maximum likelihood estimators and worst case optimal algorithms for system identification abstract for system identification problems with stochastic noise maximum likelihood estimators are frequently used if noise is deterministic worst case optimal algorithms should be considered in this paper we study the following problem under what circumstances are maximum likelihood estimators optimal in the worst case

quadric inclusion programs an lmi approach to mathcal h_ infty model identification practical application of mathcal h infty robust control relies on the system identification of a valid model set described by a linear system in feedback with a stable norm bounded uncertainty this model set should explain all possible or at least all previously measured behavior for the controlled plant such models can be viewed as norm bounded inclusions in the frequency domain this note introduces the quadric inclusion program that can identify inclusions from the input output data as a convex problem we prove several key properties of this algorithm and give a geometric interpretation for its behavior while we stress that the inclusion fitting is outlier sensitive by design we offer a method to mitigate the effect of measurement noise we apply this method to robustly approximate the simulated frequency domain data using orthonormal basis functions the result compares favorably with a least squares approach that satisfies the same data inclusion requirements

robust identification in h abstract we consider system identification in the banach space h in the framework proposed by helmicki jacobson and nett it is shown that there is no robustly convergent linear algorithm for identifying exponentially stable systems in the presence of noise which is not tuned to prior information about the unknown system or noise various nonlinear algorithms some closely related to one of gu and khargonekar are analysed and results on trigonometric interpolation used to provide new error bounds an application of these techniques to approximation is given and finally some numerical results are provided for illustration

identification with nonparametric uncertainty an identification technique that is robust to nonparametric uncertainty i e model mismatch is presented the identifier produces either a parameter set estimate or a frequency response set estimate the estimates result from the inclusion of a model of the nonparametric uncertainty in the plant model the frequency response set estimate is shown to always contain the frequency response of the plant as long as certain modeling conditions are met this type of identifier would be useful in applications such as control where a property such as stability or performance level must be achieved in the face of low order modeling and its associated nonparametric uncertainty

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

a generalization of the structured singular value and its application to model validation the structured singular value spl mu gives a means of assessing the stability and performance of a system under a class of norm bounded structured perturbations this paper introduces a generalization of spl mu which guarantees stability with respect to perturbations where some elements of the perturbation structure are bounded from above and some are bounded from below this generalization is used to solve the frequency domain robust control model validation problem model validation determines whether or not there exist a model in the model set and a bounded noise signal that are consistent with the experimental input output measurements a computable convex upper bound to the generalized spl mu problem is presented and the relationship between the generalized spl mu problem and the standard spl mu and model validation problems is discussed

identification of probabilistic system uncertainty regions by explicit evaluation of bias and variance errors a procedure is developed for identification of probabilistic system uncertainty regions for a linear time invariant system with unknown dynamics on the basis of time sequences of input and output data the classical framework is handled in which the system output is contaminated by a realization of a stationary stochastic process given minor and verifiable prior information on the system and the noise process frequency response pulse response and step response confidence regions are constructed by explicitly evaluating the bias and variance errors of a linear regression estimate in the model parametrizations use is made of general forms of basis functions conservatism of the uncertainty regions is limited by focusing on direct computational solutions rather than on closed form expressions using an instrumental variable method for identification the procedure is suitable also for input output data obtained from closed loop experiments

optimal estimation theory for dynamic systems with set membership uncertainty an overview in many problems such as linear and nonlinear regressions parameter and state estimation of dynamic systems state space and time series prediction interpolation smoothing and functions approximation one has to evaluate some unknown variable using available data the data are always associated with some uncertainty and it is necessary to evaluate how this uncertainty affects the estimated variables typically the problem is approached assuming a probabilistic description of uncertainty and applying statistical estimation theory an interesting alternative approach referred to as set membership or unknown but bounded ubb error description has been investigated since the late 1960s in this approach uncertainty is described by an additive noise which is known only to have given integral typically l2 of l1 or componentwise l bounds in this paper we review the main results of this theory with special attention to the most recent advances obtained in the case of componentwise bounds

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

brief paper risk sensitivity conditions for stochastic uncertain model validation the paper presents sufficient and necessary conditions that verify the relevance of an assumed linear stochastic system model for problems in which probabilistic characteristics of the plant are not known exactly the approach is to establish the existence of an admissible probability space on which the output of the candidate stochastic system model is consistent in a stochastic sense with the noisy output of the plant

