Cost Functions for Scheduling Tasks in Cyber-physical Systems

In Cyber Physical Systems (CPS), computational delays can cause the controlled plant to exhibit degraded control. The traditional approach to scheduling in such systems has been to define controller task deadlines, based on the dynamics of the controlled plant. Controller tasks are then scheduled to meet these deadlines; meeting the deadline is considered the sole criterion for scheduling success. This traditional approach has the advantage of simplicity, but overlooks the fact that the quality of control depends on the actual task response times. Two different schedules, each satisfying the task deadlines, can provide very different levels of control quality, if their task response times are different. In this paper, we consider using cost functions of task response time to capture the impact of computational delay on the quality of control. Since the controller workload typically consists of multiple tasks, these cost functions are multivariate in nature. Furthermore, since these tasks are generally coupled, the response time of one control task can affect the sensitivity of the controlled plant to the response times of other tasks. In this paper, we first demonstrate how a multivariate cost function can be formulated to quantify the effect of computational delays in vehicles. We then develop cost-sensitive real-time control task scheduling algorithms. We use as an application example an automobile: the controller workload consists of steering and torque control. Our results indicate that cost-function-based scheduling provides superior control to the traditional deadline-only-based approach.