程序代写代做代考 Exercise 3

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.