Question 14.1: Consider a heating system for a one-room sealed-up house, sh...

The heat balance equation is

mc\frac{dT_{1r}(t) }{dt} =Q_{in}(t)-Q_{loss}(t) .       (14.31)

The rate of heat losses, Q_{loss} , is given by

Q_{loss}(t)=U_{o}\left[T_{1r}(t)-T_{2r}(t)\right]  ,       (14.32)

where U_{o} is a heat loss coefficient. The heat balance equation becomes

mc\frac{dT_{1r}(t) }{dt} =Q_{in}(t)-U_{o}\left[T_{1r}(t)-T_{2r}(t)\right] .   (14.33)

Transferring Equation (14.33) from the time domain into the domain of complex variable s yields

or

T_{1r} (s)=\frac{Q_{in}(s)-U_{o}T_{2r}(s)}{mcs + U_{o}}  .       (14.34)

The block diagram of the system represented by Eq. (14.34) is shown in Fig. 14.6.

When Figs. 14.4(a) and 14.6 are compared, the disturbance and process transfer functions for the open-loop system can be identified as

G_{V} (s)=\frac{1}{\frac{mc}{U_{o} } s+1 }  ,

G_{OL} (s)=\frac{\frac{1}{U_{o} } }{\frac{mc}{U_{o} }s+1 }  .

Using Eq. (14.28), one can calculate the steady-state disturbance sensitivity as

S_{DO}=\underset{s\rightarrow 0}{\lim } \frac{1}{\frac{mc}{U_{o} }s+1 } =1 .   (14.35)

Equation (14.35) indicates that the change of ambient temperature by ΔT will cause the change in the house temperature by the same value ΔT.

Now consider a closed-loop system in which the rate of heat supply is controlled to cause the house temperature to approach the desired level, T_{d}. The block diagram of the closed-loop temperature control system is shown in Fig. 14.7. The rate of heat supply Q_{in}(t) is assumed to be proportional to the temperature deviation:

Q_{in } (t) = k_{f}  [T_{d}  (t) − T_{1r}  (t)] .

By use of Eq. (14.30), the disturbance sensitivity in this closed-loop system is found to be

S_{DC}=\frac{1}{1+\frac{k_{f}}{U_{o}} }  ,

where  k_{f}/U_{o} is the open-loop gain of the system. Because both   k_{f}  and U_{o} are positive, the sensitivity of the closed-loop system to variations in ambient air temperature is smaller than that of the open-loop system. The greater the open-loop gain, the less sensitive the closed-loop system is to disturbances.