9 American Control Conference Hyatt Regency Riverfront, St. Loui, MO, USA June -, 9 WeC5.5 PID Controller Synthei with Shifted Axi Pole Aignment for a Cla of MIMO Sytem A. N. Gündeş and T. S. Chang Abtract For certain clae of linear, time-invariant, multiinput multi-output plant, a ytematic ynthei i developed for tabilization uing Proportional+Integral+Derivative (PID) controller, where the cloed-loop pole can be aigned to the left of an axi hifted away from the origin. The real-part of the cloed-loop pole can be maller than the real-part of the larget tranmiion-zero of the plant. Plant clae that admit PID controller with thi property include table and untable multi-input multi-input plant with tranmiion-zero in the left-half complex-plane. I. INTRODUCTION Proportional+Integral+Derivative (PID) controller are preferred in many control deign ince they are imple, have low-order, provide integral-action and hence, achieve aymptotic tracking of tep-input reference (e.g., []). Although the implicity of PID controller i deirable due to eay implementation and tuning, the order contraint preent a major retriction that only certain clae of plant can be controlled by uing PID controller. Rigorou PID ynthei method baed on modern control theory are explored recently in e.g., [7], [], [6]. Sufficient condition for PID tabilizability of linear, time-invariant (LTI), multi-input multi-output (MIMO) plant were given in [6] and everal plant clae that admit PID controller were identified. An important criterion for control deign i to aign the cloed-loop pole ufficiently far from the imaginary-axi of the complex-plane in order for the ytem to have mall timecontant and ufficient damping. Therefore, it i deirable for the cloed-loop pole to have real-part le than h for a pre-pecified poitive contant h. Thi deign objective i achievable for certain LTI, MIMO plant clae a identified here. Aignment of the imaginary part of the cloed-loop pole i not within the objective or the cope of thi work. All plant clae that admit PID controller are necearily trongly tabilizable, although trong tabilizability i not ufficient for exitence of PID controller [6]. The integralcontant of the PID controller can be non-zero only if the plant ha no zero at the origin. Stable plant are obviouly trongly tabilizable and they admit PID controller. The additional objective of aigning value le than h to the real-part of the cloed-loop pole can be achieved only for certain value of h [3]. The retriction on h i removed for table plant that have no finite zero with real-part larger than the given h; the cloed-loop pole can be aigned to the left of thi h for any choen value of h a hown A. N. Gündeş and T. S. Chang are with the Department of Electrical and Computer Engineering, Univerity of California, Davi, CA 9566 angunde@ucdavi.edu, chang@ece.ucdavi.edu in [3]. The untable plant clae invetigated here have no finite zero with real-part larger than the given h. For thee plant clae, Propoition, and 3 preent ytematic PID controller ynthei method for cloed-loop pole aignment to the left of the finite zero with the larget negative real-part. The main reult preented in Section II tart with the problem tatement and baic definition. The three plant clae under conideration are tudied under three ubection. An illutrative MIMO example i alo given in Section II for the cae of one plant zero at infinity baed on the linearized model of a batch proce [8]. The only goal of the deign in thi