CAD.2:4 - Solution
CAD.2:4.1 - Define the exchanged behavior
Use MMP.18 to establish meanings, units, coordinate frames, observation times, validity intervals and the operation performed between messages. A target, an input, an output trajectory and a constraint are different commands. State which one the lower computation receives.
Choose how the upper model represents the lower behavior. For a planning state z and plant state x, a reconstruction Pz can define an error e=x-Pz. A measured projection z=\Pi x is a different map. Even if \Pi P=I, initializing z=\Pi x leaves e=(I-P\Pi)x, which need not vanish.
If the controlled quantity is y=Cx and the planner predicts y_p=C_pz, then
y-y_p=Ce+(CP-C_p)z.
Output agreement requires the second term to vanish or be included in the bound. State reconstruction, output matching and physical identity are separate questions.
CAD.2:4.2 - Derive the error instead of assuming tracking
For a useful linear branch, express both models at the fast sampling instants:
x^+=Ax+Bu+w,\qquad z^+=A_pz+B_pv.
Here w is an admitted bounded mismatch or disturbance, not a name for all unknown behavior. Select an interface law
u=K(x-Pz)+Qz+Rv.
Substitute x=Pz+e before simplifying. The actual error equation is
e^+=Fe+Hz+Gv+w, \quad F=A+BK,\quad H=AP+BQ-PA_p,\quad G=BR-PB_p.
This calculation makes three design choices visible. K changes error correction; P and Q determine whether the nominal state evolution can match; R determines input matching. Solving H=0 is useful when possible but is not automatically possible or necessary. If it fails, retain Hz and bound or redesign it. A stable F does not remove forced error from H, G or w.
Use MMP.9 to construct and qualify a reduced model before relying on it here. A fitted upper model with unexamined hidden-state memory is not made adequate by a convenient matrix dimension.
CAD.2:4.3 - Add the clock, held values and reset events
Specify the fast interval, slow update interval, which variables evolve between slow updates, and the event order at a shared instant: observation, computation, command acceptance and actuation. Derive the state just before and just after each event.
For example, if z and v are held for N fast steps, first derive the held-z mode by setting A_p=I, B_p=0 in the residual equation. With F, H and G for that mode, repeated substitution gives
e_N=F^Ne_0+\sum_{i=0}^{N-1}F^i(Hz+Gv+w_{N-1-i}).
When the next planning state is z_{\mathrm{new}}, the same physical x gives
e_{\mathrm{new}}=e_N+P(z-z_{\mathrm{new}}).
A reset is thus an error input, even when the tracker is stable. If the planner instead propagates z every fast step, use that evolution in :4.2; do not also treat z as held.
Model a known delay through the state and commands actually available to the controller. Replacing x_k by x_{k-d} changes the law; augmenting the state with the necessary history is one way to derive its dynamics. If an estimator supplies x, include estimation error and its qualification. A full-state result cannot simply be relabeled output feedback.
A variable delay or missed update needs an explicit admissible timing set and an error model covering it. A bound for one fixed period does not establish a bound for arbitrary switching among periods.
CAD.2:4.4 - Return an interface that can be tested and changed
Give CAD.3 the combined evolution, initial-error condition, admissible command class, mismatch class and exact moments at which each required constraint must hold. Retain the meaning and time convention of every exchanged quantity. These are engineering design values; no special record format is required.
A failed error bound has several possible repairs: change the reconstruction, retain an omitted mode, alter feedforward, reduce command jumps, supply an interpolated segment, shorten latency or choose another split. The error equation identifies which change addresses the failure. Do not reflexively raise every update rate.
Test the actual exchange implementation with boundary events: simultaneous update, maximum permitted delay, reset, command reversal and missing command. These tests can falsify the model of the interface; they do not prove its physical assumptions.