a regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its l norm the i th singular value of a transfer matrix need not be a differentiable function of frequency where its multiplicity is greater than one we show that near a local maximum however the largest singular value has a lipschitz second derivative but need not have a third derivative using this regularity result we give a quadratically convergent algorithm for computing the l norm of a transfer matrix

robust stability of interval matrices a computational approach a simple computational procedure is given to determine the robust stability of matrices containing interval parameter uncertainty the computation consists of determining the maximal phase difference of the characteristic polynomial evaluated over the vertices of the uncertain parameter set at each point of the stability boundary and verifying that this maximal phase difference is less than radians this follows as a direct application of the mapping theorem which allows us effectively to treat rank one perturbations of matrices we discuss the conservatism of the method and show how the solution may be improved to any desired level of accuracy we give several numerical examples of systems with state space perturbations and compare our results with those obtained by other methods

robust stability of delay difference systems under fractional perturbations in infinite dimensional spaces in this article we study the robust stability of difference systems with delays under fractional perturbations in infinite dimensional spaces first the estimates of the complex stability radius are addressed second it is shown that for positive linear systems the complex real and positive stability radii coincide and can be computed by a simple formula finally a simple example is given to illustrate the obtained results

on structured perturbation of hermitian matrices abstract this paper concerns a quantity which is equal to the norm of the smallest structured perturbation to a hermitian matrix that makes the perturbed matrix singular this quantity of course then gives an indication on how much such structured perturbation the hermitian matrix can tolerate before becoming singular for some structures this quantity can be computed explicitly for some more general structures only the lower bound on this quantity is given

new algorithms for computing the real structured pseudospectral abscissa and the real stability radius of large and sparse matrices we present two new algorithms for investigating the stability of large and sparse matrices subject to real perturbations the first algorithm computes the real structured pseudospectral abscissa and is based on the algorithm for computing the pseudospectral abscissa proposed by guglielmi and overton siam j matrix anal appl 32 2011 pp 1166 1192 it entails finding the rightmost eigenvalues for a sequence of large matrices and we demonstrate that these eigenvalue problems can be solved in a robust manner by an unconventional eigenvalue solver we also develop an algorithm for computing the real stability radius of a real and stable matrix which utilizes a recently developed technique for detecting the loss of stability in a large dynamical system both algorithms are tested on large and sparse matrices

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

real perturbation values and real quadratic forms in a complex vector space abstract a sequence of real numbers connected to a complex matrix is introduced it is shown how these real perturbation values can be computed and that they have several properties similar to the singular values the so called real pseudospectra and real stability radii can be computed using the real perturbation values the main result concerns the signature of real quadratic forms in complex vector spaces

a globally convergent method to compute the real stability radius for time delay systems abstract this paper presents a novel algorithm to compute the real stability radius for a linear delay system of retarded type with multiple delays the real stability radius is the distance to instability measured as the minimal real valued perturbation that renders the system unstable our method is based on characterizing this distance to instability as the inverse of the global maximum of a real structured singular value function we develop a criss cross type algorithm that globally converges to this maximum and whose convergence rate seems to be superlinear and sometimes quadratic in numerical experiments the algorithm exploits that the intersections of these singular value functions with constant functions can be written as purely imaginary eigenvalues of certain delay eigenvalue problems dep with positive and negative delays this is an extension of the well known linear case without delays where this results in algebraic eigenvalue problems in addition a novel numerical solver to compute all the imaginary eigenvalues of this dep is also presented it combines an approximation using spectral discretizations and an automatic procedure to determine the required number of discretization points finally due to the presence of multiple eigenvalues at the maximum these approximations are corrected with a block newton algorithm for nonlinear eigenvalue problems

randomized algorithms for probabilistic robustness with real and complex structured uncertainty there has been a growing interest in developing randomized algorithms for probabilistic robustness of uncertain control systems unlike classical worst case methods these algorithms provide probabilistic estimates assessing for instance if a certain design specification is met with a given probability one of the advantages of this approach is that the robustness margins can be often increased by a considerable amount at the expense of a small risk in this sense randomized algorithms may be used by the control engineer together with standard worst case methods to obtain additional useful information the applicability of these probabilistic methods to robust control is presently limited by the fact that the sample generation is feasible only in very special cases which include systems affected by real parametric uncertainty bounded in rectangles or spheres sampling in more general uncertainty sets is generally performed through overbounding at the expense of an exponential rejection rate in the paper randomized algorithms for stability and performance of linear time invariant uncertain systems described by a general m spl delta configuration are studied in particular efficient polynomial time algorithms for uncertainty structures spl delta consisting of an arbitrary number of full complex blocks and uncertain parameters are developed

