reference r(t) r(t) Mux Scope Kd derivative error Plant Y(s) c(t) U(s) Adder Kp Ki integrator e) For the analysis of each controller include the specification of the transient response, this is, provide the values for Ts and %OS. For the steady-state analysis provide the steady-state value and the steady-state error. f) Generate a Table to compare the performance of the 3 controllers. Discuss your results. Which controller is better and why? Given the following transfer function for a DC motor (output is the angular position in radians), design a PID controller using the Ziegler-Nichols tuning method (2nd method: closed-loop method). Transfer function of the DC motor Y(s) 0.1464 GDc(s) = U(s) 7.89×10 s³ +8.25×10 *s² +0.00172s where Y(s) is the angular displacement of the motor shaft and U(s) is the armature voltage Note that you are designing a Position Control System. Once you have designed the controller, and you have implemented a feedback control system (unit feedback) your reference will be the desired angular position in radians. For the design follow the next steps. a) Find the value of the critical Gain Kcr and the Critical Period Pcr. Note: to find these two parameter you can used: i) a theoretical method that involves applying the Routh Hurwitz criteria OR ii) using simulation (Simulink) construct the closed-loop feedback control system with a proportional controller (Gain Kp) and enter a step to your CL system. Start varying the gain Kp until the response of the system starts to oscillate. At this moment, you are finding the Critical Gain Kcr. Then, measure the period of the oscillations and this will be the value of the Critical Period Pcr. b) Once you I have the values of Kcr and Pcr you use the Table 10-2 (bellow) to select the gains needed for the design of the PID controller. c) Using this Table you will design 3 different controllers for the DC motor. The 3 controllers are: Proportional controller i. ii. PI controller iii. PID controller d) Analyze the performance of your controllers. You can use Simulink to analysis the transient response and the steady-state response of each controller. A Simulink block diagram for a feedback control system with a PID controllers is as follows: