online model selection based on the variational bayes the bayesian framework provides a principled way of model selection this framework estimates a probability distribution over an ensemble of models and the prediction is done by averaging over the ensemble of models accordingly the uncertainty of the models is taken into account and complex models with more degrees of freedom are penalized however integration over model parameters is often intractable and some approximation scheme is needed recently a powerful approximation scheme called the variational bayes vb method has been proposed this approach defines the free energy for a trial probability distribution which approximates a joint posterior probability distribution over model parameters and hidden variables the exact maximization of the free energy gives the true posterior distribution the vb method uses factorized trial distributions the integration over model parameters can be done analytically and an iterative expectation maximization like algorithm whose convergence is guaranteed is derived in this article we derive an online version of the vb algorithm and prove its convergence by showing that it is a stochastic approximation for finding the maximum of the free energy by combining sequential model selection procedures the online vb method provides a fully online learning method with a model selection mechanism in preliminary experiments using synthetic data the online vb method was able to adapt the model structure to dynamic environments

constraint handling in genetic search using expression strategies a traditional penalty function formulation for treatment of nonlinear constrained optimization problems in genetic search has been shown to be extremely sensitive to user specified schedules of selecting penalty parameters the sensitivity of such an approach is manifested in a biasing of the search toward suboptimal designs and a general increase in the number of function evaluations required to obtain a converged design alternative methods are described for handling constraints that are motivated by the fact that the structure of both feasible and infeasible designs is generally present in the population of designs at any generation of search a preconditionin g of the infeasible designs prior to the genetic transformations by an expression operation that is conceptually analogous to the theory of dominant and recessive genes in genetics is shown to be highly effective in evolving feasible solutions and with savings of computational resource two alternative implementations of this approach are described and a comparison is made of numerical efficiency vis a vis the penalty function based approach

recursive noise adaptive kalman filtering by variational bayesian approximations this article considers the application of variational bayesian methods to joint recursive estimation of the dynamic state and the time varying measurement noise parameters in linear state space models the proposed adaptive kalman filtering method is based on forming a separable variational approximation to the joint posterior distribution of states and noise parameters on each time step separately the result is a recursive algorithm where on each step the state is estimated with kalman filter and the sufficient statistics of the noise variances are estimated with a fixed point iteration the performance of the algorithm is demonstrated with simulated data

neural crack identification in steady state elastodynamics an inverse crack identification problem with harmonic excitation in linear elastodynamics is treated here by means of back propagation neural network methods and boundary element techniques the problem concerns the determination of the existence and the characteristics of a hidden crack within an elastic structure by means of measurements of the structural response on the accessible boundary for given external time periodic loadings the direct problem is solved by a boundary element formulation in the frequency domain which leads to a system of linear equations with frequency dependent matrices thus for a given frequency certain similarities with linear elastostatics exist feed forward multilayer neural networks trained by back propagation are used to learn the inverse input output relation of the structural system then the inverse problem is solved by a simple application of the neural network recalling production ability

robust model identification of actuated vortex wakes we present a low order modeling technique for actuated flows based on the regularization of an inverse problem the inverse problem aims at minimizing the error between the model predictions and some reference simulations the parameters to be identified are a subset of the coefficients of a polynomial expansion which models the temporal dynamics of a small number of global modes these global modes are found by proper orthogonal decomposition which is a method to compute the most representative elements of an existing simulation database in terms of energy it is shown that low order control models based on a simple galerkin projection and usual calibration techniques are not viable they are either ill posed or they give a poor approximation of the solution as soon as they are used to predict cases not belonging to the original solution database in contrast numerical evidence shows that the method we propose is robust with respect to variations of the control laws applied thus allowing the actual use of such models for control

particle filters for state space models with the presence of unknown static parameters particle filters for dynamic state space models handling unknown static parameters are discussed the approach is based on marginalizing the static parameters out of the posterior distribution such that only the state vector needs to be considered such a marginalization can always be applied however real time applications are only possible when the distribution of the unknown parameters given both observations and the hidden state vector depends on some low dimensional sufficient statistics such sufficient statistics are present in many of the commonly used state space models marginalizing the static parameters avoids the problem of impoverishment which typically occurs when static parameters are included as part of the state vector the filters are tested on several different models with promising results

new square root smoothing algorithms this paper presents new square root smoothing algorithms for the three best known smoothing formulas 1 rauch tung striebel rts formulas 2 desai weinert yusypchuk dwy formulas called backward rts formulas and 3 mayne fraser mf formulas called two filter formulas the main feature of the new algorithms is that they use unitary rotations to replace all matrix inversion and backsubstitution steps common in earlier algorithms with unitary operations this feature enables more efficient systolic array and parallel implementations and leads to algorithms with better numerical stability and conditioning properties

a zonal galerkin free pod model for incompressible flows abstract a domain decomposition method which couples a high and a low fidelity model is proposed to reduce the computational cost of a flow simulation this approach requires to solve the high fidelity model in a small portion of the computational domain while the external field is described by a galerkin free proper orthogonal decomposition pod model we propose an error indicator to determine the extent of the interior domain and to perform an optimal coupling between the two models this zonal approach can be used to study multi body configurations or to perform detailed local analyses in the framework of shape optimisation problems the efficiency of the method to perform predictive low cost simulations is investigated for an unsteady flow and for an aerodynamic shape optimisation problem

