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Lecture Control system design: The stability of linear feedback systems - Nguyễn Công Phương

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Lecture Control system design: The stability of linear feedback systems - Nguyễn Công Phương

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Lecture Control system design: The stability of linear feedback systems include all of the following content: The concept of stability, the Routh – Hurwitz stability criterion, the stability of state variable systems, system stability using control design software.

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Nội dung Text: Lecture Control system design: The stability of linear feedback systems - Nguyễn Công Phương

  1. Nguyễn Công Phương CONTROL SYSTEM DESIGN The Stability of Linear Feedback Systems
  2. Contents I. Introduction II. Mathematical Models of Systems III. State Variable Models IV. Feedback Control System Characteristics V. The Performance of Feedback Control Systems VI. The Stability of Linear Feedback Systems VII. The Root Locus Method VIII.Frequency Response Methods IX. Stability in the Frequency Domain X. The Design of Feedback Control Systems XI. The Design of State Variable Feedback Systems XII. Robust Control Systems XIII.Digital Control Systems sites.google.com/site/ncpdhbkhn 2
  3. The Stability of Linear Feedback Systems 1. The Concept of Stability 2. The Routh – Hurwitz Stability Criterion 3. The Stability of State Variable Systems 4. System Stability Using Control Design Software sites.google.com/site/ncpdhbkhn 3
  4. The Concept of Stability (1) • Stability is of the utmost importance. • A close – loop feedback system that is unstable is of little value. • A stable system is a dynamic system with a bounded response to a bounded input. sites.google.com/site/ncpdhbkhn 4
  5. The Concept of Stability (2) http://www.ctc.org.uk/cyclists-library/bikes-and-other- cycles/cycle-styles/city-bike sites.google.com/site/ncpdhbkhn 5
  6. The Concept of Stability (4) Bk s  Ck M N M N    Ae D e 1 Ai  i t  k t Y ( s)     y (t )  1   sin(k t   k ) s  i i k k 1 s  2 k s  ( k  k ) 2 2 2 s i 1 i 1 k 1 j 1 1 10 1 0 0 0 0 -1 -1 -1 -10 0 5 10 0 5 10 0 5 10 0 5 10 1 1 10 1 0 0 0 0 -1 -1 -1 -10 0 5 10 0 5 10 0 5 10 0 5 10 1 1 2 10  0.5 0.5 1 5 0 0 0 0 0 5 10 0 5 10 0 5 10 0 5 10 sites.google.com/site/ncpdhbkhn 6
  7. The Stability of Linear Feedback Systems 1. The Concept of Stability 2. The Routh – Hurwitz Stability Criterion 3. The Stability of State Variable Systems 4. System Stability Using Control Design Software sites.google.com/site/ncpdhbkhn 7
  8. The Routh – Hurwitz Stability Criterion (1) q( s )  an s n  an 1s n 1  an 2 s n 2  ...  a1s  a0  0 sn an an  2 an  4  1 an an 2 an 1an 2  an an 3 bn 1   , s n 1 an 1 an  3 an  5  an 1 an 1 an 3 an 1 s n 2 bn 1 bn 3 bn 5  1 an an 4 bn 3  , s n 3 cn 1 cn 3 cn 5  an 1 an 1 an 5      1 an 1 an 3 s 0 hn 1 cn 1  , bn 1 bn 1 bn 3 The Routh – Hurwitz criterion states that the number of roots of q(s) with positive real parts is equal to the number of changes in sign of the first column of the Routh array sites.google.com/site/ncpdhbkhn 8
  9. The Routh – Hurwitz Stability Criterion (2) q( s )  an s n  an 1s n 1  an 2 s n 2  ...  a1s  a0  0 1. No element in the 1st column is sn an an  2 an  4  zero. s n 1 an 1 an  3 an  5  2. There is a zero in the 1st column, but some other elements of the row s n 2 bn 1 bn 3 bn 5  containing the zero in the 1st column are nonzero. s n 3 cn 1 cn 3 cn 5  3. There is a zero in the 1st column, and the other elements of the row      containing the zero are also zero. s0 hn 1 4. Repeated roots of the characteristic equation on the jω – axis. The Routh – Hurwitz criterion states that the number of roots of q(s) with positive real parts is equal to the number of changes in sign of the first column of the Routh array sites.google.com/site/ncpdhbkhn 9
