Design of Optimal PID, Fuzzy and New Fuzzy-PID Controller for CANSAT Carrier System Thrust Vector

Size: px

Start display at page:

Download "Design of Optimal PID, Fuzzy and New Fuzzy-PID Controller for CANSAT Carrier System Thrust Vector"

Joan Davidson
5 years ago
Views:

1 In J Advanced esgn and Manufacung Technology, Vol 8/ No 2/ June esgn of Opmal PI, Fuzzy and New Fuzzy-PI Conolle fo CANSAT Cae Sysem Thus Veco A Kosa * epamen of New Scences and Technologes, Unvesy of Tehan, Ian E-mal: kosa_a@uac *Coespondng auho H Jahanshah & A A Razav epamen of New Scences and Technologes, Unvesy of Tehan, Ian E-mal: had_jahanshah@uac, aazav68@uac Receved: 4 Febuay 2015, Revsed: 7 Mach 2015, Acceped: 20 May 2015 Absac: In hs pape, mul-objecve opmzaon based on Genec Algohm s used o fnd he desgn vaables of PI, fuzzy and new Fuzzy-PI conolles applyng fo a hus veco conol of CANSAT cae sysem Moon veco conol s consdeed accodng o he dynamc govenng equaon of he sysem whch s deved usng Newon s mehod and defned msson n delveng payload no he specfc hegh and flgh pah angle The cos funcons of he sysem ae poson eo fom he se pon and devaon of he veco angle of cae sysem wh cae body, whee hese cos funcons mus be mnmzed smulaneously Resuls demonsae ha hs new Fuzzy-PI conolle s supeo o ohe conolles whch ae exeed n he hus veco conol of a CANSAT cae sysem Ths Fuzzy-PI s capable of dong he msson wh decease n selng me and se me wh espec o he convenen mnmzed objecve funcon values Keywods: Mul-objecve opmzaon, Genec algohm, PI-conolle, Fuzzy conolle, Fuzzy-PI conolle, CANSAT cae sysem Refeence: Kosa, A, Jahanshah, H, and Razav, A A, esgn of Opmal PI, Fuzzy and New Fuzzy-PI Conolle fo CANSAT Cae Sysem Thus Veco, In J of Advanced esgn and Manufacung Technology, Vol 8/No 2, 2015, pp 1-9 Bogaphcal noes: A R Kosa eceved hs Ph n Aeospace Engneeng fom Shaf Unvesy of Technology n 2003 He s cuenly Asssan Pofesso a he epamen of Aeospace and Mechaonc and Sysem Engneeng, Faculy of New Scence and Technologes, Unvesy of Tehan, Tehan, Ian Hs cuen eseach nees ncludes opmal conol and Pah Plannng H Jahanshah s MSc Suden of Aeospace engneeng a he Unvesy of Tehan, Ian He eceved hs BSc n Mechancal engneeng fom Sjan Unvesy of Technology Hs cuen eseach focuses on Fuzzy se and sysems A A Razav s MSc suden of Aeospace engneeng a he Unvesy of Tehan, Ian He eceved hs BSc n Mechancal engneeng fom Fedows Unvesy of Mashhad