example are cloed-loop tability with cloed-loop pole to the left of a line at h; due to the integral-action in the controller, the teady-tate error due to tep input reference i zero and hence, aymptotic tracking i alo achieved. The choice of the free parameter can be optimized with a choen cot function. Section III give concluding remark. Although we dicu continuou-time ytem here, all reult alo apply to dicrete-time ytem with appropriate modification. Notation: Let C, R, R + denote complex, real, poitive real number. For h R + {}, let U h := { C Re() h} { }. If h =, U h = U := { C Re() } { } i the extended cloed right-half complex plane. Let R p denote real proper rational function of. For h, S h R p i the ubet with no pole in U h. The et of matrice with entrie in S h i denoted by M(S h ) ; S m m h i ued intead of M(S h ) to indicate the matrix ize explicitly. A matrix M M(S h ) i called S h -table; M M(S h ) i called S h -unimodular iff M M(S h ). The H -norm of M() M(S h ) i M := up Uh σ(m()), where σ i the maximum ingular value and U h i the boundary of U h. We drop () in tranfer-matrice uch a G() where thi caue no confuion. We ue coprime factorization over S h ; i.e., for G R m m p, G = Y X denote a denote a right-coprime-factorization (RCF) where X, Y, N g,d g M(S h ), dety ( ), detd g ( ). For MIMO tranfer-function, we refer to tranmiion-zero imply a zero; blocking-zero are a ubet of tranmiion-zero. If G R m m p i full (normal) rank, then z o U h i called a tranmiion-zero of G = Y X if rankx(z o ) < m; left-coprime-factorization (LCF) and G = N g D g z b U h i called a blocking-zero of G = Y X if X(z b ) = and equivalenty, G(z b ) =. 978--444-454-/9/$5. 9 AACC 849
II. MAIN RESULTS Conider the LTI, MIMO unity-feedback ytem Sy(G,C) hown in Fig., where G R p m m and C R p m m are the plant and controller tranfer-function. Aume that Sy(G,C) i well-poed, G and C have no untable hidden-mode, and G R p m m i full rank. can be placed to the left of a hifted-axi that goe through h may be retricted. Let ŝ and Ĝ, Ĉ pid be defined a ŝ := + h, equivalently, =: ŝ h ; (3) Ĝ(ŝ) := G ; (4) v r e C w G y Fig.. Unity-Feedback Sytem Sy(G, C). We conider the realizable form of proper PID controller given by (), where K p, K i, K d R m m are the proportional, integral, derivative contant, repectively, and τ R + (ee [4]): C pid () = K p + K i + τ + K d. () For implementation, a (typically fat) pole i added to the derivative term o that C pid in () i proper. The integral-action in C pid i preent when K i. Subet of PID controller are obtained by etting one or two of the three contant equal to zero: () become a PI-controller C pi when K d =, an ID-controller C id when K p =, a PD-controller C pd when K i =, a P-controller C p when K d = K i =, an I-controller C i when K p = K d =, a D-controller C d when K p = K i =. Definition : a) Sy(G,C) i aid to be S h -table if the cloed-loop tranfer-function from (r,v) to (y, w) i in M(S h ). b) C i aid to S h -tabilize G if C i proper and Sy(G,C) i S h -table. c) G R m m p i aid to admit a PID controller uch that the cloed-loop pole of Sy(G,C ) are in C \ U h if there exit C = C pid a in () uch that Sy(G,C pid ) i S h -table. We ay that G i S h -tabilizable by a PID controller, and C pid i an S h -tabilizing PID controller. Let G = Y X be any LCF of G, C = N c Dc be any RCF of C; for G R m m p, X,Y M(S h ), dety ( ) n, and for C R u n y p, N c,d c M(S h ), detd c ( ). Then C i a S h -tabilizing