on the structure of h sup infinity control systems and related extensions an existence condition of a h sup infinity controller which achieves a prescribed norm bound of the closed loop transfer function is derived in the frequency domain based on a generalization of the notion of j lossless systems this condition is regarded as a frequency domain representation of the well known existence condition in the state space represented in terms of the two algebraic riccati equations a notion of j orthogonal complement which is introduced as a generalization of the usual orthogonal complement play an important role in clarifying the fundamental frequency domain structure of the model matching problem and simplifying the computation of controllers the results are extended to the nonstandard case where the direct feedthrough from the input to the error or from the exogenous signal to the output is no longer of full rank it is shown that the proper controller achieving the prescribed norm bound exists even in this case in nonstandard cases the controller order can be smaller than the plant order

a mirror symmetric solution to the quantum toda lattice we give a representation theoretic proof of a conjecture from rietsch adv math 217 2401 2442 2008 providing integral formulas for solutions to the quantum toda lattice in general type this result generalizes work of givental for sl n b in a uniform way to arbitrary type and can be interpreted as a kind of mirror theorem for the full flag variety g b we also prove the existence of a totally positive and totally negative critical point of the superpotential in every mirror fiber

mixed time frequency domain based robust identification in this paper we propose a new robust identification framework that combines both frequency and time domain experimental data the main result of the paper shows that the problem of obtaining a nominal model consistent with the experimental data and bounds on the identification error can be recast as a constrained finite dimensional convex optimization problem that can be efficiently solved using linear matrix inequalities techniques this approach based upon a generalized interpolation theory contains as special cases the caratheodory fejer purely time domain and nevanlinna pick purely frequency domain problems the proposed procedure interpolates the frequency and time domain experimental data while restricting the identified system to be in an a priori given class of models resulting in a nominal model consistent with both sources of data thus it is convergent and optimal up to a factor of two with respect to central algorithms

simultaneous identification of nominal model parametric uncertainty and unstructured uncertainty for robust control abstract the problem attacked in this paper is to obtain the smallest model set which is described by nominal model parametric uncertainty bound unstructured uncertainty bound and consistent with a number of noise free input output data under the condition that the denominator of the nominal model is prescribed for the compatibility with the available robust control theory the unstructured uncertainty is measured by h norm while the parametric uncertainty is by parameter variation interval it is shown that the unstructured uncertainty bound will generally increase if the nominal model error uncertainty is completely regarded as unstructured moreover the identification problem is reduced to a convex optimization problem which is computationally tractable

control oriented system identification a worst case deterministic approach in h sub infinity the authors formulate and solve two related control oriented system identification problems for stable linear shift invariant distributed parameter plants in each of these problems the assumed a priori information is minimal consisting only of a lower bound on the relative stability of the plant an upper bound on a certain gain associated with the plant and an upper bound on the noise level the first of these problems involves identification of a point sample of the plant frequency response from a noisy finite output time series obtained in response to an applied sinusoidal input with frequency corresponding to the frequency point of interest this problem leads naturally to the second problem which involves identification of the plant transfer function in h sub infinity from a finite number of noisy point samples of the plant frequency response concrete plans for identification algorithms are provided for each of these two problems

modelling of uncertain systems via linear programming abstract real plants are in general time varying and uncertain yet most industrial control loops are designed using linear time invariant lti plant models the present work formalizes the problem of best approximate lti modelling of bibo stable linear time varying ltv systems in a way that is compatible with the notion of induced l norm used in robust l 1 control this setup is closely associated with certain identification methods and controlmotivated model validation invalidation procedures that can be efficiently implemented with special purpose linear programming lp techniques results are given for approximate modelling of bibo stable ltv and lti systems using lti models especially bibo stable fixed pole state space model parametrizations it is shown that such parametrizations are satisfactory from an approximate modelling point of view and can be used in output error type lp identification techniques and in lp model validation procedures various useful constraints on model parameters etc can be included as long as they are linear in the unknown parameters and both insensitive statistically robust and sensitive criteria can be used in identification and model validation simulation examples are included to illustrate that such techniques indeed give good results and can be used to solve problems of realistic size

