robust kalman bucy filter development of a robust estimator for uncertain stochastic systems under persistent excitation is presented the given continuous time stochastic formulation assumes norm bounded parametric uncertainties and excitations when there are no system uncertainties the performance of the proposed robust estimator is similar to that of the kalman bucy filter and the proposed approach asymptotically recovers the desired optimal performance in the presence of uncertainties and or persistent excitation

robust mse equalizer design for mimo communication systems in the presence of model uncertainties this paper considers a robust mean square error mse equalizer design problem for multiple input multiple output mimo communication systems with imperfect channel and noise information at the receiver when the channel state information csi and the noise covariance are known exactly at the receiver a minimum mean square error mmse equalizer can be employed to estimate the transmitted signal however in actual systems it is necessary to take into account channel and noise estimation errors we consider here a worst case equalizer design problem where the goal is to find the equalizer minimizing the equalization mse for the least favorable channel model within a neighborhood of the estimated model the neighborhood is formed by placing a bound on the kullback leibler kl divergence between the actual and estimated channel models lagrangian optimization is used to convert this min max problem into a convex min min problem over a convex domain which is solved by interchanging the minimization order the robust mse equalizer and associated least favorable channel model can then be obtained by solving numerically a scalar convex minimization problem simulation results are presented to demonstrate the mse and bit error rate ber performance of robust equalizers when applied to the least favorable channel model

technical communique robust state prediction for descriptor systems this paper deals with the problem of state prediction for descriptor systems subject to bounded uncertainties the problem is stated in terms of the optimization of an appropriate quadratic functional this functional is well suited to derive not only the robust predictor for descriptor systems but also that for usual state space systems numerical examples are included in order to demonstrate the performance of this new filter

technical communique robust state estimation for uncertain discrete time stochastic systems with missing measurements in this paper results of robust estimation of zhou 2010a are extended to state estimation with missing measurements a new procedure is derived which inherits the main properties of that of zhou 2010a in this extension a covariance matrix used in the recursions is replaced by its estimate which makes its asymptotic property investigation mathematically difficult though introducing a monotonic function and using the so called squeeze rule this new robust estimator is proved to converge to a stable system numerical simulation results indicate that the proposed estimator may have an estimation accuracy better than the estimator of wang yang daniel and liu 2005

bounding bar d distance by informational divergence a method to prove measure concentration there is a simple inequality by pinsker between variational distance and informational divergence of probability measures defined on arbitrary probability spaces we shall consider probability measures on sequences taken from countable alphabets and derive from pinsker s inequality bounds on the bar d distance by informational divergence such bounds can be used to prove the concentration of measure phenomenon for some nonproduct distributions

a semi definite programming approach for robust tracking tracking problems are prevalent in the present day gps and video systems the problem of target tracking is a specific case of dynamic linear system estimation with additive noise the most widely used filter for these systems is the kalman filter kf the optimality of the kf and similar bayesian filters is guaranteed under particular probabilistic assumptions however in practice and specifically in applications such as tracking these probabilistic assumptions are not realistic indeed the system noise is typically bounded and in fact might be adversarial for such cases robust estimation approaches such as mathcal h infty h filtering and set value estimation were introduced with the aim of providing filters with guaranteed worst case performance in this paper we present an innovative approximated set value estimator sve which is obtained through a semi definite programming problem we demonstrate that our problem is practically tractable even for long time horizons the framework is extended to include the case of partially statistical noise thus combining the kf and sve frameworks a variation of this filter which applies a rolling window approach is developed achieving fixed computational cost per iteration and coinciding with the classical sve when window size is one finally we present numerical results that show the advantages of this filter when dealing with adversarial noise and compare the performance of the various robust filters with the kf

sparse exchangeable graphs and their limits via graphon processes in a recent paper caron and fox suggest a probabilistic model for sparse graphs which are exchangeable when associating each vertex with a time parameter in r here we show that by generalizing the classical definition of graphons as functions over probability spaces to functions over finite measure spaces we can model a large family of exchangeable graphs including the caron fox graphs and the traditional exchangeable dense graphs as special cases explicitly modelling the underlying space of features by a finite measure space s s and the connection probabilities by an integrable function w s s 0 1 we construct a random family gt t 0 of growing graphs such that the vertices of gt are given by a poisson point process on s with intensity t with two points x y of the point process connected with probability w x y we call such a random family a graphon process we prove that a graphon process has convergent subgraph frequencies with possibly infinite limits and that in the natural extension of the cut metric to our setting the sequence converges to the generating graphon we also show that the underlying graphon is identifiable only as an equivalence class over graphons with cut distance zero more generally we study metric convergence for arbitrary not necessarily random sequences of graphs and show that a sequence of graphs has a convergent subsequence if and only if it has a subsequence satisfying a property we call uniform regularity of tails finally we prove that every graphon is equivalent to a graphon on r equipped with lebesgue measure

the heat flow on metric random walk spaces abstract in this paper we study the heat flow on metric random walk spaces which unifies into a broad framework the heat flow on locally finite weighted connected graphs the heat flow determined by finite markov chains and some nonlocal evolution problems we give different characterizations of the ergodicity and prove that a metric random walk space with positive ollivier ricci curvature is ergodic furthermore we prove a cheeger inequality and as a consequence we show that a poincare inequality holds if and only if an isoperimetric inequality holds we also study the bakry emery curvature dimension condition and its relation with functional inequalities like the poincare inequality and the transport information inequalities