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

convex necessary and sufficient conditions for frequency domain model in validation under sltv structured uncertainty this paper deals with the problem of model in validation of discrete time causal linear time invariant lti stable models subject to slowly linear time varying structured uncertainty using frequency domain data corrupted by additive noise it is well known that in the case of structured lti uncertainty the problem is np hard in the number of uncertainty blocks the main contribution of this paper shows that on the other hand if one considers arbitrarily slowly time varying uncertainty and noise in l sub 2 then tractable convex necessary and sufficient conditions for in validation can be obtained additional results include a discussion of the case where the noise is characterized in terms of the l sub spl infin norm

brief paper validity of the standard cross correlation test for model structure validation in the standard prediction error framework of system identification statistical properties of estimated models are typically derived under the assumption that the true system is in the model class the standard model structure validation test for plant models is the sample cross correlation test between the residuals of the model and the input it turns out that the standard test itself is valid only under exactly those assumptions it is meant to verify i e the system is in the model class it is shown that for reliable results of the validation test a vector valued test is required and that accurate noise modelling is indispensable for reliable model structure validation this shows the limitation of separate validation of plant and noise model structures improvements of the test are presented and it is motivated by the fact that reserving data only to be used for model validation is not efficient

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

model validation of multirate systems from time domain experimental data the model validation problem using time domain experimental data is studied for multirate linear fractional uncertain models in this note as a technical tool the caratheodory fejer cf interpolation problem with a nest operator constraint is first investigated this problem is itself of interest mathematically and has potential applications in addressing other problems in control signal processing and circuit theory a necessary and sufficient solvability condition for this interpolation problem is given the validation tests are then presented based on this condition and the lifting technique tractable convex optimization methods can be used to solve the validation problems

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

recursive system identification in the presence of noise and model uncertainties the main contribution of this paper is a recursive algorithm for parametric system identification in the presence of both noise and model uncertainties the estimates provided by this algorithm are not invalidated after a learning period by the observed input output data and the assumed system and uncertainty structures a complementary off line algorithm derived from the on line algorithm is also presented

analytic solutions of a class of linear and nonlinear functional equations abstract a functional analysis method is used to prove the existence and the uniqueness of solutions of a class of linear and nonlinear functional equations in the hilbert space h2 and the banach space h1 in the case of the nonlinear functional equation a bound of the solution is also given

a new approach to an analysis of henry type integral inequalities and their bihari type versions abstract in this paper we propose a new method to solve integral inequalities of henry gronwall type and their bihari nonlinear version nonlinear integral inequalities with weakly singular kernels and with multiple integrals as well as a modification of the ou iang pachpatte inequality are also treated

worst case system identification in h sub spl infin validation of a priori information essentially optimal algorithms and error bounds resolves several important open issues pertaining to a worst case control oriented system identification problem known as identification in h sub spl infin first a method is presented for developing confidence that certain a priori information available for identification is not invalid this method requires the solution of a certain nondifferentiable convex program second an essentially optimal identification algorithm is constructed this algorithm is worst case strongly optimal to within a factor of two finally new upper and lower bounds on the optimal identification error are derived and used to estimate the identification error associated with the given algorithm interestingly the development of each of these results draws heavily upon the classical nevanlinna pick interpolation theory as such the authors results establish a clear link between the areas of system identification and optimal interpolation theory both the formulation and techniques in this paper are applicable to problems where the frequency data available for identification may essentially be arbitrarily distributed

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 measure of robust stability for an identified set of parametrized transfer functions defines a measure of robustness for a set of parameterized transfer functions as delivered by classical prediction error identification and that contains the true system at a prescribed probability level this measure of robustness is the worst case vinnicombe distance between the model and the plants in the uncertainty region we show how it can be computed exactly using lmi based optimization in addition we show that this measure is directly connected to the size of the set of controllers that are guaranteed to stabilize all plants in the uncertainty region i e the smaller the worst case vinnicombe distance for an uncertainty region the larger the set of model based controllers that are guaranteed to stabilize all systems in this uncertainty region

subspace based multivariable system identification from frequency response data two noniterative subspace based algorithms which identify linear time invariant mimo multi input multioutput systems from frequency response data are presented the algorithms are related to the recent time domain subspace identification techniques the first algorithm uses equidistantly in frequency spaced data and is strongly consistent under weak noise assumptions the second algorithm uses arbitrary frequency spacing and is strongly consistent under more restrictive noise assumptions promising results are obtained when the algorithms are applied to real frequency data originating from a large flexible structure

center stable manifold for planar fractional damped equations in this paper we discuss the existence of a center stable manifold for planar fractional damped equations by constructing a suitable lyapunovperron operator via giving asymptotic behavior of mittagleffler function we obtain an interesting center stable manifold theorem finally an example is provided to illustrate the result

