determinative vertices of interval family with ωθ stability abstract in this article we pay attention to the stability of interval polynomials with regard to the region an analysis on how to generate the finite set whose stability implies robust stability of the entire family is given those results are intuitive and convenient to operate

parameter identification of nonlinear aeroelastic system with time delayed feedback control in this paper an enhanced response sensitivity approach is proposed for parameter identification of nonlinear time delay aeroelastic systems two common aeroelastic models are investigated namely

a new l curve for ill posed problems the truncated singular value decomposition is a popular method for the solution of linear ill posed problems the method requires the choice of a truncation index which affects the quality of the computed approximate solution this paper proposes that an l curve which is determined by how well the given data right hand side can be approximated by a linear combination of the first few left singular vectors or functions be used as an aid for determining the truncation index

comments with reply on robust stability of an interval plant with respect to a convex region in the complex plane by j shaw and j jayasuriya the commenter points out that the results of j shaw and j jayasuriya ibid vol 38 no 2 pp 284 287 feb 1993 were published by f j kraus and w truol int j contr vol 52 no 4 pp 967 983 1991 for robust stability of polynomial families with affline linear parameter dependence the authors agree with the commenter and point out that their own results derive from a paper they submitted for publication in 1990 but which has not yet been printed which contained arguments similar to those of the commenters

a class of weak kharitonov regions for robust stability of linear uncertain systems kharitonov s theorems are generalized to the problem of so called weak kharitonov regions for robust stability of linear uncertain systems given a polytope of characteristic polynomials p and a stability region d in the complex plane p is called d stable if the zeros of every polynomial in p are contained in d it is of interest to know whether the d stability of the vertices of p implies the d stability of p a simple approach is developed which unifies and generalizes many known results on this problem

properties of the entire set of hurwitz polynomials and stability analysis of polynomial families it is proved in this paper that all hurwitz polynomials of order not less than n form two simply connected borel cones in the polynomial parameter space based on this result edge theorems for hurwitz stability of general polyhedrons of polynomials and boundary theorems for hurwitz stability of compact sets of polynomials are obtained both cases of families of polynomials with dependent and independent coefficients are considered different from the previous ones our edge theorems and boundary theorems are applicable to both monic and nonmonic polynomial families and do not require the convexity or the connectivity of the set of polynomials moreover our boundary theorem for families of polynomials with dependent coefficients does not require the coefficient dependency relation to be affine

robust stability of an interval plant with respect to a convex region in the complex plane the determination of robust stability of interval polynomials with respect to an arbitrary closed convex contour in the left half complex plane is considered it is shown that the problem can be solved by checking only the root locations of a subset of the exposed edges of the family the number of exposed edges to be checked depends on the damping margin defined by the contour and the degree of interval polynomials but does not depend on the exact shape of the contour a simple easy to apply graphical approach determines the subset of the exposed edges an example illustrates the merit of the approach

robustness analysis a case study the stability test of polynomials whose coefficients depend multilinearly on interval parameters is considered the authors describe and compare four brute force solution approaches these are eigenvalue calculation zero exclusion from a specified value set algebraic tests of real and complex hurwitz roots and the parameter space method they are applied to a simple example with two parameters and third order polynomial an interesting feature of the example is that it can have an isolated unstable point the example may be useful as a benchmark for future approaches to the multilinear problem all four methods are shown to be feasible for the simple example but they require effort

directional stability radius a stability analysis tool for uncertain polynomial systems coefficients of characteristic polynomials for stable parametrically uncertain systems are allowed to perturb to some extent for stability stability radius is a useful tool to assess the allowance of the stability for the systems to enhance its usefulness we modify stability radius so that it takes into account of given restricted perturbations which we call directional stability radius for an application we show shifted hurwitz stability conditions and a stability analysis method for interval polynomial systems using the directional stability radii

on a root distribution criterion for interval polynomials h kokame and t mori 1991 and c b soh 1990 derived conditions under which an interval polynomial has a given number of roots in the open left half plane and the other roots in the open right half plane however the one shot test approach using sylvester s resultant matrices and bezoutian matrices implies that the implemented conditions are only sufficient not necessary for an interval polynomial to have at least one root in the open left half plane and open right half plane alternative necessary and sufficient conditions which only require the root locations of four polynomials to check the root distribution of an interval polynomial are presented

