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

existence of conjugate points for second and fourth order differential equations the paper presents sufficient conditions on the coefficients of second and fourth order differential equations to ensure that there exists at least one pair of conjugate points on an interval a b a b oscillation criteria related to the equation p x y q x y 0 0 x q x

robust state estimation for singularly perturbed systems abstractthis paper deals with the design of interval observers for singularly perturbed linear systems the full order system is first decoupled into slow and fast subsystems then using the cooperativity theory an interval observer is designed for the slow and fast subsystems assuming that the measurement noise and the disturbances are bounded and the singular perturbed parameter is uncertain this decoupling leads to two observers that estimate the lower and upper bounds for the feasible state domain a numerical example shows the efficiency of the proposed technique

an accurate scheme for mixed mode fracture analysis of functionally graded materials using the interaction integral and micromechanics models summary the interaction integral is a conservation integral that relies on two admissible mechanical states for evaluating mixed mode stress intensity factors sifs the present paper extends this integral to functionally graded materials in which the material properties are determined by means of either continuum functions e g exponentially graded materials or micromechanics models e g self consistent mori tanaka or three phase model in the latter case there is no closed form expression for the material property variation and thus several quantities such as the explicit derivative of the strain energy density need to be evaluated numerically this leads to several implications in the numerical implementation the sifs are determined using conservation integrals involving known auxiliary solutions the choice of such auxil

brief paper interval observer design for consistency checks of nonlinear continuous time systems this paper deals with fault detection for nonlinear continuous time systems a procedure based on interval analysis is proposed to build a guaranteed qlpv quasi linear parameter varying approximation of the nonlinear model the interval qlpv approximation makes it possible to derive two point observers which estimate respectively the lower and the upper bound of the state vector using cooperativity theory a set guaranteed to contain the actual value of the residual is then designed the modelling uncertainties and measurement errors are taken into account at the design stage the proposed methodology is illustrated through numerical simulations

end point behaviour of solutions to hypersingular integral equations we consider one dimensional hypersingular integral equations over finite intervals the integral must be interpreted as a finite part integral such equations arise naturally in various physical situations involving thin rigid bodies or cracks examples are given a method is developed for determining the behaviour of the solution to a hypersingular integral equation near the end points of the interval of integration the method uses the mellin transform several examples are worked out in detail

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

an interfacially cracked orthotropic rectangular bi material subjected to antiplane shear loading abstract an interfacially cracked rectangular bi material composed of two dissimilar orthotropic elastic media is analyzed the crack is parallel to or perpendicular to the boundary surfaces under the action of antiplane shear loading the problem is reduced to a singular integral equation of the first kind by employing the fourier series method for four types of boundary conditions including the f f type c c type f c type and c f type numerical results of stress intensity factors are obtained by using the lobatto chebyshev method the results show that in addition to the geometric factor stress intensity factors are dependent on the material properties the influences of the material properties on stress intensity factors are examined and compared under different boundary conditions

distributed set membership observers for interconnected multi rate systems abstract this paper presents a new distributed observer for interconnected multi rate systems the developed observer belongs to the family of set membership estimators and the use of zonotopes is proposed to mathematically describe the sets a choice motivated by the available mathematical background in operations such as intersections combinations and linear manipulations the main features of the proposed distributed observer are a the actual state of any subsystem always belongs to the computed sets b the volume of these sets is minimized in real time c under equivalent assumptions the performance of the observer approaches to that of an analogous distributed kalman filter

strain gradient elasticity for antiplane shear cracks a hypersingular integrodifferential equation approach we consider casal s strain gradient elasticity with two material lengths ell ell associated with volumetric and surface energies respectively for a mode iii finite crack we formulate a hypersingular integrodifferential equation for the crack slope supplemented with the natural crack tip conditions the full field solution is then expressed in terms of the crack profile and the green function which is obtained explicitly for ell 0 we obtain a closed form solution for the crack profile the case of small ell is shown to be a regular perturbation the question of convergence as ell ell to 0 is studied in detail both analytically and numerically

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

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

validated solutions of initial value problems for ordinary differential equations compared to standard numerical methods for initial value problems ivps for ordinary differential equations odes validated methods for ivps for odes have two important advantages if they return a solution to a problem then 1 the problem is guaranteed to have a unique solution and 2 an enclosure of the true solution is produced the authors survey taylor series methods for validated solutions of ivps for odes describe several such methods in a common framework and identify areas for future research

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

brief paper rigorous parameter reconstruction for differential equations with noisy data we present a method that given a data set a finitely parametrized system of ordinary differential equations odes and a search space of parameters discards portions of the search space that are inconsistent with the model ode and data the method is completely rigorous as it is based on validated integration of the vector field as a consequence no consistent parameters can be lost during the pruning phase for data sets with moderate levels of noise this yields a good reconstruction of the underlying parameters several examples are included to illustrate the merits of the method

the crack problem for nonhomogeneous materials under antiplane shear loading a displacement based formulation abstract this paper presents a displacement based integral equation formulation for the mode iii crack problem in a nonhomogeneous medium with a continuously differentiable shear modulus which is assumed to be an exponential function i e g x g0exp x this formulation leads naturally to a hypersingular integral equation the problem is solved for a finite crack and results are given for crack profiles and stress intensity factors the results are affected by the parameter describing the material nonhomogeneity this study is motivated by crack problems in strain gradient elasticity theories where higher order singular integral equations naturally arise even in the slope based formulation

biem analysis of dynamically loaded anti plane cracks in graded piezoelectric finite solids anti plane cracks in finite functionally graded piezoelectric solids under time harmonic loading are studied via a non hypersingular traction based boundary integral equation method biem the formulation allows for a quadratic variation of the material properties in two directions the boundary integral equation bie system is treated by using the frequency dependent fundamental solution based on radon transforms its numerical solution provides the displacements and tractions on the external boundary as well as the crack opening displacements from which the mechanical stress intensity factor sif and the electrical displacement intensity factor edif are determined several examples for single and multiple straight and curved cracks demonstrate the applicability of the method and show the influence of the different system parameters

