Correct Answer - Option 1 :
\({{\rm{K}}_{\rm{P}}} > \frac{{{{\rm{K}}_{\rm{I}}}}}{2} > 0\)
Concept:
According to RH stability criteria there are two necessary conditions:
First is there should not be any sign changes in the first column of the tabulation and Second Is there should not be any missing coefficients in the equation 1 + G(s) H(s).
The RH criteria can be solved as follows:
\({s^3} - - - {\rm{\;\;\;}}{{\rm{a}}_0} \searrow {\rm{\;\;\;\;\;}} \swarrow {{\rm{a}}_2}{\rm{\;\;\;\;\;\;}}{{\rm{a}}_4}\)
\({s^2} - - - {\rm{\;\;\;}}{{\rm{a}}_1}{\rm{\;}} \nearrow {\rm{\;\;\;}} \nwarrow {{\rm{a}}_3}{\rm{\;\;\;\;\;\;\;}}0\)
\({s^1} - - - {\rm{\;\;\;}}\frac{{{{\rm{a}}_1}{{\rm{a}}_2} - {{\rm{a}}_0}{{\rm{a}}_3}}}{{{{\rm{a}}_1}}}{\rm{\;}} = {\rm{A\;\;\;\;\;\;\;\;\;\;}}0{\rm{\;}}\)
\({s^0} - - - {\rm{\;\;\;}}\frac{{{\rm{A}}{{\rm{a}}_3}}}{{\rm{A}}} = {{\rm{a}}_3}\)
Calculation:
Given the open-loop transfer function
\({\rm{G}}\left( {\rm{s}} \right) = \frac{{\left( {{\rm{s}}{{\rm{K}}_{\rm{P}}} + {{\rm{K}}_{\rm{I}}}} \right)}}{{{{\rm{s}}^2}{\rm{\;}}\left( {{\rm{s}} + 2} \right)}}\)
The characteristic equation of the above transfer function will be
⇒ q(s) = s3 + 2s2 + sKP + KI = 0
From RH-criteria
\(\begin{array}{*{20}{c}} {{s^3}}\\ {{s^2}}\\ {{s^1}}\\ {{s^0}} \end{array}\left| {\begin{array}{*{20}{c}} 1&2\\ K_P&{K_I}\\ {\frac{2K_P-K_I}{K_P}}&0\\ K_I&{} \end{array}} \right.\)
From the Routh criterion, the condition for stability is
⇒ 2KP > KI and KI > 0
∴ \({{\rm{K}}_{\rm{P}}} > \frac{{{{\rm{K}}_{\rm{I}}}}}{2} > 0\)