CAD.3:5 - Archetypal Grounding

For the held-reference case of CAD.2, $a=1/2,N=4,\delta=0.1$ give $E=8/75$ . Since $|u|=(1-a)|e|$ , a sufficient input bound is $4/75\approx0.05333<0.06$ . Requiring $|r|\leq1-E=67/75$ also suffices for $|x|\leq1$ . This position tightening is conservative: the scalar convex update can support less restrictive position arguments. It is not claimed to be the maximal command domain.

Now double the requested average progression from 0.025 to 0.05 per fast tick while retaining four-tick held updates. Then $\delta=0.2$ , $E=16/75$ , and the sufficient input bound is $8/75\approx0.10667$ . More decisively, an initial jump from $x=r=0$ to $r=0.2$ commands $u=0.1$ , directly violating the limit. Merely saturating u changes the error dynamics and requires a new analysis.

For continuous joined segments with $|d_k|\leq0.05$ , $E=0.1$ and $|u|\leq0.05$ . Choose $|r_k|\leq0.9$ and initialize $|x_0-r_0|\leq0.1$ . Then the stated state and input limits hold at every fast sample. Generate the next reference by clipping $r_k+d_k$ to the interval $[-0.9,0.9]$ , with $|d_k|\leq0.05$ . The choice $d=0$ is always available, so the reference domain has an explicit continuation. This proves model feasibility even at a boundary; it does not promise arrival at every requested target or continuous physical safety between samples.