sensor placement for linear parabolic PDEs using spatial H2 norm

Chemical Engineering Science 61 (2006) 7351 – 7367 www.elsevier.com/locate/ces

Optimal actuator/sensor placement for linear parabolic PDEs using spatial H2 norm Antonios Armaou a,∗ , Michael A. Demetriou b a Department of Chemical Engineering, The Pennsylvania State University, University Park, PA 16802, USA b Department of Mechanical Engineering, Worcester Polytechnic Institute, Worcester, MA 01609, USA

Received 29 October 2005; received in revised form 30 May 2006; accepted 6 July 2006 Available online 24 July 2006

Abstract The present work focuses on the optimal, with respect to certain criteria, placement of control actuators and sensors for transport-reaction processes which are mathematically modeled by linear parabolic partial differential equations. Using modal decomposition to discretize the spatial coordinate, and the notions of spatial and modal controllability and observability, the semi-infinite optimization problem is formulated as a nonlinear optimization problem in an abstract space, which is subsequently solved using standard search algorithms. The proposed method is successfully applied to a representative process, modeled by a one-dimensional parabolic PDE, where the optimal location of a single and multiple point actuators are computed. 䉷 2006 Published by Elsevier Ltd. Keywords: Transport processes; Optimization; Spatial norms; Spillover effects; Actuator network; Actuator placement; Sensor network; Sensor placement

1. Introduction A large number of industrially important chemical processes exhibit variation of the process variables in space, such as transport-reaction processes, where an interplay between reaction and momentum, mass and energy transport phenomena exists. Examples from the microelectronic industry include chemical vapor deposition and plasma etching processes (Armaou and Christofides, 1999; Varshney and Armaou, 2006). Process models can be derived from dynamic conservation equations and usually involve parabolic partial differential equation (PDE) systems. The issue of control and optimization of transport-reaction processes has received considerable attention in the last two decades (Christofides, 2001; Biegler et al., 2003; Armaou and Christofides, 2002; Karafyllis and Daoutidis, 2002; Christofides and Daoutidis, 1997). One research direction focuses on the development of reduced-order models based on the property

∗ Corresponding author. Tel.: +1 814 865 5316; fax: +1 814 865 7846.

E-mail addresses: [email protected], [email protected] (A. Armaou), [email protected] (M.A. Demetriou). 0009-2509/$ - see front matter 䉷 2006 Published by Elsevier Ltd. doi:10.1016/j.ces.2006.07.027

of parabolic PDEs that the eigenspectrum of the spatial operator can be partitioned into a finite size set of eigenvalues that are close to the imaginary axis and an infinite size set of eigenvalues that are far in the left half plane, implying that the dominant behavior of the system can be accurately captured by a finite number of eigenmodes (Temam, 1988). Controllers are subsequently designed based on the reduced-order models (Christofides, 2001; El-Farra et al., 2003; Armaou and Christofides, 2000). An important issue for the controller design methodology is the actuator placement such that the system exhibits desired system theoretic properties such as enhanced controllability. Another approach for controller design is based on the notions of passivity (Ydstie and Alonso, 1997) and on the definitions of spatial H2 norms (Jovanovi´c et al., 2003; Jovanovi´c and Bamieh, 2005). The conventional approach to actuator placement is to select the locations based on open-loop considerations to ensure that the necessary controllability, reachability or power factor requirements are satisfied (Tali-Maamar et al., 1994; Waldraff et al., 1998; Omatu and Seinfeld, 1983; Iwasaki et al., 2003; Slater and Inman, 1997; Burke and Hubbard, 1990; Liang et al., 1995; Arbel, 1981; Gaudiller and Hagopian, 1996; Wang and Wang, 2001; Bruant and Proslier, 2005). For further work

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on different aspects of actuator and sensor placement based on controllability/observability, the reader is directed to the survey papers of van de Wal and de Jager (2001), of Kubrusly and Malebranche (1985) and the book by Uci´nski (2004). The issue of integrated observer design and sensor placement has been investigated in Alonso et al. (2004), while the issue of integrating feedback control and optimal actuator placement has been examined for linear (Dochain et al., 1997; Omatu and Seinfeld, 1989; Demetriou, 1999) and quasi-linear (Antoniades and Christofides, 2001, 2002) parabolic PDEs, in actively controlled structures (Rao and Pan, 1991; Sunar and Rao, 1997; Chen et al., 2004; Frecker, 2003; Shelley and Clark, 2000) and in acoustic and structural acoustic systems (Fahroo and Demetriou, 2000; Demetriou and Fahroo, 1999; Fahroo and Demetriou, 1999). Using quadratic performance measures, parameterized by the actuator and sensor locations, the optimal position was found as the one yielding the minimum of the optimal control measures. For the linear parabolic PDEs this amounts to parameterizing the solution to the associated algebraic Riccati equation by the candidate actuator locations and minimizing the optimal value of the trace of the locationparameterized Riccati solution (for an in-depth exposure see Omatu and Seinfeld, 1989). However, when using such performance measures, one might obtain an actuator location that in a sense is “averaged” over all frequencies; consider for example the specific case of the one-dimensional heat equation with Dirichlet boundary conditions and with constant diffusivity parameter. When the solution to the ARE is parameterized by the actuator locations, which are assumed to have a spatial distribution given by the spatial Dirac delta function (point actuator), one obtains the midpoint of the spatial domain as the optimal actuator location with the smallest performance index. This location is also the zero of the second eigenfunction, which means that the actuator will have no authority to handle exogenous signals containing the second eigenfrequency. Results have also been reported on the identification of optimal strategies for the activation of actuators (from a pool of available ones) according to predetermined criteria for distributed processes (Demetriou, 2001a,b), and the integration of these strategies with controller design (Demetriou and Kazantzis, 2004a,b). The present work deals with the optimal placement of control actuators for transport-reaction processes, mathematically modeled by linear parabolic PDEs. Using modal decomposition to discretize the spatial coordinate, and based on the definitions of spatial and modal controllability (Halim and Moheimani, 2003; Moheimani et al., 2003), the semi-infinite optimization problem is formulated as a nonlinear optimization problem in appropriate L2 spaces. Standard optimal search algorithms are subsequently used to obtain the optimal location. The approach considered here for actuator placement has a different flavor than earlier methods, in that it caters to specific low frequencies/modes of the system while at the same time takes into consideration medium-range frequencies often neglected in traditional model reduction techniques. The measure for actuator placement considers the enhancement of the actuator location over the dominant modes, while at the same time offers robustness with respect to the intermediate range of modes which

certainly ameliorate spillover effects. A summarized version of this work appeared in Demetriou and Armaou (2005), which consider the problem of selecting a single actuator, but it did not address the issue of location redundancy and the problem of placing multiple actuators. The mathematical formulation of the problem under consideration is presented in the next section. The notion of the spatial norms along with the associated spatial and modal controllabilities that are used as an optimization measure of the actuator placement problem are given in Section 3. The optimization algorithms and the modifications that account for spillover effects are also given in Section 3. The dual problem of sensor placement is summarized in Section 4. The proposed method is applied to a representative diffusion process, where the location of single and multiple point actuators are considered in Section 5 with conclusions following in Section 6. 2. Mathematical preliminaries We consider the problem of computationally identifying the optimal locations of actuators for processes that can be mathematically described by parabolic partial differential equation (PDE) systems of the form j x(t,
) = A(
)x(t,
) + b(
;
a )u(t), jt