regularization parameter determination for discrete ill posed problems the straightforward solution of discrete ill posed linear systems of equations or least squares problems with error contaminated data does not in general give meaningful results because the propagated error destroys the computed solution the problems have to be modified to reduce their sensitivity to the error in the data the amount of modification is determined by a regularization parameter it can be difficult to determine a suitable value of the regularization parameter when no knowledge of the norm of error in the data is available this paper proposes a new simple technique for determining a value of the regularization parameter that can be applied in this situation it is based on comparing computed solutions determined by tikhonov regularization and truncated singular value decomposition analogous comparisons are proposed for large scale problems the technique for determining the regularization parameter implicity provides an estimate for the norm of the error in the data

does it suffice to check a subset of multilinear parameters in robustness analysis for polynomials with general multilinear dependence on uncertain real parameters it does not suffice to check a subset for q for stability

the backus gilbert method and their minimum norm solution abstractthis paper has two objectives one is to review the backus gilbert bg method backus and gilbert investigate continuous inverse problems in a radially symmetric earth using an approach completely different from that of standard inversion methods if m is an earth property of interest the bg method provides average values of m over some radial intervals if an interval w is relatively short and centered near a particular radius r0 the bg average will be a local average of m near r0 with a resolving length of w backus and gilbert also introduce the concept of trade off between resolution and error which is essential in inverse theory regardless of the inversion method in spite of its importance inverse theory texts neither describe the bg method in full nor show how to apply it in all its generality because the three papers that introduce the bg method are mathematically difficult this paper provides background material and a detailed derivation of the method as well as two completely wor

robust adaptive beamforming based on conjugate gradient algorithms the mismatches of signal and array geometry will seriously degrade the performance of adaptive beamformer in this paper we propose two methods for robust adaptive beamforming based on the conjugate gradient cg algorithm the proposed beamformers offer a significant improvement in the computational complexity while providing the same performance of the best robust beamformers at present the first method belongs to the diagonal loading technique we derive a diagonal loading cgls algorithm cg applied to normal equations and propose a simple method to choose the loading level based on a coarse estimation of the desired signal power this parameter free method can effectively reduce the signal self cancellation at high signal to noise ratio the second method belongs to the regularization technique since the cg algorithm has a regularizing effect with iteration number being the regularization parameter the stopping criterion plays an important role on the robustness we develop three fast stopping criteria for cg iteration which reduce the stopping complexity from o n or o n 2 to o 1 the former two are the fast versions of existing methods and the later one is new moreover the new criterion based on fast ritz value estimation has better performance than others

on p dunford integrable functions with values in banach spaces abstract let be a complete probability space x a banach space and 1 p in this paper we discuss several aspects of p dunford integrable functions f x special attention is paid to the compactness of the dunford operator of f we also study the p bochner integrability of the composition u f y where u is a p summing operator from x to another banach space y finally we also provide some tests of p dunford integrability by using w thick subsets of x

robust stability of control systems extreme point results for the stability of edges for the investigation of the robust stability of control systems with structured uncertainties many results have been presented recently that lead to stability tests of the edges of a polytope in this paper some results are discussed where the stability of an edge is guaranteed by the stability of the vertices of the edge given the characteristic polynomial of a closed loop system with structured uncertainties in the parameters for special types of uncertainties the family of polynomials is defined by a polytope in the space of parameters in the frequency domain the corresponding value set of the family for s jw fixed is a polygon whose edges are formed by the so called exposed edges of the polytope for the investigation of the stability of the polynomial family these exposed edges must be tested kraus and truol 1991 a this test can be simplified if an edge has the vertex property i e if the stability of the edge is guaranteed by the stability of the two vertices

new results on the separation of matrix eigenvalues and the clustering of matrix elements the problem of separation of matrix eigenvalues is considered in this paper necessary and sufficient conditions are given for a matrix to have all eigenvalues contained in an open set defined on the complex plane by a separable polynomial spl gamma z sub 1 z sub 2 the problem of clustering of matrix elements is also considered in this case a set of matrices is identified having all eigenvalues in a given subset of the complex plane the presented results can be useful in both robust stability analysis and the design of control systems

robust controller synthesis using kharitonov s theorem the authors present a method based upon kharitonov s theorem for synthesizing robustly stabilizing feedback controllers the approach is based upon a multiple plant model formulation with a quadratic cost functional sufficient conditions are obtained for characterizing robustly stabilizing static output feedback proportional controllers for mimo multiple input multiple output plants with denominator polynomial uncertainty

