Control Sytem Root locu chibum@eoultech.ac.kr
Outline Concet of Root Locu Contructing root locu Control Sytem
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
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
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
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
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
Root Locu Magnitude condition Angle condition G G k k 0,,, - Im G, 3, 5, Re G Control Sytem
Recall Comlex Function Control Sytem
Examle Ex. Oen-loo tranfer function Characteritic Eqn. G 0 0 0 5 Cloed-loo ole 0,, j j Cloed-loo ole j Control Sytem
Examle Characteritic Eqn. At ole G 0 k Control Sytem
Examle generalized method? Control Sytem
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 0 0 0 : root locu arameter 0
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
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
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
Examle Ex. Characteritic Eqn. Ste: Ste: 4 G 0 4 Pole: = 0,-4 Zero: = - 0 = - Control Sytem
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
Examle Ex. GH 4 oen-loo ole: 0,-,-4,-4 oen-loo zero: - n=4, m= Center of aymtote: A 4 4 3 Control Sytem
Examle Angle of aymtote: A k 5,, 4 3 3 Control Sytem
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
Contructing Root Locu Ex. Oen-loo G 4 G 4 0 Rewritten 4 0 4 Control Sytem
Examle Ex. G H 3 0 3 d d d d 3 3 8 0 6 0.46, 0.77 0. 79 j Control Sytem
Contructing Root Locu Ste5: Find the croover oint where the locu cro the imaginary axi Routh-Hurwitz criterion Let = jω in Δ Control Sytem
Examle Ex. Find the croover oint of following oen loo tranfer function G and correonding G 3 5 Characteritic eqn. 3 5 0 Routh Array 0 C 0 0 3 8 5 0 5 Find croover oint j j 3 8 j 5 j 8 j5 0 5 Control Sytem - 5 -
Contructing Root Locu Ste6: Find the dearture angle @ ole and the arrival angle @ 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
Control Sytem Contructing Root Locu H G : angle from zero z : angle from ole : angle from ole D z z H G n n
Examle Ex. G H j j D j 80 o o 90 angle from ole @ -- j o 45 angle from zero @ - 35 o 35 o j 45 o - 90 o -j Control Sytem
Examle Ex. G H j j o o o o A j 80 90 90 45 5 angle zero @ from -j angle ole@ from 0 angle ole@ from - o - 45 o 5 o j 90 o 90 o Control Sytem
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
Contructing Root Locu MATLAB command rlocfind rlocu rltool Control Sytem
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
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
Outline Root locu examle Control Sytem
Examle Ex G H 0 Ste: Oen loo ole: 0,0,- Oen loo zero: - Ste: Real axi ortion 0,0, -,- Control Sytem
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
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
Examle Control Sytem
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
Control Sytem
Examle Ex G H 3 64 8 4 0 Ste: Oen loo ole: 0, -4, 4j4 Ste: Real axi ortion: 0, -4 Ste3: Center of aymtote: A 0 4 4 j4 4 4 j4 3 Angle of aymtote: A k 4, 4 3, 4 5, 4 7 4 Control Sytem
Ste4: Breakaway oint 4 3 64 8 d 3 4 36 8 8 d.5767, 3.77 j.5533 Ste5: Croover oint Routh Array 4 4 64 3 8 53.33 8-0.5 0 3 64 8 0 Characteritic Eqn. 0<<568.89 for tability croover oint : j3.66 Control Sytem
Control Sytem Examle Ste6: angle of dearture angle of arrival 4 3 4 4 tan 4 3 3 3 3 4 3 D
Examle Ste7: Control Sytem
Examle Ex. R 0 Y + + + 4 - + 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
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
Outline Tyical root locu configuration Conditional tability Non-minimum hae ytem Control Sytem
Tyical Root Locu Configuration Control Sytem
Tyical Root Locu Configuration Control Sytem
Tyical Root Locu Configuration Caution: the relative oition of the oen-loo ole and zero. Control Sytem
Conditional Stability There may be range of over which the ytem i table/untable. Conditional Stability Ex. Y Control Sytem
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
Outline Addition of oen-loo ole Addition of oen-loo zero Root locu of multivariable Control Sytem
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
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
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
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