(1)

where x(·,
) ∈ R is the state, t ∈ R+ is the time,
∈  is the spatial coordinate and  is a bounded smooth domain in Rn (n = 1, 2 or 3), and u(t) ∈ RK is the manipulated variable vector. A is a second-order (strongly) elliptic operator (Dautray and Lions, 2000a) of the form   n n

j j(
) A = j k (
) j
j j
k j =1 k=1 n


+

j (
)

j =1

j(
) + 0 (
)(
), j
j

for
∈ , and b(
;
a ) = [b1 (
;
a,1 ) b2 (
;
a,2 ) · · · bK (
;
a,K )], where bj (
;
a,j ) denotes the spatial distribution of the j th actuating device (e.g., boundary, distributed and/or pointwise) placed at location
a,j , the j th component  of the vector
a = {
a,1 ,
a,2 , . . . ,
a,K } ∈  ;  ⊆ K  denotes the domain of permissible actuator locations. All three cases of boundary conditions for the above PDE system may be considered: mixed (Robin), Neumann or Dirichlet (Dautray and Lions, 2000b), where the boundary, denoted by j, can be decomposed to j = a ∪ b with a denoting the part of the boundary where the actuator(s) may be placed and b (=j\a ) the remainder of the boundary where actuators are not desired or allowed to be placed. Defining an appropriate Hilbert space H = L2 () with inner product  1 , 2 L2 = ∗1 (
)2 (
) d
, (2) 

A. Armaou, M.A. Demetriou / Chemical Engineering Science 61 (2006) 7351 – 7367

 and norm L = 1 , 2 L2 , the PDE system of (1) can be equivalently written in the following abstract form (Dautray and Lions, 2000a; Curtain and Zwart, 1995): x(t) ˙ = Ax(t) + Bu(t),

(3)

where x(t, ·) = x(t) ∈ H is the state and B denotes the input operator either on the part of the boundary a where actuation is desired or permissible, or on the permissible part of the interior of the domain  . The PDE of (1) can be solved independently for each mode by using the orthogonality properties of the eigenfunctions of spatial operator A (Courant and Hilbert, 1989). In this case  

∗j (
)Ai (
) d
= i j i ,

 

∗j (
)i (
) d
= j i ,

(4)

where i denotes the ith eigenvalue, and j i denotes the Kronecker delta. We assume that the eigenvalues are ordered such that i+1
i , ∀i = 1, . . . , ∞. Using the computed eigenfunctions as a basis function set for H, x(t,
) can be equivalently expressed via the expansion x(t,
) =

∞


The resulting K-input/distributed(infinite)-output transfer function is then given by G(s,
;
a ) =

∞


i (
)

i=1

=

∞


1 Bi (
a ) (s − i )

i (
)Gi (s;
a ).

(9)

i=1

A property of strongly elliptic operators is that their eigenspectrum can be decomposed into a finite number of eigenvalues that are close to the imaginary axis and an infinite one that lies far in the left half plane, which implies that the long-term dynamics of the process can be accurately captured by only a finite number of eigenmodes, while the infinite complement eigenmodes relax to their steady-state values fast. Based on this property, we assume that only N eigenmodes need be considered. As a result, we restrict the expansion of (9) to the first N terms GN (s,
;
a ) = =

i (
)xi (t).

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N
i=1 N


i (
)

1 Bi (
a ) (s − i )

i (
)Gi (s;
a ).

(10)

i=1

(5)

i=1

3. Actuator placement Employing Laplace transforms (Debnath, 1995) with L[xi (t)] = Xi (s), the system is represented in the s domain  1 ∗ (
)b(
;
a ) d
U (s) (6) (s − i )  i  with X(s,
;
a ) = ∞ i=1 i (
)Xi (s;
a ). We define the Kdimensional row vector  Bi (
a ) ∗i (
)b(
;
a ) d
   ∗ i (
)b1 (
;
a,1 ) d
∗i (
)b2 (
;
a,2 ) d
=    ... ∗i (
)bK (
;
a,K ) d
, Xi (s;
a ) =



and the transfer function matrix of the ith eigenmode as Gi (s;
a )

1 Bi (
a ). (s − i )

=

∞
i=1 ∞
i=1

i (
)Xi (s;
a ) =

(7)

∞


The spatial H2 norm (Moheimani et al., 2003) of the transfer function G(s,
;
a ) in (9) is defined as  ∞ 1 G2H2  tr{G∗ (j ,
;
a )G(j ,
;
a )} d
d . 2 −∞  (11)

G2H2 = =

∞


i (
)Gi (s,
a ) U (s)

1 Bi (
a ) U (s). (s − i )

(8)

Gi (s,
a )22

i=1

∞ 
i=1

i=1

i (
)

3.1. Spatial norms and controllability measures

Using the orthogonality property of the eigenfunctions, the above norm simplifies to

The Laplace transform of the spatial distributed state can then be represented as the infinite sum X(s,
;
a ) =

We may now pose the computation of the optimal actuator locations problem as an optimization problem, which is constrained in a set of admissible actuator locations and which ensures certain open-loop objectives are satisfied. We initially introduce the necessary mathematical background leading to the definition of the objective function and constraint functions.

∞ −∞

tr{G∗i (j ;
a )Gi (j ;
a )} d
.

(12)

With the aid of this spatial H2 norm, a number of measures of actuator placement effectiveness can be proposed. We define



1



(13) Bi (
a )

fi (
a )Gi (s;
a )2 =

. (s − i ) 2

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A. Armaou, M.A. Demetriou / Chemical Engineering Science 61 (2006) 7351 – 7367

Following Halim and Moheimani (2003), we define the ith modal controllability at actuator locations
a as follows. Definition 1 (ith modal controllability). The ith modal controllability at actuator locations
a = {
a,1 ,
a,2 , . . . ,
a,K } is defined as fi (
a ) Mi (
a ) = × 100%, i = 1, 2, . . . , N (14) max∈ fi () and describes the total controller authority of all actuators at locations
a over the ith mode. If the modal controllability at specific actuator locations
a of a given mode is zero, it means that none of the controllers has any authority over that mode. This also coincides with the notion of approximate controllability for the class of PDEs with Riesz-spectral operators (Curtain and Zwart, 1995). The requirement for approximate controllability in this case is b, i L2 = 0

∀i.

Indeed, when the ith mode has a zero modal controllability at location
a , it means that fi (
a ) ≡ 0 and hence Bi (
a ) = 0, or that  i (
)∗ b(
;
a ) d
= 0. 

