mimo system identification state space and subspace approximations versus transfer function and instrumental variables the identification of multi input multi output mimo linear systems has previously received a new impetus with the introduction of the state space ss approach based on subspace approximations this approach has immediately gained popularity owing to the fact that it avoids the use of canonical forms requires the determination of only one structural parameter and has been empirically shown to yield mimo models with good accuracy in many cases however the ss approach suffers from several drawbacks there is no well established rule tied to this approach for determining the structural parameter and perhaps more important the ss parameter estimates depend on the data in a rather complicated way which renders almost futile any attempt to analyze and optimize the performance of the estimator in this paper we consider a transfer function tf approach based on instrumental variables iv as an alternative to the ss approach we use the simplest canonical tf parameterization in which the denominator is equal to a scalar polynomial times the identity matrix the analysis and optimization of the statistical accuracy of the tf approach is straightforward additionally a simple test tailored to this approach is devised for estimating the single structural parameter needed a simulation study in which we compare the performances of the ss and the tf approaches shows that the latter can provide more accurate models than the former at a lower computational cost

parameter estimation problems with singular information matrices the case of a singular fisher information matrix fim represents a significant complication for the theory of the cramer rao lower bound crb that is usually handled by resorting to the pseudoinverse of the fisher matrix we take a different approach in which the crb is derived as the solution to an unconstrained quadratic maximization problem which enables us to handle the singular case in a simple yet rigorous manner when the fisher matrix is singular except under unusual circumstances any estimator having the specified bias derivatives that figure in the crb must have infinite variance

exploring estimator bias variance tradeoffs using the uniform cr bound we introduce a plane which we call the delta sigma plane that is indexed by the norm of the estimator bias gradient and the variance of the estimator the norm of the bias gradient is related to the maximum variation in the estimator bias function over a neighborhood of parameter space using a uniform cramer rao cr bound on estimator variance a delta sigma tradeoff curve is specified that defines an unachievable region of the delta sigma plane for a specified statistical model in order to place an estimator on this plane for comparison with the delta sigma tradeoff curve the estimator variance bias gradient and bias gradient norm must be evaluated we present a simple and accurate method for experimentally determining the bias gradient norm based on applying a bootstrap estimator to a sample mean constructed from the gradient of the log likelihood we demonstrate the methods developed in this paper for linear gaussian and nonlinear poisson inverse problems

estimates for numerical approximations of rank one convex envelopes summary we present a convergence analysis of an algorithm for the numerical computation of the rank one convex envelope of a function f mathrm m m times n rightarrow mathbb r a rate of convergence for the scheme is established and numerical experiments are presented to illustrate the analytical results and applications of the algorithm

on the computation of the rank one convex hull of a function we report on several numerical experiments where the rank one convexification of an energy density is computed the explicit examples cover a whole spectrum of typical situations one may encounter one of those is especially relevant for the computation of microstructures in crystalline solids

on the stabilization of the disk beam system via torque and direct strain feedback controls this note deals with the stabilization problem of a rotating disk beam system the proposed feedback law consists of a nonlinear torque control exerted on the disk whereas a direct strain control is applied at the beam with a rigid offset as a consequence we end up with a nondissipative nonlinear hybrid pde ode system and hence the existence and stability of solutions of the system cannot be addressed by using standard methods such as energy multiplier or frequency domain to overcome this situation we adopt the riesz basis methodology and show that the closed loop system is well posed and stable as long as the angular velocity of the disk is less than a well defined value related to the beam the results are illustrated through a wide set of numerical examples

phase noise in mimo systems bayesian cramer rao bounds and soft input estimation this paper addresses the problem of estimating time varying phase noise caused by imperfect oscillators in multiple input multiple output mimo systems the estimation problem is parameterized in detail and based on an equivalent signal model its dimensionality is reduced to minimize the overhead associated with phase noise estimation new exact and closed form expressions for the bayesian cramer rao lower bounds bcrlbs and soft input maximum a posteriori map estimators for online i e filtering and offline i e smoothing estimation of phase noise over the length of a frame are derived simulations demonstrate that the proposed map estimators mean square error mse performances are very close to the derived bcrlbs at moderate to high signal to noise ratios to reduce the overhead and complexity associated with tracking the phase noise processes over the length of a frame a novel soft input extended kalman filter ekf and extended kalman smoother eks that use soft statistics of the transmitted symbols given the current observations are proposed numerical results indicate that by employing the proposed phase tracking approach the bit error rate performance of a mimo system affected by phase noise can be significantly improved in addition simulation results indicate that the proposed phase noise estimation scheme allows for application of higher order modulations and larger numbers of antennas in mimo systems that employ imperfect oscillators

