A.3.3.TR:5.4 - A switch changes the law within the prediction interval

Consider a stated temperature model with T in degrees Celsius and heater mode OFF or ON. While OFF, temperature falls at 1 degree/minute; while ON, it rises at 2 degrees/minute. OFF at T<=18 switches immediately to ON. ON at T>=20 switches immediately to OFF. A switch changes the mode and preserves T.

Start at T=19, OFF. The requested result is temperature and mode at 1.5 minutes. In OFF mode, flow is permitted while T>18. The trajectory T(t)=19-t reaches 18 at minute 1; the event then switches the mode to ON. The new law gives T(t)=18+2*(t-1) until T reaches 20. At 1.5 minutes the result is 19 degrees, ON.

At minute 2, the upper-threshold event switches to OFF; the next segment uses T(t)=20-(t-2). This continuation follows from retaining the mode and restarting the evolution after each event.

Keeping the initial OFF law through the whole first interval would give 17.5 degrees. Checking the switch only at the interval’s end describes a controller that samples at those times, with different behavior from the supplied immediate-event model. A real controller’s sampling or delay therefore belongs in its modeled timing.