Linear Control System EE 711. Design. Lecture 8 Dr. Mostafa Abdel-geliel

Transcription

1 Linear Conrol Sysem EE 7 MIMO Sae Space Analysis and Design Lecure 8 Dr. Mosafa Abdel-geliel

2 Course Conens Review Sae Space SS modeling and analysis Sae feed back design Oupu feedback design Observer design Sae feedback and observer Tracking problem Minimum realizaion Applicaion

3 Sae Space Definiion Seps of conrol sysem design Modeling: Equaion of moion of he sysem Analysis: es sysem behavior Design: design a conroller o achieve he required specificaion Implemenaion: Build he designed conroller Validaion and uning: es he overall sysem In SS :Modeling, analysis and design in ime domain

4 SS-Definiion In he classical conrol heory, he sysem model is represened by a ransfer funcion The analysis and conrol ool is based on classical mehods such as roo locus and Bode plo I is resriced o single inpu/single oupu sysem I depends only he informaion of inpu and oupu and i does no use any knowledge of he inerior srucure of he plan, I allows only limied conrol of he closed-loop behavior using feedback conrol is used

5 Modern conrol heory solves many of he limiaions by using a much richer descripion of he plan dynamics. The so-called sae-space descripion provide he The so-called sae-space descripion provide he dynamics as a se of coupled firs-order differenial equaions in a se of inernal variables known as sae variables, ogeher wih a se of algebraic equaions ha combine he sae variables ino physical oupu variables.

6 SS-Definiion The Philosophy of SS based on ransforming he equaion of moions of order nhighes derivaive order ino an n equaion of s order Sae variable represens sorage elemen in he sysem which leads o derivaive equaion beween is inpu and oupu; i could be a physical or mahemaical variables # of sae#of sorage elemensorder of he sysem For eample if a sysem is represened by This sysem of order 3 hen i has 3 sae and 3 sorage elemens

7 SS-Definiion The concep of he sae of a dynamic sysem refers o a minimum se of variables, known as sae variables, ha fully describe he sysem and is response o any given se of inpus The sae variables arean inernal descripion of he sysem which compleely characerize he sysem sae a any ime, and from which any oupu variables yi may be compued.

8 The Sae Equaions A sandard form for he sae equaions is used hroughou sysem dynamics. In he sandard form he mahemaical descripion of he sysem is epressed as a se of n coupled firs-order ordinary differenial equaions, known as he sae equaions, in which he ime derivaive of each sae variable is epressed in erms of he sae variables,..., n and he sysem inpus u,..., ur.

9 I is common o epress he sae equaions in a vecor form, in which he se of n sae variables is wrien as a sae vecor [,,..., n ] T, and he se of r inpus is wrien as an inpu vecor u [u, u,..., u r ] T. Each sae variable is a ime varying componen of he column vecor. In his noe we resric aenion primarily o a descripion of sysems ha are linear and ime-invarian LTI, ha is sysems described by linear differenial equaions wih consan coefficiens. ٩

10 where he coefficiens a ij and b ij are consans ha describe he sysem. This se of n equaions defines he derivaives of he sae variables o be a weighed sum of he sae variables and he sysem inpus. ١٠

11 where he sae vecor is a column vecor of lengh n, he inpu vecor u is a column vecor of lengh r, A is an n n square mari of he consan coefficiens a ij, and B is an n r mari of he coefficiens b ij ha weigh he inpus. A sysem oupu isdefined o be any sysem variable of ineres. A descripion of a physical sysem in erms of a se of sae variables does no necessarily include all of he variables of direc engineering ineres. An imporan propery of he linear sae equaion descripion is ha all sysem variables may be represened by a linear combinaion of he sae variables i and he sysem inpus ui. ١١

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

13 where y is a column vecor of he oupu variables y i, C is an m nmari of he consan coefficiens c ij ha weigh he sae variables, and D is an m r mari of he consan coefficiens d ij ha weigh he sysem inpus. For many physical sysems he mari D is he null mari, and he oupu equaion reduces o a simple weighed combinaion of he sae variables: ١٣

14 Eample Find he Sae equaions for he series R-L-C elecric circui shown in Soluion: capacior volage v C and he inducor curren i L are sae variables ١٤

15 Prove Appling KVL on he circui di v R * i v L K s 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 s u ]

16 Eample Draw a direc form realizaion of a block diagram, and wrie he sae equaions in phase variable form, for a sysem wih he differenial equaion Soluion y, y&, and 3 & y 3u, we define sae variables as hen he sae space represenaion is & 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 Sysem

