balanced truncation for temporal and spatial lpv interconnected systems based on the full block s procedure abstractthis paper presents a technique for model order reduction of exponentially stable temporal and spatial lpv linear parameter varying interconnected systems represented in linear fractiona

linear time variable systems balancing and model reduction a uniformly balanced realization for linear time variable systems is defined this representation is characterized by the fact that its controllability and observability gramians are equal and diagonal existence and uniqueness of the uniformly balanced realization is studied such a framework has many remarkable properties and leads to a novel method for approximating time variable systems where the subsystems of the balanced realization can be taken as a reduced model the reduced model is examined from the point of view of stability controllability and observability

optimal control system design of a nuclear reactor by generalized spectral factorization optimal feedback control system design of a nuclear power plant has usually been conducted in the time domain by solving the riccati equation in this paper the problem will be treated in the s domain by means of the macfarlane extension criterion mec the algorithm is based on generalized spectral factorization gsf and the mec is resolved by the newton raphson method featuring rapid convergence and high precision whether the plant is stable or unstable the initial matrix can be chosen effectively and a general computer code for finding the optimal feedback gain matrix of the multi input system is presented the control system design of a boiling water reactor bwr shows that the results obtained from the proposed method are consistent with those from solving the conventional riccati equation a satisfactory closed loop multivariable system with a prescribed degree of stability can also be designed in the s domain the methodology of this paper can also be generalized to treat similar control sys

model reduction of nonstationary lpv systems this paper focuses on the model reduction of nonstationary linear parameter varying nslpv systems we provide a generalization of the balanced truncation procedure for the model reduction of stable nslpv systems along with a priori error bounds then for illustration purposes this method is applied to reduce the model of a two mass translational system furthermore we give an approach for the model reduction of stabilizable and detectable systems which requires the development and use of coprime factorizations for nslpv models for the general class of eventually periodic lpv systems which includes periodic and finite horizon systems as special cases our results can be explicitly computed using semidefinite programming

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

an error bound for a discrete reduced order model of a linear multivariable system the design of feasible controllers for high dimension multivariable systems can be greatly aided by a method of model reduction in order for the design based on the order reduction to include a guarantee of stability it is sufficient to have a bound on the model error previous work has provided such a bound for continuous time systems for algorithms based on balancing in this note an linftybound is derived for model error for a method of order reduction of discrete linear multivariable systems based on balancing

existence of global weak solutions for a 2d viscous shallow water equations and convergence to the quasi geostrophic model we consider a two dimensional viscous shallow water model with friction term existence of global weak solutions is obtained and convergence to the strong solution of the viscous quasi geostrophic equation with free surface term is proven in the well prepared case the ill prepared data case is also discussed

hamilton s law of variable mass system and time finite element formulations for time varying structures based on the law summary traditional principles of mechanics are primarily conceived for constant mass systems which are only valid if mass is gained or lost at null velocity with respect to an inertial reference frame for variable mass systems thus the numerical algorithms for time varying structures based on these principles are only suitable for this special case in this paper hamilton s law of variable mass system is derived based on meshchersky s fundamental equation and two classes of novel time finite element formulations for linear systems with arbitrary continuous time varying parameters are developed based on the previous law the formulations are verified extensively through numerical examples in which the convergence and effectiveness of algorithms are evaluated numerical examples demonstrate that compared with the algorithms for time varying structures that developed based on traditional principles of mechanics the proposed algorithms provide extended capabilities in both time varying mass problems that mass is gained or lost at any velocity such as rocket problem and moving mass problems such as vehicle bridge interaction problem besides the time varying stiffness and damping problems the proposed algorithms have a wider range of application in particular hamilton s law of variable mass system provides a solid theoretical foundation for further research on the algorithm design for time varying structures copyright 2014 john wiley sons ltd

cauchy problem for viscous rotating shallow water equations abstract we consider the cauchy problem for a viscous compressible rotating shallow water system with a third order surface tension term involved derived recently in the modeling of motions for shallow water with free surface in a rotating sub domain marche 2007 19 the global existence of the solution in the space of besov type is shown for initial data close to a constant equilibrium state away from the vacuum unlike the previous analysis about the compressible fluid model without coriolis forces see for instance danchin 2000 10 haspot 2009 16 the rotating effect causes a coupling between two parts of hodge s decomposition of the velocity vector field and additional regularity is required in order to carry out the friedrichs regularization and compactness arguments