information and asymptotic efficiency in parametric nonparametric models asymptotic lower bounds for estimation of the parameters of models with both parametric and nonparametric components are given in the form of representation theorems for regular estimates and asymptotic minimax bounds the methods used involve i the notion of a hellinger differentiable root density where part of the differentiation is with respect to the nonparametric part of the model to obtain appropriate scores and ii calculation of the effective score for the real or vector finite dimensional parameter of interest as that component of the score function orthogonal to all nuisance parameter scores perhaps infinite dimensional the resulting asymptotic information for estimation of the parametric component of the model is just 4 times the squared l 2 norm of the effective score a corollary of these results is a simple necessary condition for adaptive estimation adaptation is possible only if the scores for the parameter of interest are orthogonal to the scores for the nuisance function or nonparametric part of the model examples considered include the one sample location model with and without symmetry mixture models the two sample shift model and cox s proportional hazards model

orientation and stabilization of a flexible beam attached to a rigid body planar motion the author considers a flexible structure modeled as a rigid body which rotates in inertial space a light flexible beam is clamped to the rigid body at one end and free one is clamped at the other it is assumed that the flexible beam performs only planar motion the equations of motion are obtained by using free body diagrams two control problems are posed namely the orientation and stabilization of the system it is shown that suitable boundary controls applied to the free end of the beam and suitable control torques applied to the rigid body solve the problems posed above the proofs are obtained by using the energy of the system as a lyapunov functional

a generalized least absolute deviation method for parameter estimation of autoregressive signals this paper proposes a generalized least absolute deviation glad method for parameter estimation of autoregressive ar signals under non gaussian noise environments the proposed glad method can improve the accuracy of the estimation of the conventional least absolute deviation lad method by minimizing a new cost function with parameter variables and noise error variables compared with second and high order statistical methods the proposed glad method can obtain robustly an optimal ar parameter estimation without requiring the measurement noise to be gaussian moreover the proposed glad method can be implemented by a cooperative neural network nn which is shown to converge globally to the optimal ar parameter estimation within a finite time simulation results show that the proposed glad method can obtain more accurate estimates than several well known estimation methods in the presence of different noise distributions

the geometry of weighted low rank approximations the low rank approximation problem is to approximate optimally with respect to some norm a matrix by one of the same dimension but smaller rank it is known that under the frobenius norm the best low rank approximation can be found by using the singular value decomposition svd although this is no longer true under weighted norms in general it is demonstrated here that the weighted low rank approximation problem can be solved by finding the subspace that minimizes a particular cost function a number of advantages of this parameterization over the traditional parameterization are elucidated finding the minimizing subspace is equivalent to minimizing a cost function on the grassmann manifold a general framework for constructing optimization algorithms on manifolds is presented and it is shown that existing algorithms in the literature are special cases of this framework within this framework two novel algorithms a steepest descent algorithm and a newton like algorithm are derived for solving the weighted low rank approximation problem they are compared with other algorithms for low rank approximation as well as with other algorithms for minimizing a cost function on a grassmann manifold

cramer rao bound for doa estimators under the partial relaxation framework derivation and comparison a class of computationally efficient doa estimators under the partial relaxation pr framework has recently been proposed conceptually different from conventional doa estimation methods in the literature the estimators under the pr framework rely on the non complete relaxation of the array manifold while performing a spectral search in the field of view this particular type of relaxation essentially implies a modified signal model with partial information loss due to the relaxation the information loss and its impact on the doa estimation performance have not yet been analytically quantified in the literature in this paper the information loss induced by the relaxation of the array manifold is investigated through the cramer rao bound crb the closed form expression of the crb for doa estimation under the pr model on the one hand provides insight on the information loss in the asymptotic region where the number of snapshots tends to infinity on the other hand the proposed crb characterizes the lower bound for the doa estimation performance of all pr estimators we prove that under the assumptions of gaussian source signal and noise the crb of the pr signal model is lower bounded by the conventional stochastic crb we also prove that the previously proposed weighted subspace fitting estimator under the pr framework asymptotically achieves the crb of the pr signal model furthermore it is shown that the asymptotic mean squared errors of all weighted subspace fitting estimators under the pr framework for any positive definite weighting matrix are identical