robust h sub 2 and h sub spl infin filters for uncertain lft systems in this note robust h sub 2 and h sub spl infin filter design problems are considered the uncertainties unstructured or structured are norm bounded and represented by linear fractional transformation lft the main result is that after upper bounding the objectives the problems of minimizing the upper bounds are converted to finite dimensional convex optimization problems involving linear matrix inequalities lmis these are extensions of well known results for systems with polytopic uncertainty it is shown that for the unstructured norm bounded uncertainty case the upper bounds are directly minimized without further overbounding yielding less conservative results than previously available an elementary numerical example is given to illustrate the results

multiobjective filter design for uncertain stochastic time delay systems this note addresses the problem of robust multiobjective filtering for discrete time delay systems with mixed stochastic and deterministic uncertainties in addition to unmodeled nonlinearities a procedure is developed for the design of linear and exponentially stable filters with a bounded error variance exponential rate of decay and robust performance for the error system

robust receding horizon state estimation for uncertain discrete time linear systems an approach to robust receding horizon state estimation for discrete time linear systems is presented estimates of the state variables can be obtained by minimizing a worst case quadratic cost function according to a sliding window strategy this leads to state the estimation problem in the form of a regularized least squares one with uncertain data the optimal solution involving on line scalar minimization together with a suitable closed form approximation are given the stability properties of the estimation error for both the optimal filter and the approximate one have been studied and conditions to select the design parameters are proposed simulation results are reported to show the effectiveness of the proposed approach

robust h reduced order filtering for uncertain bilinear systems this paper investigates both the h and robust h reduced order unbiased filtering problems for respectively a nominal bilinear system and a bilinear system affected by norm bounded structured uncertainties in all the system matrices first an algebraic framework is used to solve the unbiasedness condition and second a change of variable is introduced on the inputs of the system to reduce the conservatism inherent to the requirement of exponential convergence of the filter then the reduced order filtering solution is obtained through lmi with an equality constraint by transforming the problem into a robust state feedback in the nominal case and a robust static output feedback in the presence of uncertainties in the last case an additional bilinear matrix equality must also be solved

parameter estimation with multiple sources and levels of uncertainties least squares designs are sensitive to errors in the data which can be due to several factors including the approximation of complex models by simpler ones the presence of unavoidable experimental errors when collecting data or even due to unknown or unmodeled effects we formulate a new design criterion that treats multiple sources of uncertainties in the data with possibly varied degrees of intensity we show that the solution has a regularized form with one regularization parameter for each source of uncertainty the parameters turn out to be model dependent and can be determined optimally as the nonnegative roots of certain coupled equations applications in array signal processing and image processing are considered

localization and tensorization properties of the curvature dimension condition for metric measure spaces abstract this is an addendum to the paper k bacher k t sturm localization and tensorization properties of the curvature dimension condition for metric measure spaces j funct anal 259 2010 28 56 we prove the tensorization property for the curvature dimension condition add some detailed calculations including explicit dependence of constants and comment on assumptions and conjectures concerning the local to global statement in bacher and sturm 2010 1 and villani 2009 6 respectively

recursive state estimation for uncertain systems with an integral quadratic constraint this paper considers a robust state estimation problem for a class of uncertain systems where the noise and uncertainty are modeled deterministically via an integral quadratic constraint the robust state estimation problem involves constructing the set of all possible states at the current time consistent with given output measurements and the integral quadratic constraint

sensitivity penalization based robust state estimation for uncertain linear systems this technical note deals with robust state estimation when parametric uncertainties nonlinearly affect a plant state space model based on a simultaneous minimization of nominal estimation errors and their sensitivities an analytic solution is derived for the optimal estimator which can be recursively realized this estimator has a form similar to the robust estimator of and its computational complexity is comparable to the kalman filter under certain conditions this robust estimator is proved to converge to a stable system its estimation errors have a bounded covariance matrix and the estimate is asymptotically unbiased numerical simulations show that the obtained estimator has nice estimation performances

kantorovich duality for general transport costs and applications we introduce a general notion of transport cost that encompasses many costs used in the literature including the classical one and weak transport costs introduced by talagrand and marton in the 90 s and prove a kantorovich type duality theorem as a by product we obtain various applications in different directions we give a short proof of a result by strassen on the existence of a martingale with given marginals we characterize the associated transport entropy inequalities together with the log sobolev inequality restricted to convex concave functions some explicit examples of discrete measures satisfying weak transport entropy inequalities are also given

robust mimo equalization for non parametric channel model uncertainty in this paper three mimo robust equalization problems are considered for non parametric classes of channel models defined by weighted h2 or h balls of frequency responses and performance criteria based on h2 variance or h norms of error signals the approach pursued here centers on characterizing the worst case performance of candidate equalizers or upper bounds on it by means of dual lagrangian functionals then for linearly parametrized finite dimensional classes of candidate equalizers the corresponding robust equalization problems are converted into semi definite linear programming problems for which approximate solutions can be effectively computed a simple numerical example is presented involving h2 model uncertainty and error variance performance to illustrate for various levels of uncertainty the changes in the worst case performances of the nominally optimal equalizer and of the one in a specific linear class which minimizes the worst case error variance

