Control System I EE 411. Lecture 11 Dr. Mostafa Abdel-geliel

Transcription

1 Conrol Syem I EE 4 Sae Space Analyi Lecure Dr. Moafa Abdel-geliel

2 Coure Conen Sae Space SS modeling of linear yem SS Repreenaion from yem Block Diagram SS from Differenial equaion phae variable form Canonical form Parallel Cacade Sae raniion mari and i properie Eigen value and Eigen Vecor SS oluion

3 Sae Space Definiion Sep of conrol yem deign Modeling: Equaion of moion of he yem Analyi: e yem behavior Deign: deign a conroller o achieve he required pecificaion Implemenaion: Build he deigned conroller Validaion and uning: e he overall yem In SS :Modeling, analyi and deign in ime domain

4 SS-Definiion In he claical conrol heory, he yem model i repreened by a ranfer funcion The analyi and conrol ool i baed on claical mehod uch a roo locu and Bode plo I i rericed o ingle inpu/ingle oupu yem I depend only he informaion of inpu and oupu and i doe no ue any knowledge of he inerior rucure of he plan, I allow only limied conrol of he cloed-loop behavior uing feedback conrol i ued

5 Modern conrol heory olve many of he limiaion by uing a much richer decripion of he plan dynamic. The o-called ae-pace decripion provide he The o-called ae-pace decripion provide he dynamic a a e of coupled fir-order differenial equaion in a e of inernal variable known a ae variable, ogeher wih a e of algebraic equaion ha combine he ae variable ino phyical oupu variable.

6 SS-Definiion The Philoophy of SS baed on ranforming he equaion of moion of order nhighe derivaive order ino an n equaion of order Sae variable repreen orage elemen in he yem which lead o derivaive equaion beween i inpu and oupu; i could be a phyical or mahemaical variable # of ae#of orage elemenorder of he yem For eample if a yem i repreened by Thi yem of order 3 hen i ha 3 ae and 3 orage elemen

7 SS-Definiion The concep of he ae of a dynamic yem refer o a minimum e of variable, known a ae variable, ha fully decribe he yem and i repone o any given e of inpu The ae variable arean inernal decripion of he yem which compleely characerize he yem ae a any ime, and from which any oupu variable yi may be compued.

8 The Sae Equaion A andard form for he ae equaion i ued hroughou yem dynamic. In he andard form he mahemaical decripion of he yem i epreed a a e of n coupled fir-order ordinary differenial equaion, known a he ae equaion, in which he ime derivaive of each ae variable i epreed in erm of he ae variable,..., n and he yem inpu u,..., ur.

9 I i common o epre he ae equaion in a vecor form, in which he e of n ae variable i wrien a a ae vecor [,,..., n ] T, and he e of r inpu i wrien a an inpu vecor u [u, u,..., u r ] T. Each ae variable i a ime varying componen of he column vecor. In hi noe we reric aenion primarily o a decripion of yem ha are linear and ime-invarian LTI, ha i yem decribed by linear differenial equaion wih conan coefficien. ٩

10 where he coefficien a ij and b ij are conan ha decribe he yem. Thi e of n equaion define he derivaive of he ae variable o be a weighed um of he ae variable and he yem inpu. ١٠

11 where he ae vecor i a column vecor of lengh n, he inpu vecor u i a column vecor of lengh r, A i an n n quare mari of he conan coefficien a ij, and B i an n r mari of he coefficien b ij ha weigh he inpu. A yem oupu idefined o be any yem variable of inere. A decripion of a phyical yem in erm of a e of ae variable doe no necearily include all of he variable of direc engineering inere. An imporan propery of he linear ae equaion decripion i ha all yem variable may be repreened by a linear combinaion of he ae variable i and he yem inpu ui. ١١

12 An arbirary oupu variable in a yem of order n wih r inpu may be wrien: ١٢

13 where y i a column vecor of he oupu variable y i, C i an m nmari of he conan coefficien c ij ha weigh he ae variable, and D i an m r mari of he conan coefficien d ij ha weigh he yem inpu. For many phyical yem he mari D i he null mari, and he oupu equaion reduce o a imple weighed combinaion of he ae variable: ١٣

14 Eample Find he Sae equaion for he erie R-L-C elecric circui hown in Soluion: capacior volage v C and he inducor curren i L are ae variable ١٤

15 Prove Appling KVL on he circui di v R * i v L K c d The relaion of capacior volage and curren hen i c dv d c dvc & i d c c from equaion di & v c d & [ L y v c R * i R * v u ]

16 Eample Draw a direc form realizaion of a block diagram, and wrie he ae equaion in phae variable form, for a yem wih he differenial equaion Soluion y, y&, and 3 & y 3u, we define ae variable a hen he ae pace repreenaion i & y& ١٦ & & 3 y && y &&& y 3u& 3u 7&& y 9y& 3u y 3 7u 6u 6u

