ROOT LOCUS. Poles and Zeros

Transcription

1 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity ROOT LOCUS The Root Locu i the ath of the root of the characteritic equation traced out in the - lane a a ytem arameter i changed. Pole and Zero For a function F, the zero of the function are the value of for which the function i zero. The ole are the value of for which the function i infinity. The zero are denoted by o and the ole are denoted by x. For examle F ha two zero at and 3, and two ole at and 5 Im. Re Alo, for the zero can be calculated a the root of 3 F Root Locu ite.google.com/ite/ziyadmaoud 83

2 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity and the ole can be calculated a the root of and j3 Im Re Root Locu for Feedback Sytem Conider a baic feedback ytem with the contant K a the arameter of interet. R + - K G Y The tranfer unction of the ytem i H Root Locu ite.google.com/ite/ziyadmaoud 84

3 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity KG T KG H The erformance of the ytem deend on the root of the characteritic equation KG H 0 The roduct G H i called the oen-loo tranmittance of the ytem. Now let u rereent G H a The characteritic equation become or N G H D N K 0 D D KN 0 At K 0, the root of the characteritic equation are the ame a the root of D 0, which are the ole of the oen-loo tranmittance KG H. The characteritic equation can alo be rearranged a D N 0 K At K, the root of the characteritic equation are the ame a the root of 0 which are the zero of the oen-loo tranmittance G H. N, Therefore, we can tate that the locu of the root of the characteritic equation KG H 0 begin at the ole of G H and end at the zero of G H a K increae from zero to infinity. Root Locu ite.google.com/ite/ziyadmaoud 85

4 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity Examle Conider the following ytem: R + - K 4 Y The tranfer function of the ytem i KG T KG H K 4 K The root of the characteritic equation 4 K 0 will determine the erformance characteritic of the ytem. For K 0, the characteritic equation become 4 0. The root are at and 4. The characteritic equation can be rearranged a 4 0 characteritic equation become 0. The root are at 0 and K.. At K, the The ame reult can be achieved by determining the ole and zero of the tranmittance G H. G H 4 Root Locu ite.google.com/ite/ziyadmaoud 86

5 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity The ole are at and 4. The zero are at 0 and Examle. Conider the following ytem: R + - K 4 Y The oen-loo tranmittance i G H 4 The ytem ha a zero a and two ole at j The tranfer function of the ytem i. KG T KG H K 4 K The following table how the root of the characteritic equation at elected value of the gain K. Root Locu ite.google.com/ite/ziyadmaoud 87

6 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity K Root 0 j j j , Double root , K 4 K K 6.47 Im K 0 K Re K 0 K Root Locu ite.google.com/ite/ziyadmaoud 88

7 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity Root Locu Rule Conider The characteritic equation of a imle feedback ytem or KGH 0 KGH The oen-loo tranmittance GH i generally a comlex function. Therefore, the characteritic equation can alo be written a and hence KGH 0 j KGH KGH 80 k360, k 0,,, The oen-loo tranmittance function can alo be written a For examle N GH D z z z3 zm 3 n can be written a GH Root Locu ite.google.com/ite/ziyadmaoud 89

8 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity Root Locu ite.google.com/ite/ziyadmaoud GH where 4 and 5 are the zero and,, and 3 are the ole of the tranmittance function GH. Comlex quantitie can be written i a olar form a j ce jb a u where b a u c and a b tan The tranmittance function can be written a n m j n j j jz m jz jz e e e e z e z e z GH where n r r m i i n m z K z z z K KGH and k z z z z KGH n r r m i i which i known a the angle criterion.

9 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity Examle Conider a feedback ytem with function G and H. The oen-loo tranmittance GH. The ytem ha no zero and two ole at 0 and. In the -lane, conider an arbitrary tet oint. For the oint to be on the root locu of the ytem the following condition mut be atified: Im - 0 Re and K KGH k 360 Root Locu ite.google.com/ite/ziyadmaoud 9

