set inversion via interval analysis for nonlinear bounded error estimation abstract in the context of bounded error estimation one is interested in characterizing the set of all the values of the parameters to be estimated that are consistent with the data in the sense that the errors between the data and model outputs fall within prior bounds while the problem can be considered as solved when the model output is linear in the parameters the situation is far less advanced in the general nonlinear case in this paper the problem of nonlinear bounded error estimation is viewed as one of set inversion an original algorithm is proposed based upon interval analysis that makes it possible to characterize the feasible set for the parameters by enclosing it between internal and external unions of boxes the convergence of the algorithm is proved and the algorithm is applied to two test cases the results obtained are compared with those provided by signomial analysis

closed loop identification for the best asymptotic performance of adaptive robust control discrete linear systems subjected to both additive and parametric deterministic disturbances are considered the certainty equivalence principle is used for the design of adaptive control two different estimation algorithms are proposed to meet different control objectives in the regulation problem with the worst case limsup performance index

critical properties of the classical xy and classical heisenberg models a renormalization group study abstract by using two approaches of renormalization group rg mean field rg mfrg and effective field rg efrg we study the critical properties of the simple cubic lattice classical xy and classical heisenberg models the methods are illustrated by employing its simplest approximation version in which small clusters with one n 1 and two n 2 spins are used the thermal and magnetic critical exponents y t and y h and the critical parameter k c are numerically obtained and are compared with more accurate methods monte carlo series expansion and e expansion the results presented in this work are in excellent agreement with these sophisticated methods we have also shown that the exponent y h does not depend on the symmetry n of the hamiltonian hence the criteria of universality for this exponent is only a function of the dimension d

renormalisation group recursions by mean field approximations within a renormalisation group strategy different rescaled mean field approximations for the magnetisation are combined good qualitative estimates are obtained for the critical properties of classical and quantum spin systems a particularly simple realisation of the method yields recursions for arbitrary dimensionality d the critical couplings are accurate for all d and have the correct asymptotic behaviour for d approaching infinity

guaranteed robust nonlinear minimax estimation minimax parameter estimation aims at characterizing the set of all values of the parameter vector that minimize the largest absolute deviation between the experimental data and the corresponding model outputs it is well known however to be extremely sensitive to outliers in the data resulting e g of sensor failures in this paper a new method is proposed to robustify minimax estimation by allowing a prespecified number of absolute deviations to become arbitrarily large without modifying the estimates by combining tools of interval analysis and constraint propagation it becomes possible to compute the corresponding minimax estimates in an approximate but guaranteed way even when the model output is nonlinear in its parameters the method is illustrated on a problem where the parameters are not globally identifiable which demonstrates its ability to deal with the case where the minimax solution is not unique

reduced solutions of douglas equations and angles between subspaces in this paper we study some particular solutions of douglas type equations by means of generalized inverses and angles we apply this result to characterize positive solutions and some special projections which are symmetrizable for a semidefinited positive bounded linear operator

new extremal characterizations of generalized inverses of linear operators abstract in 1956 r penrose studied best approximate solutions of the matrix equation ax b he proved that a b where a is the moore penrose inverse is the unique matrix of minimal frobenius norm among all matrices which minimize the frobenius norm of ax b in particular a is the unique best approximate solution of ax i the vector version of penrose s result that is the fact that the vector a b is the best approximate solution in the euclidean norm of the vector equation ax b has long been generalized to infinite dimensional hilbert spaces in this paper an infinite dimensional version of penrose s full result is given we show that a straightforward generalization is not possible and provide new extremal characterizations in terms of the hermitian order of a and of the classes of generalized inverses associated with minimal norm solutions of consistent operator equations or with least squares solutions for a certain class of operators we can phrase our characterizations in terms of a whole class of norms including the hilbert schmidt and the trace norms thus providing new extremal characterizations even in the matrix case we treat both operators with closed range and with not necessarily closed range finally we characterize a as the unique inner inverse of minimal hilbert schmidt norm if a 2

