Problem $4$ where $K_1$ is a positive constant. We also assume that the change in the inflow rate $q_i$ is

negatively proportional to the change in the valve opening $y,$ or

$q_i=-K_{v}y$

where $K_v$ is a positive constant.

Assuming the following numerical values for the system,

$C=2m^2,R=\frac{0\mathrm{.}5sec}{m^2},K_v=\frac{1m^2}{sec},a=0\mathrm{.}25m,b=0\mathrm{.}75m,K_1=4se{c}^{-1}$

obtain the transfer function $\frac{H(s)}{Q_d(s)}.$

Consider the liquid$-$level control system shown $in$ Figure. The inlet valve $is$ controlled $bya$

Guardado $en$ Este $PC$ I controller. Assume that the steady$-$state inflow rate $\frac{?}{\text{b}\text{a}\text{r}}(Q)is$ and steady$-$state

outflow rate $is$ also $\frac{?}{\text{b}\text{a}\text{r}}(Q),$ the steady$-$state head $is\frac{?}{\text{b}\text{a}\text{r}}(H),$ steady$-$state pilot valve displacement $is$

$\stackrel{}{x}=0,$ and steady$-$state valve position $is\frac{?}{\text{b}\text{a}\text{r}}(Y).We$ assume that the set point $\frac{?}{\text{b}\text{a}\text{r}}(R)$ corresponds $to$

the steady$-$state head $\frac{?}{\text{b}\text{a}\text{r}}(H).$ The set point $is$ fixed. Assume also that the disturbance inflow

rate $q_d,$ which $is$ a small quantity, $is$ applied $to$ the water tank $att=0.$ This disturbance

causes the head $to$ change from $\frac{?}{\text{b}\text{a}\text{r}}(H)to\frac{?}{\text{b}\text{a}\text{r}}(H)+h.$ This change results $in$ a change $in$ the outflow

rate $by{q}_0.$ Through the hydraulic controller, the change $in$ head causes a change $in$ the

inflow rate from $\frac{?}{\text{b}\text{a}\text{r}}(Q)to\frac{?}{\text{b}\text{a}\text{r}}(Q)+q_i.(The$ integral controller tends $to$ keep the head constant $as$

much $as$ possible $in$ the presence $of$ disturbances.$)We$ assume that all changes are $of$

small quantities.

$We$ assume that the velocity $of$ the power piston $(valve)is$ proportional $to$ pilot$-$valve

displacement $x,or$

$\frac{dy}{dx}=K_{1}x$