%  Example 8.12
% second order response
%  varying wn, zeta 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 = .5;
zeta = .4;
num = wn^2; den = [1 2*zeta*wn wn^2];
y1 = step(num,den,t);
wn = 1;
num = wn^2; den = [1 2*zeta*wn wn^2];
y2 = step(num,den,t);
wn = 2;
num = wn^2; den = [1 2*zeta*wn wn^2];
y3 = step(num,den,t);
plot(t,y1,'-',t,y2,'--',t,y3,'-.');
xlabel('Time (sec)')
ylabel('y(t)')
title('Second order step response, zeta = 0.4')
% the following inserts a legend

hold on
plot([9 11],[.5 .5],[9 11],[.4 .4],'--',[9 11],[.3 .3],'-.');
text(12,.5,'wn = 0.5')
text(12,.4,'wn = 1')
text(12,.3,'wn = 2')
hold off

% The following portion requires the Symbolic Math Toolbox.
syms Y y s
zeta = 0.4; wn = 0.5;  % rerun with different values of wn
Y = wn^2/(s^3+2*zeta*wn*s^2+wn^2*s);
y = ilaplace(Y);
ezplot(y,[0 20])
axis([0 20 0 1.5])
xlabel('Time (sec)')
ylabel('y(t)')
title('Second order step response, zeta = 0.4')