Timing Debugging for Cyber-Physical Systems

This paper is concerned with the following question: Given a set of control tasks that are not schedulable, i.e., their required timing properties cannot be satisfied, what should be changed? While the real-time systems literature proposes many different schedulability analysis techniques, it surprisingly provides almost no guidelines on what should be changed to make a task set schedulable, when it is not. We show that when the tasks in question are control tasks, this timing debugging question in the context of cyber-physical systems (CPS) may be answered by exploiting the dynamics of the physical systems that these control tasks are expected to influence. Towards this, we study a very simple setup, viz., when a set of periodic tasks with implicit deadlines is not schedulable, by how much should the periods be changed in order to make the task set schedulable? Among the many ways in which the periods can be modified, our proposed strategy is to change the periods in a manner such that while the task set becomes schedulable, the poles of the closed-loop system experience the minimal shift. Since the poles influence the closed loop dynamics of the system, we thereby ensure that we obtain a system with the desired timing properties whose dynamics is very similar to the dynamics of the original (non-schedulable) system. We formulate this CPS timing debugging strategy as an optimization problem and illustrate it with a concrete example.