loewner and bezout matrices abstract the notion of intertwining matrices introduced recently by the authors is used to study loewner matrices as in the case of hankel and bezout matrices compatible with a polynomial f of degree not necessarily equal to the size of the matrices the theory requires a careful handling of polynomials with a root at infinity the authors establish first some technical results essentially of a projective character concerning divisibility of such polynomials these are then used to represent products of the form b1b 12b3b 14 b 12rb2r 1 where the bj are bezoutian matrices corresponding to a polynomial f again in the form of a bezoutian matrix corresponding to the same f this result can be deduced from the barnett formula if the degree of f equals the size of the matrices but requires the intertwining theory if this assumption is dropped among other results a strengthening and a conceptual proof of a recent result of z vavrin on inverses of loewner matrices is given in the last section the authors investigate analogies of barnett type formulae for bezoutian matrices b f g where the degree of f is allowed to be smaller than the size of the matrix the barnett formula does not extend immediately but it can be shown that an analogous formula holds asymptotically in the general case

frequency domain design of pid controllers for stable and unstable systems with time delay an approach for tuning pid type controllers is developed for single input single output linear time invariant systems based on an extension to the method of d partition this method permits design for simultaneous minimum gain and phase margin requirements it also allows design for specified maximum gain and phase cross over frequencies of the controlled system the technique can be applied to systems with stable or unstable plants as well as to irrational systems with significant time delay another advantage of the method is that it can be used for various controller configurations including derivative in the feedback path three examples illustrate the tuning method

lowner based blind signal separation of rational functions with applications a new blind signal separation bss technique is proposed enabling a deterministic separation of signals into rational functions rational functions can take on a wide range of forms such as the well known pole like shape the approach is a possible alternative for the well known independent component analysis when the theoretical sources are not independent such as for frequency spectra or when only a small number of samples is available the technique uses a low rank decomposition on the tensorized version of the observed data matrix the deterministic tensorization with lowner matrices is comprehensively analyzed in this paper uniqueness properties are investigated and a connection with the separation into exponential polynomials is made finally the technique is illustrated for fetal electrocardiogram extraction and with an application in the domain of fluorescence spectroscopy enabling the identification of chemical analytes using only a single excitation emission matrix

inversion formulas and fast algorithms for lowner vandermonde matrices abstract matrices consisting of two parts one of vandermonde and the other of lowner type are considered inversion formulas also for one sided inverses are presented moreover in the nonsingular case the parameters of the inversion formulas are described in terms of the evaluation of numerator and denominator polynomials which constitute the solution of associated rational interpolation problems these results can be used to establish an algorithm for the solution of lowner vandermonde systems of equations with o n log 2 n complexity

the general rational interpolation problem in the scalar case and its hankel vector abstract this paper deals with the general rational interpolation problem grip in the scalar case in the recent work of antoulas ball kang and willems the general solution to the grip has been derived in the framework of linear fractions using the so called generating matrix as the main tool within this framework the contribution here consists in a new approach of computing a particular 2 2 polynomial generating matrix based on the deep connection between loewner and hankel matrices it turns out that given a fixed grip we can readily obtain a unique vector called the hankel vector such that both of the allowable mcmillan degrees of interpolants to this grip and the corresponding polynomial generating matrix are completely determined by this hankel vector combined with its characteristic degrees and characteristic polynomials

rational approximations to the zeta function this article describes a sequence of rational functions which converges locally uniformly to the numerators and denominators of these rational functions can be expressed as characteristic polyn

brief paper a parameter space approach to constrained variance pid controller design this paper presents a parameter space approach to constrained variance and minimum variance pid controller design for lti models the technique is based on rational transfer functions of the plant and noise models loci corresponding to a fixed variance can be mapped into parameter planes for pid and pi type controllers to graphically display regions which satisfy the constraint thereby guiding a design to appropriate controller gains requirements for tracking regulation can be readily achieved without excessively increasing the output variance the usual advantages of parameter space techniques apply where other design criteria may be superimposed allowing multiple objectives to be achieved non conservatively the superposition of the parameter space boundaries from existing robust control techniques thus allows non conservative robust minimum variance pid design a design example compares the technique to an algebraic minimum variance design using an integrator when tracking is required

on invertibility of nonsquare generalized bezoutians abstract conditions for inverting h bezoutians of nonsquare matrix polynomials are studied necessary and sufficient conditions for invertibility of the anderson jury bezoutian are obtained an application to the inversion of block hankel matrices is given