when are the complex and the real stability radii equal characterization of those real stable matrices for which the unstructured complex and real stability radii are equal is presented special cases of this equality are discussed as application of this result it is shown that for 2 2 matrices with nonreal eigenvalues neither the case of equality nor the case of strict inequality is generic

stability radii of positive linear functional differential equations under multi perturbations we study stability radii of linear retarded systems described by general linear functional differential equations a lower and an upper bound for the complex stability radius with respect to multi perturbations are given furthermore in some special cases concerning the structure matrices the complex stability radius can precisely be computed via the associated transfer function then the class of positive linear retarded systems is studied in detail it is shown that for this class complex real and positive stability radius under multi perturbations or multi affine perturbations coincide and can be computed by simple formulae expressed in terms of the system matrices

bayesian prediction of deterministic functions with applications to the design and analysis of computer experiments abstract this article is concerned with prediction of a function y t over a multidimensional domain t given the function values at a set of sites t 1 t 2 t n in t and with the design that is with the selection of those sites the motivating application is the design and analysis of computer experiments where t determines the input to a computer model of a physical or behavioral system and y t is a response that is part of the output or is calculated from it following a bayesian formulation prior uncertainty about the function y is expressed by means of a random function y which is taken here to be a gaussian stochastic process the mean of the posterior process can be used as the prediction function t and the variance can be used as a measure of uncertainty this kind of approach has been used previously in bayesian interpolation and is strongly related to the kriging methods used in geostatistics here emphasis is placed on product linear and product cubic correlation func

two classes of efficient digital controller structures with stability consideration in this note the efficient controller structure problem is investigated subject to minimizing finite word length fwl effects two classes of sparse controller structures characterized with a set of free parameters are derived the stability behavior of each class is analyzed with a newly defined stability related measure the optimal structure problems are studied in terms of maximizing the stability related measure for the two classes of sparse structures as well as the fully parametrized state space realizations a design example is given which shows that the optimized sparse structures can beat the fully parametrized optimal realizations in terms of both stability performance and computation efficiency

on the robustness of symmetric systems this note concerns the robustness of lti symmetric systems under symmetric or diagonal perturbations it is shown that for symmetric systems the robust stability condition for dynamic perturbations given by the small gain theorem is also necessary if the perturbation matrix is assumed to be diagonal but the robust stability condition for parametric perturbations given by the real stability radius is no longer necessary if the perturbation matrix is merely assumed to be symmetric

on the real stability radius of sparse systems abstract in this paper we study robust stability of sparse lti systems using the stability radius sr as a robustness measure we consider real perturbations with an arbitrary and pre specified sparsity pattern of the system matrix and measure their size using the frobenius norm we formulate the sr problem as an equality constrained minimization problem using the lagrangian method for optimization we characterize the optimality conditions of the sr problem thereby revealing the relation between an optimal perturbation and the eigenvectors of an optimally perturbed system further we use the sylvester equation based parametrization to develop a penalty based gradient newton descent algorithm which converges to the local minima of the optimization problem finally we illustrate how our framework provides structural insights into the robust stability of sparse networks

a formula for computation of the real stability radius this paper presents a readily computable formula for the real stability radius with respect to an arbitrary stability region in the complex plane

efficient least angle regression for identification of linear in the parameters models least angle regression as a promising model selection method differentiates itself from conventional stepwise and stagewise methods in that it is neither too greedy nor too slow it is closely related to l1 norm optimization which has the advantage of low prediction variance through sacrificing part of model bias property in order to enhance model generalization capability in this paper we propose an efficient least angle regression algorithm for model selection for a large class of linear in the parameters models with the purpose of accelerating the model selection process the entire algorithm works completely in a recursive manner where the correlations between model terms and residuals the evolving directions and other pertinent variables are derived explicitly and updated successively at every subset selection step the model coefficients are only computed when the algorithm finishes the direct involvement of matrix inversions is thereby relieved a detailed computational complexity analysis indicates that the proposed algorithm possesses significant computational efficiency compared with the original approach where the well known efficient cholesky decomposition is involved in solving least angle regression three artificial and real world examples are employed to demonstrate the effectiveness efficiency and numerical stability of the proposed algorithm

