Stabilizing gain design for PFC (Predictive Functional Control) with estimated disturbance feed-forward

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1 Stblzg g sg o PFC Pctv Fctol Cotol wth stt stbc -ow. Zbt R. Hb. och Dtt o Pocss Egg Plt Dsg Lboto o Pocss Atoto Colog Uvst o Al Scc D-5679 öl Btzo St. -l: hl.zbt@sl.h-ol. {obt.hb l.och}@ h-ol. Abstct: PFC Pctv Fctol Cotol s sl bt v ctv SISO Sgl-It Sgl- Ott ctv cotoll. A stbc stto s oos bs o PFC lgoth cs o osbl stbcs. Th lgoth coss both xtl stbc sgl tl stctl stbcs ocss/ol stch. Estt stbc -ow cotol c l to stblt bcs o th tol clos-loo o th stto. Stblzg tch b ltg th stt stbc s vstgt o oc ocsss clg t. wos: Pctv ctol cotol stbc stto -ow cotol stblt. INTRODUCTION PFC Pctv Fctol Cotol s sl ctv cotol lgoth Rchlt t l. 978 whch os ot tx vso o cl zto o cost cto. PFC lgoths c b sl lz o ll ts o SISO Sgl-It Sgl-Ott ocsss. Ftho th lt vbl /o ts ct s wll oth vbls c b cost. PFC c b xt b -ow cotol th cs o sbl stbcs. Th o IC Itl ol Cotol s to stt th stbc th xtl o o ol stch btw ocss ol s th cotoll to s ths sgl th cotol lgoth. D. Rchlt co t sch to stt th stbc wth PFC to s th stt sgl o -ow cotol. Th vtg o th tho s tht t coss both xtl ot-sbl stbcs tl stctl ol stch. Fo ths so th xsso globl stbc stto. I th cs o xtl stbcs th tho wos sl w s th stbc wol hv b s -ow cotoll. F-ow cotol wth s stbc s oloo cotol. Howv -ow cotol wth stt stbc hs tol clos loo lcs th stblt. Tho so octo st b o to hc th stblt. Th slst tch s sg ttto cto ltl b th stt stbc Rchlt O Dov 9. Th s stct s ollows. I Scto th PFC lgoth s shotl show. I Scto th stbc stto sg PFC lgoth s st. I Scto 4 th stt stbc -ow cotol s show. Scto 5 sts sct stblt cto o th -ow g o both sg ot sg st-o stblzg lt. A cs st o x wth htg ct llstts th thotcl slts Scto 6.. PREDICTIVE FUNCTIONAL CONTROL Th cl o PFC s tht th cotoll vbl chvs th c tcto t th tgt ot o ots sg o chg o l b o chgs th lt vbl. Th s chg th cotoll vbl g th cto hozo o th ctl t s clclt o th s chg o th c tcto th ct chg o th ol ott. Th lt vbl c b clclt sl o th chg o th c tcto th ct chg o th ol ott g th cto hozo s Fg.. Fg.: PFC cl Th s chgs th cotoll vbl g cto st c b sosg tht chs th c tcto t th tgt ot st h:

2 wh s th ss costt c sgl. Th c tcto c b chos xotl cto o slct. Th th cotol o s csg ootoosl: wh λ s th cto to o th tcto s o. Th c tcto ovs th sttlg t t 95% =T c o th clos loo cotol sst T c t x wh Δt s th slg t. Fo th s chg s s ollows: Th chgs o c b ct sg ootol sto ol to wthot t chos o slct sct-t s 4 wh s th ol ott s th ol t s th sct-t ol t s th sttc g o th ol. Sosg tht th ctl t sgl s t costt g th cto hozo th ct ol ott bcos t sts: Th th ct chg bcos: 5 Sl coso btw th ct chg o th c tcto th ct chg o 5 slts th lt vbl: 6 wh: 6b I th ocss hs t th 6 hs to b lc b 7 I cs o hgh-o oc ocsss th ts cto c b tto lll cocto o sto ocsss 8 wth th cosog ts o th -th sb-ocss. I th ocss hs ltl ols th t bt v sl ols hv to b ssg to ch ltl ol. Th bsc lgoth c b sl xt o ths cs s wll h Rgwoo 8: 9 wh s th o o th ocss 9b. PFC WITH LOBAL DISTURBANCE ESTIATOR Th ollowg globl stbc obsv ws vlo b D. Rchlt bs o PFC s th sch Fg.. Th vtg o ths sch s tht t ss s tools s th l stll PFC whch c b sl lt. Fg. : Rchlt s sch o PFC wth stbc stto A slt ocss ol s cotoll b st PFC th stto. Both th l cotoll th stto cotoll s th s ocss ol wthot t. Th cotoll vbl s l s th c sgl o th stto PFC. As th stto cotol loo s ot stb th c btw both lt/cotol sgls l to th xtl stbc ctg o th ocss t th ocss ol th cotolls ct. Costl ths c s th stt stbc ctg to th

