Constraints-Driven Optimal Actuation Policies for Diffusion-Reaction

Ind. Eng. Chem. Res. 2008, 47, 105-115

105

Constraints-Driven Optimal Actuation Policies for Diffusion-Reaction Processes with Collocated Actuators and Sensors Stevan Dubljevic* Department of Chemical and Biomolecular Engineering, and, CardioVascular Research Laboratories, DaVid Geffen School of Medicine, UniVersity of California, Los Angeles, California 90095-1760

This work introduces modal model predictive control (MMPC) design methodology in the framework of optimal actuation policies, which results due to the presence of input and state constraints for a class of distributed parameter systems modeled by parabolic partial differential equations (PDEs). The predictive control law accounts for input and state constraints and with respect to actuator/sensor position placement, it generates an optimal actuation policy that switches applied control action among available prespecified actuator locations. The proposed constrained predictive control law utilizes a low-order modal representation in the optimization functional while higher modes are included only in the PDE state constraints. Accordingly, the proposed control law is formulated by the minimization algorithm whereby optimization is performed over all available preset collocated actuator/sensor positions. In this sense, the minimizing algorithm provides a control law that chooses among best collocated actuator/sensor positions available, with respect to the lowest optimal cost among these positions. An example of a diffusion-reaction process, with spatially uniform unstable steady state, subject to flux boundary conditions, is considered. 1. Introduction The fundamental problem, in terms of identification, state estimation, and control of distributed systems, is an issue of effective utilization of actuators and sensors to improve process performance characteristics. In particular, the important aspect of many engineering applications is the use of a large number of sensor and actuator networks that provide efficient and effective monitoring and control of processes. The use of mobile sensors and moving actuators is widely used in the automotive and process industries, where many actuating and sensing devices are attached to robotic manipulators that ultimately reduce the operation and production costs. Moreover, current advanced sensing state-of-the-art equipment is easily mounted on or added to already existing process architectures, to enhance process performance characteristics, reduce power consumption, and address more-stringent final product requirements, which then inevitably brings constraints into controller realization. A large number of relevant industrial processes are described by transport-reaction models that admit a form of parabolic partial differential equation (PDE). Control methodologies for nonlinear/linear parabolic PDEs are based on the structural feature of parabolic systems in which dominant dynamics can be successfully captured by a few dominant modes. Typically, the issue of stabilization for a parabolic system is resolved by the state space decomposition, based on system modes, as the “relocation’’ of unstable modes using well-known finite dimensional controllers stabilizes the original system under distributed feedback.1-8 This technique relies on the assumption of dissipative operator spectrum decomposition, which implies that the dominant dynamics described by a finite number of the operator’s possibly unstable modes, once stabilized, and along with the exponential stable infinite-dimensional modal complement, provide asymptotic stabilizability.5,6,9 Control methodologies for the class of linear parabolic PDEs are well-estab* To whom correspondence should be addressed. Mailing address: UCLA Division of Cardiology, BH-307 CHS, 650 Charles E. Young Drive S., Los Angeles, CA 90095-1760. Tel: 310-794-3658. Fax: 310206-4107. E-mail address: [email protected].

lished,1,4,7 and optimal control methodologies, which originated from optimal control methods for finite dimensional systems, have also been developed for infinite dimensional systems.7,10,11 However, even in the simplest cases of the finite time linear quadratic optimal control problem, the solution is provided by tedious numerical solution of the operator’s differential Riccati equation.11,7 In addition to the issue of the optimal control of a given actuator/sensor architecture,12 considerable research efforts have been directed toward the optimal placement of actuators and sensors,13-15 nonlinear control of parabolic PDEs with input constraints,16 coordinated feedback and switching for control of parabolic PDEs,17 computationally efficient methods for the solution of dynamic constraint optimization problems arising in the context of control of PDEs,18 nonlinear model predictive control of distributed parameter systems,19 and switching policies among the actuators within an optimal control setting.20,21 However, in all considered research of linear parabolic PDEs within an optimal control setting, an important issue of input and state constraints has not been successfully resolved, because their inclusion in the linear optimal control problem formulation creates computationally demanding non-convex optimization problems.4,19,22 The issue of preserving beneficial properties of a linear PDE, on one side, and an optimal control design with the inclusion of state and input constraints, on the other, has been explored by Christofides and co-workers through the synthesis of model predictive controllers (see refs 23-25 for surveys of the results on model predictive control (MPC) and references in this area) for linear parabolic PDEs26-29 and in Hoo and Zheng.30 In addition, an important aspect of successful controller synthesis in the scope of distributed parameter system within an optimal setting is the placement of actuators, because inadequate actuator placements may affect some important system properties (such as its controllability and stabilizability) and/or deteriorate performance characteristics of the modelbased controller. In particular, the optimality, with respect to the appropriate selection of sensors and actuators can improve the overall performance characteristics and possibly save the energy consumed for control purposes.20,31-34 Motivation for the use of collocated actuator/sensors can be justified by