parameter estimation algorithms for a set membership description of uncertainty abstract the problem of parameter estimation when the data are corrupted by unknown but bounded errors mainly consists in characterizing the minimal parameter set consistent with the measurements the model and the error description two different approaches are reviewed and compared in this paper the first one based on a recursive parameter ellipsoidal bounding algorithm the other on an orthotopic bounding set obtained by solving linear programming problems the original ellipsoidal algorithm modified for higher efficiency has been compared with the linear programming method although the orthotopic description may result in a more accurately defined parameter region it is time consuming when a large number of measurements is present conversely the ellipsodal bounding algorithm often provides a loose approximation to the parameter region but allows a fast data preprocessing producing a smaller number of constraints that can be subsequently fed to the linear programming algorithm a combined use of the two procedures is outlined and tested on simulated and real data the latter being relative to the tuning of an a d converter resulting in a remarkable reduction in the computing time

parameter range reduction from partial data in systems of differential algebraic equations abstract this paper presents an algorithm for parameter range reduction from partial data in systems of differential algebraic equations interval arithmetic and linear multistep discretizations are used to prune regions of parameter space that are inconsistent with available time series data methods of splitting parameter space to ensure further progress are discussed the algorithm outputs a reduced range and an approximation for each parameter value these outputs are shown to have beneficial effect as input to traditional parameter estimation schemes the range reduction algorithm is tested on three systems with partial data sets

brief guaranteed parameter bounding for nonlinear models with uncertain experimental factors in the context of bounded error estimation it is customary to assume that the error between the model output and output data should lie between some known prior bounds in this paper it is also assumed that the factors characterizing the experiments that have been carried out e g measurement times are uncertain with known prior bounds an algorithm based on interval analysis is used to characterize the set of all values of the parameter vector to be estimated that are consistent with these hypotheses this is performed in a guaranteed way even when the model output is a nonlinear function of the parameters and factors characterizing the experiments

constraint reasoning based on interval arithmetic the tolerance propagation approach abstract interval constraint satisfaction interval labeling systems have traditionally been based on local waltz filtering techniques that cannot in general determine global solutions in contrast this paper documents a related technique tolerance propagation tp that generalizes the idea of numerical exact value propagation into interval propagation in tp consistency techniques based on the topology of the constraint net can be combined with techniques of interval arithmetic in a new fruitful way in particular by tp it is possible to determine global solutions for interval constraint satisfaction problems with arbitrary accuracy and without losing all attractions of simple local computations

zonotopic guaranteed state estimation for uncertain systems this paper presents a new approach for guaranteed state estimation based on zonotopes for linear discrete time multivariable systems with interval multiplicative uncertainties in the presence of bounded state perturbations and noises at each sample time the presented approach computes a zonotope which contains the real system state a p radius based criterion is minimized in order to decrease the size of the zonotope at each sample time and to obtain an increasingly accurate state estimation the proposed approach allows one to efficiently handle the trade off between the complexity of the computation and the accuracy of the estimation an illustrative example is analyzed in order to highlight the advantages of the proposed state estimation technique

minimizing euclidian state estimation error for linear uncertain dynamic systems based on multisensor and multi algorithm fusion in this paper a multisensor linear dynamic system with model uncertainty and bounded noises is considered based on previously developed set valued estimation methods in terms of convex optimization we propose several efficient algorithms of centralized sensor fusion distributed sensor fusion and multi algorithm fusion to minimize the euclidian estimation error of the state vector obviously an ellipsoid box with a larger size cannot be in general guaranteed to contain another ellipsoid box with a smaller size since centers and shapes of the two ellipsoids boxes may be different from each other this fact and the complementary advantages of multiple sensors and multiple algorithms motivate us to construct multiple estimation ellipsoids boxes squashed along each entry of the state vector as much as possible respectively by using the technique of multiple differently weighted objectives then intersection fusion of these estimation ellipsoids boxes yields a final euclidian error minimized state estimate numerical examples show that the new method with multi algorithm at both the sensors and the fusion center can significantly reduce the euclidian estimation error of the state

