OUTPUT FEEDBACK SLIDING MODE CONTROLLER DESIGN VIA H THEORY

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1 4 Asia Jural f Crl, Vl. 5, N. 1, pp. 4-31, March 003 OUPU FEEDBACK SLIDING MODE CONROLLER DESIGN VIA H HEORY Jea-Li Cha ABSRAC Fr a liear sysem wih mismached disurbaces, a slidi mde crller usi ly upu feedback is develped i his paper. hruh applicai f he H crl hery, he desied swichi surface ca achieve rbus sabiliai ad uaraee a level f disurbace rejeci duri slidi mde. Alhuh he sysem exhibis disurbaces, a sae esimar is used which, usi ly measured upus, ca asympically esimae he sysem saes. he crl law is desied wih respec he esimaed sials. Fially, a umerical example is preseed demsrae he prpsed crl scheme. KeyWrds: Slidi mde crl, mismached disurbace, sae esimar, H apprach. I. INRODUCION he slidi mde crl (SMC is a useful l fr deali wih sysem uceraiies ad exeral disurbaces [1]. Earlier resuls fr he SMC were based he assumpi ha all he sysem saes are available [-5]. Hwever, crller desiers fe ecuer sysems fr which ly upu ifrmai is baiable. deal wih hiiuai, several upu feedback slidi mde crl (OFSMC alrihms have recely bee prpsed [6-9]. I hese papers [6-9], he ucerai erm r disurbace mus saisfy he s-called machi cdii [10]. I fac, hwever, mismached uceraiy r disurbace exiss i may pracical sysems. here currely exis effecive crl echlies desied hadle sysems wih mismached disurbaces, fr example, H crl [11-1], adapive crl [13], ad SMC [4-5]. Whe a SMC sysem suffers frm mismached disurbaces, a cmpe f he disurbace will affec sysem perfrmace while i slidi mde [4]. I [4-5], he researchers csidered ly he sae feedback SMC fr sysems wih mismached ucerai erms. I his paper, we will sudy he OFSMC prblem ad replace he machi cdii. Mauscrip received Nvember 30, 000; revised May 16, 001; acceped Jauary 15, 00. he auhrs is wih Deparme f Elecrical Eieeri, Orieal Isiue f echly, Pa-Chia, aipei Cuy 0, aiwa. Sice he sysem saes are available ad ukw disurbaces exis i he sysem, rbus sae esimai is eeded. his prblem is kw i he lieraure as ha f rbus faul deeci r ukw ipu bservers [17-19]. I his paper, we will apply he desi echique i OFSMC [6-9] fid a pracical rasfrmai marix effecively esimae he rue saes. I addii, a swichi surface desi mehd will be prpsed, which cmbies H crl hery ad ur prir wrk [15]. he mai feaure is ha he swichi surface is deermied by slvi he s-called alebraic Riccai equai arisi i H crl hery [11,1]. his mehd ca uaraee rbus sabiliy ad disurbace aeuai fr he sysem i slidi mde. A similar sraey ca be fud i he wrk by Nami ad Nishimura [15-16]. hey applied he dyamic slidi mde ad used H crl hery bai he frequecyshaped swichi surface. Hwever, i ur mehd, he sysem dyamics are exeded. Based he esimaed sials, he al crl law is desied. Ne ha he esimar ad crller desi ca be ake as w separae seps. his ccep iimilar ha f a idirec adapive crl sysem [13]. I he ex seci, a class f sysems be crlled is irduced al wih sme impra assumpis relaed he sysem marices. Seci 3 preses he swichi surface desi ad Seci 4 describes he sae esimar a l wih is vel rasfrmai marix. Afer shwi ha he sysem aes are well esimaed, we prpse a crller desi i Seci 5.

