Topic #38 Transfer function to state-space Reference textbook : - PowerPoint PPT Presentation

ME 779 Control Systems Topic #38 Transfer function to state-space Reference textbook : Control Systems, Dhanesh N. Manik, Cengage Publishing, 2012 1

Transfer function to state-space Phase-variable and controller canonical forms         m m 1 m 2 K s b s b s b s b ( ) Y s    m 1 m 2 1 0       n n 1 n 2 U s ( ) s a s a s a s a   n 1 n 2 1 0 2

Transfer function to state-space Phase-variable and controller canonical forms Z s ( ) K        n n 1 n 2 U s ( ) s a s a s a s a   n 1 n 2 1 0 Y s ( )        m m 1 m 2 s b s b s b s b   m 1 m 2 1 0 Z s ( ) 3

Transfer function to state-space Phase-variable and controller canonical forms          1 2 n n n s a s a s a s a Z s ( ) KU s ( )   n 1 n 2 1 0     n n 1 n 2 d z d z d z dz        ( ) a a a a z Ku t     n 1 n 2 1 0 n n 1 n 2   dt dt dt dt 4

Transfer function to state-space Phase variable form dz    x x x z 1 2 dt 1 dz 2 d z    x x x 2 2 3 2 dt dt 3 2 d z d z    x x x 3 4 3 dt 3 2 dt  n 1 d z    n 1 x x d z    n 1 n n 1 dt x  n n 1 dt n d z  x n n dt 5

Transfer function to state-space Phase variable form     n n 1 n 2 d z d z d z dz        Ku t ( ) a a a a z     n 1 n 2 1 0 n n 1 n 2   dt dt dt dt        Ku t ( ) a x a x a x a x    n 1 n n 2 n 1 1 2 0 1 6

Transfer function to state-space Phase variable form  x x 1 2  x x 2 3  x x 3 4  x x  n 1 n        ( ) x Ku t a x a x a x a x    n n 1 n n 2 n 1 1 2 0 1 7

Transfer function to state-space Phase variable form         x 0 1 0 0 x 0 1 1         x 0 0 1 0 x 0         2 2                 x 0 u t ( ) 3         x 0 0 0 1          n 1                     x a a a a x K  0 1 2 1 n n n Upper companion matrix 8

Transfer function to state-space Phase variable form     m m 1 m 2 d z d z d z dz        y t ( ) b b b b z     m 1 m 2 1 0 m m 1 m 2   dt dt dt dt        y t ( ) x b x b x b x b x     m 1 m 1 m m 2 m 1 1 2 0 1 9

Transfer function to state-space Phase variable form   x 1   x   2   x 3            y t ( ) b b b b 1 0 0 x   0 1 2 m 1 m 1   x   m     x    n 1     x n 10

Transfer function to state-space Phase variable form 11

Transfer function to state-space EXAMPLE Phase variable form Obtain the phase-variable representation of the following transfer function     2 20 s 2 s 5 Y s ( )      4 3 2 U s ( ) s 3 s 5 s 6 s 7 12

Transfer function to state-space EXAMPLE Phase variable form         x 0 1 0 0 x 0 1 1               x 0 0 1 0 x 0     2 2       u t ( )   x 0 0 0 1 x 0       3 3                     x 7 6 5 3 x 20 4 4   x 1     x  2   y t ( ) [5 2 1 0] x   3     x 4 13

Transfer function to state-space Controller canonical form  n 1 d z  n x d z          1 x Ku t ( ) a x a x a x a x n 1 dt    1 n 1 1 n 2 2 1 n 1 0 n n dt n  n 1 d z d z    x x x  2 1 1 n 2 dt n dt 2 d z   x x dz   n 1 n 2 2  dt x  n 1 dt dz   x x  n n 1  dt x z n 14

Transfer function to state-space Controller canonical form             x x a a a a K    1 n 1 n 2 n 3 0 1         x 1 0 0 0 x 0         2 2                 x 0 u t ( ) 3         x 0 0 1 0 0          n 1                 x x 0 0 0 1 0 0 n n        y t ( ) x b x b x b x b x         n m m 1 n m 1 m 2 n m 2 1 n 1 0 n 15

Transfer function to state-space Controller canonical form   x 1   x   2     x     n m 1     x     n m   y t ( ) 0 0 0 1 b b b b   m 1 m 2 1 0 x     n m 1   x     n m 2     x    n 1     x n 16

Transfer function to state-space Controller canonical form 17

Transfer function to state-space EXAMPLE Controller canonical form Obtain the controller canonical representation of the following transfer function     2 20 s 2 s 5 Y s ( )      4 3 2 U s ( ) s 3 s 5 s 6 s 7 18

Transfer function to state-space EXAMPLE Controller canonical form             x x 3 5 6 7 20 1 1               x 1 0 0 0 x 0     2 2       u t ( )   x 0 1 0 0 x 0       3 3                 x 0 0 1 0 x 0 4 4   x 1     x  2   y t ( ) [0 1 2 5] x   3     x 4 19

Transfer function to state-space Observer canonical form   b b 1 b     m 1 0  1  K       n m n m 1 n 1 n Y s ( ) s s s s  a a a a U s ( )       n 1 n 2 1 0 1  2 n 1 n s s s s 20

Transfer function to state-space Observer canonical form   a a a a         n 1 n 2 1 o ( ) 1 Y s    2 n 1 n s s s s   1 b b b        m 1 1 0 KU s ( )       n m n m 1 n 1 n s s s s 21

Transfer function to state-space Observer canonical form 1 1         Y s ( ) b KU s ( ) a Y s ( ) b KU s ( ) a Y s ( )  0 0 1 1 n n 1 s s a a 1          m 1 n 1 KU s ( ) a Y s ( ) Y s ( ) Y s ( )    m 1 n m n m s s s 22

Transfer function to state-space Observer canonical form 23

Transfer function to state-space Observer canonical form    x a x x  1 n 1 1 2    x a x x  2 n 2 1 3     x a x x u t K ( )    n m m 1 n m 1     x a x x b u t K ( )  n 1 1 1 n 1    x a x b u t K ( ) n 0 1 0 24

Transfer function to state-space Observer canonical form          1 0 0 0 0 0 0 x a x  1 n 1 1          x a 0 1 0 0 0 0 x 0          2 n 2 2         0 0 0 0 0 0                  x a 0 0 0 1 0 0 x 1 K u t ( )     n m m n m         0 0 0 0 0                  x a 0 0 0 0 0 1 x b   1 1 1 1 n n                  x a 0 0 0 0 0 0 x b n 0 n 0 25

Transfer function to state-space Observer canonical form   x 1   x   2          y t ( ) 1 0 0 0 x    n 2   x    n 1     x n 26