constrained zonotopes this article introduces a new class of sets called constrained zonotopes that can be used to enclose sets of interest for estimation and control the numerical representation of these sets is sufficient to describe arbitrary convex polytopes when the complexity of the representation is not limited at the same time this representation permits the computation of exact projections intersections and minkowski sums using very simple identities efficient and accurate methods for computing an enclosure of one constrained zonotope by another of lower complexity are provided the advantages and disadvantages of these sets are discussed in comparison to ellipsoids parallelotopes zonotopes and convex polytopes in halfspace and vertex representations moreover extensive numerical comparisons demonstrate significant advantages over other classes of sets in the context of set based state estimation and fault detection

a dual method for parameter identification under deterministic uncertainty proposes a numerical method for reduction of a priori bounds on the values of uncertain plant parameters the method successively eliminates parts of the parameter domain which are inconsistent with the plant measurement the size of the eliminated parts is determined by a measure of inconsistency evaluated with the help of the support functions of the set of feasible states and of the set of possible measurements we specify our approach on an identification problem for a linear control system with bounds on the magnitude and or on the energy for the disturbances in the dynamics and in the measurement

set theoretic estimation based on a priori knowledge of the noise distribution a new algorithm for estimation of a linear in parameters model is developed and tested by simulation the method is based on the assumption of independent identically distributed noise samples with a triangular density function such a noise model well approximates the symmetrically distributed sources of noise frequently encountered in practice and the inclusion of a distribution assumption allows the computation of a pseudo mean estimate to complement the set solution the proposed algorithm recursively incorporates incoming observations with decreasing computational complexity as the number of updates increases simulations demonstrate that the algorithm has very favorable convergence rates and estimation accuracy and is very robust to deviations from the assumed noise properties comparisons with other set theoretic algorithms and with conventional rls are given

factorizations of linear relations abstract given two linear relations a and b we characterize the existence of a linear relation operator c such that a b c respectively a c b these factorizations extend and improve well known results by douglas and sebestyen

guaranteed characterization of exact non asymptotic confidence regions as defined by lscr and sps in parameter estimation it is often desirable to supplement the estimates with an assessment of their quality a new family of methods proposed by campi et al for this purpose is particularly attractive as it makes it possible to obtain exact non asymptotic confidence regions under mild assumptions on the noise distribution a bottleneck of this approach however is the numerical characterization of these confidence regions so far it has been carried out by gridding which provides no guarantee as to its results and is only applicable to low dimensional spaces this paper shows how interval analysis can contribute to removing this bottleneck

sequential approximation of feasible parameter sets for identification with set membership uncertainty in this paper the problem of approximating the feasible parameter set for identification of a system in a set membership setting is considered the system model is linear in the unknown parameters a recursive procedure providing an approximation of the parameter set of interest through parallelotopes is presented and an efficient algorithm is proposed its computational complexity is similar to that of the commonly used ellipsoidal approximation schemes numerical results are also reported on some simulation experiments conducted to assess the performance of the proposed algorithm

set membership state and parameter estimation for systems described by nonlinear differential equations this paper investigates the use of guaranteed methods to perform state and parameter estimation for nonlinear continuous time systems in a bounded error context a state estimator based on a prediction correction approach is given where the prediction step consists in a validated integration of an initial value problem for an ordinary differential equation ivp for ode using interval analysis and high order taylor models while the correction step uses a set inversion technique the state estimator is extended to solve the parameter estimation problem an illustrative example is presented for each part

weighted least squares solutions of the equation axb c 0 abstract let h be a hilbert space l h the algebra of bounded linear operators on h and w l h a positive operator such that w 1 2 is in the p schatten class for some 1 p given a b l h with closed range and c l h we study the following weighted approximation problem analyze the existence of 0 1 m i n x l h a x b c p w where x p w w 1 2 x p we also study the related operator approximation problem analyze the existence of 0 2 m i n x l h a x b c w a x b c where the order is the one induced in l h by the cone of positive operators in this paper we prove that the existence of the minimum of 0 2 is equivalent to the existence of a solution of the normal equation a w a x b c 0 we also give sufficient conditions for the existence of the minimum of 0 1 and we characterize the operators where the minimum is attained