a parametric extension of mixed time frequency robust identification a parametric extension to the time frequency robust identification framework is presented the results can be applied to stable linear time invariant systems on which time and or frequency experiments have been performed the parametric portion of the model should be affine in the unknown parameters which includes practical applications such as flexible structures the consistency problem is cast as a constrained finite dimensional convex optimization problem that can be formulated as a linear matrix inequality the proposed procedure provides an interpolating identification algorithm convergent and optimal up to a factor of two with respect to central algorithms

the caratheodory fejer problem and h sub spl infin l sub 1 identification a time domain approach in this paper we study a worse case robust control oriented identification problem this problem is in the framework of h sub spl infin identification but the formulation here is more general the available a priori information in our problem consists of a lower bound on the relative stability of the plant an upper bound on a certain gain associated with the plant and an upper bound on the noise level the plant to be identified is assumed to lie in a certain subset in the space of h sub spl infin characterized by the assumed a priori information the available experimental information consists of a corrupt finite output time series obtained in response to a known nonzero but otherwise arbitrary input the proposed algorithm is in the class of interpolatory algorithms which are known to possess desirable optimality properties in reducing the identification error this algorithm is obtained by solving an extended caratheodory fejer problem via standard convex programming methods both the algorithm and error bounds ran be applied to l sub 1 identification problems as well

errata to totally positive toeplitz matrices and quantum cohomology of partial flag varieties remark 4 3 if p is the parabolic subgroup thengj is a well defined regular function on the bruhat cell bwpb b precisely in the case m i n1 nk proof of theorem 4 2 1 is proved in 33 see also lemma 2 3 in 35 we will deduce 2 very explicitly from the ask presentation we begin by defining a particular system of coordinates on the affine space bwpb b for indexing purposes introduce sets 1 2 defined by r m z n1 r n and 1 m nl if nl r nl 1 1 nl m z where l 1 k and 1 m nl and 2 1 consider the polynomial rings c c g m r m and c i c g m r m i for i 1 2 we have n nl 1 nl matrices u l 1 over c 1 defined by

vibrations of an axially accelerating beam with small flexural stiffness transverse vibrations of an axially moving beam are considered the axial velocity is harmonically varying about a mean velocity the equation of motion is expressed in terms of dimensionless quantities the beam effects are assumed to be small since in this case the fourth order spatial derivative multiplies a small parameter the mathematical model becomes a boundary layer type of problem approximate solutions are searched using the method of multiple scales and the method of matched asymptotic expansions results of both methods are contrasted with the outer solution

robust identification and interpolation in h abstract we consider system identification in h in the framework proposed by helmicki jacobson and nett an algorithm using the jackson polynomials is proposed that achieves an exponential convergence rate for exponentially stable systems it is shown that this and similar identification algorithms can be successfully combined with a model reduction procedure to produce low order models connections with the nevanlinna pick interpolation problem are explored and an algorithm is given in which the identified model interpolates the given noisy data some numerical results are provided for illustration finally the case of unbounded random noise is discussed and it is shown that one can still obtain convergence with probability 1 under natural assumptions

continuous time control model validation using finite experimental data the application of robust control theory requires models containing unknown bounded perturbations and unknown bounded input signals model validation is a means of assessing the applicability of a given model with respect to experimental data this paper develops a theoretical framework and a computational solution for the model validation problem in the case where the model including unknown perturbations and signals is given in the continuous time domain yet the experimental datum is a finite sampled signal the continuous nature of the unknown components is treated directly with a sampled data lifting theory this gives results which are valid for any sample period and any datum length explicit calculation of whether sufficient data for invalidation has been obtained arises naturally in this framework a common class of robust control models is treated and leads to a convex matrix optimization problem a simulation example illustrates the approach

a mirror symmetric construction of qht g p q abstract let g be a simple simply connected complex algebraic group we give a lie theoretic construction of a conjectural mirror family associated to a general flag variety g p and show that it recovers the peterson variety presentation for the t equivariant quantum cohomology rings q h t g p q with quantum parameters inverted for sl n b we relate our construction to the mirror family defined by givental and its t equivariant analogue due to joe and kim

a class of algorithms for identification in h abs traet in this paper a class of algorithms for the problem of system identification in are investigated these algorithms are characterized by a two stage structure and involve a class of window functions some conditions in terms of properties of the window functions are derived which guarantee robust convergence of the algorithms identification errors are analyzed for several common window functions this leads to some insights into the trade off between the error induced by approximation and that due to noise

