A root locus for the system is shown A Use the Routh method

A root locus for the system is shown.

A: Use the Routh method to find the gain which makes the system marginally stable.

B: We will assume the closed loop poles with bigger w_n are dominant. Find K to make the closed loop damping = 0.3

C: From your root locus, what is the closed loop transfer function in factored form? One pair of complex poles has real part -1.11.

D: What is the rise time: What is the steady state error to r(t)=t²?

For the givan G(s)

characteristic equation is S⁴-S³(K-1)+KS²+11KS+6K

Routh method

S^{4 1 K 6K}

S^{3 (K-1) 11K 0}

S² K²-12K/K-1 6K

S ( (K²-12K/K-1)11K-6K²+6)/(K²-12K/K-1))

S⁰ 6K

if the system is marginally stable first element should be zero

K²-12K/K-1 =0

K=12

-For the marginally stable system K=12