CAD.5:4.3 - Construct an internal state that can be updated

Let (D) be the largest allowed measurement age. A general sufficient representation retains the compatible joint set of recent states

\xi_k=(x_k,x_{k-1},\ldots,x_{k-D}),\qquad \xi_k\in\Xi_k,

together with the applied-input history, sample identities and the schedule or arrival mode needed for subsequent updates. At startup retain only states at nonnegative times and use (X_0); do not invent a prehistory. The set (\Xi_k) represents states compatible with the initial condition, model, applied inputs and all observations already incorporated.

Predict the joint set by the map

T(\xi_k,u^a_k,w_k) =(A x_k+B u^a_k+w_k,\;x_k,\ldots,x_{k-D+1}).

For a packet arriving at the next action time, intersect this predicted set with the measurement constraint on its historical component (x_\tau). With several packets, intersect all their valid constraints. The projection onto the first component gives the possible current states (X_k=\pi_0\Xi_k).

This is a constructive sufficient internal state, not a claim of minimum order. It preserves the correlation between a past observation and the present through intervening dynamics and actions. Separate marginal intervals need not preserve that correlation; using them as a Cartesian outer approximation is safe only when the ensuing larger uncertainty is carried into control.

A practical realization can use a replay buffer, an observer with equivalent sufficient memory, or a recursive outer set such as an interval, polytope or zonotope. In each case, the represented set must contain every state compatible with the stated model and data. Approximation may enlarge that set; it may not discard a possible state merely to obtain a feasible control.