PARETO OPTIMAL ROBUST FEEDBACK LINEARIZATION CONTROL OF A NONLINEAR SYSTEM WITH PARAMETRIC UNCERTAINTIES

Transcription

1 INTERNATIONA JOURNA ON SMART SENSING AND INTEIGENT SYSTEMS VO. 7, NO., MARCH 4 PARETO OPTIMA ROBUST FEEDBACK INEARIZATION CONTRO OF A NONINEAR SYSTEM WITH PARAMETRIC UNCERTAINTIES A. Hajloo,, M. saad, N. Naran-Zadeh Departent of Mechancal and Indstral Engneerng, Concorda Unversty, Montreal, Canada. Departent of Mechancal Engneerng, Islac Azad Unversty, Taestan, Iran. Departent of Mechancal Engneerng, Engneerng Faclty, Glan Unversty, Rasht, Iran. Eals: a_haj@encs.concorda.ca; gdarzsaad@gal.co; nnzadeh@glan.ac.r Sbtted: Dec. 5, 4 Accepted: Feb. 7, 4 Pblshed: Mar., 4 Abstract- The proble of lt-objectve robst feedbac lnearzaton controller desgn of nonlnear syste wth paraetrc ncertantes s solved n ths paper. The an objectve of ths paper s to prope an optal technqe to desgn a robst feedbac lnearzaton controller wth lt-objectve genetc algorth. A nonlnear syste s consdered as a benchar and feedbac lnearzaton controller s desgned for deternstc and probablstc odel of the benchar. Three and for conflctng objectve fnctons are sed n Pareto desgn of feedbac lnearzaton controller for deternstc and probablstc desgn, respectvely. The slaton resl reveal the effectveness of the proped ethod. Inde ters: Pareto, Feedbac nearzaton, Paraetrc Uncertanty, Robst Desgn. 4

2 A. Hajloo, M. Saad, N. Naran-Zadeh, PARETO OPTIMA ROBUST FEEDBACK INEARIZATION CONTRO OF A NONINEAR SYSTEM WITH PARAMETRIC UNCERTAINTIES I. INTRODUCTION Feedbac lnearzaton s a nonlnear control ethod whch s sed a coordnate transforaton to change a nonlnear syste nto a lnear one []. The obtaned lnear syste can be controlled by a lnear control ethod, sch as state feedbac control. Therefore, the feedbac lnearzaton ethod can be classfed nto two par: one part cancels ot the nonlneartes of the plant, and other part controls the lnearzed syste []. There are any ethods to desgn sch controllers whch can be regarded as optzaton probles of certan perforance easres of the controlled systes []. A very effectve eans of solvng sch opt desgn of controllers s genetc algorths (GAs) [4, 5]. The splcty and global characterstcs of sch evoltonary ethods have been the an reasons for ther etensve applcatons n off-lne opt control syste desgn. However, a ajor restrcton of feedbac lnearzaton ethod s that the syste shold be precsely nown, n order to cancel the nonlneartes eactly []. In real control engneerng probles, however, eact odels of the systes are not often nown [6]. Addtonally, there are any sorces of ncertantes sch as paraetrc ncertantes that prevent the effectveness of sch feedbac lnearzaton [7]. Therefore, t s strongly needed to desgn a robst feedbac lnearzaton ethod whch can stand the ncertantes and provdes satsfactory both robst stablty and robst perforance n the presence of ncertantes and odelng error. In ths regard, any researches have been accoplshed n robst feedbac lnearzaton ethod [8, 9]. Mt of the are based on the worst case robst analyss and synthess whch se the t pessstc vale of the perforance for a partclar eber of the set of ncertan odels []. The conservats nvolved n sch approaches s not very desrable n robst control desgn and the t lely plan wth respect to ncertantes shold be consdered []. There have been any effor for desgnng robst controllers redcng the conservats or accontng ore for the t lely plan wth respect to ncertantes. Ths dea resl n propagatng the probablstc ncertantes throgh the ncertan odel whch blds the set of plan as the actal dynac systes each wth a separate probablty densty fncton (PDF) []. Therefore, sch nforaton regardng the lelhood of each plant allows a relablty-based desgn n whch probablty s ncorporated n the robst desgn []. In ths ethod, robstness and perforance 5