controller for G if and only if i S h -unimodular [], [5]. M := Y D c + XN c () The problem addreed here i the following: Suppoe that h R + i a given contant. I there a PID controller C pid that tabilize the ytem Sy(G,C pid ) with a guaranteed tability margin, i.e., with real-part of the cloed-loop pole of the ytem Sy(G, C pid ) le than h? It i clear that thi goal i not achievable for ome plant. Even when it i achievable, thoe h R + for which the cloed-loop pole Ĉ pid (ŝ) := C pid := K p + ŝ h K i + τ + K d. (5) Then C pid () S h -tabilize G() if and only if Ĉpid(ŝ) S - tabilize Ĝ(ŝ). For any ρ R +, an RCF of Ĉ pid (ŝ) i given by Ĉ pid = ( ŝ + ρ Ĉpid ) ( ŝ + ρ I ). (6) We conider plant clae that admit PID controller and identify value of h R uch that the cloed-loop pole lie to the left of h. A neceary condition for exitence of PID controller with nonzero integral-contant K i i that the plant G() ha no zero (tranmiion-zero or blockingzero) at = [6]. Therefore, all plant under conideration are aumed to be free of zero at the origin (of the -plane). Let G ph be the et of S h -table m m plant that have no (tranmiion or blocking) zero at = ; i.e., for a given h R + {}, let G ph S h m m be defined a G ph := { G() S h m m det G() }. (7) For G() G ph, with Ĝ(ŝ) := G, det G() i equivalent to detĝ(h). The plant G G ph may have tranmiion-zero or blocking-zero anywhere in C other than =. There exit S h -tabilizing PID controller for thi clae of plant for certain value of h R + [3]. In thi paper we focu on plant that may have pole in the region U h. We conider the following three plant clae, which have no retriction in the pole anywhere in the complex-plane C and no retriction on the zero in the region C \ U h : ) The firt cla of plant G zh i the et of m m plant that have no (tranmiion or blocking) zero in U h ; i.e., for a given h R + {}, let G zh R m m p be defined a G zh := { G() R p m m G () S h m m }. (8) In the ingle-input ingle-output (SISO) cae, thi cla repreent plant without zero in U h that have relative degree zero. Some plant in the et G zh are not S h -table; therefore, thee plant either have pole in U, or they are S -table but ome pole have negative real-part larger than the pecified h. Obviouly, the plant in G zh atify the neceary condition for exitence of PID controller with nonzero integral-contant K i ince the fact that they have no zero in U h implie that they have no zero at =. ) The econd cla of plant G i the et of m m trictly-proper plant that have no (tranmiion or blocking) zero in U h except at infinity with multiplicity one, a 85
defined below: For a given h R + {}, let G R m m p be defined a G := {G() R m m p + a G () S m m h,a > h}. (9) In the SISO cae, thi cla repreent plant without zero in U h that have relative degree one. Some plant in the et G are not S h -table; thee plant either have pole in U, or they are S -table but ome pole have negative real-part larger than the pecified h. Obviouly, the plant in G atify the neceary condition for exitence of PID controller with nonzero integral-contant K i ince the fact that they have no zero in U h (other than at infinity) implie that they have no zero at =. 