Integrating nonlinear output feedback control and optimal actuator=sensor placement for transport-reaction processes

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1 Cheical Engineering Science 56 (21) Integrating nonlinear output feedback control and optial actuator=sensor placeent for transport-reaction processes Charalabos Antoniades, Panagiotis D. Christodes Departent of Cheical Engineering, University of California, 45 Hilgard Avenue, Box , Los Angeles, CA , USA Received 28 March 2; accepted 2 February 21 Abstract This paper proposes a general and practical ethodology for the integration of nonlinear output feedback control with optial placeent of control actuators and easureent sensors for transport-reaction processes described by a broad class of quasi-linear parabolic partial dierential equations (PDEs) for which the eigenspectru of the spatial dierential operator can be partitioned into a nite-diensional slow one and an innite-diensional stable fast copleent. Initially, Galerkin s ethod is eployed to derive nite-diensional approxiations of the PDE syste which are used for the synthesis of stabilizing nonlinear state feedback controllers via geoetric techniques. The optial actuator location proble is subsequently forulated as the one of iniizing a eaningful cost functional that includes penalty on the response of the closed-loop syste and the control action and is solved by using standard unconstrained optiization techniques. Then, under the assuption that the nuber of easureent sensors is equal to the nuber of slow odes, we eploy a procedure proposed in Christodes and Baker (1999) for obtaining estiates for the states of the approxiate nite-diensional odel fro the easureents. The estiates are cobined with the state feedback controllers to derive output feedback controllers. The optial location of the easureent sensors is coputed by iniizing a cost function of the estiation error in the closed-loop innite-diensional syste. It is rigorously established that the proposed output feedback controllers enforce stability in the closed-loop innite-diensional syste and that the solution to the optial actuator=sensor proble, which is obtained on the basis of the closed-loop nite-diensional syste, is near-optial in the sense that it approaches the optial solution for the innite-diensional syste as the separation of the slow and fast eigenodes increases. The proposed ethodology is successfully applied to a representative diusion reaction process and a nonisotheral tubular reactor with recycle to derive nonlinear output feedback controllers and copute optial actuator=sensor locations for stabilization of unstable steady states.? 21 Elsevier Science Ltd. All rights reserved. Keywords: Nonlinear control; Static output feedback control; Optial actuator=sensor placeent; Diusion convection reaction processes 1. Introduction Transport-reaction processes with signicant diusive and dispersive echanis are typically characterized by severe nonlinearities and spatial variations, and are naturally described by quasi-linear parabolic partial differential equations (PDEs). Nonlinearities usually arise fro coplex reaction echaniss and Arrhenius dependence of the reaction rates on teperature, and spatial variations occur due to the presence of signicant Corresponding author. Tel.: ; fax: E-ail address: pdc@seas.ucla.edu (P. D. Christodes). diusion and convection phenoena. Typical exaples of transport-reaction processes are tubular reactors, packed-bed reactors, and cheical vapor deposition reactors. Parabolic PDE systes typically involve spatial differential operators whose eigenspectru can be partitioned into a nite-diensional slow one and an innite-diensional stable fast copleent (Friedan, 1976; Balas, 1979). This iplies that the dynaic behavior of such systes can be approxiately described by nite-diensional systes. Therefore, the standard approach to the control of parabolic PDEs involves the application of Galerkin s ethod to the PDE syste to derive ODE systes that describe the dynaics of the doinant (slow) odes of the PDE syste, which 9-259/1/$ - see front atter? 21 Elsevier Science Ltd. All rights reserved. PII: S9-259(1)123-3

2 4518 C. Antoniades, P. D. Christodes / Cheical Engineering Science 56 (21) are subsequently used as the basis for the synthesis of nite-diensional controllers (e.g., Balas, 1979; Ray, 1981). A potential drawback of this approach is that the nuber of odes that should be retained to derive an ODE syste that yields the desired degree of approxiation ay be very large, leading to coplex controller design and high diensionality of the resulting controllers. Motivated by this, recent eorts on control of parabolic PDE systes have focused on the proble of synthesizing low-order controllers on the basis of ODE odels obtained through cobination of Galerkin s ethod with approxiate inertial anifolds (see the recent book Christodes, 21 for details and references). Even though the developed ethods allow to systeatically design nonlinear controllers for transport-reaction processes, there is no work on the integration of nonlinear controllers with optial placeent of control actuators and easureent sensors for transport-reaction processes so that the desired control objectives are achieved with inial energy use. Regarding the proble of optial placeent of control actuators, the conventional approach is to select the actuator locations based on open-loop considerations to ensure that the necessary controllability requireents are satised. More recently, eorts have been ade on the proble of integrating feedback control and optial actuator placeent for certain classes of linear distributed paraeter systes including investigation of controllability easures and actuator placeent in oscillatory systes (Arbel, 1981), optial placeent of actuators for linear feedback control in parabolic PDEs (Xu, Warnitchai, & Igusa, 1994; Deetriou, 1999) and in actively controlled structures (Rao, Pan & Venkayya, 1991; Choe & Baruh, 1992). On the other hand, the proble of selecting optial locations for easureent sensors in distributed paraeter systes has received very signicant attention over the last 2 years. The essence of this proble is to use a nite nuber of easureents to copute the best estiate of the entire distributed state for all positions and ties eploying a state observer in the presence of easureent noise. Eorts for the solution of this proble have focused on linear systes (Yu & Seinfeld, 1973; Chen & Seinfeld, 1975; Kuar & Seinfeld, 1978a; Oatu, Koide, & Soeda, 1978; Morari & O Dowd, 198) and the application of the results to optial state estiation in tubular reactors (Colantuoni & Padanabhan, 1977; Kuar & Seinfeld, 1978b; Harris, MacGregor, & Wright, 198; Alvarez, Roagnoli, & Stephanopoulos, 1981; Waldra, Dochain, Bourrel, & Magnus, 1998). The central idea to the solution involves the use of a spatial discretization schee to obtain a luped approxiation of the distributed paraeter syste followed by the forulation and solution of an optial state estiation proble which involves coputing sensor locations so that an appropriate functional that includes penalty on the estiation error and the easureent noise is iniized. Signicant research eorts have also been ade on the integrated optial placeent of controllers and sensors for various classes of linear distributed paraeter systes (see, for exaple, Aouroux, Di Pillo, & Grippo, 1976; Ichikawa & Ryan, 1977; Courdesses, 1978; Malandrakis, 1979; Oatu & Seinfeld, 1983 and the review paper of Kubrusly & Malebranche, 1985). Despite the progress on optial sensor placeent and the availability of results on the integration of linear feedback control with actuator placeent for linear parabolic PDEs, there are no results on the integration of nonlinear output feedback control with optial placeent of control actuators and easureent sensors for transport-reaction processes described by nonlinear parabolic PDEs. This paper proposes a general and practical ethodology for the integration of nonlinear output feedback control with optial placeent of control actuators and easureent sensors for transport-reaction processes described by a broad class of quasi-linear parabolic partial dierential equations (PDEs) for which the eigenspectru of the spatial dierential operator can be partitioned into a nite-diensional slow one and an innite-diensional stable fast copleent. Initially, Galerkin s ethod is eployed to derive nite-diensional approxiations of the PDE syste which are used for the synthesis of stabilizing nonlinear state feedback controllers via geoetric techniques. The optial actuator location proble is subsequently forulated as the one of iniizing a eaningful cost functional that includes penalty on the response of the closed-loop syste and the control action and is solved by using standard unconstrained optiization techniques. Then, under the assuption that the nuber of easureent sensors is equal to the nuber of slow odes, we eploy a procedure proposed in Christodes and Baker (1999) for obtaining estiates for the states of the approxiate nite-diensional odel fro the easureents. The estiates are cobined with the state feedback controllers to derive output feedback controllers. The optial location of the easureent sensors is coputed by iniizing a cost function of the estiation error in the closed-loop innite-diensional syste. It is rigorously established that the proposed output feedback controllers enforce stability in the closed-loop innite-diensional syste and that the solution to the optial actuator=sensor proble, which is obtained on the basis of the closed-loop nite-diensional syste, is near-optial in the sense that it approaches the optial solution for the innite-diensional syste as the separation of the slow and fast eigenodes increases. The proposed ethodology is successfully applied to a diusion reaction process and a nonisotheral tubular reactor with recycle to derive nonlinear output feedback controllers and copute optial actuator=sensor locations for stabilization of unstable steady states.

