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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.

The massive data generated by large-scale dynamic systems makes its optimization facing a tough challenge. Traditional White Box-based methods directly model the internal operating mechanism of the system, so massive amounts of measured data need to be handled, which is costly and time-consuming. The poor interpretability of the Black Box-based methods makes it difficult to adapt to the dynamic environment. Thus we propose a novel Gray Box-based approach namely Deep Reinforcement Learning-enabled Constraint Set Inversion Algorithm (DRESIA), which establishes a quantitative model of the nonlinear interoperability effects of system internal states which simplifies the White Box's complex mechanism of reconstruction and prediction and retains the interpretability of the model, therefore improves the prediction efficiency of feasible region while also improving the generalization ability. It further improves the dynamic adaptability of the modeling environment, which provides a new performance balancing scheme for system modeling. Under the premise that the large-scale 5G Cyber-Twin system satisfies the given Quality of Service (QoS) requirements, we perform DRESIA to realize the efficient and dynamic optimal search of feasible region, the results show that the DRESIA reduces the computational cost, and balances the accuracy and robustness of the feasible region, which validate the effectiveness and superiority of Gray Box-based approach.

[no abstract available]

International audience; Controlling multi-DOF (Degree of Freedom) micropositioning systems always represents great challenge becauseof the high sensitivity to the environment at this scale and the cross-coupling effects present between the different axes. A robust Luenberger observer-based state feedback design using interval analysis and regional pole assignment technique are introduced to control such systems. This robust control design keeps the same structure of the classical state-feedback with the usual Luenberger observer. However, the synthesises of the observer and the feedback controller are performed by means of interval techniques to find the set of gains that are robustagainst system uncertainties and that satisfy some predefined performances. For this matter, an algorithm based on Set Inversion Via Interval Analysis (SIVIA) combined with interval eigenvalues computation is proposed to find these robust gains. The control approach is validated in simulation and then tested experimentally to control a multi-DOF positioning structure.

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.

This paper focuses on the guaranteed identification of viscous friction parameters for a nonlinear inverted pendulum. The method is based on the interval analysis (IA) and set-inversion tools to determine the set of all the feasible friction parameters from a prior domain of interest, i.e. initial interval vector or box, that are consistent with all the experimental and theoretical datasets including their uncertainties. The capabilities of our proposed guaranteed identification are compared with the more commonly used approach based on the least square method identification (LSMI), which is used especially to adjust the inertial and geometric parameters of our experimental plant. Both of them have been investigated through several experiments on a real inverted pendulum and simulations with uncertain ODEs via the DynIbex library.

In the paper, we present a new interval-based set inversion algorithm which takes into account the continuity of the problem. In the case where the set Y to be inverted has some volume, we show that inverting the boundary ∂ Y of Y is sufficient to reconstruct the preimage X = f − 1 ( Y ) . The inversion of ∂ Y separates the domain of f into two regions : one inside X and one outside. To detect which part is inside or outside, we show that we can retro-propagate the information coming from Y at negligible cost. The efficiency of the approach is illustrated on a localization problem.

In this paper we propose an efficient sampling strategy to solve an inversion problem subjected to functional uncertainties. More precisely, we aim at characterizing a control variable region defined by exceedance above a prescribed threshold of specific Quantities of Interest (QoT). This study is motivated by an automotive industrial application consisting in the identification of the set of values of control variables of a gas after-treatment device, in line with pollutant emission standards of a vehicle under driving profile uncertainties. In that context, driving profile uncertainties are modelled by a functional random variable and the constrained response in the inversion problem is formulated as the expectation over this functional random variable only known through a set of realizations. As often in industrial applications, this problem involves time-consuming computational models. We thus propose an approach that uses Gaussian Process meta-models built on the joint space of control and uncertain input variables. Specifically, we define a learning criterion based on uncertainty in the excursion of the Gaussian Process and derive tractable expressions for variance reduction in such a framework. Applications to analytical examples, followed by the automotive industrial test case show the accuracy and the efficiency brought by the procedure we propose.

