The characteristic equation of the above closed-loop system is:

Concept:

Characteristic equation “q(s)” is Numerator part of 1 + G(s)H(s) = 0

Where G(s)H(s) is open loop transfer function

Let,

\(G\left( s \right)H\left( s \right) = \frac{{z\left( s \right)}}{{p\left( s \right)}}\)     -----(1)

So, 1 + G(s)H(s) = 0 can be written as:

\(1 + \frac{{z\left( s \right)}}{{p\left( s \right)}} = 0\)

⇒ q(s) = p(s) + z(s) = 0 is characteristic equation.       -----(2)

Analysis:

G(s) = 4 × (3/s+1)

H(s) = 10/(s+10)

The characteristic equation is given as:

1+ G(s) H(s) = 0

\(1 + (\frac{12}{s+1} \times \frac{10}{s+10}) = 0\)

(s+1)(s+10) + 120 = 0

s² + 11s + 130 = 0