survey paper transfer functions of distributed parameter systems a tutorial in this tutorial article the rich variety of transfer functions for systems described by partial differential equations is illustrated by means of several examples under various boundary conditions an important feature is the strong influence of the choice of boundary conditions on the dynamics and on system theoretic properties such as pole and zero locations properness relative degree and minimum phase it is sometimes possible to design a controller using the irrational transfer function and several such techniques are outlined more often the irrational transfer function is approximated by a rational one for the purpose of controller design various approximation techniques and their underlying theory are briefly discussed

dynamic feedback stabilization of an unstable wave equation abstract in this paper we consider the stabilization of an unstable wave equation through a dynamic boundary compensator the measurement which is one internal point of the state of the wave equation is fluxed into the compensator while the output of the compensator is forced into the boundary of the wave equation the well posedness of the closed loop system is proved by using the semigroup method by choosing appropriate controller parameters and adopting nyquist criterion for distributed parameter systems the eigenvalues of the closed loop systems are shown to be inside the left half complex plane then the closed loop system is proved to be exponentially stable by using the riesz basis approach numerical simulations are presented to verify the effectiveness of the proposed compensator

a nyquist stability criterion for distributed parameter systems a nyquist graphical stability criterion is developed for distributed parameter possible unstable single loop systems practical conditions are presented for existence uniqueness causality and asymptotic or exponential stability of the closed loop impulse response the hypotheses are given for the transfer function only and do not require any knowledge of its time domain impulse response

generalized bremmer series with rational approximation for the scattering of waves in inhomogeneous media the bremmer series solution of the wave equation in generally inhomogeneous media requires the introduction of pseudodifferential operators in this paper sparse matrix representations of these pseudodifferential operators are derived the authors focus on designing sparse matrices keeping the accuracy high at the cost of ignoring any critical scattering angle phenomena such matrix representations follow from rational approximations of the vertical slowness and the transverse laplace operator symbols and of the vertical derivative as they appear in the parabolic equation method sparse matrix representations lead to a fast algorithm an optimization procedure is followed to minimize the errors in the high frequency limit for a given discretization rate the bremmer series solver consists of three steps directional decomposition into up and downgoing waves one way propagation and interaction of the counterpropagating constituents each of these steps is represented by a sparse matrix equation the resulting algorithm provides an improvement of the parabolic equation method in particular for transient wave phenomena and extends the latter method systematically for backscattered waves

a hybrid scheme for absorbing edge reflections in numerical modeling of wave propagation we propose an efficient scheme to absorb reflections from the model boundaries in numerical solutions of wave equations this scheme divides the computational domain into boundary transition and inner areas the wavefields within the inner and boundary areas are computed by the wave equation and the one way wave equation respectively the wavefields within the transition area are determined by a weighted combination of the wavefields computed by the wave equation and the one way wave equation to obtain a smooth variation from the inner area to the boundary via the transition zone the results from our finite difference numerical modeling tests of the 2d acoustic wave equation show that the absorption enforced by this scheme gradually increases with increasing width of the transition area we obtain equally good performance using pseudospectral and finite element modeling with the same scheme our numerical experiments demonstrate that use of 10 grid points for absorbing edge reflections attains nearly perfect absorption

absorbing boundary conditions for the schrodinger equation a large number of differential equation problems which admit traveling waves are usually defined on very large or infinite domains to numerically solve these problems on smaller subdomains of the original domain artificial boundary conditions must be defined for these subdomains one type of artificial boundary condition which can minimize the size of such subdomains is the absorbing boundary condition the imposition of absorbing boundary conditions is a technique used to reduce the necessary spatial domain when numerically solving partial differential equations that admit traveling waves such absorbing boundary conditions have been extensively studied in the context of hyperbolic wave equations in this paper general absorbing boundary conditions will be developed for the schrodinger equation with one spatial dimension using group velocity considerations previously published absorbing boundary conditions will be shown to reduce to special cases of this absorbing boundary condition the well posedness of the initial boundary value problem of the absorbing boundary condition coupled to the interior schrodinger equation will also be discussed extension of the general absorbing boundary condition to higher spatial dimensions will be demonstrated numerical simulations using initial single gaussian double gaussian and a narrow gaussian pulse distributions will be given with comparision to exact solutions to demonstrate the reflectivity properties of various orders of the absorbing boundary condition

