### 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