Chapter 6, Problem 6.2E

Interpretation Introduction

(a)

Interpretation:

The given transfer function should be written in a standard constant form of time or gain.

$\frac{\text{4}\left(\text{s+2}\right)}{\left(\text{0}\text{.5s+1}\right)\left(\text{2s+1}\right)}{\text{e}}^{\text{-5s}}$.

Concept introduction:

A transfer function model characterizes the dynamic relationship of two process variables:

1. A dependent variable (or output variable).
2. An independent variable (or input variable).

Transfer function models are only directly applicable to processes that exhibit linear dynamic behavior, such as a process that can be modeled by a linear ODE. If the process is nonlinear, a transfer function can provide an approximate linear model.

Interpretation Introduction

(b)

Interpretation:

The value of gain, poles and zeroes of standard transfer function should be determined.

Concept introduction:

• Gain: At steady state, the proportional value which shows the relationship between the magnitude of input and output signal, is known as Gain of the Transfer Function.
• Poles: The values of $\left(\text{s}\right)$ at which the denominator of the transfer function is equal to zero, is known as Poles of Transfer function.
• Zeroes: The values of $\left(\text{s}\right)$ at which the Numerator of the transfer function is equal to zero, is known as Zeroes of Transfer Function.

Interpretation Introduction

(c)

Interpretation:

The values of gain, poles and zeroes if the time delay function is replaced by $\left(\frac{1}{1}\right)$ should be determined.

Concept introduction:

The Pade approximation for a time delay is a ratio of two polynomials in $\left(\text{s}\right)$ with coefficients selected to match the terms of a truncated Taylor series expansion of $\left({\text{e}}^{\text{-θs}}\right)$.

• Gain: At steady state, the proportional value which shows the relationship between the magnitude of input and output signal, is known as Gain of the Transfer Function.
• Poles: The values of $\left(\text{s}\right)$ at which the denominator of the transfer function is equal to zero, is known as Poles of Transfer function.
• Zeroes: The values of $\left(\text{s}\right)$ at which the Numerator of the transfer function is equal to zero, is known as Zeroes of Transfer Function.