LMI-Based Synthesis of Robust Iterative Learning Controller with Current Feedback for Linear Uncertain Systems

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1 Iteratioal LMI-Based Sythesis Joural of of Cotrol Robust Automatio Iterative Learig ad Systems Cotroller vol. with 6 o. Curret 2. Feedback 7-79 Aril for Liear 28 Ucertai Systems 7 LMI-Based Sythesis of Robust Iterative Learig Cotroller with Curret Feedback for Liear Ucertai Systems Jiamig Xu Migxua Su ad Li Yu Abstract: his aer addresses the sythesis of a iterative learig cotroller for a ass of liear systems with orm-bouded arameter ucertaities. We take ito accout a iterative learig algorithm with curret cye feedback i order to achieve both robust covergece ad robust stability. he sythesis roblem of the develoed iterative learig cotrol (ILC) system is reformulated as the γ -subotimal H cotrol roblem via the liear fractioal trasformatio (LF). A sufficiet covergece coditio of the ILC system is reseted i terms of liear matrix iequalities (LMIs). Furthermore the ILC system with fast covergece rate is costructed usig a covex otimizatio techique with LMI costraits. he simulatio results demostrate the effectiveess of the roosed method. Keywords: H cotrol ILC LF LMI.. INRODUCION Iterative learig cotrol (ILC) is a techique to cotrol a system carryig out a task over a fiite time iterval reetitively such that the system outut accurately tracks a secified referece trajectory. Motivated by huma learig the basic idea of iterative learig cotrol is to use a iterative rocedure to calculate the iut sigal from the revious oeratio data such that the trackig error is gradually reduced. Sice the itroductio of iterative learig cotrol (ILC) methodology by Arimoto et al. [] the geeral area of ILC has bee the subject of itese research effort both i terms of the uderlyig theory ad egieerig ractice [2]. Iterative learig cotrol was iitially develoed as a feedforward actio alied directly to the oe-loo system. Although the ure feedforward learig cotrol scheme is theoretically accetable it is ulikely to be alied to real systems without a feedback cotrol. Oe reaso is that it may geerate harmful effects if the oe-loo system is ustable or exists ucertaity. I additio the trackig error may ossibly grow quite large i the early stages of learig though it evetually coverges after a umber of trials. hus i Mauscrit received February 8 26; revised July 9 27; acceted December Recommeded by Editorial Board member Myotaeg Lim uder the directio of Editor Jae Weo Choi. his work was suorted by the Natioal Natural Sciece Foudatio of Chia uder Grats ad Jiamig Xu Migxua Su ad Li Yu are with the College of Iformatio Egieerig Zhejiag Uiversity of echology 8 Chaowag St. Hagzhou 34 Chia ( s: {xujm mxsu lyu}@zjut.edu.c). real eviromets a curret feedback cotrol is commoly emloyed alog with the iterative learig cotrol for system robustess ehacemet ad better erformace [3]. he combiatio of ILC with curret feedback is tyically doe as a feedforward-feedback cofiguratio the curret feedback cotroller esures osed-loo stability ad suresses exogeous disturbaces ad the iterative learig cotroller rovides imroved trackig erformace over a secific referece trajectory. Most of the existig results have cocetrated o derivig ew algorithms ad aalyzig their covergece roerties. Few results o ILC sythesis have bee reorted esecially uder model ucertaities i literature. Ama et al. roosed a two-ste desig rocedure based o H otimizatio [4]. De Roover sythesized a iterative learig cotroller based o H cotrol uder ustructured ucertaities [5]. Moo ad Doh [67] derived a sufficiet coditio for covergece of the iterative rocess i the resece of lat ucertaity. A iterative learig cotroller that satisfies the covergece coditio ca be obtaied by µ - sythesis rocedure called D-K iteratio [8]. However the D-K iteratio caot guaratee the global covergece ad usually leads to a high-order cotroller that is ot easy to imlemet i the ractical situatio. It has recetly bee emhasized by Boyd et al. that may roblems arisig i system theory ca be cast ito the form of liear matrix iequalities (LMIs) [9] which belog to the grou of covex roblems ad thus oe ca efficietly fid feasible ad global solutios to them via iterior-oit methods. o the best of our kowledge the ILC desig roblem for liear ucertai systems has ot

2 72 Jiamig Xu Migxua Su ad Li Yu bee ivestigated via LMI aroaches. I this aer based o the frequecy domai reresetatio we deal with the sythesis roblem of the ILC algorithm with curret feedback that assures covergece for a ass of liear systems with ormbouded time-varyig arameter ucertaity. he roblem is first reformulated as a γ -subotimal H cotrol roblem via the liear fractioal trasformatio (LF). he a sufficiet covergece coditio is established for the ILC system i terms of LMIs. he solutios of the LMIs ca be used to costruct a suitable ILC algorithm. Furthermore a covex otimizatio roblem with LMI costraits is formulated to desig the ILC algorithm that achieves fast covergece seed of the resultig ILC system. 2. PROBLEM SAEMEN Cosider a iterative learig cotrol system with curret feedback show i Fig. yd () t is the desired outut trajectory over a fiite iterval t [ ] yk () t uk () t ad ek ( t ) are the system outut the cotrol iut ad the trackig error at the k th iteratio resectively. P( s ) is the cotrolled lat ad ca be described by the followig state sace model xt () = Axt () + But () = [ A + Axt ] () + [ B + But ] () yt () = Cxt () () xt () R ut () R yt () R deote the state the cotrol iut ad the system outut resectively A B C are kow real costat matrices with aroriate dimesios A B are matrix-valued fuctios reresetig time-varyig arameter ucertaities i the system model ad are assumed to be of the followig form: [ A B] = DF( t)[ E E ] (2) 2 D E E2 are kow real costat matrices with aroriate dimesios which rereset the i j structure of ucertaities ad Ft R () is a ukow matrix fuctio with Lebesgue measurable elemets ad satisfies F () t F() t I (3) I deotes the idetity matrix of aroriate dimesio. Cosider the ILC algorithm with curret feedback give i the frequecy domai as Uk+ () s = Lu() s Uk() s + Le() s Ek() s + CsE () () s k+ (4) Lu ( s ) ad Le ( s ) are the learig cotrollers Cs () is the curret feedback cotroller. o imlemet i the ractical situatios all cotrollers should be i RH sace. I the formulatio of the sythesis roblem of the ILC system we make the followig assumtios. Assumtio : he iitial state of the lat P( s ) is ivariat with resect to k so Y () s = Y () s for k = 2... trasform of y k (). k Yk () s deotes the Lalace Assumtio 2: Let Lu () s be lu ( s) I lu () s is a tye of low-ass filter with a cut-off frequecy ω c beig higher tha the trackig badwidth such that lu ( j ω) = ω [ ω c ] ad lu ( j ω) < ω> ω c. Lemma [7]: Suose that Assumtios ad 2 are satisfied ad Yd () s Y () s H2 he the ILC system i Fig. is L 2 coverget with remaiig error if the H orm of stable Ls ( ) is less tha Ls () = ( + PsCs () ()) ( L() s PsL () ()). s (5) Moreover the remaiig error E ( s) is E () s = lim( Yd() s Yk()) s k (6) = ( Ls ()) Dd()( s Yd() s Y ()) s H2 D () s = ( + P() s C()) s ( I L ()). s (7) d For give Lu ( s ) the sythesis roblem of the roosed ILC system is coverted ito solvig Cs ( ) ad Le ( s ) from the followig (sub)otimal H sythesis roblem: u u e ( I + P( s) C( s)) ( L ( s) P( s) L ( s)) = γ <. (8) u e Fig.. A ILC system with curret feedback. Note that the smaller is γ the faster is the