10.1021/ie070546v CCC: $40.75 © 2008 American Chemical Society Published on Web 12/04/2007

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Ind. Eng. Chem. Res., Vol. 47, No. 1, 2008

practical and economical reasons; for example, in cases of the temperature control heating/cooling coil or the probe-actuator, they can also serve as sensing devices (e.g., thermocouples are easily mounted on them), because they comprise part of the majority of the state-of-the-art equipment that considers heattransfer problems. Another important issue from the control practitioner point of view is that control constraints associated with actuators may change due to performance depreciation of worn-out actuator equipment. Such a case then leads into constraints control problem realizations that require fast and quick fix-up in a timely manner, without changing or inducing large-scale maintenance costs. Along the same line, a change in the performance specifications, which is reflected as constraints in the state/output variables, must be easily taken into account and applied in the context of the distributed parameter systems setting. Therefore, an important set of issues of a large number of spatially distributed actuators/sensors present, optimality of the applied control action, control inputs, and the presence of PDE state constraints should be simultaneously taken into account in the controller realization. In this work, we provide the controller realization that addresses the control features of the aforementioned distributed parameter systems. A novel controller synthesis framework is developed, wherein the actuator activation policy and the state feedback controller structure are integrated by means of the modal MPC that obeys the input and PDE state constraints for reaction-diffusion processes modeled by linear PDEs. The proposed approach focuses on the development of a methodology that allows multiple actuators within the optimal-based decision to activate only one specific actuator while the others are kept inactive. In this sense, this work complements the work on the actuator activation policy for the performance enhancement of controlled diffusion processes by Demetriou and coworkers,20,21 where the issue of the state and input constraints and their influence on switching policies has not been addressed. In this paper, the modal MPC synthesis is applied to the finite dimensional approximation of the original distributed parameter system. Furthermore, the optimization of the location parametrized quadratic performance index is performed with respect to the optimal input sequence that satisfies both input and PDE state constraints, and it is also minimized, with respect to the location of the actuator/sensor position. In that sense, the actuation guiding policy (switching among positions of actuator locations) is merely driven by the inherent dynamics of the system and by input and PDE state constraints in the optimal manner. Finally, a representative case study of a diffusionreaction process described by a linear parabolic PDE with the Newmann-type boundary conditions is presented and the performance-enhancing capabilities of the proposed method are evaluated by means of simulations. 2. Preliminaries 2.1. Parabolic Partial Differential Equations (PDEs). A large number of process systems (for example, a thin, narrow, homogeneous, continuous metal strip that is fed into a furnace by means of a variable-speed transport mechanism,7 or the control of a short, homogeneous packed-bed chemical reactor in which a zero-order exothermic reaction is taking place and is modeled by the axial dispersion model4) admit a form of parabolic PDEs. In this work, a class of distributed parameter systems that is represented by the one-dimensional linear parabolic PDEs is considered and given in the following form:

dZ ∂ 2Z ) bh 2 + cjZ + bdιudι(t) dt ∂σ y(t) )

∫0l cd Z dσ j

(1) (2)

with boundary and initial conditions of

bh1

∂Z | ) cj1Z ∂σ σ)0

(3a)

bh2

∂Z | ) cj2Z ∂σ σ)l

(3b)

Z(σ,0) ) Z0(σ)

(3c)

subject to the following input and state constraints:

umin e udι(t) e umax ι ι Zmin w e

∫0l rw(σ)Z(σ, t) dσ e Zmax w

(for ι ) 1, ..., ma) (for w ) 1, ..., g)

(4) (5)

