Anticipatory Iterative Learning Control for Nonlinear Systems with Arbitrary Relative Degree

Transcription

1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 46, NO 5, MAY [14] G De Nicolao, L Magni, and R Scaolini, Sabilizing receding-horizon conrol o nonlinear ime-varying sysems, IEEE Trans Auoma Conr, vol 43, pp , July 1998 [15] T Parisini and R Zoppoli, A receding horizon regulaor or nonlinear sysems and a neural approximaion, Auomaica, vol 31, pp , 1995 [16] J A Primbs, V Nevisić, and J C Doyle, A receding horizon generalizaion o poinwise min-norm conrollers, IEEE Trans Auoma Conr, vol 45, pp , June 2 [17] A Schwarz, Theory and implemenaion o numerical mehods based on Runge Kua inegraion or opimal conrol problems, PhD disseraion, Universiy o Caliornia, Berkeley, 1996 [18] P Scokaer, D Mayne, and J Rawlings, Subopimal model predicive conrol (easibiliy implies sabiliy), IEEE Trans Auoma Conr, vol 44, pp , Mar 1999 [19] A J van der Scha, On a sae space approach o nonlinear H conrol, Sys Conrol Le, vol 116, pp 1 8, 1991 [2], L -Gain and Passiviy Techniques in Nonlinear Conrol London, UK: Springer-Verlag, 1994, vol 218, Lecure Noes in Conrol and Inormaion Sciences wih relaive degree one, by imposing somewha sric resricion on sysem dynamics, or example, he passiviy propery [11] and he boundedness o derivaive o he inpu-oupu coupling marix [12],[13] Mos recenly, in [1], a undamenal concep is inroduced in parallel o he wo basic schemes: D-ype and P-ype ILCs This design approach has he anicipaory characerisic o he D-ype ILC and he simpliciy like P-ype ILC Resuls have been developed again or nonlinear coninuous-ime sysems wih relaive degree one and experimenal resuls are obained in roboic sysems This approach is also sudied in he orm o noncausal ilering [9] In his noe, he anicipaory learning algorihm [1] is applied o sysems wih arbirary relaive degree A deiniion o exended relaive degree is presened o explore a causal propery o he sysems under consideraion The racking error convergence resuls are esablished II PROBLEM FORMULATION Consider he class o nonlinear coninuous-ime sysems described by he sae-space equaions Anicipaory Ieraive Learning Conrol or Nonlinear Sysems wih Arbirary Relaive Degree Mingxuan Sun and Danwei Wang Absrac In his noe, he anicipaory ieraive learning conrol is exended o a class o nonlinear coninuous-ime sysems wihou resricion on relaive degree The learning algorihm calculaes he required inpu acion or he nex operaion cycle based on he pair o inpu acion aken and is resulan variables The racking error convergence perormance is examined under inpu sauraion being aken ino accoun The learning algorihm is shown eecive even i diereniaion o any order rom he racking error is no used Index Terms Convergence, learning conrol, nonlinear sysems, relaive degree I INTRODUCTION Recenly, rigorous analyses o coninuous-ime ieraive learning conrol (ILC) have been developed, see, or example, [2] [1] In paricular, a undamenal characerisic o a class o learning conrol design mehodologies is examined in [5], which clariies he necessiy o he use o error derivaive or sysems wihou direc ransmission erm In [6], his characerisic is urher clariied or nonlinear coninuous-ime sysems error derivaives, he highes order is equal o he relaive degree o he sysems, are used o updae he conrol inpu