Robust Output Command Tracking for Linear Systems with Nonlinear Uncertain Structure with Application to Flight Control

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1 Poceedings of he 44h IEEE Confeence on Decision and Conol and he Euopean Conol Confeence 5 Seville Spain Decebe -5 5 hb5.6 Robus Oupu Coand acking fo Linea Syses wih Nonlinea Unceain Sucue wih Applicaion o Fligh Conol M. G. Skapeis Mebe IEEE F. N. Kouboulis Mebe IEEE and N. S. Roussos Absac he poble of asypoic obus oupu coand acking is sudied fo he case of linea syses wih nonlinea unceain sucue. he poble is solved using a copensao wih saic sae feedback uniy oupu feedback and inegal acion. he conolle is esiced o be independen fo he unceainies. he oupu of he closed loop syse acks polynoial ype efeence signals. Sufficien condiions fo he poble o have a soluion ae esablished. he esul of asypoic obus oupu coand acking is applied o conol he sho peiod longiudinal oion of an aicaf. In he aheaical descipion of he aicaf an unceain sabiliy deivaive is included. he seps o deeine he conolle ae analyically pesened. he effeciveness of he conolle ove a wide ange of he syse unceainy wind disubances and acuao failue is illusaed hough siulaion o he nonlinea odel. I. INRODUCION HE poble of oupu acking has aaced conside aenion duing he las decades (see []-[6] and he efeence heein). I is a coon pacice o anslae boh he non unceain and he obus acking poble o a sabilizabiliy poble (see []-[3]). Paiculaly fo he case of obus acking a vaiey of appoaches paiculaly opial and adapive has been poposed (e.g. [4]-[6]). he poble of acking appeas o be of excepional inees o fligh conol syses and indusial applicaions []. In he pesen pape we focus on he geneal syse caegoy of linea syses wih nonlinea unceain sucue. Fuheoe he deived esuls ae successfully applied o conol sho peiod longiudinal oion of an aicaf. he conibuion of he pesen pape consiss in esablishing sufficien condiion fo he afoeenioned syse caegoy. he aeial of he pape is oganized in o wo pas. In Pa A he poble of obus oupu coand acking fo polynoial efeence inpus is solved fo linea SISO syses wih nonlinea unceain sucue. he poble is solved using a copensao wih saic sae feedback uniy oupu feedback and inegal acion. he oiginal unceain syse is augened wih eo dynaics and he oveall syse is obusly sabilized using he appoach in [7] and [8]. his way sufficien condiions fo he poble o have a soluion ae esablished. In Pa B he esuls of asypoic Manuscip eceived Mach 7 5. M. G. Skapeis F. N. Kouboulis and N. S. Roussos ae wih Halkis Insiue of echnology Depaen of Auoaion 344 Psahna Evoias Halkis Geece (e-ails: kouboulis@eihal.g skapeis@eihal.g Roussos_n@sudens.eihal.g) oupu obus acking ae applied o conol he sho peiod longiudinal oion of an aicaf having unceain sabiliy deivaive. An analyic algoih fo he deivaion of he conolle is pesened. Siulaion esuls execued fo he nonlinea aicaf aheaical descipion illusae he effeciveness of he conolle ove a wide ange of he syse unceainy wind disubances and acuao failue. PAR A: CONROL DESIGN PROCEDURE II. PROBLEM FORMULAION Conside he linea ie-invaian SISO syse wih non linea unceain sucue descibed by x () A( q) x() b( q) u() y() c( q) x() () n whee x () is he sae veco u () is he inpu and y () is he oupu of he syse. he aices q bq ( ) ( q) n and ( ) ( ) Aq ( ) ( ) n n cq q n ae funcion aices depending upon he unceainy veco q q ql ( denoes he unceain doain). he se ( q) is he se of nonlinea funcions of q. he unceainies q ql do no depend upon he ie. Wih egad o he nonlinea sucue of he syse aices A( q ) bq ( ) and cq ( ) no liiaion o specificaion is assued (i.e. boundness coninuiy ec.).. Polynoial Refeence Signals Conside he case whee he efeence oupu y () is he oupu of a efeence odel descibed by x () A x () y () c x () x x () whee y () x () and x is an abiay veco of iniial condiions and whee A c ( Clealy i holds ha y ) (). Define he acking eo e y y (3) Diffeeniaing he eo -ies we ge /5/$. 5 IEEE 75