3 C. Antoniades, P. D. Christodes / Cheical Engineering Science 56 (21) Preliinaries 2.1. Description of parabolic PDE systes We consider transport-reaction processes described by quasi-linear parabolic PDE systes of the x x = + x B@2 + wb(z)u + f(x); 2 y i c = y = c i (z)k x(z; t)dz; s (z)! x(z; t)dz; i =1;:::;l; =1;:::;p; subject to the boundary x C 1 x(; t)+d 1 (; t)=r x C 2 x(; t)+d 2 (; t)=r 2; (1) (2) and the initial condition: x(z; ) = x (z); (3) where x(z; t) =[x 1 (z; t) x n (z; t)] T R n denotes the vector of state variables, z [; ] R is the spatial coordinate, t [; ) is the tie, u=[u 1 u 2 u l ] T R l denotes the vector of anipulated inputs, yc i R denotes the ith controlled output, and y R denotes the th easured 2 x= 2 denote the rstand second-order spatial derivatives of x, f(x) is a nonlinear vector function, w; k! are constant vectors, A; B; C 1 ;D 1 ;C 2 ;D 2 are constant atrices, R 1 ;R 2 are colun vectors, and x (z) is the initial condition. b(z) is a known sooth vector function of z of the for b(z) =[b 1 (z) b 2 (z) b l (z)], where b i (z) describes how the control action u i (t) is distributed in the interval [; ]; c i (z) is a known sooth function of z which is deterined by the desired perforance specications in the interval [; ], and s (z) is a known sooth function of z which depends on the shape (point or distributing sensing) of the easureent sensors in the interval [; ]. Whenever the control action enters the syste at a single point z, with z [; ] (i.e. point actuation), the function b i (z) is taken to be nonzero in a nite spatial interval of the for [z ; z + ], where is a sall positive real nuber, and zero elsewhere in [; ]. Fig. 1 shows the location of the anipulated inputs, controlled outputs, and easured outputs in the case of a prototype exaple. Throughout the paper, we will use the order of agnitude notation O(). In particular, () = O() if there exist positive real nubers k 1 and k 2 such that: () 6 k 1 ; k 2. Referring to the syste of Eq. (1), several rearks are in order: (a) the spatial dierential operator is linear; this assuption is valid for diusion convection reaction processes where the diusion coecient and the Fig. 1. Location of the anipulated inputs, controlled outputs, and easured outputs in the case of a prototype exaple. conductivity can be taken independent of teperature and concentrations, (b) the anipulated input enters the syste in a linear and ane fashion; this is typically the case in any practical applications where, for exaple, the wall teperature is chosen as the anipulated input, and (c) the nonlinearities appear in an additive fashion (e.g., coplex reaction rates, Arrhenius dependence of reaction rates on teperature). In the reainder of this section, we precisely characterize the class of parabolic PDE systes of the for of Eq. (1) which we consider in the anuscript. To this end, we forulate the parabolic PDE syste of Eq. (1) as an innite-diensional syste in the Hilbert space H([; ]; R n ) (this will also siplify the notation of the paper, since the boundary conditions of Eq. (2) will be directly included in the forulation; see Eq. (8) below), with H being the space of n-diensional vector functions dened on [; ] that satisfy the boundary condition of Eq. (2), with inner product and nor (! 1 ;! 2 )= (! 1 (z);! 2 (z)) R n dz;! 1 2 =(! 1 ;! 1 ) 1=2 ; (4) where! 1 ;! 2 are two eleents of H([; ]; R n ) and the notation ( ; ) R n denotes the standard inner product in R n. Dening the state function x on H([; ]; R n )as x(t)= x(z; t); t ; z [; ]; (5) the operator A in H([; ]; R n )as Ax = x + x B@2 ; { 2 x D(A)= x H([; ]; R n ); C 1 x(; t)+d x (; t)=r x C 2 x(; t)+d 2 (; t)=r 2 ; (6) and the input, controlled output, and easured output operators as Bu = wbu; Cx =(c; kx); Sx =(s;!x); (7) }

