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

convergent sdp relaxations in polynomial optimization with sparsity we consider a polynomial programming problem p on a compact basic semialgebraic set k subset r n described by m polynomial inequalities g j x geq0 and with criterion f in r x we propose a hierarchy of semidefinite relaxations in the spirit of those of waki e siam j optim 17 2006 pp 218 242 in particular the sdp relaxation of order r has the following two features a the number of variables is o kappa 2r where kappa max kappa 1 kappa 2 with kappa 1 resp kappa 2 being the maximum number of variables appearing in the monomials of f resp appearing in a single constraint g j x geq0 b the largest size of the linear matrix inequalities lmis is o kappa r this is to compare with the respective number of variables o n 2r and lmi size o n r in the original sdp relaxations defined in j b lasserre siam j optim 11 2001 pp 796 817 therefore great computational savings are expected in case of sparsity in the data g j f i e when k is small a frequent case in practical applications of interest the novelty with respect to h waki s kim m kojima and m maramatsu siam j optim 17 2006 pp 218 242 is that we prove convergence to the global optimum of p when the sparsity pattern satisfies a condition often encountered in large size problems of practical applications and known as the running intersection property in graph theory in such cases and as a by product we also obtain a new representation result for polynomials positive on a compact basic semialgebraic set a sparse version of putinar s positivstellensatz m putinar indiana univ math j 42 1993 pp 969 984

a new approximate algorithm for the chebyshev center the parameter or state estimation with bounded noises is getting increasingly important in many applications of practical systems with some uncertainties the problem to estimate a deterministic parameter or state which is known to lie in an intersection of some ellipsoids can be formulated to find the chebyshev center of the intersection set in the case of l 2 norm of the estimation error in this paper an appropriate positive semidefinite relaxation of non convex optimization problem is derived and then a new algorithm for robust minimax estimation is provided some examples are given to compare the approximate estimate with the existing relaxed chebyshev center

feasible parameter set approximation for linear models with bounded uncertain regressors nonconvex feasible parameter sets are encountered in set membership identification whenever the regressor vector is affected by bounded uncertainty this occurs for example when considering standard output error models or when the available measurements are provided by binary or quantized sensors in this paper a unifying framework is proposed to deal with several identification problems involving a nonconvex feasible parameter set and a procedure is proposed for approximating the minimum volume orthotope containing the feasible set the procedure exploits different relaxations for autoregressive and input parameters based on the solution of a sequence of linear programming problems the proposed technique is shown to provide tight bounds in some special cases moreover it is extended to cope with bounds not aligned with the parameter coordinates in order to obtain polytopic approximations of the feasible set a number of numerical tests on randomly generated models and data sets demonstrates the accuracy of the computed set approximations

heavy traffic limits for queues with many exponential servers two different kinds of heavy traffic limit theorems have been proved for s server queues the first kind involves a sequence of queueing systems having a fixed number of servers with an associated sequence of traffic intensities that converges to the critical value of one from below the second kind which is often not thought of as heavy traffic involves a sequence of queueing systems in which the associated sequences of arrival rates and numbers of servers go to infinity while the service time distributions and the traffic intensities remain fixed with the traffic intensities being less than the critical value of one in each case the sequence of random variables depicting the steady state number of customers waiting or being served diverges to infinity but converges to a nondegenerate limit after appropriate normalization however in an important respect neither procedure adequately represents a typical queueing system in practice because in the heavy traffic limit an arriving customer is either a

identification and application of bounded parameter models abstract the place of parameter bounding algorithms in identification methodology is discussed such algorithms provide a radical alternative to the computation of parameter point estimates and covariances instead of p d f or mean and covariance for the noise and prior parameter estimates they require bounds in an essentially deterministic model formulation the paper is a preliminary review of some standard tasks namely experiment design testing for outliers toleranced prediction and worst case control design in the context of parameter bounding identification computational requirements of these tasks are examined and the limitations of existing parameter bounding algorithms are noted

discrete transforms semidefinite programming and sum of squares representations of nonnegative polynomials we present a new semidefinite programming formulation of sum of squares representations of nonnegative polynomials cosine polynomials and trigonometric polynomials of one variable the parametrization is based on discrete transforms specifically the discrete fourier cosine and polynomial transforms and has a simple structure that can be exploited by straightforward modifications of standard interior point algorithms

