Chapter 8, Problem 8.7E

(a)

Interpretation Introduction

Interpretation:

The expression for controller output is to be derived.

Concept Introduction :

The response of the transfer function is derived by taking the inverse Laplace of the function.

The transfer function of a PID controller is as follows:

$\frac{\text{P}\left(\text{s}\right)}{\text{E}\left(\text{s}\right)}\text{=}{\text{K}}_{\text{c}}\left[\text{1}\text{+}\frac{\text{1}}{{\text{τ}}_{\text{i}}\text{s}}\text{+}{\text{τ}}_{\text{d}}\text{s}\right]$

Here,

P(s) represents the output variable in s domain.

E(s) represents the input variable (error) in s domain.

K_c represents the gain of the controller.

t_i represents the integral time constant.

t_d represents the derivative time constant.

Inverse Laplace of the function used is used in the given controller is given as follows:

$\begin{array}{c}{\text{L}}^{-\text{1}}\left[\frac{\text{1}}{\text{s}}\right]\text{=}\text{1}\\ {\text{L}}^{-\text{1}}\left[\frac{\text{1}}{{\text{s}}^{\text{n}}}\right]\text{=}\frac{{\text{t}}^{\text{n}-\text{1}}}{\left(\text{n}-\text{1}\right)\text{!}}\text{,}\text{n}\text{=}\text{1,2,3,}\mathrm{....}\end{array}$

(b)

Interpretation Introduction

Interpretation:

The expression for controller output excluding the derivative controller is to be derived.

Concept Introduction :

The transfer function of the derivative controller equates to zero and the PID controller becomes the PI controller.

(c)

Interpretation Introduction

Interpretation:

The controller output is to be plotted and analyzed qualitatively.

Concept Introduction :

Derivative controller speeds up the response and takes action in an anticipatory manner.

PID controller decreases the time of response while PI delays the response comparatively.