comments on a regulating and tracking pi d controller eitelberg 1987 presents a modification of the classical pid controller which amounts to a two degree of freedom structure beside the fact that this modification is already known from the literature its potentials are not fully exploited by eitelberg based on a closed loop interpretation of the approach a systematic tuning procedure and an additional extension for pi controllers are presented

a subclass of symmetric loewner matrices abstract the subclass of real symmetric loewner matrices with fixed interpolation nodes and row sums zero is studied it is shown that it forms a commutative semigroup with respect to multiplication and contains with each element its moore penrose inverse as well

confluent cauchy and cauchy vandermonde matrices abstract the s c confluent cauchy and confluent cauchy vandermonde matrices are introduced corresponding to an interpolation problem with multiple interpolation nodes inversion formulas are derived and a factorization of confluent loewner matrices by means of confluent cauchy vandermonde matrices is described

hankel and loewner matrices abstract mutual relations between the hankel toeplitz bezout and loewner matrices as well as further connections to rational interpolation and projective geometry are investigated

a stabilized superfast solver for indefinite hankel systems abstract we present a stabilized superfast solver for indefinite hankel systems whose size is a power of 2 the hankel system is transformed into a loewner system which is solved by using an inversion formula for loewner matrices this explicit formula for the inverse of a loewner matrix contains certain parameters that are computed by solving two linearized rational interpolation problems on the unit circle the heart of our hankel solver is a superfast algorithm to solve these interpolation problems this algorithm is stabilized via pivoting iterative improvement and by giving the so called difficult interpolation points an adequate treatment we have implemented our algorithm in fortran 90 numerical examples illustrate the effectiveness of our approach

fourth order blind identification of underdetermined mixtures of sources fobium for about two decades numerous methods have been developed to blindly identify overdetermined p spl les n mixtures of p statistically independent narrowband nb sources received by an array of n sensors these methods exploit the information contained in the second order so the fourth order fo or both the so and fo statistics of the data however in practical situations the probability of receiving more sources than sensors increases with the reception bandwidth and the use of blind identification bi methods able to process underdetermined mixtures of sources for which p n may be required although such methods have been developed over the past few years they all present serious limitations in practical situations related to the radiocommunications context for this reason the purpose of this paper is to propose a new attractive bi method exploiting the information contained in the fo data statistics only that is able to process underdetermined mixtures of sources without the main limitations of the existing methods provided that the sources have different trispectrum and nonzero kurtosis with the same sign a new performance criterion that is able to quantify the identification quality of a given source and allowing the quantitative comparison of two bi methods for each source is also proposed in the paper finally an application of the proposed method is presented through the introduction of a powerful direction finding method built from the blindly identified mixture matrix

more on loewner matrices abstract the study of loewner matrices is continued in a direction close to the recent work by fiedler ptak and vav in all results are stated for the general case of multiple interpolation nodes i e for generalized loewner matrices both the loewner bezoutian connection of antoulas and anderson and the loewner hankel connection of vav in in the general case which are re proved in a simple way provide a good opportunity for finding various properties of loewner matrices some of them being generalizations of results published by other authors for simple nodes only there are also results which are new especially the barnett like formulas and an intertwining relation characteristic for loewner matrices as well as the loewner matrix hankel vector connection in order to handle loewner matrices further on the authors introduce and deal with a class of so called r matrices it turns out that some known results concerning bezoutians hankel matrices and their products can in a natural way be lifted to the level of r matrices and of the products of loewner and r matrices

a framework for the solution of the generalized realization problem in this paper we present a novel way of constructing generalized state space representations e a b c d of interpolants matching tangential interpolation data the loewner and shifted loewner matrices are the key tools in this approach

rational interpolation and state variable realizations abstract the problem is considered of passing from interpolation data for a real rational transfer function matrix to a minimal state variable realization of the transfer function matrix the tool is a loewner matrix which is a generalization of the standard hankel matrix of linear system realization theory and which possesses a decomposition into a product of generalized observability and controllability matrices