3 LMI-Based Sythesis of Robust Iterative Learig Cotroller with Curret Feedback for Liear Ucertai Systems 73 resectively. he the lat G is described i state sace coordiates: Fig. 2. Diagram reresetatio of F l ( GK ). covergece rate of the error because ( E ( s) E ( s) L( s) E ( s) E ( s) k+ 2 k 2 (9) γ ca be used to idicate the covergece rate of the ILC system. o solve the roblem described i (8) for ractical situatios we adot the aroach suggested i Zhou et al. [8]. herefore the ILC sythesis roblem is reformulated i the stadard lat format deicted i Fig. 2. Withi this framework tools are available for comutig a stabilizig K that miimizes zw (with zw beig the trasfer fuctio from w to z ). o formulate the ILC sythesis i this stadard lat framework this trasfer fuctio from w to z is cosidered give by the lower liear fractioal trasformatio (LF) of G ad K. If G is artitioed as: G G G := () G 2 2 G 22 the the lower LF of G ad K deoted F l ( GK ) is defied as: F ( GK ):= G + G KI ( G K) G =.() l Clearly L(s) ca be described i this form by takig: ad Lu () s P() s G G 2 G := = Lu () s P() s G 2 G 22 I zw (2) K := C() s L (). s (3) e It should be oted that for solutio of the sythesis roblem usig the stadard lat format if the learig cotroller Lu ( s ) is described by the followig state sace equatio: x u = Auxu + Buuu y = C x + D u u u u u u (4) x 2 u R uu R yu R deote the state the cotrol iut ad measure outut x u = Ax+ Bw + B2u z = Cx+ Dw y = C x+ D w 2 2 (5) x = [ x xu ] R ( = + 2) u = u R 2 w = u u R z R ad y R deote the state the cotrol iut the disturbace iut the cotrolled outut ad the measure outut of the system G resectively; ad A B A = B = B2 = C C Cu A u B u = C Cu Du C2 = D = Du D2 =. I As a cosequece the sythesis roblem of the ILC system is reduced to a γ -subotimal H cotrol roblem i.e. to desig a outut cotroller (3) such that = γ < zw for the system (5). I this aer we solve the roblem via a LMI aroach. I the roof the mai results we will eed the followig lemmas. Lemma 2 (the Bouded Real Lemma) []: Cosider a cotiuous time trasfer fuctio ( s ) of realizatio s () = D+ CsI ( A) B the followig statemets are equivalet: (i) D+ C( si A) B < γ ad A is stable; (ii) here exists a symmetric ositive defiite solutio X to the LMI: A X + XA XB C B X γi D <. C D Lemma 3 []: Give matrices M N ad symmetric matrix S of aroriate dimesios the S + MFN + N F M < for ay ucertai matrix F satisfyig F F I if ad oly if there exists a scalar ε > such that S + εmm + ε N N <. Lemma 4 []: Give a symmetric matrix m Π R ad two matrices Ψ Φ of colum dimesio m cosider the roblem of fidig some matrix Θ of comatible dimesio such that

4 74 Jiamig Xu Migxua Su ad Li Yu Π+ Ψ ΘΦ+ Φ Ψ<. Deoted by N Ψ N Φ ay matrices whose colums form bases of the ull sace of Ψ ad Φ resectively. he the above iequality is solvable for Θ if ad oly if Ψ Ψ Φ Φ N ΠN < N ΠN <. Lemma 5 [2]: Give symmetric ositive defiite matrices XY R the there exist matrices M N R ad symmetric matrices ZW R satisfyig X M X M Y N > = M Z M Z N W if ad oly if X I > ad rak( I XY ). I Y 3. MAIN RESULS heorem : Cosider the system (5) the cotiuous time γ -subotimal H roblem is solvable if ad oly if there exist a scalar ε > symmetric matrices RS satisfyig A R + RA + εdd RE RC B ER εi Ω Ω CR D B D < (6) SA + A S + ε E E SB C SD B S D Ξ Ξ C D D S ε I < (7) R I I S (8) A E A B D = B 2 = D A u = E E 2 = E = 2 Ω = diag{ NR I} Ξ = diag{ NS I} N R ad N S deote bases of the ull saces of [ B 2 E 2 ] ad [ C2 D 2] resectively. I additio there exist γ -subotimal cotrollers of order h< (reduced order) if ad oly if (6)-(8) hold for a scalar ε > ad some R S which further satisfy: rak( I RS) h. (9) Proof: Give ay roer real-ratioal cotroller K of realizatio h h K K K K K K = D + C ( si A ) B ( A R ) (2) a realizatio of the osed-loo trasfer fuctio from w to z is obtaied as: l GK = D + C si A B F ( ) ( ) (2) A + B D C B C A C C 2 K 2 2 K = = BKC2 A K B + B2DK D 2 = =. BK D 2 B D D Gatherig all cotroller arameters ito the sigle variable AK B K Θ := (22) C D K K ad itroducig the shorthad: A B B2 A= 2 B = B = Ih I h C = D 2 C = D 2 the osed-loo matrices A B ca be writte as: A = A+ B ΘC B = B + B Θ D. (23) From the Bouded Real Lemma (2) is a h order γ -subotimal cotroller if ad oly if the LMI ( A+ B2Θ C) X X ( B+ B2Θ D2) C X ( A B2Θ C) + + ( B B2Θ D2) + X D C D < (24) holds for some X > i ( + h ) ( + h R ).

