Chapter 5.2, Problem 34E

Feedback System with Pooling Delay. Many physical and biological systems involve time delays. A pure time delay has its output the same as its input but shifted in time. A more common type of delay is pooling delay. An example of such a feedback system is shown in Figure(5.3) on page 251. Here the level of fluid in tank B determines the rate at which fluid enters tank A. Suppose this rate is given by $R_1\left(t\right)=\alpha \left[V-V_2\left(t\right)\right]$, where $\alpha$ and $V$ are positive constants and $V_2\left(t\right)$ is the volume of fluid in tank B at time $t$.

a. If the outflow rate $R_3$ from tank B is constant and the flow rate $R_2$ from tank A into tank B is $R_2\left(t\right)=K{V}_1\left(t\right)$, where $K$ is a positive constant and $V_1\left(t\right)$ is the volume of fluid in tank A at time $t$, then show that this feedback system is governed by the system

$\frac{d{V}_1}{dt}=\alpha \left(V-V_2\left(t\right)\right)-K{V}_1\left(t\right)$,

$\frac{d{V}_2}{dt}=K{V}_1\left(t\right)-R_3$.

b. Find a general solution for the system in part (a) when

$\alpha =5{\left(\min \right)}^{-1}$, $V=20\text{L}$, $K=2{\left(\min \right)}^{-1}$, and $R_3=10\text{L}/\text{min}$.

c. Using the general solution obtained in part (b), what can be said about the volume of fluid in each of the tanks as $t\to +\infty$?