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where M is the controllability matrix, equation (8.88) W a 1 a 2 . . . a n ± 1 1 a 2 a 3 . . . 1 0 . . . a n ± 1 1 . . . 0 0 1 0 . . . 0 0 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 0 0 . . . 0 1 M ± 1 ± ( A ) (8 : 103) where M is the controllability matrix and ± ( A ) A n ² n ± 1 A n ± 1 ² ² ² 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 ± 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 0 1 0 ± 4 ± ² x 1 x 2 ± ² 0 1 ± ² u (8 : 108) y [ 1 0 ] x 1 x 2 ± ² (8 : 109) The characteristic equation for the open-loop system is j s I ± A j s 0 0 s ± ² ± 0 1 0 ± 4 ± ² ³ ³ ³ ³ ³ ³ ³ ³ s 2 4 s 0 s 2 a 1 s a 0 (8 : 110) Thus a 1 4, a 0 0 The required closed-loop characteristic equation is ( s 2)( s 2) 0 or s 2 4 s 4 0 (8 : 111) i.e. s 2 ² 1 s ² 0 0 (8 : 112) hence ² 1 4, ² 0 4 (a) Direct comparison method : From equations (8.99) and (8.111) j s I ± A BK j s 2 4 s 4 (8 : 113) s 0 0 s ± ² ± 0 1 0 ± 4 ± ² 0 1 ± ² k 1 k 2 ³ ³ ³ ³ ³ ³ ³ ³ s 2 4 s 4 s ± 1 0 s 4 ± ² 0 0 k 1 k 2 ± ² ³ ³ ³ ³ ³ ³ ³ ³ s 2 4 s 4 s ± 1 k 1 s 4 k 2 ³ ³ ³ ³ ³ ³ ³ ³ s 2 4 s 4 s 2 (4 k 2 ) s k 1 s 2 4 s 4 (8 : 114) From equation (8.114) k 1 4 (4 k 2 ) 4 i : e : k 2 0 (8 : 115) 252 Advanced Control Engineering
(b) Controllable canonical form method : From equation (8.100) K [ ² 0 ± a 0 : ² 1 ± a 1 ] T ± 1 [4 ± 0 : 4 ± 4] T ± 1 [ 4 0 ] T ± 1 (8 : 116) now T MW where M [ B : AB ] AB 0 1 0 ± 4 ± ² 0 1 ± ² 1 ± 4 ± ² giving M 0 1 1 ± 4 ± ² (8 : 117) Note that the determinant of M is non-zero, hence the system is controllable. From equation (8.102) W a 1 1 1 0 ± ² 4 1 1 0 ± ² Hence T MW 0 1 1 ± 4 ± ² 4 1 1 0 ± ² 1 0 0 1 ± ² I (8 : 118) Thus proving that equation (8.108) is already in the controllable canonical form. Since T ± 1 is also I , substitute (8.118) into (8.116) K [ 4 0 ] I [ 4 0 ] (8 : 119) (c) Ackermann's formula : From (8.103) K [ 0 1 ] M ± 1 ± ( A ) (8 : 120) From (8.117) M ± 1 1 ± 1 ± 4 ± 1 ± 1 0 ± ² 4 1 1 0 ± ² (8 : 121) From (8.104) ± ( A ) A 2 ² 1 A ² 0 I State-space methods for control system design 253

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inserting values ± ( A ) 0 1 0 ± 4 " # 2 4 0 1 0 ± 4 " # 4 1 0 0 1 " # 0 ± 4 0 16 " # 0 4 0 ± 16 " # 4 0 0 4 " # 4 0 0 4 " # (8 : 122) Insert equations (8.121) and (8.122) into (8.120) K [ 0 1 ] 4 1 1 0 ± ² 4 0 0 4 ± ² [ 0 1 ] 16 4 4 0 ± ² K [ 4 0 ] (8 : 123) These results agree with the root locus diagram in Figure 5.9, where K 4 produces two real roots of s ± 2, s ± 2 (i.e. critical damping).
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