on the observer design for descriptor systems for the descriptor system ex ax bu y cx du an observer of the order p rank e p sub 1 0

on the finite element method for singularly perturbed reaction diffusion problems in two and one dimensions second order elliptic boundary value problems which are allowed to degenerate into zero order equations are considered the behavior of the ordinary galerkin finite element method without special arrangements to treat singularities is studied as the problem ranges from true second order to singularly perturbed

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

pole assignment and observer design for continuous descriptor systems the problem of pole assignment and asymptotic observer design for multi input multi output continuous descriptor generalized state space systems is studied the problem is solved by introducing a new canonical form for controllable descriptor systems the internal stability of the descriptor variable feedback and observer configuration is also considered i e the extension of the separation principle to descriptor systems is given

on the accuracy of the finite element method plus time relaxation if u denotes a local spatial average of u then u u u is the associated fluctuation consider a time relaxation term added to the usual finite element method the simplest case for the model advection equation u t a u x t is u h t a u h v h u h v h f x t v h we analyze the error in this and more importantly higher order extensions and show that the added time relaxation term not only suppresses excess energy in marginally resolved scales but also increases the accuracy of the resulting finite element approximation

design of a class of luenberger observers for descriptor systems a new approach to the design of a class of luenberger observers for descriptor systems is presented the design is performed in a straightforward manner the conditions required in the design are given in terms of the original system matrices the relationship among these and other well known conditions is highlighted

the regularized chapman enskog expansion for scalar conservation laws rosenao r has recently proposed a regularized version of the chapmanenskog expansion of hydrodynamics this regularized expansion resembles the usual navier stokes viscosity terms at low wave numbers but unlike the latter it has the advantage of being a bounded macroscopic approximation to the linearized collision operator in this paper we study the behavior of the rosenau regularization of the chapman enskog expansion r c e in the context of scalar conservation laws we show that this r c e model retains the essential properties of the usual viscosity approximation e g existence of travelling waves monotonicity upper lipschitz continuity etc and at the same time it sharpens the standard viscous shock layers we prove that the regularized r c e approximation converges to the underlying inviscid entropy solution as its mean free path e 0 and we estimate the convergence rate

on the vreman filter based stabilization for the advection equation this report investigates a stabilization method for first order hyperbolic differential equations if the usual unstabilized finite element method is used to solve numerically first order hyperbolics on unstructured meshes using general elements it is expected a loss of accuracy by one power of the mesh size and also for non smooth solutions the computed approximation might contain spurious oscillations to stabilize the finite element method the authors consider adding to the first order hyperbolic system as a stabilization term the vreman filtering analog of the vms variational multiscale method vreman 2003 22 the resulting model is shown to have the expected highest attainable convergence rate o h k 0 5 when the filter radius is optimally chosen the theoretical rates are also checked numerically and comparison with another known scheme is provided

a subgrid scale deconvolution approach for shock capturing we develop a method for the modeling of flow discontinuities which can arise as weak solutions of inviscid conservation laws due to its similarity with recently proposed approximate deconvolution models for large eddy simulation the method potentially allows for a unified treatment of flow discontinuities and turbulent subgrid scales a filtering approach is employed since for the filtered evolution equations the solution is smooth and can be solved for by standard central finite difference schemes without special consideration of discontinuities a sufficiently accurate representation of the filtered nonlinear combination of discontinuous solution components which arise from the convection term can be obtained by a regularized deconvolution applied to the filtered solution for stable integration the evolution equations are supplemented by a relaxation regularization based on a secondary filter operation and a relaxation parameter an estimate for the relaxation parameter is provided the method is related to the spectral vanishing viscosity method and the regularized chapman enskog expansion method for conservation laws we detail the approach and demonstrate its efficiency with the inviscid and viscous burgers equations the isothermal shock problem and the one dimensional euler equations

truncation of scales by time relaxation abstract we study a time relaxation regularization of flow problems proposed and tested extensively by stolz and adams the aim of the relaxation term is to drive the unresolved fluctuations in a computational simulation to zero exponentially fast by an appropriate and often problem dependent choice of its coefficient this relaxation term is thus intermediate between a tunable numerical stabilization and a continuum modeling term our aim herein is to understand how this term by itself acts to truncate solution scales and to use this understanding to give insight into parameter selection

on the robust popov criterion for interval lur e system the classical popov criterion is generalized to include problems with parametric uncertainty as well as nonlinear sector bounded perturbations the parametric dependence is modeled by an interval plant description a robust popov criterion is obtained for a plant controller configuration stronger external results are obtained for a special class of controllers an example illustrating the numerical implementation of these results is presented

a connection between subgrid scale eddy viscosity and mixed methods we consider a new mixed method related to the evss method in computational viscoelasticity for the convection dominated convection diffusion equation in which stabilization is added then removed through the extra mixed variables this consistent stabilization is equivalent to an artificial viscosity operator acting only on the fluctuations in u h by suitable choice of the mixed spaces a method of guermond is recovered exactly we show that for a different natural choice a new method results with global error estimates similar to both guermond s method and the streamline diffusion supg method