robust filtering for linear systems with convex bounded uncertain time varying parameters this note addresses the design of robust h2 filters for linear systems with a state space model subject to time varying uncertain parameters with limited variation the uncertain parameters and their rate of variation are assumed to belong to a given convex bounded polyhedral domain a method based on a parameter dependent lyapunov function is proposed for designing a linear stationary asymptotically stable filter with a guaranteed average error variance irrespective of the uncertain parameters the proposed design is formulated in terms of linear matrix inequalities

estimation and control with bounded data uncertainties the paper describes estimation and control strategies for models with bounded data uncertainties we shall refer to them as bdu estimation and bdu control methods for brevity they are based on constrained game type formulations that allow the designer to explicitly incorporate into the problem statement a priori information about bounds on the sizes of the uncertainties in this way the effect of uncertainties is not unnecessarily over emphasized beyond what is implied by the a priori bounds consequently overly conservative designs as well as overly sensitive designs are avoided a feature of these new formulations is that geometric insights and recursive techniques which are widely known and appreciated for classical quadratic cost designs can also be pursued in this new framework also algorithms for computing the optimal solutions with the same computational effort as standard least squares solutions exist thus making the new formulations attractive for practical use moreover the framework is broad enough to encompass applications across several disciplines not just estimation and control examples will be given of a quadratic control design an h control design a total least square design image restoration image separation and co channel interference cancellation a major theme in this paper is the emphasis on geometric and linear algebraic arguments which lead to useful insights about the nature of the new formulations despite the interesting results that will be discussed several issues remain open and indicate potential future developments these will be briefly discussed

diversity effects in kalman filtering over rayleigh fading channels a scalar first order gauss markov signal is sent over d parallel i i d rayleigh fading channels using uncoded transmission assuming perfect channel estimation at the receiver the optimal mmse estimator is the kalman filter with random instantaneous estimation error variance statistical characteristics of the instantaneous estimation error variance are studied to characterize the estimation quality one such characteristic is the estimation error outage which probabilistically depicts how often the value of the instantaneous estimation error variance exceeds a certain threshold in this paper we study the effect of channel diversity on the estimation error outage probability we show that for a certain range of outage thresholds the estimation error outage probability decreases with inverse of d th power of the average signal to noise ratio at the receiver side this means that analogous to the context of digital communication over fading channels our result implies a diversity order of d in estimating the input process

moderate deviations of a random riccati equation the paper characterizes the invariant filtering measures resulting from kalman filtering with intermittent observations in which the observation arrival is modeled as a bernoulli process with packet arrival probability our prior work showed that for 0 the sequence of random conditional error covariance matrices converges weakly to a unique invariant distribution this paper shows that as approaches one the family 0 satisfies a moderate deviations principle with good rate function i 1 as 1 the family converges weakly to the dirac measure p concentrated on the fixed point of the associated discrete time riccati operator 2 the probability of a rare event an event bounded away from p under decays to zero as a power law of 1 as 1 and 3 the best power law decay exponent is obtained by solving a deterministic variational problem involving the rate function i for specific scenarios the paper develops computationally tractable methods that lead to efficient estimates of rare event probabilities under

ricci curvature of markov chains on metric spaces abstract we define the coarse ricci curvature of metric spaces in terms of how much small balls are closer in wasserstein transportation distance than their centers are this definition naturally extends to any markov chain on a metric space for a riemannian manifold this gives back after scaling the value of ricci curvature of a tangent vector examples of positively curved spaces for this definition include the discrete cube and discrete versions of the ornstein uhlenbeck process moreover this generalization is consistent with the bakry emery ricci curvature for brownian motion with a drift on a riemannian manifold positive ricci curvature is shown to imply a spectral gap a levy gromov like gaussian concentration theorem and a kind of modified logarithmic sobolev inequality the bounds obtained are sharp in a variety of examples

lmi based minimax estimation and filtering under unknown covariances in this paper we consider a minimax approach to the estimation and filtering problems in the stochastic framework where covariances of the random factors are completely unknown the term random factors refers either to unknown parameters and measurement noise in the estimation problem or to disturbance process and the initial state of a linear discrete time dynamic system in the filtering problem we introduce a notion of the attenuation level of random factors as a performance measure for both a linear unbiased estimate and a filter this is the worst case variance of the estimation error normalised by the sum of variances of all random factors over all nonzero covariance matrices it is shown that this performance measure is equal to the spectral norm of the transfer matrix and therefore the minimax estimate and filter can be computed in terms of linear matrix inequalities lmis moreover the explicit formulae for both the minimax estimate and the minimal value of the attenuation level are prese

regularized robust filters for time varying uncertain discrete time systems this note develops robust filters for time varying uncertain discrete time systems the developed filters are based on a data regularization solution and they enforce a minimum state error variance property simulation results confirm their superior performance over other robust filter designs

