Question

Use MATLAB to plot a root locus and design a control system. 1. download and review the project file at root locus.ma Create a root locus plot to determine design a control system for the following system which has a standard negative unity feedback system. G(s) = K (s2 - 4s +20) /[(s+2)(s+4)] Question 1 10 pts Damping ratio goal for the control system gain Kis to maintain a 45% damping ratio, or zeta = 0.45. Select the gain, K, using the root locus software. O 0.211 O 0.417 O 0.987 O 1.97

% MATLAB allows root loci to be plotted with the

% rlocus(GH) command, where G(s)H(s) = numgh/dengh and GH is an LTI transfer-

% function object. Points on the root locus can be selected interactively

% using [K,p] = rlocfind(GH) command. MATLAB yields gain(K) at

% that point as well as all other poles(p) that have that gain. We can zoom

% in and out of root locus by changing range of axis values using

% command axis([xmin,xmax,ymin,ymax]). root locus can be drawn

% over grid that shows constant damping ratio(z) and constant natural

% frequency(wn) curves using sgrid(z,wn) command. To plot multiple

% z and wn curves, use z = zmin:zstep:zmax and wn = wnmin:wnstep:wnmax to

% specify a range of values.

clf % Clear graph.

numgh insert yours; % Define numerator of G(s)H(s).

dengh= insert yours; % Define denominator of G(s)H(s).

'G(s)H(s)' % Display label.

GH=tf(numgh,dengh) % Create G(s)H(s) and display.

rlocus(GH) % Draw root locus.

z=0.2:0.05:0.5; % Define damping ratio values: 0.2 to

% 0.5in steps of 0.05.

wn=0:1:10; % Define natural frequency values: 0

% to 10 in steps of 1.

sgrid(z,wn) % Generate damping ratio and natural

% frequency grid lines for root

% locus.

title('Root Locus') % Define title for root locus.

'Hit Enter to Continue to Close-Up'

pause %Pause

rlocus(GH) % Draw close-up root locus.

axis([-3 1 -4 4]) % Define range on axes for root locus

% close-up view.

title('Close-up') % Define title for close-up root

% locus.

z=0.45; % Define damping ratio line for

% overlay on close-up root locus.

wn=0; % Suppress natural frequency overlay

% curves.

sgrid(z,wn) % Overlay damping ratio curve on

% close-up root locus.

for k=1:10 % Loop allows 10 points to be selected

% (z=0.45, jw crossing, breakaway)

[K,p]=rlocfind(GH) % Generate gain, K, and closed-loop

% poles, p, for point selected interactively on

% root locus.

end % End loop.

Answer 1

Code:

% MATLAB allows root loci to be plotted with the

% rlocus(GH) command, where G(s)H(s) = numgh/dengh and GH is an LTI transfer-

% function object. Points on the root locus can be selected interactively

% using [K,p] = rlocfind(GH) command. MATLAB yields gain(K) at

% that point as well as all other poles(p) that have that gain. We can zoom

% in and out of root locus by changing range of axis values using

% command axis([xmin,xmax,ymin,ymax]). root locus can be drawn

% over grid that shows constant damping ratio(z) and constant natural

% frequency(wn) curves using sgrid(z,wn) command. To plot multiple

% z and wn curves, use z = zmin:zstep:zmax and wn = wnmin:wnstep:wnmax to

% specify a range of values.

clf % Clear graph.
numgh=[1,-4,20]; % Define numerator of G(s)H(s).
dengh=[1,6,8]; % Define denominator of G(s)H(s).
disp('G(s)H(s)'); % Display label.
GH=tf(numgh,dengh); % Create G(s)H(s) and display.
disp(GH);
rlocus(GH);
set(gca,'FontSize',20);% Draw root locus.
z=0.2:0.05:0.5; % Define damping ratio values: 0.2 to 0.5in steps of 0.05.
wn=0:1:10; % Define natural frequency values: 0 to 10 in steps of 1.
sgrid(z,wn) % Generate damping ratio and natural frequency grid lines for root locus.
title('Root Locus','FontSize',20); % Define title for root locus.
pause; %Pause
%%
figure;
rlocus(GH) % Draw close-up root locus.
axis([-3 1 -4 4]) % Define range on axes for root locus close-up view.
title('Close-up'), % Define title for close-up root locus.
z=0.45; % Define damping ratio line for overlay on close-up root locus.

wn=0; % Suppress natural frequency overlay curves.

sgrid(z,wn) % Overlay damping ratio curve on close-up root locus.

for k=1:10 % Loop allows 10 points to be selected

% (z=0.45, jw crossing, breakaway)

[K,p]=rlocfind(GH); % Generate gain, K, and closed-loop
disp(K)
% poles, p, for point selected interactively on

% root locus.

end % End loop.

Output:

Figure:1-Root Locus:

Root Locus 5 0.5 0.450.40.350.30.250.2 4 3 2 5 3 Imaginary Axis (seconds) -2 -3 0.5 0.450.40.350.30.250.2 56 -5 -4 -3 -2 0 1 2 3 4 Asal Auds (d)
Figure-2: selected points and Gain:

Clup Imaginary Axis (seconds) Asal Auths (wide)

We found for zeta = 0.45, K= 0.417