How to optimize nonlinear multiple input model with multiple constr... (2024)

% LIVE OPTIMISATION EDITOR

function f = objectiveFcn(optimInput)

z1 = optimInput(1);

z2 = optimInput(2);

z3 = optimInput(3);

f = -2*((0.442*z2-0.207*z3-0.367*exp(-z3)*exp(-z1)+1.33*z3*exp(-z3) +0.348) - (0.192*z1 - 0.624*exp(-3*z3)+ 0.507*z2^(1/2)- 0.11* abs(z3)^2*abs(z2)* abs(2*z3 +z1) + 0.66)*(0.916886995261484 - 0.26709662073194) + 0.26709662073194)*1.76 - ((0.192*z1 - 0.624*exp(-3*z3)+ 0.507*z2^(1/2)- 0.11* abs(z3)^2*abs(z2)* abs(2*z3 +z1) + 0.66) * (1.00459544182241 - 0.200345860452685) + 0.200345860452685)*2.958;

end

function [c,ceq] = constraintFcn(optimInput)

% Note, if no inequality constraints, specify c = []

% Note, if no equality constraints, specify ceq = []

z1 = optimInput(1);

z2 = optimInput(2);

z3 = optimInput(3);

c(1) = z1 - 50;

c(2) = 10 - z1;

c(3) = z2 - 170;

c(4) = 130 - z2;

c(5) = z3 - 89;

c(6) = 11 - z3;

ceq = [];

end

% LAGRANGE FORMULA

syms z1 z2 z3 lambda

%conversion rates from normalization to (%)

fur_min = 0.0020923694565838 ;

fur_max = 0.17896582936453 ;

glu_min = 0.26709662073194 ;

glu_max = 0.916886995261484 ;

xyl_min = 0.200345860452685 ;

xyl_max = 1.00459544182241 ;

% functions (normalized)

glu_n = 0.442*z2-0.207*z3-0.367*exp(-z3)*exp(-z1)+1.33*z3*exp(-z3) +0.348 ; % glucose function (normalized)

xyl_n = 0.192*z1 - 0.624*exp(-3*z3)+ 0.507*z2^(1/2)- 0.11* abs(z3)^2*abs(z2)* abs(2*z3 +z1) + 0.66 ; % xylose function (normalized)

fur_n = 0.332*z3*z2^2 - 0.0289*z3 + 0.332*z2*z1^2 + 0.332*z1*z2*z3 + 0.00825 ; % furfural function (normalized)

glu_p = glu_n * (glu_max - glu_min) + glu_min ; % glucose function in (%)

xyl_p = xyl_n * (xyl_max - xyl_min) + xyl_min ; % xylose function in (%)

fur_p = fur_n * (fur_max - fur_min) + fur_min ; % furfural function in (%)

glu = glu_p * 2.958 ; % glucose function in (g)

xyl = xyl_p * 1.584 ; % xylose function in (g)

% constraints

fur = (fur_p * 1.76 / 0.9 * 10) <= 3 ; % furfural CONSTRAINT in (g/l)

% hb_z1 <= 50

% lb_z1 >= 10

% hb_z2 <= 170

% lb_z2 >= 130 % HOW DO I PUT THESE CONSTRAINTS IN?

% hb_z3 <= 89

% lb_z3 >= 11

% lb_glu >= 0

% hb_glu <= 1

% lb_xyl >= 0

% hb_glu <= 1

% lagrange functions

f = 2*glu + xyl % maximazation function

L = f - lambda * lhs(fur) % lagrange formula

% calculations

dL_dz1 = diff(L,z1) == 0; % derivative of L with respect to z1 (time)

dL_dz2 = diff(L,z2) == 0; % derivative of L with respect to z2 (temp)

dL_dz3 = diff(L,z3) == 0; % derivative of L with respect to z3 (conc)

dL_dlambda = diff(L,lambda) == 0; % derivative of L with respect to lambda

% outcome

system = [dL_dz1; dL_dz2; dL_dz3; dL_dlambda]; % build the system of equations

[z1_val_n, z2_val_n, z3_val_n,lambda_val] = solve(system, [z1 z2 z3 lambda], 'Real', true) % solve the system of equations and display the results