Finite-Dimensional Control of Parabolic PDE Systems Using Approximate Inertial Manifolds

Transcription

1 JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 16, ARTICLE NO AY Finite-Dimenional Control o Parabolic PDE Sytem Uing Approximate Inertial Maniold Panagioti D Chritoide and Prodromo Daoutidi Department o Chemical Engineering and Material Science, Unierity o Minneota, Minneapoli, Minneota Submitted by Mar J Bala Received September 16, 1996 Thi paper introduce a methodology or the ynthei o nonlinear inite-dimenional output eedbac controller or ytem o quai-linear parabolic partial dierential equation Ž PDE, or which the eigenpectrum o the patial dierential operator can be partitioned into a inite-dimenional low one and an ininitedimenional table at complement Combination o Galerin method with a novel procedure or the contruction o approximate inertial maniold or the PDE ytem i employed or the derivation o ordinary dierential equation Ž ODE ytem Ž whoe dimenion i equal to the number o low mode that yield olution which are cloe, up to a deired accuracy, to the one o the PDE ytem, or almot all time Thee ODE ytem are ued a the bai or the ynthei o nonlinear output eedbac controller that guarantee tability and enorce the output o the cloed-loop ytem to ollow up to a deired accuracy, a prepeciied repone or almot all time 1997 Academic Pre 1 INTRODUCTION Parabolic PDE ytem arie naturally a model o diuion-convection-reaction procee 16 and typically involve patial dierential operator whoe eigenpectrum can be partitioned into a inite-dimenional low one and an ininite-dimenional table at complement 11, 1 Thi implie that the dynamic behavior o uch ytem can be approximately decribed by inite-dimenional ytem Motivated by thi, the tandard approach to the control o parabolic PDE involve the application o Galerin method to the PDE ytem to derive ODE ytem that decribe the dynamic o the dominant Ž low mode o the PDE ytem, which are ubequently ued a the bai or the ynthei o inite-dimen X97 $500 Copyright 1997 by Academic Pre All right o reproduction in any orm reerved 398

2 CONTROL OF PARABOLIC PDE SYSTEMS 399 ional controller 1, 16 However, there are two ey controller implementation and cloed-loop perormance problem aociated with thi approach Firt, the number o mode that hould be retained to derive an ODE ytem that yield the deired degree o approximation may be very large, leading to high dimenionality o the reulting controller 1 Second, there i a lac o a ytematic way to characterize the dicrepancy between the olution o the PDE ytem and the approximate ODE ytem in inite time, which i eential or characterizing the tranient perormance o the cloed-loop PDE ytem A natural ramewor to addre the problem o deriving low-dimenional ODE ytem that accurately reproduce the olution o diipative PDE i baed on the concept o inertial maniold Ž IM Žee 18 and the reerence therein An IM i a poitively invariant, inite-dimenional Lipchitz maniold, which attract every trajectory exponentially I an IM exit, the dynamic o the parabolic PDE ytem retricted on the inertial maniold i decribed by a et o ODE called the inertial orm Hence, tability and biurcation tudie o the ininite-dimenional PDE ytem can be readily perormed on the bai o the inite-dimenional inertial orm 18 However, the explicit derivation o the inertial orm require the computation o the analytic orm o the IM Unortunately, IM have been proven to exit only or certain clae o PDE Žor example, the KuramotoSivahiny equation and ome diuion-reaction equation 18, and even then it i almot impoible to derive their analytic orm In order to overcome the problem aociated with the exitence and contruction o IM, the concept o approximate inertial maniold Ž AIM ha been introduced 9, 10 and ued or the derivation o ODE ytem whoe dynamic behavior approximate the one o the inertial orm In the area o control o nonlinear parabolic PDE ytem, ew paper have appeared in the literature dealing with the application o IM or the ynthei o inite-dimenional controller In particular, in 17 the problem o tabilization o a parabolic PDE with boundary inite-dimenional eedbac wa tudied; a tandard oberver-baed controller augmented with a reidual mode ilter 3 wa ued to induce an inertial maniold in the cloed-loop ytem, and thu reduce the tabilization problem or the PDE ytem to a tabilization problem or the inite-dimenional inertial orm In 5, the theory o inertial maniold wa utilized to determine the extent to which linear boundary proportional control inluence the dynamic and teady-tate repone o the cloed-loop ytem In thi paper, we introduce a methodology or the ynthei o nonlinear inite-dimenional output eedbac controller or ytem o quai-linear parabolic PDE Singular perturbation method are initially employed to etablih that the dicrepancy between the olution o an ODE ytem o dimenion equal to the number o low mode, obtained through Galerin

