Question

3. Use MATLAB to plot the root locus of S +4 s 6s +13 H(s)1 Provide the commands you used and a copy of root locus figure. Also calculate the angles that the root locus leaves the complex poles. Use sgrid to plot lines of 0.7,0.8,0.9, and 0.99 and wn circles of 2,4, and 6. Provide command and plot of root locus with sgrid. Click on the root locus to determine the gain (K) where ζ-0.9 and ζ-0.99 intersect the root locus. Simulate the closed- loop system in Simulink with both gain values. Provide a screen shot of the Simulink model and the plot for both gain values. How did the system response change when the gain was changed?

Answer 1

das 2 -180-90+63. 4 4 -3 IS3·4718 2 2.

matlab code:

clc;

clear all;

s=tf('s');

g=(s+4)/(s^2+6*s+13);% plant

rlocus(g);

sgrid([0.7 0.8 0.9 0.99],[2 4 6])

Root Locus 2.5 0.9 Gain: 295 Damping: 0.899 Overstd 0.161 Frequency (ads): 4.9 0.794 Damping: 0.982 OFrequency radvS):6.14 10 1.5 09 03 12 10 Real Axis (seconds
The gain for gheta = 0.9 is 2.95

The gain for gheta = 0.99 is 6.18

close loop response using simulink:

simulink blocks

Function Block Parameters: Gc2 Transfer Fcn The numerator coefficient can be a vector or matrix expression. The denominator coefficient must be a vector. The output width equals the number of rows in the numerator coefficient. You should specify the coefficients in descending order of powers of s. Parameters Numerator coefficients 11 4] Denominator coefficients 1 6 13] Absolute tolerance auto State Name: (e.g., 'position') OK Cancel Help Apply

Source Block Parameters: Step5 Step Output a step. Parameters Step time Initial value Final value Sample time Interpret vector parameters as 1-D Enable zero-crossing detection OK Cancel Help Apply

s+4 s2+6S+13 Gc2 .95 Scope Step5 Gain2 Gain1

S+4 $2+6s+13 18 Scope Step5 Gain2 Gc2 Gain1
outputs:

Scope File Tools View Simulation Help 1.4 1.2 10 eady T-10.000

Scope File Tools View Simulation Help .5 .4 3 .2 eady T=10 .000