where Z(σ,t) ∈ [0,l] denotes the state variable, t ∈ [0,∞), udι(t) ∈ R denotes the dι-th constrained manipulated input; [umin ι , max uι ] ∈ R represents the lower and upper limits associated with max the input applied at the dι-th actuator location, and [Zmin w , Zw ] ∈ R represents the lower and upper state constraints enforced at the w-th constraints location, y(t) is the output variable obtained by the dι-th sensor, bh and cj are constants, bhη, cjη ∈ R (for η ) 1, 2, with the condition |bh|η + |cj|η * 0), and Z0(σ) ∈ L2([0,l]). The functions bdι(σ) and cdj(σ) ∈ L2([0,l]) describe how the control action udι(t) and sensing are distributed within the spatial domain. We choose a particular type of actuation and sensing function, which is given as an approximation of the pointwise actuation and sensing. This choice is motivated by the type of device used in the state-of-the-art process industry, in particular, temperature control and monitoring, where the probe is used to reach the point within the domain where measurements are taken, or by sensing by optical methods and locally, at the point, some other property that is translated into the temperature signal. In the same vein, the state constraints distribution function is taken to be the same in nature as the pointwise actuation and sensing. Motivation for this consideration is that in an actual realization of the proposed controller synthesis methodology, it can give an advantage to the controller practitioner, who may check whether state constraints are obeyed at the measurement locations and it can give a valuable insight into the model mismatch and/or unknown disturbances present in the system. In eq 5, the function rw(σ) ∈ L2([0,l]) is a “state constraint distribution” function, which describes how the w-th state constraint is enforced within the spatial domain. Whenever the state constraint is applied at a single point of the spatial domain σcw, the function rw(σ) is taken to be nonzero in a finite spatial interval of the form [σcw - µ,σcw + µ], where µ is a small positive real number, and zero elsewhere. The state space of interest is H ) L2([0,l]), which is the Hilbert space of measurable square-integrable real-valued functions f: [0,l] f R, so that ∫[0,l] |f(σ)|2 dσ < ∞ with the standard weighted inner product (‚,‚)R and norm |‚| defined on it.

Ind. Eng. Chem. Res., Vol. 47, No. 1, 2008 107

The PDE of eqs 1 and 2 is formulated as an abstract evolutionary equation in the well-defined Hilbert space, with the state z(‚,t) ) {Z(σ,t), 0 eσ e l}, as follows:

z˘ (t) ) Az(t) + Bιu(t), z(0) ) z0 y(t) ) Cjz(t)

(6a,b) (6c)

where the operator A is defined as

Aφ ) bh

d2φ + cjφ dσ2

(for 0 < σ < l)

(7)

where φ(σ) is a smooth function on [0,l], with the following dense domain:

D(A) ) {φ ∈ H: φ,φ′ are absolute constants, Aφ ∈ H, bh1φ′(0) ) cj1φ(0), bh2φ′(l) ) cj2φ(l)} (8) The input operator is given as

Bιu(t) ) bdι(‚)udι(t)

(9)

and the output operator and PDE state constraints are given as

Cjz(t) ) (cdj(‚), Z(σ,t))

(10)

max Zmin w e (rw(‚), Z(σ,t)) e Zw

(11)

Using the aforementioned definitions, the system of eqs 1-5 can be written as a family of switched systems, which is parametrized by the actuator and sensing distribution function, and takes the following abstract evolutionary equation form:

z˘ (t) ) Az(t) + Bιu(t), z(0) ) z0

(12)

(for j ) 1, ..., ms)

(13)

y(t) ) Cjz(t) e u(t) e umax umin ι ι max Zmin w e Swz(t) e Zw

(for ι ) 1, ..., ma) (for w ) 1, ..., g)

(14) (15)

The spectrum of the Riesz spectral operator A is obtained by solving the following eigenvalue problem:

Aφκ ) bh

d2φκ dσ2

+ cjφκ ) λκφκ

(16)

subject to

bh1

dφκ dφκ (0) ) cj1φκ(0), bh2 (l) ) cj2φκ(l) dσ dσ

(17)

where λκ denotes an eigenvalue and φκ denotes an eigenfunction. The point spectrum of A,35 σ(A), in the case of self-adjoint operators, which are considered in this work, consists of real eigenvalues with finite multiplicity, and λκ+1 < λκ, ∀ κ ) 1, ..., ∞. Remark 1: It is important to emphasize that the input and output operators Bι and Cj in eqs 12 and 13 are parametrized by the spatial location of the ιth actuator and jth sensor, which invokes conditions on the generic property of the controllability and observability of the evolutionary equations eqs 12 and 13. Namely, in the context of the proposed predictive control optimal switching policy with collocated actuators and sensors

developed in the ensuing sections, it is assumed that the set of all available considered actuator locations preserves approximate controllability condition for the (A, Bι) pair, and approximate observability condition for the (A, Cj) pair, as C′j ) Bι and ι ) j.7 Because the duality of approximate controllability and approximate observability holds, only the properties of approximate controllability are addressed in the ensuing text. Remark 2: The test of approximate controllability is simplified to the requirement that

〈bdι(‚),φκ〉 * 0

∀κ, κ < ∞

which induces a set of candidate locations within the domain,7,20 with the condition; that is,

P ) {pι(σ) ∈ [0, l]:

∫0l φκ(σ)bd (σ) dσ * 0 ι

∀ κ}

(18)

