We consider a real-time system of multiple tasks, each task having a plant to control. The overall quadratic control cost is to be optimized. We exploit the periodicity of the task response time, which corresponds to a periodic delay pattern in the feedback control loop. Perturbed periods are used as a tool to find a finite hyperperiod. We present an analytical procedure to design a periodic linear-quadratic-Gaussian (LQG) controller for tasks with fixed execution times as well as a numerical solution to the periodic--stochastic LQG problem for tasks with variable execution times. The controllers are evaluated using simulations in real-time scheduling and control co-design examples.
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