3) The third cla of plant G i the et of m m trictly-proper plant that have no (tranmiion or blocking) zero in U h except at infinity with multiplicity two, a defined below: For a given h R + {}, let G R m m p be defined a G := {G() R p m m ( + a) G () S h m m for any a > h}. () In the SISO cae, thi cla repreent plant without zero in U h that have relative degree two. Some plant in the et G are not S h -table; thee plant either have pole in U, or they are S -table but ome pole have negative real-part larger than the pecified h. Obviouly, the plant in G atify the neceary condition for exitence of PID controller with nonzero integral-contant K i ince the fact that they have no zero in U h (other than at infinity) implie that they have no zero at =. The et G ph G and the et G ph G correpond to S h -table plant with no pole in U h, and no zero in U h other than (one or two, repectively) zero at infinity. A. Plant with no zero in U h Conider the cla G zh of m m plant with no (tranmiion or blocking) zero in U h a decribed in (8). The plant G G zh may not be S h -table but G M(S h ); an LCF of G() i G = Y X = (G ) I. () The plant in G zh are trongly tabilizable, and they admit S -tabilizing PID controller [6]. Propoition how that thee plant alo admit S h -tabilizing PID controller for any pre-pecified h R +, and propoe a ytematic PID controller ynthei. Propoition : (PID for plant with no U h -zero): Let G G zh. Then there exit an S h -tabilizing PID controller C pid. Furthermore, C pid can be deigned a follow: Chooe any noningular ˆK p R m m. Chooe any K d R m m, and τ R + atifying τ < /h. Chooe any α R + atifying α > h. Define Φ(ŝ) a Φ(ŝ) := ˆK p [ Ĝ (ŝ) + τ + K d ]. () Let K p = β ˆK p, K i = α β ˆK p, where β R + atifie β > Φ(ŝ). (3) Then an S h -tabilizing PID controller C pid i given by C pid = β ˆK p + α β ˆK p + τ + K d. (4) For K d =, (4) i a PI-controller. Proof of Propoition : Subtitute ŝ = + h a in (3)-(5). Then an LCF of Ĝ(ŝ) i Ĝ(ŝ) = Ŷ ˆX := ( Ĝ (ŝ)) I. Write the controller C pid () given in (4) a C pid () = ( + α C pid)( I + α ) = (β ˆK p + ( + α) (τ + ) K d)( I + α ). (5) Subtitute ŝ = + h into (5) to obtain an RCF of Ĉpid(ŝ) a in (6), with ρ = α h. Then Ĉ pid (ŝ) = ( β ˆK p + (ŝ h + α) (τ + ) K d)( ŝ h + α I), (6) where ( τh) R + and (α h) R + by aumption. By (), Ĉ pid (ŝ) in (6) tabilize Ĝ(ŝ) if and only if M β(ŝ) i S -unimodular: M β (ŝ) = Ŷ (ŝ) ŝ h + α I + ˆX(ŝ) ŝ h + αĉpid(ŝ) = Ĝ (ŝ) I + = β ˆK p + = β ˆK p (I+ β ŝ h + α (ŝ h + α) [ Ĝ (ŝ) + ˆK p ŝ h + αĉpid(ŝ) τ + K d ] (ŝ h + α) [Ĝ (ŝ)+ τ + K d]) = β ˆK p (I + β Φ(ŝ) (ŝ h + α) ), (7) where ˆK p i unimodular and G () M(S h ) by aumption. If α > h a aumed, then β > Φ(ŝ) implie β Φ(ŝ) (ŝ h + α) β Φ(ŝ) (ŝ h + α) = β Φ(ŝ) < ; (8) hence, M β (ŝ) in (7) i S -unimodular. Therefore, Ĉ pid (ŝ) an S -tabilizing controller for Ĝ(ŝ); hence, C pid i an S h - tabilizing controller for G. B. Strictly-proper plant with no other zero in U h Conider the cla G of m m trictly-proper plant that have no other (tranmiion or blocking) zero in U h a decribed in (9). The plant G G are not all S h -table but +a G M(S h ) for any a > h. An LCF of G() i G = Y X = ( + a G ) ( + a I ) ; (9) 85
in (9), G( ) =, and Y ( ) = ( + a)g() = G(). The plant in G are trongly tabilizable, and they admit S -tabilizing PID controller [6]. Propoition how that thee plant alo admit S h -tabilizing PID controller for any pre-pecified h R +, and propoe a ytematic PID controller ynthei procedure. Propoition : (PID for plant with one zero at infinity): Let G G. Then there exit an S h -tabilizing PID controller, and C pid can be deigned a follow: Let Y ( ) := G(). Chooe any K d R m m, and τ R + atifying τ < /h. Chooe any α R + atifying α > h. Define Ψ(ŝ) a Ψ(ŝ) := [Ĝ (ŝ)+ τ + K d] (ŝ h + α) Y ( ) I. () Let K p = δy ( ), K i = α δy ( ), where δ R + atifie δ > Ψ(ŝ) + h. () Then an S h -tabilizing PID controller C pid i given by C pid = δ Y ( ) + α δ Y ( ) + τ + K d. () For K d =, () i a PI-controller. Proof of Propoition : Subtitute ŝ = + h a in (3)-(5). Then an LCF of Ĝ(ŝ) i Ĝ(ŝ) = Ŷ ˆX := ( Ĝ (ŝ)) ( I ). Write the controller C pid() ŝ h+a ŝ h+a given in () a C pid () = ( + δ C pid)( I + δ ) = ( + α) ( δy ( ) + + δ ( + δ) (τ + ) K d)( I + δ ). (3) Subtitute ŝ = + h into (3) to obtain an RCF of Ĉpid(ŝ) a in (6), with ρ = δ h. Then (ŝ h + α) Ĉ pid (ŝ) = ( ŝ h + δ δ Y ( ) + (ŝ h + δ) (τ + ) K d )( ŝ h + δ I), (4) where ( τh) R + and (δ h) R + by aumption. By (), Ĉ pid (ŝ) in (4) tabilize Ĝ(ŝ) if and only if M δ(ŝ) i S -unimodular: M δ (ŝ) = Ŷ (ŝ) (ŝ h + δ) I + ˆX(ŝ) (ŝ h + δ)ĉpid(ŝ) = Ŷ (ŝ) (ŝ h + δ) I + (ŝ h + a) I (ŝ h + δ)ĉpid(ŝ) δ = [ ŝ h + δ I + [Ŷ (ŝ)(ŝ h + a) (ŝ h + δ)(ŝ h + α) (τ + ) K d]y ( ) (ŝ h + α) ] (ŝ h + a) Y ( ) (ŝ h + a) = [I + [(Ŷ (ŝ)y ( ) (ŝ h + δ) (ŝ h + α) I) + (τ + )(ŝ h + α) K dy ( ) (ŝ h + α) ]] (ŝ h + a) Y ( ) = [I + h + α) Ψ(ŝ)](ŝ Y ( ). (5) (ŝ h + δ) (ŝ h + a) Then M δ (ŝ) in (5) i S -unimodular for δ R + atifying () ince δ > Ψ(ŝ) + h implie (ŝ h + δ) Ψ(ŝ) (ŝ h + δ) Ψ(ŝ) = δ h Ψ(ŝ) <. Therefore, Ĉ pid (ŝ) an S -tabilizing controller for Ĝ(ŝ); hence, C pid i an S h -tabilizing controller for G. In Example, we conider the two-input two-output linearized proce model of an untable batch reactor (alo conidered in e.g., [8], [9]): Example : Conider the linearized model of an untable batch reactor a ẋ = Ax + Bu, y = Cx + Du, where.38.77 6.75 5.676 A =.584 4.9.675.67 4.73 6.654 5.893,.48 4.73.343.4 [ ] B = 5.679.36 3.46, C =, D =..36 The tranfer-function G i given by: G = [ ] g + h g + h, d g g where d = 4 +.668 3 + 5.7538 88.9 + 5.546, g =.8 + 9.56 + 33.6673, g = (.54 +.94 + 6.766), g = 5.679 3 + 4.6665 68.834 6.84, g = 9.434 + 5.53, h =.8 + 9.7745, h = 3.46 3.549 +.688 5.579. The pole of G are at {.99,.635, 5.566, 8.6659}. The finite tranmiion-zero are at {.96, 5.36, 5.598} and G ha a blocking-zero at infinity. Therefore, we can deign PID controller uch that the cloed-loop pole have realpart le than h for any h <.96 following the procedure in Propoition. Suppoe h =. Since the tranmiion 85
zero 5.598 and the pole 5.566 are cloe to each other, G () can be approximated by a one order lower tranfer matrix. By ubtituting = ŝ h, Ĝ (ŝ) = G become Ĝ (ŝ) = [ ] G G, G G.5.5 In, Out.5.5 In, Out where G = 9.434ŝ + 5.799, G = 3.46ŝ 3 + 3.364ŝ + 33.946ŝ 8.495, G = 5.679ŝ 3 5.695ŝ + 37.64ŝ +.9845, G =.8ŝ + 9.48ŝ+34.6, = 7.866ŝ +75.534ŝ+3.86. For implicity, we chooe K d = and deign a PIcontroller; the term K d can be varied depending on other deign pecification that are to be atified. Chooing [ α = > h] atifying α > h, with Y ( ) = 3.46, we compute Ψ(ŝ) = 9.9385. Therefore, () i atified for δ >.9385. We chooe δ = 5.679 and obtain the PI-controller from () a C pi = δ Y ( )+ αδ Y ( ) = (+ [ ] ).3. 3.844 The correponding cloed-loop pole with thi controller are at { 8.77 ± j7.9389, 5.564 ± j7.8895, 7.698, 5.565,.7}, and all have negative real-part le than h =. Due to the integralaction in the deigned controller, the contant reference input applied at r are tracked aymptotically at the output y with zero teady-tate error. Other value can be choen for the variou parameter if additional performance pecification are to be atified in addition to cloed-loop tability with ufficient damping and aymptotic tracking of contant reference