5 LMI-Based Sythesis of Robust Iterative Learig Cotroller with Curret Feedback for Liear Ucertai Systems 75 Sice there exist time-varyig arameter ucertai matrices i the system model P(s) the the matrices A B 2 ca be writte ito: A = A + DF E B2 = B2 + DF E2 (25) A B 2 D A = 2 B = D I = F = diag{ F FF} E E2 E =. E = 2 Substitutig (25) ito (24) usig Lemma 3 ad the Schur comlemet argumet the the LMI (24) holds for all F satisfyig F F I if ad oly if there exist a scalar ε > Θ ad X such that X Π+ Ψ ΘΦ+ Φ Θ Ψ < (26) X A X + X A XB C XD E B X D = C D D X ε I E εi X = B 2 X E 2 = C D2 Π Ψ Φ. We ca ow ivoke Lemma 4 to elimiate Θ ad obtai solvability coditios deedig oly o ε X ad the lat arameters. Secifically let N Ψ X ad N Φ deote matrices whose colums form bases of the ull sace of Ψ X ad Φ resectively. he by Lemma 4 (26) holds for some Θ if ad oly if Ψ Ψ Φ Φ N ΠN < N ΠN <. (27) Now if we defie Ψ = [ B2 E 2] ad Y= diagx { IIII } the Ψ = Ψ Y. Hece ΨX X = Ψ is a base of the ull sace of Ψ X N Y N wheever N Ψ is a base of the ull sace of Ψ. Cosequetly ΨX N ΠN < is equivalet to Ψ Ψ ΨX N ΛN < Λ = ( Y ) Π( Y ) X A + A X B = B C X D D E X Furthermore (27) is equivalet to Ψ Ψ Φ Φ D X C D X E ε I εi. N ΛN < N ΠN <. (28) Fidig a ositive defiite matrix X satisfyig (28) is awkward sice it ivolves both X ad its iverse simultaeously. his ca be doe by artitioig X ad X as X S N R M := X := N M h. (29) RS R ad M N R Cosider the first costrait N Ψ ΛN Ψ <. With the artitio (29) Λ ca be writte ito AR + RA AM B RC MA M C B D Λ = CR CM D D ER EM (3) D RE M E. ε I ε I εi εi

6 76 Jiamig Xu Migxua Su ad Li Yu Meawhile from I Ψ = B2 E2 (3) it follows that base of the ull sace of Ψ are of the form N Ψ W = I W2 I (32) NR := W W 2 is ay base of the ull sace of B2 E 2. Observig that the secod row of N Ψ is idetically zero ad ivokig Schur comlemet argumet agai the coditio N Ψ ΛN Ψ < ca be reduced to (6). Similarly the coditio NΦΛN Φ < is equivalet to (7). Hece X satisfies (28) if ad oly if RS satisfy (6)-(7). Moreover by Lemma 5 X > is equivalet to RS satisfyig (8)-(9). heorem 2: Suose that Assumtios ad 2 are satisfied ad Yd () s Y () s H2. he the ILC system i Fig. is robustly 2 coverget with remaiig error ad robustly stable for ay ucertai matrix F satisfyig F F I if there exist scalars < γ < ε > ad symmetric matrices RS satisfyig (6)-(9). Proof: he costrait coditio of γ ( < γ < ) guarateed the covergece of the ILC via Lemma. Furthermore by usig Lemma ad heorem we ca directly obtai the results of the heorem 2. Now we roose the followig desig rocedure for the cotroller (3). heorem 3: Cosider the ILC system i Fig. ad give ε > if the followig otimizatio roblem mi RS st..