Advanced Robust PDC Fuzzy Control of Nonlinear Systems

Transcription

1 Advanced obust PDC Fuzzy Contol of Nonlnea Systems M Polanský Abstact hs pape ntoduces a new method called APDC (Advanced obust Paallel Dstbuted Compensaton) fo automatc contol of nonlnea systems hs method mpoves a qualty of obust contol by ntepolatng of obust and optmal contolle he weght of each contolle s detemned by an ognal ctea functon fo model valdty and dstubance appecaton APDC method s based on nonlnea akag-sugeno (-S) fuzzy systems and Paallel Dstbuted Compensaton (PDC) contol scheme he elaxed stablty condtons of APDC contol of nomnal system have been deved he advantages of pesented method ae demonstated on the nvese pendulum benchmak poblem Fom compason between thee dffeent contolles (obust optmal and APDC) follows that APDC contol s almost optmal wth the obustness close to the obust contolle he esults ndcate that APDC algothm can be a good altenatve not only fo a obust contol but n some cases also to an adaptve contol of nonlnea systems Keywods obust contol optmal contol akag Sugeno (S) fuzzy models lnea matx nequalty (LMI) obseve Advanced obust Paallel Dstbuted Compensaton (APDC) D I INODUCION IFFEEN desgn technques wee developed fo modellng and contol of nonlnea uncetan systems ey nteestng appoach was done n the fuzzy modellng and contol especally wth akag-sugeno (-S) fuzzy modellng [6] and elated Paallel Dstbuted Compensaton (PDC) contol algothm [5] Decson fuzzy vaables ae desgned to dvde the state space of the system nto aeas whee the lnea local models descbe the nonlnea system In the ove-lapped pats of these aeas local models and assocated contolles ae ntepolated accodng to fuzzy membeshp functons -S model based PDC contol of nonlnea systems s qute popula now fo ts smple and effectve desgn based usually on Lnea Matx Inequaltes (LMIs) he obustness of the contolle s vey mpotant popety f model paametes ae uncetan o change n tme Some obust contol technques fo systems descbed by akag- Sugeno fuzzy model wee publshed A dawback of obust contolle s that t s usually slowe than the contolle that s optmally desgned fo an exact model of the system and wok Mchal Polanský s wth the Depatment of Contol and Instumentaton Faculty of Electcal Engneeng and Communcaton Bno Unvesty of echnology Kolení 4 6 Bno Czech epublc (tel: fax : e-mal : polansky@feecvutbcz) on t Pesented method elmnates ths dawback and nceases the qualty of obust contol by ntepolatng of a obust and an optmal contolle accodng to some fuzzy ules Stablty condtons of an oveall system ae based on Lyapunov theoy and can be poven by pescbed LMIs he ctea functon fo model valdty and dstubance appecaton s also pesented hee By settng of some paametes t allows us to lay accent ethe on the obustness o on the qualty of contol pocess he nveted pendulum contol system s adopted to show the pefomance and obustness of the new method and compae t wth optmal and obust contolle ey pomsng esults have been acqued especally n the case of nput dstubance whee the contol s almost optmal and nose attenuaton s same as wth obust contolle II AKAGI-SUGENO FUZZY MODELLING AND CONOL OF NONLINEA SYSEMS A -S model he standad akag-sugeno fuzzy model conssts of the set of fuzzy ules wth lnea consequent that descbe system n local aeas Fo ou pupose we suppose followng fom: ule : IF z M and z ( M and and z n ( s M n HEN x ( Ax( B w( B y( C ( () x whee ( z zn ( ( y yl ( ( x ( xn ( ( u u ( z ae some pemse vaables y s an output vecto x s a state vecto u s a contol nput vecto and m w ( w ( wp ( s a dstubance vecto = denotes the aea s numbe s the numbe of aeas and thus also of fuzzy ules M s a fuzzy set ( M ( z( ) s the gade of membeshp of pemse vaable z ( n the aea numbe ) m s the numbe of nputs and l s the numbe of n n outputs of -S fuzzy system Matxes A n m B n m B l n C descbe the system n the aea numbe 859