4 452 C. Antoniades, P. D. Christodes / Cheical Engineering Science 56 (21) the syste of Eqs. (1) (3) takes the for ẋ = Ax + Bu + f(x); x() = x ; (8) y c = Cx; y = Sx; where f(x(t)) = f(x(z; t)) and x =x (z). We assue that the nonlinear ters f(x) are locally Lipschitz with respect to their arguents and satisfy f() =. For A, the eigenvalue proble is dened as A j = j j ; j=1;:::; ; (9) where j denotes an eigenvalue and j denotes an eigenfunction; the eigenspectru of A, (A), is dened as the set of all eigenvalues of A, i.e. (A)={ 1 ; 2 ;:::;}. Assuption 1 (Christodes and Daoutidis, 1997) that follows states that the eigenspectru of A can be partitioned into a nite-diensional part consisting of slow eigenvalues and a stable innite-diensional copleent containing the reaining fast eigenvalues, and that the separation between the slow and fast eigenvalues of A is large. Assuption 1. (1) Re{ 1 } Re{ 2 } Re{ j }, where Re{ j } denotes the real part of j. (2) (A) can be partitioned as (A) = 1 (A) + 2 (A), where 1 (A) consists of the rst (with - nite) eigenvalues, i.e. 1 (A)={ 1 ;:::; }, and Re{1} Re{ = } O(1). (3) Re +1 and sall positive nuber. Re {} Re{ +1} = O() where 1 is a The assuption of nite nuber of unstable eigenvalues is always satised for nonlinear parabolic PDE systes (Friedan, 1976), while the assuption of discrete eigenspectru and the assuption of existence of only a few doinant odes that describe the dynaics of the nonlinear parabolic PDE syste are usually satised by the ajority of diusion convection reaction processes (see the exaples of Sections 6 and 7) Galerkin s ethod We will now review the application of standard Galerkin s ethod to the syste of Eq. (8) to derive an approxiate nite-diensional syste. Let H s, H f be odal subspaces of A, dened as H s = span{ 1 ; 2 ;:::; } and H f = span{ +1 ; +2 ;:::} (the existence of H s, H f follows fro Assuption 1). Dening the orthogonal projection operators P s and P f such that x s = P s x, x f = P f x, the state x of the syste of Eq. (8) can be decoposed as x = x s + x f = P s x + P f x: (1) Applying P s and P f to the syste of Eq. (8) and using the above decoposition for x, the syste of Eq. (8) can be equivalently written in the following for: dx s = A s x s + B s u + f s (x s ;x f ); f = A f x f + B f u + f f (x s ;x f y c = Cx s + Cx f ; y = Sx s + Sx f ; x s () = P s x() = P s x ; x f () = P f x() = P f x ; (11) where A s = P s AP s, B s = P s B, f s = P s f, A f = P f AP f, B f =P f B and f f =P f f and the f =@t is used to denote that the state x f belongs in an innite-diensional space. In the above syste, A s is a diagonal atrix of diension of the for A s = diag{ j }, f s (x s ;x f ) and f f (x s ;x f ) are Lipschitz vector functions, and A f is an unbounded dierential operator which is exponentially stable (following fro part (3) of Assuption 1 and the selection of H s ; H f ). Neglecting the fast and stable innite-diensional x f -subsyste in the syste of Eq. (11), the following -diensional slow syste is obtained: d x s = A s x s + B s u + f s ( x s ; ); dt ỹ c = C x s ; ỹ = S x s ; (12) where the tilde sybol in x s, ỹ c and ỹ denotes that the state x s, the controlled output ỹ c, and the easured output ỹ are associated with the approxiation of the slow x s -subsyste. Reark 1. We note that the above odel reduction procedure which led to the approxiate ODE syste of Eq. (11) can also be used, when epirical eigenfunctions of the syste of Eq. (8) coputed through Karhunen Loeve expansion (see Christodes, 21 for details) are used as basis functions in H s and H f instead of the eigenfunctions of A. Furtherore, we note that due to the separation of the fast and slow odes of the spatial dierential operator (which is characterized by ; part (3) of Assuption 1), the coupling of the x s and x f subsystes in the interconnection of Eq. (11) through the ters f s (x s ;x f ) and f f (x s ;x f ) in a bounded region of the state space is weak (i.e., it scales with and disappears as ); this property is the basis for using the nite-diensional syste of Eq. (12) for nonlinear output feedback controller design and optial actuator=sensor placeent. 3. Proble stateent and solution fraework In this paper, we address the proble of coputing optial locations of point control actuators and point easureent sensors associated with nonlinear output

5 C. Antoniades, P. D. Christodes / Cheical Engineering Science 56 (21) feedback control laws of the following general for: u = F(y ); (13) where F(y ) is a nonlinear vector function and y denotes the vector of easured outputs, so that the following properties are enforced in the closed-loop syste: (a) exponential stability, and (b) the solution to the optial actuator=sensor location proble, which is obtained on the basis of the closed-loop nite-diensional syste, is near-optial in the sense that it approaches the optial solution for the innite-diensional syste as the separation of the slow and fast eigenodes increases. To address this proble, we will initially synthesize stabilizing nonlinear state feedback controllers via geoetric techniques on the basis of nite-diensional approxiations of the PDE syste obtained via Galerkin s ethod. The optial actuator location proble will be subsequently forulated as the one of iniizing a eaningful cost functional that includes penalty on the response of the closed-loop syste and the control action and will be solved by using standard unconstrained optiization techniques. Then, under the assuption that the nuber of easureent sensors is equal to the nuber of slow odes, we will eploy a procedure proposed in Christodes and Baker (1999) for obtaining estiates for the states of the approxiate nite-diensional odel fro the easureents. The estiates will be cobined with the state feedback controllers to derive output feedback controllers. The optial location of the easureent sensors will be coputed by iniizing a cost function of the estiation error in the closed-loop innite-diensional syste. It will be established by using singular perturbation techniques that the desired properties are enforced in the closed-loop syste, provided that the separation of the slow and fast eigenodes is suciently large. 4. Integrating nonlinear control and optial actuator placeent 4.1. Nonlinear state feedback controller synthesis In this section, we assue that easureents of the states of the PDE syste of Eq. (12) are available and address the proble of synthesizing nonlinear static state feedback control laws of the general for u = F(z a ; x s ); (14) where F(z a ; x s ) is a nonlinear vector function and z a denotes the vector of the actuator locations, that guarantee exponential stability of the closed-loop nite-diensional syste. To this end, we will need the following assuption (see Reark 3 below for a discussion on this assuption). Assuption 2. l = (i.e., the nuber of control actuators is equal to the nuber of slow odes), and the inverse of the atrix B s exists. Proposition 1 that follows provides the explicit forula for the state feedback controller that achieves the control objective. Proposition 1. Consider the nite-diensional syste of Eq. (12) for which Assuption 2 holds. Then; the state feedback controller: u = Bs 1 (( s A s ) x s f s ( x s ; )); (15) where s is a stable atrix; guarantees global exponential stability of the closed-loop nite-diensional syste. Reark 2. The structure of the closed-loop nitediensional syste under the controller of Eq. (15) has the following for: x s = s x s ; (16) and thus, the response of this syste depends only on the stable atrix s and the initial condition, x s (), and is independent of the actuator locations. Reark 3. The requireent l = is sucient and not necessary, and it is ade to siplify the solution of the controller synthesis proble. Full linearization of the closed-loop nite-diensional syste through coordinate change and nonlinear feedback can be achieved for any nuber of anipulated inputs (i.e., for any l [1;]), provided that an appropriate set of involutivity conditions is satised by the corresponding vector elds of the syste of Eq. (12) (see Isidori, 1989 for details) Coputation of optial location of control actuators In this subsection, we copute the actuator locations so that the state feedback controller of Eq. (15) is near-optial for the full PDE syste of Eq. (11) with respect to a eaningful cost functional which is dened over the innite tie-interval and iposes penalty on the response of the closed-loop syste and the control action. To this end, we initially focus on the ODE syste of Eq. (12) and consider the following cost functional: J s = (( x T s (x s ();t);q s x s (x s ();t)) + u T ( x s (x s ();t);z a )Ru( x s (x s ();t);z a )) dt; (17) where Q s and R are positive denite atrices. The cost of Eq. (17) is well dened and eaningful since it iposes penalty on the response of the closed-loop

