On the optimum timing of observations for linear control systems with unknown initial state

A linear system with a cost function that is quadratic in the control and terminal position error is given. The initial state is unknown and noise-corrupted observations are taken. Due to the cost of taking the observations, they are limited to a given number. Using the tools of optimum stochastic control theory, the optimum timing of the available observations is determined (minimizing the expected value of the loss function). For the cases considered, the locations can be determined a priori, and do not depend on the values of previous observations. Graphs of the optimum observation locations for several scalar cases are given. Graphs of the true cost, as a function of the locations of the observations, are also given for several scalar cases. The change in cost for small variations from the optimum location is usually not significant.