Chapter 3 : Transfer Functions Block Diagrams Signal Flow Graphs

Transcription

1 Chapter 3 : Tranfer Function Block Diagram Signal Flow Graph 3.. Tranfer Function 3.. Block Diagram of Control Sytem 3.3. Signal Flow Graph 3.4. Maon Gain Formula 3.5. Example 3.6. Block Diagram to Signal Flow Graph 3.7. State Diagram By Electrical Engineering Department College of Engineering King Sau Univerity EE35: Control Sytem Chapter 3 page:

2 Tranfer Function Definition The tranfer function G of an LTI ytem i efine a the Laplace tranform of the impule repone with all initial conition et to zero. U G Conier that U = L[ut], =L[yt] an G=L[gt] The tranfer function G i relate to the Laplace tranform of the input an the output through the following relation: U=L[t]= G=. Therefore, the tranfer function G can alo be coniere a the Laplace tranform of the output when the input i the unit impule function t. Becaue thi fact, the tranfer function i alo calle impule repone. G If ut = t = impule Function U t t 0 t 0 t 0 EE35: Control Sytem Chapter 3 page:

3 Tranfer Function Definition cont. The tranfer function G can alo be erive from the ytem ifferential equation a follow: n n y t a n n- n- y t... a y t a 0 y t b m m m u t b m m- n- u t... b Auming zero initial conition an taking the Laplace tranform of both ie, we get: n n... m m a a b b b U n... 0 m m 0 0 u t G U bm n b a m m n n m... b... a Remember that : L[f t]=sf-f0 L[f n t]= n F n- f0 n- f 0 -. f n EE35: Control Sytem Chapter 3 page: 3

4 Tranfer Function Propertie. Tranfer function are efine only for a linear Time-Invariant LTI Sytem. They are not efine for non-linear ytem.. The tranfer function between the input an the output i the Laplace tranform of the impule repone. Alo, it i the ratio of the Laplace tranform of the output to the Laplace tranform of the input. 3. All initial conition of the ytem are zero. 4. The tranfer function i inepenent of the input 5. The tranfer function i a function only of the complex variable an no other variable If the orer of the tranfer function numerator i equal to that of the enominator, the tranfer function i calle proper. If the orer of the numerator i le then that of the enominator, the tranfer function i calle trictly proper tranfer function. If the orer of the numerator i greater then that of the enominator, the tranfer function i calle improper. EE35: Control Sytem Chapter 3 page: 4

5 Tranfer Function The characteritic equation of a linear ytem i efine a the equation obtaine by etting the enominator polynomial of the tranfer function to zero: n n an... a a0 0 Example Conier the following ifferential equation: 3 3 yt yt 5 yt 8yt 5 ut u t 7u t Taking the Laplace tranform auming zero initial conition [ 3 5 8] [5 7] 5 7 G EE35: Control Sytem Chapter 3 page: 5

6 Tranfer Function Thi tranfer function i trictly proper an it characteritic equation i : The zero are the root of the numerator The pole are the root of the enominator Example G 5 Proper tranfer function: 5 G Improper tranfer function: 3 3 G Strictly proper tranfer function: EE35: Control Sytem Chapter 3 page: 6

7 Block Diagram of Control Sytem Example rt et=rt-yt rt et=rt.yt R - yt R - yt R - B U G R? H B= H. U=R-B =GU=GRS-GR [GH]=GR G R G H EE35: Control Sytem Chapter 3 page: 7

8 Block Diagram of Control Sytem Chapter 3 page: 8 EE35: Control Sytem Example of ytem with two input R an D 0 H H G G G R D - B G H U H G R D.Fin /D when R=0.Fin /R when D=0 3.Deuce the total repone of the control ytem when R an D 0 0 H H G G D R R H H G G G D H H G G

9 Signal Flow Graph Baic Element of a Signal Flow Graph Definition : A Signal Flow graph i a implifie verion of block iagram. It repreent a graphical input-output relationhip between the variable of a et of linear algebraic equation Noe: A Junction point to repreent the variable Branch: A line egment to connect two noe The branch i aociate with a gain an a irection Example: y = a y y a y EE35: Control Sytem Chapter 3 page: 9