spectral analysis and the riemann hypothesis the explicit formulas of riemann and guinand weil relate the set of prime numbers with the set of nontrivial zeros of the zeta function of riemann we recall alain connes spectral interpretation of the critical zeros of the riemann zeta function as eigenvalues of the absorption spectrum of an unbounded operator in a suitable hilbert space we then give a spectral interpretation of the zeros of the dedekind zeta function of an algebraic number field k of degree n in an automorphic setting if k is a complex quadratic field the torical forms are the functions defined on the modular surface x such that the sum of this function over the gauss set of k is zero and eisenstein series provide such torical forms in the case of a general number field one can associate to k a maximal torus t of the general linear group g the torical forms are the functions defined on the modular variety x associated to g such that the integral over the subvariety induced by t is zero alternately the torical forms are the functions which are orthogonal to orbital series on x we show here that the riemann hypothesis is equivalent to certain conditions bearing on spaces of torical forms constructed from eisenstein series the torical wave packets furthermore we define a hilbert space and a self adjoint operator on this space whose spectrum equals the set of critical zeros of the dedekind zeta function of k

on two variable rational interpolation abstract the goal of this contribution is to investigate interpolation of two variable rational functions the tool is the two variable loewner matrix which is an extension of its single variable counterpart the main property of the loewner matrix is that its rank encodes the information concerning minimal complexity interpolants both polynomial and generalized state space descriptor realizations of interpolants are presented examples illustrate how two variable rational functions can be recovered from appropriate measurements

a unified approach to loewner and hankel matrices abstract we introduce so called generalized loewner matrices the definition is almost the same as antoulas and anderson s our class of generalized loewner matrices contains both classical loewner and hankel matrices as special cases the properties of such matrices are studied together with the associated interpolation problem the detailed characterization of its nonsolvable cases is given by means of the s c minimal generalized solution by means of rank properties of the loewner matrix and by means of the s c l polynomial introduced by fiedler

a tensor based method for large scale blind source separation using segmentation many real life signals are compressible meaning that they depend on much fewer parameters than their sample size in this paper we use low rank matrix or tensor representations for signal compression we propose a new deterministic method for blind source separation that exploits the low rank structure enabling a unique separation of the source signals and providing a way to cope with large scale data we explain that our method reformulates the blind source separation problem as the computation of a tensor decomposition after reshaping the observed data matrix into a tensor this deterministic tensorization technique is called segmentation and is closely related to hankel based tensorization we apply the same strategy to the mixing coefficients of the blind source separation problem as in many large scale applications the mixture is also compressible because of many closely located sensors moreover we combine both strategies resulting in a general technique that allows us to exploit the underlying compactness of the sources and the mixture simultaneously we illustrate the techniques for fetal electrocardiogram extraction and direction of arrival estimation in large scale antenna arrays

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

robust gamma stability analysis in a plant parameter space abstract given a characteristic polynomial whose coefficients depend polynomially on l uncertain parameters the following robustness problem arises determine whether all the roots of the polynomial are located in a prescribed region in the complex plane for all admissible parameter values to this end the boundary of is mapped into the parameter space a necessary and sufficient condition for stability of an operating domain in parameter space is that it contains at least one stable point and is not intersected by the image of this condition may be tested graphically by gridding l 2 parameters and projecting all boundaries into a two dimensional subspace of the parameter space finally the method is applied to a track guided bus with uncertain mass and velocity

tensor based large scale blind system identification using segmentation many real life signals can be described in terms of much fewer parameters than the actual number of samples such compressible signals can often be represented very compactly with low rank matrix and tensor models the authors have adopted this strategy to enable large scale instantaneous blind source separation in this paper we generalize the approach to the blind identification of large scale convolutive systems in particular we apply the same idea to the system coefficients of finite impulse response systems this allows us to reformulate blind system identification as a structured tensor decomposition the tensor is obtained by applying a deterministic tensorization technique called segmentation on the observed output data exploiting the low rank structure of the system coefficients enables a unique identification of the system and estimation of the inputs we obtain a new type of deterministic uniqueness conditions moreover the compactness of the low rank models allows one to solve large scale problems we illustrate our method for direction of arrival estimation in large scale antenna arrays and neural spike sorting in high density microelectrode arrays

inversion of a block lo wner matrix in this paper we give a fast algorithm to compute the parameters of an inversion formula for any nonsingular block lowner matrix the connection with matrix rational interpolation is given

properties of stabilizing pid gain set in parameter space the properties of the pid gain set for which all the closed loop poles lie in the left half of the complex plane are examined on the parameter plane a new method of determining the increase in the number of unstable poles when the gain crosses the boundary of stability regions is proposed this method is useful for determining the number of unstable poles in each region these results are extended to gamma stability based on conformal mapping

