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In this paper, we are interested in identifying the parameters of a bioreactor in the case of a nitrification process. To represent the uncertainties that affect these parameters, we fo cus on the set approach based on interval arithmetic, in particular set inversion, to obtain guaranteed results. First, a method of studying observability and identifiability by an algebraic method is carried out. The LSCR (Leave out Sign dominant Correlat ion Regions) method used in this article for the identification of parameters is based on the construction of non asymptotic confidence regions for the parameters of the dynamic system This method, using the calculation of the correlation functions, makes it possible to construct regions containing the real value of the parameters to be identified with a guaranteed probability and with a minimum knowledge of noise. For guaranteed results, set inversion has been associated with this approach.

International audience; — Model-based positioning methods involve nonlinear equations as is the case when using satellite pseudoranges on global navigation satellite systems (GNSSs) and local measurements on road features. As these are nonlinear models, classical estimation methods cannot provide guaranteed position estimation and can converge to local optima, sometimes far away from the global optimum or the true value. Based on interval analysis, set inversion, and constraints propagation on real values provide a framework that guarantees to find the true position with a characterized confidence domain. This paper describes an error bounded set membership algorithm that computes the absolute position of a road vehicle by using raw GNNS pseudoranges, lane boundary measurements, and a 2D road network map as geometric constraints. The algorithm is based on set inversion using interval analysis, and bounds are set on the measurements by taking into account a chosen risk. The GNSS pseudoranges errors are modeled carefully, and road constraints are formalized to provide additional information in the data fusion process. The proposed algorithm, named lane boundary augmented set-membership GNSS positioning (LB-ASGP), provides a novel and inexpensive approach to improve position estimation performance for road vehicles guaranteeing the enclosure of the computed solution with high confidence. Results from simulations and field experiments show that the LB-ASGP significantly reduces GNSS errors in the direction perpendicular to the lane thanks to the lane boundary measurements. Index Terms— ITS, interval analysis, bounded-error, GNSS and sensor data fusion.

In recent years, many applications, as well as theoretical properties of interval analysis have been investigated. Without any claim for completeness, such applications and methodologies range from enclosing the effect of round-off errors in highly accurate numerical computations over simulating guaranteed enclosures of all reachable states of a dynamic system model with bounded uncertainty in parameters and initial conditions, to the solution of global optimization tasks. By exploiting the fundamental enclosure properties of interval analysis, this paper aims at computing invariant sets of nonlinear closed-loop control systems. For that purpose, Lyapunov-like functions and interval analysis are combined in a novel manner. To demonstrate the proposed techniques for enclosing invariant sets, the systems examined in this paper are controlled via sliding mode techniques with subsequently enclosing the invariant sets by an interval based set inversion technique. The applied methods for the control synthesis make use of a suitably chosen Gr&ouml, bner basis, which is employed to solve B&eacute, zout&rsquo, s identity. Illustrating simulation results conclude this paper to visualize the novel combination of sliding mode control with an interval based computation of invariant sets.

For automatic formation flight and applications such as automatic aerial refueling, relative positions between aircraft need to be accurately estimated. However, to ensure flight safety, guaranteed consistent uncertainty estimates are of equally crucial importance. This paper presents a novel Iterated Extended Set Membership Filter (IESMF) providing ellipsoidal regions guaranteed to contain another aircraft's position in a formation. Two easily implementable modifications to the Extended Set Membership Filter are shown to greatly reduce its conservatism. The filter runs in fixed time and handles asynchronous observations. As another novelty, we take into account parametric uncertainty of the observation equation. The proposed filter is applied to a realistic UAS localization benchmark simulation. We compare the new filter to a computationally very strong but much more expensive nonlinear set inversion algorithm and find that similar performances can be achieved. Also, simulation results illustrate the effective reduction of filter conservatism by a few iterative updates.

Reliable computation of confidence domains for point position, and inconsistency check of the observations is of great importance for any navigation system, especially for safety critical applications. In this work, non-stochastic error bound for reaming observation errors are introduced as intervals. Using concepts from interval mathematics four different deterministic bounding methods will be applied on GPS positioning, namely: Least Squares Adjustments based on Interval Analysis (LSA-IA), extension of LSA-IA by means of Zonotopes, Linear Programing (LP) applied to the navigation equation linearized by Taylor expansion, and finally solving the non-linear navigation equation by Set Inversion Via Interval Analysis (SIVIA). These methods provide different type of confidence regions and inconsistency measures. Each method will be investigated alone, then conducting a comparison study of the mathematical properties of these methods, and providing geometrical interpretation with respect to the navigation problem (e.g. form and orientation of the bounding area w.r.t. line-of-sight and number of satellites in view). Thanks to a reduced position scatter and the implicit inconsistence measure, the LP approach seems especially promising for further investigation towards an alternative integrity concept.