17 [ ] [] u y u & Then he model will be [ ], 7 3, D C B A where

18 ١٨ Elecro Mechanical Syem

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21 Sae Space Repreenaion The complee yem model for a linear ime-invarian yem coni of: i aeofnaeequaion,definedinermofhemariceaandb,and ii a e of oupu equaion ha relae any oupu variable of inere o he ae variableandinpu,andepreedinermofhecanddmarice. The ak of modeling he yem i o derive he elemen of he marice, and o wrie he yem model in he form: The marice A and B are properie of he yem and are deermined by he yem rucure and elemen. The oupu equaion marice C and D are deermined by he paricular choice of oupu variable.

22 Block Diagram Repreenaion of Linear Syem Decribed by Sae Equaion Sep : Draw n inegraor S block, and aign a ae variable o he oupu of each block. Sep : A he inpu o each block which repreen he derivaive of i ae variable draw a umming elemen. Sep 3: Ue he ae equaion o connec he ae variable and inpu o he umming elemen hrough caling operaor block. Sep 4: Epand he oupu equaion and um he ae variable and inpu hrough a e of caling operaor o form he componen of he oupu. ٢٢

23 Eample Draw a block diagram for he general econd-order, ingle-inpu ingle-oupu yem:

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25 The overall modeling procedure developed in hi chaper i baed on he following ep:. Deerminaion of he yem order n and elecion of a e of ae variable from he linear graph yem repreenaion.. Generaion of a e of ae equaion and he yem A and B marice uing a well defined mehodology. Thi ep i alo baed on he linear graph yem decripion. 3. Deerminaion of a uiable e of oupu equaion and derivaion of he appropriae C and D marice.

26 Conider he following RLC circui We can chooe ae variable o be vc, il, Alernaively, we may chooe ˆ vc, ˆ vl. Thi will yield wo differen e of ae pace equaion, bu boh of hem have he idenical inpu-oupu relaionhip, epreed by V R. U LC RC Can you derive hi TF? ٢٦

27 Linking ae pace repreenaion and ranfer funcion Given a ranfer funcion, here ei infiniely many inpuoupu equivalen ae pace model. We are inereed in pecial forma of ae pace repreenaion, known a canonical form. I i ueful o develop a graphical model ha relae he ae pace repreenaion o he correponding ranfer funcion. The graphical model can be conruced in he form of ignalflow graph or block diagram. ٢٧

28 We recall Maon gain formula when all feedback loop are ouching and alo ouch all forward pah, Conider a 4 h- order TF 3 4 a a a a b U Y G feedback loop gain um of forward pah gain Sum of N q q k k k k k L P P T ٢٨ We noice he imilariy beween hi TF and Maon gain formula above. To repreen he yem, we ue 4 ae variable Why? a a a a b a a a a U

29 Signal-flow graph model Thi 4 h -order yem can be repreened by Y b U a3 a a a G How do you verify hi ignal-flow graph by Maon gain formula? ٢٩

30 Block diagram model Again, hi 4 h -order TF can be repreened by he block diagram a hown a a a a b a a a a b U Y G ٣٠ can be repreened by he block diagram a hown

31 Wih eiher he ignal-flow graph or block diagram of he previou 4 h -order yem, we define ae variable a,,,, y & & & ٣١ we define ae variable a hen he ae pace repreenaion i,,,, b y & & & b y u a a a a & & & &

32 Wriing in mari form we have Du C y Bu A & ٣٢ [ ],, 3 D b a a a a C B A

33 When udying an acual conrol yem block diagram, we wih o elec he phyical variable a ae variable. For eample, he block diagram of an open loop DC moor i ٣٣ We draw he ignal-flow diagraph of each block eparaely and hen connec hem. We elec y, i and 3 /4r-/u o form he ae pace repreenaion.

34 Phyical ae variable model ٣٤ The correponding ae pace equaion i ] [ y r &

35 ٣٥ Elecro Mechanical Syem

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38 ٣٨ Conrol Flow

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42 Sae-Space Repreenaion in Canonical Form. - Conrollable Canonical Form Special Cae ٤٢

43 Aume hen

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45 General Cae ٤٥

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47 Conrollable Canonical Form General cae ٤٧

48 - Obervable Canonical Form u Prove d d n n y rearrange ٤٨ d d d d n n n d d d y... a y b u b u b n n n n d d d n n a y a n n d d Inegrae boh ide n ime - bn an n n y n n n d d d b u bu a y bu a y... b u a y n n n n n d d d y bu bu a y d bu a y d... b u a ʃ - b a ʃ Xn- - b a ʃ n n n u... b u bo n n y d y

