MODERN CONTROL SYSTEMS

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1 MODERN CONTROL SYSTEMS Lecture 1 Root Locu Emam Fathy Department of Electrical and Control Engineering emfmz@aat.edu 1

2 Introduction What i root locu? Root locu i the locu (graphical preentation) of the cloed-loop pole a a pecific parameter (uually gain, K) i varied from 0 to infinity. Why do we need to ue root locu? We ue root locu to analyze the tranient repone qualitatively. (E.g. the effect of varying gain upon percent overhoot, ettling time and peak time). We can alo ue root locu to check the tability of the ytem.

3 Introduction we ue root locu to analyze the feedback control ytem. K in the feedback ytem i called a gain. Gain i ued to vary the ytem in order to get a different output repone.

4 Example How the dynamic of the ytem (camera) change a K i varied?

5 Drawing the root locu Characteritic equation: S S + K = 0

6

7 Drawing the root locu Next tep i to plot the pole value on the -plane by varying the gain, K, value.

8 Drawing the root locu Join the pole with olid line and you will get the hape of the locu (path)

9 Drawing the root locu The proce of drawing a root locu i time conuming. If the ytem i complex. An alternative approach i to ketch the root locu intead of drawing the root locu.

10 Sketching the root locu In order to ketch the root locu we mut follow thee ix rule.

11 Example 1) Sketch the root locu of the following ytem: 2) Determine the value of K uch that the damping ratio ζ of a pair of dominant complex conjugate cloed-loop i 0.5.

12 Contruction of root loci Step-1: The firt tep in contructing a root-locu plot i to locate the open-loop pole and zero of G()H() in -plane. G( ) H ( ) ( K 1)( 2) Pole-Zero Map n= > number of pole p 1 = 0; p 2 = -1; p 3 = -2 m=0----> number of zero Imaginary Axi Real Axi 12

13 Contruction of root loci Step-2: Determine the root loci on the real axi. The loci on the real axi are to the left of an ODD number of REAL pole and/or REAL zero of G()H() Red line on -plane are part of the real-axi where the root locu exit. Imaginary Axi Pole-Zero Map Real Axi 13

14 Contruction of root loci Step-3: Determine the aymptote of the root loci. Number of aymptote = n - m Interection point of aymptote with real axi: n p i m n m z i Angle of aymptote with real axi: (0 1 2) o 180 2k n m 1, k=0,1,2,,(n-m-1) 14

15 2 when when when 60 k k k k o 180, 60, 60

16 Contruction of root loci Step-3: Determine the aymptote of the root loci. 1 Pole-Zero Map 60, 60, Imaginary Axi Real Axi 16

17 implicit tep 17

18 Break Away Point

19 Contruction of root loci Step-4: Determine the breakaway point or break-in point. The breakaway or break-in point are the cloed-loop pole that atify: dk 0 d It hould be noted that not all the olution of dk/d=0 correpond to actual breakaway point. G( ) H ( ) K ( 1)( 2) The characteritic equation of the ytem i K 1 G( ) H ( ) 1 ( 1)( 2) 0 19

20 K ( 1)( 2) K ( 1)( 2) 1 The breakaway point can now be determined a dk d dk d dk d d d d d Set dk/d=0 in order to determine breakaway point. ( 1)( ) ,

21 Break Away Point

22 Contruction of root loci Step-5: Interection point with the imaginary axi. An alternative approach i to let =jω in the characteritic equation, equate both the real part and the imaginary part to zero, and then olve for ω and K. For preent ytem the characteritic equation i K 0 ( j) 3 3( j) 2 2 j K 0 ( K 2 3 ) 3 j(2 ) 0 26

23 Contruction of root loci Step-5: Determine the point where root loci cro the imaginary axi. 2 3 ( K 3 ) j(2 Equating both real and imaginary part of thi equation to zero 3 (2 ) ) 0 0 Which yield 2 ( K 3 ) 0 27

24 28

25 5 Root Locu Imaginary Axi Real Axi 29

26 Example#1 Conider following unity feedback ytem. Determine the value of K uch that the damping ratio of a pair of dominant complex-conjugate cloed-loop pole i 0.5. G( ) H ( ) ( K 1)( 2) 30

27 Example#1 The damping ratio of 0.5 correpond to co co 1 co 1 (0.5) 60 31

28 ? 32

29 Example#1 The value of K that yield uch pole i found from the magnitude condition K ( 1)( 2) j

30 34

31 Example#1 The third cloed loop pole at K= can be obtained a K 1 G( ) H ( ) 1 0 ( 1)( 2) ( 1)( 2) ( 1)( 2)

32 36

33 Home Work Conider following unity feedback ytem. Determine the value of K uch that the natural undamped frequency of dominant complex-conjugate cloed-loop pole i 1 rad/ec. G( ) H ( ) ( K 1)( 2) 37

34 Example#2 Sketch the root locu of following ytem and determine the location of dominant cloed loop pole to yield maximum overhoot in the tep repone le than 30%. 38

35 Step-1: Pole-Zero Map Example#2 1 Pole-Zero Map Imaginary Axi Real Axi 39

36 Example#2 Step-2: Root Loci on Real axi 1 Pole-Zero Map Imaginary Axi Real Axi 40

37 Example#2 Step-3: Aymptote 1 Pole-Zero Map Imaginary Axi Real Axi 41

38 Example#2 Step-4: breakaway point 1 Pole-Zero Map Imaginary Axi Real Axi 42

39 Example#2 8 Root Locu 6 4 Imaginary Axi Real Axi 43

40 Example#2 Mp<30% correpond to M p e % e

41 Example#2 8 Root Locu Imaginary Axi Real Axi 45

42 Example#2 8 Root Locu Imaginary Axi Sytem: y Gain: 28.9 Pole: i Damping: Overhoot (%): 30.5 Frequency (rad/ec): Real Axi 46