2 J.L. Cha: Oupu Feedback Slidi Mde Crller Desi via H hery 5 demsrae he develped crller, a umerical example is ive i Seci 6. Fially, Seci 7 ives ccludi remarks. II. PROBLEM SAEMEN Csider a MIMO liear sysem expressed by he fllwi sae-space equais: x( = Ax( + Bu( + Gd(, (1a y( = Cx (, (1b where x R are he saes f he sysem, u R m are crl ipus, y R l are measured upus, ad d( R p are disurbaces. he sysem mariceaisfy A R, B R m, C R l, ad G R p, wih rak(b = m, rak(c = l, ad rak(g = p. Fr he pla, we assume he fllwi hld wih respec he sysem: (A1 he pair (C, A, B is crllable ad bservable. (A he disurbaces d( are L buded [11], ad hey saisfy d ( β, ( where β is a kw csa. (A3 rak(cg = rak(g = p. (A4 he ivaria ers f he pair (C, A, G mus lie i he lef-half cmplex plae. Assumpis (A1 ad (A are sadard i SMC desi [1-9], ad assumpis (A3 ad (A4 are impra fr he sae esimar desi [17-19]. Sice he slidi mde crls ly cacel u he mached disurbace [1-10], hw esimae he effec f he mismached disurbace he sysem (1 is he mai prblem. I he ex seci, a rasfrmai marix mehd ha decmpses he disurbace erm Gd i is mached par ad mismached par is esablished. Based his rasfrmai, we he prpse a mehd fr baii he swichi surface desi by usi H crl hery achieve disurbace aeuai. he prblem f disurbace aeuai ivlves desii a crller such ha he clsed-lp sysem is ierally sable, ad such ha here exiss 0 γ < fr he fllwi iequaliy [11,1]: y( γ d ( γβ. (3 Mrever, sice he sysem (1 exhibis disurbace, a pracical sae esimar mehd is als preseed such ha xˆ( x ( fr, where x ˆ( is he esimaed sae. III. SWICHING SURFACE DESIGN VIA H HEORY Sice he sysem (1 is crllable, accrdi liear crl hery, a ai marix K R m ca be baied by assii desired eievalues. We decmpse hese eievalues i w ses, { λ1,, λ m} ad { ω1,, ω m }. his implies ha { λ1,, λ m}, ad { ω1, ω m } is he se f desired eievalues fr he marix A BK. Mrever, hese eievalues mus saisfy he fllwi cdiis: (C1 Ay eievalue i { ω1,, ω m } is i he specrum f A. (C If a cmplex eievalue is i { λ1,, λ m}, s is is cjuae. s (C3 he umber f repeaed eievalues i { λ1,, s λm, ω1,, ωm} is reaer ha m. Ne ha he eievalues { ω1,, ω m } are ecessarily lcaed i he lef-half cmplex plae. Siswa ad Fallside [0] have shw ha if he umber f repeaed eievalues f A BK is reaer ha he rak f B, he A BK is diaaliable. Hece, based heir resuls ad cdii (C3, A BK ca be diaalied as ( A BK W = W J, (4 r r r ( A BK W = W J, (5 s ( m where W r C m ad W s C are rih eievecr marices which crrespd { λ1,, λ m} ad { ω1,, ω m }. I addii, J r ad J s are diaal marices wih elemes { λ1,, λ m} ad { ω1,, ω m }, i.e., J r = dia{ λ1,, λ m} ad J s = dia{ ω1,, ω m }. Le V r W = [ Wr W s] ad V = W = V, where V r s ( m C m ad V s C. I fac, V is he lef eievecr marix fr A BK. Similar (4 ad (5, we bai V ( A BK = J V, (6 r r r V ( A BK = J V. (7 s Rearrai (7 yields s s VA s JV =( VBK s. (8 Lemma 1 [14]. he marix uder cdii (C1. m m VB s C is siular Lemma [14]: he marix W r ca be decmpsed as Wr = UrΓ r, (9

