Lecture 11. Course Review. (The Big Picture) G. Hovland Input-Output Limitations (Skogestad Ch. 3) Discrete. Time Domain

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1 MER4 Advaced Cotrol Lecture Coure Review (he ig Picture MER4 ADVANCED CONROL EMEER, 4 G. Hovlad 4 Mai heme of MER4 Frequecy Domai Aalyi (Nie Chapter Phae ad Gai Margi Iput-Output Limitatio (kogetad Ch. 3 Frequecy Domai Deig (Nie Chapter Clatet MER4 ADVANCED CONROL EMEER, 4 iliear raformatio Overhoot, ettlig time to phae margi, badwidth Cotiuou ime Domai Aalyi (Nie Chapter Dicrete ime Domai Aalyi (Nie Chapter 3 Pole placemet imilarity raformatio uti Cotiuou ime Domai Deig (Nie Chapter LQR Fial Eam

2 Advatage / Diadvatage MER4 ADVANCED CONROL EMEER, 4 Frequecy Domai: ( Eay to obtai model eperimetally through frequecy repoe. (- Model do ot reveal the phyical tructure, oly iput-output relatiohip. Cotiuou ime Domai: ( Model baed o atural law (phyic, chemitry, etc. ome tate are more importat tha other ad cotroller ca be deiged accordigly. (- More difficult to etimate model parameter. Dicrete ime Domai: ( Allow aalyi of amplead-hold effect. (- Require cotroller deig i z- plae. ( uti approimatio allow u to re-ue tool from Frequecy Domai. Overview: Frequecy Domai Aalyi Polar Plot Aymptotic ode Plot: t ad d order zero ad pole he Nyquit tability Criterio ketchig Nyquit Diagram: Polar Plot poitive, egative jw Gai ad Phae Margi from Nyquit Plot tability, Gai ad Phae Margi from ode Plot MER4 ADVANCED CONROL EMEER, 4 M & N Circle, Nichol Diagram: Ope v Cloed-Loop teady-tate Error from ode Plot ytem with ime Delay Iput-Output Limitatio: he 8 Rule of kogetad & Potlethwaite

3 Commo Aymptote MER4 ADVANCED CONROL EMEER, 4 Figure.9 Normalied ad caled ode plot for a. G( ; b. G( /; c. G( ( a; d. G( /( a he Fial tatemet of the Nyquit Criterio MER4 ADVANCED CONROL EMEER, 4 he umber of cloed-loop pole, Z, i the right half-plae equal the umber of opeloop pole, P, that are i the right half-plae miu the umber of couter-clockwie revolutio, N, aroud - of the mappig GH Z P N Oly if Z, the ytem i table Remember thi! Zero of G( H( Pole of G( / [ G( H( ]

4 Eample Z P - N Frequecy Repoe Polar Plot MER4 ADVANCED CONROL EMEER, 4 Figure.5 Mappig eample: a. cotour doe ot ecloe cloed-loop pole; b. cotour doe ecloe cloed-loop pole Cla Quetio: Are the cloed-loop ytem a,b table or utable? ketchig the Nyquit Diagram (.4 MER4 ADVANCED CONROL EMEER, 4 Figure.6 a. urbie ad geerator; b. block diagram of peed cotrol ytem for Eample.4

5 Vector Evaluatio of the Nyquit Diagram MER4 ADVANCED CONROL EMEER, 4 a. vector o cotour at low frequecy; b. vector o cotour aroud ifiity; c. Nyquit diagram Cla Quetio: I the cloed-loop ytem table? Eample.7 G( ( K ( 4( w jw(6 w G( jw 6( w w (6 w Double-check thi calculatio yourelf! Cla Quetio (ue oly poitive jw ai: MER4 ADVANCED CONROL EMEER, 4 a Fid the rage of gai for tability ad itability b For margial tability fid the radia frequecy of ocillatio

6 Eample.7 - olutio MER4 ADVANCED CONROL EMEER, 4 G( ( K ( 4( w jw(6 w G( jw 6( w w (6 w Gai ad Phae Margi via Nyquit (.6 Figure.35 Nyquit diagram howig gai ad phae margi MER4 ADVANCED CONROL EMEER, 4

7 able 7.: teady-tate Error MER4 ADVANCED CONROL EMEER, 4 MER4 ADVANCED CONROL EMEER, 4 teady-tate Characteritic

