CAD.5:4.1 - Fix the action time and the control-relevant distinctions

Use the sampled plant model

x_{k+1}=A x_k+B u^a_k+w_k,\qquad w_k\in W,\qquad x_0\in X_0.

Here (u^a_k) is the input actually applied during the update, not merely the requested command. Specify state or output constraints, input limits and the duration for which they must hold. The discrete guarantee covers sample instants; physical constraints between samples require a qualified intersample model and bound. A tracking problem can use deviation coordinates about a known reference; changes of that reference must remain in the model.

For the direct construction below, (A,B,W,X_0) are known, (W) and (X_0) are bounded, and the executed input is available to the estimator. Known actuator queues can first be included in the plant state. Unknown actuation error must be bounded and included in the model; an unbounded discrepancy between requested and applied input defeats the stated guarantee.

State what is available before selecting (u_k). A packet that arrives after that selection cannot support that action. Include sampling phase, transmission, computation and actuation timing in this account. Specify a maximum age of usable observations and a maximum gap between required samples: a delivery-delay bound alone does not guarantee that a sensor samples often enough.

Ask which possible situations the controller must distinguish. If two situations consistent with all available information require disjoint admissible action sets, no deterministic causal controller using only that information can meet the requirement for both. Conversely, do not demand that every state coordinate be accurately observed if their remaining differences do not prevent an acceptable action and continuation.

Begin with the least elaborate plausible arrangement. Include a qualified current sensor, a fixed observer already meeting the task, or no observation of a harmless stable component. These are substantive alternatives, not merely baselines to be defeated.