absorbing boundary conditions for elastic media a class of new absorbing boundary conditions for numerical modelling of two dimensional linear time dependent elasticity is introduced many existing absorbing boundary conditions are included as special cases the mechanism used to formulate these new conditions is the complete absorption at a certain number of prescribed incidence angles successful applications to finite difference models of elastic wave scattering from defect like features in solids are presented

dynamic analysis of unbounded domains by a reduced set of base functions the scaled boundary finite element method a semi analytical method based on finite element technology is well suited to model unbounded domains only the boundary is discretized no fundamental solution is necessary general anisotropic material is analyzed without any increase in computational effort the goal of this paper is to increase the computational efficiency of the scaled boundary finite element analysis of large scale problems a reduced set of weighted block orthogonal base functions is introduced the scaled boundary finite element equation is reformulated in generalized coordinates reducing the number of degrees of freedom on boundary the solution procedure in frequency domain is presented to enforce the radiation condition at infinity an asymptotic expansion for displacement at high frequency is developed numerical examples have demonstrated that the computational time is reduced by more than an order of magnitude

riesz basis approach to the stabilization of a flexible beam with a tip mass using an abstract condition of riesz basis generation of discrete operators in the hilbert spaces we show in this paper that a sequence of generalized eigenfunctions of an euler bernoulli beam equation with a tip mass under boundary linear feedback control forms a riesz basis for the state hilbert space in the meanwhile an asymptotic expression of eigenvalues and the exponential stability are readily obtained the main results of siam j control optim 36 1998 pp 1962 1986 are concluded as a special case and the additional conditions imposed there are removed

boundary stabilization of a 1 d wave equation with in domain antidamping we consider the problem of boundary stabilization of a 1 d one dimensional wave equation with an internal spatially varying antidamping term this term puts all the eigenvalues of the open loop system in the right half of the complex plane we design a feedback law based on the backstepping method and prove exponential stability of the closed loop system with a desired decay rate for plants with constant parameters the control gains are found in closed form our design also produces a new lyapunov function for the classical wave equation with passive boundary damping

absorbing boundary conditions for acoustic media by decomposing the acoustic wave equation into incoming and outgoing components an absorbing boundary condition can be derived to eliminate reflections from plane waves according to their direction of propagation this boundary condition is characterized by a first order differential operator the differential operator or absorbing boundary operator is the basic element from which more complicated boundary conditions can be constructed the absorbing boundary operator can be designed to absorb perfectly plane waves traveling in any two directions by combining two or more absorption operators boundary conditions can be created which absorb plane waves propagating in any number of directions absorbing boundary operators simplify the task of designing boundary conditions to reduce the detrimental effects of outgoing waves in many wave propagation problems

two dimensional empirical mode decomposition by finite elements empirical mode decomposition emd is a powerful tool for analysis of non stationary and nonlinear signals and has drawn significant attention in various engineering application areas this paper presents a finite element based emd method for two dimensional data analysis specifically we represent the local mean surface of the data a key step in emd as a linear combination of a set of two dimensional linear basis functions smoothed with bi cubic spline interpolation the coefficients of the basis functions in the linear combination are obtained from the local extrema of the data using a generalized low pass filter by taking advantage of the principle of finite element analysis we develop a fast algorithm for implementation of the emd the proposed method provides an effective approach to overcome several challenging difficulties in extending the original one dimensional emd to the two dimensional emd numerical experiments using both simulated and practical texture images show that the proposed method works well

rigorous absorbing boundary conditions for 3 d one way wave extrapolation absorbing boundary conditions play an important role in one way wave extrapolations by reducing reflections at grid edges clayton and engquist s 2 d absorbing boundary conditions for one way wave extrapolation by depth stepping in the frequency domain are extended to three dimensions using paraxial approximations of the scalar wave equation internal consistency is retained by incorporating the interior extrapolation equation with the absorbing boundary conditions numerical schemes are designed to make the proposed absorbing boundary conditions both mathematically correct and efficient with negligible extra cost synthetic examples illustrate the effectiveness of the algorithm for extrapolation with the 3 d 45 one way wave equation

