A Variable Structure Model Reference Adaptive Control For MIMO Systems

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1 Proceedgs of the Iteratoal ultcoferece of Egeers ad Coputer Scetsts 8 Vol II IECS 8 9- arch 8 Hog Kog A Varale Structure odel Referece Adaptve Cotrol For IO Systes Ardeshr Kara ohaad Astract A Varale Structure odel Referece Adaptve Cotroller usg state varales s proposed for ult put ult output systes. Adaptato law s of varale structure type ad swtchg fuctos are desged ased o stalty reureets. Gloal epoetal stalty s proved ased o Lyapuov crtero. raset ehavor s aalyzed usg sldg ode cotrol ad shows perfect odel followg at a fte te. Keywords Adaptve cotrol IO syste odel referece Varale structure. I. INRODUCION odel Referece Adaptve Cotroller wth covetoal cotuous adaptato laws has ee vestgated etesvely the lteratures two a raches: oe assug full state accesslty [7 ad the other assug accesslty of put ad output [7. Cotuous adaptato laws are the for of pure tegral actos ad have soe proles such as: raset ehavor s dffcult to aalyss. Oly gloal (ut ot asyptotc stalty has ee guarateed. Udesrale traset resposes ad tracg perforace. Lac of roustess. he varale structure systes (VSS have ee studed great detals the lteratures [ he asc cocept of the varale structure cotrol s that of sldg ode cotrol. Swtchg cotrol fuctos are geerally desged to geerate sldg surfaces or sldg odes the state space [8. Whe ths s attaed the swtchg fuctos eep the traectory o the sldg surfaces ad the closed loop syste ecoes sestve to a certa etet to paraeter varatos ad dsturaces. he varale structure odel referece adaptve cotroller has ee vestgated the lteratures two a raches: oe assug full state accesslty [5 ad the other assug accesslty of put ad output [39. varale structure systes (VSS have ee studed great detals the lteratures. auscrpt receved Deceer 3 7. A. Kara ohaad s wth the Departet of echacal Egeerg Shahrood Uversty of echology Shahrood IRAN e-al: aara@yahoo.co. II. PROBLE DEFINIION Cosder a lear te varat plat wth uow paraeters whch ther ouds are ow. Let the plat e of -th order wth accessle states ad descred y the dfferetal euato X & AX + BU ( atr A s uow ad of full ra atr B s uow ad parttoed as B [.... U s a -desoal cotrol vector ad (AB s cotrollale. he referece odel s characterzed y the lear te varat dfferetal euato & ( X A X + B V A s a asyptotcally stale atr B s a ow atr ad parttoed as... ad V s a - [ B desoal put vector wth ouded eleets. he purpose s to fd cotrol U such that the state error e X (3 X epoetally teds to zero a fte te. III. VARIABLE SRUCURE ADAPIVE CONROLLER he atr B of the referece odel ca e chose as B B Q (4 Q s a ow dagoal atr. It s further assued that a uow atr A B Θ ests such that + A (5 ISBN: IECS 8

2 Proceedgs of the Iteratoal ultcoferece of Egeers ad Coputer Scetsts 8 Vol II IECS 8 9- arch 8 Hog Kog A. Stalty Ad Swtchg Fucto Desg he cotrol U to the plat s geerated troducg cotrol law U Φ (Ψ X + V (6 feedac atr Ψ wth the eleets ψ ad atr Φ wth the eleets are adusted usg VS approach y desgg swtchg fuctos ψ ad as descred the followgs. Sutractg euato ( fro ( ad usg euatos (4 (5 ad (6 the error euato s otaed as & e + B (Ψ Θ X + B (Q Φ U (7 e A Cosder a Lyapuov fucto of the for Λ e Pe (8 P s a postve defte syetrc atr whch satsfes the Lyapoov euato A P + Q (9 Q s a postve defte syetrc atr. Dfferetatg (8 wth respect to te alog the traectory (7 yelds Λ & e Qe + e P B ( Ψ Θ X + e P B (Q Φ U Assug We have Û ( Ψ X + V Λ & e Qe U Φ Û [ e P ( ψ [ e P ( + + ( Now troducg the swtchg fuctos ψ as ψ > sg (e P ( sg (e P sg( > ad susttute to ( yelds Λ & e Q e ( { [ e P + (e P } [ e P + ( e P sg( (3 he ters the suatos are always postve therefore Λ & < ad regardg (8 t ca e cocluded that e decreases at least epoetally. B. Estece Of Sldg ode Here t s show that the hyper surfaces S e P (4 are always sldg hyper surfaces for the syste. Let eae the followg reachg codtos [5 9. S& > (5 < Regardg euatos (7 ad ( we have S S & e + S PB ( Ψ Θ X + S PB ( Q Φ e + S P ( ψ + S S S Susttutg fro ( ad ( S U [ [ P ( e + S + S [ P [ sg (S { P [ ( sg(s sg( } Ad the swtchg fuctos as ISBN: IECS 8

