Root Locus Contents. Root locus, sketching algorithm. Root locus, examples. Root locus, proofs. Root locus, control examples

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1 Root Locu Content Root locu, ketching algorithm Root locu, example Root locu, proof Root locu, control example Root locu, influence of zero and pole Root locu, lead lag controller deign 9 Spring ME45 - GGZ Week -: Root Locu Page

2 Root Locu What i it? W. R. Evan developed in 948. Pole location characterize the feedback ytem tability and tranient propertie. Conider a feedback ytem that ha one parameter (gain) K > to be deigned. K L() L(): open-loop TF Root locu graphically how how pole of CL ytem varie a K varie from to infinity. 9 Spring ME45 - GGZ Week -: Root Locu Page

3 Root Locu A Simple Example K L() L( ) ( ) Characteritic eq. K ( ) K K :,- K : -, - K > : complex number ± Cloed-loop loop pole K Im Re 9 Spring ME45 - GGZ Week -: Root Locu Page 3

4 Root Locu A Complicated Example K L() L( ) ( )( 3) Characteritic eq. K ( )( 3) ( )( 3) K( )??? It i hard to olve thi analytically for each K. I there ome way to ketch a rough root locu by hand? (In Matlab, you may ue command rlocu.m.) 9 Spring ME45 - GGZ Week -: Root Locu Page 4

5 Root Locu Rule Root locu i ymmetric w.r.t. the real axi. The number of branche order of L() Mark pole of L with x and zero of L with o. Im L( ) ( )( 3) Re 9 Spring ME45 - GGZ Week -: Root Locu Page 5

6 Root Locu Rule RL include all point on real axi to the left of an odd number of real pole/zero. RL originate from the pole of L and terminate at the zero of L, including infinity zero. Im Re Indicate the direction with an arrowhead. 9 Spring ME45 - GGZ Week -: Root Locu Page 6

7 Root Locu Rule (Aymptote) Number of aymptote relative degree (r) of L: : Angle of aymptote are r n{ m{ deg(den) deg( num) π ( k ), k,,, L, r r 9 Spring ME45 - GGZ Week -: Root Locu Page 7

8 Root Locu Rule (Aymptote) (cont d) Interection of aymptote pole zero r L( ) ( )( 3) pole zero r ( ( ) ( 3)) ( ) Aymptote (Not root locu) Im Re 9 Spring ME45 - GGZ Week -: Root Locu Page 8

9 Root Locu Rule 3 Breakaway point are among root of Point where two or more branche meet and break away. dl( ) d L( ) ( )( 3) dl( ) d For each candidate, check the poitivity of [ ( )( 3)].4656,.767 ±. 795 j (Matlab CMD: root([ 4 5 3]) K.486,.797 ± j K L( ) 9 Spring ME45 - GGZ Week -: Root Locu Page 9

10 Root Locu Rule 3 (cont d - ) Quotient rule: d N( ) D( ) d d( N( )) D( ) N( ) d ( D( )) d( D( )) d d ( )( d 3) ( 5 6) ( ( ( )(3 )( 3)) 6) 3 8 ( ( )( 3)) 6 9 Spring ME45 - GGZ Week -: Root Locu Page

11 Root Locu Rule 3 (cont d - ) Im Re Breakaway point 9 Spring ME45 - GGZ Week -: Root Locu Page

12 Root Locu Matlab Command rlocu.m 8 Root Locu 6 4 L( ) ( )( 3) Imaginary Axi Real Axi 9 Spring ME45 - GGZ Week -: Root Locu Page

13 Root Locu A Simple Example Reviited K L() L( ) ( ) Aymptote Relative degree Aymptote interection Breakaway point ( ) Im Re dl( ) d ( ) ( ( )) 9 Spring ME45 - GGZ Week -: Root Locu Page 3

14 Root Locu Rule Summary Root locu What i root locu How to roughly ketch root locu Sketching root locu relie heavily on experience. PRACTICE! To accurately draw root locu, ue Matlab. Next, more example 9 Spring ME45 - GGZ Week -: Root Locu Page 4

15 Root Locu Practice Practice (hand drawing and Matlab) L( ) L ( ) L ( ) 3 L( ) ( 4) L( ) ( ) L( ) ( )( 5) 9 Spring ME45 - GGZ Week -: Root Locu Page 5

16 Root Locu What I It (Review)? Conider a feedback ytem that ha one parameter (gain) K > to be deigned. K L() L(): open-loop TF Root locu graphically how how pole of a CL ytem varie a K varie from to infinity. 9 Spring ME45 - GGZ Week -: Root Locu Page 6