on bezoutian reduction with the vandermonde matrix abstract the vandermonde matrix reduces by congruence the bezoutian matrix when the zeros of the highest degree associated polynomial are simple in this paper a generalization of this result to the case of multiple zeros is presented

note on products of bezoutians and hankel matrices abstract a quantitative theorem due to fiedler and ptak on products of bezoutians and their inverses is extended to a more general case using as the main tool an alternative to fiedler s assertion on the quasidirect decomposition of hankel matrices

brief fast calculation of stabilizing pid controllers a fast method for the computation of stabilizing pid controllers is proposed it is shown in particular that for a fixed value of the proportional term k p the resulting stabilizing pid compensators form a finite set of disjoint polyhedral sets in the parameter space a complete characterization and a practical way of finding such stabilizing regions are given moreover the bounds on stabilizing k p parameters are derived the method therefore has several advantages over recent results and is very promising for real time and or robust control applications

polynomials and hankel matrices abstract compatibility of a hankel n n matrix h and a polynomial f of degree m m n is defined if m n compatibility means that hc t f c f h where c f is t companion matrix of f with a suitable generalization of c f this theorem is generalized to the case that m n

polynomials compatible with a symmetric loewner matrix abstract several mutually equivalent conditions are proved for a polynomial of degree n and a symmetric n n matrix which form a basis for so called compatibility of these two objects one of the conditions yields a new parametric expression for a symmetric loewner matrix

random matrix theory and ζ 1 2 it we study the characteristic polynomials z u of matrices u in the circular unitary ensemble cue of random matrix theory exact expressions for any matrix size n are derived for the moments of z and z z and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log z as n in the limit we show that these two distributions are independent and gaussian costin and lebowitz 15 previously found the gaussian limit distribution for im log z using a different approach and our result for the cumulants proves a conjecture made by them in this case we also calculate the leading order n asymptotics of the moments of z and z z these cue results are then compared with what is known about the riemann zeta function s on its critical line re s 1 2 assuming the riemann hypothesis equating the mean density of the non trivial zeros of the zeta function at a height t up the critical line with the mean density of the matrix eigenvalues gives a connection between n and t invoking this connection our cue results coincide with a theorem of selberg for the value distribution of log 1 2 it in the limit t they are also in close agreement with numerical data computed by odlyzko 29 for large but finite t this leads us to a conjecture for the moments of 1 2 it finally we generalize our random matrix results to the circular orthogonal coe and circular symplectic cse ensembles

tuning of pid type controllers for stable and unstable systems with time delay abstract a graphical technique for tuning pid type controllers based on the method of d partition is proposed for single input single output linear time invariant systems the tuning method shifts all the roots of the characteristic equation of the controlled system to a certain desirable region in the left half of the complex plane lhp to achieve a specified least absolute and least relative stability margin the method can be used to tune a precompensator in a two degree of freedom structure so as to improve tracking behaviour the method can be used for both stable and unstable systems for systems with significant time delay and for various controller configurations including derivative in the feedback path

fourth order cumulant based blind identification of underdetermined mixtures in this paper we study two fourth order cumulant based techniques for the estimation of the mixing matrix in underdetermined independent component analysis the first method is based on a simultaneous matrix diagonalization the second is based on a simultaneous off diagonalization the number of sources that can be allowed is roughly quadratic in the number of observations for both methods explicit expressions for the maximum number of sources are given simulations illustrate the performance of the techniques

analysis of sparse representation and blind source separation in this letter we analyze a two stage cluster then l1 optimization approach for sparse representation of a data matrix which is also a promising approach for blind source separation bss in which fewer sensors than sources are present first sparse representation factorization of a data matrix is discussed for a given overcomplete basis matrix the corresponding sparse solution coefficient matrix with minimum l1 norm is unique with probability one which can be obtained using a standard linear programming algorithm the equivalence of the l1 norm solution and the l0 norm solution is also analyzed according to a probabilistic framework if the obtained l1 norm solution is sufficiently sparse then it is equal to the l0 norm solution with a high probability furthermore the l1 norm solution is robust to noise but the l0 norm solution is not showing that the l1 norm is a good sparsity measure these results can be used as a recoverability analysis of bss as discussed the basis matrix in this article is estimated using a clustering algorithm followed by normalization in which the matrix columns are the cluster centers of normalized data column vectors zibulevsky pearlmutter boll and kisilev 2000 used this kind of two stage approach in underdetermined bss our recoverability analysis shows that this approach can deal with the situation in which the sources are overlapped to some degree in the analyzed domain and with the case in which the source number is unknown it is also robust to additive noise and estimation error in the mixing matrix finally four simulation examples and an eeg data analysis example are presented to illustrate the algorithm s utility and demonstrate its performance

