N min Σσκ-1Δ æk,uk s.t. k=1 xk+1 = Akxk + Bkuk + ck Akxk+Bkuk Ck - -Zibk Tmine kok≤ Tmax C¯¯zbk [1 − (Zk - Zbk)], ||ak||2 ≤ ok - ||Srk bj₂+drk + hj ≤0, xo = [r], vŢ,mo]', rN = 0, T T 2lbk≤zk VN = 0 where xk = [57, vk, zk]" and uk = [a‚σk]¯. Zlbk = Zlb(tk). 'k' (objective function) (14a) (dynamical constraints) (14b) (control constraints) (14c) (state path constraints) (14d) (boundary constraints) (14e)
I need help with a MATLAB code. I am trying to solve this question. Based on the Mars powered landing scenario
solve Eq. (14) via convex programming. Report the consumed fuel, and discuss the results with relevant plots. I am using the following MATLAB code and getting an error. I tried to fix the error and I get another one saying something about log and exp not being convex. Can you help fix my code and make sure it works.
The error is CVX Warning:
Models involving "log" or other functions in the log, exp, and entropy
family are solved using an experimental successive approximation method.
This method is slower and less reliable than the method CVX employs for
other models. Please see the section of the user's guide entitled
The successive approximation method
for more details about the approach, and for instructions on how to
suppress this warning message in the future.
Error using .* (line 173)
Disciplined convex programming error:
Cannot perform the operation: {log-affine} .* {real affine}
Error in * (line 36)
z = feval( oper, x, y );
^^^^^^^^^^^^^^^^^^^
Error in untitled7 (line 60)
T_min * exp(-x(7, k)) <= u(1, k) <= T_max * exp(-x(7, k)) * (1 - (x(7, k) - z_lb));
^
>>
clc;
clear all;
% Define the given parameters
g = -3.7114e-3; % Gravitational acceleration in km/s^2
alpha = 0.5086; % Fuel consumption rate in s/km
T_min = 4.97; % Minimum thrust in kg·km/s^2
T_max = 13.26; % Maximum thrust in kg·km/s^2
delta_t = 1.0; % Time step in seconds
N = 78; % Number of time steps
r0 = [1.5; 0; 2.0]; % Initial position in km
v0 = [-0.075; 0.03; 0.1]; % Initial velocity in km/s
m0 = 2000; % Initial mass in kg
gamma_min = 3; % Minimum glide slope angle in degrees
% Initial conditions
x0 = [r0; v0; log(m0)]; % Initial state: [position, velocity, log(mass)]
z_lb = 0;
% Define matrices for the optimization
cvx_begin
variables x(7, N) u(4, N) log_mass(1, N) exp_mass(1, N) % Auxiliary variables for log and exp terms
% Objective function: minimize the total fuel consumption
minimize( sum(sum(u)) ) % This would correspond to the sum of fuel used
for k = 1:N-1
% Dynamic constraints (from Eq. (14b)) - Define A_k, B_k, and c_k
% Compute A_k, B_k, c_k for each time step
[A_k, B_k, c_k] = computeMatrices(x(:, k), g);
% State transition equation (Eq. (14b))
x(:, k+1) == A_k * x(:, k) + B_k * u(:, k) + c_k; % You will need to compute A_k, B_k, c_k
% Control constraints (from Eq. (14c))
T_min * exp(-x(7, k)) <= u(1, k) <= T_max * exp(-x(7, k)) * (1 - (x(7, k) - z_lb));
norm(u(:, k), 2) <= u(1, k); % Thrust bound
% Path constraints (from Eq. (14d))
norm(S_k * x(1:3, k) - b_k) + d_k' * x(1:3, k) + h_j <= 0; % State path constraints
x(7, k) >= z_lb(k); % Mass bound (from boundary conditions)
end
% Boundary conditions (from Eq. (14e))
x(:, 1) == x0; % Initial state constraint
x(1:3, N) == 0; % Final position is zero
x(4:6, N) == 0; % Final velocity is zero
cvx_end
% Fuel consumption (objective value)
fuel_consumed = cvx_optval;
% Plot results
% Example for plotting control inputs:
figure;
plot(1:N, u(1, :), 'LineWidth', 2);
title('Thrust Magnitude Over Time');
xlabel('Time step');
ylabel('Thrust Magnitude (kg·km/s^2)');
% Example for plotting fuel consumption
disp('Total Fuel Consumed:');
disp(fuel_consumed);
%% Function to compute A_k, B_k, and c_k
function [A_k, B_k, c_k] = computeMatrices(x_k, g)
% A_k is a 7x7 matrix
A_k = cvx(zeros(7, 7)); % Initialize as a zero matrix
% Fill A_k dynamics
A_k(1:3, 4:6) = eye(3); % position dynamics
A_k(4:6, 7) = -g * ones(3,1); % gravity term on velocity
% B_k is 7x4 matrix
B_k = cvx(zeros(7, 4));
B_k(4:6, 1:3) = eye(3); % control inputs affect velocity
% c_k vector (nonlinear effects)
c_k = cvx(zeros(7, 1));
c_k(4:6) = -g * x_k(7); % nonlinear gravity term depending on mass
end
![N
min Σσκ-1Δ
æk,uk
s.t.
k=1
xk+1 = Akxk + Bkuk + ck
Akxk+Bkuk Ck
-
-Zibk
Tmine kok≤ Tmax C¯¯zbk [1 − (Zk - Zbk)], ||ak||2 ≤ ok
-
||Srk bj₂+drk + hj ≤0,
xo = [r], vŢ,mo]', rN = 0,
T T
2lbk≤zk
VN = 0
where xk = [57, vk, zk]" and uk = [a‚σk]¯. Zlbk = Zlb(tk).
'k'
(objective function) (14a)
(dynamical constraints) (14b)
(control constraints) (14c)
(state path constraints) (14d)
(boundary constraints) (14e)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad0d55fe-d83b-4711-86a1-cee8ecea510f%2Ffedec73a-df4b-4525-a130-804fa91d46b3%2Fm2bfakl_processed.png&w=3840&q=75)
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