2 he defuzzfed output can be then epesented as follows: whee x ( y( )[ A x( B ) C x( n w( B } () M ( z ( ) h (z( ) n (3) M ( z ( ) If z( s n specfed ange then ) hs epesentaton of defuzzfed model can be easly mplemented nto Matlab model B Fuzzy State Obseve Hee we can suppose that the dstubance sgnal s not measued Fuzzy state obseve s then descbed by smla fuzzy ules: IF z M and z ( M and and z n ( s M n HEN x ˆ( Axˆ( B G[ y( yˆ( ] yˆ ( C ˆ( (4) x whee G s an obseve gan n the aea numbe x ˆ ( xˆ xˆ ( s estmated state vecto and n ) y ˆ ( yˆ yˆ ( t s a vecto of estmated outputs l Afte defuzzyfcaton we get the followng equatons: x ˆ( h ( z( ) Axˆ( B G [ y( yˆ ( ] y ˆ( ) C xˆ( (5) C PDC Fuzzy Contol Algothm he contol sgnal of PDC contolle s computed as ) K x( (6) m n whee K s a constant feedback gan If some state vaables have to be estmated by fuzzy obseve then the contol output s computed n the followng way: ) K xˆ( (7) Optmal PDC contolle K O u O ( x : x( Auxlay PDC contolle K A u A () t x + Plant y( obust PDC contolle K u () t x f( y( y( : y( Fuzzy model and obseve x( f( y( Fg Closed loop wth APDC contol algothm 86

3 I ADANCED OBUS PAALLEL DISIBUED COMPENSAION he basc dea of ths method s to ncease qualty of obust contol by ntepolaton of obust and optmal contolle he ntepolaton algothm uses the same pncples as the PDC contol only t ntepolates the whole PDC contolles It s on the fgue he obustness s n the sense of H nom that detemne dstubance attenuaton of the contol system he optmal contolle s desgned to mnmze pescbed ntegal cteon whee we can expess equed ato between speed and enegy consumpton Auxlay contolles ae used to mantan monotonc change of these popetes dung ntepolaton In the followng K Ag denotes the gan of contolle numbe g g n the aea s a numbe of contolles Let s K A K and K A KO be the gan of obust and optmal PDC contolle espectvely Dung contol we wll ntepolate obust PDC contolle K K K A wth that ethe the optmal o the fst auxlay contolle hen K A wth K A3 and so on he oveall contol output s computed as h Ag g ( z ( ) ) K Ag x( whee weghtng functons h ( z ( ) ae taken fom the -S model () Weghtng functons hag ( z ( ) ae deved n the ctea functon fo model valdty and dstubance appecaton hs computaton must guaantee that A (8) h ( z Ag ( ) (9) h ( z ( ) () Ag g Fo bette nsght denote the weght of obust and optmal PDC contolle as: h ( z ( ) h z ( ) a h ( z ( ) h ( z ( ) () A O A CIEIA FUNCION FO MODEL ALIDIY AND DISUBANCE APPECIAION Ou am s to estmate qualty of a -S model and level of dstubance sgnals Fo ths we wll defne two sgnals: f ( ) A xˆ( B t () m c( ) G y( yˆ( ) (3) Both sgnals f (t m ) and c ( can be easly compaed Sgnal f m( coesponds to estmated tme devatve of the state vaables of -S model and c ( detemnes absolute value of the estmaton eo multpled by the obseve gan Sgnals c ( c ( cn( and f m ( fm fmn( ae used fo computaton of decson vaable z ( n kcc ( z ( z ( (4) d n fm ( kcc ( km whee k c n ae postve eal coeffcents that allow us to lay accent ethe on the obustness o qualty of the contol pocess By the exponent we can ncease senstvty to dstubance that can speed up settng of the obust contolle he constant k only ensue that thee m wont be dvson by zeo when fm ( kcc ( he low pass flte wth tme constant d s ntoduced to ncease contnuty and pevent an algebac loop Nomally the weght of obust contolle h ( z ( ) z ( I SABILIY ANALYSIS heoem : Equlbum of a -S fuzzy system wth APDC contol (8) wth standad fuzzy patton (SFP) s asymptotcally stable n lage f thee exsts a common matx P k > symmetc matx g fo each MOG S k F Q kag and matces k q such that QkAg Q kag P P F Q lk k q (5) Ag k k Ag kag Ag Ag Pk Pk FAg FAg kag kag lk [ QkA ] kai kag [ QkA ] kai F F Q Q whee F Ag A B K Ag he poof can be found n [3] (6) (7) he postve defnte matxes P k k q ae elated to a numbe q of the Most Ovelapped ules Goup (MOG) aeas that ae defned as the aeas wth the hghest numbe of ovelapped local aeas n Standad Fuzzy Patton (SFP) he SFP of two dmensonal state space wth MOGs S S S 3 and S 4 s on the fgue he unon of all MOG aeas s the q unvesum of the fuzzy system Sk S k Fg Standad fuzzy patton of -S system 86