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21 Sae Space Represenaion The complee sysem model for a linear ime-invarian sysem consiss of: i aseofnsaeequaions,definedinermsofhemaricesaandb,and ii a se of oupu equaions ha relae any oupu variables of ineres o he sae variablesandinpus,andepressedinermsofhecanddmarices. The ask of modeling he sysem is o derive he elemens of he marices, and o wrie he sysem model in he form: The marices A and B are properies of he sysem and are deermined by he sysem srucure and elemens. The oupu equaion marices C and D are deermined by he paricular choice of oupu variables.

22 Block Diagram Represenaion of Linear Sysems Described by Sae Equaions Sep : Draw n inegraor S blocks, and assign a sae variable o he oupu of each block. Sep : A he inpu o each block which represens he derivaive of is sae variable draw a summing elemen. Sep 3: Use he sae equaions o connec he sae variables and inpus o he summing elemens hrough scaling operaor blocks. Sep 4: Epand he oupu equaions and sum he sae variables and inpus hrough a se of scaling operaors o form he componens of he oupu. ٢٢

23 Eample Draw a block diagram for he general second-order, single-inpu single-oupu sysem:

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25 The overall modeling procedure developed in his chaper is based on he following seps:. Deerminaion of he sysem order n and selecion of a se of sae variables from he linear graph sysem represenaion.. Generaion of a se of sae equaions and he sysem A and B marices using a well defined mehodology. This sep is also based on he linear graph sysem descripion. 3. Deerminaion of a suiable se of oupu equaions and derivaion of he appropriae C and D marices.

26 Consider he following RLC circui We can choose sae variables o be vc, il, Alernaively, we may choose ˆ vc, ˆ vl. This will yield wo differen ses of sae space equaions, bu boh of hem have he idenical inpu-oupu relaionship, epressed by V s R. U s LCs RCs Can you derive his TF? ٢٦

27 Linking sae space represenaion and ransfer funcion Given a ransfer funcion, here eis infiniely many inpuoupu equivalen sae space models. We are ineresed in special formas of sae space represenaion, known as canonical forms. I is useful o develop a graphical model ha relaes he sae space represenaion o he corresponding ransfer funcion. The graphical model can be consruced in he form of signalflow graph or block diagram. ٢٧

28 We recall Mason s gain formula when all feedback loops are ouching and also ouch all forward pahs, Consider a 4 h- order TF 3 4 a s a s a s a s b s U s Y s G feedback loop gain sum of forward pah gain Sum of N q q k k k k k L P P T ٢٨ We noice he similariy beween his TF and Mason s gain formula above. To represen he sysem, we use 4 sae variables Why? s a s a s a s a s b a s a s a s a s s U

29 Signal-flow graph model This 4 h -order sysem can be represened by Y s b s U s a3s as as as G s How do you verify his signal-flow graph by Mason s gain formula? ٢٩

30 Block diagram model Again, his 4 h -order TF can be represened by he block diagram as shown s a s a s a s a s b a s a s a s a s b s U s Y s G ٣٠ can be represened by he block diagram as shown

31 Wih eiher he signal-flow graph or block diagram of he previous 4 h -order sysem, we define sae variables as,,,, y & & & ٣١ we define sae variables as hen he sae space represenaion is,,,, 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 sudying an acual conrol sysem block diagram, we wish o selec he physical variables as sae variables. For eample, he block diagram of an open loop DC moor is 5 5 s 5s s 6 s s 3s ٣٣ We draw he signal-flow diagraph of each block separaely and hen connec hem. We selec y, i and 3 /4r-/u o form he sae space represenaion.