3d scalar wave absorbing boundary conditions with optimal coefficients in the frequency space domain abstractto improve the computational efficiency for the solution of the 3d helmholtz equation in the frequency space domain high order compact forms of finite differences are preferred we applied a pointwise pade approximation to develop a 3d 27 point fourth order compact finite difference fd stencil in the grid interior with a space differentiated source term for the scalar wave equation this has similar high accuracy 4 5 grid points per the shortest wavelength to another 27 point fourth order fd stencil using a parsimonious mixed grid and staggered grid combination but is much simpler for absorbing boundary conditions abcs a damping zone is expensive and a perfectly matched layer can not be straightforwardly introduced into the compact fd form for the second order wave equation thus we developed 3d one way wave equation owwe abcs with adjustable coefficients they have different angle approximations and fd forms for the six faces twelve edges and eight corners in 3d models to fit wi

an fdtd algorithm with perfectly matched layers for general dispersive media a three dimensional 3 d finite difference time domain fdtd algorithm with perfectly matched layer pml absorbing boundary condition abc is presented for general inhomogeneous dispersive conductive media the modified time domain maxwell s equations for dispersive media are expressed in terms of coordinate stretching variables we extend the recursive convolution rc and piecewise linear recursive convolution plrc approaches to arbitrary dispersive media in a more general form the algorithm is tested for homogeneous and inhomogeneous media with three typical kinds of dispersive media i e lorentz medium unmagnetized plasma and debye medium excellent agreement between the fdtd results and analytical solutions is obtained for all testing cases with both rc and plrc approaches we demonstrate the applications of the algorithm with several examples in subsurface radar detection of mine like objects cylinders and spheres buried in a dispersive half space and the mapping of a curved interface because of their generality the algorithm and computer program can be used to model biological materials artificial dielectrics optical materials and other dispersive media

adaptive boundary control for unstable parabolic pdes part i lyapunov design we develop adaptive controllers for parabolic partial differential equations pdes controlled from a boundary and containing unknown destabilizing parameters affecting the interior of the domain these are the first adaptive controllers for unstable pdes without relative degree limitations open loop stability assumptions or domain wide actuation it is the first necessary step towards developing adaptive controllers for physical systems such as fluid thermal and chemical dynamics where actuation can be only applied non intrusively the dynamics are unstable and the parameters such as the reynolds rayleigh prandtl or peclet numbers are unknown because they vary with operating conditions our method builds upon our explicitly parametrized control formulae to avoid solving riccati or bezout equations at each time step most of the designs we present are state feedback but we present two benchmark designs with output feedback which have infinite relative degree

iterative finite difference solution analysis of acoustic wave equation in the laplace fourier domain we have investigated numerical characteristics of iterative solutions to the acoustic wave equation in the laplacefourier lf domain we transformed the time domain acoustic wave equation into the lf domain the transformed equation was discretized with finite differences and was solved with iterative methods finite difference modeling experiments demonstrate that iterative methods require an infinitesimal stopping tolerance to accurately compute the pressure field especially at long offsets to understand the requirement for such infinitesimal tolerance values we analyzed the evolution of intermediate solution vectors residual vectors and search direction vectors during the iteration the analysis showed that the requirement arises from the fact that in the solution space the amplitude of the pressure field varies more than sixty orders of magnitude on the common log scale accordingly we propose a rule of thumb for choosing a proper stopping tolerance value we also examined numerical dispersion errors in terms of the grid sampling resolutions per skin depth and wavelength we found that despite the similarity of the form of the acoustic wave and electromagnetic diffusion equations the former is different from the latter due to the fact that in the lf domain the skin depth of the acoustic wave equation is decoupled from its wavelength this aspect requires that in the lf domain its grid size be determined by considering the minimum grid sampling resolutions based not only the wavelength but also the skin depth

absorbing boundary conditions for elastic waves absorbing boundary conditions are needed for computing numerical models of wave motions in unbounded spatial domains the boundary conditions developed here for elastic waves are generalizations of ones developed earlier for acoustic waves these conditions are based on compositions of simple first order differential operators the formulas can be applied without modification to problems in both two and three dimensions the boundary conditions are stable for all values of the ratio of p wave velocity to s wave velocity and they are effective near a free surface and in a horizontally stratified medium the boundary conditions are approximated with simple finite difference equations that use values of the solution only along grid lines perpendicular to the boundary this property facilitates implementation especially near a free surface and at other corners of the computational domain

