Function for linearization of nonlinear system of equations

• V =F(rp,x1,x2,theta(t), psi(t), phi(t),wx(t),wy(t) ) is a non linear equation and the objective is to minimize output vector V for the given optimization variables rp, x1 and x2. Can anyone recommend me a method to linearize it using matlab ?

[Alan Stevens]

Why do you want to linearize it? The function fminsearch can work with nonlinear equations.

[HN]

Alan Stevens , rp,x1,x2 are bounded. So, I wanted to use constrained optimization. I couldn't implement fmincon because V is a vector. I wanted to get the minimum of V(1),V(2),V(3) simultaneously. Can I still use fminsearch ? (rp,x1,x2) = (rp,xi_2,xi_3).

Below is the function I wanted to minimize . Thank you.

function V = Opt_QN(rp,xi_2,xi_3) 
L=1;     
z=0.7071068; 
ts=1/2;
t=0:ts:1;
for k=1:length(t)
    xi_2=beta(k);
    rp=radius(k);
    for j=1:length(t)
        xi_1=0;
        xi_2=alpha(j);
        xi_3=beta(j);
        th(j)=-0.2*cos(2*pi*t(j));
        psi(k)=0.2*sin(2*pi*t(k)); 
        phi(j)=atan2(sin(psi(k))*sin(th(j)),(cos(psi(k))+cos(th(j))));
        R=Rot('y',th(j))*Rot('x',psi(k))*Rot('z',phi(j));
        x(k)=1/2*rp*(-cos(phi(j))*cos(psi(k))+cos(phi(j))*cos(th(j))+sin(phi(j))*sin(psi(k))*sin(th(j)));
        y(k)=-rp*cos(psi(k))*sin(phi(j));
        zc(k)=z;
        delta(k)=sqrt(x(k)^2+y(k)^2)
        p=[x(k);y(k);zc(k)];
       
        
    end 
end 
end 

[Alan Stevens]

You pass rp, xi_2 and xi_3 to the function, but almst immediately reassign them, so the input values do nothing!

Your output is a vector, so what is your minimization measure?

If you have access to the Optimisation toolbox you will have more options than fminsearch.

[HN]

Alan Stevens, I tried to use fmincon but I can't trust the result. It is also difficult to find global minimum and corresponding optimum parameters. Moreover, the computation time is quite long. Did I do something wrong in the double four loop shown below to call the function ? Yes, I have the toolbox.

%main program  
clc
close all 
%% UB LB setting  
x2L = deg2rad(90);
x2U = deg2rad(150);
x3U = deg2rad(-90);
x3L = deg2rad(-150);
x2=x2L:(x2U-x2L)/ts:x2U;
x3=x3L:(x3U-x3L)/ts:x3U;
A = [];
b = [];
Aeq = [];
beq = [];
%%
LB = [deg2rad(90), deg2rad(-150)]; % 120 =2.0944 , 150 =-2.6180
UB= [deg2rad(150), deg2rad(-90)];
options = optimoptions('fmincon','Algorithm','interior-point','SpecifyObjectiveGradient',true,'SpecifyConstraintGradient',true,'Display' ,'iter');
options = optimoptions(options,'SpecifyObjectiveGradient',false,'SpecifyConstraintGradient',false);

% %% Plot 
[X,Y] = meshgrid(x2,x3);
for idx =1:length(x2)
    for jdx=1:length(x3)  
        x12 = [x1(idx) , x2(jdx)];
        fun = @(x)PRSopt_QN(x)
       [V,S]=fun(x12);
        [Vec,Fval,exitflag,output,lambda,grad,hessian] = fmincon(fun,x12,[],[],[],[],LB,UB, [],options);
        Min(idx,jdx)=Fval;
        [xs,index]= sort(Min)
        result = Min(sort(index(1:x1)));
        Opt(:,idx)=Vec ;
        Average(idx,jdx)=V;
    end 
end 
 Mini=min(Min,[],'all'); 
 figure,
 contour3(X,Y,Average*100,'color','blue','ShowText','on');   
 title('Function evaluation')
 hLeg=legend({''},'FontSize',12);
 set(hLeg,'Interpreter','latex');
 box off;
 xlabel('\xi_1')
 ylabel('\xi_2')
 zlabel('mm/s')
 
 figure,contour3(X,Y,Min,'color','blue','ShowText','on');   
 title('Function evaluation')
 hLeg=legend({''},'FontSize',12);
 set(hLeg,'Interpreter','latex');
 box off;
 xlabel('\xi_1')
 ylabel('\xi_2')
 zlabel('mm/s')
 
 figure, surf(X,Y,Min)
 figure, surf(X,Y,Average*100)
 hLeg=legend({''},'FontSize',12);
 set(hLeg,'Interpreter','latex');
 box off;
 title('Optimized linear velocity ')
 xlabel('\xi_1')
 ylabel('\xi_2')
 zlabel('mm/s')
 % [vout, fout] = fmincon(fun,x);

[HN]

Alan Stevens, the output V elements are hetrogeneous physical models. So, I am confused how can I set minimization measure. My plan was to minimize each elements separately.

Thank you

[John D'Errico]

You can "trust" fmincon way more than you can trust fminsearch, lol.

[HN]

John D'Errico , I am just afraid if my assignment/difinition of input variables is not suitable to fmincon. Would you please share me your rough assesment on the following lines of calling the fucntion ? Thank you

% %% Plot  
x2L = deg2rad(90);
x2U = deg2rad(150);
x3U = deg2rad(-90);
x3L = deg2rad(-150);
x2=x2L:(x2U-x2L)/ts:x2U;
x3=x3L:(x3U-x3L)/ts:x3U;

[X,Y] = meshgrid(x2,x3);
for idx =1:length(x2)
    for jdx=1:length(x3)  
        x12 = [x1(idx) , x2(jdx)];
        fun = @(x)PRSopt_QN(x)
       [V,S]=fun(x12);
        [Vec,Fval,exitflag,output,lambda,grad,hessian] = fmincon(fun,x12,[],[],[],[],LB,UB, [],options);
        Min(idx,jdx)=Fval;
        [xs,index]= sort(Min)
        result = Min(sort(index(1:x1)));
        Opt(:,idx)=Vec ;
        Average(idx,jdx)=V;
    end 
end 

[Walter iacomacci]

Have you tried making the Block diagram of that differential equation with Simulink?

If you have a model in Simulink there is a pretty useful command called linmod which is easy to use, the thing is that it will linearize the model for you in terms of a State-space model (A,B,C,D) matrices which I dont know if works for you.

[HN]

Walter iacomacci Thank you,

I don't have model in simulink. I hope I can convert the program above easily. right ?

[Walter iacomacci]

If you have the actual differential equation you can recreate it easily in Simulink. Here is an example:

Which is a Differential Equation System of a Circuit, the Block diagram looks like this with Vin=30

Ignore the gain with 1/4 going to VL it is just a relation to obtain the values in time for a voltage in the inductor.

If you have only 1 differential equation is much easier. Let me know if you need help.