Lecture 5. Material from Various Sources, Mainly Nise Chapters 5 and 6. Introduction to State Space Modelling and Controller Design
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1 ETR4 Advced otrol ectre 5 teril fro Vrio Sorce, il Nie hpter 5 d 6 ETR4 ADVANED ONTRO SEESTER, 4 Itrodctio to Stte Spce odellig d otroller Deig G. Hovld 4 Z. Dog Wh Stte-Spce ethod Ope p cotrol poiilitie for Nolier te ltiple-ipt, ltiple-otpt te Elegt Oerver Deig ch the Kl Filter ETR4 ADVANED ONTRO SEESTER, 4
2 Stte Spce odellig - Ai Wh e tte pce Wht i tte pce How to for tte pce odel How to covert trfer fctio to tte pce d vice ver How to olve the tte eqtio ig plce d the iplictio for tilit li ETR4 ADVANED ONTRO SEESTER, 4 ETR4 ADVANED ONTRO SEESTER, 4 Stte Spce odellig - Ai cot d The lticl, cotio tie oltio for the tie repoe. The copter iltio oltio for the tie repoe. How to repreet tte pce te igl flow grph. How the igl flow grph c e ed to covert etwee lterte repreettio i tte pce.
3 lic v. oder otrol otrol Ste Deig tte-vrile ethod oder tte pce root loc freqec repoe clicl trfer fctio ETR4 ADVANED ONTRO SEESTER, 4 Deig ojective e i oth ethod e.g. trcig, regltio i ittio where tte-vrile c e ered, ttepce deig ethod re ore powerfl lic v. oder otrol Advtge of tte-pce deig ecoe ovio i the reltive iplicit with which lti-ipt lti-otpt te c e treted. Alo tte-vrile repreettio provide ifortio o iterl repreettio ot ville fro ol iptotpt dt. Prc : Iverted Pedl ETR4 ADVANED ONTRO SEESTER, 4 Swig-Up otroller: ed opetor Blcig otroller: Stte-Spce Switchig ogic: otroller Pltfor: Stteflow P
4 Ste Stte Vrile Ste vrile: vrile tht repod to ipt or iitil coditio i te. Stte vrile: the llet et of lierl idepedet te vrile ch tht the vle of the eer of the et t tie t log with ow forcig fctio copletel deterie the vle of ll te vrile for ll t t. Iverted pedl tte: two gle q, q d two velocitie d/dtq, d/dtq ETR4 ADVANED ONTRO SEESTER, 4 ETR4 ADVANED ONTRO SEESTER, 4 Stte Spce I the et or pce cotiig the tte Ide ce fro tte-vrile decriptio of differetil eqtio rewritig of high order differetil eqtio ito firt order for otrol te deig ig the tte-vrile decriptio i clled tte-pce deig
5 Eple Give the te: It c e rewritte phe-vrile for: ETR4 ADVANED ONTRO SEESTER, 4 X AXB Nericl Itegrtio - Siltig dic te with ATAB e.g. ETR4 ADVANED ONTRO SEESTER, 4 di Ri dt vt
6 Nericl Itegrtio i Siltig dic te with ATAB e.g. di Ri dt vt ETR4 ADVANED ONTRO SEESTER, 4 Approite the Differetitio: i [ v t Ri t] t Updte lw: i i Dipl the idctor voltge triet ETR4 ADVANED ONTRO SEESTER, 4 rlci. % rli. : eple of iltig tie repoe of R loop. ; % Idctce of H R ; % Reitce of Oh i; % Set the iitil coditio overwrite i t; dt.5; % Set the tep ize v; % Thi i the tep ipt for : % Over tie tep di / * v - R*i * dt; % rerrge loop eq i i di; % get et vle of i t t dt; % get et vle of t.9 ed.8 plot t,v-r*i;.7 grid;.6.5 title'voltge cro idctor'; Voltge V.4... Voltge cro idctor 4 5 Tie
7 e.g.. Stte Spce Repreettio Stte vrile: loop crret it d cpcitor chrge qt R it ETR4 ADVANED ONTRO SEESTER, 4 di dt We ow: Ri idt V i dq dt V V c t Write the tte pce repreettio? e.g.. Stte Spce Repreettio R it ETR4 ADVANED ONTRO SEESTER, 4 Re-write: di dt Ri idt V di R q i V dt dq i dt di V q Ri V dt q i V V V V c t A B D A : te tri; B : ipt tri; : ipt/cotrol, : otpt; : otpt tri, D : feed-forwrd ter c e tri for IO te.
