An electro-hydraulic positioning system is shown in Figure

Chapter , Problem 11.38

(choose chapter or problem)

An electro-hydraulic positioning system is shown in Figure P11.38. Use the following values.

\(\begin{aligned}K_a & =10 \mathrm{~V} / \mathrm{A} \quad K_1=10^{-2} \mathrm{in} . / \mathrm{V} \\K_2 & =3 \times 10^5 \mathrm{sec}^{-3} \quad K_3=20 \mathrm{~V} / \mathrm{in} . \\\zeta & =0.8 \quad \omega_n=100 \mathrm{rad} / \mathrm{sec} \quad \tau=0.01 \mathrm{sec}\end{aligned}\)

a. Develop a state-variable model of the plant with the controller current \(i_c\) as the input and the displacement y as the output.

b. Assuming that proportional control is used so that \(G_c(s)=K_P\), develop a state model of the system with \(y_r\) as the input and y as the output. Draw the root locus and use it to determine whether or not the system can be made stable with an appropriate choice for the value of \(K_P\).