Optimization with many nonlinear Constraints

Hello,

I want to use my optimization Problem with the Optimization Toolbox and the fmincon function.

This is my Optimization problem with my linear constraints and lower and upper bounds

objective=@z;

x(1:4)=[Fx(1), Fx(2), Fx(3), Fx(4)];
x(5:8)=[alpha(1), alpha(2), alpha(3), alpha(4)];

%initial guess
x0=[0, 0, 0, 0, 0, 0, 0, 0]


%variable bounds
lb=[Fx_min(:,1), Fx_min(:,2), Fx_min(:,3), Fx_min(:,4), d_min_vorne+beta(:,1), d_min_vorne+beta(:,2), d_min_hinten+beta(:,3), d_min_hinten+beta(:,4)];
ub=[Fx_max(:,1), Fx_max(:,2), Fx_max(:,3), Fx_max(:,4), d_max_vorne-beta(:,1), d_max_vorne-beta(;,2), d_max_hinten-beta(:,3), d_max_hinten-beta(:,4)];


%linear constraints
%Definition alpha
Aeq(2:5)=[0 0 0 0 1 0 0 0; 0 0 0 0 0 1 0 0; 0 0 0 0 0 0 1 0; 0 0 0 0 0 0 0 1];
beq(2:5)=[d(1)-beta(1); d(2)-beta(2); d(3)-beta(3); d(4)-beta(4)];

    
%Alternative
%Folgen Solltrajektorie Fx
Aeq(:,1)=[1 1 1 1 0 0 0 0];
beq(:,1)=Fx_v;

A=[];
b=[];


%nonlinear constraints
nonlincon=@nlcon;

%optimize with fmincon
[x,fval,exitflag,output]=fmincon(objective,x0,A,b,Aeq,beq,lb,ub,@nonlcon);

I have my objective function with

i=1:4;
if Fx(:,i) < 0
   FxB(:,i)=(-Fx(:,i)+c_w(:,i)*alpha(:,i)*sin(d(:,i)))/cos(d(:,i));
   FxA(:,i)=0;
else
    FxA(:,i)=(Fx(:,i)+c_w(:,i)*alpha(:,i)*sin(d(:,i)))/cos(d(:,i));
    FxB(:,i)=0;
end

%delta Lenkwinkel
delta_d(i)=d(i,t-1)-d(i,t);

%die drei Ziele
S_1(i)=(A_i/SOC(i))*FxA(i);
S_2(i)=((B_i*M(i))/SOC(i))*(1/FxB(i));
S_3(i)= C_i*delta_d(i);

% die Summen der drei Zielfunktionen
syms i
z_1 = symsum(S_1,i,1,4);
z_2 = symsum(S_2,i,1,4);
z_3 = symsum(S_3,i,1,4);

%objektive function 
create function z = z_1 + z_2 + z_3;

and my nonlinear constraints

function[C,Ceq]=nlcon(FxA,alpha);

i=1:4;

%Folgen der Solltrajektorie
Mz(i) = -w_vl*(cos(d(1))*Fx(1)-sin(d(1))*Fy(1))+w_vr*(cos(d(2))*Fx(2)-sin(d(2))*Fy(2))-w_hl*(cos(d(3))*Fx(3)-sin(d(3))*Fy(3))+w_hr*(cos(d(4))*Fx(4)-sin(d(4))*Fy(4))+l_v*(sin(d(1))*Fx(1)-cos(d(1))*Fy(1)+sin(d(2))*Fx(2)-cos(d(2))*Fy(2))-l_h*(sin(d(3))*Fx(3)-cos(d(3))*Fy(3)+sin(d(4))*Fy(4)-cos(d(4))*Fy(4));
Ceq(:,1)=Fy_v-symsum(sin(d(i))*((Fx(i)+c_w(i)*alpha(i)*sin(d(i)))/cos(d(i)))-cos(d(i))*c_w(i)*alpha(i),i,1,4);
Ceq(:,2)=Mz_v-symsum(Mz(i),i,1,4);