3 400 CHRISTOFIDES AND DAOUTIDIS method, and the PDE ytem i proportional to the degree o eparation o the at and low mode o the patial operator Then, a procedure, motivated by the theory o ingular perturbation, i propoed or the contruction o AIM or the PDE ytem The AIM are ued or the derivation o ODE ytem o dimenion equal to the number o low mode, that yield olution which are cloe, up to a deired accuracy, to the one o the PDE ytem, or almot all time Thee ODE ytem are ued a the bai or the ynthei o nonlinear output eedbac controller that guarantee tability and enorce the output o the cloed-loop ytem to ollow up to a deired accuracy, a prepeciied repone or almot all time PRELIMINARIES We conider quai-linear parabolic PDE ytem o the orm x x x A B wbž z u Ž x t z z H i y z i1 i c Ž z x dz, i 1,,l z i Ž 1 ubject to the boundary condition x C xž,t D Ž,t R z x C xž,t D Ž,t R z and the initial condition xž z,0 x0ž z, Ž 3 where xž z,t x 1 z,t xn z,t T denote the vector o tate variable,, i the domain o deinition o the proce, z, i the 1 patial coordinate, t 0, i the time, u u u u l T l denote the vector o manipulated input, and y i denote a controlled output x z, x z denote the irt- and econd-order patial derivative o x, Ž x i a vector unction, w, are contant vector, A, B, C 1, D 1, C, D are contant matrice, R 1, R are column vector, and x Ž z i the initial condition bž z 0 i a nown mooth vector unction o z o the orm bž z b 1 Ž z b Ž z b l Ž z, where b i Ž z decribe how the

4 CONTROL OF PARABOLIC PDE SYSTEMS 401 control action u i Ž t i ditributed in the patial interval z, z, i i1, and c i Ž z i a nown mooth unction o z which i determined by the deired perormance peciication in the interval z, z i i1 Whenever the control action enter the ytem at a ingle point z, with z z, z 0 0 i i1 Ž ie, point actuation, the unction b i Ž z i taen to be nonzero in a inite patial interval o the orm z, z 0 0, where i a mall poitive real number, and zero elewhere in z, z i i1 Throughout the paper, we will ue the order o magnitude notation OŽ In particular, O i there exit poitive real number and uch that 1 1, We ormulate the ytem o Eq ŽŽ 1 3 in a Hilbert pace H Ž,, n, with H being the pace o n-dimenional vector unction deined on, that atiy the boundary condition o Eq Ž, with inner product and norm H Ž, Ž z, Ž z n dz Ž,, where Ž n 1, are two element o H, ; and the notation, n denote the tandard inner product in n Deining the tate unction x on H Ž,, n a xž t xž z,t, t0, z,, Ž 5 Ž n the operator A in H,, a x x Ax A B, z z x n xdž A ½x H Ž, ; ; C1x D1 R 1, Ž 6 z x Cx D R z 5 and the input and output operator a Bu wbu, C x Ž c, x, Ž 7 1 where c cc c l, the ytem o Eq ŽŽ 1 3 tae the orm x Ax Bu Ž x y Cx Ž 8 xž 0 x 0,

5 40 CHRISTOFIDES AND DAOUTIDIS where ŽxŽ t ŽxŽ z,t and x x Ž z 0 0 We aume that the nonlinear term Ž x atiie Ž 0 0 and i alo locally Lipchitz continuou, ie, there exit poitive real number a 0, K0 uch that or any x 1, x H that atiy maxx, x 4 a, we have that 1 0 Ž x Ž x K x x Ž For A, the tandard eigenvalue problem i o the orm A j, j j j1,,, Ž 10 where j denote an eigenvalue and j denote an eigenunction; the eigenpectrum o A, Ž A, i deined a the et o all eigenvalue o A, ie, Ž A,,, 4 1 Aumption 1 that ollow tate our hypothee or the propertie o Ž A Aumption 1 Ž 1 Re 1 Re Re j, where Re j denote the real part o j Ž A can be partitioned a Ž A Ž A Ž A, where Ž A 1 1 conit o the irt m Ž with m inite eigenvalue, ie, Ž A,, m, and Re Re OŽ 1 m 1 Ž 3 Re 0 and Re Re 0Ž m1 m m1 where Re Re 1 i a mall poitive parameter 1 m1 Aumption 1 tate that the eigenpectrum o A can be partitioned into a inite-dimenional part coniting o m low, poibly untable, eigenvalue, Ž A,, m and a table ininite-dimenional part containing the remaining at eigenvalue, Ž A,, 4 m1, and that the eparation between low and at eigenvalue o A i large The aumption o the inite number o untable eigenvalue i alway atiied or parabolic PDE 11, while the exitence o only a ew dominant mode that capture the dominant dynamic o a parabolic PDE ytem i well-etablihed or the majority o diuion-convection-reaction procee Žee, or example, the application in the boo 16, and the paced-bed reactor example tudied in 8 Aumption 1 guarantee 11 that A generate a trongly continuou emigroup o bounded linear operator Ut Ž which implie that the generalized olution o the ytem o Eq Ž 8 i given by Ž Ut H t Ž x UŽ t x UŽ t Bu x d Ž 11 atiie the growth property 0 0 at U t Ke 1 1, t0, 1