( i) (6)(7)(8) ( ii) < γ < γ (33) has a otimal solutio γ RS the the arameter matrix Θ of the cotroller (3) of order h= is derived i terms of the solvability to the LMI (26). Proof: Sice RS R ad rak ( I RS) the cotroller (3) of order h= satisfy the rak costrait (9). O the bases of heorem 2 the otimal covergece rate γ ad matrices R S ca be obtaied by solvig the otimizatio roblem (33). h Furthermore comutig two matrices M N R such that MN = I RS (34) a adequate X is the obtaied as solutio of the liear equatio: S I I R X = N M (35) ad the existece of a solutio Θ to the iequality (26) ca be guarateed i virtue of heorem. Remark : Sice the matrix iequalities (6) ad (7) cotai ε ad ε resectively the otimizatio roblem (33) ca be solved by two stes as follows: Ste : o solve the followig otimizatio roblem mi R ε γ st..() i (6) ( ii) < γ < a otimal solutio γ R ad ε ca be obtaied. Ste 2: o solve the feasible roblem about the matrix iequalities (7) ad (8) the variables R ad ε are resectively substituted by R ad ε obtaied i Ste a feasible solutio S ca be give. 4. AN ILLUSRAIVE EXAMPLE o illustrate a alicatio of the ILC techique osider a aealig rocess model [3]. he thermal rocessig setu is illustrated schematically i Fig. 3. A cotrolled heater lam heats the art i the furace ad the furace chamber. For such thermal rocessig has two states: art temerature P ad furace temerature F. his rocess is a oliear system. For the alicatio of the learig algorithm this rocess is liearized aroud the statioary oit = 5 C P ad F = 5 C leadig a cotiuous-time liear ucertai model described as () i which A = B =.. C = D= E = E2 = 3 9.

7 LMI-Based Sythesis of Robust Iterative Learig Cotroller with Curret Feedback for Liear Ucertai Systems 77 hus it follows from the above equatio (5) (3) ad (2) that Fig. 3. hermal rocess overview s +. 2 s s Cs () = s s s (37) s + 539s 2. 7 s 2. 3 Le () s = s s s (38) ad the Bode Diagram of Ls ( ) as Fig. 5. Fig. 6 shows trackig errors at st iteratio (oly curret feedback cotroller) 2d iteratio 3rd iteratio ad th iteratio. By addig the learig cotrollers the trackig errors are dimiished as the iteratio umber icreases. More quatitative iformatio ca be obtaied from Fig. 7 showig the root mea square (RMS) values of the trackig errors resectively. I additio i the Fig. 7 the solid lie dash lie ad dot lie deote the corresodig results about the omial lat model the lat model for Ft () = ad the lat model for Ft ( ) = Fig. 4. Referece temerature trajectory. ad the system outut y = P is the art temerature that is assumed to be directly measured. I this simulatio the cotrol goal is to ram the art temerature from maitai it at 4 C C to 4 C i sec for sec the ram the temerature u to 8 C i sec stay there for 2sec ad fially ram the temerature dow to 4 C i sec ad stay there for sec as Fig. 4. lu () s is chose as a first order low-ass filter ad Lu () s is described by the state sace equatio (4) i which Fig. 5. he bode diagram of L(s). A = 5 B = C = 5 D =. (36) u u u u By alyig heorem 3 ad solvig the corresodig otimizatio roblem (33) we obtai γ = A Θ = C K K B D K K = Fig. 6. rackig errors at st (dash) 2d (dot) 3rd (dash-dot) ad th (solid) iteratio.

