Low order controller design for systems with time delays

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1 20 50th IEEE Conference on Deciion and Contro and European Contro Conference (CDC-ECC) Orando, FL, USA, December 2-5, 20 Low order controer deign for ytem with time deay A N Gündeş and H Özbay Abtract Finite-dimeniona controer ynthei method are deveoped for ome cae of inear, time-invariant, ingeinput inge-output, or muti-input muti-output ytem, which are ubject to time deay The propoed ynthei procedure give ow-order tabiizing controer that ao achieve integraaction o that contant reference input are tracked aymptoticay with zero teady-tate error I INTRODUCTION A wide range of dynamica phenomena cannot be modeed ufficienty accuratey a finite-dimeniona inear timeinvariant (LTI) ytem due to time deay The effect of thee deay often cannot be ignored and have to be incuded in the mode 4, 7 Thi paper preent finite-dimeniona tabiizing controer ynthei method for ome cae of LTI, inge-input inge-output (SISO) or muti-input mutioutput (MIMO) ytem that are ubject to time deay The propoed controer are impe and have ow-order, and they ao provide integra-action o that tep-input reference are tracked aymptoticay with zero teady-tate error The pant cae conidered in Section III-A and Section III-B have no retriction on their poe Thee pant may be tabe or untabe The (tranmiion) zero in the open eft-haf compex pane (OLHP) are unretricted and there may be any number of zero at infinity a we The dua cae in Section III-C conider pant with no retriction on the number or ocation of the (tranmiion) zero, but the poe are either in the OLHP or at the origin = 0 Section III-A examine SISO deayed pant of retarded type (eg, 4,, 2) and Theorem deveop a impe controer ynthei procedure, which i generaized and extended to MIMO ytem in Theorem 2, Section III-B Stabiity of deay ytem of retarded type and of neutra type wa tudied extenivey and many deay-independent and deay-dependent tabiity reut are avaiabe 7, 0 The tuning and interna mode contro technique ued in proce contro ytem generay appy to deay ytem 2 Infinite dimeniona integra action controer have been deigned in to maximize the aowabe controer gain uing the robut contro technique for infinite dimeniona ytem 5 For MIMO tabe pant ubject to inputoutput deay, proportiona-derivative (PD) and proportionaintegra-derivative (PID) controer were deigned in 8 for pant that have no more than two untabe poe coe Department of Eectrica and Computer Engineering, Univerity of Caifornia, Davi, CA 9566 angunde@ucdaviedu H Özbay i with the Department of Eectrica and Eectronic Engineering, Bikent Univerity, Ankara, Turkey hitay@bikentedutr //$ IEEE 5633 to the origin Arbitrary deay term in addition to inputoutput deay were conidered in 9 with decentraized controer tructure Retricting the deigned controer to be PD and PID impoe thee retriction on the number of untabe pant poe 3 The reut in thi paper appy to much wider cae of SISO and MIMO ytem by aowing the order of the controer to exceed that of PD or PID The advantage of integra-action and impe ow-order impementation are ti part of the ynthei technique