Chapter 5, Problem 5.16E

Interpretation Introduction

(a)

Interpretation:

The expression for the time to first peak of the underdamped second-order system is to be derived.

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.

Interpretation Introduction

(b)

Interpretation:

The expression for the fraction overshoot of the underdamped second-order system is to be derived.

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.

The maximum value of the response of a system to achieve its peak from the desired response of the given system is known as overshoot. It exceeds its final steady-state value.

Interpretation Introduction

(c)

Interpretation:

The expression for the decay ratio of the underdamped second-order system is to be derived.

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.

The ratio of the successive peaks sizes is termed as the decay ratio (DR).

Interpretation Introduction

(d)

Interpretation:

The expression for the settling time $\left(t_s\right)$ of the underdamped second-order system is to be derived. Also, it is to be stated if a single expression can be used for $t_s$ over the full range of $\zeta \left(0<\zeta <1\right)$.

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.

Settling time is defined as the time which is required for the process output to reach and remain in a band with its width equal to $\pm 5\%$ of the total change occurring in $y$ for $95\%$ of the response time.

Trigonometric identity used is:

$\sin \left(A+B\right)=\sin A\cos B+\cos A\sin B$