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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 = | | *((xtf−xfmeas)2 + (ytf−yfmeas)2) |
subject to:
[x0 y0 u0 v0] = [0.4 −0.3 0 0] |
[xfmeas yfmeas] = [−0.231625 −0.443109] |
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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