In this paper, we generalize the idea of star-shaped set inversion fractals using iterations known from fixed point theory. We also extend the iterations from real parameters to so-called q-system numbers and proposed the use of switching processes. All the proposed generalizations allowed us to obtain new and diverse fractal patterns that can be used, e.g. as textile and ceramics patterns. Moreover, we show that in the chaos game for iterated function systems-which is similar to the inversion fractals generation algorithm-the proposed generalizations do not give interesting results.

The minimal model of glucose-insulin dynamics is currently being used in several diabetes-related applications, such as investigating the glucose metabolism, and in the developments of model predictive controllers and fault detection techniques for automatic blood glucose control i.e., artificial pancreas. Different approaches have been proposed to identify this model, but none of them is capable of providing guaranteed robust enclosures for its parameters, something very desired in applications such as the artificial pancreas, where robustness is paramount. This paper presents a novel approach for guaranteed set-membership parameter estimation of the minimal model based on the well-renowned Set Inversion via Interval Analysis SIVIA algorithm. Because the computational complexity of this algorithm is the main barrier for its applicability, an efficient vectorial implementation of SIVIA was employed. Clinical data from a standard intravenous glucose tolerance test were used to prove the validity of the presented approach. Finally, Modal Interval Analysis was used to reduce the numerical overestimation due to the dependency problem of interval arithmetic and significantly speeding up the computations. Copyright © 2015 John Wiley & Sons, Ltd.

Set inversion is a classical problem in control theory that has many important applications in various fields of science and engineering. The state-of-the-art method for solving this problem, Set Inverter Via Interval Analysis (SIVIA), usually does not work well in high dimensions and often fails to recover sets with complicated structures. In this work, we propose a new approach to the problem of set inversion, which employs techniques from machine learning to resolve these issues. Our algorithm can handle problems in high dimensions and achieve the same level of accuracy with fewer data points compared to SIVIA. We illustrate the performance of our method in various simulation studies and apply it to investigate the dynamics of the 17th-century plague in Eyam village, England.

International audience; Cooperation between road vehicles through information exchange is a promising way to enhance their absolute and relative positions. This paper presents an approach for generating, sharing and applying Global Navigation Satellite System (GNSS) pseudorange corrections through a V2X communication network. Conventionally, differential corrections are generated by fixed base stations with known positions and sent to mobile users. Here, the proposed cooperative method has no central server and the estimation of the raw measurements errors is done in a fully distributed way. Using a model of the correlation of the pseudorange errors and through the knowledge of the local motions of the vehicles obtained by Dead Reckoning (DR) or tracking, a non linear observability shows that the estimation problem is solvable. A cooperative and fully distributed estimation method is then presented using Set Inversion and Constraint propagation techniques. Positions, pseudorange estimated errors and DR data are shared in the network of vehicles and confidence is handled by intervals, in a bounded error context. This allows computing highly reliable confidence domains with no direct range measurements, which is crucial for applications involving close proximity navigation. Indeed, the proposed data fusion framework does not require any linearization of the equations and is insensitive to the data incest problem since the same information can be exploited several times in the computation process without making the estimation over-converge. Results using real measurements are presented to illustrate the performance of the proposed cooperative method in comparison with standalone estimation. A classical sequential Bayesian method has also been implemented on the same data set and compared in terms of accuracy and confidence with a ground truth system.

International audience; Fully distributed localization methods with no central server are relevant for autonomous vehicles that need real-time cooperation. In this paper, mobile vehicles share estimates of GNSS pseudoranges common errors also known as biases. The biases that affect the pseudoranges are mainly due to signal propagation and inaccurate ephemeris data. By describing the measurements models as geometric constraints on intervals, cooperative localization turns into distributed set inversion problem. The solution of this problem is guaranteed to contain the true vehicles positions. We consider vehicles which cooperate and exchange information in order to improve the absolute and relative estimation by fusing pseudoranges corrections shared between them. Results using real measurements are presented to illustrate the performance of the proposed approach in comparison with a standalone method.

In the paper, an approach, based on bounded-error estimation via interval analysis, is presented to identify viscous damping coefficient and Coulomb friction coefficient of such-rod pumping system of directional well. The approach assumes the error of colleted output data of suck-rod pumping system is bounded. Hence, the problem of parameter estimation can be viewed as that of set inversion, which will be effectively solved via interval analysis. The main advantage of the proposed method in comparison with the traditional estimation methods is that, as it is global, it bypasses the problem of initialization. Moreover, the method avoids amount error due to transformation of measurement output data. An identification example is given to show the correctness and efficiency of the proposed method.

