a geometric approach to variance analysis in system identification this paper addresses the problem of quantifying the model error variance error in estimates of dynamic systems it is shown that under very general conditions the asymptotic in data length covariance of an estimated system property represented by a smooth function of estimated system parameters can be interpreted in terms of an orthogonal projection of a certain function associated with the property of interest onto a subspace determined by the model structure and experimental conditions the presented geometric approach simplifies structural analysis of the model variance and this is illustrated by analyzing the influence of inputs and sensors on the model accuracy

identification for control optimal input design with respect to a worst case nu gap cost function parameter identification experiments deliver an identified model together with an ellipsoidal uncertainty region in parameter space the objective of robust controller design is thus to stabilize all plants in the identified uncertainty region the subject of the present contribution is to design an identification experiment such that the worst case nu gap over all plants in the resulting uncertainty region between the identified plant and plants in this region is as small as possible the experiment design is performed via input power spectrum optimization two cost functions are investigated which represent different levels of trade off between accuracy and computational complexity it is shown that the input optimization problem with respect to these cost functions is amenable to standard numerical algorithms used in convex analysis

plant friendly input design convex relaxation and quality a common practice in a system identification exercise is to perturb the system of interest and use the resulting data to build a model the problem of interest in this contribution is to synthesize an input signal that is maximally informative for generating good quality models while being plant friendly i e least hostile to plant operation in this contribution limits on input move sizes are the plant friendly specifications the resulting optimization problem is nonlinear and nonconvex hence the original plant friendly input design problem is relaxed which results in a convex optimization problem we formulate a semidefinite programme using the theory of generalized tchebysheff inequalities to derive tight bounds on the quality of relaxation simulations show that the relaxation results in more plant friendly input signals

ma estimation in polynomial time the parameter estimation of moving average ma signals from second order statistics was deemed for a long time to be a difficult nonlinear problem for which no computationally convenient and reliable solution was possible we show how the problem of ma parameter estimation from sample covariances can be formulated as a semidefinite program that can be solved in a time that is a polynomial function of the ma order two methods are proposed that rely on two specific over parametrizations of the ma covariance sequence whose use makes the minimization of a covariance fitting criterion a convex problem the ma estimation algorithms proposed here are computationally fast statistically accurate and reliable none of the previously available algorithms for ma estimation methods based on higher order statistics included shares all these desirable properties our methods can also be used to obtain the optimal least squares approximant of an invalid estimated ma spectrum that takes on negative values at some frequencies which was another long standing problem in the signal processing literature awaiting a satisfactory solution

identifiability and estimation of structural vector autoregressive models for subsampled and mixed frequency time series causal inference in multivariate time series is challenging because the sampling rate may not be as fast as the time scale of the causal interactions so the observed series is a subsampled version of the desired series furthermore series may be observed at different sampling rates yielding mixed frequency series to determine instantaneous and lagged effects between series at the causal scale we take a model based approach that relies on structural vector autoregressive models we present a unifying framework for parameter identifiability and estimation under subsampling and mixed frequencies when the noise or shocks is non gaussian by studying the structural case we develop identifiability and estimation methods for the causal structure of lagged and instantaneous effects at the desired time scale we further derive an exact expectation maximization algorithm for inference in both subsampled and mixed frequency settings we validate our approach in simulated scenarios and on a climate and an econometric dataset

brief control relevant experiment design for multivariable systems described by expansions in orthonormal bases the problem of control relevant experiment design is considered we first develop the problem in abstract form as an optimization of an unspecified closed loop performance index then we make the problem specific by considering systems that can be described via a finite expansion of orthonormal bases and designing the controller by specifying the closed loop system s sensitivity function although the resulting optimization problem is nonconvex and may admit local solutions simulations of high purity binary distillation columns are presented to demonstrate that significant improvements in controller performance may be obtained even by using the local solution

covariance analysis in siso linear systems identification in this paper we analyze the asymptotic covariance of models of causal single input single output linear time invariant systems expressions for the asymptotic co variance of system properties es

