Robust Adaptive Inner-Loop Design for Vehicles with Uncertain Dynamics

Transcription

1 8 Aericn Conrol Conerence Wesin Sele Hoel, Sele, Wshingon, USA June -3, 8 ha6 ous Adpive Inner-Loop esign or Vehicles wih Uncerin ynics Eugene Lvresky Asrc We presen rous dpive ugenion design or clss o nd order uncerin nonliner cded syses hese dynics generlize he odels h re oen used or he design o inner-loop ligh conrollers he proposed conrol rchiecure ugens seline dynic inversion conroller wih direc dpive coponen nd vrile srucure syse, (VSS While he dpive ugenion is designed o inin rcking perornce in he presence o he syse uncerinies, he VSS coponen proecs he syse rjecories ro leving llowle suse in he syse se spce he design is pplied o consruc ngle-o-ck (AOA cond rcking syse or shor period dynics o ixed wing ircr I I INOUCION n his pper, we consider nd order uncerin dynicl syses in he cded or: x F ( x, z + B x + ( x, z ( x F ( x, x, z + x + ( x, x, z where x ( x x is he syse se vecor, z is he known ounded exernl signl, ( F, F re known sedependen uncions, B is known nonzero consn, x is he syse conrol inpu, nd (, re unknown, coninuously dierenile uncions h represen he syse uncerinies he conrol ojecive is ounded rcking in he presence o he syse uncerinies (, Speciiclly, he conrol gol is o design he conrol inpu x so h he syse s se coponen x rcks ny given ounded ie-vrying cond x (, in he presence o he syse uncerinies, while keeping ll he signls in he closed-loop syse ounded, uniorly in ie Our ineres in considering his priculr clss o syses ses ro ligh conrol reled pplicions, where innerloop conrollers or ixed wing ircr re oen designed sed on he so-clled sipliied odels, [ 4] he ler re in he or o ( nd represen he ircr decoupled s responses in pich, roll, nd yw xes he s conrol chllenge here is o design n inner-loop conroller h inins vehicle rcking perornce in he presence o uncerin erodynic eecs, cuor ilures, nd unknown environenl disurnces his is ccoplished using ligh proven dpive design ehods ro [, 3] E Lvresky is wih he Boeing Copny, Huningon Bech, CA 9647, e-il: eugenelvresky@oeingco his eril is sed upon work suppored y he Unied Ses Air Force under Conrcs No FA955-4-C-47 nd FA955-7-C-5 Copyrigh 8 he Boeing Copny All righs reserved Moreover, n ircr ligh conroller us e designed o keep he vehicle dynics in pre-speciied region o he corresponding se spce his region i oen reerred o s he operionl ligh envelope For exple, n AOA cond rcking conroller us include n AOA proecion syse, whose purpose is o inin he ircr AOA wihin pre-speciied rnge, ouside o which losso-conrol is expeced In essence, such n AOA conroller would hve o lend he wo su-syses, he AOA rcker nd he AOA liier, wih only one or he oher eing cive ny given ie Coining hese susyses ino single inner-loop conroller, while using heoreiclly jusiied design ehods wih perornce nd siliy gurnees, consiues he nd conrol chllenge o ddress he ler, we will eploy he design h ws originlly developed in [7] Furherore, oen in rel-world ligh conrol pplicions, n inner-loop syse us provide deque dping in he presence o high order dynics, such s he syse srucurl odes, s well s oher unodelled eecs owrds h end, we pose he 3 rd conrol chllenge which consiss o dding dping o he syse dynics low requencies only, nd wihou exciing he high requency odes Our proposed rous dpive conroller ddresses nd solves ll o he 3 conrol chllenges he res o he pper is orgnized s ollows Secion II presens he proposed rous dpive conrol rchiecure Online pproxiion o he syse uncerinies is diussed in Secion III Suicien condiions h gurnee ounded rcking nd unior ulie oundedness o ll signls in he corresponding closed-loop syse re sed in Secion IV Bsed on hese resuls, in Secion V we peror AOA cond rcking design, wih ligh envelope proecion logic, nd dpive dping his conroller is consruced or shor period dynics o ixed wing ircr he pper ends wih conclusions h re given in Secion VI II MOEL EFEENCE CONOL ACHIECUE We will eploy odel reerence sed conrol design rework he reerence odel is chosen o e nd order, wih he desired dping rio ξ nd he nurl requency ω his odel is driven y ounded possily ievrying reerence cond, x ω x x ( s + ξωs+ ω iereniing he irs se coponen in ( yields: ( F F F x ( F + B x + z+ BF + Bx + + x + z+ B x z x z z ( x, x, z, z d( x, x, z, z or, equivlenly /8/$5 8 AACC 3

