Question

29. 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, Xv(s), is given by (Qu, 2010) Gs) X-(S) where Ai -effective area of a the valve's chamber, Krate of change of the load flow rate with a change in displacement, and the natural fre- quency of the hydraulic system. Find the state-space representation of the system, where the state variables are the phase variables associated with the piston.

Answer 1

Given, x_p(s)/x_v(s) = k_q w_h^2/A_1/s(s^2 + 2{w_h s+ w^2_n) sate space form has [x^middot] = [A]_ n times n [x] + [B]_n times l [u] [y] = [c]_m times n [x] + [0]_m times l [U] x is state variable u is input y output n = I stotal no of state variable l = is total no of input m = is total no of output Direct Decomposition method using masson's gain formula = sigma p_k Delta_k/Delta Delta_k = 1 - [sum of all loop not touching k^th forward path] + [sum of product of two non touching loop not touching k^th forward path]+ --- Delta = 1 - [sum of all loop] + [sum of product of two non touching loop] - [sop of 3 non touching loop]\r\n converting the transformer function into nasson gain form: - x_p(s)/x_v(s) = k_qw^2_h/A_1/[s^3 + 2{w_As^2 + w^2_n s] (k_qw^2_h/A_1/s^3) [divide by s^3 on both \r\n