  10. The Routh – Hurwitz Stability Ex. 1 Criterion (3) q( s )  a2 s 2  a1s  a0 sn an an  2 an  4  s2 a2 a0 s2 a2 a0 s n 1 an 1 an  3 an  5  s1 a1 0 s1 a1 0 s n 2 bn 1 bn 3 bn 5  s0 b1 0 s0 a0 0 s n 3 cn 1 cn 3 cn 5       1 an an 2 1 a2 a0 bn 1   b1   a0 an 1 an 1 an 3 a1 a1 0 s0 hn 1 The Routh – Hurwitz criterion states that the number of roots of q(s) with positive real parts is equal to the number of changes in sign of the first column of the Routh array The system is stable if a2, a1 & a0 are all positive or all negative sites.google.com/site/ncpdhbkhn 10
  11. The Routh – Hurwitz Stability Ex. 2 Criterion (4) q( s )  s 3  s 2  2 s  50 sn an an  2 an  4  s3 1 2 s3 1 2 s n 1 an 1 an  3 an  5  s2 1 50 s2 1 50 s n 2 bn 1 bn 3 bn 5  s1 b1 b0 s1 48 0 s n 3 cn 1 cn 3 cn 5  s0 c1 c0 s0      50 0 s0 hn 1 1 1 2 1 1 0 1 1 50 1 1 0 b1   48, b0   0, c1   50, c0  0 1 1 50 1 1 0 48 48 0 48 48 0 The Routh – Hurwitz criterion states that the number of roots of q(s) with positive real parts is equal to the number of changes in sign of the first column of the Routh array sites.google.com/site/ncpdhbkhn 11
  12. The Routh – Hurwitz Stability Ex. 3 Criterion (5) q( s )  a3s 3  a2 s 2  a1s  a0 s3 a3 a1 s2 a2 a0 a2 a1  a0a3 b1  , c1  a0 s1 b1 0 a2 s0 c1 0 a3  0 a3  0 a  0 a3  0 a  0  2 a  0  2  2    a2 a1  a0a3  b1  0  0 a2 a1  a0a3  0 a2 c1  0  a0  0 a0  0 q( s )  s 3  2 s 2  6s  10, a2 a1  a0a3  2  6  10  1  2 sites.google.com/site/ncpdhbkhn 12
  13. The Routh – Hurwitz Stability Ex. 4 Criterion (6) q( s )  s 5  2 s 4  2 s 3  4 s 2  11s  10 s5 1 2 11 s5 1 2 11 s4 2 4 10 s4 2 4 10 s3 0 6 0 s3  6 0 s2 c1 10 0 s2 c1 10 0 s1 d1 0 0 s1 d1 0 0 s0 10 s0 10 4  12 12 6c  10 c1   4  , d1  1  6  10   c1 Two sign changes  two roots with positive real part  the system is unstable sites.google.com/site/ncpdhbkhn 13
  14. The Routh – Hurwitz Stability Ex. 5 Criterion (7) q( s )  s 4  s 3  s 2  s  K s4 1 1 K s4 1 1 K s3 1 1 0 s3 1 1 0 s2 0 K 0 s2  K 0 s1 c1 0 0 s1 c1 0 0 s1 K s1 K  K K c1  1   One sign change  one root with positive real part  the system is unstable for all values of K sites.google.com/site/ncpdhbkhn 14
  15. The Routh – Hurwitz Stability Ex. 6 Criterion (8) q( s )  s 5  s 4  4 s 3  24 s 2  3s  63  ( s  s  s  21)( s  3) 3 2 3 s5 1 4 3 s5  s 4  4 s 3  24 s 2  3s  63 s 2  3 s4 1 24 63 s5  3s 3 s 3  s 2  s 21 s3 20 60 0 s 4  s 3  24 s 2  3s  63 s2 21 63 0 s4  3s 2 s1 s 3  21s 2  3s  63 0 0 0 s3  3s  U ( s )  21s 2  63  21( s 2  3) 21s 2  63 21s 2  63 0 sites.google.com/site/ncpdhbkhn 15
  16. The Routh – Hurwitz Stability Ex. 6 Criterion (9) q( s )  s 5  s 4  4 s 3  24 s 2  3s  63  ( s  s  s  21)( s  3) 3 2 3 s5 1 4 3 s4 1 24 63 s3 1 1 20 60 3 s 0 2 s2 1 21 s 21 63 0 s1 0 0 0 s1 20 0 s0 21 0 Two sign changes  two roots with positive real part  the system is unstable sites.google.com/site/ncpdhbkhn 16
  17. The Routh – Hurwitz Stability Ex. 7 Criterion (10) K (s  a) 1 R( s ) K (s  a) 1 Y ( s) G( s)  s  1 s( s  2)( s  3) s 1 s( s  2)( s  3) ( ) G( s) K (s  a) T ( s)   4 1  G ( s ) s  6s 3  11s 2  ( K  6) s  Ka  60  K  q( s )  s  6s  11s  ( K  6) s  Ka  6 0 4 3 2  s4  b3 ( K  6)  6 Ka 1 11 Ka  0  b 3 s3 6 K 6 0  Ka  0  s2 b3 Ka 0 s1 c3 0  K  60 0 0   (60  K )( K  6) s1 Ka  a  36 K 60  K b ( K  6)  6 Ka b3  , c3  3 6 b3 sites.google.com/site/ncpdhbkhn 17
  18. The Stability of Linear Feedback Systems 1. The Concept of Stability 2. The Routh – Hurwitz Stability Criterion 3. The Stability of State Variable Systems 4. System Stability Using Control Design Software sites.google.com/site/ncpdhbkhn 18
  19. The Stability of State Variable Ex. 1 Systems (1) 1 3  x1  3x1  x2 1 K 1  U ( s) X1 ( s)  x2  x2  Kx1  Ku 1/ s X 2 ( s) X2 1/ s 1 L1  s 1 , L2  3s 1 , L3   Ks 2 N  1 L n 1 n  n ,m Ln Lm   n ,m , p Ln Lm Lp  ... nontouching nontouching  1  ( L1  L2  L3 )  ( L1L2 )  1  2s 1  ( K  3) s 2  q( s )  s 2  2 s  ( K  3)  K 3 0 K  3 sites.google.com/site/ncpdhbkhn 19
  20. The Stability of State Variable Ex. 2 Systems (2)  a   0 1 0 dx  u     0 x   0 1   1  dt     u2     0 0 0   0 0   a   0      0   det( I  A )  det  0  0       0      0      0 0      0          [ 2  (   )  (   2 )]  q( )   [ 2  (   )  (   2 )]     0      0 2 sites.google.com/site/ncpdhbkhn 20

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