Fuzzy PID Iterative learning control for a class of Nonlinear Systems with Arbitrary Initial Value Xiaohong Hao and Dongjiang Wang

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1 7h Ieraioal Coferece o Educaio Maageme Compuer ad Medicie (EMCM 216) Fuzzy PID Ieraive learig corol for a class of Noliear Sysems wih Arbirary Iiial Value Xiaohog Hao ad Dogjiag Wag School of Compuer ad commuicaio. Lazhou Uiversiy of echology Lazhou 735Chia Keywords: Noliear sysem; Arbirary iiial value; Fuzzy PID Ieraive learig corol; Norm heory. Operaor heory Absrac. Aimig a a class of oliear sysems ruig repeaedly i he sudy of he covergece of ieraive learig corol a PID fuzzy ieraive learig corol algorihm has bee proposed i he arbirary iiial value usig fuzzy PID corol algorihm usig radiioal experiece PID parameers by referece ad experiecig real-ime correcio of PID parameers geeraig higher precisio fuzzy PID learig law. The use of operaor heory ad he heory of covergece orm has bee proved by simulaio experime oliear sysems uder arbirary iiial codiios he use of fuzzy PID ieraive learig corol algorihm he sysem coverges faser higher rackig accuracy he oupu of he error curve faser eds o zero. The resuls of he demosraes he effeciveess of he algorihm. Iroducio Ieraive learig corol is a brach of iellige corol. I has a good effec o he research of o-liear ad srog-couplig corol sysems wih repeiive moio. By cosaly adjusig he ipu of he corol sysem he rackig performace of rackig sysem is improved. A complee rackig of he desired oupu by a corolled objec over a fiie ime ierval. PID corol robus corol ad opimal corol cao achieve high-precisio rackig corol because of he uceraiies such as iiial error ad iiial sae deviaio which affec mos sysem modelig accuracy. Arimoo [1] ad ohers have sudied he covergece of he ieraive learig corol sysem o a cerai rage of guesses bu cao guaraee ha he learig law coverges o he rue value. Lee [2] ad ohers gave he D-ype PD-ype ieraive learig corol covergece o a cerai exe. Che [3] for a give ieraive learig corol sysem for he iiial value of he problem hrough he sysem ipu ad iiialize he sysem o achieve he same ime ieraive rackig. Su Migxua e al. [4] discussed he problem of covergece for a give D-ype PD-ype ieraive corol sysem whe he iiial value of he sysem is biased. Re Xuemei ec. [5] gives he simulaeous ipu ad iiial value of learig he iiial sae of he sysem a he begiig of learig wihou he reques he resuls show ha he feasibiliy of his mehod. Yao Zhogshu e al [5] proposed he ieraive learig coroller i he frequecy domai i he desig of he idea give i ay iiial ieraive learig corol algorihm uder he sufficie codiios for covergece. Lu e al. [6] sudied he ieraive learig corol of he effec of acceleraig he suppressio of radom iiial sae errors. Su e al [7] sudied he ope-closed-loop PD-ype ieraive learig corol for oliear ime-delay sysems. I order o improve he corol precisio of he corolled sysem ad accelerae he covergece speed i he ieraio domai he ieraive learig corol ad he fuzzy corol are combied o opimize he parameers of he ieraive learig coroller i order o achieve beer corol effec o he oliear ad ucerai complex corol sysem. Fuzzy corol is difficul o esablish a accurae mahemaical model of he sysem has srog robusess; may scholars will ieraive learig corol ad fuzzy corol combied o achieve a more ideal corol effec. Pok [8] proposed a fuzzy ieraive learig corol algorihm he algorihm will be filered afer he previous ieraio corol sigal ad wih he curre error ad error accumulaio of he derivaive as fuzzy ieraive learig coroller oupu simulaio experimes verify ha he ew The algorihm improves he accuracy of sysem rackig. Yu [9] used ieraive learig algorihm o Copyrigh 217 he Auhors. Published by Alais Press. This is a ope access aricle uder he CC BY-NC licese (hp://creaivecommos.org/liceses/by-c/4./). 34

