The specific form of the new matrix l in 12 suggests

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 reduced-order observer, namely: ˆˆ z = Ax2 + Ly + Hu (14) where L = A 21 − KA11 (15) A block-diagram representation of this option is given in Figure 1 (b) The selection of the gain matrix K of the reduced-order observer may be accomplished by any of the methods that can be used to select the gains of the full-order observer as discussed in the previous article. In particular, pole-placement, using any convenient algorithm is feasible. The gain matrix can also be obtained as the solution of a reduced-order Kalman filtering problem, taking into account the cross-correlation 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 - Reduced-Order 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 noise-free: 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 - Reduced-Order State Observers - Bernard Friedland which means that the steady-state 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 steady-state 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 half-plane. If not, (24) is not the correct steady-state 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 steady-state variance” observer dynamics matrix (25) have an interesting interpretation: as shown in by...
View Full Document

Ask a homework question - tutors are online