CAD.2:5 - Archetypal Grounding

CAD.2:5.1 - A held reference

For the constructed plant x_{k+1}=x_k+u_k, choose

u_k=(1-a)(r_k-x_k),\qquad a=\tfrac12.

With a held reference r, e_{k+1}=ae_k. Let the reference change by \Delta r_j at the start of each N-step block. The error immediately after that change obeys

e_{j+1,0}=a^Ne_{j,0}-\Delta r_j.

For |\Delta r_j|\leq\delta, the interval |e|\leq E is invariant at block starts if

E\geq a^NE+\delta,\quad\text{hence}\quad E=\frac{\delta}{1-a^N}.

It also bounds intermediate fast instants because |a|\leq1. The initialization condition is |e_{0,0}|\leq E. With N=4,\delta=0.1, E=8/75. The reference jump, not instability of the fast law, determines this worst-case envelope.

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.