variance error quantification for identified poles and zeros this paper deals with quantification of noise induced errors in identified discrete time models of causal linear time invariant systems where the model error is described by the asymptotic in data length variance of the estimated poles and zeros the main conclusion is that there is a fundamental difference in the accuracy of the estimates depending on whether the zeros and poles lie inside or outside the unit circle as the model order goes to infinity the asymptotic variance approaches a finite limit for estimates of zeros and poles having magnitude larger than one but for zeros and poles strictly inside the unit circle the asymptotic variance grows exponentially with the model order we analyze how the variance of poles and zeros is affected by model order model structure and input excitation we treat general black box model structures including armax and box jenkins models

paper unprejudiced optimal open loop input design for identification of transfer functions the problem to estimate transfer functions of linear systems is considered the quality of the resulting estimate depends among other things on the input used during the identification experiment we measure the quality using a quadratic norm in the frequency domain the problem to determine optimal inputs i e inputs that minimize the chosen norm subject to constrained input variance has long been studied we point out that such procedures may involve a prejudice that the system is to be found in a certain model set that may have some surprising effects we discuss how such a prejudice can be reduced by allowing the possibility that the true system cannot be exactly described in the chosen model set we also calculate explicit expressions for the resulting unprejudiced optimal inputs these expressions relate the signal to noise ratio as a function of frequency to the chosen weighting function in the quadratic norm we also point out the role of the employed noise model for the design

brief paper conditions when minimum variance control is the optimal experiment for identifying a minimum variance controller it is well known that if we intend to use a minimum variance control strategy which is designed based on a model obtained from an identification experiment the best experiment which can be performed on the system to determine such a model subject to output power constraints or for some specific model structures is to use the true minimum variance controller this result has been derived under several circumstances first using asymptotic in model order variance expressions but also more recently for armax models of finite order in this paper we re approach this problem using a recently developed expression for the variance of parametric frequency function estimates this allows a geometric analysis of the problem and the generalization of the aforementioned finite model order armax results to general linear model structures

variance analysis of identified linear miso models having spatially correlated inputs with application to parallel hammerstein models this contribution concerns variance analysis of linear multi input single output models when the inputs are temporally white but where different inputs may be correlated an expression is provided for the variance of a linearly parametrized estimate of the frequency response function from one block i e from one input to the output in particular this expression reveals that the variance increases in one block when the number of estimated parameters in another block is increased but levels off when the number of parameters in the other block reaches the number of parameters in the block in question it also quantifies exactly how correlation between inputs affects the resulting accuracy and a graphical representation is provided for this purpose the results are applicable to parallel miso hammerstein models when the nonlinearities are known and generalize an existing variance expression for this type of model

a useful input parameterization for optimal experiment design optimal inputs are usually determined by minimizing a scalar valued function of the inverse fisher information matrix the function should be monotonically increasing this is the case e g for the trace and the determinant the minimization must be performed under some constraints to prevent the input or output amplitude to blow up in this note it is proved that assuming open loop experiments the optimal input signal can be realized as a certain arma process of low order or at least can be approximated with any degree of accuracy by such a process this allows the optimal input design problem to be reformulated as a standard static optimization problem of low dimension

on the georgiou lindquist approach to constrained kullback leibler approximation of spectral densities we consider the georgiou lindquist constrained approximation of spectra in the kullback leibler sense we propose an alternative iterative algorithm to solve the corresponding convex optimization problem the lagrange multiplier is computed as a fixed point of a nonlinear matricial map simulation indicates that the algorithm is extremely effective

variance error quantifications that are exact for finite model order this paper is concerned with the frequency domain quantification of noise induced errors in dynamic system estimates preceding and seminal work on this problem has suggested general expressions that are approximations whose accuracy increases with observed data length and model order in the interests of improved accuracy this paper provides new expressions whose accuracy depends only on data length they are therefore exact for arbitrarily small true model order other authors have recognized the importance of such expressions and have derived them for the case of finite impulse response like model structures in which denominators are fixed at true values and only numerator terms are estimated this paper progresses beyond this situation to address the more general output error and box jenkins structures in which full dynamics models both numerator and denominator terms and noise models may be estimated a key aspect of the work here is that it establishes that the variance quantification problem is equivalent to that of deriving the reproducing kernel for a subspace that depends on the model structure being employed

application of entropy and energy measures of fuzziness to processing of ecg signal the paper deals with the application of entropy and energy measure of fuzziness to processing of ecg signal after formulation of the problem the notion of entropy measure of fuzziness is recalled next the idea of a fuzzy signal created from the original signal is proposed and the entropy measure of fuzziness determined for each sample is calculated the application of this measure to building a detection function of ecg signal is presented the theoretical considerations have been illustrated with digital processing of the real ecg signal