2 x ( x, x, z, z + B x + d( x, x, z, z Noe h in (3, he uncion (,,, B re known, while d( x, x,, (3 x x z z nd he z z represens he consn unknown syse unceriny Consider dynic inversion sed conroller in he or: ( ( x x (4 s x B x K x x KP x x KI v where x( x ( ( x( τ x ( τ dτ, nd v denoes n s ugenion coponen, deined ler in (7 Le e x x e he rcking error signl Susiuing (4 ino (3, resuls in he closed-loop rcking error dynics: e K e KPe KI e( τ dτ + d + K v (5 ( x, x, z, z Inroduce x ( x x z z hen (5 cn e wrien s: e K e K e K e ( τ dτ + ( x v (6 P I A his ie, conrol signl v is deined o pproxie / doine he syse uncerinies on-line v ( γ ( x ( x + γ ( x v + w (7 d where ( x ( x, x, z, z is he on-line dpive pproxior, w d is he so-clled dpive dping er o e deined ler using re led-lg iler, nd v represens he swiching coponen o he conrol lw In γ x is he so-clled odulion uncion his is ddiion, coninuous se-dependen p which llows he conroller o soohly rnsiion eween he swiching nd he dpive odes o operion Consrucion o he odulion uncion is perored nex Le represen copc region o pproxiion or he dpive coponen, nd le e is copc suse he odulion uncion γ is deined s:, x γ, x < γ <, x (8 I provides coninuous rnsiion ro he dpive coponen in (7 o he swiching coponen v, i nd when he vecor x leves he suse, u eore i reches he oundry o he suicienly sll preer deines he widh o he nnulus region he in gol o he dpive coponen in (7 is o cncel / doine he unceriny ( x in (5 y using is on-line esied vlue ( x, or ll x Wih (7, rcking error dynics ke he or: ( (9 e K e KP e KI e τ dτ γ γ ( v wd e where e is he unceriny esiion error eiled design o he swiching coponen v will e presened ler in he pper Using pole plceen, seline PI eedck gins re chosen o enorce he desired nd order dynics (, ugened y inegred rcking error he eedck gins re: K ξω+ k, KP ω( ω+ ξk, KI ω k ( Susiuing ( ino (9 yields: e ( ξ ω+ k e ω( ω+ ξ k e ω k e( τ dτ ( ( γ e γ ( v wd egrouping he ers in ( resuls in: e + ke ξ ω( e + ke ω e+ k e( τ dτ ( ( γ e γ ( v wd Inroduce he so-clled ilered rcking error: e e + k e ( τ dτ (3 Using ( nd (3, ilered rcking error dynics cn e wrien s: e ξωe ω e ( γ e γ ( v wd (4 As seen ro (4, he er ξ ω e provides seline dping o he error dynics Oen, in rel-world pplicions, he vlue o he seline dping er is opiized such h iu possile dping is chieved in he presence o high order dynics, such s he syse srucurl odes, s well s oher unodelled eecs In order o dd exr dping, wihou exciing he high requency odes, he orer is inroduced ino he syse low requencies only owrds his end, deine: η s e e (5 s+ s+ nd choose he dpive dping er s: w K + k η (6 d where K represens he seline dping gin, k is he dpive increenl dping gin, nd > is he desired crossover requency, ove which he increenl dping us resor ck o is seline vlue elion (5 cn e wrien in se-spce or η ( η e (7 Augening he error dynics (4 wih (7, he syse exended error dynics cn e wrien s: e e (8 ( e ω ξω K e γ e + γ v + k η η η e Are where e ( e e η e re represens he exended ilered rcking error vecor Nex, condiions us e ound such h rix A ecoes Hurwiz he rix chrcerisic re polynoil cn e copued s ollows: 33

