The assumption that y x2 makes the resulting

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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 reduced-order 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 block-diagram representation of the reduced-order 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 - Reduced-Order State Observers - Bernard Friedland Figure 1: Reduced-order observer (Two forms). ˆ The matrices A, L, H, and K are chosen, as in the case of the full-order 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 - Reduced-Order State Observers - Bernard Friedland (10) As in the case of the full-order 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 full-order observer; (12) is a new requirement for the additional matrix L that is required by the reduced-order 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) left-half plane; A 22 and A12 in the reduced-order observer take the roles of A and C in the full-order observer;...
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