on the kalman yakubovich popov lemma the purpose of this note is to present a new elementary proof for the multivariable k y p lemma a minimum of linear algebra and finite dimensional convexity theory is used

a time domain constrained fuzzy clustering method and its application to signal analysis this paper introduces a new fuzzy clustering method with time domain constraints which is used to signal analysis proposed method makes it possible to include natural constraints for signal analysis using fuzzy clustering that is the neighboring samples of signal belong to the same cluster this method can be called time domain constrained fuzzy clustering this paper introduces two approaches to include the above kind of constraints the first approach leads to the time domain constrained fuzzy c regression models method the second approach leads to the insensitive version of the above method which results in additional robustness for outliers and non gaussian noise finally simulations on synthetic as well as real life signals are realized to evaluate the performance of the time domain constrained fuzzy clustering methods a comparison with the traditional fuzzy c regression models is also made large scale simulations demonstrate the competitiveness of the proposed methods for signal analysis with respect to the traditional fuzzy clustering methods

system identification using kautz models in this paper the problem of approximating a linear time invariant stable system by a finite weighted sum of given exponentials is considered system identification schemes using laguerre models are extended and generalized to kautz models which correspond to representations using several different possible complex exponentials in particular linear regression methods to estimate this sort of model from measured data are analyzed the advantages of the proposed approach are the simplicity of the resulting identification scheme and the capability of modeling resonant systems using few parameters the subsequent analysis is based on the result that the corresponding linear regression normal equations have a block toeplitz structure several results on transfer function estimation are extended to discrete kautz models for example asymptotic frequency domain variance expressions

measuring fuzzy uncertainty first this paper reviews several well known measures of fuzziness for discrete fuzzy sets then new multiplicative and additive classes are defined we show that each class satisfies five well known axioms for fuzziness measures and demonstrate that several existing measures are relatives of these classes the multiplicative class is based on nonnegative monotone increasing concave functions the additive class requires only nonnegative concave functions some relationships between several existing and the new measures are established and some new properties are derived the relative merits and drawbacks of different measures for applications are discussed a weighted fuzzy entropy which is flexible enough to incorporate subjectiveness in the measure of fuzziness is also introduced finally we comment on the construction of measures that may assess all of the uncertainties associated with a physical system

non asymptotic quality assessment of generalised fir models with periodic inputs in any real life identification problem only a finite number of data points is available on the other hand almost all results in stochastic identification pertain to asymptotic properties that is they tell us what happens when the number of data points tends to infinity in this paper we consider the problem of assessing the quality of the estimates identified from a finite number of data points we focus on least squares identification of generalised fir models and develop a method to produce a bound on the uncertainty in the parameter estimate the method is data driven and based on tests involving permuted data sets moreover it does not require that the true system is in the model class

closed loop experiment design for linear time invariant dynamical systems via lmis all stationary experimental conditions corresponding to a discrete time linear time invariant causal internally stable closed loop with real rational system and feedback controller are characterized using the youla kucera parametrization finite dimensional parametrizations of the input spectrum and the youla kucera parameter allow a wide range of closed loop experiment design problems based on the asymptotic in the sample size covariance matrix for the estimated parameters to be recast as computationally tractable convex optimization problems such as semi definite programs in particular for box jenkins models a finite dimensional parametrization is provided which is able to generate all possible asymptotic covariance matrices as a special case the very common situation of a fixed controller during the identification experiment can be handled and optimal reference signal spectra can be computed subject to closed loop signal constraints finally a brief numerical comparison with closed loop experiment design based on a high model order variance expression is presented

technical communique optimal closed loop identification test design for internal model control in this work optimal closed loop test design for control is studied simple design formulas are derived based on the asymptotic theory of ljung the control scheme used is internal model control imc and the design constraint is the power of the process output or that of the reference signal the test signal is applied at the setpoint the results give clear guidelines for closed loop identification applications

the dimensionality of the aliasing problem in models with rational spectral densities this paper reconsiders the aliasing problem of identifying the parameters of a continuous time stochastic process from discrete time data it analyzes the extent to which restricting attention to processes with rational spectral density matrices reduces the number of observationally equivalent models it focuses on rational specifications of spectral density matrices since rational parameterizations are commonly employed in the analysis of the time series data this abstract was borrowed from another version of this item