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

on the convergence and stability of a robust state estimator convergence and stability of the robust state estimator obtained in is reinvestigated in this technical note some new relations have been established for matrix updates in the recursive state estimation it is proved that under certain stabilizability and detectability conditions this robust estimator converges to a stable time invariant system provided that plant nominal parameters are time invariant and the filter design parameter is fixed these results are consistent with existent ones but different from them at the point that there are no orthogonality constraints on uncertainty related system matrices and therefore widen theoretical guarantees for the effectiveness of the estimation procedure

mass transportation and rough curvature bounds for discrete spaces abstract we introduce and study rough approximate lower curvature bounds for discrete spaces and for graphs this notion agrees with the one introduced in j lott c villani ricci curvature for metric measure spaces via optimal transport ann of math 169 2009 in press and k t sturm on the geometry of metric measure spaces i acta math 196 2006 65 131 in the sense that the metric measure space which is approximated by a sequence of discrete spaces with rough curvature k will have curvature k in the sense of j lott c villani ricci curvature for metric measure spaces via optimal transport ann of math 169 2009 in press k t sturm on the geometry of metric measure spaces i acta math 196 2006 65 131 moreover in the converse direction discretizations of metric measure spaces with curvature k will have rough curvature k we apply our results to concrete examples of homogeneous planar graphs

brief paper minimax state estimation for linear discrete time differential algebraic equations this paper presents a state estimation approach for an uncertain linear equation with a non invertible operator in hilbert space the approach addresses linear equations with uncertain deterministic input and noise in the measurements which belong to a given convex closed bounded set a new notion of a minimax observable subspace is introduced by means of the presented approach new equations describing the dynamics of a minimax recursive estimator for discrete time non causal differential algebraic equations daes are presented for the case of regular daes it is proved that the estimator s equation coincides with the equation describing the seminal kalman filter the properties of the estimator are illustrated by a numerical example

competitive robust estimation for uncertain linear dynamic models in this paper two types of robust linear estimation problems for dynamic channel model uncertainty are considered with the aim of characterizing in computationally effective ways competitive robust estimators i e robust estimators that improve on the pointwise performance of minimax mse mean squared error estimators over the uncertain model set at the expense of a moderate increase in the worst case mse the first one corresponds to the minimization of the worst case value of an approximate regret function defined by a quadratic approximation of the lower mse envelope on the uncertain model set the second one corresponds to minimizing the nominal mse error while ensuring that the worst case estimation error does not exceed a prescribed value for uncertain classes defined by h 2 norm balls it is shown that these two types of estimation problems can be recast as semidefinite programming problems sdps for short numerical examples are presented for both the case of linear finite dimensional model classes firs of a given length and the case of nonparametric uncertain sets of causal real rational frequency responses suggesting that these two types of estimators can be attractive alternatives to the min max mse estimator for the case of spectral norm in the finite dimensional case or h infty norm in the nonparametric case the worst case mse and approximate regret for each candidate estimator are replaced by upper bounds obtained by lagrangian relaxation and somewhat conservative versions of the estimation problems previously mentioned are posed it is shown that these problems can also be recast as sdps

robust transport over networks we consider transportation over a strongly connected directed graph the scheduling amounts to selecting transition probabilities for a discrete time markov evolution which is designed to be consistent with initial and final marginal constraints on mass transport we address the situation where initially the mass is concentrated on certain nodes and needs to be transported over a certain time period to another set of nodes possibly disjoint from the first the evolution is selected to be closest to a prior measure on paths in the relative entropy sense such a construction is known as a schrodinger bridge between the two given marginals it may be viewed as an atypical stochastic control problem where the control consists in suitably modifying the prior transition mechanism the prior can be chosen to incorporate constraints and costs for traversing specific edges of the graph but it can also be selected to allocate equal probability to all paths of equal length connecting any two nodes i e a uniform distribution on paths this latter choice for prior transitions relies on the so called ruelle bowen random walker and gives rise to scheduling that tends to utilize all paths as uniformly as the topology allows thus this ruelle bowen law mathfrak m mathrm rb taken as prior leads to a transportation plan that tends to lessen congestion and ensures a level of robustness we also show that the distribution mathfrak m mathrm rb on paths which attains the maximum entropy rate for the random walker given by the topological entropy can itself be obtained as the time homogeneous solution of a maximum entropy problem for measures on paths also a schrodinger bridge problem albeit with prior that is not a probability measure finally we show that the paradigm of schrodinger bridges as a mechanism for scheduling transport on networks can be adapted to graphs that are not strongly connected as well as to weighted graphs in the latter case our approach may be used to design a transportation plan which effectively compromises between robustness and other criteria such as cost indeed we explicitly provide a robust transportation plan which assigns maximum probability to minimum cost paths and therefore compares favorably with optimal mass transportation strategies