2 2 In J Advanced esgn and Manufacung Technology, Vol 8/ No 2/ June INTROUCTION CANSAT s ae geng moe popula each day n aeospace engneeng cuculums because hey enable sudens o have paccal expeence on vual saelle launch opeaons and expeence he whole pocess of desgn, es, launch and ecovey emand fo anng of spacecaf engneeng s nceasng As paccal space engneeng s a elavely vey expensve feld, hee s an ongong seach fo affodable anng appoaches One of hese appoaches s CANSAT Launch concep whch was noduced by an Amecan Pofesso, Robe Twggs [1] He s a pofesso emeus a Sanfod Unvesy, who along wh Jod Pug-Sua of Calfona Polyechnc Sae Unvesy s esponsble fo ceang he CubeSa sandad fo mnauzed saelles and a sandad fo deploymen of he saelles He hough he could be egaded as he fahe of he mnaue saelles due o hs woks wh he ponees of hs dea [2] CANSAT s a desgn-bul-launch oganzaon whch s made by sudens The man goal of he CANSAT s o educae fuue engnees and scenss Ths pogam spead all ove he wold fom he USA and akes s name (CANSAT) fom he combnaon of "can" and "saelle" [3] The key feaues of a CANSAT desgn may be expessed as: Hgh Technology, Affodably and Lgh wegh The name of he saelle s deemned by he sze of he can "CANSAT" s a nanoscale saelle model, weghng 350 o 1050g These saelles ae launched by he ocke and can each a hegh of 150 o 2000m Afe geng sepaaed fom he saelle ockes, begns o pefom s dues n he couse of a fee fall These asks ae adaped o he ecen space mssons lke eafomng and a bag mssons [2] Havng he vaous sensos, ccus and mechancal componens conanng hese saelles, he CANSAT s he smalles sucue ha could be called a mnaue saelle I should be noed ha vehcles each vey low aludes, up o a few housand mees, whch s much lowe han ha he smalles soundng ockes can each [4] Ths small saelle s lef a a few hunded mees above he gound suface hen euns o he eah by he ecovey sysem Ths hegh s suffcen fo he payload opeaonal es and fgung ou ha CANSAT can pefom mssons smla o hose of eal saelles bu n a small lmed scale The CANSAT can handle dffeen knds of msson The mos mpoan known mssons exsng fo hs sysem whch ae suggesed fo elevan compeons nclude: Amosphee monong Imagng In addon o he pmay mssons of such flyng vehcles, menoned above, ecoveng and landng on a cean locaon o expandng anenna mechansm ae pas of he seconday msson of hs sysem In pevous compeons, CANSAT was usually caed by balloons and hen eleased a a specfc hegh, bu n new ounamens balloon s eplaced by he cayng sysem In hs case cayng sysems ae launched vecally and CANSAT s eleased a he specfc hegh wh specfc pah angle o he eah local hozon, hen a s hghes pon he cayng umbella s opened and lands slowly So, pah conolle desgn of cayng sysem fo delveng CANSAT s payload a he specfed hegh and deemned flgh condon plays an mpoan ole n he sysem desgn of hs msson Some eseaches caed ou n he feld of dffeen cayng pah conolle desgns (excep CANSAT), ae manly based on nonlnea conol mehods In ecen yeas, he developmen of Fuzzy-PI has become one of he majo eseach aeas n dffeen engneeng poblems The mos sgnfcan applcaons and sudes abou fuzzy sysems have concenaed on conol aeas, such as hose n [5], [6] Fuzzy sysems ae knowledge-based o ule-based sysems fomed va human knowledge and heuscs They have been appled o a wde ange of eseachng felds such as conol, communcaon, medcne, managemen, busness, psychology, and so foh The mos sgnfcan applcaons and sudes of he fuzzy sysems have concenaed on he conol aea, such as hose n Refs [7-12] The developmen of fuzzy PI conolles fo vaous engneeng poblems has been a majo eseach acvy n ecen yeas Along he way, he heusc paamees of Fuzzy-PI conolles have o be deemned va an appopae appoach A vey effecve way o choose hese facos s he use of evoluonay algohms, such as he genec algohm (GA) A consaned opmzaon of a smple Fuzzy-PI sysem s desgned fo he onlne mpovemen of PI conol pefomance whle unnng he poducve conol [13] In Ref [14], uan, L and eng poposed an nheen sauaon of he Fuzzy-PI conolle evealed due o he fne fuzzy ules In Ref [15], Oh, Jang, and Pedycz developed a desgn mehodology fo a fuzzy P cascade conolle fo a ball-beam sysem usng pacle swam opmzaon (PSO) In Ref [16], an on-lne unng mehod s poposed fo fuzzy PI conolles va ule weghng Boubeakh, Tadjne, Gloennec, and Labod poposed a new auo-unng fuzzy P and PI conolles usng enfocemen-leanng (QL) algohm fo SISO (sngle-npu sngle-oupu) and TITO (wo npu wooupu) sysems [17] Ne and Tan pesened an mpoved veson of he sable fuzzy adapve conol