Furthermore, we also consider the cumulative spatial controllability (Moheimani et al., 2003). Definition 2 (Cumulative spatial controllability). The cumulative spatial controllability defined for the first N system modes is given by  N 2 i=1 fi (
a ) SN (
a ) × 100%  N 2 () max∈ f i=1 i  N 2 i=1 Gi (s;
a )2 = × 100%. (15)  N 2 max∈ G (s; ) i i=1 2 We note that the term

N


 Gi (s;
a )22 i=1

corresponds to the truncation of the H2 spatial norm to the first N modes, since G(s,
;
a )2H2 = =

∞


Gi (s;
a )22 =

i=1 N


Gi (s;
a )22 +

i=1

=

N
i=1

∞


fi2 (
a )

i=1 ∞


Gi (s;
a )22

i=N+1

fi2 (
a ) +

∞


fi2 (
a )

i=N+1

= GN (s,
;
a )2H2 + S(
a ),

(16)

 ∞ 2 2 where S(
a ) ∞ i=N +1 Gi (s;
a )2 = i=N +1 fi (
a ) denotes the higher mode contribution. Therefore, the cumulative spatial controllability is also given in terms of the truncated spatial H2 norm via SN (
a )=

GN (s,
;
a )H2 GN (s,
;
a )H2 . =  max∈ GN (s,
; )H2 N 2 max∈ i=1 fi ()

Following Halim and Moheimani (2003), the cumulative spatial controllability at location
a is a measure of controller authority placed at location
a over the entire spatial domain in an average sense (averaged over the first N modes). Even though the above definitions allow us to formulate the optimal actuator placement problem, in order to gain better control over the placement of each individual actuator, we also define the ith modal controllability of the j th actuator at location
a,j . Definition 3 (ith modal controllability of the j th actuator at location
a,j ). The ith modal controllability of the j th actuator at location
a,j is defined as fi,j (
a ) × 100%, i = 1, 2, . . . , N, max∈ fi,j ()





1

∗

. fi,j (
a )

 (
)b (
;
) d
(17) j a,j i

(s −  )

i  2

Mi,j (
a ) =

The ith modal controllability of the j th actuator at location
a,j describes the controller authority of the j th actuator over the ith mode. If the modal controllability at a specific actuator location of a given mode is zero, it means that the specific controller has no authority over that mode. Note that from the definition of fi,j (
a ) and fi (
a ) we have fi2 (
a ) =

K


2 fi,j (
a ).

j =1

Finally, we define the component spatial controllability. Definition 4 (Component spatial controllability). The component spatial controllability is defined as  N 2 i=1 fi,j (
a ) SN (
) × 100%. (18)  a j N 2 max∈ f () i=1 i,j Remark 1. The cumulative and component spatial controllabilities are C2 functions (since they are linear combinations of C2 functions). As a consequence (Bertsekas, 1999), the Hessian and Jacobian matrices of SN (·) and SN j (·) with respect to the locations are well defined for
∈ . Remark 2. An issue which often arises during the search of optimal actuator locations using modal methods is the weight in the objective function that should be assigned to each mode. In the current formulation the weight that is assigned to each mode is dependent upon the eigenvalue of the specific eigenmode. As

A. Armaou, M.A. Demetriou / Chemical Engineering Science 61 (2006) 7351 – 7367

a result, modes that are less stable have a greater contribution to the spatial controllability and as a consequence the optimal actuator placement will gravitate toward locations that assign greater importance to the specific modes. 3.2. Multiple location issues A problem commonly encountered during the placement of multiple actuators is the issue of coupling effects between actuators. An excellent exposure of the optimization and computational issues associated with multiple actuators, including that of actuator clusterization, can be found in the book by Uci´nski (2004). While the actuator placement problem can be formulated as an optimization one for a single actuator using the definitions of the previous subsection, multiple actuators pose additional challenges. One such issue is placing two actuators at or near the same location. Additionally, one must avoid placing actuators at locations that provide certain spatial symmetry (i.e., they affect the system modes in a similar fashion) while placed at different locations. For example, in the case of a one-dimensional heat equation with constant coefficients and Dirichlet boundary conditions, presented in Section 5, an actuator placed at L/3 has the same effects as the one placed at 2L/3 (where L is the length of the process domain). Both locations control the same modes and both render the 3rd, 6th, . . . , (3j )th, j = 1, 2, . . ., modes uncontrollable. To address this issue, with the aid of (17), we define the spatial controllability matrix M ∈ RN×K as ⎡ M (
) M (
) . . . M (
) ⎤ 1,1

a

1,2

a

1,K

a

⎢ M2,1 (
a ) M2,1 (
a ) . . . M2,K (
a ) ⎥ ⎥. M(
a ) = ⎢ .. .. .. .. ⎦ ⎣ . . . . MN,1 (
a ) MN,2 (
a ) . . . MN,K (
a )

(19) The proximity, in an L2 sense, of the actuator locations can be identified through the singular values of M, which are equal to the number of locations K (assuming that the number of modes that are of interest are higher than the number of available actuators N K). Specifically, if the columns of M are linearly dependent then at least one singular value attains the value of zero. The proximity of the columns of M to linear dependence can be identified through the ratio of the largest singular value to the smallest one, which represents the condition number of M (Golub and VanLoan, 1996). This implies that there is a direct relationship between the value of the condition number and the redundancy in the actuator network. Selection of bound c on the condition number that represents the desired minimum distance between actuators, 
, in both a physical and an L2 sense, can be identified through the solution of the following two-level optimization problem:   c = min

{j,k}∈Na

s.t.

min cond(M(
a )) jk


a ∈


a = {
a,j
a,k },
a,k =
a,j + 
, max Mi (
a,k )  , max Mi (
a,j )  ,

i=1,...,N

i=1,...,N

(Pc-I)

7355

where the spatial controllability matrix M is computed for the specific {j, k} pair of actuators, cond(M) represents the conjk dition number, and  ⊆  is the two actuator permissible locations space. The threshold , (0%

100%), denotes the user-defined modal controllability level. The finite countable set of index pairs {j, k} that need be considered, Na , is dependent upon the minimum distance constraints that are imposed between actuators with the same and different distribution functions. For example, if K point actuators are considered, then only one pair of actuators at most need be considered. Remark 3. The constraint in the inner optimization problem accounts for locations where modal controllability of all the modes is zero, thus avoiding locations for which M(
a ) is illdefined. Remark 4. In certain cases, the coupling of the actuators due to the location selection is desired as it gives a level of redundancy in the actuator authority over the modes of the system, thus allowing for robustness with respect to actuator failure (outage). The use of spatial H2 norm and the modal norm in that case is also desirable, to identify actuator networks for which even though no individual actuators occupy the same location, the system however remains controllable in the event of a number of actuator failure. Such a concept was explored in Demetriou (2001a,c, 2003a) for fault tolerant control of flexible structures and in Demetriou (2003b) for parabolic distributed parameter systems. Remark 5. The condition number of the modal controllability matrix, M, is an exhaustive measure of redundancy, beyond simple pairwise actuator location comparisons. Used as a constraint to identify similar locations in an L2 sense, it ensures that there is no redundancy in the complete set of actuators. Conversely, the proposed formulation can be employed within an optimization framework to identify optimal actuator locations that confer user-defined levels of redundancy to the system. The methodological tools to enforce such a redundancy are beyond the scope of the present work and are the focus of a complimentary article. 3.3. Spillover effects While the notions of spatial and modal controllability allow one to choose actuator locations that cater to specific (primarily dominant low) modes, care must be exercised in order to avoid choosing locations that might excite medium range modes, that lead to performance deterioration; medium range modes are modes that are at immediate proximity of the first N modes. The effect of the actuators on the higher modes is mathematically described by the term S(
a ) in (16), which, in general, cannot be computed since it is an infinite sum of modal controllabilities. Capitalizing on the property of parabolic systems that the higher modes become progressively more stable (i+1
i and i → ∞ as i → ∞), we assume that the spillover effect on a finite number of higher modes (medium range modes) i (
),