an interpretation of the moore penrose generalized inverse of a singular fisher information matrix it is proved that in a non bayesian parametric estimation problem if the fisher information matrix fim is singular unbiased estimators for the unknown parameter will not exist cramer rao bound crb a popular tool to lower bound the variances of unbiased estimators seems inapplicable in such situations in this correspondence we show that the moore penrose generalized inverse of a singular fim can be interpreted as the crb corresponding to the minimum variance among all choices of minimum constraint functions this result ensures the logical validity of applying the moore penrose generalized inverse of an fim as the covariance lower bound when the fim is singular furthermore the result can be applied as a performance bound on the joint design of constraint functions and unbiased estimators

stochastic analysis of the lms algorithm for system identification with subspace inputs this paper studies the behavior of the low rank least mean squares lms adaptive algorithm for the general case in which the input transformation may not capture the exact input subspace it is shown that the independence theory and the independent additive noise model are not applicable to this case a new theoretical model for the weight mean and fluctuation behaviors is developed which incorporates the correlation between successive data vectors as opposed to the independence theory model the new theory is applied to a network echo cancellation scheme which uses partial haar input vector transformations comparison of the new model predictions with monte carlo simulations shows good to excellent agreement certainly much better than predicted by the independence theory based model available in the literature

numerical computation of rank one convex envelopes we describe an algorithm for the numerical computation of the rank one convex envelope of a function f mm m times n rightarrow rr we prove its convergence and an error estimate in l infty

a numerical iterative scheme for computing finite order rank one convex envelopes abstract it is known that the ith order laminated microstructures can be resolved by the kth order rank one convex envelopes with k i so the requirement of establishing an efficient numerical scheme for the computation of the finite order rank one convex envelopes arises in this paper we develop an iterative scheme for such a purpose the first order rank one convex envelope r1f is approximated by evaluating its value on matrixes at each grid point in rmn and then extend to non grid points by interpolation the approximate kth order rank one convex envelope rkf is obtained iteratively by computing the approximate first order rank one convex envelope of the numerical approximation of rk 1f compared with o h1 3 obtained so far for other methods the optimal convergence rate o h is established for our scheme and numerical examples illustrate the computational efficiency of the scheme

optimal reduced rank estimation and filtering this paper provides a unified view of and a further insight into a class of optimal reduced rank estimators and filters an alternating power ap method for computing the optimal reduced rank estimators and filters is derived and analyzed the ap method is a generalization of the conventional power method for subspace computation which is shown to be globally and exponentially convergent under weak conditions when the rank reduction is relatively large the ap method is computationally more efficient than the conventional methods the ap method is useful for adaptive computation of the canonical components of a desired reduced rank estimate which in turn facilitates the detection of a time varying rank the study shown in this paper is particularly useful for applications that involve a large number of sources and a large number of receivers where rank reduction is either inherent in the multivariate system or required to reduce the model complexity and or the computational load

minimum variance in biased estimation with singular fisher information matrix this paper extends the work of y c eldar ldquominimum variance in biased estimation bounds and asymptotically optimal estimators rdquo in ieee trans signal process vol 52 pp 1915 1929 jul 2004 which deals with only nonsingular fisher information matrix in order to guarantee the uniform cramer rao bound to be a finite lower bound and also to have a feasible solution to the optimization problem in the work of y c eldar it is proved that the norms of bias gradient matrices of all biased estimators must have a nonzero exact lower bound which mainly depends on the rank of the singular fisher information matrix the smaller the rank of the singular fisher information matrix is the larger the lower bound of norms of bias gradient matrices of all biased estimators is for a specific frobenius norm the exact lower bound is simply the difference between the parameter dimension and the rank of the singular fisher information matrix

statistical analysis of subspace based estimation of reduced rank linear regressions a number of signal processing and system identification problems include linear regressions with a reduced rank regression matrix a typical step in subspace based algorithms is to apply the singular value decomposition svd to compute a low rank factorization however it is not clear how certain weighting matrices should be defined for best possible accuracy we present a statistical analysis of the estimate of the reduced rank regression matrix and we discuss a couple of approaches for finding weighting matrices