the estimation of prediction error having constructed a data based estimation rule perhaps a logistic regression or a classification tree the statistician would like to know its performance as a predictor of future cases there are two main theories concerning prediction error 1 penalty methods such as cp akaike s information criterion and stein s unbiased risk estimate that depend on the covariance between data points and their corresponding predictions and 2 cross validation and related nonparametric bootstrap techniques this article concerns the connection between the two theories a rao blackwell type of relation is derived in which nonparametric methods such as cross validation are seen to be randomized versions of their covariance penalty counterparts the model based penalty methods offer substantially better accuracy assuming that the model is believable

stability radii for linear time varying differential algebraic equations with respect to dynamic perturbations this paper is concerned with the robust stability for linear time varying differential algebraic equations we consider the systems under the effect of uncertain dynamic perturbations a formula of the structured stability radius is obtained the result is an extension of a previous result for time varying ordinary differential equations proven by birgit jacob b jacob a formula for the stability radius of time varying systems j differential equations 142 1998 167 187

generalized linear models with functional predictors summary we present a technique for extending generalized linear models to the situation where some of the predictor variables are observations from a curve or function the technique is particularly useful when only fragments of each curve have been observed we demonstrate on both simulated and real data sets how this approach can be used to perform linear logistic and censored regression with functional predictors in addition we show how functional principal components can be used to gain insight into the relationship between the response and functional predictors finally we extend the methodology to apply generalized linear models and principal components to standard missing data problems

on computing the infinity norm an iterative algorithm is given for computing the infinity norm of the transfer matrix of a bounded linear time invariant finite dimensional system the procedure which is based on the numerically stable qz algorithm enables the infinite norm to be computed with a guaranteed accuracy no ad hoc procedures such as those for selecting a suitable grid of frequencies for example are used the degree of accuracy is under the direct control of the user with each additional decimal place of accuracy requiring approximately 3 3 calls to the qz algorithm experience has shown the algorithm to function reliably and in reasonable time for models of order as large as 56

reduction of controller fragility by pole sensitivity minimization presents a method for the reduction of controller fragility the method is based on the sensitivity of closed loop poles to perturbations in the controller parameters by means of a state space parameterization of the controller the closed loop pole sensitivity can be reduced a controller fragility measure based on the closed loop pole sensitivity is proposed conditions for the optimal state space realization of the controller are presented along with a numerical method for obtaining the solution

sure independence screening in generalized linear models with np dimensionality ultrahigh dimensional variable selection plays an increasingly important role in contemporary scientific discoveries and statistical research among others fan and lv 2008 propose an independent screening framework by ranking the marginal correlations they showed that the correlation ranking procedure possesses a sure independence screening property within the context of the linear model with gaussian covariates and responses in this paper we propose a more general version of the independent learning with ranking the maximum marginal likelihood estimates or the maximum marginal likelihood itself in generalized linear models we show that the proposed methods with fan and lv 2008 as a very special case also possess the sure screening property with vanishing false selection rate the conditions under which that the independence learning possesses a sure screening is surprisingly simple this justifies the applicability of such a simple method in a wide spectrum we quantify explicitly the extent to which the dimensionality can be reduced by independence screening which depends on the interactions of the covariance matrix of covariates and true parameters simulation studies are used to illustrate the utility of the proposed approaches in addition we supported in part by grant nsf grants dms 0714554 and dms 0704337 the bulk of the work was conducted when rui song was a postdoctoral research fellow at princeton university the authors would like to thank the associate editor and two referees for their constructive comments that improve the presentation and the results of the paper ams 2000 subject classifications primary 68q32 62j12 secondary 62e99 60f10

a geometric approach to detecting influential cases amari s dual geometries are used to study measures of influence in exponential family regression the dual geometries are presented as a natural extension of the eucidean geometry used for the normal regression model these geometries are then used to extend cook s distance to generalized linear models and exponential family regression some of these extensions lead to measures already considered while other extensions lead to new measures of influence the advantages of one of these new measures are discussed

l1 regularization path algorithm for generalized linear models summary we introduce a path following algorithm for l1 regularized generalized linear models the l1 regularization procedure is useful especially because it in effect selects variables according to the amount of penalization on the l1 norm of the coefficients in a manner that is less greedy than forward selection backward deletion the generalized linear model path algorithm efficiently computes solutions along the entire regularization path by using the predictor corrector method of convex optimization selecting the step length of the regularization parameter is critical in controlling the overall accuracy of the paths we suggest intuitive and flexible strategies for choosing appropriate values we demonstrate the implementation with several simulated and real data sets