3 INTERNATIONA JOURNA ON SMART SENSING AND INTEIGENT SYSTEMS VO. 7, NO., MARCH 4 are stochastc varables []. Stochastc behavor of the syste can be slated by Monte-Carlo Slaton (MCS) [6]. Robstness and perforance can be consdered as objectve fnctons wth respect to the controller paraeters n an optzaton proble. Wang et al. [9] sed genetc algorths to desgn robst nonlnear controls, n whch both stablty and perforances etrcs was agented for a sngle objectve optzaton proble. Snce conflctons est between robstness and perforance etrcs, chong approprate weghtng factor n the ct fncton consstng of weghted qadratc s of the non-coensrable objectves s nherently dffclt and cold be regarded as a sbjectve desgn concept. Moreover, the trade-offs ested between the objectves cannot be eplored and t wold be, therefore, psble to choe an approprate opt desgn reflectng the copre of the desgner s choce concernng the absolte vales of objectve fnctons. Therefore, ths proble shold be forlated as a lt objectve optzaton proble (MOP) so that the trade-offs between objectves can be fond conseqently [4, 5]. The a of ths paper s to se a lt-objectve optzaton approach to robstly desgn a feedbac lnearzed controller for an ncertan nonlnear syste. A nonlnear two asses and sprng syste s consdered and the nonlnear controller wll be desgned for both deternstc and probablstc statons. In ths regard, three non-coensrable objectve fnctons are consdered for deternstc desgn and for non-coensrable objectve fnctons are consdered for probablstc one. The coparson wll be accoplshed to show the robstness and sperorty of the probablstc desgn over the deternstc desgn n the presence of paraetrc ncertantes of the nonlnear odel. The organzaton of the paper s as follows. After a general ntrodcton of the effect of falt on the power syste, the seflness and reqreent of a falt crrent lter s presented to the stden whch has been dscssed n secton II. The tradtonal ways of fng falt crren n power syste has been dscssed n secton III. In secton IV, operatng prncple, desgn detals, and eperental resl of agnetc crrent lter has been presented. The analyss and slaton resl of hgh teperatre spercondctng falt crrent lter has been dscssed n secton V. The lectre has been conclded n secton VI. 6

4 A. Hajloo, M. Saad, N. Naran-Zadeh, PARETO OPTIMA ROBUST FEEDBACK INEARIZATION CONTRO OF A NONINEAR SYSTEM WITH PARAMETRIC UNCERTAINTIES II. ROBUST FEEDBACK INEARIZATION DESIGN Consder a nonlnear syste as follow ( ) n f ( ) g( ) ( t) q( ) w( t), () y h( ) where f (.), g (.), and q (.) are nonlnear fnctons, ( n) T [,,..., ] [,,..., ] s the state vector of the syste, w s the bonded dstrbance, and and y are the npt and otpt of the syste, respectvely. In the feedbac lnearzaton ethod, f the nonal syste s feedbac lnearzable, there es an algebrac transforaton whch s sed to lnearze nonlnear syste []. The cled-loop syste nder the feedbac lnearzaton ethod s represented n the Fgre. Ths control strctre conss of two loops, wth the nner loop achevng the lnearzaton of the npt-state (or npt-otpt) relaton, and the oter loop achevng the stablzaton of the cled-loop dynacs []. n T R n Fgre. Feedbac nearzaton [] The lnearzed syste eqaton wth respect to dstrbance w(t) can be wrtten as z Az bv Cw b (z) w, () where A, nn b. () n The new lnearzng state z can be defned as z n T [ h ( ), f h( ),, f h( )], and h() f s the e dervatve of h( ) f along the vector feld f, and C s the n constant atr and (.) s a nonlnear fncton []. The lnear control law can be deterned by a lnear control desgn sch 7

5 INTERNATIONA JOURNA ON SMART SENSING AND INTEIGENT SYSTEMS VO. 7, NO., MARCH 4 as pole placeent ethod. The nonlnear feedbac control law based on the feedbac lnearzaton ethod can be then obtaned as where () and () ( ) ( ) v, (4) can be derved as n f h( ) ( ), (5) n h( ) g f ( ). (6) h( ) g n f A cled-loop syste n the feedbac lnearzaton ethod s stable wth respect to the dstrbance w(t) f a ptve-defnte atr P can be fond fro the Rccat eqaton wth the desgn paraeters Q and r [7] T T A P PA r Pbb P Q, (7) where, Q s a ptve-defnte atr and r s the ptve real nber. Therefore, the lnear control law can be gven as v r b T Pz b T Pz. (8) In order to optally desgn a feedbac lnearzaton controller the desgn vector d { Q, r, } shold be fond approprately. If there s any ncertanty n the odel paraeters n feedbac lnearzaton ethod, t wll case error n the new state and the control npt, and the otpt of the syste ay deterorate accordngly []. Therefore, t s benefcal for control engneers to desgn controllers robstly. In the stochastc robst control desgn, an ncertan nonlnear syste wth the set of ncertan paraeters, p, s consdered so that the deternstc nonlnear syste gven n () can be rewrtten n a new for wth paraetrc ncertantes as follows ( ) n f (, p) g(, p) ( t) q(, p) w( t) y h(, p). (9) In the stochastc robst desgn, the ncertan paraeters, p, consdered as probablstc ncertantes whch have PDFs, for eaple, noral dstrbton. Therefore, the propagaton of the ncertan paraeters throgh the syste provdes soe probablstc etrcs sch as rando varables (e.g., settlng te, a overshoot ), and rando processes (e.g., step response, 8