a nonlinear pod galerkin reduced order model for compressible flows taking into account rigid body motions abstract the construction of a nonlinear reduced order model for fluid structure interaction problems is investigated in this paper for unsteady compressible flows excited by the rigid body motion of a structure the reduction is achieved by means of a galerkin projection of the navier stokes equations on the first pod modes resulting from the proper orthogonal decomposition in the first part of the paper the projection technique is carried out on a purely aerodynamic case in order i to validate an efficient iterative technique based on an updated qr decomposition to compute the pod modes and ii to discuss the merits of different correction methods introduced to improve the long term stability of the reduced order model the second and most original part of the paper deals with the construction of the reduced set of equations which arise from the projection of the compressible navier stokes equations formulated in a suitable moving frame representing the rigid body motion the expressions of the resulting non autonomous terms appearing in the reduced order model have also been optimized to reduce the computational costs

robust finite horizon minimax filtering for discrete time stochastic uncertain systems we study a finite horizon robust minimax filtering problem for time varying discrete time stochastic uncertain systems the uncertainty in the system is characterized by a set of probability measures under which the stochastic noises driving the system are defined the optimal minimax filter has been found by applying techniques of risk sensitive lqg control the structure and properties of resulting filter are analyzed and compared to h and kalman filters

interval estimation methods for discrete time linear time invariant systems this paper investigates interval estimation methods for discrete time linear time invariant systems we propose a novel interval estimation method by integrating robust observer design with reachability analysis by introducing h infty design into interval estimation the proposed method is effective in improving the accuracy of interval estimation moreover the comparisons and relationships between the existing methods and the proposed method are discussed in detail finally simulation results are presented to demonstrate the effectiveness of the proposed method

optimal control of the cylinder wake in the laminar regime by trust region methods and pod reduced order models in this paper optimal control theory is used to minimize the total mean drag for a circular cylinder wake flow in the laminar regime re 200 the control parameters are the amplitude and the frequency of the time harmonic cylinder rotation in order to reduce the size of the discretized optimality system a proper orthogonal decomposition pod reduced order model rom is derived to be used as state equation we then propose to employ the trust region proper orthogonal decomposition trpod approach originally introduced by fahl m fahl trust region methods for flow control based on reduced order modeling ph d thesis trier university 2000 to update the reduced order models during the optimization process a lot of computational work is saved because the optimization process is now based only on low fidelity models a particular care was taken to derive a pod rom for the pressure and velocity fields with an appropriate balance between model accuracy and robustness the key enablers are the extension of the pod basis functions to the pressure data the use of calibration methods for the pod rom and the addition in the pod expansion of several non equilibrium modes to describe various operating conditions when the trpod algorithm is applied to the wake flow configuration this approach converges to the minimum predicted by an open loop control approach and leads to a relative mean drag reduction of 30 at reduced cost

variational bayesian adaptive cubature information filter based on wishart distribution this paper presents a noise adaptive variational bayesian cubature information filter based on wishart distribution in the frame of recursive bayesian estimation the noise adaptive information filter propagating the information matrix and information state is derived and the integration of recursive bayesian estimation is approximated by cubature integration rule then the inverse of measurement noise matrix is modeled as a wishart distribution so the joint distribution of posterior state and measurement noise can be approximated by the product of independent gaussian and wishart furthermore the corresponding square root version is also derived to improve numerical characteristics simulation results with unknown and correlated measurement noise demonstrate the effectiveness of the proposed algorithms

robust least squares estimation with a relative entropy constraint given a nominal statistical model we consider the minimax estimation problem consisting of finding the best least squares estimator for the least favorable statistical model within a neighborhood of the nominal model the neighborhood is formed by placing a bound on the kullback leibler kl divergence between the actual and nominal models for a gaussian nominal model and a finite observations interval or for a stationary gaussian process over an infinite interval the usual noncausal wiener filter remains optimal however the worst case performance of the filter is affected by the size of the neighborhood representing the model uncertainty on the other hand standard causal least squares estimators are not optimal and a characterization is provided for the causal estimator and the corresponding least favorable model the causal estimator takes the form of a risk sensitive estimator with an appropriately selected risk sensitivity coefficient

performance analysis of the kalman filter with mismatched noise covariances the kalman filter is a powerful state estimator and has been successfully applied in many fields to guarantee the optimality of the kalman filter the noise covariances need to be exactly known however this is not necessarily true in many practical applications usually they are either completely unknown or at most partially known in this technical note we study performance of the kalman filter with mismatched process and measurement noise covariances for this purpose three mean squared errors mses are used namely the ideal mse imse the filter calculated mse fmse and the true mse tmse the main contribution of this work is that the relationships between the three mses are disclosed from two points of views the first view is about their ordering and the second view is about the relative closeness from the fmse and tmse to the imse using the first view it is found that for the case with positive definite deviation from the truth the fmse is the worst and the imse is the best and for the case with negative definite deviation the tmse is the worst and the best is the fmse using the second view it is found that the tmse is relatively closer to the imse than the fmse if the deviation is larger than certain threshold and the tmse will be farther away otherwise numerical examples further verify these conclusions

on the reduction of the continuous time generalized algebraic riccati equation an effective procedure for solving the singular lq problem with smooth solutions abstract this paper presents a reduction technique for the continuous time constrained generalized riccati equation arising in the context of the singular linear quadratic lq optimal control problem this technique allows to express the solutions of the constrained generalized riccati equation in terms of the solutions of a reduced order standard riccati equation this result is used to provide a solution to the singular lq problem with closed loop stability in the case when the allowed controls are restricted to be regular for any initial condition

