Dcs lec03 - z-analysis of discrete time control systems

Digital Control
Faculty of Engineering
Helwan University

1. رَدـْقـِن ،،،املاننا قدصنْْقِنرَد
LECTURE (3)
Z-Analysis of Discrete Time
Control Systems
ASCO. Prof. Amr E. Mohamed

2. Agenda
Mathematicalrepresentationofimpulsesampling.
Theconvolutionintegralmethodforobtainingthez-transform.
Examples
Properties
InverseZ-Transform
LongDivision
PartialFraction
SolutionofDifferenceEquation
Mappingbetweens-planetoz-plane
2

3. Impulse Sampling
FictitiousSampler.
Theoutputofthesamplerisatrainofimpulses.
Let’sdefinethetrainofimpulses
??????
??????�=
??????=0
∞
????????????????????????�−????????????
3

4. Impulse Sampling
Thesamplermaybeconsideredamodulator
??????
∗
�=??????�??????
??????�=
??????=0
∞
????????????????????????�−????????????
??????
∗
�=??????0??????�+??????????????????�−??????+⋯+????????????????????????(�−????????????)+⋯
4

5. Impulse Sampling
TheLaplacetransformof??????
∗
�
??????
∗
�=??????0??????�+????????????�
−????????????
+⋯+??????????????????�
−??????????????????
+⋯
??????
∗
�=
??????=0
∞
??????????????????�
−??????????????????
Ifwedefine??????=�
????????????
⟹�=
1
??????
ln??????
??????
∗
�
??????=
1
??????
ln??????
=????????????=
??????=0
∞
????????????????????????
−??????
TheLaplacetransformofsampledsignal??????
∗
�hasbeenshowntobethe
sameasz-transformofthesignal??????�if�
????????????
isdefinedasz.
5

6. Data Hold Circuits
Dataholdisaprocessofgeneratingacontinuous-timesignalℎ(�)from
adiscretetimesequence??????
∗
�.
Aholdcircuitapproximatelyreproducesthesignalappliedtothe
sampler.
ℎ????????????+�=�
??????�
??????
+�
??????−1�
??????−1
+⋯+�
1�+�
0
Notethatthesignalℎ????????????mustequal??????????????????,hence
ℎ????????????+�=�
??????�
??????
+�
??????−1�
??????−1
+⋯+�
1�+??????????????????
Thesimplestdata-holdisobtainedwhen�=0[Zero-OrderHold(ZOH)]
ℎ????????????+�=??????????????????
When�=1[First-OrderHold(FOH)]
ℎ????????????+�=�
1�+??????????????????
6

7. Zero-Order Hold (ZOH)
ℎ????????????+�=?????????????????? ���0≤�≤??????
7

8. Zero-Order Hold (ZOH)
Theoutputℎ(�)
ℎ
1�=??????0[1�−1(�−??????)]+????????????[1�−??????−1(�−2??????)]+⋯
ℎ
1�=
??????=0
∞
??????????????????[1�−????????????−1(�−(??????+1)??????)]
Since,ℒ1�−????????????=
??????
−??????????????????
??????
ThentheLaplacetransformoftheoutputℎ
1�is�
1�
�
1�=
??????=0
∞
??????????????????
�
−??????????????????
−�
−(??????+1)????????????
�
=
1−�
−????????????
�

??????=0
∞
??????????????????�
−??????????????????
8

9. Zero-Order Hold (ZOH)
Since,ℒℎ
1�=�
1�=ℒℎ
2�=�
2�
�
2�=
1−�
−????????????
�

??????=0
∞
??????????????????�
−??????????????????
=
1−�
−????????????
�
??????
∗
(�)=�
ℎ0�??????
∗
(�)
ThenthetransferfunctionoftheZOHis
�
ℎ0�=
1−�
−????????????
�
Thus,therealsamplerandzero-orderholdcanbereplacedbya
mathematicallyequivalentcontinuoustimesystemthatconsistsofan
impulsesamplerandatransferfunction
1−??????
−????????????
??????
.
9

10. First Order Hold (FOH)
Theequationofthefirstorderholdisℎ????????????+�=�
1�+??????????????????���0≤�≤??????
Byapplyingtheconditionℎ(??????−1)??????=??????(??????−1)??????
Theconstant�
1canbedeterminedasfollows:
ℎ(??????−1)??????=−�
1??????+??????????????????=??????(??????−1)??????
�
1=
??????????????????−??????(??????−1)??????
??????
10

11. First Order Hold (FOH)
Hence,
ℎ(??????−1)??????=??????????????????+
??????????????????−??????(??????−1)??????
??????
Supposethattheinput??????�isunit-stepfunction
ℎ�=1+
�
??????
1�−
�−??????
??????
1�−??????−1(�−??????)
11