computing constrained cramer rao bounds we revisit the problem of computing submatrices of the cramer rao bound crb which lower bounds the variance of any unbiased estimator of a vector parameter mbi we explore iterative methods that avoid direct inversion of the fisher information matrix which can be computationally expensive when the dimension of mbi is large the computation of the bound is related to the quadratic matrix program where there are highly efficient methods for solving it we present several methods and show that algorithms in prior work are special instances of existing optimization algorithms some of these methods converge to the bound monotonically but in particular algorithms converging nonmonotonically are much faster we then extend the work to encompass the computation of the crb when the fisher information matrix is singular and when the parameter mbi is subject to constraints as an application we consider the design of a data streaming algorithm for network measurement

low rank detection of multichannel gaussian signals using block matrix approximation the exact design of an m channel matched filter with l taps requires the solution of an ml spl times ml block system of linear equations with toeplitz entries practical cases where m 50 and l 100 are not uncommon when the individual sensors of an array are closely spaced the temporal and spatial correlation of the resulting vector noise process may be modeled in a separable fashion in this case the noise covariance block matrix attains a special structure where all block entries are just weighted versions of each other it is shown that in this case the complexity of the detector design can be reduced drastically by a factor of ml compared to a conventional multichannel matched filter design procedure it is further shown that the separable vector noise model facilitates a complete exploitation of the rank properties of noise and data matrices a constructive procedure for the design of low rank detectors in the multichannel case is derived these detectors consist of two consecutive blocks a data and noise dependent compression stage which maps the significant signal energy into a subspace of minimal dimension followed by a minimal set of independent matched filters which point in the subspace directions in which the signal is much stronger than the noise this low rank detector concept enables discrimination with little or no performance penalty at a minimal computational cost

maximum likelihood array processing in spatially correlated noise fields using parameterized signals this paper deals with the problem of estimating signal parameters using an array of sensors this problem is of interest in a variety of applications such as radar and sonar source localization a vast number of estimation techniques have been proposed in the literature during the past two decades most of these can deliver consistent estimates only if the covariance matrix of the background noise is known in many applications the aforementioned assumption is unrealistic recently a number of contributions have addressed the problem of signal parameter estimation in unknown noise environments based on various assumptions on the noise herein a different approach is taken we assume instead that the signals are partially known the received signals are modeled as linear combinations of certain known basis functions the exact maximum likelihood ml estimator for the problem at hand is derived as well as computationally more attractive approximation the cramer rao lower bound crb on the estimation error variance is also derived and found to coincide with the crb assuming an arbitrary deterministic model and known noise covariance

nonexistence of informative unbiased estimators in singular problems in many nonparametric problems such as density estimation nonparametric regression and so on all the existing informative estimators are biased asymptotic or finite sample there has long been a suspicion that either informative unbiased estimators do not exist for such problems or they must be quite complicated in this paper we clarify the nonexistence of informative unbiased estimators in all singular problems both for fixed sample size and asymptotically this includes most problems with optimal rate of convergence slower than n1 2 we also discuss situations in regular problems where such nonexistences can occur

the appearance of microstructures in problems with incompatible wells and their numerical approach the goal of the paper is to analyze the creation of microstructure in problems of calculus of variations with wells more precisely we consider a case with strong incompatibility between the wells this forces the minimizing sequences to use other gradients than the wells in a puzzling way using a p 1 method we are then able to single out discrete minimizing sequences and to give energy estimates in terms of the mesh size

reduced rank linear regression and weighted low rank approximations this paper addresses parameter estimation in reduced rank linear regressions this estimation problem has applications in several subject areas including system identification sensor array processing econometrics and statistics a new estimation procedure based on instrumental variable principles is derived and analyzed the proposed method is designed to handle noise that is both spatially and temporally autocorrelated an asymptotical analysis shows that the proposed method outperforms previous methods when the noise is temporally correlated and that it is asymptotically efficient otherwise a numerical study indicates that the performance is significantly improved also for finite sample set sizes in addition the cramer rao lower bound crb on unbiased estimator covariance for the data model is derived a statistical test for rank determination is also developed an important step in the new algorithm is the weighted low rank approximation wlra as the wlra lacks a closed form solution in its general form two new noniterative and approximate solutions are derived both of them asymptotically optimal when part of the estimation procedure proposed here these methods are also interesting in their own right since the wlra has several applications