3 6 Asia Jural f Crl, Vl. 5, N. 1, March 003 ( m where U r R ( m ( m is full rak ad Γ r C is iverible. Frm VW = I, we kw ha VW s r = 0. Sice VW s r = 0 ad Wr = UrΓ r, i fllws ha VU s r = 0. Hece, U r is kw spa he ull space fr V s. Frm Lemma 1, we kw ha B is idepede f he ull space f V s. Hece, B is idepede f U r, i.e., R( U R( B =, (10 r where R(U r ad R(B represe he rae spaces f U r ad B, respecively. Defie [ ] M = B U (11 r, where M R. Sice B ad U r are bh full rak, we ca cclude frm (10 ad (11 ha M is iverible [14]. Mrever, is iverse marix M ca be defied as M B = U r, (1 m ( m where B R ad U r R ca be ake as 1 he eeralied iverses f B ad U r. Frm M M = I, he fllwi equais ca be baied: B B = I, U U = I, m r r m BUr = 0, Ur B= 0. Nw, he swichi surface is chse as (13 σ = Sx = B x + EU r x, (14 where R m m σ, m ( m S = B + EU r R, ad E R is a ai marix be deermied laer. Siificaly, frm (13 ad (14 we have ad SB = B B + EU r B = I m (15 S B + EU r U r U r Im 0 = = I 0 Im [ B Ur BE] = [ B Ur BE]. (16 Nex, we perfrm he fllwi sysem rasfrmai: where σ S = ad = + ( x x B σ U r BE η, η U r η = U x R r m (17. Sysem (1 ca he be rewrie as σ ( = SABσ( + SA( U r BE η ( + u( + SGd(, (18 η( = U ABσ( + U A( U BE η ( + U Gd(, (19 r r r r y( = CBσ( + C( U BE η (. (0 r If he sysem is i slidi mde, σ = 0, ad σ = 0, frm (18-(0, he he reduced-rder sysem ca be baied as η ( = U r AU rη( U r ABEη( + U rgd( = Aη( + B1d( BEη(, y( = CU rη( CBEη( = C η( D Eη(, (1 ( ( m ( m ( m p where A = Ur AUr R, B1 = UrG R, ( r m m l ( m B = U AB R, C = CUr R, ad D = l m CB R. Nw, we will csruc he marix E such ha A B E iable ad he disurbace aeuai y γ d is uaraeed. his is a sadard H crl prblem [11,1], ad he fllwi herem ives he eire slui. herem 1 [11,1]. Csider he sysem (1-( ad assume ha he fllwi hld: (1 he pair (A, B is crllable, ad he pair (C, A is bservable. A jω I B ( rak =, fr ay frequecy ω. C D If fr a ive value f 0 γ <, here exiss a real, symmeric slui P 0 he alebraic Riccai equai (ARE, ha is, ad ( ( + ( ( P A B D D D C A B D D D C P 1 + P B 1B1 B( D D B P γ + C I D ( D D D C = 0, ( (3 E = ( D D D C + ( D D B P, (4 he A BE iable ad he cdii y γ d ca be saisfied. I addii, ly mached disurbaces exis i

4 J.L. Cha: Oupu Feedback Slidi Mde Crller Desi via H hery 7 sysem (1; he swichi surface ca be described by σ = Sx = B x. Fr his case, he sabiliy aalysis i slidi mde ca be fud i ur earlier wrk [14]. IV. SAE ESIMAOR Csider he pla ive i (1. he sadard Lueberer bserver is suiable fr his pla due he ukw ipu erm d(. Rbus esimai f saes i he presece f ukw ipu has bee discussed i may papers [17-19]. I hieci, we shall apply he desi echique i upu slidi mde crl [6-9] fid a suiable rasfrmai marix which ca be used apprpriaely rasfrm he riial sysem i a ew mdel caii w sub-sysems; e is relaed he disurbaces, ad he her is. Based his ew mdel, he reduced-rder sae-esimar will he be desied. Lemma 3 [6]. If assumpis (A3 