10 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity Then, can be on the real axi between and where 80 and 0, or on a vertical line through where 80. Im Re Rule The root locu tart at the ole where K 0 and end at the zero where K. The number of root locu branche mut be equal to the number of root of the characteritic equation. Real-Axi Segment Conider a ytem with a real-axi ole or zero a hown. The angle from the ole or zero to a tet oint i 0 and to a tet oint i 80. Thi mean that ole or zero to the right of a tet oint contribute an angle of 80 to KGH while ole or zero to the left of a tet oint contribute an angle of 0 to KGH. Root Locu ite.google.com/ite/ziyadmaoud 9

11 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity Im 80 Re For examle, in the following figure, in the ection to the right of, all angle contribution of the ole and zero are 0 which doe not atify the angle criterion KGH 80 k 360. Point in the ection between and have 80 contribution from and 0 from the ret of the ole and zero which mean that KGH zero ole k 360 z 3 z Point in the ection between and z have 80 contribution from and and 0 from the ret of the ole and zero which mean that KGH zero ole k 360 Point in the ection between z and 3 have 80 contribution from,, and z and 0 from the ret of the ole and zero which mean that Root Locu ite.google.com/ite/ziyadmaoud 93

12 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity and o on. Rule KGH zero ole k 360 The root locu include all oint on the real-axi to the left of an odd number of ole lu zero of the tranmittance function GH. Examle Plot the real-axi egment of the root locu of 3 GH. 4 The zero of the ytem are z 3 The ole of the ytem are 0 3,4 4 j Note that the reence of the comlex ole or zero doe not affect the real-axi egment. Thi i becaue they aear in ymmetric air and their contribution to the angle criterion cancel each other. Root Locu ite.google.com/ite/ziyadmaoud 94

13 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity Aymtote When the number of ole i greater than the number of zero of a ytem, the locu will roceed to zero at the infinity. Conider a remote oint. A aroache infinity, the angle contribution of all finite ole and zero become nearly equal. The angle criterion i Then KGH m i zero n r ole m i n r m n 80 k 360 Root Locu ite.google.com/ite/ziyadmaoud 95

14 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity Rule 3 80 k 360 n m A K aroache infinity, the branche of the root locu become nearly traight line with angle 80 n m k 360 where n i the number of ole and m i the number of zero. Centroid of Aymtote Rule 4 Aymtote are centered at a centroid that i imilar to center of gravity of ole and zero. Examle ole n m Conider the ytem of the reviou examle The aymtote angle are zero 3 GH 4 80 k 360 n m 80 k ,80, 300 Root Locu ite.google.com/ite/ziyadmaoud 96

15 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity The aymtote centroid i ole n m zero 0 4 j 4 j 3 3 z 4 Examle Conider the ytem of the reviou examle 3 GH 6 8 The zero of the ytem are z 0 Root Locu ite.google.com/ite/ziyadmaoud 97

16 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity The ole of the ytem are The aymtote angle are The aymtote centroid i 80 k 360 n m,3 3 j3 80 k , ole zero j j n m 3 4 Root Locu ite.google.com/ite/ziyadmaoud 98

17 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity Root Locu ite.google.com/ite/ziyadmaoud 99 Breakaway Point The breakaway oint i the oint at which the root locu leave the real-axi. At that oint, two or more branched meet. Therefore, there will be a multilicity of root at that oint. Conider the cae one air of breakaway branche, the characteritic equation of a baic feedback ytem i 0 KN D D N K KGH Q When the characteritic equation ha a double root at b, then we can rewrite the equation a Q Q a b The derivative of the characteritic equation i Q Q Q Q Q a b a b a b a b

18 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity Then, Q alo ha a root at b Q can be written a But then. Subtitute K in the Q equation, we get Rule 5 Q D KN 0 Q D KN 0 K D N N D N D 0 The root locu break away or arrive at the real-axi at a breakaway oint determined by the root of Examle Conider GH 3 The aymtote angle are N D N D 0 The aymtote centroid i 80 k 360 n m 80 k ,80, 300 Root Locu ite.google.com/ite/ziyadmaoud 00

19 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity For the given GH, ole n m zero D. N, and The breakaway oint are determined by the root of 3 N D N D The root of the above equation are. 43 and Point. 43 fall between the ole and, which mean that it i the breakaway oint. Root Locu ite.google.com/ite/ziyadmaoud 0

