  Linear Time Invariant (LTI) systems are ubiquitous in control applications.
Unbounded-time reachability analysis that can cope with industrial-scale models
with thousands of variables is needed. To tackle this problem, we use abstract
acceleration, a method for unbounded-time polyhedral reachability analysis for
linear systems. Existing variants of the method are restricted to closed
systems, i.e., dynamical models without inputs or non-determinism. In this
paper, we present an extension of abstract acceleration to linear loops with
inputs, which correspond to discrete-time LTI control systems under guard
conditions. The new method relies on a relaxation of the solution of the linear
dynamical equation that leads to a precise over-approximation of the set of
reachable states, which are evaluated using support functions. In order to
increase scalability, we use floating-point computations and ensure soundness
by interval arithmetic. Our experiments show that performance increases by
several orders of magnitude over alternative approaches in the literature. In
turn, this tremendous gain allows us to improve on precision by computing more
expensive abstractions. We outperform state-of-the-art tools for unbounded-time
analysis of LTI system with inputs in speed as well as in precision.