3 In J Advanced esgn and Manufacung Technology, Vol 8/ No 2/ June sucue, whch compses an appoxmaon of he deal conolle and a supevsoy conolle [18] In hs pape a PI conolle, hen a fuzzy conolle and fnally a new Fuzzy-PI conolle s exeed on he hus veco of he CANSAT cae sysem In all of hese conolles, genec algohm s appled o fnd he bes conolle gans 2 CANSAT CARRYING SYSTEM YNAMIC MOELLING The fs sep n desgn of pah conol s denfcaon and exacon of dynamc govenng equaons of he sysem In mechancal sysems (such as aeospace sysems), hee ae vaous mehods such as Newon, Lagange, Keen and so foh fo modellng of sysem dynamcs The bes known of hese mehods s Newon's mehod whch s used n hs pape Fgue 1 shows a smplfed model of a cae sysem as θ s he cae longudnal veco angle wh decon of vecal o he eah and Φ s he cae hus veco angle wh decon of cae body Snce n he CANSAT sepaaon nsance fom s cae, he CANSAT velocy veco mage magnude on he hozon plane needs o be zeo, and also n he CANSAT opeaonal me, hs saelle s conneced wh gound saon hough ado waves, o educe he sgnal o nose ao (fuhe dsance equal o sgnal aenuaon and nose ncease), he conol pupose of hs poblem s eseng he angle of θ on zeo Theefoe, usng Newon's laws, govenng dynamc equaons of he sysem can ealze as follows: M CM I Whee M CM, I and α ae he momenum aound cene of mass, oaonal nea, and cae s angula acceleaon aound he axs pependcula o he plane, especvely Fg 1 Cae sysem scheme Fnally govenng dynamc equaon of he sysem s made as follows: 1 lm( a g) an( ) 2I Whee b s calculaed fom he followng equaon: 1 b lm( a g) 2I If we ake an( ) as u, equaon (1) becomes as follows: bu whee a s he acceleaon along pependcula axs, m s he mass of cae, g s he gavy acceleaon, and l s he lengh of cae The sae vaables ae he sysem obsevable sae veco whch s descbed as follows: x [ x1, x2, x3] [,, ] 3 OPTIMIZATION APPROACH Thee ae seveal mehods fo engneeng poblems opmzaon Ou seleced echnque s usng he genec algohm One advanage of genec algohm mehod s dealng wh funcon quany So, fo choosng he funcon, hee s no need o know s exac equaon and seems enough o fnd a way o calculae funcon value by eneng poblem vaables, unlke ohe opmzaon mehods ha eque havng he objecve funcon equaon exacly Eg f we could fnd he funcon value fo a specal case wh dffeen knd of vaables hough gaphcal mehods, hee s no need o know he elaon beween funcon and vaables, because he genec algohm mehod gves us he opmum value wh good accuacy Also, f we had a sofwae fo analyzng engneeng poblem whch could ge he vaables and gave he answe, we could oban he opmum value wh genec algohm only by knowng he value of funcon fo npu vaables wh no equemen of awaeness of soluon pocesses Ths s he unque advanage whch could be aely found n ohe opmzaon mehods In each opmzaon poblem we deal wh some goals whch ae menoned as objecve funcons These funcons ae hose vaables ha need o be mnmzed o maxmzed If we have only one objecve funcon, we encoune sngle-objecve opmzaon poblem, bu f we have moe han one objecve funcons we nvolve n mul-objecve o MOGA poblem, whee genec algohm es o opmze he desed values smulaneously MOGA s he abbevaon fom of