stability in parametric resonance of axially moving viscoelastic beams with time dependent speed stability in transverse parametric vibration of axially accelerating viscoelastic beams is investigated the governing equation is derived from newton s second law the kelvin constitution relation and the geometrical relation when the axial speed is a constant mean speed with small harmonic variations the governing equation can be regarded as a continuous gyroscopic system under small periodically parametric excitations and a damping term the method of multiple scales is applied directly to the governing equation without discretization the stability conditions are obtained for combination and principal parametric resonance numerical examples are presented for beams with simple supports and fixed supports respectively to demonstrate the effect of viscoelasticity on the stability boundaries in both cases

nonparametric methods for the estimation of l2 gain and gains at sample frequencies using bandpass filters this paper describes a method of estimating the gains at sample frequencies and the l2 gain for a linear time invariant discrete time system using a truncated step response the gain is estimated using a bandpass filter at each sample frequency and the filter parameter is optimized by maximizing the estimate of the l2 gain near the peak gain this filtering method is extended to multivariable systems another method of making an optimal bandpass filter is derived from the iterative experiment method the filtering method is compared with other estimation methods in terms of noise sensitivity through a series of numerical experiments

uniform sample generations from contractive block toeplitz matrices this note deals with generating a series of random matrices uniformly and independently from contractive lower triangular block toeplitz ltbt matrices all the contractive ltbt matrices are parameterized nonlinearly but recursively in a closed form by a sequence of contractive unstructured matrices moreover the conditional probability density function pdf is obtained in a closed form for all the matrix parameters as well as the squares of their singular values it has been made clear that the conditional pdfs of the matrix parameters are unitarily invariant a procedure is developed to accurately produce the required random matrix samples

adaptive quantification of model uncertainties by rational approximation an adaptive rational approximation approach for quantifying the effect of uncertainty is presented it is shown that a tight frequency upper bound on the uncertainty is obtainable by adaptive rational approximation the identifier which consists of the plant identifier and the uncertainty identifier is discussed the plant identifier gives a nominal model which has lower complexity than that of the true plant errors between the estimated nominal model and the true plant are characterized by a sequence of rational functions which converges to the accurate upper bound of the uncertainty in the frequency domain since approximation and identification are grouped together the whole procedure is completely automatic this allows robust control and adaptive control to be combined

a nonlinear functional approach to lft model validation model validation provides a useful means of assessing the ability of a model to account for a specific experimental observation and has application to modeling identification and fault detection in this paper we consider a new approach to the model validation problem by deploying quadratic functionals and more generally nonlinear functionals to specify noise and dynamical perturbation sets specifically we consider a general linear fractional transformation framework for the model structure and use constraints involving nonlinear functional inequalities to specify model non linearities and unknown perturbations and characteristics of noise and disturbance signals sufficient conditions for invalidation of such models are provided in terms of semidefinite programming problems

time domain identification for robust control abstract in this paper we are concerned with a problem of robust control oriented system identification in the time domain based on the well known schur takagi aak theorem we propose a linear algorithm to obtain the nominal model of the plant to be identified and the minimal bound of the uncertainty of the nominal model error which is measured by h norm it is also shown that in the model set defined by the nominal model and the uncertainty bound there exists at least one model which matches the prescribed input output data given in the time domain

a continuous time extension condition this paper presents a continuous time extension condition which has applications in the areas of identification and model validation of continuous time systems consider two signals specified on a finite interval the result gives a necessary and sufficient condition for the existence of a contractive causal linear time invariant operator mapping one signal to the other this result is a continuous time analog of the classical caratheodory extension theorem

suboptimal algorithms for worst case identification in h sup spl infin and model validation new algorithms based on convex programming are proposed for worst case system identification the algorithms are optimal within a factor of two asymptotically further model validation or data consistency is embedded in the identification process explicit worst case identification error bounds in the h sup spl infin norm are also derived for both uniformly and nonuniformly spaced frequency response samples

h sub spl infin identification of multivariable systems by tangential interpolation methods the purpose of this paper is to present an extension to some of the current work on worst case identification problems to multivariable systems we consider an h sub spl infin identification problem for a class of linear shift invariant multi input multi output systems our main results are an interpolatory algorithm and a number of bounds on the identification error this algorithm operates on available input and output data in the time domain and is constructed by solving an extended matrix tangential caratheodory fejer problem similar to its counterpart for scalar systems this interpolatory algorithm possesses certain desirable optimality properties and can be obtained via standard convex programming methods

