265_adv_control_eng

5 solution from equation 889 the observability matrix

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Unformatted text preview: P where (A) is defined in equation (8.104). Example 8.12 (See also Appendix 1, examp812.m) A system is described by ! ! ! ! • x1 0 1 x1 0 ˆ ‡ u • À2 À3 x2 1 x2 ! x1 y ˆ [1 0] x2 (8:134) //SYS21/D:/B&H3B2/ACE/REVISES(08-08-01)/ACEC08.3D ± 258 ± [232±271/40] 9.8.2001 2:34PM 258 Advanced Control Engineering Design a full-order observer that has an undamped natural frequency of 10 rad/s and a damping ratio of 0.5. Solution From equation (8.89), the observability matrix is h i 1 N ˆ CT X AT CT ˆ 0 0 1 ! (8:135) N is of rank 2 and therefore non-singular, hence the system is completely observable and the calculation of an appropriate observer gain matrix Ke realizable. Open-loop eigenvalues: j s I À A j ˆ s 2 ‡ 3s ‡ 2 ˆ s 2 ‡ a1 s ‡ a0 (8:136) Hence a0 ˆ 2, a1 ˆ 3 And the open-loop eigenvalues are s 2 ‡ 3s ‡ 2 ˆ 0 (s ‡ 1)(s ‡ 2) ˆ 0 s ˆ À1, s ˆ À2 (8:137) Desired closed-loop eigenvalues: s2 ‡ 2!n s ‡ !2 ˆ 0 n s2 ‡ 10s ‡ 100 ˆ s2 ‡ 1 s ‡ 0 ˆ 0 (8:138) Hence 0...
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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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