Negative feedback
Outline of the problems
We will model the temperature in an
apartment that is being heated by a thermostat–controlled
heater/cooler. This system has been designed to attempt to drive the
temperature to one specified target value $T_c$. In each setup the
time dependence of the temperature will be governed by a system of
differential equations, and we will address the following
Basic questions
- Nondimensionalize the
equations.
- Can we solve the differential equations?
- What is the long term behaviour of the temperature $T$? Will
it converge to the desired temperature $T_c$?
- If we want to make the temperature converge faster, how should
we change the parameters in the problem?
Single room
Consider a one–room apartment with a heater/cooler, and a thermometer. The
thermometer is coupled to the heater cooler in such a way that it will activate
the heater when the temperature $T$ is below the target value $T_c$, while it
will activate the cooler when the temperature is above $T_c$.
A very simple model for this is to assume that the heat production of the
heater/cooler is proportional to $T-T_c$, and that by Newton's cooling law the
rate of change of the temperature of the room is also proportional to
$T-T_c$.
This leads to a differential equation
\begin{equation}
\frac{dT}{dt} = -\kfb (T-T_c)
\label{eq:one-room-ode}
\end{equation}
where $\kfb$ is the feedback rate.
Problem 1. Answer the basic questions
for this one room apartment.
Double room
Consider a two room apartment with a heater/cooler in the living room, and a
thermometer in the bedroom. The temperature in the living area is $T_1$, and the
temperature in the bedroom is $T_2$.
The thermometer is again coupled to the heater/cooler in such a way that it will
activate the heater when the temperature $T$ is below the target value
$T_c$, while it will activate the cooler when the temperature is above
$T_c$.
From here on we assume that $T_c=0$.
Since the two rooms are connected heat will be exchanged between them, and we
will assume that the rate at which heat flows from room 1 to room 2 is
proportional to the temperature difference $T_1-T_2$.
We get the following system of ODE for $T_1$ and $T_2$:
\begin{align*}
\frac{dT_1}{dt} &= - \kfb T_2 + \kex (T_2-T_1) \\
\frac{dT_2}{dt} &= \kex (T_1-T_2)
\end{align*}
Here $\kex$ is the rate of heat exchange across the wall.
Write this system in matrix form $\dot\vx = A\vx$, where
\[
\vx =
\begin{bmatrix}
T_1 \\ T_2
\end{bmatrix}
\]
and find the $2\times 2$ matrix $A$.
Then answer the basic questions.
Triple room
\begin{align*}
\frac{dT_1}{dt} &= - \kfb T_3 + \kex (T_2-T_1) \\
\frac{dT_2}{dt} &= \kex (T_1-T_2) + \kex (T_3-T_2) \\
\frac{dT_3}{dt} &= \kex (T_1-T_3)
\end{align*}
Again, write this system in matrix form $\dot\vx = A\vx$,
where
\[
\vx =
\begin{bmatrix}
T_1 \\ T_2 \\ T_3
\end{bmatrix}
\]
and find the $3\times 3$ matrix $A$.
Then answer the basic questions.
