%
% A.A. Rodriguez
% Feedback Control Systems
% Copyright (c) 1998
%

%
% This macro examines the step response of a second order system:
%
%        H(s) = 2/a (s + a) / s^2 + 2s + 2
%
% for different values of a > 0.
%


%
% For large a, the step response approaches that of the second order system
%
%        H(s) = 2 / s^2 + 2s + 2
%
% As a is decreased, the derivative action of the numerator becomes 
% more and more pronounced, resulting in more and more overshoot.
%


clf
t = [0:0.1:10];                                                        % Create time vector


%
% Generate Step Responses
%
a100=100;
y100 = step( tf([2/a100 2],[1 2 2]),  t);             
plot(t,y100)
grid                                                                 % activate grid on plot
title('Step Response: H(s) = (2/a) (s+a)/(s^2+2s+2), a =[100 ]')
ylabel('y(t)')
xlabel('time (seconds)')
text(2.25,0.7, 'For large a, system responds like H(s) = 2/(s^2 + 2s + 2).')
text(2.5,0.95, 'a = 100')
pause

a10=10;
y10 = step( tf([2/a10 2],[1 2 2]),  t);             
plot(t,y100,t,y10)
grid                                                                 % activate grid on plot
title('Step Response: H(s) = (2/a) (s+a)/(s^2+2s+2),  a =[ 100, 10 ]')
ylabel('y(t)')
xlabel('time (seconds)')
text(2.25,0.7, 'For large a, system responds like H(s) = 2/(s^2 + 2s + 2).')
text(2.5,0.95, 'a = 100')
text(1.0,0.95, 'a = 10')
text(1.1,0.45, 'As a increases, derivative action of zero becomes less pronounced.')
text(1.1,0.25,  'As a decreases, derivative action of zero becomes more pronounced.')
pause


a5=5;
y5 = step( tf([2/a5 2],[1 2 2]),  t);             
plot(t,y100,t,y10, t, y5)
grid                                                                 % activate grid on plot
title('Step Response: H(s) = (2/a) (s+a)/(s^2+2s+2),  a =[ 100, 10, 5 ]')
ylabel('y(t)')
xlabel('time (seconds)')
text(2.25,0.7, 'For large a, system responds like H(s) = 2/(s^2 + 2s + 2).')
text(2.5,0.95, 'a = 100')
text(1.0,0.95, 'a = 5')
text(1.1,0.45, 'As a increases, derivative action of zero becomes less pronounced.')
text(1.1,0.25,  'As a decreases, derivative action of zero becomes more pronounced.')
pause


a2=2;
y2 = step( tf([2/a2 2],[1 2 2]),  t);             
plot(t,y100,t,y10, t, y5,t, y2)
grid                                                                 % activate grid on plot
title('Step Response: H(s) = (2/a) (s+a)/(s^2+2s+2),  a =[ 100, 10, 5, 2 ]')
ylabel('y(t)')
xlabel('time (seconds)')
text(2.25,0.7, 'For large a, system responds like H(s) = 2/(s^2 + 2s + 2).')
text(2.5,0.95, 'a = 100')
text(1,0.95, 'a = 2')
text(1.1,0.45, 'As a increases, derivative action of zero becomes less pronounced.')
text(1.1,0.25,  'As a decreases, derivative action of zero becomes more pronounced.')
pause


a1=1;
y1 = step( tf([2/a1 2],[1 2 2]),  t);             
plot(t,y100,t,y10, t, y5,t, y2, t, y1)
grid                                                                 % activate grid on plot
title('Step Response: H(s) = (2/a) (s+a)/(s^2+2s+2),  a =[ 100, 10, 5, 2, 1 ]')
ylabel('y(t)')
xlabel('time (seconds)')
text(2.25,0.7, 'For large a, system responds like H(s) = 2/(s^2 + 2s + 2).')
text(2.5,0.95, 'a = 100')
text(0.5,0.95, 'a = 1')
text(1.1,0.45, 'As a increases, derivative action of zero becomes less pronounced.')
text(1.1,0.25,  'As a decreases, derivative action of zero becomes more pronounced.')
text(1.1,0.1, 'Derivative action of zero becomes most pronounced for a > 1.')
pause

a05=0.5;
y05 = step( tf([2/a05 2],[1 2 2]),  t);             
plot(t,y100,t,y10, t, y5,t, y2, t, y1,t, y05)
grid                                                                 % activate grid on plot
title('Step Response: H(s) = (2/a) (s+a)/(s^2+2s+2),  a =[ 100, 10, 5, 2, 1, 0.5 ]')
ylabel('y(t)')
xlabel('time (seconds)')
text(2.25,0.7, 'For large a, system responds like H(s) = 2/(s^2 + 2s + 2).')
text(2.5,0.95, 'a = 100')
text(1.25,1, 'a = 2')
text(0.75,1.2, 'a = 1')
text(0.75,1.5, 'a = 0.5')
text(1.1,0.45, 'As a increases, derivative action of zero becomes less pronounced.')
text(1.1,0.25,  'As a decreases, derivative action of zero becomes more pronounced.')
text(1.1,0.1, 'Derivative action of zero becomes most pronounced for a < 1.')
pause

return