Chapter 5, Problem 5.13E

Interpretation Introduction

(a)

Interpretation:

The time required for the given process to settle within $5\%$ of the total change in the output for a step change of magnitude $M$ in the input is to be calculated.

Concept introduction:

For the second-order transfer function of the form,

$G\left(s\right)=\frac{Y\left(s\right)}{U\left(s\right)}=\frac{K}{{\tau }^2{s}^2+2\zeta \tau s+1}$   ...... (1)

Where, $K,\tau ,\text{ and }\zeta$ are the system parameters, response $\left(y\left(t\right)\right)$ for the step input of magnitude $M$ for the critically damped system $\left(\zeta =1\right)$ is given by the expression:

$y\left(t\right)=KM\left[1-\left(1+\frac{t}{\tau }\right)e^{-t/\tau }\right]$   ...... (2)

Interpretation Introduction

(b)

Interpretation:

The time period by which $y\left(t\right)$ lags behind $u\left(t\right)$ once the output changes linearly with time is to be determined.

Concept introduction:

For the second-order transfer function of the form,

$G\left(s\right)=\frac{Y\left(s\right)}{U\left(s\right)}=\frac{K}{{\tau }^2{s}^2+2\zeta \tau s+1}$   ...... (1)

Where, $K,\tau ,\text{ and }\zeta$ are the system parameters. For a critically damped system, $\zeta =1$.

To determine the coefficients of a repeated factor in the partial fraction, a differential approach is used. If the polynomial in denominator $D\left(s\right)$ contains repeated factor ${\left(s+b\right)}^r$, then write:

$\begin{array}{c}Q\left(s\right)=\frac{N\left(s\right)}{D\left(s\right){\left(s+b\right)}^r}\\ ={\left(s+b\right)}^{r-1}{\alpha }_1+{\left(s+b\right)}^{r-2}{\alpha }_2+\cdots +{\alpha }_r+{\left(s+b\right)}^r\end{array}$   ...... (5)

${\alpha }_r$ is calculated by directly setting $s=-b$.

The value of coefficients ${\alpha }_{r-i}$ is given by:

${\alpha }_{r-i}={\frac{1}{i!}\frac{d^{\left(i\right)}Q\left(s\right)}{d{s}^{\left(i\right)}}|}_{s=-b}\left(i=0,\dots ,r-1\right)$   ...... (6)

Interpretation Introduction

(c)

Interpretation:

The results in part (a) and part (b) are to be verified using computer simulation.

Concept introduction:

For the second-order transfer function of the form,

$G\left(s\right)=\frac{Y\left(s\right)}{U\left(s\right)}=\frac{K}{{\tau }^2{s}^2+2\zeta \tau s+1}$   ...... (1)

Where, $K,\tau ,\text{ and }\zeta$ are the system parameters,response $\left(y\left(t\right)\right)$ for the step input of magnitude $M$ for the critically damped system $\left(\zeta =1\right)$ is given by the expression:

$y\left(t\right)=KM\left[1-\left(1+\frac{t}{\tau }\right)e^{-t/\tau }\right]$   ...... (2)

To determine the coefficients of a repeated factor in the partial fraction, a differential approach is used. If the polynomial in denominator $D\left(s\right)$ contains repeated factor ${\left(s+b\right)}^r$ ,then write:

$\begin{array}{c}Q\left(s\right)=\frac{N\left(s\right)}{D\left(s\right){\left(s+b\right)}^r}\\ ={\left(s+b\right)}^{r-1}{\alpha }_1+{\left(s+b\right)}^{r-2}{\alpha }_2+\cdots +{\alpha }_r+{\left(s+b\right)}^r\end{array}$   ...... (5)

${\alpha }_r$ is calculated by directly setting $s=-b$.

The value of coefficients ${\alpha }_{r-i}$ is given by:

${\alpha }_{r-i}={\frac{1}{i!}\frac{d^{\left(i\right)}Q\left(s\right)}{d{s}^{\left(i\right)}}|}_{s=-b}\left(i=0,\dots ,r-1\right)$ …… (6)