robust state space filtering under incremental model perturbations subject to a relative entropy tolerance this paper considers robust filtering for a nominal gaussian state space model when a relative entropy tolerance is applied to each time increment of a dynamical model the problem is formulated as a dynamic minimax game where the maximizer adopts a myopic strategy this game is shown to admit a saddle point whose structure is characterized by applying and extending results presented earlier in robust least squares estimation with a relative entropy constraint b c levy and r nikoukhah ieee trans inf theory vol 50 no 1 89 104 jan 2004 for static least squares estimation the resulting minimax filter takes the form of a risk sensitive filter with a time varying risk sensitivity parameter which depends on the tolerance bound applied to the model dynamics and observations at the corresponding time index the least favorable model is constructed and used to evaluate the performance of alternative filters simulations comparing the proposed risk sensitive filter to a standard kalman filter show a significant performance advantage when applied to the least favorable model and only a small performance loss for the nominal model

a computational study of transformation methods for optimal design in this paper computational aspects of transformatio n methods are studied transformation methods which include sequential unconstrained minimization techniques sumts and multiplier methods are based on solving a sequence of unconstrained minimization problems an efficient technique is given to compute gradients of the transformation function an operations count is given to demonstrate savings of the suggested technique over two other techniques used in the literature computer programs implementing the use of this technique in sumts and multiplier algorithms are developed applications of these programs on a set of structural design problems are given multiplier methods are found to be very stable and reliable even on some relatively difficult problems and they perform better than sumts some ways of improving the efficiency of transformation methods are given together with possible extensions of using the suggested approach to compute gradients of implicit functions

on the robustness of the bayes and wiener estimators under model uncertainty abstract this paper deals with the robust estimation problem of a signal given noisy observations we assume that the actual statistics of the signal and observations belong to a ball about the nominal statistics this ball is formed by placing a bound on a suitable divergence or distance between the actual and the nominal statistics then the robust estimator is obtained by minimizing the mean square error according to the least favorable statistics in that ball therefore we obtain a divergence based minimax approach to robust estimation choosing a set of divergences called tau divergence family we show that the bayes estimator based on the nominal statistics is the optimal solution moreover in the dynamic case the optimal offline estimator is the noncausal wiener filter based on the nominal statistics

convergence analysis of a family of robust kalman filters based on the contraction principle in this paper we analyze the convergence of a family of robust kalman filters for each filter of this family the model uncertainty is tuned according to the so called tolerance parameter assuming that the corresponding state space model is reachable and observable we show that the n fold composition of the corresponding riccati like mapping is strictly contractive provided that the tolerance is sufficiently small and accordingly the filter converges

model reduction for large scale systems with high dimensional parametric input space a model constrained adaptive sampling methodology is proposed for the reduction of large scale systems with high dimensional parametric input spaces our model reduction method uses a reduced basis approach which requires the computation of high fidelity solutions at a number of sample points throughout the parametric input space a key challenge that must be addressed in the optimization control and probabilistic settings is the need for the reduced models to capture variation over this parametric input space which for many applications will be of high dimension we pose the task of determining appropriate sample points as a pde constrained optimization problem which is implemented using an efficient adaptive algorithm that scales well to systems with a large number of parameters the methodology is demonstrated using examples with parametric input spaces of dimension 11 and 21 which describe thermal analysis and design of a heat conduction fin and compared with statistically based sampling methods for these examples the model constrained adaptive sampling leads to reduced models that for a given basis size have error several orders of magnitude smaller than that obtained using the other methods

a unifying review of linear gaussian models factor analysis principal component analysis mixtures of gaussian clusters vector quantization kalman filter models and hidden markov models can all be unified as variations of unsupervised learning under a single basic generative model this is achieved by collecting together disparate observations and derivations made by many previous authors and introducing a new way of linking discrete and continuous state models using a simple nonlinearity through the use of other nonlinearities we show how independent component analysis is also a variation of the same basic generative model we show that factor analysis and mixtures of gaussians can be implemented in autoencoder neural networks and learned using squared error plus the same regularization term we introduce a new model for static data known as sensible principal component analysis as well as a novel concept of spatially adaptive observation noise we also review some of the literature involving global and local mixtures of the basic models and provide pseudocode for inference and learning for all the basic models

stiffness matrix adjustment using mode data a procedure is introduced that uses in addition to mode data structural connectivity information to optimally adjust deficient stiffness matrices the adjustments performed are such that the percentage change to each stiffness coefficient is minimized the physical configuration of the analytical model is preserved and the adjusted model will exactly reproduce the modes used in the identification the theoretical development is presented and the procedure is demonstrated by numerical simulation of a test problem

fast kalman like filtering for large dimensional linear and gaussian state space models this article considers the filtering problem for linear and gaussian state space models with large dimensions a setup in which the optimal kalman filter kf might not be applicable owing to the excessive cost of manipulating huge covariance matrices among the most popular alternatives that enable cheaper and reasonable computation is the ensemble kf enkf a monte carlo based approximation in this article we consider a class of a posteriori distributions with diagonal covariance matrices and propose fast approximate deterministic based algorithms based on the variational bayesian vb approach more specifically we derive two iterative kf like algorithms that differ in the way they operate between two successive filtering estimates one involves a smoothing estimate and the other involves a prediction estimate despite its iterative nature the prediction based algorithm provides a computational cost that is on the one hand independent of the number of iterations in the limit of very large state dimensions and on the other hand always much smaller than the cost of the enkf the cost of the smoothing based algorithm depends on the number of iterations that may in some situations make this algorithm slower than the enkf the performances of the proposed filters are studied and compared to those of the kf and enkf through a numerical example