lft uncertain model validation with time and frequency domain measurements in this paper the authors study a model validation problem pertaining to linear fractional uncertainty models they extend the previous validation approaches based upon either time or frequency measurements to one using simultaneously time and frequency domain data they show that this problem can be reduced to two independent convex feasibility tests each of which corresponds to the time or frequency domain data alone and each can be verified numerically using available algorithms and software programs the merit of such a mixed time and frequency domain approach therefore lies in that not only can it accommodate the two distinctively different types of measurement data simultaneously but also from a computational standpoint it will not suffer from a complexity growth beyond that associated with the existing time domain tests

modeling of linear fading memory systems motivated by questions of approximate modeling and identification we consider various classes of linear time varying bounded input bounded output bibo stable fading memory systems and the characterizations are proved these include fading memory systems in general almost periodic systems and asymptotically periodic systems we also show that the norm and strong convergence coincide for bibo stable causal fading memory systems

validation of linear fractional uncertain models solutions via matrix inequalities a time domain approach is provided in this paper to tackle the problem of model validation pertaining to uncertain models described by linear fractional transforms both unstructured and structured dynamic uncertainties are considered it is shown that in the first case the problem can be solved by finding a feasible solution to a convex optimization problem while in the second case it amounts to solving a biaffine matrix inequality problem to which only a convex optimization based necessary condition is given

persistent identification of time varying systems identification of time varying systems especially slowly time varying systems is of importance in the development of a comprehensive theory of adaptation the persistent identification measures employed in this paper capture a main characterization in such identification problems namely one input signal must be used for identification of all possible observation windows this paper establishes several essential features in persistent identification problems which highlight their potential utility in adaptation 1 they have computable upper and lower bounds for typical classes of prior uncertainty sets 2 any full rank n periodic signals are optimal and the simple least squares estimates are optimal identification algorithms 3 optimal probing inputs can be approximately generated in a closed loop configuration when the plant and the controller are slowly time varying and 4 n periodic signals are asymptotically optimal for slowly time varying systems the main results of this paper have been successfully combined with a certain slow h sup spl infin design to derive an adaptive stabilization scheme

on computational complexity of invalidating structured uncertainty models abstract in this paper we study a class of uncertainty model validation invalidation problems for linear discrete time systems we consider models described by a linear fractional transformation lft of modeling uncertainties which are structured time invariant or time varying and are bounded in h or l 1 norms the experimental data available for use in invalidation are either input output time series observations or frequency response measurements we analyze the computational complexity associated with these problems while earlier work showed that for lft models with an unstructured uncertainty the invalidation problems can be reduced to one of solving linear matrix inequalities and hence may be tackled readily using well developed numerical methods in this paper we provide a simple proof to show that in the presence of structured uncertainties they are all np hard with respect to the number of uncertainty blocks suggesting that the problems are inherently intractable from a computational standpoint additionally we also demonstrate that the problems do become tractable when the lft model description is changed to an additive one leading to lmi based invalidation tests similar to those obtained elsewhere these results indicate that the main source of the computational difficulty lies in the lft description together with the structured nature of the uncertainties

frequency domain tests for validation of linear fractional uncertain models a frequency domain approach is adopted in this paper to tackle the problem of validating uncertainty models described by linear fractional transforms this problem amounts to verifying the consistency of certain given mathematical models to experimental information obtained from a physical plant using either input output frequency domain measurements or frequency samples of the plant linear fractional models with both unstructured and structured uncertainties are considered the problem is resolved in the former case and solved approximately in the latter both results lead to tests that are readily computable via convex optimization methods and can be implemented using standard algorithms in comparison to previously available algorithms based on time domain information the main advantage of these tests is that they have a considerably lower level of computational complexity it is shown that the validation problem reduces to one of nevanlinna pick boundary interpolation and it can be solved by computing independently a sequence of convex programs of a lower dimension each of which corresponds to only one frequency sample

transition behaviour from string to beam for an axially accelerating material an approximate analytical expression for the natural frequency is given for the problem for velocity profiles harmonically varying about a mean velocity stability borders are determined analytically for fluctuation frequency versus amplitudes beam effects cause the stability boundaries to shift to higher frequency values

