Adaptive Dynamic Surface Control: Stability Analysis Based on LMI

Transcription

1 Itratoal Joural of Cotrol Scc a Egrg 0 (3): DOI: 0.593/.cotrol Aatv Dyamc Surfac Cotrol: Stablty Aalyss Bas o LMI Bogsob Sog Dartmt of Mchacal Egrg Aou Uvrsty Suwo Kora (ROK) Abstract I ths ar quaratc stablty of aatv yamc surfac cotrol for a class of olar systms strct-fback form s aalyz th framwork of lar matrx qualty. Whl th xstc of cotrollr gas a fltr tm costats for sm-global stablty was thortcally rov th ltratur t s ot suffct to scrb how a st-ot valu a aramtr uat laws affct stablty a aramtr covrgc. hus t s cssary to rov a systmatc aalyss mtho to guarat both stablty a aramtr covrgc. By rvg th augmt clos-loo rror yamcs lar ffrtal cluso form a suffct coto of th cotrollr gas for stablty a aramtr covrgc s rv th form of lar matrx qualty. Fally th quaratc Lyauov fucto for ts quaratc stablty s comut umrcally va covx otmzato. Kywors Aatv Dyamc Surfac Cotrol Quaratc Stablty Paramtr Covrgc Lar Matrx Iqualty. Itroucto h yamc surfac cotrol (DSC) s a yamc xtso of multl slg surfac (MSS) cotrol to ovrcom th rawback of "xloso of comlxty" backstg as wll as MSS cotrol[]. h us of a srs of yamc fltrs abls th cotrollr to b sg squtally a sml. Furthrmor th xstc of cotrollr gas for sm-global stablty was thortcally rov []. Rctly a aalyss a sg mtho th framwork of covx otmzato has b trouc to allow us to f a quaratc Lyauov fucto umrcally for a class of olar systms call strct-fback form[]. hs cotrol aroach was xt to a class of olar systms whr th ucrtaty s larly aramtrz.g. af () whr a s a ukow costat a f s a kow olar fucto. h aatv backstg mtho has b vlo[3] a xt to a class of tm-lay olar systms[4 5]. As trouc abov th aatv DSC to solv th roblm of "xloso of comlxty" has b vlo for a class of olar systms a tm-lay systms[6 7]. Furthrmor DSC has b comb wth aatv ural twork cotrol schm th ltratur[8 9]. Howvr th usful tools such as lar matrx qualts (LMIs) ar har to aly to olar systm wth larly aramtrz ucrtats. hr s lttl work th ltratur for LMIs to b us for stablty aalyss of aatv olar cotrol roblms. * Corrsog author: bsog@aou.ac.kr (Bogsob Sog) Publsh ol at htt://oural.saub.org/cotrol Coyrght 0 Sctfc & Acamc Publshg. All Rghts Rsrv h followg xaml llustrats th sg rocur of aatv yamc surfac cotrol [6]: x ax : x + af( x) () u x whr a s ukow but bou by a kow ostv costat c such that a [ 0c ] a f x s a kow olar fucto a locally Lschtz o D x R x < r r > 0.. { } f x r : γ x whr γ s a Lschtz costat. h cotrol obctv s x() t x whr x s a costat wth x 0 whch s call a st-ot cotrol roblm. Frst f th frst rror surfac as S x x. Aftr takg ts rvatv alog th tractors of () S x + af th sythtc ut whch s to rv S 0 s x af ˆ KS () whr K s a cotrollr ga a â s th stmat of th ukow aramtr a followg th uat law as roos [6]: â ρsf (3) whr ρ s a ostv costat. h f th sco slg surfac as S x x whr x quals x ass through a frst-orr fltr.. τ + x x x (0) x(0) (4) whr τ s a fltr tm costat. Smlarly th rvatv of S alog th tractors of () s