input design as a tool to improve the convergence of pem the prediction error method pem is related to an optimization problem built on input output data collected from the system to be identified it is often hard to find the global solution of this optimization problem because the corresponding objective function presents local minima and or the search space is constrained to a nonconvex set the shape of the cost function and hence the difficulty in solving the optimization problem depends directly on the experimental conditions more specifically on the spectrum of the input output data collected from the system therefore it seems plausible to improve the convergence to the global minimum by properly choosing the spectrum of the input in this paper we address this problem we present a condition for convergence to the global minimum of the cost function and propose its inclusion in the input design we present the application of the proposed approach to case studies where the algorithms tend to get trapped in nonglobal minima

identification for control of multivariable systems controller validation and experiment design via lmis this paper presents a new controller validation method for linear multivariable time invariant models classical prediction error system identification methods deliver uncertainty regions which are nonstandard in the robust control literature our controller validation criterion computes an upper bound for the worst case performance measured in terms of the h norm of a weighted closed loop transfer matrix achieved by a given controller over all plants in such uncertainty sets this upper bound on the worst case performance is computed via an lmi based optimization problem and is deduced via the separation of graph framework our main technical contribution is to derive within that framework a very general parametrization for the set of multipliers corresponding to the nonstandard uncertainty regions resulting from pe identification of mimo systems the proposed approach also allows for iterative experiment design the results of this paper are asymptotic in the data length and it is assumed that the model structure is flexible enough to capture the true system

from finite covariance windows to modeling filters a convex optimization approach the trigonometric moment problem is a classical moment problem with numerous applications in mathematics physics and engineering the rational covariance extension problem is a constrained version of this problem with the constraints arising from the physical realizability of the corresponding solutions although the maximum entropy method gives one well known solution in several applications a wider class of solutions is desired in a seminal paper georgiou derived an existence result for a broad class of models in this paper we review the history of this problem going back to carath eodory as well as applications to stochastic systems and signal processing in particular we present a convex optimization problem for solving the rational covariance extension problem with degree constraint given a partial covariance sequence and the desired zeros of the shaping filter the poles are uniquely determined from the unique minimum of the corresponding optimization problem in this way we obtain an algorithm for solving the covariance extension problem as well as a constructive proof of georgiou s existence result and his conjecture a generalized version of which we have recently resolved using geometric methods we also survey recent related results on constrained nevanlinna pick interpolation in the context of a variational formulation of the general moment problem

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

the entropy value for ecg of the sport horses in this study we compared the amplitude and the duration of p and t waves of sport horses before a training program with after a training program waves are transformed to the entropies by the same methods of engonul et al the entropy of the waves p and t in electrocardiogram and some comments int j math model comput vol 6 pp 159 173 2016 we have benefitted from the definition of fuzzy sets and have assigned an entropy for each p and t waves of sport horses then we have calculated cesaro entropies for each wave in a particular segment for ten beats in addition the graphical representations are obtained for healthy and diseased horses by formalization from the paper the entropy of the waves p and t in electrocardiogram and some comments by engonul et al

brief paper identification and control joint input design and h state feedback with ellipsoidal parametric uncertainty via lmis one obstacle in connecting robust control with models generated from prediction error identification is that very few control design methods are able to directly cope with the ellipsoidal parametric uncertainty regions that are generated by such identification methods in this contribution we present a joint robust state feedback control input design procedure which guarantees stability and prescribed closed loop performance using models identified from experimental data this means that given h specifications on the closed loop transfer function are translated into sufficient requirements on the input signal spectrum used to identify the process the condition takes the form of a linear matrix inequality

frequency domain uncertainty and the graph topology a new metric on linear time invariant systems is defined this metric is no greater than the gap metric and is in fact the smallest metric for which a certain robust stabilization result holds unlike other known metrics which induce the graph topology it has a clear frequency response interpretation this allows questions regarding robustness in the face of parametric uncertainty to be considered in terms of this metric

brief paper accuracy of linear multiple input multiple output mimo models obtained by maximum likelihood estimation in this paper we study the accuracy of linear multiple input multiple output mimo models obtained by maximum likelihood estimation we present a frequency domain representation for the information matrix for general linear mimo models we show that the variance of estimated parametric models for linear mimo systems satisfies a fundamental integral trade off this trade off is expressed as a multivariable water bed effect an extension to spectral estimation is also discussed