4 4 In J Advanced esgn and Manufacung Technology, Vol 8/ No 2/ June 2015 mul objecve based on genec algohm The objecve funcons n hs opmzaon ae poson eo fom he se poson, and devaon of veco angle of cae sysem wh cae body, descbed as follows: O F1 d O F2 d 4 PI CONTROLLER Sysem sae veco s x 1, x2, x3 whch ae negal of cae longudnal veco angle wh decon of vecal o he gound, cae longudnal veco angle wh decon of vecal o he gound and devaon of cae longudnal veco angle wh decon of vecal o he gound, especvely In PI conolle, advanced conolle can be ceaed consdeng paamees such as se pon changes wh espec o he cuen value of he pocess wh facos lke nensy and amoun of changes wh espec o me In PI conolles, popoonal-devave-negal algohm s used As comes fom s name, he lee P goes fo popoonal, I means negal and means evave Popoonal conolle deceases selng me and eo bu nceases oveshoo evave conolle deceases amoun of oveshoo and selng me and has less nfluence on se me and seady sae eo Inegal conolle deceases se me bu nceases oveshoo and selng me and emoves seady sae eo So a fs, he esponse of open loop and he values whch should be mpoved ae deemned We use Popoonal conolle fo se me mpovemen, devave conolle fo oveshoo mpovemen and negal conolle fo seady sae eo The gans of popoonal, devave and negal exeed n hs conolle accodng o he defned objecve funcons ae calculaed by usng genec algohm 5 FUZZY CONTROLLER Conolles wh fuzzy logc bass ae consdeed a subse of nellgen conol sysems Acually, nellgen conol sysems ae combnaon of conol engneeng and afcal nellgence (pogammng, easonng and leanng), whee fuzzy conolles ae no excluded fom hs law In ohe wods, fuzzy conolles ae song combnaon of nonlnea conols (conol engneeng) and fuzzy logc (nfeence) Thee ae some nfeence mehods n fuzzy logc One of hese mehods s Mamdan fuzzy nfeence sysem A Mamdan fuzzy nfeence sysem s made up of some lmed f-hen ules Fuzzy logc The fac ha cae hus foce mus be n a decon so ha could apply a oque n oppose of cae s head angle decon unl cae angle o he vecal lne become zeo, s he smples lngusc ule comes o he mnd Ths smple ule leads o a peodc sysem f he sysem nea s ovelooked ue o he Newon s laws, an objec keep movng a s unfom speed when hee s no foce on Theefoe, due o he nea law, sysem keeps movng a he same speed and cosses he zeo pon unl hus apples an oppose oque o he cae To solve hs poblem, cae angle velocy s eneed n he poblem n appled ules of makng decson So, he man ules of cae pah conol ae ewen as follows: 1- If he cae angle s negave and cae angle velocy s negave, hen hus veco s posve bg 2- If he cae angle s negave and cae angle velocy s nea o zeo (whou consdeng posve and negave), hen hus veco s posve small 3- If he cae angle s zeo (whou consdeng posve and negave) and cae angle velocy s negave hen hus veco s posve small 4- If he cae angle s zeo (whou consdeng posve and negave) and cae angle velocy s zeo (whou consdeng posve and negave) hen hus veco s posve zeo 5- If he cae angle s zeo (whou consdeng posve and negave) and cae angle velocy s posve hen hus veco s negave small 6- If he cae angle s posve and cae angle velocy s nea o zeo (whou consdeng posve and negave), hen hus veco s negave small 7- If he cae angle s posve and cae angle velocy s posve, hen hus veco s negave bg Fuzzfcaon Snce govenng equaon of cae dynamc sysem s connuous and dffeenable, s bee o use Gaussan funcons n fuzzfcaon The pupose of desgn s o make cae angle o he vecal lne zeo, so s necessay o have less ceon devaon n Gaussan funcon n he saemen of f cae angle s zeo han he ohe lngusc saemens The fuzzfcaon membeshp funcons of cae angle o he vecal axs ae shown n Fg 2