2 48 Bogsob Sog: Aatv Dyamc Surfac Cotrol: Stablty Aalyss Bas o LMI S x x u x x a th cotrol ut s rv as u + x KS (5) whr s calculat from (4) such that x x. τ It s trstg to rmark that th calculato of bcoms smlr u to th cluso of th frst-orr fltr whl t rsults "xloso of comlxty" backstg. A xt qusto s how to sg a st of cotrollr gas to guarat stablty.g. K τ a ρ th xaml. It was rov [6] that thr xst a st of cotrollr gas (K a τ ) to guarat th stablty for stablzato a st-ot cotrol roblms. Howvr th rformac of th aatv DSC s o ρ crtcally[0]. If a small magtu of ρ s chos th aato of a (3) wll b slow a th trast rror wll b larg. O th othr ha too larg magtu of ρ wll la to oscllatory stmato of th aramtr thus rsultg th oscllatory rror. Suos a () K.5 () a (5) a τ 0.05 (4). Wh ρ s assg as a 0 rsctvly th tm rsoss of x a â ar show s Fg.. As xla abov th largr magtu of ρ rsults fastr covrgc of stmato rror of â a trackg rror. Howvr wh ρ 70 th oscllatory stmato of th aramtr s show Fg.. hus th trackg rror os ot covrg to zro. Furthrmor f x s chag to a ffrt costat although t wll b scuss latr Scto 4 th ffrt tm rsos (.g. oscllatory stmato) of â may b show for th sam st of K τ a ρ. Motvat by ths xaml t s uclar what valus of ρ a x guarat stablty a covrgc of th aramtr stmato rror for th gv st of a cotrollr ga (K) a a tm costat (τ). h ma cotrbuto of ths ar s to rv th augmt clos-loo rror yamcs clug aramtr stmato rrors a fltr rrors lar ffrtal cluso form a to rv th suffct coto for stablty a aramtr covrgc. Furthrmor th suffct coto allows us to chck stablty of th clos-loo systm a covrgc of stmat aramtrs by solvg th LMI umrcally. hrough ths ar w wll us th followg otato: m 0 R s a zro vctor a 0 R s a zro matrx wth arorat msos. I R s a squar tty matrx a I m R s a tty matrx th ss that m all agoal lmts ar o whatvr th mso of th matrx s. If x R s a vctor ag(x) s a agoal matrx wth th vctor x formg th agoal a ag(x) (or ag(x-)) s a squar matrx of sz (+) wth th vctor x formg th th sur-agoal (or sub-agoal) stas for a ostv ft (or smft) matrx r(x) s th sum of all agoal trs X.. Problm Statmt Cosr th class of olar systm as follows: + + af ( x... x)... (6) u+ a f( x... x) whr a s a ukow aramtr but bou by a kow ostv costat c such that a [0 c] f a [ f / x] ar cotuous o D R a f : D R o D D s a kow olar fucto strct-fback form th ss that th f oly o x... x. It s ml that f s locally Lschtz a [ f ( x)/ x] s bou o D []. hrfor thr xsts a costat γ > 0 such that f f f... γ (7) x x x for all x o D. 3. Aatv Dyamc Surfac Cotrol Fgur. m rsos of x a stmat of a wth rsct to ρ 3.. Dsg Procur A outl of th staar sg rocur for th aatv DSC scrb [6 ] s as follows: Df th th rror surfac as S : x x for... whr x s th costat valu. Aftr takg th tm rvatv of S alog th tractors of (6) : af + x + h surfac rror S wll covrg to zro f SS < 0 howvr thr s o rct cotrol ovr th surfac yamcs. If x + s cosr as th forcg trm for th rror surfac yamcs th th slg coto outs som bouary layr s satsf f x+ x+ whr x + x af ˆ KS (8) a th uat law for th aramtr stmat s as follows: aˆ ρ Sf. (9) whr K s a cotrollr ga a ρ s a ostv ga.