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

identification and model reduction of time varying discrete time systems abstract linear discrete time variable systems are considered and an important class of uniform realizations is defined the necessary and sufficient conditions for a pulse response to be uniformly realizable are obtained two model reduction schemes one via balanced realizations and the other via hankel matrix are proposed the latter approach can be considered as an identification algorithm for the two dimensional pulse response sequence h k i it is shown that the two approaches yield an identical reduced model which is always asymptotically stable

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

a case study in model reduction of linear time varying systems in this paper the balanced truncation procedure is applied to time varying linear systems both in continuous and in discrete time the methods are applied to a linear approximation of a diesel exhaust catalyst model the reduced order systems are obtained by using certain projections instead of direct balancing an approximate zero order hold discretization of continuous time systems is described and a new a priori approximation error bound for balanced truncation in the discrete time case is obtained the case study shows that there are several advantages to work in discrete time it gives simpler implementation with fewer computations

structure preserving model reduction for spatially interconnected systems with experimental validation on an actuated beam abstracta technique for model reduction of exponentially stable spatially interconnected systems is presented where the order of the reduced model is determined by the number of truncated small generalised singular values of the structured solutions to a pair of lyapunov inequalities for parameter invariant spatially interconnected systems the technique is based on solving a pair of lyapunov inequalities in continuous time and space domain with a rank constraint using log det and cone complementarity methods an improved error bound can be obtained the approach is extended to spatially parameter varying systems and a balanced truncation approach using parameter dependent gramians is proposed to reduce the conservatism caused by the use of constant gramians this is done by considering two important operators which can be used to represent multidimensional systems temporal and spatial linear parameter varying interconnected systems the results are illustrated with their application to an experi

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

generating functions for connected embeddings in a lattice v application to the simple cubic and body centred cubic lattices for pt iv see ibid vol 19 p 2431 1986 the method of partial generating functions is used to derive for the simple cubic lattice the number of connected strong embeddings through 13 sites the number of connected weak embeddings through 14 bonds and three new bond perimeter polynomials d10 d11 d12 for the bond percolation problem for the body centred cubic and simple cubic lattices an expression is derived for the mean number of clusters for the site percolation problem in powers of the probabilities of occupation of a and b sites

perimeter polynomials for bond percolation processes perimeter polynomials are given for the bond percolation problem on the following lattices the triangular up to d9 simple quadratic d13 honeycomb d17 face centred cubic d7 body centred cubic d8 simple cubic d9 and diamond d13 the total number of bond clusters grouped by size is given to two further orders in each case

the lp decay estimates for the chemotaxis shallow water system abstract in this paper we consider the large time behavior of solution for the chemotaxis shallow water system in two dimension by the method of frequency decomposition first we construct the green s function of the linearized system and get the corresponding estimates for low frequency part then we introduce an energy method we named double cycle method to get the decay rate for high frequency part due to the dissipative mechanism of this system the high frequency part decays faster than the low frequency part

series expansion evidence supporting the alexander orbach conjecture in two dimensions the authors extend the series expansion of fisch and harris 1978 for the resistive susceptibility chi r p by a further six terms on the square lattice this leads to a more precise estimate of the corresponding exponent gamma r 3 65 or 0 02 they also obtain the exact relation chi r p 2e tau mod f where tau is the sum of the relaxation times for charge diffusion on the cluster containing the origin and the expectation value is subject to the condition that the latter is finite a known scaling relation for the fracton dimension df in terms of gamma r and the static exponents is derived without the usual reference to the infinite cluster using the estimate of gamma r they find df 1 334 or 0 007 which is consistent with the ao conjecture df 4 3 they also note that gamma r is the same for directed and undirected percolation to within the accuracy of the calculations