12. First Order Hold (FOH)
TheLaplacetransformofthelastequation
��=
1
�
+
1
??????�
2
−
1
??????�
2
�
−????????????
−
1
�
�
−????????????
=
1−�
−????????????
�
+
1−�
−????????????
??????�
2
��=1−�
−????????????
??????�+1
??????�
2
TheLaplacetransformoftheinput??????
∗
�is
??????
∗
�=
??????=0
∞
1????????????�
−??????????????????
=
1
1−�
−????????????
12

13. First Order Hold (FOH)
Since,��=1−�
−????????????
????????????+1
????????????
2
=�
ℎ1�??????
∗
�
Hence,thetransferfunctionoftheFOHis
�
ℎ1�=
��
??????
∗
�
=1−�
−????????????2
??????�+1
??????�
2
??????
??????�??????=
�−??????
−????????????
??????
�
????????????+�
??????
13

14. Obtaining the z-Transform by the Convolution Integral Method
Calculating??????
∗
�fromtheoriginal
signal??????�
Bysubstituting??????for??????
????????????
toobtain??????(??????)fromthesampledsignal??????
∗
�
Forsimplepole
Formultiplepoleofordern
14

15. Example
Solution:
15

16. Example
Solution:
16

17. Reconstructing Original Signals from Sampled Signals
SamplingTheorem
Ifthesamplingfrequencyissufficientlyhighcomparedwiththehighest-
frequencycomponentinvolvedinthecontinuous-timesignal,theamplitude
characteristicsofthecontinuous-timesignalmaybepreservedinthe
envelopeofthesampledsignal.
Toreconstructtheoriginalsignalfromasampledsignal,thereisacertain
minimumfrequencythatthesamplingoperationmustsatisfy.
Weassumethat??????(�)doesnotcontainanyfrequencycomponentsabove??????
1
rad/sec.
17

18. Reconstructing Original Signals from Sampled Signals
SamplingTheorem
If??????
??????,defiedas2??????/??????isgreaterthan2??????
1,where??????
1isthehighest-
frequencycomponentpresentinthecontinuous-timesignal??????(�),thenthe
signal??????(�)canbereconstructedcompletelyfromthesampledsignal??????(�).
Thefrequencyspectrum:
18

19. Reconstructing Original Signals from Sampled Signals
19

20. Reconstructing Original Signals from Sampled Signals
IdealLow-passfilter
Theidealfilterattenuatesallcomplementarycomponentstozeroandwill
passonlytheprimarycomponent.
Ifthesamplingfrequencyislessthantwicethehighest-frequency
componentoftheoriginalcontinuous-timesignal,eventheidealfilter
cannotreconstructtheoriginalcontinuous-timesignal.
20

21. Reconstructing Original Signals from Sampled Signals
IdealLow-passfilterisNOTphysicallyrealizable
Fortheidealfilteranoutputisrequiredpriortotheapplicationoftheinput
tothefilter
21

22. Reconstructing Original Signals from Sampled Signals
FrequencyresponseoftheZero-OrderHold.
ThetransferfunctionoftheZOH
22

23. Reconstructing Original Signals from Sampled Signals
FrequencyresponsecharacteristicsoftheZero-OrderHold.
23

24. Reconstructing Original Signals from Sampled Signals
FrequencyresponsecharacteristicsoftheZero-OrderHold.
ThecomparisonoftheidealfilterandtheZOH.
ZOHisaLow-passfilter,althoughitsfunctionisnotquitegood.
TheaccuracyoftheZOHasanextrapolatordependsonthesampling
frequency.
24

25. Reconstructing Original Signals from Sampled Signals
Folding
Thephenomenonoftheoverlapinthefrequencyspectra.
Thefoldingfrequency(Nyquistfrequency):??????
??????
??????
??????=
1
2
??????
??????=
??????
??????
Inpractice,signalsincontrolsystemshavehigh-frequencycomponents,and
somefoldingeffectwillalmostalwaysexist.
25

26. Reconstructing Original Signals from Sampled Signals
Aliasing
Thephenomenonthatfrequencycomponentn??????
??????±??????
2showsupat
frequency??????
2whenthesignal??????(�)issampled.
Toavoidaliasing,wemusteitherchoosethesamplingfrequencyhigh
enoughoruseaprefilteraheadofthesamplertoreshapethefrequency
spectrumofthesignalbeforethesignalissampled.
26