3 Itratoal Joural of Cotrol Scc a Egrg 0 (3): h xt st s to forc x+ x+ so f S+ : x+ x( + ) whr x ( + ) quals x + ass through a frst-orr fltr τ + ( + ) + x( + ) x + x( + ) (0) : x+ (0) (0) Aftr cotug ths rocur for f S : x x. Aftr takg ts rvatv th cotrol ut s u af ˆ ( x... x) KS () whr s calculat from (0) a th uat law of aˆ s followg x x aˆ ρs f. τ 3.. Augmt Error Dyamcs h augmt clos loo rror yamcs s rv for aalyss of stablty a aramtr covrgc. Aftr subtractg a ag x + a x ( + ) a usg u () th clos-loo yamcs (6) ca b wrtt as [ x+ x( + ) ] + [ x( + ) x+ ] + x+ + af aˆ f KS + af for.... Usg (8) a th fto of rror surfacs th abov quatos ca b scrb as a fucto of rrors as follows: S+ + + KS + h () KS + h whr + x( + ) x + s th fltr rror a h ( a aˆ ) f af s th aramtr stmato rror multl by f. I ato w to cosr th augmt rror yamcs u to cluso of a st of th frst orr low ass fltrs a th uat law for th stmat. Aftr takg a rvatv of + for th fltr rror yamcs s + ( + ) + + / τ+ + (3) whr th last qualty coms from (0). By takg a rvatv of (8) w ca wrt x + as ( af ˆ ) K t + x ( af ˆ ) KS / τ ( af ˆ ) KS t t (4) for.... Sc th rvatv of h s wrtt as h {( a aˆ ) f} ( af ˆ ) + af (5) t t (4) s rwrtt as h af KS (6) + + h af K τ th fltr rror yamcs (3) s K + h + af (7) τ + + KS + h + af for... τ τ+ Equato (5) wth th uat law (9) s wrtt as h af ˆ + af ρ f S + af (8) Fally th ovrall rror yamcs () (7) a (8) ca b summarz as KS + S h KS + h h ρ f S + a f (9) K + h + af τ k k+ k+ Kk k + h k + ak f k τk τk+ whr - a k -. Furthrmor th abov rror yamcs ca b wrtt matrx form as follows: I A I I ( ) S 0 I 0 h A h K0 I( ) Γ (0) af I 0 + af 0 I whr th vctors ar f as S [ S S ] R [ ] R h [ h h ] R af [ a f a f ] R af [ af a f ] R a th submatrcs ar K 0 0 K A K + ag([ ]) R K A ag( ρ f ρ f ) ( ) K ag( K K) K0 [ ag( K K )0 ] R ( ) ( ) Γ ag R τ τ I + ag. τ τ Sc th frst block matrx (0) s vrtbl such that I I 0 I 0 0 I 0 K 0 I( ) K0 whr 0 τ ττ 3 τ3 ττ τ3τ τ

4 50 Bogsob Sog: Aatv Dyamc Surfac Cotrol: Stablty Aalyss Bas o LMI aftr multlyg th vrs matrx to both ss (0) th augmt clos-loo rror yamcs ar wrtt as A z+ B () cl w z [ S h ] R : R z a th matrcs ar whr th rror stat 3 [ af af ] R : R I A I I ( ) cl K0 Γ A I A A I I A3 A3 A 33 ( ) A B I 0 I( ) whr th submatrcs ar A3 ( K0 A I ) A K ( ) :( ) A ( K I Γ ) ( K Γ ) K ag( K... K ). :( ) It s ot that th thr row of Bw s ( I a + a) f af. ( ) ˆ By fg [ af af ] () s rwrtt as follows: A z+ B () cl whr Bw I 0 0 Nxt w to trm th ur bou of (). Usg th assumto (7) th ur bou of for... s af a k a [ ] k xk x x λγ [ ] f f f f ˆ ˆ [ ] + a f a k λγ k xk λγ [ ] for -. Usg () s wrtt a fucto of z as follows: KS + S+ + h KS + S h+ / τ for... KS / τ hrfor thr xsts a matrx C z such that C z. (3) Fally th augmt rror yamcs (9) wth th ur bou of () ca b wrtt agoal orm-bou lar ffrtal