CAD.5:4.4 - Preserve the information needed by the next update

At each action time, use the following causal order:

1. Propagate the previous representation using the input actually executed and the allowed disturbance.
2. Incorporate each newly available packet at its sample time, then obtain the current-state projection.
3. Choose the estimate or set-based action from that projection, apply it by the qualified deadline, and retain the input information needed by the next update.

A state older than (D) can be projected out once no allowed future packet refers to it. This is different from throwing away the dependence between retained states. Duplicate packets do not supply additional independent information. In a bounded-set construction, independent noise need not be assumed: common error sources must be represented jointly or enclosed conservatively. With an uncertain timestamp, use the union of the admissible timestamp hypotheses, not their intersection.

If the compatible set becomes empty, the current model, bounds, packet, timestamp or executed-input record is inconsistent. Do not silently reset to a convenient point. Use a separately qualified fallback, obtain a new trusted initialization, or declare the demanded continuation unsupported. Packet loss or delay outside the allowed family likewise invalidates the relevant bound until a qualified recovery applies.

Reduce the internal representation when its update and control use remain sufficient. One exact reduction is available when (s=T_rx) has closed dynamics (T_rA=A_rT_r), the needed observations factor as (C_j=C_{rj}T_r), and the control law and constrained quantities depend only on (s). Then use (s^+=A_rs+T_rB u^a+T_rw) and the corresponding observation sets. If an omitted component influences a required future quantity, its influence must instead be retained or bounded.

Thus two histories may be merged only when the merged representation still supports every required future update and action under the allowed future observations and disturbances. Equal current estimates are not enough. A.3.3.PI develops this general sufficiency test; here the joint-state construction provides a direct control-specific realization.