2 Define he vaiables ( ) ( ) ( ) e y c q x (4) ( ) ( ) z x u u (5) Accoding o (4) and (5) he syse () can be augened wih he acking eo dynaics as follows: d x A q x b q u (6) d and ( ) I A q ( ) c( q) b q b q. n( ) A( q) o he augened syse (6) apply he saic sae feedback conol law () ( ) whee x e e e z abiay iniial condiions i.e he conol law u f x can be expessed in es of he oiginal syses as follows u () f e( ) d f e( ) d f e( ) d f x( ) (9) whee f ( i... ) ae he eleens of f. i I. SOLVABILIY CONDIIONS he chaaceisic polynoial (8) can be ewien as follows de adj () pcl s q f sin A q f sin A q b q Define: a q a q an q () whee a ( q) ( i... n ) ae he coefficiens of he i chaaceisic polynoial of he open loop syse (). Also define he polynoial aix q P s qq[ s s ] adjsin Aq b q () whee ( q) n is he axiu degee of he polynoial aix adj sin Aq bq and whee q q ( q) q ; iq i q i n q (3) u f x fe fz (7) Since de n si n Aq s de sin Aq a q[ s s ] () ( ) whee e e e e f and accoding o definiions () () and (3) he n and f. he obus oupu coand acking is augened closed loop chaaceisic polynoial can equivalenly be expessed as follows: foulaed in he following Lea: Lea : he oble of obus oupu coand n pcl s q f aq f q acking is solvable i.e. he oupu of he unceain syse [ s s ] (4) () follows he oupu of he efeence syse () while he acking eo (3) deceases asypoically o zeo if hee whee q n n q q o equivalenly as exiss a saic sae feedback conol law of he fo u f x such ha he chaaceisic polynoial n pcl s q f [ s s ] q f a q (5) p cl s q f desin A qb q f (8) Define is obusly sable. n Poof: If hee exiss a conol law u f x aking aˆ ( q) n( q) n A ( q) he augened syse obusly sable hen he acking eo (6) aˆ ( q) ˆ ( q) e is deceasing asypoically o zeo as fo ˆ ( q) li y ( ) y( ). q a a whee aˆ ( q) aˆ ( q) a n( q) q a n q n q ˆ ( q) ( q) q ( q) n q ( q) q ( q) n q ˆ ( q) ( q) q ( q) n q n( q) q q 753

3 Based on he above definiions and he esuls in [7] and [8] he following heoe is pesened. heoe. he poble of obus oupu coand acking fo he unceain syse () via he conolle (9) is solvable if he following condiions ae saisfied (i) he eleens of A qae coninuous funcions of q fo evey q (ii) hee exiss n ow subaix of A * A q which is posiive anisyeic. q le Poof: Accoding o Lea he poble of obus oupu coand acking fo he unceain syse () via he conolle (9) is solvable if he closed loop polynoial of he exended case is obusly sable. Accoding o he esuls in [7] and [8] he unceain polynoial is Huwiz invaian if condiions (i) and (ii) of heoe ae saisfied. Reak. he class of he syses ha saisfy condiion (ii) A of heoe can be widen if insead of q he aix A q is consideed whee is an appopiae inveible and independen fo q aix. Fo he definiion of posiive anisyeic aices see [7-8]. An analyic algoih fo he copuaion of he veco f ha peseve Huwiz invaiabiliy can be found in [7-9]. he esuls of he pesen secion can fuhe be exended if one consides ha he diension of he syse depends upon he unceainies n n( q) using he esuls in [8] fo unceainy dependen syse ode. Reak. Fo he case of a consan efeence oupu y () he augened syse wih he acking eo dynaics is educed o he following siple sae space equaions: d x A q x b e q u ; u u x d z cq x Aq Aq b q z bq. he conol law can also be ewien as follows n u () f e( ) d fx () whee f f and he esuling closed loop sucue is shown in Figue. y( s ) es () us () Unceain ys () f s Syse Fig.. Closed loop sucue PAR B: AIRCRAF APPLICAION II. AIRCRAF MODEL he longiudinal dynaics of an aicaf ae chaaceized f xs () by he phugoid (long peiod) and sho peiod odes. Assuing ha he aicaf hoizonal velociy coponen U eains consan and dopping he pich angle fo he saes he following non-linea sho peiod dynaics of an AFI/F-6 (see Fig. ) ae pesened [] []: Lcos a Dsin a a cos aq cos a a U M q a e e ue I (7) whee a is he angle of aack q a is he pich ae and e is he elevao deflecion angle and u e is he elevao coand. he aeodynaic foces D L and he oen M ae defined as follows: CLqc D qscdaacdee LqS Ca qa CLee V (8) Cqc M qsc Caa qa C ee V he paaees of he aicaf appeaing in equaions (7) and (8) ae defined in able I. ABLE I VARIABLE DEFINIION Sybol Definiion Values V aicaf velociy V W U ( /sec) p densiy of ai kg / slug / f q dynaic pessue q.5 pv ( N/ ) S suface (3 f ) c ean a/d chod (.3 f) ass kg( slug) g gaviy acceleaion 9.8 / sec (3. f/ sec ) U hoizonal velociy 84.4 / sec(933 f/ sec) I oen of ineia kg (53876 slug f ) C a/d a foce due o q 3.6( uniless) Lq CL e a/d a foce due o e.55( uniless) uniless C a/d a oen due o a a.46( ) C a/d a oen due o q.38( uniless) q C e a/d a oen due o e.6933( uniless) Cda a/d a foce due o a.56( uniless) Cd e a/d a foce due o e.99( uniless) a a/d sands fo aeodynaic. he paaee C is an unceain paaee q valued in he doain q C C C in ax. he elevao deflecion is consained accoding o he inequaliies e 5deg and e 6deg/ sec. he lineaized equaions of he AFI F-6 can be wien in he sae space fo as follows: 754