6 A. Hajloo, M. Saad, N. Naran-Zadeh, PARETO OPTIMA ROBUST FEEDBACK INEARIZATION CONTRO OF A NONINEAR SYSTEM WITH PARAMETRIC UNCERTAINTIES Bode or Nyqst dagra ) n a control syste desgn [], [6, 7]. Ths staton leads to varatons n the perforance of an syste. Therefore, t s very desrable to fnd robst desgn whe perforance varaton n the presence of ncertantes s low. Generally, there est two approaches addressng the stochastc robstness sse, naely, robst desgn optzaton (RDO) and relablty-based desgn optzaton (RBDO) [8]. Both approaches represent nondeternstc optzaton forlatons n whch the probablstc ncertanty s ncorporated nto the stochastc optal desgn process. In ths wor, the relablty-based desgn optzaton ethod s sed to desgn a robst nonlnear controller. In ths ethod, rando saples are generated assng soe pre-defned probablstc dstrbtons for ncertan paraeters. The syste s then slated wth each of these randoly generated saples and the percentage of cases prodced n falre regon defned by a lt state fncton approately reflec the probablty of falre. et X be a rando varable, then the prevalng odel for ncertantes n stochastc randoness s the probablty densty fncton (PDF), f X or eqvalently by the clatve dstrbton fncton (CDF), F X, where the sbscrpt X refers to the rando varable. Ths can be gven by X X f F d () where (.) s the probablty that an event (X ) wll occr. X In the relablty-based desgn, t s reqred to defne relablty-based etrcs va soe neqalty constran. Therefore, n the presence of ncertan paraeters of plant (p) whe PDF or CDF can be gven by fp(p) or Fp(p), respectvely, the relablty reqreen can be gven as p g p, P () f 9

7 INTERNATIONA JOURNA ON SMART SENSING AND INTEIGENT SYSTEMS VO. 7, NO., MARCH 4 where, P denotes the probablty of falre of the th relablty easre, g (p) s the lt state f fncton whch separates the falre regon g ( p) fro the safe regon g ( p). In the relablty-based desgn shold be nzed. Therefore, tang nto consderaton the stochastc dstrbton of ncertan paraeters (p) as for each probablty fncton as p g p f p p f p p, eqaton () can now be evalated P dp, () f g p Ths ntegral s, n fact, very coplcated partclarly for systes wth cople g(p) [9] and MCS s alternatvely sed to approate eqaton (4). Based on ths ethod the probablty sng saplng technqe can be estated sng P f g N () N p I p F,, p, where, F (.) s the th nonlnear syste that s slated by MCS. Also, I gp s an ndcator whch cases the probablty of falre to be obtaned nercally, and defned as g( p) I g p. (4) g( p) Therefore, the probablty of falre whch eans the falre occrred s the nber of saples n the falre regon dvded by the total nber of saples. Evdently, sch estaton of P f approaches to the actal vale n the lt as N [9]. However, there have been any research actvtes on saplng technqes to redce the nber of saples eepng a hgh level of accracy. Alternatvely, the qas-mcs has now been ncreasngly accepted as a better saplng technqe whch s also nown as Haersley Seqence Saplng (HSS) [9]. In ths wor, HSS has been sed to generate saples for probablty estaton of falres. In a RBDO proble, the probablty of falre of soe etrcs shold be nzed sng an optzaton ethod. In a lt-objectve optzaton of a RBDO proble presented n ths paper, however, there are dfferent conflctng robst etrcs that shold be nzed sltanely. In ths wor, lt-objectve Pareto genetc algorth of MATAB s sed for RBDO of robst feedbac lnearzaton control desgn.