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 nonlinear bounded error state estimation of continuous time systems this paper presents a first study on the application of interval analysis and consistency techniques to state estimation of continuous time systems described by nonlinear ordinary differential equations the approach is presented in a bounded error context and the resulting methodology is illustrated by an example

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

interval bounds on the solutions of semi explicit index one daes part 1 analysis this article presents two methods for computing interval bounds on the solutions of nonlinear semi explicit index one differential algebraic equations daes part 1 presents theoretical developments while part 2 discusses implementation and numerical examples the primary theoretical contributions are 1 an interval inclusion test for existence and uniqueness of a solution and 2 sufficient conditions in terms of differential inequalities for two functions to describe componentwise upper and lower bounds on this solution point wise in the independent variable the first proposed method applies these results sequentially in a two phase algorithm analogous to validated integration methods for ordinary differential equations the second method unifies these steps to characterize bounds as the solutions of an auxiliary system of daes efficient implementations of both are described using interval computations and demonstrated on numerical examples

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

zonotopes and kalman observers state bounding observation based on zonotopes is the subject of this paper dealing with zonotopes is motivated by set operations resulting in simple matrix calculations with regard to the often huge number of facets and vertices of the equivalent polytopes discrete time ltv lpv systems with state and measurement uncertainties are considered based on a new zonotope size criterion called f w radius and by merging optimal and robust observer gain designs a zonotopic kalman filter zkf is proposed with a proof of robust convergence the notion of covariation is introduced and results in an explicit bridge between the zonotopic set membership and the stochastic paradigms for kalman filtering no intersection is used and the influence of the reduction operator limiting to a tunable maximum the size of the matrices involved in the zonotopic set computations is fully taken into account in the lmi based robust stability analysis a numerical example illustrates the effectiveness of the proposed zkf

weak solutions for nonlinear antiplane problems leading to hemivariational inequalities abstract we analyze the antiplane shear deformation of an elastic cylinder in frictional contact with a rigid foundation for static processes under the small deformation hypothesis based on the knaster kuratowski mazurkiewicz technique in the theory of the hemivariational inequalities we prove that the model has at least one weak solution moreover we present several examples of constitutive laws and friction laws for which our theoretical results are valid finally we comment on the conditions which guarantee the uniqueness of the solution

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

interval state estimation for a class of nonlinear systems the goal of this technical note is to design interval observers for a class of nonlinear continuous time systems the first part of this work shows that it is usually possible to design an interval observer for linear systems by means of linear time invariant changes of coordinates even if the system is not cooperative this result is extended to a class of nonlinear systems using partial exact linearisations the proposed observers guarantee to enclose the set of system states that is consistent with the model the disturbances and the measurement noise moreover it is only assumed that the measurement noise and the disturbances are bounded without any additional information such as stationarity uncorrelation or type of distribution the proposed observer is illustrated through numerical simulations

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

bounds on the reachable sets of nonlinear control systems the computation of rigorous enclosures of the reachable sets of nonlinear control systems is considered with a focus on applications for which speed is crucial low computational costs make interval methods based on differential inequalities an attractive option unfortunately such methods are prone to large overestimation and often produce useless results in practice from physical considerations however it is common that some crude set is known to contain the reachable set we establish a general bounding result based on differential inequalities which enables the effective use of such sets during the bounding procedure in the case where this set is a convex polyhedron an efficient implementation using interval computations is developed using readily available physical information from practical examples this method is shown to provide significant advantages over alternative methods in terms of both efficiency and accuracy

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

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

loop detection of mobile robots using interval analysis this paper proposes an original set membership approach for loop detection of mobile robots in the situation where proprioceptive sensors only are available to detect loops the new concepts of thet plane which is a two dimensional space with time coordinates are introduced intervals of functions or tubes are then used to represent uncertain trajectories and tests are provided in order to eliminate parts of the t plane that do not correspond to any loop an experiment with an actual underwater robot is proposed in order to illustrate the principle and the efficiency of the approach

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

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

brief paper robust set membership state estimation application to underwater robotics this paper proposes a new observer for estimating the state vector of a nonlinear system this observer which is robust with respect to outliers assumes that the measurement errors as well as the number of outliers that could occur within a given time window are bounded the principle of the approach is to use interval analysis to deal properly with the nonlinearities involved in the system without any linearization or approximation and to propagate through the time in a forward and backward manner the assumptions made about outliers a test case related to the localization and control of an underwater robot is also proposed to illustrate the efficiency of the approach

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

the effect of variable change on oscillation and disconjugacy criteria with applications to spectral theory and asymptotic theory this study of the effect of variable change on differential operators was motivated by recent papers of tipler 131 and hinton and lewis 37 which appeared in the same issue of the journal of d erential equations tipler gave a geometric proof 13 p 1671 of the following result of hawking and penrose see 29 301 let f t be a real valued continuous nonnegative function on w co such that f b 0 for some point b then there exists a pair of conjugate points on co co for the differential equation