8 78 Jiamig Xu Migxua Su ad Li Yu Fig. 7. Root mea square values of trackig errors versus iteratio umber. resectively. hese simulatios show that the resultig ILC system with the curret feedback cotroller (37) ad the learig cotrollers (36) ad (38) is robustly coverget ad robustly stable CONCLUSIONS It has bee show i this aer that the desig of the iterative learig cotrol algorithm ca be geeralized to the desig of a γ -subotimal H cotroller by choosig a aroriate comlex matrix G ad reformulatig the ILC sythesis roblem i the stadard lat format. A sufficiet covergece coditio for the roosed ILC rocess i the resece of lat ucertaity is give i terms of liear matrix iequalities. Based o the derived coditio we showed that the iterative learig cotrol desig roblem ca be reformulated as a geeral robust cotrol roblem ad thus ca be solved by the liear matrix iequality aroach. REFERENCES [] S. Arimoto S. Kawamura ad F. Miyazaki Betterig oeratio of robots by learig Joural Robotic Systems vol. o [2] R. W. Logma Iterative learig cotrol ad reetitive cotrol for egieerig ractice Iteratioal Joural of Cotrol vol. 73 o [3] K. S. Lee S. H. Bag ad K. S. Chag Feedback-assisted iterative learig cotrol based o a iverse rocess model Joural of Process Cotrol vol. 4 o [4] N. Ama D. H. Owes E. Rogers ad A. Wahl A H aroach to liear iterative learig cotrol desig Iteratioal Joural of Adative Cotrol ad Sigal Processig vol. o [5] D. D. Roover Sythesis of a robust iterative learig cotroller usig a H aroach Proc. of the 35th Cof. o Decisio ad Cotrol [6] J. H. Moo. Y. Doh ad M. J. Chug A robust aroach to iterative learig cotrol desig for ucertai systems Automatica vol. 34 o [7]. Y. Doh J. H. Moo K. B Ji ad M. J. Chug Robust ILC with curret feedback for ucertai liear systems Iteratioal Joural of Systems Sciece vol. 3 o [8] K. Zhou J. C. Doyle ad K. Glover Robust ad Otimal Cotrol Pretice Hall Eglewood Cliffs NJ 996. [9] S. Boyd L. E. Ghaoui E. Fero ad V. Balakrisha Liear Matrix Iequalities i System ad Cotrol heory Philadelhia SIAM 994. [] L. Yu ad J. Chu A LMI aroach to guarateed cost cotrol of liear ucertai timedelay systems Automatica vol. 35 o [] P. Gahiet ad P. Akaria A liear matrix iequality aroach to H cotrol Iteratioal Joural of Robust ad Noliear Cotrol vol. 4 o [2] A. Packard K. Zhou P. Padey ad G. Becker A collectio of robust cotrol roblems leadig to LMI s Proc. of the 3th Cof. o Decisio ad Cotrol [3] D. Gorievsky Loo shaig for iterative cotrol of batch rocesses IEEE Cotrol Systems Magazie vol. 22 o Jiamig Xu received the B.Eg. degree from Nachag Uiversity i 998 ad the M.S. degree from Zhejiag Uiversity of echology Hagzhou Chia i 23. He is curretly a Lecturer i the College of Iformatio Egieerig Zhejiag Uiversity of echology Chia. His research iterests iude iterative learig cotrol ad robust cotrol as well as their alicatios.

9 LMI-Based Sythesis of Robust Iterative Learig Cotroller with Curret Feedback for Liear Ucertai Systems 79 Migxua Su received the B.Eg. degree from the Xi a Uiversity of echology Chia i 982 the M.Sc.(Eg.) degree from the Beijig Istitute of echology Chia i 987 ad the Ph.D. degree from the Nayag echological Uiversity Sigaore i 22. From 982 to 984 he was with the Hefei Geeral Machiery Research Istitute Chia he became a Assistat Egieer of the Istitute i 983. He joied the Deartmet of Electrical Egieerig Xi a Istitute of echology Chia i 987 becomig a Associate Professor i 994 ad the Deuty Director of the Deartmet i 997. He held Research Fellow ositios at the Nayag echological Uiversity ad the Natioal Uiversity of Sigaore from 2 to 24. Sice 24 he has bee with the Zhejiag Uiversity of echology Chia ad is curretly a Professor of the College of Iformatio Egieerig. His curret research iterests iude iterative learig cotrol ad reetitive cotrol as well as their alicatios. Li Yu received the B.S. degree i Cotrol heory from Nakai Uiversity i 982 ad the M.S. ad Ph.D. degrees from Zhejiag Uiversity Hagzhou Chia. He is curretly a Professor i the College of Iformatio Egieerig Zhejiag Uiversity of echology Chia. His research iterests iude robust cotrol timedelay systems decetralized cotrol.