In addition, please note that there are disjoint continuums of points that fulfill eq 18, so that one must consider only a finite number ma of these points or explore controllability measure when λκ < 0, ∀ κ, by choosing optimal actuator locations from the above set P so that the controllability Gramian is maximized.20 2.2. Modal Decomposition. In this section, we apply standard modal decomposition to the infinite-dimensional system of eq 12 to obtain a finite-dimensional system. Let us define the spectral projection operator Ps, which induces the following decomposition of the separable Hilbert space H into two subspaces Hs and Hf, H ) Hs x Hf, so that Hs ) PsH, and Hf ) (I - Ps)H ) PfH. The state z(t) of the system of eq 12 can be decomposed as

z˘ s(t) ) Aszs(t) + Bsιu(t), zs(0) ) Psz(0) ) Psz0

(19a,b)

z˘ f(t) ) Afzf(t) + Bfιu(t), zf(0) ) Pfz(0) ) Pfz0

(20a,b)

y(t) ) Csjzs(t) + Cfjzf(t)

(21)

where As ) PsA, Bsι ) PsBι, Af ) PfA, Bfι ) PfBι, Csj ) CjPs,Cfj ) CjPf. In the aforementioned system, As is a diagonal matrix of dimension m × m of the form As ) diag{λνκ} (λνκ are possible unstable eigenvalues of As, κ ) 1, ..., m and ν ) S) and Af is an infinite-dimensional operator, which is exponentially stable (which follows from the fact that λm+1 < 0). In the remainder of the paper, we will refer to the zs(t) and zf(t) subsystems in eqs 19 and 20 as slow and fast subsystems, respectively. Because of the structure of eqs 19-21, and from the properties of operators As and Af, coupling among modes is realized only through the input injection u(‚). It is important to emphasize that the property of evolutionary PDE state equations (eqs 12 and 13) to be formulated in the form of eqs 19-21 is crucial for our ability to synthesize a low-order modal model predictive controller. Namely, as will be demonstrated in the ensuing subsection, the construction of a convex optimization problem in the form of quadratic programming (QP) requires a structure where modes are only coupled through the input injection. We consider a high fidelity approximation (ℵth-order approximation) of the zs(t) and zf(t) subsystems described by eqs 19-21, which can be transformed to an appropriate discrete equivalent of the continuous dynamics, when the ideal sampler is used:36

108


zs(k + 1) ) A˜szs(k) + Bsιu(k)

(22)

zf(k + 1) ) A˜fzf(k) + Bfιu(k)

(23)

y(k) ) C˜sjzs(k) + C˜fjzf(k)

(24)

where A˜s, A˜f, B˜si, B˜fi, C˜si, and C˜fi are (m × m), ((ℵ - m) × (ℵ - m)), (m × 1), ((ℵ - m) × 1), (1 × m), and (1 × (ℵ - m)) matrices, respectively. The transformations that relate the continuous system, described by eqs 19-21, and the discrete system representation, described by eqs 22-24, are given as follows:

A˜ν ) eAνδ Bν ) TeAνδBν C˜ν ) Cν

(for ν ) s, f)

uN

|∆u(k + i|k)| e ∆umax

(for ν ) s, f)

u

where δ is a time constant of the ideal sampler that produces a given continuous signal z(t), the sequence of which is defined by z(k) ) z(kδ).36 Because formulations in the ensuing sections require the enforcement of state constraints in the domain [0,l], the state of the PDE at the point or region of interest where state constraints are going to be enforced is given by the following expression:

(25)

where S˜sw and S˜fw are discrete equivalents of Ssw ) SwPs and Sfw ) SwPf, respectively, and the operator Sw is given by eq 11. Remark 3: One must be aware that, because of the ℵthorder finite-dimensional approximation of the entire infinitedimensional state in the PDE state constraints, modes higher than ℵ are neglected in the controller synthesis. This mismatch may eventually cause deterioration in the controller performance if modes higher than ℵ start to affect the state constraints condition. This phenomena is referred as a “spillover” and should not be neglected in the synthesis of the modal model predictive controller at its realization and implementation stages. Remark 4: Another important point in the realization of modal MPC synthesis is that the discrete systems dynamics of slow and fast subsystems are coupled only through the input injection, which will allow us to construct a modal model predictive controller that can account for the infinite dimensionality of the fast modal states. 2.3. Model Predictive Control. In the formulation utilized in this work, a linear time-invariant discrete model of the system is considered by the controller and it is given in the following form:

z(k + 1) ) Az(k) + Bu(k)

(26a)

y(k) ) Cz(k)

(26b)

where z(k) ∈ Rn, u ∈ Rm, and y ∈ Rp. A stabilizing regulator can be determined as the solution of minimization of the following infinite horizon open-loop quadratic objective function at the time k: ∞

Φ(k) )

J(k) ) min Φ(k) + T(k)Q(k)

