I\'ve attached the Matlab code below for this problem. Please answer each part.

Question

I've attached the Matlab code below for this problem. Please answer each part.

mycruisecontrol.m------------

%Cruise control example
clear all;
m = 2400; % mass of car in kg
v(1) = 10; % initial velocity in m/s
u(1) = 0; % initial throttle input

deltaT = 0.05; % time step
maxT = 45; % max time
t = [0 : deltaT : maxT];

vd = 20; % desired velocity in m/s
e(1) = vd - v(1); % initial error in velocity
eI(1) = e(1)*deltaT; % initial error integral

% Controller
Ki = 0.02;               % integral gain
Kp = 1;                   % proportional gain

for i=2:length(t)

%based on current speed, pick a gear (see figure 3.2)
if v(i-1) < 12
gear = 1;
elseif v(i-1) < 18
gear = 2;
elseif v(i-1) < 27
gear = 3;
elseif v(i-1) < 40
gear = 4;
else
gear = 5;
end

% add any hill rules here
theta = 0.0; %radians or degrees?
  
% change desired velocity here
if t(i-1)>20
vd = 17.5;
end
  
%calculate control input
u(i) = Kp*e(i-1) + Ki*eI(i-1); % PI control input
% Saturate the input to the range of 0 to 1
u(i) = min(u(i), 1); u(i) = max(0, u(i));

dv = cruisedyn(v(i-1), u(i), gear, theta, m);
  
%update my states and results
a(i) = dv;
v(i) = v(i-1)+dv*deltaT;
e(i) = vd - v(i); %error in velocity
eI(i) = e(i)*deltaT + eI(i-1); %error in integral
  

end

%plot whatever outputs I am interested in.
plot( t, v, 'b' );
hold on;
plot( t, u, 'r' );
legend ('output1', 'output2')

----------------cruisedyn.m----------

% cruisedyn - dynamic model for cruise control example
%
% Usage: acc = cruisedyn(v, u, gear, theta, m)
%
% This function returns the acceleration of the vehicle given the
% current velocity (in m/s), throttle setting (0 <= u <= 1), gear (1-5),
% road slope (default 0) and car mass (default 1000 kg).

% RMM, 2 Jul 06

function dv = cruisedyn(v, u, gear, theta, m)

% Check the parameters to make sure that they are OK
% MISSING

% Parameters for defining the system dynamics
alpha = [40, 25, 16, 12, 10];       % gear ratios
Tm = 190;               % engine torque constant, Nm
wm = 420;               % peak torque rate, rad/sec
beta = 0.4;               % torque coefficient
Cr = 0.01;               % coefficient of rolling friction
rho = 1.3;               % density of air, kg/m^3
Cd = 0.32;               % drag coefficient
A = 2.4;               % car area, m^2
g = 9.8;               % gravitational constant

% Set defaults if all arguments aren't passed
if (nargin < 3) gear = 1; end;   % gear ratio
if (nargin < 4) theta = 0; end;   % road angle
if (nargin < 5) m = 1000; end;   % mass of the car

% Check to make sure arguments are reasonable
if (gear < 1 || gear > 5)
fprintf(2, 'cruisedyn: gear ratio out of range');
acc = 0; return;
end

if (theta < -pi/4 || theta > pi/4)
fprintf(2, 'cruisedyn: road slope out of range ');
acc = 0; return;
end

if (m < 500 || m > 10000)
fprintf(2, 'cruisedyn: vehicle mass out of range ');
acc = 0; return;
end

% Saturate the input to the range of 0 to 1
u = min(u, 1); u = max(0, u);

% Compute the torque produced by the engine, Tm
omega = alpha(gear) * v;       % engine speed
torque = u * Tm * ( 1 - beta * (omega/wm - 1)^2 );
F = alpha(gear) * torque;

% Check to make sure engine settings are reasonable
if (omega > 700)
fprintf(2, 'cruisedyn: engine rpm too high (%g) ', omega*30/pi);
end

if (torque < 0)
fprintf(2, 'cruisedyn: torque is negative (u=%g, omega=%g) ', u, omega);
end

% Compute the external forces on the vehicle
Fr = m * g * Cr;           % Rolling friction
Fa = 0.5 * rho * Cd * A * v^2;       % Aerodynamic drag
Fg = m * g * sin(theta);       % Road slope force
Fd = Fr + Fa + Fg;           % total deceleration

