optimising a parameter not directly involved in the objective function to be minimised

someone kindly help me write a code to optimise h using fminsearch such that SSE is minimised. h is is not directly invloved in solving for SSE but is involved in the first differential equation whose output, 'L' is used in the subquest non-differential equations.

I am trying to replicate a publication in which it's mentioned that, 'equation 1 is a 1st order ODE solved by 4th order Runge-kutta solver which facilitates simultaneous solution of equation 2-4 at each time step'. I wonder how to simulataneously solve the differential equation and non-differential equation using ODE45. At the moment am only solving for equation 1 using the ODE45 and then solve for the non-differential equations.

Thank you.

%constants; KA, KV, p, D, Cs, mo, Ni and V

% Cb_exp is a column vector

dLdt = -(KA / (3 * p * KV)) * (D / h) .* (Cs - Cb_exp); %equation 1
 
 
mi_j = p * KV *((L).^3)*Ni; % Equation 2
Cb_cal = (mo - mi_j) / V; % Equation 3
S = KA * (L).^2 * Ni / V; % Equation 4


SSE= sum(Cb_cal-Cb_exp)^2

[Torsten]

As far as I understand your equations, mi_j, Cb_cal and S can be directly computed once you know L. So what's the problem ? You don't need to "solve" for them with the integrator you use.

[Pc]

That's right @Torsten, was just wondering if there was a way since the publication mentioned, they simulataneously solved it using ODE45. Must be my misinterpretation. How about the optimisation of h? do you have any suggestions for that?

Answer 1

[Torsten]

t_exp = ...;
Cb_exp = ...;    % experimental data 
KA =...;
KV =...;
p =...;
D =...;
Cs =...;
mo =...;
Ni =...;
V =...;    % constants
h0 = ...;  % Guess value for h
h = lsqnonlin(@(h)fun(h,t_exp,Cb_exp,KA, KV, p, D, Cs, mo, Ni,V),h0)

function res = fun(h,t_exp,Cb_exp,KA, KV, p, D, Cs, mo, Ni,V)
  tspan = t_exp;   % Assumes t_exp starts with t = 0
  L0 = ...;        % Initial value for L at t=0
  fun_ode = @(t,L) -(KA / (3 * p * KV)) * (D / h) * (Cs - interp1(t_exp,Cb_exp,t));
  [~,L] = ode45(@fun_ode,tspan,L0);
  mi_j = p * KV * L.^3 * Ni;
  Cb_cal = (mo - mi_j) / V;
  res = Cb_cal - Cb_exp;
end

[Pc]

Hi @Torsten, I get the following error from the code you shared.

t_exp = [0 30 60 90 120 180 240 300 360 420 480 540 600]';
Yexpt= [0.0000 0.1321 0.2447 0.3343 0.4122 0.5298 0.6246 0.6869 0.7348 0.7729 0.7931 0.8114 0.8272]';
V=100;
V=V*1e12;

Cb_exp= (Yexpt.*0.00425)./V;
Cs=4.25e-15;
mo=1;
v=200.3;
mw=206.28;
p=mo/((mo/mw)*v);
p = p * 1e-12; 
Lc=50;
D=8.88*10^-6;
D=D* 1e8; 
KA=pi; 
KV=pi/6;
Ni=18806.1232307428;

h0 = Lc/2; 

h = lsqnonlin(@(h)fun(h,t_exp,Cb_exp,KA, KV, p, D, Cs, mo, Ni,V),h0)

Unrecognized function or variable 'fun_ode'.

Error in odearguments (line 92)
f0 = ode(t0,y0,args{:}); % ODE15I sets args{1} to yp0.

Error in ode45 (line 104)
odearguments(odeIsFuncHandle,odeTreatAsMFile, solver_name, ode, tspan, y0, options, varargin);

Error in solution>fun (line 29)
[~,L] = ode45(@fun_ode,tspan,L0);

Error in solution>@(h)fun(h,t_exp,Cb_exp,KA,KV,p,D,Cs,mo,Ni,V) (line 22)
h = lsqnonlin(@(h)fun(h,t_exp,Cb_exp,KA, KV, p, D, Cs, mo, Ni,V),h0)

Error in lsqnonlin (line 242)
initVals.F = feval(funfcn{3},xCurrent,varargin{:});

function res = fun(h,t_exp,Cb_exp,KA, KV, p, D, Cs, mo, Ni,V)
  tspan = t_exp;  
  L0 = 462;        % Initial value for L at t=0
  fun_ode = @(t,L) -(KA / (3 * p * KV)) * (D / h) * (Cs - interp1(t_exp,Cb_exp,t));
  [~,L] = ode45(@fun_ode,tspan,L0);
  mi_j = p * KV * L.^3 * Ni;
  Cb_cal = (mo - mi_j) / V;
  res = Cb_cal - Cb_exp;
end

[Torsten]

[~,L] = ode45(fun_ode,tspan,L0);
res = 1e16*(Cb_cal - Cb_exp);

instead of

[~,L] = ode45(@fun_ode,tspan,L0);
res = Cb_cal - Cb_exp;