CAD.2:5.2 - A continuous reference segment

Now let the fast computation generate r from a timed segment and let d_k=r_{k+1}-r_k, with |d_k|\leq v_{\max}. The same plant and gain give

e_{k+1}=ae_k-d_k,\qquad E=\frac{v_{\max}}{1-a}.

The planner can still update every four ticks. What changed is the command trajectory between updates. At v_{\max}=0.05, E=0.1. Segment endpoints must join continuously; an additional jump would add to d and invalidate the bound.

If a new segment is unavailable, holding the current reference gives d=0, which remains within this command class. This is an error-bound argument for that fallback, not evidence that holding position satisfies every mission requirement.