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

global existence and well posedness of the 2d viscous shallow water system in sobolev spaces with low regularity abstract in this paper we consider the cauchy problem for the 2d viscous shallow water system in h s r 2 s 1 we first prove the local well posedness of this problem by using the littlewood paley theory the bony decomposition and the theories of transport equations and transport diffusion equations then we get the global existence of the system with small initial data in h s r 2 s 1 our obtained result improves considerably the recent result in 14

branched polymers with a prescribed number of cycles the authors consider the number of lattice animals with exactly c cycles for weakly embeddable bond clusters they show rigorously that the growth constant is independent of c and they derive upper and lower bounds on the critical exponents for each value of c they use series analysis methods to estimate the critical exponents for c 1 and 2 in two and three dimensions and find that the critical exponent does depend on the number of cycles evidence from series analysis results and from expansions in inverse powers of the dimension suggests that in the case of strong embeddings i e site clusters the growth constant is independent of c and the corresponding values of the critical exponents are identical to the values for weak embeddings i e bond clusters they discuss the relationship of these results to the field theory prediction that the critical exponent is independent of cycle fugacity

phragmen lindelof alternative results for the shallow water equations for transient compressible viscous flow abstract in this paper the spatial behavior of the viscous shallow water equations in a two dimensional channel domain is studied a differential inequality for suitable energy associated with the solutions of the viscous shallow water equations in a semi infinite channel is derived from which we show the phragmen lindelof alternative results the main tool used is the energy method our results can be viewed as a version of saint venant s principle to the transient compressible viscous flow

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

on the benefits of the ldlt factorization for large scale differential matrix equation solvers abstract we propose efficient algorithms for solving large scale matrix differential equations in particular we deal with the differential riccati equations dre and state the applicability to the differential lyapunov equations dle we focus on methods based on standard versions of ordinary differential equations in the matrix setting the application of these methods yields algebraic lyapunov equations ales with a certain structure to be solved in every step the alternating direction implicit adi algorithm and krylov subspace based methods allow to exploit this special structure however a direct application of classic low rank formulations requires the use of complex arithmetic using an l d l t type decomposition of both the right hand side and the solution of the equation we avoid this problem thus the proposed methods are a more practical alternative for large scale problems arising in applications also they make the application of higher order methods feasible the numerical results show the better performance of the proposed methods compared to earlier formulations

a posteriori error bounds for discrete balanced truncation balanced truncation of discrete linear time invariant systems is an automatic method once an error tolerance is specified and yields an a priori error bound which is why it is widely used in engineering for simulation and control we present some new insight into this method we derive a discrete version of antoulas s mathcal h 2 norm error formula cite p 218 ant05 and show how to adapt it to some special cases this error bound is an a posteriori computable upper bound for the mathcal h 2 norm of the error system defined as the system whose transfer function corresponds to the difference between the transfer function of the original system and the transfer function of the reduced system the main advantage of our results is that we use the information already available in the balanced truncation algorithm in order to compute the mathcal h 2 norm instead of computing one gramian of the corresponding error system there is always a computational restriction on solving high dimensional stein equations for gramians the a posteriori bound gives insight into the quality of the reduced system and can be used to solve many problems accompanying the order reduction operation

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

global existence for the dirichlet problem for the viscous shallow water equations abstract a global existence and uniqueness theorem of strong solutions for the initial boundary value problem with dirichlet boundary conditions is established for small forcing and small initial data an exponentialc0decay rate is also established and the solution is shown to be classical fort 0 the positivity of the fluid heighthfort 0 is also proved

