% part 6 of the mini project: 
% Prabir Barooah, March 20, 2009

clear all
%conversion: 
mph2mpersec = 0.44704; %m/sec
mpersec2mph = 1/mph2mpersec;

m = 1300; %kg
b = 42926.2; %N/volt
a = 150.91; %30.17; % N/m^2 s^2 

v_0 = 60*mph2mpersec; %m/s
theta_0 = a*v_0^2/b; %volt

%more constants
p= 2*a*v_0/m;
k = b/m;

%PI controller - previously designed
kp = 0.01;
ki = 0.4341;

% the closed loop transfer function from r to y 
P = tf(k,[1 p]);
C = tf([kp ki],[1 0]);
H_ry = feedback(series(P,C),1);

time = [0:0.01:10]';
y = step(H_ry,time);
scale = max(abs(y))/v_0
% so the step size required to make max|y/v_0| = 0.1 is
step_size_threshold = 0.1/scale

%now plot the response to a step of this size to check visually that indeed
%|y/v| (at some time t) becomes greater than 0.1
u = step_size_threshold *ones(length(time),1);
y_with_large_step = lsim(H_ry,u,time);
figure;
plot(time,y_with_large_step/v_0);
ylabel('y/v_0');
xlabel('time (sec)');