6 CONTROL OF PARABOLIC PDE SYSTEMS 403 where K 1, a1 are real number, with K1 1 and a1 Re 1 I a1 i trictly negative, we will ay that A generate an exponentially table emigroup Ut Ž Throughout the manucript, we will ocu on local exponential tability, and not on wea Ž aymptotic tability 11, becaue o it robutne to bounded perturbation Žeg, uncertain variable, diturbance, which are alway preent in mot practical application 3 We will now review the application o Galerin method to the ytem o Eq Ž 8 to derive an approximate inite-dimenional ytem Let H, H be modal ubpace o A, deined a H pan,,, 4 1 m and H pan,,, 4 Ž m 1 m the exitence o H, H ollow rom Aump- tion 1 Deining the orthogonal projection operator P and P uch that x Px, x Px, the tate x o the ytem o Eq Ž 8 can be decompoed a x x x PxPx Ž 13 Applying P and P to the ytem o Eq Ž 8 and uing the above decompoition or x, the ytem o Eq Ž 8 can be equivalently written in the orm dx A x B u Ž x, x x A x Bu Ž x, x t Ž 14 y Cx Cx xž 0 Px Ž 0 Px 0, xž 0 Px0 Px, 0 where A P AP, B P B, P, AP AP, BP B, and Pand the notation x t i ued to denote that the tate x belong in an ininite-dimenional pace In the above ytem, A i a diagonal matrix o dimenion m m o the orm A diag 4, Ž x, x and Ž x, x j are Lipchitz vector unction, and A i an unbounded dierential opera- tor which generate a trongly continuou exponentially table emigroup Žollowing rom part Ž 3 o Aumption 1 and the election o H, H Neglecting the at mode, the ollowing inite-dimenional ytem i derived, dx A x B u Ž x,0 Ž 15 y Cx, where the ubcript in y denote that the output i aociated with the low ytem The above ytem can be directly ued or controller deign employing tandard control method or ODE 1, 16, 7

7 404 CHRISTOFIDES AND DAOUTIDIS Remar 1 We note that the large eparation o low and at mode o the patial operator in parabolic PDE enure that a controller which exponentially tabilize the cloed-loop ODE ytem, alo tabilize the cloed-loop ininite-dimenional ytem 1 Thi i in contrat to the application o thi approach to hyperbolic PDE where the eigenmode cluter along nearly vertical aymptote in the complex plane and thu, the controller ha to be modiied to compenate or the detabilizing eect o the reidual mode 3 3 ACCURACY OF ODE SYSTEM OBTAINED FROM GALERKIN S METHOD In thi ection, we ue ingular perturbation method to etablih that i the inite-dimenional ytem o Eq Ž 15 i exponentially table, then the ytem o Eq Ž 14 i alo exponentially table and the dicrepancy between the olution o the x -ubytem o the ytem o Eq Ž 14 and the olution o the ytem o Eq Ž 15 i proportional to the pectral eparation o the low and at eigenvalue Uing that Re Re, the ytem o Eq Ž 14 1 m1 can be written in the orm dx A x B u Ž x, x Ž 16 x A x BuŽ x, x, t where A i an unbounded dierential operator deined a A A Since i a mall poitive number le than unity Žollowing rom Aumption 1, part Ž 3 and the operator A, A generate emigroup with growth rate which are o the ame order o magnitude, the ytem o Eq Ž 16 i in the tandard ingularly perturbed orm, with x being the low tate and x being the at tate Introducing the at time-cale t and etting 0, we obtain the ollowing ininite-dimenional at ubytem rom the ytem o Eq Ž 16 : x A x Ž 17 From the act that Re m 1 0 and the deinition o, we have that the above ytem i globally exponentially table Setting 0 in the ytem o Eq 16 and uing the act that the invere operator A 1 exit and i alo bounded Ž it ollow rom the act that zero i in the reolvent o A, we have that x 0 Ž 18

8 CONTROL OF PARABOLIC PDE SYSTEMS 405 and thu the inite-dimenional low ytem tae the orm dx A x B u Ž x,0 Ž 19 We note that the above ytem i identical to the one obtained by applying the tandard Galerin method to the ytem o Eq Ž 8, eeping the irt m ODE and completely neglecting the x -ubytem Aumption that ollow tate a tability requirement on the ytem o Eq Ž 19 Aumption The inite-dimenional ytem o Eq Ž 19 with ut 0 i exponentially table, ie, there exit poitive real number Ž K, a, a 4, with a K 1 uch that or all x H that atiy x 4 a 4, the ollowing bound hold: a t x t Ke x 0, t0 0 Propoition 1 that ollow etablihe that the olution o the open-loop ytem o Eq Ž 19 Ž 17, ater a hort inite time interval required or the trajectorie o the ytem o Eq Ž 16 to approach the quai teady-tate o Eq Ž 18, conit o an OŽ approximation o the olution o the open-loop ytem o Eq Ž 16 The proo i given in the Appendix PROPOSITION 1 Conider the ytem o Eq Ž 16 with už t 0 and uppoe that Aumption 1 and hold Then, there exit poitie real number,, * uch that i x Ž 0, x Ž 0, and Ž0, * 1 1, then the olution x Ž t, x Ž t o the ytem o Eq Ž 16 atiie or all t 0, x Ž t x Ž t O / t xž t xž OŽ, Ž 1 where x Ž t, x Ž t are the olution o the low and at ubytem o Eq Ž 19 Ž 17 with už t 0, repectiely Remar The counterpart o the reult o Propoition 1 in inite-dimenional pace i well nown ŽTihonov theorem 19, while a imilar reult ha alo been etablihed or linear ininite-dimenional ytem The main technical dierence in etablihing thi reult between linear and quai-linear ininite-dimenional ytem i that, or quai-linear ytem the proo i baed on Lyapunov argument, while or linear ytem the proo i obtained uing combination o etimate o the tate, obtained rom the application o variation o contant ormula Thi i a

