Exercise 3
(a)
Let be the measurement at time .
Let be the number of measurements. In this case, .
The residual at time is
The objective function is
Our goal is to minimize , it is a least-squares problem.
The Jacobian matrix is
(b)
Gauss-Newton
Parameters
Name
Value
x0
[1,1,1]’
descent
‘gauss’
alpha0
0.05
tol
0.00001
maxIter
10000
Result
3.3976
147.2555
1.9922
88.0913
Plot
Levenberg-Marquardt
Parameters
Name
Value
x0
[1,1,1]’
Delta
1
eta
0.001
tol
0.00001
maxIter
10000
Result
3.3984
147.2763
1.9922
88.0908
Plot
Discussion
We can see that the parameters estimated by Gauss-Newton and Levenberg-Marquardt are very similar. The objective value achieved by Levenberg-Marquardt is a little lower than Gauss-Newton (88.0908 compared with 88.0913).
From the fit plots, we also can see their estimation have no obvious difference, both are good fit the noisy measurements. The estimated paraeters are close to the actual value.