Thick ellipsoids were recently introduced by the authors to represent uncertainty in state variables of dynamic systems, not only in terms of guaranteed outer bounds but also in terms of an inner enclosure that belongs to the true solution set with certainty. Because previous work has focused on the definition and computationally efficient implementation of arithmetic operations and extensions of nonlinear standard functions, where all arguments are replaced by thick ellipsoids, this paper introduces novel operators for specifically evaluating quasi-linear system models with bounded parameters as well as for the union and intersection of thick ellipsoids. These techniques are combined in such a way that a discrete-time state observer can be designed in a predictor-corrector framework. Estimation results are presented for a combined observer-based estimation of state variables as well as disturbance forces and torques in the sense of an unknown input estimator for a hovercraft.

Stability contractors, based on interval analysis, were introduced in recent work as a tool to verify stability domains for nonlinear dynamic systems. These contractors rely on the property that - in case of provable asymptotic stability - a certain domain in a multi-dimensional state space is mapped into its interior after a certain integration time for continuous-time processes or after a certain number of discretization steps in a discrete-time setting. However, a disadvantage of the use of axis-aligned interval boxes in such computations is the omnipresent wrapping effect. As shown in this contribution, the replacement of classical interval representations by ellipsoidal domain enclosures reduces this undesirable effect. It also helps to find suitable ratios for the edge lengths if interval-based domain representations are investigated. Moreover, ellipsoidal domains naturally represent the possible regions of attraction of asymptotically stable equilibrium points that can be analyzed with the help of quadratic Lyapunov functions, for which stability criteria can be cast into linear matrix inequality (LMI) constraints. For that reason, this paper further presents possible interfaces of ellipsoidal enclosure techniques with LMI approaches. This combination aims at the maximization of those domains that can be proven to be stable for a discrete-time range-only localization algorithm in robotics. There, an Extended Kalman Filter (EKF) is applied to a system for which the dynamics are characterized by a discrete-time integrator disturbance model with additive Gaussian noise. In this scenario, the measurement equations correspond to the distances between the object to be localized and beacons with known positions.

In this paper, we propose a new approach to prove stability of non-linear discrete-time systems. After introducing the new concept of stability contractor, we show that the interval centred form plays a fundamental role in this context and makes it possible to easily prove asymptotic stability of a discrete system. Then, we illustrate the principle of our approach through theoretical examples. Finally, we provide two practical examples using our method : proving stability of a localisation system and that of the trajectory of a robot.

Recent advances in computational power, algorithms, and sensors allow robots to perform complex and dangerous tasks, such as autonomous missions in space or underwater. Given the high operational costs, simulations are run beforehand to predict the possible outcomes of a mission. However, this approach is limited as it is based on parameter space discretization and therefore cannot be considered a proof of feasibility. To overcome this limitation, set-membership methods based on interval analysis, guaranteed integration, and contractor programming have proven their efficiency. Guaranteed integration algorithms can predict the possible trajectories of a system initialized in a given set in the form of tubes of trajectories. The contractor programming consists in removing the trajectories violating predefined constraints from a system’s tube of possible trajectories. Our contribution consists in merging both approaches to allow for the usage of differential constraints in a contractor programming framework. We illustrate our method through examples related to robotics. We also released an open-source implementation of our algorithm in a unified library for tubes, allowing one to combine it with other constraints and increase the number of possible applications.

In this paper, we propose to study the stability of a navigation method that allows a robot to move in an unstructured environment without compass by measuring a scalar function ϕ which only depends on the position. The principle is to ask the robot to roll along an isovalue of ϕ . Using an interval method, we prove the stability of our closed loop system in the special case where ϕ is linear.