inherently incomplete finite element model and its effects on model updating some of the questions regarding the use of measured data to improve a finite element model of a structure to make it better able to predict the dynamic behavior of the structure are addressed the issues discussed include the following all such models are truly reduced models the parameters of a reduced model of a linear structure vary with frequency a linear model of a linear structure has a limited frequency range of applicability and there are an infinite number of reasonable models that can match measured data these and other related concepts are discussed and demonstrated of course because there is no unique solution questions arise as to how to select the best one and are there different best models for different applications it is concluded that new paths of research in this important field are necessary

optimal boundary feedback flow stabilization by model reduction abstract in this paper we consider the optimal boundary feedback stabilization of fluid flows governed by navier stokes equations using model reduction the model reduction is carried out using a combination of proper orthogonal decomposition pod and galerkin projection the resulting reduced order model is employed in the optimal linear quadratic regulator lqr design to derive a feedback control the feedback control is then used in the nonlinear navier stokes equation to stabilize the system sufficient conditions are derived for the linear control to stabilize the nonlinear system feasibility of the proposed boundary feedback stabilization design is numerically demonstrated on the problem of stabilizing the flow past an airfoil the control is effected through unsteady blowing on the airfoil surface the computational results show that the closed loop system amplitudes converge to the desired equilibrium state showing the stability predicted by the theory in addition they show that downstream directed blowing on the upper surface of the airfoil near the leading edge is more efficient in stabilizing the state

variational learning for switching state space models we introduce a new statistical model for time series that iteratively segments data into regimes with approximately linear dynamics and learns the parameters of each of these linear regimes this model combines and generalizes two of the most widely used stochastic time series models hidden markov models and linear dynamical systems and is closely related to models that are widely used in the control and econometrics literatures it can also be derived by extending the mixture of experts neural network jacobs jordan nowlan hinton 1991 to its fully dynamical version in which both expert and gating networks are recurrent inferring the posterior probabilities of the hidden states of this model is computationally intractable and therefore the exact expectation maximization em algorithm cannot be applied however we present a variational approximation that maximizes a lower bound on the log likelihood and makes use of both the forward and backward recursions for hidden markov models and the kalman filter recursions for linear dynamical systems we tested the algorithm on artificial data sets and a natural data set of respiration force from a patient with sleep apnea the results suggest that variational approximations are a viable method for inference and learning in switching state space models

improved simulated annealing search for structural optimization an optimization algorithm based on simulated annealing is proposed in which the domain of the search is successively reduced based on a probability concept until the stopping criteria are satise ed by introducing the ideas of probability cumulative distribution function and stable energy the selection of initial temperature and equilibrium criterion in the process of simulated annealing becomes easy and effective numerical studies using a set of standard test functions and an example of a 10 bar truss show that the approach is effective and robust in solving both functional and structural optimization problems he achievement of optimum design is a goal naturally attractive to the designer the e eld of structural design is generally characterized by a large number of variables which are usually discrete in dimensions or properties and these variables should satisfy all of the constraints to be a feasible structural design some structuraloptimizationproblemsarefairlyamenableto amathematical programming approach in such instances the design space is continuousand convex the searchmay be deterministic and methods are employed using for example gradient concepts however there is a large class of structural optimization problems with nonconvexities in the design space and with a mix of continuous and discrete variables under such circumstances standard mathematical programming techniques are usually inefe cientbecause they are computationally expensive and are almost assured of locating the relative optimum close to the starting design to overcome these dife culties the stochastic search in structural optimization is considered many methods have become possible with the powerful computing facilities available in recent years among the stochastic algorithms pure random search 1 is the simplest strategy for optimal design some modie ed versions have been suggested such as single start multistart and random directions 2 3 methods in this class generally are quite simple to implement but the appropriate stopping rules are very dife cult to derive recently two classes of powerful search methods which have their philosophical basis in processes found in nature have been widely used in structural optimization the e rst class of methods including genetic algorithms 4 and simulated evolution 5 is based on the spirit of darwinian theory of evolution the second class of methods is generally referred to as simulated annealing techniques 6 because they are qualitatively derived from a simulation of the behaviorofparticlesinthermalequilibriumatagiventemperature because of their global capabilities research on the utilization of these search methods in design optimization has been undertaken 7i 11 somehybridtechniqueshavealsobeendevelopedbycombiningfeaturesofthesetwoalgorithms suchasusingageneticalgorithmtodetermine better annealing schedules 12 and introducing a boltzmanntype mutation or selection process into simulated evolution 13 14 in this paper we propose a method based on simulated annealing that searches from a population as in the method of simulated evolution instead of from a single point the algorithm is called the region reduction simulated annealing rrsa method because it locates the optimum by successively eliminating the regions with low probability of containing the optimum a brief review of basic simulated annealing is given in sec ii which is helpful for development and explanation of the proposed

dynamic modeling of neuronal responses in fmri using cubature kalman filtering this paper presents a new approach to inverting fitting models of coupled dynamical systems based on state of the art cubature kalman filtering crucially this inversion furnishes posterior estimates of both the hidden states and parameters of a system including any unknown exogenous input because the underlying generative model is formulated in continuous time with a discrete observation process it can be applied to a wide variety of models specified with either ordinary or stochastic differential equations these are an important class of models that are particularly appropriate for biological time series where the underlying system is specified in terms of kinetics or dynamics i e dynamic causal models we provide comparative evaluations with generalized bayesian filtering dynamic expectation maximization and demonstrate marked improvements in accuracy and computational efficiency we compare the schemes using a series of difficult nonlinear toy examples and conclude with a special focus on hemodynamic models of evoked brain responses in fmri our scheme promises to provide a significant advance in characterizing the functional architectures of distributed neuronal systems even in the absence of known exogenous experimental input e g resting state fmri studies and spontaneous fluctuations in electrophysiological studies importantly unlike current bayesian filters e g dem our scheme provides estimates of time varying parameters which we will exploit in future work on the adaptation and enabling of connections in the brain