5 In J Advanced esgn and Manufacung Technology, Vol 8/ No 2/ June efuzzfcaon As menoned n above pa, govenng equaon of cae dynamc sysem s connuous and dffeenable So, s bee o use Gaussan funcons (whch ae dffeenable) n defuzzfcaon Moeove, usng Gaussan funcons esul n smooh pefomance of oupu ahe han usng ohe funcons such as angula o apezodal funcons ue o symmec popey of he sysem, Gaussan funcons ae consdeed homogeneous and symmec The defuzzfcaon membeshp funcons of hus angle ae shown n Fg 3 Fg 2 Fuzzfcaon of cae angle o he vecal axs Fg 3 efuzzfcaon of hus angle Takng Laplace ansfom fom Eq 4, and selecng conol paamee U as follows: U( s) ( w 0 w s)( ( s)) Closed loop ansfe funcon of he sysem becomes: ( s) U( s) s 2 0 bw s bw 0 bw s bw Fuzzy gan used n hs conolle accodng o he defned objecve funcons, poson eo fom he se pon, and devaon of he veco angle of cae sysem wh cae body s calculaed by genec algohm 6 FUZZY-PI CONTROLLER In he poposed new Fuzzy-PI conolle, wo fuzzy nfeence moos ae ulzed The fs s he Sngle Inpu Fuzzy Infeence Moo (SIFIM) ha has only one npu Snce all of he desed values n he sablzaon conol ae zeos, he vaables ae evesely npued no he Nom block Fo each nomalzed vaable (Nom block oupu), a SIFIM s defned The second s he Pefee Fuzzy Infeence Moo (PFIM) ha epesens he conol poy ode of each Nom block oupu j j j m : j 1 SIFIM { R : f x A henu C } SIFIM- menon he sngle npu nfeence moo whch accep he h j npu among npus and R s he j h ule of he sngle npu nfeence moo j A and C j ae he membeshp funcons Each npu em usually has a dffeen ole n mplemenaon of conol To expess he dffeen effecs of each npu em n he mplemenaon of sysem, sngle npu fuzzy nfeence moo defnes a dynamc mpoance degee ( w each npu em w ) fo w B w Whee w, B and w ae conol paamees whch ae descbed by fuzzy ules SIFIM- block calculae f as follows: VB f1 PO f2 ZB f f VB PO ZB 3 The membeshp funcons of SIFIMs ae shown n Fg 4 f 1, f 2 and f 3 (fuzzy ules of SIFIMs) noed n he above equaon, ae exploed fom he Table 1 Ohe ype of fuzzy nfeence moos ae PFIMs PFIMs guaanee CANSAT cae conol sysem when s deved fom desed values n one o moe of he coodnae sysem decons PFIM- calculae w as follows: W1 HS W2 HM W3 HB W1 W2 W3 HS HM HB The membeshp funcons of SIFIMs ae shown n Fg 4, and fuzzy ules of PFIMs ae shown n Table 2 Afe calculang f and w, s possble o defne new Fuzzy-PI conolle as he followng: f Kˆ ˆd Kˆ ˆ Kˆ p d dˆ d dˆ Whee f s he conol acon ˆ d, ˆ, ae he fuzzy d foms of d d,,, especvely, and should be d

6 6 In J Advanced esgn and Manufacung Technology, Vol 8/ No 2/ June 2015 obaned by SIFIM In ohe wods, ˆ ˆ ˆ d d f1; f 2; f3 Fuhemoe, n Eq 13, d ˆ Kˆ K p, and Kˆ d ae he fuzzy vaables calculaed by he followng equaons: ˆ b K K KW 1 ˆ b K K KW 2 ˆ b Kd Kd KdW 3 b b b Whee K, K and K d ae he base vaables K, K, and K d ae he egulaon vaables The base and egulaon vaables can be obaned by al and eo pocess Howeve, he bes soluon o have an opmal conol, s he use of opmzaon appoaches such as evoluonay algohms, especally genec algohm Fg 4 Membeshp funcons of SIFIM f Table 1 Fuzzy ules of SIFIMs Then VB f 1 1 PO f 2 0 ZB f SIMULATION AN RESULT Fo PI, fuzzy and Fuzzy-PI conolle, gan values and objecve funcon values elaed o hese gans ae shown n Tables 3, 4, 5, 6, 7 and 8, especvely The me esponses of cae poson o he vecal decon, cae angula velocy and cae hus veco angle wh cae body decon fo PI, fuzzy and Fuzzy-PI conolles ae shown n Fgues 6, 7, 8, 9, 10, 11, 13, 14 and 15 especvely Also, Smulaon of fuzzy conolle fo cae sysem n MATLAB, s shown n Fg 12 The sysem paamees used fo smulaon ae, m 100 kg, g 10 m s 2, l 1m, 100 m 2 a 2, I 1000kgm and he nom block faco s s 1 ad The nal values ae x [ x1, x2, x3] [0,02,0] and he algohm confguaon of he genec algohm (exs n he envonmen of MATLABR2012a) s as follows The cossove facon = 08, populaon sze = 200, selecon funcon = ounamen, muaon funcon =consan dependen, cossove funcon= nemedae, cossove ao= 1, mgaon decon= fowad, mgaon facon=02, mgaon neval= 20, dsance measue funcon=dsance cowdng, Paeo fon populaon funcon=035, and soppng cea s defned as funcon oleance = 10-4 Table 3 esgn vaables of opmum pon fo PI conolle esgn vaable value K p K d K Table 4 Objecve funcons of opmum pon fo PI conolle objecve funcon O F1 d O F2 d Fg 5 Membeshp funcons of PFIM Table 2 Fuzzy ules of PFIMs f Then HS W 1 1 HM W HB W 3 1 Table 5 esgn vaables of opmum pon fo fuzzy conolle esgn vaable w w Table 6 Objecve funcons of opmum pon fo fuzzy conolle objecve funcon O F1 d O F2 d 01961