3 Proceedgs of the Iteratoal ultcoferece of Egeers ad Coputer Scetsts 8 Vol II IECS 8 9- arch 8 Hog Kog S e S [ P sg (SS S [ [ + sg(e P P sg (S S S ε e [ P sg (S S [ [ + sg(e P P sg (S S ε s a postve costat. Now we ote that Pe Pe he atr Pee P S S Pe e P (6 ee s a postve se-defte atr hece s also postve se-defte ad therefore & Pe e P should has the sae sg as P or should e zero. Hece the ters the suato (6 are always postve or partly zero sce e epoetally teds to zero ad f we defe a ew vector Z [ X Û t ca e cocluded that f we have Z (t > λ > t > (7 t the there ests > t such that S S & < t (8 Codto S & > the relato (5 s cosdered here. Oe ca wrtes S& Pe& e + PB ( Ψ Θ X + PB ( Q Φ U e P ( ψ S Susttutg fro ( ad ( S& [ [ P ( e P sg (S + P ( sg(s sg( sce [ [ { [ + } (9 e epoetally teds to zero ad cosder to euato (6 t ca e see at least oe of the ters the suato s ozero ad we have S & > for t > ( Relatos (8 ad ( ea that for all t > the surface S e P s a sldg surface. C. Average Cotrol Whe sldg ode s occurred actually the average cotrol U ca e used [8 whch eleets are the outputs of frst order flters τ u + u u... ( te costats τ are suffcetly sall. he loc dagra s preseted fgure. IV. SIULAION RESULS I ths secto sulato results are preseted to show the perforace of the proposed schees ad coparg these wth the covetoal schees. he eaple cossts of a te varat syste wth paraeter varato. Eaple Cosder a te-varat syste descred y X & AX + BU ( ISBN: IECS 8

4 Proceedgs of the Iteratoal ultcoferece of Egeers ad Coputer Scetsts 8 Vol II IECS 8 9- arch 8 Hog Kog A.s t.57( +.s t B α α - 9.8( +.s t.s t α (3 Choose a referece odel descred y euato X& A X B V (4 A B (5 hece fro (4 ad (5 t ca e cocluded that Q α ψ ψ α ψ.98s(t ψ.85.s(t 3 ψ.57(.s(t ψ 5.s(t (6 3 therefore wth regard to euato ( ad ( we ca choose Θ.. 5. (7 hus the VS-RA Cotroller ca e desged usg euatos ( ad (6 for ufltered or euatos ( ad (3-3 for fltered cotrol. Syste was sulated resposes was copared wth the resposes of syste wth covetoal adaptato law as B pe X ad cotrol law & U X + Q V. he put sgal was + s(π t V + s(π t ad tal codtos was X( [ ad X ( [. Resposes are preseted fgure for fltered cotrol. V. CONCLUSIONS A varale structure odel referece adaptve cotroller has ee proposed for IO systes. Structure of the swtchg fuctos was desged ased o epoetal stalty reureets. agtude of the swtchg fuctos the ca e detered usg reachg codtos of sldg ode. Relato (5 s used to detere the ouds o eleets ad these eleets ust e lted. hs cotroller has soe sgfcat advatages copared to the covetoal odel referece adaptve cotroller. Gloal epoetal stalty s proved wthout reureets o persstece of ectato. he t was show that t s always possle to troduce sldg ode to the syste. raset ehavor was aalyzed ad showed perfect odel followg at a fte te. Isestvty wth respect to put dsturaces was vestgated ad showed preferece to the covetoal schees. Sulato was preseted to clear the theoretcal results. REFERENCES [ Chag-u C. Kg-Chua S. A-Che W. ad L-Che F. (996 " A Roust RAC Usg Varale Structure Desg for ult varale plats" Autoatca 3o. 6 pp [ Hsu L. Arauo A.D. ad Costa R.R. (994 " Aalyss ad desg of I/O ased varale structure adaptve cotrol" IEEE ras. Autoat. Cotrol pp. 4-. [3 Hsu L. (99 " Varale structure odel referece adaptve cotrol usg oly I/O easureet: Geeral case" IEEE ras. Autoat. Cotrol pp [4 Hug J.Y. Gao W. Hug J.C. (993 " Varale structure cotrol: A survey" IEEE ras. O Idust. Elect. Vol. 4 o. pp. -. [5 Kara ohaad A. (4 Varale Structure odel Referece Adaptve Cotrol for SIO Systes proceedgs of ICCAS4 August 5-7 Bago hlad. [6 Its U. (976" cotrol systes of varale structure" Joh Wley & sos Ic. New yor. [7 Naradra K.S. Aasway A.. (989 " Stale Adaptve Systes" Pretce-Hall It. Ic.. [8 Ut V.I. (978 " Sldg odes ad ther applcato varale structure systes" New yor: IR. [9 Cuha J. P. V. S. Hsu L. Costa R. R. & Lzarralde F. (3. Outputfeedac odel-referece sldg ode cotrol of ucerta ultvarale systes. IEEE rasactos o Autoatc Cotrol 48( [ Ia A. K. Costa R. R. ad Hsu L. ultvarale RAC usg Nussau ga. I proceedgs of Iteratoal syposu o adaptve ad tellget systes ad cotrol.. ISBN: IECS 8

5 Proceedgs of the Iteratoal ultcoferece of Egeers ad Coputer Scetsts 8 Vol II IECS 8 9- arch 8 Hog Kog Fg. he loc dagra. X - - U te [sec te [sec. 3 X te [sec. U X Referece odel VS RAC FS RAC te [sec Fg. Resposes ad cotrols. ISBN: IECS 8