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A. Armaou, M.A. Demetriou / Chemical Engineering Science 61 (2006) 7351 – 7367

i =N +1, . . . , M need be considered. The spatial controllability for these medium range modes is similarly defined as: Definition 5 (Cumulative spillover spatial controllability). The cumulative spillover effect of K actuators placed at locations
a to M intermediate modes is defined as  M 2 i=N+1 fi (
a ) N SM (
a ) × 100%. (20)  M 2 max∈ i=N+1 fi () Similarly, the component spillover of the j th actuator is defined as: Definition 6 (j th component spillover spatial controllability). The spillover effect of the j th actuator at location
a,j to M intermediate modes is defined as  M 2 i=N+1 fi,j (
a ) N SM j (
a ) × 100%. (21)  M 2 () max∈ f i=N+1 i,j Remark 6. The search for a precise bound on the condition number that represents the desired minimum distance between actuators, can also be refined to exclude regions with high spillover effects.   c = min

{j,k}∈Na

s.t.

min cond(M(
a )) jk


a ∈


a = {
a,j
a,k },
a,k =
a,j + 
, max Mi (
a,j )  , SN M j (
a,j )
,

i=1,...,N

max Mi (
a,k )  ,

i=1,...,N

SN M k (
a,k )
,

constrained optimization problem is thus formulated as
opt = arg max


a ∈

s.t.

where is the maximum allowable level for spatial controllability of the higher modes to implicitly enforce spillover reduction. Remark 7. The optimal actuator solution involves knowledge of modal controllabilities as a function of the spatial coordinate (by definition C2 functions). To reduce the computational requirements of the method we sample the modal controllability functions over the spatial domain and subsequently use cubic splines (Höllig, 2003) for the intermediate regions. Cubic splines are used due to their C3 properties, guaranteeing Jacobian and Hessian continuity when gradient-based algorithms are used (see also Remark 1). To ensure correct optimal location placement, modal controllabilities may be locally sampled again during the search, when higher accuracy is required. 3.4. Program formulation It is desired to find K actuator locations, denoted in vector form by
a ∈  , that maximize the cumulative spatial controllability of the system while at the same time maintain a reasonable level of controller authority over each mode. The

cond(M(
a ))
c, max Mi,j (
a )  i j =1,...,K

∀i = 1, 2, . . . , N. (Pc-III)

The use of component modal controllabilities over total modal controllabilities in the optimization problem is preferred, to account for possible solutions where one actuator is redundant (i.e., with low modal controllability values for all the modes). In this case the optimization problem is infeasible. To implicitly address the issue of actuator spillover that inadvertently affects the closed-loop process response, we need to quantitatively include an additional constraint that guarantees a maximum level of spatial controllability for higher-frequency modes. The optimization problem of (Pc-III) is now reformulated as
opt = arg max


a ∈

s.t.

SN (
a ) cond(M(
a ))
c, max Mi,j (
a )  i j =1,...,K SN M j (
a )


∀i = 1, 2, . . . , N,

∀j = 1, 2, . . . , K, (Pc-IV)

where is the maximum allowable level for spatial controllability of the higher modes to implicitly enforce spillover reduction. An issue of the specific formulation is that even though it gives us precise control over the spillover effects that our choice of locations has, it may render the optimization problem infeasible. To circumvent this issue, we reformulate the problem as
opt = arg max

(Pc-II)

SN (
a )


a ∈

s.t.

{SN (
a ) − s SN M (
a )} cond(M(
a ))
c, max Mi,j (
a )  i

j =1,...,K SN M j (
a )


∀i = 1, 2, . . . , N,

∀j = 1, 2, . . . , K, (Pc-V)

where, the constraint on spillover effect now becomes a soft constraint, when the hard constraint on the spillover effects are inactive by setting = 100%. The relative importance of minimizing the spillover effect, compared to obtaining locations that maximize the spatial controllability is now controlled through the selection of the weight s . The formulated optimization problems of (Pc-III) and (PcIV) can be solved using standard search algorithms such as Newton-based, interior-point or direct-search methods (Kelley, 1999; Bertsekas, 1999). Due to the nonlinear nature of the objective function and the inequality constraints, global optimization methods are preferable, including BB (Floudas, 2000), particle swarm optimization (Krusienski, 2004) and simulated annealing. Another issue is the “rugged” topology of the objective function (accentuated in (Pc-V)); algorithms that efficiently handle such objective functions have been developed, including funneling (Lucia et al., 2004), derivative free and direct search (Kelley, 1999; Kolda et al., 2003; Polak, 1987)

A. Armaou, M.A. Demetriou / Chemical Engineering Science 61 (2006) 7351 – 7367

7357

optimization algorithms. Furthermore, through sampling and estimation, one can identify surrogate models, subsequently used to identify local or global (e.g., see Jones et al., 1998; Meyer et al., 2002; Kelley and Sachs, 2003) optima. In the event the set a is disjoint, branch and bound algorithms (Bertsekas, 1999) and genetic algorithms (Rao and Pan, 1991) can also be used to obtain the optimal actuator locations. In the current work a two-level optimization scheme was employed, comprising of an inner sequential quadratic programming algorithm (guaranteeing local optimality), and a simulated annealing outer structure, to identify the optimal locations.

vector is defined as   c(
;
s )i (
) d
= cj (
;
s,j )i (
) d
. Ci (
s ) =

Remark 8. Similar to the optimization problem of (Pc-V), we may impose soft constraints on the minimum distance between the actuators, avoiding clustering effects, through the augmentation of the term − c cond(M(
a )) in the objective function, resulting in max
a ∈ {SN (
a )− s SN M (
a )− c cond(M(
a ))}. The relative importance of clustering to maximizing the spatial controllability is controlled through the appropriate choice of values for c . Note that due to the wide range of values that the condition number may have and the fact that it is always larger than one, it is preferable to use the logarithm of the condition number. Alternatively, the inverse of the condition number may be used in a minor modification of the formulated problem, motivated by Remark 9.

