Find the optimal state and optimal control based on minimizing the performance index

Question:

Find the optimal state and optimal control based on minimizing the performance index 𝐽=∫ (𝑥(𝑡) − 1/2 (𝑢(𝑡)^2) ) 𝑑𝑡 , 0 ≤ 𝑡 ≤ 1 subject to 𝑢(𝑡) = 𝑥̇(𝑡) + 𝑥(𝑡) with the condition 𝑥(0) = 0, 𝑥(1) = 1 2 (1 − 1 /e )^2 where 𝐽𝑒𝑥𝑎𝑐𝑡 = 0.08404562020 In this example the initial approximation is 𝑥1 (𝑡) = 1/2 (1 − 1 /e ) ^2

My finding:

I saw there are warnings in global variables and was thinking of alternative of global variables. Please explain this giving an example.

ERROR:

solve_system_of_equations

Error using fmincon (line 641)

Supplied objective function must return a scalar value.

Error in solve_system_of_equations (line 40)

x = fmincon(fun,x0,A,bb,Aeq,beq,lb,ub);

CODE:

function F = cost_function(x)

global def;

global m;

s=def.k ;

C2=x(1,(s+1):2*s);

u=(C2*m.H) ;

F= x - 1/2*(u*u');

function F = system_of_equations(x)

global def;

global m;

global init;

global P_alpha_1;

s=def.k ;

C1=x(1,1:s);

C2=x(1,(s+1):2*s);

x1=(C1*P_alpha_1*m.H) + init(1);

u=(C2*m.H) ;

D_alpha1_x1= C1*m.H;

%

M=Haar_matrix(s);

HC=M(:,s);

%%control law1

F = horzcat( D_alpha1_x1 + x1 - u , ...

(C1*P_alpha_1*HC) + init(1) - (1/2*((1-exp(-1))^2))) ;

end

alpha_1=1;

k=8; %no. of Haar wavelets

b=2; %Total number of days to plot

initialize(alpha_1,k,b )

global m;

global init;

global def;

global P_alpha_1;

global P_alpha_2;

P_alpha_1=fractional_operation_matrix(k,alpha_1,b,m.H);

s=def.k;

x0=zeros(3,3*s);

% system_of_equations(x)

A = [];

bb = [];

Aeq = [];

beq = [];

lb = [];

ub = [];

fun = @cost_function;

nonlcon=@system_of_equations;

x = fmincon(fun,x0,A,bb,Aeq,beq,lb,ub,nonlcon)

Answer 1

[Abolfazl Chaman Motlagh]

your cost_function doesn't return a scalar. it should return one number for every input.

it seems that x is a 1xn vector. and u is a vector with same size as x. (for every t i guess!) but in last line of cost_function you write F = x - 1/2*(u*u') . the second term is a scalar beacuse it is inner product of u with itself make this term become . but x is still a vector so the result (F) become 1xn vector. F should be integration if x and u are vector with same size and space. you should sum it over time. (with dt !) : becoming :

F = sum(x + 0.5*u.^2)

also this might not be a good choice becuse it doen't contain any information about t and the integration is over t. if grid of t is uniform and the intervals are equal summation and numerical integration is different in just a constant coefficient which doesn't change the problem. but integration over t might be better soltion in general:

F = trapz(x+0.5*(u.^2)); %numerical integration with Trapezoidal method

if you have the value of t : (t as vector)

F = trapz(t,x+0.5*(u.^2));