Notation: Let C, R, R + denote compex, rea, and poitive rea number The extended coed right-haf compex pane i U = { C Re() 0} { }; R p denote rea proper rationa function (of ); S R p i the tabe ubet with no poe in U; M(S) i the et of matrice with entrie in S ; I r i the r r identity matrix The pace H i the et of a bounded anaytic function in C + For h H, the norm i defined a h = e up C + h(), where e up denote the eentia upremum A matrix-vaued function H i in M(H ) if a it entrie are in H ; in thi cae H = e up C + σ(h()), where σ denote the maximum inguar vaue Since a norm of interet here are H norm, we drop the norm ubcript, ie, From the induced L 2 gain point of view, a ytem whoe tranfer-matrix i H i tabe iff H M(H ) A quare tranfer-matrix H M(H ) i unimoduar iff H M(H ) We drop () in tranfer-matrice uch a G(); ue δ(n) to denote the degree of the poynomia n(); ue a m or a a 2 a m to denote the (m m) ona matrix, whoe ona entrie are a,,a m For G R m m p we ue coprime factorization over S ; ie, G = Y X denote a eft-coprime-factorization (LCF), where X,Y S m m, dety ( ) 0 For the deayed pant cae, we ue coprime factorization over H ; ie, Ĝ = Ŷ X denote a eft-coprime-factorization (LCF), where X,Ŷ H m m II PROBLEM DESCRIPTION Conider the feedback ytem Sy(Ĝ,C) in Fig ; C R m m p i the tranfer-function of the controer and Ĝ i the tranfer-function of the pant with time deay It i aumed that the feedback ytem i we-poed and the deay-free part of the pant (ie, the pant without the time deay term) and the controer have no untabe hiddenmode With u, v, w, y a the input and output vector, the coed-oop tranfer-matrix Ĥ from (u,v) to (w,y) i C(I + Ĥ = ĜC) C(I + ĜC) Ĝ () ĜC(I + ĜC) (I + ĜC) Ĝ Let the (input-error) tranfer-function from u to e be denoted

2 v u e C w Ĝ y Fig The feedback ytem Sy( b G, C) by H eu and et the (input-output) tranfer-function from u to y be denoted by H yu ; then H eu = (I +ĜC) = I ĜC(I +ĜC) = I H yu (2) Definition : a) The feedback ytem Sy(Ĝ,C) hown in Fig, i tabe if the coed-oop map Ĥ i in M(H ) b) The controer C tabiize Ĝ if C i proper and Sy(Ĝ,C) i tabe c) The ytem Sy(Ĝ,C) i tabe and ha integraaction if the coed-oop tranfer-function from (u,v) to (w,y) i tabe, and the (input-error) tranfer-function H eu ha bocking-zero at = 0 d) The controer C i aid to be an integra-action controer if C tabiize Ĝ and D(0) = 0 for any RCF C = ND Let Ĝ = Ŷ X, where Ŷ, X M(H ) Let C = ND be an RCF, where D,N S m m, detd( ) 0 Then C tabiize Ĝ if and ony if M M(H ), where M := Ŷ D + X N (3) Suppoe that the ytem Sy(Ĝ,C) i tabe and that tep input reference are appied at u(t) The teady-tate error e(t) due to tep input at u(t) goe to zero a t if and ony if H eu (0) = 0 Therefore, by Definition -(c), the tabe ytem Sy(Ĝ,C) achieve aymptotic tracking of contant reference input with zero teady-tate error if and ony if it ha integra-action By (3), write H eu = (I + ĜC) = DM Ŷ Then by Definition -(d), Sy(Ĝ,C) ha integraaction if C = ND i an integra-action controer ince D(0) = 0 impie H eu (0) = (DM Ŷ )(0) = 0 The ytem Sy(Ĝ,C) woud ao have integra-action if every entry of the MIMO pant ha poe at = 0 ince Ŷ (0) = 0 impie H eu (0) = 0 even