A sampled feedback recurrence has to start somewhere, so the initial state changes the early error path even when target and correction stay fixed. The three sequences are exact only for the declared target, correction fraction, initial state, ideal linear update, and deterministic sampling; real control loops add sensor and actuator dynamics, delay, saturation, quantization, noise, temperature drift, and voltage/time measurement uncertainty.
Holding target and correction fixed shows the sampled recurrence from three starting states. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
Starting from zero reaches six after two corrections
With target 8 volts and correction 1/2, initial state 0 volts gives sequence 0, 4, 6 volts.
0,4,6
Starting from four reaches seven after two corrections
The same correction from initial state 4 volts gives sequence 4, 6, 7 volts.
4,6,7
Starting from six reaches fifteen halves
Initial state 6 volts gives sequence 6, 7, 15/2 volts.