state sensitivity evaluation within ud based array covariance filters this technical note addresses the ud factorization based kalman filtering kf algorithms using this important class of numerically stable kf schemes we extend its functionality and develop an elegant and simple method for computation of sensitivities of the system state to unknown parameters required in a variety of applications for instance it can be used for efficient calculations in sensitivity analysis and in gradient search optimization algorithms for the maximum likelihood estimation the new theory presented in this technical note is a solution to the problem formulated by bierman in which has been open since 1990s as in the cited paper our method avoids the standard approach based on the conventional kf and its derivatives with respect to unknown system parameters with its inherent numerical instabilities and hence improves the robustness of computations against roundoff errors

a moment of indecision we consider inference from a single observation x about a location parameter 0 that is given 0 0 the density function of x has the form f x 0 for some known function f the prior distribution for 0 has density g 6 it is well known that if f and g are both normal densities then the posterior density is normal with a variance independent of x and less than the variance of either f or g very different behaviour can occur if either component is not normal dawid 1973 studied the limiting behaviour of the posterior distribution as the observation x tends to infinity he showed that for certain forms of f and g depending on the thickness of their tails one component of the posterior will in the limit completely dominate the other then the limiting posterior density is either the prior density g 6 or the likelihood f x 0 in particular he showed that if one of these components is a normal density and the other a t density then the normal component will dominate o hagan 1979 proved that the normal density cannot be dominated by any other dawid s work showed that with a thick tailed f or g the limiting behaviour of the posterior distribution given an extreme observation could be very different from the case when both f and g are normal in the present note we consider the approach to that limit and prove that under certain circumstances the posterior variance undergoes a sharp peak before settling down to its limiting value we interpret this phenomenon intuitively as a manifestation of indecision the observation is large enough for the prior and the data to conflict i e the marginal probability of x being in a neighbourhood of its observed value is very small but not large enough for that conflict to be clearly resolved by one source of information dominating the other box 1980 uses such conflict between prior and data as a means for criticizing the model but we will suppose that the model is not questioned

intrinsically bayesian robust kalman filter an innovation process approach in many contemporary engineering problems model uncertainty is inherent because accurate system identification is virtually impossible owing to system complexity or lack of data on account of availability time or cost the situation can be treated by assuming that the true model belongs to an uncertainty class of models in this context an intrinsically bayesian robust ibr filter is one that is optimal relative to the cost function in the classical sense and the prior distribution over the uncertainty class in the bayesian sense ibr filters have previously been found for both wiener and granulometric morphological filtering in this paper we derive the ibr kalman filter that performs optimally relative to an uncertainty class of state space models introducing the notion of bayesian innovation process and the bayesian orthogonality principle we show how the problem of designing an ibr kalman filter can be reduced to a recursive system similar to the classical kalman recursive equations except with effective counterparts such as the effective kalman gain matrix after deriving the recursive ibr kalman equations for discrete time we use the limiting method to obtain the ibr kalman bucy equations for continuous time finally we demonstrate the utility of the proposed framework for two real world problems sensor networks and gene regulatory network inference

robust filtering for discrete time systems with bounded noise and parametric uncertainty this note presents a new approach to finite horizon guaranteed state prediction for discrete time systems affected by bounded noise and unknown but bounded parameter uncertainty our framework handles possibly nonlinear dependence of the state space matrices on the uncertain parameters the main result is that a minimal confidence ellipsoid for the state consistent with the measured output and the uncertainty description may be recursively computed in polynomial time using interior point methods for convex optimization with n states l uncertain parameters appearing linearly in the state space matrices with rank one matrix coefficients the worst case complexity grows as o l n l sup 3 5 with unstructured uncertainty in all system matrices the worst case complexity reduces to o n sup 3 5

a framework for state space estimation with uncertain models develops a framework for state space estimation when the parameters of the underlying linear model are subject to uncertainties compared with existing robust filters the proposed filters perform regularization rather than deregularization it is shown that under certain stabilizability and detectability conditions the steady state filters are stable and that for quadratically stable models the filters guarantee a bounded error variance moreover the resulting filter structures are similar to various time and measurement update prediction and information forms of the kalman filter albeit ones that operate on corrected parameters rather than on the given nominal parameters simulation results and comparisons with spl hscr sub spl infin guaranteed cost and set valued state estimation filters are provided

kalman filter sensitivity evaluation with orthogonal and j orthogonal transformations this technical note addresses the array square root kalman filtering smoothing algorithms with the conventional orthogonal and j orthogonal transformations in the adaptive filtering context j orthogonal matrices arise in computation of the square root of the covariance or smoothed covariance by solving an equation of the form cct ddt bbt the latter implies an application of the qr decomposition with j orthogonal transformations in each iteration step in this paper we extend functionality of array square root kalman filtering schemes and develop an elegant and simple method for computation of the derivatives of the filter variables to unknown system parameters required in a variety of applications for instance our result can be implemented for an efficient sensitivity analysis and in gradient search optimization algorithms for the maximum likelihood estimation of unknown system parameters it also replaces the standard approach based on direct differentiation of the conventional kalman filtering equations with their inherent numerical instability and hence improves the robustness of computations against roundoff errors

multivariate spectral estimation based on the concept of optimal prediction in this technical note we deal with a spectrum approximation problem arising in three like multivariate spectral estimation approaches the solution to the problem minimizes a suitable divergence index with respect to an a priori spectral density we derive a new divergence family between multivariate spectral densities which takes root in the prediction theory under mild assumptions on the a priori spectral density the approximation problem based on this new divergence family admits a family of solutions moreover an upper bound on the complexity degree of these solutions is provided

