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Unformatted text preview: Friedland in 1989 (among others) these
eigenvalues are the transmission zeros of the plant with respect to the noise input to the
process. Hence the variance of the estimation error converges to zero if the plant is
“minimum phase” with respect to the noise input.
For purposes of robustness the noise distribution matrix F should include a term
proportional to the control distribution matrix B , i.e., F = F + q 2BB′ In this case, the zerovariance observer gain would satisfy H = B2 − KB1 = 0 (26) as q → ∞. If (26) is satisfied the observer poles are located at the transmission zeros of the plant.
Thus, in order to use the gain givenby (26), it is necessary for the plant to be minimumphase with respect to the input. In 1982 Rynaski has defined observers meeting this
requirement as robust observers which, as discussed below, have remarkable robustness
characteristics.
When a reducedorder observer is used, it is readily established that the closedloop
dynamics are given by ⎡ x ⎤ ⎡ A − BG
⎢e ⎥ = ⎢ 0
⎣ 2⎦ ⎣ BG2
⎤
A 22 − KA12 ⎥
⎦ ⎡x⎤
⎢e ⎥
⎣ 2⎦ (27) and hence that the eigenvalues of the closedloop system are given by s I − A + BG s I − A 22 + KA12 = 0
Thus the separation principle also holds when a reducedorder observer is used. ©Encyclopedia of Life Support Systems (EOLSS) (28) CONTROL SYSTEMS, ROBOTICS AND AUTOMATION  Vol. VIII  ReducedOrder State Observers  Bernard Friedland Robustness can be assessed by carrying out an analysis for a reducedorder observer
similar to the analysis for a fullorder observer. It is found that the characteristic
polynomial for the closedloop control system, when a reducedorder observer is used
and the actual control distribution matrix B = B + δB differs from the nominal
(design) value B , is given by sI − A = sI − F + ΔG2 ΔG −BG2 sI − A c (29) where Δ = Kδ B1 − B2 U
SA NE
M SC
PL O
E–
C EO
H
AP LS
TE S
R
S (30) It is seen that the characteristic polynomial of the closedloop system reduces to that of
(28) when Δ=0 (31) It is noted that (31) can hold in a singleinput system in which the loop gain is the only
variable parameter. In this case δ B1 = ρ B1 , δ B2 = ρ B2 (32) and thus ( ) Δ = ρ K B 1 − B 2 = −ρ H ( ) Hence, if the observer is designed to satisfy (26) H = B2 − KB 1 = 0 the separation
principle holds for arbitrary changes in the loop gain, thus justifying Ryn...
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This document was uploaded on 03/09/2014 for the course EL 5021 at Institute of Technology.
 Spring '14

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