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

optimal nonparametric identification from arbitrary corrupt finite time series formulates and solves a worst case system identification problem for single input single output linear shift invariant distributed parameter plants the available a priori information in this problem consists of time dependent upper and lower bounds on the plant impulse response and the additive output noise the available a posteriori information consists of a corrupt finite output time series obtained in response to a known nonzero but otherwise arbitrary input signal the authors present a novel identification method for this problem this method maps the available a priori and a posteriori information into an uncertain model of the plant which comprises a nominal plant model a bounded additive output noise and a bounded additive model uncertainty the upper bound on the model uncertainty is explicit and expressed in terms of both the l sub 1 and h sub spl infin system norms the identification method and the nominal model possess certain well defined optimality properties and are computationally simple requiring only the solution of a single linear programming problem

optimal asymptotic identification under bounded disturbances the intrinsic limitation of worst case identification of linear time invariant systems using data corrupted by bounded disturbances when the unknown plant is known to belong to a given model set is studied this is done by analyzing the optimal worst case asymptotic error achievable by performing experiments using any bounded input and estimating the plant using any identification algorithm it is shown that under some topological conditions on the model set there is an identification algorithm which is asymptotically optimal for any input and the optimal asymptotic error is characterized as a function of the inputs these results which hold for any error metric and disturbance norm are applied to three specific identification problems identification of stable systems in the l sub 1 norm identification of stable rational systems in the h sub infinity norm and identification of unstable rational systems in the gap metric for each of these problems the general characterization of optimal asymptotic error is used to find near optimal inputs to minimize the error

performance robustness of discrete time systems with structured uncertainty given an interconnection of a nominal discrete time plant and a stabilizing controller together with structured norm bounded nonlinear time varying perturbations necessary and sufficient conditions for robust stability and performance of the system are provided it is shown that performance robustness is equivalent to stability robustness in the sense that both problems can be dealt with in the framework of a general stability robustness problem the resulting stability robustness problem is shown to be equivalent to a simple algebraic one the solution of which provides the desired necessary and sufficient conditions for performance stability robustness these conditions provide an effective tool for robustness analysis and can be applied to a large class of problems in particular it is shown that some known results can be obtained immediately as special cases of these conditions

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

probabilistic estimates for mixed model validation problems with cal h _ infty type uncertainties a mixed deterministic probabilistic model validation problem is investigated in this technical note which consists in an additive uncertain model with model uncertainty characterized by the h norm the data available for validation are time domain experimental data corrupted by a random noise sequence our aim is to compute the probability for such an uncertain model to be validated by the data and our main results are bounds on this probability that are computable based on the distribution of chi square random variables when the noise is a gaussian variable and solvable as an lmi problem when only statistical information such as the expectation and covariance of the noise are known

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

barrier certificates for nonlinear model validation methods for model validation of continuous time nonlinear systems with uncertain parameters are presented in this paper the methods employ functions of state parameter time termed barrier certificates whose existence proves that a model and a feasible parameter set are inconsistent with some time domain experimental data a very large class of models can be treated within this framework this includes differential algebraic models models with memoryless dynamic uncertainties and hybrid models construction of barrier certificates can be performed by convex optimization utilizing recent results on the sum of squares decomposition of multivariate polynomials

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

control oriented model validation and errors quantification in the spl lscr sub 1 setup a priori information required for robust synthesis includes a nominal model and a model of uncertainty the latter is typically in the form of additive exogenous disturbance and plant perturbations with assumed bounds if these bounds are unknown or too conservative they have to be estimated from measurement data in this paper the problem of errors quantification is considered in the framework of the spl lscr sub 1 optimal robust control theory associated with the spl lscr sub spl infin signal space the optimal errors quantification is to find errors bounds that are not falsified by measurement data and provide the minimum value of a given control criterion for model with unstructured uncertainty entering the system in a linear fractional manner the optimal errors quantification is reduced to quadratic fractional programming for system under coprime factor perturbations the optimal errors quantification is reduced to linear fractional programming

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

method of centers for minimizing generalized eigenvalues abstract we consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices each of which depends affinely on the decision variables although this problem may appear specialized it is in fact quite general and includes for example all linear quadratic and linear fractional programs many problems arising in control theory can be cast in this form the problem is nondifferentiable but quasiconvex so methods such as kelley s cutting plane algorithm or the ellipsoid algorithm of shor nemirovsky and yudin are guaranteed to minimize it in this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently the algorithm is a variation on huard s method of centers using a self concordant barrier for matrix inequalities developed by nesterov and nemirovsky nesterov and nemirovsky have also extended their potential reduction methods to handle the same problem since the problem is quasiconvex but not convex devising a nonheuristic stopping criterion i e one that guarantees a given accuracy is more difficult than in the convex case we describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper in some cases than a more general result to nesterov and nemirovsky the algorithm is demonstrated on an example determining the quadratic lyapunov function that optimizes a decay rate estimate for a differential inclusion

