Theory of Machines and Automatic Control Winter 2018/2019

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1 Theory of Machines and Automatic Control Winter 2018/2019 Lecturer: Sebastian Korczak, PhD, Eng. Institute of Machine Design Fundamentals - Department of Mechanics

2 Lecture 13 Stability criteria. Gain margin and phase margin. System correction TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 2

3 STABILITY CRITERIA General stability criterion Hurwitz criterion Nyquist stability criterion TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 3

4 General stability criterion LTI SISO system is assymptotically stable if real part of every pole of the system's transfer function is less than zero. G= (s z 1)(s z 2 )...(s z m ) ( s p 1 )(s p 2 )...(s p n ) Re p 1 <0 Re p 2 <0... Re p n < TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 4

5 Hurwitz criterion mathematics a necessary and sufficient condition whether all the roots of the polynomial are in the left half of the complex plane TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 5

6 Hurwitz criterion mathematics control theory a necessary and sufficient condition whether all the roots of the polynomial are in the left half of the complex plane a necessary and sufficient condition whether all the poles of transfer function of a linear system have negative real parts TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 6

7 Hurwitz criterion LTI SISO system with a transfer function H = b m s m +b m 1 s m b 1 s+b 0 a n s n +a n 1 s n a 1 s+a 0 = (s z 1)(s z 2 )...(s z m ) (s p 1 )(s p 2 )...(s p n ) is stable if: TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 7

8 Hurwitz criterion LTI SISO system with a transfer function H = b m s m +b m 1 s m b 1 s+b 0 a n s n +a n 1 s n a 1 s+a 0 = (s z 1)(s z 2 )...(s z m ) (s p 1 )(s p 2 )...(s p n ) is stable if: 1 a n >0, a n 1 >0,..., a 1 >0, a 0 > TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 8

9 Hurwitz criterion LTI SISO system with a transfer function H = b m s m +b m 1 s m b 1 s+b 0 a n s n +a n 1 s n a 1 s+a 0 = (s z 1)(s z 2 )...(s z m ) (s p 1 )(s p 2 )...(s p n ) is stable if: 1 a n >0, a n 1 >0,..., a 1 >0, a 0 >0 2 M n =[ a n 1 a n a n 3 a n 2 a n 1 a n 0 0 a n 5 a n 4 a n ] a 0 a 1 a a TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 9

10 Hurwitz criterion LTI SISO system with a transfer function H = b m s m +b m 1 s m b 1 s+b 0 a n s n +a n 1 s n a 1 s+a 0 = (s z 1)(s z 2 )...(s z m ) (s p 1 )(s p 2 )...(s p n ) is stable if: 1 2 Δ i a n >0, a n 1 >0,..., a 1 >0 =[, a 0 >0 det Δ 2 >0 det Δ 3 >0... M n det Δ n 1 >0 - leading principal minor of order i Δ 2 Δ 3 Δ n 1 a n 1 a n a n 3 a n 2 a n 1 a n 0 0 a n 5 a n 4 a n ] a 0 a 1 a a TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 10

11 Hurwitz criterion Hurwitz criterion Routh criterion (1895) (1876) Liénard Chipart criterion modification of Hurwitz criterion TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 11

12 Hurwitz criterion Example 1 G= 5 s+3 10 s 2 +3 s TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 12

13 G= 2 s 2 s 3 +s+20 Hurwitz criterion Example TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 13

14 G= 3 s 5 s 3 +4 s 2 +3 s+10 Hurwitz criterion Example TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 14

15 G= 1 3 s 4 +4 s 3 +6 s 2 +4 s+5 Hurwitz criterion Example TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 15

16 k s 4 s 3 +3 s 2 +k s+1 Hurwitz criterion Example 5 Choose k parameter to satisfy Hurwitz criterion TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 16

17 Hurwitz criterion Example 6 Choose k parameter to satisfy Hurwitz criterion 2 2 s 3 +ks 2 +(1+k ) s+3 Homework TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 17

18 Hurwitz criterion Example 7 Choose T parameter to satisfy Hurwitz criterion G r = 4 s T s+1 1 G o = s 3 +2 s 2 +s TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 18