a new family of high resolution multivariate spectral estimators in this paper we extend the beta divergence family to multivariate power spectral densities similarly to the scalar case we show that it smoothly connects the multivariate kullback leibler divergence with the multivariate itakura saito distance we successively study a spectrum approximation problem based on the beta divergence family which is related to a multivariate extension of the three spectral estimation technique it is then possible to characterize a family of solutions to the problem an upper bound on the complexity of these solutions will also be provided finally we will show that the most suitable solution of this family depends on the specific features required from the estimation problem

secant method adjustment for structural models on orbit testing will be required for final tuning and validation of any mathematical model of large space structures identification methods using limited response data to produce optimally adjusted property matrices seem ideal for this purpose but difficulties exist in the application of previously published methods to large space truss structures this article presents new stiffness matrix adjustment methods that generalize optimal update secant methods found in quasi newton approaches for nonlinear optimization many aspects of previously published methods of stiffness matrix adjustment may be better understood within this new framework of secant methods one of the new methods preserves realistic structural connectivity with minimal storage requirements and computational effort a method for systematic compensation for errors in measured data is introduced that also preserves structural connectivity two demonstrations are presented to compare the new methods results to those of previously published techniques

enhanced structural damage detection using alternating projection methods alternating projection algorithms are examined for the solution of damage detection problems in structures the damage detection problem is formulated as a feasibility problem to find a damaged stiffness matrix that is close to the refined stiffness matrix of the undamaged structure and that satisfies the necessary symmetry sparsity positive definiteness eigenequation and damage localization constraints alternating projection methods are proposed to utilize the orthogonal projections onto these constraint sets in an iterative fashion to find a solution that best satisfies these constraints in addition directional alternating projections that exploit the geometry of the damage detection feasibility problem are introduced to improve the computational efficiency of the approach the techniques are applied to detect damage in a simulated cantilever beam model and in the nasa eight bay truss damage detection experimental test bed

wheat stem moduli in vivo via reference basis model updating a novel structural analysis method was developed and used to infer the in vivo material properties of an important crop species in particular reference basis model updating was used to reconcile the predictions of an idealized analytical model with the observed forced vibration response of living wheat and thereby determine the effective longitudinal moduli of the tissues comprising the vibrating structure the ensuing study of two varieties of wheat found that the stem tissue moduli decreased nonlinearly with height these findings were in good qualitative agreement with the limited data available in the literature and the updating procedure was highly effective in this unusual nondestructive testing application to our knowledge this is the first time that the reference basis method has been used to infer the spatially varying material properties of a living plant moreover similar approaches could be used to characterize spatial variations in the material properties of other lightweight composite structures of arbitrary geometry

closed loop stability analysis of discrete time negative imaginary systems abstract we derive necessary and sufficient conditions for stability analysis of a positive feedback interconnection of a discrete time negative imaginary system and a discrete time strictly negative imaginary system general stability analysis results for continuous time negative imaginary systems connected in positive feedback have recently been proposed those recent results extend previous theorems by removing restrictive assumptions on the infinite frequency gains imposed in the earlier literature and by extending the class of negative imaginary systems for which the results are applicable to include systems with free body dynamics i e poles at the origin here we present the discrete time counterparts of the aforementioned recently developed results which specialise to simple and easy to check conditions under specific assumptions last we illustrate some of the results by several examples

numerical aspects of different kalman filter implementations a theoretical analysis is made of the error propagation due to numerical roundoff for four different kalman filter implementations the conventional kalman filter the square root covariance filter the square root information filter and the chandrasekhar square root filter an experimental analysis is performed to validate the new insights gained by the theoretical analysis

some matrix variate distribution theory notational considerations and a bayesian application summary we introduce and justify a convenient notation for certain matrix variate distributions which by its emphasis on the important underlying parameters and the theory on which it is based eases greatly the task of manipulating such distributions important examples include the matrix variate normal t f and beta and the wishart and inverse wishart distributions the theory is applied to compound matrix distributions and to bayesian prediction in the multivariate linear model

wishart distributions for decomposable graphs when considering a graphical gaussian model n g markov with respect to a decomposable graph g the parameter space of interest for the precision parameter is the cone p g of positive definite matrices with fixed zeros corresponding to the missing edges of g the parameter space for the scale parameter of n g is the cone q g dual to p g of incomplete matrices with submatrices corresponding to the cliques of g being positive definite in this paper we construct on the cones q g and p g two families of wishart distributions namely the type i and type ii wisharts they can be viewed as generalizations of the hyper wishart and the inverse of the hyper inverse wishart as defined by dawid and lauritzen ann statist 21 1993 1272 1317 we show that the type i and ii wisharts have properties similar to those of the hyper and hyper inverse wishart indeed the inverse of the type ii wishart forms a conjugate family of priors for the covariance parameter of the graphical gaussian model and is strong directed hyper markov for every direction given to the graph by a perfect order of its cliques while the type i wishart is weak hyper markov moreover the inverse type ii wishart as a conjugate family presents the advantage of having a multidimensional shape parameter thus offering flexibility for the choice of a prior both type i and ii wishart distributions depend on multivariate shape parameters a shape parameter is acceptable if and only if it satisfies a certain eigenvalue property we show that the sets of acceptable shape parameters for a noncomplete g have dimension equal to at least one plus the number of cliques in g these families as conjugate families are richer than the traditional diaconis ylvisaker conjugate families which all have a shape parameter set of dimension one a decomposable graph which does not contain a three link chain as an induced subgraph is said to be homogeneous in this case our wisharts are particular cases of the wisharts on homogeneous cones as defined by andersson and wojnar j theoret probab 17 2004 781 818 and the dimension of the shape parameter set is even larger than in the nonhomogeneous case it is indeed equal to the number of cliques plus the number of distinct minimal separators using the model where g is a three link chain we show by computing a 7 tuple integral that in general we cannot expect the shape parameter sets to have dimension larger than the number of cliques plus one

