Optimal Resource Allocation for Stochastic Systems Performance Optimisation of Control Tasks undergoing Stochastic Execution Times

The problem addressed in this paper is the optimal allocation of a CPU to a number of software control tasks. Each task is used to implement a feedback controller for a linear and time invariant system and is activated with a fixed period. On every periodic activation, the task executes a job, which collects the output of the system, and produces the control values after executing for a random computation time. If a job's duration exceeds a deadline, then the job is cancelled and the control values are not updated. The systems to be controlled are affected by process noise. Therefore the performance of each control loop can be evaluated through the steady state covariance of the system's state, which depends on the probability with which the task implementing the controller drops its jobs. We show that by making a proper choice for the scheduling algorithm, this probability can be straightforwardly computed as a function of the scheduling parameters. This observation enables the construction of a very efficient procedure for finding the scheduling parameters that attain the optimal tradeoff between the performance of the different control loops.