17 Root Locu Rule (Mark Pole/Zero) Root locu i ymmetric w.r.t. the real axi. The number of branche order of L() Mark pole of L with x and zero of L with o. L( ) ( )( 5) Im Re 9 Spring ME45 - GGZ Week -: Root Locu Page 7

18 Root Locu Rule (Real Axi) RL include all point on real axi to the left of an odd number of real pole/zero. RL originate from the pole of L and terminate at the zero of L, including infinity zero. Im Re Indicate the direction with an arrowhead. 9 Spring ME45 - GGZ Week -: Root Locu Page 8

19 Root Locu Rule (Aymptote) Number of aymptote relative degree (r) of L: : Angle of aymptote are r n{ m{ deg(den) deg( num) π ( k ), k,,, L, r r 9 Spring ME45 - GGZ Week -: Root Locu Page 9

20 Root Locu Rule (Aymptote) (cont d) Interection of aymptote pole zero r L( ) ( )( Aymptote (Not root locu) 5) pole zero r Im ( ( ) ( 5)) () 3 Re 9 Spring ME45 - GGZ Week -: Root Locu Page

21 Root Locu Rule 3 (Breakaway) Breakaway point are among root of Point where two or more branche meet and break away. L( ) ( )( 5) dl( ) d dl( ) d 3 [ ( ± 3 For each candidate, check the poitivity of K K )( 5)] K L( ) 9 Spring ME45 - GGZ Week -: Root Locu Page

22 Root Locu Rule 3 (Breakaway) (cont d - ) L( ) ( )( 5) Im What i thi value? For what K? Re Routh-Hurwitz! Breakaway point ( K.47.3) 9 Spring ME45 - GGZ Week -: Root Locu Page

23 Root Locu Find K for critical tability Characteritic equation K ( )( 5) K Routh array K 6 K 5 K Stability condition < K < 3 When K ± 5 j 9 Spring ME45 - GGZ Week -: Root Locu Page 3

24 Root Locu Root Locu Example Im L( ) ( )( 5) 5 ( K j 3) Re Breakaway point ( K.47.3) - 5 j ( K 3) 9 Spring ME45 - GGZ Week -: Root Locu Page 4

25 Root Locu Example with Complex Pole L( ) zero : pole : - ± j 3 How to compute angle of departure? After Step,,,3, we obtain Im Re dl( ) d Breakaway point ( ( ) ) ± 9 Spring ME45 - GGZ Week -: Root Locu Page 5

26 Root Locu Rule 4 (Angle of Departure) Angle condition: For to be on RL, p j 3 θ "" Im φ L ( ) ( ( z ) ) ( φ θ θ 8 z p )( p p ) ( o ( ) p ) p j 3 θ z Re If"" i o φ, θ 9 o θ 5 cloe to p, o 9 Spring ME45 - GGZ Week -: Root Locu Page 6

27 Root Locu Rule 4 (Angle of Departure) Im Breakaway point Re 9 Spring ME45 - GGZ Week -: Root Locu Page 7

28 Root Locu Summary Example for root locu. Gain computation for marginal tability, by uing Routh- Hurwitz criterion Angle of departure (Angle of arrival can be obtained by a imilar argument.) Next, ketch of proof for root locu algorithm 9 Spring ME45 - GGZ Week -: Root Locu Page 8

29 Root Locu Exercie L( ) L( ) ( )( )( 3) L( ) ( L ) ( )( ) 9 Spring ME45 - GGZ Week -: Root Locu Page 9

30 Root Locu Exercie L( ) ( )( ) L( ) ( )( ) L( ) ( )( ) L( ) ( 4 5)( 5) 9 Spring ME45 - GGZ Week -: Root Locu Page 3

31 Root Locu Exercie 3 L( ) ( 3)( ) L( ) ( )( )( ) L( ) L( ) ( )( 4 5) 9 Spring ME45 - GGZ Week -: Root Locu Page 3

32 Root Locu Exercie 4 4 L( ) ( )( 3)( 4 5) L( ) ( 4 5)( 6 ) L( ) ( ( ) )( 3) L( ) ( )( ( ) 3) 9 Spring ME45 - GGZ Week -: Root Locu Page 3

