Reliable Decentralized PID Stabilization of MIMO Systems

Transcription

1 Proceedig of the 6 America Cotrol Coferece Mieapoli, Mieota, USA, Jue 4-6, 6 FrB.5 Reliable Decetralized PID Stabilizatio of MIMO Sytem A. N. Mete, A. N. Güdeş, ad A. N. Palazoğlu Abtract Sytematic method are propoed for reliable decetralized PID-cotroller ythei. Thee cotroller achieve cloed-loop tability ad aymptotic trackig of tep-iput referece at each output chael whe all cotroller are operatioal, ad they maitai tability whe oe of the cotroller fail completely. I. INTRODUCTION I may practical cotrol applicatio, Proportioal + Itegral + Derivative (PID) cotroller are widely ued ad preferred due to their implicity. For multi-iput multi-output (MIMO) ytem, the decetralized cotroller tructure provide imple implemetatio. I thi paper we propoe ytematic method for the ythei of reliable decetralized PID-cotroller for two-chael liear, time-ivariat MIMO ytem. Thee cotroller achieve cloed-loop tability whe both chael are operatioal, ad maitaied eve if oe of the cotroller fail completely. Due to the itegral-actio i the PID-cotroller, aymptotic trackig of tep-iput referece i achieved at each output chael. Reliable cotrol ha bee coidered uder full-feedback ad decetralized cotroller tructure i e.g., [3], [6], [7], [9]. We oly coider PID-cotroller, which ca tailize oly certai clae of plat. Although ued widely, mot PID deig approache lack ytematic procedure ad rigorou cloed-loop tability proof. Rigorou ythei method are tudied i recet literature, e.g., [8]. Decetralized PID deig were coidered for two-by-two plat i []. We preet ytematic ythei procedure for everal plat clae that ca be tabilized uig PID-cotroller. The propoed deig achieve cloed-loop tability ad aymptotic trackig. Performace iue are ot coidered but the freedom offered i the deig parameter may be ued to atify other criteria. The PID deig are (partially) reliable A. N. Mete ad A. N. Güdeş are with the Departmet of Electrical ad Computer Egieerig, Uiverity of Califoria, Davi, CA amete@ucdavi.edu, agude@ucdavi.edu A. N. Palazoğlu i with the Departmet of Chemical Egieerig ad Material Sciece, Uiverity of Califoria, Davi, CA apalazoglu@ucdavi.edu agait the failure of oe cotroller. If the plat i table, (fully) reliable cotroller deig are alo preeted agait the failure of either oe of the two cotroller. A cotroller that fail i et equal to zero; i.e., the failure i recogized ad the failed cotroller i take out of ervice. Although the failed chael doe ot achieve aymptotic trackig with zero teady-tate error, cloed-loop tability i till maitaied. We illutrate the deig method with two example. A partially reliable PID-cotroller i deiged for the liearized model of a ugar mill proce, which ha pole at the origi ad a o-miimum phae zero, [4]. A fully reliable PIDcotroller i deiged to achieve aymptotic trackig of the deired tep-iput referece at two output for a implified model repreetig a particular patiet uder aetheia, []. Notatio: Let CI, IR, IR + deote complex, real, poitive real umber; U = { CI Re() } { } i the exteded cloed right-half plae; I i the idetity matrix; R p deote real proper ratioal fuctio of ; S i the table ubet with o U-pole; M(S) i the et of matrice with etrie i S. The H -orm of M() M(S) i M := up σ(m()); σ i the maximum igular value ad U U i the boudary of U. We drop () i trafer fuctio uch a G() wheever thi caue o cofuio. We ue coprime factorizatio over S ; i.e., for G R p, G = XY = Ỹ X give a right-coprime-factorizatio (RCF) ad a left-coprime-factorizatio (LCF); X, X,Y,Ỹ S, det Y ( ), det Ỹ ( ). II. PRELIMINARIES We coider the liear time-ivariat (LTI) decetralized feedback ytem S(G, C D ) with two multi-iput multioutput (MIMO) chael a how i Fig.. We aume that the feedback ytem i well-poed, the plat ad cotroller have o utable hidde-mode. The plat G R p ad the decetralized cotroller C D R p are partitioed a: [ ] G G G =, C D = diag [ C, C ], () G G where each chael ha a may iput a output, i.e., G j R j j p, ad rakg =. The plat are either table, or /6/$. 6 IEEE 536

2 if they are utable, the oe of the iput chael or oe of the output chael ha table trafer-fuctio. Sice the iput ad output ca be re-ordered, we aume that thee table trafer-fuctio are aociated with the firt chael. Furthermore, the utable pole of the ecod chael are reflected i G. Let G = X Y = Ỹ X be a RCF ad LCF of G. Therefore we aume that G atifie the followig plat aumptio: i)g S ; ii) G R p may be utable; iii) either G S ; ad G Y S or G S ad ỸG S. Let r := [r,r ] T, v := [v,v ] T, y := [y,y ] T, w := [w,w ] T deote the iput ad output vector. Defiitio.: [5] a) The feedback ytem S(G, C D ) i table iff the trafer-fuctio from (r, v) to (y, w) i table. b) The cotroller C D tabilize G iff C D i proper ad S(G, C D ) i table. c) The cotroller C D that tabilize G i partially reliable iff the ytem S(G,,C ) i table, i.e., the trafer-fuctio from (r,v) to (y, w ) i table. d) The cotroller C D that tabilize G i fully reliable iff the ytem S(G,,C ) i table, i.e., the trafer-fuctio from (r,v) to (y, w ) i table, ad the ytem S(G, C, ) i table, i.e., the trafer-fuctio from (r,v) to (y, w ) i table. Fully reliable decetralized cotroller exit if ad oly if G i table. Partially reliable decetralized cotroller exit for the two utable plat cae, where the trafer-fuctio of the firt chael iput or output are table. Theorem.: [6] a) Let G M(R p ) a i () atify the plat aumptio. Let rakg =, rakg =. The decetralized C D = diag [ C,C ] tabilize G ad i partially reliable if ad oly if i) C tabilize G ad ii) C tabilize W = G G C (I +G C ) G M(S). b) Let G M(S), rakg j = j. The decetralized C D = diag [ C,C ] tabilize G ad i fully reliable if ad oly if i) C tabilize G ad ii) C imultaeouly tabilize ad W = G G C (I + G C ) G M(S). G By Theorem., partially reliable decetralized cotroller ca be deiged for the cla of plat uder coideratio. The cotroller C j R j j p, j =,, will be deiged i the followig proper PID-cotroller form, [4]: C j = K pj + K ij + K dj τ j +, () where, for j =,, K pj,k ij,k dj IR j j are called the proportioal, the itegral, ad