MMP.8:5.1 - Choose a preload before or after learning an external load

An ideal static arrangement has a downward load w, an adjustable upward preload a and a residual y=w-a. A later use requires abs(y)<=1/4. The load is either 1 or 2, and a can be any value from 0 to 2. The relation is a supplied illustrative mechanical model; the pattern’s work is to formulate the choice and its information.

If w=1, acceptable choices form [3/4,5/4]; if w=2, they form [7/4,2] after the actuator limit is applied. Both sets are nonempty, so each known load has a feasible choice. Their intersection is empty. No one preload chosen before distinguishing the loads can meet the requirement for both.

Suppose a reading available before adjustment reports which of the two loads is present. The policy a=w then meets the requirement. A reading received only after the preload is locked cannot support that policy at the relevant decision.

Now the tolerance is relaxed to 3/5. The acceptable intervals overlap from 7/5 to 8/5, so a=3/2 works before any reading. Additional measurement is unnecessary for this revised requirement. The change is in the required result, not the sophistication of the computation.