2: A segway control system has transfer function ?(?)= 1?249 1. The gyro sensor for the angle can be sampled as fast as 1ms and the loop computations require approximately 3ms. Estimate approximately the constraints on the selection of the achievable crossover frequency with a strictly causal DT PID. 2. Use the same sampling rate, design a strictly causal DT PID controller for the maximum crossover frequency with PM = 45deg. Hint: Consider a PID where the time constant of the pseudo-differentiator is placed at Ts/2 so that, after Tustin discretization, it corresponds to an one-sample delay. Also recall that the ZOH can be approximated by a half-sample delay. Finally, the strictly causal PID implementation calls for an extra one-sample delay. On the other hand, the zeros of the PID are typically limited to a phase contribution of 75deg/zero to maintain a reasonable integral action (i.e., avoid "canceling" the integrator). Problem 3: Consider the system with transfer function ?(?)= 1(? 1)2 which is to be controlled by a DT-PI controller of the form ?(?)= ?(??)?(?1) so that the loop crossover frequency is 1 rad/s and the phase margin is 45o and the sampling rate is 10Hz. 1. Design a