CAD.5:4.7 - Connect the information result to a layered control construction

The method can be used without a layered architecture. When CAD.2 supplies a nominal state (z), command (v), projection (P) and control interface, replace its exact-state feedback by the implemented estimate:

u=K(\hat x-Pz)+Qz+Rv,\qquad e=x-Pz,\qquad \eta=x-\hat x.

For (x^+=Ax+Bu+w) and (z^+=A_pz+B_pv), the derived error dynamics are

e^+=F e+H z+G v+w-BK\eta,

where (F=A+BK), (H=AP+BQ-PA_p), and (G=BR-PB_p). The additional forcing (-BK\eta) is the effect of acting on the estimate.

Use the effective disturbance set (W\oplus(-BK)V_\eta) in CAD.3’s invariant-error construction. For every allowed nominal state and command, state admissibility requires ({Pz}\oplus E\subseteq X), while the implemented input requires

\{Qz+Rv\}\oplus K E\oplus(-K)V_\eta\subseteq U.

These sums are conservative enclosures, not assertions that tracking and estimation errors are independent. A joint set can retain their correlation when the separate bounds are too costly.

CAD.4 then compares whole arrangements using the estimator’s memory, communication, worst-case completion time, admissible packet timing and resulting error set. A new sensor changes (V_\eta), which can change the tracking-error set (E), the allowed nominal motion and the value of the whole architecture. A larger planner can also increase sensing or estimation latency through a shared resource, requiring the same calculation in reverse.

The added result is the construction of observation availability, internal information state and estimation uncertainty. CAD.1–4 continue to own decomposition, operational interfaces, tracking bounds and whole architectural comparison; this method does not replace them.