cluso (LDI) form as z follows[3]: A z+ B cl t C z t z 3.3. Quaratc Stablty (4) Sc A cl (4) s ot tm varat u to A s () both A a A 3 ca b comos to a stay-stat trm a a tm varyg trm ur th assumto that z 0 as t for th gv st of cotrollr gas. hat s A ag( ρ ρ) ag( f f ) Φ ag{ f + ( f f ) f + ( f f )} A + A A ( K A I A ) whr 3 0 ( ) ( K0 A I( ) A ) I( ) A A3 + A3 a a [0 c ] Φ ag( ρ ρ ) A ag f f A ag( f f f f ) A K A I A Φ ( ) Φ 3 ( 0 ( ) ) 3 ( ) A I A f f ( x) f f( x af ) f f ( x af ) s a omal costat.g. a c /or a rough stmat of a. hrfor A cl ca b wrtt as A I I ( ) 0 Acl A + A[ I ] (5) A3 A3 A33 A3 A + A Usg (5) (4) ca b cosr as a omal clos-loo rror yamcs subct to a vashg rturbato trm as follows: Acl z+ Bw ( A + A ) z+ Bw A z+ Bw+ B (6) Az + [ Bw B] Az + B + whr [ ] R : R 0 B Az A [ I ] z A 3 0 Φ ag( f f f f ) S Φ ag( f f f f ) S.

5 Itratoal Joural of Cotrol Scc a Egrg 0 (3): Sc s a fucto of S a s bou o D thr xsts a matrx C z such that Czz for -. Fally th augmt rror yamcs (4) ca b wrtt as Az + B (7) t C z t z Sc th augmt rror yamcs (7) s wrtt agoal ormbou LDI ts quaratc stablty ca b al as follows[3]: Dfto. Suos A (7) s Hurwtz for th gv st of cotrollr gas { K K τ τ} a uat law gas { ρ ρ } h augmt rror yamcs (7) s. quaratcally stabl f thr xsts a ostv ft matrx P such that V ( z) ( z Pz) t t (8) ( Az+ B ) Pz+ z PAz ( + B ) < 0. If th rror yamcs s quaratcally stabl z 0 s a xotally stabl qulbrum ot o D. h suffct coto abov for quaratc stablty ca b xrss trms of lar matrx qualty (LMI) as scrb th followg thorm. horm. Suos that th agoal orm-bou rror yamcs (7) s gv for gv st of cotrollr gas a A s Hurwtz. h rror yamcs (7) s quaratcally stabl o D f thr xst P > 0 a Σ ag( σ σ σ ) 0such that A + PA + Cz ΣBCz PB < 0 (9) B P Σ whr Cz [ Cz Cz ] Σ B ag( σi σi σ ) I s th agoal block matrx. h quvalc btw (8) a (9) ca b rfrr to[3]. Rmark. If thr xsts th soluto for (9) z 0 s xotally stabl. hat s x x a af 0 as t. Morovr f f satsfs th so call rsstt xctato [6].. thr xst strctly ostv costats a a such that for ay t > 0 t+ f r a t a 0 as t. Othrws t s ot guarat for th stmat aramtr to covrg to th corrct valu although x x as t. 4. Illustratv Examl Cosr () aga wth th ukow aramtr a a th cotrol obctv s x x. Suos th oma D { x R x r r } a aˆ [0 c] whr c. h f x r γ aˆ c a c. x If th cotrollr rv Scto s al th clos-loo rror yamcs s followg as (9): KS+ S + + h KS h ρ f S+ af (30) K + h + af τ. whr x x a h af. Equato (30) ca b wrtt matrx form as follows: 0 K 0 0 K af z z + (3) 0 ρ f 0 0 af K 0 0 / τ 0 4 whr z [ S S h ] R. As rv () (3) ca b wrtts as K 0 K af z z + ρ f 0 0 af ˆ K + ρ f 0 K K K / τ Acl z+ B. whr [ af af ˆ ] R a th ur bou of s trm as follows: af a x a x KS+ S + h+ cγ czz Czz af ˆ aˆ x C z z whr c [ K ] z a Cz Cz cγ cz. hrfor th augmt rror yamcs ca b wrtt LDI form as Acl z+ B t Cz z t (3) For stablty aalyss (3) s cosr as a rturb systm as follows: Acl z+ B + B (33) t Cz z t whr th matrcs ar K 0 0 K 0 A B ρ f ( x ) 0 ρ K + ρ f ( x ) K K K / τ ρ f ( x) f ( x ) S s a vashg rturbato a ts ur bou s f( x) + f( x ) f( x) f( x ) S M γ S czz Cz3z whr M s a maxmum of f o D.. M r z [ 0] c a Cz3 4γ rmcz. Fally th augmt rror yamcs s wrtt agoal orm-bou LDI as

