  We describe new methods for deciding the stability of switching systems. The
methods build on two ideas previously appeared in the literature: the polytope
norm iterative construction, and the lifting procedure. Moreover, the
combination of these two ideas allows us to introduce a pruning algorithm which
can importantly reduce the computational burden. We prove several appealing
theoretical properties of our methods like a finiteness computational result
which extends a known result for unlifted sets of matrices, and provide
numerical examples of their good behaviour.
