$($a$)$ Name and briefly explain the different modes of operation of a PID

controller.

$(3)$

$($b$)$ Show that the disturbance step response of a first order process under

proportional control $($Fig$.$ Q$3$a$)$ has the form of a first order transfer function.

Where $()$ is the process output, $()$ is the disturbance. Note that you

should assume negligible set$-$point $()$ change. i$.$e$.$ equal to zero.

Analyse how the process will respond by considering the expressions for

the gain $1$ and time constant $1$ of the closed$-$loop process.

$(7)$

$($c$)$ State the two main methods of creating a dynamic model of a chemical

process. Describe each method in turn and state one advantage of the

method.

$(8)$

$($d$)$ Determine, using an empirical approach, the transfer functions of the first

order processes A$,$ B and C$,$ whose unit step responses are given in Fig.