a homotopy method for solving bilevel programming problem abstract a homotopy method for solving bilevel programming problem blpp is proposed the algorithm is based on constructing the homotopy equation for blpp one satisfiable sufficient condition is given to ensure that the point of homotopy path globally converges to a k k t point of blpp numerical results are presented to show the effectiveness of this method

kharitonov regions it suffices to check a subset of vertex polynomials the authors deal with the d stability property of interval polynomials in particular they show that certain d domains are kharitonov regions that is the d stability of interval polynomials is implied by the d stability of all its vertex polynomials they then proceed to show that it suffices to check the d stability of a subset of the vertex polynomials

generalized pi controllability integral controllable systems can be stabilized by all sufficiently low gain integral controllers the concept of integral controllability is extended to two generalized stability regions and to proportional controllability and pi controllability with respect to those regions furthermore closed eigenvalue formulae are derived for the maximal low integral gain and the maximal low proportional gain for generalized stability related eigenvalue formulae are then used in combination to find all low gain pi controllers that place the closed loop eigenvalues in the desired region

a class of stability regions for which a kharitonov like theorem holds families of complex polynomials whose coefficients lie within given intervals are discussed in particular the problem of determining if all polynomials in a family have the property that all of their roots lie within a given region is discussed towards this end a notion of a kharitonov region is defined roughly speaking a kharitonov region is a region in the complex plane with the following property given any suitable family of polynomials in order to determine if all polynomials in the family have all of their roots in the region it suffices to check only the vertex polynomials of the family the main result is a sufficient condition for a given region to be a kharitonov region

comments on comments on stability analysis of a family of matrices by j h kim and z bien a flaw is pointed out in the proof of a necessary and sufficient condition for robust stability of a polytope of matrices given by j h kim and z bien ibid vol 387 no 9 pp 1470 1472 sep 1992 in their comments on a paper by the present authors ibid vol 35 no 11 pp 1257 1259 nov 1190 hence the necessary condition as proposed may not be correct

damping margins of interval polynomials the zero locations of interval polynomials are examined in particular it is shown that a family of interval polynomials will have zeros only in the left sector if the real and imaginary parts of four specially constructed complex polynomials have an interlacing real zero property this is significant for the analysis of uncertain systems as the computation cost associated with checking the zero locations of interval polynomials will be greatly reduced the results presented can be readily extended to more general stability regions where the real and imaginary parts of the polar plot are polynomial functions

sirt and cg type methods for the iterative solution of sparse linear least squares problems abstract sirt and cg type methods have been successfully employed for the approximate solution of least squares problems arising in tomography in this paper we study and compare their convergence and regularization properties it is pointed out that sirt methods apply an uncontrollable implicit rescaling which affects the statistical characteristics of the system whereas cg type methods do not for a large class of model problems it is shown that virtually the same solutions as obtained by sirt methods can be obtained by applying a cg type method to a properly rescaled system but with an amount of work proportional to the square root of the amount of work with sirt

structured and simultaneous lyapunov functions for system stability problems abstract it is shown that many system stability and robustness problems can be reduced to the question of when there is a quadratic lyapunov function of a certain structure which establishes stability of x ax for some appropriate a the existence of such a lyapunov function can be determined by solving a convex program we present several numerical methods for these optimization problems a simple numerical example is given

exploiting randomness in acoustic impulse responses to achieve headphone compensation through deconvolution this paper presents a method of headphone earphone equalization based upon deconvolution of the headphone impulse response from other acoustic filters in the processing chain the methods presented are thus applicable to areas such as spatial audio where input signals are processed with binaural impulse responses the extraction of low order from higher order acoustic impulse responses is justified based upon an application of the theory pertaining to the clustering of the zeros of random coefficient polynomials about the unit circle an argument is presented supporting the applicability of this theory to acoustic impulse responses such an equalization method guarantees a considerable reduction in computational effort over traditional separate compensating filters a comparison is presented of the implementation of the equalization using the homomorphic method and least squares method with and without regularization to a traditional separate least squares inverse filter the proposed approach is shown to have a performance comparable to a separate least squares designed compensator often giving superior performance despite a large reduction in complexity

the 2 norm of random matrices abstract numerical experiments show that it is possible to derive simple estimates for the expected 2 norm of random matrices a with elements from a normal distribution with zero mean and standard deviation and from a poisson distribution with mean value these estimates are max m n 2 max m n and e 2 mn respectively where m and n are the dimensions of a

