Intervals and Contractors for Dynamical Systems – CONTREDO

The CONTREDO project (Intervals et contractors for dynamic systems) aims at designing a software tool based on intervals to handle dynamic systems.

Intervals take naturally into account rounding errors in the computations and bounded uncertainties of the parameters. Intervals thus make it possible to handle in a reliable way dynamic systems for which initial conditions, parameters, control errors and measurements taken along time are known with a bounded uncertainty.

Our software tool will propose a general language for defining dynamic systems and original algorithmic operators. The language will enable to define a so-called trajectory constraint satisfaction problem (TCSP), i.e. a system containing ODEs, but also other constraints on the state variables: delay constraints, inter-temporal constraints, non numerical constraints, and so on. The language should allow the manipulation of entities that are not native in standard interval methods, like functions, state variables, i.e. trajectories comprised in a domain forming a "tube", and also variables representing the derivative functions.

The CONTREDO project will also work on obtaining efficient solving algorithms. First, the operators defined in the language will be associated to original primitive \textit{contractors} (operators reducing the variable domains in polynomial time). This will enable to contract the domains of the different entities defined in a dynamic system (TCSP), in particular the tubes of the trajectories. Second, this TCSP solver will use new sophisticated contractors to enhance its performance results. Standard guaranteed integration methods, like VNODE-LP, will be encapsulated inside contractors. We will also propose contractors dedicated to specific constraints like delay or inter-temporal constraints. Third, advanced features will be studied to improve the performances, including clever time discretization, bisection or "shaving" interval methods, or symbolic reformulation of expressions (improving interval computations).

The project will be validated on several applications. The project feasibility will be first demonstrated on existing maritime robotics applications. This will check the expressiveness power of the language and performance of algorithms. The main application carried out by our industrial partner MBDA deals with the validation and computation of temporal invariance sets (V-stability) and the calculation of an attraction basin. This problem has numerous civil (e.g., autonomous vehicle in urban area, with its security aspects) and military (missile guidance) aeronautic applications. The project will also study the challenging model predictive control problem, which has applications in surgical robotics (adaptive needle insertion).

This software tool will be implemented in the free C++ library IBEX (Interval-Based EXplorer) developed by several partners. IBEX is already used by robotic and control practitioners for handling dynamic systems, but no tool currently offers high level features taking into account the dynamics of applications while managing additional constraints and uncertainties on initial conditions and command errors. This is the ambition behind CONTREDO that gathers the main force task who has laid the algorithmic foundations for such a tool.