6 5 Bogsob Sog: Aatv Dyamc Surfac Cotrol: Stablty Aalyss Bas o LMI Az + Bw B Az B + t C z t 3 z h gvalus of A (34) ar caculat as followg: λ + K 0 λ K + t( λi A ) t ρ f 0 λ 0 K ρ f K K λ K + / τ (34) λ + K t( λ+ K)t ρ f λ 0 0 K ρ f K λ K + / τ whr f f( x). h 3 λ + K 0 τλ + λ + Kλ + ρ f 0 It s ot that th co charactrstc quato ca b rv usg Symbolc Math oolbox of MALAB. Usg th Routh stablty crtro th qualty coto for A to b Hurwtz s rv to b K > 0 K > τρ f (35) Suos K.5 a τ h th qualty coto (35) bcoms ρ f < K / τ 50 If thr valus of ρ ar cosr.. ρ {070} 3 thr rags of x for A to b Hurwtz ar obta as follows: /4 f ( x ) x < 50 / ρ x < (50 / ρ ). (36) hat s x <.659 for ρ x <.4953 for ρ 0 a x < for ρ Wh x a ρ s thr or 0 th matrx A (3) s Hurwtz for both cass a LMI (9) ca b solv va covx otmtho call CVX[4] s us to solv th fasblty roblm by calculatg P a Σ (9) umrcally th framwork of MALAB. As rct through stablty aalyss x x a â a as t as show Fg.. For ρ 3 70 th gvalus of A ar λ ( A) {0.434 ± }. Sc th matrx A s ot Hurwtz ths rsults th oscllatory stmat of a a thus oscllatory trackg of x (rfr to Fg ). If x s chag to.5 th gvalus of A wth rsct to ρ ar λ ( A ) {.65 ± } for ρ λ ( A ) {0.039 ± } for ρ 0. It s show Fg. that th tm rsoss of for x a â ar oscllatory for ρ whl th trackg rror a aramtr stmato rror covrgs to zro for ρ. If τ bcoms smallr as 0.0 th qualty coto (36) s mof as /4 x < (50 / ρ). (37) hus th matrx A s Hurwtz for all ρ a thr xst a soluto for LMI (9). h corrsog tm rsoss of x a â ar show Fg. 3. It s rmark that a smallr ga of τ allows us to larg th rag of x for A to b Hurwtz. Howvr t s wll kow that /τ th frst-orr fltr s a cut-off frqucy a th os thus may ot b attuat f τ s too small. Fgur. m rsos of x a â wth rsct to ρ for x.5 Fgur 3. m rsos of x a â wth rsct to ρ forτ Coclusos h aalyss mtho for stablty a aramtr covrgc of aatv yamc surfac cotrol was roos by rvg th augmt clos-loo rror yamcs lar ffrtal cluso form. h suffct coto for stablty s rv for th gv cotrollr gas th form of lar matrx qualty. It allows us to aalyz both quaratc stablty a aramtr covrgc by comutg a quaratc Lyauov fucto umrcally va covx otmzato. ACKNOWLEDGEMENS hs rsarch was suort by th Iustral Stratgc chology Dvlomt Program ( Dvlomt of Satal Awarss a Autoomous Drvg chology for Automatc Valt Parkg) fu by th Mstry of Kowlg Ecoomy (MKE Kora). REFERENCES

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