7 In J Advanced esgn and Manufacung Technology, Vol 8/ No 2/ June Fg 6 Tme esponse of angula poson of PI conolle fo opmum pon Fg 10 Tme esponse of angula velocy of fuzzy conolle fo opmum pon Fg 7 Tme esponse of angula velocy of PI conolle fo opmum pon Fg 11 Tme esponse of hus veco angle of fuzzy conolle fo opmum pon Fg 8 Tme esponse of hus veco angle of PI conolle fo opmum pon Fg 12 Smulaon of fuzzy conolle fo cae sysem n MATLAB/SIMULINK Fg 9 Tme esponse of angula poson of fuzzy conolle fo opmum pon Table 7 esgn vaables of opmum pon fo Fuzzy-PI conolle esgn vaable b K b K b Kd K K Kd 09018

8 8 In J Advanced esgn and Manufacung Technology, Vol 8/ No 2/ June 2015 Table 8 Objecve funcons of opmum pon fo Fuzzy- PI conolle objecve funcon O F1 d O F2 d Fg 13 Tme esponse of angula poson of Fuzzy-PI conolle fo opmum pon mehod The objecve funcons fo hs sysem ae poson eo fom se pon and devaon of veco angle of cae sysem o cae body An negal em s augmened o he sae vaables due o seady sae eo elmnaon and se me decease In PI and fuzzy and Fuzzy-PI conolle, desgn vaables ae calculaed by usng genec algohm wh espec o defned objecve funcons Mamdan nfeence sysem wh some defned f-hen ules and Gaussan membeshp funcons fo fuzzfcaon and defuzzfcaon pas ae used n fuzzy conolle The new Fuzzy-PI conolle ulzes wo aveage nfeence engnes called SIFIM and PFIM whee he fs engne ge one npu fom each Nom block and gve f as oupus, and he second one guaanee CANSAT cae conol sysem n lage devaon fom desed values n one o moe of coodnae sysem decons and gve he W as oupus The epoed esuls demonsaed ha he poposed mehodology fo Fuzzy-PI conolle can conol CANSAT cae sysem effecvely ahe han appled PI and fuzzy conolles I s ecommended o denfy accuae dynamc of sysem and choose convenen fuzzy ules o mpove pefomance of conolles REFERENCES Fg 14 Tme esponse of angula velocy of Fuzzy-PI conolle fo opmum pon Fg 15 Tme esponse of hus veco angle of Fuzzy-PI conolle fo opmum pon 8 CONCLUSION In hs wok, he mul-objecve opmzaon was successfully used fo an opmum desgn of PI, fuzzy and new Fuzzy-PI conolles fo he CANSAT cae sysem whee he dynamc s deved by usng Newon s [1] Aydem, M E, usun, R C, and Pehlevan, M, Gound Saon esgn Pocedues fo CANSAT, he 6h Inenaonal Confeence on Recen Advanced n Space Technologes (RAST), Isanbul, Tukey, June 2013, pp [2] Soye, S, Small Space Can: CANSAT, n 5h Inenaonal Confeence on Recen Advanced n Space Technologes (RAST), Isanbul, Tukey, June 2011, pp [3] Çabuloğlu, C, Aykş, H, Yapacak, R, Çalşkan, E, Ağbuş, Ö, Abu, Ş, Soye, S, Tükmen, H, Ay, S, Kaaaş, Y, Aydem, M E, and Ҫeleb, M, Msson Analyss and Plannng of a CANSAT, The 5h Inenaonal Confeence on Recen Advanced n Space Technologes (RAST), Isanbul, Tukey, June 2011, pp [4] Oknnsk, A, Macnak, B, Bakowak, B, Kanewsk,, Mayszewsk, J, Kndack, J, and Wolansk, P, evelopmen of he Polsh Small Soundng Rocke Pogam, Aca Asonauca, Vol 108, 2015, pp [5] Zadeh, L A, Fuzzy algohms, Infomaon and Conol, Vol 12, 1968, pp [6] Zadeh, L A, Oulne of a new appoach o he analyss of complex sysems and decson pocesses, IEEE Tansacons on Sysems, Man and Cybenecs, Vol 3, 1973, pp [7] Nasse, H, Kefe-Kamal, E H, Hu, H, Belouea, S, and Bakanov, E, Acve vbaon dampng of

