Use MATLAB or another suitable software package for all of t

Use MATLAB or another suitable software package for all of the following problems. You should produce a diary file that shows the input the numerical results, and (where appropriate) computer-generated plots. Obtain a root-locus plot for the system described by KF(s) = K/(s + 1)(s + 6 - j2) (s + 6 + j2) Determine the value of K that will result in a complex closed-loop pole having a real part of-2. Find the other closed-loop poles that correspond to this value of gain.

Solution

% Filename:     rootlocus.m

%

% Description: M-file demonstrating the use of matlab\'s

%               rlocus() and rlocfind() functions on the

%               example characteristic equation

%               K/(s+1)(s+6-j^2)( s+6+j^2)

%

% clear matlab memory and close all figures

clear all; close all;

% define numerator and denominator of L(s) and print to check

numL = [1]; denL = [1 1 0];

disp([\'L(s) = \', poly2str(numL,\'s\'), \'/\', poly2str(denL, \'s\')]);

disp(\' \');

%***** First Approach to Root Locus - Loop Over K *****

% open figure 1

figure(1);

% find open loop zeros and plot as o\'s,

% find open loop poles and plot as x\'s

olzeros = roots(numL);

olpoles = roots(denL);

plot(real(olzeros),imag(olzeros),\'ro\'); hold on;

plot(real(olpoles),imag(olpoles),\'rx\');

% loop over K, find and plot closed-loop poles

for K = 0.1:0.1:10,

    clpoles = roots([1, 1+K, 2*K]);

    plot(real(clpoles),imag(clpoles),\'bx\');

end

hold off;

xlabel(\'Real Axis\'); ylabel(\'Imaginary Axis\');

title(\'Root Locus for CLTF K/(s+1)(s+6-j^2)( s+6+j^2)

)\');

%***** Second Approach to Root Locus - Use rlocus() and rlocfind() *****

% open figure 2

figure(2);

% define transfer function object for L(s)

sysL = tf(numL, denL);

% plot closed-loop system\'s root locus

rlocus(sysL);

title(\'Root Locus for Characteristic Equation 1 + K/(s+1)(s+6-j^2)( s+6+j^2)

\');

% loop a user-specified number of times to find poles and corresponding

% gains from root-locus plot

nK = input(\'Enter number of CL-poles on which you wish to click to find corresponding gain, K: \');

n = 0;

while(n < nK)

    [K, poles] = rlocfind(sysL)

    n = n + 1;

end

Matlab Response Generated:

>> rootlocus

KF(s) =    K/(s+1)(s+6-j^2)( s+6+j^2)

Enter number of CL-poles on which you wish to click to find corresponding gain, K: 3

Select a point in the graphics window

selected_point =

-0.5829 - 0.0047i

K =

    0.1716

poles =

-0.5858 + 0.0037i

-0.5858 - 0.0037i

Select a point in the graphics window

selected_point =

-2.0047 + 1.4115i

K =

    3.0095

poles =

-2.0048 + 1.4142i

-2.0048 - 1.4142i

Select a point in the graphics window

selected_point =

-3.4076 - 0.0047i

K =

    5.8284

poles =

   -3.4189

   -3.4095