ILC using he highes-order error derivaives only is ermed D-ype ILC Numerical calculaions migh be required o obain error derivaives or he implemenaion However, he signals obained by numerical diereniaion will be very noisy i he measuremen is conaminaed wih noise ILC wihou using diereniaion is reerred o as P-ype ILC Several echnical analyzes o P-ype ILC are presened or nonlinear coninuous-ime sysems Manuscrip received March 14, 2; revised Sepember 9, 2 Recommended by Associae Edior C Wen The auhors are wih he School o Elecrical and Elecronic Engineering, Nanyang Technological Universiy, Nanyang, , Singapore Publisher Iem Ideniier S (1) _x() = (x()) + B(x())u() (1) y() =g(x()) (2) x 2 R n, u 2 R r and y 2 R m denoe he sae, conrol inpu and oupu o he sysem, respecively The uncions (1) 2 R n, B(1) = [b 1 (1); ;b r (1)] 2 R n2r and g(1) =[g 1 (1); ;g m (1)] T 2 R m are smooh in heir domain o deiniion and are known o cerain properies only This sysem perorms repeiive operaions wihin a inie ime inerval [;T] For each ixed x(), S denoes a mapping rom (x();u(); 2 [;T]) o (x(); 2 [;T]) and O a mapping rom (x();u(); 2 [;T]) o (y(); 2 [;T]) In hese noaions, x(1) = S(x();u(1)) and y(1) = O(x();u(1)) The conrol problem o be solved is ormulaed as ollows Given a realizable rajecory y d (); 2 [;T] and a olerance error bound " >, ind a conrol inpu u(); 2 [;T], by applying an ILC echnique, so ha he error beween he oupu rajecory y() and he desired one y d () is wihin he olerance error bound, ie, ky d () y()k < "; 2 [;T], k1kis he vecor norm deined as kak = max 1in ja ij or an n-dimensional vecor a = [a 1 ; ;a n ] T Throughou he paper, or a marix A = a ij g 2 R m2n, he induced norm kak = max 1im 6 n j=1 ja ij j To solve his problem, we use he ILC in he orm o he ollowing anicipaory updaing law [1]: v k+1 () = u k()+ k ()e k ( + ); i 2 [;T ] v k (T ); i 2 (T ; T ] (3) u k () =sa (v k ()) (4) > small number; k number o operaion cycle; e k () =y d () y k () oupu or racking error; k () 2 R r2m learning gain marix piecewise coninuous and bounded This updaing law is based on he causal relaionship beween he conrol inpu and he sysem oupu o be speciied in he nex secion The ime shi ahead in he racking error insalls he anicipaory characerisic in he updaing law, acuaor sauraion is aken ino /1$1 21 IEEE

2 784 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 46, NO 5, MAY 21 accoun The inpu sauraion uncion sa : R r sa(v) =[sa(v 1 ); ; sa(v r )] T, R r is deined as r and/or L b L 1 g q(x()) = ; 1 q m, and he number o oupus can be greaer han he number o inpus sa(v p )= vp; i jvpj p sgn(v p) p; i jv pj > p or he inpu sauraion bound p > ;p = 1; ;r Deine = max 1pr p g Assumpions are as ollows: A1) or each ixed x k (), he mappings S and O are one o one; A2) he desired rajecory y d ()); 2 [;T] is achievable by an inpu wihin sauraion bounds, ie, u d () =sa (u d ()); 2 [;T]; A3) here exiss a compac se X R n such ha he sysem sae x(); 2 [;T] produced by any inpu u() 2 U; 2 [;T] belongs o X, U R r is a bounded se, ie, x() 2 X; 2 [;T]; A4) he operaions sar rom he iniial condiion x k () = x d () x d () is he iniial condiion corresponding o he desired rajecory Remark 21: For a realizable rajecory y d (); 2 [;T], (A1) implies ha here exiss a unique conrol inpu u d (); 2 [;T] such ha y d () =g(x d ()) and _x d () = (x d ()) + B(x d ())u d () x d (); 2 [;T] is he corresponding sae ILC wih an inpu sauraor can be sill eecive by assuming A2) as argued in [1], [14] The sauraion bounds can be se in accordance wih acuaor limiaions Assumpion A3) is reasonable or sysems which have no inie escape ime on [;T] and mos pracical sysems driven by a bounded inpu will no diverge in a inie-ime inerval due o energy limiaion The ollowing deiniion exends he relaive degree concep in [15] Here, he derivaive o a scalar uncion g(x) along a vecor (x) is deined as L g(x) =(@g(x)=@x)(x) The repeaed derivaives along he same vecor are L i g(x) = L (L i1 g(x));l g(x) = g(x) In addiion, he derivaive o g(x) aken irs along (x) and hen along a vecor b(x) is L b L g(x) =(@(L g(x))=@x)b(x) Deiniion 21: Exended relaive degree o he sysem (1) and (2) is associaed wih a se o inegers 1; ; mg such ha L b L i g q (x()) = ; i q 2; 1 p r 1 q m and he m 2 r marix, shown in he equaion a he boom o he page, has ull-column rank or 2 [;T] and x() 2 X Remark 22: The relaive degree o a coninuous-ime sysem is he imes o diereniaion o he oupu so ha he erms involving he inpu appear [15] The exended relaive degree o he same sysem is he inegraion imes o cerain erms so ha he oupu y( + ) is dynamically relaed wih he inpu u() Deiniion 21 allows ha or some saes a some insans, L b L 1 g q(x()) = ; 1 p III CONVERGENCE ANALYSIS In his secion, we shall examine he convergence perormance when he proposed updaing law (3) (4) is applied o sysems (1) (2) wih exended relaive degree 1 ; ; m g For simpliciy, he resul is presened or he single-inpu single-oupu (SISO) case o he nonlinear sysems beore i is exended o he muliple-inpu mulipleoupu (MIMO) case A Single-Inpu Single-Oupu Sysems The SISO nonlinear sysem under consideraion akes he orm o (1) (2) wih u() and y() being he scalar inpu and he scalar oupu, respecively, B(1) =b(1) 2 R n, and g(1) 2 R being smooh in heir domain o deiniion The updaing law is (3) (4) wih k () = k () being he scalar learning gain Remark 31: The relaive degree o he SISO nonlinear sysem is he ineger such ha [15] L b L i g(x) =; i 2 L b L 1 g(x) 6=: However, he SISO nonlinear sysem has exended relaive degree, i and + L b L i g(x()) = ; i 2 L b L 1 g(x( )) d d 1 6= L b L 1 g(x()) = is allowed or some saes a some insans Obviously, he sysem has exended relaive degree i he relaive degree o he sysem is Remark 32: I he SISO sysem has exended relaive degree, he sysem oupu a he insan + ; 2 [;T ] can be wrien as y( + ) =g(x()) + + L g(x( 1 )) d 1 he second erm can be expressed as, keeping in mind o he exended relaive degree o he sysem + L g(x( 1 )) d 1 = L g(x()) + + L 2 g(x( 2 )) d 2 d 1 : + + [L b L 1 g 1(x( )); ;L b L 1 g 1(x( ))] d d 1 [L b L 1 g m(x( )); ;L b L 1 g m(x( ))] d d 1

3 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 46, NO 5, MAY Similarly, he second erm can be rewrien repeiively unil 2 diereniaion as + = 2 ( 2) L2 g(x()) + L 2 g(x( 2)) d 2 d 1 + L 1 g(x( 1)) d 1 d 1: Then, he conrol appears in he inegraion as ollows: + = 1 ( 1) L1 g(x()) + + L 1 g(x( 1)) d 1 d 1 [L g(x()) + L b L 1 g(x( ))u( )] d d 1 : Thus, he sysem oupu a he insan + can be inally wrien as y( + ) =g(x()) + L g(x()) ( 1) L1 g(x()) + + L g(x( )) + L b L 1 g(x( ))u( )g d d 1 : (5) Equaion (5) shows ha u();y( + )g is a pair o dynamically relaed cause and eec