265_adv_control_eng

265_adv_control_eng - /SYS21/D/B&H3B2/ACE/REVISES/ACEC08.3D...

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where M is the controllability matrix, equation (8.88) W ± a 1 a 2 ... a n ÿ 1 1 a 2 a 3 10 . . . a n ÿ 1 1 00 2 6 6 6 6 6 4 3 7 7 7 7 7 5 (8 : 102) Note that T ± I when the system state equation is already in the controllable canonical form. (c) Ackermann's formula : As with Method 2, Ackermann's formula (1972) is a direct evaluation method. It is only applicable to SISO systems and therefore u ( t )and y ( t ) in equation (8.87) are scalar quantities. Let K ± 01 ²³ M ÿ 1 ± ( A )( 8 : 103) where M is the controllability matrix and ± ( A ) ± A n ´ ² n ÿ 1 A n ÿ 1 ´µµµ´ ² 1 A ´ ² 0 I (8 : 104) where A is the system matrix and ² i are the coefficients of the desired closed-loop characteristic equation. Example 8.11 (See also Appendix 1, examp811.m ) A control system has an open-loop transfer function Y U ( s ) ± 1 s ( s ´ 4) When x 1 ± y and x 2 ± _ x 1 ,expressthestateequationinthecontrollablecanonicalform. Evaluate the coefficients of the state feedback gain matrix using: (a) The direct comparison method (b) The controllable canonical form method (c) Ackermann's formula such that the closed-loop poles have the values s ±ÿ 2, s 2 Solution From the open-loop transfer function  y ´ 4 _ y ± u (8 : 105) Let x 1 ± y (8 : 106) Then _ x 1 ± x 2 _ x 2 4 x 2 ´ u (8 : 107) State-space methods for control system design 251
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Equation (8.106) provides the output equation and (8.107) the state equation _ x 1 _ x 2 ±² ± 01 0 ÿ 4 x 1 x 2 ² 0 1 u (8 : 108) y ± [1 0] x 1 x 2 (8 : 109) The characteristic equation for the open-loop system is j s I ÿ A s 0 0 s ÿ 0 ÿ 4 ³ ³ ³ ³ ³ ³ ³ ³ ± s 2 ² 4 s ² 0 ± s 2 ² a 1 s ² a 0 (8 : 110) Thus a 1 ± 4, a 0 ± 0 The required closed-loop characteristic equation is ( s ² 2)( s ² 2) ± 0 or s 2 ² 4 s ² 4 ± 0( 8 : 111) i.e. s 2 ² ± 1 s ² ± 0 ± 8 : 112) hence ± 1 ± 4, ± 0 ± 4 (a) Direct comparison method : From equations (8.99) and (8.111) j s I ÿ A ² BK s 2 ² 4 s ² 4( 8 : 113) s 0 0 s ÿ 0 ÿ 4 ² 0 1 k 1 k 2 ³´ ³ ³ ³ ³ ³ ³ ³ ³ ± s 2 ² 4 s ² 4 s ÿ 1 0 s ² 4 ² 00 k 1 k 2 ³ ³ ³ ³ ³ ³ ³ ³ ± s 2 ² 4 s ² 4 s ÿ 1 k 1 s ² 4 ² k 2 ³ ³ ³ ³ ³ ³ ³ ³ ± s 2 ² 4 s ² 4 s 2 ² (4 ² k 2 ) s ² k 1 ± s 2 ² 4 s ² 8 : 114) From equation (8.114) k 1 ± 4 (4 ² k 2 ) ± 4i : e : k 2 ± 0 (8 : 115) 252 Advanced Control Engineering
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(b) Controllable canonical form method : From equation (8.100) K ± [ ± 0 ÿ a 0 : ± 1 ÿ a 1 ] T ÿ 1 ± [4 ÿ 0 : 4 ÿ 4] T ÿ 1 ± [4 0] T ÿ 1 (8 : 116) now T ± MW where M ± [ B : AB ] AB ± 01 0 ÿ 4 ±² 0 1 ± 1 ÿ 4 giving M ± 1 ÿ 4 (8 : 117) Note that the determinant of M is non-zero, hence the system is controllable. From equation (8.102) W ± a 1 1 10 ± 41 Hence T ± MW ± 1 ÿ 4 ± ± I (8 : 118) Thus proving that equation (8.108) is already in the controllable canonical form. Since T ÿ 1 is also I , substitute (8.118) into (8.116) K ± I ± (8 : 119) (c) Ackermann's formula : From (8.103) K ± [0 1] M ÿ 1 ² ( A )( 8 : 120) From (8.117) M ÿ 1 ± 1 ÿ 1 ÿ 4 ÿ 1 ÿ ± (8 : 121) From (8.104) ² ( A ) ± A 2 ² ± 1 A ² ± 0 I State-space methods for control system design 253
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inserting values ± ( A ) ± 01 0 ÿ 4 "# 2 ² 4 0 ÿ 4 ² 4 10 ± 0 ÿ 4 6 ² 04 0 ÿ 16 ² 40 ± (8 : 122) Insert equations (8.121) and (8.122) into (8.120) K ± [0 1] 41 ±² ± 16 4 K ± [4 0] (8 : 123) These results agree with the root locus diagram in Figure 5.9, where
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265_adv_control_eng - /SYS21/D/B&H3B2/ACE/REVISES/ACEC08.3D...

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