The transfer function is given below:
\(\frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{{2s + 5}}{{{s^5} + 1.5{s^4} + 2{s^3} + 4{s^2} + 5s + 10}}\)
The number of roots with positive real part isConcept:
The characteristic equation of a system from the given CLTF is obtained as:
1 + GH = 0
\({a_n}{s^n} + {a_{n  1}}{s^{n  1}} + \ldots + {a_1}s + {a_a} = 0\)
Application:
The characteristic Equation for the given CLTF will be:
C.E. = s^{5} + 1.5s^{4} + 2s^{3} + 4s^{2} + 5s + 10
Forming the Routh array, we get:
s^{5} 
1 
2 
5 
s^{4} 
1.5 
4 
10 
s^{3} 
0.66 
1.66 
0 
s^{2} 
0.227 
10 

s^{1} 
27.4 
0 

s^{0} 
10 
As there are two sign changes in the first column of the Routh array, the system is unstable with 2 poles on the Right side of the jω axis, i.e. 2roots with a positive real part.