International audience; This paper addresses the control of manipulation force in a piezoelectric tubeactuator (piezotube) subjected to temperature variation and input constrains.To handle this problem a robust output-feedback design is proposed using aninterval state-space model, which permits consideration of the parameter uncertainties caused by temperature variation. The design method is robust in the sense that the eigenvalues of the interval system are designed to be clustered inside desired regions. For that, an algorithm based on Set Inversion Via IntervalAnalysis (SIVIA) combined with interval eigenvalues computation is proposed. This recursive SIVIA-based algorithm allows to approximate with subpaving the set solutions of the feedback gain[K]that satisfy the inclusion of the eigenvalues of the closed-loop system in the desired region, while at the same time ensuring the control inputs amplitude is bounded by specified saturation. The effectiveness of the control strategy is illustrated by experiments on a real piezotube of which the environmental temperature is varied.

International audience; In this paper we propose a new approach for nonlinear parameter estimation under additive Gaussian noise. We provide an algorithm based on interval analysis and set inversion which computes an inner and an outer approximation of a set enclosing the parameter vector with a given probability. The principle of the approach is illustrated by examples related to parameter estimation and range-only localization.

In this paper we propose an efficient sampling strategy to solve inversion problem under functional uncertainty. This approach aims to characterize region of a control space defined by exceedance above prescribed threshold. This study is motivated by an application on identifying the set of control parameters leading to meet the pollutant emission standards of a vehicle under driving profile uncertainties. In that context, the constrained response in the inversion problem is here formulated as the expectation over the functional random variable only known through a set of realizations and the unknown set is thus associated with the control variables. As often in industrial applications, this problem involves high-fidelity and time-consuming computational models. We thus proposed an approach that makes use of Gaussian Process meta-models built on the joint space of control and uncertain input variables. Specifically, we define a design criterion based on uncertainty in the excursion of the Gaussian Process and derive tractable expressions for the variance reduction in such a framework. Applications to analytical examples, followed by the automotive industrial test case show the accuracy and the efficiency brought by the proposed procedure.

A methodology is developed for the estimation of internal axial force and boundary restraints within in-service, prismatic axial force members of structural systems using interval arithmetic and contractor programming. The determination of the internal axial force and end restraints in tie rods and cables using vibration-based methods has been a long standing problem in the area of structural health monitoring and performance assessment. However, for structural members with low slenderness where the dynamics are significantly affected by the boundary conditions, few existing approaches allow for simultaneous identification of internal axial force and end restraints and none permit for quantifying the uncertainties in the parameter estimates due to measurement uncertainties. This paper proposes a new technique for approaching this challenging inverse problem that leverages the Set Inversion Via Interval Analysis algorithm to solve for the unknown axial forces and end restraints using natural frequency measurements. The framework developed offers the ability to completely enclose the feasible solutions to the parameter identification problem, given specified measurement uncertainties for the natural frequencies. This ability to propagate measurement uncertainty into the parameter space is critical towards quantifying the confidence in the individual parameter estimates to inform decision-making within structural health diagnosis and prognostication applications. The methodology is first verified with simulated data for a case with unknown rotational end restraints and then extended to a case with unknown translational and rotational end restraints. A laboratory experiment is then presented to demonstrate the application of the methodology to an axially loaded rod with progressively increased end restraint at one end.

Footprint evaluation is an important tool for assessing the appropriation of ecological assets, GHG emissions, freshwater consumption parameters, etc., within a specified region. However, traditional evaluation of footprints for mega cities or urban agglomerations requires overmuch different types of high-quality data. There is a great need of seeking a smart model/approach with declined data requirements for evaluation of footprints where part of data can hardly be accessed. Here we propose a new ensemble inversion model (EIM) based on integrated multitask machine learning (MML) and multi-modeling stacking (MMS) algorithms for smart evaluation and prediction of water, carbon and ecological footprints. The accuracy and generalization capability of the model are illustrated through three largest urban agglomerations in the middle reaches of the Yangtze River (MRYR). The testing results show that the EIM achieves similar prediction performance compared to traditional footprints calculation methods (R2 = 0.91, RMSE = 0.18, MAE = 0.11), yet greatly reduces the amount of required data by approximately 80%. Moreover, the accuracy of the EIM is improved by more than 20%, compared with other models using a single inversion algorithm. The modeling results also show that 1) water, carbon and ecological footprints are significantly positively correlated, and 2) an annual increase of 4.8% can be found in terms of the urban environmental pressure index (UEPI), and its projection is even less optimistic for the future.

