Introduction to Modern Control Theory

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1 Itroductio to Moder Cotrol Theory MM : Itroductio to Stte-Spce Method MM : Cotrol Deig for Full Stte Feedck MM 3: Etitor Deig MM 4: Itroductio of the Referece Iput MM 5: Itegrl Cotrol d Rout Trckig //4 log d Digitl Cotrol

2 MM Itroductio to Stte-Spce Method Wht tte-pce ethod? How to get the tte-pce decriptio? 3 Property lyi Bed o SS Model Redig Mteril: FC: p469-49, DC: p- Eecercie: ee the ditriuted pper //4 log d Digitl Cotrol

3 Wht the StteS tte-spce Method? I the tte-pce ethod the differetil equtio decriig dyic yte re orgized et of firt-order differetil equtio i the vector-vlued tte, d the olutio i viulized trjectory of thi tte vector i pce dvtge: To tudy ore geerl odel To del with ultile iput d ultiple output yte To coect iterl d eterl decriptio ee pge of FC //4 log d Digitl Cotrol 3

4 Stte-Spce Spce Decriptio The tte-pce repreettio i give y the equtio: d t t Bu t dt Stte equtio Y t C t Du t Output equtio where t i vector repreetig the tte eg, poitio d velocity vrile i echicl yte ut i clr repreetig the iput eg, cooly force or torque i echicl yte, d yt i clr repreetig the output The trice, B, d C deterie the reltiohip etwee the tte d iput d output vrile Stte pce repreettio c lo e ued for yte with ultiple iput d output MIMO, Geerl decriptio //4 log d Digitl Cotrol 4

5 How to get Stte Spce odel? Vi odellig techique Vi trfer fuctio 3 Vi other SS decriptio 4 Vi yte idetifictio //4 log d Digitl Cotrol 5

6 SS Decriptio fro Modelig Eple : upeio odel y w Newto lw for trltiol otio //4 log d Digitl Cotrol 6 r

7 //4 log d Digitl Cotrol 7 Secod-order differetil equtio Stte-pce decriptio y y y r y y w w SS Decriptio of Supeio yte SS Decriptio of Supeio yte r y y y y w w

8 //4 log d Digitl Cotrol 8 Modelig DC Motor Modelig DC Motor Workig echi of DC otor t torque cott i rture curret e electrootive force ef cott Differetil equtio decriptio e t e i T θ t e t e t v R R J iplified v i R dt di L i J : θ θ θ θ θ See FC p47-49

9 How to get Stte Spce odel? Vi odellig techique Vi trfer fuctio 3 Vi other SS decriptio 4 Vi yte idetifictio //4 log d Digitl Cotrol 9

10 //4 log d Digitl Cotrol Trfer fuctio decriptio: Cotrol coicl SS for: G L L Cotrol Coicl Reliztio Cotrol Coicl Reliztio [ ] where D C B DU C Y BU L L L M M O M M L L,,,,

11 //4 log d Digitl Cotrol Eple: Modl coicl for G Modl Coicl Reliztio Modl Coicl Reliztio [ ],,, 3 4 D C B U D C Y U B Syte pole pper the eleet log the digol of The uertor ter i the prtil-frctio epio pper i the C

12 3 Mtl Ipleettio Cotrol coicl: [,B,C,D] tfnum,den Modl coicl for: CSYS cosys,type cy coy,'odl' retur reliztio cy i odl for, tht i, where the rel eigevlue pper o the digol of the tri d the cople cojugte eigevlue pper i -y- lock o the digol of TYPE: 'copio': G Nu[ ]; De[ 7 ]; [,B,C,D]tfNu,De CycoSy, 'odl' CycoSy, 'copio' Sy,B,C,D //4 log d Digitl Cotrol

13 How to get Stte Spce odel? Vi odellig techique Vi trfer fuctio 3 Vi other SS decriptio 4 Vi yte idetifictio //4 log d Digitl Cotrol 3

14 3 Trfortio etwee SS Model Priciple: Stte coordite trfortio for tte-pce odel yt y,t //4 log d Digitl Cotrol 4

15 //4 log d Digitl Cotrol 5 Step oe: coputer the cotrollility tri T C [ B B - B ] ctr,b, ctry Step two: coputer the row t through t [ ]T C - Step three: coputer the etire trfor tri Step four: coputer ew yte trice uig T d T - 3 Trfor to Cotrol Coicl For 3 Trfor to Cotrol Coicl For t t t T M D D CT C B T B T T c c c c,,,

16 3 Trfor to Modl Coicl For Eigevlue d eigevector of tri λ i ν i ν i for i,, Trfor to odl coicl for Y C BU DU Y Step : clculte the eigevlue d eigevector of Step : cotruct the trfor tri with prter α i T[α ν α ν α ν ] Step 3: elect the cle fctor α i of the eigevector uch tht ll the eleet of B re uity: B T - B Step 4: ue the deteried T to clculte C B D T T, B T B, C CT, D //4 log d Digitl Cotrol 6 U U D

17 How to get Stte Spce odel? Vi odellig techique Vi trfer fuctio 3 Vi other SS decriptio 4 Vi yte idetifictio //4 log d Digitl Cotrol 7

18 4 Syte Idetifictio for SS odel Wht i Syte Idetifictio? Syte Idetifictio llow you to uild theticl odel of dyic yte ed o eured dt How i tht doe? Eetilly y djutig preter withi give odel util it output coicide well poile with the eured output Wht the CD tool? idet i Mtl Syte idetifictio coure i IRS7 eeter //4 log d Digitl Cotrol 8

19 MM Itroductio to Stte-Spce Method Wht tte-pce ethod? How to get the tte-pce decriptio? 3 Property lyi Bed o SS Model //4 log d Digitl Cotrol 9

20 3 lye Bed o SS Model Eigevlue of the yte eig Triio zero of the yte tzero,b,c,d BU Y C DU I det C B D Trfer fuctio decriptio of the yte Stility criterio MIMO yte Siultio SS yte //4 log d Digitl Cotrol

21 3 Trfer Fuctio fro SS Model Trfor fuctio Y C BU DU G Y U [,B,C,D] tfnum,den [NUM,DEN] tf,b,c,d C I B D Stte-pce decriptio fequecy repoe decriptio Y C BU DU G det I det C I D B Eigevlue of eig Pole of G rootdeoitor Triio zero tzero,b,c,d Zero of G rootuertor //4 log d Digitl Cotrol

22 Eercie Oe I Eercie : coider two e coected vi lole prig, ovig t urfce with o fricitio F ue,, derive the tte pce odel of thi yte, uig tht the force F i iput, d poitio i output; coputer the eigevlue of thi yte d drw out their loctio i the cople ple; 3 coputer the trfer fuctio of thi yte ed o the tte pce equtio; 4 fid out the correpodig dicrete tie yte decriptio uig zero order hold d plig period T //4 log d Digitl Cotrol

23 Eercie Oe II Eercie : coider the followig yte Y [ ] 4 3 u Rewrite ove yte i odl coicl for; Rewrite ove yte i cotrol coicl for; 3 Ue tl fuctio coy d [u,de]tf,b,c,d, [c,bc,cc,dc]tfu,de to check your reult //4 log d Digitl Cotrol 3

MM1. Introduction to State-Space Method

MM1. Introduction to State-Space Method MM Itroductio to Stt-Spc Mthod Wht tt-pc thod? How to gt th tt-pc dcriptio? 3 Proprty Alyi Bd o SS Modl Rdig Mtril: FC: p469-49 C: p- /4/8 Modr Cotrol Wht th SttS tt-spc Mthod? I th tt-pc thod th dyic