(28)

subject to

(for ν ) s, f)

max Zmin w e S˜swzs(k) + S˜fwzf(k) e Zw

) u(k + i|k) - u(k + i - 1|k) is the change of the input vector at the time k, z(k + i|k) and u(k + i|k) denote the variable z(‚) and u(‚) at a sampling time k + i predicted at the sampling time k, and (‚)(k) ) (‚)(k|k). At a time k + N, the control input vector u(k + i,k) is set to zero and maintained at this value for all i g N in the open-loop objective function value calculation, so that, at sampling time k, the control move u(k) equals the first element u(k|k) of the sequence uN ) [u(k|k), ..., u(k + N 1|k)], which is the minimizer of the optimization problem:

z(k + i|k)TCTQCz(k + i|k) + ∑ i)1

u(k + i|k)TRu(k + i|k) + ∆u(k + i|k)TP∆u(k + i, k) (27) where Q ) QT g 0, R ) RT > 0, P ) PT g 0, term ∆u(k + i|k)

min

e u(k + i|k) e u

u(k + i|k) ) 0

max

(for i ) 0, 1, ..., N) (for i ) 0, 1, ..., N - 1)

(for i ) N, N + 1, ..., ∞)

Gz(k + i|k) e g + (k)

(for i ) 0, 1, ..., ∞)

where G ∈ RnGxn, g ∈ RnG, and (k) g 0. Input constraints represent physical limitations on actuators that cannot be violated under any circumstances, whereas the output/state constraints can be softened by slack variables (k) and can be temporally violated, if necessary. It has been shown in the work of Muske and Rawlings37 that the constraints must be satisfied on a finite horizon to guarantee satisfaction on the infinite horizon. Through straightforward algebraic manipulations of the quadratic objective function given in eq 28, one can easily obtain the following quadratic program for uN in the case of unstable plant dynamics:

h uN + 2(uN)T(G h zk - F h uk-1) min (uN)T H uN

(29)

subject to

zu(i|k) ) 0

(for i ) N)

Du(k + i|k) e d

(for i ) 0, 1, ..., N - 1)

Hz(k + i|k) e h

(for i ) 0, 1, ..., N - 1)

where zu refers to unstable modes that are stabilized at the end of the receding horizon and the construction of the matrices H h, G h, F h , D, d, H and h are obtained as it is given the work of Muske and Rawlings.37 The input constraints represent physical limitations on the actuators that cannot be violated under any circumstances, whereas the output constraints are softened by the slack variable and can be temporarily violated. The feasible region for the optimization problem described by eq 17, which has unstable modes, is a complex function of {A,B,C}, the initial condition z(0), and a receding horizon length. An admissible region ΩN is defined as a set of initial conditions z0 for which there exists {u(k)N-1} ∈ U and u(k) ) 0, ∀ k g N, such that limkf∞ z(k) f 0. The condition of constrained stabilization requires that the system be stabilizable if z(0) belongs to Ω∞, such that sufficient condition for stabilizability on the finitelength horizon is z0 ∈ ΩN ⊂ Ω∞. In other words, asymptotic stability of the closed-loop receding horizon controller is implied by feasibility of optimization quadratic program given in eq 29. Stability properties of the state feedback controller given by eq 29 are conditionally connected with the feasibility issue of the constrained optimization problem;38-40 therefore, the issue of handling feasibility in the realm of the linear MPC formulation will be explored within the framework of handling state


constraints of the infinite-dimensional PDE. It is not by chance that the framework of handling feasibility of the optimization problem also can be used to address the issue of formulation of an optimal control problem applied to infinite-dimensional system dynamics that possesses a certain structure in its dynamics representation. 3. Actuator/Sensor Scheduling Policy by Modal Model Predictive Control (MPC) In this section, a new optimal predictive control law that accounts for the input and state constraints with an optimal actuation/sensing policy is considered. It explores the optimality in the sense of the best location utilized within a finite set of available locations, with respect to actuation/sensing under the presence of state and input constraints. In particular, we consider the fixed actuator and sensor architecture with associated input constraints, while the PDE state constraints are fixed at prespecified locations, where, at a time instance associated with the control input duration, only one of the prespecified actuators is active while the others remain dormant, even though this realization can be easily reconfigured to a movable-actuatorfixed-sensor architecture. Such flexibility of the actuator architecture must be allowed, either due to the process specifications or the actuator’s ability to provide the control signal with desirable speed and accuracy.21,41 To proceed with formulation of the predictive control law that is able to consider fixed and movable-actuator fixed-sensing architecture, we introduce necessary assumptions into the control implementation architecture.21 Namely, we assume that there is a finite number of admissible collocated locations where actuation/sensing is applied, as denoted by