the derivative cotat, repectively, ad τ j IR, τ j >. The itegral-actio i C j i preet whe K ij. Subet of PID-cotroller are obtaied by ettig oe or two of the three cotat equal to zero; () become a PI-cotroller whe K dj =ad a I-cotroller whe K pj = K dj =. III. PID STABILIZATION CONDITIONS AND DESIGN PID-cotroller caot tabilize certai utable plat. We ow give coditio for reliable PID tabilizability. Lemma 3.: (Exitece coditio for partially reliable decetralized PID-cotroller): a) Let G M(R p ) a i () atify the plat aumptio. Let rakg =, rakg =. If there exit partially reliable decetralized PID-cotroller C D = diag[ C,C ] with ozero itegral cotat K ij IR j j, the G ad G have o tramiio-zero at =ad G i trogly tabilizable. b) Let G M(S), rakg =, rakg =. There exit partially reliable decetralized PID-cotroller C D = diag[ C, C ] with ozero itegral cotat K ij IR j j if ad oly if G ad G have o tramiio-zero at =. Lemma 3.: (Exitece coditio for fully reliable decetralized PID-cotroller): Let G M(S), rakg j = j. a) There exit fully reliable decetralized PID-cotroller C D = diag[ C,C ] with ozero itegral cotat K ij IR j j oly if G, G,G have o tramiio-zero at =. b) There exit fully reliable decetralized PID-cotroller C D = diag[ C,C ] with ozero itegral cotat K ij IR j j if G, G,G have o tramiio-zero at = ad W ()G () =[I G ()G () G ()G () ] i ymmetric, poitive-defiite. Remark: If at leat oe of C,C,G,G ha blockigzero i U (icludig ifiity), the fully reliable decetralized PID-cotroller exit oly if G ad G j, j =,, have o tramiio-zero at = ad det W ()G () = det [ I G ()G () G ()G () ] >. By Lemma 3., a eceary coditio for exitece of PID-cotroller i that G i trogly tabilizable. Obviouly the, table plat admit PID-cotroller. A PID-cotroller ythei method applicable to MIMO plat baed o the mall-gai approach i give i Propoitio 3.. Propoitio 3.: Let H S j j. Let rakh() = j. Chooe ay ˆK p, ˆK d IR j j, τ>. The, for ay β IR + atifyig (4), a PID-cotroller that tabilize H i give by (3); for ˆK d =, (3) i a PI-cotroller; for ˆK d = ˆK p =, (3) i a I-cotroller: C pid = β ˆK p + βh() + β ˆK d τ+, (3) 537

3 β< H()( ˆK p + ˆK d () I τ+ )+H()HI. (4) We propoe PID-cotroller for three clae of utable trogly tabilizable plat with retrictio o the blockigzero i U ad complete freedom i the U-pole. ) Utable plat with o zero i the utable regio: Coider utable G with o zero i the utable regio U, icludig ifiity. For thee plat, G S.I the SISO cae, G ha relative-degree equal to zero. A PIDcotroller ythei i give i Propoitio 3.. Propoitio 3.: Let G R p, rakg () =. Let G have o tramiio-zero i U (icludig ifiity). Chooe K p,k d IR, τ > uch that det R( ) = det [ G ( ) +K p +τ K d], where R := G ()+ K p + K d τ +. The, for ay γ IR + atifyig (6), a PIDcotroller that tabilize G i give by (5); if K d =, (5) i a PI-cotroller; if K p = K d =, (5) i a I-cotroller: C = K p + γ[g ( ) + K p + τ K d] + K d τ +, (5) γ> [ R (G ( ) + K p + τ K d ) I ]. (6) ) Utable plat with oe poitive real-axi zero icludig ifiity: Coider utable G with oe real-axi blockig-zero i