constructing convex directions for stable polynomials new constructions of convex directions for hurwitz stable polynomials are obtained the technique is based on interpretations of the phase derivative conditions in terms of the sensitivity of the root locus associated with the even and odd parts of a polynomial

a generalization of the routh hurwitz stability criteria and an application to a problem in robust controller design the classical results of routh and hurwitz on the stability of polynomials are generalized from the open left half plane to certain other subregions g of the complex plane by our method arbitrary conic sections can be handled the results are interpreted as criteria for g stability of linear systems a numerical algorithm to compute the criteria is given an application to a problem in robust controller design is outlined

improved finite word length characteristics in digital control using delta operators this paper examines some of the consequences of finite word lengths in digital control it is shown that in many cases of practical importance the usual shift operator formulation is inferior to an alternative formulation which we designate the delta operator approach this latter approach is shown to give better coefficient representation and less roundoff noise in many cases we thus argue that the shift operator and its associated z transform can be replaced by delta operators and their associated transform which we designate a transform an added advantage of this approach is that discrete designs and transforms converge to their continuous time counterparts as the sampling rate is increased

estimating solutions of first kind integral equations with nonnegative constraints and optimal smoothing a method is presented for estimating solutions of fredholm integral equations of the first kind given noisy data regularization is effected by a smoothing term which is the l 2 norm of the estimate we propose a scheme by which an approximately optimal amount of smoothing may be computed based only on the data and the assumed known noise variances numerical examples are given for estimating inverse laplace transforms

strong kharitonov theorems for low order polynomials the root clustering property of low order interval polynomials is examined and it is shown that the number of polynomials required to check root clustering within the left sector may be reduced if the order of the interval polynomial is low specifically the reduction is possible if n phi

on damping ratio of polynomials with perturbed coefficients c b soh and c s berger 1988 derived a sufficient condition for a family of interval polynomials to have a damping ratio of phi using kharitonov s theorem for complex polynomials this paper points out that the transformations used by soh and berger to obtain the sufficient conditions also guarantee a simplification of kharitonov s theorem for complex polynomials that is the number of required polynomials to be hurwitz is half the number specified by soh and berger

nonconvex kharitonov regions an interesting nonconvex kharitonov region considered by m fu 1991 is reviewed it is shown that by means of a counterexample that the region identified is not a kharitonov region a different nonconvex kharitonov region is proposed

a wide class of ultrabornological spaces of measurable functions abstract our main result states that a bornological locally convex space having a suitable boolean algebra of projections is ultrabornological this general theorem whose proof is a variation of the sliding hump techniques used in diaz et al arch math basel 60 1993 73 78 diaz et al resultate math 23 1993 242 250 drewnowski el al proc amer math sec 114 1992 687 694 drewnowski et al atti sem mat fis univ modena41 1993 317 329 is applied to prove that some non complete normed spaces such as the spaces of dunford pettis or mcshane integrable functions as well as other interesting spaces of weakly or strongly measurable functions are ultrabornological we also give applications to vector valued sequence spaces in particular we prove that lp x 1 p

brief paper smooth trivial vector bundle structure of the space of hurwitz polynomials due to the importance of the robustness of stable systems the properties of the set of hurwitz real polynomials are interesting to study among these properties is that the set is an open and not a convex set and recently it was proved that the set of hurwitz polynomials of degree n with positive coefficients h n is contractible now we will prove that the set h n is a smooth trivial vector bundle over h n k of rank k

on kharitonov type theorems for real polynomials when is degree drop admissible a root locus approach is adopted to show that extension of kharitonov type theorem for real interval polynomial with degree drop is not only applicable to the open left half plane but also other regions such as the left shifted open left half plane and the left sector

analysis of discrete ill posed problems by means of the l curve when discrete ill posed problems are analyzed and solved by various numerical regularization techniques a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector in particular the graph associated with tikhonov regularization plays a central role the main purpose of this paper is to advocate the use of this graph in the numerical treatment of discrete ill posed problems the graph is characterized quantitatively and several important relations between regularized solutions and the graph are derived it is also demonstrated that several methods for choosing the regularization parameter are related to locating a characteristic l shaped corner of the graph

on an improved levin oscillatory quadrature method abstract performance of an improved levin quadrature method for oscillatory integrals is studied in the study the behavior of the target system of linear equations is analyzed and an error reduction factor is proposed to measure the behavior s impact on the integral result numerical investigations show that the error reduction factor is extremely small for ill conditioned case and the ill conditioning has little impact on the final integral result therefore the concerned quadrature method is numerically very stable and it has addressed the levin method s problem of being susceptible to the ill conditioning