International audience; This paper deals with the set inversion problem X = f −1 (Y) in the case where f : R n → R m depends on a parameter vector p ∈ R q which is known to be inside a box [p]. We show that for a large class of problems, we can obtain an accurate approximation of the solution set, without bisecting in the p-space. To do this, symbolic methods are required to cast our initial problem into a chain of set-inversion problems, the links of which have some nice properties with respect to p. As an application, we consider the problem of computing the set of all initial states of an uncertain discrete-time state system that reach a target set Y in a given time.

This publication is devoted to the design of an interval-based set-membership state estimator that can be applied to totally observable linear parameter-varying (LPV) systems. Hereto, a set-inversion procedure to determine state intervals that are consistent with a sequence of input and output values as well as two intersections with predicted values are used to reduce the pessimism of interval arithmetic. A numerical example illustrating the performance of the method is presented. The benefit of this method is that it provides an a priori known accuracy of the result based on the assumption of unknown but bounded uncertainties.

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.

The problem of guaranteed parameter estimation (GPE) consists in enclosing the set of all possible parameter values, such that the model predictions match the corresponding measurements within prescribed error bounds. One of the bottlenecks in GPE algorithms, commonly exploiting set inversion, is the construction of enclosures for the image-set of factorable functions. In this paper, we introduce a novel set-based computing method called interval superposition arithmetics (ISA) for the construction of enclosures of such image sets and its use in GPE algorithms. The main benefits of using ISA in the context of GPE lie in the improvement of enclosure accuracy and in the implied reduction of the number of set-membership tests in the set-inversion algorithm.

In this article we address the problem of cooperative pose estimation in a group of unmanned aerial vehicles (UAVs) in a bounded-error context. The UAVs are equipped with cameras to track landmarks, and with a communication and ranging system to cooperate with their neighbors. Measurements are represented by intervals, and constraints are expressed on the robots poses (positions and orientations). Pose domains subpavings are obtained by using set inversion via interval analysis. Each robot of the group first computes a pose domain using only its sensors measurements. Then, through position boxes exchanges, the positions are cooperatively refined by constraint propagation in the group. Results with real robot data are presented, and show that the position accuracy is improved thanks to cooperation.

International audience; This paper addresses the problem of cooperative localization in a group of unmanned aerial vehicles (UAV) in a bounded error context. The UAVs are equipped with cameras to tracks landmarks, and a communication and ranging system to cooperate with their neighbours. Measurements are represented by intervals, and constraints are expressed on the robots poses (positions and orientations). Each robot first computes a pose domain using only its sensors measurements, by using set inversion via interval analysis [1]. Then, through position boxes exchange, positions are cooperatively refined by constraint propagation in the group. Results are presented with real robot data, and show position accuracy improvement thanks to cooperation.

International audience; The dynamical systems in various science and engineering problems are often governed by nonlinear equations (differential equations). Due to insufficiency and incompleteness of system information, the parameters in such equations may have uncertainty. Interval analysis serves as an efficient tool for handling uncertainties in terms of closed intervals. One of the major problems with interval analysis is handling “dependency problems” for computation of the tightest range of solution enclosure or exact enclosure. Such dependency problems are often observed while dealing with complex nonlinear equations. In this regard, initially, two test problems comprising interval nonlinear equations are considered. The Set Inversion via Interval Analysis (SIVIA) along with the Monte Carlo approach is used to compute the exact enclosure of the test problems. Further, the efficiency of the proposed approach has also been verified for solving nonlinear differential equation (Van der Pol oscillator) subject to interval initial conditions.

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.

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.

International audience; No abstract

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; A set-membership (SM) approach is proposed to design reliable static stabilizing controllers for non-linear dynamical systems. First, the control design problem is reformulated as a parameter identification issue in an unknown-but-bounded error framework. Then, a new dual set integration method for reachability computation is introduced. Finally, the latter dual method is used in conjunction with set inversion techniques via interval analysis, in order to develop an SM parameter estimation algorithm to solve the control design problem. The identified feedback control must achieve two aims. The first aim is that, starting from a given bounded set of initial states, the state trajectories generated by the controlled non-linear system have to reach in a finite-time a desired target set. The second aim is to ensure asymptotic stability of the controlled system over the target set. The effectiveness of the proposed method is illustrated through a complex non-linear system.