measures of entropy and fuzziness related to aggregation operators we suggest some new measures for the calculation of the entropy of a probability distribution one motivation for this is to provide tools for the application of the maximum entropy principle the first class of entropy measures is based on the ordered weighted averaging operator another class of these measures is based on the distance of the probability distribution from the uniform distribution a third class of entropy measures is based on the t conorms we then suggest some measures of fuzziness associated with a fuzzy set

input design for structured nonlinear system identification this paper is concerned with the input design problem for a class of structured nonlinear models this class contains models described by an interconnection of known linear dynamic systems and unknown static nonlinearities many widely used model structures are included in this class the model class considered naturally accommodates a priori knowledge in terms of signal interconnections under certain structural conditions the identification problem for this model class reduces to standard least squares we treat the input design problem in this situation an expression for the expected estimate variance is derived a method for synthesizing an informative input sequence that minimizes an upper bound on this variance is developed this reduces to a convex optimization problem features of the solution include parameterization of the expected estimate variance by the input distribution and a graph based method for input generation

a globally convergent matricial algorithm for multivariate spectral estimation in this paper we first describe a matricial newton type algorithm designed to solve the multivariable spectrum approximation problem we then prove its global convergence finally we apply this approximation procedure to multivariate spectral estimation and test its effectiveness through simulation simulation shows that in the case of short observation records this method may provide a valid alternative to standard multivariable identification techniques such as matlab s pem and matlab s n4sid

information matrix and d optimal design with gaussian inputs for wiener model identification we present a closed form expression for the fischer s information matrix associated with the identification of wiener models in the derivation we assume that the input signal is gaussian the analysis allows the linear sub system in the wiener model to have a generic rational transfer function of arbitrary order it also allows the static nonlinearity of the wiener model to be a polynomial of arbitrary degree in addition we show how this analysis can be used to design tractable algorithms for d optimal input design the idea is further extended to design optimal inputs consisting of a sequence of gaussian signals with different mean values and variances by combining gaussian inputs with different means we can tune the amplitude distribution of the input to achieve the best identification accuracy in d optimal sense the analytical results are also illustrated with some numerical simulations

identification of arx systems with non stationary inputs asymptotic analysis with application to adaptive input design a key problem in optimal input design is that the solution depends on system parameters to be identified in this contribution we provide formal results for convergence and asymptotic optimality of an adaptive input design method based on the certainty equivalence principle i e for each time step an optimal input design problem is solved exactly using the present parameter estimate and one sample of this input is applied to the system the results apply to stable arx systems with the input restricted to be generated by white noise filtered through a finite impulse response filter or a binary signal obtained from the latter by a static nonlinearity

technical communique asymptotic variance expressions for closed loop identification asymptotic variance expressions are analyzed for models that are identified on the basis of closed loop data the considered methods comprise the classical direct method as well as the more recently developed indirect methods employing coprime factorized models dual youla kucera parametrizations and the two stage approach the variance expressions are compared with the open loop situation and evaluated in terms of their relevance for subsequent model based control design

system identification using laguerre models the traditional approach of expanding transfer functions and noise models in the delay operator to obtain linear in the parameters predictor models leads to approximations of very high order in cases of rapid sampling and or dispersion in time constants by using prior information about the time constants of the system more appropriate expansions related to laguerre networks are introduced and analyzed it is shown that the model order can be reduced compared to arx fir ar modeling by using laguerre models furthermore the numerical accuracy of the corresponding linear regression estimation problem is improved by a suitable choice of the laguerre parameter consistency error bounds persistence of excitation conditions and asymptotic statistical properties are investigated this analysis is based on the result that the covariance matrix of the regression vector of a laguerre model has a toeplitz structure

a generalized entropy criterion for nevanlinna pick interpolation with degree constraint we present a generalized entropy criterion for solving the rational nevanlinna pick problem for n 1 interpolating conditions and the degree of interpolants bounded by n the primal problem of maximizing this entropy gain has a very well behaved dual problem this dual is a convex optimization problem in a finite dimensional space and gives rise to an algorithm for finding all interpolants which are positive real and rational of degree at most n the criterion requires a selection of a monic schur polynomial of degree n it follows that this class of monic polynomials completely parameterizes all such rational interpolants and it therefore provides a set of design parameters for specifying such interpolants the algorithm is implemented in a state space form and applied to several illustrative problems in systems and control namely sensitivity minimization maximal power transfer and spectral estimation