fixed order fir approximation of linear systems from quantized input and output data the problem of identifying a fixed order fir approximation of linear systems with unknown structure assuming that both input and output measurements are subjected to quantization is dealt with in this paper a fixed order fir model providing the best approximation of the input output relationship is sought by minimizing the worst case distance between the output of the true system and the modeled output for all possible values of the input and output data consistent with their quantized measurements the considered problem is firstly formulated in terms of robust optimization then two different algorithms to compute the optimum of the formulated problem by means of linear programming techniques are presented the effectiveness of the proposed approach is illustrated by means of a simulation example

heavy traffic limits for the g h2 n m queue we establish heavy traffic stochastic process limits for queue length waiting time and overflow stochastic processes in a class ofg gi n m queueing models withn servers andm extra waiting spaces we let the arrival process be general only requiring that it satisfy a functional central limit theorem to capture the impact of the service time distribution beyond its mean within a markovian framework we consider a special class of service time distributions denoted byh2 which are mixtures of an exponential distribution with probabilityp and a unit point mass at 0 with probability 1 p these service time distributions exhibit relatively high variability having squared coefficients of variation greater than or equal to one as in halfin and whitt 1981 heavy traffic limits for queues with many exponential servers oper res 29 567 588 puhalskii and reiman 2000 the multiclassgi ph n queue in the halfin whitt regime adv appl probab 32 564 595 and garnett mandelbaum and reiman 2002 designing a call center with impatient customers manufacturing service oper management 4 208 227 we consider a sequence of queueing models indexed by the number of servers n and letn tend to infinity along with the traffic intensities nso that v n 1 n for 8 8 to treat finite waiting rooms we letm n v n for 0 8 with the specialh2 service time distribution the limit processes are one dimensional markov processes behaving like diffusion processes with different drift and diffusion functions in two different regions above and below zero we also establish a limit for theg m n m m model having exponential customer abandonments

parameter bounds for armax models from records with bounded errors in variables the errors in variables problem is that of parameter estimation where all observed variables are corrupted by noise so far the computation of parameter bounds for dynamical models has been performed using equation error and output error approaches in this paper parameter bounds for autoregres sive moving average exogenous armax models are derived on the assumption that both input and output are affected by bounded noise parameter bounding by both the bounded equation error and the bounded errors in variables approach is outlined together with the approximation of the feasible parameter region by an orthotope outer bounding set and is tested on simulated data

identification of parameter bounds for armax models from records with bounded noise the computation of bounds on the parameters rather than point estimates and covarianccs is considered for autoregressive moving average exogenous armax models the bounds derive from observations affected by bounded noise the influence of noise structure on the performance of the best known parameter bounding algorithm is examined a problem analogous to the bias in least squares estimates when regressors and noise are mutually correlated is also found to arise in parameter bounding an explanation is offered and the possibility of armax parameter bounds being non convex is pointed out

quaternions and matrices of quaternions abstract we give a brief survey on quaternions and matrices of quaternions present new proofs for certain known results and discuss the quaternionic analogues of complex matrices the methods of converting a quaternion matrix to a pair of complex matrices and homotopy theory are emphasized

fluid limits for processor sharing queues with impatience we investigate a processor sharing queue with renewal arrivals and generally distributed service times impatient jobs may abandon the queue or renege before completing service the random time representing a job s patience has a general distribution and may be dependent on its initial service time requirement a scaling procedure that gives rise to a fluid model with nontrivial yet tractable steady state behavior is presented this fluid model captures many essential features of the underlying stochastic model and it is used to analyze the impact of impatience in processor sharing queues

set membership error in variables identification through convex relaxation techniques in this technical note the set membership error in variables identification problem is considered that is the identification of linear dynamic systems when both output and input measurements are corrupted by bounded noise a new approach for the computation of parameter uncertainty intervals is presented first the identification problem is formulated in terms of nonconvex optimization then relaxation techniques based on linear matrix inequalities are employed to evaluate parameter bounds by means of convex optimization the inherent structured sparsity of the original identification problems is exploited to reduce the computational complexity of the relaxed problems finally convergence properties and complexity of the proposed procedure are discussed advantages of the presented technique with respect to previously published results are discussed and shown by means of two simulated examples

