Implementation of Integral Cost function in Matlab

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Please, I need ideas how to simulate the model in the attached document. writing code for equation 39 - 41 is trivial, however, I am not sure how to write the code for equation 42.
I implemented the code below for one time step and assuming that the control input u = 0.1. The question is it correct to compute the optimal cost function like this or there is a better way. Please, find attached the model.
tspan = [0,1];
x0 = 1;
u = 0.1;
tau = 1;
xn = 1;
[time, dxdt, J] = plant_dynamics(tspan,x0,u, tau, xn);
%% System Dynamics
function [time, dxdt, J] = plant_dynamics(tspan,x0,u, tau, xn)
[time, dxdt] = ode23(@solve_ode,tspan,x0);
J = xn + integral_cost(dxdt, u);
function dx = solve_ode(t,x)
A = 1 + tau/12000;
B = 1 + 0.25 * sin(2*pi*t/3000);
dx = A*x + B * u;
end
function int_J = integral_cost(dxdt, u)
x = dxdt;
integral_J = x.^2 + u.^2;
int_J = trapz(integral_J);
end
end

Accepted Answer

VBBV
VBBV on 18 Aug 2022
Edited: VBBV on 18 Aug 2022
tspan = [0,1];
x0 = 1;
u = 0.1;
tau = 1;
xn = 1; % noise
[time, dxdt, J] = plant_dynamics(tspan,x0,u, tau, xn);
subplot(211)
plot(time,dxdt); title('Plant response')
subplot(212)
plot(time,J);title(' cost function (J) varying with noise input ')
%% System Dynamics
function [time, dxdt, J] = plant_dynamics(tspan,x0,u, tau, xn)
[time, dxdt] = ode23(@solve_ode,tspan,x0);
for k = 1:length(dxdt)
J(k,:) = (rand(1)*xn *dxdt).^2 + integral_cost(dxdt, u,xn); % add noise here
end
function dxdt = solve_ode(t,x)
A = 1 + tau/12000;
B = 1 + 0.25 * sin(2*pi*t/3000);
dxdt = A*x + B * u ;
end
function int_J = integral_cost(dxdt, u,xn)
x = dxdt;
integral_J = x.^2 + u.^2;
int_J = trapz(integral_J,xn);
end
end

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