Frequency domain design of observers for linear systems using a ...

1
Third component of the estimation error
0.9
0.8
0.7
0.6
Amplitude
0.5
0.4
0.3
0.2
0.1
0
0 1 2 3 4 5 6 7 8
Time (sec)
Figure 3.: The evolution of the third component of the estimation error
which is stable. So,
⎡
⎣
s+2
s+4
−3
s+2
2
s+1
ˆM 1(s)
=
1
s+4
s+5
s+2
−2
s+1
−1
s+4
−1
s+2
s+2
s+1
−1
and therefore, ˆM 1 (s) needed in the frequency domain design reads
⎡
⎢
⎣
s 3 +11s 2 +36s+32
s 3 +9s 2 +27s+24
3s 2 +16s+16
s 3 +9s 2 +27s+24
−2s 2 −12s−16
s 3 +9s 2 +27s+24
ˆM −1
1 (s)
=
−s 2 −2s
s 3 +9s 2 +27s+24
s 3 +6s 2 +14s+12
s 3 +9s 2 +27s+24
2s 2 +10s+12
s 3 +9s 2 +27s+24
⎤
⎦
s 2 +5s+4
s 3 +9s 2 +27s+24
s 2 +6s+5
s 3 +9s 2 +27s+24
s 3 +8s 2 +20s+13
s 3 +9s 2 +27s+24
⎤
⎥
⎦
(42)
(43)
2) From (27), ˆN1 (s) = A l B l I 0 with B l = B. So,
⎡ ⎤
0
ˆN 1 (s) = ⎣ 0 ⎦ (44)
1
s+1
3) From (31), ˆN2 (s) = A l G l I 0 with G l = G. So,
⎡
⎤
ˆN 2 (s) = ⎣
2
s+4
−1
s+2
1
s+1
−2 2
s+4 s+4
1 1
s+2 s+2
−1
s+1
0
⎦ (45)
Following (43), (44) and (45) the proposed frequency domain observer (22)
holds.
Finally we propose to draw , as the time domain case, the behavior of the
frequency estimation error.
For that, from (22), it is readily follows that
ˆx(s) = M 0 u(s) (46)
8