27. The Pulse Transfer Function -Convolution Summation
27

28. The Pulse Transfer Function –Starred Laplace transform
28

29. The Pulse Transfer Function –Starred Laplace transform
29

30. The Pulse Transfer Function of Cascade Elements
30

31. The Pulse Transfer Function of Cascade Elements
31

32. The Pulse Transfer Function of The Closed Loop Systems
32

33. The Pulse Transfer Function of The Closed Loop Systems
33

34. Closed-Loop Pulse Transfer Function of a Digital Control System
34

35. Closed-Loop Pulse Transfer Function of a Digital Control System
35

36. Closed-Loop Pulse Transfer Function of a Digital PID Controller
36

37. Closed-Loop Pulse
Transfer Function
of a Digital PID
Controller
37

38. Closed-Loop Pulse Transfer Function of a Digital PID Controller
38

39. PID Controller
39

40. Obtaining Response Between Consecutive Sampling Instants
Thez-transformanalysiswillnotgiveinformationonthereponse
betweentwoconsecutivesamplinginstants.
Threemethodsforprovidingaresponsebetweenconsecutivesampling
instantsarecommonlyavailable:
1)Laplacetransformmethod
2)Modifiedz-Transformmethod
3)State-Spacemethod.
40

41. Laplace Transform Method
41

42. Realization of Digital Controllers and Digital Filters
Adigitalfilterisacomputationalalgorithmthatconvertsaninput
sequenceofnumbersintoanoutputsequenceinsuchawaythatthe
characteristicsofthesignalarechangedinsomeprescribedfashion.
Adigitalfilterprocessesadigitalsignalbypassingdesirablefrequency
componentsofthedigitalinputsignalandrejectingundesirableones.
Ingeneral,adigitalcontrollerisaformofdigitalfilter.
Ingeneral,“Realization”meansdeterminingthephysicallayoutforthe
appropriatecombinationofarithmeticandstorageoperation.
Realizationtechniquesare
Directrealization.
Standardrealization.
Seriesrealization.
Parallelrealization.
Laderrealization.
42

43. Direct realization.
Thegeneralformofthedigitalfilteris
Then
43

44. Standard Programming
Indirectprogramming,thenumeratorusesasetof�delayelements
andthedenominatorusesadifferentsetof�delayelements.Thusthe
totalnumberofdelayelementsusedindirectprogrammingis(�+�).
Thestandardprogrammingusesaminimumnumberofdelayelements
(�).
44

45. Standard Programming
45

46. Standard Programming
46

47. Note:
Inrealizingdigitalcontrollersordigitalfilters,itisimportanttohavea
goodlevelofaccuracy.Basically,threesourcesoferrorsaffectthe
accuracy:
1)Thequantizationerrorduetothequantizationoftheinputsignalintoa
finitenumberofdiscretelevels.Thequantizationnoisemaybeconsidered
whitenoise;thevarianceofthenoiseis??????
2
=??????
2
/12.
2)Theerrorduetotheaccumulationofround-offerrorsinthearithmetic
operationsinthedigitalsystem.
3)Theerrorduetoquantizationofthecoefficients�
??????and�
??????ofthepulse
transferfunction.Thiserrormaybecomelargeastheorderofthepulse
transferfunctionisincreased.
•Thatis,inahigher–orderdigitalfilterindirectstructure,smallerrorsinthe
coefficientscauselargeerrorsinthelocationsofthepolesandzerosofthe
filter.
47

48. Decomposition Techniques
Notethatthethirdtypeoferrorlistedmaybereducedby
mathematicallydecomposingahigher-orderpulsetransferfunctioninto
acombinationoflower-orderpulsetransferfunctions.
Fordecomposinghigher-orderpulsetransferfunctionsinordertoavoid
thecoefficientsensitivityproblem,thefollowingthreeapproachesare
commonlyused.
1)SeriesProgramming.
2)ParallelProgramming
3)LaderProgramming
48

49. Series Programming
The�(??????)maybedecomposedasfollows:
Thentheblockdiagramforthedigitalfilter�(??????)isaseriesconnection
ofpcomponentdigitalfiltersasshown.
49

50. Parallel Programming
The�(??????)isexpandedusingthepartial
fractionsasfollows:
Thentheblockdiagramforthedigital
filter�(??????)isaparallelconnectionof
pcomponentdigitalfiltersasshown.
50

51. LaderProgramming
The�(??????)isdecomposedintoacontinued-fractionsformasfollows:
TheG(z)maybewrittenasfollows:
51

52. LaderProgramming (Cont.)
Thentheblockdiagramforthedigitalfilter�(??????)isaLaderconnection
ofpcomponentdigitalfiltersasshown.
52

53. 53