Question

Problem 3. For the above feedback system, the bode diagram of the stable open-loop transfer function G(s) is plotted below: (a) Find the approximate gain margin and phase margin of the system? Is the closed-loop system stable? (b) Suppose in the closed-loop system (s) is replaced with KG(8). What is the range of K so that the closed-loop system is stable? (C) Determine the system type of G(s). (d) Estimate the steady-state errors of the closed-loop system for tracking the unit step input and the unit ramp input, respectively (e) It is known that G(s) has five poles. From the plot, can you determine its number of zeros

Answer 1

Question a

The gain margin is the gain at phase crossover frequency. And phase crossover frequency is the frequency where the phase angle is -180.

The phase margin is the phase +180 at gain crossover frequency. And gain crossover frequency is the frequency where the gain is 0 dB.

From the bode plot shown below, the gain and phase margins are computed.

50 Bode Diagram of G(s) -50 -100 160 -90 -180 -270 10 10 10 10 10 10 Phase (deg) Magntude (dB)

The gain margin : 12 dB

phase margin: 108 degrees.

The gain and phase margins are positive. Therefore the system is stable.

Question b

from the above question we obtained the gain margin which is the maximum value of gain K before the system becomes unstable.

Therefore K = 12 dB = 4.

Question c

From the magnitude response it is observed that the slope of curve at zero frequency is close to -20 dB /decade.

Therefore the system is of type 1.

Question d

For type 1 system, the steady state error to step input is zero.

For type 1 system the steady state error to unit ramp input is non-zero.