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.