8 Eample.4 ype : K p 7.78 ype : K v.55 MER4 ADVANCED CONROL EMEER, 4 ype : K a 3 9 Overview: Frequecy Domai Deig imple P-Cotroller ad ode Plot Lag Compeator Deig from ode Plot Icreaig Low Frequecy Gai Zero oe decade below badwidth Lead Compeator Deig from ode Plot hiftig adwidth to a Higher Frequecy ad/or Icreaed Phae-Margi MER4 ADVANCED CONROL EMEER, 4 Lag-Lead Compeator Deig from ode Plot he eefit of Lag ad Lead Combied

9 Chapter.5: Lag-Lead Compeatio Lag Compeatio improve maily teady-tate error (ie. gai at low frequecie ca be raied ad/or tabilie the ytem Lead Compeatio improve maily traiet behaviour (ie. croover frequecy or phae margi ca be raied MER4 ADVANCED CONROL EMEER, 4 Lag-Lead Compeatio i a attempt to achieve both improvemet at oce Lag-Lead Compeatio Notice gai cacel out! G ( G c lead ( G lag ( γ γ Deig parameter, ad γ Note relatio betwee γ ad /β MER4 ADVANCED CONROL EMEER, 4 ee able 9. i Nie

lag-lead compeator ha egligible effect at low frequecie. Hece, et gai K to atify teadytate error.

10 Lag-Lead Compeatio: ode Plot MER4 ADVANCED CONROL EMEER, 4 MER4 ADVANCED CONROL EMEER, 4 Deig Procedure: Lag-Lead Compeatio t tep: Fid the cloed-loop badwidth w bw required to meet the ettlig time, peak time or rie time (ee Lecture Note d tep: he lag-lead compeator ha egligible effect at low frequecie. Hece, et gai K to atify teadytate error. 3 rd tep: Plot ode for gai value of tep. Determie the ucompeated ytem' phae margi. 4 th tep: Fid phae margi to determie overhoot requiremet ( d order approimatio.

11 Deig Procedure: Lag-Lead Compeatio MER4 ADVANCED CONROL EMEER, 4 5 th tep: elect a croover frequecy w c ear (below w bw 6 th tep: At w c determie additioal amout of phae required. Add a margi of 5 o - o. hi amout determie γ. Ue γ /β ad lead compeatio figure.8 or formula 7 th tep: Deig lag compeator firt. elect upper break frequecy decade below w c 8 th tep: Deig lead compeator. w c w ma /( β give 9 th tep: Plot ode for compeated ytem ad repeat -8 if eceary Eample.4 Give the ytem below MER4 ADVANCED CONROL EMEER, 4 where G( ( K ( deig a lead-lag compeator uig ode plot to yield a 3.5% overhoot, peak time of ecod ad K v. 4

12 Eample.4 3.5% overhoot ζ.54 which yield PM 55 o p. w bw.9 rad/ec MER4 ADVANCED CONROL EMEER, 4 K v G( ( yield K 4 48 K ( 4 Eample.4: ode plot for K48 tep 5: elect w c.8 (<.9 MER4 ADVANCED CONROL EMEER, 4 Phae-76 o Lead Compeatio required: 5 o 5 o

13 MER4 ADVANCED CONROL EMEER, 4 Eample.4: (γ ad Upper break decade below w c. Hece /.8 or o at w c.8 rad/ec yield β.94 ad γ/β.6 β β β φ β ( i ma ma ma jw G w c MER4 ADVANCED CONROL EMEER, 4 Eample.4: /.56 whe w c w ma Fial lead-lag deig: where.79, 5.56, γ.6 β β β φ β ( i ma ma ma jw G w c ( ( ( G G G lag lead c γ γ

14 Characteritic of Lead-Lag Compeated ytem he reult below could ot have bee achieved with lag or lead compeatio aloe! MER4 ADVANCED CONROL EMEER, 4 he deig approach i approimate. Redeig ometime eceary Overview: Dicrete ime Domai Aalyi Modellig the Zero-Order-Hold he z-raform ad it Ivere tability Aalyi i z-domai iliear raformatio ad Routh-Hurwitz tability et teady-tate Error from Dicrete rafer Fuctio MER4 ADVANCED CONROL EMEER, 4 he uti Approimatio Allow Deig i -domai

