EE3530-3.6-9

EE3530-3.6-9 - 3.7 Stability Let the transfer function be...

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Unformatted text preview: 3.7 Stability Let the transfer function be given by T ( s ) = Y ( s ) R ( s ) = b s m + b 1 s m- 1 + + b m s n + a 1 s n- 1 + + a n The characteristic equation is c ( s ) = s n + a 1 s n- 1 + + a n = 0 Stability: all roots of the characteristic equations are in the open left half plane. Routh Criterion Consider a polynomial a ( s ) = a s n + a 1 s n- 1 + + a n , a > . (1) Theorem If a ( s ) is stable, then a i > 0, i = 1 , , n . This is because a ( s ) = a ( s + 1 )( s + 2 ) ( s + m )[( s + 1 ) 2 + 2 1 ] [( s + l ) 2 + 2 l ] where i and j are positive. Example Consider a polynomial a ( s ) = s 3 + s 2 + 4 s + 30 = ( s + 3)( s 2- 2 s + 10) . This polynomial has all positive coefficients but it is not stable. Its roots are- 3 and 1 3 j . 1 Routh Table s n r 00 = a r 01 = a 2 r 02 = a 4 r 03 = a 6 s n- 1 r 10 = a 1 r 11 = a 3 r 12 = a 5 r 13 = a 7 s n- 2 r 20 r 21 r 22 r 23 s n- 3 r 30 r 31 r 32 r 33 . . . . . . . . . . . . . . . s 2 r ( n- 2)0 r ( n- 2)1 s 1 r ( n- 1)0 s r n r ij =- 1 r ( i- 1)0 fl fl fl fl fl fl fl fl r ( i- 2)0 r ( i- 2)( j +1) r ( i- 1)0 r ( i- 1)( j +1) fl fl fl fl fl fl fl fl = r ( i- 1)0 r ( i- 2)( j +1)- r ( i- 2)0 r ( i- 1)( j +1) r ( i- 1)0 . r 20 =- 1 r 10 fl fl fl fl fl fl fl fl r 00 r 01 r 10 r 11 fl fl fl fl fl fl fl fl = r 10 r 01- r 00 r 11 r 10 r 21 =- 1 r 10 fl fl fl fl fl fl fl fl r 00 r 02 r 10 r 12 fl fl fl fl fl fl fl fl = r 10 r 02- r 00 r 12 r 10 r 22 =- 1 r 10 fl fl fl fl fl fl fl fl r 00 r 03 r 10 r 13 fl fl fl fl fl fl fl fl = r 10 r 03- r 00 r 13 r 10 , . . . r n =- 1 r ( n- 1)0 fl fl fl fl fl fl fl fl r ( n- 2)0 r ( n- 2)1 r ( n- 1)0 fl fl fl fl fl fl fl fl = r ( n- 2)1 . 2 Routh Stability Criterion The following three statements are equivalent: 1. a ( s ) is stable. 2. All elements of the Routh table are positive, i.e., r ij > 0, i = , 1 , . . . , n , j = 0 , 1 , . . . , b n- i 2 c . 3. All elements in the first column of the Routh table are positive, i.e., r i > 0, i = 0 , 1 , . . . , n ....
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EE3530-3.6-9 - 3.7 Stability Let the transfer function be...

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