highly accurate absorbing boundary conditions for wide angle wave equations we develop a new class of absorbing boundary conditions abcs to prevent unwanted artifacts and wraparounds associated with aperture truncation in migration modeling using high order one way wave equations the fundamental approach behind the proposed development is the efficient discretization of the half space beyond the boundary of interest using midpoint integrated imaginary finite elements an idea recently utilized in the development of effective one way wave equations the proposed absorbing boundary conditions essentially add absorbing layers at the aperture truncation points we derive the absorbing boundary conditions analyze their properties and develop a stable explicit finite difference scheme to solve the downward continuation problem modified by these boundary conditions with the help of numerical examples we conclude that with as few as three absorbing layers i e two additional gridpoints the waves can be absorbed completely thus preventing associated artifacts

unified formulation of radiation conditions for the wave equation a family of radiation boundary conditions for the wave equation is derived by truncating a rational function approximation of the corresponding plane wave representation and it is demonstrated how these boundary conditions can be formulated in terms of fictitious surface densities governed by second order wave equations on the radiating surface several well established radiation boundary conditions appear as special cases corresponding to different choice of the coefficients in the rational approximation the relation between these choices is established and an explicit formulation in terms of selected directions with ideal transmission is presented a mechanical interpretation of the fictitious surface densities enables identification of suitable conditions at corners and boundaries of the radiating surface numerical examples illustrate excellent results with one or two fictitious layers with suitable corner and boundary conditions

on convergence of the nonlinear active disturbance rejection control for mimo systems in this paper the global and semiglobal convergence of the nonlinear active distur bance rejection control adrc for a class of multi input multi output nonlinear systems with large uncertainty that comes from both dynamical modeling and external disturbance are proved as a result a class of linear systems with external disturbance that can be dealt with by the adrc is classified the adrc is then compared both analytically and numerically to the well known in ternal model principle a number of illustrative examples are presented to show the efficiency and advantage of the adrc in dealing with unknown dynamics and in achieving fast tracking with lower overstriking

new full wave phase shift approach to solve the helmholtz acoustic wave equation for modeling we have developed a method of solving the helmholtz equation based on a new way to generalize the one way wave equation impose correct boundary conditions and eliminate exponentially growing evanescent waves the full two way nature of the helmholtz equation is included but the equation is converted into a pseudo one way form in the framework of a generalized phase shift structure consisting of two coupled first order partial differential equations for wave propagation with depth a new algorithm based on the particular structure of the coupling between p z and 2p z2 is introduced to treat this problem by an explicit approach more precisely in a depth marching strategy the wave operator is decomposed into the sum of two matrices the first one is a propagator in a reference velocity medium whereas the second one is a perturbation term which takes into account the vertical and lateral variation of the velocity the initial conditions are generated by solving the lippmann schwinger integral e

on convergence of tracking differentiator the tracking differentiator was first proposed by han in 1989 and the proof of convergence was presented the first time in han and wang han j q and wang w 1994 nonlinear tracking differentiator journal of systems science and mathematical science 14 177 183 in chinese unfortunately the proof there is incomplete this problem has been open for over two decades in this article we give a rigorous proof under some additional conditions an application for online estimation of the unknown frequencies for the finite sum of the sinusoidal signals is presented the numerical simulations illustrate the effectiveness of the estimation for both linear and nonlinear tracking differentiators

wavefield extrapolation by nonstationary phase shift the phase shift method of wavefield extrapolation applies a phase shift in the fourier domain to deduce a scalar wavefield at one depth level given its value at another the phase shift operator varies with frequency and wavenumber and assumes constant velocity across the extrapolation step we use nonstationary filter theory to generalize this method to nonstationary phase shift nsps which allows the phase shift to vary laterally depending upon the local propagation velocity for comparison we derive an analytic form for the popular phase shift plus interpolation pspi method in the limit of an exhaustive set of reference velocities nsps and this limiting form of pspi can be written as generalized fourier integrals which reduce to ordinary phase shift in the constant velocity limit in the x domain these processes are the transpose of each other however only nsps has the physical interpretation of forming the scaled linear superposition of laterally variable impulse responses i e huygen