2 modify he fuzzy corol rules. Zhag Lipig [1] proposed a closed-loop ieraive learig coroller for oliear dyamic model combiig fuzzy-fuzzy corol ad ieraive learig corol ad improved he fuzzy corol Smoohess of he oliear mappig ad has good robusess. The radiioal PID ieraive learig corol coroller is o ideal for oliear sysem ime-varyig sysems ad ime-delay sysems. I order o solve his problem a oliear coroller is eeded. I his paper a fuzzy coroller based o fuzzy-pid corol is proposed o solve he above problems. I his paper a fuzzy coroller based o fuzzy-pid is proposed which is a oliear coroller for corollig complex sysems. Algorihm usig he radiioal empirical PID parameers as a referece ad real-ime correcio of empirical PID parameers o geerae a more accurae fuzzy PID learig law. So ha he sysem covergece faser higher rackig accuracy. ILC Problem Descripio The dyamic equaios of a class of oliear ime-varyig sysems are give as follows: x f ( x( ) ) B( )u ( ) y ( ) g x ( x( )) (1) amog [ T ] x( ) R Is he sae quaiy u( ) R For he corol volume y( ) R For he oupu f B C Is he marix of he correspodig dimesio Le Ieraive learig corol sysem iiial value of each ieraio arbirary I ime [ T ] Applyig PID Type Ieraive Learig Law r m By adjusig he corol ipu uk ( ) So ha he oupu sequece { yk ( )} Cosise covergece i he desired rajecory. I he imes ieraio he sysem equaio is expressed as x k ( ) f ( x( ) ) B( )uk ( ) yk ( ) g x ( xk ( )) (2) Fuzzy PID ieraive learig law is as follows: uk 1 ( ) uk ( ) ( k p )ek 1 ( ) ( ki ) d (ek ( )) ek 1 ( )d ( ) ( kd ) d ( ) (3) I formula for he learig gai marix he oupu error is: ek ( ) yd ( ) yk ( ) (4) The sysem iiial sae adops he ieraive learig law: xk 1 () xk () B() L()ek () (5) Fuzzy PID Ieraive Learig Corol Fuzzy PID Coroller. I order o improve he precisio of sysem corol ad obai good corol effec for smooh operaio a fuzzy PID ieraive learig corol algorihm is adoped. The radiioal empirical PID parameers are used as referece ad he empirical PID parameers are correced i real ime o geerae more accurae Fuzzy logic learig law so ha he sysem has a good dyamic rackig performace ad improve he ieraive covergece rae of he sysem. PID fuzzy parameer is o fid he hree parameers of PID fuzzy relaioship bewee E ad EC i he ruig by cosaly esig E ad EC accordig o he fuzzy priciple of he hree parameers modified o mee he differe E ad EC whe he Corol parameers required. 1. Proporioal lik kp 35

3 The effec of proporioal coefficie is o speed up he respose speed of he sysem. The higher he respose speed of he sysem he faser he respose speed of he sysem he higher he adjusme accuracy of he sysem bu easy o produce overshoo ad eve lead o sysem isabiliy. The value is oo small i will reduce he adjusme accuracy so ha he correspodig slow hereby exedig he adjusme ime. k 2. Iegral lik i The effec of he iegral coefficie is o elimiae he seady-sae error of he sysem. The higher he iegral coefficie is he faser he saic error is elimiaed bu if i is oo large he iegral sauraio will occur a he begiig of he respose process which will cause he overshoo of he respose process. If oo small will make he sysem saic error is difficul o elimiae affec he sysem regulaio accuracy. k 3. Differeial lik d The role of differeial coefficie is o improve he dyamic performace of he sysem is role i he respose process o suppress deviaio i oher direcios chage ad will reduce he sysem's ierferece performace. I ca display he variaio of he sigal deviaio ad ca iroduce a correcio amou i he course of he deviaio chage o corol he sysem acio ad preve he adjusme ime icrease whe he deviaio is oo large hus effecively reducig he adjusme ime of he sysem The Fuzzy Corol Rule Se Is Esablished. The fuzzy corol rule able show i he able is esablished ad he correspodig fuzzy variable is deermied accordig o he defiiio i he able. The core of he desig of fuzzy corol is he echical kowledge ad pracical experiece of he desiger esablish he appropriae fuzzy rules able ge he fuzzy rules of hree parameers able. Defie he fuzzy corol ipu ad oupu fuzzy subse are { ZE } k (1)The fuzzy rule able of p Table 2.1 k p fuzzy rule able (2)The fuzzy rule able of ki Table 2.2 ki fuzzy rule able (3)The fuzzy rule able of kd 36