an interpretation of the dual problem of the three like approaches spectral estimation can be performed using the so called three like approach such method leads to a convex optimization problem whose solution is characterized through its dual problem in this paper we show that the dual problem can be seen as a new parametric spectral estimation problem this interpretation implies that the three like solution is optimal in terms of closeness to the correlogram over a certain parametric class of spectral densities describing arma models enriching in this way its meaningfulness

a contraction analysis of the convergence of risk sensitive filters a contraction analysis of risk sensitive riccati equations is proposed when the state space model is reachable and observable a block update implementation of the risk sensitive filter is used to show that the n fold composition of the riccati map is strictly contractive with respect to the thompson s part metric of positive definite matrices when n is larger than the number of states the range of values of the risk sensitivity parameter for which the map remains contractive can be estimated a priori it is also found that a second condition must be imposed on the risk sensitivity parameter and on the initial error variance to ensure that the solution of the risk sensitive riccati equation remains positive definite at all times the two conditions obtained can be viewed as extending to the multivariable case an earlier analysis of whittle for the scalar case

joint maximum a posteriori state path and parameter estimation in stochastic differential equations abstract in this article we introduce the joint maximum a posteriori state path and parameter estimator jme for continuous time systems described by stochastic differential equations sdes this estimator can be applied to nonlinear systems with discrete time sampled measurements with a wide range of measurement distributions we also show that the minimum energy state path and parameter estimator mee obtains the joint maximum a posteriori noise path initial conditions and parameters these estimators are demonstrated in simulated experiments in which they are compared to the prediction error method pem using the unscented kalman filter and smoother the experiments show that the mee is biased for the damping parameters of the drift function furthermore for robust estimation in the presence of outliers the jme attains lower state estimation errors than the pem

multiple acceptable solutions in structural model improvement this paper examines the question as to whether it is possible to obtain the correct discrete linear analytical model of an actual structure by the use of analysis and test data it is shown that no such correct model exists an illustration of an application is shown where multiple acceptable and very different analytical models are all representative of the structure the implications regarding methods of improving analytical models and detecting damage are discussed

cubature kalman filters in this paper we present a new nonlinear filter for high dimensional state estimation which we have named the cubature kalman filter ckf the heart of the ckf is a spherical radial cubature rule which makes it possible to numerically compute multivariate moment integrals encountered in the nonlinear bayesian filter specifically we derive a third degree spherical radial cubature rule that provides a set of cubature points scaling linearly with the state vector dimension the ckf may therefore provide a systematic solution for high dimensional nonlinear filtering problems the paper also includes the derivation of a square root version of the ckf for improved numerical stability the ckf is tested experimentally in two nonlinear state estimation problems in the first problem the proposed cubature rule is used to compute the second order statistics of a nonlinearly transformed gaussian random variable the second problem addresses the use of the ckf for tracking a maneuvering aircraft the results of both experiments demonstrate the improved performance of the ckf over conventional nonlinear filters

elliptically contoured random variables and their application to the extension of the kalman filter abstract we summarize some properties of elliptically contoured e c random variables and indicate their possible application in engineering practice in particular after presenting some new results we apply them to modify the kalman filter the modified version is robust against the extension of the noises model from normal to elliptically contoured

calibrated reduced order pod galerkin system for fluid flow modelling to improve the behaviour of reduced order proper orthogonal decomposition pod galerkin systems two numerical methods are proposed these methods determine free parameters in the pod galerkin system from flow simulations via a minimization problem they give rise to linear systems and their computational costs are reasonable both methods are assessed for two flow configurations a two dimensional flow around a square cylinder for a reynolds number of 100 and a three dimensional flow past a backward facing step for a reynolds number of 7432 based on the step height and the streamwise velocity at the middle of the inlet for both configurations the methods are effective since accurate calibrated reduced order pod galerkin systems are obtained

cholesky decomposition of a hyper inverse wishart matrix the canonical parameter of a covariance selection model is the inverse covariance matrix 1 whose zero pattern gives the conditional independence structure characterising the model in this paper we consider the upper triangular matrix obtained by the cholesky decomposition 1 t this provides an interesting alternative parameterisation of decomposable models since its upper triangle has the same zero structure as 1 and its elements have an interpretation as parameters of certain conditional distributions for a distribution for the strong hyper markov property is shown to be characterised by the mutual independence of the rows of this is further used to generalise to the hyper inverse wishart distribution some well known properties of the inverse wishart distribution in particular we show that a hyper inverse wishart matrix can be decomposed into independent normal and chi squared random variables and we describe a family of transformations under which the family of hyper inverse wishart distributions is closed

a non linear observer for unsteady three dimensional flows a method is proposed to estimate the velocity field of an unsteady flow using a limited number of flow measurements the method is based on a non linear low dimensional model of the flow and on an expansion of the velocity field in terms of empirical basis functions the main idea is to impose that the coefficients of the modal expansion of the velocity field gives the best approximation of the available measurements while at the same time satisfying the non linear low order model as closely as possible practical applications may range from feedback flow control to the monitoring of the flow in non accessible regions the proposed technique is applied to the flow around a confined square cylinder both in two and three dimensional flow regimes comparisons are provided with existing linear and non linear estimation techniques