In this paper, we consider Filtering of discrete time linear models with bounded uncertainties on parameters and measurement noise modeled by gaussian distribution. This work is based on a previous work Chen et al. [1997] in which the classical Kalman Filtering technique (Kalman [1960]) has been extended to interval linear models with known classical statistical assumptions. As the expressions for deriving the Kalman flter involve matrix inversion, one must find a way to implement or avoid this tricky algebraic operation within an interval framework. Our contribution consists in proposing an alternative approach to solve the interval matrix inversion problem without loss of solution while controlling the inherent pessimism of interval calculus. Several techniques are proposed to limit highly overestimation eects propagating within the interval Kalman flter recursive structure. In particular the gain of the flter is obtained by a calculus based on the set inversion algorithm SIVIA. A dedicated example is developed to compare dierent approaches.

A numerical approach to design control invariant sets for constrained nonlinear discrete-time systems with guaranteed optimality is proposed in this paper. The addressed approach is based on the fact that zonotopes are more flexible for representing sets than boxes in interval analysis. Then the solver of set inversion via interval analysis is extended to set inversion via zonotope geometry by introducing the novel idea of bisecting zonotopes. The main feature of the extended solver of set inversion is the bisection and the evolution of a zonotope rather than a box. Thus the shape of admissible domains for set inversion can be broadened from boxes to zonotopes and the wrapping effect can be reduced as well by using the zonotope evolution instead of the interval evolution. Combined with global optimization via interval analysis, the extended solver of set inversion via zonotope geometry is further applied to design control invariant sets for constrained nonlinear discrete-time systems in a numerical way. Finally, the numerical design of a control invariant set and its application to the terminal control of the dual-mode model predictive control are fulfilled on a benchmark Continuous-Stirred Tank Reactor example.

In this paper, a new algorithm based on Set Inversion techniques and Modal Interval Analysis is presented. This algorithm allows one to solve problems involving quantified constraints over the reals through the characterization of their solution sets. The presented methodology can be applied to a wide range of problems involving uncertain (non)linear systems. Finally, an advanced application is solved.

International audience; This paper proposes a set-membership method to characterize a probabilistic set,i.e., a set enclosing the true value for the parameter vector of a parametric system witha given probability. The approach assumes that all errors are independent and an intervalfor the error is known. To each error interval, a probability to be an outlier is provided.It is shown that characterizing the probabilistic set is a set inversion problem. The maincontribution of the paper is to provide a method to characterize the inner part of theprobabilistic set. As an illustration, an application to the static localization of a mobilerobot is considered.

International audience; The bounded-error approach to parameter estimation, mainly developed in the context of control and signal processing, is applied in the electrochemistry field in order to obtain reliable estimates for kinetic parameters. The method is based on the assumption that an uncertainty bar is available for each measurement. A set guaranteed to contain all values of the parameter vector that lead to model outputs consistent with these error bars is then computed, based on interval analysis and set inversion. The resulting technique is applied on simulated and experimental data for several classical electrochemical models. Its merits are compared to those of a more traditional approach based on least square estimation by iterative local optimization. A first obvious difference is that the point estimate provided by the latter method may or may not belong to the set estimated by the former, because it does not take the bounds on the uncertainty into account. Moreover, as our approach is global, it escapes the difficult problem of initialization encountered with iterative optimization methods. Finally, the set obtained is a natural characterization of the uncertainty on the estimated parameters.

International audience; Characterizing the set of all parameter vectors such that their image by a vector function belongs to a given set is a set-inversion problem. The algorithm SIVIA (Set Inversion Via Interval Analysis) makes it possible to perform this task in an approximate but guaranteed way. In the examples treated so far, the function to be inverted was given either explicitly or by a sequential algorithm. In this paper, this approach is extended to the case of branching algorithms involving if statements. As an illustration, the static localization of a robot from bounded-error range measurements is considered. The notion of remoteness, introduced for an archetypal but realistic sonar model, allows this problem to be cast into the set-inversion framework.

Discrete-event systems are driven by events and generate events. To describe their evolution, the dater approach associate to each event a sequence of dates, namely a dater, corresponding to the dates at which the event occurs.