%maximales regeneratives Bremsmoment
if 
    Fx_v-symsum(Fx(i),i,1,4)=0;
    Fy_v-symsum(sin(d(i))*((Fx(i)+c_w(i)*alpha(i)*sin(d(i)))/cos(d(i)))-cos(d(i))*c_w(i)*alpha(i))=0;
    Mz_v-symsum(Mz(i),i,1,4)=0;

then C(:,1)=-(FxB(i)+abs(M(i)/r_dyn));
else C(:,1)=-FxB(i)-Fx_max;
       
    
%aktive Anregung 
%Eingabe k=1 oder k=0, 
%Eingabe j=1...4, u=1...4, 
%Eingabe deltaT_aA=Differenzmoment
for k=1;
    Ceq(:,3)=r_dyn*FxA(:,j)-r_dyn*FxA(:,u)-deltaT_aA;
end

%Lenkpriorisierung
for SOC(i)<=SOC_krit;
        C=FxA(i);

I guess I have everything whats needed but I still get many errors back, because of the dimensions of the Vektors.

Can someone help me whats my fault?

Answer 1

[Walter Roberson]

then is not a valid MATLAB keyword in your if/then/else construct.

MATLAB comparison uses the == operator, not the = operator.

To test that three conditions all hold, use the && operator between them.

symbolic variables can never be used to index arrays. symsum(Fx(i),i,1,4) will not work. Try sum(Fx(1:4)) instead.

It is not obvious that your nonlinear constraint function has access to the variable Fx

[Sarah Kern]

Thank you very much.

I changed everything but its still not working.

You have another idea what else could be wrong?

[Walter Roberson]

What is your revised code?

[Sarah Kern]

So I rewrote some parts, now it looks like this

My problem looks like

objective = @z;

load parameter();

load Inputs;

x(1:4) = [FxW(1), FxW(2), FxW(3), FxW(4)];

x(5:8) = [alpha(1), alpha(2), alpha(3), alpha(4)];

%initial guess

x0=[0, 0, 0, 0, 0, 0, 0, 0];

%variable bounds

%case decision braking or driving backwards (decision input: flag=0 oder 1)

%braking mode

if flag == 1 %if it is possible the regenerative braking torque is the limit

if %if it is not possible to follow the target trajectory the maximum of the manual braking is used

Fx_v-sum((FxW(i)/cos(d(i)))-c_w(i)*sin(d(i))*alpha(i),(1:4)) == 0;

&& Fy_v-sum(Fy_v-sum(sin(d(i))*FxA(i)-cos(d(i))*c_w(i)*alpha(i),(1:4)) == 0;

&& Mz_v-sum(Mz(i),(1:4)) == 0;

% max. regenerative braking torque

lb(1:4) = [M_reg(1)/r_dyn, M_reg(2)/r_dyn, M_reg(3)/r_dyn, M_reg(4)/r_dyn]; %regenerative braking torque must be negative

% mechanical braking

else lb(1:4) = [FxB_max, FxB_max, FxB_max, FxB_max]; %FxB_max is negative

end % close conditions for braking loop

%driving mode

else lb(1:4) = [-P_max(:,1), -P_max(:,1), -P_max(:,1), -P_max(:,1)];

end

lb(5:8)= [d_min_vl-beta(:,1), d_min_vr-beta(:,2), d_min_hl-beta(:,3), d_min_hr-beta(:,4)]; %d and betato the left side are positive, to the right side are negative

ub = [P_max(:,1)/(r_dyn*w), P_max(:,2)/(r_dyn*w), P_max(:,3)/(r_dyn*w), P_max(:,4)/(r_dyn*w), d_max_vorne-beta(:,1), d_max_vorne-beta(;,2), d_max_hinten-beta(:,3), d_max_hinten-beta(:,4)];