9 406 CHRISTOFIDES AND DAOUTIDIS conequence o the act that or quai-linear ytem it i not poible to derive a coordinate change that tranorm the ytem o Eq Ž 16 into a cacaded interconnection where the at mode are decoupled rom the low mode, which allow u to derive an exponentially decaying etimate, or uiciently mall, or the at tate, which i independent o the one o the low tate, and thu to prove the reult through a direct combination o thee etimate Remar 3 We note that it i poible, uing tandard reult rom center maniold theory or ininite-dimenional ytem o the orm o Eq Ž 8 6, to how that i the ytem o Eq Ž 19 i aymptotically table, then the ytem o Eq Ž 8 i alo aymptotically table and the dicrepancy between the olution o the ytem o Eq Ž 19 and the x -ubytem o the ytem o Eq 16 i aymptotically Ž a t proportional to Although thi reult i important becaue it allow etablihing aymptotic tability o the cloed-loop ininite-dimenional ytem by perorming a tability analyi on a low-order inite-dimenional ytem, it doe not provide any inormation about the dicrepancy between the olution o thee two ytem or inite t 4 CONSTRUCTION OF ODE SYSTEMS OF DESIRED ACCURACY VIA AIM In thi ection, we propoe an approach originating rom the theory o inertial maniold or the contruction o ODE ytem o dimenion m which yield olution that are arbitrarily cloe Žcloer than OŽ to the one o the ininite-dimenional ytem o Eq Ž 8, or almot all time An inertial maniold M or the ytem o Eq Ž 8 i a ubet o H, which atiie the ollowing propertie 18 : Ž i M i a inite-dimenional Lipchitz maniold; Ž ii M i a graph o a Lipchitz unction Ž x, u, l mapping H Ž0, * into H and or every olution x Ž t, x Ž t o Eq Ž 16 with x Ž 0 Žx Ž 0,u,, then x Ž t x Ž t,u,, t0; and Ž iii M attract every trajectory exponentially The evolution o the tate x on M i given by Eq Ž, while the evolution o the tate x i governed by the inite-dimenional inertial orm dx A x B u Ž x, Ž x,u, Ž 3

10 CONTROL OF PARABOLIC PDE SYSTEMS 407 Auming that ut Ž i mooth, dierentiating Eq Ž, and utilizing Eq Ž 16, Ž x, u, can be computed a the olution o the partial dierential equation Ax Bu Ž x, x u A x BuŽ x, x x u Ž 4 l which ha to atiy or all x H, u, Ž0, * However, even or parabolic PDE or which it i nown that M exit, the derivation o an explicit analytic orm o Ž x, u, i an extremely diicult Ž i not impoible ta Motivated by thi, we will now propoe a procedure, motivated by ingular perturbation 15, to compute approximation o Ž x, u, Ž approximate inertial maniold and approximation o the inertial orm, o deired accuracy To thi end, conider an expanion o Ž x, u, and u in a power erie in, u u0 u1 u u OŽ 1 Ž x,u, 0 Ž x,u 1 Ž x,u Ž x,u Ž 5 Ž x,u OŽ 1, where u, are mooth unction Subtituting the expreion o Eq Ž 5 into Eq Ž 4, and equating term o the ame power in, one can obtain approximation o Ž x, u, up to a deired order Subtituting the expanion or Ž x, u, and u up to order into Eq Ž 3, the ollowing approximation o the inertial orm i obtained: dx Ax B u0 u1 u u Ž x, 0 Ž x,u 1 Ž x,u Ž x,u Ž x,u Ž 6 In order to characterize the dicrepancy between the olution o the open-loop inite-dimenional ytem o Eq Ž 6 and the olution o the x -ubytem o the open-loop ininite-dimenional ytem o Eq Ž 16, we will impoe a tability requirement on the ytem o Eq Ž 6 Aumption 3 The inite-dimenional ytem o Eq Ž 6 with ut 0 i exponentially table, ie, there exit poitive real number Ž K, a, a 4, with a K 1 uch that or all x H that atiy x 4 a 4, the ollowing bound hold: a t x t Ke x 0, t0 7