8 A. Hajloo, M. Saad, N. Naran-Zadeh, PARETO OPTIMA ROBUST FEEDBACK INEARIZATION CONTRO OF A NONINEAR SYSTEM WITH PARAMETRIC UNCERTAINTIES III. ROBUST CONTRO DESIGN FOR A NONINEAR SPRING-MASS SYSTEM In ths secton, a lt-objectve robst controller desgn s descrbed. The proble s a two ass and sprng benchar whch ncldes two asses (, ), a lnear sprng ( ), and a nonlnear cbc sprng ( ).The control sgnal apples to and nt plse dstrbance eer on (Fgre ). The state space representaton of the syste s [7] Fgre. A nonlnear sprng-ass syste 4 4, w (5) y. (6) In ths syste the ncertan paraeters nclde p,,, ]. The nonal vale of the [ are,. 5, and. 5. The correspondng constran are consdered as follows.5.5,.5,.5.5.,.5. (7) The npt-otpt lnearzaton of ths sngle npt/otpt syste can be accoplshed by sng z. The eqatons of ths procedre whch are adopted fro [7] are gven as z h(, ) 4 z f h,

9 INTERNATIONA JOURNA ON SMART SENSING AND INTEIGENT SYSTEMS VO. 7, NO., MARCH 4 h z f, 4 4 h z f. (8) As a reslt of the above transforaton, the lnear eqaton can be wrtten as z z, w z z, z 4 z, hw h h z f q f g f 4 4, (9) where,, h f () h f g, () h f q. () Then transfored syste can be forlated as, w w v b C b Az z () where C. (4)

10 A. Hajloo, M. Saad, N. Naran-Zadeh, PARETO OPTIMA ROBUST FEEDBACK INEARIZATION CONTRO OF A NONINEAR SYSTEM WITH PARAMETRIC UNCERTAINTIES Then the control law of the syste can be wrtten as 4 T T h r b Pz b. f g f h Pz (5) In ths stdy, the approprate vales of the paraeters of the control law are desgned by ltobjectve optzaton. For objectve fnctons are consdered to desgn robst optal nonlnear controller for ths benchar. In the frst place, the t portant goal of the robst controller desgn s the robst stablty whch ples that all the cled-loop systes rean stable n the presence of any ncertanty. Defnton. For the npt control sgnal (t) that leads the syste to stablty, there shold est a class K fncton and a class K fncton sch that [] y( t) (, t) ( ), (6) holds for all soltons. Ths syste s npt-to-otpt stable []. denotes Ecldean nor and denotes nfnty nor. Ths, n the case of stochastc robst desgn, the lt state fncton to defne the probablty of nstablty wll be represented by g ns ( p ) (, t) ( ) y( t). (7) The probablty of nstablty can now be copted as ns N I g ns F,, p. (8) The robst perforance s another crteron for desgnng nonlnear controllers. In ths stdy robst perforance ncldes settlng te and overshoot. Ther lt state fnctons are defned as g (p) 7 Ts, (9) g (p), () where, Ts and OS are settlng te and overshoot of the otpt (), respectvely. Therefore, the probablty of the falre for each of the can be defned as N I g F,, p, ()

11 INTERNATIONA JOURNA ON SMART SENSING AND INTEIGENT SYSTEMS VO. 7, NO., MARCH 4 N I g F,, p. () The control effort plays a pvotal role n the control desgn, and n any practcal cases, there s a ltaton on control effort whch st be consdered by control engneers. The lt state fncton of ths crteron s g (p), () and the falre probablty of the control effort can be forlated as N I g F,, p. (4) These robst etrcs shold be nzed sltanely sng lt-objectve optzaton. In the net secton the resl of sch lt-objectve optzaton procedre wll be gven. IV. RESUT In order to show the spreacy of the robst desgn, the lt-objectve optzaton wll be accoplshed for both deternst and probablstc feedbac lnearzaton desgn. In the frst place, t s obv that dsplayng ore than two objectve fnctons to deonstrate the trade-off s not feasble. In ths way, several ltdensonal vsalzaton ethods are proped [, ]. One of the ethods whch leads to coprehensve analyss of the Pareto front s called evel Dagras ethod [] whch s sed n ths paper to vsalze the Pareto fron of the lt-objectve optzaton. In ths ethod, each pont of Pareto front st be noralzed between and based on n and a vales [] J J M a J, J n J,,..., n, (5) J J, (6) M J J ovded that the orgn of the n-densonal space s consdered as the deal pont, the dstance of the each Pareto front pont can be sed for coparson. In ths paper, Ecldean nor of all objectve fnctons ( J n J ) s sed for ths prpe. To represent the Pareto front, Y as s specfed for Ecldean nor of all objectve fnctons and X as s specfed for each 4