diagnostics for nonparametric regression models with additive terms abstract recent developments of multivariate smoothing methods provide a rich collection of feasible models for nonparametric multivariate data analysis among the most interpretable are models with additive terms construction of various models and algorithms for computing the models have been the main concern of the existing literature in this area few results are available on the validation of computed fits and many applications of nonparametric methods unfortunately end up interpreting the noise this article proposes and illustrates some simple retrospective diagnostics to help data analysts in detecting possible aliasing effects in computed nonparametric fits and in building parsimonious models in an interactive fashion it also discusses the concepts and rationale behind the proposal including concurvity diagnostics versus tests and so forth for their ready availability interaction splines are used in the illustrations

nearest matrix with two prescribed eigenvalues abstract given a square complex matrix a and two complex numbers z1 and z2 we find the distance from a to the set of matrices that have z1 z2 as some of their eigenvalues we use the distance between two matrices associated with the spectral norm

a pair of matrices sharing common lyapunov solutions a closer look abstract let a b be a pair of matrices with regular inertia if ha a h and hb b h are both positive definite for some hermitian matrix h then all matrices in conv a a 1 b b 1 have identical regular inertia this in turn implies that both conv a b and conv a b 1 consist of non singular matrices in general neither of the converse implications holds in this paper we seek situations where they do hold in particular when a and b are real 2 2 matrices several aspects of the above statements for n n matrices are discussed a connection to the characterization of the convex hull of matrices with regular inertia is introduced differences between the real and the complex case are indicated

robust matrix root clustering analysis through extended kyp lemma this paper is dedicated to robust matrix eigenvalue clustering in a subregion mathcal d of the complex plane a norm bounded uncertainty is considered cd is any intersection and or union of half planes discs and outsides of discs this work aims at computing the associated complex cd stability radius this challenge was already tackled in a recent paper but the technique therein may fail or lead to conservative results when the subregions overlap the present paper proposes to circumvent those limits thanks to an extended version of the kalman yakubovich popov kyp lemma which is deduced from the so called generalized s procedure

the distance from a matrix polynomial to matrix polynomials with a prescribed multiple eigenvalue abstract for a matrix polynomial p and a given complex number we introduce a spectral norm distance from p to the matrix polynomials that have as an eigenvalue of geometric multiplicity at least and a distance from p to the matrix polynomials that have as a multiple eigenvalue then we compute the first distance and obtain bounds for the second one constructing associated perturbations of p

generalized functional linear models we propose a generalized functional linear regression model for a regression situation where the response variable is a scalar and the predictor is a random function a linear predictor is obtained by forming the scalar product of the predictor function with a smooth parameter function and the expected value of the response is related to this linear predictor via a link function if in addition a variance function is specified this leads to a functional estimating equation which corresponds to maximizing a functional quasi likelihood this general approach includes the special cases of the functional linear model as well as functional poisson regression and functional binomial regression the latter leads to procedures for classification and discrimination of stochastic processes and functional data we also consider the situation where the link and variance functions are unknown and are estimated nonparametrically from the data using a semiparametric quasi likelihood procedure an essential step in our proposal is dimension reduction by approximating the predictor processes with a truncated karhunen loeve expansion we develop asymptotic inference for the proposed class of generalized regression models in the proposed asymptotic approach the truncation parameter increases with sample size and a martingale central limit theorem is applied to establish the resulting increasing dimension asymptotics we establish asymptotic normality for a properly scaled distance between estimated and true functions that corresponds to a suitable l 2 metric and is defined through a generalized covariance operator as a consequence we obtain asymptotic tests and simultaneous confidence bands for the parameter function that determines the model the proposed estimation inference and classification procedures and variants with unknown link and variance functions are investigated in a simulation study we find that the practical selection of the number of components works well with the aic criterion and this finding is supported by theoretical considerations we include an application to the classification of medflies regarding their remaining longevity status based on the observed initial egg laying curve for each of 534 female medflies