4.1. Observability measures

Remark 9. Even though the problem formulation necessitates the computationally intensive step of calculating the singular values of M(
a ) at each iteration, for the usual size of the problem under consideration, the computational requirements of the used search and the numerical algebra algorithms are well within the capabilities of current processors. Furthermore, a number of algorithms exist that may numerically estimate the inverse of the condition number, further reducing the computational cost. 4. Sensor placement considerations





j

Employing Laplace transforms for the state vector  1 Xi (s;
s ) = c(
;
s )i (
) d
x0 , (s − i )  we define the transfer function matrix of the ith eigenmode as Hi (s;
s )Ci (
s )

1 . (s − i )

Defining the open-loop transfer function between the infinite-dimensional state and the sensor output, we may define the ith modal observability at sensor locations
s = {
s,1 ,
s,2 , . . . ,
s,K } as follows (cf. Definition 1). Definition 7 (ith modal observability). The ith modal observability at sensor locations
s is defined as Li (
s ) =

gi (
s ) × 100%, max∈ gi ()

i = 1, 2, . . . , N,

with



1 gi (
s )Hi (s;
s )2 =

Ci (
s ) s − 

(22)







i 2

and describes the capability of all sensors at locations
s to measure the ith mode. Furthermore, we also consider the cumulative spatial observability (Moheimani et al., 2003). Definition 8 (Cumulative spatial observability). The cumulative spatial observability defined for the first N system modes is given by  N 2 i=1 gi (
s ) N T (
s ) × 100%. (23)  N 2 max∈ i=1 gi ()

In general the sensor placement problem is the dual problem to the actuator placement one. We thus pose the computation of the optimal sensor locations problem as an optimization problem, which is constrained in a set of admissible sensor locations to ensure certain open-loop objectives are satisfied. Within the context of sensor placement, the system dynamics are given by

The cumulative spatial observability of the sensor network, located at
s is a measure of the observation capability of the sensors placed at location
s over the entire spatial domain in an average sense (averaged over the first N modes).

j x(t,
) = A(
)x(t,
), x(0,
) = x0 , jt  y(t) = c(
;
s )x(t,
) d
,

Similar to the dual problem of actuator placement, we also define the ith modal observability of the j th sensor at location
s,j , in order to further refine the placement problem for each individual sensor.



where y ∈ RK is the vector of measured variables, c(
;
s ) = [cj (
;
s,j )] denotes a vector with j th component the sensor shape function of the j th sensor, located at
s,j ;
s is a vector with j th component the location of the j th sensor,
s,j . To obtain the transfer function matrix, the K-dimensional column

Definition 9 (ith modal observability of the j th sensor at location
s,j ). The ith modal observability of the j th sensor at location
s,j is defined as Li,j (
s ) =

gi,j (
s ) × 100%. max∈ gi,j ()

(24)

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The modal spatial observability describes the capability of the j th sensor, placed at location
s,j , to measure the ith mode. To circumvent the issue of coupling effects between sensors, (e.g., placing sensors at locations that provide certain spatial symmetry i.e., they observe similar combinations of system modes) while placed at different locations, we define the spatial observability matrix L ∈ RN×K with components: [L(
s )]i,j = Li,j (
s ).

(25)

Selection of bound c on the condition number that represents the desired minimum distance between sensors, 
, in both a physical and an L2 sense, can be identified through the solution of a two-level optimization problem similar to (Pc-I). Finally, to address the performance deterioration of the sensor network, due to the contribution of the higher modes to the measurements, and capitalizing on the property of parabolic PDEs that the contribution of the higher modes becomes progressively more negligible, we define the following measure of the effect of a finite number of higher modes (medium range modes) i (
), i = N + 1, . . . , M on the sensor measurements. Definition 10 (Cumulative spillover spatial observability). The cumulative spillover effect of M intermediate modes to the measurement ability of K sensors placed at locations
s is defined as  M 2 i=N+1 gi (
s ) (
) TN × 100%. (26)  M s M 2 () max∈ g i=N+1 i Similarly, the component spillover of the j th sensor is defined as: Definition 11 (j th component spillover spatial observability). The spillover effect of M intermediate modes to the measurement ability of the j th sensor at location
s,j is defined as  M 2 i=N+1 gi,j (
s ) N TM j (
s ) × 100%. (27)  M 2 max∈ i=N+1 gi,j () Based on the above definitions, we now formulate the optimization problem as one where it is desired to find K sensor locations, denoted in vector form by
s ∈  , that maximize the cumulative spatial observability of the system while at the same time maintain a reasonable level of observability to each mode. Furthermore, the issue of performance deterioration due to higher mode contributions to the measurements is addressed via the inclusion of the spatial observability in the objective function (with an appropriate weight s ). The optimization problem thus takes the form
opt = arg max


s ∈

s.t.

{TN (
s ) − s TN M (
s )} cond(L(
s ))
c, max Li,j (
s )  i

j =1,...,K

∀i = 1, 2, . . . , N, (Pc-VI)

where now i is the level of modal observability and   c = min

{j,k}∈Ns

min cond(L(
s )) . jk


s ∈

5. Numerical example We consider a representative spatially distributed process, mathematically described by the following linear onedimensional PDE: K


j j2 x(t,
) = 0.01 2 x(t,
) + bj (
;
j )uj (t), jt j
j =1

subject to Neumann boundary conditions:   jx  jx  = 0, = 0, j

=0 j

=L where the spatial domain is defined as  = [0, L], with L = 2. The spatial distribution of an actuating device placed at location
a is   1
−
a b(
;
a ) =  ,   where (·) is the standard rectangle function and  > 0 is a small number, >1. The permissible actuator domain is thus defined as  = [0 + /2, L − /2] with the span of the actuating device  = L/500. We initially assume that only one such actuator is available, followed by an analysis for two and subsequently three actuators. For the specific system, the associated eigenvalues and normalized eigenfunctions can be computed analytically (Courant and Hilbert, 1989) and are of the form  2 i i = −0.01 L    2 i
and i (
) = cos , i = 1, . . . , ∞. L L Based on the values of the associated eigenvalues, we assume that for the optimal actuator location, the first five modes are significant and should be considered for the calculation of the objective function SN (
), (i.e., N =5; 5 =−0.617). We anticipate that the subsequent eigenvalues are sufficiently stable to preclude explicit consideration; to reduce spillover effects and avoid excessive excitation of the higher modes, the subsequent 10 higher modes are considered (i.e., M =15 with 15 =−5.55). The subsystem of the higher modes that is neglected is sufficiently stable and has a fast transient, with 16 = −6.317 and  ≡ 1 /16 =0.0039. Figs. 1a and b depict the modal controllabilities for the first five modes. Due to the specific form of the system’s eigenfunctions, we observe that the modal controllabilities are symmetric with respect to the center of the spatial domain
= 1. We initially consider the placement of only one actuator K =1. We require that the minimum level of modal controllability for the first five modes to be i = 70% and the contributions

A. Armaou, M.A. Demetriou / Chemical Engineering Science 61 (2006) 7351 – 7367

7359

% Spatial controllability

Modal controllabilities 100

100

M1(ξ) M2(ξ) M3(ξ)

90

90

80

80

70

70

60

60

50

50

40

40

30

30

20

20

spatial controllability S5(ξ)

10

10

modal controllability level β=70% optimal location

0

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

spatial distance ξ

(a)

0.8

1

1.2

1.4

1.6

1.8

2

spatial distance ξ

Fig. 2. Percentage spatial controllability, S5 (
); K = 1, i = 70%, = 60%.