% Now compute the acceleration
dv = (F - Fd) / m;

Problem 3. Modify the code again to look at the response to a hill. (a) Set the vehicle mass to 1200 kg, desired speed equal to initial speed at 20, gear at 3, simulation time to 0-60 seconds, uphill angle at 0, controller gain kp 1, ki 30.1. At t 20s, the uphill angle changes to T/15. Run this simulation and submit your plot. (b) Show the effects of increasing and decreasing the mass of the car by 25%. Submit your plots. What happens when you increase the mass? (c) Keep the mass at 1200 kg. Redesign the controller gain kp and ki (using trial and error is fine) so that it returns to within 1% of the desired speed within 5 s of encountering the beginning of the hill. Submit your plot.

Explanation / Answer

% QUESTION A

clc;
close all;

%Cruise control example
clear all;
m = 2400; % mass of car in kg
%v(1) = 10; % initial velocity in m/s
v(1) = 20; % initial velocity in m/s
u(1) = 0; % initial throttle input

deltaT = 0.05; % time step
%maxT = 45; % max time
maxT = 60; % max time
t = [0 : deltaT : maxT];

vd = 20; % desired velocity in m/s
e(1) = vd - v(1); % initial error in velocity
eI(1) = e(1)*deltaT; % initial error integral

% Controller
% Ki = 0.02;               % integral gain
% Kp = 1;                   % proportional gain
Ki = 0.1;               % integral gain
Kp = 1;                   % proportional gain

% add any hill rules here
theta = (0<t<20)*0.0 + (t>=20)*(pi/15) %radians or degrees?
figure
plot(t,theta);

for i=2:length(t)
%based on current speed, pick a gear (see figure 3.2)
if v(i-1) < 12
gear = 1;
elseif v(i-1) < 18
gear = 2;
elseif v(i-1) < 27
gear = 3;
elseif v(i-1) < 40
gear = 4;
else
gear = 5;
end

% change desired velocity here
if t(i-1)>20
vd = 17.5;
end

%calculate control input
u(i) = Kp*e(i-1) + Ki*eI(i-1); % PI control input
% Saturate the input to the range of 0 to 1
u(i) = min(u(i), 1); u(i) = max(0, u(i));

dv = cruisedyn(v(i-1), u(i), gear, theta(i-1), m);

%update my states and results
a(i) = dv;
v(i) = v(i-1)+dv*deltaT;
e(i) = vd - v(i); %error in velocity
eI(i) = e(i)*deltaT + eI(i-1); %error in integral

end

%plot whatever outputs I am interested in.
figure
plot( t, v, 'b' );
hold on;
plot( t, a, 'r' );
legend ('output1', 'output2')

% QUESTION B

clc;
close all;

%Cruise control example
clear all;
m = 900; % mass of car in kg
%v(1) = 10; % initial velocity in m/s
v(1) = 20; % initial velocity in m/s
u(1) = 0; % initial throttle input

deltaT = 0.05; % time step
%maxT = 45; % max time
maxT = 60; % max time
t = [0 : deltaT : maxT];

vd = 20; % desired velocity in m/s
e(1) = vd - v(1); % initial error in velocity
eI(1) = e(1)*deltaT; % initial error integral

% Controller
% Ki = 0.02;               % integral gain
% Kp = 1;                   % proportional gain
Ki = 0.1;               % integral gain
Kp = 1;                   % proportional gain

% add any hill rules here
theta = (0<t<20)*0.0 + (t>=20)*(pi/15) %radians or degrees?

for i=2:length(t)
%based on current speed, pick a gear (see figure 3.2)
if v(i-1) < 12
gear = 1;
elseif v(i-1) < 18
gear = 2;
elseif v(i-1) < 27
gear = 3;
elseif v(i-1) < 40
gear = 4;
else
gear = 5;
end

% change desired velocity here
if t(i-1)>20
vd = 17.5;
end

%calculate control input
u(i) = Kp*e(i-1) + Ki*eI(i-1); % PI control input
% Saturate the input to the range of 0 to 1
u(i) = min(u(i), 1); u(i) = max(0, u(i));

dv = cruisedyn(v(i-1), u(i), gear, theta(i-1), m);

