Unformatted text preview: f the state vector that cannot be measured directly. The assumption that y = x2 makes the resulting equations simpler, but it is not
necessary. Equivalent results can be obtained for any observation matrix C of rank m .
In terms of x1 and x2 the plant dynamics are written x1 = A11x1 + A12 x2 + B1u (3) x2 = A 21x1 + A 22 x2 + B 2u (4) Since x1 is directly measured, no observer is required for that substate, i.e. ˆ
x1 = x1 = y (5) For the remaining substate, we define the reducedorder observer by ˆ
x2 = Ky + z (6) where z is the state of a system of order n − m : ˆ
z = Az + Ly + Hu (7) A blockdiagram representation of the reducedorder observer is given in Figure 1(a) ©Encyclopedia of Life Support Systems (EOLSS) U
SA NE
M SC
PL O
E–
C EO
H
AP LS
TE S
R
S CONTROL SYSTEMS, ROBOTICS AND AUTOMATION  Vol. VIII  ReducedOrder State Observers  Bernard Friedland Figure 1: Reducedorder observer (Two forms). ˆ
The matrices A, L, H, and K are chosen, as in the case of the fullorder observer, to
ensure that the error in the estimation of the state converges to zero, independent of
x, y, and u .
Since there is no error in estimation of x1 , i.e., ˆ
e1 = x1 − x1 = 0 (8) by virtue of (5), it is necessary only to ensure the convergence of ˆ
e 2 = x2 − x 2 (9) to zero.
From (4) – (7)
ˆ
ˆ
ˆ
e 2 = ( A 21 − KA11 + AK − L ) x1 + ( A 22 − KA12 − A ) x2 + Ae 2 + ( B2 − KB1 − H ) u ©Encyclopedia of Life Support Systems (EOLSS) CONTROL SYSTEMS, ROBOTICS AND AUTOMATION  Vol. VIII  ReducedOrder State Observers  Bernard Friedland (10)
As in the case of the fullorder observer, to make the coefficients of x1 , x2 , and u
vanish it is necessary that the matrices in (5) and (7) satisfy ˆ
A = A 22 − KA12 (11) ˆ
L = A 21 − KA11 + AK (12) H = B2 − KB1 (13) U
SA NE
M SC
PL O
E–
C EO
H
AP LS
TE S
R
S Two of these conditions (11) and (13) are analogous to conditions for a fullorder
observer; (12) is a new requirement for the additional matrix L that is required by the
reducedorder observer.
When these conditions are satisfied, the error in estimation of x2 is given by ˆ
e2 = Ae2 Hence the gain matrix K must be chosen such that the eigenvalues of ˆ
A = A 22 − KA12 lie in the (open) lefthalf plane; A 22 and A12 in the reducedorder
observer take the roles of A and C in the fullorder observer;...
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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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