  To design critical systems engineers must be able to prove that their system
can continue with its mission even after losing control authority over some of
its actuators. Such a malfunction results in actuators producing possibly
undesirable inputs over which the controller has real-time readings but no
control. By definition, a system is resilient if it can still reach a target
after a partial loss of control authority. However, after such a malfunction, a
resilient system might be significantly slower to reach a target compared to
its initial capabilities. To quantify this loss of performance we introduce the
notion of quantitative resilience as the maximal ratio of the minimal times
required to reach any target for the initial and malfunctioning systems. Naive
computation of quantitative resilience directly from the definition is a
complex task as it requires solving four nested, possibly nonlinear,
optimization problems. The main technical contribution of this work is to
provide an efficient method to compute quantitative resilience of control
systems with multiple integrators and nonsymmetric input sets. Relying on
control theory and on two novel geometric results we reduce the computation of
quantitative resilience to a linear optimization problem. We illustrate our
method on two numerical examples: a trajectory controller for low-thrust
spacecrafts and a UAV with eight propellers.
