A Generic Ellipsoid Abstract Domain for Linear Time Invariant Systems

Pierre Roux (1) , Romain Jobredeaux (2) , Pierre-Loïc Garoche (1) , Eric Féron (2)

1 ONERA - The French Aerospace Lab [Toulouse]
2 GA TECH - Daniel Guggenheim School of Aerospace Engineering

Embedded system control often relies on linear systems, which admit quadratic invariants. The parts of the code that host linear system implementations need dedicated analysis tools since intervals or linear abstract domains will give imprecise results, if any at all, on these systems. Reference~\cite{feret:esop04} proposes a specific abstraction for digital filters that addresses this issue on a specific class of controllers. This paper aims at generalizing the idea, relying on existing methods from Control Theory to automatically generate quadratic invariants for linear time invariant systems, whose stability is provable. This class encompasses $n$-th order digital filters and, in general, controllers embedded in critical systems. While control theorists only focus on the existence of such invariants, this paper proposes a method to effectively compute tight ones. The method has been implemented and applied to some benchmark systems, giving good results. It also considers floating points issues and validates the soundness of the computed invariants.

Mots clés: stable linear systems; ellipsoids; quadratic invariants; Lyapunov functions; semi-definite programming; floating point errors; abstract interpretation.; abstract interpretation

Domaines: Systèmes embarqués

Pierre Roux, Romain Jobredeaux, Pierre-Loïc Garoche, Eric Féron. A Generic Ellipsoid Abstract Domain for Linear Time Invariant Systems. 2011. ⟨hal-00608673v2⟩