12 A. Hajloo, M. Saad, N. Naran-Zadeh, PARETO OPTIMA ROBUST FEEDBACK INEARIZATION CONTRO OF A NONINEAR SYSTEM WITH PARAMETRIC UNCERTAINTIES objectve fncton; therefore, each objectve fncton has own graphcal representaton whlst Y as of each graph wold be the sae. a. Deternstc Desgn The three objectve fnctons are now consdered sltanely n a Pareto optzaton process to obtan soe portant trade-offs aong the conflctng objectves. In a deternstc desgn approach, the vector of objectve fnctons to be optzed n a Pareto sense s gven as follow J Ts, OS,. (7) In the deternstc desgn, n order to prevent nstablty, f a syste s nstable, each objectve fncton s coerced to. The evoltonary process of the lt-objectve optzaton s accoplshed wth a poplaton sze of whch has been chen wth crsover probablty Pc and taton probablty P as.8 and., respectvely. A total nber of non-donated opt desgn pon have been obtaned for optal feedbac lnearzed controller. The resl of the -objectve optzaton process are shown n Fgre. It s evdent fro fgre, that there s conflcton between settlng te and control effort, also, between overshot and control effort. The V shape of the Pareto fron confrs ths staton. Tae for llstraton, the syste whe settlng te s low has the worse vale of control effort. Ths staton s very evdent n the evel Dagras of both overshoot and nfnty nor of control effort, becase the syste wth lowest vale of overshoot has the a vale of the control effort and vce versa. Therefore the prng resl show the conflcton between each par of objectve fnctons whch can be fond fro ths fgre. Also, the crcled pont whch s the vertces of the V layot of each Pareto front (the nearest pont to deal) s very cle to the n vale of each objectve fncton, hence, t can be chen as a pvotal trade-off pont. The objectve fnctons and desgn varables for n vale of each objectve fncton and for n vale of the -nor fncton are gven n Table. 5

13 INTERNATIONA JOURNA ON SMART SENSING AND INTEIGENT SYSTEMS VO. 7, NO., MARCH Pareto pon n Ts n OS n.5.5 Pareto pon n Ts n OS n n J n J J.9 J Ts OS.5.5 Pareto pon n Ts n OS n n J J Fgre : -Nor evel Dagras of Pareto fron for deternst desgn Table. The vales of objectve fnctons and the correspondng desgn varables of the selected opt pon. Opt Pon Desgn Varables Objectve Fnctons q q q q 4 r Ts OS J n Ts n OS n n J

14 A. Hajloo, M. Saad, N. Naran-Zadeh, PARETO OPTIMA ROBUST FEEDBACK INEARIZATION CONTRO OF A NONINEAR SYSTEM WITH PARAMETRIC UNCERTAINTIES The response of each consdered optal controller aganst nt plse dstrbance s shown n Fgre 4. In addton, the correspondng control effor are shown n Fgre n Ts n OS n n J (t) Te (sec) Fgre 4: The responses of the opt selected pon wth respect to the nt plse 4 n Ts n OS n n J (t) Te (sec) Fgre 5: The control effor of the opt selected pon wth respect to the nt plse Accordng to fgres 4 and 5, desgnng hgh-perforance and ct effectve controllers s a very cople sbject whch cold be practcally psble, becase the systes whch have a good te response need hgh ct control effort, and, on the other hand, the syste whe control effort s low has the adverse te response. 7

15 INTERNATIONA JOURNA ON SMART SENSING AND INTEIGENT SYSTEMS VO. 7, NO., MARCH 4 Each state of the controlled syste wth the crcled pont ( n J or the trade-off desgn) s shown n Fgre 6. It s obv that the syste s npt-state stable syste becase all of the states are asyptotcally stable, and they have acceptable settlng tes (t) (t) (t) 4 (t) States b. obablstc Desgn Te (sec) Fgre 6: Te response of each state of the pont wth n -nor In ths secton the stochastc desgn wll be accoplshed to obtan robst feedbac lnearzaton controller. In ths way, for objectve fnctons are now consdered sltanely n a Pareto optzaton process to obtan soe portant trade-offs aong the conflctng objectves. In a robst desgn approach, the vector of objectve fnctons to be optzed n a Pareto sense s gven as follow J,,,, (8) ns whe eleen are copted by eqatons (8), (), (), and (4), respectvely, n the qas- Monte Carlo slaton process. In ths stdy, two nds of ncertanty are consdered, naely nfor dstrbton and Gassan dstrbton. In nfor dstrbton the range of syste paraeters have been lsted n eqaton (7), and for Gassan dstrbton, ean and varance of 8