8 e.g.. Stte Spce Repreettio ETR4 ADVANED ONTRO SEESTER, 4 di R q i V dt dq i dt di V q Ri V dt V R V c t it q i V V A B D [ R] [ ] R tl Siltio of e.g. ETR4 ADVANED ONTRO SEESTER, 4 % Set p the iitil coditio for the te ; % Idctce of H R ; % Reitce of R ; % pcitce of F :, [; ]; % Ae zero iitil coditio t; % Tie trt fro dt.; % Set the tep ize/plig period v; % oider tep ipt A [, ; -/*, -R/ ]; % Ste tri B [; /]; % Ipt tri [-/, -R]; % Otpt tri D ; % Feed-forwrd cott % The followig c e ed for tte pce repreettio for : %Over tie tep d A*:, B*v * dt; % tte eqtio :, :, d; % pdte.8 t t dt; % pdte t.6 ed.4 * D*v; % Otpt vrile. % Otpt relt plot t, ; -. grid; title'voltge cro idctor'; -.4 lel'tie '; lel'voltge V'; Voltge cro idctor Tie Se : tepa, B,, D Voltge V
9 Sili Ipleettio of Stte-Spce odel tri Gi ETR4 ADVANED ONTRO SEESTER, 4 X AXB X B SISO or IO Altertive: SISO Soe reo for ig tte-vrile for to td ore geerl odel tte-vrile c eil iclde o-lier well tie-vrig te. Tie-vrig te t t t ETR4 ADVANED ONTRO SEESTER, 4,, t t t where trfer fctio decriptio i ot poile tte vrile decriptio i trightforwrd.
10 Soe reo for ig tte-vrile for Itrodce ide of geoetr ito d.e. plot of poitio v velocit i phe ple, which how otio of te i two-dieio. Stte-pce geerlize thi ide to higher dieio. Vilize oltio of d.e. trjector i tte-pce of pproprite dieio. ETR4 ADVANED ONTRO SEESTER, 4 to coect iterl d eterl decriptio tte-vrile how the iterl dic of the te. Throgh the tri repreettio, iple reltiohip c e etlihed to the eterl ipt-otpt decriptio. Trfer-fctio decriptio foce o ipt-otpt repreettio ol A power te eple Geerl for chie dic eqtio ETR4 ADVANED ONTRO SEESTER, 4
11 A power te eple: Phe portrit I - Ocilltio ETR4 ADVANED ONTRO SEESTER, 4 ETR4 ADVANED ONTRO SEESTER, 4 A power te eple: Phe Portrit Stle
12 Forig the Stte-Spce Repreettio Select Stte Vrile: t let the ii er of tte vrile order of the differetil eqtio decriig the te. ll tte vrile t e lierl idepedet of ech other oe vrile c ot e lier coitio of other vrile. ETR4 ADVANED ONTRO SEESTER, 4 Forig the Stte-Spce Repreettio Stte eqtio: A : te tri; B : ipt tri; : ipt/cotrol, : otpt; : otpt tri, D : feed-forwrd ter c e tri for IO te. ETR4 ADVANED ONTRO SEESTER, 4... Otpt eqtio: D A B
13 ETR4 ADVANED ONTRO SEESTER, 4 overtig Trfer Fctio ito Stte Spce Trfer fctio: ro-ltipl: te the ivere plce trfor ig zero iitil coditio: U Y U Y dt d dt d dt d ETR4 ADVANED ONTRO SEESTER, 4 overtig Trfer Fctio ito Stte Spce phe-vrile dt d dt d dt d et we hve:
14 ETR4 ADVANED ONTRO SEESTER, 4 overtig Trfer Fctio ito Stte Spce phe-vrile O tri for: U Y ETR4 ADVANED ONTRO SEESTER, 4 overt Trfer Fctio ito Stte Spce: Y/U / X Y U X t How?