4 x () A( C ) x() bu() y() cx() x( ) x (9) whee a x qa e c y () a u () ue and b.5c.5cclq ps.5cl e.5cc.5.5 ( ) a c Cq cc e AC I I I x b V U a Fig.. Aicaf Scheaic z b W III. ROBUS RACKING CONROLLER In his secion a obus acking conolle fo asypoic sep acking of he angle of aack will be designed. Accoding o (6) fo he following augened sae space descipion of he aicaf odel is deived: whee x AC xbu () e c x z z x and AC AC b b () o syse () apply he following saic sae feedback law f f f f3 f4 ; fi () he augened syse closed loop chaaceisic unceain polynoial is deived o be 4 3 pcl s C f s C f s (3) C f s C f s C f whee C f 3 I f4i.5ci.5c Cq C f C f C f C f CL e fi C f4 I.5c Cq C f c[5.ca C e f3u c5cq f4.5 CLqCa ] C f C f cuc f 3C f I C f CL e f I c.5ccq f 5Ca f U 3 C f.5cccq f4 ps Ca f45cclq f4 ps C f 5C e f cclq f ps C f 5cfpS Ce cclqceps.5cclecqps U C f Accoding o definiions () and () he following aices ae defined: a C a C a C a C (4) C 3 4 C (5) 3 C 4 C 3.5cC q.5c whee a C I a C [ c5ccq.5cau.5cclqca C I.5 c Cq ] cps a C Ca5cCLqCa ps.5cccq C L e cc e C 3 L e I 5cCq CI 4 C cps Ce 5cCLqCepS.5cCLeCqpS U 3 5cCL eca CC e 3 C a 5 Lq a.5 q cps C cc C ps cc C ps U 4 C cps Ce 5cCLqCepS.5cCLeCqpS U 3. heoe : he poble of obus oupu sep coand acking fo he sho peiod odel of he aicaf is always solvable. Poof: Accoding o definiions (4) and (5) he aix 755

5 A C is consuced. he eleens of A C ae coninuous funcions of C fo evey C C C. Choose he following 5 4 ow in ax A C (whee ) I /( ) C I /( ) * A C C 3C 4 I 5C 6C (6) I whee C.5IC c Cq C c5ccq.5cau.5cclqca C I.5c C subaix of C IC c Cq q C L e I 8 C L e I a Lq a q 5 C.5c 4C cc C cc C ps U L e.5 4 a Lq a q 6 C C I c C cc C cc C ps U 7 5CLeI c Ce cclqce.5cclecq ps U 9.5c 4Ce cclqce cclecq ps U.5c 4Ce cclqce cclecq ps U. he aix.5c 4Ce cclqce cclecq ps U A C using he hee posiive up augenaions: C is posiive anisyeic since i can be consuced * C C A C whee C 8 3 I 4 C 7 8 I 3 C 3C 4. I 6C he veco cc C is a Huwiz invaian coe since he associae polynoial of cc ( cc s ) is posiive Huwiz invaian. Hence condiion (ii) of heoe is also saisfied. IV. COMPUAION OF HE CONROLLER PARAMEERS he conolle f f f f3 f4 will be copued using he following algoih [7]-[9]: 756 Sep (Consucion of he augenaion aices): he c C C C coe of A C is using an up posiive augenaion he aix C. Fo is consuced. Using anohe up posiive augenaion he C is consuced. Using anohe up aix 3 augenaion he aix C A C is 4 consuced. Le. Sep (Deeinaion of he egion of fo which C is posiive Huwiz invaian): Accoding o he fo of he associaed polynoial i is obseved ha obus sabiliy is guaaneed. Hence he egion of is. Le and choose.9. Sep 3 (Deeinaion of he egion of such ha C 3 is posiive Huwiz invaian): he especive associaed polynoial is obusly sable Le 3 and choose.. Sep 4 (Deeinaion of he egion of 3 such ha 4 C 3 is posiive Huwiz invaian): he especive associae polynoial is obusly sable Choose 3.4. Sep 5 (Deivaion of he gain veco): he gain veco f ha obusly sabilizes he associaed polynoial of A C f is f 3 and consequenly he gain veco ha obusly sabilizes he associae polynoial of A C is e e e e f f. e3 e3 e3 e3 e3 Accoding o (9) he conol signal is defined as e e e e u () e( ) d x () e e e (7) e e V. SIMULAION RESULS he paaees values fo an aicaf flying a. f and a.9 M ae defined in able I. Le he desied anoeuve be a loop wih a seady angle of aack of.5 adians. Suppose ha he aicaf is flying hough a windy and gusy envionen as shown in Fig. 3(b) [] [3]. Addiionally fo a ie peiod e 3sec he elevao acuao pesens up o 8% educed efficiency as shown in Fig. 3(a). Applying he conol law (7) o he non linea aicaf odel he sae esponses ae quie saisfacoy as illusaed in Fig. 4(a) Fig. 4(c) fo he iniu axiu and noinal values of he unceain paaee C. he angula velociy of he acuao is shown in Fig. 4(d) while he acking eo is shown in Fig. 4(e).