closed loop hyperstability of interval plants this note examines the conditions for robust bounded input bounded output closed loop stability of interval plants when there is a nonlinear element in the feedback path specifically the authors present the robust generalized circle criterion and popov s criterion

comments with reply on stability of a polytope of matrices counterexamples by b r barmish et al the commenters show that the counterexample given in the above mentioned paper ibid vol 33 no 6 p 569 72 june 1988 does not invalidate the conjecture about the necessary and sufficient conditions for the stability of interval matrices comments on the relationship between interval matrices and polytopes of polynomials are also given the authors reply that their counterexample is valid and attribute the misunderstanding to insufficiently precise wording in their paper

recovering the density of a string from only lowest frequency data a classical inverse problem is can you hear the density of a string clamped at both ends the answer as narrowly stated is negative one has to give an additional data sequence such as the energy in each mode norming constants or information on the string slope at the endpoints or alternatively change the boundary conditions and obtain the new frequencies what if neither of these options is possible one solution recently proposed was to add a known mass density and remeasure the dirichlet spectrum w rundell and p e sacks siam j appl math 73 2013 pp 1020 1037 in this work we take a related approach we are able to add any number of known mass densities but we can only measure the fundamental frequency that is the lowest eigenvalue in each case we will show that our problem can be closely approximated by a related problem for which there is uniqueness and we discuss a suite of reconstruction methods

schur stability of interval polynomials a method for checking the schur stability of interval polynomials is presented the numbers of critical vertex and edge polynomials that are sufficient for inferring robust schur stability are obtained it is shown that the critical edges can be greatly reduced to the order of 2n sup 2

sar image regularization with fast approximate discrete minimization synthetic aperture radar sar images like other coherent imaging modalities suffer from speckle noise the presence of this noise makes the automatic interpretation of images a challenging task and noise reduction is often a prerequisite for successful use of classical image processing algorithms numerous approaches have been proposed to filter speckle noise markov random field mrf modelization provides a convenient way to express both data fidelity constraints and desirable properties of the filtered image in this context total variation minimization has been extensively used to constrain the oscillations in the regularized image while preserving its edges speckle noise follows heavy tailed distributions and the mrf formulation leads to a minimization problem involving nonconvex log likelihood terms such a minimization can be performed efficiently by computing minimum cuts on weighted graphs due to memory constraints exact minimization although theoretically possible is not achievable on large images required by remote sensing applications the computational burden of the state of the art algorithm for approximate minimization namely the alpha expansion is too heavy specially when considering joint regularization of several images we show that a satisfying solution can be reached in few iterations by performing a graph cut based combinatorial exploration of large trial moves this algorithm is applied to joint regularization of the amplitude and interferometric phase in urban area sar images

expansion of det a b and robustness analysis of uncertain state space systems in this note an expansion for the determinant of the sum of two matrices det a b is given a robustness analysis approach for state space systems with uncertain parameters is investigated by using kharitonov s theorem some useful results are obtained several illustrative examples are given

asymptotic theory of filtering for linear operator equations with discrete noisy data on considere les equations integrales de fredholm de premiere espece a noyaux continus avec echantillonnage des donnees et alteration par le bruit blanc on etablit une condition suffisante de convergence des methodes de filtrage

optimally scaled vector regularization method to solve ill posed linear problems instead of the tikhonov regularization method which with a scalar being the regularization parameter liu et al 1 have proposed a novel regularization method with a vector as being the regularization parameter as a continuation we further propose an optimally scaled vector regularization method osvrm to solve the ill posed linear problems which is better than the tikhonov regularization method the presently proposed vector regularization method is shown to be well conditioning to overcome the ill posedness of the linear equations system the osvrm causes a significant improvement of stability and accuracy in the numerical solution of ill posed linear problem and its convergence speed is as fast as by solving the well posed linear problem some tests of the linear inverse problems confirm the efficiency and accuracy of the osvrm

robust hurwitz l sup p stability of polynomials with complex coefficients necessary and sufficient conditions for robust stability of a family of polynomials with complex coefficients are provided this work generalizes a frequency domain criteria for robust stability of continuous linear systems reported on by y z tsypkin and b t polyak 1991

a survey of extreme point results for robustness of control systems abstract this paper surveys a subset of the body of research which was sparked by kharitonov s theorem the focal point is extreme point results for robust stability and robust performance that is we give conditions under which satisfaction of a performance specification is ascertained for a family of systems by only checking a finite subset of the extreme members of this family the results which are surveyed apply mainly to systems with structured real parametric uncertainty in addition a number of counterexamples are given to illustrate cases for which an extreme point result does not hold for such cases a solution via the so called edge theorem is often possible

absolutely summing operators and integration of vector valued functions abstract let be a complete probability space and u x y an absolutely summing operator between banach spaces we prove that for each dunford integrable i e scalarly integrable function f x the composition u f is scalarly equivalent to a bochner integrable function such a composition is shown to be bochner integrable in several cases for instance when f is properly measurable birkhoff integrable or mcshane integrable as well as when x is a subspace of an asplund generated space or a subspace of a weakly lindelof space of the form c k we also study the continuity of the composition operator f u f some other applications are given

stability conditions for polytopes of polynomials in considering the stability or performance of linear systems with uncertain parameters one is led to consider the simultaneous stability of families of polynomials for a general stability region in the complex plane the author gives a stability criterion for polytopes of polynomials in terms of the stability of a minimal number of corners and edges of the polytope the testing set of edges and corners depends entirely on the edge directions of the polytope hence the results are particularly applicable to problems with fixed directions but varying lengths and positions of the edges applications to robust pole placement of uncertain systems and computation of worst case h sub infinity norms are demonstrated

a note on two methods related to stability robustness of polynomials in a sector relative stability two recent results on the stability robustness of uncertain systems polynomials are addressed one gives a vertex test for left sector relative stability where the number of the required vertices is four for a real case and eight for a complex case the other gives an elegant frequency domain graphical hodograph approach for left plane stability where the plotting of only one two for a complex case hodograph plus simple boundary conditions is both necessary and sufficient to obtain the maximum coefficient perturbation bounds it is shown that a similar graphical test exists for the left sector stability case

efficient maximum entropy reconstruction of nuclear magnetic resonance t1 t2 spectra this paper deals with the reconstruction of t1 t2 correlation spectra in nuclear magnetic resonance relaxometry the ill posed character and the large size of this inverse problem are the main difficulties to tackle while maximum entropy is retained as an adequate regularization approach the choice of an efficient optimization algorithm remains a challenging task our proposal is to apply a truncated newton algorithm with two original features first a theoretically sound line search strategy suitable for the entropy function is applied to ensure the convergence of the algorithm second an appropriate preconditioning structure based on a singular value decomposition of the forward model matrix is used to speed up the algorithm convergence furthermore we exploit the specific structures of the observation model and the hessian of the criterion to reduce the computation cost of the algorithm the performances of the proposed strategy are illustrated by means of synthetic and real data processing

generalized edge theorem abstract in this paper we shall show that a family of analytic functions constructed from the convex hull of a finite number of vertex functions will have no zeros within a simply connected region in the complex plane if and only if all its edge functions have no zeros within the simply connected region the result is in fact a generalization of the edge theorem which has been derived for polynomial functions 1 2 7

parameter determination for tikhonov regularization problems in general form abstract tikhonov regularization is one of the most popular methods for computing an approximate solution of linear discrete ill posed problems with error contaminated data a regularization parameter 0 balances the influence of a fidelity term which measures how well the data are approximated and of a regularization term which dampens the propagation of the data error into the computed approximate solution the value of the regularization parameter is important for the quality of the computed solution a too large value of 0 gives an over smoothed solution that lacks details that the desired solution may have while a too small value yields a computed solution that is unnecessarily and possibly severely contaminated by propagated error when a fairly accurate estimate of the norm of the error in the data is known a suitable value of often can be determined with the aid of the discrepancy principle this paper is concerned with the situation when the discrepancy principle cannot be applied it then can be quite difficult to determine a suitable value of we consider the situation when the tikhonov regularization problem is in general form i e when the regularization term is determined by a regularization matrix different from the identity and describe an extension of the cose method for determining the regularization parameter in this situation this method has previously been discussed for tikhonov regularization in standard form i e for the situation when the regularization matrix is the identity it is well known that tikhonov regularization in general form with a suitably chosen regularization matrix can give a computed solution of higher quality than tikhonov regularization in standard form

some discrete time counterparts to kharitonov s stability criterion for uncertain systems in 1 kharitonov gave an elegant and simple stability criterion for continuous time systems this note reports on similar results for discrete time systems

