Question 10.4.1: Hydraulic Implementation of Proportional Control Figure 10.4...

When the input motion y occurs, the beam pivots about its lower end. For small motions x = L_{3}θ and the beam geometry is such that
z = \frac{L_{1}  +  L_{2}}{L_{2}} y  −  \frac{L_{1}}{L_{2}} x          (1)
The proof of this equation for small angular motions is as follows. First, suppose that x is fixed to be 0. Then from similar triangles,
\frac{z}{L_{1}  +  L_{2}} = \frac{y}{L_{2}}       or     z = (L_{1} + L_{2}) \frac{y}{L_{2}}
Now suppose that y is fixed to be 0. From similar triangles,
\frac{z}{L_{1}} = − \frac{x}{L_{2}}       or           z = − L_{1} \frac{x}{L_{2}}
Since the general motion is the superposition of the two individual motions, adding these two relations gives equation (1).
For the servomotor, as shown in Example 7.4.7, if the rudder inertia is small,
\frac{X(s)}{Z(s)} = \frac{C_{1}}{As}
These equations give the following transfer function.
\frac{\Theta(s)}{Y (s)} = \frac{L_{1}  +  L_{2}}{L_{1} L_{3}} \frac{1}{\tau s  +  1}
where
τ = \frac{L_{2} A}{L_{1} C_{1}}
If the servomotor gain C_{1}/A is large, then \tau \approx 0, and the transfer function becomes
\frac{\Theta(s)}{Y (s)} =\frac{L_{1}  +  L_{2}}{L_{1} L_{3}} \equiv K_{P}
Thus the system implements proportional control with proportional gain K_{P} if the motion y represents the error signal and θ represents the controller output.