Valve Transfer Function

Using the Bilinear Transform, the transfer function becomes

$$\frac{X(z)}{F(z)+kx_0} = \frac{1+2z^{-1}+z^{-2}}{a_0+a_1z^{-1}+a_2z^{-2}},$$

where

$$a_0 = mc^2 + mgc + k,$$

$$a_1 = -2(mc^2-k),$$

$$a_2 = mc^2 -mgc + k.$$

The corresponding difference equation is

$$x(n) = \frac{1}{a_0}[F_k(n) + 2F_k(n-1) + F_k(n-2)) -$$

$$a_1x(n-1) -a_2x(n-2)],$$

where $F_k(n) = F(n) + kx_0$.