structure selection for bounded parameter models consistency conditions and selection criterion strong consistency conditions for structure selection in bounded parameter models are studied a certain robust selection criterion based on the volume of the exact parameter bounding polytope is proposed for linear regression models the effectiveness of the polytope volume criterion is demonstrated on a model nonlinear in its variables but linear in its parameters its strong consistency is proved for a large class of noise distributions the usual assumptions on the noise namely independence constant variance or martingale difference properties need not be made but asymptotic independence is assumed

on the value of information in system identification bounded noise case assuming instantaneous bounds on the noise system parameter identification is formulated as membership set estimation problem sequential algorithms are constructed to estimate the membership sets of the parameters which are consistent with the measurements and the noise constraints the important new feature of the proposed algorithms is their ability to ignore redundant data the efficient data extraction property of the new algorithms is achieved with small computational effort and with improved performance when compared to the least square algorithm the convergence properties and the notion of identifiability in the set theoretic context are also studied

global numerical approach to nonlinear discrete time control interval analysis is used to characterize the set of all input sequences with a given length that drive a nonlinear discrete time state space system from a given initial state to a given set of terminal states no requirement other than computability i e ability to be evaluated by a finite algorithm is put on the nature of the state equations the method is based on an algorithm for set inversion and approximates the solution set in a guaranteed way

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

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

numerical solution for bounding feasible point sets finding feasible points is important in optimization there are currently two major classes of algorithms to deal with the problem of feasible points the first class of algorithms of local nature is to find an approximate feasible point given a neighbourhood of an approximate feasible point the second class of algorithms is to prove whether a feasible point exists inside this neighbourhood to the best of our knowledge no methods have been practically implemented to efficiently find the smallest boxes for bounding the feasible points defined by a system of nonlinear and nonconvex inequalities unless the feasible set is convex in this paper we will present a numerical method to find the smallest boxes for bounding the feasible point sets defined by a nonlinear and nonconvex inequality and or a system of nonlinear and nonconvex inequalities two examples have been synthetically constructed and used to show that the proposed numerical method can indeed correctly find all the smallest bounding boxes at any given accuracy efficiently a brief comparison with relevant techniques will be discussed our method may also be thought of as the first solid theoretical basis for multisection and multisplitting in global optimization when compared with those empirical ones in the literature

operator theoretic and computational approaches to ill posed problems with applications to antenna theory a general framework for regularization and approximation methods for ill posed problems is developed three levels in the resolution processes are distinguished and emphasized philosophy of resolution regularization approximation schema and regularization algorithms dilemmas and methodologies of resolution of ill posed problems and their numerical implementations are examined in this framework with particular reference to the problem of finding numerically minimum weighted norm least squares solutions of first kind integral equations and more generally of linear operator equations with nonclosed range a common problem in all these methods is delineated each method reduces the problem of resolution to a nonstandard minimization problem involving an unknown critical parameter whose optimal value is crucial to the numerical realization and amenability of the method the nonstandardness results from the fact that one does not have explicitly or a priori the function to be minimized it has to built up using additional information convergence rate estimates and robustness conditions etc several results are developed that complement recent advances in numerical analysis and regularization of inverse and ill posed identification and pattern synthesis problems an emphasis is placed on the role of constraints function space methods the role of generalized inverses and reproducing kernels in the regularization and stable computational resolution of these problems the results will be applied specifically to problems of antenna synthesis and identification however the thrust of the paper is devoted to the interdisciplinary character of operator theoretic and numerical methods for ill posed problems

