To measure or to control: optimal control with scheduled measurements and controls

We consider a linear discrete-time optimal control problem where the controller has to choose between measurement and control. The controller is restricted in such a way that at a given time instance, it may either send a control packet to the actuator, or it can request a measurement packet from the sensor. We show that under a quadratic cost structure that does not penalize the control directly, the optimal control is a linear threshold policy, where the controller decides to measure or control by comparing its last measurement of the plant state against a pre-determined threshold