Julien Alexandre Dit Sandretto

Control of nonlinear switched systems based on validated simulation

We present an algorithm of control synthesis for nonlinear switched systems, based on an existing procedure of state-space bisection and made available for nonlinear systems with the help of validated simulation. The use of validated simulation also permits to take bounded perturbations and varying parameters into account. The whole approach is entirely guaranteed and the induced controllers are correct-by-design.

An Improved Algorithm for the Control Synthesis of Nonlinear Sampled Switched Systems

A novel algorithm for the control synthesis for nonlinear switched systems is presented in this paper. Based on an existing procedure of state-space bisection and made available for nonlinear systems with the help of guaranteed integration, the algorithm has been improved to be able to consider longer patterns of modes with a better pruning approach. Moreover, the use of guaranteed integration also permits to take bounded perturbations and varying parameters into account. It is particularly interesting for safety critical applications, such as in aeronautical, military or medical fields. The whole approach is entirely guaranteed and the induced controllers are correct-by-design. Some experimentations are performed to show the important gain of the new algorithm.

Appropriate Design Guided by Simulation: An Hovercraft Application

A design methodology based on simulation of dynamical behavior is presented in this paper. The particularity of our method is that it exploits the set-membership simulation. Indeed, this method allows one to consider an interval of values for each parameter of the dynamical model. Finding the parameters validating the requirements is then a filter, based on a Branch and Prune algorithm. We prefer an approach of appropriate design that is a design which validates the physical constraints coming from requirements, to an optimal design which does not lead to satisfy imperative requirements. Our method is described and applied on the complex problem of design of an hovercraft under dynamic requirements.

Validated computation of the local truncation error of Runge–Kutta methods with automatic differentiation

A novel approach to bound the local truncation error of explicit and implicit Runge–Kutta methods is presented. This approach takes its roots in the modern theory of Runge–Kutta methods, namely the order condition theorem, defined by John Butcher in the 1960s. More precisely, our work is an instance, for Runge–Kutta methods, of the generic algorithm defined by Ferenc Bartha and Hans Munthe-Kaas in 2014 which computes B-series with automatic differentiation techniques. In particular, this specialized algorithm is combined with set-membership framework to define validated numerical integration methods based on Runge–Kutta methods.

An Interval-based Sliding Horizon Motion Planning Method

Abstract A new algorithm of motion planning based on set-membership approach is presented. The goal of this algorithm is to find a safe and optimal path taking into account various sources of bounded uncertainties on the dynamical model of the plant, on the model of the environment, while being robust with respect to the numerical approximations introduced by numerical integration methods. The main approach is based on a sliding horizon method to predict the behavior of the system allowing the computation of an optimal path. As an example, the motion planning algorithm is applied to an Autonomous Underwater Vehicle (AUV) case study, showing the benefit of the proposed approach.

Solving over-constrained systems of non-linear interval equations – And its robotic application

This paper presents and describes in details an original method developed to solve over-constrained systems of non-linear interval equations that arise namely in parameter identification problems deriving from physical models and uncertain measurements. Our approach consists of computing an interval enclosure of the least square solution set and an inner box of tolerable solution set. This method is applied in a detailed example and some interesting results obtained for the calibration of a cable-driven robot are shown.

Formal Verification of Robotic Behaviors in Presence of Bounded Uncertainties

Robotic behaviors are mainly described by differential equations. Those mathematical models are usually not precise enough because of inaccurately known parameters or model simplifications. Nevertheless, robots are often used in critical contexts as medical or military fields. So, uncertainties in mathematical models have to be taken into account in order to produce reliable and safe analysis results. A framework based on interval analysis is proposed to safely verify and analyze robotic behaviors with bounded uncertainties. It follows an interval constraint programming approach, combined with validated numerical integration methods to deal with differential equations. A case study on robust path planning is presented to emphasize the efficiency of the complete framework.

Studying Sequences of Jumps in Hybrid Systems to Detect Zeno Phenomenon

We propose a numerical analysis of sequences of points of interest associated to the dynamics of hybrid systems. These sequences are made of instants of switching mode or instants when a particular quantity vanishes. This analysis allows one to discover instant of accumulation points, a.k.a. Zeno phenomenon. Some examples are given to show the potential of this approach.

Tuning PI controller in non-linear uncertain closed-loop systems with interval analysis

The tuning of a PI controller is usually done through simulation, except for few classes of problems, e.g., linear systems. With a new approach for validated integration allowing us to simulate dynamical systems with uncertain parameters, we are able to design guaranteed PI controllers. In practical, we propose a new method to identify the parameters of a PI controller for non-linear plants with bounded uncertain parameters using tools from interval analysis and validated simulation. This work relies on interval computation and guaranteed numerical integration of ordinary differential equations based on Runge-Kutta methods. Our method is applied to the well-known cruise-control problem, under a simplified linear version and with the aerodynamic force taken into account leading to a non-linear formulation.