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.

Contents

Problem description

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

$$ J = \sum_{i=1,2,3,5} ( x_1(t_i) - x_1^{m}(t_i) )^2 $$

subject to:

$$ \frac{dx_1}{dt} = x_2 $$

$$ \frac{dx_2}{dt} = 1-2*x_2-x_1 $$

where

$$ x_0 = [p_1 \ p_2] $$

$$ t_i = [1 \ 2 \ 3 \ 5] $$

$$ x_1^{m}(t_i) = [0.264 \ 0.594 \ 0.801 \ 0.959] $$

$$ -1.5 <= p_{1:2} <= 1.5 $$

% 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');