16 A. Hajloo, M. Saad, N. Naran-Zadeh, PARETO OPTIMA ROBUST FEEDBACK INEARIZATION CONTRO OF A NONINEAR SYSTEM WITH PARAMETRIC UNCERTAINTIES each paraeter are chen whch have the sae range as the of the nfor dstrbton. The paraeters that descrbe the Gassan dstrbton are lsted n Table. The evoltonary process of the lt-objectve optzaton for both nfor and Gassan dstrbtons are accoplshed wth a poplaton sze of whch has been chen wth crsover probablty Pc and taton probablty P as.8 and., respectvely. The optzaton process of the robst controller s accoplshed for Monte Carlo evalatons sng HSS dstrbton for each canddate control law drng the evoltonary process. Table. Uncertanty odel Mean Varance b. Mlt-objectve optzaton resl for Gassan dstrbton A total nber of 7 non-donated opt desgn pon have been obtaned for robst feedbac lnearzed controller wth the consdered Gassan dstrbton for ncertan paraeters. Snce, the vale of probablty of nstablty ( ns ) of all non-donated opt pon has been fond eqal to zero, the resl of the 4-objectve optzaton process, conseqently, correspond to a -objectve optzaton process whch s shown n Fgre 7. Also, the crcled pont n Pareto front whch has the lowest vale of the -nor, can be selected as a sgnfcant trade-off pont. Ths pont whch s located on the vertces of the V layot for of the evel Dagra has the low vale of and the oderate vale of two other objectve fnctons. For coparson, the objectve fnctons and desgn varables for n vale of each objectve fncton and for n vale of -nor are gven n Table. It s obv fro the nercal resl n Table that the systes whe have the % falre n the control sgnal reqreent. and are low, 9

17 INTERNATIONA JOURNA ON SMART SENSING AND INTEIGENT SYSTEMS VO. 7, NO., MARCH 4.5 Pareto pon n.5 Pareto pon n.5 n.5 n n n n J J n J J Pareto pon n n J n n J Fgre 7: -Nor evel Dagras of Pareto fron for probablstc desgn wth Gassan dstrbton Table. The vales of objectve fnctons and the correspondng desgn varables of the selected opt pon. Opt Pon Desgn Varables q q q q 4 r Objectve Fnctons J n n n n J

18 A. Hajloo, M. Saad, N. Naran-Zadeh, PARETO OPTIMA ROBUST FEEDBACK INEARIZATION CONTRO OF A NONINEAR SYSTEM WITH PARAMETRIC UNCERTAINTIES The stochastc response of each entoned opt pont s shown n Fgre 8. In ths fgre, pper nterval and lower nterval llstrate that all slated response for saples are lad between the. Also, the stochastc response of the proped controller fro [7] s shown n Fgre n Upper Interval Mean ower Interval.4. n Upper Interval Mean ower Interval.8.8 Otpt.6 Otpt Te (sec) Te (sec).5 n Upper Interval Mean ower Interval.4. n J Upper Interval Mean ower Interval Otpt.5 Otpt Te (sec) Ref. [7] Te (sec) Upper Interval Mean ower Interval Otpt Te (sec) Fgre 8: obablstc responses

19 INTERNATIONA JOURNA ON SMART SENSING AND INTEIGENT SYSTEMS VO. 7, NO., MARCH 4 Accordng to Fgre 8, both pon wth the lowest vale of and have the sae stochastc te responses wth respect to the nt plse. In contrast, the syste whe s the lowest has the worse stochastc te responses n coparson wth other pon. Also, the pont wth the lowest vale of -nor whch can be consdered as an otandng trade-off pont has the acceptable stochastc te responses. In addton, t can be seen that the stochastc te response of the proped controller by [7] s worse than that of proped controller by ths paper. The PDFs of each stochastc varable for each selected opt pont are shown n Fgre 9. It can be conclded fro Fgre 9 that the syste wth the lowest vale of has the settlng te bgger than 7 seconds. As t was shown n Fgre 8, both systes wth the lowest vale of and have the sae stochastc te response and sae effort are sae for both of the. b. Mlt-objectve optzaton resl for nfor dstrbton and the a control A total nber of 65 non-donated opt desgn pon have been obtaned for robst feedbac lnearzed controller wth the consdered nfor dstrbton for ncertan paraeters. Unle the Gassan dstrbton, the vale of probablty of nstablty ( ns ) of soe nondonated opt pon has not been eqal to zero. The resl of the 4-objectve optzaton process are shown n Fgre. The a probablty of nstablty n ths desgn s eqal to 6%, t eans that aong HSS est 7 nstable plan. e as a pror desgn, there s no conflcton between and It s evdent fro Fgre that there s conflcton between robst stablty and robst perforance, de to the fact that the systes wth low the other hand, the systes wth the low vale of ns have worse vale of and and have the worse vale of., on ns. In addton, both and have conflcton wth. Therefore, the systes whe and are low have the hgh vale of worse vale of and., and the syste wth the lowest vale of has the The crcled pont n Pareto front whch has the lowest vale of the -nor and s located on the vertces of the V layot for of the evel Dagra has the oderate vale of each objectve fncton; therefore, t can be consdered as an otandng opt pont. For coparson, the