34 Physical sae variable model ٣٤ The corresponding sae space equaion is ] [ y r &

35 ٣٥ Elecro Mechanical Sysem

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38 ٣٨ Conrol Flow

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42 Sae-Space Represenaions in Canonical Forms. - Conrollable Canonical Form Special Case ٤٢

43 Assume hen

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45 General Case ٤٥

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47 Conrollable Canonical Form General case ٤٧

48 - Observable 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 side n imes - 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- cascade Form 5 5 s 5s s 6 s s 3s ٥٢

53 ٥٣ The corresponding sae space equaion is ] [ y r &

54 Eamples - Consider he sysem given by Obain sae-space represenaions in he conrollable canonical form, observable canonical form, and diagonal canonical form. Conrollable Canonical Form: ٥٤

55 Eamples Observable Canonical Form: Diagonal Canonical Form: ٥٥

56 Eamples ٥٦

57 Eamples ٥٧

58 Eigenvalues of an n X n Mari A. The eigenvaluesare also called he characerisic roos. Consider, for eample, he following mari A: The eigenvaluesof A are he roos of he characerisic equaion, or,, and 3. ٥٨

59 Jordan canonical form If a sysem has a muliple poles, he sae space represenaion can be wrien in a block diagonal form, known as Jordan canonical form. For eample, Three poles are equal ٥٩

60 Sae-Space and Transfer Funcion The SS form Can be ransformed ino ransfer funcion Tanking he Laplace ransform and neglec iniial condiion hen sx s AX s BU s and Ys C Xs DUs hen sx s AX s BU s ٦٠

61 condiion hen by neglecing inial s B s A si s B s A s s U X U X X s B A si s U X D B A si s G s s s D s B A si C / C Ys sub in U Y U U

62 Sae-Transiion Mari We can wrie he soluion of he homogeneous sae equaion Laplace ransform The inverse Laplace ransform Noe ha ٦٢

63 Hence, he inverse Laplace ransform of Sae-Transiion Mari Where Noe ha ٦٣

64 If he eigenvalues of he mari A are disinc, han will conain he n eponenials ٦٤

65 ٦٥ Properies of Sae-Transiion Marices.

66 ٦٦ Obain he sae-ransiion mari of he following sysem:

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68 ٦٨

69 he NON- homogeneous sae equaion and premuliplying boh sides of his equaion by Inegraing he preceding equaion beween and gives or ٦٩

70 ٧٠ uni-sep funcion

71 ٧١

72 Prove Transfer funcion of he given ss Soluion G s C si A B D ٧٢

73 Relaion of Differen SS Represenaions of he Same Sysem For a given sysem Gs has wo differen ss represenaions : M Rep.: D C B A u y u & : M Rep.: D C B A u z y u z z & Le ZT Where T is he ransformaion 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 Le i and i are he sae Le i and vcare he sae [ ].. i i y u i i i i [ ].. i i y u v i v i c c be he ransfromaion mari T will i i v i c

76 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

77 Sae-Space DiagonalizaionFuncion ٧٧ Eignvalues and eignvecors Definiion: for a given mari A, if hereis a real comple λ and a corresponding vecor v, such ha A v λv Thenλis called eignvalue and vis he eignvecor i.e. A λi v And since v Then i.e A λi de A λi

78 Eigenvalues of an n X n Mari A. The eigenvaluesare also called he characerisic roos. Consider, for eample, he following mari A: The eigenvaluesof A are he roos of he characerisic equaion, or,, and 3. ٧٨

79 Eample 8 A 8 A - I he soluion of hen he eign valueis λ λ λ λ A I 8 λ λ λ λ λ λ λ and hen

80 8 4.. A - I -4 v v e i v a λ λ A - I v v e i v a λ λ Eignvecors are obained as 4 4 v le v v v le v v ] [ n V v v v L Eign vecor mari

81 For all eign values and vecors Av λ v ; i,, K, n i i i These equaions can be wrien in mari form AV VΛ where hus 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

82 hus...! 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 sysem has a sysem mari A and a sae vecor X The diagonal sysem mari Adand sae 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

83 Eample : find he ransformaion ino diagonal form and he sae ransiion 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 Discus how o obain he ransformaion mari beween wo represenaion

84 ٨٤ Diagonal Canonical Form

85 Alernaive Form of he Condiion for Complee Sae Conrollabiliy. If he eigenvecors ofa are disinc, hen i is possible o find a ransformaion mari Psuch ha ٨٥

86 Conrollabiliy and Observabiliy Deermine and conrol he sysem sae from he observaion of he oupu over a finie ime inerval. The conceps of conrollabiliy and observabiliywere inroduced by Kalman. They play an imporan role in he design of conrol sysems in sae space. In fac, he condiions of conrollabiliy and observabiliymay govern he eisence of a complee soluion o he conrol sysem design problem. ٨٦