6 4522 C. Antoniades, P. D. Christodes / Cheical Engineering Science 56 (21) nite-diensional syste and the control action. However, a potential proble of this cost is its dependence on the choice of a particular initial condition, x s (), and thus, the solution to the optial placeent proble based on this cost ay lead to actuator locations that perfor very poorly for a large set of initial conditions. To eliinate this dependence and obtain optiality over a broad set of initial conditions, we follow Levine and Athans (1978) and consider an average cost over a set of linearly independent initial conditions, xs(), i i =1;:::;,of the following for: Ĵ s = 1 (( x T s (x i s();t);q s x s (xs();t)) i + u T ( x s (x i s();t);z a )Ru( x s (x i s();t);z a )) dt: (18) Referring to the above cost, we rst note that the penalty on the response of the closed-loop syste Ĵ xs = 1 ( x T s (x i s();t);q s x s (xs();t)) i dt (19) is nite because the solution of the closed-loop syste of Eq. (16) is exponentially stable by appropriate choice of s. Moreover, Ĵ xs is independent of the actuator locations (Reark 2), and thus, the optial actuator placeent proble reduces to the one of iniizing the following cost which only includes penalty on the control action: Ĵ us = 1 u T ( x s (x i s();t);z a )Ru( x s (xs();t);z i a )dt Ĵ us is a function of ultiple variables, z a =[z a1 z a2 z al ], and thus, it obtains its local iniu values when its gradient with respect to the actuator locations is equal to zero, us a = us us al ] T =[ ] T ; (2) and zaz a Ĵ us (z a ) where zaz a Ĵ us is the Hessian atrix of Ĵ us and z a is a solution of the syste of nonlinear algebraic equations of Eq. (2) (which includes l equations with l unknowns). The solution z a for which the above conditions are satised and Ĵ us obtains its sallest value (global iniu) corresponds to the optial actuator locations for the closed-loop nite-diensional syste. Theore 1 that follows establishes that these locations are near-optial for the closed-loop innite-diensional syste (the proof of this theore can be found in the Appendix). Theore 1. Consider the innite-diensional syste of Eq. (11) for which Assuption 1 holds; and the nite-diensional syste of Eq. (12); for which Assuption 2 holds. Then; there exist positive real nubers 1 ; 2 and such that if x s () 6 1 ; x f () ; and (; ]; then the controller of Eq. (13): (a) guarantees exponential stability of the closed-loop innite-diensional syste, and (b) the optial locations of the point actuators obtained for the closed-loop nite-diensional syste are near-optial for the closed-loop innite-diensional syste in the sense that: Ĵ = 1 ((xs T (x i s();t);q s x s (xs();t)) i +(x T f(x i s();t);q f x f (x i f();t)) + u T (x s (x i s();t);z a )Ru(x s (x i s();t);z a )) dt Ĵ s as ; (21) where Q f is an unbounded positive denite operator and Ĵ is the average cost function associated with the controller of Eq. (15) and the innite-diensional syste of Eq. (11). Reark 4. Note that even though the response of the closed-loop nite-diensional syste of Eq. (16) is independent of the actuator locations, the response of the closed-loop innite-diensional syste does depend on the actuators locations. However, this dependence is scaled by, and therefore, it decreases as we increase the nuber of odes included in the nite-diensional syste used for controller design. Reark 5. In general, the solution to the syste of Eq. (2) can be coputed through cobination of nuerical integration techniques and ultivariable Newton s ethod. Reark 6. The results of this section, state feedback control and optial actuator placeent, can be generalized to the case where the nite-diensional approxiation of the syste of Eq. (11) is obtained through cobination of Galerkin s ethod with approxiate inertial anifolds (Christodes, 21). This would lead to a higher than O() closeness between the solution of the nite-diensional closed-loop syste and the solution of the innite-diensional closed-loop syste (see Christodes (21) for a detailed study of this issue), which, in turn, would allow to obtain a better characterization and a signicant iproveent of the closed-loop innite-diensional syste perforance. However, since the objective of the present work is output feedback control with optial actuator=sensor placeent, we do not pursue this approach because the error that will be introduced in the estiates of the slow states fro the output easureents will be of O() (see Assuption 3 below), and thus, it is not possible to obtain a better than O() characterization of the closed-loop innite-diensional

7 C. Antoniades, P. D. Christodes / Cheical Engineering Science 56 (21) syste perforance under the static output feedback controller presented in the next section. 5. Integrating nonlinear output feedback control and optial actuator=sensor placeent The nonlinear controller of Eq. (15) was derived under the assuption that easureents of the state variables, x(z; t), are available at all positions and ties. However, fro a practical point of view, easureents of the state variables are only available at a nite nuber of spatial positions, while in addition, there are any applications where easureents of process state variables cannot be obtained on-line (for exaple, concentrations of certain species in a cheical reactor ay not be easured on-line). Motivated by these practical probles, we address in this section: (a) the synthesis of nonlinear output feedback controllers that use easureents of the process outputs, y, to enforce stability in the closed-loop innite-diensional syste, and (b) the coputation of optial locations of the easureent sensors. Specically, we consider output feedback control laws of the general for u(t)=f(y ); (22) where F(y ) is a nonlinear vector function and y is the vector of easured outputs. The synthesis of the controller of Eq. (22) will be achieved by cobining the state feedback controller of Eq. (15) with a procedure proposed in Christodes and Baker (1999) for obtaining estiates for the states of the approxiate ODE odel of Eq. (12) fro the easureents. To this end, we need to ipose the following requireent on the nuber of easured outputs in order to obtain estiates of the states x s of the nite-diensional syste of Eq. (12), fro the easureents y, =1;:::;p. Assuption 3. p= ( i.e.; the nuber of easureents is equal to the nuber of slow odes), and the inverse of the operator S exist, so that ˆx s = S 1 y. We note that the requireent that the inverse of the operator S exists can be achieved by appropriate choice of the location of the easureent sensors (i.e., functions s (z)). The optial locations for the easureent sensors can be coputed by iniizing an average cost function of the estiation error of the closed-loop innite-diensional syste of the for Ĵ (e)= 1 ( x s (x i s();t) ˆx s (xs();t) i 2 )dt; (23) where x s is the slow state of the closed-loop innitediensional syste of Eq. (11), ˆx s = S 1 y, and e(t)= x s ˆx s 2 is the estiation error. In contrast to the solution of the optial location proble for the control actuators, the solution to this optiization proble requires the solution of the closed-loop innite-diensional syste in order to copute x s, and ˆx s (fro the easureents y, =1; 2;:::;p), and thus, it is ore coputationally deanding. Theore 2 that follows establishes that the proposed output feedback controller enforces stability in the closed-loop innite-diensional syste and that the solution to the optial actuator=sensor proble, which is obtained on the basis of the closed-loop nite-diensional syste, is near-optial in the sense that it approaches the optial solution for the innite-diensional syste as the separation of the slow and fast eigenodes increases. The proof of this theore can be found in the Appendix. Theore 2. Consider the full syste of Eq. (11) for which Assuption 1 holds; and the nite-diensional syste of Eq. (12); for which Assuptions 2 and 3 hold; under the nonlinear output feedback controller: ˆx s = S 1 y ; u = B 1 s (( s A s )ˆx s f s (ˆx s ; )): (24) Then; there exist positive real nubers 1 ; 2 and such that if x s () 6 1 ; x f () ; and (; ]; then the controller of Eq. (24): (a) guarantees exponential stability of the closed-loop syste; and (b) the locations of the point actuators and easureent sensors are near-optial in the sense that the cost function associated with the controller of Eq. (24) and the syste of Eq. (11) satises Ĵ = 1 ((x T s (x i s();t);q s x s (x i s();t)) +(x T f(x i s();t);q f x f (x i f();t)) + u T (x s (x i s());t;z a )Ru(x s (x i s());t;z a )) dt Ĵ s as ; (25) where Ĵ and Ĵ s are the average cost functions of the innite-diensional syste of Eq. (11) and the nite-diensional syste of Eq. (12); respectively; under the output feedback controller of Eq. (24). Reark 7. We note that the controller of Eq. (24) uses static feedback of the easured outputs y ;=1;:::;p, and thus, it feeds back both x s and x f (this is in contrast to the state feedback controller of Eq. (15) which only uses feedback of the slow state x s ). However, even though the use of x f feedback could lead to destabilization of the stable fast subsyste, the large separation of the slow

