Dsign a closed-loop controller for a servo system with

Question

Assignment: Project

Title: Design of Control Systems

Problem:

Design a closed-loop controller for a servo system (motor) with tachometer feedback. The output; c(t), corresponds to the position of the motor, while the tachometer feedback provides a proportional signal to the velocity of the motor, K_h ω(t). A proportional controller with gain K is inserted in the forward loop to adjust the transient performance of the system. A bock diagram for this system is shown below:

Task (1):

Obtain the closed-loop transfer function of the system, C(s)/R(s), in terms of both K and K_h; then, determine the ranges of stability for both K and K_h, given the constraint that the tachometer gain, K_h, must be positive. Draw a two-dimensional graph (K vs. K_h) showing the corresponding stability range.

Task (2):

Assuming K = 5, find the value of K_h such that the system will be on the verge of instability (undamped oscillation). Simulate your design and calculate both the magnitude and frequency of the output oscillation.

Task (3):

Choose two suitable values for the controller gain, K and K_h, such that the system will be approximated by a dominant second order system with 0.7 < ζ < 0.8 and 1.5 < ω_n < 2.0 rad/s. Plot the pole-zero map of the corresponding closed-loop system.

Task (4):

Replace the controller gains of the original block diagram with a PD controller and try to find suitable values for the proportional and derivative gains such that the new response is very close to the one obtained in task. Compare between the two control structures.

Task (5):

Develop a state space model of the servo system, assuming the position (output) and the velocity as two states for the system. Choose the third state to be the current of the armature (see chapter three for details).You may rearrange the block diagram of the system to explicitly show the three states of the system. Prove that the transfer function of the state space model is equivalent to the one obtained in task (1).

Hint: Find suitable values for the motor parameters that suit the block diagram.