Modal controllabilities 100 90

% Spillover spatial controllability

80

100

70

90

60

80

50

70

40

60

30

50

20

40

10

30

0

20 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

spatial distance ξ

(b)

5 spillover spatial controllability S15 (ξ)

spatial controllability level c=60% optimal location

10 0

Fig. 1. Percentage modal controllability of (a) modes 1–3 and (b) modes 4 and 5.

of the next 10 modes to the spatial norm to be limited below

= 60% (i.e., limiting the spillover effect of the actuator). The associated constrained optimization problem of (Pc-IV) then takes the specific form
opt = arg max


a ∈

s.t.

S5 (
a ) Mi (
a ) 70%, S515 (
a )
.

i = 1, 2, . . . , 5, (Pc-VII)

Considering the inequality constraint of controllability level of 70% in the optimization program of (Pc-VII), it can be observed that the set of admissible locations (the set of locations for which all modal controllabilities are above 70%, shown in Figs. 1a and b) now becomes  = [0.002, 0.1588] ∪ [1.8412, 1.998]. Due to the symmetry of the process and the

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

spatial distance ξ

Fig. 3. Percentage spillover spatial controllability S515 (
); K = 1, i = 70%,

= 60%.

objective function with respect to the center of the spatial domain
= 1, we focus our attention to region [1.8412, 1.998]. If we do not consider spillover effects (i.e., = 100%), the solution to the optimization problem of (Pc-VII) is computed to be location
opt = 1.998. We observe in Fig. 2 that this value maximizes the objective function, with S5 (1.988)=100%. The issue with the chosen location though is that it also maximizes the spillover effect, with S515 (1.998) = 100%. The spillover spatial controllability S515 (
) is shown in Fig. 3. Constraining the acceptable level of spillover to 60%, we observe that the set of spatial locations that ensure that S515 (
)
60% is given by [0.0699, 0.1589] ∪ [1.8411, 1.9301].

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A. Armaou, M.A. Demetriou / Chemical Engineering Science 61 (2006) 7351 – 7367

modes above 85%. Additionally, the spatial controllability of the sixth to 15th modes S515 (
) is also maintained at a level below 60%. In fact, at the optimal actuator location, one has S515 (
opt ) = 60%. Note that due to symmetry of both process and objective and constraint functions with respect to the center of the process
= 1, position
= 0.0699 is also an optimal location. We subsequently focus on the identification of optimal locations when multiple actuators are available; the issue of actuator redundancy becomes relevant. We initially consider two actuators, K = 2 with a minimum distance requirement between the two actuators of 
= 0.024. The constrained optimization problem of (Pc-IV) attains the specific form

condition number value 100 δξ =0.025 δξ =0.05 δξ =0.01

80

60

40

20


opt = arg max


a ∈

0 0

0.5

1

1.5

2

ξ

Fig. 4. Condition number value of M as a function of minimum actuator distance 
.

Table 1 Controllability values at optimal actuator location for K = 1


opt S5 (
opt ) S515 (
opt )

1.930 98.03% 60.00 %

M1 (
opt ) M2 (
opt ) M3 (
opt ) M4 (
opt ) M5 (
opt )

99.40% 97.62% 94.66% 90.59% 85.42%

Optimization parameters: i = 70% (∀i = 1, . . . , 5), = 60%.

Table 2 Controllability at optimal locations for K = 2; nominal optimization parameters: i = 60% (∀i = 1, . . . , 5), = 60% Case

c = 10

c = 100

c = 103


opt S5j (%) S515j (%) M1 (%) M2 (%) M3 (%) M4 (%) M5 (%)

[1.857 1.930] 92.52 98.03

[0.08099 0.06964] 97.36 98.03

[0.07087 0.06964] 97.96 98.03

54.01 97.49 90.08 78.15 62.29 43.30

51.42 99.19 96.78 92.81 87.34 80.45

59.00 99.38 97.53 94.47 90.25 84.91

60.00 99.40 97.62 94.66 90.59 85.42

60.00 99.40 97.62 94.66 90.59 85.42

60.00 99.40 97.61 94.66 90.59 85.42

Its intersection with  would then yield the admissible set as = [0.0699, 0.1588] ∪ [1.8412, 1.9301] (Fig. 4 and Table 1). Solving the optimization problem of (Pc-VII), the optimal actuator location was computed to be
= 1.9301, shown in Fig. 2, which depicts the spatial controllability S5 (
) as a function of the spatial distance
. Table 2, summarizes the levels of controllabilities at the optimal location
opt = 1.9301. This optimal location maintains a spatial controllability above 98% while maintaining a modal controllability of the first five

s.t.

S5 (
a ) cond(M(
a ))
c, max Mi,j (
a )  i

j =1,2 S515j (
a )
,

∀i = 1, . . . , 5,

j = 1, 2. (Pc-VIII)

The minimum level of modal controllability for the first five modes was chosen to be i = = 60%, ∀i = 1, . . . , 5. Figs. 1a and b depict the modal controllabilities of the first five modes. We observe that for every value of
in the set of admissible locations there is at least one component modal controllability above the limit i = 60%. Considering thus the inequality constraints of controllability level of 60% in the optimization program of (Pc-VIII), the set of admissible locations remains the whole process domain (i.e., the associated inequality constrains will always be inactive during the search); this is not the case in general (e.g., for i 75% the modal controllability requirements constraint the set of feasible actuator locations). In the following paragraphs we will present the effects of spillover and actuator proximity (through the variation of parameters and c, respectively) on the identified optimal locations for the two actuator problem. If we disregard spillover effects (i.e., setting = 100% in the formulation of (Pc-VIII)), the optimal solution for c = 10 is computed to be
opt = [1.998 1.871] (we will elaborate on the choice of c in the sequel). This value maximizes the objective function, with S5 = 96.9%; in Fig. 5 we observe the component spatial controllabilities at the optimal locations obtain the values S5j = {100, 97.8}%. S5 does not become 100% due to the (active) constraint, M(
1 ,
2 )
c, which enforces a distance of 
= 0.127. The issue with the chosen location is that it also maximizes the spillover effects with S515 = 87% (S515j = {100, 48}%). Due to symmetry of both process and objective function with respect to the center of the process
= 1, positions
= 0.002 and 0.129 also attain the same objective value, thus giving four possible combinations for optimal locations. Note that use of the condition number constraint precludes us from obtaining the optimal solution of
1 = 0.002,
2 = 1.998, (an undesirable combination of locations, having exactly the same modal controllability values), since in that case the controllability matrix is singular.

A. Armaou, M.A. Demetriou / Chemical Engineering Science 61 (2006) 7351 – 7367

7361

% Spatial controllability

% Spatial controllability 100

100

90

90

80

80

70

70

60

60

50

50

40

40 30

30

S 5(ξ)

S 5(ξ)

20

M level β=60% optimal locations

10

20

M level β=60%

10

optimal locations

0

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0

2

spatial distance ξ

Fig. 5. Percentage component spatial controllability, S5j (
); K =2, i =60%,

= 100%, c = 10.