on the frequency domain accuracy of closed loop estimates it has been argued that the frequency domain accuracy of high model order estimates obtained on the basis of closed loop data is largely invariant to whether direct or indirect approaches are used the analysis underlying this conclusion has employed variance expressions that are asymptotic both in the data length and the model order and hence are approximations when either of these are finite however recent work has provided variance expressions that are exact for finite possibly low model order and hence can potentially deliver more accurate quantification of estimation accuracy this paper and a companion one revisits the study of identification from closed loop data in light of these new quantifications and establishes that under certain assumptions there can be significant differences in the accuracy of frequency response estimates these discrepencies are established here and in the companion paper to be dependent on what type of direct indirect or joint input output identification strategy is pursued

brief robustness analysis tools for an uncertainty set obtained by prediction error identification this paper presents a robust stability and performance analysis for an uncertainty set delivered by classical prediction error identification this nonstandard uncertainty set which is a set of parametrized transfer functions with a parameter vector in an ellipsoid contains the true system at a certain probability level our robust stability result is a necessary and sufficient condition for the stabilization by a given controller of all systems in such uncertainty set the main new technical contribution of this paper is our robust performance result we show that the worst case performance achieved over all systems in such an uncertainty region is the solution of a convex optimization problem involving linear matrix inequality constraints note that we only consider single input single output systems

a convex optimization approach to arma modeling we formulate a convex optimization problem for approximating any given spectral density with a rational one having a prescribed number of poles and zeros n poles and m zeros inside the unit disc and their conjugates the approximation utilizes the kullback leibler divergence as a distance measure the stationarity condition for optimality requires that the approximant matches n 1 covariance moments of the given power spectrum and m cepstral moments of the corresponding logarithm although the latter with possible slack the solution coincides with one derived by byrnes enqvist and lindquist who addressed directly the question of covariance and cepstral matching thus the present paper provides an approximation theoretic justification of such a problem since the approximation requires only moments of spectral densities and of their logarithms it can also be used for system identification

spectral estimation via selective harmonic amplification the state covariance of a linear filter is characterized by a certain algebraic commutativity property with the state matrix of the filter and also imposes a generalized interpolation constraint on the power spectrum of the input process this algebraic property and the relationship between state covariance and the power spectrum of the input allow the use of matrix pencils and analytic interpolation theory for spectral analysis several algorithms for spectral estimation are developed with resolution higher than state of the art

the fundamental role of general orthonormal bases in system identification the purpose of this paper is threefold firstly it is to establish that contrary to what might be expected the accuracy of well known and frequently used asymptotic variance results can depend on choices of fixed poles or zeros in the model structure secondly it is to derive new variance expressions that can provide greatly improved accuracy while also making explicit the influence of any fixed poles or zeros this is achieved by employing certain new results on generalized fourier series and the asymptotic properties of toeplitz like matrices in such a way that the new variance expressions presented here encompass pre existing ones as special cases via this latter analysis a new perspective emerges on recent work pertaining to the use of orthonormal basis structures in system identification namely that orthonormal bases are much more than an implementational option offering improved numerical properties in fact they are an intrinsic part of estimation since as shown here orthonormal bases quantify the asymptotic variability of the estimates whether or not they are actually employed in calculating them

monotonic convergence of a general algorithm for computing optimal designs 1 a general class of algorithms optimal experimental design approximate theory is a well developed area and we refer to kiefer 1974 silvey 1980 p zm n 1986 and pukelsheim 1993 for a general introduction and basic results we consider computational aspects of optimal designs focusing on a finite design space x x xn suppose the probability density or mass function of the response is specified as p y x where 0i 9m t is the parameter of interest let a denote the m m expected fisher information matrix from a unit assigned to x with the j k entry the expectation is with respect to p y x

