High-performance tape transport systems are designed with a small capstan to pull the tape past the read/write heads and with take-up reels turned by DC motors. The tape is to be controlled at speeds up to 200 inches per second, with start-up as fast as possible, while preventing permanent distortion of the tape. Since we wish to control the speed and the tension of the tape, we will use a DC tachometer for the speed sensor and a potentiometer for the position sensor. We will use a DC motor for the actuator. Then the linear model for the system is a unity feedback system with

where Y(s) is position. The specifications for the system are (1) Settling time of T_s  ≤ 12 ms (2) A percent overshoot to a step position command of P.O. ≤ 10% (3) A steady-state velocity error of less than 0.5%. Determine a compensator to achieve these specifications

Y(s) E(s) = G(s) K(s + 4000) s(s+1000) (s+ 3000) (s + 4000s + 8000000)'