on the reduced order causal observer for generalized control systems abstract this paper considers the problem of a causal state observer for a square singular system necessary and sufficient conditions for solvability of the problem have been established the approach of this study leads to a new insight concerning synthesis of a causal observer and establishes new results with regard to the structure and the dimension of a reduced order observer for singular systems

on an optimal finite element scheme for the advection equation in this report it is presented a numerical finite element scheme for the advection equation that attains the optimal l 2 convergence rate o h k 1 when order k finite elements are used improving the order o h k 0 5 of other previous regularization methods this result is also confirmed by the numerical test performed in the last section the scheme assumes unstructured grids periodic boundary conditions a constant advection field and a bit two units stronger regularity on the exact solution than in the classical suboptimal finite element theory

edge stabilization for galerkin approximations of convection diffusion reaction problems n this paper we recall a stabilization technique for finite element methods for convection diffusion reaction equations originally proposed by j douglas jr and t dupont in computing methods in applied sciences second internat sympos versailles 1975 207 216 lecture notes in phys 58 springer berlin 1976 mr0440955 55 13823 the method uses least square stabilization of the gradient jumps across element boundaries we prove that the method is stable in the hyperbolic limit and prove optimal a priori error estimates we address the question of monotonicity of discrete solutions and present some numerical examples illustrating the theoretical results

on the design of observers for generalized state space systems using singular value decomposition in this paper a method is developed for the design of luenberger type observers for linear time invariant control systems whose state equation is of the form ex ax bu where e is a singular square matrix the method is based on the singular value decomposition of the matrix e and on the reduction of the equation ex ax bu to a system consisting of a differential equation of the form w1 f1w1 f2 w2 g1u and an algebraic equation of the form h1w1 h2w2 g1u 0 if w2 can be eliminated from the differential equation by the aid of the algebraic equation and the original output equation of the system the method yields a reduced order observer for the generalized state space system

numerical analysis of filter based stabilization for evolution equations we consider filter based stabilization for evolution equations in general and for the navier stokes equations in particular the first method we consider is to advance in time one time step by a given method and then to apply an uncoupled and modular filter to get the approximation at the new time level this filter based stabilization although algorithmically appealing is viewed in the literature as introducing far too much numerical dissipation to achieve a quality approximate solution we show that this is indeed the case we then consider a modification evolve one time step filter deconvolve then relax to get the approximation at the new time step we give a precise analysis of the numerical diffusion and error in this process and show it has great promise confirmed in several numerical experiments

observers for descriptor systems for a singular non causal descriptor system of order n a p k lh order observer is proposed where p n q q is the number of infinite non dynamic modes of the system and k is the rank of a matrix obtained from a transformed system derived by using the singular value decomposition some useful results are given which establish the existence of this observer and which resolve problems encountered with pth order observers proposed in the literature robustness is achieved by choosing optimum gain matrices which reduce the sensitivity of observer performance to perturbations

geometric design techniques for observers in singular systems abstract we present geometric design techniques for observer design in generalized state space or singular systems the unknown input conditioned invariant subspaces are defined for singular systems and are related to the solution of a generalized lyapunov equation the approach emphasizes solution procedures based on the lyapunov equation and a singular system structure algorithm this paper extends the singular system observer theory both by providing a geometric basis for it and showing the best that may be accomplished if the system is not observable

stabilized finite element methods for the generalized oseen problem the numerical solution of the non stationary incompressible navierstokes model can be split into linearized auxiliary problems of oseen type we present in a unique way different stabilization techniques of finite element schemes on isotropic meshes first we describe the state of the art for the classical residual based supg pspg method then we discuss recent symmetric stabilization techniques which avoid some drawbacks of the classical method these methods are closely related to the concept of variational multiscale methods which seems to provide a new approach to large eddy simulation finally we give a critical comparison of these methods

design of observers for time varying discrete time descriptor systems in this note linear discrete time time varying descriptor systems i e control systems of the form ek 1 xk 1 akxk bkuk yk ckxk are considered where ek 1 and ak 1 are constant not necessarily square matrices the problem is to design a reduced order observer for the system using matrix generalized inverses and singular value decomposition a procedure is developed which either leads to a routine observer design problem involving an ordinary discrete time state equation or proves that no observer exists for the given descriptor system the method of this note is partly based on an earlier paper of the authors on observer design for continuous time time invariant systems of the form e dx dt ax bu y cx where e and a are square matrices el tohami et al 1983 and makes use of some results of luenberger on discrete time time varying descriptor systems luenberger 1977

parametric absolute stability of lur e systems the concept of parametric stability is extended to include lur e type nonlinear control systems with uncertain parameters and constant reference inputs conditions for parametric absolute stability are derived which guarantee that the system remains stable despite uncertainty of equilibrium location caused by parametric uncertainties and values of the reference input the conditions can be tested by computing value sets using the polygon interval arithmetic algorithm