4 k (x) s a chaactestc functon of each MOG S k It ndcates f the wokng pont of the state space belongs to the MOG numbe k It s defned as xsk k ( x( ) k ( x( ) (8) othe q k Moe nfomaton and defntons about SFP can be found n [5] whee s also the basc veson of stablty condtons Note that the condtons wth PSQ Lyapunov functons cannot be dectly conveted to a contolle desgn method due to dscontnuty of ( x( ) Fom the Sepaaton popety fom Ma et Sun [4] follows that the contol system wth obseve and PDC contolle s stable f both of them ae stable sepaately Snce APDC (8) behave as a specal case of PDC contol then the sepaaton popety wll hold also fo APDC contol system II APDC DESIGN POCESS APDC contol algothm solve ethe fo mpovng of exstng obust contol systems o fo a new contolle desgn whch s show n ths chapte It conssts of optmal and obust contolle desgn obseve desgn smulaton and settng of ctea functon paametes A Optmal contolle desgn Pesented method have been publshed by L Wang Bushnell Hong and anaka n [] he optmzaton contol obectve s to mnmze an ntegal cteon J k y ( Wy( u ( (9) whee W=W > detemne the weght of output eo and = > detemne the weght of enegy consumptons he method poduces a sub-optmal contolle that s close to an optmal one n the case of mpotant weght of enegy consumpton he poof can be found n [] heoem : Equlbum of a -S fuzzy system wth a PDC contol (6) s asymptotcally stable n lage f thee exsts a common postve defnte matx Z > and matces M such that the followng LMI condtons hold AZZA BM M B / W CZ / M ZC W / W CZ I / W C Z / M / M x () () x Z () / ZC W I / ZC W I M I / / M I M I / / () () whee n () and n () A ZA ZZA ZA B M B M M B M B he ntegal cteon J then wll be less than he suboptmal contolle gan can be computed as K he obectve s to mnmze B obust contolle desgn MZ (3) he method have been publshed by Lee Jeung and Pak n [] he obectve s to mnmze H nom that detemne a dstubance attenuaton and s fequently used also as a obustness measue he poof can be found n [] y( w( (4) he method mnmzes that s an uppe bound of It s summased n heoem 7 yw heoem 3: Equlbum of a -S fuzzy system wth a PDC contol (6) s asymptotcally stable n lage wth a decay ate and f thee exsts a common postve yw defnte matx Z > > scala and matces M such that the followng LMI condtons hold B ZC B I (5) CZ I B B ZC ZC B B I (6) CZC Z I whee AZZA BM M B Z he gan of the obust contolle the wll be: K M Z (7) he optmzaton obectve s to mnmze C Obseve desgn An ognal method fo fuzzy obseve desgn s poposed hee he motvaton fo development of a new method s a need of fast obseve wth a mnmal estmaton eo Although the contol stablty s guaanteed wthn the unvese of a fuzzy system [4] the estmaton affects the ntegal ctea and H nom hen t s mpotant that poles of the obseve loop ae moe negatve than the poles of the contolle loop he desgn method allows settng of obseve loop poles nto a specfed aea D { a bc:( aqd ) b D } qd D It also ntoduces H nom that detemnes attenuaton of the dstubance sgnal w( on the output estmaton eo sgnal y( yˆ( he poof can be found n [3] 86