49 General Form

50 3- Diagonal Canonical Form ٥٠

51 General Form ٥١

52 4- cacade Form ٥٢

53 ٥٣ The correponding ae pace equaion i ] [ y r &

54 Eample - Conider he yem given by Obain ae-pace repreenaion in he conrollable canonical form, obervable canonical form, and diagonal canonical form. Conrollable Canonical Form: ٥٤

55 Eample Obervable Canonical Form: Diagonal Canonical Form: ٥٥

56 Eample ٥٦

57 Eample ٥٧

58 Eigenvalue of an n X n Mari A. The eigenvalueare alo called he characeriic roo. Conider, for eample, he following mari A: The eigenvalueof A are he roo of he characeriic equaion, or,, and 3. ٥٨

59 Jordan canonical form If a yem ha a muliple pole, he ae pace repreenaion can be wrien in a block diagonal form, known a Jordan canonical form. For eample, Three pole are equal ٥٩

60 Sae-Space and Tranfer Funcion The SS form Can be ranformed ino ranfer funcion Tanking he Laplace ranform and neglec iniial condiion hen X AX BU and Y C X DU hen X AX BU ٦٠

61 condiion hen by neglecing inial B A I B A U X U X X B A I U X D B A I G D B A I C / C Y ub in U Y U U

62 Sae-Traniion Mari We can wrie he oluion of he homogeneou ae equaion Laplace ranform The invere Laplace ranform Noe ha ٦٢

63 Hence, he invere Laplace ranform of Sae-Traniion Mari Where Noe ha ٦٣

64 If he eigenvalue of he mari A are diinc, han will conain he n eponenial ٦٤

65 ٦٥ Properie of Sae-Traniion Marice.

66 ٦٦ Obain he ae-raniion mari of he following yem:

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69 he NON- homogeneou ae equaion and premuliplying boh ide of hi equaion by Inegraing he preceding equaion beween and give or ٦٩

70 ٧٠ uni-ep funcion

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72 Prove Tranfer funcion of he given Soluion G C I A B D ٧٢

73 Relaion of Differen SS Repreenaion of he Same Syem For a given yem G ha wo differen repreenaion : M Rep.: D C B A u y u : M Rep.: D C B A u z y u z Le ZT Where T i he ranformaion mari beween and z For eample y y y y y y y y ake & && && & & 3 z, z, z,

74 y y y y y y y y ake & && && & & 3 3 z, z, z, z z hen 3 3 z z z z T z z z 3 3 T

75 Sub. By ztin rep. muiply by T - D C T y B T A T T T T z T B A T T z u u u & & & D C T y u Compare wih M;rep. M : Rep.: D C B A u y u hen D D T C C B T B T A T A D D T C C TB B T TA A

76 Sae-Space DiagonalizaionFuncion ٧٦ Eignvalue and eignvecor Definiion: for a given mari A, if herei a real comple λ and a correponding vecor v, uch ha A v λv Thenλi called eignvalue and vi he eignvecor i.e. A λi v And ince v Then i.e A λi de A λi

77 Eigenvalue of an n X n Mari A. The eigenvalueare alo called he characeriic roo. Conider, for eample, he following mari A: The eigenvalueof A are he roo of he characeriic equaion, or,, and 3. ٧٧

78 Eample 8 A 8 A - I he oluion of hen he eign valuei λ λ λ λ A I 8 λ λ λ λ λ λ λ and hen

79 8 4.. A - I -4 v v e i v a λ λ A - I v v e i v a λ λ Eignvecor are obained a 4 4 v le v v v le v v ] [ n V v v v L Eign vecor mari

80 For all eign value and vecor Av λ v ; i,, K, n i i i Thee equaion can be wrien in mari form AV VΛ where hu V v λ Λ M [ v v L λ M ] n L L O L A VΛV Λ V AV M λn diag { λ, i,, L, n} i

81 hu...! Λ V Ve e A A I e A A φ,,...,,...! n i e diag e e e I e i Λ Λ Λ Λ λ λ λ φ O e n λ O Then for a given yem ha a yem mari A and a ae vecor X The diagonal yem mari Adand ae Xd ; D D T C C B V B AV V A mari vecor eign V T T T AT T A d d d d d d Λ

82 Eample : find he ranformaion ino diagonal form and he ae raniion mari of eample 4 4 Λ Λ Λ V Ve e e e e A V Ve e A A e e e e e e Dicu how o obain he ranformaion mari beween wo repreenaion

83 ٨٣ Diagonal Canonical Form

84 Alernaive Form of he Condiion for Complee Sae Conrollabiliy. If he eigenvecor ofa are diinc, hen i i poible o find a ranformaion mari Puch ha ٨٤