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

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

an l p theory of sparse graph convergence i limits sparse random graph models and power law distributions we introduce and develop a theory of limits for sequences of sparse graphs based on lp graphons which generalizes both the existing l1 theory of dense graph limits and its extension by bollob as and riordan to sparse graphs without dense spots in doing so we replace the no dense spots hypothesis with weaker assumptions which allow us to analyze graphs with power law degree distributions this gives the rst broadly applicable limit theory for sparse graphs with unbounded average degrees in this paper we lay the foundations of the lp theory of graphons characterize convergence and develop corresponding random graph models while we prove the equivalence of several alternative metrics in a companion paper

conditional central algorithms for worst case set membership identification and filtering this paper deals with conditional central estimators in a set membership setting the role and importance of these algorithms in identification and filtering is illustrated by showing that problems like worst case optimal identification and state filtering in contexts in which disturbances are described through norm bounds are reducible to the computation of conditional central algorithms the conditional chebyshev center problem is solved for the case when energy norm bounded disturbances are considered a closed form solution is obtained by finding the unique real root of a polynomial equation in a semi infinite interval

sectional curvature of polygonal complexes with planar substructures in this paper we introduce a class of polygonal complexes for which we consider a notion of sectional combinatorial curvature these complexes can be viewed as generalizations of 2 dimensional euclidean and hyperbolic buildings we focus on the case of non positive and negative combinatorial curvature as geometric results we obtain a hadamard cartan type theorem thinness of bigons gromov hyperbolicity and estimates for the cheeger constant we employ the latter to get spectral estimates show discreteness of the spectrum in the sense of a donnelly li type theorem and present corresponding eigenvalue asymptotics moreover we prove a unique continuation theorem for eigenfunctions and the solvability of the dirichlet problem at infinity

convergent sequences of dense graphs i subgraph frequencies metric properties and testing we consider sequences of graphs gn and define various notions of convergence related to these sequences left convergence defined in terms of the densities of homomorphisms from small graphs into gn right convergence defined in terms of the densities of homomorphisms from gn into small graphs and convergence in a suitably defined metric in part i of this series we show that left convergence is equivalent to convergence in metric both for simple graphs gn and for graphs gn with nodeweights and edgeweights one of the main steps here is the introduction of a cut distance comparing graphs not necessarily of the same size we also show how these notions of convergence provide natural

correspondence comments on robust state estimation for uncertain discrete time stochastic systems with missing measurements automatica 47 2011 1520 1524 we show that some results presented in the aforementioned article are rather limited one of the major restrictions on applicability of the obtained results is the ergodic requirement on the received plant output measurements which is generally not satisfied by a time varying system another major restriction is that the comparisons are not appropriate for its numerical simulations that is using the state estimation results of an estimator that utilizing current and past observations to compare with those of a one step state predictor is not appropriate to evaluate the filter performances objectively we do a numerical simulation with the robust state estimator and the well known kalman filter that does not take either parametric errors or random measurement loss into account and the simulation results show that there is no significant difference in filtering performance between these two estimators

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

robust steady state filtering for systems with deterministic and stochastic uncertainties for uncertain systems containing both deterministic and stochastic uncertainties we consider two problems of optimal filtering the first is the design of a linear time invariant filter that minimizes an upper bound on the mean energy gain between the noise affecting the system and the estimation error the second is the design of a linear time invariant filter that minimizes an upper bound on the asymptotic mean square estimation error when the plant is driven by a white noise we present filtering algorithms that solve each of these problems with the filter parameters determined via convex optimization based on linear matrix inequalities we demonstrate the performance of these robust algorithms on a numerical example consisting of the design of equalizers for a communication channel

optimal robust filtering for systems subject to uncertainties in this paper we deal with an optimal filtering problem for uncertain discrete time systems parametric uncertainties of the underlying model are assumed to be norm bounded we propose an approach based on regularization and penalty function to solve this problem the optimal robust filter with the respective recursive riccati equation is written through unified frameworks defined in terms of matrix blocks these frameworks do not depend on any auxiliary parameters to be tuned simulation results show the effectiveness of the robust filter proposed

transportation cost information inequalities and applications to random dynamical systems and diffusions we first give a characterization of the l1 transportation cost information inequality on a metric space and next find some appropriate sufficient condition to transportation cost information inequalities for dependent sequences applications to random dynamical systems and diffusions are studied

an l p theory of sparse graph convergence ii ld convergence quotients and right convergence we extend the lplp theory of sparse graph limits which was introduced in a companion paper by analyzing different notions of convergence under suitable restrictions on node weights we prove the equivalence of metric convergence quotient convergence microcanonical ground state energy convergence microcanonical free energy convergence and large deviation convergence our theorems extend the broad applicability of dense graph convergence to all sparse graphs with unbounded average degree while the proofs require new techniques based on uniform upper regularity examples to which our theory applies include stochastic block models power law graphs and sparse versions of ww random graphs

robust kalman filtering for uncertain discrete time systems this paper is concerned with the problem of a kalman filter design for uncertain discrete time systems the system under consideration is subjected to time varying norm bounded parameter uncertainty in both the state and output matrices the problem addressed is the design of a linear filter such that the variance of the filtering error is guaranteed to be within a certain bound for all admissible uncertainties furthermore the guaranteed cost can be optimized by appropriately searching a scaling design parameter

