An engineering laboratory has presented a plan to operate an Earth-orbiting satellite that is to be controlled from a ground station. A block diagram of the proposed system is shown. It takes T seconds for a signal to reach the spacecraft from the ground station and the identical delay for a return signal. The proposed ground-based controller is a proportional- derivative (PD) controller, where
Gr(s) = KD + KDs.
(a) Assume no transmission time delay (i.e., T = 0), and design the controller to the following specifications: (1) percent overshoot less than 20% to a unit step input and (2) time to peak less than 30 seconds.
(b) Compute the phase margin with the controller in the loop but assuming a zero transmission time delay. Estimate the amount of allowable time delay for a stable system from the phase margin calculation.
(c) Using a second-order Pade approximation to the time delay, determine the maximum allowable delay Tmw for system stability by developing a m file script that employs the pade function and computes the closed-loop system poles as a function of the time delay T. Compare your answer with the one obtained in part (b).