33 Root Locu Characteritic equation and RL Characteritic equation KL( ) K L( ) L( ) K Root locu i obtained by for a fixed K >, finding root of the characteritic equation, and weeping K over real poitive number. A point i on the root locu, if and only if L() evaluated for that i a negative real number. 9 Spring ME45 - GGZ Week -: Root Locu Page 33

34 Root Locu Angle and Magnitude Condition Characteritic eq. can be plit into two condition. Angle condition L o ( ) 8 (k ), k, ±, ± Odd number, K Magnitude condition L( ) K For any point, thi condition hold for ome poitive K. 9 Spring ME45 - GGZ Week -: Root Locu Page 34

35 Root Locu A Simple Example L( ) ( ) Select a point -j L( j) ( (- ) j j)( i on root locu. K L( ) j) L( j) 8 Select a point -j L( 9 Spring ME45 - GGZ Week -: Root Locu Page 35 j) Im Re ( ) L( ) 8 ( j)( j) j( j) j i NOT on root locu.

36 Root Locu Step-by by-step: Step Root locu i ymmetric w.r.t. the real axi. Characteritic equation i an equation with real coefficient. Hence, if a complex number i a root, it complex conjugate i alo a root. The number of branche order of L() If L() n()/d(), then Characteritic eq. i d()kn(), which ha root a many a the order of d(). Mark pole of L with x and zero of L with o. L( ) ( p z )( p ) p p z Im Re 9 Spring ME45 - GGZ Week -: Root Locu Page 36

37 Root Locu Step-by by-step: Step - RL include all point on real axi to the left of an odd number of real pole/zero. Tet point p p z Im L( ) Re ( z) ( p) ( p) Not atify angle condition! Im p p z L( ) Re ( z) ( p) ( p) Satify angle condition! 9 Spring ME45 - GGZ Week -: Root Locu Page 37

38 Root Locu Step-by by-step: Step - (cont d) RL include all point on real axi to the left of an odd number of real pole/zero. Tet point p p z Im L( ) Re ( z) ( p) ( p) Not atify angle condition! Im p p z L( ) Re ( z) ( p) ( p) Satify angle condition! 8 9 Spring ME45 - GGZ Week -: Root Locu Page 38

39 Root Locu Step-by by-step: Step - RL originate from the pole of L, and terminate at the zero of L, including infinity zero. L( ) } n( ) K d( ) d( ) Kn( ) K n( ) d( ) K K n( ) d( ) n( ) d( ) : Pole of L() : Zero of L() 9 Spring ME45 - GGZ Week -: Root Locu Page 39

40 Root Locu Step-by by-step: Step - Number of aymptote relative degree (r) of L: Angle of aymptote are r : deg(den) π ( k ), k r deg(num),,k r r r 3 r 4 9 Spring ME45 - GGZ Week -: Root Locu Page 4

41 Root Locu Step-by by-step: Step - (cont d) For a very large, Characteritic equation i approximately n n n r L( ) n r L L n r KL( ) K Kn r r Kn (auming n < > ) r π ( k ), k,,,... π ( k ), k,,,... r 9 Spring ME45 - GGZ Week -: Root Locu Page 4

42 Root Locu Step-by by-step: Step - Interection of aymptote pole r zero p i r z i Proof for thi i omitted and not required in thi coure. Intereted tudent hould read page 363 in the book by Dorf & Bihop. 9 Spring ME45 - GGZ Week -: Root Locu Page 4

43 9 Spring ME45 - GGZ Page 43 Week -: Root Locu Breakaway point are among root of Suppoe that Suppoe that b i a breakaway point. i a breakaway point. Root Locu Root Locu Step Step-by by-step: Step 3 Step: Step 3 ) ( d dl ) ( ) ( ) ( ) ( b Kn b d b Kn b d & & ) ( ) ( ) ( ) ( b n b n b d b d & & ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( b n b n b d b d b d b n b d b d b n b d b n d dl b & & & &

44 9 Spring ME45 - GGZ Page 44 Week -: Root Locu RL depart from a pole p j with angle of departure RL arrive at a zero z j with angle of arrival (No need to memorize thee formula.) (No need to memorize thee formula.) Root Locu Root Locu Step Step-by by-step: Step 4 Step: Step 4 i j i i i j i j d p p z p, 8 ) ( ) ( θ i j i i i j i j a z z p z, 8 ) ( ) ( θ

45 Root Locu Step-by by-step: Step 4 (cont d) Sketch of proof for angle of departure Im For to be on root locu, due to angle condition ( p z) θ ( p p ) 8 Re θ φ θ d 8 9 Spring ME45 - GGZ Week -: Root Locu Page 45