10 Signal Flow Graph Baic Element of a Signal Flow Graph Input Noe ource : A noe with only outgoing branche. Output Noe : A noe with only incoming branche. Path: Any collection of continuou ucceion of branche travere in the ame irection. Forware Path: A path that tart at the input noe an en at the output noe without paing any noe more than once. Loop: A path that tart an en at the ame noe without paing any noe more than once. Loop Gain : The prouct of the branch gain encountere in travering a path. EE35: Control Sytem Chapter 3 page: 0

11 Signal Flow Graph Gain Formula Maon Gain Formula N : Total number of forware path between y in an y out. M k : gain of the k th forwar path between y in an y out. = - um of all ifferent loop gain um of the gain prouct of all combination of poible non touching loop um of the gain prouct of all combination of 3 poible non touching loop i calle the eterminant of the graph k : for the part of the SFG with the loop touching the k th path remove that i non touching with the k th forwar path EE35: Control Sytem Chapter 3 page:

12 Signal Flow Graph Example R E G - H R E G -H From R to there i only One forwar path M =G There are no non touching loop. Alo, the forwar path i touch with the only loop = an = GH T= /R=M / = G/GH EE35: Control Sytem Chapter 3 page:

13 Signal Flow Graph Example cont. /R -/R -R 3 -/R R 3 /R R 4 v i v i v 3 v 3 The i one forwar path : M = R 3 R 4 /R R There are 3 feeback loop with a 3 gain : -R 3 /R, -R 3 /R an R 4 /R There are non-touching loop with a gain R 3 R 4 /R R All loop touch the forwar path = T=v 3 /v =M / EE35: Control Sytem Chapter 3 page: 3

14 Signal Flow Graph Example cont. G 6 R G G G 3 G 4 G 5 There are forwar path M = G G G 3 G 4 G 5 M = G 6 G 4 G 5 -H -H All loop touch the firt forwar path = The firt loop oen t touch the econ path = G H There are two non touching loop = G H G 4 H G G 4 H H T= M M / EE35: Control Sytem Chapter 3 page: 4

15 Signal Flow Graph Example cont. Conier a reaonable complex ytem that woul be ifficult to reuce by clock iagram technique G 7 G 8 R G G G 3 G 4 G 5 G 6 -H -H 4 -H There are 3 forwar path: -H 3 M = G G G 3 G 4 G 5 G 6 ; M = G G G 7 G 6 ; M 3 = G G G 3 G 4 G 8 The gain of the feeback loop are: L = -G G 3 G 4 G 5 H ; L = -G 5 G 6 H ; L 3 = -G 8 H ; L 4 = -G 7 H G ; L 5 = -G 4 H 4 L 6 = -G G G 3 G 4 G 5 G 6 H 3 ; L 7 = -G G G 7 G 6 H 3 ; L 8 = -G G G 3 G 4 G 8 H 3 EE35: Control Sytem Chapter 3 page: 5

16 Signal Flow Graph Example cont. The cofactor are: = = an = -L 5 =G 4 G 4 L 5 oe not touch loop L 4 an L 7 ; Loop L 3 oe not touch loop L 4 an all other loop touch The eterminant i: = L L L 3 L 4 L 5 L 6 L 7 L 8 L 5 L 7 L 5 L 4 L 3 L 4 T= /R= M M M 3 3 / EE35: Control Sytem Chapter 3 page: 6

17 Block Diagram to Signal Flow Graph The SFG can be contructe from the block iagram a how in the following example: H G 3 R G G - -H -H R E 3 G G -H To get the tranfer function, we ue the Maon Gain Formula. EE35: Control Sytem Chapter 3 page: 7

18 State Diagram Conier the following equation: x t x t t x t x x t0 0 X x t 0 X X t 0 S - X X EE35: Control Sytem Chapter 3 page: 8

19 State Diagram Chapter 3 page: 9 EE35: Control Sytem Conier the following ifferential equation an aume zero initial conition : 3 t r t y t y t y Taking the Laplace tranform, we get : 3 R 3 R S - R -3 S - - To get the tranfer function, we ue the Maon Gain Formula. 3 M