pi(z) ) {p1(z), p2(z), ..., pma(z)} ∈ P An important factor in the implementation of a movableactuator-fixed-sensing architecture is the time required by the actuation device to transverse from one location pj(z) to pk(z) location, and, in this context, we assume that this time is negligible and may be assumed to be zero (see Assumption 2 (“Zero Transverse Time”) in the work reported by Demetriou and Iftime21). However, an important attribute of the proposed MPC formulation described by eq 28 is the flexible ability to account for the speed of the actuation device to transverse from one location to another by adding the additional term in the performance functional, which will represent the weight associated with the transfer of the actuation device from the given current location to all other available actuator locations. 3.1. Low-Order MPC formulation. We consider the family of linear time-invariant systems given by eqs 22-24, parametrized by the position where actuation/sensing are applied within the domain, on the basis of which the predictive control law is designed. Predictive control effectively considers the lowdimensional approximation of the PDE state in the cost functional that is parametrized by the sensor and actuator positions, and treats high-dimensional approximation only in the construction of the state constraints whereby construction of the higher modes of the MMPC algorithm are treated as slack variables. Therefore, in this way, one may easily identify the reasons for infeasibility of the MPC optimization problem that result either from the input constraints (hard constraints) or the state constraints (soft constraints). A collocated actuator/sensor position parametrized MPC algorithm (that is, C˜ Tsj ) B ˜ sj) that is constructed based only on the slow modes in the optimization functional and in the state constraints admits the following form:

N-1

min u

[zs(k + i|k)TC ˜ sjT QCsjzs(k + i|k) + ∑ i)0

u(k + i|k)TRu(k + i|k)] + zs(k + N|k)TQ h pjzs(k + N|k) (30) ˜ sju(i|k) zs(i + 1|k) )A˜szs(i|k) + B

(31)

subject to

e uj(i|k) e umax umin j j max Zmin w e S˜swzs(i|k) e Zw

zus(N) ) 0

(for j ) 1, 2, ..., l) (for w ) 1, 2, ..., g)

(for i ) 0, 1, ..., N - 1)

where u ) [u(i|k), ..., u(i + N - 1|k)] is the vector of control moves computed over the control horizon N, R > 0 is a strictly positive definite and C˜ sjT Q C˜ sj ) Q g 0 is a positive semidefinite matrix, while matrix Q h pj denotes the terminal penalty. Stabilization of unstable slow modes (zus) is ensured at the end of the horizon, assuming that the constrained optimization problem, which has been cast as quadratic programming (QP),42 provides a feasible solution, with respect to initial conditions zs(0). Despite its low-order characteristics, a major drawback of this formulation is the fact that the resulting MPC law (which is described by eqs 30 and 31 to be feasible, when implemented on the full system described by eqs 22 and 23) enforces a closed-loop stability but it does not enforce fullstate constraints satisfaction, because it neglects the evolution of fast states. Therefore, it is necessary to account for fast-mode dynamics to satisfy constraints and still maintain a low-order MPC formulation. 3.2. Low-Order MPC Formulation with State Constraint Satisfaction. To account for the effect of fast states on PDE state constraints, the formulation of eqs 30 and 31 can be modified by incorporating fast states into the state constraints equation.38,40 The control action at time instance k in this formulation is computed by solving the following optimization problem: N-1

min u

[zs(k + i|k)TC ˜ sjT QCsjzs(k + i|k) + u(k + i|k)TRu × ∑ i)0

(k + i|k)] + zs(k + N|k)TQ h pjzs(k + N|k) + Γpj(zf(k + N|k)) (32) ˜ sju(i|k) zs(i + 1|k) )A˜szs(i|k) + B

(33a)

zf(i + 1|k) )A˜fzf(i|k) + B ˜ fju(i|k)

(33b)

subject to

e uj(i|k) e umax umin j j S˜swzs(i|k) e Zmax w - S˜fwzf(i|k)

(for j ) 1, 2, ..., l) (for w ) 1, 2, ..., g)

-S˜swzs(i|k) e - Zmin w + S˜fwzf(i|k) zus(N) ) 0

(for i ) 0, 1, ..., N - 1)

The constructed MMPC law formulation, described by eqs 32 and 33, at each time of the predictive control implementation, resolves the constrained optimization program, based on slow modes dynamics, whereas state constraints are represented in the form of two contributions: one that is associated with modal states of zs (designated as a “slow’’ dimensional system) and