input. Example : The ytematic deign procedure in Propoition ha ome free parameter that may be poible to chooe in order to fulfill additional performance criteria. In thi example, we will how a preliminary reult regarding uing thee free parameter along the ame line a in []. Conider the ame ytem a in Example. In Fig., the tep repone of the cloed loop ytem i hown by the dotted line. Suppoe that we want to get le overhoot with a lower riing time. We can chooe a model ytem T m () uch that T m () = T m () = and T m () = T m () equal the ame prototype econd order model plant, with ζ =.7 and ω n =.56; i.e., T m = + ζω + ωn. The tep repone for T m () i hown in Fig. a the dahed line. To make a comparion, we ue the ame h = a in Example. To maintain the PID controller tructure, we chooe a mall τ =.5. Denote the tep repone of the model plant T m a m (t). The goal i to make the actual cloed-loop tep repone a cloe a poible to m (t). That i, we conider the cot function error =.5 i= j=.5 ω n ( oij (t) mij (t)) dt, (6).5.5.5.5.5.5 In, Out.5.5.5.5.5.5.5.5.5.5 In, Out.5.5.5.5 Fig.. Step repone for three tranfer matrice in Example where o (t) denote the tep repone for a choice of (α, δ, K d ). The goal i to minimize error by chooing the bet (α, δ, K d ), ubject the imple linear contraint (α > h and the complex nonlinear contraint (). The MATLAB function fmincon i ued to olve the problem by uing the controller in Example a the initial point. The optimal controller ha the following coefficient in (), and how the derivative part doe help to improve the performance: [ ].8.69 α =.3368, δ =.685, K d =..848.597 The tep repone of the optimal deign correpond to the given model plant i hown in Fig. by the olid line. We can ee that it i cloer to the given model plant than the original deign. C. Strictly-proper plant with two zero at infinity Conider the cla G of m m trictly-proper plant that have no other (tranmiion or blocking) zero in U h a decribed in (). The plant G G are not all S h -table but (+a) G M(S h ) for any a > h. An LCF of G() i G = Y X = ( ( + a) G ) ( ( + a) I ) ; (7) in (7), G( ) =, and Y ( ) = ( + a) G() = G(). The plant in G are trongly tabilizable, and they admit S -tabilizing PID controller [6]. Propoition 3 how that thee plant alo admit S h -tabilizing PID controller for any pre-pecified h R +, and propoe a ytematic PID controller ynthei procedure. Propoition 3: (PID for plant with two zero at infinity): Let G G. Then there exit an S h -tabilizing PID controller, and C pid can be deigned a follow: Let Y ( ) := G(). Chooe any α R + atifying α > h. Define Γ(ŝ) a Γ(ŝ) := (ŝ h + α) Ĝ (ŝ)y ( ) ŝi. (8) 853
Let µ R + atify µ > Γ(ŝ) + h. (9) Then an S h -tabilizing PID controller C pid i given by C pid = (µ h) ( + α) ( + µ h) Y ( ). (3) Proof of Propoition 3: Subtitute ŝ = + h a in (3)-(5). Then an LCF of Ĝ(ŝ) i Ĝ(ŝ) = Ŷ ˆX := ( (ŝ h+a) Ĝ (ŝ)) ( I). Write the controller (ŝ h+a) C pid () given in (3) a ( + µ h) ( + µ h) C pid () = ( ( + µ) C pid )( ( + µ) I) ( + α) + µ h) = ((µ h) Y ( ))(( ( + µ) ( + µ) I). (3) Subtitute ŝ = + h into (3) to obtain an RCF of Ĉpid(ŝ) a Ĉ pid (ŝ) = ( (µ h) (ŝ h + α) (ŝ + µ h) (ŝ h + µ) Y ( ))( (ŝ h + µ) I), (3) where (µ h) R + by (9). By (), Ĉ pid (ŝ) in (3) tabilize Ĝ(ŝ) if and only if M µ(ŝ) i S -unimodular: (ŝ + µ h) M µ (ŝ) = (Ŷ (ŝ) + ˆX(ŝ)Ĉpid(ŝ)) (ŝ h + µ) (ŝ + µ h) = Ŷ (ŝ) (ŝ h + µ) I + (ŝ h + a) I (µ h) (ŝ h + α) (ŝ h + µ) Y ( ) (ŝ h + α) = (ŝ h + a) [ (µ h) (ŝ h + µ) I + (ŝ + (µ h)) (ŝ h + a) Ŷ (ŝ) (ŝ h + µ) (ŝ h + α) Y ( ) ]Y ( ) (ŝ h + α) (ŝ + (µ h)) = [I + (ŝ h + a) (ŝ h + µ) Γ(ŝ) ]Y ( ). (33) Then M µ (ŝ) in (33) i S -unimodular for µ R + atifying (9) ince µ > Γ(ŝ) + h implie (ŝ + (µ h)) (ŝ + (µ h)) (ŝ h + µ) Γ(ŝ) (ŝ h + µ) Γ(ŝ) = Γ(ŝ) <. µ h Therefore, Ĉ pid (ŝ) i an S -tabilizing controller for Ĝ(ŝ); hence, C pid i an S h -tabilizing controller for G. Example 3: Conider the MIMO ytem [ G = (+3) (+)( 4)( 8) (+5) (+3)(+6)(+7) (+5)(+) (+4) (+9)( 6+) ], (34) which ha no finite zero with real-part larger than.364. Thu we can chooe h =. By chooing α =, we can compute Γ(ŝ) = 79.849 from (8). Chooe µ = 6 to atify the inequality in (9). The maximum of the realpart of the cloed-loop pole i le than.39. Thu the requirement i fulfilled. III. CONCLUSIONS Sytematic PID controller deign were propoed for LTI, MIMO plant, where cloed-loop pole are placed in the left-half complex-plane to the left of the plant zero with the larget negative real-part. The plant under conideration may be table or untable; there are no retriction on the plant pole and no retriction on the zero in the region of tability with real-part le than h. However, the zero in the untable region are retricted. We howed that for plant that have no zero in the untable region and either one or two zero at infinity (a decribed in (9) and (7), repectively) it i poible to deign PID controller uch that the cloed-loop pole have negative real-part le than any precribed h. The ynthei method focue on the objective of hifting the real-part of the cloed-loop pole away from the origin, which enure tability margin; aignment of the imaginary-part i not within the deign objective and cope. Due to uing PID controller with non-zero integral-term, the cloed-loop ytem are guaranteed to achieve aymptotic tracking of contant reference input with zero teady-tate error. The propoed ynthei method allow freedom in the choice of parameter, which may be ued to atify additional performance pecification. REFERENCES [] K. J. Atröm, and T. Hagglund, PID Controller: Theory, Deign, and Tuning, Second Edition. Reearch Triangle Park, NC: Intrument Society of America, 995. [] T. S. Chang, Performance improvement for PID controller deign with guaranteed tability margin for certain clae of MIMO ytem, Proc. World Congre on Engineering and Computer Science, WCECS 8. [3] T. S. Chang, A. N. Gündeş, PID controller ynthei with pecified tability requirement for ome clae of MIMO ytem, Engineering Letter, vol. 6, no., pp. 56-65, 8. [4] G. C. Goodwin, S. F. Graebe, and M. E. Salgado, Control Sytem Deign, Prentice Hall, New Jerey,. [5] A. N. Gündeş, and C. A. Deoer, Algebraic Theory of Linear Feedback Sytem with Full and Decentralized Compenator, Lect. Note in Contr. and Inform. Science, 4, Springer, 99. [6] A. N. Gündeş, A. B. Özgüler, PID tabilization of MIMO plant, IEEE Tran. Automatic Control, 5:5-58, 7. [7] M.-T. Ho, A. Datta, and S. P. Bhattacharyya, An extenion of the generalized Hermite-Biehler theorem: relaxation of earlier aumption, Proc. 998 American Contr. Conf., 36 39, 998. [8] N. Munro, Deign of controller for open-loop untable multivariable ytem uing invere Nyquit array, Proc. IEE, vol. 9, no. 9, pp. 377-38, 97. [9] H. H. Roenbrock, Computer-Aided Control Sytem Deign, London; New York: Academic Pre, 974 [] G. J. Silva, A. Datta, and S. P. Bhattacharyya, PID Controller for Time-Delay Sytem, Birkhäuer, Boton, 5. [] M. Vidyaagar, Control Sytem Synthei: A Factorization Approach. MIT Pre, 985. 854