3 λ λi A ω λ ξω K λ + ( re de + (9 3 λ + λ ( ξω+ + λ( ξω+ K + ω + ω In order or he chrcerisic polynoil in (9 o hve ll is roos in he le hl plne, i is suicien o ipose he ollowing relions: ω > >, K ξω ( ω < ( ξω + K + ω ( ξω + erk he s inequliy in ( is lredy sisied, since deines he desired crossover requency or he re iler in (7 he nd condiion in ( plces lower ound on he vlue o he seline re dping gin I is cler h he inequliy is sisied or ny posiive gin K Finlly, he 3 rd inequliy ollows ro he previous wo erk I he ie derivive o he ilered rcking error e ( is driven o ecoe sll hen he originl rcking error signl e ( will lso ecoe sll his seen direcly ollows ro he deiniion in (3 In c, he ler cn e wrien s: e ke+ e ( Consequenly, i here exiss such h e ( ε or ll +, hen s k ( ε k ε e e e + e k k ( erk 3 I he ilered rcking error e ( is driven o ecoe sll hen he originl rcking error signl e ( will lso ecoe sll Using (, yields: k( k ( τ e( e ( e( e ( + e ( k e e ( τ dτ (3 Consequenly, i here exiss such h e ( ε or ll +, hen s k( k( e( e e( + e ( + + ε( e ε (4 In oher words, i e ( ε hen s he ollowing sypoic relion kes plce: e ε + o (5 ( ( III ON-LINE UNCEAINY APPOXIMAION On-line pproxiion o he uncerin uncion ( x in (5 is perored on copc x region, nd using liner-in-preers riicil neurl nework (NN, wih rdil sis uncions (BF in is inner-lyer, [6] he online uncion pproxiion is: x θ Φ x (3 where θ is he on-line esied vecor o preers nd Φ ( x represens ixed BF regressor vecor o he corresponding diension I is ssued h he unceriny cn e pproxied y n BF NN, wihin preried olernce, nd on he copc x region : ( x ( θ Φ ( x + ε( x (3 In (3, θ is he rue unknown consn preer nd ε d is he unknown ounded pproxiion error: ε ( x ε (33 Surcing (3 ro (3, he on-line uncion pproxiion error cn e expressed in ers o he online preer esiion error: ( e θ θ Φ ε Δθ Φ ε (34 Δθ Susiuing (34 ino (8, closed-loop ilered rcking error dynics cn e derived e A e γ Δθ Φ ε + γ v + k η (35 ( re re IV PAAMEE AAPAION AN CLOSE-LOOP SYSEM YNAMICS Choose syeric posiive-deinie rix Q > nd solve he ollowing lgeric Lypunov equion: PAre + Are P Q (4 Since A re is Hurwiz, he Lypunov equion hs unique posiive-deinie syeric soluion P Use he ler o or Lypunov uncion cndide: (, ( V e θ e Pe θ Δ +Δ Γ Δ θ + γ k (4 where syeric posiive-deinie rix Γ nd posiive lr γ will e used o deine he res o dpion iereniing (4 long he rjecories o he syse (35 rjecories, yields: e Qe + ( γ e Pre ( Δθ Φ + ε (43 e Pre k η θ + Δ Γ θ + γ k k γ ( v e Pre egrouping he ers urher yields: e ( Qe + e Pre γ ε + γ v e Pre (44 ( + Δθ Φ γ e P +Γ θ + k e P η+ γ k re re In order o ke he ie derivive V in (44 o e,, e θ k negive ouside o copc ( Δ suse o, choose he ollowing preer dpion lws: Proj θ θ, e Pre ( γ Γ Φ e ( Proj k γ k, ηe P re e ( (45 In (45, Proj denoes he Projecion Operor, which orces he dpive preers o evolve in pre-speciied copc, θ k region, [8] Furherore, i is esy o see h ( 34

4 ( re e e P p e p e p η (46 Siilr o [7], he swiching coponen o he conroller is deined s: v K sgn e ( K sgn ( p e + p e + p3η (47 where K is suicienly lrge posiive consn gin Susiuing (45, (46 nd (47 ino (44, yields V e Qe + γ ( Ksgne ( e ( + e ( ( γ ε (48 ecll h he odulion uncion γ is deined s in (8 Consider hree disinc cses Cse I x hen γ nd choosing K (49 relion (48 ecoes: e Qe + ( Ksgne ( e ( (4 e Qe + ( sgne ( K e ( λin ( Q e + ( K e Pre λin ( Q e < Consequenly, he rcking error will decy unil he syse se eners he region o pproxiion Cse I x hen ccording o (8 γ, nd hereore V e Qe + e Pre ε λin ( Q e + e Pre ε (4 where λ in ( Q is he iniu eigenvlue o Q, ε ε( x Also noe h ecuse o he Projecion x Operor, nors o he preer esiion errors will sy uniorly ounded, h is: Δθ Δ θ < k k < (4 ( ( where, ( Δθ k re he preer ounds Using (4, Unior Ulie Boundedness (UUB [5] o he closed-loop syse rjecories cn now e eslished owrds h end, deine he ollowing