schultz matrix iteration based method for stable solution of discrete ill posed problems abstract in this paper we propose an iterative method for solving discrete ill posed problems based on matrix iterations generated by schultz method and known to converge to the moore penrose pseudoinverse of a matrix practically letting the schultz matrix iteration be x k we construct the vector x k x k b where b is a data vector hence by construction the iterates converge to the minimum 2 norm solution of a least squares problem with coefficient matrix a and data vector b we derive theoretical properties of the sequence x k and show that it is quadratically convergent in the case of corrupted data we analyze the semi convergence behavior of the iterates and conclude that the iteration must be truncated to control the propagation of the noise error as a result we derive an error estimate for the case where the truncation parameter is chosen by the discrepancy principle in addition combining a projected approach with the new method we propose variants of the method that are well suited for large scale problems several numerical results are presented to illustrate the effectiveness of the method on well known test problems

an adaptive pruning algorithm for the discrete l curve criterion we describe a robust and adaptive implementation of the l curve criterion the algorithm locates the corner of a discrete l curve which is a log log plot of corresponding residual norms and solution norms of regularized solutions from a method with a discrete regularization parameter such as truncated svd or regularizing cg iterations our algorithm needs no predefined parameters and in order to capture the global features of the curve in an adaptive fashion we use a sequence of pruned l curves that correspond to considering the curves at different scales we compare our new algorithm to existing algorithms and demonstrate its robustness by numerical examples

inversion of nuclear magnetic resonance echo data based on maximum entropy abstractnuclear magnetic resonance nmr t2 inversion is an ill posed problem in which regularization techniques are usually adopted to suppress the oscillations caused by noise in the solutions the maximum entropy concept provides an unbiased way to obtain information from incomplete data and it implicitly imposes a positive constraint on probability distribution so we used the maximum entropy method to invert nmr echo data we have developed a simple and effective method for solving the objective function of the maximum entropy method first the solution was replaced by a positive function to achieve the positive constraint of the solution the objective function was converted to an unconstrained one and then the levenberg marquardt method was used to solve the newly obtained unconstrained objective function to suppress the highly tilted tail at the short relaxation time of the t2 distribution a modified or normalized shannon entropy function was used to replace the standard shannon entropy funct

robust stability of control systems with polytopical uncertainty a nyquist approach for technical plants a bounded continuum of parameter values describes admissible process models a robust controller guarantees the observance of minimal requirements for all of the allowed process models in this paper a frequency domain approach is proposed that permits the instability of a family of polynomials to be checked this method can be extended in an easy manner to investigate the instability of a closed loop the plant is thereby described by a family of models with a polytopical region of uncertainty in the parameter space as an important special case of the general d stability the hurwitz stability of the closed loop is investigated and a minimal set of necessary and sufficient conditions for robust hurwitz stability is given a generalization for other stability domains is possible besides an algebraic criterion based on a zero inclusion check for a minimal set of exposed edges of the polytopical region a graphic criterion of nyquist type can also be used

hurwitz testing sets for parallel polytopes of polynomials in considering robustness of linear systems with uncertain paramenters one is lead to consider simultaneous stability of families of polynomials efficient hurwitz stability tests for polytopes of polynomials have earlier been developed using evaluations on the imaginary axis this paper gives a stability criterion for parallel polytopes in terms of hurwitz stability of a number of corners and edges the testing set of edges and corners depends entirely on the edge directions of the polytope hence the results are particularly applicable in simultaneous analysis of several polytopes with equal edge directions it follows as a consequence that kharitonov s four polynomial test for independent coefficient uncertainties is replaced by a test of 2q polynomials when the stability region is a sector eiv 0 r q v and r q is a rational number

stability analysis of a family of matrices the authors extend the use of the lyapunov equation method to check the stability of a family of matrices the family of interval matrices is defined as the convex hull of a finite number of matrices and is a special case where each element has independent perturbation while the stability of interval matrices cannot be inferred from the stability of its vertex matrices the authors show that the family of matrices will be stable if the symmetric matrices associated with the vertex matrices have matrix measures less than two

relations between svd and gsvd of discrete regularization problems in standard and general form abstract we consider an algorithm due to elden that transforms general discrete regularization problems into standard form we derive a simple relationship between the svd associated with the standard form problem and the gsvd associated with the general problem and we discuss the accuracy of the gsvd computed via this transformation

