An inverted pendulum control system has transfer function P(s) = 1/s^2 - 100. Discuss how the estimate of the maximum crossover frequency would change if we get a better sensor and we are able to sample twice as fast. Use sampling T_s = 0.01s, crossover wc = 15rad/s, PM = 40 degree to design a strictly causal DT PID controller. What happens if in #3 we choose wc = 5rad/s? (For simplicity, work with a CT controller to satisfy the PM equation: does the controller yield a stable closed-loop system?) Determine intervals in which solutions are sure to exist. (Enter your answer using interval notation.)