copc suse in he e region: Sr e r ε λin ( Q e Pe Pre eine inil level se { } (43 h conins S r Since λ in P e e Pe λ P e (44 hen choosing λ P r (45 iplies h or ll e r e Pe λ ( P e λ ( P r (46 Hence, he se S r is conined in he level se Suppose h ll iniil vlues o he ilered rcking error e ( sr in copc se S { e } { e Pe B} Le B e he il level se which elongs o S In order o inin closed-loop syse siliy, speciic relion eween he oundries or he ses, B, S r, nd S us e iposed hese ses will e used o prove h he closed-loop syse rjecories re UUB Grphicl represenion o he our ses is given in Figure 3 Figure 3: UUB Ses Choose: B λin ( P (47 hen i e Pe B hen using (44 yields: λin ( P e e Pe B λin ( P (48 Consequenly e, h is he ilered rcking error is in S Becuse o (4 nd (43, he ie derivive V is negive ouside o S r Consequenly, he ilered rcking error e will ener level se in inie ie, nd will rein in he se ro hen on Consequenly, he closed-loop syse rjecories re UUB Cse c I x hen oh he dpive nd he swiching coponens o he conroller re cive In his cse, using (48 one ges e Qe + γ ( Ksl sgne ( e ( + e (( γ ε (49 λin ( Q e + γ ( Ksl + ( γ ε e Pre λin ( Q e + ( γ e Pre ε Since y deiniion γ λin ( Q e + ( γ e Pre ε (4 λin ( Q e + e Pre ε Siilr o Cse, one cn show h he syse rjecories will ener he su-region in inie ie, er which he Cse c condiions ke plce he hree cses prove UUB o he closed-loop syse rjecories Moreover, due o he use o he Projecion Operor in (45, ll he esied preers θ ( re ounded Hence he rcking prole is solved he corresponding ol explici odel ollowing conrol signl cn e wrien using relions (4, (7, (6, (3, nd (47 x( x ( x B x ( x K( x x KP( x x KI s (4 Bseline ynic Inversion Conroller ( γ x B θ ( Φ( x γ ( x B K sgn p e + p e ( B K + k η Addiionl e ping Adpive Augenion Modulion Funcion Swiching Coponen erk 4 Using (5, he dping er in (6 cn e wrien s: 35

5 s + s+ k k w K + k x x K + k k x x + s + ( s ( s s Gs Led-Lg Filer (4 I he dpion re γ in (45 is se o zero hen he exr dping er ecoes pr o he seline conroller hus, he ol conrol cond cn e reoruled s: PI Conroller Led-Lg K I x B x ( x K s KP K G( s ( x x s Bseline ynic Inversion Conroller γ ( x B K sgn pe + pe Adpive Augenion Modulion Swiching Coponen Funcion ( Adpive e ping ( γ x B θ ( ( x Φ B k G s x x (43 Fro (43, i ollows h he ol conrol signl is coprised o he ive jor ers: seline dynic inversion conroller, dpive ugenion, c swiching coponen, d odulion uncion, nd e dpive dping V ESIGN EXAMPLE: ANGLE OF AACK ACKING In his secion, we pply he developed rous dpive design ehodology o consruc AOA cond rcking syse or ixed wing ircr, whose shor period dynics, wih li nd piching oen uncerinies, cn e wrien in he cded or (, [, 4] L + Qgrv + q+δl (5 q M + Mq q + MIC + q +Δ M (, q where is he ircr AOA, q is he ngulr pich re, L is he known li curve slope, er, L Q grv is he known grviy M is he known piching oen, M q is he known consn pich dping, M IC is he known piching oen increen due o ineril cross-coupling eecs, q is he conded pich ccelerion (conrol inpu, nd inlly Δ M (, q represens he piching oen unceriny he AOA desired / reerence odel dynics is chosen in he or o (: ω (5 s + ξωs+ ω Noe, h in ddiion o he known quniies in (5, he syse dynics conin uncerinies in he ircr li orce Δ L( nd in he piching oen Δ M (, q According o (4, he re dping er is chosen s: + (53 Δ is he li orce unceriny, ( K k ( ( ( ( w q q K q q + k q q s s+ s+ + Bseline Adpive he seline porion q l o he ol pich ccelerion cond, wih he dping er included, cn now q e wrien s: l ( q q (54 q q M Mq q MIC + ξω+ K q q + ω s+ s where q + L Qgrv is he reerence odel pich re signl he pich re error cn e wrien in ers o he AOA rcking error s: q q ( s+ L ( (55 In order o peror dpive / swiching ugenion design, we copre (5 wih he generic cded dynics ( x, x q, z ( Qgrv MIC, x q (56 F L + Qgrv, B, F M + Mq q+ MIC Δ L, ΔM (, q In his cse, seline PI eedck gins re: K K ξω+ k (57 K KP ω( ω+ ξ k I K KI ω k nd he inegror pole k is: k L (58 Using (3, he ilered rcking error signl, gives s+ k s+ L q q e e (59 s s s Hence, he ilered rcking vecor is: q q e e e e q q q q (5 s+ s s+ he regressor vecor Φ is chosen o depend on AOA only hen preer dpion lws re wrien sed on (45 q q Proj, ( θ Γ θ Φ q q q q P γ s s + Proj q q k, γ k η q q q q P s s+ (5 where P P > is he unique posiive deinie syeric soluion o he lgeric Lypunov equion (4 wih he Hurwiz reerence odel rix A re s speciied in (8 In sury, using (6 nd (3, dpive pich ccelerion cond kes he or: d ( q θ Φ + k ( ( q q (5 s+ Also relion (47 deines he swiching coponen o he pich ccelerion cond: q Ksl sgn ( p e + p e + p3η (53 Noe h he swiching gin K needs o e chosen o doine he syse unceriny: F + z + B L Δ L +ΔM (, q (54 x z In oher words, K ΔM (, q L ΔL (55 in qin q q In (55, he suse ( q q in in represens he ircr operionl AOA / pich re envelope 36

6 For sipliciy, le s ssue h he piching oen unceriny depends on AOA only, h is Δ M Δ M ( hen he swiching ode gin us e chosen such h K ΔM L ΔL (56 in Le ( in denoe AOA pproxiion region, where in in nd re he AOA iniu nd iu rek poins, correspondingly Piece-wise liner odulion uncion γ ( cn e esily wrien o sisy (8:, in in, in < in γ, in < <, < < +, + Figure 5 shows skech o he odulion uncion in erk 5 Figure 5: Piece-wise liner odulion uncion A uli-diensionl odulion uncion γ ( x (57 cn e creed s ollows Suppose h x is cener poin o he sphere { x x } Also suppose h he se represens he pproxiion region or BF-s Le > e sll posiive consn nd deine { x x } he odulion uncion is deined s:, x γ γ ( x, x, x (58 Forlly, he odulion uncion cn e wrien s: x x γ ( x, in, + (59 In order o see h (59 iplies (58 i is suicien o siply skech γ ( x versus x x Figure 5 shows he d γ ( x in in Figure 5: Muli-diensionl odulion uncion Using (59 yields hree relions h orlly prove he vlidiy o he odulion uncion choice + x x x x γ ( x + x x x x x γ ( x + x x x x x γ ( x + x x x erk 5 Suppose h x ( nd x ( Choose L weighed nor wih he weighs se o nd Se, nd wrie he corresponding diensionl odulion uncion + γ (,, in,+ In his cse +, In sury, ol pich ccelerion cond consiss o he ive ers: he seline dynic inversion cond l q, he dpive ugenion q d, c he swiching coponen q, d he AOA odulion uncion γ (, nd e he dpive dping er q l d q q γ q γ q + q ( VI CONCLUSIONS Moived y ligh conrol pplicions, in his pper we presened rous dpive conrol design ugenion o seline dynic inversion conroller In order o proec he syse rjecories ro leving n llowle suse in he syse se spce, VSS coponen ws dded Also, requency-dependen dpive dping er ws incorpored ino he syse he proposed design ws pplied o consruc AOA cond rcking syse or shor period dynics o ixed wing ircr EFEENCES [] KA Wise, E Lvresky, nd N Hovkiyn, Adpive Conrol o Fligh: heory, Applicions, nd Open Proles, In Proceedings o Aericn Conrol Conerence, Minnepolis, MN, 6 [] A Young, C Co, N Hovkiyn, E Lvresky, An Adpive Approch o Nonine Conrol esign or Aircr Applicions, AIAA , In Proceedings o AIAA Guidnce, Nvigion nd Conrol Conerence, Keysone, CO, 6 [3] E Lvresky, KA Wise, Adpive Fligh Conrol or Mnned / Unnned Miliry Aircr, In Proceedings o Aericn Conrol Conerence, Porlnd, O, 5 [4] B Sevens, F Lewis Aircr conrol nd siulion, Wiley, New York, 99 [5] HK Khlil Nonliner Syses, 3 rd Ediion, Prenice Hll Inc, [6] S Hykin, Neurl Neworks: A Coprehensive Foundion, nd ediion, Prenice Hll Inc, 999 [7] Snner nd JJ Sloine, Gussin neworks or direc dpive conrol, IEEE rnions on Neurl Neworks, 3(6, pp , 99 [8] J B Poe nd L Prly Adpive nonliner regulion: esiion ro he Lypunov equion IEEE rns Auo Conr, 37(6, pp 79-74, 99 37