U ; G may have other tramiio-zero i the table regio. I the SISO cae, G ha relative-degree equal to or. We coider two cae where the real-axi U-zero i either ) large, icludig ifiity, or ) mall. ( /z) ( /z) Cae ) Let G =[ + a G ] + a I =: Ỹ X, z IR, <z, a IR +. Propoitio 3.3 coider plat with oe large real-axi zero atifyig z> Ψ defied i (7). A zero at ifiity atifie thi boud. I the SISO cae, if G i trictly-proper, thi cla correpod to miimum-phae utable plat with relative-degree oe. Propoitio 3.3: Let G R p, rakg () =. Let G have o tramiio-zero at =. Let z IR, ( /z) < z. Let + a G M(S), for a IR, a>. Let Ỹ ( ) =( /z) G (). If z> i fiite, let K d =. If the zero i at ifiity, chooe ay K d IR, τ >. Defie Ψ:=( /z) [G ()+ K d τ + ]Ỹ ( ) I. (7) If z> Ψ the, for ay α IR + atifyig (9), let C pd be a i (8): α Cpd = +α/z Ỹ ( )+ K d τ +, (8) α > Ψ ( Ψ /z ). (9) Defie H pd := G (I + C pd G ). The, for ay γ IR + atifyig (5), a PID-cotroller that tabilize G i give by (); if K d =, () i a PI-cotroller: C pid = C pd + γ [G () + α Ỹ ( )]. +α/z () ( z) Cae ) Let G = [ a + G ( z) ] a + I =: Ỹ X, z IR, z >, a IR +. Propoitio 3.4 give a PIDcotroller ythei for plat with oe mall real-axi zero atifyig <z< Φ defied i (). Propoitio 3.4: Let G R p, rakg () =. Let G have o tramiio-zero at =. Let G have ( z) o pole at =. Let z IR, z>. Let a + G M(S), for a IR +. Let Ỹ () = z G (). Chooe ay K d IR, τ >. Defie ( z)g Φ:= I + ( z)k dỹ (). () τ + If <z< Φ the, for ay α IR + atifyig (3), let C pd be a i (): α C pd = +αz Ỹ () + K d τ +, () α>( Φ z ). (3) Defie H pd := G (I + C pd G ). The, for ay γ IR + atifyig (5), a PID-cotroller that tabilize G i give by (4); if K d =, (4) i a PI-cotroller: C pid = C pd + γ [ G () + α ( + αz) Ỹ () ], (4) γ< H pdh pd () I = (I + G C pd ) [ G G () I] H pd K d τ +. (5) 3) Utable plat with two poitive real-axi zero icludig ifiity: Coider utable G with two real-axi blockig-zero i U ; G may have other tramiio-zero i the table regio. I the SISO cae, G ha relative-degree equal to or. We coider two cae where z,z are either ) both large, icludig ifiity, or both mall. Cae ) Let ( /z l ) G =[ ( + a l ) G ] ( /z l ) ( + a l ) I =: Ỹ X, l= l= z l IR, <z l, a l IR +, l =,. Propoitio 3.5: Let G R p, rakg () =. Let z l IR, < z l, l =,. Let ( /z l ) l= ( + a l ) G M(S), for a l IR +. Let Ỹ ( ) =( /z ) ( /z ) G (). Chooe ay δ>. Defie Ψ := ( /z )( /z ) G ( + δ) I. (6) If Ψ <z, the chooe ay α IR + atifyig α > Ψ ( Ψ /z ). (7) Defie ( + δ) Ψ := ( + α/z ) [I+ α( + δ)g ()Ỹ ( ) ( + α/z )( /z ) ] I. (8) 538

4 If (+α/z ) Ψ <z, the chooe ay β IR + atifyig Ψ ( Ψ /z ) <β< α z z. (9) With η := ( αβ ), τ = η[β( + α )+δ( + β )], z z z z K p = η τ δαβỹ ( ), K d =( δ τ )K p, let C pd = K p + K d τ + = η τ αβ( + δ) Ỹ ( ). τ + () The, for ay γ IR + atifyig (5), a PID-cotroller that tabilize G i give by (): C pid = C pd + γ [ G () + η τ δαβỹ ( )]. Cae ) Let () ( z l ) ( z l ) G =[ (a l +) G ] (a l +) I =: Ỹ X, l= l= z l IR, z l >, a l IR +, l =,. Propoitio 3.6: Let G R p, rakg () =. Let G have o pole at =. Let z l IR, z l >. Let ( z l ) l= (a l +) G M(S), for a, b IR, a, b >. Let Ỹ () = z G (). Chooe ay δ>. Defie Φ := ( ( z )( z ) G (δ +) I). () If <z < φ, the chooe ay α IR + atifyig α > ( Φ z ). (3) Defie Φ := (δ +) ( ( + αz ) [I+ α(δ +)G ()Ỹ () ( + αz )( z ) ] I). (4) If <z < ( + αz ) Φ, the chooe ay β IR + atifyig ( Φ z ) < β < ( αz z ). (5) With η := ( αβz z ), τ = η [ β ( + αz )+δ( + βz )], K p = η αβỹ (), K d =(δ τ ) K p, let C pd = K p + K d τ + = η αβ( δ+) Ỹ (). τ + (6) If β = (αz z ), the (6) i a PI-cotroller C pi = αβ[ β ( + αz )+δ( + βz )] ( δ+)/. The, for ay γ IR + atifyig (5), a PID-cotroller that tabilize G i give by (7): C pid = C pd + γ [ G () + η αβỹ () ]. (7) IV. RELIABLE DECENTRALIZED DESIGN I thi ectio, for partially reliable cotroller deig, aume rakg =, rakg = ad G ad G have o tramiio-zero at =. Baed o Theorem., firt deig a PID-cotroller C that tabilize G by uig the ythei method i Sectio III. The deig a PIDcotroller C that tabilize the table W = G G C (I+ G C ) G followig the ythei i Propoitio 3.. For table G, we ca deig partially reliable decetralized PID-cotroller if ad oly if rakg() =, rakg () =. We follow the ythei method of Propoitio 3. to deig a PID-cotroller C that tabilize G ad the a PID-cotroller C that tabilize the table traferfuctio W. By Lemma 3., fully reliable decetralized PIDcotroller deig require rakg() =, rakg () =, rakg () =. We further aume the ufficiet coditio that W ()G () >. A fully reliable decetralized PIDcotroller ythei i give i Propoitio 4.: Propoitio 4.: Let G S. Let rakg() =, rakg j () = j, j =,. Let W ()G () := I G ()G ()G ()G () be ymmetric, poitivedefiite. Chooe ay ˆK p, ˆK d IR, τ >. For ay β IR + atifyig (9), let C be give by (8): C = β ˆKp + β G () + β ˆK d τ +, (8) β < G ()( ˆK p + ˆK d τ + )+G ()G () I. (9) Let W := G G C (I + G C ) G. Chooe ay ˆK p, ˆK d IR, τ >. For ay β IR + atifyig (3), let C be a i (3): C = β ˆKp + β G () + β ˆK d τ +, (3) β < mi { G ()( ˆK p + ˆK d τ + )+G ()G () I, W ( ˆK p + ˆK d τ + )+[W () W ()]G () }. (3) The C D = diag [ C,C ] i a fully reliable decetralized PID-cotroller. For ˆK dj =, (8) ad (3) are PI-cotroller; for ˆK dj = ˆK pj =, (8) ad (3) are I-cotroller. Example 4. illutrate partially reliable decetralized PID deig for a liearized model[ of a ugar mill proce, [4]. ] 5.5(+) 5+ (+) Example 4.: Let G = =.3 5+ [ ][ ].5(+) G (+a)(+) I ; With Y =.3 +a G, (+a) (+a) a>, G atifie the plat aumptio: G,G M(S), G Y M(S). The oly U-pole of G i at =, which alo appear a a pole of G. The plat G ad G have o tramiio-zero at =; G ha a tramiio-zero at =.37 U(ad aother at =.5). The oly zero of G i at ifiity. Followig Propoitio 3.3, deig C : Chooe K d =, τ =.. The (9) hold for α>.3; take α =.6. The (5) hold for γ<.6; take γ =.4. The PID-cotroller C =

