transcendental and interpolation methods in simultaneous stabilization and simultaneous partial pole placement problems in this paper we investigate the existence of a compensator which simultaneously renders a given r tuple of multiinput multioutput p times m linear dynamical systems internally stable in particular we parametrize a set of simultaneously stabilizable r tuples of plants and show that provided r leqq max m p the above set is semialgebraic and dense in the space sigma of r tuples of plants furthermore we also consider an extension of the classical pole placement and stabilization problems and investigate the simultaneous partial pole placement problem consequently we consider a suitable topology in and obtain a necessary condition and a sufficient condition for the generic partial pole placement problem surprisingly enough the problem of simultaneously stabilizing a triplet of m times m plants chosen generically is shown equivalent to the problem of partially pole placing a single m times m plant by a stable minimum phase compensator on the other hand the problem of simultaneous

on the partial stochastic realization problem we describe a complete parameterization of the solutions to the partial stochastic realization problem in terms of a nonstandard matrix riccati equation our analysis of this covariance extension equation cee is based on a complete parameterization of all strictly positive real solutions to the rational covariance extension problem answering a conjecture due to georgiou 1987 in the affirmative we also compute the dimension of partial stochastic realizations in terms of the rank of the unique positive semidefinite solution to the cee yielding some insights into the structure of solutions to the minimal partial stochastic realization problem by combining this parameterization with some of the classical approaches in partial realization theory we are able to derive new existence and robustness results concerning the degrees of minimal stochastic partial realizations as a corollary to these results we note that in sharp contrast with the deterministic case there is no generic value of the degree of a minimal stochastic realization of partial covariance sequences of fixed length

three dimensional frequency domain full waveform inversion with an iterative solver with the acquisition of wide aperture seismic data sets full waveform inversion is an attractive method for deriving velocity models three dimensional implementations require an efficient solver for the wave equation computing 3d time harmonic responses with a frequency domain solver is complicated because a large linear system with negative and positive eigenvalues must be solved time domain schemes are an alternative nevertheless existing frequency domain iterative solvers with an efficient preconditioner are a viable option when full waveform inversion is formulated in the frequency domain an iterative solver with a multigrid preconditioner is competitive because of a high order spatial discretization numerical examples illustrated the efficiency of the iterative solvers three dimensional full waveform inversion was then studied in the context of deep water ocean bottom seismometer acquisition three dimensional synthetic data inversion results showed the behavior of full waveform inversion with respect to the initial model and the minimum frequency available in the data set results on a 3d real ocean bottom seismometer data set demonstrated the relevance of full waveform inversion especially to image the shallow part of the model

on the partial realization problem abstract in this paper we take a unified approach to the partial realization problem in which we seek to incorporate ideas from numerical linear algebra most of which were originally developed in other contexts we approach the partial realization problem from several different angles and explore the connections to such topics as factoriza tion of hankel matrices block tridiagonalization generalizations of the lanczos process for biorthogonalization the euclidean algorithm and the principal part con tinued fractions of arne magnus the pade table and the berlekamp masseyalgo rithm in this way we are able to clarify some previous results by rissanen kalman and others and place them in a broader context this leads to several results and concepts which we think are new our analysis is restricted to the scalar case but some definitions and formulations have been rigged to facilitate an extension to the matrix case

semidefinite characterization of sum of squares cones in algebras we extend nesterov s semidefinite programming characterization of squared functional systems and faybusovich s abstraction to bilinear symmetric maps to cones of sum of squares elements in general abstract algebras using algebraic techniques such as isomorphism linear isomorphism tensor products sums and direct sums we show that many concrete cones are in fact sum of squares cones with respect to some algebra and thus are representable by the cone of positive semidefinite matrices we also consider nonnegativity with respect to a proper cone mathcal k and show that in some cases cones of mathcal k nonnegative functions are either sum of squares or at least semidefinite representable for example we show that some well known chebyshev systems when extended to euclidean jordan algebras induce cones that are semidefinite representable finally we will discuss some concrete examples and applications including minimum ellipsoid enclosing given space curves minimization of eigenvalues of po