8 4524 C. Antoniades, P. D. Christodes / Cheical Engineering Science 56 (21) and fast odes of the spatial dierential operator (i.e., assuption that is suciently sall) and the fact that the controller does not include ters of the for O(1=) do not allow such a destabilization to occur. In the reainder of the paper, we show two applications of the proposed approach for optial actuator=sensor placeent to a typical diusion reaction process and a nonisotheral tubular reactor with recycle. 6. Application to a diusion--reaction process 6.1. Process description control proble forulation Consider a long, thin rod in a reactor (Fig. 2). The reactor is fed with pure species A and a zeroth order exotheric catalytic reaction of the for A B takes place on the rod. Since the reaction is exotheric, a cooling ediu which is in contact with the rod is used for cooling. Under the assuptions of constant density and heat capacity of the rod, constant conductivity of the rod, and constant teperature at both ends of the rod, the atheatical odel which describes the spatioteporal evolution of the diensionless rod teperature consists of the following parabolic x + 2 T e =(1+ x) + U (b(z)u(t) x) T e ; (26) subject to the Dirichlet boundary conditions: x(; t) = ; x(; t) = (27) and the initial condition: x(z; ) = x (z); (28) where x denotes the diensionless teperature in the reactor, T denotes a diensionless heat of reaction, denotes a diensionless activation energy, U denotes a diensionless heat transfer coecient, and u denotes the anipulated input (teperature of the cooling ediu). The following typical values were given to the process paraeters: T =5:; U =2:; =4:: (29) For the above values, the operating steady state x(z; t)= is an unstable one (Fig. 3 shows the prole of evolution of open-loop rod teperature starting fro initial conditions close to the steady state x(z; t) = ; the process oves to another stable steady state characterized by a axiu in the teperature prole, hot-spot, in the iddle of the rod). A 3th order Galerkin truncation of the syste of Eqs. (26) (28) was used in our siulations in order to accurately describe the process (further increase on the order of the Galerkin truncation was found to give Fig. 2. Catalytic rod. Fig. 3. Prole of evolution of rod teperature in the open-loop syste. negligible iproveent on the accuracy of the results). The control objective is to stabilize the rod teperature prole at the unstable steady state x(z; t) =. The eigenvalue proble for the spatial dierential operator of the process: Ax x ; 2 x D(A)={x H([;]; R); x(; t)=; x(; t)=} (3) can be solved analytically and its solution is of the for j = j 2 ; j (z)= 2 j (z)= sin(j z); j =1;:::; ; (31) where j ; j ; j, denote the eigenvalues, eigenfunctions and adjoint eigenfunctions of A i, respectively. In the reainder of this section, we use the proposed ethod to copute the optial locations in the case of using two and three control actuators and easureent sensors.

9 C. Antoniades, P. D. Christodes / Cheical Engineering Science 56 (21) Two actuator=sensor exaple Initially, we assue that two point control actuators and easureent sensors are used to stabilize the syste. Therefore, following Assuptions 2 and 3, we use Galerkin s ethod to derive a second order ODE approxiation of the PDE of Eq. (26) which is used for controller design and optial actuator=sensor placeent. The for of the approxiate ODE syste is given below: [ ] x [ ] [ ] s1 1 = U x s1 x 2 U s2 x s2 [ ][ ] 1 (z + a1 ) 1 (z a2 ) u1 U 2 (z a1 ) 2 (z a2 ) u 2 [ ] f1 ( x + s ; ) T ; (32) f 2 ( x s ; ) where z a1 and z a2 are the locations of the two point actuators and the explicit fors of the ters f 1 ( x s ; ) and f 2 ( x s ; ) are oitted for brevity. The easured output y (t) R 2 is dened as: [ ] [ y1 x(zs1 ;t) = x(z s2 ;t) y 2 ] ; (33) where z s1 and z s2 are the locations of the two point sensors. For the syste of Eq. (32), the nonlinear state feedback controller of Eq. (15) takes the for [ ] u1 u = = B 1 (z a )F( x s ) = 1 u 2 U [ ] 1 1 (z a1 ) 1 (z a2 ) 2 (z a1 ) 2 (z a2 ) ([ 1 + U 2 + U + T [ (f1 ( x s ; ); 1 ) (f 2 ( x s ; ); 2 ) ][ ] ( xs1 ; 1 ) ( x s2 ; 2 ) ]) : (34) Substituting the above controller into the syste of Eq. (32), we obtain the following closed-loop ODE syste: [ x s1 x s2 ] = [ ][ xs1 x s2 ] ; (35) where and are positive real nubers. Since the response of the above syste depends only on the paraeters ; and the initial condition x s, and is independent of the actuator locations,we copute the optial actuator locations by iniizing the following cost functional, which only includes penalty on the control action: Ĵ us = 1 2 F T ( x s (x i 2 s();t))(b 1 (z a )) T RB 1 (z a ) F( x s (x i s();t)) dt: (36) Table 1 Results for two control actuators Case Actuator locations ˆ J u Optial :39; : Linearized :32; : :2; : :3; : Using the following values for the paraeters = =1, xs 1 ()= 1, and xs 2 ()= 2, and taking R; Q s ;Q f to be unit atrices of appropriate diensions, the optial actuator locations were found to be: z a1 =:39 and z a2 =:66. Finally, we copute the optial sensor locations by iniizing the following cost functional of the estiation error: Ĵ (e)= 1 2 ( x s (x 2 s()) i ˆx s (xs()) i 2 )) dt; (37) where x s is obtained fro the siulation of the full-order closed-loop syste of Eq. (11), and ˆx s is obtained fro the easured outputs of the full-order closed-loop syste as shown below: ] = [ ˆxs1 ˆx s2 [ 1 (z s1 ) 2 (z s1 ) 1 (z s2 ) 2 (z s2 ) ˆ J x ˆ J ] 1 [ ] y1 (z s1 ;t) : (38) y 2 (z s2 ;t) We found the optial location of easureent sensors to be: z s1 =:31 and z s2 =:72. Finally, by cobining the state feedback controller of Eq. (34) with the easureents, and the optial actuator=sensor locations, we derive the following nonlinear output feedback controller: [ ] [ ] 1 [ ] ˆxs1 1 (z = s1 ) 2 (z s1 ) y1 (z s1 ;t) ˆx s2 1 (z s2 ) 2 (z s2 ) y 2 (z s2 ;t) u = 1 [ ] 1 1 (z a1 ) 1 (z a2 ) U 2 (z a1 ) 2 (z a2 ) ([ ][ ] 1 + U (ˆxs1 ; 1 ) 2 + U (ˆx s2 ; 2 ) [ ]) (f1 (ˆx + s ; ); 1 ) T : (39) (f 2 (ˆx s ; ); 2 ) We perfored several siulation runs to evaluate the perforance of the proposed ethod for coputing optial locations of control actuators and easureent sensors. We initially apply the state feedback controller of Eq. (34) to the 3th order Galerkin truncation of the syste of Eqs. (26) (28) and investigate the inuence of the dierent actuator locations on the various cost functions. Table 1 shows the values of the costs Ĵ u ; Ĵ x, and Ĵ of the full-order closed-loop syste under the state feedback controller of Eq. (34), in the case of optial actuator placeent, and for the sake of coparison, the values of these costs in the case of alternative actuator placeents including optial actuator placeent based on the linearized syste (second line). The cost for the control