In this communication, vehicle localization in a 2D-mapped environment from inaccurate telemetric measurements is stated as a set-inversion problem. A novel methodology based upon a mechanical interpretation of range measurements and interval analysis is presented. The approach aims at characterizing the set Q of all vehicle configurations which are consistent with all range measurements, their bounded associated inaccuracies and the environment model. It provides a global, guaranteed and accurate characterization of the solution set Q. As no a priori matching hypotheses are needed and as the solution set Q can be non connected, the method naturally produces the multiple hypotheses of feature-measurement association.

International audience; This paper deals with computational aspects of interval kalman filtering of discrete time linear models with bounded uncertainties on parameters and gaussian measurement noise. In this work, we consider an extension of conventional Kalman filtering to interval linear models [1]. As the expressions for deriving the Kalman filter involve matrix inversion which is known to be a difficult problem. One must hence find a way to implement or avoid this tricky algebraic operation within an interval framework. To solve the interval matrix inversion problem and other problems due to interval calculus, we propose an original approach combining the set inversion algorithm SIVIA and constraint satisfaction propagation. Several contractors are proposed to limit overestimation effects propagating within the interval Kalman filter recursive structure. Thus the description of our approach is followed by an application and we compare the proposed approach with interval kalman filtering developped in [1].

Background On delayed-enhancement (DE-CMR), both fat and infarcted myocardium appear bright making them difficult to differentiate. Standard chemical shift fat-suppression (FatSatCS) consists of a fat-frequency selective saturation followed by readout. This approach is suboptimal in clinical DE-CMR sequences which typically have long readout times (100-200ms) and as a result, fat magnetization has significantly recovered (T1 = 290ms, 3T) when the center of k-space is acquired for linear reordering. Centric reordering can overcome this limitation, but at the cost of blurring artifacts. The phase sensitive inversion recovery (PSIR) variant of DE-CMR avoids the need to precisely set inversion time through the acquisition of a reference data set interleaved between image acquisitions. We hypothesized that modifying the reference acquisition via a Dixon-type approach could provide PSIR images that simultaneously show dark fat without added acquisition time. The result is a virtual chemically selective inversion (FatSatVCSI), without increased SAR and at no cost to signal-to-noise from pulse imperfections.

Robust state estimation for a subclass of nonlinear systems is considered in this paper through a comparison of two methodologies. The comparison is based on modus operandi complexity, calculation time and interval observation quality. The first approach is based on a fundamental property of flat systems which states that the state of the system can be written as a function of the so-called flat outputs and their derivatives up to some order. 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 probabilistic distribution. Therefore, the interval state estimation is expressed as a set inversion problem which is solved using interval analysis. The second approach is based on coordinate transformation in order to obtain a partially linear cooperative presentation of the nonlinear system for which a closed-loop interval observer is designed. Both methods ensure to enclose the set of system states that is consistent with the model and the measurement noise bounds. The performance of each technique is discussed. Numerical examples are given throughout the paper to illustrate the performances of the proposed techniques.

This paper deals with parameter estimation in the bounded-error context. A new approach, based on interval analysis, is proposed to compute guaranteed estimates of suitable characteristics of the set of all values of the parameter vector such that the error between the experimental data and the model outputs belongs to some predefined feasible set. This approach is especially suited to models whose output is nonlinear in their parameters, a situation where most available methods fail to provide any guarantee as to the global validity of the results obtained. After a brief presentation of interval analysis, an algorithm is proposed, which makes it possible to obtain guaranteed estimates of characteristics of such as its volume or the smallest axis-aligned box that contains it. Properties of this algorithm are established, and illustrated on a simple example.

WOS; International audience; The problem of characterizing in a guaranteed way the set of all feasible set-points of a control problem is known to be difficult. In the present work, the problem to be solved involves non-linear equality constraints with variables affected by logical quantifiers. This problem is not solvable by current symbolic methods like quantifier elimination, which is commonly used for solving this class of problems. We propose the utilization of guaranteed set-computation techniques based on interval analysis, in particular a solver referred to as Quantified Set Inversion (QSI). As an application example, the problem of simultaneously controlling the speed and the orientation of a sailboat is presented. For this purpose, the combination of QSI solver and feedback linearization techniques is employed.

International audience; This paper is devoted to set membership parameter estimation for non-linear complex-valued models. In such a context, the error between the measured data and the output model is supposed to be bounded with known prior bounds. The proposed approach is based on set inversion via complex interval analysis. A first contribution of this paper is the extension of logarithm and exponentiel functions to polar intervals. In a second part, it is shown on an engineering application devoted to thermal parameter estimation, that the proposed interval complex representation is useful.