numerical analysis of a nonlinear time relaxation model of fluids abstract this report investigates a time relaxation model for the regularization of the strong solution of the navier stokes equations following the earlier works of adams stolz and kleiser 37 38 ervin layton and neda 11 layton and neda 26 the nonlinear term investigated herein aims to better control the generalized turbulent fluctuations thus providing better stabilization effect on the underlying model we study the numerical properties of the resulting model and prove that the addition of the nonlinear term does not affect the overall order of convergence to the strong solution of the navier stokes equation our theoretical results based on the finite element method are confirmed by the numerical tests performed in the last section of the paper

a reduced order observer for descriptor systems abstract in this paper a reducer order observer is analyzed for so called generalized state space or descriptor systems based on the staircase form for generalized state space models a recursive algorithm is presented to construct a reduced order observer for a given observable descriptor system in the algorithm only the eigenfrequencies of the observer have to be specified

design of observers for descriptor systems using a descriptor standard form methods are developed for constructing luenberger type observers for linear time invariant descriptor systems based on the descriptor standard form derived here by means of a simple matrix calculus full order observers reduced order observers and function observers of a general type are proposed some necessary and sufficient conditions for the function observers to estimate linear functions of the state are discussed then the so called separation property of combined observer controller systems is established

observer design for generalized state space systems with unknown inputs abstract the following four major aspects for the observer design problem for generalized state space systems with unknown inputs are resolved necessary and sufficient conditions for the problem to have a solution the order of the minimal observer the properties of the closed loop system separation principle and general analytical expressions of the minimal order observer matrices all above results are first in the field

galerkin methods for two point boundary value problems for first order systems galerkin methods for two point boundary value problems for first order systems are studied for a model problem rigorous upper and lower estimates upon the error are derived these estimates show the previous work to be sharp in some cases and pessimistic in others a nonstandard least squares type of method is then analyzed this method gives the optimal accuracy of the usual least squares method when the system does not satisfy a regularity assumption the nonstandard method gives slightly more accurate answers measured in l 2 the linear system arising from the nonstandard method has condition number o h 1 contrasted with o h 2 for the usual least squares method

comments on on the robust popov criterion for interval lur e systems this paper supplements the remarks made in a paper by dahleh et al ieee trans automat contr vol 38 p 1400 5 1993 regarding the popov sector of lur e systems with interval plants a numerical example is provided to show that strict inequalities hold among three sectors those obtained by the theorem of the paper by applying popov theorem to each member system and by applying that to each of the systems with kharitonov plants this counterexample negates a tempting conjecture on strong robust popov criterion

stable galerkin methods for hyperbolic systems this paper considers the stability of the galerkin method for first order hyperbolic systems just as in finite difference theory many galerkin methods stable and convergent for the cauchy problem are unstable for hyperbolic boundary value problems because of improper treatment of the boundary conditions the usual galerkin method is seen to be unstable for a linear constant coefficient well posed hyperbolic system on 0 1 a method of treating boundary conditions is then proposed which will yield a stable and convergent method for any well posed linear hyperbolic system in one dimension this idea generalizes in a natural way to problems in more than one space dimension examples in one and two dimensions are considered this paper extends the work of gunzburger math comp 31 1977 pp 671 675 to the general hyperbolic system

design of a minimal order observer for singular systems this paper considers the design of an observer for generalized state space systems using the concept of generalized inverses in contrast to the luenberger observer theory a new method is proposed which does not presuppose the observer structure it is shown that under certain conditions it is possible to construct a minimal order observer for this class of systems illustrative examples are included

the d step conjecture and its relatives the d step conjecture arose from an attempt to understand the computational complexity of edge following algorithms for linear programming such as the simplex algorithm it can be stated in terms of diameters of graphs of convex polytopes in terms of the existence of nonrevisiting paths in such graphs in terms of an exchange process for simplicial bases of a vector space and in terms of matrix pivot operations first formulated by w m hirsch in 1957 the conjecture remains unsettled though it has been proved in many special cases and counterexamples have been found for slightly stronger conjectures if the conjecture is false as we believe to be the case then finding a counterexample will be merely a small first step in the line of investigation related to the conjecture this report summarizes what is known about the d step conjecture and its relatives a considerable amount of new material is included but it does not seem to come close to settling the conjecture of special interest is the first example of a polytope that is not vertex decomposable showing that a certain natural approach to the conjecture will not work also significant are the quantitative relations among the lengths of paths associated with various forms of the conjecture

observers for discrete singular systems the author considers the design of observers for the discrete singular system ex k 1 ax k bu k y k cx k placing special emphasis on the problems of state reconstruction and minimal time state reconstruction it is shown that for a singular system finite poles can be moved to infinity by state feedback and the state can be reconstructed by causal observers

symbolic computation approach for absolute stability a criterion for absolute stability of nonlinear control systems which is equivalent to popov s criterion is given the condition for absolute stability can be tested without graphics symbolic computation is employed for the test a maple program and some examples are given