10 4526 C. Antoniades, P. D. Christodes / Cheical Engineering Science 56 (21) Table 2 Results for two control actuators and easureent sensors Case Sensor locations ˆ J (e) ˆ J u ˆ J x ˆ J Optial :31; : e :48; : e :31; : e Fig. 4. Closed-loop nor of the control eort, u, for the two actuator=sensor exaple, for x s() = 1, for the optial case (solid line), the linearized case (long-dashed line), case 3 (short-dashed line), and case 4 (dotted line). Eqs. (26) (28) with actuator locations z a1 =:39 and z a2 =:66 and dierent sensor locations. Table 2 shows the values of the costs Ĵ (e); Ĵ u ; Ĵ x, and Ĵ of the full-order closed-loop syste under the output feedback controller of Eq. (39), in the case of optial sensor placeent, and for the sake of coparison, the values of these costs in the case of two other sensor locations. The estiation error of the sensor locations of :31 and :72 coputed by the proposed approach is clearly saller than the other two cases. In Fig. 6, we display the closed-loop nor of the estiation error versus tie, for the optial actuator=sensor locations, for x s () = 1 (solid line) and x s () = 2 (dashed line). We can see that for both initial conditions the estiation error is very sall. Finally, Figs. 7 and 8 show the proles of the evolution of the teperature of the rod, under output feedback control, for the optial actuator=sensor locations, for x s () = 1 (Fig. 7), and for x s () = 2 (Fig. 8). We can see that the proposed controller with optial actuator=sensor locations, stabilizes the syste to the spatially unifor operating steady state very quickly, for both cases Three actuator=sensor exaple Fig. 5. Closed-loop nor of the control eort, u, for the two actuator=sensor exaple, for x s() = 2, for the optial case (solid line), the linearized case (long-dashed line), case 3 (short-dashed line), and case 4 (dotted line). action used to stabilize the syste at x(z; t) = when the actuators are optially placed, is clearly saller than the case of actuator placeent based on the linearized syste (by 1.1%), case 3 (by 84.5%), and case 4 (by 1.3%). Figs. 4 and 5 show the nor of the control action, u, for x() = 1 (Fig. 4) and x() = 2 (Fig. 5), for the optial case (solid line), the linearized case (long-dashed line), case 3 (short-dashed line), and case 4 (dotted line). Clearly, the control action used for stabilization in the case of optial actuator placeent is saller than all the other cases. We also tested the proposed optial sensor locations z s1 =:31 and z s2 =:72. To this end, we ipleent the nonlinear output feedback controller of Eq. (39) on the 3th order Galerkin truncation of the syste of In the second set of the siulation runs, we assue that three control actuators and easureent sensors are available for stabilization. Following Assuption 2, we use Galerkin s ethod to derive a third-order approxiation of the PDE syste which is used for controller design and optial actuator=sensor placeent. To reduce the size of the paper, we proceed with the presentation of the results. Initially, we synthesized and ipleented a state feedback controller on the 3th order Galerkin truncation of the syste of Eqs. (26) (28) in order to copute the optial actuator locations. Table 3 shows the values of the costs Ĵ u ; Ĵ x, and Ĵ of the full-order closed-loop syste under state feedback control, in the case of optial actuator placeent, and for the sake of coparison, the values of these costs in the case of alternative actuator placeents. Clearly, in the case of optial actuator placeent at :17; :5, and :81, the cost of the control action used to stabilize the syste at x(z; t) = is saller than case 2 (by 32.9%), and case 3 (by 66.5%). Figs show the nor of the control action, u, for x() = 1 (Fig. 9), x()= 2 (Fig. 1), and x()= 3 (Fig. 11), for the optial case (solid line), case 2 (long-dashed line),

11 C. Antoniades, P. D. Christodes / Cheical Engineering Science 56 (21) Table 3 Results for three control actuators Case Actuator locations ˆ J u ˆ J x ˆ J Optial :17; :5; : :1; :5; : :2; :6; : Fig. 6. Closed-loop nor of the estiation error e versus tie, for the two actuator=sensor exaple, and for the optial actuator=sensor locations, for x s() = 1 (solid line) and x s() = 2 (dashed line). Fig. 9. Closed-loop nor of the control eort, u, for the three actuator=sensor exaple, for x s() = 1, for the optial case (solid line), the case 2 (long-dashed line), case 3 (short-dashed line), and case 4 (dotted line). Fig. 7. Prole of evolution of the teperature of the rod, for the two actuator=sensor exaple, under output feedback control, for the optial actuator=sensor locations, for x s() = 1. Fig. 1. Closed-loop nor of the control eort, u, for the three actuator=sensor exaple, for x s() = 2, for the optial case (solid line), the case 2 (long-dashed line), case 3 (short-dashed line), and case 4 (dotted line). Fig. 8. Prole of evolution of the teperature of the rod, for the two actuator=sensor exaple, under output feedback control, for the optial actuator=sensor locations, for x s() = 2. and case 3 (short-dashed line). In the case of optial actuator placeent, the control action spent is saller. Subsequently, we synthesized and ipleented a nonlinear output feedback controller on the 3th order Galerkin truncation of the syste of Eqs. (26) (28) with actuator locations z a1 =:17; z a2 =:5 and z a3 =:81 and coputed the optial sensor locations. Table 4 shows the values of the costs Ĵ (e); Ĵ u ; Ĵ x, and

12 4528 C. Antoniades, P. D. Christodes / Cheical Engineering Science 56 (21) Fig. 11. Closed-loop nor of the control eort, u, for the three actuator=sensor exaple, for x s() = 3, for the optial case (solid line), the case 2 (long-dashed line), and case 3 (short-dashed line). Fig. 13. Prole of evolution of the teperature of the rod, for the three actuator=sensor exaple, under output feedback control, for the optial actuator=sensor locations, for x s() = 1. Table 4 Results for three control actuators and easureent sensors Case Location of sensors ˆ J (e) ˆ J u ˆ J x ˆ J Optial :13; :4; : e :13; :43; : e :13; :42; : e Fig. 14. Prole of evolution of the teperature of the rod, for the three actuator=sensor exaple, under output feedback control, for the optial actuator=sensor locations, for x s() = 2. Fig. 12. Closed-loop nor of the estiation error e versus tie, for the three actuator=sensor exaple, and for the optial actuator=sensor locations, for x s() = 1 (solid line), x s() = 1 (long-dashed line) and x s() = 3 (short-dashed line). Ĵ of the full-order closed-loop syste under output feedback control, in the case of optial sensor placeent, and for the sake of coparison, the values of these costs in the case of two other sensor locations. The estiation error of the proposed sensor locations at :13; :4, and :73, is clearly saller than the other two cases. Fig. 12 shows the closed-loop nor of the estiation error versus tie, for the optial actuator=sensor locations, for x s () = 1 (solid line), for x s () = 2 (dashed Fig. 15. Prole of evolution of the teperature of the rod, for the three actuator=sensor exaple, under output feedback control, for the optial actuator=sensor locations, for x s() = 3. line), and for x s () = 3 (dotted line). We can see that for all three initial conditions the estiation error is very sall. Finally, Figs display the proles of the evolution of the teperature of the rod, under output feedback control, for the optial actuator=sensor locations, for x s () = 1 (Fig. 13), for x s () = 2 (Fig. 14),