on the convergence of an extended state observer for nonlinear systems with uncertainty abstract the extended state observer first proposed by jingqing han in j q han a class of extended state observers for uncertain systems control decis 10 1 1995 85 88 in chinese is the key link toward the active disturbance rejection control that is taking off as a technology after numerous successful applications in engineering unfortunately there is no rigorous proof of convergence to date in this paper we attempt to tackle this long unsolved extraordinary problem the main idea is to transform the error equation of objective system with its extended state observer into a asymptotical stable system with a small disturbance for which the effect of total disturbance error is eliminated by the high gain

absorbing boundary conditions for acoustic and elastic waves in stratified media abstract absorbing boundary conditions are needed for computing numerical models of wave motions in unbounded spatial domains prior progress on this problem for acoustic and elastic waves has generally been concerned with waves propagating through uniform media the present paper is concerned with waves in stratified media which are of interest for example in geophysical problems suppose that the medium consists of homogeneous layers separated by parallel horizontal interfaces and suppose that absorbing boundary conditions are needed along a vertical computational boundary the boundary conditions that are described in this paper are based on a quantity known as the ray parameter according to snell s law this parameter remains the same when a plane wave propagates through a stratified medium and undergoes reflection refraction and in the case of elastic waves conversion once can therefore use the same absorbing boundary conditions in all layers for acoustic waves the absorption properties are the same in all layers for elastic waves the absorption properties vary somewhat from one layer to another however one still obtains good absorption in all layers even in the presence of strong contrasts between layers the boundary conditions are also effective in absorbing rayleigh waves which propagate along free surfaces of elastic media the boundary formulas developed here can be applied without modification to problems in both two and three dimensions

adaptive boundary control for unstable parabolic pdes part ii estimation based designs the certainty equivalence approach to adaptive control is commonly used with two types of identifiers passivity based identifiers and swapping identifiers the passive also known as observer based approach is the prevalent identification technique in existing results on adaptive control for pdes but has so far not been used in boundary control problems the swapping approach prevalent in finite dimensional adaptive control is employed here for the first time in adaptive control of pdes for a class of unstable parabolic pdes we prove a separation principle result for both the passive and swapping identifiers combined with the backstepping boundary controllers the result is applicable in any dimension for physical reasons we restrict our attention to dimensions no higher than three the results of the paper are illustrated by simulations

artificial boundary conditions for 2d problems in geophysics abstract the numerical solution of a class of two dimensional problems in geophysics is considered the earth is modeled as a semi infinite acoustic or elastic medium in order to make the computational domain finite a semi circular artificial boundary ob is introduced thus truncating the original semi infinite domain an exact nonlocal relation is then derived on ob this relation is used as a boundary condition on ob which is combined with the finite element scheme employed in the computational domain the approach is applied to the helmholtz equation modeling time harmonic acoustic waves and to problems in elastostatics numerical examples validate the method and demonstrate its performance

a boundary condition in pade series for frequency domain solution of wave propagation in unbounded domains a boundary condition satisfying the radiation condition at infinity is frequently required in the numerical simulation of wave propagation in an unbounded domain in a frequency domain analysis using finite elements this boundary condition can be represented by the dynamic stiffness matrix of the unbounded domain defined on its boundary a method for determining a pade series of the dynamic stiffness matrix is proposed in this paper this method starts from the scaled boundary finite element equation which is a system of ordinary differential equations obtained by discretizing the boundary only the coefficients of the pade series are obtained directly from the ordinary differential equations which are not actually solved for the dynamic stiffness matrix the high rate of convergence of the pade series with increasing order is demonstrated numerically this technique is applicable to scalar waves and elastic vector waves propagating in anisotropic unbounded domains of irregular geometry it can be combined seamlessly with standard finite elements copyright 2006 john wiley sons ltd

attenuating boundary conditions for numerical modeling of acoustic wave propagation four types of boundary conditions dirichlet neumann transmitting and modified transmitting are derived by combining the damped wave equation with corresponding boundary conditions the dirichlet attenuating boundary condition is the easiest to implement for an appropriate choice of attenuation parameter it can achieve a boundary reflection coefficient of a few percent in a one wavelength wide zone the neumann attenuating boundary condition has characteristics similar to the dirichlet attenuating boundary condition but it is numerically more difficult to implement both the transmitting boundary condition and the modified transmitting boundary condition need an absorbing boundary condition at the termination of the attenuating region the modified transmitting boundary condition is the most effective in the suppression of boundary reflections for multidimensional modeling there is no perfect absorbing boundary condition and an approximate absorbing boundary condition is used