if the controer D(0) 0 Therefore, it i not a neceary condition to have integra-action controer for the ytem to have integra-action when Ŷ (0) = 0 However, for robut deign, integra-action i achieved with poe dupicating the dynamic tructure of the exogenou igna that the reguator ha to proce; thee integra-action controer obey the we-known interna mode principe 6 We aume throughout that Ĝ ha no tranmiion-zero at = 0 Thi condition i a neceary condition for exitence of integra-action controer: Let the (m m) matrix Ĝ() have (norma) rankg() = m If Ĝ admit an integra-action controer, then it ha no tranmiion-zero at = 0 III CONTROLLER SYNTHESIS We propoe finite-dimeniona tabiizing controer ynthei for certain cae of pant that have time deay The dicuion in Section III-A appie to a ca of SISO deay ytem In Section III-B, the reut are extended to a ca of MIMO pant with poe anywhere in the compex pane, 5634 but zero retricted to be in the OLHP Section III-C incude MIMO deay ytem whoe zero are unretricted, but the poe are either at the origin or in the tabe region A SISO pant of retarded type We conider SISO deay pant decribed a x() ν Ĝ() = y() + q(), q() = e hi q i (), (4) i= where x(),y(),q i () are poynomia with rea coefficient, δ(x) δ(y) > δ(q i ), the integer h i > 0, i =,,ν We aume that the finite zero of Ĝ are in the OLHP, ie, the poynomia x() i tricty Hurwitz Let r := δ(y) δ(x) 0 Let ξ() be any monic r-th order tricty Hurwitz poynomia; for exampe, ξ = ( + a) r for any a R + Define Y n := y() x()ξ(), Y d := q() x()ξ(), Ŷ = Y n + Y d, X := ξ() (5) Then X,Y n S, Y d H, and Ĝ = (Ŷ ) X = (Y n + Y d ) X Theorem preent a finite-dimeniona controer ynthei for coed-oop tabiity Thi deign give integraaction controer of order r when the reative degree of x()/y() i r, or of order when the reative degree of x()/y() i zero Theorem : (SISO tabiizing controer ynthei): Let Ĝ() be a in (4) For any monic r-th order tricty Hurwitz poynomia ξ(), et Ĝ() = Ŷ X = (Y n + Y d ) X be a in (5) a) If r = 0, then chooe any g R + Let o R + be uch that o > gĝ = + + gŷ (6) Then the controer C o in (7) tabiize Ĝ: ( + g) C o = o (7) b) If r, then chooe any monic, tricty Hurwitz poynomia ξ() of order r Define Θ a Θ() := Y n ( ) ξ()ĝ() Let R + be uch that = (Y n () + Y d ())Y n ( ) (8) > r Θ() (9) Then the controer C in (0) tabiize Ĝ: r ξ() C = ( + ) r r Y n( ) (0) Remark: In Theorem, the SISO controer C o in (7) for r = 0, C in (0) for r are biproper, and each ha a poe at = 0 providing integra-action The remaining (r ) poe of r-th order controer C in (0) are a in the OLHP Proof of Theorem : a) If r = 0, then X = in Ĝ = Ŷ X Let N =, D = Co ; then C o = ND i a

3 coprime factorization of the propoed controer in (7) By (3), C o tabiize Ĝ if and ony if M H, where M = X N + Ŷ D = + Ŷ () o ( + g) o (+g) Since o atifie (6), Ŷ () <, which i a ufficient condition for M H Therefore, C o tabiize Ĝ ince M H b) Define := ( + ) r r Let N = r, D = () ξ() Y n( ) Then C = ND i a coprime factorization of the propoed controer in (0) By (3), C tabiize Ĝ if and ony if M M(H ), where M = X N + Ŷ D = r ξ() + Ŷ ()() ξ() Y n( ) r = ( ( + ) r + () ( + ) r Ŷ ()Y n( ) ( + )r ) ξ() = ( + ( + ) r Ŷ ()Y n( ) ( + )r ) ξ() + )r = ( + Θ())( ( + ) r ξ() A ufficient condition for M to be in M(H ) i that (+) Θ() < We firt how that r ( + ) r = ( + )r r ( + ) r = r () ( ) For r, (+) r r =0 (+) = r r impie that the norm in () i greater than or equa to r/ We prove the norm in () i e than or equa to r/ by iteration: For r =, () hod ince (+) = / For r = 2, 2 (+)2 (+) = (+) = 2/ For r = 3, 2 + = / impie ( + )3 3 ( + ) 3 ( + )( + ) 2 2 ( + ) 2 = ( + ) + ( + )2 2 ( + ) 2 + ( + )2 2 ( + ) 2 + = 2 + = 3, hence, () hod Continuing imiary, uppoe that () hod for r and how that it hod for (r + ): ( + )r+ r+ ( + ) r+ = ( + )( + )r r ( + ) ( + ) r + ( + ) r + ( + ) r + = r + = r +, hence, () hod In (8), Y n ()Y n ( ) = impie Y n ()Y n ( ) I S Since δ(q) < δ(y) = δ(xξ), we have Y d ()Y n ( ) H Therefore, Θ() = Y n ()Y n ( ) I + Y d ()Y n ( ) H Since atifie (9), (+) Θ() r r Θ() < Therefore, C in (0) tabiize Ĝ ince M H 5635 B MIMO pant with unretricted poe We conider (m m) MIMO pant with deay, where the deay are a in the denominator matrix Ŷ M(H ) of Ĝ = Ŷ X Therefore, X i deay-free and we denote it by X M(S) We aume that Ĝ can be written a Ĝ = Ŷ X ; Ŷ = Y n + Y d, Y n () M(S), ν dety n ( ) 0, Y d = e hi Q i (), Q i ( ) = 0 (2) i= We aume that the tranmiion-zero of Ĝ are a in the OLHP and at infinity, ie, rankx( ) < m but rankx() = m for a C + With n k and d k a poynomia, write X nk () () = (3) d k () k, {,,m} Since the tranmiion-zero of Ĝ are a in the OLHP, X ha no poe in the coed right-haf compex pane C + (ie, d k are tricty Hurwitz) but may have poe at infinity, ie, X may be improper Define the integer r k and r a { δ(nk ) δ(d r k := k ), if δ(n k ) > δ(d k ) 0, if δ(n k ) δ(d k ) r := max k m r k, =,,m (4) Let ξ () be any monic r -th order tricty Hurwitz poynomia, =,,m; eg, ξ () = ( + a) r for a R + Define () := ξ () ξ 2 () ξ m () (5) If r = 0, then ξ = Athough X may be improper, X n i tabe ince k () d k ()ξ () S Define Ŷ ( ) := (X()Ĝ() ) ; ie, Y j ( ) = (Ĝ()X ()) By (2), Ŷ ( ) = Y n( ) For thi ca of MIMO pant, Theorem 2 preent a finite-dimeniona controer ynthei with integra-action Theorem 2: (MIMO tabiizing controer ynthei): Let Ĝ = Ŷ X = (Y n + Y d ) X be a in (2) Define Θ a Θ() := Ŷ ()Y n( ) I (6) For =,,m, define ρ a {, if r = 0 ρ := r, if r Let R + be uch that For =,,m, define a (7) > max ρ Θ (8) () := ( + ) ρ ρ (9) Then the controer C in (20) tabiize Ĝ: C = X ρ ρ2 () () 2 () ρm m () Y n ( ) (20)

4 m Remark: In Theorem 2, the ona term of ρ in the controer C in (20) a have poe at = 0 and hence, C ha integra-action The term correponding to r = 0 are in the form The term correponding to r are in the form ( r (+) r ), with one poe at = 0, and the r remaining (r ) poe a in the OLHP Proof of Theorem 2: For =,,m, define η a η := ( + ) ρ r ; (2) η = (+) if r = 0, and η = if r If r = 0, then et ξ = If r, then chooe any monic, tricty Hurwitz poynomia ξ () of order r With a in (5), et N = X η ρ η 2 ρ2 η m ρm m = X ρ ξ ()( + ) ρ, r D = Y n ( ) η () m = Y n ( ) () ξ ()( + ) ρ r m (22) Then C = ND i a right-factorization of the propoed controer in (20) By (3), C tabiize Ĝ if and ony if M M(H ), where M = X N + Ŷ D = XX ρ η m + Ŷ ()Y n( ) η m ρ m = ξ ()( + ) ρ r m + Ŷ ()Y n( ) () ξ ()( + ) ρ r ρ m = ( ( + ) ρ m + Ŷ ()Y n( ) () ( + ) r m ( + ) r ) ξ () m = (I + Ŷ ()Y n( ) () I ( + ) ρ ) ( + ) r m ξ () m () ( + ) r m = (I+Θ() ( + ) ρ ) ξ m The entrie of () (+) for r ρ = 0 have norm (+) = The entrie for r have norm ( + )r r ( + ) r = r = ρ a hown in the proof of Theorem Therefore, m () ρ ( + ) ρ = max m In (6), Y n ()Y n ( ) χ M(S) ince Y n ()Y n ( ) = I Since Q i ( ) = 0 by (2), 5636 Y d ()Y n ( ) M(H ); hence, Θ H Therefore C tabiize Ĝ ince M M(H ) if m () ρ Θ() ( + ) ρ Θ() max <, which hod ince atifie (8) C MIMO pant with unretricted tranmiion-zero We conider (m m) MIMO pant with deay, where the deay are a in the numerator matrix X M(H ) of Ĝ = Ŷ X, ie, the denominator matrix Ŷ i deay-free and we denote it by Y M(S) Therefore, we aume that a eft-coprime factorization of Ĝ can be written a Ĝ = Y X ; Xij = e hij X ij ; i,j =,,m; (23) the integer h ij 0; Y M(S) i deay-free; X M(H ) and X ij denote it ij-th entry Suppoe that each ij-th entry X ij of X may contain any arbitrary deay term and that the deay are known If the finite-dimeniona part Y of the deayed pant Ĝ i tabe, then (23) impie that the entrie of Ĝ may contain a different arbitrary known deay term Let Ĝ have fu (norma) rank m Let Ĝ have no tranmiion zero at = 0, equivaenty, rank X(0) = m We ao aume that Y may have poe anywhere in the OLHP, but the ony U-poe of are a at = 0, ie, the ony C + -poe of Y are at the origin The entrie of Y may have different mutipicitie of poe at = 0 and ome entrie may have ony poe in the tabe region C\U Write Y () = Y k () k,,,m (24) For =,,m, define the integer γ k 0 be the number of poe of Y k () at = 0, and define γ a γ := max k m γ k ; (25) ie, γ 0 i the arget number of poe at = 0 of the entrie in the -th coumn of Y () For =,,m, athough Y k () S, (Y k () γ (+β) ) S for any β R γ + For thi ca of (MIMO or SISO) pant, Theorem 3 preent a finite-dimeniona controer ynthei; Coroary incude integra-action in the tabiizing controer ynthei Theorem 3: (MIMO tabiizing controer ynthei): Let Ĝ = Y X be a in (23) Define Φ a Φ() := X()X(0) I (26) Chooe β R + uch that β < Φ() (27) max γ For =,,m, define ψ () := ( + β) γ γ (28) Then the controer C in (29) tabiize Ĝ: ψ () 0 0 γ ψ 0 2() C = X(0) 0 γ 2 Y () (29) 0 0 ψ m() γm

5 Coroary : (Integra-action controer ynthei): Under the aumption of Theorem 3, chooe β R + uch that β < For =,,m, define Φ() (30) + max γ ψ () := ( + β) +γ +γ (3) Then the integra-action controer C in (32) tabiize Ĝ: ψ () 0 0 +γ ψ C = X(0) 0 2() 0 +γ 2 Y () (32) Proof of Theorem 3: Let Ψ() := 0 0 ψ() (+β) γ Define N := X(0) Ψ(), D := Y γ (+β) γ ψ 2() (+β) γ 2 ψ m() +γm ψ m() (+β) γm (33) γ 2 (+β) γm γ 2 (+β) (34) γm Since the order of the poynomia ψ () i (γ ), the tricty-proper term S are tabe and hence, ψ () (+β) γ N = X(0) Ψ() M(S) Since (Y k () γ (+β) γ ) S, the matrix D M(S) Therefore, C = ND i a rightfactorization of the propoed controer in (29) By (3), C tabiize Ĝ if and ony if M M(H ), where M = X N + Y D = X()X(0) Ψ() + Y ()Y () γ (+β) γ = X()X(0) Ψ + m γ (+β) γ m = X()X(0) IΨ() = X()X(0) IΨ() = Φ()Ψ() A ufficient condition for M to be in M(H ) i that Φ()Ψ() < To find Ψ(), we firt how that ψ () ( + β) γ = ( + β)γ γ ( + β) γ = β γ (35) For ( γ = 0, (35) obviouy hod For γ, (+β) γ γ ) (+β) = βγ γ impie that the norm in (35) = i greater than or equa to β γ We prove the norm in (35) i e than or equa to β γ For γ =, (35) hod ince β +β = β For γ = 2, (+β)2 2 (+β) = 2β +β2 2 (+β) = 2 2β and hence, (35) hod For γ = 3, +β = impie (+β)3 3 (+β) 3 +β (+β)2 2 (+β) +β = (2β+β) = 2 3β and hence, (35) hod Continuing imiary, uppoe that (35) hod for γ and how that it hod for (γ + ): ( + β)γ+ γ+ ( + β) γ + + β ψ () ( + β) γ + β = (βγ + β) = β(γ + ) and hence, (35) hod Now (35) impie Ψ() = β max γ Since β atifie (27), Φ()Ψ() Φ() Ψ() = β max γ Φ() < Therefore, C tabiize Ĝ ince M M(H ) Proof of Coroary : Let ψ Ψ() := () ψ 2() ψ m() (+β) +r (+β) +r 2 (+β) +rm (36) With D a in (34), et Ñ = X(0) Ψ(), D = (+β) D Then C = Ñ D i a right-factorization of the propoed controer in (32); ince D(0) = 0, by Definition -(d), C i an integra-action controer We how that by (3), C tabiize Ĝ if and ony if M M(H ), where M = X Ñ + Y D = X Ñ + (+β) Y D = X()X(0) Ψ() + (+β) r m (+β) r = X()X(0) I Ψ() = 5637 X()X(0) I Ψ() = Φ() Ψ() A ufficient condition for M M(H ) i Φ() Ψ() <, where, by (35), Ψ() = β ( + max γ ) Since β atifie (30), Φ() Ψ() Φ() Ψ() = β ( + max γ ) Φ() < Therefore, the integra-action controer C tabiize Ĝ ince M M(H ) Exampe : Conider IV EXAMPLES ( + ) Ĝ() = ( ) + 2( )e h + 5e h2 x() = y() + q(), h 2 = π 2 h (37) The pant Ĝ i in the ca conidered in Section III-A Since the reative degree r =, the controer a in (0) i a firt order controer with integra action (ie a PI controer) Let ξ() = ( + b) for a free parameter b > 0; define Y n () = ( ) ( + )( + b), Y d() = 2( )e h + 5e h2 ( + )( + b) With Θ a in (8), Θ() = (2( )e h + 5e h2 ) (3 + b) b + 2, ( + )( + b) the minimum vaue Θ() of atifying (9) i hown in Fig 2 for variou h For h 0, 25, if we chooe b = 2, then = 8 atifie (9) The controer in (0) i C() = 8(+2) Thi feedback ytem i tabe if the tranformed characteritic equation + (+) Θ() = 0 ha no root in the coed right haf pane Since Θ < and ( + ) = /, the ma gain theorem impie tabiity Now change the pant Ĝ in (37) to Ĝ w = x() ( p) w y() + q(), w > 0, p R, and x(),y(),q() are the ame a in (37) The reative degree become r = w+, and the deayed part

6 min increaing direction of h h = 0, 05, 0, 20, b Fig 2 Exampe : b veru Θ for variou h Y d () of Ŷ () remain the ame The new Y n i Y nw = ( p)w y() ( + )( + b) w, and Y nw ( ) = Y n ( ) = We re-cacuate Θ() = Y nw ()Y n ( ) I + Y d ()Y n ( ) Y nw ()Y n ( ) I + Y d ()Y n ( ) =: Θ n () + Θ d () A condition on more conervative than (9) i > r Θ n () + r Θ d () For exampe, if p = 05 and w =, then r = 2 and Θ() = 45 for h 0, 25 Therefore, (9) hod if we chooe = 30 The controer in (0) i then given by C() = 900(+2) (+60) Exampe 2: For h 2 = π 2 h, conider Ĝ() = Ŷ () X() = + 2e h e 0 h The MIMO pant Ĝ i in the ca in Section III-B We have Ŷ ( ) = Y n ( ) =, r 0 = r 2 = We compute Θ() = 05 2e h (+e h ) + (2+3e h 2 ) + It can be hown that for a h 00, 25, (8) i atified for = 5 Hence C() a in (20) tabiize the given Ĝ: C() = (+05)(+2) ( ) +05 (+2)( 2 ++5) ( ) Exampe 