An optimal LQG controller for stochastic event-triggered scheduling over a lossy communication network

We consider a networked control loop in which the sensors acquire partial state information and communicate to a remote controller through a lossy communication network. A scheduler, collocated with the sensors, decides to transmit a locally estimated state to the controller based on an event-triggered transmission policy with stochastic thresholds. Assuming that the local estimator either senses the communication channel or receives an ideal acknowledgment from the remote estimator, then the optimal control law can be shown to be a linear function of the conditional expectation of the state. However, the probability distribution of the state conditioned on the information available to the controller based on the mentioned transmission policy and network is not Gaussian, but rather described by a sum of Gaussians with an increasing number of terms at every time-step. We show that the optimal LQG control law can be determined without tracking this probability distribution for finding its expected value. Moreover, we establish that the stochastic event-triggered scheduler can be appropriately regulated in order to achieve a desired triggering probability at every time-step.