20 A. Hajloo, M. Saad, N. Naran-Zadeh, PARETO OPTIMA ROBUST FEEDBACK INEARIZATION CONTRO OF A NONINEAR SYSTEM WITH PARAMETRIC UNCERTAINTIES objectve fnctons and desgn varables for n vale of each objectve fncton and for n vale of -nor are gven n Table 4. Table 4. The vales of objectve fnctons and the correspondng desgn varables of the selected Opt Pon Desgn Varables opt pon. q q q q 4 r ns Objectve Fnctons n ns n n n n J J The nercal resl n Table and Table 4 show that the te response perforance of the systes wth noral dstrbton are better that the of nfor dstrbton, becase of the fact that the nfor dstrbton ncertantes are ore strngent than noral ones. The stochastc response of each entoned opt pont s shown n Fgre. Ther stochastc behavors are slated for saples, and pper bond and lower bond show the regon of the responses. Also, the stochastc response of the proped controller fro [7] s shown n Fgre 8. Fro Fgre, the systes wth the lowest vale of ns and have the worse behavors of the stochastc te responses wth respect to the nt plse. Also, the pont wth the lowest vale of -nor whch can be consdered as an otandng trade-off pont has the acceptable stochastc te responses. It s obv fro Fgre that the proped controller by [7] has the worse te response n coparson wth proped controller n ths paper. In addton, accordng to the Fgre 8 and Fgre, the systes wth the Gassan dstrbton have the better te responses n coparson wth the systes wth the nfor dstrbton. It sees de to the fact that the nfor dstrbton s ore strngent than Gassan one. The PDFs of each stochastc varable for each selected opt pont are shown n Fgre. It can be conclded fro Fgre that the systes wth the lowest vale of ns and have the nacceptable settlng te and overshoot. In addton, the systes wth the lowest vale of and need bg control effort sgnal.

21 INTERNATIONA JOURNA ON SMART SENSING AND INTEIGENT SYSTEMS VO. 7, NO., MARCH 4 In order to copare the robstness of each desgn, the trade-off pont of each robst desgn s consdered. Controller A and B are consdered as the trade-off pon of the systes wth Gassan probablty dstrbton and nfor probablty dstrbton desgn, respectvely. Table 5 shows the coparson of the robstness of the two controllers. Each evalaton data s based on, qas Monte Carlo slatons, aganst ether nfor or Gassan evalaton dstrbtons. Also, the resl of the proped controllers by Wang et al. [7] are gven n Table 5. The controller C was desgned for Gassan probablty dstrbton, and D was desgned for nfor probablty dstrbton. Table 5. Coparson between dfferent desgns. ns Gassan Unfor Gassan Unfor Gassan Unfor Gassan Unfor Controller A Controller B Controller C Controller D As shold be epected, f the controller s desgned by the nfor probablty dstrbton, t ac ore robst than the controller wth Gassan one. Tae for llstraton, controller B whch has been desgned nder nfor dstrbton, prodces lower ct fnctons n coparson wth controller A. Bt controller A whe probablty dstrbton s Gassan has better than controller A when they are evalated by Gassan probablty dstrbton. Althogh the controllers whch were desgned by Wang et al. al [7] have the low vale, the correspondng and ns and ns and are worse than the proped controllers. Becase the paraeters of the controller proped by [7] were desgned by nzng a sngle objectve fncton n whch the weghtng factor for Therefore they have low vale of vale of ns has also low vale of. ns has been chen as and others chen as.. ns and t was shown n ths paper that the syste wth low 4

22 A. Hajloo, M. Saad, N. Naran-Zadeh, PARETO OPTIMA ROBUST FEEDBACK INEARIZATION CONTRO OF A NONINEAR SYSTEM WITH PARAMETRIC UNCERTAINTIES V. CONCUSIONS Ths paper proped an optal robst desgn ethod for the feedbac lnearzaton ethod sng a lt-objectve optzaton ethod. In ths wor, robst stablty and robst perforance etrcs are sed to defne the objectve fnctons. The paraeters of the nonlnear syste are as ncertan paraeters wth the nown probablty dstrbton. Two types of ncertantes were consdered for the nonlnear systes. The lt-objectve optzaton of the nonlnear controller led to the dscoverng soe portant trade-offs aong the objectve fnctons. The desgn procedre n ths paper can be appled very easly to any nonlnear systes wth feedbac lnearzaton control. The lt-objectve GAs of ths wor for the Pareto optzaton of feedbac lnearzaton controllers sng soe non-coensrable objectve fnctons s very prng and can be generally sed n the opt desgn of real-world cople control systes wth ncertantes. REFERENCES [] J.J.E. Slotne and W., Appled Nonlnear Control, entce-hall Internatonal Inc., USA, 99. [] S. H. Za, Systes and Control, Oford Unversty ess, New Yor,. [] J.. Chen, W. Chang, Feedbac lnearzaton control of a two-ln robot sng a ltcrsover genetc algorth, Epert Systes wth Applcatons 6, pp , 9. [4] M. Keshr, A. F. Jahro, A. Mohebb, M. Had Aoozgar, & W. F. Xe, MODEING AND CONTRO OF BA AND BEAM SYSTEM USING MODE BASED AND NON- MODE BASED CONTRO APPROACHES, Internatonal Jornal on Sart Sensng & Intellgent Systes, Vol. 5, N.,. [5] D.E. Goldberg, Genetc Algorths n Search, Optzaton, and Machne earnng, Addson-Wesley,