3 ocss s t. A tl scto o th stbc stto s gv Rchlt O Dov 9. Th stto cotoll s tg ts : T c. Th cl o th stto ths sch s to th cotol sgl whch oc th stto ocss ott to ch th cotoll vbl g th cto hozo t th t. Th s chgs g sts : wh x t T c s th cto to o th bs btw. H s th o-l ol ott o th stto loo - =. Th s stto clos loo sttlg t t 95% =T c hs to b chos v sll to ov st stto o th stbc. Th ct chg o th stto ol ott s: Usg th-o ol wth th-o ocss - s sl - wol l to btt cotol bhvo. Both xtl stbc sgl ocss g vto ss th sltos. Th cotol sco s: - slg t t=s slto t = 46s - t t=s stws cs o o to - t t=6s stws xtl stbc t t=s ocss g cs b 5%. Fg. shows th stto o th stbc o th gv ocsss wthot -ow cotol. Th stto ts : T c = s = o st-o ocss T c = 5s = o th-o ocss. Cog slts wh s th o o th ol Dl st-o ocss st-o ol Th stt stbc s st 4 I th ollowg sltos th ollowg ocsss PFC tg ts sos s Tbl. Tbl : Pocss cotoll ts Pocss/ol T c s Dl st-o: s Dl th-o: s.s 5s 5s Dl st-o ol o ths ocsss s ss th PFC lgoth wth ol g = t costt T = s t T = s. A st-o oxtg ol ws s wth th th-o ocss o to show th obst bhvo o th lgoth. b Dl th-o ocss st-o ol Fg. : PFC wth stbc stto wthot ow cotol Th lots Fg. / /b show tht th l xtl stbcs stt wll o cos ol t th t. Th stctl stbc cog o th ocsss g chg s stt s xtl stbc sgl t th ocss t. Th stt stbc hs th s ct s th ocss ts chg. Th lot Fg. /b shows stt stct stbc t bot t= s l bcs o th stch btw th th-o ocss ts oxtg st-o ol.

4 4. PFC WITH ESTIATED DISTURBANCE FEED FORWARD CONTROL To cost th stbc cts o th ocss PFC wth stt stbc -ow lgoth c b s s Fg.. PFC cotol to wth stbc -ow s: st 5 Th lot shows tht th xtl stbc sgl s cost coltl. Th sll o-sbl stct stbcs cost s slowl s th xtl stbcs b th cotol wthot stbc -ow lgoth. Fg. 5 shows th slto o PFC wth stt stbc -ow lgoth. wh st sts th s o stt stbc sgl: st 6 Fg. 4 shows th slto o PFC wth -ow lgoth bs o th s stbc. Dl st-o ocss st-o ol Dl st-o ocss st-o ol b Dl th-o ocss st-o ol Fg. 4: PFC wth s stbc -ow cotol b Dl th-o ocss st-o ol Fg. 5: PFC wth stt stbc -ow cotol Plot 5/ shows tht th stt xtl stctl stbcs cost cs o th st-o ocss howv wth hgh oscllto wh th stctl stbc occ. Plot 5/b shows stblt o th ocss cotol sltg o th stch btw th th-o ocss ts st-o oxtg ol. It s hsz tht coct th-o ol ws s wth th th-o ocss th th cotol lt sgls wol ot oscllt so ch. I th xt scto w shll show how th cotol bhvo c b ov v th cs o sch ol stch. Stblzg th ocss cotol wth stt stbc -ow lgoth c b lll b lg th stt stbc thogh lt s co b Rchlt O Dov 9.

