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85  Pendulum Gravity Estimation

User’s Guide for DIRCOL

Problem 2.5 Pendulum

85.1  Problem Formulation

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

J = 
1
2
*((xtfxfmeas)2 + (ytfyfmeas)2


subject to:

dx
dt
 = u 
dy
dt
 = v 
du
dt
 = lambda*x/m 
dv
dt
 = lambda*y/mg 
x2 + y2 − L2 = 0 
[x0  y0  u0  v0] = [0.4  −0.3  0  0] 
[xfmeas  yfmeas] = [−0.231625  −0.443109] 
L = 0.5 
m = 0.3 


Reference: [33]

85.2  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);

85.3  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
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2011-02-05
=====================================================================================
Problem: ---  1: Pendulum Gravity               f_k       0.000000017178935982
                                       sum(|constr|)      0.000001384358714611
                              f(x_k) + sum(|constr|)      0.000001401537650593
                                              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: 0.265625 sec. Elapsed time: 0.266000 sec.
Problem type appears to be: qpcon
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2011-02-05
=====================================================================================
Problem: ---  1: Pendulum Gravity               f_k       0.000000000017699633
                                       sum(|constr|)      0.000000032137877804
                              f(x_k) + sum(|constr|)      0.000000032155577437
                                              f(x_0)     -0.124997846074063950

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: 0.437500 sec. Elapsed time: 0.484000 sec.
end

85.4  Show result

disp(sprintf('Gravity estimated to %g',gopt));
Gravity estimated to 9.82655

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