# Optimal Control and Estimation

Optimal Control and Estimation1 Problem IUsing a least-square algorithm, t quadratic and cubic polynomials to thefollowing 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 IIRepeat Problem I, assuming that each point has a di erent weight in theleast square estimation procedure. In particular, the rst data point is 20times better than the last, the second data point is 19 times better, and soon.3 Problem IIIOne more piece of data is to be added to the sequence in Problem I. Assumingequal weighting of all points. How would a new reading of 25 a ect thequadratic curve t? Use a recursive least-square estimator to nd the answer.4 Problem IVApply a recursive least square estimator to the entire time series of ProblemI, that is, compute a running estimate beginning with the rst point andending with the last.5 Problem VThe vectorxis related to the vector yby the following equation, x1x2 = 0 13 4 y1y2 : (2)1Given the following noisy measurementsz =y+n, what is the least-square estimate ofx?z1= 0;1;7;8;5;7;9;10;6;4z2= 10;7;4;5;5;3;0;2;2;4!

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