vandermonde factorization and canonical representations of block hankel matrices abstract we study to which extent well known facts concerning vandermonde factorization or canonical representation of scalar hankel matrices transfer to block hankel matrices with p q blocks it is shown that nonsingular block hankel matrices can be factored like in the scalar case into nonconfluent vandermonde matrices and that the theorem on full rank factorization of arbitrary hankel matrices transfers in a weak version to the 2 2 block case but not to larger block sizes in general the minimal rank of a vandermonde factorization both with finite nodes and affine is described in terms of the hankel matrix the main tools are realization partial realization and moebius transformations

intertwining and testing matrices corresponding to a polynomial abstract given a natural number n and a polynomial of degree m n a description is given of a class r n of n by n matrices the intertwining matrices for with the following properties let h n be the family of all n by n hankel matrices compatible with and consider the relation hr t rh then an n by n matrix h belongs to h n if and only if this relation holds for all r r n a characterization is also given of testing matrices a matrix r is said to be a testing matrix if the fulfillment of the above relation already implies that h h n

a fast method to diagonalize a hankel matrix abstract we consider a vandermonde factorization of a hankel matrix and propose a new approach to compute the full decomposition in o n 2 operations the method is based on the use of a variant of the lanczos method to compute a tridiagonal matrix whose eigenvalues are the modes generating the entries in the hankel matrix by adapting existing methods to solve for these eigenvalues and then for the coefficients one arrives at a method to compute the entire decomposition in o n 2 operations the method is illustrated with a simple numerical example

rational interpolation and recursive solution of lowner vandermonde systems of equations abstract rational interpolation problems for functions having numerator degree higher than denominator degree are connected with solutions of systems of equations the coefficient matrix of which has a mixed structure the first columns are of vandermonde type whereas the last columns form a lowner matrix here three term recursions for the rational interpolants are developed which can be translated into recurrence formulas for the solutions of homogeneous systems with such a coefficient matrix on this base an o n2 algorithm for the solution of n n nonhomogeneous lowner vandermonde systems is obtained

a regulating and tracking pi d controller traditionally pi controllers operate solely on the reference error signal modified reference error in the case of pid controllers in the forward path of the classical feedback loop they are designed either as regulating e g to reduce process disturbances or as tracking controllers e g to follow reference changes if both characteristics are important then a compromise design has to be found here a special connection of the classical pi d controller is proposed which allows a greater degree of independence in the design and practical tuning of the regulating and tracking features

exploiting efficient representations in large scale tensor decompositions decomposing tensors into simple terms is often an essential step toward discovering and understanding underlying processes or toward compressing data however storing the tensor and computing its

generalized cauchy vandermonde matrices abstract matrices of the form c v consisting of a generalized cauchy matrix and a generalized vandermonde matrix are considered using the displacement structure of these matrices inversion formulas and criteria are presented the interpretation of linear systems with such a coefficient matrix as tangential interpolation problems leads to the concept of fundamental matrix which is basic in this approach for fundamental matrices recursion formulas are established from them fast inversion algorithms emerge that work for arbitrary nonsingular matrices of this kind

recursive solution of lowner vandermonde systems of equations ii abstract an o n 2 algorithm for the solution of a linear system the n n coefficient matrix of which consists of a vandermonde and a lowner part is presented in the first part of the algorithm the corresponding vandermonde system is solved in the second part the last rows of the vandermonde matrix are successively replaced by rows of the lowner structure the algorithm is convenient if the vandermonde part is dominant the main tool is the connection of such systems with rational interpolation problems

blind source separation by sparse decomposition in a signal dictionary the blind source separation problem is to extract the underlying source signals from a set of linear mixtures where the mixing matrix is unknown this situation is common in acoustics radio medical signal and image processing hyperspectral imaging and other areas we suggest a two stage separation process a priori selection of a possibly overcomplete signal dictionary for instance a wavelet frame or a learned dictionary in which the sources are assumed to be sparsely representable followed by unmixing the sources by exploiting the their sparse representability we consider the general case of more sources than mixtures but also derive a more efficient algorithm in the case of a nonovercomplete dictionary and an equal numbers of sources and mixtures experiments with artificial signals and musical sounds demonstrate significantly better separation than other known techniques