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

parameter identification in the presence of non parametric dynamic uncertainty abstract this paper provides identification laws which yield monotonic parameter error reduction in the absence of external noise inputs but in the presence of non parametric dynamic uncertainty unmodelled dynamics this paper considers two non parametric uncertainty sets motivated by robust control the weighted ball in h and the weighted ball in the induced l2 norm the non parametric uncertainty produces signals which corrupt the signals used in identification this paper shows that the corruption is non conservatively bounded in an l2 sense this l2 bound is then compared with the signal energy with the result used to enable disable the parameter adjustment of a gradient least squares adjustment mechanism it is proved that the resulting parameter adjustment yields non increasing parameter errors with strictly decreasing parameter errors whenever the parameter errors are distinguishable from zero given the system input and output histories

modeling of normalized coprime factors with spl nu metric uncertainty modeling of uncertain systems with normalized coprime factor description is investigated where the experimental data is given by a finite set of frequency response measurement samples of the open loop plant that is linear time invariant and possibly infinite dimensional the objective is not only to identify the nominal model but also to quantify the modeling error with sup norm bounds in frequency domain the uncertainty to be identified and quantified is chosen as the spl nu metric proposed by vinnicombe 1993 because of its compatibility with h sub spl infin based robust control an algorithm is developed to model the normalized coprime factors of the given plant using techniques of discrete fourier analysis and balanced stochastic truncation and is shown to be robust in the presence of the worst case noise upper bounds are derived for the associated modeling error based on the minimum a priori information of the underlying model set and of the noise level in the measurement data a simulation example is used to illustrate the effectiveness of the proposed algorithm

an lmi approach to control oriented identification and model in validation of lpv systems this note proposes a control oriented identification framework for a class of linear parameter varying systems that takes into account both the dependence of part of the model on time varying parameters as well as the possible existence of a nonparametric component the main results of the note show that the problems of obtaining and validating a model for these systems can be recast as linear matrix inequality feasibility problems moreover as the information is completed the algorithm is shown to converge in the l sub 2 induced topology to the actual plant additional results include deterministic bounds on the identification error these results are illustrated with a practical example arising in the context of active vision

an interior point method for minimizing the maximum eigenvalue of a linear combination of matrices we consider the problem of finding lambda opt mathop inf limits x in r m lambda max a 0 sum limits i 1 m x i a i 1 and an optimal solution xopt if it exists here the a i are given symmetric matrices in ir n n for 0 i m and max a is the largest eigenvalue of a matrix a problem 1 can be rewritten as a convex differentiable problem with a positive definite constraint positive definiteness can easily be verified numerically via cholesky decomposition moreover the cone of positive definite matrices a is characterized by the domain of the smooth convex barrier function log det a

long time behavior for a nonlinear fractional model the asymptotic behavior for solutions of a weighted cauchy type nonlinear fractional problem is investigated we find bounds for solutions on infinite time intervals and also provide sufficient conditions assuring decay to zero this work improves earlier results by the same authors

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

l 1 optimal feedback controllers for mimo discrete time systems the problem considered in this paper is the design of a closed loop system consisting of a mimo discrete time plant and compensator in such a way that the system is internally stable and optimally tracks all persistent bounded inputs the solution consists of two parts first the calculation of the minimum value of the performance index a weighted transfer function which is done by solving a linear programming problem and second construction of the optimal transfer function by solving a set of linear equations it is shown that in general the discrete time problem will have a rational solution and thus appears to be of considerable practical significance

consistency of open loop experimental frequency response data with coprime factor plant models the model data consistency problem for coprime factorization considered is given some possibly noisy frequency response data obtained by running open loop experiments on a system show that these data are consistent with a given family of perturbed coprime factor models and a time domain spl lscr sub spl infin noise model in the noise free open loop case the model data consistency problem boils down to the existence of an interpolating function in spl rscr spl hscr sub spl infin that evaluates to a finite number of complex matrices at a finite number of points on the imaginary axis a theorem on boundary interpolation in spl rscr spl hscr sub spl infin is a building block that allows one to devise computationally simple necessary and sufficient tests to check if the perturbed coprime factorization is consistent with the data for standard coprime factorizations the test involves the computation of minimum norm solutions to underdetermined complex matrix equations the schmidt mirsky theorem is used in the case of special factorizations of flexible systems for spl lscr sub spl infin noise corrupting the frequency response measurements a complete solution to the open loop noisy siso problem using the structured singular value spl mu is given

