Optimization of system ODE

Hello Matlabers!

I'm using fsolve function for optimization of system of ODE. And I'm facing difficulty now. Could someone advise me with this please? I'm just beginner in using Matlab and I do appreciate your advise.

The functions file is expressed below.

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function fval= SimFnc(X)
%Initial values
dt=1/32;
g=9.8065;
t=1:2003; %number of points
%Importing data from STAT
STAT=importdata('STAT.txt');
nx= STAT(:,10);
ny= STAT(:,11);
nz= STAT(:,12);
Wx_d= STAT(:,7);
Wx_r=deg2rad(Wx_d);
% Wx  = interp1(Wx_r,t);
Wy_d= STAT(:,8);
Wy_r=deg2rad(Wy_d);
% Wy  = interp1(Wy_r,t);
Wz_d= STAT(:,9);
Wz_r=deg2rad(Wz_d);
Theda_iz=STAT(:,13);
Gamma_iz=STAT(:,14);
Theda_izr=deg2rad(Theda_iz);
Gamma_izr=deg2rad(Gamma_iz);
Alpha_iz=STAT(:,29);
Alpha_izr=deg2rad(Alpha_iz);
Beta_iz=STAT(:,30);
Beta_izr=deg2rad(Beta_iz);
V_iz =STAT(:,28);
%Initial values of Variables
Alpha(1)=0.1587/57.2958;
Beta(1)=0.0143/57.2958;
V(1)=58.0154;
Theda(1)=0.0025/57.2958;
Gamma(1)=0.0013/57.2958;
Erx=X(1);
Ery=X(2);
Erz=X(3);
Erv=X(4);
Erw=X(5);
%Calculating by Euler's method
for i=1:2002
    ax=g*(nx(i)-sin(Theda(i)));
    ay=g*(ny(i)-cos(Theda(i))*cos(Gamma(i)));
    az=g*(nz(i)+cos(Theda(i))*sin(Gamma(i))); 
    csB=cos(Beta(i));
    csBinv=1/csB;
    ssB=sin(Beta(i)); 
    csA=cos(Alpha(i));
    ssA=sin(Alpha(i));
    axv=ax/V(i);
    ayv=ay/V(i);
    azv=az/V(i);
    Theda(i+1)=Theda(i)+dt*(((Wy_r(i)*Ery*sin(Gamma(i)))+(Wz_r(i)*Erz*cos(Gamma(i)))));

    Gamma(i+1)=Gamma(i)+dt*((Wx_r(i)*Erx-(tan (Theda(i))*(Wy_r(i)*Ery*cos(Gamma(i))-Wz_r(i)*Erz*sin(Gamma(i))))));
    
    Alpha(i+1)= Alpha(i)+ dt*( Wz_r(i)*Erz-csBinv*Erv(ssA*(axv-Wy_r(i)*Ery*ssB)+csA*(ayv+Wx_r(i)*Erx*ssB)) );       
                
    Beta(i+1)=Beta(i)+dt*( azv*csB-csA*(axv*Erv*ssB-Wy_r(i))+ssA*(ayv*ssB+Wx_r(i)*Erx) );           
              
    V(i+1)=V(i)+dt*(ax*csA*csB-ay*Erw*ssA*csB+az*ssB );
           
fval(1,1,:)=Theda;
fval(2,1,:)=Gamma;
fval(3,1,:)=Alpha;
fval(4,1,:)=Beta;
fval(5,1,:)=V;       
end
end

The optimization file is expressed below.

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%Solving NLAE
clc
clear 
zg=input('Enter initial guess');
format short g
%options = optimoptions('fsolve','Display','iter');
%options=optimoptions('fsolve','Algorithm','trust-region');
%options=optimoptions('fsolve','Algorithm','trust-region-dogleg');
options=optimoptions('fsolve','Algorithm','levenberg-marquardt');
%Sol=fsolve(@SimFnc,zg,options)
% @fncSi = @(t,y)myfun(t,y,Wx,Wy,Wz);
[X,fval,exitflag,output,hessian] =fsolve(@SimFnc,zg,options)

[Sargondjani]

It is not clear what your problem is. Also it is helpful if you provide a denser version of your question, 90% of your code just seems to define some parameters. (ie. provide a minimum working example).

[Nicholas Moung]

You are right. I have done the integration in decrete time and I need to identify some parameters (Erx=X(1); Ery=X(2); Erz=X(3); Erv=X(4); Erw=X(5) of the system by using fsolve function. I'm looking forward to hearing your further advise. Thank you!

Answer 1

[Sargondjani]

It is still not clear what your problem is. Do you get an error? Do you get a different solution than you expected?

Anyway fsolve will try to find the zeros of the output, but your output are large vectors, so they can not all be zero by setting 5 parameters. In that case it is probably better to use the function 'lsqnonlin'.

Other notes:

• you should preallocate variables: Theda = NaN(1,2002+1); before the loop;
• if you use lsqnonlin, i would recommend to use output arguments in a matrix (not 3d array): fval(1,:) = Theda;

[Nicholas Moung]

Hello Sir!

I did everything you suggested and I'm still getting some errors as expressed below.

Array indices must be positive integers or logical values.

Error in SimFnc5 (line 74)

Alpha(i+1)= Alpha(i)+ dt*( Wz_r(i)*Erz-csBinv*Erv(ssA*(axv-Wy_r(i)*Ery*ssB)+csA*(ayv+Wx_r(i)*Erx*ssB)) );

Error in fsolve (line 242)

fuser = feval(funfcn{3},x,varargin{:});

Error in Sol_SimFnc (line 12)

[X,fval,exitflag,output,hessian]=fsolve(@SimFnc5,zg,options)%jacobian,

Caused by:

Failure in initial objective function evaluation. FSOLVE cannot continue.

Could you please tell me what to do further? Thank you so much!

Answer 2

[Star Strider]

See the documentation section on Optimizing a Simulation or Ordinary Differential Equation