A Control system with PD controller is shown. If velocity error constant is K_V = 1000 and the damping ratio is 0.5 then the values of K_P and K_D Should be:

Concept:

Velocity error coefficient is given

\({K_V} = \mathop {\lim }\limits_{s \to 0} s(G\left( s \right)H\left( s \right)\)

Calculations:

\({K_V} = \mathop {\lim }\limits_{s \to 0} s(G\left( s \right)H\left( s \right)\)

\(1000 = \mathop {\lim }\limits_{s \to 0} \;\frac{{s\left( {{K_P} + {K_D}s} \right)100}}{{s\left( {s + 10} \right)}}\)

1000 = 10 kp

Kp = 100

Characteristic Equation:

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

\(1 + \frac{{\left( {100 + {K_D}s} \right)100}}{{s\left( {s + 10} \right)}} = 0\)

⇒ s² + (10+100k_D)s+10⁴ = 0

Comparing with standard second order equation:

2ξw_n = 100 k_D + 10

\(2\left( {\frac{1}{2}} \right)\left( {100} \right) = {K_D} + 10\) 

90 = 100 k_D