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hjb_minimum_fuel_discrete_time.m
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hjb_minimum_fuel_discrete_time.m
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clc;
clear;
close all;
%% parameters
mq = 0.3812;
Ixx = 2.661e-5;
Iyy = 2.661e-5;
Izz = 2.661e-5;
k = 2e-6;
L = 0.035;
b = 7e-9;
kd = 0;
rho = 0.037;
r = 0.03;
z_offset = 0.02;
g = 9.8;
parameters = [mq, Ixx, Iyy, Izz, k, L, b, kd, rho, r, z_offset, g];
xd = 2;
yd = 2;
zd = 1;
%% Matrices
%states = [wx wy wz phi theta psi x y z vx vy vz]
A1 = [0 g 0
-g 0 0
0 0 0];
A = [zeros(3, 12);
eye(3), zeros(3, 9);
zeros(3, 9), eye(3);
zeros(3), A1, zeros(3, 6)];
B1 = [L, 0, -L, 0;
0, L, 0, -L;
b / k, -b / k, b / k, -b / k];
J = diag([Ixx, Iyy, Izz]);
B = [(J^(-1)) * B1;
zeros(8, 4);
ones(1, 4) / mq] * 2 * k * eye(4) * ((mq * g)^0.5/2);
Q = diag([1, 1, 100, 1, 1, 100, 1, 1, 100, 1, 1, 10000]);
R = eye(4);
H = zeros(12);
Rinv = R^ - 1;
%% calculate recutti
Ts = 1e-3;
Tf = 30;
t = 0:Ts:Tf;
N = numel(t);
P = zeros(12, 12, N);
P(:, :, N) = H;
sysd = c2d(ss(A, B, eye(12), zeros(12, 4)), Ts);
Ad = sysd.A;
Bd = sysd.B;
Qd = Q * Ts;
Rd = R * Ts;
Rdinv = R^ - 1;
for i = N:-1:2
P(:, :, i - 1) = Ad.' * P(:, :, i) * Ad - Ad.' * P(:, :, i) * Bd * (Rd + Bd.'*P(:, :, i)*Bd)^-1 * Bd.' * P(:, :, i) * Ad + Qd;
end
%%%%
%% HJB
% linear
x = zeros(12, N);
x(:, 1) = [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0].';
u = zeros(4, N);
K = zeros(4, 12, N);
for i = 1:N - 1
K(:, :, i) = (Rd + Bd.' * P(:, :, i + 1) * Bd)^(-1) * Bd.' * P(:, :, i + 1) * Ad;
u(:, i) = -K(:, :, i) * (x(:, i) - [0, 0, 0, 0, 0, 0, xd, yd, zd, 0, 0, 0].');
x(:, i + 1) = x(:, i) + Ts * (A * x(:, i) + B * u(:, i));
end
figure
subplot(2, 1, 1);
plot(t(1:round(N)), x([7, 8, 9, 1, 2, 3], 1:round(N)), 'LineWidth', 2);
grid on
grid minor
xlabel('time (sec)');
ylabel('States');
legend('$x$', '$y$', '$z$', '$\phi$', '$\theta$', '$\psi$', 'Interpreter', 'latex')
subplot(2, 1, 2);
plot(t(1:round(N)), u(:, 1:round(N)) + ((mq * g / 4) / k).^0.5, 'LineWidth', 2);
grid on
grid minor
xlabel('time (sec)');
ylabel('Inputs');
legend('$u_1$', '$u_2$', '$u_3$', '$u_4$', 'Interpreter', 'latex')
% nonlinear
x = zeros(12, N);
x(:, 1) = [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0].';
u = zeros(4, N);
Rep = 100;
for i = 1:N - 1
u(:, i) = -K(:, :, i) * (x(:, i) - [0, 0, 0, 0, 0, 0, xd, yd, zd, 0, 0, 0].');
w1 = u(1, i) + (mq * g / 4 / k)^0.5;
w2 = u(2, i) + (mq * g / 4 / k)^0.5;
w3 = u(3, i) + (mq * g / 4 / k)^0.5;
w4 = u(4, i) + (mq * g / 4 / k)^0.5;
xtmp = x(:, i);
for j = 1:Rep