ad (A4 hld, he a p l marix F R ca be fud such ha he p -er eievalues fr A G( FCG FCA are lcaed i he lef-half cmplex plae. Mrever, [6] prvided a mehd fr seleci he marix F. Ne ha he marix F mus be chse such ha FCG is siular. Lemma 4. he eievalues fr w marices A G(FCG FCA ad A AG(FCG FC are he same. Prf: By meas f calculais, we have I 0 1 λi A G I 0 p ( λ FC I FC 0 FCG FCA I p λ (5 λi A + G( FCG FCA λg = 0 FCG ad I AG( FCG λi A G I G λ λ p 0 I FC 0 0 I p λi A + AG( FCG FC 0 =. λfc FCG (6 Ieresily, if he deermias bh sides frm he abve equais (5 ad (6 are calculaed, he i fllws ha m λi A G λ de FC 0 = de I A + G( FCG FCA de( FCG ( λ (7 ad m λi A G λ de FC 0 (8 = de I A + AG( FCG FC de( FCG. ( λ Frm (7 ad (8, we ca cclude ha he eievalues fr he w marices A G( FCG FCA ad A AG( FCG FC are he same. Hece, frm Lemma 4, A AG( FCG FC has p -er eievalues λ i, i = 1,,, p, saisfyi Re{ λ i } < 0, ad here exiss a full rak marix ( p W C such ha A AG FCG FC W = W J (9 ( (, ( p ( p where J C is he Jrda frm fr he eievalues { λ,, }. Sice G is full rak ad 1 λ p ( 1 A AG( FCG FC G = 0, (30 G ca be reaed as he rih eievecr marix f A AG( FCG FC crrespdi p er eievalues. Cmbii (9 ad (30, we have 1 ( [ ] [ ] J 0 A AG FCG FC W G = W G. 0 0 (31 Frm (31, [ W ] G is he rih eievecr marix ad is iverible. Similar Lemma, he marix W ca be decmpsed i W = UΓ, (3 ( p where U R ( p ( p is full rak ad Γ C is iverible. Ne ha [ U G ] is a siular ma- W G is iverible. Le rix because [ ] U = G [ U G], p ( P where G R ad U R. Clearly, (33 UU + GG = I, (34 GG= I, UU = I, p p U G = 0, G U = 0. Frm (9, (3, ad (35, i is easy verify ha ( 1 (35 U A AG( FCG FC = Γ J Γ U = RU, (36

5 8 Asia Jural f Crl, Vl. 5, N. 1, March ( p ( p where R = J R wih eievalues { λ1,, λ p} Γ Γ is a sable marix. Frm (34 ad (35, i is easy check ha ( FCG FC ( U ad G I G( FCG FC U = I.(37 Nw, we defie he sysem rasfrmai as 1 ( FCG FC ( FCG Fy = = = x U Ux ( ( x = G + I G FCG FC U 1 ( ( = G FCG Fy + I G( FCG FC U, (38 (39 p p where 1 = ( FCG FCx R ad = Ux R. Evidely, i rder esimae x, i is ecessary se up a reduced-rder bserver fr based (39 ly. I he fllwi, a sae-esimar fr is prpsed. Firs, pre-muliplyi (1 by U ad usi (36, we have = U Ax+ U Bu+ U Gd ( = U A AG( FCG FC x + U AG( FCG Fy 1 + U Bu = R + U AG( FCG Fy + U Bu. (40 Frm (40, i is clear ha he disurbace erm d des exis i he dyamics f. Hece, ur vel reduced-rder bserver is csruced as ˆ = Rˆ + U AG( FCG Fy + U Bu, (41 where ẑ is he esimae f. We ex defie he esimai errr as e = ˆ, ad frm (40 ad (41, we have e = Re. (4 Sice he marix R iable, i fllws ha e ( = ( ˆ ( 0 fr. his meas ha ẑ successfully esimaes. Fially, we defie he esimae f x as ( xˆ = G( FCG Fy + I G( FCG FC U ˆ, ( where x ˆ R. Le he sae errr be e = xx. ˆ Frm (39 ad (43, we have ( e = I G( FCG FC U e = Me, ( ( p M = I G( FCG FC U R. Sice he cdii e ( 0 fr is uaraeed, i fllws frm (44 ha xˆ( x ( fr. where ( V. CONROLLER DESIGN Sice he rue saes x ca be measured