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

technical communique model reduction of periodic systems a lifting approach this note furthers existing results on the model reduction of stable periodic systems it utilizes for that matter a lifting technique to potentially attain less conservative error bounds

convex invertible cones and the lyapunov equation abstract convex cones of matrices which are closed under matrix inversion are defined and their structure is studied various connections with the algebraic lyapunov equation of general inertia are explored

two dimensional lattice tree exponents and amplitudes simulation algorithms versus series we use a local monte carlo algorithm to simulate lattice trees in two dimensions for the site and bond problem we investigate the properties of radius of gyration perimeter to site ratio and vertex degree in a tree adding some new results in the site problem compare our results on their noncritical properties with those obtained from earlier reversible and slightly nonreversible algorithms and combine our determinations with new exact series expansion data on the controversy surrounding the possible lack of universality for the first confluent singularity for the gyration radius we feel that conclusions must be guarded

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

an improved error estimate for reduced order models of discrete time systems the authors derive under weaker conditions a discrete time counterpart of k glover s 1984 error estimate for reduced order models following the same lines a simple proof of glover s result for continuous time systems can be given

error bounds for balanced model reduction of linear time varying systems error bounds are developed for balanced truncation of linear time varying systems leading to an extension of the twice the sum of the tail formulas well known in the time invariant case the approach relies on an operator theoretic framework for analysis of linear time varying systems this provides a multivariable notion of frequency for such systems which are thus characterized by rational functions of many complex variables allowing the problem to be formulated in the linear fractional framework using a time varying version of standard necessary conditions for reduced order modeling based on convex operator inequalities we show that these error bounds for balanced truncation are related to the closest possible reduced order modeling error in a sense which parallels the time invariant case

initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids the equations of motion of compressible viscous and heat conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region the global solution in time is proved to exist uniquely and approach the stationary state ast provided the prescribed initial data and the external force are sufficiently small

a schur method for balanced truncation model reduction it is shown that a not necessarily balanced state space realization of the moore reduced model can be computed directly without balancing via projections defined in terms of arbitrary bases for the left and right eigenspaces associated with the large eigenvalues of the product pq of the reachability and controllability grammians two specific methods for computing these bases are proposed one based on the ordered schur decomposition of pq and the other based on the cholesky factors of p and q the algorithms perform reliably even for nonminimal models

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

order propagation near the percolation threshold a new monte carlo method for studying bond percolation clusters is developed and used to identify new critical quantities associated with the percolation threshold the bonds in each cluster are partitioned into three distinct connectivity classes red singly connected backbone bonds blue multiply connected backbone bonds and yellow non backbone bonds often called dangling ends among the new cluster properties studied are the mean number of red bonds a critical quantity diverging at p c with exponent y r approximately 1 and the length of the shortest connected path through the cluster which is critical with exponent y min 1 35 0 02 for all cluster properties studies the authors also compute averages over only the largest clusters the corresponding critical exponents are found to be significantly different from those obtained by averaging over clusters of all sizes

generating functions for connected embeddings in a lattice iii bond percolation for pt ii see ibid vol 19 p 1027 32 1986 the method of partial generating functions is developed to obtain bond perimeter polynomials for percolation processes four new polynomials d9 d12 are given for the body centered cubic lattice

well posedness for the viscous rotating shallow water equations with friction terms we consider the cauchy problem for viscous rotating shallow water equations with friction terms the global existence of the solution in some hybrid spaces is shown for the initial data close to a constant equilibrium state away from the vacuum

site percolation threshold for honeycomb and square lattices a new method of estimating the critical percolation threshold is proposed based on stauffer s cluster number scaling hypothesis and the universality with respect to lattice structure of the corresponding stauffer scaling function this method is illustrated by obtaining estimates of the site percolation threshold for the honeycomb lattice pc 0 6962 or 0 0006 and for the square lattice pc 0 5923 or 0 0007 the error bars or confidence limits of our estimates are substantially smaller than previous series estimates and for the honeycomb lattice would have to be multiplied by the factor 18 to include the value pc 2 12 recently proposed by kondor 1980 to be exact an additional result is r 4 95 or 0 15 where r lims to infinity rs and rs is the ratio of the cluster number scaling function at its maximum to its value at pc for clusters of size s