13 C. Antoniades, P. D. Christodes / Cheical Engineering Science 56 (21) and for x s () = 3 (Fig. 15). We can see that the proposed controller with optial actuator=sensor locations, stabilizes the syste to the spatially unifor operating steady state very quickly, for all three cases. 7. Application to a nonisotheral tubular reactor with recycle We consider a nonisotheral tubular reactor shown in Fig. 16, where an irreversible rst-order reaction of the for A B takes place. The reaction is exotheric and a cooling jacket is used to reove heat fro the reactor. The outlet of the reactor is fed to a separator where the unreacted species A is separated fro the product B. The unreacted aount of species A is then fed back to the reactor through a recycle loop. Under standard odeling assuptions, the dynaic odel of the process can be derived fro ass and energy balances and takes the following diensionless x x x 1 Pe T + B 2 T B C exp x1=(1+ x1) (1 + x 2 ) + T (b(z)u(t) x 1 ); x 2 x + 2 x 2 Pe C 2 B C exp x1=(1+ x1) (1 + x 2 ); subject to the boundary x 1 x 2 x 1 (1;t) = Pe T (x 1 (;t) (1 r)x 1f (t) r x 1 (1;t)); = Pe C (x 2 (;t) (1 r)x 2f (t) r x 2 (1;t)); x 2 (1;t) =; (41) where x 1 and x 2 denote diensionless teperature and concentration of species A in the reactor, respectively, x 1f and x 2f denote diensionless inlet teperature and inlet concentration of species A in the reactor, respectively, Pe T and Pe C are the heat and theral Peclet nubers, respectively, B T and B C denote a diensionless heat of reaction and a diensionless pre-exponential factor, respectively, r is the recirculation coecient (it varies fro zero to one, with one corresponding to total recycle and zero fresh feed and zero corresponding to no recycle), is a diensionless activation energy, T is a diensionless heat transfer coecient, u is a diensionless jacket teperature (chosen to be the anipulated input), and b(z) is the actuator distribution function. Note here that for the purposes of this analysis, we will assue that there is no recycle loop dead tie. In order to transfor the boundary condition of Eq. (41) to a hoogeneous one, we insert the nonhoogeneous part of the boundary condition into the Fig. 16. A tubular reactor with recycle. dierential equation and obtain the following PDE representation of the x x x 1 Pe T 2 + B T B C exp x1=(1+ x1) (1 + x 2 )+ T (b(z)u(t) x 1 ) + (z )((1 r)x 1f + r x 1 (1;t)); x x 2 Pe C 2 B C exp x1=(1+ x1) (1 + x 2 ) + (z )((1 r)x 2f + r x 2 (1;t)); (42) where ( ) is the standard Dirac function, subject to the hoogeneous boundary x 1 x 1 (1;t) = Pe T x 1 (;t); x 2 x 2 (;t) = Pe C x 2 (;t); =: (43) The following values for the process paraeters were used in our calculations: Pe T =7:; Pe C =7:; B C =:1; B T =2:5; T =2:; =1:; r =:5; =5:: (44) For the above values, the operating steady state of the open-loop syste is unstable (the linearization around the steady state possesses one real unstable eigenvalue, =:328, and innitely any stable eigenvalues), thereby iplying the need to operate the process under feedback control. We note that in the absence of recycle loop (i.e., r = ), the above process paraeters correspond to a stable steady state for the open-loop syste. The spatial dierential operator of the syste of Eq. (42) is of the for [ ] A1 x A x = 1 A 2 x 2 2 x 1 = Pe T x 1 2 x 2 Pe C x 2 : (45)

14 453 C. Antoniades, P. D. Christodes / Cheical Engineering Science 56 (21) Fig. 17. Spatioteporal evolution of x 1 in the open-loop syste. The solution of the eigenvalue proble for A i can be obtained by utilizing standard techniques fro linear operator theory (see, for exaple, Ray (1981)) and is of the for ij = a2 ij Pe + Pe ; ; 2; j=1;:::; ; 4 ( ij (z)=b ij e Pez=2 cos( a ij z)+ Pe ) sin( a ij z) ; 2a ij i =1; 2; j=1;:::; ; ij (z)=e Pez ij (z); ; 2; j=1;:::; ; (46) where Pe = Pe T = Pe C, and ij ; ij ; ij, denote the eigenvalues, eigenfunctions and adjoint eigenfunctions of A i, respectively. a ij ;B ij can be calculated fro the following forulas: tan( a ij )= Pe a ij ; ; 2; a 2 j=1;:::; ij ( Pe 2 )2 { 1 ( B ij = cos( a ij z)+ Pe ) } 2 1=2 sin( a ij z) dz ; 2a ij i =1; 2; j=1;:::; : (47) A 4th order Galerkin truncation of the syste of Eqs. (42) (44) was used in our siulations in order to accurately describe the process (further increase on the order of the Galerkin truncation was found to give negligible iproveent on the accuracy of the results). Fig. 17 shows the open-loop prole of x 1 along the length of the reactor, which corresponds to the operating unstable steady-state. Therefore, the control proble is to anipulate the wall teperature, u(t), in order to stabilize the reactor at the desired operating steady-state, and the control output was dened as y c (t) = 1 e Pez 11 (z)x 11 dz. The process was initially (t = ) assued to be at the unstable steady state, and the desired reference input value was set at v =:12. Based on siulations of the open-loop process dynaics, we take as the slow odes of the process the rst eight teperature odes plus the rst thirty concentration odes and use Galerkin s ethod to derive a 38th-order ODE syste eployed for controller synthesis. We use one control actuator to stabilize the syste (note that this is possible since the assuption = l is sucient and not necessary; see also discussion in Reark 3), and the actuator distribution function was taken to be b(z)=(z z act ) (one point control actuator placed at z = z act ). Since it is not feasible in practice to easure the concentration of species A in the reactor at 3 spatial positions, we use eight point teperature sensors to obtain estiates of the rst eight odes of the reactor teperature (i.e., the easureent sensor shape function takes the for s(z)=[(z z s1 ) (z z s2 ) ::: (z z s8 )] T ) and design a nonlinear Luenberger-type state observer consisting of 3 ordinary dierential equations to obtain estiates of the rst 3 concentration odes fro the teperature easureents (see, for exaple, Christodes (1998) for details on how such an observer can be designed). Since the process is initially (t = ) assued to be at the unstable steady state, we will copute the optial location of control actuator and easureent sensors with respect to this initial condition. Furtherore, since the reference input value is set at v=:12, we dene the costs J; J x, and J u as follows: J = J x + J u = ((x s x sf ) T Q s (x s x sf ) +(x f x ff ) T Q f (x f x ff )) dt + (u u f ) T R(u u f )dt; (48) where x sf ;x ff, and u f are the values of x s ;x f, and u, respectively, at the desired operating steady state, to ensure that the costs becoe zero when the process is stabilized at the steady state. Table 5 shows the values of the costs J u ;J x, and J for state feedback control, in the case of optial actuator placeent, and for the sake of coparison, the values of these costs in the case of alternative actuator placeents. Fig. 18 shows the control action u, for the optial case (solid line), case 2 (long-dashed line), case 3 (short-dashed line), case 4 (dotted line), and case 5 (dashed-dotted line). Clearly, the control action spent to stabilize the syste at the desired operating prole in Table 5 Results for dierent actuator location Case Actuator location J u(1 5 ) J x(1 5 ) J (1 5 ) Optial , ,79.3