11 408 CHRISTOFIDES AND DAOUTIDIS Propoition that ollow etablihe that the dicrepancy between the olution obtained rom the open-loop ytem o Eq Ž 6 and the expanion or Ž x, u, o Eq Ž 9, and the olution o the ininite-dimen- Ž 1 ional open-loop ytem o Eq 16 i o O, or almot all time The proo i given in the Appendix PROPOSITION Conider the ytem o Eq Ž 16 with už t 0 and uppoe that Aumption 1 and 3 hold Then, there exit poitie real number,, * uch that i x Ž 0, x Ž 0, and Ž0, * 1 1, then the olution x Ž t, x Ž t o the ytem o Eq Ž 16 atiie or all t t, b xž t x Ž t OŽ 1 Ž 8 1 x Ž t x Ž t OŽ, where t i the time required or x Ž t to approach x Ž t, x Ž b t i the olution 1 o Eq 6 with u t 0, and x t x,0 Ž x,0 Ž x,0 Remar 4 The reult o Propoition provide the mean or characterizing the dicrepancy between the olution o the open-loop ininitedimenional ytem o Eq Ž 8, xt ŽŽand thu the olution o the parabolic PDE ytem o Eq Ž 1 with ut Ž 0, and the olution xt Ž x Ž t x Ž t Ž 1 Ž x t x t,0 Žx Ž t, 0, which i obtained rom the OŽ approximation o the open-loop inertial orm Žie, Eq Ž 6 with ut Ž 0 In particular, ubtituting Eq Ž 8 into the equation xt Ž x Ž t Ž Ž Ž Ž Ž 1 x t, we have that xt x t x t O or t t b Utilizing the deinition o order o magnitude, we inally obtain the ollowing characterization or the dicrepancy between xt Ž and xt: Žxt Ž xt Ž 1 1 or t t b, where 1 i a poitive real number Remar 5 Following the propoed approximation procedure, it can be hown that the OŽ approximation o Ž x,0, i 0 Ž x,0 0 and the correponding approximate inertial orm i identical to the ytem o Eq Ž 19 Ž obtained via Galerin method with ut 0 Thi ytem doe not utilize any inormation about the tructure o the at ubytem, thu yielding olution which are only OŽ cloe to the olution o the open-loop ytem o Eq ŽŽ 8 Propoition 1 On the other hand, the OŽ approximation o Ž x,0, can be hown to be o the orm Ž x,0, Ž x,0 Ž x,0 A Ž x,0 Ž 9 The correponding open-loop approximate inertial orm doe utilize inormation about the tructure o the at ubytem, and thu allow u to obtain olution which are OŽ cloe to the olution o the open-loop ytem o Eq ŽŽ 8 Propoition

12 CONTROL OF PARABOLIC PDE SYSTEMS 409 Remar 6 The tandard approach ollowed in the literature or the contruction o AIM or ytem o the orm o Eq Ž 14 with ut 0 Žee, or example, 4 i to directly et x t 0, olve the reulting algebraic equation or x, and ubtitute the olution or x to the x -ubytem o Eq Ž 14, to derive the ODE ytem dx 1 Ax ž x, Ž A Ž x / Ž 30 It i traightorward to how that the low ytem o Eq Ž 30 i identical to the one obtained by uing the OŽ approximation or Ž x,0, or the contruction o the approximate inertial orm Remar 7 The expanion o u in a power erie in i motivated by our intention to modiy the ynthei o the eedbac controller appropri- Ž 1 ately uch that the dicrepancy between the output o the O approximation o the cloed-loop inertial orm and the output o the Ž 1 cloed-loop PDE ytem will be o O or almot all time Žee alo Remar 8 5 FINITE-DIMENSIONAL CONTROL In thi ection, we ue the reult o Propoition to etablih that a nonlinear inite-dimenional output eedbac controller, that guarantee tability and enorce output tracing in the ODE ytem o Eq Ž 6, exponentially tabilize the cloed-loop PDE ytem and enure that the dicrepancy between the output o the cloed-loop ODE ytem and the Ž 1 output o the cloed-loop PDE ytem i o O, provided that i uiciently mall The inite-dimenional output eedbac controller which achieve the deired objective or the ytem o Eq Ž 6 i contructed through a tandard combination o a tate eedbac controller with a tate oberver In particular, we conider a tate eedbac control law o the general orm u u u u 0 1 p0ž x Q0Ž x p1ž x Q1Ž x Ž 31 p x Q x, where p Ž x,, p Ž x are mooth vector unction, Q Ž x,,q Ž x 0 0 are mooth matrice, and l i the contant reerence input vector Žee Remar 8 or a procedure or the ynthei o the control law, ie, the explicit computation o p Ž x,, p Ž x,q Ž x,,q Ž x The ol- 0 0

