PROPT Pendulum Gravity Estimation

This page is part of the PROPT Manual. See PROPT Manual.

User's Guide for DIRCOL

Problem 2.5 Pendulum

Problem Formulation

Find u over t in [0; 2 ] to minimize

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J = \frac{1}{2}*((x_{t_f}-x_{f_{meas}})^2 + (y_{t_f}-y_{f_{meas}})^2) }

subject to:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dx}{dt} = u }

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dt} = v }

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{du}{dt} = lambda*x/m }

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dv}{dt} = lambda*y/m-g }

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 + y^2 - L^2 = 0 }

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [x_0 \ y_0 \ u_0 \ v_0] = [0.4 \ -0.3 \ 0 \ 0] }

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [x_{f_{meas}} \ y_{f_{meas}}] = [-0.231625 \ -0.443109] }

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = 0.5 }

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = 0.3 }

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

Problem setup

toms t g

% Initial guess
gopt = 20;
xopt = 0.4-(0.4+0.231625)*t/2;
yopt = -0.3-(-0.3+0.443109)*t/2;
uopt = 0;
vopt = 0;
lambdaopt = -5;

for n=[20 51]

    p = tomPhase('p', t, 0, 2, n);
    setPhase(p);
    tomStates x y u v
    tomControls lambda

    % Initial guess
    x0 = {g == gopt
        icollocate({
        x == xopt
        y == yopt
        u == uopt
        v == vopt})
        collocate(lambda == lambdaopt)};

    % Box constraints
    cbox = {1 <= g <= 100};

    % Boundary constraints
    cbnd = initial({x == 0.4; y == -0.3
        u == 0; v == 0});

    L   = 0.5;
    m   = 0.3;
    xmeas = -0.231625;
    ymeas = -0.443109;

    % ODEs and path constraints
    ceq = collocate({
        dot(x) == u
        dot(y) == v
        dot(u) == lambda.*x/m
        dot(v) == lambda.*y/m-g
        x.^2 + y.^2 - L^2 == 0});

    % Objective
    objective = 1/2*((final(x)-xmeas)^2+(final(y)-ymeas)^2);

Solve the problem

    options = struct;
    options.name = 'Pendulum Gravity';
    solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
    gopt = subs(g, solution);
    xopt = subs(x, solution);
    yopt = subs(y, solution);
    uopt = subs(u, solution);
    vopt = subs(v, solution);
    lambdaopt = subs(lambda, solution);

Problem type appears to be: qpcon
Time for symbolic processing: 0.14939 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Pendulum Gravity               f_k       0.000000017178935982
                                       sum(|constr|)      0.000001384358823592
                              f(x_k) + sum(|constr|)      0.000001401537759574
                                              f(x_0)     -0.124997863253000020

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

FuncEv    1 ConstrEv   98 ConJacEv   98 Iter   43 MinorIter   96
CPU time: 0.078000 sec. Elapsed time: 0.088000 sec. 

Problem type appears to be: qpcon
Time for symbolic processing: 0.15159 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Pendulum Gravity               f_k       0.000000000017699578
                                       sum(|constr|)      0.000000032138058518
                              f(x_k) + sum(|constr|)      0.000000032155758096
                                              f(x_0)     -0.124997846074064280

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

FuncEv    1 ConstrEv   13 ConJacEv   13 Iter   12 MinorIter  190
CPU time: 0.202801 sec. Elapsed time: 0.200000 sec. 

end

Show result

disp(sprintf('Gravity estimated to %g',gopt));

Gravity estimated to 9.82655