History information states

The history, $({\tilde{u}}_{k-1},{\tilde{y}}_k)$, in combination with the initial condition, ${\eta_0}$, yields the history I-state, which is denoted by ${\eta}_k$. This corresponds to all information that is known up to stage $k$. In spite of the fact that the states, $x_1$, $\ldots$, $x_k$, might not be known, the history I-states are always known because they are defined directly in terms of available information. Thus, the history I-state is

$${\eta}_k = ({\eta_0},{\tilde{u}}_{k-1},{\tilde{y}}_k) .$$ (11.14)

When representing I-spaces, we will generally ignore the problem of nesting parentheses. For example, (11.14) is treated a single sequence, instead of a sequence that contains two sequences. This distinction is insignificant for the purposes of decision making.

The history I-state, ${\eta}_k$, can also be expressed as

$${\eta}_k = ({\eta}_{k-1},u_{k-1},y_k) ,$$ (11.15)

by noticing that the history I-state at stage $k$ contains all of the information from the history I-state at stage $k-1$. The only new information is the most recently applied action, $u_{k-1}$, and the current sensor observation, $y_k$.