20 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity Examle Conider 0 GH 5 The zero of the ytem are z 0 The ole of the ytem are The aymtote angle are For the given 80 k 360 n m GH, k N, and D 5 5 The breakaway oint are determined by the root of. N D N D The root of the above equation are. 99 and Both oint are true breakaway oint ince both oint fall on the real axi egment of the root locu. Point.99 fall between the ole 0 and 5, and oint fall between the zero z 0 and the zero at. The final root locu of the ytem i hown below. Root Locu ite.google.com/ite/ziyadmaoud 0

21 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity Examle Conider GH The ytem ha no zero and four ole at -, -3, -4, and -6. The aymtote angle are The aymtote centroid i 80 k 360 n m 80 k ,35,5,35 ole n m zero Root Locu ite.google.com/ite/ziyadmaoud 03

22 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity For the given, and GH, N 4 3 D The breakaway oint are determined by the root of N D N D The root of the above equation are. 697, 3. 5 and Point. 697 and are true breakaway oint ince both oint fall on the real axi egment of the root locu. Root Locu ite.google.com/ite/ziyadmaoud 04

23 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity Imaginary Axi Croing Imaginary axi croing oint are very imortant becaue they rereent the witching oint between table and untable reone. Rule 6 The Routh-Hurwitz criterion can be alied to the characteritic equation of a ytem to determine the value of the feedback gain K at which the root of the ytem are on the imaginary axi. Examle Conider GH The ytem ha no zero and three ole at, -, and -3. The aymtote angle are The aymtote centroid i 80 k 360 n m 80 k ,80,300 ole n m zero For the given N, and 4 6 GH, D 3. The breakaway oint are determined by the root of N D N D Root Locu ite.google.com/ite/ziyadmaoud 05

24 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity The root of the above equation are and The oint i the true breakaway oint ince it fall on the real axi egment of the root locu. The root locu aymtotic branche at 60 and 300 cro the imaginary axi toward the right half ide of the -lane a hown in the figure below. The characteritic equation of the ytem can be derived a D KN K 0 Alying the Routh-Hurwitz criterion to the above characteritic equation a follow determine the range of K for which the root are on the left half lane: K 4 K 6 K K 0 K 6 The root of the characteritic equation are on the left half lane for 6 K 0. Thi lead to the concluion that the root locu of the ytem croe the imaginary axi when the value of K i 6 and 0. To determine the oint of imaginary axi croing, we ubtitute the croing value of K in the characteritic equation and olve for the root at the croing oint. For K 6, the characteritic equation become which mean that the root are ,-.679,-3.73 Root Locu ite.google.com/ite/ziyadmaoud 06

25 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity thi mean that the root locu croe the imaginary axi at the oint 0. For K 0, the characteritic equation become The root of the characteritic equation are j,-4 Thi mean that the root locu croe the imaginary axi at The final hae of the root locu i hown below. j. Angle of Dearture and Angle of Arrival Angle of dearture are ued to determine the loe of a root locu at it tarting oint ole while the angle of arrival are ued to determine the loe of a root locu at it ending oint zero. Root Locu ite.google.com/ite/ziyadmaoud 07

26 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity Conider the ole in the -lane a hown below. To determine the angle of dearture of the root locu that tart at, we will elect a tet oint which i very cloe to the ole. For a cloe enough, the line from to rereent the tarting egment of the root locu of. The angle criterion at i then zero ole k 360 k k 360 At the limit, and become the ame oint and the angle,,, and become the angle of meaured with reect to the ret of the ole and zero. Therefore, in general, Root Locu ite.google.com/ite/ziyadmaoud 08

27 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity Rule 7 the angle of dearture of a root locu at a ole i defined by di m k zero n l li ole 80 k 360 and imilarly, the angle of arrival of a root locu at a zero i defined by Examle Conider GH aj 4 8 n l ole m k k j zero 80 k 360 The ytem ha two zero at j, and two ole at j. Root Locu ite.google.com/ite/ziyadmaoud 09

28 Automatic Control Sytem, 343 Deartment of Mechatronic Engineering, German Jordanian Univerity For the ole at j, the angle of dearture can be calculated a di m k zero n l li ole 80 k 360 For the zero at d j, the angle of arrival can be calculated a aj n l ole m k k j zero 80 k 360 a The final root locu become Root Locu ite.google.com/ite/ziyadmaoud 0