dispersion analysis of an average derivative optimal scheme for laplace domain scalar wave equation laplace domain modeling is an important foundation of laplace domain full waveform inversion however dispersion analysis for laplace domain numerical schemes has not been completely established this hampers the construction and optimization of laplace domain modeling schemes by defining a pseudowavelength as a scaled skin depth i establish a method for laplace domain numerical dispersion analysis that is parallel to its frequency domain counterpart this method is then applied to an averagederivative nine point scheme for laplace domain scalar wave equation within the relative error of 1 the laplace domain average derivative optimal scheme requires four grid points per smallest pseudowavelength whereas the classic five point scheme requires 13 grid points per smallest pseudowavelength for general directional sampling intervals the average derivative optimal scheme is more accurate than the classic five point scheme for the same sampling intervals by using much smaller sampling intervals the classic fivepoint scheme can approach the accuracy of the averagederivative optimal scheme but the corresponding cost is much higher in terms of storage requirement and computational time

a new formulation of q markov covariance equivalent realization the q markov covariance equivalent realization q markov cover is a state space realization which matches the first q markov parameters and covariance parameters of a system king desai and skelton 5 developed a generalized approach to generate q markov covers desai and skelton 3 presented a recursive computational algorithm for generating q markov covers when q is very large based on the works of the above two papers we developed a new formula for generating q markov covers and also an algorithm to compute them when q is very large the new formula has been applied for model identification of large flexible structures by liu and skelton 6 8 compared with previously developed formulas and algorithms the new formula and algorithm are relatively straightforward and easy to program this formula may simplify the further analytical study of q markov covers and its application to system identification

linear matrix inequality formulation of spectral mask constraints with applications to fir filter design the design of a finite impulse response fir filter often involves a spectral mask that the magnitude spectrum must satisfy the mask specifies upper and lower bounds at each frequency and hence yields an infinite number of constraints in current practice spectral masks are often approximated by discretization but in this paper we derive a result that allows us to precisely enforce piecewise constant and piecewise trigonometric polynomial masks in a finite and convex manner via linear matrix inequalities while this result is theoretically satisfying in that it allows us to avoid the heuristic approximations involved in discretization techniques it is also of practical interest because it generates competitive design algorithms based on interior point methods for a diverse class of fir filtering and narrowband beamforming problems the examples we provide include the design of standard linear and nonlinear phase fir filters robust chip waveforms for wireless communications and narrowband beamformers for linear antenna arrays our main result also provides a contribution to system theory as it is an extension of the well known positive real and bounded real lemmas

convex optimization problems involving finite autocorrelation sequences we discuss convex optimization problems in which some of the variables are constrained to be finite autocorrelation sequences problems of this form arise in signal processing and communications and we describe applications in filter design and system identification autocorrelation constraints in optimization problems are often approximated by sampling the corresponding power spectral density which results in a set of linear inequalities they can also be cast as linear matrix inequalities via the kalman yakubovich popov lemma the linear matrix inequality formulation is exact and results in convex optimization problems that can be solved using interior point methods for semidefinite programming however it has an important drawback to represent an autocorrelation sequence of length n it requires the introduction of a large number n n 1 2 of auxiliary variables this results in a high computational cost when general purpose semidefinite programming solvers are used we present a more efficient implementation based on duality and on interior point methods for convex problems with generalized linear inequalities