46 Root Locu Step-by by-step: Step 4 (cont d-) Sketch of proof for angle of arrival Im For to be on root locu, due to angle condition 3 φ ( z z ) ( z pi ) 8 i Re z φ θ θ θ3 φ A 8 9 Spring ME45 - GGZ Week -: Root Locu Page 46

47 Root Locu Summary (How to Draw) Four tep drawing proce: Root locu i ymmetric w.r.t. the real axi; the number of branche equal to the order of L(); and mark pole of L with x and zero of L with o. RL include all point on real axi to the left of an odd number of real pole/zero; and it originate from the pole of L, and terminate at the zero of L, including infinity zero. Number of aymptote equal to relative degree (r) of L; and angle of aymptote are π(k)/r (k,, ) Breakaway point are among root of dl()/d RL depart from a pole p j with angle of departure θ ( p RL arrive at a zero z j with angle of arrival θ Next, we will move on to root locu application to control example. d a i i ( z j j z i ) p ) i i, i j i, i j ( p ( z j j z i p ) 8 i ) 8 9 Spring ME45 - GGZ Week -: Root Locu Page 47

48 Root Locu Control Example R () () E Y () K ( 5) K t a) Set K t. Draw root locu for K >. b) Set K. Draw root locu for K t >. c) Set K 5. Draw root locu for K t >. 9 Spring ME45 - GGZ Week -: Root Locu Page 48

49 Root Locu (a) K t (Control Example ) L( ) ( 5) Im Re There i no tabilizing gain K! 9 Spring ME45 - GGZ Week -: Root Locu Page 49

50 Root Locu (b) K (Control Example ) R () () E Y () ( 5) K t Characteritic eq. 3 5 K t K t ( 5) Kt ( 5) L( ) 3 9 Spring ME45 - GGZ Week -: Root Locu Page 5

51 Root Locu (b) K (Control Example ) L( ) 3 5 Im Re By increaing Kt, we can tabilize the CL ytem. 9 Spring ME45 - GGZ Week -: Root Locu Page 5

52 Root Locu (b) K (Control Example ) (Find K t for MS) Characteritic equation 3 K 5 K 3 5 t t Routh array When K t 5 K t K t Stability condition 5 ± j 9 Spring ME45 - GGZ Week -: Root Locu Page 5

53 Root Locu (b) K (Control Example ) ( (Control Example ) (Matlab rlocu.m ) Root Locu Damping ratio If K,, we cannot achieve Imaginary Axi for any K t > Real Axi Spring ME45 - GGZ Week -: Root Locu Page 53

54 Root Locu (c) K 5 (Control Example ) () R E () Y () 5 ( 5) K t Characteritic eq. 3 5 K 5 t ( 5) 5 K t ( 5) K t L( ) 3 9 Spring ME45 - GGZ Week -: Root Locu Page 54

55 Root Locu (c) K 5 (Control Example ) ( (Control Example ) (Matlab rlocu.m ).5.38 Root Locu Root Locu Imaginary axi Imaginary Axi axi Real axi Axi Real axi 9 Spring ME45 - GGZ Week -: Root Locu Page 55

56 Root Locu Control Example K L() L( ) ( T )( ) a) Set T. Draw root locu for K >. L( ) ( )( ) b) Vary T to ee the effect of a zero on root locu. 9 Spring ME45 - GGZ Week -: Root Locu Page 56

57 Root Locu (a) (Control Example ) Root locu for L( ) ( )( ) Im j Re Breakaway point K j ( K 6) 9 Spring ME45 - GGZ Week -: Root Locu Page 57

58 Root Locu (b) (Control Example ) When K i fixed and T i a poitive parameter, the characteritic equation can be written a T K ( )( ) )( ) K TK { ( with T Term without T K T ( )( ) K 9 Spring ME45 - GGZ Week -: Root Locu Page 58

59 Root Locu (b) (Control Example ) (cont d) Root locu for variou K & T Zero of L(): T Generally, addition of a zero improve tability. 9 Spring ME45 - GGZ Week -: Root Locu Page 59

60 Root Locu Control Example Summary Multiple parameter deign example Next, lead compenator deign baed on root locu More Example For the feedback ytem, Set a, and draw RL for K >. Set K 9, and draw RL for a >. K ( a) 9 Spring ME45 - GGZ Week -: Root Locu Page 6