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

a robust estimator for stochastic systems under unknown persistent excitation a robust estimator for uncertain stochastic systems under unknown persistent disturbance is presented the given discrete time stochastic formulation neither requires a known bound on the magnitude of the unknown excitation nor assumes stability of the system however the proposed estimator assumes certain structural conditions on system uncertainties though the proposed estimator is developed based on stochastic lyapunov analysis its structure and performance are comparable to that of unbiased minimum variance filters based on the disturbance decoupling technique unlike unbiased minimum variance filters implementation of the developed estimator only requires adding an auxiliary term to the nominal steady state kalman filter and it does not involve any similarity transformation or propagation of matrix difference equations

mixed norm h 2 h regulation and estimation the discrete time case abstract a discrete time h2 static output feedback design problem involving a constraint on h disturbance attenuation is addressed and state space formulae are derived the dual problem of discrete time dynamic estimation with an h error bound is also addressed these results are analogous to results obtained previously for the continuous time problem

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

robust estimation for discrete time markovian jump linear systems this technical note deals with recursive robust estimation of markov jump linear systems subject to unobserved chain state it is developed an augmented system modeled via norm bounded uncertainties upper and lower bounds of an uncertain quadratic cost function are computed in order to define the variances of the augmented system numerical examples are provided to show the effectiveness of the proposed estimators

author s reply to the comments on robust kalman filter for descriptor systems in the comments provided by zhou and zhang 2009 the authors showed that through off line convex optimization the filter performance of the article by ishihara et al 2006 can be improved through an appropriate combination of ellipsoids determined by the actual plant states and structured model uncertainties these guidelines are interesting and useful to improve the performance of the filter we proposed

robust h2 filtering for uncertain linear systems lmi based methods with parametric lyapunov functions abstract this paper addresses the design of robust h 2 filters for linear continuous time systems subject to parameter uncertainty in the state space model the uncertain parameters are supposed to belong to a given convex bounded polyhedral domain two methods based on parameter dependent lyapunov functions are proposed for designing linear stationary asymptotically stable filters that assure asymptotic stability and a guaranteed h 2 performance irrespective of the uncertain parameters the proposed filter designs are given in terms of linear matrix inequalities which depend on a scalar parameter that should be searched for in order to optimize the filter performance

h infinity kalman estimation for rectangular descriptor systems with unknown inputs this note considers h filtering and prediction for rectangular descriptor systems with unknown inputs that affect both the system and the output an optimal h descriptor kalman estimator hdke which can simultaneously solve the h filtering and prediction problems for rectangular descriptor systems with unknown inputs is developed based on the maximum likelihood descriptor kalman filtering method the hdke serves as a unified solution to solve h and kalman filtering for descriptor systems and standard systems with or without unknown inputs to reduce the computational complexity problem some efficient alternatives to the developed hdke are further proposed the relationship between the hdke and the existing literature results is also addressed an illustrative example is given to show the usefulness of the proposed results

robust minimum variance filtering this paper deals with the robust minimum variance filtering problem for linear systems subject to norm bounded parameter uncertainty in both the state and the output matrices of the state space model the problem addressed is the design of linear filters having an error variance with a guaranteed upper bound for any allowed uncertainty two methods for designing robust filters are investigated the first one deals with constant parameter uncertainty and focuses on the design of steady state filters that yield an upper bound to the worst case asymptotic error variance this bound depends on an upper bound for the power spectrum density of a signal at a specific point in the system and it can be made tighter if a tight bound on the latter power spectrum can be obtained the second method allows for time varying parameter uncertainty and for general time varying systems and is more systematic we develop filters with an optimized upper bound for the error variance for both finite and infinite horizon filtering problems

approximate optimal periodic scheduling of multiple sensors with constraints a constrained periodic multiple sensor scheduling problem is considered in this paper for each sensor constraints on dwell time and activation times are imposed at each time instant only one sensor can update its measurement with the estimator and the objective is to minimize the average state estimation error an approximation framework is proposed to calculate the objective function which transforms the original scheduling problem into an approximate optimal scheduling problem aosp an upper bound on the approximation error is presented to evaluate the performance of the framework to solve the aosp a necessary condition is first proposed on the optimal schedules when no constraints on activation times exist a dynamic programming based algorithm is devised to identify the optimal schedule with polynomial computational complexity when activation time constraints exist we show that the aosps can be solved by solving traveling salesman problems examples are provided to illustrate the proposed results

on modified logarithmic sobolev inequalities for bernoulli and poisson measures we show that for any positive functionfon the discrete cube 0 1 n ent np f pqe np1f df 2where npis the product measure of the bernoulli measure with probability of successp as well as related inequalities which may be shown to imply in the limit the classical gaussian logarithmic sobolev inequality as well as a logarithmic sobolev inequality for poisson measure we further investigate modified logarithmic sobolev inequalities to analyze integrability properties of lipschitz functions on discrete spaces in particular we obtain under modified logarithmic sobolev inequalities some concentration results for product measures that extend the classical exponential inequalities for sums of independent random variables

robust and stable predictive control with bounded uncertainties min max optimization is often used for improving robustness in model predictive control mpc an analogy to this optimization could be the bdu bounded data uncertainties method which is a regularization technique for least squares problems that takes into account the uncertainty bounds stability of mpc can be achieved by using terminal constraints such as in the crhpc constrained receding horizon predictive control algorithm by combining both bdu and crhpc methods a robust and stable mpc is obtained which is the aim of this work bdu also offers a guided method of tuning the empirically tuned penalization parameter for the control effort in mpc

