Control Systems. Root locus.

Transcription

1 Control Sytem Root locu

2 Outline Concet of Root Locu Contructing root locu Control Sytem

3 Root Locu Stability and tranient reone i cloely related with the location of ole in -lane How the ole of a given ytem migrate about the -lane a the arameter are varied. Root locu method Evan in 948 Control Sytem

4 Examle DC motor with P control J B T y = J B T θ T J B Oen Loo Tranfer Function: Cloed Loo Tranfer Function: J B J B Control Sytem

5 Control Sytem Examle Cloed loo ole are Cloed loo ole Natural frequency & Daming ratio J B J n, n n n B J R Y 4, n n J J B B

6 Examle For a given lant, ie. J,B i fixed, how will affect the location of ole? Im Re, B B J 4J B 4J 0 two real root B 4J 0 double root B 4J 0 comlex conjugate root cf n J, B J Control Sytem

7 Root Locu Path of ole traced out in -lane a a ytem arameter varie from 0 to T G G Characteritic equation i G 0 Cloed loo ole atifie G G G k in - lane Control Sytem

8 Root Locu Magnitude condition Angle condition G G k k 0,,, - Im G, 3, 5, Re G Control Sytem

9 Recall Comlex Function Control Sytem

10 Examle Ex. Oen-loo tranfer function Characteritic Eqn. G Cloed-loo ole 0,, j j Cloed-loo ole j Control Sytem

11 Examle Characteritic Eqn. At ole G 0 k Control Sytem

12 Examle generalized method? Control Sytem

13 Control Sytem Contructing Root Locu Ste0: Preare the characteritic equation Aume the arameter of interet i the gain 0 Ex. m n z z z G n m : root locu arameter : root locu arameter 0

14 Control Sytem Contructing Root Locu Ste: Locate the ole & zero of G on the comlex lane the root locu branche tart from the oen loo ole =0 and terminate at the zero = 0 0 : n m n z z Oen Loo Pole Starting oint Oen Loo Zero Terminating oint 0 0 : m m n m n z z z z z z

15 Contructing Root Locu # of earate loci = # of OL ole Starting oint of root loci =0 Oen loo ole n Terminating oint of root loci = Finite zero m : Oen loo zero Infinite zero n-m Control Sytem

16 Contructing Root Locu Ste: Locate the egment of the real axi that are root loci The root locu on the real axi alway lie in a ection of the real axi to the left of an odd number of ole and zero. >0 Root loci mut be ymmetrical with reect to the real axi. Control Sytem

17 Examle Ex. Characteritic Eqn. Ste: Ste: 4 G 0 4 Pole: = 0,-4 Zero: = - 0 = - Control Sytem

18 Contructing Root Locu Ste3: Find the center and the angle of aymtote Root loci emanate from real axi along aymtote centered at A and roceed to the zero at with angle of A #of ditinct aymtote i N=n-m i.e., # of zero at n = # of finite ole of G m = # of finite zero of G n-m = Relative degree of ytem The center of aymtote: on the real axi A n m n m z j i j i n z z n m z m The angle of the aymtote: A k n m k 0,,, n m Control Sytem

19 Examle Ex. GH 4 oen-loo ole: 0,-,-4,-4 oen-loo zero: - n=4, m= Center of aymtote: A Control Sytem

20 Examle Angle of aymtote: A k 5,, Control Sytem

21 Contructing Root Locu Ste4: Find the breakaway oint on the real axi if any G X d d Find Y 0 X ' Y X Y ' Y uch that Y X d d 0 0 Control Sytem

22 Contructing Root Locu Ex. Oen-loo G 4 G 4 0 Rewritten Control Sytem

23 Examle Ex. G H d d d d , j Control Sytem

24 Contructing Root Locu Ste5: Find the croover oint where the locu cro the imaginary axi Routh-Hurwitz criterion Let = jω in Δ Control Sytem

25 Examle Ex. Find the croover oint of following oen loo tranfer function G and correonding G 3 5 Characteritic eqn Routh Array 0 C Find croover oint j j 3 8 j 5 j 8 j5 0 5 Control Sytem - 5 -

26 Contructing Root Locu Ste6: Find the dearture ole and the arrival zero G k k 0,,, Angle of dearture from a comlex ole j D = π - angle of vector to j from other ole + angle of vector to j from zero Angle of arrival at a comlex zero z i A = π - angle of vector to z i from other zero + angle of vector to z i from ole Only need to deal with finite zero the zero will be taken care of by aymtote Control Sytem