110


another that is a complementary contribution of zf (“fast’’ modes of the system that accounts for the PDE state’s infinite dimensionality. By doing this, the “fast’’ modes evolution is taken into account exactly in state constraints and enforced in the construction of the control input in the MPC control law implementation, so that possible state constraints violations are avoided, and this ensures that, if a MPC optimization problem is initially feasible, it will successfully stabilize the system in a closed loop, account for the performance criterion, and continue to be feasible. Subsequent feasibility is guaranteed by the claim that there exists a sufficiently long horizon for the feasibility to be achieved and the length of the horizon (N) can be computed as given in ref 42. Proper account for the influence of “fast’’ modal states evolution is twofold, in a sense that it allows us to directly account for them in the state constraints evolution, and also allows for the characterization of an appropriate measure of peaking magnitude of the fast modes and therefore explicitly accounts for penalizing the peaking through an added penalty term Γpj(zf) in the objective functional that is described by eq 32.40 Considering the constrained quadratic optimization problem, as given in eq 28, the fast modes are represented by the slack variables, and that is the reason why the infinite-dimensional feature of the PDE constraints can be captured in this MMPC formulation. 3.3. Actuator/Sensor Activation Policy. In the realization of the model predictive controller where the decision to activate a single actuator out of the available ma actuators is guided by the state and input constraints, the switching rules between the activated actuator and the dormant one are based on the minimal cost criteria and the ability of a predictive controller to steer a state to a steady state, without any violation of input and state constraints. It can be demonstrated that the closed-loop performance is further enhanced through the selection of an optimal location pi of the actuator device by the simple optimization algorithm, providing the location where the cost equations (eqs 32 and 33 (or eqs 30 and 31)) are the smallest. The countable set of all possible switches with the finite number of admissible locations is given as a set of ma2 - ma elements, which are defined as

Σ ) {σ|σij, i * j, i ) 1, ..., ma; j ) 1, ..., ma} In the operational mode, the controller activates the most optimal actuator at the present time while others are kept dormant and then activation of the optimally placed device is continued at each time instance, which results in the optimal switching control policy. The activation policy and guidelines as to the selection of the actuator is based on performance of a predictive controller where, at each time instance, the cost functional in the predictive controller (eqs 32 and 33 (or eqs 30 and 31)) is re-evaluated for each of the finite number of actuator locations, so that the location used for the actuator activation has the smallest cost functional. Therefore, the controller activation policy can be formulated as follows: Actuator/Sensor ActiVation Policy Algorithm (1) At time instance k and for each of the ma-prespecified actuator/sensor positions, pi ) [p1, p2, ..., pma], a standard receding horizon MPC program, described by eqs 30 and 31 (or eqs 32 and 33) is constructed. (2) At the time instance k, the quadratic constrained predictive control program described by eqs 30 and 31 (or eqs 32 and 33) is solved and, from the obtained control policies, implement the one that minimizes the cost functional, which is given by

p* ) arg min J(z(k),k,u(‚),pi) pi

(34)

and activate the corresponding actuator that remains active over the time interval [k, k + 1]. (3) Repeat step (1) for the next time interval. The stability property of the aforementioned actuators/sensor activation policy algorithm through MPC law (eqs 32 and 33 (or eqs 30 and 31)) is ensured by asymptotic stabilization over the horizon length of unstable modes of the operator A˜ s using the MPC algorithm42 and the following theorem provides conditions for the closed-loop stabilization of the system by the actuator/sensor activation policy algorithm. Theorem 1: Given a family of stabilizable actuator/sensor location parametrized linear parabolic PDEs, given by {A,Bι,Cj}, and an initial condition zs(0) ∈ ΩN, there exists an optimal switching sequence σs ∈ Σ, such that Z(σ,t) ) 0 is an asymptotically stable solution for the feasible quadratic optimization programs described by eqs 30 and 31 (or eqs 32 and 33). Proof: If the constrained optimization problem described by eqs 30 and 31 (or eqs 32 and 33) is not feasible, the controller is not defined. We demonstrate that, at time k, an optimization functional J(k,pi) is finite, and at time k + 1, this functional is greater than J(k + 1,pi) for at least zT(k|k)CTQCz(k|k) and uT(k|k)Ru(k|k). In addition, we note that, in the case of switching, the following relations hold: J(k + 1,pj) e J(k,pi), because of cost minimization, with respect to the actuator/sensor locations, or J(k + 1,pj) ) minpi J(k + 1,pi). If switching does not occur, the relation J(k + 1,pi) e J(k,pi) holds, so that switching will not occur unless J(k + 1,pi) > J(k + 1,pj). In this way, the switching sequences of the cost J(k,p) are bounded from below and are conVergent, which implies z(k) f 0 and u(k) f 0 as k f ∞. 9 The aforementioned actuator activation policy within the MMPC framework has three essential merits. First, it exploits the particular linear structure of the modal representation of linear parabolic PDE state in the infinite dimensional setting. Second, it utilizes the best features of the MPC-constructed constraints relaxation optimization algorithm, based on the use of the penalty function.38 Finally, it addresses the optimality from the standpoint of the best location for the implementation of an actuator/sensor device. In this work, only state feedback structure is considered as the control law utilizes the knowledge of an entire modal state evolution in its structure. The extension of proposed control law in the case of output feedback realization will heavily rely on the accurate estimation of the modal dynamics and this issue is not considered in this work. Remark 5: Two presented MMPC formulations that are parametrized with the sensor/actuator location given by eqs 30 and 31 (and eqs 32 and 33) are complementary. Namely, the control law formulation described by eqs 30 and 31, being feasible, can provide a set of initial conditions from which the controller can stabilize low-dimensional PDE state approximation with the given horizon length, input, and low-dimensional formulated PDE state constraints. Therefore, given the inclusion of fast modes in the control law realization described by eqs 32 and 33 and possible infeasibility, which initially may be found with identical given parameters (limits on input, state constraints, and horizon length) as shown in eqs 30 and 31, it will be exclusively attributed to the inclusion of the evolution of fast modes in the state constraints in eqs 32 and 33. Remark 6: At the level of implementation of the actuator/ sensor activation policy algorithm, one must be cautious about