19 G z = Hurwitz criterion Example 7 Choose T parameter to satisfy Hurwitz criterion G r = 4 s T s+1 1 G o = s 3 +2 s 2 +s+1 G r G o 1+G r G o G p = 4 s a 4 s 4 +a 3 s 3 +a 2 s 2 +a 1 s+a 0 a 4 =T, a 3 =2 T +1, a 2 =T +2, a 1 =T +5, a 0 = TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 19

20 G z = Hurwitz criterion Example 7 Choose T parameter to satisfy Hurwitz criterion G r = 4 s T s+1 1 G o = s 3 +2 s 2 +s+1 G r G o 1+G r G o G p = a 4 >0, a 3 >0, a 2 >0, a 1 >0, a 0 >0 T >0 4 s a 4 s 4 +a 3 s 3 +a 2 s 2 +a 1 s+a 0 a 4 =T, a 3 =2 T +1, a 2 =T +2, a 1 =T +5, a 0 = TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 20

21 G z = Hurwitz criterion Example 7 Choose T parameter to satisfy Hurwitz criterion G r = 4 s T s+1 1 G o = s 3 +2 s 2 +s+1 G r G o 1+G r G o G p = a 4 >0, a 3 >0, a 2 >0, a 1 >0, a 0 >0 T >0 a 3 a 4 2] a 1 a =T 2 +2>0 T R Δ 2 =[ 4 s a 4 s 4 +a 3 s 3 +a 2 s 2 +a 1 s+a 0 a 4 =T, a 3 =2 T +1, a 2 =T +2, a 1 =T +5, a 0 =1 Δ 3 =[a n 1 a n 0 a n 3 a n 2 a n 1 a n 5 a n 4 a n 3]=T 3 +T 2 2T +9>0 T > TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 21

22 Nyquist stability criterion x + G 1 y G 2 G z = y x = G 1 1+G 1 G TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 22

23 Nyquist stability criterion x + G 1 y G 2 G z = y x = G 1 1+G 1 G 2 Unstable if: G 1 G 2 = TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 23

24 Nyquist stability criterion x + a G z = y x = G 1 G 2 G 1 y 1+G 1 G 2 x a G 1 G 2 y G open = a x =G 1G 2 Unstable if: G 1 G 2 = TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 24

25 Nyquist stability criterion x + a G z = y x = G 1 G 2 G 1 y 1+G 1 G 2 x a G 1 G 2 y G open = a x =G 1G 2 Im G open Unstable if: G 1 G 2 = 1 ω ω + ω=0 Re G open TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 25

26 Nyquist stability criterion x + a G z = y x = G 1 G 2 G 1 y 1+G 1 G 2 x a G 1 G 2 y G open = a x =G 1G 2 Im G open Unstable if: G 1 G 2 = 1 ω ω + ω=0 Re G open TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 26

27 Nyquist stability criterion (particular) The closed-loop system is stable if: 1) open-loop transfer function is stable AND 2) open-loop transfer function not enclosing the point (-1,j0). Im G open Im G open -1 Re G open -1 Re G open stable unstable TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 27

28 Nyquist criterion Example 8 Choose k parameter to satisfy Nyquist criterion x + G 1 = 2 s 3 +3 s 2 +s+1 y G 2 =k TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 30

29 Nyquist criterion Example 8 Choose k parameter to satisfy Nyquist criterion x + G 1 = 2 s 3 +3 s 2 +s+1 y G 2 =k G open =G 1 G 2 = 2 k s 3 +3 s 2 + s TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 31

30 Nyquist criterion Example 8 Choose k parameter to satisfy Nyquist criterion x + G 1 = 2 s 3 +3 s 2 +s+1 y G 2 =k G open =G 1 G 2 = 2 k s 3 +3 s 2 + s+1 - stable from Hurwitz TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 32

31 Nyquist criterion Example 8 Choose k parameter to satisfy Nyquist criterion x + G 1 = 2 s 3 +3 s 2 +s+1 y G 2 =k G open =G 1 G 2 = 2 k s 3 +3 s 2 + s+1 - stable from Hurwitz 2 k 6 k ω 2 P (ω)= (1 3 ω 2 ) 2 +(ω ω 3 ), Q(ω)= 2 k ω 3 2 k ω 2 (1 3ω 2 ) 2 +(ω ω 3 ) TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 33