direcly, he esimaed saes ˆx shuld be used i he crl alrihm. Hece, he verall crl scheme becmes ˆ ˆ r ˆ σ ˆ = Sx = B x + EU x, (45 u=saxˆ ( β SG + ρ + k, ( (46 xˆ = G( FCG 1 Fy + I G( FCG 1 FC U ˆ, (47 ˆ = Rˆ + U AG( FCG Fy + U Bu (48 1 where k > 0, ive belw, is a csa. Frm (44 ad (45, we have σ = σ ˆ +SMe. Sice he sysem (4 is sable, here exis real psiive umbers α ad γ such ha e ( γ e α e (0 fr > 0. (49 herem. Csider he pla (1 ad he crl alrihm described i (45-(48. If he crl ipus are desied as i (46 wih k > γ SAM SMR e (0, (50 he we have σ ( 0 fr. Prf. Usi (1 ad (44, we ca yield he derivaive f ˆx as xˆ = Ax+ Bu+ Gd MRe = Axˆ + Bu + Gd + ( AM MR e (51 Frm (45 ad (51, he dyamics fr ˆ σ ca be baied as σ ˆ = SAxˆ + u + SGd + ( SAM SMR e = ( β SG + ρ + k + SGd + ( SAM SMR e Pre-muliplyi (5 by σ ˆ bais ( β SG ρ k σ ˆ = SGd + ( SAM SMR e ( ρ + k (5

6 J.L. Cha: Oupu Feedback Slidi Mde Crller Desi via H hery 9 + γ e α SAM SMR e (0 ρ (53 Hece, he sysem will reach σ ˆ = 0 i fiie ime []. Sice σ = σ ˆ +SMe ad e ( 0 fr, we ca cclude σ ( 0 fr. Fr alleviai uwaed chaeri, he erm ca be mdified by usi he saurai fuc- σ ˆ i sa( σ ˆ, ε, where ε > 0 [3]. I is easy cclude ha ce he sysem es i he slidi layer σ ˆ < ε, is rajecry will be cmpleely resriced i his layer []. where VI. SIMULAION RESULS Csider he fllwi pla irduced as: x ( = Ax( + Bu( + Gd( y( = Cx( A= , B = 0, C =, = G 1 ( π π (54 d = si( + cs( (55 4 he disurbace erm d( is L -buded, ad i saisfies d 1 = β. Mrever, i is easy check ha he sysem (54 saisfies he fur assumpis (A1-(A4. Firs, we chse λ 1 = 3 + i, λ =3 i, ad ω 1 = 1. Sice hese eievalues fr A are {6.67, 0.75± i}, he hree impra cdiis (C1-(C3, irduced i seci 3, are saisfied. Wih he help f he MALAB sfware packae, we fud B ad U be [ ] = , B ( U r = (57 Sei γ = 0.1 ad slvi he ARE equai (3 yields he symmeric, psiive-defiie marix P ad he marix E as r P = > 0 ad E = [ ]. (58 Hece, he desi al is fid he crl ipus u, usi he sysem upus y ly, such ha y γ d = 0.1. (59 By meas f calculais, he marix ca be baied as [ ] S = B + EU r S = (60 Sice e f he sysem saes x are measurable, we ca bei desi he sae esimar. We chse F = [ ] (61 ad verify ha he eievalues fr A AG( FCG FC are {0, ± i}. Afer perfrmi he seps prpsed i Seci 4, we frd ha he crrespdi marices fr he sae-esimar are U =, ( R = , M = ( Hece, he reduced-rder bserver is csruced as ˆ = Rˆ + U AG( FCG Fy + U Bu, (64 ad he esimaed saes ˆx are ˆ = + ˆ x G( FCG Fy M. (65 he crl alrihms are desied as σ ˆ = Sxˆ, (66 ( u=saxˆ k + β SG + ρ sa( σ ˆ, ε, (67 where ρ = 0.1, k = 3, ad ε = 5. Fiures 1-5 shw simulai resuls baied usi he iiial sae x(0 = [0 0]. he ime respses f y ad y are shw i Fi. 1 ad Fi., respecively. he crl ipu u ihw i Fi. 3. Fiure 4 shws he rajecry f σ. Frm Fi., e ca see ha y 0.1 ad, hece, ha he crl al (59 is achieved. he rajecries f rue saes ad esimaed saes are shw i Fi. 5, which clearly shws ha he esimaed saes x ˆ( apprach he rue saes x ( as.

