CAD.Reference:3.1 - How information changes a command margin

CAD.5 constructs compatible state histories from the measurements actually available before the action. Past inputs must be the inputs actually applied; a planned input or a measured value treated as current can produce a different information set. A run-wide bound (V_\eta) must cover startup and every allowed delivery mode, not only a favorable observation window.

When the interface is

u=K(\hat x-Pz)+Qz+Rv,

CAD.5:4.7 derives the estimate-based version of CAD.2’s tracking equation:

e^+=Fe+Hz+Gv+w-BK\eta.

CAD.3 therefore receives the effective forcing (W\oplus(-BK)V_\eta). For every admitted nominal state and command, its state and input tests become

\{Pz\}\oplus E\subseteq X,\qquad \{Qz+Rv\}\oplus KE\oplus(-K)V_\eta\subseteq U.

The separate set sums are sufficient enclosures, not assertions that tracking and estimation errors are independent. A tighter joint representation may recover a useful design that these separate margins reject. A failure of a sufficient enclosure is consequently different from CAD.5’s indistinguishability proof that no action can meet the demand.

This is a two-way design connection. A tighter required command domain can require fresher observations. A sensor or estimator change can enlarge (V_\eta), shrink the usable command domain and invalidate a nominal performance claim. CAD.4 includes the cost of obtaining and updating that information. If a larger planner delays the estimator or communication, the old age bound is no longer a premise of the same realization. Reuse an unaffected result, but return the changed timing to CAD.5 and the resulting error to CAD.3.