guaranteed tuning with application to robust control and motion planning many design problems e g in control theory amount to tuning a parameter vector c so as to guarantee that specifications are met for all feasible values of some unknown perturbation vector p a new prototype algorithm for solving this guaranteed tuning problem is proposed and its convergence properties are established it applies when the design specifications translate into a finite number of possibly nonlinear inequalities three test cases taken from the field of control are considered namely the design of a pid controller robust to structured uncertainty the control of a nonlinear discrete time model with uncertain parameters and initial state and a problem of motion planning with obstacles to be avoided

equivalence of the o n vector ferromagnetic and antiferromagnetic models abstract the effective field renormalization group efrg approach is used to find the neel temperature t n of the o n vector model with antiferromagnetic af interaction the efrg method is illustrated by employing approximations in which clusters with one n 1 and two n 2 spins are used the critical temperature t n is obtained as a function of component n and coordination z numbers for all values of n and z we show that t n t c where t c is the curie temperature for the ferromagnetic f case as a comparison the results of the quantum heisenberg model n 3 with f and af interactions are also presented and we find that t n t c which is different from the classical result t c t n

mean field renormalisation group for the spin 1 2 anisotropic heisenberg model the mean field renormalisation group approach is applied to the spin 1 2 anisotropic heisenberg model the critical temperature and the critical exponent are obtained as a function of the anisotropy for the two dimensional model on square triangular and hexagonal lattices the three dimensional model on a simple cubic lattice is also analysed

brief paper an algorithm for bounded error identification of nonlinear systems based on dc functions this paper presents a guaranteed method for the parameter estimation of nonlinear models in a bounded error context this method is based on functions which consists of the difference of two convex functions called dc functions the method considers dc representations of the functional form of the dynamic system to obtain an outer bound of the set of parameters that are consistent with the measurements the system and the considered bounded error at each iteration the proposed algorithm solves several convex optimization problems to discard from the initial search region subregions that are proved not consistent this operation is repeated while the obtained solution is improved four examples are provided to clarify the proposed identification algorithm

technical communique interval constraint propagation with application to bounded error estimation for a large class of bounded error estimation problems the posterior feasible set s for the parameters can be defined by nonlinear inequalities the set inversion approach combines classical interval analysis with branch and bound algorithms to characterize s unfortunately as bisections have to be done in all directions of the parameter space this approach is limited to problems involving a small number of parameters techniques based on interval constraint propagation make it possible to drastically reduce the number of bisections in this paper these techniques are combined with set inversion to bracket s between inner and outer subpavings union of nonoverlapping boxes when only interested in the feasible intervals for the parameters the set inversion approach becomes inefficient and a new algorithm able to compute these intervals is given this algorithm uses a new interval based local research to compute the smallest box that contains s it is then compared with existing methods on an example taken from the literature

slide window bounded error time varying systems identification this technical note presents a new identification method for discrete linear systems with time varying parameters based on bounded error approach it is assumed a bounded additive error on measurements and a bound on parameter variation each time instant the identification method uses a slide window along historical data and an explicit expression to provide an outer bound of the set of model parameters that is consistent with measurements model structure and bounds on error and variations considered furthermore the center estimation of the obtained outer solution set is a suitable nominal estimation of the parameters this center is the optimal solution of a least squares problem that penalizes the prediction error of the model and the parameter variations the explicit expression that provides the outer bound of the parameters and the optimal property of the center estimation are the main contributions of the technical note in order to clarify the proposed method an application example and a comparison with a previous recursive bounded error method are included

adaptive robust control for siso systems abstract this paper addresses the problem of nonconservative adaptive robust regulating a siso system allowing for model perturbations in the coprime factor representation of the system as well as external additive bounded disturbance signals the perturbations and disturbances are assumed to belong to a particular parameterized class and are norm bounded the norm bounds are not available for control design but the class knowledge is available the performance measure is the worst possible steady state absolute value of the controlled output over the class of allowable disturbances and model perturbations an identification based adaptive control method is proposed where the identification criterion coincides with the control objective the proposed adaptive scheme achieves near optimal performance simulations comparing the new scheme with an adaptive controller based on classical recursive least squares identification are presented