on intrinsic cramer rao bounds for riemannian submanifolds and quotient manifolds we study cramer rao bounds crb s for estimation problems on riemannian manifolds in s t smith covariance subspace and intrinsic cramer rao bounds ieee trans signal process vol 53 no 5 1610 1630 2005 the author gives intrinsic crb s in the form of matrix inequalities relating the covariance of estimators and the fisher information of estimation problems we focus on estimation problems whose parameter space bar cal p is a riemannian submanifold or a riemannian quotient manifold of a parent space cal p that is estimation problems on manifolds with either deterministic constraints or ambiguities the crb s in the aforementioned reference would be expressed w r t bases of the tangent spaces to bar cal p in some cases though it is more convenient to express covariance and fisher information w r t bases of the tangent spaces to cal p we give crb s w r t such bases expressed in terms of the geodesic distances on the parameter space the bounds are valid even for singular fisher information matrices in two examples we show how the crb s for synchronization problems including a type of sensor network localization problem differ in the presence or absence of anchors leading to bounds for estimation on either submanifolds or quotient manifolds with very different interpretations

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

a minimal state approach to dynamic stabilization of the rotating disk beam system with infinite memory this note is dedicated to the investigation of the stabilization problem of the well know rotating disk beam system we suggest a linear feedback law which consists of a torque control applied on the disk and a dynamic force control with infinite memory term exerted on the beam thereafter sufficient conditions on the angular velocity of the disk and the memory kernel are derived to assure the existence and uniqueness of solutions of the closed loop system the main ingredient of the proof of such a well posedness result is to adopt the method used for viscoelastic systems which mainly consists in utilizing the concept of minimal state variables moreover it is shown that the beam vibrations are forced to exponentially decay to zero while the disk keeps rotating with a fixed angular velocity this desirable property which can be interpreted as the exponential stability of the system is achieved by means of lyapunov energy method

maximum likelihood estimation the cramer rao bound and the method of scoring with parameter constraints maximum likelihood ml estimation is a popular approach to solving many signal processing problems many of these problems cannot be solved analytically and so numerical techniques such as the method of scoring are applied however in many scenarios it is desirable to modify the ml problem with the inclusion of additional side information often this side information is in the form of parametric constraints which the ml estimate mle must now satisfy we unify the asymptotic constrained ml cml theory with the constrained cramer rao bound ccrb theory by showing the cml estimate cmle is asymptotically efficient with respect to the ccrb we also generalize the classical method of scoring using the ccrb to include the constraints satisfying the constraints after each iterate convergence properties and examples verify the usefulness of the constrained scoring approach as a particular example an alternative and more general cmle is developed for the complex parameter linear model with linear constraints a novel proof of the efficiency of this estimator is provided using the ccrb

dynamic boundary control of a euler bernoulli beam a flexible beam clamped to a rigid base at one end and free at the other end is considered to stabilize the beam vibrations a dynamic boundary force control and a dynamic boundary torque control applied at the free end of the beam are proposed it is proved that with the proposed controls the beam vibrations decay exponentially the proof uses a lyapunov functional based on the energy functional of the system

generic weighted filtering of stochastic signals in this paper a new theory of optimal weighted nonlinear filtering is presented two filter models are considered the first model is based on a representation of the filter in the polynomial like form with q terms where each term consists of weighted matrices and the matrix determined from the error minimization problem the second model extends the first one to the case of the filter concatenation the filter models are given in terms of pseudo inverse matrices i e the requirement of invertibility for covariance matrices is omitted thus our filters always exist we develop methods which allow us to exploit advantages associated with the proposed nonlinear filter models the methods consist of the orthogonalization procedure and the reduction of the original problem to q individual minimization problems for smaller matrices this leads to a considerable reduction in the required computational work the error associated with the first filter model decreases when the number q of terms of filter increases its compression ratio can be adjusted by varying a particular value of ranks in each of its q terms this means that the proposed filer structure provides the two degrees of freedom the second filter model provides another degree of freedom a number k of filters in the concatenation variations of the degrees of freedom improve the performance of the proposed filters in particular the error associated with the filter concatenation decreases as the filter number k in the concatenation increases