totally positive toeplitz matrices and quantum cohomology of partial flag varieties we show that the set of totally positive unipotent lower triangular toeplitz matrices in gln form a real semi algebraic cell of dimension n 1 furthermore we prove a natural cell decomposition for its closure the proof uses properties of the quantum cohomology rings of the partial flag varieties of gln c relying in particular on the positivity of the structure constants which are enumerative gromov witten invariants we also give a characterization of total positivity for toeplitz matrices in terms of the quantum schubert classes this work builds on some results of dale peterson s which we explain with proofs in the type a case

a time domain approach to model validation in this paper we offer a novel approach to control oriented model validation problems the problem is to decide whether a postulated nominal model with bounded uncertainty is consistent with measured input output data our approach directly uses time domain input output data to validate uncertainty models the algorithms we develop are computationally tractable and reduce to generally nondifferentiable convex feasibility problems or to linear programming problems in special cases we give analytical solutions to these problems

unfalsified plant model parameterization from closed loop experimental data abstract the problem of parameterizing all unfalsified plant models from closed loop time domain identification experimental data is investigated in this paper the assumed a priori plant information is an upper bound of the l norm and the number of the unstable poles of its transfer function moreover it is assumed that magnitude bounds of the disturbances that perturb the closed loop system have been supplied under the condition that the stabilizing controller is known it is shown that all the unfalsified plant transfer functions can be parameterized by a linear fractional transformation of a fixed transfer function matrix and an l norm bounded structure fixed but uncertain transfer function matrix these results are very similar to those when plant open loop time domain identification experimental data are supplied however further efforts are still needed in order to apply these results to robust controller design this results mainly from the computational complexity and the high dimensions of the transfer function matrices involved in the parameterization

comments on robust kalman filter for descriptor systems in the above paper a kalman type robust filter is derived for descriptor systems on the basis of the framework suggested in the performance of the resulting descriptor filter in however is generally not very good as structured perturbations in the corresponding regularized least squares problem are directly replaced by unstructured ones in this comment we show that through off line convex optimization the filter s performance can be improved without sacrificing its recursiveness the idea is to use the ellipsoid with the smallest volume to replace the intersection of several ellipsoids that are determined by the actual plant states and structured model uncertainties and to re scale the inputs and outputs of a model uncertainty block with the semi axes of the optimal ellipsoid using the same numerical example and design parameter the variance of the estimation error can be reduced by approximately 10 moreover it has been shown by simulations that when a standard state space model is adopted and model uncertainties do not include zero the method of does not always outperform that of

estimation of model quality this paper provides an introduction to recent work on the problem of quantifying errors in the estimation of models for dynamic systems this is a very large field we therefore concentrate on approaches that have been motivated by the need for reliable models for control system design this will involve a discussion of efforts that go under the titles of estimation in th worst case estimation estimation in l1 and stochastic embedding of undermodelling a central theme of this survey is to examine these new methods with reference to the classic bias variance tradeoff in model structure selection

structure of model uncertainty for a weakly corrupted plant in this paper we investigate the structure of the transfer function set which includes all the transfer functions deduced from the plant available information it is shown that when an upper bound of the plant transfer function s h sup spl infin norm has been supplied and the noise contaminating the time domain identification experiment data is not too significant such a transfer function set can be parameterized by a linear fractional transformation of two transfer function matrices one of them is a fixed transfer function matrix which is completely determined by the plant available information and the noise magnitude the other is a norm bounded structure fixed free transfer function matrix moreover it is shown that the problem of analytically obtaining the fixed complexity nominal model that best approximates this transfer function set is as difficult as the spl mu synthesis problem

on control system design using random samples of contractive block toeplitz matrices contractive lower triangular block toeplitz ltbt matrices have recently been parametrized in a closed form by a sequence of unstructured contractive matrices a problem naturally arises between uniform samplings over contractive ltbt matrices and their matrix parameters that is which one is more appropriate in control system analysis and synthesis this paper makes it clear that uniform sampling over matrix parameters results in a contractive ltbt matrix with a more rapid concentration speed towards the boundary of the uncertainty set this is not an attractive property in control related problems numerical comparisons also show that uniform sampling directly over contractive ltbt matrices outperforms that over the matrix parameter set in robustness analysis for a closed loop system as well as in strongly stable closed loop system design with complexity constraints on the controller as a sacrifice uniform sampling over contractive ltbt matrices is more mathematically complicated and more time cons