[T_ddot, W_dot] = NonlinearDynamics(w1, w2, w3, w4, xtmp(7:9), xtmp(10:12), xtmp(4:6), xtmp(1:3), parameters);
xtmp = xtmp + Ts / Rep * ([W_dot; W2E(xtmp(1:3), xtmp(4:6)); xtmp(10:12); T_ddot]);
end
x(:, i + 1) = xtmp;
end
figure
subplot(2, 1, 1);
plot(t(1:round(N)), x([7, 8, 9, 1, 2, 3], 1:round(N)), 'LineWidth', 2);
grid on
grid minor
xlabel('time (sec)');
ylabel('States');
legend('$x$', '$y$', '$z$', '$\phi$', '$\theta$', '$\psi$', 'Interpreter', 'latex')
subplot(2, 1, 2);
plot(t(1:round(N)), u(:, 1:round(N)) + ((mq * g / 4) / k).^0.5, 'LineWidth', 2);
grid on
grid minor
xlabel('time (sec)');
ylabel('Inputs');
legend('$u_1$', '$u_2$', '$u_3$', '$u_4$', 'Interpreter', 'latex')
figure
plot(t(1:round(N)), reshape(P(:, :, 1:round(N)), 144, round(N)), 'LineWidth', 2);
grid on
grid minor
xlabel('time (sec)');
ylabel('$P$', 'interpreter', 'latex');
%% functions
function [T_ddot, W_dot] = NonlinearDynamics(w1, w2, w3, w4, T, T_dot, Theta, W, parameters)
%% parameters
mq = parameters(1);
Ixx = parameters(2);
Iyy = parameters(3);
Izz = parameters(4);
k = parameters(5);
L = parameters(6);
b = parameters(7);
kd = parameters(8);
rho = parameters(9);
r = parameters(10);
z_offset = parameters(11);
g = parameters(12);
%% states
x = T(1);
y = T(2);
z = T(3);
phi = Theta(1);
theta = Theta(2);
psi = Theta(3);
%% Matrices
Rx = [1 0 0; 0 cos(phi) -sin(phi); 0 sin(phi) cos(phi)];
Ry = [cos(theta) 0 sin(theta); 0 1 0; -sin(theta) 0 cos(theta)];
Rz = [cos(psi) -sin(psi) 0; sin(psi) cos(psi) 0; 0 0 1];
R = Rz * Ry * Rx;
I = diag([Ixx, Iyy, Izz]);
%% Input Modification
in = zeros(4, 1);
Ground_Effect_Coef = @(z) 1;
z1 = [0 0 1] * ([x; y; z] + R * [0; L; z_offset]);
z2 = [0 0 1] * ([x; y; z] + R * [-L; 0; z_offset]);
z3 = [0 0 1] * ([x; y; z] + R * [0; -L; z_offset]);
z4 = [0 0 1] * ([x; y; z] + R * [L; 0; z_offset]);
in(1) = Ground_Effect_Coef(z1) * w1 * abs(w1);
in(2) = Ground_Effect_Coef(z2) * w2 * abs(w2);
in(3) = Ground_Effect_Coef(z3) * w3 * abs(w3);
in(4) = Ground_Effect_Coef(z4) * w4 * abs(w4);
%% Translational Dynamics
TB = k * [0; 0; sum(in)]; %trust force with ground effect
FD = -kd * T_dot; %frictional force
T_ddot = [0; 0; -g] + (R * TB + FD) / mq; %Translational Dynamics
%% Rotational Dynamics
tau = [L * k * (in(1) - in(3)); L * k * (in(2) - in(4)); b * (in(1) - in(2) + in(3) - in(4))]; %Rotational Torque with ground effect
W_dot = I \ (tau - cross(W, I * W)); %Rotational Dynamics
end
function Theta_dot = W2E(W, Theta)
phi = Theta(1);
theta = Theta(2);
psi = Theta(3);
E = [1, sin(phi) * tan(theta), cos(phi) * tan(theta);
0, cos(phi), -sin(phi);
0, sin(phi) / cos(theta), cos(phi) / cos(theta); ];
Theta_dot = E * W;
end