7 30 Asia Jural f Crl, Vl. 5, N. 1, March im (sec Fi. 1. Oupus y 1 ad y im (sec (a im (sec Fi.. he respse f y im (sec (b im (sec Fi. 3. Crl ipu u ime (sec (c Fi. 5. (a Sae x 1 ad esimaed sae ˆx 1, (b Sae x ad esimaed sae ˆx, (c Sae x 3 ad esimaed sae ˆx im (sec Fi. 4. he rajecry f σ. VII. CONCLUSIONS I his paper, we have preseed a upu feedback slidi mde crl fr liear sysems wih mismached disurbaces via H hery. Applyi he H apprach ad slvi he alebraic Riccai equai, we have fud ha he desired swichi surface ca uaraee sme level f rbus sabiliy ad disurbace rejeci fr he sysem ce i is i slidi mde. I addii, a sae bserver has bee develped effecively esimae he sysem saes. Hece, he verall slidi mde crl desi ca be divided i w separae

8 J.L. Cha: Oupu Feedback Slidi Mde Crller Desi via H hery 31 seps. Firs, he bserver is used esimae he saes, ad he he crl law is desied wih respec he esimaed saes. ACKNOWLEDGMEN his research wauppred by he Naial Sciece Cucil, aiwa, R.O.C., uder Crac NSC E REFERENCES 1. Hu, J.Y., W. Ga, ad J.C. Hu, Variable Srucure Crl: A Survey, IEEE ras. Id. Elecr., Vl. 40, pp. - ( Baily, E. ad A. Arapsahis, Simple Slidi Mde Crl Scheme Applied Rb Maipular, I. J. Cr., Vl. 45, pp ( Slie, J.J.E. ad S.S. Sasry, racki Crl f Nliear Sysems Usi Slidi Surfaces wih Applicai Rb Maipulars, I. J. Cr., Vl. 38, pp ( Spure, S.K. ad R. Davies, A Nliear Crl Sraey fr Rbus Slidi Mde Perfrmace i he Presece f Umached Uceraiy, I. J. Cr., Vl. 57, pp ( Glas, M.P. ad S.H. Zak, Pracical Sabiliai f Nliear/Ucerai Dyamic Sysems wih Buded Crllers, I. J. Cr., Vl. 6, PP ( Edwards, C. ad Spure, S.K., Slidi Mde Crl: hery ad Applicais, aylr & Fracis, Ld ( El-Khaali, R. ad R. Decarl, Oupu Feedback Variable Srucure Crl Desi, Aumaica, Vl. 31, pp ( Yallapraada, S.V., B.S. Heck, ad J.D. Fiey, Reachi ad Cdiis fr Variable Srucure Crl wih Oupu Feedback, J. Guid., Cr., Dy., Vl. 19, pp ( Zak, S.H. ad S. Hui, Oupu Feedback Variable Srucure Crllers ad Sae Esimars fr Ucerai/Nliear Dyamic Sysems, IEE Prc. P. D, Crl hery Appl., Vl. 140, pp ( Draevic, B., he Ivariace Cdiis i Variable Srucure Sysems, Aumaica, Vl. 5, pp ( Zhu, L., J.C. Dyle, ad K. Glver, Rbus ad Opimal Crl, Preice Hall ( Ya, C.D. ad C.B. Yeh, Liear/Nliear H Crl hery, Chwa Ld., aiwa ( Kaufma, H.I., K. Barkaa, ad K. Sbel, Direc Adapive Crl Alrihms hery ad Applicai, Sprier-Verla, New Yrk ( Cha, J.L. ad Y.P. Che, Slidi Vecr Desi Based Ple-Assime Mehd, Asia J. Cr., Vl., pp ( Nami, K., Nishimura, H., ad ia, H., H /µ Crl Based Frequecy Shaped Slidi Mde Crl fr Flexible Srucures, JSME I.. J. Ser. C, Vl. 39, pp ( Nami, K., Hirasawa, S., Slidi Mde Psiii Crl f Sile Lik Flexible Arm Usi H Crl Hyperplae wih Q parameeriai, JSME I. J. Ser. C, Vl. 4, pp ( Hu, M. ad P.C. Müller, Desi f Observers fr Liear Sysems wih Ukw Ipus, IEEE ras. Aum. Cr., Vl. 37, pp ( Crless, M. ad J. u, Sae ad Ipu Esimai fr a Class f Ucerai Sysems, Aumaica, Vl. 34, pp ( Edwards, C., S.K. Spure, ad R.J. Pa, Slidi Mde Observers fr Faul Deeci ad Islai, Aumaica, Vl. 36, pp ( Siswa, V. ad F. Fallside, Eievalue/Eievecr Assime by Sae Feedback, I. J. Cr., Vl. 3, pp (1977. Jea-Li Cha received he B. S. ad M. S. derees i Crl Eieeri, he Ph. D. deree i Elecrical ad Crl Eieeri frm Naial Chia u Uiversiy, aiwa, i 199, 1994, ad 1999, respecively. He was wih he Mechaical Research Labrary, Idusrial echly Research Isiue (aiwa, duri I 1999, he jied he Deparme f Elecrical Eieeri, Orieal Isiue f echly (aiwa, as a Assisa Prfessr. His research ieress iclude variable srucure crl, mi crl, ad sial prcessi.