Question

Design a control system gain to yield 1.52% overshoot and evaluate the step response. 1. download the project m file from root locus2.m Design a control system gain, K, for the following standard negative unity feedback system whose feedforward transfer function is: G(s) = K(s+1.5)/[ s (5+1) (s+10) ] The criteria for the control system is to maintain the following 1. maximum 1.52% overshoot to a step response 2. maximum 20 second settling time 3. maximum 10 second peak time 4. no steady state error to a unit step input

% We can couple the design of gain on the root locus with a

% step-response simulation for the gain selected. We introduce the command

% rlocus(G,K), which allows us to specify the range of gain, K, for plotting the root

% locus. This command will help us smooth the usual root locus plot by equivalently

% specifying more points via the argument, K. Notice that the first root locus

% plotted without the argument K is not smooth. We also introduce the command

% x = input('prompt'), which allows keyboard entry of a value for x in response to a

% prompt. We apply this command to enter the desired percent overshoot. We also add

% a variable's value to the title of the root locus and step-response plots by

% inserting another field in the title command and use num2str(value) to convert

% value from a number to a character string for display. Let us apply the concepts

% to Example 8.8 in the text.

clear % Clear variables from workspace.

clf % clear graph.

numg=[1 1.5]; % Define numerator of G(s).

deng=poly([0 -1 -10]); % Define denominator of G(s).

'G(s)' % Display label.

G=tf(numg,deng) % Create and display G(s).

rlocus(G) % Draw root locus (H(s)=1).

title('Original Root Locus') % Add title.

'Hit Enter to plot smoothed root locus'

pause

K=0:0.005:50; % Specify range of gain to smooth

% root locus.

rlocus(G,K) % Draw smoothed root locus (H(s)=1).

title('Smoothed Root Locus') % Add title.

pos=input('Type %OS '); % Input desired percent overshoot

% from keyboard.

z=-log(pos/100)/sqrt(pi^2+[log(pos/100)]^2)

% Calculate damping ratio.

sgrid(z,0) % Overlay desired damping ratio line

% on root locus.

title(['Root Locus with ',num2str(pos),'% overshoot line'])

% Define title for root locus

% showing percent overshoot used.

for k=1:10

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

end; % poles, p, for point selected

% interactively on the root locus.

'Hit Enter to plot step response'

pause

'T(s)' % Display label.

T=feedback(K*G,1) % Find closed-loop transfer function

% with selected K and display.

step(T) % Generate closed-loop step response

% for point selected on root locus.

title(['Step Response for K=',num2str(K)])

% Give step response a title which

% includes the value of K.

Answer 1

clc;clear all; close all;
num=[1 1.5];%(s+1.5)
den= poly([0 -1 -10]);%(s)*(s+1)*(s+10)
fprintf('open loop tranfer function\n')
G=tf(num,den)
i=1;
K(i)=7;
T=feedback(K(i)*G,1); % Find closed-loop transfer function
t=0:0.005:6;
[y,t] = step(T,t);
ymax= max(y);
while round((ymax-y(end))*100/y(end),3)~=1.52
i=i+1;
K(i)=K(i-1)-0.001;
T=feedback(K(i)*G,1);
t=0:0.005:9;
[y,t] = step(T,t);
ymax= max(y);
end
fprintf('The value of gain K is:%1.3f\n',K(i-1))
fprintf('close loop tranfer function\n')
T=feedback(K(i-1)*G,1); % Find closed-loop transfer function
t=0:0.005:9;
[y,t] = step(T,t);
ymax= max(y);
figure
step(T);
ylabel('Response x(t)')
[ymax,tp] = max(y);
fprintf('Peak time=%1.3fsec\n',t(tp))
fprintf('Peak Response magnitude=%1.3f\n',ymax)
fprintf('percent of overshoot=%1.3fpercent\n',(ymax-y(end))/y(end)*100)
hold on
plot(t(tp),ymax,'*')
s = 601;
while y(s) > 0.998 & y(s) < 0.102;
s = s - 1;
end;
fprintf('Settling time=%1.3fsec\n',t(s))
plot(t(s),y(s),'*')
legend('Response','Peak point','Settling point')
title(['Root Locus with ',num2str(1.52),'% overshoot line'])
fprintf('Steady state error is:e=%1.3f\n',1-y(end))

Root Locus with 1.52% overshoot line 1.2 0.8 Response * Peak point * Settling point Response x(t) 0. 0.4 0.2 3 6 7 8 9 N 5 Time (seconds)

open loop tranfer function

G =

s + 1.5
-------------------
s^3 + 11 s^2 + 10 s

Continuous-time transfer function.

The value of gain K is:5.652
close loop tranfer function
Peak time=4.780sec
Peak Response magnitude=1.015
percent of overshoot=1.521percent
Settling time=3.000sec
Steady state error is:e=-0.000
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