rigidity of the bonnet myers inequality for graphs with respect to ollivier ricci curvature abstract we introduce the notion of bonnet myers and lichnerowicz sharpness in the ollivier ricci curvature sense our main result is a classification of all self centered bonnet myers sharp graphs hypercubes cocktail party graphs even dimensional demi cubes johnson graphs j 2 n n the gosset graph and suitable cartesian products we also present a purely combinatorial reformulation of this result we show that bonnet myers sharpness implies lichnerowicz sharpness and classify all distance regular lichnerowicz sharp graphs under the additional condition 1 b 1 1 we also relate bonnet myers sharpness to an upper bound of bakry emery curvature which motivates a general conjecture about bakry emery curvature

new conditions for the convergence of h sub spl infin filters and predictors the finite horizon h sub spl infin filtering and prediction problems in the discrete time case admit solutions only if a suitable difference riccati equation is solved by a matrix sequence satisfying feasibility conditions in this work sufficient conditions ensuring the existence of filters and predictors over an arbitrarily long time interval are derived moreover it is shown that under these conditions finite horizon estimators tend to stable stationary ones as the time horizon increases

displacement convexity of entropy and related inequalities on graphs we introduce the notion of an interpolating path on the set of probability measures on finite graphs using this notion we first prove a displacement convexity property of entropy along such a path and derive prekopa leindler type inequalities a talagrand transport entropy inequality certain hwi type as well as log sobolev type inequalities in discrete settings to illustrate through examples we apply our results to the complete graph and to the hypercube for which our results are optimal by passing to the limit we recover the classical log sobolev inequality for the standard gaussian measure with the optimal constant

optimal design of robust predictors for linear discrete time systems the one step ahead prediction problem for systems subject to parameter uncertainty in the system dynamics and noise statistics is considered the objective is the design of a robust predictor that minimizes an upper bound of the error covariance sufficient and necessary conditions for the existence of such an optimal robust estimator are given the computation of the predictor is based on the stabilizing solution of a suitable h type riccati equation in the uncertainty free case the robust predictor reduces to the standard kalman predictor

quadratic stability with real and complex perturbations it is shown that the equivalence between real and complex perturbations in the context of quadratic stability to linear fractional unstructured perturbations does not hold when the perturbations are block structured for a limited class of problems quadratic stability in the face of structured complex perturbations is equivalent to a particular class of scaled norms and hence appropriate synthesis techniques coupled with diagonal constant scalings can be used to design quadratically stable systems

ollivier ricci curvature for general graph laplacians heat equation laplacian comparison non explosion and diameter bounds abstract discrete time random walks on a finite set naturally translate via a one to one correspondence to discrete laplace operators typically ollivier curvature has been investigated via random walks we first extend the definition of ollivier curvature to general weighted graphs and then give a strikingly simple representation of ollivier curvature using the graph laplacian using the laplacian as a generator of a continuous time markov chain we connect ollivier curvature with the heat equation which is strongly related to continuous time random walks in particular we prove that a lower bound on the ollivier curvature is equivalent to a certain lipschitz decay of solutions to the heat equation this is a discrete analogue to a celebrated ricci curvature lower bound characterization by renesse and sturm our representation of ollivier curvature via the laplacian allows us to deduce a laplacian comparison principle by which we prove non explosion and improved diameter bounds

technical communique robust state estimation and model validation for discrete time uncertain systems with a deterministic description of noise and uncertainty the paper presents a new approach to robust state estimation for a class of uncertain discrete time systems with a deterministic description of noise and uncertainty the main result is a recursive scheme for constructing an ellipsoidal state estimation set of all states consistent with the measured output and the given noise and uncertainty description the paper also includes a result on model validation whereby it can be determined if the assumed model is consistent with measured data

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

an observer based controller for nonlinear systems a gain scheduling approach abstract this paper presents an observer based controller for discrete time nonlinear systems whose control inputs are subject to a set of equality and or inequality constraints firstly a regularized least square estimator is developed to estimate the states of the nonlinear system then the estimated states are used to generate a feedback control law using the jacobean linearization in conjunction with gain scheduling the stability of the closed loop control system is rigorously investigated applications of the proposed observer based control scheme to two numerical examples are presented then the developed controller is used to control a synchronous machine which is subject to constraints on the control the simulation results show the effectiveness of the proposed observer based control scheme

optimal robust linear quadratic regulator for systems subject to uncertainties in this technical note a robust recursive regulator for linear discrete time systems which are subject to parametric uncertainties is proposed the main feature of the optimal regulator developed is the absence of tuning parameters in online applications to achieve this purpose a quadratic cost function based on the combination of penalty function and robust weighted least squares methods is formulated the convergence and stability proofs for the stationary system and a numerical comparative study among the standard linear quadratic regulator guaranteed cost and h controllers are provided