23 INTERNATIONA JOURNA ON SMART SENSING AND INTEIGENT SYSTEMS VO. 7, NO., MARCH 4 [6]. R. Ray, & R. F. Stengel, A Monte Carlo approach to the analyss of control syste robstness, Atoatca, 9, pp. 9 6, 99. [7] Q. Wang and R.F. Stengel, Robst control of nonlnear systes wth paraetrc ncertanty, Atoatca, Vol. 8, pp ,. [8] J. Zhang, & J. Ca, Error Analyss and Copensaton Method Of 6-as Indstral Robot, Internatonal Jornal on Sart Sensng & Intellgent Systes, Vol. 6, No. 4,. [9] Q. Wang, & R. F. Stengel, Robst nonlnear control of a hypersonc arcraft, Jornal of Gdance, Control, and Dynacs, (4), pp ,. [] A.Savn, I.R. Peterson, V.A. Ugronovs, Robst Control Desgn Usng H_nfnty ethods, Sprnger-Verlag, ondon,. [].G. Crespo, Optal perforance, robstness and relablty base desgns of systes wth strctred ncertanty, Aercan Control Conference, Denver, Colorado, USA, pp ,. [].G. Crespo, S.P. Kenny, Robst Control Desgn for systes wth probablstc Uncertanty, NASA report, TP-5-5, March 5. [] Q. Wang and R.F. Stengel, Searchng for Robst Mnal-order Copensators, Jornal of Dynac Systes, Measreent, and Control, Vol., pp. -6,. [4] A. Hajloo, N. Naran-zadeh and A. Moen, Pareto Opt Desgn of Robst Controllers for Systes wth Paraetrc Uncertantes, Robotcs, Atoaton and Control, ISBN , I-Tech, Venna, Astra, 8. [5] A. Hajloo, N. Naran-zadeh, & A. Moen, Pareto optal robst desgn of fractonalorder pd controllers for systes wth probablstc ncertantes, Mechatroncs, Vol., No. 6, pp ,. [6] A. Hajloo, N. Naran-zadeh, A. Jaal, A. Bagher, A. Alast, Pareto Opt Desgn of Robst PI Controllers for Systes wth Paraetrc Uncertanty, Internatonal Revew of Mechancal Engneerng(I.R.M.E.), Vol. No. 6 (7),

24 A. Hajloo, M. Saad, N. Naran-Zadeh, PARETO OPTIMA ROBUST FEEDBACK INEARIZATION CONTRO OF A NONINEAR SYSTEM WITH PARAMETRIC UNCERTAINTIES [7] W. A. Btt, OBSERVER BASED DYNAMIC SURFACE CONTRO OF A HYPERSONIC FIGHT VEHICE, Internatonal Jornal on Sart Sensng & Intellgent Systes, Vol. 6, N.,. [8] M. Papadraas, N.D. agar, and V. Plevrs, Strctral optzaton consderng the probablstc syste response, Theoretcal Appled Mechancs, Vol., No. -4, Belgrade, pp. 6-9, 4. [9] B.A. Sth, S.P. Kenny and.g. Crespo, obablstc Paraeter Uncertanty Analyss of Sngle npt Sngle Otpt Control Systes, NASA report, TM-5-8, March 5. [] E.D. Sontag, frther fac abot npt to state stablzaton, IEEE Transactons on Atoatc and Control, Vol. 4, pp , 99. [] G. Agrawal, C.. Bloeba, K. ews, Inttve desgn selecton sng vsalzed n- densonal Pareto fronter, n: 46th AIAA/ASME/ASCE/AHS/ASC Strctres, Strctral Dynacs and Materals Conference, Astn, TX, 5. [] X. Blasco, J.M. Herrero, J. Sanchs, M. Martnez, A new graphcal vsalzaton of n- densonal Pareto front for decson-ang n ltobjectve optzaton, Inforaton Scences, Vol. 78 (8),