27 Control Sytem Contructing Root Locu H G : angle from zero z : angle from ole : angle from ole D z z H G n n

28 Examle Ex. G H j j D j 80 o o 90 angle from -- j o 45 angle from - 35 o 35 o j 45 o - 90 o -j Control Sytem

29 Examle Ex. G H j j o o o o A j angle from -j angle ole@ from 0 angle ole@ from - o - 45 o 5 o j 90 o 90 o Control Sytem

30 Contructing Root Locu Ste7: Comlete the ketch and locate cloed loo ole of interet From to find cloed loo ole Plug in numerical value of to find =+j N G H D 0 D N From cloed loo ole to find Ue magnitude criterion n j M i z i j * Control Sytem

31 Contructing Root Locu MATLAB command rlocfind rlocu rltool Control Sytem

32 Rule for Contructing Root Locu Note: If the numerator and the denominator of the oen-loo tranfer function have common factor, they will cancel each other. Pole-zero cancellation. The root locu lot of GH will then not how all the root of the characteritic equation. Only thoe that haven t been cancelled are hown. Control Sytem

33 Cancellation of Pole of G with Zero of H If the denominator of G and the numerator of H have common factor, they will cancel each other. Polezero cancellation. Ex. Y Y G H Control Sytem

34 Outline Root locu examle Control Sytem

35 Examle Ex G H 0 Ste: Oen loo ole: 0,0,- Oen loo zero: - Ste: Real axi ortion 0,0, -,- Control Sytem

36 Examle Ste3: Center of aymtote A n i i m j n m 3 Angle of aymtote 3 A k, n m z j 0.5 Control Sytem

37 Examle Ex. Y G H 3 0 Ste: Oen loo ole and zero Ste: Real axi ortion Ste3: Center and angle of aymtote Ste4: Breakaway oint Ste5: Croover oint Ste6: Angle of dearture and arrival Control Sytem

38 Examle Control Sytem

39 Examle Ex. Y G H 0 Ste: Oen loo ole and zero Ste: Real axi ortion Ste3: Center and angle of aymtote Ste4: Breakaway oint Ste5: Croover oint Ste6: Angle of dearture and arrival Control Sytem

40 Control Sytem

41 Examle Ex G H Ste: Oen loo ole: 0, -4, 4j4 Ste: Real axi ortion: 0, -4 Ste3: Center of aymtote: A j4 4 4 j4 3 Angle of aymtote: A k 4, 4 3, 4 5, Control Sytem

42 Ste4: Breakaway oint d d.5767, 3.77 j.5533 Ste5: Croover oint Routh Array Characteritic Eqn. 0<< for tability croover oint : j3.66 Control Sytem

43 Control Sytem Examle Ste6: angle of dearture angle of arrival tan D

44 Examle Ste7: Control Sytem

45 Examle Ex. R 0 Y Ste: Oen loo ole and zero Ste: Real axi ortion Ste3: Center and angle of aymtote Ste4: Breakaway oint Ste5: Croover oint Ste6: Angle of dearture and arrival Control Sytem

46 Examle Ex. R + - k 3 Y Ste: Oen loo ole and zero Ste: Real axi ortion Ste3: Center and angle of aymtote Ste4: Breakaway oint Ste5: Croover oint Ste6: Angle of dearture and arrival Control Sytem

47 Outline Tyical root locu configuration Conditional tability Non-minimum hae ytem Control Sytem

48 Tyical Root Locu Configuration Control Sytem

49 Tyical Root Locu Configuration Control Sytem

50 Tyical Root Locu Configuration Caution: the relative oition of the oen-loo ole and zero. Control Sytem

51 Conditional Stability There may be range of over which the ytem i table/untable. Conditional Stability Ex. Y Control Sytem

52 Non-minimum Phae Sytem If a ytem ha at leat one zero in the RHP Non-minimum hae ytem Gain hould be limited Control Sytem

53 Outline Addition of oen-loo ole Addition of oen-loo zero Root locu of multivariable Control Sytem

54 Addition of Oen-loo Pole Adding ole to oen loo tranfer function. Y Original: τ = 0, τ = 0 Add one: τ 0, τ = 0 Add two: τ 0, τ 0 Control Sytem

55 Addition of Oen-loo Zero Adding zero to oen loo tranfer function Y Add two: τ z 0, τ z 0 Original: τ z = 0, τ z = 0 Add one: τ z 0, τ z = 0 Control Sytem

56 Addition of Oen-loo Pole and Zero Adding a ole to the LHP uhe art of the locu into the RHP. Adding a zero to the LHP ull art of the locu into the LHP. Control Sytem

57 Root locu of multivariable Root locu for variable, a Characteritic eqn.: + a + = 0 R a Y a = 0, =variable + = 0 = cont, a = variable + a + = 0 Control Sytem