15 MER4 ADVANCED CONROL EMEER, 4 z-domai tability : : : ( < < > > j j e e e e e e e e z e z α α α α ω α ω α α α α ω Regio (margially table Regio C (utable Regio A (table MER4 ADVANCED CONROL EMEER, 4 z-domai tability directly from -domai here i o equivalet Routh-Hurwitz tability criterio for dicrete cotrol ytem A imple traformatio allow u to check dicrete tability by traformig to -domai ad applyig the Routh-Hurwitz criterio a ormal. iliear traformatio z z z

16 iliear raformatio α ( α z ( α z jω ( α ( α jω jω ω ω MER4 ADVANCED CONROL EMEER, 4 z z z < > whe whe whe α < α > α Hece, the biliear traformatio preerve the behaviour above! Note: the traformatio hould ot be ued for other purpoe tha determiig dicrete tability!! Eample 3.8: Dicrete tability Cloed-Loop Dicrete deomiator: z 3 z.z. Ue the Routh-Hurwitz criterio to determie tability 3 z MER4 ADVANCED CONROL EMEER, 4 How may pole outide the uit circle i z-domai doe the ytem have?

17 -domai deig, z-domai implemetatio MER4 ADVANCED CONROL EMEER, 4 We wat to re-ue all the techique we have developed for cotiuou ytem. Ideally, we wat to deig everythig i -domai ad the covert to z-domai a the lat tep before implemetatio o the real-time cotroller. he uti approimatio will allow u to do thi z z z hi traformatio yield a digital trafer fuctio whoe output repoe at the amplig itat i approimately the ame a the aalog trafer fuctio. -domai to z-domai: uti Approimatio z z G c ( 977( 6 ( 9. ample time, chooe. MER4 ADVANCED CONROL EMEER, 4 ( z 977( 6 G ( ( z z c z 9. z 977(z 6z 6 z 9.z (6z 94 9.z z 674 z.746

18 Overview: Cotiuou ime Domai Aalyi tate-pace Model: ACD Form Cotrollability by Ipectio: Parallel Form he Cotrollability Matri imilarity raformatio he raformatio Matri P via the Cotrollability Matrice MER4 ADVANCED CONROL EMEER, 4 Obervability by Ipectio: Parallel Form he Obervability Matri he raformatio Matri P via the Obervability Matrice ummary tate pace Form Figure 5.3 tate-pace form for C( R( 3 ( 4( 6 y c( t MER4 ADVANCED CONROL EMEER, 4 Cotrollability /Obervability by ipectio

19 MER4 ADVANCED CONROL EMEER, 4 Phae-Variable Form u a a a M M L L M O M L L & & M & & A Matri Vector ( ( ( a k k a k a K A L L M O M L L Major advatage of the phaevariable ad cotroller caoical form MER4 ADVANCED CONROL EMEER, 4 Phae-Variable Form Pole of ucotrolled ytem: a a a L ytem parameter to adjut Pole of cotrolled ytem: ( ( ( k a k a k a L

20 MER4 ADVANCED CONROL EMEER, 4 P from cotrollability matri C M For the origial ytem For the traformed ytem Hece, P C MZ C MX - I Matlab: P ctrb(az,z iv(ctrb(a, ] [ A A A C MZ L ] [ ] ( ( [ A A A P AP P P AP P APP P P C MX L L PP - term diappear MER4 ADVANCED CONROL EMEER, 4 P-matri that give phae-variable form If we have the origial ytem ad the phae-variable form P ctrb(az,z iv(ctrb(a, will traform ay ytem z to the phae-variable form!!!! u A z z z z & u A u a a a & M M L L M O M L L & & M& &

21 Overview: Cotiuou ime Domai Deig Cotroller Deig i Phae-Variable or Cotroller Caoical Form he raformatio Matri P ad the Cotroller Gai K Oberver Deig i Oberver Caoical Form he raformatio Matri P ad the Oberver Gai L teady-tate Error Deig ad Augmeted tate Itegral tate MER4 ADVANCED CONROL EMEER, 4 LQR Cotroller: Optimal Cotrol. uig Parameter Q ad R LQR Cotroller Gai ad the Riccati Equatio he Kalma Filter. Optimal Etimatio. uig Parameter Q ad R. Kalma Filter Gai ad the Riccati Equatio LQG Cotrol: Combiatio of LQR ad the Kalma Filter Limitatio of Frequecy Domai echique We wat to place all pole, ot oly the d order domiat I frequecy domai, we have three parameter: gai, compeator pole ad compeator zero. MER4 ADVANCED CONROL EMEER, 4 hree parameter are ot ufficiet to place all pole for ytem of high order Eample: For the 5 th order ytem Y ( R( deig a cotroller that yield % overhoot ad peak time. ec for a tep-repoe. 5 a a a 3 a a

