optimization of delayed differential equations (dde)

Hello,

I try to optimise (finding the parameter's value) in the system of dde, unfortunately I can't find any example how to do it. I used to optimization of ode using ode45 solver with lsqlin for instance. I will appreciate the help. The dde23 seems not to be working with lsqlin. I will appreciate any help.

[Torsten]

Do you have a code for your model equations set up with dde23 ?

If yes, you should include it and tell which parameter(s) you are trying to optimize on the basis of which input data.

Answer 1

[Torsten]

Make the best of it.

pTrue = [1 1 1 1 1 1 1 1 1];
time = 0:7;
Y = ...;
Am1=[1,1.1,1.4,0.7,0.8,1.6,2,1.5];
Bm1=[1.5,1.0,0.4,1.7,0.9,1.3,1.5,1.4];
Am=@(t)interp1(time,Am1,t)
Bm=@(t)interp1(time,Bm1,t)
p = lsqnonlin(@(p) Errors(p,time,Y,Am,Bm),0.8*pTrue)

function res = Errors(p,time,Y,Am,Bm)
  lags=[1,2];
  [T,SOL]=dde23(@(t,y,Z)ddefunEx(t,y,Z,p,Am,Bm), lags, [1,1,1,1], time);
  res = Y-SOL;
  res = res(:);
end

function dydt = ddefunEx(t,y,Z,p,Am,Bm)
  ylag1 = Z(:,1);
  ylag2 = Z(:,2);
  E1p=y(1)
  E2p=y(2)
  A1=y(3)
  B1=y(4)
  n_A=p(1);%0.6;
  n_B=p(2);%0.1;
  KE1=p(3);%0.2;
  KE2=p(4);%0.2;
  Vp_A=p(5);%1.2;
  Vp_B=p(6);%1.5;
  alpha_p=p(7);%0.4;
  kA=p(8);%0.4;
  kB=p(9);%1.2;
  %Auxiliary equations
  ALPHA1=n_A*E1p*A1/(KE1*(1+A1))-A1;
  ALPHA2=n_B*E2p*B1/(KE2*(1+B1))-B1;
  dydt= [Vp_A*Am(t)-alpha_p*ylag1(1);
         Vp_B*Bm(t)-alpha_p*ylag1(2);
         kA*Am(t)-ylag2(3)-ALPHA1+ALPHA2;
         kB*Bm(t)-ylag2(4)+ALPHA1-ALPHA2];
end

[Malgorzata Wieteska]

Thank you

[Priya Verma]

these command is not running in my matlab...please share all code..

[Priya Verma]

how to plot graphs between lags and variables for dde ?...please reply ..

[Torsten]

What do you mean by your question ? The line

[T,SOL]=dde23(@(t,y,Z)ddefunEx(t,y,Z,p,Am,Bm), lags, [1,1,1,1], time);

gives you a solution SOL at times T that you can plot. How do you think that the lags come into play ?

Answer 2

[Malgorzata Wieteska]

% I'm embedding simplified version of the system. I run it the function ddedunEx to get simulated data (Y) and then try to optimise for the values of the parameters against obtained earlier data.

function dydt = ddefunEx(t,y,Z,p)

%global p

ylag1 = Z(:,1);

ylag2 = Z(:,2);

time=0:7;

E1p=y(1)

E2p=y(2)

A1=y(3)

B1=y(4)

Am1=[1,1.1,1.4,0.7,0.8,1.6,2,1.5];

Bm1=[1.5,1.0,0.4,1.7,0.9,1.3,1.5,1.4];

Am=interp1(time,Am1,t)

Bm=interp1(time,Bm1,t)

n_A=p(1);%0.6;

n_B=p(2);%0.1;

KE1=p(3);%0.2;

KE2=p(4);%0.2;

Vp_A=p(5);%1.2;

Vp_B=p(6);%1.5;

alpha_p=p(7);%0.4;

kA=p(8);%0.4;

kB=p(9);%1.2;

%Auxiliary equations

ALPHA1=n_A*E1p*A1/(KE1*(1+A1))-A1;

ALPHA2=n_B*E2p*B1/(KE2*(1+B1))-B1;

dydt= [Vp_A*Am-alpha_p*ylag1(1);

Vp_B*Bm-alpha_p*ylag1(2);

kA*Am-ylag2(3)-ALPHA1+ALPHA2;

kB*Bm-ylag2(4)+ALPHA1-ALPHA2;];

end

%lags=[1,2];

%tspan=0:7;

%sol = dde23(@ddefunEx, lags, [1,1,1,1], tspan);

%figure(2)

%plot(sol.x,sol.y)

%%%%%%%%%%%%% Obtaining data

%Y=sol.y+0.05*randn(size(sol.x));

function res=Errors(p,Y)

tspan=0:7;;

x0=[4;1];

lags=[1,2];

[T,X]=dde23(@ddefunEx, lags, [1,1,1,1], tspan);

res1=(X(:,1)-Y(:,1));

res2=(X(:,2)-Y(:,2));

res3=(X(:,3)-Y(:,3));

res4=(X(:,4)-Y(:,4));

res=abs(res1)+abs(res2) +abs(res3)+abs(res4);

%pTrue=[0.6;0.1;0.2;0.2];

%[pOpt,resnorm,res,exitflag,~,lambda,J]=...

% lsqnonlin(@(p) Errors(p,Y),0.8*pTrue);

[Priya Verma]

these command is not running in my matlab...please share all code..

[Priya Verma]

what are define Am1 and Bm1 ?