on the convergence of an efficient algorithm for kullback leibler approximation of spectral densities this paper deals with a method for the approximation of a spectral density function among the solutions of a generalized moment problem a la byrnes georgiou lindquist the approximation is pursued with respect to the kullback leibler pseudo distance which gives rise to a convex optimization problem after developing the variational analysis we discuss the properties of an efficient algorithm for the solution of the corresponding dual problem based on the iteration of a nonlinear map in a bounded subset of the dual space our main result is the proof of local convergence of the latter established as a consequence of the central manifold theorem supported by numerical evidence we conjecture that in the mentioned bounded set the convergence is actually global

extended yule walker identification of varma models with single or mixed frequency data chen and zadrozny 1998 developed the linear extended yule walker xyw method for determining the parameters of a vector autoregressive var model with available covariances of mixed frequency observations on the variables of the model if the parameters are determined uniquely for available population covariances then the var model is identified the present paper extends the original xyw method to an extended xyw method for determining all arma parameters of a vector autoregressive moving average varma model with available covariances of single or mixed frequency observations on the variables of the model the paper proves that under conditions of stationarity regularity miniphaseness controllability observability and diagonalizability on the parameters of the model the parameters are determined uniquely with available population covariances of single or mixed frequency observations on the variables of the model so that the varma model is identified with the single or mixed frequency covariances

the structure of state covariances and its relation to the power spectrum of the input we study the relationship between power spectra of stationary stochastic inputs to a linear filter and the corresponding state covariances and identify the structure of positive semidefinite matrices that qualify as state covariances of the filter this structure is best revealed by a rank condition pertaining to the solvability of a linear equation involving the state covariance and the system matrices we then characterize all input power spectra consistent with any specific state covariance the parametrization of input spectra is achieved through a relation to solutions of an analytic interpolation problem which is analogous but not equivalent to a matricial nehari problem

closed loop identification revisited identification of systems operating in closed loop has long been of prime interest in industrial applications the problem offers many possibilities and also some fallacies and a wide variety of approaches have been suggested many quite recently the purpose of the current contribution is to place most of these approaches in a coherent framework thereby showing their connections and display similarities and differences in the asymptotic properties of the resulting estimates the common framework is created by the basic prediction error method and it is shown that most of the common methods correspond to different parameterizations of the dynamics and noise models the so called indirect methods e g are indeed direct methods employing noise models that contain the regulator the asymptotic properties of the estimates then follow from the general theory and take different forms as they are translated to the particular parameterizations we also study a new projection approach to closed loop identification with the advantage of allowing approximation of the open loop dynamics in a given and user chosen frequency domain norm even in the case of an unknown nonlinear regulator

choosing between open and closed loop experiments in linear system identification this correspondence shows that open loop experiments are optimal for a broad class of systems when the system input is constrained in addition we show that for a general class of systems when the output power is constrained closed loop experiments are optimal both results use a strong notion of optimality and use expressions for estimation accuracy which are nonasymptotic in model order but asymptotic in data length

the interpolation problem with a degree constraint the author previously 1983 1987 showed that there is a correspondence between nonnegative hermitian trigonometric polynomials of degree spl les n and solutions to the standard nevanlinna pick caratheodory interpolation problem with n 1 constraints which are rational and also of degree spl les n it was conjectured that the correspondence under suitable normalization is bijective and thereby that it results in a complete parametrization of rational solutions of degree spl les n the conjecture was proven by byrnes et al 1995 along with a detailed study of this parametrization however byrnes et al used a slightly restrictive assumption that the trigonometric polynomials are positive and accordingly the corresponding solutions have positive real part the purpose of the present note is to extend the result to the case of nonnegative trigonometric polynomials as well we present the arguments in the context of the general nevanlinna pick caratheodory fejer interpolation

input design via lmis admitting frequency wise model specifications in confidence regions a framework for reformulating input design problems in prediction error identification as convex optimization problems is presented for linear time invariant single input single output systems this framework unifies and extends existing results on open loop input design that are based on the finite dimensional asymptotic covariance matrix of the parameter estimates basic methods for parametrizing the input spectrum are provided and conditions on these parametrizations that guarantee that all possible covariance matrices for the asymptotic distribution of the parameter estimates can be generated are provided a wide range of model quality constraints can be handled in particular different frequency by frequency constraints can be used this opens up new applications of input design in areas such as robust control furthermore quality specifications can be imposed on all models in a confidence region thus allowing for statements such as with at least 99 probability the model quality specifications will be satisfied