bayesian kalman filtering with elliptically contoured errors summary the basic recursive equations of the kalman filter for independent normally distributed error sources in both the observation and the system equations are shown to hold for a larger class of distributions it is shown that for error sources following jointly a general elliptical distribution the conditional posterior distribution of the parameters and the one step ahead predictive distribution at every stage t are also elliptical this generalizes some recent results of meinhold singpurwalla for error sources distributed as multivariate student t and some older results of zellner for the linear regression model

minimization by random search techniques we give two general convergence proofs for random search algorithms we review the literature and show how our results extend those available for specific variants of the conceptual algorithm studied here we then exploit the convergence results to examine convergence rates and to actually design implementable methods finally we report on some computational experience

robustness and risk sensitive filtering this paper gives a precise meaning to the robustness of risk sensitive filters for problems in which one is uncertain as to the exact value of the probability model it is shown that risk sensitive estimators including filters enjoy an error bound which is the sum of two terms the first of which coincides with an upper bound on the error that one would obtain if one knew exactly the underlying probability model while the second term is a measure of the distance between the true and design probability models the paper includes a discussion of several approaches to estimation including h sub spl infin filtering

robust kalman filtering under model perturbations we consider a family of divergence based minimax approaches to perform robust filtering the mismodeling budget or tolerance is specified at each time increment of the model more precisely all possible model increments belong to a ball which is formed by placing a bound on the tau divergence family between the actual and the nominal model increment then the robust filter is obtained by minimizing the mean square error according to the least favorable model in that ball it turns out that the solution is a family of kalman like filters their gain matrix is updated according to a risk sensitive like iteration where the risk sensitivity parameter is now time varying as a consequence we also extend the risk sensitive filter to a family of risk sensitive like filters according to the tau divergence family

robust estimation in non linear state space models with state dependent noise in this paper we present a robust estimation algorithm for non linear state space models driven by state dependent noise the algorithm is robust to outliers in the data we derive the algorithm step by step from first principles from theory to implementation the implementation is straightforward and consists mainly of two components 1 a slightly modified version of the rauch tung striebel recursions and 2 a backtracking line search strategy since it preserves the underlying chain structure of the problem its computational complexity grows linearly with the number of data the algorithm is iterative and is guaranteed to converge under mild assumptions to a local optimum from any starting point we validate our approach via experiments on synthetic data from a multi variate stochastic volatility model

robustification of kalman filter models abstract kalman filter models based on the assumption of multivariate gaussian distributions are known to be nonrobust this means that when a large discrepancy arises between the prior distribution and the observed data the posterior distribution becomes an unrealistic compromise between the two in this article we discuss a rationale for how to robustify the kalman filter specifically we develop a model wherein the posterior distribution will revert to the prior when extreme outlying observations are encountered and we point out that this can be achieved by assuming a multivariate distribution with student t marginals to achieve fully robust results of the kind desired it becomes necessary to forsake an exact distribution theory approach and adopt an approximation method involving poly t distributions a recursive mechanism for implementing the multivariate t based kalman filter is described its properties are discussed and the procedure is illustrated by an example

modal parameter identification using simulated evolution a global optimization algorithm based on the darwinian evolutionary theory is implemented for the identification of modal parameters of linear vibrating structures the capability of the evolutionary search method in locating the global minimum among numerous local minima is demonstrated the modal parameter identification procedure is based on the concept of modal sweep according to which modes of a structure are identified individually and iteratively a fast recursive algorithm is also developed for forward response calculation of a linear oscillator to lessen the computational burden of using simulated evolution in the iterative identification process the proposed modal parameter identification approach does not require any initial estimate of parameter values simulation results show that the approach is effective and reliable in identifying dominant modes under noisy conditions

a variational bayesian approach to robust sensor fusion based on student t distribution in this paper a robust sensor fusion method is proposed where the measurement noise is modeled by a student t distribution the student t distribution has a heavy tail compared to the gaussian distribution and is robust to outliers we formulate sensor fusion as a state space estimation problem in the bayesian framework both batch and recursive variational bayesian vb algorithms are developed to perform this non gaussian state space estimation problem to obtain the fusion results computer simulations show that the proposed approach has an improved fusion performance and a lower computation cost compared to methods based on gaussian and finite gaussian mixture models

approximate inference in state space models with heavy tailed noise state space models have been successfully applied across a wide range of problems ranging from system control to target tracking and autonomous navigation their ubiquity stems from their modeling flexibility as well as the development of a battery of powerful algorithms for estimating the state variables for multivariate models the gaussian noise assumption is predominant due its convenient computational properties in some cases anyhow this assumption breaks down and no longer holds we propose a novel approach to extending the applicability of this class of models to a wider range of noise distributions without losing the computational advantages of the associated algorithms the estimation methods we develop parallel the kalman filter and thus are readily implemented and inherit the same order of complexity we derive all of the equations and algorithms from first principles in order to validate the performance of our approach we present specific instances of non gaussian state space models and test their performance on experiments with synthetic and real data

asymptotic properties of linear filter for deterministic processes abstract it is known that kalman bucy filter is stable with respect to initial conditions under the assumptions of uniform complete controllability and uniform complete observability in this paper we prove the stability of kalman bucy filter for the case of noise free dynamical system i e for deterministic processes the earlier stability results cannot be applied for this case as the system is not controllable we further show that the optimal linear filter for a general class of non gaussian initial conditions is asymptotically proximal to kalman bucy filter it is also shown that the filter corresponding to non zero system noise in the limit of small system noise approaches the filter corresponding to zero system noise in the case of gaussian initial conditions