32 Nyquist criterion Example 8 2 k 6 k ω 2 P (ω)= (1 3 ω 2 ) 2 +(ω ω 3 ), Q(ω)= 2 k ω 3 2 k ω 2 (1 3ω 2 ) 2 +(ω ω 3 ) TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 34

33 Nyquist criterion Example 8 2 k 6 k ω 2 P (ω)= (1 3 ω 2 ) 2 +(ω ω 3 ), Q(ω)= 2 k ω 3 2 k ω 2 (1 3ω 2 ) 2 +(ω ω 3 ) 2 Q(ω) ω=1 ω= ω=0 1 k 0 2 k P(ω) 3 3k ω= TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 35

34 Nyquist criterion Example 8 2 k 6 k ω 2 P (ω)= (1 3 ω 2 ) 2 +(ω ω 3 ), Q(ω)= 2 k ω 3 2 k ω 2 (1 3ω 2 ) 2 +(ω ω 3 ) 2 Q(ω) ω=1 ω= ω=0 1 k 0 2 k P(ω) closed-loop system stable for 0 < k <1 3 3k ω= TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 36

35 Gain margin Closed-loop system will loose its stability if we add additional gain (in serial) greater or equals to gain margin. Δ M Q(ω) 0 L(ω) [db] ω [rad/s] Δ M -1 P(ω) φ(ω) [rad] π ω [rad/s] TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 37

36 Phase margin Closed-loop system will loose its stability if we add additional delay (in serial) greater or equals to phase margin. Q(ω) 0 L(ω) [db] ω [rad/s] Δ φ -1 P(ω) φ(ω) [rad] π Δ φ ω [rad/s] TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 38

37 Stability vs Bode plot Bodego plot (gain + delay) has no physical meaning if the system is unstable! Example: G= 1 s+1 Q(ω) P(ω) L(ω) [db] ω [rad/s] φ(ω) [rad] ω [rad/s] G= 1 s 1 Q(ω) L(ω) [db] ω [rad/s] φ(ω) [rad] ω [rad/s] P(ω) TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 39

38 Summing of Bode plots example G = 10 s 2 +s = 10 1 s+1 1 s TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 40

39 Summing of Bode plots example G = 10 s 2 +s = 10 1 s+1 1 s Q(ω) P(ω) Q(ω) 10 P(ω) Q(ω) 1 Q(ω) P(ω) 10 P(ω) TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 41

40 Summing of Bode plots example G = 10 s 2 +s = 10 1 s+1 1 s 20 0 L(ω) [db] ω [rad/s] 0 L(ω) [db] 20 0, ω [rad/s] L(ω) [db] , ω [rad/s] φ(ω) [rad] 0 ω [rad/s] φ(ω) [rad] π 2 0, ω [rad/s] φ(ω) [rad] π 2 0, ω [rad/s] TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 42

41 Summing of Bode plots example G = 10 s 2 +s = 10 1 s+1 1 s L(ω) [db] , ω [rad/s] φ(ω) [rad] π 2 0, ω [rad/s] π TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 43

42 Nyquist stability criterion control loop with P controller x + G y G closed = G 1+ G G opened = G =k P TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 44

43 Nyquist stability criterion control loop with P controller x + G y G closed = G 1+ G G opened = G =k P G= 1 Ts+1 Im -1 1 Re TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 45

44 Nyquist stability criterion control loop with P controller x + G y G closed = G 1+ G G opened = G =k P G= 1 Ts+1 Im G opened =k p 1 Ts+1 Im -1 1 Re Re -1 k p TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 46

45 Nyquist stability criterion control loop with P controller x + G y G closed = G 1+ G G opened = G =k P G= 1 Ts+1-1 Im G opened =k p 1 Ts+1 1 Re Re -1 Im G opened is always stable G closed is always stable steady state error ratio: k P k P TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 47 k p

46 Nyquist stability criterion control loop with P controller x + G y G closed = G 1+ G G opened = G =k P TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 48

47 Nyquist stability criterion control loop with P controller x + G y G closed = G 1+ G G opened = G =k P G= Im 1 T 1 2 s 2 +T 2 s+1 1 Re TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 49