guaranteed active failure detection and isolation for linear dynamical systems a method for active failure detection and isolation in linear systems subject to inequality bounded perturbations is presented this method under a detectability condition can guarantee detection and isolation of failures to make the failures detectable the injection of a test signal is proposed a method for testing whether the test signal is proposed a method for testing whether the test signal does make the failures detectable is presented under certain conditions a method fro designing test signals which achieve detectability is also given

set membership identification of nonlinear systems in the paper the problem of identifying nonlinear dynamic systems described in nonlinear regression form is considered using finite and noise corrupted measurements most methods in the literature are based on the estimation of a model within a finitely parametrized model class describing the functional form of involved nonlinearities a key problem in these methods is the proper choice of the model class typically realized by a search from the simplest to more complex ones linear bilinear polynomial neural networks etc in this paper an alternative approach based on a set membership framework is presented not requiring assumptions on the functional form of the regression function describing the relations between measured input and output but assuming only some information on its regularity given by bounds on its gradient in this way the problem of considering approximate functional forms is circumvented moreover noise is assumed to be bounded in contrast with statistical methods which rely on assumptions such as stationarity ergodicity uncorrelation type of distribution etc whose validity may be difficult to test reliably and is lost in presence of approximate modeling in this paper necessary and sufficient conditions are given for the validation of the considered assumptions an optimal interval estimate of the regression function is obtained providing its uncertainty range for any assigned regressor values the set estimate allows to derive an optimal identification algorithm giving estimates with minimal guaranteed l p error on the assigned domain of the regressors the properties of the optimal estimate are investigated and its worst case l p identification error is evaluated the presented approach is tested and compared with other nonlinear methods on the identification of a water heater a mechanical system with input saturation and a vehicle with controlled suspensions

state estimation for markovian jump linear systems with bounded disturbances in this paper we investigate the state estimation problem for a class of markovian jump linear systems mjlss in the presence of bounded polyhedral disturbances a set membership estimation algorithm is first proposed to find the smallest consistent set of all possible states which is shown to be expressed by a union of multiple polytopes the posterior probabilities of the system jumping modes are then estimated by introducing the lebesgue measure based on which the optimal point estimate is further provided moreover a similarity relationship for polytopes is defined and an approximate method is presented to calculate the minkowski sum of polytopes which can help reduce the computational complexity of the overall estimation algorithm

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

behavior critical for bond diluted n vector model in the effective field theory abstract in the present work the bond diluted n vector model on the simple cubic lattice is studied by applying the effective field theory for a cluster with two spins the critical temperature and percolation concentrations for xy and heisenberg models as well the phase diagram in the t p plane respectively are obtained and the results when compared with the standard mean field approximation are an improvement even for small size clusters

asymptotically convergent modified recursive least squares with data dependent updating and forgetting factor for systems with bounded noise continual updating of estimates required by most recursive estimation schemes often involves redundant usage of information and may result in system instabilities in the presence of bounded output disturbances an algorithm which eliminates these difficulties is investigated based on a set theoretic assumption the algorithm yields modified least squares estimates with a forgetting factor it updates the estimates selectively depending on whether the observed data contain sufficient information the information evaluation required at each step involves very simple computations in addition the parameter estimates are shown to converge asymptotically at an exponential rate to a region around the true parameter

thick set inversion abstract this paper deals with the set inversion problem x f 1 y in the case where the function f r n r m and the set y are both uncertain the uncertainty is treated under the form of intervals more precisely for all x f x is inside the box f x and the uncertain set y is bracketed between an inner set y and an outer set y the introduction of new tools such as thick intervals and thick boxes will allow us to propose an efficient algorithm that handles the uncertainty of sets in an elegant and efficient manner some elementary test cases that cannot be handled easily and properly by existing methods show the efficiency of the approach

