Figure PI2.2 shows a pneumatic positioning system, where the displacement x is controlled by varying the pneumatic pressure P1. Assume that the pressure P2 is constant, and consider the specific plant
Gp (s) = X (s) / P1 (s) = K / s2 + 2s
With K = 4, the damping ratio is ( = 0.5, the natural frequency is (n = 2 rad/s, and the steady-state ramp error is 0.5.
Figure P12.2

a. Design an electrical compensator to obtain (n = 4 while keeping ( = 0.5. Obtain the compensator's resistances if C= 1 μF.
b. Suppose that with K= 4, the original system gives a satisfactory transient response, but the ramp error must be decreased to 0.05. Design a compensator to do this.

Pa P2 7n1 PI Pa