% L91_cruise.m - cruise control example for Lecture 9.1
% RMM, 24 Nov 02

% Parameter definitions
m = 1000;				% mass of the car, kg
b = 50;					% damping coefficient, N sec/m
a = 0.2;				% engine lag coefficient
r = 5;					% transmission gain

% Dynamics
veh = tf([1/m], [1 b/m]);		% vehicle
eng = tf([r], [1 a]);			% engine

%% Slide 6: PID design for cruise controller

% Compute the step response and slope of the step response
[y, t] = step(veh*eng, 50); 
slope = (y(2:length(y))-y(1:length(y)-1))./(t(2:length(y))-t(1:length(y)-1));
plot(t, y, t(1:length(y)-1), slope*10); 

% Add the axis lines
hold on; plot([-10 40], [0 0], 'k-', [0 0], [-0.1 0.5], 'k-');

% Figure out the Nichols lines [easier to do by hand, but I need the plot]
[smax, imax] = max(slope);		% find point of max slope
line = y(imax) + slope(imax)*([0:0.1:50]-t(imax));
hold on; plot([0:0.1:50], line, 'r-');
axis([0 50 -0.1 0.5]); print -dmeta L9_7a.wmf
hold off;

% Now compute the PID controller
L = t(imax) - y(imax)/slope(imax);	% x-axis intercept
a = -(y(imax) - slope(imax)*t(imax));	% y-axis intercept

% Use Ziegler-Nichols rules to choose gains
Kc = 1.2/a;
Ti = 2*L;
Td = 0.5*L;
pid = Kc * (1 + tf([1], [Ti 0]) + tf([Td 0], [1]));

% Now plot out the closed loop response curves
L = veh*eng*pid; T = L/(1+L);
bode(veh*eng, pid, veh*eng*pid); print -dmeta L9_7b.wmf
step(T, 60); print -dmeta L9_7c.wmf

%% Slide 11 - rlocus for cruise
%% Note: for the plots used in L9.1, many were generated with rltool
L = veh*eng*pid; 
rlocus(L); print -dmeta L9_11a.wmf
% rltool(veh*eng, pid); print -dmeta L9_11b.wmf