15 ETR4 ADVANED ONTRO SEESTER, 4 overt Trfer Fctio ito Stte Spce: Y/U / t [ ] ETR4 ADVANED ONTRO SEESTER, 4 overt Trfer Fctio ito Stte Spce - Sr U Y [ ] O
16 Trfer Fctio - Bloc digr Stte Spce A dic te decried differetil eqtio c e repreeted trfer fctio; Ad c e coverted ito tte pce vi loc digr with fdetl ildig loc itegrtor it ETR4 ADVANED ONTRO SEESTER, 4 Trfer Fctio Stte-Spce odel Ipt-otpt ehvior i ideticl, however, the derlig odel trctre i lot with trfer fctio overt Stte Spce ito Trfer Fctio Uig plce trfor: ETR4 ADVANED ONTRO SEESTER, 4 A B D Y T I A B D U How?
17 Net Topic ETR4 ADVANED ONTRO SEESTER, 4 plce oltio to tte eqtio tilit Stilit i Stte Spce Sigl flow grph Altertive repreettio i tte pce Phe-vrile for cde for Prllel for otroller coicl for Oerver coicl for I ectre 6 we will deig cotroller d oerver ed o thee two for plce Soltio to Stte Eqtio d Stilit plce Trfor: X X AX BU Y X DU A B D ETR4 ADVANED ONTRO SEESTER, 4 Solvig for X d Y: X I A [ X BU ] dj I A [ X BU ] det I A Y X DU
18 plce Soltio to Stte Eqtio d Stilit I-A - dji-a/deti-a Aig zero iitil coditio X: Y dj I A T B D U det I A dj I A B Ddet I A det I A ETR4 ADVANED ONTRO SEESTER, 4 Pole of the te re fored : Root of the deoitor, i.e. deti-a The eigevle of the te tri A, i.e. deti-a. [,de]tfa,b,,d, Stte Spce Repreettio to Trfer fctio e.g..6 X [ ]X X ETR4 ADVANED ONTRO SEESTER, 4 dj i j where ij det ji e.g. [ ]
19 ETR4 ADVANED ONTRO SEESTER, 4 Stte Spce Repreettio to Trfer fctio e.g..6 [ ]X X X A I det A I A I dj A I [ ] B T dj det ji j i ij ] [ where e.g. Soltio: 4 D B A I U Y T ETR4 ADVANED ONTRO SEESTER, 4 plce trfor oltio of tte eqtio eercie 4.9 Solve for t d fid eigevle d te pole. [ ]X X X 5 X e t
20 plce trfor oltio of tte eqtio eercie 4.9 Solve for t d fid eigevle d te pole. X X 5 [ ]X t e X ETR4 ADVANED ONTRO SEESTER, 4 ETR4 ADVANED ONTRO SEESTER, 4 tl iltio The te i tle: t[:.:]; -.5.*ep-t-.*ep-*t7.5.*ep-*t; plott, grid
21 Stilit i Stte Spce ETR4 ADVANED ONTRO SEESTER, 4 A λ λ : eigevle : eigevector correpodig to eigevle λ ATAB: [, λ] eiga digol tri λ of eigevle d fll tri whoe col re the correpodig eigevector Stilit oditio for AB, D If ll eigevle of A re locted i the left hd ide of the -ple, the te i tle. Otherwie, the te i tle Roth Tle reviio * chrcteritic eqtio of cloedloop TF: 4 4 ETR4 ADVANED ONTRO SEESTER, 4 Tle 6. opleted Roth tle Roth-Hrwitz criterio: o No. of root of the poloil i the right hlf-ple er of ig chge i the firt col. o No ig chge i t col tle.
