CAD.4:5 - Archetypal Grounding
CAD.4:5.1 - Compare rates at the same requested progression
For the discrete actuator example, assume a dedicated planner resource needs two fast ticks per result. The fast controller has a separate resource. Commands are prepared for their applicability interval; initialization supplies the first segment. This is a declared toy resource model, not measured processor performance.
At progression 0.025 per fast tick, a held reference changes by \delta=N/40. At progression 0.05 it changes by \delta=N/20. The sufficient error and input bounds are derived from CAD.2, with a=1/2.
| Command and demand | N | Error bound E | Input bound | Two-tick planning time fits? |
|---|---|---|---|---|
| Held, 0.025 per tick | 1 | 0.05 | 0.025 | No |
| Held, 0.025 per tick | 2 | 0.066667 | 0.033333 | Yes |
| Held, 0.025 per tick | 4 | 0.106667 | 0.053333 | Yes |
| Held, 0.05 per tick | 1 | 0.1 | 0.05 | No |
| Held, 0.05 per tick | 2 | 0.133333 | 0.066667 | Yes |
| Held, 0.05 per tick | 4 | 0.213333 | 0.106667 | Yes |
| Joined segment, increment at most 0.05 per tick | 4 | 0.1 | 0.05 | Yes |
At the original demand, two held-command arrangements pass this sufficient input test; their compute use and tracking behavior can be compared. At doubled demand, neither of the considered held arrangements that fits the planning time passes it. The N=4 case also has the direct violating first jump shown in CAD.3. For N=2, two successive positive changes of 0.1 from rest produce an input of 1/16 = 0.0625 at the second change, also above the limit.
The segment arrangement admits the higher demand under the stated sampled model and margins. It adds a clocked reference generator and continuity checks. It does not demonstrate that this arrangement dominates all possible controllers or that more than two layers would help.
CAD.4:5.2 - Carry the mathematical result into execution
The planner proposes a next endpoint and applicability time. The reference generator checks that the segment starts at the current reference, remains in [-0.9,0.9], and advances by at most 0.05 per fast tick. The fast law uses that generated reference, not the endpoint held for four ticks. An unavailable segment leaves the reference unchanged; the initialization error must already satisfy the bound.
A hardware test that reveals a one-tick delay in actuation changes x_{k+1}=x_k+u_k into a model with an input delay. The table is no longer sufficient evidence. Return to CAD.2 to derive the delayed dynamics and to CAD.3 for the resulting margins. Retain the comparison of command meanings and the source-independent arithmetic; do not retain the unqualified physical feasibility conclusion.