comments regarding stability margin evaluation for uncertain linear systems and counterexample the stability bound proposed in the above mentioned paper by gong and thompson ieee trans automat contr vol 39 1994 cannot be applied to all matrices having unstable unitary factor in their polar decompositions the case study pursued in that paper and the succeeding papers is completed in this note in a rational way some basic results about the eigenvalues of unitary polar factor are also presented

on the degrees of freedom of the lasso we study the effective degrees of freedom of the lasso in the framework of stein s unbiased risk estimation sure we show that the number of nonzero coefficients is an unbiased estimate forthe degrees of freedom of the lasso a conclusion that requires no special assumption on the predictors in addition the unbiased estimator is shown to be asymptotically consistent with these results on hand various model selection criteria cp aic and bic are available which along with the lars algorithm provide a principled and efficient approach to obtaining the optimal lasso fit with the computational effort of a single ordinary least squares fit

a differential geometric approach to generalized linear models with grouped predictors we propose an extension of the differential geometric least angle regression method to perform sparse group inference in a generalized linear model an efficient algorithm is proposed to compute the solution curve the proposed group differential geometric least angle regression method has important properties that distinguish it from the group lasso first its solution curve is based on the invariance properties of a generalized linear model second it adds groups of variables based on a group equiangularity condition which is shown to be related to score statistics an adaptive version which includes weights based on the kullback leibler divergence improves its variable selection features and is shown to have oracle properties when the number of predictors is fixed

note on structured indefinite perturbations to hermitian matrices in a recent paper overton and van dooren have considered structured indefinite perturbations to a given hermitian matrix we extend their results to skew hermitian hamiltonian and skew hamiltonian matrices as an application we give a formula for computation of the smallest perturbation with a special structure which makes a given hamiltonian matrix own a purely imaginary eigenvalue

adaptive model selection and assessment for exponential family distributions in many scientific and engineering problems selecting the optimal model from a large pool of candidate models is important particularly in data mining in the literature model assessment in the context of nonnormal distributions has not yet received a lot of attention indeed many existing model selection criteria such as the bayes information criterion and cp may not be suitable for a situation in which the conditional mean and variance of the response are dependent such as in generalized linear model regression in this article we propose a new adaptive model selection criterion and construct an approximately unbiased kullback leibler loss estimator for model assessment in the context of exponential family distributions this permits comparing any arbitrary complex modeling procedures our proposal uses a concept called generalized degrees of freedom that generalizes the concept originally proposed for the normal distribution the proposed procedure is implemented for the binomial and poisson distr

how biased is the apparent error rate of a prediction rule abstract a regression model is fitted to an observed set of data how accurate is the model for predicting future observations the apparent error rate tends to underestimate the true error rate because the data have been used twice both to fit the model and to check its accuracy we provide simple estimates for the downward bias of the apparent error rate the theory applies to general exponential family linear models and general measures of prediction error special attention is given to the case of logistic regression on binary data with error rates measured by the proportion of misclassified cases several connected ideas are compared mallows s cp cross validation generalized cross validation the bootstrap and akaike s information criterion

on the structured distance to uncontrollability this article is concerned with the structured distance to uncontrollability of a linear time invariant system and relates this concept to a variation of the value the developed framework is applied to derive computational expressions for the class of real perturbations as well as for hermitian symmetric and skew symmetric perturbations in a relatively simple manner examples demonstrate that the structured distance can differ from the standard unstructured distance to uncontrollability by an arbitrary amount it is also shown how systems of higher order can be addressed

on the distance to uncontrollability and the distance to instability and their relation to some condition numbers in control according to the methodology of 6 many measures of distance arising in problems in numerical linear algebra and control can be bounded by a factor times the reciprocal of an appropriate condition number where the distance is thought of as the distance between a given problem to the nearest ill posed problem in this paper four major problems in numerical linear algebra and control are further considered the computation of system hessenberg form the solution of the algebraic riccati equation the pole assignment problem and the matrix exponential the distances considered here are the distance to uncontrollability and the distance to instability

on measuring and correcting the effects of data mining and model selection abstract in the theory of linear models the concept of degrees of freedom plays an important role this concept is often used for measurement of model complexity for obtaining an unbiased estimate of the error variance and for comparison of different models i have developed a concept of generalized degrees of freedom gdf that is applicable to complex modeling procedures the definition is based on the sum of the sensitivity of each fitted value to perturbation in the corresponding observed value the concept is nonasymptotic in nature and does not require analytic knowledge of the modeling procedures the concept of gdf offers a unified framework under which complex and highly irregular modeling procedures can be analyzed in the same way as classical linear models by using this framework many difficult problems can be solved easily for example one can now measure the number of observations used in a variable selection process different modeling procedures such as a tree based regression and a