61 Root Locu Cloed Loop Deign uing RL Deign Target! C() Controller G() Plant Fixed! Place cloed-loop pole at deired location by tuning the gain C() K. (for time domain pec) If root locu doe not pa the deired location, then rehape the root locu by adding pole/zero to C(). (How?) Compenation 9 Spring ME45 - GGZ Week -: Root Locu Page 6

62 Root Locu Effect of Adding Pole Pulling root locu to the RIGHT Le table Slow down the ettling Im Im Im Re Re Re Add a pole Add a pole 9 Spring ME45 - GGZ Week -: Root Locu Page 6

63 Root Locu Effect of Adding Zero Pulling root locu to the LEFT More table Speed up the ettling Im Re Im Im Add a zero Im Re Re Re 9 Spring ME45 - GGZ Week -: Root Locu Page 63

64 Root Locu Adding Pole/Zero Remark Adding only zero often problematic becaue uch controller amplifie the high-frequency noie. Adding only pole C( ) K( z), ( z > often problematic becaue uch controller generate a le table ytem (by moving the cloed-loop pole to the right). ) C( ) K /( p), ( p > ) Thee fact can be explained by uing frequency repone analyi. Add both zero and pole! 9 Spring ME45 - GGZ Week -: Root Locu Page 64

65 Root Locu Lead and Lag Compenator C() Controller G() Plant z C( ) K, ( z >, p > p ) Lead compenator Im Lag compenator Im Re Re Why thee are called lead and lag? We will ee that from frequency repone in thi cla. 9 Spring ME45 - GGZ Week -: Root Locu Page 65

66 Root Locu Lead Compenator Poitive angle contribution Tet point Im C Lead ( ) θ > Lead Re C Lead ( ) θ θ z z p p -p ( θ Lead z > ) -z ( p ) 9 Spring ME45 - GGZ Week -: Root Locu Page 66

67 Root Locu Lag Compenator Negative angle contribution C Lag ( ) θ < Lag Tet point Im -z -p C Lag ( ) θ θ z z p p ( θ Lag z < ) ( p ) Re 9 Spring ME45 - GGZ Week -: Root Locu Page 67

68 Root Locu Rule of Lead/Lag Compenator Lead compenator Improve tranient repone Improve tability C Lead ( ) K z p Lag compenator Reduce teady tate error Lead-lag compenator C Lag ( ) K Take into account all the above iue. C z p ( ) C ( ) C ( ) LL Lead Lag 9 Spring ME45 - GGZ Week -: Root Locu Page 68

69 Root Locu Example: Radar Tracking Sytem 4 ( ) 9 Spring ME45 - GGZ Week -: Root Locu Page 69

70 Root Locu RTS: Lead Compenator Deign () Conider a ytem G( ) 4 ( ) C() Controller G() Plant Analyi of CL ytem for C() Damping ratio ζ.5 Undamped natural freq. ω n rad/ Deired pole Im 3 j Performance pecification Damping ratio ζ.5 Undamped natural freq. ω n 4 rad/ CL pole with C() ) Re 9 Spring ME45 - GGZ Week -: Root Locu Page 7

71 Root Locu RTS: Angle and Mag Condition A point to be on root locu it atifie Angle condition Odd number o L( ) 8 (k ), k, ±, ±,... For a point on root locu, gain K i obtained by Magnitude condition L( ) K 9 Spring ME45 - GGZ Week -: Root Locu Page 7

72 Root Locu RTS: Lead Compenator Deign () Evaluate G() at the deired pole. G( 3 j) ( 4 3 j) 3 j 3 3 j o If angle condition i atified, compute the correponding K. o In thi example, Deired pole Im Angle condition i not atified. G( 3 j) Angle deficiency φ 3 Re 3 j 9 Spring ME45 - GGZ Week -: Root Locu Page 7

73 Root Locu RTS: Lead Compenator Deign (3) To compenate angle deficiency, deign a lead compenator C( ) K z p atifying C( 3 j) 3( : φ) Deired pole Im GC( 3 j) 8 3 j There are many way to deign uch C()! Re 9 Spring ME45 - GGZ Week -: Root Locu Page 73

74 Root Locu RTS: Lead Compenator Deign (4) Poitive angle contribution Tet point Im C Lead ( ) θ > Lead Triangle relation Re θ θ π θ ) π p Lead ( z -p -z θ θ θ z p Lead 9 Spring ME45 - GGZ Week -: Root Locu Page 74

