On the Nonexistence of Event Triggers That Preserve Gaussian State in Presence of Packet-Drop

Remote estimation is fundamental to studying a cyber-physical system. The occurrence of packet drop and the usage of event-based trigger have also been studied separately as practical additions to the estimation problem. Because the Kalman filter can be successfully modified slightly to accommodate each of these two additions, it is desired to attain a modification of the Kalman update equations that incorporates both. More precisely, it will be ideal to design an event-based trigger under a Bernoulli packet drop such that the resulting distribution of the state variable remains Gaussian. The article shows that this is impossible. We further explore a larger family of probability distributions, the p-generalized normal distribution, and show that the existence of the trigger is impossible in most cases. The case under which the trigger does exist is identified.