Unformatted text preview: once the gain matrix
K is chosen, there is no further freedom in the choice of L and H .
The specific form of the new matrix L in (12) suggests another option for
implementation of the dynamics of the reducedorder observer, namely: ˆˆ
z = Ax2 + Ly + Hu (14) where L = A 21 − KA11 (15) A blockdiagram representation of this option is given in Figure 1 (b) The selection of the gain matrix K of the reducedorder observer may be accomplished
by any of the methods that can be used to select the gains of the fullorder observer as
discussed in the previous article. In particular, poleplacement, using any convenient
algorithm is feasible.
The gain matrix can also be obtained as the solution of a reducedorder Kalman filtering
problem, taking into account the crosscorrelation between the observation noise and the
process noise. Suppose the dynamic process is governed by ©Encyclopedia of Life Support Systems (EOLSS) CONTROL SYSTEMS, ROBOTICS AND AUTOMATION  Vol. VIII  ReducedOrder State Observers  Bernard Friedland x1 = A11x 1 + A 12 x 2 + B1 u + F1 v (16) x2 = A 21 x1 + A 22 x 2 + B 2 u + F2 v (17) with the observation being noisefree: y = x1 (18) In this case the gain matrix is given by ′
′
K = ( PA12 + F2QF1 ) R −1 (19) U
SA NE
M SC
PL O
E–
C EO
H
AP LS
TE S
R
S where ′
R = F1QF1 and P is the ( n − m) × ( n − m) covariance matrix of the estimation error e 2 , as given by ′
P = AP + PA′ − PA12 R −1A12P + Q (20) where ′
A = A 22 − F2QF1 R −1A12 (21) ′
′
′
Q = F2QF2 − F2QF1 R −1F1QF2 (22) The initial condition on (20) is () P t 0 = P0 the covariance matrix of the initial uncertainty of the substate x2 . Note that (20) becomes homogeneous when Q=0 (23) In this case it is possible that lim P (t ) := P ( ∞ ) = 0 t →∞ ©Encyclopedia of Life Support Systems (EOLSS) (24) CONTROL SYSTEMS, ROBOTICS AND AUTOMATION  Vol. VIII  ReducedOrder State Observers  Bernard Friedland which means that the steadystate error in estimating x2 converges to zero! We can’t
expect to achieve anything better than this. Unfortunately, P ( ∞ ) = 0 is not the only
possible steadystate solution to (23). To test whether it is, it is necessary to check
whether the eigenvalues of the resulting observer dynamics matrix ˆ
A = A 22 − F2F1−1A12 (25) lie in the open left halfplane. If not, (24) is not the correct steadystate solution to (20). U
SA NE
M SC
PL O
E–
C EO
H
AP LS
TE S
R
S The eigenvalues of the “zero steadystate variance” observer dynamics matrix (25) have
an interesting interpretation: as shown in by...
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 Spring '14

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