%update my states and results
a(i) = dv;
v(i) = v(i-1)+dv*deltaT
e(i) = vd - v(i); %error in velocity
eI(i) = e(i)*deltaT + eI(i-1); %error in integral

end

%plot whatever outputs I am interested in.
figure
plot( t, v, 'b' );
hold on;
plot( t, a, 'r' );
legend ('output1', 'output2')

clc;
close all;

%Cruise control example
clear all;
m = 1500; % mass of car in kg
%v(1) = 10; % initial velocity in m/s
v(1) = 20; % initial velocity in m/s
u(1) = 0; % initial throttle input

deltaT = 0.05; % time step
%maxT = 45; % max time
maxT = 60; % max time
t = [0 : deltaT : maxT];

vd = 20; % desired velocity in m/s
e(1) = vd - v(1); % initial error in velocity
eI(1) = e(1)*deltaT; % initial error integral

% Controller
% Ki = 0.02;               % integral gain
% Kp = 1;                   % proportional gain
Ki = 0.1;               % integral gain
Kp = 1;                   % proportional gain

% add any hill rules here
theta = (0<t<20)*0.0 + (t>=20)*(pi/15); %radians or degrees?

for i=2:length(t)
%based on current speed, pick a gear (see figure 3.2)
if v(i-1) < 12
gear = 1;
elseif v(i-1) < 18
gear = 2;
elseif v(i-1) < 27
gear = 3;
elseif v(i-1) < 40
gear = 4;
else
gear = 5;
end

% change desired velocity here
if t(i-1)>20
vd = 17.5;
end

%calculate control input
u(i) = Kp*e(i-1) + Ki*eI(i-1); % PI control input
% Saturate the input to the range of 0 to 1
u(i) = min(u(i), 1); u(i) = max(0, u(i));

dv = cruisedyn(v(i-1), u(i), gear, theta(i-1), m);

%update my states and results
a(i) = dv;
v(i) = v(i-1)+dv*deltaT
e(i) = vd - v(i); %error in velocity
eI(i) = e(i)*deltaT + eI(i-1); %error in integral

end

%plot whatever outputs I am interested in.
figure
plot( t, v, 'b' );
hold on;
plot( t, u, 'r' );
legend ('output1', 'output2')

% QUESTION C

clc;
close all;

%Cruise control example
clear all;
m = 1200; % mass of car in kg
%v(1) = 10; % initial velocity in m/s
v(1) = 20; % initial velocity in m/s
u(1) = 0; % initial throttle input

deltaT = 0.05; % time step
%maxT = 45; % max time
maxT = 5; % max time
t = [0 : deltaT : maxT];

vd = 20; % desired velocity in m/s
e(1) = vd - v(1); % initial error in velocity
eI(1) = e(1)*deltaT; % initial error integral

% Controller
% Ki = 0.02;               % integral gain
% Kp = 1;                   % proportional gain
Ki = 0.01;               % integral gain
Kp = 0.5;                   % proportional gain
%Changed Ki = 0.01 and Kp = 0.5

% add any hill rules here
theta = (0<t<20)*0.0 + (t>=20)*(pi/15); %radians or degrees?
figure
plot(t,theta);

for i=2:length(t)
%based on current speed, pick a gear (see figure 3.2)
if v(i-1) < 12
gear = 1;
elseif v(i-1) < 18
gear = 2;
elseif v(i-1) < 27
gear = 3;
elseif v(i-1) < 40
gear = 4;
else
gear = 5;
end

% change desired velocity here
if t(i-1)>20
vd = 17.5;
end

%calculate control input
u(i) = Kp*e(i-1) + Ki*eI(i-1); % PI control input
% Saturate the input to the range of 0 to 1
u(i) = min(u(i), 1); u(i) = max(0, u(i));

dv = cruisedyn(v(i-1), u(i), gear, theta(i-1), m);

%update my states and results
a(i) = dv;
v(i) = v(i-1)+dv*deltaT
e(i) = vd - v(i); %error in velocity
eI(i) = e(i)*deltaT + eI(i-1); %error in integral

end

%plot whatever outputs I am interested in.
figure
plot( t, v, 'b' );
hold on;
plot( t, u, 'r' );
legend ('output1', 'output2')

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