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.

SUMMARY The main objective of this paper was to estimate tip-tilt disturbances in adaptive optics systems. In a bounded-error context, set inversion methods based on interval analysis are used to guarantee both state and parameter estimation of tip-tilt disturbances. Consequently, two methods are performed. The first method is based on contraction–bisection, and the second one is based only on contraction. Both methods are thus compared, and results are discussed according to computational time and pessimism introduced on each estimated parameter. Copyright © 2013 John Wiley & Sons, Ltd.

International audience; The knowledge of localization uncertainties is of prime importance when the navigation of intelligent vehicles has to deal with safety issues. This paper presents a robust estimation method that is able to quantify the localization confidence based on interval analysis and constraint propagation. First, tightly coupled position domains are computed by constraint propagation on Global Positioning System (GPS) measurements and a precise 3-D map of the drivable area. Since GPS is prone to satellite masking and wrong measurements in urban areas, a second stage provides localization integrity and information availability by the use of a position and proprioceptive data history. A robust constraint propagation algorithm is employed to compute the current vehicle pose. It is able to handle erroneous positions with a chosen integrity risk. Experiments carried out in urban canyons illustrate the performance of the method in comparison with a particle filter. Despite bad satellite visibility, full positioning availability is obtained, and errors are less than 5.1 m during 95% of the trial. In opposition to the particle filter, confidence domains are consistent with ground truth, which confirms the high integrity of the method.

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.

This paper presents a technique for designing a robust polynomial RST controller for parametric uncertain systems. The uncertain parameters are assumed to be bounded by intervals. The computation of the controller is addressed by introducing the interval arithmetic. The controller synthesis is formulated as a set inversion problem that can be solved using the SIVIA algorithm. The proposed method is afterwards applied to design a robust controller for a piezoelectric microactuator. The experimental results show the efficiency of the proposed method. Finally, a fine stability analysis is performed to analytically prove the robustness of the designed controller.

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.

WOS; International audience; This paper describes a solver programming method, called "contractor programming", that copes with two issues related to constraint processing over the reals. First, continuous constraints involve an inevitable step of solver design. Existing softwares provide an insufficient answer by restricting users to choose among a list of fixed strategies. Our first contribution is to give more freedom in solver design by introducing programming concepts where only configuration parameters were previously available. Programming consists in applying operators (intersection, composition, etc.) on algorithms called "contractors" that are somehow similar to propagators. Second, many problems with real variables cannot be cast as the search for vectors simultaneously satisfying the set of constraints, but a large variety of different outputs may be demanded from a set of constraints (e.g., a paving with boxes inside and outside of the solution set). These outputs can actually be viewed as the result of different "contractors" working concurrently on the same search space, with a bisection procedure intervening in case of deadlock. Such algorithms (which are not strictly speaking solvers) will be made easy to build thanks to a new branch & prune system, called "paver". Thus, this paper gives a way to deal harmoniously with a larger set of problems while giving a fine control on the solving mechanisms. The contractor formalism and the paver system are the two contributions. The approach is motivated and justified through different cases of study. An implementation of this framework named Quimper is also presented.

A general framework for computing robust controllable sets of constrained nonlinear un- certain discrete-time systems as well as controlling such complex systems based on the computed robust controllable sets is introduced in this paper. The addressed one-step control approach turns out to be a robust model predictive control scheme with feasible unit control horizon and contractive con- straint. The solver of 1-dimensional quantified set inversion in modal interval analysis is extended to 2-dimensional cases for computing robust controllable sets off-line with a clear semantic interpreta- tion, where both universal and existential quantifiers are concerned simultaneously. An interval-based solver of constrained minimax optimization is also proposed to compute one-step control inputs online in a reliable way, which guarantee to drive the system state contractively along the computed robust controllable sets to a selected terminal robust control invariant set. Mathematics Subject Classification. 65G40, 93B05, 93B51.

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.