Quantitative Resilience of Linear Driftless Systems.

This paper introduces the notion of quantitative resilience of a control system. Following prior work, we study linear driftless systems enduring a loss of control authority over some of their 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 a malfunction, a resilient system might be significantly slower to reach a target compared to its initial capabilities. We quantify this loss of performance through the new concept of quantitative resilience. We define such a metric as the maximal ratio of the minimal times required to reach any target for the initial and malfunctioning systems. Naïve 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. Relying on control theory and on two novel geometric results we reduce the computation of quantitative resilience to a single linear optimization problem. We demonstrate our method on an opinion dynamics scenario.

Resilient Reachability for Linear Systems.

A fault-tolerant system is able to reach its goal even when some of its components are malfunctioning. This paper examines tolerance to a specific type of malfunction: the loss of control authority over actuators. Namely, we investigate whether the desired target set for a linear system remains reachable under any undesirable input. Contrary to robust control, we assume that the undesirable inputs can be observed in real time, and subsequently allow the control inputs to depend on these undesirable inputs. Building on previous work on reachability with undesirable inputs, this paper develops a reachability condition for linear systems, and obtains a formula that describes reachability of the goal set for driftless linear systems by computing the minimum of a concave-convex objective function. From this formulation we establish two novel sufficient conditions for resilient reachability.