75 Root Locu RTS: Lead Compenator Deign (5) How to elect pole and zero: Draw horizontal line PA Draw line PO Draw biector PB APB BPO APO A Deired pole P Im 3 j Draw PC and PD CPB BPD φ B C D -p( p(-5.4) O -z( z(-.9) Re Pole and zero of C() are hown in the figure. 9 Spring ME45 - GGZ Week -: Root Locu Page 75

76 Root Locu RTS: Lead Compenator Deign (6) Compenator realization: One example, uing operational amplifier C C R R 4 v i (t) R - R 3 - v o (t) V V o i ( ) ( ) R 4 R RC R3 R RC 9 Spring ME45 - GGZ Week -: Root Locu Page 76

77 Root Locu RTS: Lead Compenator Deign (7) Tranfer function K z C( ) V V o i ( ) ( ) R R C R R C ( ( R C R C ) ) R R 4 3 R C 4 R C 3 ( ( R C R C ) ) Lead compenator Lag compenator p R C R C Im Re R C R C Im Re 9 Spring ME45 - GGZ Week -: Root Locu Page 77

78 Root Locu RTS: Lead Compenator Deign (8) Sytem repone (uncompenated and compenated) Compenated ytem.4 Uncompenated ytem (C(( C()) Lead compenator give fater tranient repone (horter rie and ettling time) improved tability Spring ME45 - GGZ Week -: Root Locu Page 78

79 Root Locu RTS: Lead Compenator Deign (9) Error contant (after lead compenation) G( ) C Lead ( ) 4 (.9) ( ) ( 5.4) Step-error contant K p : limg( ) C ( ) Lead Unit ramp input Ramp-error contant K v : lim G( ) C ( ) Lead 5. NOT SATISFACTORY! Ramp repone Lag compenator can reduce teady-tate tate error. 9 Spring ME45 - GGZ Week -: Root Locu Page 79

80 Root Locu RTS: Lag Compenator Deign () How to deign lag compenator? Lag compenator We want to increae ramp-error contant K v : lim G( ) C Take, for example, z p. C Lag ( ) Lead ( ) C We do not want to change CL pole location o much (already atifactory tranient). Lag z p ( ) 5. z p > 5 G( ) C Lead C Lag ( ) ( ) G( ) CLead ( ) CLag ( ) 9 Spring ME45 - GGZ Week -: Root Locu Page 8

81 Root Locu RTS: Lag Compenator Deign () Guideline to chooe z and p The zero and the pole of a lag compenator hould be cloe to each other, for ) ( C Lag The pole of a lag compenator hould be cloe to the origin, to have a large ratio z/p, leading to a large ramp-error contant K v. However, the pole of a lag compenator too cloe to the origin may be problematic: Difficult to realize (recall op-amp realization) Slow ettling (due to cloed-loop pole near the origin) 9 Spring ME45 - GGZ Week -: Root Locu Page 8

82 Root Locu RTS: Lag Compenator Deign (3) Root locu with lag compenator Without compenator With compenator θ θ θ 8 θ θ θ θ z θ p 9 Spring ME45 - GGZ Week -: Root Locu Page 8

83 Root Locu RTS: Lag Compenator Deign (4) How to deign lag compenator? For the deired CL pole 3 j C Lag ( ) p p, p p Take a mall p (by trial-and-error!) p.5 p p.97,.88 p p o Lead-lag controller C LL ( ) Spring ME45 - GGZ Week -: Root Locu Page 83

84 Root Locu RTS: Lag Compenator Deign (5) Root locu With lead compenator 5 Ro o t Lo c u With lead-lag lag compenator 5 Ro o t Lo c u 5 Deired pole 5 Imaginary Axi -5 Imaginary Axi Re a l A x i Re al A x i 9 Spring ME45 - GGZ Week -: Root Locu Page 84

85 Root Locu RTS: Lag Compenator Deign (6) Comparion of tep repone Uncompenated With lead compenator With lead-lag lag compenator Spring ME45 - GGZ Week -: Root Locu Page 85

86 Root Locu RTS: Lag Compenator Deign (7) Comparion of ramp repone 5 4 Unit ramp input 3 Uncompenated With lead compenator With lead-lag lag compenator Spring ME45 - GGZ Week -: Root Locu Page 86

87 Root Locu Summary Controller deign baed on root locu Lag compenator deign Lag compenator improve teady tate error. Lead-lag compenator deign Lead-lag compenator improve tability, tranient and teady-tate repone. Next, frequency repone and Bode plot 9 Spring ME45 - GGZ Week -: Root Locu Page 87

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