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# Optimal Control and Estimation

Optimal Control and Estimation

1 Problem I
Using a least-square algorithm, t quadratic and cubic polynomials to the
following time series of the variable z:
1;27;33;45;12;16;83;67;54;39;23;6;14;15;19;31;37;44;56;60: (1)
Compute the mean-square error in both cases, and plot the results.
2 Problem II
Repeat Problem I, assuming that each point has a dierent weight in the
least square estimation procedure. In particular, the rst data point is 20
times better than the last, the second data point is 19 times better, and so
on.
3 Problem III
One more piece of data is to be added to the sequence in Problem I. Assuming
equal weighting of all points. How would a new reading of 25 aect the
quadratic curve t? Use a recursive least-square estimator to nd the answer.
4 Problem IV
Apply a recursive least square estimator to the entire time series of Problem
I, that is, compute a running estimate beginning with the rst point and
ending with the last.
5 Problem V
The vectorxis related to the vector yby the following equation,

x
1
x
2

=

0 1
3 4

y
1
y
2

: (2)
1
Given the following noisy measurementsz =y+n, what is the least-square estimate ofx?
z
1
= 0;1;7;8;5;7;9;10;6;4
z
2
= 10;7;4;5;5;3;0;2;2;4

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