CAD.5:5.2 - Why the individual channels do not suffice

With F alone, the second coordinate is never observed and does not influence the observed first coordinate. Consider two runs with opposite second-coordinate disturbances and otherwise identical information. The controller chooses the same second-coordinate commands in both. After five updates, half their separation is

0.02(1+0.8+0.8^2+0.8^3+0.8^4)=0.067232>0.06.

No common center can place both states inside the required interval. This excludes every causal controller using F alone under the stated assumptions, not just (u=-A\hat x).

With S alone, the latest two first-coordinate disturbances are unobserved when the next command is chosen. Even with perfect older observations, the next first-coordinate interval has unavoidable half-width

0.04(1+1.2+1.2^2)=0.1456>0.08.

The same indistinguishability argument rules out the required all-time bound. Improving the old measurement’s accuracy cannot recover these disturbances before their consequences matter.

F+S is therefore sufficient where either offered channel alone is insufficient, under the same sensor budget and timing assumptions. This establishes a task-dependent benefit of combining channels. It does not establish that all heterogeneous sensing is superior or that the offered catalog is globally optimal.