The effect of the spillover spatial controllability S515j (
) on the optimal actuator placement problem is shown in Fig. 6a and b presenting the component spatial and spillover spatial controllabilities, respectively. We note that the component spatial controllability and component spillover spatial controllability profiles as a function of
do not depend on the specific actuator index j, since the actuators considered have the same distribution function. Constraining the acceptable level of spillover to 60%, we observe that the set of spatial locations that ensure that S515j (
)
60% is given by [0.070, 0.159] ∪ [1.841, 1.930]. Its intersection with  would then yield the admissible set as = [0.070, 0.159] ∪ [1.841, 1.930]. Solving the optimization problem of (Pc-VIII), for optimization parameter values ( = 60% and c = 10), the optimal actuator locations were computed to be
opt = [1.857 1.930], shown in Fig. 6a which depicts the component spatial controllability S5j (
) as a function of the spatial distance
. Table 2, column 2, presents the values of spillover at
opt . We observe that the location of one actuator is constrained by the spillover constraint, while the other is constrained by the condition constraint. In addition, Table 2, column 2, summarizes the levels of modal controllabilities at the optimal location
opt . We observe that at least one of the actuators maintains a high level of authority on each mode for all the modes under consideration, well above the limit of = 60%. To better present the condition number constraint effect on obtaining optimal locations, let us disregard the constraint on M in (Pc-VIII) and impose the constraints
1 =
2 (the two actuators cannot occupy the same physical location) and |
1 −
2 |
(the two actuators must have a minimum distance of 
= 0.024). Note that this is a specific to the selected actuators constraint. The optimization problem solution is then [0.0699 1.9301], and the spatial controllability and component spillover controllability obtain the values S5j = 98.03% and

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1.8

2

spatial distance ξ

(a)

% Spillover spatial controllability 100 90 80 70 60 50 40 30

5 (ξ) S15

20

S level c=60% optimal location

10 0 0

(b)

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

spatial distance ξ

Fig. 6. (a) Component spatial controllability, S5j (
), and (b) component spillover spatial controllability, S515j (
) at optimal location for parameter values K = 2, i = 60%, = 60%, c = 10.

S515j = {60%, 60%}, respectively. In an L2 sense however, the two locations are identical, as elaborated upon previously, having the same effect on the slow eigenmodes. To address this issue, we considered the (admittedly more complex) condition number constraint. The optimization problem of (Pc-I) was solved for 
= 0.024 to obtain the condition number bound of c = 10. To further illustrate the relationship between actuator distance and condition number of M, in Fig. 4 we present the condition number value of a two-actuator controllability matrix (19) as a function of the spatial coordinate for different values of 
. We observe that the condition number of the spatial controllability matrix remains in a ball around a specific value irrespective of position, except near the process boundaries where

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A. Armaou, M.A. Demetriou / Chemical Engineering Science 61 (2006) 7351 – 7367

we observe a manyfold increase. This is due to all of the slow modal controllabilities approaching the same value (as can be seen in Figs. 1a and b), thus making the two locations practically indistinguishable from a modal controllability point of view. This is exactly a phenomenon that we attempt to avoid through the use of the condition number. Furthermore, we observe that as the location distance between two actuators, 
, decreases, the condition number monotonically increases (for specific location of the first actuator
1 ), as expected. The effect of the condition constraint on the optimal actuator locations (obtained through the solution of (Pc-VIII)) can be further demonstrated in Table 2, columns 3 and 4, where the value of c is increased to 102 and 103 , respectively, while spillover and modal controllability constraints are kept constant. The distance between the actuators decreases to 
= 0.0114 and then to 
= 0.0123 for c = 102 and 103 , respectively. We observe that while the value of spillover actually decreases for the second actuator for c = 102 , the modal controllability vectors become very similar, which implies that the two actuators have a similar effect on the first five modes. This becomes even more pronounced for c = 103 , as can be seen in column 4 of Table 2. The closeness of the actuators can be seen in Figs. 7a and 8a for c = 102 and 103 , respectively, while the component spillover spatial controllability is presented in Figs. 7b and 8b for c = 102 and 103 , respectively. The applicability and the limitations of the proposed optimization program when three actuators are available (i.e., K = 3) were also exposed when applied to the specific process. The optimization program of (Pc-IV) was infeasible for three actuators using the nominal parameters = 60%, c = 10, since the minimum space required for the three actuators was larger than the admissible set, which is a limitation of the original formulation of (Pc-IV). Note that, if the physical distance constraint between the three actuators was employed, the problem would be feasible, however, the resulting controllability matrix M would then be almost singular, implying that in an L2 sense the actuators would lie very close to each other. Increasing the condition constraint to c = 106 , lead to the clustering of the three actuators at
opt , shown in Table 3, column 1. Even then, the three actuators were not equidistant, due to the larger similarity of the modal controllability vectors as we approach the process boundary. The component spatial and spillover spatial controllabilities are presented in Figs. 9a and b, respectively. Note that due to symmetry of both the process and the objective with respect to
= 1, eight possible combinations of actuator locations will attain the same objective value. Solving the optimization program for spillover constraint = 65%, also led to the optimal locations presented in Table 3, column 2. We observe that at location
= 0.4801, S515j < 65%, due to the condition constraint. Figs. 10a and b present the spatial controllability and spillover spatial controllability, respectively. The optimal actuator locations when = 70% are presented in Figs. 11a and b which also present the spatial controllability and spillover spatial controllability, respectively. To circumvent the issues of problem infeasibility for (PcVIII), the optimization problem of (Pc-VIII) can be alterna-

% Spatial controllability 100 90 80 70 60 50 40 30

S 5(ξ) M level β=60% optimal locations

20 10 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1.8

2

spatial distance ξ

(a)

% Spillover spatial controllability 100 90 80 70 60 50 40 30

5 (ξ) S15

20

S level c=60%

10

optimal location

0 0

(b)

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

spatial distance ξ

Fig. 7. (a) Component spatial controllability, S5j (
), and (b) component spillover spatial controllability, S515j (
) at optimal location for parameter values K = 2, i = 60%, = 60%, c = 100.

tively recast as one with soft constraints on the spillover effects (similar to (Pc-V)). Specifically, we set = 100% and augmented the objective function in (Pc-VIII) with the term − s S515 ; the problem thus became feasible. Solving the problem for s = 0.1, we obtained the optimal locations
opt = [1.9333 1.8372 1.7476], with cumulative spatial and spillover spatial controllabilities S5 = 90.68% S515 = 67.74%, respectively. In Table 4 line 1, we present the optimization results. We observe that the component spillover controllabilities were now allowed to obtain any value and while two are below the 65% limit, one of them is well above it. The optimal actuator locations are also presented in Figs. 12a and b, describing the spatial controllability and spillover spatial controllability

A. Armaou, M.A. Demetriou / Chemical Engineering Science 61 (2006) 7351 – 7367

profiles, respectively. As s is increased, the choice of locations reduces the spillover controllability, with an adverse effect on spatial controllability. For example, for s = 0.5, we obtain the optimal locations
opt = [0.17048 1.7503 1.9065],

7363

with cumulative spatial and spillover spatial controllabilities S5 = 89.9192% S515 = 63.4357%, respectively. The locations are the ones where one location is at the lowest possible spillover spatial controllability, while the other ones are at the