generalized functional linear models with semiparametric single index interactions we introduce a new class of functional generalized linear models where the response is a scalar and some of the covariates are functional we assume that the response depends on multiple covariates a finite number of latent features in the functional predictor and interaction between the two to achieve parsimony the interaction between the multiple covariates and the functional predictor is modeled semiparametrically with a single index structure we propose a two step estimation procedure based on local estimating equations and investigate two situations a when the basis functions are predetermined e g fourier or wavelet basis functions and the functional features of interest are known and b when the basis functions are data driven such as with functional principal components asymptotic properties are developed notably we show that when the functional features are data driven the parameter estimates have an increased asymptotic variance due to the estimation error of the basis functio

an eigenvalue based approach for the robust stabilization of linear time delay systems we present a new numerical procedure for the robust stabilization of linear delay differential equations based on the position of the eigenvalues in the complex plane we assume static perturbations on the system matrices and express the robustness of the stability in terms of complex stability radii in the numerical procedure these stability radii are maximized as a function of the controller parameters this corresponds to a synthesis problem which is solved by a quasi continuous shaping of some frequency response plots the structure of the algorithm is analogous to the continuous pole placement algorithm for the non robust stabilization of linear delay differential equations

nearest southeast submatrix that makes multiple a prescribed eigenvalue part 1 abstract given four complex matrices a b c and d where a c n n and d c m m and given a complex number z 0 what is the spectral norm distance from d to the set of matrices x c m m such that z 0 is a multiple eigenvalue of the matrix a b c x this problem is solved when z 0 is not an eigenvalue of a we also give a conjecture when z 0 is an eigenvalue of a

real controllability stabilizability radius of lti systems in this note we present a simple and readily computable formula for the calculation of the real controllability stabilizability and observability detectability radii for linear time invariant systems

a generalized dantzig selector with shrinkage tuning summary the dantzig selector performs variable selection and model fitting in linear regression it uses an l 1 penalty to shrink the regression coefficients towards zero in a similar fashion to the lasso while both the lasso and dantzig selector potentially do a good job of selecting the correct variables they tend to overshrink the final coefficients this results in an unfortunate trade off one can either select a high shrinkage tuning parameter that produces an accurate model but poor coefficient estimates or a low shrinkage parameter that produces more accurate coefficients but includes many irrelevant variables we extend the dantzig selector to fit generalized linear models while eliminating overshrinkage of the coefficient estimates and develop a computationally efficient algorithm similar in nature to least angle regression to compute the entire path of coefficient estimates a simulation study illustrates the advantages of our approach relative to others we apply the methodology to two datasets

estimating the distance to uncontrollability a fast method and a slow one abstract for a linear control system a b the distance to uncontrollability is characterized by min ec n a i b where n a i b is the smallest singular value of the augmented matrix a i b two methods are developed to estimate the distance to uncontrollability giving both a lower bound and an upper bound one method is fast requiring only one spectral decomposition of a and computations of three smallest singular values and being used for well conditioned a the other is slow requiring computations of a large number of the smallest singular values but it produces bounds as tight as possible and also a region containing global minimizers newton s method can be used to compute the global minimizers if one really wants

a general formula for eigenvalue assignment by static output feedback with application to robust design in this note generalizing an idea of a previous paper we derive a new expression of the characteristic polynomial of a linear system subject to static output feedback from this expression an explicit form of the assignment equations in the unknown elements of the feedback matrix is obtained the equations are completely general and seem much simpler than many others proposed until now in special cases a closed or almost a closed form solution is obtained the assignment equations are then used within an algorithm for global optimization to solve the problem of exactly assigning the eigenvalues of the nominal plant while maximizing the norm of perturbation matrices for which the perturbed eigenvalues remain in specified regions of the complex plane

distance bounds for prescribed multiple eigenvalues of matrix polynomials abstract in this paper motivated by a problem posed by wilkinson we study the coefficient perturbations of a square matrix polynomial to a matrix polynomial that has a prescribed eigenvalue of specified algebraic multiplicity and index of annihilation for an n n matrix polynomial p and a given scalar c we introduce two weighted spectral norm distances e r and e r k from p to the n n matrix polynomials that have as an eigenvalue of algebraic multiplicity at least r and to those that have as an eigenvalue of algebraic multiplicity at least r and maximum jordan chain length exactly k respectively then we obtain a lower bound for e r k and derive an upper bound for e r by constructing an associated perturbation of p

