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

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Kunal Jain 2022 年 2 月 22 日

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)

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### 回答 (1 件)

Abolfazl Chaman Motlagh 2022 年 2 月 22 日
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));

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