A scalable algebraic method to infer quadratic invariants of switched systems

We present a new numerical abstract domain based on ellipsoids designed for the formal verification of switched linear systems. Unlike the existing approaches, this domain does not rely on a user-given template. We overcome the difficulty that ellipsoids do not have a lattice structure by exhibiting a canonical operator over-approximating the union. This operator is the only one which permits to perform analyses that are invariant with respect to a linear transformation of state variables. Moreover, we show that this operator can be computed efficiently using basic algebraic operations on positive semidefinite matrices. We finally develop a fast non-linear power-type algorithm, which allows one to determine sound quadratic invariants on switched systems in a tractable way, by solving fixed point problems over the space of ellipsoids. We test our approach on several benchmarks, and compare it with the standard techniques based on linear matrix inequalities, showing an important speedup on typical instances.