You come in one morning and find that the temperature is 21°C. To correct this, you move the thermostat down by 1°C to 19°C. But the next day the temperature has dropped to 18.5°C.
Expanding oscillations can result from improperly tuned feedback systems. Suppose that a thermostat is supposed to keep a room at 20°C. Follow the steps to figure out what is happening in each of the given cases.
a. Suppose the temperature produced is a linear function of the temperature on the thermometer. Find this function.
b. You continue to respond to a temperature x°C above 20 by setting the thermostat to 20 - x°C, and to a temperature x°C below 20° by setting the thermostat to 20 + x°C. Find the temperature for the next few days.
c. Denote the temperature on day t by Tt. Find a formula for the thermostat setting zt in response.
d. Use the answer to a to find Tt + 1. Write the updating function.
e. Use the stability condition to describe what will happen in this room.