5 5. STABILIZIN PFC WITH ESTIATED DISTURBANCE FEED-FORWARD CONTROL To hc th stblt o th clos-loo sst cotol wth stt stbc -ow lgoth th stt stbc s ow thogh lt. Zo-o lt wth g s ss st th ollowg lss. Th cotol sgl to sg s: st 7 Assg ol t ocss g vto s stct stbc th ocss ls-ts cto hs th o:. Pls-ts cto o th cotol sgl wth stt stbc -ow s th Ax s: st st } { 8 Pls-ts cto o th stto cotol sgl s: 9 Pls-ts cto o th stt stbc s: st st Th chctstc to o th clos-loo cotol sst c b toc o 8 s: } { Th sct stblt coto o th clos-loo sst c b toc th c o o s: Th c sos o th ocss c b tcll ccog to th t ocss ol o o-tcll ccog to th ls sos I o th st sos s tg to ccot th vtos o th ocss g s ollows: s I whs th ls sos ts to zo t stt s. Assg lt wth sttc g to th osclltos o th stt stbc sgl th sct stblt coto o th clos-loo sst c b toc th c o o s: } { 4 Th stblzg g 4 c b hgh th th g whs o oc lt st-o lt sll th bsolt vl. Ths s xct bcs th lt s stblzg th cotol o th wthot lt. Th slt stblzg g cotos 4 l o th slt ocsss. Th sg gs st Tbl. Tbl : Dsg stblzg lt s g Dl ocss Flt Zo-o Fst-o Fst-o Th-o

6 Fg. 6 shows PFC wth stt stbc -ow sg g wthot lt T s s Tbl. Th lots Fg. 6 Fg. 6b show tht th sg stblzg g slts osclltg sgls wth v sll g whch shows tht s v clos to th bo stblzg g t s sll th th ctcl vl. Alg sll lt s g s th osclltos lg hgh lt s g ovs th ocss cotol tow stblt. Fg. 6c shows PFC wth stt stbc -ow o th l th-o ocss ssg ol tho ol wth t. Th g.5.66 whch ls to stblt Fg. 6b s l Fg. 6c. Also hgh vls l to show tht wth ol ol th stblzg g c b cs whs t s stbl o. 75 stbl o. Fg. 7 shows th slto o PFC wth stt stbc -ow sg st-o stblzg lt wth g s Tbl t costt T s. Th lot shows tht th sg stblzg g slts osclltg sgls wth v sll g whch ostts tht s sll th v clos to th c Dl th-o ocss th-o ol Fg. 6: PFC wth stt stbc -ow cotol sg stblzg g Dl st-o ocss st-o ol Dl st-o ocss st-o ol b Dl th-o ocss st-o ol b Dl th-o ocss st-o ol Fg. 7: PFC wth stt stbc -ow cotol sg st-o stblzg lt

7 lt vl o th stblzg g. Alg sll lt s g s th osclltos lg hgh lt s g ovs th ocss cotol tow stblt. 6. CASE STUDY xg tg o ls s tl oto th chcl st. Tho ths ocss hs b chos s cs st to llstt th w sg tho. A l s to th x. Ths l s st t. Th tt s cotoll v htg ct whch s sl wth wt. Th sch o th x s show Fg. 8. Fg. 8: Sch o th x Th ollowg sbols ts s: _ : lt tt o l x _ot : otlt tt o l x F _ F _ot : lt otlt low o l x F / h.56 / s ; low o l x V : vol o x c _ 7 g/ : ht cct o l x 84g/ : st o l x : tt o l x _ : wt lt tt to ct _ot : wt otlt tt o ct F _ot F _ot : wt lt otlt low ct F / h.8 / s : wt low ct V = : vol o ct c _ 48 g/ : ht cct o wt g/ : st o wt A= : cto ht ts sc. U 5 W/ : thl tsttc UA U A 5 W/ W/ Th ct s scb b 5. t V c_ UA t t c_ F _ t t t 5 Fo 5 w obt th ollowg tl to t T t t t t Ths to lcts th ollowg ts o th ct: T V c _ UA c_ UA UA c_ F 5s 585 : t costt F 5 : g o t c_ F 9748: g o t _ UA c F Th x s scb b 6. t V c UA t t _ t c F t t 6 B tsog o 6 w gt th ollowg tl to: t T t t t t Ths to lcts th ollowg ts o th x: V c_ T 7s 5 : t costt UA c F _ UA UA c F _ : g o t 469 c_ F : g o t 85 _ UA c F Th ocss ol s show Fg. 9 wth th cotoll. Fg. 9: Cotol sch o th x ol Th ocss ws slt b slg t o t. Fst-o ol wth sttc g. 47 t costt T 6 / ws ss PFC th tg ts w: Tc 6. Estt stbc -ow cotol ws l wth PFC stto ts: T. c St ot lvl ϑ : 4 C 4 C ws l t t=5 t 5 stbc sgl st ϑ _ : 4 C 4 C C ws s th lt l tt to th x. Fg. shows sltos o th tt cotol wthot wth stt stbc -ow cotol sg stblzg g tht s o st-o lt Fg. shows th sltos sg sg ctcl stblzg g. 976 ; ths lot shows goo stbc cto wth csg osclltos. Sll g.75 s slt Fg. 9b; th lot shows goo stbc cto wth sll osclltg sgls. Hgh

