265_adv_control_eng

45 reduced order state observers a full order state

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Unformatted text preview: ed by the introduction of the state observer. Since the observer is normally designed to have a more rapid response than the control system with full order observed state feedback, the pole-placement roots will dominate. ” Using the state vectors x(t) and x(t) the state equations for the closed-loop system are From equations (8.151) and (8.152) • ” x ˆ Ax À BKx (8:158) • ” ” ” x ˆ (A À Ke C)x À BKx ‡ Ke Cx ” ˆ (A À Ke C À BK)x ‡ Ke Cx (8:159) and from equation (8.129) Thus the closed-loop state equations are ! • x A • ˆ Ke C ” x ÀBK A À Ke C À BK ! x ” x ! (8:160) //SYS21/D:/B&H3B2/ACE/REVISES(08-08-01)/ACEC08.3D ± 262 ± [232±271/40] 9.8.2001 2:34PM 262 Advanced Control Engineering 8.4.5 Reduced-order state observers A full-order state observer estimates all state variables, irrespective of whether they are being measured. In practice, it would appear logical to use a combination of measured states from y ˆ Cx and observed states (for those state variables that are either not being measured, or not being measured with sufficient accuracy). If the state vector is of nth order and the measured output vector is of mth order, then it is only necessary to design an (n À m)th order state observer. Consider the case of the measurement of a single state variable x1 (t). The output equation is therefore y ˆ x1 ˆ Cx ˆ [ 1 Partition the state vector xˆ x1 xe 0 F F F 0 ]x (8:161) ! where xe are the state variables to be observed. Partition the state equations ! ! ! ! • x1 a11 A1e x1 b1 ˆ ‡ u • Ae1 Aee x2 Be xe (8:162) (8:163) If the desired eigenvalues for the reduced-order observer are s ˆ 1e , s ˆ 2e , F F F , s ˆ (nÀ1)e Then it can be shown that the characteristic equation for the reduced-order observer is jsI À Aee ‡ Ke A1e j ˆ (s À 1e ) F F F (s À (nÀ1)e ) ˆ snÀ1 ‡ (nÀ2)e snÀ2 ‡ Á Á Á ‡ 1e s ‡ oe (8:164) In equation (8.164) Aee replaces A and A1e replaces C in the full-order observer. The reduced-order observer gain matrix Ke can also be obtained using appropriate substitutions into equations mentioned earlier. For example, equation (8.132) becomes P Q oe À aoe T U 1e À a1e T U (8:165) K e ˆ Qe T F U F R S F (nÀ2)e À a(nÀ2)e where aoe , F F F , a(nÀ2)e are the coefficients of the open-loop reduced order characteristics equation jsI À Aee j ˆ snÀ1 ‡ a(nÀ2)e snÀ2 ‡ a1e s ‡ aoe (8:166) Qe ˆ (We NT )À1 e (8:167) and...
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This note was uploaded on 01/18/2014 for the course MECH 108 taught by Professor Sali mon during the Winter '12 term at Bingham University.

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