3: Conider the foowing pant Ĝ, which i in the ca conidered in Section III-C: Ĝ() = K( + z)e h γ ( + p )( + p 2 ), where γ, p,p 2 R +, z R \ {0}, and h 0; Note that z may be poitive or negative, ie, Ĝ may have a finite zero in the right-haf compex pane Write Ĝ = Y X a Ĝ = Y X γ K( + z)e h = ( ( + b) ) ( γ ( + b) γ ( + p )( + p 2 ) ) 5638 for any b R + With X(0) = bγ p p 2 Kz, and Φ a in (26), et β R + atify (27), ie, β < γ Φ, where Φ = b γ p p 2 K( + z)e h Kz( + b) γ ( + p )( + p 2 ) Then C = X(0) (+β) γ γ Y = a p p2 (+β)γ γ γ K z (+b) a γ in (29) i a controer that tabiize Ĝ The controer C i tabe, and it order i γ, the ame a the number of poe of Ĝ at = 0, which i e than the pant order Let β R + atify (30), ie, β < +γ Φ Then an integraaction controer C a in (32) that tabiize Ĝ i C = X(0) (+β) γ + γ + Y = bγ p p 2(+β) γ+ γ+ γ + K z (+b) For γ exampe, if γ =, then C and C become C = X(0) β b p p2 β Y = K z (+b), where β < Φ, and C = X(0) (2β +β 2 ) Y = b p p2(2β +β2 ) 2 K z (+b), where β < 2 Φ V CONCLUSIONS We propoed finite-dimeniona controer deign for certain cae of SISO and MIMO ytem ubject to deay Thee deign achieve coed-oop tabiity and integraaction The controer order matche the reative degree of the finite-dimeniona part of the pant for the pant in Section III-A-III-B or the number of pant poe at the origin in Section III-C Performance pecification beyond aymptotic tracking of contant reference are not within the cope of thi tudy Future work wi focu on expanding the pant cae to thoe that aow finite right-haf pane zero whie not retricting the ocation of untabe poe REFERENCES C Bonnet, J R Partington, Anayi of fractiona deay ytem of retarded and neutra type, Automatica, vo 38, pp 3338, C Bonnet, JR Partington, Stabiization of ome fractiona deay ytem of neutra type, Automatica, vo 43, pp , RF Curtain, K Gover, Robut tabiization of infinite-dimeniona ytem by finite-dimeniona controer, Sytem Contro Letter, 7 (), pp 447, RF Curtain, H Zwart, An Introduction to Infinite-dimeniona Linear Sytem Theory, Text in Appied Mathematic, Vo 2, Springer, New York, C Foia, H Özbay, A Tannenbaum, Robut Contro of Infinite Dimeniona Sytem, LNCIS 209, Springer-Verag, London, B A Franci and W A Wonham, The interna mode principe for inear mutivariabe reguator, Appied Mathematic & Optimization, 2:2, pp 70-95, K Gu, V L Kharitonov, J Chen, Stabiity of Time-Deay Sytem, Birkhäuer, Boton, A N Gündeş, H Özbay, and A B Özgüer, PID controer ynthei for a ca of untabe MIMO pant with I/O Deay, Automatica, vo 43, no, pp 35-42, A N Gündeş, H Özbay, Reiabe decentraized contro of deayed MIMO pant, Int Jour Contro, vo 83, no 3, pp , S I Nicuecu, Deay Effect on Stabiity: A Robut Contro Approach, LNCIS, vo 269, Heideberg: Springer-Verag, 200 H Özbay, A N Gündeş, Integra action controer for ytem with time deay, in Topic in Time Deay Sytem Anayi, Agorithm and Contro, LNCIS 388, J J Loieau, W Michie, S-I Nicuecu, R Sipahi (Ed), pp , Springer-Verag, London, S Skogetad, Simpe anaytic rue for mode reduction and PID controer tuning, Jour Proce Contro, vo 3, pp , G J Siva, A Datta, S P Bhattacharyya, PID Controer for Time- Deay Sytem, Birkhäuer, Boton, 2005