weighted procrustes problems abstract let h be a hilbert space l h the algebra of bounded linear operators on h and w l h a positive operator such that w 1 2 is in the p schatten class for some 1 p given a l h with closed range and b l h we study the following weighted approximation problem analyze the existence of m i n x l h a x b p w where x p w w 1 2 x p in this paper we prove that the existence of this minimum is equivalent to a compatibility condition between r b and r a involving the weight w and we characterize the operators which minimize this problem as w inverses of a in r b

identification of nonlinear systems in the presence of unknown but bounded disturbances this paper deals with recursive parameter estimation of nonlinear systems in the presence of unknown but bounded disturbances the proposed technique is performed by the use of a modified extended least squares algorithm so that the estimated parameters are consistent with the measurements and the noise constraints while local asymptotic convergence under very general conditions is assured one of the main features is to show from the stability analysis connection between some arbitrary factors and basin of attraction high performances from the design and implementation point of view will be shown through a numerical example under severe conditions

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

estimation of model quality this paper provides an introduction to recent work on the problem of quantifying errors in the estimation of models for dynamic systems this is a very large field we therefore concentrate on approaches that have been motivated by the need for reliable models for control system design this will involve a discussion of efforts that go under the titles of estimation in th worst case estimation estimation in l1 and stochastic embedding of undermodelling a central theme of this survey is to examine these new methods with reference to the classic bias variance tradeoff in model structure selection

guaranteed nonlinear parameter estimation in knowledge based models knowledge based models are ubiquitous in pure and applied sciences they often involve unknown parameters to be estimated from experimental data this is usually much more difficult than for black box models only intended to mimic a given input output behavior the output of knowledge based models is almost always nonlinear in their parameters so that linear least squares cannot be used and analytical solutions for the model equations are seldom available moreover since the parameters have some physical meaning it is not enough to find some numerical values of these quantities that are such that the model fits the data reasonably well one would like for instance to make sure that the parameters to be estimated are identifiable if this is not the case all equivalent solutions should be provided the uncertainty in the parameters resulting from the measurement noise and approximate nature of the model should also be characterized this paper describes how guaranteed methods based on interval analysis may contribute to these tasks examples in linear and nonlinear compartmental modeling widely used in biology are provided

quantum enveloping algebras with von neumann regular cartan like generators and the pierce decomposition quantum bialgebras derivable from uq sl2 which contain idempotents and von neumann regular cartan like generators are introduced and investigated various types of antipodes invertible and von neumann regular on these bialgebras are constructed which leads to a hopf algebra structure and a von neumann hopf algebra structure respectively for them explicit forms of some particular r matrices also invertible and von neumann regular are presented and the latter respects the pierce decomposition

a state set estimation algorithm for linear systems in the presence of bounded disturbances this paper concerns the problem of state set estimation for discrete time linear systems the linear system is given in state space form and the system s uncertainty is described by bounded scalar variations we present a recursive algorithmic procedure of computing guaranteed estimated state sets in particular a polyhedron variation algorithm is presented the new algorithm is based on linear programming and sensitivity analysis techniques for improved computational efficiency and practicability

an easy design for interval observers this paper deals with the problem of interval estimation the proposed approach is based on the use of successive derivatives in continuous time or forward measurements in discrete time in an alg

prognosis of uncertain linear time invariant discrete systems using unknown input interval observer in this paper a model based prognosis method where the degradation cannot be directly measured but only detectable through the drift of a model parameter is considered this parameter drift is vie

exact recursive polyhedral description of the feasible parameter set for bounded error models a method is described which exactly characterizes the set of all the values of the parameter vector of a linear model that are consistent with bounded errors on the measurements it provides a parameterized expression of this set which can be used for robust control design or for optimizing any criterion over the set this approach is based on a new variant of the double description method for determining the edges of a polyhedral cone it can be used in real time and provides a suitable context for implementation on a computer whenever a new measurement modifies the set the characterization is updated the technique is illustrated with a simple example