Thus, he updaing law (3) (4) is eec-driven, and i has he anicipaory naure capuring he rend/direcional inormaion However, no error diereniaion is required in he updaing law Theorem 31: Given a desired rajecory y d (); 2 [;T] or he SISO sysem (1) (2) wih exended relaive degree, le he sysem saisy assumpions A1) A4) and use he updaing law (3) (4) I 1 k () + L b L 1 g(x k ( )) d d 1 < 1 (6) he sysem oupu converges o he desired rajecory in he sense o sup ke k ()k max 2[;T ] ; is a posiive consan o be deined Proo: We irs evaluae he error u d ()v k () or 2 [;T] I ollows rom (3) and (5) ha: 1v k+1 () = 1 k () + L b L 1 g(x k ( )) d d 1 1u k () k ()( k ()+ k ()+$ k ()) 1v k () =u d () v k (); 1u k () =u d () u k () and k () =g(x d ()) g(x k ()) + (L g(x d ()) L g(x k ())) + k () = $ k () = + 1 ( 1) (L1 g(x d ()) L 1 + g(x k ())) [L g(x d( )) L g(x k( )) +(L b L 1 g(x d ( )) L b L 1 g(x k ( )))u d ( )] d d 1 + 1u k ()) d d 1: Taking norms and applying he bounds yield L b L 1 g(x k ( ))(1u k ( ) k1v k+1 ()k k1u k ()k + c (k k ()k + k k ()k + k$ k ()k) c is he norm bound or k () Noe ha he uncions (1);b(1);L i g(1); i and L b L 1 g(1) are local Lipschiz in x 2 X since hey are smooh uncions Boh L b L 1 g(1) and b(1) are bounded on X due o he same reason In he res o he proo, l ;l b ;l g, l bg, c bg and c b denoe he Lipschiz consans and he norm bounds, respecively Thereore k k ()k 1+ 1 k k ()k (l g + l bg c ud ) k$ k ()k c bg ( 1) + k1x k ( )k d d 1 l g k1x k ()k k1u k ( ) 1u k ()k d d 1 1x k () = x d () x k () and c ud = sup 2[;T ] ku d ()k Deining gives rise o k1v k+1 ()k c 1 = c ( 1) l g c 2 = c (l g + l bg c ud ) and c 3 = c c bg k1u k ()k + c 1 k1x k ()k + + c 2 k1x k ( )k d d c 3 k1u k ( ) 1u k ()k d d 1 : (7) To evaluae he sae errors in he righ-hand side o (7), we inegrae he sae equaions o obain k1x k ()k (k(x d (s)) (x k (s))k + kb(x d (s)) b(x k (s))kku d (s)k + kb(x k (s))kk1u k (s)k) ds:

4 786 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 46, NO 5, MAY 21 Deining c 4 = l + l b c ud and using he Bellman Gronwall lemma, we have k1x k ()k c b Subsiuing (8) ino (7) produces e c (s) k1u k (s)k ds: (8) k1v k+1 ()k k1u k ()k + c 1c b e c (s) k1u k (s)k ds + c ( s) + c 2c b e 1k1u k (s)k ds d d1 + + c 3 k1u k ( ) 1u k ()k d d1: (9) Muliplying boh sides o (9) by e ( > ) gives e k1v k+1 ()k e k1u k ()k + c 1c b e (c )(s) e s k1u k (s)k ds + e + c 2c b k1u k (s)k ds d d 1 + e + c 3 k1u k ( ) 1u k ()k d d1: The sauraion eaure leads o k1u k (s) 2 and k1u k ( ) 1u k ()k 4, and under he assumpion A2), k1u k ()k k1v k ()k Thus sup e k1v k+1 ()kg 2[;T ] sup e k1v k ()kg + c 5 2[;T ] >c 4 (c )T 1 e = + c 1 c b c 4 c 5 =2c 2 c b e c T 1 c 4 +4c 3 : (1) Since < 1, i is possible o ind a > c 4 suicienly large such ha < 1 Then, (1) is a conracion in sup 2[;T] e k1v k ()kg which leads o sup e k1v k ()kg 2[;T ] c 5 1 From (8), and using he similar manipulaions, we obain sup e k1x k ()kg 2[;T ] (c )T 1 e c b c 4 c 5 1 : (11) : (12) For 2 (T ; T ], (8) sill holds so ha which resuls in T k1x k ()k c b e c (s) k1v k (s)k ds + c b e c (s) k1u k (s)k ds T sup e k1x k ()kg 2(T ;T ] (c )T 1 e c b c 4 or > c 4 Thereore sup e k1v k ()kg +2c b 2[;T ] sup e k1x k ()kg 2(T ;T ] (c )T 1 e c b c 4 c c b: (13) Now, he resul is esablished or he oupu error e k (), 2 [;T], ollowing (12) and (13) = e T l g c b sup ke k ()k max 2[;T ] (c )T 1 e c 5 c : ; (14) This complees he proo Remark 33: Theorem 31 gives an explici suicien condiion guaraneeing he convergence o racking error, and provides a guide or he learning conrol design The design needs a sysem model, bu model discrepancy is allowed Thus, his major advanage o ieraive learning conrol mehodology remains in he proposed learning algorihm For example, we consider he case is se o be small enough so ha he condiion (6) can be approximaely wrien as 1 k () + d 1 1 k ()d k () L b L 1 g(x k ( )) d (15) d k () =L b L 1 g(x k ())Id k () is modeled o be ^d k () and we assume ha ^d k () = k ()d k ()( k () > ) We choose k () = 1 ^d () so ha k 1 k ()d k () = 1 = 1 1 k 1 ^d k ()d k() () is an adjusable parameer The condiion k1 1 k ()( =)k < 1 will hold i < < 2 k ()(= ) B Muli-Inpu Muli-Oupu Sysems The obained convergence resul or he SISO sysems can be exended o he MIMO sysems Theorem 32: Given a desired rajecory y d (); 2 [;T] or he sysem (1) (2) wih exended relaive degree 1 ; ; m g, le he sysem saisy assumpions A1) A4) and be under he acion o he

5 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 46, NO 5, MAY updaing law (3) (4) The sysem oupu converges o he desired rajecory in he sense o sup ke k ()k max 2[;T ] wih a posiive consan o be deined, i max 1qm q ; ki k ()D k ()k < 1 (16) he equaion shown a he boom o he page holds rue Proo: Parallel o (5), he qh oupu componen a he insan + can be wrien as y q;k ( + ) =g q(x k ()) + L g q(x k ()) L 1 ( q 1) g q (x k ()) + + L gq(x k( )) +[L b L 1 g q (x k ( ); ); ;L b L 1 g q (x k ( ); )]u k ( )g d d1: (17) Using (17), he error 1v k () or 2 [;T ] saisies 1v k+1 () =(I k ()D k ())1u k () k ()( k ()+ k ()+$ k ()) k () =[ 1;k (); ; m;k ()] T k () =[ 1;k (); ; m;k ()] T $ k () =[$ 1;k (); ;$ m;k ()] T ; and q;k () =g q (x d ()) g q (x k ()) + [L g q(x d ()) L g q(x k ())] [L g q (x d ()) L 1 g q (x k ())] ( q 1) +h q;k () = $ q;k () = L gq(x d( )) L gq(x k( )) +([L b L 1 g q (x d ( )) ;L b L 1 g q (x d ( ))] [L b L 1 g q(x k ( )) ;L b L 1 g q(x k ( ))])u d ( )g d d1 + [L b L 1 g q(x k ( )); ;L b L 1 g q(x k ( ))] 1 (1u k ( ) 1u k ()) d d 1 : Taking norms and applying he bounds yield k1v k+1 ()k k1u k ()k + c (k k ()k + k k ()k + k$ k ()k) c is he norm bound or k () To proceed, we need he Lipschiz condiions and he bounds Denoe by l ;l B;l g and l bg he Lipschiz consans or he uncions (1);B(1);L i g q (1); i q ; 1 q m and L b L 1 g q(1); 1 p r; 1 q m, c bg and c B he norm bounds or [L b L 1 g q (1); ;L b L 1 g q (1)], 1 q m and B(1), respecively Thereore k q;k ()k l g k1x k ()k 1 ( q 1) + k q;k ()k (l g + rl bg c ud ) k1x k ( )k d d1 + k$ q;k ()k c bg Deining k1u k ( ) 1u k ()k d d 1 c 1 = c max 1+ 1qm l g ( q 1) c 2 = c (l g + rl bg c ud ); and c 3 = c c bg we have (18), shown a he boom o he page Inegraing he sae D k () = + + [L b L 1 g 1 (x k ( )); ;L b L 1 g 1 (x k ( ))] d d 1 [L b L 1 g m (x k ( )); ;L b L 1 g m (x k ( ))] d d 1 : k1v k+1 ()k k1u k ()k + c 1 k1x k ()k + c 2 + c k1x k ( )k d k1u k ( ) 1u k ()k d d 1 k1u k ( ) 1u k ()k d d 1 d1 k1x k ( )k d d 1 : (18)