on the application of a hybrid ellipsoidal rectangular interval arithmetic algorithm to interval kalman filtering for state estimation of uncertain systems modelling uncertainty is a key limitation to the applicability of the classical kalman filter for state estimation of dynamic systems for such systems with bounded modelling uncertainty the interval kalman filter ikf is a direct extension of the former to interval systems however its usage is not yet widespread owing to the over conservatism of interval arithmetic bounds in this paper the ikf equations are adapted to use an ellipsoidal arithmetic that in some cases provides tighter bounds than direct rectangular interval arithmetic in order for the ikf to be useful it must be able to provide reasonable enclosures under all circumstances to this end a hybrid ellipsoidal rectangular enclosure algorithm is proposed and its robustness is evidenced by its application to two characteristically different systems for which it provides stable estimate bounds whereas the rectangular and ellipsoidal approaches fail to accomplish this in either one or the other case

robust discrete time minimum variance filtering the bounded variance filtered estimation of the state of an uncertain linear discrete time system with an unknown norm bounded parameter matrix is considered an upper bound on the variance of the estimation error is found for all admissible systems and estimators are derived that minimize the latter bound we treat the finite horizon time varying case and the infinite time case where the nominal system model is time invariant in the special stationary case where it is known that the uncertain system is time invariant we provide a robust filter for all uncertainties that still keep the system asymptotically stable

information filtering and array algorithms for descriptor systems subject to parameter uncertainties this paper presents information filters in riccati recursions and in array algorithms for descriptor systems subject to parameters uncertainties the filters are developed in filtered and predicted forms the inversion of the state matrix is avoided in the new information recursive formulas therefore it turns out clear that the invertibility of the state matrix usually considered in the state space information recursions is not necessary a numerical example is provided to illustrate the performance of the proposed robust array algorithms

periodically time varying cal h _ infty memory filter design for discrete time lti systems with polytopic uncertainty this technical note focuses on the development of a general periodically time varying memory filter ptvmf to improve the robust h filtering performance for discrete time linear time invariant lti systems with polytopic parameter uncertainties a sufficient condition to design the h ptvmf is given in terms of convex optimizations expressed as linear matrix inequalities lmis finally examples demonstrate its potential less conservatism

the optimal projection equations for reduced order discrete time state estimation for linear systems with multiplicative white noise the optimal projection equations obtained in 2 3 for reduced order discrete time state estimation are generalized to include the effects of state and measurement dependent noise to provide a model of parameter uncertainty in contrast to the single matrix riccati equation arising in the full order kalman filter case the optimal steady state reduced order discrete time estimator is characterized by three matrix equations one modified riccati equation and two modified lyapunov equations coupled by both an oblique projection and stochastic effects

convergent sequences of dense graphs ii multiway cuts and statistical physics we consider sequences of graphs gn and dene various notions of convergence related to these sequences including left convergence dened in terms of the densities of homomorphisms from small graphs into gn and right convergence dened in terms of the densities of homomorphisms from gn into small graphs we show that right convergence is equivalent to left convergence both for simple graphs gn and for graphs gn with nontrivial nodeweights and edgeweights other equivalent conditions for convergence are given in terms of fundamental notions from combinatorics such as maximum cuts and szemer edi partitions and fundamental notions from statistical physics like energies and free energies we thereby relate local and global properties of graph sequences quantitative forms of these results express the relationships among dierent measures of similarity of large graphs

a uniqueness result concerning a robust regularized least squares solution in solving a robust version of regularized least squares with weighting a certain scalar valued optimization problem is required in order to determine the regularized robust solution and the corresponding robustified weighting parameters this letter establishes that the required optimization problem does not have local non global minima over the interval of interest this property is proved by resorting to a useful schur complementation argument the result is reassuring in that it demonstrates that the robust design procedure is well defined and that its optimal global solution can be determined without concerns about local minima

transport inequalities gradient estimates entropy and ricci curvature we present various characterizations of uniform lower bounds for the ricci curvature of a smooth riemannian manifold m in terms of convexity properties of the entropy considered as a function on the space of probability measures on m as well as in terms of transportation inequalities for volume measures heat kernels and brownian motions and in terms of gradient estimates for the heat semigroup 2004 wiley periodicals inc

large deviations of subgraph counts for sparse erdős renyi graphs abstract for any fixed simple graph h v e and any fixed u 0 we establish the leading order of the exponential rate function for the probability that the number of copies of h in the erd s renyi graph g n p exceeds its expectation by a factor 1 u assuming n h p 1 with h 1 2 where 1 is the maximum degree of h this improves on a previous result of chatterjee and the second author who obtained h c e for a constant c 0 moreover for the case of cycle counts we can take as large as 1 2 we additionally obtain the sharp upper tail for schatten norms of the adjacency matrix as well as the sharp lower tail for counts of graphs for which sidorenko s conjecture holds as a key step we establish quantitative versions of szemeredi s regularity lemma and the counting lemma suitable for the analysis of random graphs in the large deviations regime

robust stabilization of uncertain linear systems quadratic stabilizability and h sup infinity control theory the problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time domain results on the quadratic stabilization of uncertain systems and the frequency domain results on h sup infinity optimization a complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given relations between these robust stabilization problems and h sup infinity control theory are explored it is also shown that in a number of cases if a robust stabilization problem can be solved via lyapunov methods then it can be also be solved via h sup infinity control theory based methods

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form