CAD.5:5.3 - Increase the delay, then change the design or the demand

Suppose bus contention changes S’s age from two ticks to four, while F and the requirements remain unchanged. The reconstructed second-coordinate radius becomes

r_2(4)=0.8^4(0.005)+0.02(1+0.8+0.8^2+0.8^3)=0.061088.

The same controller’s next-state bound is (0.8r_2(4)+0.02=0.0688704), which exceeds (0.06). More decisively, four unobserved disturbances plus the next one yield the half-width (0.067232) from :5.2 even if S’s old measurement is exact. F still reveals nothing about the second coordinate. Thus a different filter or gain using the same available information cannot restore this hard bound.

The engineer must change a consequential condition. Restore the qualified two-tick transport arrangement, obtain a timely observation of the second coordinate, or revise the task or physical dynamics. If the sensor budget becomes 3, H alone supports the same feedback with next-state bounds (0.046) and (0.024); its input bounds are (0.102) and (0.052). H needs no delayed-observation reconstruction. It is now a simpler sufficient choice.

A different task can also justify less information. If the allowed second-coordinate deviation becomes (0.12), retain F and set (u_2=0). The unobserved stable second coordinate has the invariant bound

|x_2|\le\frac{0.02}{1-0.8}=0.1<0.12.

S, its delayed packet processing and its input-history reconstruction are then unnecessary. This is sufficient for the changed task; it is not a cheaper solution to the original (0.06) requirement.