6 788 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 46, NO 5, MAY 21 k1vk+1()k k1uk()k + c1cb + c2cb + c e c (s) k1uk(s)k ds e c ( s) k1uk(s)k ds d d1 e c ( s) k1uk(s)k ds d d1 k1uk( ) 1uk()k d d1 k1uk( ) 1uk()k d d1 : equaions and deining c4 = l + lb cud resul in k1xk()k cb e c (s) k1uk(s)k ds: (19) Subsiuing (19) ino (18) produces he equaion shown a he op o he nex page Parallel o he developmen in he SISO case, we have inally sup kek()k max 2[;T ] = e T (c )T 1 e lgcb c4 (c )T 1 e = + c1cb c4 e c T 1 c5 =2c2cB +4c3: c4 max 1qm c5 q +2 1 and ; (2) This complees he proo Remark 34: Theorems 31 and 32 show ha he exended relaive degree o he sysems under consideraion is no included in he proposed updaing law (3) (4) isel However, i is required implicily when we design he parameer and he learning gain k () ollowing he way speciied in Remark 33 For D-ype ILC, he error derivaive wih he order being equal o he relaive degree is used o updae he conrol inpu so ha he relaive degree is required o be known explicily There may be poins an exended relaive degree canno be deined or some class o sysems and he proposed scheme ails o work This would be he major limiaion on he learning algorihm which is applicable o he sysems wih well deined exended relaive degree [2] S Arimoo, S Kawamura, and F Miyazaki, Beering operaion o robos by learning, J Robo Sys, vol 1, no 2, pp , 1984 [3] J E Hauser, Learning conrol or a class o nonlinear sysems, in Proc 26h IEEE Con Decision Conrol, Los Angeles, CA, USA, Dec 1987, pp [4] G Heinzinger, D Fenwick, B Paden, and F Miyazaki, Sabiliy o learning conrol wih disurbances and uncerain iniial condiions, IEEE Trans Auoma Conr, vol 37, pp , Jan 1992 [5] T Sugie and T Ono, An ieraive learning conrol law or dynamical sysems, Auomaica, vol 27, no 4, pp , 1991 [6] H-S Ahn, C-H Choi, and K-B Kim, Ieraive learning conrol or a class o nonlinear sysems, Auomaica, vol 29, no 6, pp , 1993 [7] K L Moore, Ieraive learning conrol or deerminisic sysems, in Advances in Indusrial Conrol London, UK: Springer-Verlag, 1993 [8] Z Bien and J-X Xu, Ieraive Learning Conrol Analysis, Design, Inegraion and Applicaions Boson, MA: Kluwer Academic Publishers, 1998 [9] Y Chen and C Wen, Ieraive Learning Conrol: Convergence, Robusness and Applicaions London, UK: Springer-Verlag, 1999 [1] M Sun and B Huang, Ieraive Learning Conrol Beijing, China: Naional Deense Indusrial Press, 1999 [11] S Arimoo, T Naniwa, and H Suzuki, Robusness o P-ype learning conrol heory wih a orgeing acor or roboic moion, in Proc 29h IEEE Con Decision Conrol, Honolulu, HI, Dec 199, pp [12] T-Y Kuc, J S Lee, and K Nam, An ieraive learning conrol heory or a class o nonlinear dynamic sysems, Auomaica, vol 28, no 6, pp , 1992 [13] S S Saab, On he P-ype learning conrol, IEEE Trans Auoma Conr, vol 39, pp , Nov 1994 [14] T-J Jang, C-H Choi, and H-S Ahn, Ieraive learning conrol in eedback sysems, Auomaica, vol 31, no 2, pp , 1995 [15] A Isidori, Nonlinear Conrol Sysems Berlin, Germany: Springer-Verlag, 1995 IV CONCLUSION The concep o exended relaive degree is inroduced o describe he inpu oupu causaliy o a class o nonlinear coninuous-ime sysems The anicipaory ieraive learning conrol mehod is shown applicable o he sysems wih such exended relaive degree I is shown o be able o reduce he racking error ieraively under he derived suicien condiion on he anicipaory parameer and he learning gain Moreover, his approach does no require any error diereniaion REFERENCES [1] D Wang, On D-ype and P-ype ILC designs and anicipaory approach, In J Conrol, vol 73, no 1, pp 89 91, 2