5 heoem 4: State fuzzy obseve (5) mantans asymptotcally stable estmaton eo e ( x( xˆ( H nom w wth poles n the aea D and decay ate e f thee exsts matces J Q Q common postve defnte matx Y and scalas a that the followng LMIs hold Q YB B Y ei Q Q B YB Y [ Q ] I q e e YB YB I D D Y YYA J C q YA YC J D Y D (8) (9) (3) whee A YC J YA J C C C Y III SIMULAION ESULS (3) Inveted pendulum have been epesented by an exact mathematcal model and by -S state space model dvded nto thee lnea local aeas (aound angle -/3 and /3) he state vaables wee x-angle [ad] x-angula speed [ad/s] x3-cat poston [m] and x4-spee of the cat [m/s] Fo ths model wee desgned optmal and obust contolle and obseve such that the stablty condtons of APDC contol ae satsfed H nom and ntegal cteon J changes monotoncally dung ntepolaton so that auxlay contolles wasn t necessay he values wee the followng: obust contolle: max H nom s 9 J max = 593 Optmal contolle: max H nom s 366 J max = 3334 ABLE I INEGAL CIEION J IS ENEGY PA (Ju) AND OUPU PA (Jy) APDC contol Optmal contol obust contol mul J Ju Jy J Ju Jy J Ju Jy 6 Stable Unstable Unstable Stable Unstable Stable Unstable Stable he contol qualty and obustness analyss of thee systems (wth the obust optmal and APDC contolle) was pefomed he APDC algothm acheved ove 3% bette ntegal ctea J than the obust contolle and almost 4% smalle enegy consumpton dung contol of the nomnal system fom the ntal condtons x () xˆ() [ ] he 3 obustness fo paametc uncetantes was tested by multplyng of the system matces A by a coeffcent mul Impotant esults n the table I shows that APDC contol qualty nceased n a wde ange of paamete mul compae to the obust contolle Paametcal obustness emaned at almost the same level as wth the obust contolle In the anges mul (59;73) and mul (3;4) ae the esults of APDC bette than esults of both obust and optmal contolles ey nteestng esults wee obtaned also fo systems dstubed by an nput nose A whte nose of dffeent nose powe was put on nputs of all thee systems he contol esponse wth nose powe 5 fom ntal condtons x() xˆ() [ ] s shown at the fg x t Fg 3 Contol esponse of all systems n the pesence of dstubance he esults ndcate that APDC algothm connects good popetes of optmal and obust PDC contol and acheve the best pefomance Fom the ntal condtons t egulates almost optmally to the equlbum but the nose attenuaton s the same as wth the obust contolle he esponse of ntegal cteon J shows fg4 Fo APDC t s small afte egulaton and also t nceases slowly wth the nose Afte 8 seconds t s the same as wth the optmal contolle 5 5 J( 5 APDC Optmal obust APD COptmal obus t t Fg 4 Integal cteon J esponse of all systems n the pesence of stong dstubance 863

6 8 h( t Fg 5 Weghtng functon o obust contolle n the pesence of small dstubance he same behavou was obtaned wth a small nose powe Afte mnutes s J value fo APDC agan bellow the othes he weght of obust contolle n ths stuaton shows fg5 able II demonstates the dffeence among the ntegal cteons J of all thee systems afte seconds he APDC esults ae taken as % ABLE II INEGAL CIEION J AND IS PAS UNDE DISUBANCE IN SEC APDC contol Optmal contol obust contol nose pow J Ju Jy J Ju Jy J Ju Jy % % % 98% 9% 5% 3% 39% 98% % % % 5% 99% 84% 34% 37% 93% IX CONCLUSIONS he APDC contol algothm could damatcally ncease qualty and educe enegy consumpton of the contol pocess he contol s almost optmal wth the obustness close to the obust contolle By settng of ctea functon coeffcent we can lay accent ethe on qualty o the obustness of the contol pocess ey mpotant s that all contolles and obseve can be effectvely numecally found by solvng LMIs and the stablty of oveall APDC contol system can be poved he smulatons pefomed above also show that the APDC algothm can be vey useful n the pesence of nput nose of dffeent ntensty fom vey dstubed to a neglgble nose It pefoms bette not only than the obust but also than the optmal contolle Its ognal dea s smple and natual so that t could became popula n many futue applcatons EFEENCES [] K Lee E Jeung H B Pak obust fuzzy H contol fo uncetan nonlnea systems va state feedback: an LMI appoach Fuzzy Sets and Systems vol p 3 34 [] J L O Wang L Bushnell Y Hong K anaka A Fuzzy Logc Appoach to Optmal Contol of Nonlnea Systems Intenatonal Jounal of Fuzzy Systems ol No 3 [3] M Polansky Nová metoda APDC po zvýšení kvalty obustního ízení nelneáních systém Dssetaton thess Faculty of electcal engneeng and Communcaton at Bno Unvesty of echnology 5 n Czech [4] X J Ma Z Q Sun Analyss and desgn of fuzzy contolle and fuzzy obseve IEEE ans Fuzzy Systems 998 vol 6 no p 4 5 [5] Z H Xu G en Stablty analyss and systematc desgn of akag Sugeno fuzzy contol systems Fuzzy Sets and Systems 4 Avalable fom wwwscencedectcom [6] akag M Sugeno Fuzzy dentfcaton of systems and ts applcatons to modelng and contol IEEE ansactons on Systems Man and Cybenetcs 985 vol no p