State-Space Model. In general, the dynamic equations of a lumped-parameter continuous system may be represented by

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2 Sae-Space Model I geeral, he dyaic equaio of a luped-paraeer coiuou ye ay be repreeed by x & f x, u, y g x, u, ae equaio oupu equaio where f ad g are oliear vecor-valued fucio Uig a liearized echique, he dyaic equaio for liear ie ivaria ye ca be repreeed by x& y Ax Cx Bu Du p where A R, B R ad C R are ye ae arix, ipu arix ad oupu arix, repecively

3 For igle-ipu igle-oupu ye, p Thi for i call he ae pace for of he dyaic ye I a brief for, he ye dyaic behaviour i decribed by A, B,C,D or The block diagra i how a u D x C y u B x A x p p & D C B A u x y A B C D

4 Caoical For of he Sae-pace Decripio Corollable caoical for [ ] x y u x p p p p x γ γ γ γ &

5 Corollable Caoical For γ - γ u x x x γ y -p - -p -p

6 Obervable Caoical For [ ] x y u x p p p p x β β β β & x - y u - β x -p - x β β -p -p

7 Sae Traforaio The choice of ae variable for a paricular ye i o uique x & Ax Bu y Cx Du z & Az $ Bu $ y Cz $ Du $ z Px oigular liear raforaio x P z z& PAP z PBu y CP z Du

8 I geeral, P ca be ie depede, a P, ad Dicree ie ae-pace equaio ] [ Du z CP y Bu P z AP P P P z & & k Du k Cx k y k Bu k Ax k x

9 Sae-pace Decripio ad Trafer Fucio Decripio Model of dyaic copoe or ye ca be foud i he for of a e of ordiary differeial equaio The repreeaio of a ye ca be forulaed io a rafer fucio for wih zero iiial codiio or a ae-pace for becoe a e of fir order differeial equaio Trafer fucio for: how he relaiohip bewee he ye ipu ad oupu I i called he ipu-oupu decripio, or he exeral decripio, of a ye Sae-pace for: how he rucural iforaio of he ye uig ye ae I i, herefore, called he ieral decripio of he ye

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12 f Where he reolve arix Du Cx y Bu Ax x & DU CX Y BU X A I G D B C D B A I C U Y Φ de P Q A I A I adj A I Φ

13 P i he characeriic polyoial of A, ad P p p p λ λ λ Q Q Q Q where Q I Pi r AQi Q AQ PI i i i Aad Q coue ad AQ P I i CTChe 984 ad λ i deoe he eigevalue of he arix A, which correpodig o he aural behaviour ode of he ye

14 Soluio of he ae pace equaio x & Ax Bu * Takig he Laplace rafor of *, X x AX BU herefore X [ I A] x [ I A] BU ** The ivere Laplace rafor of ** reul i he oluio x e A x e A τ Bu τ dτ A A A where e exp A I A! k! i called he ae raiio arix, deoed a Φ k k, ad x Φ x Φ τ Bu τ dτ

15 f G Y U Y ξ ξ U N D d d d Y ξ ξ ξ U ξ d ξ d ξ d ξ herefore, y ξ & ξ ξ u ξ d ξ d& ξ dξ

16 Le x x x & & ξ ξ x x x x & & ξ ξ he, he ae-pace for of hi ye ca be repreeed a [ ] x y u x d d d d x &

17 Reark Pole of he rafer fucio are he eigevalue of he ye arix A, provided ha here i o pole-zero cacellaio i CI-A - BD Zero i he value of uch ha if he ipu u e u i applied o he ye, he oupu y herefore, zero i he value of, which ake rak deficie Y U X D C B A I D C B A I

18 Exaple: 3 k k G ye zero: -3 -plae ye pole: -, reidue: k, k - k 3 k 3 Pole-zero diagra

19 Tie repoe of he ye Coider a liear ie ivaria luped ye decribed a d y dy du 3 y 3 u d d d The Laplace rafor of hi equaio i obaied a he followig Y y y& 3[ Y y ] Y 3[ U u ] U Y 3 y y& 3u zero-ipu repoe due o he iiial ae 3 U zero-ae repoe due o he ipu

20 - hoogeeou oluio Zero-ipu Repoe Y 3 y y& 3u k 3 k y ke ke where k y y & k [ y y& ] The roo -, ad - of he deoiaor i called he ode of he ye Thu, he ode eeially gover he zero-ipu repoe of he ye The deoiaor polyoial i called he characeriic polyoial of a ye

21 - Trafer fucio Zero-ae Repoe The rafer fucio deerie copleely he zero-ae repoe of he LTIL ye Le U,he y ca be obaied a Y y Y 35 e 4 43 e4 { 5 due o he due o he pole of G pole of U 44 3 U 3 I hi cae, all ye pole are excied by he ipu U

22 If le U by he ipu Y y, he he ye pole - will o be o excied 7 4 I ca be cocluded ha wheher a pole will be excied or o deped o wheher U ha a zero o cacel i The Laplace rafor of he ui-ep fucio ω ad ie fucio ω have o zero Therefore, eiher ipu will excie all pole of ay liear ie ivaria luped ye They ca be applied o ye ideificaio a eig igal 7 4 e e

23 ricly proper rafer fucio: deg{d} > deg{n} The ye uppree high-frequecy oie The oupu i coiuou eve ipu i dicoiuou biproper rafer fucio: deg{d} deg{n} The raio of he iforaio-bearig igal ad oie will o be alered A dicoiuou ipu igal will reul i a dicoiuou oupu iproper rafer fucio: deg{d} < deg{n} The ye ca aplify high-frequecy oie A coiuou ipu igal ay reul i a dicoiuou oupu

24 Iag -plae LHP RHP Real Doia pole: he pole which ha loge ie coa, ear o he iagiary axi

25 Ed of hi Chaper 3 4

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