4 kd fuzzy rule able Table 2.3 k p ki k d The fuzzy rule able is esablishedaccordig o he followig mehod Accordig o he fuzzy variables accordig o he ipu ad oupu variables of he fuzzy coroller adjus he gai facor coiuously. Fuzzy rules o deermie deermie he membership fucio ad he use he ceer of graviy o is fuzzy soluio he fuzzy PID coroller ca be oupu parameers: kp q 1 kp ki k ( k p ) q 1 oupu ( k p ) k p p k p ki kd i 1 ( ki ) ki kd k ( ki ) i 1 ki i d 1 kd ( kd )kd d 1 kd (kd ) As he oupu of he fuzzy variable defuzzificaio. The resulig fuzzy learig law is: ek 1 ( )d ( ) ( kd ) k d Is he membership fucio of he k p ki k d Is he amou of oupu afer uk 1 ( ) uk ( ) ( k p )ek 1 ( ) ( ki ) d (ek ( )) d ( ) (6) Covergece Aalysis Before proof of covergece he followig defiiios ad lemmas are iroduced defiiio 1 Vecor fucio h :[ T ] R Is defied as h sup{exp( ) h( )}. is R O a orm. lemma 1:For vecor fucios h :[ T ] R if h( ) f ( ) h( ) 1 exp( ) f ( ) he lemma 2: Se he cosa sequece {bk }k (bk ) Covergece o zero. operaor k Saisfy Qk (u )( ) M (bk u ( s) ds) amog M 1 Is cosa Cr [ T ] Fucio space of coiuous fucio ad he dimesio vecor akes he maximum orm. P(u)( ) p( )u( ). lim( P Q )( P Q 1 ) ( P Q )(u)() he heorem 1 For he oliear sysem described by equaio (1) whe [ T ] The followig codiios are saisfied: whe k yk ( ) Coverge uiformly o yd ( ) prove: via(1) (3) (5) 37

5 xk 1 ( ) xk 1 () [ f ( xk 1 ( )) B( )uk 1 ( )] B( )uk ( ) ek ( ) xk () B() L()ek () f ( xk ( )) B( )( H ( ) k p )ek ( ) ( P( ) ki ) ek ( ) ek ( ) B( )( H ( ) k p )ek ( ) xk () B() L()ek () [ f ( xk 1 ( ) f ( xk ( )] ( P( ) ki ) ek ( ) (7) By he differeial mea value heorem available ek 1 ( ) ek ( ) ( yd ( ) yk 1 ( )) ( yd ( ) yk ( )) g ( xk ( )) g ( xk 1 ( )) g x ( k ( 1)( xk ( ) xk 1 ( )) (8) g ( ( )) [ f ( xk 1 ( ) f ( xk ( )] 又 ek 1 ( ) ek ( ) g x ( k ( ))B( )(L( ) kd )ek ( ) x k g x ( k ( )){ ek ( ) B( )( H ( ) k )e ( ) } p k g x ( k ( )) ( P( ) ki ) ek ( ) (9) 则 ek 1 ( ) [ I g x ( k ( )) B( )( L( ) kd )]ek ( ) g x ( k ( )) [ f ( xk 1 ( ) f ( xk ( )] g x ( k ( )){ ek ( ) B( )( H ( ) k p )ek ( ) ( P( ) ki ) ek ( ) } Defie operaors P : Cr [ T ] Cr [ T ] is p(ek ( )) [I gx ( k ( ))B( )(L( ) kd )]ek ( ) (11) Kow by he kow codiios he specral radius Of Qk (ek )( ) g x ( k ( )){ p( ) is less ha 1. ek ( ) B( )( H ( ) k p )ek ( ) } g x ( k ( )) [ f ( xk 1 ( ) f ( xk ( )] g x ( k ( )) ( P( ) ki ) ek ( ) Equaio 11 becomes (12) ek 1 ( ) Pek ( ) Q(ek )( ) ( P Q )( P Q 1 ) The firs half of he equaio (9) akes he orm ( P Q )e ( ) (13) ek ( ) B( )( H ( ) g x ( k ( )){ ek ( ) } g x ( k ( )) ( P( ) ki )ek ( ) k p g x ( k ( )) B( ) ( H ( ) k p ) ek ( ) ( bh p) ek ( ) (14) Amog hem sup g x ( k ( )) h sup H ( ) k p p sup P( ) From(1): xk 1 ( ) xk ( ) [ f ( xk 1 ( ) f ( xk ( )] B( ) L( )ek ( ) B( )H ( )ek ( ) ( P( ) ki )ek ( ) (15) O boh eds of he orm is: xk 1 ( ) xk ( ) k f xk 1 ( ) xk ( ) bl ek ( ) ek ( ) bh ek ( ) 38

