Theory of Machines and Automatic Control Winter 2018/2019

Theory of Machines and Automatic Control Winter 2018/2019 Lecturer: Sebastian Korczak, PhD, Eng. Institute of Machine Design Fundamentals - Department of Mechanics http://www.ipbm.simr.pw.edu.pl/

Lecture 13 Stability criteria. Gain margin and phase margin. System correction. 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 2

STABILITY CRITERIA General stability criterion Hurwitz criterion Nyquist stability criterion 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 3

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 <0 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 4

Hurwitz criterion mathematics a necessary and sufficient condition whether all the roots of the polynomial are in the left half of the complex plane 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 5

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 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 6

Hurwitz criterion LTI SISO system with a transfer function H = b m s m +b m 1 s m 1 +...+b 1 s+b 0 a n s n +a n 1 s n 1 +...+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: 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 7

Hurwitz criterion LTI SISO system with a transfer function H = b m s m +b m 1 s m 1 +...+b 1 s+b 0 a n s n +a n 1 s n 1 +...+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 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 8

Hurwitz criterion LTI SISO system with a transfer function H = b m s m +b m 1 s m 1 +...+b 1 s+b 0 a n s n +a n 1 s n 1 +...+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 0 0 0 0 a n 3 a n 2 a n 1 a n 0 0 a n 5 a n 4 a n 3... 0]...... 0 0 0 a 0 a 1 a 2 0 0 0 0 0 a 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 9

Hurwitz criterion LTI SISO system with a transfer function H = b m s m +b m 1 s m 1 +...+b 1 s+b 0 a n s n +a n 1 s n 1 +...+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 0 0 0 0 a n 3 a n 2 a n 1 a n 0 0 a n 5 a n 4 a n 3... 0]...... 0 0 0 a 0 a 1 a 2 0 0 0 0 0 a 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 10

Hurwitz criterion Hurwitz criterion Routh criterion (1895) (1876) Liénard Chipart criterion modification of Hurwitz criterion 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 11

Hurwitz criterion Example 1 G= 5 s+3 10 s 2 +3 s+1 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 12

G= 2 s 2 s 3 +s+20 Hurwitz criterion Example 2 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 13

G= 3 s 5 s 3 +4 s 2 +3 s+10 Hurwitz criterion Example 3 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 14

G= 1 3 s 4 +4 s 3 +6 s 2 +4 s+5 Hurwitz criterion Example 4 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 15

k s 4 s 3 +3 s 2 +k s+1 Hurwitz criterion Example 5 Choose k parameter to satisfy Hurwitz criterion 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 16

Hurwitz criterion Example 6 Choose k parameter to satisfy Hurwitz criterion 2 2 s 3 +ks 2 +(1+k ) s+3 Homework 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 17

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 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 18

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 =1 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 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 = 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 =1 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 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 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 >2.83 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 21

Nyquist stability criterion x + G 1 y G 2 G z = y x = G 1 1+G 1 G 2 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 22

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 = 1 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 23

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 = 1 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 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 Im G open Unstable if: G 1 G 2 = 1 ω ω + ω=0 Re G open 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 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 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 26

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 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 27

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 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 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 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 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 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 32

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 ) 2 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 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 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 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) 3 3k ω= 1 3 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 35

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= 1 3 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 36

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] 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 37

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] 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 38

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) 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 39

Summing of Bode plots example G = 10 s 2 +s = 10 1 s+1 1 s 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 40

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) 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 41

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 0,1 1 10 ω [rad/s] L(ω) [db] 20 0 0,1 1 10 ω [rad/s] φ(ω) [rad] 0 ω [rad/s] φ(ω) [rad] π 2 0,1 1 10 ω [rad/s] φ(ω) [rad] π 2 0,1 1 10 ω [rad/s] 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 42

Summing of Bode plots example G = 10 s 2 +s = 10 1 s+1 1 s L(ω) [db] 60 40 20 0 20 0,1 1 10 ω [rad/s] φ(ω) [rad] π 2 0,1 1 10 ω [rad/s] 40 60 π 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 43

Nyquist stability criterion control loop with P controller x + G y G closed = G 1+ G G opened = G =k P 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 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 -1 1 Re 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 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 Im G opened =k p 1 Ts+1 Im -1 1 Re Re -1 k p 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 46

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 +1 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 47 k p

Nyquist stability criterion control loop with P controller x + G y G closed = G 1+ G G opened = G =k P 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 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 1 Re 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 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 = Im k p T 1 2 s 2 +T 2 s+1 1 Re Re -1 k p 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 50

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 +1 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 51 k p G opened is always stable G closed is always stable

Nyquist stability criterion control loop with P controller x + G y G closed = G 1+ G G opened = G =k P 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 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 Im 1 Re 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 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 Im -1 Im 1 Re Re k p 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 54

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 +1 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 55 k p

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 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 56

Nyquist stability criterion control loop with PI controller x + G y ( s )=k P ( 1+ 1 T i s ) G opened = G 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 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 -1 1 Re 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 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 -1 1 Re -1 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 59

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 0 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 60

Correction of the system Correction by proportional term G ( s ) -1 Im Re 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 61

Correction of the system Correction by proportional term G ( s ) k G ( s ) -1 Im -1 Im Re Re 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 62

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 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 63

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 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 64

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 ) 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 65

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, 10:30-11:30 8.01.2019 TM&AC, Lecture 13, Sebastian Korczak, only for educational purposes 66