Answer and Step-by-step explanation:
To determine the ultimate gain and period of the system with a positive zero and different controllers, let's analyze each case:
a) Proportional Controller:
A proportional controller adds a gain factor to the system. In this case, the transfer function G(s) remains the same. The ultimate gain (Ku) is the gain value at which the system oscillates continuously without damping. To find the ultimate gain, we can perform a frequency response analysis or use a trial and error method.
b) PI Controller for τI = 2:
A PI (Proportional-Integral) controller adds both gain and integral action to the system. The transfer function of the PI controller is given by Kp + Ki/s, where Kp is the proportional gain and Ki is the integral gain.
To determine the ultimate gain and period, we can use the Ziegler-Nichols tuning method. With a PI controller, the critical gain (Kc) and ultimate period (Tu) can be found as follows:
1. Set the integral gain (Ki) to zero and increase the proportional gain (Kp) gradually until the system starts to oscillate continuously without damping.
2. Note the value of the proportional gain at which oscillations begin (Kc).
3. Measure the period of these oscillations (Tu).
The ultimate gain (Ku) is equal to Kc, and the ultimate period (Tu) is equal to 4 times Tu.
Please note that without the specific values of the transfer function coefficients and further information about the system, I cannot provide the exact ultimate gain and period. It would be best to perform the calculations using the given transfer function G(s) and specific values of the coefficients.