A generic ellipsoid abstract domain for linear time invariant systems

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 im-precise results, if any at all, on these systems. Previous work by Feret proposes a specific abstraction for digital filters that addresses this issue on a specific class of controllers. This paper aims at generalizing the idea. It works directly on system representation, relying on existing methods from control theory to automatically generate quadratic invari-ants 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 com-pute 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 sound-ness of the computed invariants. Keywords stable linear systems, ellipsoids, quadratic invariants, Lya-punov functions, semi-definite programming, floating point errors, abstract interpretation