13 410 CHRISTOFIDES AND DAOUTIDIS lowing m-dimenional tate oberver i alo conidered or the implemen- tation o the tate eedbac law o Eq 31, d A B Ž p 0Ž Q 0Ž p 1Ž Q 1Ž p Q, 1,u,u 3 Ž, u 1 Ž Ž,u 4, L y C C,u,u where H denote the oberver tate vector and L i a matrix choen o that the eigenvalue o the matrix C A Ž L L C Ž 1 Ž Ž C, u, u Ž, u 4 lie in the open let-hal o the complex plane, where denote the teady tate or the ytem o Eq Ž 3 The inite-dimenional output eedbac controller reulting rom the combination o the tate eedbac controller o Eq Ž 31 with the tate oberver o Eq Ž 3 tae the orm d A B Ž p 0Ž Q 0Ž p 1Ž Q 1Ž p Q Ž, 1 Ž,u Ž,u Ž, u 0 1 L y C C Ž,u Ž,u Ž,u Ž up0 Q0 p1 Q1 p Q, u 4 Ž 33 We note that the tatic component o the above controller doe not ue eedbac o the at tate vector x in order to avoid detabilization o the at mode o the cloed-loop ytem Aumption 4 tate the deired control objective under the controller o Eq Ž 33 Aumption 4 The inite-dimenional output eedbac controller o the Ž 1 orm o Eq 33 exponentially tabilize the O approximation o the cloed-loop inertial orm and enure that it output y i Ž t, i1,,l, are the olution o a nown l-dimenional ODE ytem o the orm Ž i Ž r i i Ž r i 1 i y, y,, y, 0, where i a vector unction and ri i an integer

14 CONTROL OF PARABOLIC PDE SYSTEMS 411 Theorem 1 provide a precie characterization o the tability and cloed-loop tranient perormance enorced by the controller o Eq Ž 33 in the cloed-loop PDE ytem Ž the proo i given in the Appendix THEOREM 1 Conider the PDE ytem o Eq Ž 8, or which Aumption 1 and 4 hold Then, there exit poitie real number 1,, * uch that i x Ž 0, x Ž 0, and Ž0, *, then the controller o Eq Ž 33 : 1 Ž a guarantee exponential tability o the cloed-loop ytem, and Ž b enure that the output o the cloed-loop ytem atiy or all t t, b y i Ž t y i Ž t OŽ 1, i 1,,l, Ž 34 i Ž Ž 1 where y t i the ith output o the O approximation o the cloed-loop inertial orm Remar 8 The contruction o the tate eedbac law o Eq Ž 31, to enure that the control objective tated in Aumption 4 are enorced in Ž 1 the O approximation o the cloed-loop inertial orm, can be perormed ollowing a equential procedure Speciically, the component u p Ž x Q Ž x can be initially yntheized on the bai o the OŽ approximation o the inertial orm ŽEq Ž 19 ; then the component u 1 p Ž x Q Ž x can be yntheized on the bai o the OŽ 1 1 approximation o the inertial orm In general, at the th tep, the component u p Ž x Q Ž x can be yntheized on the bai o the OŽ approximation o the inertial orm Ž Eq 6 The ynthei o p Ž x,q Ž x, 0,,, can be perormed, at each tep, utilizing tandard geometric control method or nonlinear ODE Žee 13, or example Remar 9 The implementation o the controller o Eq Ž 33 require to explicitly compute the vector unction Ž, u However, Ž, u ha an ininite-dimenional range and thereore cannot be implemented in prac tice Intead a inite-dimenional approximation o Ž, u, ay Ž, u t, can be derived by eeping the irt m element o Ž, u and neglecting the remaining ininite one Clearly, a m, Ž, u t approache Ž,u Thi implie that by picing m to be uiciently large, the controller o Eq Ž 33 with Ž, u intead o Ž, u t guarantee tability and enorce the requirement o Eq Ž 34 in the cloed-loop ininite-dimenional ytem Remar 10 The propoed control methodology wa implemented through imulation on a paced-bed reactor modeled by two quai-linear parabolic PDE 8 In particular, it wa hown that a inite-dimenional output eedbac controller o dimenion 4, o the orm o Eq Ž 33 with an Ž O approximation or Ž,, u and m 3 provide a uperior peror-

15 41 CHRISTOFIDES AND DAOUTIDIS mance compared to a inite-dimenional output eedbac controller o the ame dimenion, with an O approximation or,, u 6 CONCLUSIONS In thi wor, we developed a methodology or the ynthei o nonlinear inite-dimenional output eedbac controller or ytem o quai-linear parabolic PDE, or which the eigenpectrum o the patial dierential operator can be partitioned into a inite-dimenional low one and an ininite-dimenional table at complement Combination o Galerin method with a novel procedure or the contruction o AIM wa ued, or the derivation o ODE ytem o dimenion equal to the number o low mode, that yield olution which are cloe, up to a deired accuracy, to the one o the PDE ytem, or almot all time Thee ODE ytem were ued a the bai or the ynthei o nonlinear output eedbac controller that guarantee tability and enorce the output o the cloed-loop ytem to ollow up to a deired accuracy, a prepeciied repone or almot all time APPENDIX Proo o Propoition 1 The proo o the propoition will be obtained in two tep In the irt tep, we will how that the ytem o Eq Ž 16 i exponentially table, provided that the initial condition and are uiciently mall In the econd tep, we will ue the exponential tability property to prove cloene o olution Ž Eq 1 Exponential Stability Firt, the ytem o Eq Ž 16 with ut 0 can be equivalently written a dx A x Ž x,0 Ž x, x Ž x,0 x A xž x, x t Ž 35 Let, with 1 1 a4 be two poitive real number uch that i x and x 1, then there exit poitive real number Ž,, uch that 1 3 Ž x, x Ž x,0 x 1 Ž x, x x x 3 Ž 36