15 Table 6 Results for dierent sensor locations C. Antoniades, P. D. Christodes / Cheical Engineering Science 56 (21) Case Sensor locations J u(1 5 ) J x(1 5 ) J (1 5 ) 1.5,.15,.3,.45,.55,.7,.85, ,.2,.32,.44,.56,.68,.8, ,.2,.35,.45,.55,.65,.8, ,.18,.31,.43,.56,.68,.81, Fig. 18. Closed-loop control eort, u, for the optial actuator location at z act= (solid line), for the actuator location at z act=:1 (long-dashed line), at z act =:2 (short-dashed line), at z act =:3 (dotted line), and at Z act =:4 (dashed-dotted line). Fig. 19. Closed-loop control eort, u, for four dierent sensor locations; case 1 (solid line), case 2 (long-dashed line), case 3 (short-dashed line), and case 4 (dotted line). the case of optial actuator placeent at z act =, is signicantly less than the other four cases. We now proceed with the output feedback ipleentation of the state feedback controller. To this end, we pick the sensors locations so that the resulting output feedback controller guarantees stability of the closed-loop syste and the estiation error in the closed-loop syste is very sall. Table 6 shows values of the costs J u ;J x, and J for four dierent sensor locations. Figs. 19 and 2 show the prole of the control action u, and the nal steady-state prole of x 1, for case 1 (solid line), case 2 (long-dashed line), case 3 (short-dashed line), and case 4 (dotted line). Clearly, in all these cases, the stabilization of the unstable steady state is achieved with coparable cost, thereby indicating that 8 pint teperature sensors distributed appropriately along the length of the reactor suce to obtain a stabilizing output feedback controller. To deonstrate the perforance of the controller, Fig. 21 shows the evolution of the closed-loop reactor teperature for the optial actuator locations and for the sensor location of case 1. The controller stabilizes the process very close to the desired operating prole. Reark 8. Referring to the above exaples, we note that the optial sensor locations depend signicantly on the choice of boundary conditions and the location of the Fig. 2. Final steady-state prole of x 1, for four dierent sensor locations; case 1 (solid line), case 2 (long-dashed line), case 3 (short-dashed line), and case 4 (dotted line). control actuators. When the process states at the boundaries are xed at a constant value (Dirichlet-type boundary conditions), the sensors are placed away fro the boundaries since we cannot gain uch inforation about the syste behavior by placing the sensors close to the boundaries. On the other hand, if the boundary conditions are of ixed type (Robin-type boundary conditions), the states of the syste close to the boundaries signicantly

16 4532 C. Antoniades, P. D. Christodes / Cheical Engineering Science 56 (21) Fig. 21. Spatioteporal evolution of x 1 under the nonlinear output feedback controller with the optially placed actuator, and the rst conguration for the locations of the sensors. change with tie, and thus, easureents close to the boundaries provide ore inforation about the dynaics of the syste. In addition, the sensors should be placed away fro the actuator locations in order to obtain ore inforation about the dynaics of the syste, since on and near the control actuators the state of the syste is aected ore by the dynaics of the controller and less by the dynaics of the process. Reark 9. Referring to the design of the gain of the nonlinear Luenberger-type state observer used to obtain estiates of the rst 3 concentration odes fro the teperature easureents, we note that the gain was chosen so that the poles of the linearization of the observer are stable (i.e., they lie in the left-half of the coplex plane) and close in agnitude to the slow eigenvalues of the linearized open-loop process. This was done to ake sure that the state observer does not introduce additional fast dynaics (as, for exaple, would be the case if we were using an observer whose poles were of the order 1=), which could perturb the separation of the slow and fast odes of the open-loop PDE syste. 8. Conclusions In this work, we proposed a general and practical ethodology for the integration of nonlinear output feedback control with optial placeent of control actuators and easureent sensors for transport-reaction processes described by a broad class of quasi-linear parabolic PDEs. Given a class of stabilizing nonlinear state feedback controllers which were derived on the basis of nite-diensional approxiations of the PDE, the optial actuator location proble was forulated as the one of iniizing a eaningful cost functional that includes penalty on the response of the closed-loop syste and the control action and was solved by using standard unconstrained optiization techniques. Then, under the assuption that the nuber of easureent sensors is equal to the nuber of slow odes, estiates for the states of the approxiate nite-diensional odel fro the easureents were coputed and used to derive nonlinear output feedback controllers. The optial location of the easureent sensors was coputed by iniizing a cost function of the estiation error in the closed-loop innite-diensional syste. It was rigorously established that the proposed output feedback controllers enforce stability in the closed-loop innite-diensional syste and that the solution to the optial actuator=sensor proble, which is obtained on the basis of the closed-loop nite-diensional syste, is near-optial in the sense that it approaches the optial solution for the innite-diensional syste as the separation of the slow and fast eigenodes increases. The proposed ethodology was successfully applied to a diusion-reaction process and a nonisotheral tubular reactor with recycle to derive nonlinear output feedback controllers and copute optial actuator=sensor locations for stabilization of unstable steady-states. Acknowledgeents Financial support fro NSF, CTS-2626, is gratefully acknowledged. Appendix A. A.1. Proof of Theore 1 The proof of this theore will be obtained in two steps. In the rst step, we will show exponentially stability and closeness of solutions for the closed-loop syste of Eq. (11), provided that the initial conditions and are suciently sall. In the second step, we will exploit the closeness of solutions result to show that the cost associated with the closed-loop PDE syste approaches the optial cost associated with the closed-loop nite-diensional syste under state feedback control, when the initial conditions and are suciently sall, thereby establishing that the location of the control actuators obtained by using the nite-diensional syste is near-optial. Exponential stability closeness of solutions: Using that = Re 1 = Re +1 and under the controller of Eq. (15), the closed-loop syste of Eq. (11) takes the for dx s = s x s + f s (x s ;x f ) f s (x s ; ); f = A f x f + f (x s ;x f ); (A.1)