History

Suppose that the $k$th stage has passed. What information is available? It is assumed that at every stage, a sensor observation is made. This results in a sensing history,

$${\tilde{y}}_k = (y_1,y_2,\ldots,y_k).$$ (11.12)

At every stage an action can also be applied, which yields an action history,

$${\tilde{u}}_{k-1} = (u_1,u_2,\ldots,u_{k-1}).$$ (11.13)

Note that the action history only runs to $u_{k-1}$; if $u_k$ is applied, the state $x_{k+1}$ and stage $k+1$ are obtained, which lie beyond the current stage, $k$. By combining the sensing and action histories, the history at stage $k$ is $({\tilde{u}}_{k-1},{\tilde{y}}_k)$.