5 Followig Propoitio 3., deig C tabilizig W = G G C (I +G C ) G S: Chooe ˆK d =, ˆKp = 4. The β < W ˆK W ()W () I p + =.68; take β =.. The PI-cotroller C = Fig. how the tep repoe for the two output y, y, with uit-tep applied at both referece r,r. The cotroller C D =[C,C ] i active. Fig. 3 how the tep repoe whe C fail, i.e., C D = [,C ], with oly the ecod chael operatioal. The partially reliable deig guaratee cloed-loop tability whe C =but aymptotic trackig with zero teady-tate error i achieved oly i the ecod chael with itegral-actio. To illutrate the fully reliable decetralized PID-cotroller deig approach, the ythei procedure i Propoitio 4. i applied i Example 4. to deig a cotrol ytem that maipulate the flow rate of two drug, dopamie ad odium itropruide, to a critical care patiet. We ue the implified model i [] without iput delay. The aetheiologit ifue everal drug to the patiet durig urgery to maitai the output, the mai arterial preure ad cardiac output, cloe to their deired etpoit. [ ] Example 4.: Let G = M(S); the rakg() =, G (),G (), I G ()G G ()G () =. >. Deig C : Chooe ˆK p =.5, ˆKd =. With β =.5 < 4 atifyig (9), we obtai K p =.575, K i =.3. The PI-cotroller C = a i (8). Deig C that imultaeouly tabilize G ad W = : Chooe ˆK p =., ˆKd =., τ =.. With β =.5 < mi{.346,.3779} atifyig (3), we obtai K p =.5, K d =.5, K i =.47. The PID-cotroller C = (.+) a i (3). Whe C D = diag [ C,C ], the cloed-loop pole are {.8, 3.56,.554,.8956, ± j.7938}. Fig. 4 how the tep repoe of S(G, C D ) for the two output y (dahed), y (olid), with uit-tep applied at both referece r,r, with C D =[C,C ] havig both chael active. Both chael achieve aymptotic trackig with zero teady-tate error. Fig. 5 how the tep repoe whe C i take out, i.e., C D =[,C ]. The output y doe ot track the tep referece due to the lack of itegral actio i the firt chael. Fig. 6 how the tep repoe whe C i tured off, i.e., C D =[C, ]. V. CONCLUSIONS We propoed ytematic ythei of decetralized PIDcotroller that achieve cloed-loop tability ad aymptotic trackig of tep-iput referece at each output chael whe both chael are operatioal, ad maitai cloedloop tability eve whe oe of the cotroller i tured off. Although we coidered the two-chael decetralized cae here, the reult may be exteded to more chael. VI. APPENDIX: PROOFS Proof of Propoitio 3.: Let C pid be a i (3). The M pid := +β I + H +β C pid = I + β +β [ H ( ˆK p + ˆK d τ+ )+ HH I () I ] i uimodular. Therefore, C pid tabilize H. Proof of Propoitio 3.: By aumptio, K p,k d,τ are uch that R( ) exit. By (6), M pid := +γ G + +γ C = +γ R+ γ +γ R( ) =[I + +γ ( R()R ( ) I )]R( ) i uimodular. Therefore, C tabilize G. Proof of Propoitio 3.3: By (9), M pd := Ỹ + XC pd = ( /z) +a [G + C pd ]=[I + (+α/z) +α Ψ]Ỹ ( ) (+α) (+α/z)(+a) i uimodular. Therefore, C pd tabilize G ad H pd M(S); H pd () = G () + K p. Sice K i / tabilize H pd, C pid = C pd + K i / tabilize G. Proof of Propoitio 3.4: i) By (3), M pd := Ỹ + XC pd = + C pd ] = [ α( z) ( z) a+ I +(+αz) a+ (G + ( z) a+ [G Ỹ () K d τ + )Ỹ () (+αz) ] (+αz) =[I + α+ Φ]Ỹ () (α+) (+αz)(a+) i uimodular. Therefore, C pd tabilize G ad H pd := M X pd = G (I + C pd G ) M(S); H pd () = α( + αz) Ỹ () + G (). Sice K i/ tabilize H pd, C pid = C pd + K i / tabilize G. Proof of Propoitio 3.5: i) By (9), αβ(z z ) = η >. By (7), U d := (+a ) Ỹ + α( /z ) Ỹ ( ) +δ (+α) (+α/z )(+a ) [ (+α/z )( /z )( /z ) (+α)(+δ) (+α/z )(+a ) = G Ỹ ( ) + α( /z ) +α I]Ỹ ( ) = (+α) (+α/z )(+a ) [ (+α/z) +α Ψ + I]Ỹ ( ) i uimodular. Sice z > Ψ, by τ (+δ) (9), M pd := Ỹ + XC pd = η(τ [( + +) β/z )Ỹ + β(+α/z )( /z ) (+δ) Ỹ + XαβỸ ( )] = τ (+δ) η(τ +) [( + β/z )Ỹ + U d β(+α/z)( /z) (+a ) ] = (+β)(+α/z )τ (+δ) (+a )η(τ +) U d [ (+β/z )(+a ) (+β)(+α/z ) U d Ỹ + β ( /z ) +β I] = (+β)(+α/z )τ (+δ) (+a )η(τ +) U d [ (+β/z ) +β Ψ + I] i uimodular. Therefore, C pd tabilize G ad H pd := M X pd = G (I + C pd G ) M(S); H pd () = τ δαβỹ ( )/η + G (). Sice K i/ tabilize H pd, C pid = C pd + K i / tabilize G. Proof of Propoitio 3.6: By (5), η. By (3), U d := α( z )Ỹ () (+αz )(a +) + (a+) δ+ Ỹ = (α+) (+αz )(a +) [ α( z) α+ I + 53

6 Y Y Y (+αz )( z )( z ) (α+)(δ+) (+αz ) α+ = (δ+) G Ỹ () ]Ỹ () = (α+) (+αz )(a +) [I + Φ ]Ỹ () i uimodular. By (5), M pd := Ỹ + XC pd η(τ [( + βz +) )Ỹ + β(+αz )( z ) (δ+) Ỹ + XαβỸ ()] = (δ+) η(τ +) [( + βz )Ỹ + U d β(+αz)( z) (a +) ] = (β+)(+αz )(δ+) η(a +)(τ +) U d [ (+βz) β+ Φ + I] i uimodular. Therefore, C pd tabilize G ad H pd := M X pd M(S); H pd () = αβỹ ()/η + G (). Sice K i/ tabilize H pd, C pid = C pd + K i / tabilize G. Proof of Propoitio 4.: By Propoitio 3., C i (8) tabilize G M(S) ad C i (8) tabilize G M(S). By Theorem.-(b), we mut how that C tabilize W.NowC (I +G C ) M(S) ice C tabilize G, ad hece, W M(S). Furthermore, C (I+G C ) () = G () implie W () = G () G ()G ()G (). By aumptio, Θ:=W ()G () > implie I(I + β Θ) =for β >. The M w := I(I + β Θ) + WC I(I + β Θ) = I(I + β Θ) + β W ( ˆK p + ˆK d τ + G () + ) I (I + β Θ) = I + β [ W ( ˆK p + ˆK d τ + W ()G )+ () Θ ] I (I + β Θ) i uimodular. Therefore, C tabilize W. REFERENCES y time (ec) time (ec) Fig. : Example 4. tep-repoe with C D =[C,C ]. y time (ec) time (ec) Fig. 3: Example 4. tep-repoe with C D =[,C ]. [] K. J. Atröm, K. H. Johao, Q.-G. Wag, Deig of decoupled PID cotroller for MIMO ytem, Proc. America Cotrol Cof., pp. 5-,. [] B. W. Bequette, Proce Cotrol Modelig, Deig ad Simulatio, Pretice Hall, 3. [3] R. D. Braatz, M. Morari, S. Skogetad, Robut reliable decetralized cotrol, Proc. America Cotrol Cof., pp , 994. [4] G. C. Goodwi, S. F. Graebe, M. E. Salgado, Cotrol Sytem Deig, Pretice Hall. [5] A. N. Güdeş, C. A. Deoer, Algebraic Theory of Liear Feedback Sytem with Full ad Decetralized Compeator, Lect. Note i Cotrol ad Iform. Sciece, 4, Spriger, 99. [6] A. N. Güdeş, M. G. Kabuli, Reliable decetralized itegral-actio cotroller deig, IEEE Tra. Aut. Cotr., 46, pp. 96-3,. [7] M. Morari, E. Zafiriou, Robut Proce Cotrol, Pretice-Hall, 989. [8] G. J. Silva, A. Datta, S. P. Bhattacharyya, New reult o the ythei of PID cotroller, IEEE Tra. Aut. Cotr., 47:, pp. 4-5,. [9] X. L. Ta, D. D. Siljak, M. Ikeda, Reliable tabilizatio via factorizatio method, IEEE Tra. Aut. Cotr., 37:, pp , time (ec) Fig. 4: Example 4. tep-repoe with C D =[C,C ] time (ec) Fig. 5: Example 4. tep-repoe with C D =[,C ]. y y r e C w v y r C D e C w v G y Fig. : The two-chael decetralized ytem S(G, C D )..5 y time (ec) Fig. 6: Example 4. tep-repoe with C D =[C, ]. 53