delay dependent deley independent sufficient conditions are proposed in terms of linear matrix inequality lmi to guarantee the exponential stability and the weighted performance for the error system the model reduction problem is solved by using the projection approach which casts the model reduction problem into a sequential minimization problem subject to lmi constraints by employing the cone complementary linearization algorithm a numerical example is provided to illustrate the effectiveness of the proposed theory

methods of kam theory for long range quasi periodic operators on bf z nu pure point spectrum we consider the class of quasi periodic self adjoint operators x hat d x hat v x x s 1 1 1 on a multi dimensional lattice v with the matrix elements hat d mn x delta mn d x n omega hat v mn x v m n x n omega whered x 1 d x v n s 1 v n x v and v n x ee r n r 0 we prove that if e is small enough v n andd satisfy some conditions of smoothness andd is non degenerate then for a e and for a e x s 1 the operator x has pure point spectrum all its eigenfunctions belong tol 1 v

on gramians and balanced truncation of discrete time bilinear systems this paper obtains a balanced truncation model reduction method for discrete time bilinear systems after deriving the sufficient conditions for the discrete time bilinear systems to lead to bounded outputs for some bounded inputs we study the properties of the reachability and observability gramians the gramians can be computed by solving two generalized lyapunov equations by balancing the reachability and observability gramians a model reduction method is obtained some propositions of the reduced order models are given a numerical example is given to illustrate the effectiveness of the proposed method

splitting of the low landau levels into a set of positive lebesgue measure under small periodic perturbations we study the spectral properties of a two dimensional schrodinger operator with a uniform magnetic field and a small external periodic field where and are small parameters representing as the direct integral of one dimensional quasi periodic difference operators with long range potential and employing recent results of e i dinaburg about anderson localization for such operators we assume to be typical irrational we construct the full set of generalised eigenfunctions for the low landau bands we also show that the lebesgue measure of the low bands is positive and proportional in the main order to

bending vibration and buckling of simply supported thick multilayered orthotropic plates an evaluation of a new displacement model abstract based upon a piecewise linear displacement field which allows the contact conditions for the displacements and the transverse shearing stresses at the interfaces to be satisfied simultaneously the non linear in the von karman sense equations of motion for thick multilayered orthotropic plates are developed successively the equations are specified to the linear boundary value problem of the bending and to the linear eigenvalue problems of the undamped vibration and buckling of rectangular plates in order to assess the accuracy of the proposed theory the sample problem of the bending free undamped vibration and buckling of a three layered symmetric cross ply square plate simply supported on all edges is investigated for purposes of comparison numerical results from the exact elasticity theory the classical lamination kirchhoff theory and the shear deformation theory timoshenko and mindlin with three different values of the shear correction factor are also presented it is found that the proposed approach is very efficient in predicting the global responses deflection natural frequencies and buckling loads of thick multilayered plates and models effects such as the distortion of the deformed normals not attainable from the classical lamination theory as well as the shear deformation theory