distance measures for uncertain linear systems a general theory in this paper we propose a generic notion of distance between systems that can be used to measure discrepancy between open loop systems in a feedback sense under several uncertainty structures when the uncertainty structure is chosen to be four block or equivalently normalized coprime factor uncertainty then this generic distance measure reduces to the well known nu gap metric associated with this generic distance notion we also define a generic stability margin notion that allows us to give the distance measure a feedback interpretation by deriving generic robust stability and robust performance results the proposed distance notion and the corresponding results exploit a powerful generalization of the small gain theorem which handles perturbations in rfrl infin rather than only in rfrh infin when the uncertainty structure is fixed to one of the standard structures e g additive multiplicative inverse multiplicative coprime factor four block or any mixtures of the above we give a step by step procedure based on model validation ideas that shows how the generic notion of distance and the correspondingly generic winding number conditions can be reduced to simple formulae this work provides a unified framework that captures and embeds previous results in this area and also completes the picture by showing how other results of a similar nature can be obtained from the same framework the techniques used involve only basic linear algebra so they also provide a simplification of previous advanced proofs furthermore the various distance measures so created can be used for non conservative model embedding into the smallest uncertain family an illustrative example is also given that demonstrates the superior qualities above the nu gap metric of a particular distance measure obtained from this work in situations where the plant is lightly damped all systems considered in this paper are linear time invariant

the role of model validation for assessing the size of the unmodeled dynamics the problem of assessing the quality of a given or estimated model is a central issue in system identification various new techniques for estimating bias and variance contributions to the model error have been suggested in recent literature in this contribution classical model validation procedures are placed at the focus of our attention we discuss the principles by which we reach confidence in a model through such validation techniques and also how the distance to a true description can be estimated this way in particular we stress how the typical model validation procedure gives a direct measure of the model error of the model test without referring to its ensemble properties several model error bounds are developed for various assumptions about the disturbances entering the system

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

power type estimates for a nonlinear fractional differential equation abstract in this work we are concerned with a family of nonlinear ordinary differential equations of fractional order it is proved that solutions of these equations with weighted initial data exist globally and decay as a power function

on the consistency between an lft described model set and frequency domain data the main objective of this technical note is to derive a simple necessary and sufficient condition for a linear fractional transformation lft perturbed model set being consistent with frequency domain plant input output data only discrete time models and unstructured modeling errors are dealt with compared with the available results in which the eigenvalues of a matrix are involved this condition is related only to the euclidean norms of two vectors moreover these vectors linearly depend on measurement errors some of its applications to model set validation have been briefly discussed based on this condition an almost analytic solution has been established for model set validation under a deterministic framework when the measurement errors are energy bounded numerical simulations show that this consistency condition can lead to a significant computation cost reduction

brief paper non parametric methods for l2 gain estimation using iterative experiments in this paper we develop non parametric methods to estimate the l 2 gain h norm of a linear dynamical system from iterative experiments this work is mainly motivated by model error modeling where the error dynamics are more complex than can be captured by a low order parametric model the standard system identification approach to the gain estimation problem is to estimate a parametric model of the system which is then used to calculate the gain if it is possible to update the input signal during the experiment an alternative way is to iteratively optimize the input signal in order to maximize the estimated input to output gain a key observation is that the gradient of the gain with respect to the input signal can without knowing a model be found from two experiments iterative numerical methods for calculation of eigenvalues of matrices e g the power method or the lanczos method can then be applied to update the input signal sequence between experiments in order to find the maximum gain the main difficulty compared to the corresponding eigenvalue problem in numerical analysis is the effects of additive measurement noise which require modified schemes that avoid bias errors three such related methods are derived and evaluated by a numerical example partial results on convergence and statistical properties of the gain estimator are given a constrained stochastic gradient method with local optimization of step length gives the best numerical results in the case of noisy data

asymptotic properties of black box identification of transfer functions the problem of estimating the transfer function of a linear stochastic system is considered the transfer function is parametrized as a black box and no given order is chosen a priori this means that the model orders may increase to infinity when the number of observed data tends to infinity the consistency and convergence properties of the resulting transfer function estimates are investigated asymptotic expressions for the variances and distributions of these estimates are also derived for the case that the model orders increase it is shown that the variance of the transfer function estimate at a certain frequency is asymptotically given by the noise to signal ratio at that frequency mulliplied by the model order to number of data points ratio