How to solve multiple ODEs to fit empirical observations by optimizing multiple constants?
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I have 3 ODEs and 2 parameters to be optimized to fit the ODE's to given data..
eg dA/dt = -(K1+K2)*A;
dB/dt = K1*A;
dC/dt = K2*A
where t= time and K1,K2 are constants
I have been given A,B and C vs time data..I must manipulate K1 and K2 to match the data. How do I go about doing this using optimization toolbox preferably fmincon? Please suggest a sample code if possible..
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Teja Muppirala
2012-8-21
Below is an example that does exactly what you are describing. Save it into a file and run it. First I just made some sample data, and then I fit your equations to it, getting both K1 and K2, as well as initial conditions on the data.
function fitdata
% True Values
[K1,K2,A0,B0,C0] = deal(3.5,4.2,1,2,3);
[T,Y0] = ode45(@(t,y)[-(K1+K2)*y(1); K1*y(1); K2*y(1)],[0:0.005:1],[A0;B0;C0]);
Ymeas = Y0 + 0.1*randn(size(Y0));
figure;
plot(T,Ymeas);
hold on;
h = plot(T,nan*Ymeas,'k','linewidth',2);
minERR = Inf;
opts = optimset('fminunc');
opts.LargeScale = 'off';
Xest = fminunc(@(X)objfun(X),[0;0;0;0;0],opts);
Xest = num2cell(Xest);
[K1,K2,A0,B0,C0] = deal(Xest{:}),
legend({'A','B','C'});
function ERR = objfun(X);
X = num2cell(X);
[K1,K2,A0,B0,C0] = deal(X{:});
[T,Yest] = ode45(@(t,y)[-(K1+K2)*y(1); K1*y(1); K2*y(1)],[0:0.005:1],[A0;B0;C0]);
ERR = sum((Ymeas(:) - Yest(:)).^2);
if ERR < minERR
minERR = ERR;
for n = 1:3; set(h(n),'Ydata',Yest(:,n)); end
drawnow;
end;
end;
end
更多回答(3 个)
Ryan G
2012-7-31
I'm not sure how you would do this with MATLAB only but simulink design optimization would probably handle this fairly easy.
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Bjorn Gustavsson
2012-7-31
If the ODEs are that simple it should just be to integrate them analytically, then you'd simply end up with a well overdetermined least square fitting problem for K1 and K2 (perhaps you'd get A(0), B(0) and C(0) in there as unknowns too).
If the ODEs are a bit more complicated you could try a finite difference aproximation.
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Star Strider
2012-7-31
编辑:Star Strider
2012-7-31
If you are looking for a way to use an ODE solver with an objective function, I have used this strategy:
function Y = objfun(B, t) % Objective function
[T,Ymtx] = ode45(@DifEq, t, x0); % Do the ODE integration
function dY = DifEq(t, x) % Function ode45 integrates DifEq
ydot(1) = . . .;
. . .
ydot(n) = . . .;
dY = ydot
end
Y = Ymtx(:,2); % If Ymtx has more than one column, return the one you want here
end
Note that you do not have to pass the parameter vector B specifically to DifEq, since DifEq can access the B vector since it is part of objfcn.
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