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

on left eigenvalues of a quaternionic matrix abstract in this paper we introduce the concept of left and right eigenvalues for a quaternionic matrix and investigate their properties quantities and relationship

estimation theory for nonlinear models and set membership uncertainty this chapter studies the problem of estimating a given function of a vector of unknowns called the problem element by using measurements depending non linearly on the problem element and affected by unknown but bounded noise assuming that both the solution sought and the measurements depend polynomially on the unknown problem element a method is given to compute the axis aligned box of minimal volume containing the feasible solution set i e the set of all unknowns consistent with the actual measurements and the given bound on the noise the center of this box is a point estimate of the solution which enjoys useful optimality properties the sides of the box represent the intervals of possible variation of the estimates important problems like parameter estimation of exponential models time series prediction with arma models and parameter estimates of discrete time state space models can be formalized and solved by using the developed theory

littlewood s algorithm and quaternion matrices abstract a strengthened form of schur s triangularization theorem is given for quaternion matrices with real spectrum for complex matrices it was given by littlewood it is used to classify projectors a 2 a and self annihilating operators a 2 0 on a quaternion unitary space and examples of unitarily wild systems of operators on such a space are presented littlewood s algorithm for reducing a complex matrix to a canonical form under unitary similarity is extended to quaternion matrices whose eigenvalues have geometric multiplicity 1

dynamics of new product introduction in closed rental systems we study a rental system where a fixed number of heterogeneous users rent one product at a time from a collection of reusable products the online dvd rental firm netflix provides the motivation we assume that rental durations of each user are independent and identically distributed with finite mean we study transient behavior in this system following the introduction of a new product that is desired by all the users we represent the usage process for this new product in terms of an empirical distribution this allows us to characterize the asymptotic behavior of the usage process as the number of users increases without bound via appropriate versions of glivenko cantelli and donsker s theorems analyzing the usage process we demonstrate that an increase in the variability of the rental duration distribution can actually help the firm by allowing it to set lower capacity levels to provide a desired quality of service further we show that the firm is better off not imposing any deadlines for the return of the product

similarity reduction of matrix over a quaternion division ring abstract let f be a field of characteristic 2 h f a b f the quaternion division ring over f this paper discusses the similarity of polynomials over hf by using polynomials over hf this paper proves that every square matrix over hf is similar to a uniquely generalized jordan canonical form and a quaternion matrix a is similar to a matrix over f if and only if a is similar to a via a hermite similarity transformation or if and only if a is product of two quaternion hermite matrices

sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity unconstrained and inequality constrained sparse polynomial optimization problems pops are considered a correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a pop based on this graph sets of the supports for sums of squares sos polynomials that lead to efficient sos and semidefinite program sdp relaxations are obtained numerical results from various test problems are included to show the improved performance of the sos and sdp relaxations

structure identification of parameter bounding models by use of noise structure bounds the selection of a model for a given set of records is discussed for the case when the output error is required to be always within some specified bounds a new selection criterion is proposed the model should be the simplest giving sufficiently unstructured output errors the lack of structure is specified by requiring the candidate model to be capable of giving sufficiently small sample autocorrelations of the model output errors over a range of lags computational algorithms to apply the criterion are presented and illustrated by tests on ar processes

feasible parameter set for linear models with bounded errors in all variables abstract when all observed variables of a model are affected by noise parameter estimation is known as the errors in variables problem while parameter bounding methods and algorithms have been extensively developed in the case of exactly known regressor variables little attention has been paid to the bounded errors in variables problem in this paper a formal proof of a previous result on the description of the feasible parameter region for linear models in the presence of bounded errors in all variables is given topological features of the feasible parameter region such as convexity and connectedness are also discussed

bounded error identification for closed loop systems this paper presents a scheme for the identification of a system which operates in closed loop and in the presence of bounded output disturbances two algorithms are proposed to solve this identification problem the first algorithm is an optimal bounding ellipsoid obe type algorithm this first algorithm is analyzed and sufficient conditions for stability and convergence are established relaxation of these conditions leads to a second identification algorithm the implementation of that second algorithm is realized in an iterative scheme a numerical example is provided to show the efficiency of the scheme

set membership estimation of fiber laser physical parameters from input output power measurements the problem of fiber laser parameters estimation from input output transient power measurements is considered in the paper the proposed procedure is based on a suitable laser model obtained from the rate and propagation equations by proper manipulation of the simplified laser equations the estimation problem is formulated in the bounded error estimation framework where a priori information on physical parameters can be properly taken into account first the so called feasible parameter set is defined then the problem of optimally picking a single model from the set is formulated in terms of central conditional estimation and solved by exploiting recent results by the authors properties of the proposed solution are discussed the proposed algorithm is tested on a set of simulated data obtained from a commercial laser simulator