9 In J Advanced esgn and Manufacung Technology, Vol 8/ No 2/ June compose sucues usng a nonlnea fuzzy conolle, Compose Sucues, Vol 94, 2012, pp [8] LI, P, JIN, F J, Adapve Fuzzy Conol fo Unknown Nonlnea Sysems wh Peubed eadzone Inpus, Aca Auomaca Snca, Vol 36, 2010, pp [9] Lygouas, J N, Bosas, P N, Vouvoulaks, J, and Kodoganns, V, Fuzzy logc conolle mplemenaon fo a sola a-condonng sysem, Appled Enegy, Vol 84, 2007, pp [10] Jee, S, Koen, Y, Adapve fuzzy logc conolle fo feed dves of a CNC machne ool, Mechaoncs, Vol 14, 2004, pp [11] Snha, A S C, Lyshevsk, S, Fuzzy conol wh andom delays usng nvaan cones and s applcaon o conol of enegy pocesses n mcoelecomechancal moon devces, Enegy Conveson and Managemen, Vol 46, 2005, pp [12] Zupel, U, Cus, F, and Mlfelne, M, Fuzzy conol saegy fo an adapve foce conol n endmllng, Jounal of Maeals Pocessng Technology, Vol , 2005, pp [13] Mansou, S E, Kembe, G C, ubay, R, and Robeson, B, Onlne opmzaon of fuzzy-pi conol of a hemal pocess, ISA Tansacons, Vol 44, 2005, pp [14] uan, X G, L, H X, and eng, H, Robusness of fuzzy PI conolle due o s nheen sauaon, Jounal of Pocess Conol, Vol 22, 2012, pp [15] Oh, S K, Jang, H J, and Pedycz, W, Opmzed fuzzy P cascade conolle: A compaave analyss and desgn, Smulaon Modellng Pacce and Theoy, Vol 19, 2011, pp [16] Kaasakal, O, Guzelkaya, M, Eksn, I, Yesl, E, and Kumbasa, T, Onlne unng of fuzzy PI conolles va ule weghng based on nomalzed acceleaon, Engneeng Applcaons of Afcal Inellgence, Vol 26, 2013, pp [17] Boubeakh, H, Tadjne, M, Gloennec, P Y, and Labod, S, Tunng fuzzy P and PI conolles usng enfocemen leanng, ISA Tansacons, Vol 49, 2010, pp [18] Ne, M, Tan, W W, Sable adapve fuzzy P plus PI conolle fo nonlnea uncean sysems, Fuzzy Ses and Sysems, Vol 179, 2011, pp 1-19

5-1. We apply Newton s second law (specifically, Eq. 5-2). F = ma = ma sin 20.0 = 1.0 kg 2.00 m/s sin 20.0 = 0.684N. ( ) ( )

5-1. We apply Newton s second law (specifically, Eq. 5-2). F = ma = ma sin 20.0 = 1.0 kg 2.00 m/s sin 20.0 = 0.684N. ( ) ( ) 5-1. We apply Newon s second law (specfcally, Eq. 5-). (a) We fnd he componen of he foce s ( ) ( ) F = ma = ma cos 0.0 = 1.00kg.00m/s cos 0.0 = 1.88N. (b) The y componen of he foce s ( ) ( ) F = ma = ma