2 20 points Mix and match In the following problem there are

2. (20 points) Mix and match In the following problem, there are four different systems. For each system, we have plotted the Bode and Nyquist plots of the loop gain L(s), the sensitivity function S(s) response. Correctly identify which plots belong to the same system. Briefly justify your answers. The systems are: and the closed-loop step » System 1. Loop gain too small » System 2. Bad robustness (small gain/phase margin) System 3. Good response (low bandwidth) » System 4. Good response (high bandwidth) Give your answers in the form shown below: System 1: Bode (a), Nyquist (c), Sensitivity (d), Step-response (d), Justification: (your reason Bode Diagram Bode Diagram -50 -100 -135 -180 10 10 10 Frequency (rad/sec) Frequency (radsec) Bode Diagram Bode Diagram -100 -180 -180 10 10 10 Frequency (rad/sec) Frequency (rad/sec) Figure 1: Bode plots of the loop gain L for different systems.

Solution

System 1 bode b), Nyquist c) Sensitivity a) Step response d)

Lower gain or lower spread in x axis of nyquist, no overshoot over zero db in sensitivity, low starting gain in bode and lower final point for step.

System 2 bode d), Nyquist a) Sensitivity b) Step response a)

Low robustness or lower stability and lower spread in xaxis of nyquist, high overshoot over zero db in sensitivity, low starting gain and low phase margin in bode and oscillation during starting for step.

System 3 bode a), Nyquist d) Sensitivity d) Step response c)

Low bandwidth or lower spread in y axis as value increases in x axis of nyquist, lower db change with frequency in sensitivity, lower -3db gain spread in bode and higher rise time for step.

System 4 bode c), Nyquist b) Sensitivity c) Step response b)

high bandwidth or higher spread in y axis as value increases in x axis of nyquist, higher db change with frequency in sensitivity, low starting gain and higher -3db gain spread in bode and lower rise time for step.