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

robust stabilizability for a class of transfer functions this paper is concerned with the robust stabilizability for single input single output plants robust stabilizability means that a fixed controller can stabilize simultaneously all the plants in a given class which is characterized by a frequency dependent uncertainty band function around the transfer function of a nominal model a necessary and sufficient condition for robust stabilizability is derived based on the well known nevanlinna pick theory in classical analysis it is shown that the values of the uncertainty band function should be restricted within a certain range at the unstable poles of the nominal model in order for the class to be robustly stabilizable a procedure of synthesizing a robust stabilizer is given and the parametrization of all the robust stabilizers is also shown

from finite covariance windows to modeling filters a convex optimization approach the trigonometric moment problem is a classical moment problem with numerous applications in mathematics physics and engineering the rational covariance extension problem is a constrained version of this problem with the constraints arising from the physical realizability of the corresponding solutions although the maximum entropy method gives one well known solution in several applications a wider class of solutions is desired in a seminal paper georgiou derived an existence result for a broad class of models in this paper we review the history of this problem going back to carath eodory as well as applications to stochastic systems and signal processing in particular we present a convex optimization problem for solving the rational covariance extension problem with degree constraint given a partial covariance sequence and the desired zeros of the shaping filter the poles are uniquely determined from the unique minimum of the corresponding optimization problem in this way we obtain an algorithm for solving the covariance extension problem as well as a constructive proof of georgiou s existence result and his conjecture a generalized version of which we have recently resolved using geometric methods we also survey recent related results on constrained nevanlinna pick interpolation in the context of a variational formulation of the general moment problem

a covariance extension approach to identification of time series in this paper we consider a three step procedure for identification of time series based on covariance extension and model reduction and we present a complete analysis of its statistical convergence properties a partial covariance sequence is estimated from statistical data then a high order maximum entropy model is determined which is finally approximated by a lower order model by stochastically balanced model reduction such procedures have been studied before in various combinations but an overall convergence analysis comprising all three steps has been lacking supposing the data is generated from a true finite dimensional system which is minimum phase it is shown that the transfer function of the estimated system tends in h to the true transfer function as the data length tends to infinity if the covariance extension and the model reduction is done properly the proposed identification procedure and some variations of it are evaluated by simulations

modified nevanlinna pick interpolation and feedback stabilization of linear plants with uncertainty in the gain factor abstract this paper is a sequel to an earlier work in which we applied classical novnnlinna pick interpolation to the stabilization of a single input output plant with uncertainly in the gain factor by means of an asymptotically stable control and feedback sensor in this present work we show that for the proper design we need a modification of the classical interpolation procedure a point not clearly brought out in our previous paper

subspace algorithms for the stochastic identification problem abstract in this paper we derive a new subspace algorithm to consistently identify stochastic state space models from given output data without forming the covariance matrix and using only semi infinite block hankel matrices the algorithm is based on the concept of principal angles and directions we describe how they can be calculated with qr and quotient singular value decomposition we also provide an interpretation of the principal directions as states of a non steady state kalman filter bank

a convex optimization approach to the rational covariance extension problem in this paper we present a convex optimization problem for solving the rational covariance extension problem given a partial covariance sequence and the desired zeros of the modeling filter the poles are uniquely determined from the unique minimum of the corresponding optimization problem in this way we obtain an algorithm for solving the covariance extension problem as well as a constructive proof of georgiou s seminal existence result and his conjecture a stronger version of which we have resolved in byrnes et al ieee trans automat control ac 40 1995 pp 1841 1857

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

accuracy of finite difference and finite element modeling of the scalar and elastic wave equations numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media and they promise to be invaluable in solving the full inverse problem this paper quantitatively compares finite difference and finite element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time domain and frequency domain techniques in addition to versatility and ease of implementation it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy to be of value a solution technique must be able to minimize 1 numerical attenuation or amplification 2 polarization errors 3 numerical anisotropy 4 errors in phase and group velocities 5 extraneous numerical parasitic modes 6 numerical diffraction and scattering and 7 errors and transmission coefficients this paper shows that in homogeneous media the

measures on double or resonant eigenvalues for linear schrodinger operator abstract in this paper we consider linear schrodinger operator with double or resonant eigenvalues the main result is the bound of the measure in a suitable space of functions of the potentials leading to such double or resonant eigenvalues namely we present measure type estimates evaluating neighborhoods of the so called double or resonant set

a generalized entropy criterion for nevanlinna pick interpolation with degree constraint we present a generalized entropy criterion for solving the rational nevanlinna pick problem for n 1 interpolating conditions and the degree of interpolants bounded by n the primal problem of maximizing this entropy gain has a very well behaved dual problem this dual is a convex optimization problem in a finite dimensional space and gives rise to an algorithm for finding all interpolants which are positive real and rational of degree at most n the criterion requires a selection of a monic schur polynomial of degree n it follows that this class of monic polynomials completely parameterizes all such rational interpolants and it therefore provides a set of design parameters for specifying such interpolants the algorithm is implemented in a state space form and applied to several illustrative problems in systems and control namely sensitivity minimization maximal power transfer and spectral estimation

a generalized approach to q markov covariance equivalent realizations for discrete systems abstract a new algorithm is developed which constructs q markov covariance equivalent realizations q markov covers of multivariable linear discrete time systems a q markov cover is a state realization which matches exactly a finite portion of the impulse response sequence markov parameters and output autocorrelation sequence covariance parameters of a linear system the algorithm generates the q markov covers starting with only the basic data namely the q markov parameters and the q covariance parameters the algorithm leads to a parameterization of all minimal stable q markov covers which can be constructed since the data are easily computed from a given state space model the algorithm presented will be useful for both realization and model reduction of multivariable linear discrete time systems an example is presented to demonstrate the algorithm

multiplicities of the eigenvalues of the schrodinger equation in any dimension soit l equation de schrodinger sur une region bornee d r d avec des conditions aux limites de dirichlet u qu u u d 0 ou q l r d et la frontiere d de d est suffisamment lisse on montre que l ensemble q de tous les potentiels q dans l r d qui sont tels qu il y ait au moins une valeur propre multiple est de premiere categorie

on the duality between filtering and nevanlinna pick interpolation positive real rational functions play a central role in both deterministic and stochastic linear systems theory as well as in circuit synthesis spectral analysis and speech processing for this reason results about positive real transfer functions and their realizations typically have many applications and manifestations in this paper we study certain manifolds and submanifolds of positive real transfer functions describing a fundamental geometric duality between filtering and nevanlinna pick interpolation not surprisingly then this duality while interesting in its own right has several corollaries which provide solutions and insight into some very interesting and intensely researched problems one of these is the problem of parameterizing all rational solutions of bounded degree of the nevanlinna pick interpolation problem which plays a central role in robust control and for which the duality theorem yields a complete solution in this paper we shall describe the duality theorem which we motivate in terms of both the interpolation problem and a fast algorithm for kalman filtering viewed as a nonlinear dynamical system on the space of positive real transfer functions we also outline a new proof of the recent solution to the rational nevanlinna pick interpolation problem using an algebraic topological generalization of hadamard s global inverse function theorem

experimental evidence showing that stochastic subspace identification methods may fail it is known that certain popular stochastic subspace identification methods may fail for theoretical reasons related to positive realness in fact these algorithms are implicitly based on the assumption that the positive and algebraic degrees of a certain estimated covariance sequence coincide in this paper we describe how to generate data with the property that this condition is not satisfied using these data we show through simulations that several subspace identification algorithms exhibit massive failure

identifiability and well posedness of shaping filter parameterizations a global analysis approach in this paper we study the well posedness of the problems of determining shaping filters from combinations of finite windows of cepstral coefficients covariance lags or markov parameters for example we determine whether there exists a shaping filter with a prescribed window of markov parameters and a prescribed window of covariance lags we show that several such problems are well posed in the sense of hadamard that is one can prove existence uniqueness identifiability and continuous dependence of the model on the measurements our starting point is the global analysis of linear systems where one studies an entire class of systems or models as a whole and where one views measurements such as covariance lags and cepstral coefficients or markov parameters from data as functions on the entire class this enables one to pose such problems in a way that tools from calculus optimization geometry and modern nonlinear analysis can be used to give a rigorous answer to such problems in an algorithm independent fashion in this language we prove that a window of cepstral coefficients and a window of covariance coefficients yield a bona fide coordinate system on the space of shaping filters thereby establishing existence uniqueness and smooth dependence of the model parameters on the measurements from data