6 Efficiency % Ddegsec sec (a) 5 5 sec (b) Fig. 3. (a) Elevao efficiency (b) Roay guss disubance deg q degsec e deg 3 Va degsec SSeodeg sec (a) sec (b) sec (c) sec (d) sec (e) Fig. 4. Appoxiae coand following fo he angle of aack (a) esponses of pich ae (b) elevao deflecion (c) elevao angula velociy (d) acking eo (e). (Doed line Plain line Dashed line is fo he iniu he noinal and he axiu value of C especively) VI. CONCLUSION In his pape he poble of obus oupu coand acking fo polynoials efeence signals has been solved fo he fis ie fo he case of linea syses wih nonlinea unceain sucue. he poble has been solved using a independen of he unceainies conolle ha includes a saic sae feedback a uniy oupu feedback and inegal acion. Sufficien condiions fo he poble o have a soluion ae esablished. he above esuls have been applied o conol he sho peiod longiudinal oion of an aicaf Siulaion esuls o he nonlinea aicaf odel have illusaed he effeciveness of he conolle ove a wide ange of syse unceainy wind disubances and acuao failue. Befoe closing i is ipoan o enion ha based on he esuls of he pesen pape fuhe esuls egading oe geneal efeence signals ae cuenly unde invesigaion. Acknowledgen: Wih egad o he fis and he second auho he pesen wok is co-financed by he Hellenic Minisy of Educaion and Religious Affais and he ESF of he Euopean Union wihin he faewok of he Opeaional Pogae fo Educaion and Iniial Vocaional aining (Opeaion Achiedes- ). REFERENCES [] C.. Chen Linea Syse heoy and Design. Hol Rineha and Winson New Yok 984. [] I. M. Hoowiz Synhesis Feedback Syses. New Yok Acadeic 963. [3] R. C. Dof and R. H. Bishop Moden Conol Syses 9 h ed. penice Hall. [4] M. J. Coless G. Leiann and E. P. Ryan acking in he pesence of bounded unceainies pesened a he 4 h In. Conf. Conol heoy Cabidge U.K. Sep 984. [5] K. akaba Robus peview acking conol fo polyopic unceain syses in Poc. 37 h IEEE Conf. Decision Con. apa FL 998 pp [6] I. Yaesh and U. Shaked wo degees of feedo H opiizaion of ulivaiale feedback syses IEEE ansacions on Auoaic Conol vol. 36 pp [7] K Wei B. and R. Baish Making a polynoial Huwiz invaian by choice of feedback gain In. J. Con. Vol 5 pp [8] F. N. Kouboulis and M. G. Skapeis "Inpu -Oupu decoupling fo linea syses wih non-linea unceain sucue" J. of he Fanklin Insiue vol. 333(B) pp [9] F. N. Kouboulis and M. G. Skapeis Robus iangula Decoupling wih Applicaion o 4WS Cas IEEE ansacions on Auoaic Conol vol. 45 pp [] A. W. Lee and J. K. Hendick Applicaion of Appoxiae I/O Lineaizaion o Aicaf Fligh Conol Jounal of Dynaic Syses Measueens and Conol Vol. 6 Sepebe 994 pp [] M. G. Skapeis and F. N. Kouboulis "Decoupling of he longiudinal odes of advanced aicaf " AIAA J. of Guidance Conol and Dynaics. vol. 9 pp [] M. G. Skapeis "Disubance ejecion fo an aicaf flying in aospheic disubances" Dynaics and Conol vol. 5 pp [3] R. C. Nelson Fligh Sabiliy and Auoaic Conol McGRAW- HILL Inenaional Ediions

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