Chapter 3, Problem 29P

A single-pole oil cylinder valve contains a spool that regulates hydraulic pressure, which is then applied to a piston that drives a load. The transfer function relating piston displacement, Xp(s) to spool displacement from equilibrium, X_v(s), is given by (Qu, 2010):

$G\left(s\right)=\frac{X_p\left(s\right)}{X_v\left(s\right)}=\frac{K_q{\omega }_h^2/A_1}{s\left(s^2+2\varsigma {\omega }_{h}s+{\omega }_h^2\right)}$

where A1=effective area of a the valve’s chamber, Kq=rate of change of the load flow rate with a change in displacement, and ${\omega }_h=$ the natural frequency of the hydraulic system. Find the state-space representation of the system, where the state variables are the phase variables associated with the piston.