%  Example 8.11
% second order response
%  varying zeta, wn is constant

% The user should remove the portions of the code that require a
%   toolbox that he or she does not have.

% This portion of the code requires the Control Systems Toolbox.
t = 0:.05:20;
wn = 1;
zeta = .1;
num = wn^2; den = [1 2*zeta*wn wn^2];
y1 = step(num,den,t);
wn = 1;
zeta = .25;
num = wn^2; den = [1 2*zeta*wn wn^2];
y2 = step(num,den,t);
wn = 1;
zeta =.7; 
num = wn^2; den = [1 2*zeta*wn wn^2];
y3 = step(num,den,t);
plot(t,y1,'-',t,y2,'--',t,y3,'-.')
legend('zeta = 0.1','zeta = 0.25','zeta = 0.7')
xlabel('Time (sec)')
ylabel('y(t)')
title('Second order step response')
% the following inserts a legend
hold on
plot([9 11],[.5 .5],[9 11],[.4 .4],'--',[9 11],[.3 .3],'-.');
text(12,.5,'zeta = 0.1')
text(12,.4,'zeta = 0.25')
text(12,.3,'zeta = 0.7')
hold off

% This portion of the code requires the Symbolic Math Toolbox.
syms Y y s
zeta = .25;  % rerun with different values
Y = 1/(s^3+2*zeta*s^2+s);
y = ilaplace(Y);
ezplot(y,[0 20]);
axis([0 20 0 2])
xlabel('Time (sec)')
ylabel('y(t)')
title('Second order step response')