16 CONTROL OF PARABOLIC PDE SYSTEMS 413 Pic a and From Aumption and the convere Lyapunov theorem or inite-dimenional ytem 14, Theorem 410, we have that there exit a mooth Lyapunov unction V : H 0 and a et o poitive real number Ž a, a, a, a, a , uch that or all x H that atiy x a, the ollowing condition hold: 4 a x VŽ x a x 1 V V Ž x A x Ž x,0 a x 3 x Ž 37 V x a x 5 From the global exponential tability property o the at ubytem o Eq Ž 17 and the convere Lyapunov theorem or ininite-dimenional ytem 1,, we have that there exit a Lyapunov unctional W : H 0 and a et o poitive real number Ž b, b, b, b 1 3 4, uch that or all x H the ollowing condition hold: b x WŽ x b x 1 1 W b 3 W Ž x A x x x Ž 38 W x b x 4 Conider now the mooth unction L : H H, 0 LŽ x, x VŽ x WŽ x Ž 39 a a Lyapunov unction candidate or the ytem o Eq Ž 35 From Eq Ž 37 and Eq Ž 38, we have that LŽ x, x i poitive deinite and proper Ž tend to a x, or x, with repect to it argument Computing the time-derivative o L along the trajectorie o thi ytem, and uing the bound o Eq Ž 37 and Eq Ž 38 and the etimate o Eq Ž 36, the

17 414 CHRISTOFIDES AND DAOUTIDIS ollowing expreion can be eaily obtained: V W LŽ x, x x x x x V V AxŽ x,0 Ž x, x Ž x,0 x x W W A x Ž x, x x x b a x a x x x b x x x 4 3 ž / b a x Ž a b x x b x x x a 5 1b 4 a3 x a 5 1b 4 b3 x b 4 3 Ž 40 Deining abab Ž ŽŽ a b , we have that i Ž 0, then L Ž x, x 1 0, which rom the propertie o L directly implie that the tate o the ytem o Eq Ž 35 i exponentially table, ie, there exit a poitive real number uch that x t e Ž 41 1 x Cloene o Solution Firt, we deine the error coordinate e Ž x x Ž Dierentiating e Ž with repect to, the ollowing dynam- ical ytem can be obtained: e Ae Ž x, ex Ž 4 Reerring to the above ytem with 0, we have rom the propertie o the unbounded operator A and the convere theorem o 1,, that there exit a Lyapunov unctional W : H and a et o poitive real 0

18 CONTROL OF PARABOLIC PDE SYSTEMS 415 number Ž b, b, b, b 1 3 4, uch that or all e H the ollowing condition hold: 1 b e W e b e W W A eb3 e e Ž 43 W b 4 e Ž e Computing the time-derivative o WŽ e along the trajectorie o the ytem o Eq Ž 4 and uing that Ž x, e x e 4 5, where, are poitive real number Ž 4 5 which ollow rom the act that the tate Ž x, x are bounded, we have W b e b e e b b e b e Ž 44 Set b b and * min, From the above inequality, uing Theorem 410 in 14, we have that i Ž 0, *, the ollowing bound hold or e Ž or all t 0,, a 3 Žt e K3 e 0 e e, 45 where e i a poitive real number From the above inequality and the act that e Ž 0 0, the etimate x Ž t x Ž t O ollow directly Deining the error coordinate e Ž t x Ž t x Ž t and dierentiating e Ž t with repect to time, the ollowing ytem can be obtained: de A e Ž x e, x Ž x Ž 46 The repreentation o the ytem o Eq 46 in the at time-cale tae the orm de A e Ž x e, x Ž x, Ž 47 d

19 416 CHRISTOFIDES AND DAOUTIDIS where Ž e, x can be conidered approximately contant, and thu uing that e Ž 0 0 and continuity o olution or e Ž, the ollowing bound can be written or e Ž or all 0,, b e, 0,, Ž 48 6 b b where 6 i a poitive real number and b tb O1 Ž, with t OŽ 0 i the time required or x Ž t to approach x Ž b p t, ie, x Ž t or t t, 7 b, where 7 i a poitive real number The ytem o Eq Ž 46 with x Ž t x Ž t 0 i exponentially table Ž Aump- tion Moreover, ince x Ž t decay exponentially, the ytem o Eq Ž 46 i alo exponentially table i x Ž t 0 Thi implie that or the ytem de A e Ž x e,0 Ž x Ž 49 there exit a mooth Lyapunov unction V : H 0 and a et o poitive real number Ž a, a, a, a, a , uch that or all e H that atiy e a the ollowing condition hold: 4 a e VŽ e a e 1 V VŽ e A e Ž x e,0 Ž x a e 3 e Ž 50 V e a e 5 Computing the time-derivative o VŽ e along the trajectorie o the ytem o Eq Ž 46 and uing that or t t, Ž x e, x b Ž x e,0 x 8 78, where 8 i a poitive real number Žwhich ollow rom the act that the tate Ž x, x are bounded, we have or all t t, b VŽ e a e a e Ž 51 From the above inequality, uing Theorem 410 in 14, and that e Ž t b O, we have that the ollowing bound hold or e Ž t or all t t,, e Ž t, Ž 5 where i a poitive real number From inequalitie o Eq Ž 48 Ž 5 e, the etimate x Ž t x Ž t O or all t 0 ollow directly e b