87 The vecors are linearly dependen since The vecors are linearly independen since implies ha ٨٧

88 ٨٨ Noe ha: if an nnmari is nonsingular ha is, he mari is of rank n or he deerminan is nonzero hen n column or row vecors are linearly independen. If he nnmari is singular ha is, he rank of he mari is less han n or he deerminan is zero, hen n column or row vecors are linearly dependen To demonsrae his, noice ha

89 CONTROLLABILITY A sysem is said o be conrollable, if every sae variable of he process can be conrolled o reach a cerain objecive in a finie ime by some unconsrained conrol u Or A sysem is said o be conrollable a ime if here eis a piecewise unconsrained coninuous inpu u conrol vecor ha will ransfer he sysem from any iniial sae o any oher sae final sae in a finie inerval of ime; f-. Complee Sae Conrollabiliy of Coninuous-Time Sysems ٨٩

90 The sysem described by is said o be sae conrollable a if i is possible o consruc an unconsrained conrol signal ha will ransfer an iniial sae o any final sae in a finie ime inerval If every sae is conrollable, hen he sysem is said o be compleely sae conrollable. Applying he definiion of complee sae conrollabiliy jus given, ٩٠

91 The sysem is compleely sae conrollable if and only if he vecors are linearly independen is of rank n

92 ٩٢ Anoher prove...! A A I e A φ τ τ τ τ φ φ τ τ Bu e e e Bu e e Bu A A A A d Bu e A A I...[! τ τ τ A d Bu e τ τ τ The effec of inpu u implies! ] [ ]! [ τ τ τ τ τ τ τ τ u I I I B A AB B Bu A A I Bu e A O L K Then he necessary condiion o conrol A B B A AB B n L

93 Observabiliy Definiion: For a dynamic sysem described by sae variable The sae is said o be observable if for a given inpu u wihin a finie ime > f o and he oupu y and by he knowing of sysem parameers A,B,C and D he iniial sae o is deermined

94 ]...[! A d Bu e A A I C C y τ τ τ...! A A I C d Bu e C y A τ τ τ CA C...]! [ CA CA I I I d Bu e C y n A M τ τ τ A n d Bu e C y I I I CA CA C...]! [ τ τ τ M I implies ha mus be nonsingular n CA CA C M

95 If he sysem is compleely observable, hen, given he oupu y over a ime inerval is uniquely deermined from Equaion I can be shown ha his requires he rank of he nm*n mari o be n. The sysem is compleely observable if and only if he n*nm mari is of rank n or has n linearly independen column vecors. This mari is called he observabiliy mari. ٩٥

96 ٩٦ Consider he sysem described by

97 ٩٧ Show ha he following sysem is no compleely observable:

98 Diagonal represenaion of SS model and is relaion o Observabiliyand conrolabiliyand u b b b M O & λ λ λ [ ] c c c y b n n n L λ Discuss he relaion beween Observabiliy and conrolabiliy and he coefficien of B and C mari

99 Conrol Sysem Design in Sae Space This Lecure discusses sae-space design based on he poleplacemen mehod The pole-placemen mehod is somewha similar o he roolocus mehod in ha we place closed-loop poles a desired locaions. The basic difference is ha in he roo-locus design we place only he dominan closed-loop poles a he desired locaions, while in he pole-placemen design we place all closed-loop poles a desired locaions. ٩٩

100 POLE PLACEMENT we shall presen a design mehod commonly called he pole-placemen or pole-assignmen echnique. We assume ha all sae variables are measurable and are available for feedback. I will be shown ha if he sysem considered is compleely sae conrollable, hen poles of he closed-loop sysem may be placed a any desired locaions by means of sae feedback hrough an appropriae sae feedback gain mari. ١٠٠

101 Design by Pole Placemen Consider a conrol sysem where ١٠١

102 We shall choose he conrol signal o be This means ha he conrol signal u is deermined by an insananeous sae. Such a scheme is called sae feedback. The n mari K is called he sae feedback gain mari.weassume ha all sae variables are available for feedback. In he following analysis we assume ha u is unconsrained. ١٠٢

103 The soluion of his equaion is given by where is he iniial sae caused by eernal disurbances. The sabiliy and ransien response characerisics are deermined by he eigenvalues of mari A-BK. If heeignvaluesofmari A f A BK are negaive hen he sysem is sabe sokisselecedomakeheneweignvaluesasrequired ١٠٣

104 r D C B A u y u & In general Afer sae feed back ur-k Open loop sysem BK A A f D C B A f f r y r & DK C C f τ τ τ d B e e A A f f r