Question

Let G,()+3s+5) , K-1 and Ge 1 I Determine the loop transfer function L(s)-KG.G. Use 'margin' command in matlab to generate the Bode Plot for L(s). (a) What are its gain and phase margins (these should be available in the plots). (b) Convert the gain margin in dB to absolute value. (c) For what value of the gain K would the closed loop system become marginally stable? (d) Show that, for this value of K, the closed loop system does possess purely imaginary roots (you may use Root Locus, or Routh-Hurwitz, or the roots of the closed loop characteristic polynomial for this)

Answer 1

MATLAB code is given below in bold letters.

clc;
close all;
clear all;

% define the laplace variable s
s = tf('s');

% define the plant Gp
Gp = 1/(s*(s+3)*(s+5));

Gc = 1;K = 1;

L = K*Gc*Gp % loop transfer function

% margin command
figure;margin(L);grid on;

Loop transfer function is given below:

Transfer function:
1
------------------
s^3 + 8 s^2 + 15 s

Bode plot is plotted below for the loop function.

Bode Diagram Gm 41.6 dB (at 3.87 rad/s), Pm 88 deg (at 0.0666 rad/s) 50 50 :. ー)-10 . -15아.. 200 90 -135 00-180F. : : : :. 225 . 270 102 101 10 10 Frequency (rad/s)

from the above plot, it is observed that the gain margin is 41.6 dB and the phase margin is 88 deg.

gain margin = 10^(41.6/20) = 120.2264 (absolute value).

for gain K = 122.2264 the loop becomes marginally stable and this gain is called critical gain Kc.

Let's plot the root locus to show that the closed-loop poles are purely imaginary for K = Kc = 122.2264.

System:L Gain: 120 Root Locus Pole: 9.44e-016+3.87i Damping: -2.44e-016 Overshoot (%): 100 Frequency (rad/s): 387 4 -5 9 8-65-43-21 1 Real Axis (seconds)

From the root locus plot, it is observed that for K = 122.2264 the roots are purely imaginary.