%linear constraints

%Definition alpha

Aeq(1:5) = [0 0 0 0 1 0 0 0; 0 0 0 0 0 1 0 0; 0 0 0 0 0 0 1 0; 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 0 0];

beq(1:5) = [d(1)-beta(1); d(2)-beta(2); d(3)-beta(3); d(4)-beta(4); 0];

% active stimulation for beta

%if k = 1 there is a stimulation

% k, j and n are Inputs, j and n are the actors stimulated

for k = 1

Aeq(5,j) = [1]; % target moduls

&& A(5,n) = [-1];

&& beq(5) = [delta_TaA]; % delta torque

end

%steering priority

f = 1:4;

if SOC(f) <= SOC_krit; % state of charge of one actor is really low, actor is only used for steering

A(1,f) = 1;

&& b(1) = 0;

else

A = [];

b = [];

%nonlinear constraints

nonlincon=@nlcon;

%optimize with fmincon

[x,fval,exitflag,output]=fmincon(objective,x0,A,b,Aeq,beq,lb,ub,@nonlcon);

then the objective function is

%the two objective functions

%case decision

i = 1:4;

if flag = 1; % decision braking or driving mode

S_1(i) = (A_i/SOC(i))*abs(FxW(i)); %driving mode

else

S_1(i) = ((B_i*M_reg(i))/SOC(i))*(1/FxW(i)); %braking mode

end

%delta steering angle

delta_d(i) = d_delay(i)-d(i); %d_delay in a seperate file or in simulink, equals d(t-1)

%minimizing the steering movement

S_2(i) = C_i*delta_d(i);

% sum of the three objectives

syms i

z_1 = sum(S_1,(1:4));

z_2 = sum(S_2,(1:4));

%objektive function

create function z = z_1 + z_2 ;

and my nonln constraints are

function[C,Ceq] = nlcon(FxW,alpha)

i = 1:4;

Fx_v(i) = u(1);

Fy_v(i) = u(2);

Mz_v(i) = u(3);

%Following the target trajectory ( equal for braking or driving mode)

Mz(i) = -w_vl*(cos(d(1))*FxW(1)-sin(d(1))*alpha(1)*c_w)+w_vr*(cos(d(2))*FxW(2)-sin(d(2))*alpha(2)*c_w)-w_hl*(cos(d(3))*FxW(3)-sin(d(3))*alpha(3)*c_w)+w_hr*(cos(d(4))*FxW(4)-sin(d(4))*alpha(4)*c_w)+l_v*(sin(d(1))*FxW(1)-cos(d(1))*alpha(1)*c_w+sin(d(2))*FxW(2)-cos(d(2))*alpha(2)*c_w)-l_h*(sin(d(3))*FxW(3)-cos(d(3))*alpha(3)*c_w+sin(d(4))*FxW(4)*c_w-cos(d(4))*alpha(4)*c_w);

Ceq(:,1) = Fy_v-sum(sin(d(i))*FxW(i)-cos(d(i))*c_w(i)*alpha(i),(1:4));

Ceq(:,2) = Mz_v-sum(Mz(i),(1:4));

Ceq(:,3) = Fx_v-sum((FxW(i)/cos(d(i)))-c_w(i)*sin(d(i))*alpha(i),(1:4));

the file integrated in simulink with all the inputs is

function y = fcn(u)

%definition of vehicle forces and torque

u(1) = Fx_v;

u(2) = Fy_v;

u(3) = Mz_v;

%definition of current torque limits

i = 1:4;

u(4) = P_max(i);

u(5) = M_reg(i);

%active stimulation Input

u(6) = k;

u(7) = delta_T_aA;

u(8) = beta(i);

u(9) = w(i);

u(10) = d_delay(i);

u(11) = r_dyn;

u(12) = SOC(i);

y = @myfun;

Does ist look correct and can you tell me how I will get the Inputs u into my other files ? Its not working with @

If you can give me some advise thx a lot.