MODERN CONTROL SYSTEMS

Transcription

1 MODERN CONTROL SYSTEMS Lecure 9, Sae Space Repreeaio Emam Fahy Deparme of Elecrical ad Corol Egieerig hp://

2 Trafer Fucio Limiaio TF = O/P I/P ZIC Moder corol heory i applicable o: MIMO yem. liear or oliear Syem. ime ivaria or ime varyig. Coveioal corol heory i applicable o: SISO yem. Liear. ime ivaria.

3 Sae Space Repreeaio 3

4 Eample Coider he mechaical yem how i figure. We aume ha he yem i liear. The eeral force u i he ipu o he yem, ad he diplaceme y of he ma i he oupu. The diplaceme y i meaured from he equilibrium poiio i he abece of he eeral force. Thi yem i a igle-ipu, igle-oupu yem. From he diagram, he yem equaio i m y + b y + ky = u Thi yem i of ecod order. Thi mea ha he yem ivolve wo iegraor. Le u defie ae variable ad a = y = y =

5 Eample = y = y m y + b y + ky = u The we obai Or = = b m y k m y + m = u = b m k m + m u The oupu equaio i y =

6 Eample u m m b m k y I a vecor-mari form, = = b m k m + m u y = Bu A C y

7 Eampleummary The yem equaio i m y + b y + ky = u Le = y = y = = b m y k m y + m u = = b m k m + m u y = The Or u m m b m k y

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9 Sae Space Modelig Sae pace equaio ca be implified a A Bu Sae Equaio y C Du Oupu Equaio Where, Sae Vecor A Syem Mari Bp Ipu Mari u Ipu Vecor y Oupu Vecor Cq Oupu Mari D Feed forward Mari

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11 Caoical Form Caoical form are he adard form of ae pace model. Each of hee caoical form ha pecific advaage which make i coveie for ue i paricular deig echique. There are everal caoical form of ae pace model Phae variable caoical form Corollable Caoical form Obervable Caoical form Diagoal Caoical form Jorda Caoical Form I i iereig o oe ha he dyamic properie of yem remai uchaged whichever he ype of repreeaio i ued.

12 Phae Variable Caoical form Obai he ae equaio i phae variable form for he followig differeial equaio, where u i ipu ad y i oupu. d3 y d d y d + 6 dy d + 8y = u The differeial equaio i hird order, hu here are hree ae variable: = y = y 3 = Ad heir derivaive are i.e ae equaio = = 3 3 = u y

13 Phae Variable Caoical form I vecor mari form = y = y 3 = y = = 3 3 = u y u

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15 ae-pace repreeaio Coider a yem defied by where u i he ipu ad y i he oupu. Thi equaio ca alo be wrie a We will pree ae-pace repreeaio of he yem defied by above equaio i corollable caoical form ad obervable caoical form. u b u b u b u b y a y a y a y o Y U = b o + b + + b + b + a + + a + a

16 Corollable Caoical Form Y U = b o + b + + b + b + a + + a + a u a a a a u b b b b b y o

17 Corollable Caoical Form Eample Y U = u 3 3 y

18 Obervable Caoical Form Y U = b o + b + + b + b + a + + a + a u b b b b a a a a u b y o

19 y u = Corollable form: = 3 + u y = 3 3 Obervable form: = u y = 9

20 Diagoal Caoical Form Y U = b o + b + + b + b + p + p + p = b o + c + p + c + p + + c + p u p p p.. u b c c c y o

21 Eample Y U = = = + + u y

22 Jorda Caoical Form Y U = b o + b + + b + b + p 3 + p + p = b o + c + p 3 + c + p + c 3 + p + c 4 + p + + c + p y c c c b u o

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24 Sae Space o T.F Now Le u cover a pace model o a rafer fucio model. Takig Laplace raform of equaio ad coiderig iiial codiio o zero. From equaio 3 A Bu y C Du X AX BU Y CX DU 3 4 I A X BU X I A BU 5

25 Trafer Mari Sae Space o T.F Subiuig equaio 5 io equaio 4 yield DU BU A I C Y D U B A I C Y D B A I C U Y

26 Eample 3 Cover he followig Sae Space Model o Trafer Fucio Model if K=3, B= ad M=; f M v M B M K v v y

27 Eample 3 Subiue he give value ad obai A, B, C ad D marice. 3 f v v v y

28 Eample 3 3 A C B D D B A I C U Y

29 Eample 3 3 A C B D 3 U Y

30 Eample 3 3 U Y 3 U Y 3 3 U Y

31 Eample U Y 3 3 U Y 3 U Y

32 Eample 3 Y U 3 Y U 3

33 Eample Obai he rafer fucio T from followig ae pace repreeaio. Awer

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35 Sae Corollabiliy A yem i compleely corollable if here ei a ucoraied corol u ha ca rafer ay iiial ae o o ay oher deired locaio i a fiie ime, o T. ucorollable corollable

36 Corollabiliy Mari C T Sae Corollabiliy C T B AB A B A B Syem i aid o be ae corollable if rak CT

37 Sae Corollabiliy Eample Coider he yem give below y 3 u

38 Sae Corollabiliy Eample Corollabiliy mari C T i obaied a Thu B CT B AB AB CT Sice rakct herefore yem i o compleely ae corollable.

39 Sae Obervabiliy A yem i compleely obervable if ad oly if here ei a fiie ime T uch ha he iiial ae ca be deermied from he obervaio hiory y give he corol u, T. uobervable obervable

40 Obervable Mari O T Sae Obervabiliy Obervabiliy Mari OT C CA CA CA The yem i aid o be compleely ae obervable if rak O T

41 Eample Coider he yem give below y 4 u O T i obaied a OT C CA Where C 4 CA 4

42 Sae Obervabiliy Eample Therefore O T i give a OT 4 Sice rako T herefore yem i o compleely ae obervable.

43 Eample Check he ae corollabiliy, ae obervabiliy of he followig yem A, B, C

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45 Forced ad Uforced Repoe Forced Repoe, wih u a forcig fucio Uforced Repoe repoe due o iiial codiio u b b a a a a a a a a

46 Soluio of Sae Equaio Coider he ae equaio give below A Takig Laplace raform of he equaio X X AX AX I A X X I A X I A

47 Soluio of Sae Equaio X I A Takig ivere Laplace A e e A Sae Traiio Mari

48 Eample Coider RLC Circui obai he ae raiio mari ɸ. V c V o i L u C i v L R L C i v L c L c 5 3., C ad L R u i v i v L c L c 3

49 Eample u i v i v L c L c 3 3 S S A SI ] [ 3 S S S S S S S S S S Sae raiio mari ca be obaied a Which i furher implified a

50 Eample 3 S S S S S S S S S S Takig he ivere Laplace raform of each eleme e e e e e e e e

51 Home Work Compue he ae raiio mari if A 3 Soluio [ SI A ]

52 Soluio of Sae Equaio Coider he ae equaio wih u a forcig fucio A Bu Takig Laplace raform of he equaio X X AX BU AX BU I A X BU X BU I A

53 Soluio of Sae Equaio BU X I A BU X I A I A Takig he ivere Laplace raform of above equaio. u d Naural Repoe Forced Repoe

54 Eample#6 Obai he ime repoe of he followig yem: Where u i ui ep fucio occurrig a =. coider =. 3 u Soluio Calculae he ae raiio mari ] [ A SI

55 Eample#6 Obai he ae raiio equaio of he yem u d

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