Fall 2001 16.3116-20 Can use these estimate pole locations in acker, to get that 01 2「0112 0 L 0 2001 0 CA 1 Given L, A, and C, we can develop the estimator transfer function from the measurement y to the i2 01 01 01 S 0 s+22 22 +(S+ 0 S S2+-=s+ S+ ● Filter zero asymptotes to s=0asr→0 and the two poles→ . Resulting estimator looks like a" band-limited"differentiator This was expected because we measure position and want to estimate velocity. Frequency band over which we are willing to perform the dif- ferentiation determined by the relative cleanliness" of the mea- surementsFall 2001 16.31 16–20 • Can use these estimate pole locations in acker, to get that L = 
0 1 −ω2 0 0 2 + 2 √r
0 1 −ω2 0 0 + 2 r I 
C CA −1
0 1 =  2 r − ω2 0 √ 2 r −√ 2 r ω2 0 2 r − ω2 0 
1 0  0 1  −1
0 1 =  √ 2 r 2 r − ω2 0  • Given L, A, and C, we can develop the estimator transfer function from the measurement y to the ˆx2 xˆ2 y = 0 1   sI −
0 1 −ω2 0 0 +  √ 2 r 2 r − ω2 0  1 0  −1  √ 2 r 2 r − ω2 0  = 0 1   s + √ 2 r −1 2 r s −1  √ 2 r 2 r − ω2 0  = 0 1   s 1 −2 r s + √ 2 r   √ 2 r 2 r − ω2 0  1 s2 + √ 2 r s + 2 r = −2 r √ 2 r + (s + √ 2 r )(2 r − ω2 0) s2 + √ 2 r s + 2 r ≈ s − √rω2 0 s2 + √ 2 r s + 2 r • Filter zero asymptotes to s = 0 as r → 0 and the two poles → ∞ • Resulting estimator looks like a “band-limited” differentiator. – This was expected because we measure position and want to estimate velocity. – Frequency band over which we are willing to perform the dif￾ferentiation determined by the “relative cleanliness” of the mea￾surements.