8 g.5 s slt Fg. c th lot shows v stogl osclltg sgls. Fg. : Stblzg st-o lt wth = Fg. : Stblzg g = Fg. b: Stblzg g =.75 Fg. b: Stblzg st-o lt wth =.5 Fg. shows th sltos sg sg ctcl stblzg g. 988 th lot shows goo stbc cto wth sooth sgls bcs o th lt ct. Hgh g.5 s slt Fg. b; th lot shows slow stbc cto wth stog osclltg sgls. 7. CONCLUSION Fg. c: Stblzg g =.5 Fg. shows sltos o th x tt cotol wthot wth stt stbc -ow cotol sg st-o stblzg lt wth T s. PFC s sl ctv lgoth whch c b lt sl l o SISO ocsss. Th stbc stto sg PFC lgoth ws co b D. Rchlt to vo tol ltto o oth stbc stto lgoth. Th vtg o ths stbc obsv s tht th co lgoth ss th s tools l th l lt PFC. Th lgoth ws slt o l st- hgho ocsss sg st-o ol wth t. oo stto o th xtl stctl g vto stbcs w show v th cs tht th th-o ocss ws oxt b st-o ol.

9 PFC c b xt b stbc -ow lgoth sl. Tho th stt stbc ws s th -ow cotol. Alg PFC wth stt stbc -ow lgoth th cotol c bco stbl l th ol tchg s ot xct. A stblzg tch ws l b g th stt stbc thogh stblzg lt. Th stblzg lt g ws sg bs o v sct stblt coto th c o. Th thotcl slts w co b sltos. Th w sg tho ws llstt o sttg costg th tt stbcs x wth htg ct. PFC tt cotol s ot s o chcl ctos s.g. Bohch 6. Th oos tho c b s lso o costg tt stbcs occg wth chcl ctos. ACNOWLEDEENT Th thos th o D.. Rchlt o ttg ths sch wo o hs vcs. REFERENCES Bohch H. Cbss. L Lc.V. 6. Pctv ctol cotol o th tt cotol o chcl btch cto. Cots Chcl Egg h.t. Rgwoo.V. 8. Extso o st o ctv ctol cotolls to hl hgh o tl ols It.. Al. th. Cot. Sc. Vol. 8 No Rchlt. Rlt A. Tst.L. Po ol ctv hstc cotol: Alctos to stl ocsss Atotc Vol. 4 No Rchlt. O Dov D. 9. Elt Pctv Fctol Cotol. Sg Vlg Bl. APPENDIX Pls-ts cto o th lt vbl wth stt stbc -ow thogh g s: st st st st st st st } { A Pls-ts cto o th stto cotol sgl s: A Pls-ts cto o th stt stbc s: st st st st st st A Sbstttg A to A slts : } { st st

10 } { Chctstc to o th clos-loo cotol sst s } { } { } { A4 Th sct coto o th stblt th c o s: A5 Assg stblzg lt wth g th sct coto o th stblt th c o s: } { A6 Fo oc lt wth t g th bsolt vl o th ls-ts cto th c o s: A7 Fst-o lt wth t g s xl hs bsolt vl o th ls-ts cto s: Fo A7 w c s tht th lt o th stblzg g A6 s hgh o ls th lt o th stblzg g A5 } {