Optimal network transmission to minimize state-estimation error and channel usage

We consider the problem of transmitting the state value of a dynamical system through a communication network. The dynamics of the error in state estimation is modeled using a stochastic hybrid system formalism, where the error grows exponentially over time. Transmission occurs over the network at specific times to acquire the system's state, and whenever a transmission is triggered, the error is reset to a zero-mean random variable. Our goal is to uncover transmission strategies that minimize a combination of the steady-state error variance and the average number of transmissions per unit of time. We found that a constant Poisson rate of transmission results in a heavy-tailed distribution for the estimation error. Next, we consider a random non-threshold transmission rate that varies as a power law of the error. Finally, we explore a threshold-based rate in which transmission occurs exactly when the error reaches a threshold. Our results show that if the error's variance after transmission is small enough, a threshold-based strategy is the optimal paradigm. On the other hand, if this variance is large, and the error does not grow fast enough, the random non-threshold transmission strategy emerges as optimal. These analytical results are verified by simulations of the stochastic hybrid system.