22 Cotroller Deig via tate-pace (Ch..-4 MER4 ADVANCED CONROL EMEER, 4 tate-pace formulatio of the ucotrolled ytem: & A u y C tate-pace formulatio of the cotrolled ytem: & A u A ( r K ( A K r y C Eample.: tate-pace Cotroller Deig Give the plat ( 5 G( ( ( 4 deig the phae-variable feedback gai to yield 9.5% overhoot ad a ettlig time of.74 ec. MER4 ADVANCED CONROL EMEER, 4 Deired pole: ( w ζ ζw w ( p hould ot iterfere with deig requiremet!

23 Eample.: Deired Pole ode plot G ( G ( ( 5.( ( ζw ( 5 ζw w w ad MER4 ADVANCED CONROL EMEER, 4 I geeral: Ue etra pole to cacel out zero. If o cacellatio required, place pole far away from d order pole. N: Watch iput aturatio - fat pole require high feedback gai. Eample.: igal-flow Diagram G( ( 5 ( ( 4 How? MER4 ADVANCED CONROL EMEER, 4 A 4 C [ 5 ]

24 Eample.: Cotroller Gai A K k (4 k (5 k3 MER4 ADVANCED CONROL EMEER, 4 Characteritic equatio: 3 ( 5 k3 (4 k k Mut match deig requiremet: ( 3 ζw w ( p Cotroller gai by ipectio: k 43., k 3.8, k 3.9 Large pole reult i large feedback gai ad poible aturatio. Cotroller deig by traformatio: E.4 MER4 ADVANCED CONROL EMEER, 4 Covert the igal-flow diagram above to the form & A u y C

25 MER4 ADVANCED CONROL EMEER, 4 E.4: Check Cotrollability 3 ] [ A A C M I thi ytem tate cotrollable? Check by ipectio MER4 ADVANCED CONROL EMEER, 4 E.4: raform to Phae-Variable ( ( ( ] [ MX MZ M C C P A A C u & & & Phae-variable form

26 E.4: Deired Repoe.8% overhoot ad ettlig time 4. MER4 ADVANCED CONROL EMEER, 4 Deired pole: ( ( ζw w ζ w ( p 5( We ue the etra pole to cacel the zero tate-feedback Cotroller A K ( k (7 k (8 k 3 MER4 ADVANCED CONROL EMEER, 4 Deired: ( ( ζw w ( p 5( y ipectio: k, k -4, k 3 -

27 tate-feedback Cotroller: Origial K orig K P - [ - -] MER4 ADVANCED CONROL EMEER, 4 Y ( ( C ( I A D U ( Verify deig with deired pole Coceptual Differece: LQR ad Pole-Placemet Pole-Placemet tart with time-domai requiremet (ettlig time, overhoot, etc Covert to deired d order pole Deig tate-feedback cotroller gai to meet deired pole MER4 ADVANCED CONROL EMEER, 4 LQR Optimiatio problem Miimie: However, the two approache ca lead to idetical cotroller for IO ytem LQR eted eaily to MIMO ytem

28 ummary LQR Cotroller Keep the proof for the LQR cotroller i mid he proof for the (teady-tate optimal oberver i imilar MER4 ADVANCED CONROL EMEER, 4 Optimal tate feedback gai K R A A Matrice A, i liear tate-pace model R Riccati olutio Q Riccati equatio Pealty Matrice Cotroller uig Parameter Phyical Motivatio for the LQR MER4 ADVANCED CONROL EMEER, 4

29 Performace Ide J MER4 ADVANCED CONROL EMEER, 4 Q C C Riccati Equatio K ρ R [ w ] w ρ w ρ MER4 ADVANCED CONROL EMEER, 4

30 Riccati Equatio: Numerical Eample w ad we chooe ρ MER4 ADVANCED CONROL EMEER, 4 ( Fial Eam ueday November 6 at 8:am - :3am Veue: Holt Room, tudet Uio Comple, ldg Coultatio (Arrage by Ao 5 th Level: Meetig Room A54 MER4 ADVANCED CONROL EMEER, 4

State space systems analysis

State space systems analysis State pace ytem aalyi Repreetatio of a ytem i tate-pace (tate-pace model of a ytem To itroduce the tate pace formalim let u tart with a eample i which the ytem i dicuio i a imple electrical circuit with