the interpolation problem with a degree constraint the author previously 1983 1987 showed that there is a correspondence between nonnegative hermitian trigonometric polynomials of degree spl les n and solutions to the standard nevanlinna pick caratheodory interpolation problem with n 1 constraints which are rational and also of degree spl les n it was conjectured that the correspondence under suitable normalization is bijective and thereby that it results in a complete parametrization of rational solutions of degree spl les n the conjecture was proven by byrnes et al 1995 along with a detailed study of this parametrization however byrnes et al used a slightly restrictive assumption that the trigonometric polynomials are positive and accordingly the corresponding solutions have positive real part the purpose of the present note is to extend the result to the case of nonnegative trigonometric polynomials as well we present the arguments in the context of the general nevanlinna pick caratheodory fejer interpolation

semidefinite descriptions of cones defining spectral mask constraints we discuss in detail an additive structure of cones of trigonometric polynomials nonnegative on the union of a finite number of pairwise disjoint segments of the unit circle we derive new descriptions of these cones in terms of semidefinite constraints we explain the results of m krein and a nudelman m krein a nudelman the markov moment problem and extremal problems american mathematical society providence 1977 providing a description of dual cones in terms of the same type of constraints

realization of power spectra from partial covariance sequences in this paper we address the problem of realization of a spectral density function from incomplete information about the underlying stochastic process the standing assumption is the availability of an incomplete partial sequence of covariance samples of the process we study the set of rational extensions of this finite sequence to an infinite covariance function that agrees with the available samples the classical theory of orthogonal polynomials with respect to the unit circle and the theory of moments have been utilized extensively in a variety of engineering problems including the one we are dealing with these have been known to provide a unifying framework for a variety of current spectral estimation techniques maximum entropy method pisarenko s harmonic decomposition etc in this work we consider and study the set of all covariance realizations of dimension lower than or equal to the length of the partial sequence and equal to the dimension of the maximum entropy realization the me solution is a point in this set other points correspond to pole zero models a general formula is obtained for recursively updated pole zero models of dimension increasing with the data record information about the zeros is obtained from the asymptotic behavior of the partial autocorrelation coefficients our approach combines techniques from the theory of orthogonal polynomials on the unit circle the theory of moments and also techniques from degree theory topology our objective is to develop a theory that will provide a frame for constructing recursively pole zero realizations of increasing dimension

canonical correlation analysis approximate covariance extension and identification of stationary time series abstract in this paper we analyze a class of state space identification algorithms for time series based on canonical correlation analysis in the light of recent results on stochastic systems theory in principle these so called subspace methods can be described as covariance estimation followed by stochastic realization the methods offer the major advantage of converting the nonlinear parameter estimation phase in traditional arma models identification into the solution of a riccati equation but introduce at the same time some nontrivial mathematical problems related to positivity the reason for this is that an essential part of the problem is equivalent to the well known rational covariance extension problem therefore the usual deterministic arguments based on factorization of a hankel matrix are not valid for generic data something that is habitually overlooked in the literature we demonstrate that there is no guarantee that several popular identification procedures based on the same principle will not fail to produce a positive extension unless some rather stringent assumptions are made which in general are not explicitly reported in this paper the statistical problem of stochastic modeling from estimated covariances is phrased in the geometric language of stochastic realization theory we review the basic ideas of stochastic realization theory in the context of identification discuss the concept of stochastic balancing and of stochastic model reduction by principal subsystem truncation the model reduction method of desai and pal 1982 a realization approach to stochastic model reduction proc 1st decision and control conf pp 1105 1112 based on truncated balanced stochastic realizations is partially justified showing that the reduced system structure has a positive covariance sequence but is in general not balanced as a byproduct of this analysis we obtain a theorem prescribing conditions under which the subspace identification methods produce bona fide stochastic systems