The main objective of this paper is to estimate the whole set of feasible parameters of a fractional differentiation model, based on gain and phase frequency data. All parameters, including differentiation orders, are expressed as intervals and then estimated using a bounded error approach. A contraction method named forward-backward propagation is first applied to reduce the initial searching space. Then, a set inversion algorithm named SIVIA is applied on the reduced searching space to obtain the whole set of feasible parameters. One of the interesting points of this study is to show the separate contribution of gain and phase data on the final estimation.

This paper deals with interval analysis applied to linear time-delays systems. With basic examples, we describe some applications to solve various control problems, and to show that interval computation is an effective tool for time-delays systems analysis.

We present an algorithm to characterize the set S = {x epsi Rfrl : f(x) > 0} = f-1(]0,infin[m) in the framework of set inversion using interval analysis. The proposed algorithm improves on the algorithm of Jaulin et. al. The improvement exploits powerful Hansen's method for solving systems of nonlinear inequalities. We test and compare the performance of the proposed and existing algorithms in characterizing the domain for the robust stability. The results of the testing show that the proposed algorithm is computationally efficient and encloses the solution more sharply than the existing algorithm, requires less memory space and iterations.

This paper describes a new reliable method, based on modal interval analysis (MIA) and set inversion (SI) techniques, for the characterization of solution sets defined by quantified constraints satisfaction problems (QCSP) over continuous domains. The presented methodology, called quantified set inversion (QSI), can be used over a wide range of engineering problems involving uncertain nonlinear models. Finally, an application on parameter identification is presented

In this paper, initial localization problems are solved by using set-membership estimation. The method can be used with any robot and any kind of sensor(s), provided that a computable model of the environment/sensor interaction is available. With a pedagogical aim in mind, it is detailed in the case of the localization of a vehicle from range measurements in a polygonal environment. Salient properties of the method are as follows. First, it does not need any explicit management of matching hypotheses. Second, it is able to deal with ambiguous situations where several radically different vehicle configurations are consistent with the measurements. Third, it can be made robust to outliers. Fourth, it can deal with nonlinear observation models without any approximation. Fifth, the result is guaranteed in the sense that no configuration consistent with the data and the hypotheses can be missed.

We present a new inversion strategy for the early detection of breast cancer from microwave data which is based on a new multiphase level set technique. This novel structural inversion method uses a modification of the color level set technique adapted to the specific situation of structural breast imaging taking into account the high complexity of the breast tissue. We only use data of a few microwave frequencies for detecting the tumors hidden in this complex structure. Three level set functions are employed for describing four different types of breast tissue, where each of these four regions is allowed to have a complicated topology and to have an interior structure which needs to be estimated from the data simultaneously with the region interfaces. The algorithm consists of several stages of increasing complexity. In each stage more details about the anatomical structure of the breast interior is incorporated into the inversion model. The synthetic breast models which are used for creating simulated data are based on real MRI images of the breast and are therefore quite realistic. Our results demonstrate the potential and feasibility of the proposed level set technique for detecting, locating and characterizing a small tumor in its early stage of development embedded in such a realistic breast model. Both the data acquisition simulation and the inversion are carried out in 2D.

Given a nonlinear discrete-time system, previous works exist that compute invariant sets as finite unions of boxes. Set inversion algorithms based on interval arithmetic are used to obtain inner approximations of the one step set starting in an invariant target set. In this paper a complementary approach based on set simulation is proposed. An invariant set can be obtained if a set trajectory that initiates in a given set, reaches this set again in a given number of steps at most. The first advantage of the proposed method is that there is no need to know an initial invariant set. The second one is that for a given system, a high convergence rate of the trajectories tends to reduce the computational effort of the method. The main disadvantage is that the algorithm does not guarantee that an invariant set is obtained. It just guarantees a response in finite time.

Robust parameter identification consists of finding approximations for the set of parameters of a given model that are consistent with the input/output observations. When interval uncertainties are considered, consistency can be defined in different ways by applying universal or existential quantifiers to the intervals representing input, output and model parameters, leading to different physical interpretations. In this paper a new approach to robust parameter identification based on modal intervals is introduced. Several consistency tests are stated as first order logical formulae with quantifiers. The resulting identification problems are solved using the Quantified Set Inversion (QSI) algorithm developed by the authors. The presented approach is illustrated by means of a numerical example and a real problem of identifying the hydraulic model of a river.

In this paper, a new robust fault detection test based on calculating the inverse image of an interval function is presented. It is based on a consistency test which relies on tools from interval analysis such as set inversion or contractors to check if there exists a member in the family of models described with an interval model that can explain the measured data. The proposed test is compared to related approaches of fault detection using interval analysis. The main features of the new test will be extracted and advantages and drawbacks discussed. An example will be used to illustrate the behaviour.