22 Stilit i tte pce eigevle v Roth Tle e.g.6. Fid ot how pole i the left, right hlf ple d jw-i. X 8 5 [ ]X X I A det I A ETR4 ADVANED ONTRO SEESTER, 4 Roth Tle pproch: / - -6 * Eigevle pproch: eiga or olve for I-A, eigevle re:.88 ± j.4, oe eigevle t right hlf ple tle. ig chge i t col : oe right hlf ple pole : tle. Sigl Flow Grph of Stte Eqtio A ltertive repreettio to the loc digr for ed for trfer fctio repreettio. Node : igl or vrile Brche: trfer fctio All rche re ed t ode Otpt i fored fro lier coitio of tte vrile d ipt. oo illtrtio of trfer fctio odel d tte-pce odel ETR4 ADVANED ONTRO SEESTER, 4 Figre 5.7 Sigl-flow grph copoet:. te;. igl; c. itercoectio of te d igl
23 Bildig Sigl-Flow hrt ETR4 ADVANED ONTRO SEESTER, 4 ETR4 ADVANED ONTRO SEESTER, 4 overtig Bloc Digr ito Sigl-Flow hrt Figre 5.9 Sigl-flow grph developet:. igl ode;. igl-flow grph; c. iplified igl-flow grph
24 ETR4 ADVANED ONTRO SEESTER, 4 Figre 5. Stge of developet of igl-flow grph for the te ove:. plce ode;. itercoect tte vrile d derivtive; c. for d /dt ; d. for d /dt figre cotie r Sigl flow grph of tte eqtio ETR4 ADVANED ONTRO SEESTER, 4 Figre 5. cotied e. for d /dt; f. for otpt Tr orelf: Sill-Aeet Eercie 5.5 pg r Sigl flow grph of tte eqtio cot'd
25 ETR4 ADVANED ONTRO SEESTER, 4 Altertive repreettio i tte pce Phe vrile for: vrile re cceive derivtive of ech other. oefficiet of te chrcteritic eqtio pper log the lt row. For the igl flow grph: Plce ode for tte vrile d their derivtive For rche coectig ode Other for: cde for Prllel for otroller coicl for Oerver coicl for O Reo:. Phicl vrile for prticlr tte vrile. Ee of oltio. Ee of odelig 4. Etc. U Y ETR4 ADVANED ONTRO SEESTER, 4 cde For cded firt-order te 4 4 R Note: Otpt of t order te loc re tte vrile ot phe vrile. i i i R [ ] 4 4 r Oervtio: Root of the ch. Eqtio, ie. te pole, pper log the digol of A.
26 Prllel For Prtil-frctio epio how?: R ETR4 ADVANED ONTRO SEESTER, 4 A tri i prel digol, provided o pole i repeted root of ch. eq. X X X [ ] 4 4 Oervtio: Decopled tte eqtio ech eq. i t order diff. eq. i ol vrile, c e olved idepedetl. e.g. i i i i r Root of the ch. eqtio, ie. te pole, pper log the digol of A. r Specil ce prllel for Jord coicl for R Prtil-frctio epio how?: X X [ ] r ETR4 ADVANED ONTRO SEESTER, 4 X Oervtio Jord coicl for:. Ste tri A i ot digol. Ste pole log the digol. off the digol for repeted root
27 ETR4 ADVANED ONTRO SEESTER, 4 Re-er phe vrile for to get : otroller oicl For phevrile for cotroller coicl for R [ ][ ] T r Phe vrile for: [ ][ ] T r Revere order phe vrile [ ][ ] T r Rerrge i cedig order to get cotroller coicl for: Oervtio: oefficiet of ch. eq. i the top row of A. Ech tte vrile i coected the feedc to the cotrol ipt ETR4 ADVANED ONTRO SEESTER, 4 Oerver coicl for R ro ltipl d coie ter of lie power of itegrtio: [ ] r Oervtio: oefficiet of de i t col oefficiet of i B Dl of cotroller coicl for: A D A T, B D T, D B T Feedc i fro otpt to tte vrile [ ] [ ] [ ] R R R [ ] [ ] [ ] [ ] R R R or
28 ETR4 ADVANED ONTRO SEESTER, 4 Feedc cotrol te e.g odel the forwrd trfer fctio i ccde for. Add feedc d ipt pth [ ] r r c c r 5 ETR4 ADVANED ONTRO SEESTER, 4 Figre 5. Stte-pce for for 6 4 t c R Sr Stte Spce For
29 ETR4 ADVANED ONTRO SEESTER, 4 Stte pce repreettio - r [ ] [] Phe vrile for U Y [ ] [] otroller coicl for [ ] [] Oerver coicl for Ste trfer fctio: [A c,b c, c,d c ]tf, * odl coicl for: the te eigevle pper o the digol of A. ETR4 ADVANED ONTRO SEESTER, 4 Toorrow' Ttoril Eple of Ipt-Otpt otrollilit, Sogetd hpter 5.6 Sill-Aeet.4, Nie
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