43 Root Locu of 1 t Order Sytem 1 t order ytem (without zero) are repreented by following tranfer function. G ( ) H ( ) K Root locu of uch ytem i a horizontal line tarting from -α and move toward - a K reache infinity. jω - -α σ 47

44 Home Work Draw the Root Locu of the following ytem. 1) G( ) H ( ) K 2 2) G( ) H ( ) K 1 3) G( ) H ( ) K 48

45 Root Locu of 1 t Order Sytem 1 t order ytem with zero are repreented by following tranfer function. G( ) H ( ) Root locu of uch ytem i a horizontal line tarting from -α and move toward -β a K reache infinity. jω K( ) -β -α σ 49

46 Home Work Draw the Root Locu of the following ytem. 2 ) ( ) ( K H G 1 5) ( ) ( ) ( K H G K H G 3) ( ) ( ) ( 1) 2) 3) 50

47 Root Locu of 2 nd Order Sytem Second order ytem (without zero) have two pole and the tranfer function i given G( ) H ( ) K ( )( 1 2 Root loci of uch ytem are vertical line. ) jω -α 2 -α 1 σ 51

48 Home Work Draw the Root Locu of the following ytem. 2) ( ) ( ) ( K H G 2 ) ( ) ( K H G 3) 1)( ( ) ( ) ( K H G 1) 2) 3) 10 3 ) ( ) ( 2 K H G 4) 52

49 Root Locu of 2 nd Order Sytem Second order ytem (with one zero) have two pole and the tranfer function i given K( ) G( ) H ( ) ( 1 )( 2 ) Root loci of uch ytem are either horizontal line or circular depending upon pole-zero configuration. jω jω jω -α 2 -β -α 1 σ -β -α 2 -α 1 σ -α 2 -α 1 -β σ 53

50 Home Work Draw the Root Locu of the following ytem. 2) ( 1) ( ) ( ) ( K H G 2 2) ( ) ( ) ( K H G 3) 1)( ( 5) ( ) ( ) ( K H G 1) 2) 3) 54

51 Example Sketch the root-locu plot of following ytem with complex-conjugate open loop pole. 55

52 Step-1: Pole-Zero Map Example Step-2: Determine the root loci on real axi Step-3: Aymptote 56

53 Example Step-4: Determine the angle of departure from the complex-conjugate open-loop pole. The preence of a pair of complex-conjugate open-loop pole require the determination of the angle of departure from thee pole. Knowledge of thi angle i important, ince the root locu near a complex pole yield information a to whether the locu originating from the complex pole migrate toward the real axi or extend toward the aymptote. 57

54 Example Step-4: Determine the angle of departure from the complex-conjugate open-loop pole. 58

55 Step-5: Break-in point Example 59

56 60

57 Root Locu of Higher Order Sytem Third order Sytem without zero G( ) H( ) ( K )( )( ) 61

58 Root Locu of Higher Order Sytem Sketch the Root Loci of following unity feedback ytem G( ) H ( ) ( K( 3) 1)( 2)( 4) 62

59 Let u begin by calculating the aymptote. The real-axi intercept i evaluated a; The angle of the line that interect at - 4/3, given by 63

60 The Figure how the complete root locu a well a the aymptote that were jut calculated. 64

61 Example: Sketch the root locu for the ytem with the characteritic equation of; a) Number of finite pole = n = 4. b) Number of finite zero = m = 1. c) Number of aymptote = n - m = 3. d) Number of branche or loci equal to the number of finite pole (n) = 4. e) The portion of the real-axi between, 0 and -2, and between, -4 and -, lie on the root locu for K > 0. Uing Eq. (v), the real-axi aymptote intercept i evaluated a; σ a = ( 1) n m = = 3 The angle of the aymptote that interect at - 3, given by Eq. (vi), are; θ a = (2k + 1)π n m = (2k + 1)π 4 1 For K = 0, θa = 60 o For K = 1, θa = 180 o For K = 2, θa = 300 o 65

62 The root-locu plot of the ytem i hown in the figure below. It i noted that there are three aymptote. Since n m = 3. The root loci mut begin at the pole; two loci (or branche) mut leave the double pole at = -4. Uing Eq. (vii), the breakaway point, σ, can be determine a; The olution of the above equation i σ =

63 Example: Sketch the root loci for the ytem. A root locu exit on the real axi between point = 1 and = 3.6. The interection of the aymptote and the real axi i determined a, σ a = n m = = 1.3 The angle of the aymptote that interect at 1.3, given by Eq. (vi), are; θ a = (2k + 1)π n m = (2k + 1)π 3 1 For K = 0, θa = 90 o For K = 1, θa = -90 o or 270 o Since the characteritic equation i We have (a) 67

64 The breakaway and break-in point are found from Eq. (a) a, From which we get, Point = 0 correpond to the actual breakaway point. But point are neither breakaway nor break-in point, becaue the correponding gain value K become complex quantitie. 68

65 To check the point where root-locu branche may cro the imaginary axi, ubtitute = jω into the characteritic equation, yielding. Notice that thi equation can be atified only if ω = 0, K = 0. Becaue of the preence of a double pole at the origin, the root locu i tangent to the jωaxi at k = 0. The root-locu branche do not cro the jωaxi. The root loci of thi ytem i hown in the Figure. 69

66 Home Work Conider following unity feedback ytem. Determine Root loci on real axi Angle of aymptote Centroid of aymptote 70

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