control and stabilization of a rotating flexible structure abstract we consider a flexible beam clamped to a rigid base at one end and free at the other end we assume that the rigid base rotates with a constant angular velocity and that the motion of the flexible beam takes place on a plane to suppress the beam vibrations we propose dynamic control laws for boundary control force and torque both applied to the free end of the beam we show that under some conditions one of which is the strict positive realness of the actuator transfer functions which generate the boundary control force and torque the beam vibrations asymptotically decay to zero if the rigid base angular frequency is sufficiently small moreover if the transfer functions are proper but not strictly proper we show that the decay is exponential we also give a bound on the constant angular velocity above which the system becomes unstable

modeling stabilization and control of serially connected beams many flexible structures consist of a large number of components coupled end to end in the form of a chain in this paper we consider the simplest type of such structures which is formed by n seri

on non singular information matrices and local identifiability abstract the relation between non singularity of the information matrix and local identifiability of the parameters is discussed for some model structures these two properties arc shown to be equivalent however there are also cases where a singular information matrix does not contradict local identifiability

maximum likelihood parameter and rank estimation in reduced rank multivariate linear regressions this paper considers the problem of maximum likelihood ml estimation for reduced rank linear regression equations with noise of arbitrary covariance the rank reduced matrix of regression coefficients is parameterized as the product of two full rank factor matrices this parameterization is essentially constraint free but it is not unique which renders the associated ml estimation problem rather nonstandard nevertheless the problem turns out to be tractable and the following results are obtained an explicit expression is derived for the ml estimate of the regression matrix in terms of the data covariances and their eigenelements furthermore a detailed analysis of the statistical properties of the ml parameter estimate is performed additionally a generalized likelihood ratio test glrt is proposed for estimating the rank of the regression matrix the paper also presents the results of some simulation exercises which lend empirical support to the theoretical findings

new results on deterministic cramer rao bounds for real and complex parameters the cramer rao bounds crb is a lower bound of great interest for system analysis and design in the asymptotic region high signal to noise ratio snr and or large number of snapshots as it is simple to calculate and it is usually possible to obtain closed form expressions the first part of the paper is a generalization to complex parameters of the barankin rationale for deriving mse lower bounds that is the minimization of a norm under a set of linear constraints with the norm minimization approach the study of fisher information matrix fim singularity constrained crb and regularity conditions become straightforward corollaries of the derivation the second part provides new results useful for system analysis and design a general reparameterization inequality the equivalence between reparameterization and equality constraints and an explicit relationship between parameters unidentifiability and fim singularity

nonlinear black box modeling in system identification a unified overview a nonlinear black box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics there has been considerable recent interest in this area with structures based on neural networks radial basis networks wavelet networks and hinging hyperplanes as well as wavelet transform based methods and models based on fuzzy sets and fuzzy rules this paper describes all these approaches in a common framework from a user s perspective it focuses on what are the common features in the different approaches the choices that have to be made and what considerations are relevant for a successful system identification application of these techniques it is pointed out that the nonlinear structures can be seen as a concatenation of a mapping form observed data to a regression vector and a nonlinear mapping from the regressor space to the output space these mappings are discussed separately the latter mapping is usually formed as a basis function expansion the basis functions are typically formed from one simple scalar function which is modified in terms of scale and location the expansion from the scalar argument to the regressor space is achieved by a radial or a ridge type approach basic techniques for estimating the parameters in the structures are criterion minimization as well as two step procedures where first the relevant basis functions are determined using data and then a linear least squares step to determine the coordinates of the function approximation a particular problem is to deal with the large number of potentially necessary parameters this is handled by making the number of used parameters considerably less than the number of offered parameters by regularization shrinking pruning or regressor selection

nondissipative torque and shear force controls of a rotating flexible structure this paper is concerned with the feedback stabilization problem of a rotating disk with a flexible beam attached to its center we propose a torque control applied on the disk and a shear force control exerted at the free end of the beam which lead to a nondissipative closed loop system despite this situation we prove that the system can be nonuniformly exponentially stabilized provided that the angular velocity of the disk is less than the square root of the first eigenvalue of the self adjoint positive operator of the open loop system this result is illustrated by a set of numerical simulations