% Spatial controllability

% Spatial controllability 100

100

90

90

80

80

70

70

60

60

50

50

40

40 30

30 S 5(ξ) 20

M level β=60%

10

optimal locations

S 5(ξ)

20

M level β=60%

10

optimal locations

0

0 0

0.2

0.4

0.6

0.8

1

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1.4

1.6

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2

spatial distance ξ

(a)

spatial distance ξ

(a)

0

2

% Spillover spatial controllability

% Spillover spatial controllability 100

100

90

90

80

80

70

70

60

60

50

50

40

40 30

30

5 (ξ) S15

20

S level c=60%

10

optimal location

5 S15 (ξ)

20

S level c=60%

10

optimal location

0

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

spatial distance ξ

(b)

0

2

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

spatial distance ξ

(b)

Fig. 8. (a) Component spatial controllability, S5j (
), and (b) component

Fig. 9. (a) Component spatial controllability, S5j (
), and (b) component

spillover spatial controllability, S515j (
) at optimal location for parameter values K = 2, i = 60%, = 60%, c = 103 .

spillover spatial controllability, S515j (
) at optimal location for parameter values K = 3, i = 60%, = 60%, c = 106 .

Table 3 Optimal actuator locations identified using NLP formulation of Eq. (Pc-IV) for K =3; nominal optimization parameters: i =60% (∀i =1, . . . , 5), =60%, c=10 Case

= 60%, c = 106


opt S5j S515j

[0.0696 0.0716 0.0749] 98.03 97.92

97.73

[1.936 1.822 0.4801] 98.34 89.26

67.41

[1.942 1.830 0.3787] 98.62 90.06

73.80

60.00

55.79

65.00

62.71

70.00

70.00

58.45

= 65%, c = 10

65.00

= 70%, c = 10

62.98

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A. Armaou, M.A. Demetriou / Chemical Engineering Science 61 (2006) 7351 – 7367 % Spatial controllability

% Spatial controllability 100

100

90

90

80

80

70

70

60

60 50

50

40

40

30

30

S 5(ξ)

S 5(ξ) M level β=60% optimal locations

20

20

M level β=60%

10

optimal locations

10 0

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0

2

0.2

0.4

0.6

spatial distance ξ

0.8

1

1.2

1.4

1.6

1.8

2

1.8

2

spatial distance ξ

(a)

% Spillover spatial controllability

% Spillover spatial controllability 100

100

90

90

80

80

70

70

60

60

50

50

40

40

30

30

20

5 (ξ) S15

20

10

S level c=65% optimal location

10

5

S15(ξ) S level c=70% optimal location

0

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0

2

spatial distance ξ

Fig. 10. (a) Component spatial controllability, S5j (
), and (b) component spillover spatial controllability, S515j (
) at optimal location for parameter values K = 3, i = 60%, = 65%, c = 10.

closest positions allowable by the condition constraint, as can be observed in Table 4. Remark 10. Based on the results presented in Tables 2 and 3, we observe that one actuator is always placed near location
= 1.930. This is an expected result, since the specific location provides the maximum feasible value of spatial controllability and the topology of the feasible set shows that the spatial controllability is monotonically decreasing. The results presented in Table 4 do not show such a trend due to the inclusion of the spillover spatial controllability in the objective function, which results in a topology of the objective function which is more complex (as can be qualitatively seen by superimposing Figs. 12a minus s × b).

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

spatial distance ξ

(b)

Fig. 11. (a) Component spatial controllability, S5j (
), and (b) component spillover spatial controllability, S515j (
) at optimal location for parameter values K = 3, i = 60%, = 70%, c = 10.

Table 4 Optimal actuator locations identified through NLP formulation of (Pc-V) for K = 3 as a function of s ; i = 60% (∀i = 1, . . . , 5), = 100%, c = 10 Case


opt

S5 (%)

S515 (%)

S15 5j (%)

s = 0.1

s = 0.5

s = 1.0

[1.933 1.837 1.748] [1.907 1.830 1.750] [1.902 1.828 1.751]

90.68 89.92 89.78

67.74 63.44 63.22

[62.43 61.08 78.34] [44.83 63.22 77.88] [43.54 63.65 77.75]

Remark 11. Further elaborating on the results presented in Table 2, we observe that the modal controllabilities of the higher modes are consistently lower than the modal controllabilities of the lower modes, for all values of the optimization

A. Armaou, M.A. Demetriou / Chemical Engineering Science 61 (2006) 7351 – 7367

7365

feasible. The choice of which formulation should be followed depends heavily on the specific process. For reasons of brevity, this reformulation case is not presented in this work.

% Spatial controllability 100 90 80

6. Conclusions

70 60 50 40 30 S 5(ξ) 20

M level β=60%

10

optimal locations

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

spatial distance ξ

(a)

% Spillover spatial controllability 100 90 80 70 60 50

The issue of optimal, with respect to certain criteria, placement of control actuators for transport-reaction processes, mathematically modeled by linear parabolic PDEs was considered. Using modal decomposition to discretize the spatial coordinate, and the notions of spatial and modal controllability, the semi-infinite optimization problem was reformulated as a nonlinear optimization problem (NLP) in appropriate L2 spaces, with objective to identify actuator networks that maximize the spatial controllability of the slow evolving subsystem. The issue of actuator spillover to the fast evolving system modes was considered in the formulation, thus reducing the probability of higher mode destabilization. Furthermore, constraints in the optimization problem were introduced to account for the issue of multiple actuator interplay, leading to formulations that minimize redundancy in the identified actuator network. The problem of optimizing a sensor network was subsequently summarized due to the duality between the notion of observability and controllability. The proposed method was successfully applied to a representative thermal process, modeled by a one-dimensional parabolic PDE. The optimal placement of a one, two and three point actuator network was computed using standard search algorithms, and the sensitivity of the identified locations to spillover and actuator redundancy measures was investigated.

40

Acknowledgments

30

5 (ξ) S15

20

S level c=100% optimal location

10 0 0

(b)

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

spatial distance ξ

Fig. 12. (a) Component spatial controllability, S5j (
), and (b) component spillover spatial controllability, S515j (
) at optimal location for parameter values K = 3, i = 60%, = 100%, c = 10, s = 0.1.

Financial support from the Pennsylvania State University, Chemical Engineering Department, and the Pennsylvania Department of Education, Engineering School equipment grant program, is gratefully acknowledged by the first author. Financial support from the National Science Foundation, Grant CMS0408974, for the second author is gratefully acknowledged. The authors are indebted to the reviewers for useful comments that led to the significant improvement of the manuscript.

References parameters. This is expected, due to the relative importance that the modes have on the spatial controllability as described in Remark 2. Specifically for the system, the eigenvalues of the first and the fifth mode are 1 = −0.0247 and 5 = −0.6169, respectively, giving a relative importance in the spatial H2 norm of 1 /5 = 0.0400 for the specific system and actuators employed. Remark 12. Alternatively, instead reformulating the problem of (Pc-VIII) as one with soft spillover constraints, it could be reformulated as one with soft condition number constraints (as discussed in Remark 8). The problem would then again become

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