constraint propagation with interval labels abstract constraint propagation is often used in ai systems to perform inference about quantities this paper studies one particular kind of constraint propagation where quantities are labelled with signs or with intervals and these labels are propagated through recorded constraints we review the uses of such inference schemes in ai systems of various kinds and evaluate their strengths and weaknesses in particular we determine the completeness and running time of constraint propagation for various kinds of labels and constraints

estimation theory and uncertainty intervals evaluation in presence of unknown but bounded errors linear families of models and estimators the problem of parameter estimation and of the evaluation of related uncertainty intervals is considered in the case that a probabilistic description of noise and errors is not available of suitable but only a bound on them is known in the present paper only linear parametrizations and estimators are considered very simple and computationally feasible algorithms are derived for evaluating two different types of uncertainty intervals estimates uncertainty intervals parameter uncertainty intervals the relationships between the eui s and the pui s are established and the solution to the problem of the minimum uncertainty intervals estimator is given the latter can be obtained by means of a simple linear programming algorithm

bounded error tracking of time varying parameters bounded error estimation aims at characterizing the set of all parameter vectors consistent with given data and prior bounds on acceptable values for the errors in this paper two recursive polyhedric description algorithms are presented for tracking time varying parameters of models with outputs linear in their parameters the performances of the two algorithms are illustrated on an example with both smooth and abrupt variations of parameters

guaranteed nonlinear estimation using constraint propagation on sets bounded error estimation is the estimation of the parameter or state vector of a model from experimental data under the assumption that some suitably de ned errors should belong to some prior feasible sets when the model outputs are linear in the vector to be estimated a number of methods are available to enclose all estimates that are consistent with the data into simple sets such as ellipsoids orthotopes or parallelotopes thereby providing guaranteed set estimates in the nonlinear case the situation is much less developed and there are very few methods that produce such guaranteed estimates in this paper the discrete time problem is cast into the more general framework of constraint satisfaction problems algorithms rather classical in the area of interval constraint propagation are extended by replacing intervals by more general subsets of real vector spaces this makes it possible to propose a new algorithm that contracts the feasible domains for each uncertain variable optimally i e no smaller domain could be obtained and e ciently the resulting methodology is illustrated on discrete time nonlinear state estimation the state at time k is estimated either from past measurement only or from all measurements assumed to be available from the start even in the causal case prior information on the future value of the state and output vectors due for instance to physical constraints is readily taken into account

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

minimum volume ellipsoids containing compact sets application to parameter bounding abstract the problem of finding the minimum volume ellipsoid containing a compact set is shown to be equivalent to the determination of an optimal distribution of weights over several equivalent conditions for optimality of this distribution are obtained and used to construct algorithms guaranteed to converge to the optimum these algorithms can be used to approximate the posterior feasible set for the parameters of a model in the bounded error context linear and nonlinear illustrative examples are treated

brief proving set inclusion via intervals application to parametric robust stability proving that an uncertain parametric model is stable amounts to prove the inclusion of two sets the set a of all feasible parameters and the set b of all parameters for which the model is stable in this paper a new algorithm able to decide whether or not a is included in b is presented the method is based on interval analysis which is a numerical tool able to deal with inequalities in a global and guaranteed way convergence properties of the algorithm are provided the algorithm is then applied to the robust stability of a discrete time model where the information on the parameters is given through bounded error data the behavior of the algorithm with respect to the number of parameters is illustrated on a continuous time model

a model for two dimensional orientational order abstract we use a generalization of kikuchi s cluster variational method cvm in the pair approximation to study anisotropic plane rotors on a square lattice as a simple model for two dimensional nematics the system was shown to exhibit nematic ordered states at low temperature disordering for all anisotropies through a continuous transition the orientational distribution functions order parameter internal energy and specific heat were calculated within the cvm and compared with the results of a recent computer simulation estimates for the non classical exponents were obtained using a mean field renormalization group analysis a spin wave argument was also used to investigate the role of long range fluctuations near the isotropic fixed point the accuracy of these different techniques is discussed within the context of our model