differential geometry of curved exponential families curvatures and information loss the differential geometrical framework is given for analyzing statistical problems related to multi parameter families of distributions the dualistic structures of the exponential families and curved exponential families are elucidated from the geometrical viewpoint the duality connected by the legendre transformation is thus extended to include two kinds of affine connections and two kinds of curvatures the second order information loss is calculated for fisher efficient estimators and is decomposed into the sum of two non negative terms one is related to the exponential curvature of the statistical model and the other is related to the mixture curvature of the estimator only the latter term depends on the estimator and vanishes for the maximum likelihood estimator a set of statistics which recover the second order information loss are given the second order efficiency also is obtained the differential geometry of the function space of distributions is discussed

stability regions in the parameter space d decomposition revisited the challenging problem in linear control theory is to describe the total set of parameters controller coefficients or plant characteristics which provide stability of a system for the case of one complex or two real parameters and siso system with a characteristic polynomial depending linearly on these parameters the problem can be solved graphically by use of the so called d decomposition our goal is to extend the technique and to link it with general m d framework in this way we investigate the geometry of d decomposition for polynomials and estimate the number of root invariant regions several examples verify that these estimates are tight we also extend d decomposition for the matrix case i e for mimo systems for instance we partition real axis or complex plane of the parameter k into regions with invariant number of stable eigenvalues of the matrix a kb similar technique can be applied to double input double output systems with two parameters

differential geometric least angle regression a differential geometric approach to sparse generalized linear models sparsity is an essential feature of many contemporary data problems remote sensing various forms of automated screening and other high throughput measurement devices collect a large amount of information typically about few independent statistical subjects or units in certain cases it is reasonable to assume that the underlying process generating the data is itself sparse in the sense that only a few of the measured variables are involved in the process we propose an explicit method of monotonically decreasing sparsity for outcomes that can be modelled by an exponential family in our approach we generalize the equiangular condition in a generalized linear model although the geometry involves the fisher information in a way that is not obvious in the simple regression setting the equiangular condition turns out to be equivalent with an intuitive condition imposed on the rao score test statistics in certain special cases the method can be tweaked to obtain l1 penalized generalized linear model solution paths but the method itself defines sparsity more directly although the computation of the solution paths is not trivial the method compares favourably with other path following algorithms

stability radii of positive higher order difference system in infinite dimensional spaces abstract in this paper we study stability radii of positive higher order difference system under affine perturbations of the coefficient operators it is shown that the real and complex stability radii coincide moreover explicit formulas are derived for these stability radii and illustrated by some examples the results obtained are extensions of the recent work in d hinrichsen p h a ngoc n k son stability radii of positive higher order difference system systems control lett 49 2003 377 388 to banach spaces

stability radii of positive linear volterra stieltjes equations abstract we study stability radii of linear volterra stieltjes equations under multi perturbations and affine perturbations a lower and upper bound for the complex stability radius with respect to multi perturbations are given furthermore in some special cases concerning the structure matrices the complex stability radius can precisely be computed via the associated transfer functions then the class of positive linear volterra stieltjes equations is studied in detail it is shown that for this class complex real and positive stability radius under multi perturbations or multi affine perturbations coincide and can be computed by simple formulae expressed in terms of the system matrices as direct consequences of the obtained results we get some results on robust stability of positive linear integro differential equations and of positive linear functional differential equations to the best of our knowledge most of the results of this paper are new

implicit system approach to the robust stability for a class of singularly perturbed linear systems abstract asymptotic stability and the complex stability radius of a class of singularly perturbed systems of linear differential algebraic equations daes are studied the asymptotic behavior of the stability radius for a singularly perturbed implicit system is characterized as the parameter in the leading term tends to zero the main results are obtained in direct and short ways which involve some basic results in linear algebra and classical analysis only our results can be extended to other singular perturbation problems for daes of more general form