solving fredholm integrals of the first kind with tensor product structure in 2 and 2 5 dimensions we present an efficient algorithm to solve a class of two and 2 5 dimensional 2 d and 2 5 d fredholm integrals of the first kind with a tensor product structure and nonnegativity constraint on the estimated parameters of interest in an optimization framework a zeroth order regularization functional is used to incorporate a priori information about the smoothness of the parameters into the problem formulation we adapt the butler reeds dawson 1981 algorithm to solve this optimization problem in three steps in the first step the data are compressed using singular value decomposition svd of the kernels the tensor product structure of the kernel is exploited so that the compressed data is typically a thousand fold smaller than the original data this size reduction is crucial for fast optimization in the second step the constrained optimization problem is transformed to an unconstrained optimization problem in the compressed data space in the third step a suboptimal value of the smoothing parameter is chosen by the brd method steps 2 and 3 are iterated until convergence of the algorithm we demonstrate the performance of the algorithm on simulated data

a new extension to kharitonov s theorem an extension to a well known theorem due to kharitonov is presented kharitonov s theorem gives a necessary and sufficient condition for all polynomials in a given family to be hurwitz stable in kharitonov s theorem the family of polynomials considered is obtained by allowing each of the polynomial coefficients to vary independently within an interval kharitonov s theorem shows that stability of this family of polynomials can be determined by looking at the stability of four specially constructed vertex polynomials kharitonov s theorem is extended to allow for more general families of polynomials and to allow a given margin of stability to be guaranteed for the family of polynomials

a new method for nmr data inversion based on double parameter regularization nuclear magnetic resonance nmr technology plays a significant role in petroleum exploration nmr data can be processed using inversion methods to reflect the relaxation information of all the components a new double parameter regularization dpr method was proposed for the inversion of nmr data whose regularization terms consist of tikhonov regularization and maximum entropy regularization the objective function for the dpr method was solved using the levenberg marquardt method the proportional coefficient of the regularization parameter was obtained using an iteration procedure the optimum regularization parameter of the dpr method was selected using an s curve the relationship between the optimum regularization parameter and the signal to noise ratio snr of the data was evaluated moreover we compared the results of the nmr inversion obtained from the norm smooth method the maximum entropy method and the dpr method for simulated data we evaluated how the proportional coefficient of the re

root clustering for convex combination of complex polynomials in some important applications such as the edge theorem it is required that a polynomial is alpha stable along a parameter segment using the critical constraint the necessary and sufficient conditions for root clustering alpha stability of convex combinations of complex polynomials are presented the approach is general and requires that a certain real polynomial has no zeros in the open interval

vertices and segments of interval plants are not sufficient for step response analyses abstract interval plants are of interest in control theory as models of uncertain systems they are useful because many worst case analyses of these models are simple to carry out for example robust stability of an interval plant can be determined by investigating only the four kharitonov vertices of the denominator polynomial also the maximum peak of the bode magnitude plot can be found using just 16 special plant vertices these 16 vertices are connected by 32 special line segments most stability and frequency domain analyses that cannot be done using only the special vertices can be carried out using just the segments from these results it is tempting to conjecture that the 16 vertices or at least the 32 segments are adequate for step response analyses this paper presents examples showing that these conjectures are not true

generalization of strong kharitonov theorems to the left sector the zero locations of interval polynomials are examined in particular it is shown that a family of interval polynomials will have only zeros in a certain class of left sector if and only if a finite number of specially chosen vertex polynomials have only zeros in the left sector this finite number of vertex polynomials is dependent on the damping margins of the left sector but is independent of the orders of the polynomials the exact number of vertex polynomials to be checked will depend on the damping margins but it will be independent of the orders of the polynomials these specially chosen vertex polynomials are independent of the order of the polynomials the importance of the result lies with the great reduction in computation cost associated with checking the zero locations of interval polynomials

nearest stable system using successive convex approximations stability is a crucial property in the study of dynamical systems we focus on the problem of enforcing the stability of a system a posteriori the system can be a matrix or a polynomial either in continuous time or in discrete time we present an algorithm that constructs a sequence of successive stable iterates that tend to a nearby stable approximation x of a given system a the stable iterates are obtained by projecting a onto the convex approximations of the set of stable systems some possible applications for this method are correcting the error arising from some noise in system identification and a possible solver for bilinear matrix inequalities based on convex approximations in the case of polynomials a fair complexity is achieved by finding a closed form solution to first order optimality conditions