asymptotic variance expressions for estimated frequency functions expressions for the variance of an estimated frequency function are necessary for many issues in model validation and experiment design a general result is that a simple expression for this variance can be obtained asymptotically as the model order tends to infinity this expression shows that the variance is inversely proportional to the signal to noise ratio frequency by frequency still for low order models the actual variance may be quite different we derive an exact expression for the variance which is not asymptotic in the model order this expression applies to a restricted class of models ar models as well as fixed pole models with a polynomial noise model it brings out the character of the simple approximation and the convergence rate to the limit as the model order increases it also provides nonasymptotic lower bounds for the general case the calculations are illustrated by numerical examples

dynamic identification of dynamic stochastic general equilibrium models this paper studies dynamic identification of parameters of a dynamic stochastic general equilibrium model from the first and second moments of the data classical results for dynamic simultaneous equations do not apply because the state space solution of the model does not constitute a standard reduced form full rank of the jacobian matrix of derivatives of the solution parameters with respect to the parameters of interest is necessary but not sufficient for identification we use restrictions implied by observational equivalence to obtain two sets of rank and order conditions one for stochastically singular models and another for nonsingular models measurement errors mean long run and a priori restrictions can be accommodated an example is considered to illustrate the results

identification and the information matrix how to get just sufficiently rich in prediction error identification the information matrix plays a central role specifically when the system is in the model set the covariance matrix of the parameter estimates converges asymptotically up to a scaling factor to the inverse of the information matrix the existence of a finite covariance matrix thus depends on the positive definiteness of the information matrix and the rate of convergence of the parameter estimate depends on its size the information matrix is also the key tool in the solution of optimal experiment design procedures which have become a focus of recent attention introducing a geometric framework we provide a complete analysis for arbitrary model structures of the minimum degree of richness required to guarantee the nonsingularity of the information matrix we then particularize these results to all commonly used model structures both in open loop and in closed loop in a closed loop setup our results provide an unexpected and precisely quantifiable trade off between controller degree and required degree of external excitation

variance error interpolation and experiment design we investigate how the variance error associated with the prediction error identification is related to the power spectral densities of the input and the additive noise at the output let f e i w be the ratio of the input power spectral density psd to the output noise psd we characterize the set of all functions f for which the variance error remains constant this analysis results a minimal finite dimensional affine parameterization of the variance error this parameterization connects our analysis with the theory of nevanlinna pick interpolation it is shown that the set of all f for which the variance error remains constant can be characterized by the solutions of a nevanlinna pick interpolation problem this insight has interesting consequences in optimal input design where it is possible to use some recent tools in analytic interpolation theory to tune shape of the input psd to suit certain needs

asymptotic variance expressions for identified black box transfer function models identification of black box transfer function models is considered it is assumed that the transfer function models possess a certain shift property which is satisfied for example by all polynomial type models expressions for the variances of the transfer function estimates are derived that are asymptotic both in the number of observed data and in the model orders the result is that the joint covariance matrix of the transfer functions from input to output and from driving white noise source to the additive output disturbance respectively is proportional to the inverse of the joint spectrum matrix for the input and driving noise multiplied by the spectrum of the additive output noise the factor of proportionality is the ratio of model order to number of data this result is independent of the particular model structure used the result is applied to evaluate the performance degradation due to variance for a number of typical model uses some consequences for input design are also drawn

variance analysis of linear simo models with spatially correlated noise in this paper we address the identification of linear time invariant single input multi output simo systems in particular we assess the performance of the prediction error method by quantifying

brief paper on the accuracy in errors in variables identification compared to prediction error identification errors in variables estimation problems for single input single output systems with gaussian signals are considered in this contribution it is shown that the fisher information matrix is monotonically increasing as a function of the input noise variance when the noise spectrum at the input is known and the corresponding noise variance is estimated furthermore it is shown that whittle s formula for the fisher information matrix can be represented as a gramian and this is used to provide a geometric representation of the asymptotic covariance matrix for asymptotically efficient estimators finally the asymptotic covariance of the parameter estimates for the system dynamics is compared for the two cases i when the model includes white measurement noise on the input and the variance of the noise is estimated and ii when the model includes only measurement noise on the output in both cases asymptotically efficient estimators are assumed an explicit expression for the difference is derived when the underlying system is subject only to measurement noise on the output

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