The largest error between reference input and output during the t

SOLUTION

Concept:

Time response: If the response of the system varies with respect to the time is called Time response.

It is a combination of transient response and steady-state response.

Transient term: Any term which has exponential delay is called transient term and as t → ∞ then the response will be zero.

Time response of the second-order system

The second-order system nature completely depends on ζ 

The second-order system is stable for all the positive values of ζ < ∞ and ζ > 0, because poles lie in the left half of s-plane.

FIG of ζ values

The impulse response is given by:

\(\frac{C(s)}{R(s)} = \frac{{\omega_n}^{2}}{{s}^{2}+2ζ\omega_ns+{\omega_n}^{2}}\)

The generalised response for value of 0 < ζ < 1 is defined below;

\(c\left( t \right) = 1 - \frac{{{e^{ - \zeta {\omega _n}t}}}}{{\sqrt {1 - {\zeta ^2}} }}\sin \left( {\left( {{\omega _n}\sqrt {1 - {\zeta ^2}} } \right)t + {{\tan }^{ - 1}}\left( {\frac{{\sqrt {1 - {\zeta ^2}} }}{\zeta }} \right)} \right)\)

Different time specifications are defined

Rise time, Delay time, Peak overshoot, Undershoot etc..

Peak overshoot: it gives the normalized difference between time response peak to a steady-state value.

\(M_p = \frac{c(t_p)-c(\infty)}{c(\infty)}\)

∴ The largest error between the reference input and output during the transient period is Peak overshoot