robust stability with structured real parameter perturbations this paper considers the problem of robust stabilization of a linear time invariant system subject to variations of a real parameter vector for a given controller the radius of the largest stability hypersphere in this parameter space is calculated this radius is a measure of the stability margin of the closed loop system the results developed are applicable to all systems where the closed loop characteristic polynomial coefficients are linear functions of the parameters of interest in particular this always occurs for single input multioutput or single output multiinput systems where the transfer function coefficients are linear or affine functions of the parameters many problems with transfer function coefficients which are nonlinear functions of physical parameters can be cast into this mathematical framework by suitable weighting and redefinition of functions of physical parameters as new parameters the largest stability hyperellipsoid for the case of weighted perturbations and a stability polytope in parameter space are also determined based on these calculations a design procedure is proposed to robustify a given stabilizing controller this algorithm iteratively enlarges the stability hypersphere or hyperellipsoid in parameter space and can be used to design a controller io stabilize a plant subject to given ranges of parameter excursions these results are illustrated by an example

stability radius for structured perturbations and the algebraic riccati equation abstract in this paper we generalize the notion of stability radius introduced in 1 to allow for structured perturbations we then relate the stability radius to the existence of hermitian solutions of an algebraic riccati equation and give some applications of this result

an algorithm for the computation of the structured complex stability radius abstract perturbed state space models of the form x a bdc x are considered where a b c are given matrices a is stable and d is an unknown complex perturbation matrix a reliable and efficient algorithm for the computation of the norm of the minimal norm destabilizing matrix d is presented the performance of the algorithm and possible applications are illustrated

on computation of the stability radius for nonlinearly structured perturbations this paper considers the problem of finding a perturbation matrix with the least spectral norm such that a matrix valued function becomes singular where the dependence of the function on the perturbation is allowed to be nonlinear it is proved that such a problem can be approximated by a smooth unconstrained minimization problem with compact sublevel sets a computational procedure proposed based on this result is demonstrated to be effective in both linear and nonlinear cases

on the computation of the real hurwitz stability radius recently qiu et al 1995 obtained a computationally attractive formula for the evaluation of the real stability radius this formula involves a global maximization over frequency here for the hurwitz stability case we show that the frequency range can be limited to a certain finite interval numerical experimentation suggests that this interval is often reasonably small

semiparametric analysis of variance with tensor product thin plate splines we consider fitting multivariate models to response data based on analysis of variance anova like decompositions of functions of several variables f t 1 t d c f t f t t a theory for fitting some components of these models with polynomial smoothing splines exists when each t is in a subset of the real line in this case the various estimated components turn out to be certain tensor sums and products of polynomial splines this approach may not be natural when one or more of the variables are geographic in particular where nature does not know north from east

a fast algorithm to computer the h norm of a transfer function matrix abstract a fast algorithm is presented to compute the h norm of a transfer function matrix based on the relation between the singular values of the transfer function matrix and the eigenvalues of a related hamiltonian matrix the norm is computed with guaranteed accuracy a very simple method to compute a rather close lower bound on the h norm is given

stability radii of positive linear time delay systems under fractional perturbations in this paper we study stability radii of positive linear delay systems under fractional perturbations it is shown that the three stability radii complex real and positive stability radii coincide and can be computed by a simple formula finally a simple example is given to illustrate the obtained results

stability radius of linear parameter varying systems and applications abstract in this paper we present a unifying approach to the problems of computing of stability radii of positive linear systems first we study stability radii of linear time invariant parameter varying differential systems a formula for the complex stability radius under multi perturbations is given then under hypotheses of positivity of the system matrices we prove that the complex real and positive stability radii of the system under multi perturbations or affine perturbations coincide and they are computed via simple formulae as applications we consider problems of computing of strong stability radii of linear time invariant time delay differential systems and computing of stability radii of positive linear functional differential equations under multi perturbations and affine perturbations we show that for a class of positive linear time delay differential systems the stability radii of the system under multi perturbations or affine perturbations are equal to the strong stability radii next we prove that the stability radii of a positive linear functional differential equation under multi perturbations or affine perturbations are equal to those of the associated linear time invariant parameter varying differential system in particular we get back some explicit formulas for these stability radii which are given recently in p h a ngoc strong stability radii of positive linear time delay systems internat j robust nonlinear control 15 2005 459 472 p h a ngoc n k son stability radii of positive linear functional differential equations under multi perturbations siam j control optim 43 2005 2278 2295 finally we give two examples to illustrate the obtained results

real structured singular value conditions for the strong d stability recently chen et al systems control lett 24 1995 19 proposed conditions for d stability and strong d stability in terms of structured singular values in this paper simpler conditions for the strong d stability are derived