6 p ek ( ) k f xk 1 ( ) xk ( ) ek ( ) ( bh p ) ek ( ) (16) Amog hem 1 bh p ; l sup l ( ) kd ; h sup H ( ) k p The ype ca be wrie as xk 1 ( ) xk ( ) k f xk 1 ( ) xk ( ) bl ek ( ) 1 ek ( ) k f [bl ek ( ) 1 ek (v) e k f ( ) Amog hem F 1 K f ble dv (17) kf T TK f 1 e kf T g x ( k ( )) [ f ( xk 1 ( ) f ( xk ( )] The laer half of he orm K f bl ek ( ) K f F ek (s) ds K f xk 1 ( ) xk ( ) N ek ( ) amog From (18) N K f bl K f FT he above equaio Qk ek ( ) ek ( ) bh ek ( ) N ek ( ) Qk ek ( ) ek ( ) amog max(1 bh N ) whe k 时 [ T ] yk ( ) yd ( ) Prove ha he oupu rajecory approaches he desired rajecory. Simulaed Aalysis Cosider he followig oliear sysem wih repeiive moio properies: x1 ( ) 2 x1 ( ) 3si( x2 ( )) x2 ( ) cos( x1 ( )) x ( ) x1 ( ) 1 u1 ( ) x ( ) 5 2 u ( ) 2 2 y1 ( ).5 x1 ( ) x2 ( ) y2 ( ) cos( ).1 L( ).4 The iiial ieraio codiio is [ x1 () x2 ()] [1.1.] The iiial corol is [u1 ( ) u2 ( )] [1] [1] Calculaed as follows: A I g x ( x( ))B( )L( ) 1 To mee he codiios. Through malab simulaio available: 39

7 Figure 1. Figure 2. Trackig performace of sie fucio Fuzzy PID ype algorihm oupu error curve Figure 3. PID ype algorihm oupu error curve I ca be see from he compariso of Fig. 2 ad Fig. 3 ha he oupu error of he sysem ca be approximaed o zero by 14 ieraios of he fuzzy PID ieraive learig corol algorihm while he PD algorihm eeds 18 imes o reach he same effec. Cocludig Remarks I his paper he fuzzy PID ieraive learig corol problem for a class of oliear sysems uder arbirary iiial codiios is discussed. I is show ha he covergece codiio ca be relaxed uder ay iiial learig. Fuzzy PID ieraive learig corol algorihm is used. Usig he radiioal empirical PID parameers As a referece ad real-ime correcio of he empirical PID parameers o geerae a more accurae fuzzy learig law of he fuzzy law wihou he eed for rigorous iiial posiioig operaio ca ierae he sysem o solve he ieraive learig corol iiial value problem he algorihm Which ca rack he desired rajecory more quickly ad reduce 31

8 he umber of ieraios o make he sysem have good dyamic rackig performace ad improve he covergece speed of he sysem effecively. Refereces [1] ArimooS. Robusess of learig corol for robo maipulaios [J]. Proceedigs of IEEE Ieraioal Coferece o Roboics ad Auomaio Ciciai Ohio USA 199 5: [2] Lee HSBie Z. Sudy o robusess of ieraive learig corol wih o-zero iiial error[j]. Ieraioal Joural ofcoro (3): [3] Su Migxua Huag Baojia [3]. PD ype ieraive learig corol for oliear sysems [J]. Joural of auomaio (5): 711 ~ 714 [4] Che Y We C Gog Z ad Su M. A ieraive learig coroller wih iiial sae learig [J]. IEEE Trasacios o Auomaic Coro (2): [5] Re Xue-mei Huag Bao-jiaZhag Xue-zhiPD-Type Ieraive Learig Corol Fora Class of Noliear Sysem[J]. Aca Auomaic Siica (1):74-79(i Chiese) [6] Owes D HHaoe J. Ieraive Learig Corol-Ari Opimizaio Paradigm[J]. Aual Reviews i Corol (1): 57-7 [7] Lü QFag Y C Ｒe X. Acceleraio suppressio of radom errorso he iiial sae of ieraive learig corol Aca Auomaica Siica214; 4( 7) : [8] Yag XSu Y.Closed loop PD-ype ieraive learig [9] corol of oliear delay sysems ope Sciece Techology ad Egieerig211; 11( 27) : [1] Yu S JDuaLWu J H. Sudy of fuzzy learig corol for elecry-hydraulic servo corol[c] I: Ieraioal Coferece o Machie Learig ad Cybereics. Xi'a Chia: Isiue of Elecrical ad Elecroics Egieers Ic [11] Zhag L P Yag F W. A fuzzy ieraive learig corol desig for poi-o-poi corol of oliear sysem[c]. I: Proceedigs of he World Cogress o Iellige Corol ad Auomaio. Hagzhou Chia: Isiue of Elecrical ad Elecroics Egieers Ic