Luc Jaulin

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.

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.

Contractor programming

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.

Robust Autonomous Robot Localization Using Interval Analysis

This paper deals with the determination of the position and orientation of a mobile robot from distance measurements provided by a belt of onboard ultrasonic sensors. The environment is assumed to be two-dimensional, and a map of its landmarks is available to the robot. In this context, classical localization methods have three main limitations. First, each data point provided by a sensor must be associated with a given landmark. This data-association step turns out to be extremely complex and time-consuming, and its results can usually not be guaranteed. The second limitation is that these methods are based on linearization, which makes them inherently local. The third limitation is their lack of robustness to outliers due, e.g., to sensor malfunctions or outdated maps. By contrast, the method proposed here, based on interval analysis, bypasses the data-association step, handles the problem as nonlinear and in a global way and is (extraordinarily) robust to outliers.

Guaranteed nonlinear estimation using constraint propagation on sets

Bounded-error estimation is the estimation of the parameter or state vector of a model from experimental data, under the assumption that some suitably defined errors should belong to some prior feasible sets. When the model outputs are linear in the vector to be estimated, a number of methods are available to contain all estimates that are consistent with the data within simple sets such as ellipsoids, orthotopes or parallelotopes, thereby providing guaranteed set estimates. In the non-linear case, the situation is much less developed and there are very few methods that produce such guaranteed estimates. In this paper, the problem of characterizing the set of all state vectors that are consistent with all data in the case of non-linear discrete-time systems is cast into the more general framework of constraint satisfaction problems. The state vector at time k should be estimated either on-line from past measurement only or off-line from a series of measurements that may include measurements posterior to k . Even in the causal case, prior information on the future value of the state and output vectors, due for instance to physical constraints, is readily taken into account. Algorithms taken from the literature of interval constraint propagation are extended by replacing intervals by more general subsets of real vector spaces. This makes it possible to propose a new algorithm that contracts the feasible domain for each uncertain variable optimally (i.e. no smaller domain could be obtained) and efficiently.

Technical Communique: Interval constraint propagation with application to bounded-error estimation

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

Guaranteed nonlinear parameter estimation from bounded-error data via interval analysis

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.

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.

Interval analysis and dioid: application to robust controller design for timed event graphs

This paper deals with feedback controller synthesis for timed event graphs, where the number of initial tokens and time delays are only known to belong to intervals. We discuss here the existence and the computation of a robust controller set for uncertain systems that can be described by parametric models, the unknown parameters of which are assumed to vary between known bounds. Each controller is computed in order to guarantee that the closed-loop system behavior is greater than the lower bound of a reference model set and is lower than the upper bound of this set. The synthesis presented here is mainly based on dioid, interval analysis and residuation theory.

Guaranteed tuning, with application to robust control and motion planning

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