optimization based algorithms for tensor decompositions canonical polyadic decomposition decomposition in rank l_r l_r 1 terms and a new generalization the canonical polyadic and rank l r l r 1 block term decomposition cpd and btd respectively are two closely related tensor decompositions the cpd and recently btd are important tools in psychometrics chemometrics neuroscience and signal processing we present a decomposition that generalizes these two and develop algorithms for its computation among these algorithms are alternating least squares schemes several general unconstrained optimization techniques and matrix free nonlinear least squares methods in the latter we exploit the structure of the jacobian s gramian to reduce computational and memory cost combined with an effective preconditioner numerical experiments confirm that these methods are among the most efficient and robust currently available for computing the cpd rank l r l r 1 btd and their generalized decomposition

the bezoutian the hankel matrix output feedback and system eigenstructure abstract given a minimal linear time invariant system for which the transfer matrix has a right rp 1 and a left d 1 n coprime matrix fraction description an inverse relation is established between the block hankel matrix formed from the markov parameters of the eigenstructure of p and d and the generalized bezoutian matrix formed from the polynomial matrices r p n and d furthermore it is shown that all eigenstructures that are assignable by output feedback form the same markov parameters and this suggests a new approach to eigenstructure assignment

transformation techniques for toeplitz and toeplitz plus hankel matrices part i transformations transformations of the form a c1ac2 are investigated that transform toeplitz and toeplitz plus hankel matrices into generalized cauchy matrices c1 and c2 are matrices related to the discrete fourier transformation or to various real trigonometric transformations combining these results with pivoting techniques in part ii algorithms for toeplitz and toeplitz plus hankel systems will be presented that are more stable than classical algorithms

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

remarks on complexity of polynomial and special matrix computations abstract we prove some inequalities on the asymptotic orders of complexity of the problems of polynomial and rational interpolation polynomial evaluation and inversion of loewner and toeplitz or hankel matrices in this connection an open problem remains whether the whole group of these related problems has the same order of complexity or not

frequency limited balanced truncation with low rank approximations in this article we investigate model order reduction of large scale systems using frequency limited balanced truncation which restricts the well known balanced truncation framework to prescribed frequency regions the main emphasis is put on the efficient numerical realization of this model reduction approach we discuss numerical methods to take care of the involved matrix valued functions the occurring large scale lyapunov equations are solved for low rank approximations for which we also establish results regarding the eigenvalues of their solutions these results and also numerical experiments will show that the eigenvalues of the lyapunov solutions in frequency limited balanced truncation often decay faster than those in standard balanced truncation moreover we show in further numerical examples that frequency limited balanced truncation generates reduced order models which are significantly more accurate in the considered frequency region

inversion of a generalized block loewner matrix the minimal partial realization and matrix rational interpolation problem abstract inversion theorems for generalized block loewner matrices with non square blocks are presented connections are made to the general i e hermite matrix rational interpolation theory of partial realization and its extensions theoretical and computational aspects of many other special types of matrices such as block hankel block toeplitz vandermonde block bezout etc

controllers of ziegler nichols type for unstable process with time delay abstract proportional p proportional plus integral pi and proportional plus integral plus derivative pid controllers are designed for a time delayed process having latency l and a single right half plane pole at s with l 1 explicit designs are derived from the concept of phase margin and an optimum stability approach for pi controllers is based on a parameter plane study an optimum gain margin design is also given for p control

brief analytical pid parameter expressions for higher order systems most approaches to feedback synthesis are numerical design techniques that produce their results in form of numerical values for the controller parameters in contrast analytical design techniques produce the controller parameters as explicit expressions that are a function of open loop system parameters and the desired closed loop advantages include fast commissioning in terms of physical system parameters gain scheduling without extra design effort and analytical sensitivity insights so far it has not been known how to derive analytical pid expressions for higher than second order systems the paper proposes a technique to address this problem analyses its properties demonstrates computational simplicity and shows that its performance is comparable to known numerical techniques

multiplication of diagonal transforms of loewner matrices abstract a new formula for multiplication of matrices arising as appropriate diagonal transformations of loewner matrices is derived in the case of symmetric loewner matrices it is shown that special subclasses form multiplicative groups some extensions of known results about polynomials associated with hankel and loewner matrices are given also a previously found formula of the author describing the inverses of loewner matrices is shown to be a consequence of the present results