C.43:5.5 - Explain both effective and ineffective feedback in the same account

Consider a stipulated discrete-time model x_(t+1) = x_t + u_t. Its controlled quantity is deviation from zero, and its available input is u_t = -k*x_t. Assume current state observations and immediate execution, without saturation. The resulting update is x_(t+1) = (1-k)*x_t.

At k = 0.5, successive deviations are one half and one quarter of the initial deviation. At k = 2.5, they are -1.5*x_0 and 2.25*x_0: the sign alternates and the magnitude grows. Both settings realize observation-dependent feedback in this model. The second is unsuitable for reducing the deviation, even though the description contains a sensor, rule and actuator. At k = 0, the input is zero and this correcting dependence disappears. The equation describes that limit; whether holding a zero input is a useful chosen action is a separate question about the situation.

Suppose an actual updating component and parameter memory are distinct parts of the controller assembly. The component selects a new gain from observed behavior and writes it to the memory used by the acting controller. A.12 can expose this internal action. If the update changes the gain from 0.5 to 2.5, its being controlled and traceable does not make the result an improvement. The changed rule must be compared through its consequences for the subsequent operation. A recovery to an earlier qualified setting is available only if the implementation retains and can restore it under the current conditions.

Now change the observation condition: the rule receives a delayed state. The original immediate-observation update no longer describes the joined operation. Recover the delayed information and the implemented rule, then evaluate that model or change the arrangement. Do not fix the explanation by declaring the missing timely observation a zero state. These returns preserve the common control relations while changing the subject calculation actually needed.