Parameter Estimation Problem

Example 5: DYNOPT User's Guide version 4.1.0

M. Cizniar, M. Fikar, M. A. Latifi, MATLAB Dynamic Optimisation Code DYNOPT. User's Guide, Technical Report, KIRP FCHPT STU Bratislava, Slovak Republic, 2006.

Problem description

Find p1 and p2 over t in [0; 6 ] to minimize

subject to:

where

% Copyright (c) 2007-2008 by Tomlab Optimization Inc.


Problem setup

toms t p1 p2
x1meas = [0.264;0.594;0.801;0.959];
tmeas  = [1;2;3;5];

% Box constraints
cbox = {-1.5 <= p1 <= 1.5
-1.5 <= p2 <= 1.5};


Solve the problem, using a successively larger number collocation points

for n=[10 40]

    p = tomPhase('p', t, 0, 6, n);
setPhase(p);
tomStates x1 x2

% Initial guess
if n == 10
x0 = {p1 == 0; p2 == 0};
else
x0 = {p1 == p1opt; p2 == p2opt
icollocate({x1 == x1opt; x2 == x2opt})};
end

% Boundary constraints
cbnd = initial({x1 == p1; x2 == p2});

% ODEs and path constraints
x1err = sum((atPoints(tmeas,x1) - x1meas).^2);
ceq = collocate({dot(x1) == x2; dot(x2) == 1-2*x2-x1});

% Objective
objective = x1err;


Solve the problem

    options = struct;
options.name   = 'Parameter Estimation';
options.solver = 'snopt';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);

% Optimal x, p for starting point
x1opt = subs(x1, solution);
x2opt = subs(x2, solution);
p1opt = subs(p1, solution);
p2opt = subs(p2, solution);

Problem type appears to be: qp
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem:  1: Parameter Estimation               f_k       0.000000352979299345
sum(|constr|)      0.000000000000011564
f(x_k) + sum(|constr|)      0.000000352979310909
f(x_0)      0.000000000000000000

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 Iter    4 MinorIter   15

Problem type appears to be: qp
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem:  1: Parameter Estimation               f_k       0.000000355669548258
sum(|constr|)      0.000000030827626955
f(x_k) + sum(|constr|)      0.000000386497175212
f(x_0)     -1.983813647020701100

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 MinorIter   22

end

t  = subs(collocate(t),solution);
x1 = collocate(x1opt);


Plot result

figure(1)
plot(t,x1,'*-',tmeas,x1meas,'ro');
legend('x1','Meas');
title('Parameter Estimation state variable');