20 CONTROL OF PARABOLIC PDE SYSTEMS 417 Proo o Propoition The proo o the propoition will be obtained ollowing a two-tep approach imilar to the one ued in the proo o Propoition 1 Exponential Stability Thi part o the proo o the propoition i completely analogou to the proo o exponential tability in the cae o Propoition 1, and thu, it will be omitted or brevity Cloene o Solution From the irt part o the proo, we have that exponential tability i guaranteed provided that Ž 0, *, where * ia poitive real number Deining the error coordinate e Ž t x Ž t x Ž t and dierentiating e Ž t with repect to time, the ollowing ytem can be obtained, dẽ A e Ž x e, x Ž x, x, Ž where x x x x Ž x From A- umption 3 and the act that x Ž t decay exponentially to zero, we have that the ytem dẽ A e ž x e, x / Ž x, x Ž 54 i exponentially table, which implie that there exit a mooth Lyapunov unction V : H and a et o poitive real number Ž a, a, a, a, a , uch that or all e H that atiy e a 4, the ollowing condition hold: a e V e a e 1 V V Ž e A e ž x e, x / Ž x, x a e 3 ẽ Ž 55 V aˆ e 5 ẽ Computing the time-derivative o V e along the trajectorie o the ytem o Eq Ž 53 and uing that or t 0, Ž x e, x Ž x e, x Ž 1 e 1t, where, 1 1 are poitive real num- ber and i a negative real number, we have or all t 0, i ž / 1 1t Ž V e a e e a e Ž 56

21 418 CHRISTOFIDES AND DAOUTIDIS From the above inequality, uing Theorem 410 in 14 and the act that e Ž 0 0, we have that the ollowing bound hold or e Ž t or all t 0,, Ž e t Ke K, Ž 57 1t 1 1t where K, K are poitive real number Since the term Ke vanihe outide the interval 0, t Ž b where tb i the time required or x to approach x, it ollow rom Eq Ž 57 that or all t t, b 1 Ž e t K Ž 58 Ž Ž Ž 1 From the above inequality, the etimate x t x t O, or t t b, ollow directly Proo o Theorem 1 Subtituting the output eedbac controller o Eq Ž 33 into the ytem o Eq Ž 16, we get d A B Ž p 0Ž Q 0Ž p 1Ž Q 1Ž p Q Ž, 1 Ž,u Ž,u Ž, u 1 LŽ y CC Ž,u Ž,u Ž, u dx A x B Ž p 0Ž Q 0Ž p 1Ž Q 1Ž p Q x, x x A x BŽ p0 Q0 p1 Q1 t p Q x, x y i Cx Cx, i1,,l Ž 59 Perorming a two-time-cale decompoition in the above ytem, the at ubytem tae the orm x A x Ž 60

22 CONTROL OF PARABOLIC PDE SYSTEMS 419 Ž 1 which i exponentially table Furthermore, the O approximation o the cloed-loop inertial orm i given by d A B Ž p 0Ž Q 0Ž p 1Ž Q 1Ž p Q Ž, 1 Ž,u Ž,u Ž, u 0 1 Ž L y C C,u,u,u dx A x B Ž p 0Ž Q 0Ž p 1Ž Q 1Ž p Q x, 1 Ž x,u Ž x,u Ž x, u i 1 y Cx C x,u x,u x, u, Ž,u i1,,l Ž 61 Reerring to the above cloed-loop ODE ytem, Aumption 4 yield that it i exponentially table and the output y i, i 1,,l, change in a prepeciied manner A direct application o the reult o Propoition yield that there exit poitive real number 1,, * uch that i xž 0, xž 0 and Ž0, * 1, uch that the cloed-loop ininite-dimenional ytem i exponentially table and the relation o Eq Ž 34 hold ACKNOWLEDGMENT Financial upport or thi wor rom the National Science Foundation, CTS-96475, i grateully acnowledged REFERENCES 1 M J Bala, Feedbac control o linear diuion procee, Internat J Control 9 Ž 1979, M J Bala, Stability o ditributed parameter ytem with inite-dimenional controllercompenator uing ingular perturbation, J Math Anal Appl 99 Ž 1984, 80108

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