48 Nyquist stability criterion control loop with P controller x + G y G closed = G 1+ G G opened = G =k P G= Im 1 T 1 2 s 2 +T 2 s+1 G opened = Im k p T 1 2 s 2 +T 2 s+1 1 Re Re -1 k p TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 50

49 Nyquist stability criterion control loop with P controller x + G y G closed = G 1+ G G opened = G =k P G= Im 1 T 1 2 s 2 +T 2 s+1 G opened = k p T 1 2 s 2 +T 2 s+1 1 Re Re -1 Im steady state error ratio: k P k P TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 51 k p G opened is always stable G closed is always stable

50 Nyquist stability criterion control loop with P controller x + G y G closed = G 1+ G G opened = G =k P TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 52

51 Nyquist stability criterion control loop with P controller x + G y G closed = G 1+ G G opened = G =k P 1 G= T 2 3 s 3 +T 2 2 s 2 +T 1 s+1 Im 1 Re TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 53

52 Nyquist stability criterion control loop with P controller x + G y G closed = G 1+ G G opened = G =k P 1 G= T 2 3 s 3 +T 2 2 s 2 +T 1 s+1 G opened = k p T 3 2 s 3 +T 2 2 s 2 +T 1 s+1 Im -1 Im 1 Re Re k p TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 54

53 Nyquist stability criterion control loop with P controller x + G y G closed = G 1+ G G opened = G =k P 1 G= T 2 3 s 3 +T 2 2 s 2 +T 1 s+1 G opened = k p T 3 2 s 3 +T 2 2 s 2 +T 1 s+1 G closed is not always stable Im -1 Im 1 Re Re steady state error ratio: k P k P TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 55 k p

54 Nyquist stability criterion control loop with P controller x + G y G closed = G 1+ G G opened = G 1 G= T 2 3 s 3 +T 2 2 s 2 +T 1 s+1 =k P conclusion for open-loop transfer function: higher kp lower steady state error -1 lower kp better stability (higher gain margin) Im Re k p TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 56

55 Nyquist stability criterion control loop with PI controller x + G y ( s )=k P ( 1+ 1 T i s ) G opened = G TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 57

56 Nyquist stability criterion control loop with PI controller x + G y ( s )=k P ( 1+ 1 T i s ) G opened = G G= 1 Ts+1 Im -1 1 Re TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 58

57 Nyquist stability criterion control loop with PI controller x + G y ( s )=k P ( 1+ 1 T i s ) G opened = G G= 1 Ts+1 Im G opened =k p 2 Im s T 2 i +2T i T 3 i T s 2 +T 2 i s -1 1 Re TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 59

58 Nyquist stability criterion control loop with PI controller x + G y ( s )=k P ( 1+ 1 T i s ) G opened = G G= 1 Ts+1 Im G opened =k p 2 Im s T 2 i +2T i T 3 i T s 2 +T 2 i s G opened is stable, so G closed is stable -1 1 Re Re -1 k p G opened (ω=0) so steady state error TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 60

59 Correction of the system Correction by proportional term G ( s ) -1 Im Re TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 61

60 Correction of the system Correction by proportional term G ( s ) k G ( s ) -1 Im -1 Im Re Re TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 62

61 Correction of the system Correction by proportional term G ( s ) k G ( s ) -1 Im -1 Im Re Re Higher gain margin, higher phase margin, higher steady state error Lower gain margin, lower phase margin, lower steady state error TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 63

62 Correction of the system Correction by delay G ( s ) G ( s ) e τ s -1 Im -1 Im Re Re Higher gain and phase margins Lower gain and phase margins TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 64

63 Correction of the system Derivative K ( s )= 1+T s 1+a s+b s 2 Proportional-derivative K ( s )=k P T s+1 αt s+1, α<1 Integral K ( s )=1+ k 1+T s Proportional-integral K ( s )=α T s+1 αt s+1, α>1 Proportional-integral-derivative K ( s )=k (T d s+1 ) ( 1+ 1 T i s ) TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 65

64 Materials for exam lectures from 1 to 13 (>1100 slides...) Lecture 14 material repeat, supplementary info, informations about the exam, WUT questionnaires, consultations Lecture 15 modern control theory overview, experiment with control system, consultations Exam: Wednesday, 30th January, 10:30-11:30 Wednesday, 6th February, 10:30-11: TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 66

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