the control law activation scenarios that are characterized by the optimization program, which is infeasible for some locations and feasible for the remaining actuator/sensor locations. Under those scenarios, minimization of the costs, with respect to the location, is performed only for those locations that render the optimization program feasible. In particular, successive infeasibility of certain locations can impinge on the stability property of the actuator/sensor activation policy algorithm, and systematic resolution of those cases is not considered in this work. Remark 7: An important property of the control law described by eqs 32 and 33 is the fact that it can incorporate an arbitrary conservative measure of a mismatch of the real infinitedimensional PDE state constraint realization and that given by the high-fidelity approximation in eq 25. Namely, the uncounted ∞ zfi(t)φi(σ)) in state constraints is spillover term (r(σ), ∑i)ℵ+1 always a finite and well-defined quantity. Having the initial conditions for zf(0) and bounds on the control input, the evolution of the fast modal states can be characterized by the appropriately chosen norm. In other words, MMPC can really account for the infinite-dimensional nature of the PDE state constraint. In the case of MMPC for linear PDEs, the uncounted spillover term affects only the state constraints satisfaction, because it accounts for the real infinite-dimensional systems dynamics representation, which differs from the sufficiently high finite-dimensional approximation used in controller synthesis. Contrary to the case of the linear PDE, any controller synthesis via the modal decomposition technique for nonlinear PDEs inevitably introduces a coupling between high and low modal states, which implies that even the stabilization close to the spatially uniform unstable steady state may not be guaranteed by eqs 32 and 33, because of the influence of stable higher modes in the evolution of low unstable modes. Remark 8: The MMPC law described by eqs 32 and 33, with respect to the computation load and efficiency, outperforms the linear quadratic regulator (LQR) problem with input and state constraints, which inevitably leads to a nonconvex constrained optimization. Namely, the convex property of a QP problem that is constructed is easily solved compared to nonconvex optimization algorithms. On the other hand, one must be aware of a potential drawback of the implementation aspects of the actuator/sensor activation policy in cases when the subsequent linearization of a nonlinear PDE model (presented in eq 1), which must be performed at each time instance, along with the subsequent construction of the QP problem (step (1)) for all available actuator/sensor locations with a large horizon. 4. Spatially Distributed Control Example We consider the spatially controlled parabolic PDE that represents a linearized model of a rod reactor in which a zeroorder exothermic reaction occurs,4,8 given in the form

∂Z(σ,t) ∂2Z(σ,t) ) bh + cjZ(σ,t) + bdi(σ)udi(t) ∂t ∂σ2 Z h dj(t) )

∫0

1

cdj(σ)Z(σ, t) dσ

dZ(0, t) dZ(1,t) ) 0, ) 0, Z(σ,0) ) Z0(σ) dσ dσ e udi(t) e udmax udmin i i Zmin w e

∫0

1

rw(σ)Z(σ, t)dσ e Zmax w

(for di ) 1, ..., 8)

Figure 1. Open-loop behavior of a parabolic partial differential equation (PDE) (eq 35). max ) 2.0 where cj ) 0.6762 and bh ) 0.2; Zmin w ) -0.75 and Zw for w ) 1, 2, 3. The state constraint distribution function is given by the function rw(σ) ) 1/2µ for σ ∈ [σw - µ,σw + µ] and µ ) 0.005 and is zero elsewhere in σ ∈ [0,1]. The state constraints are enforced at four points:

σw ) [0.17 0.35 0.5 0.640 ] The actuator and sensing distribution functions, bdi(σ) and cdj(σ), are of the same nature (bdi(σ) ) cdj(σ)) and they are given by bdi(σ) ) 1/2µ for σ ∈ [pai - µ, pai + µ] and bdi(σ) ) 0 elsewhere in [0,1], where µ ) 0.005 is a small positive real number and the actuation/sensing is enforced at the following points:

pai ) [0.15 0.21 0.36 0.45 0.55 0.64 0.79 0.85 ] (where i ) 1, ..., 8) Control input injection u(t) ) udj(t) is constrained for all actuator locations between [umin,umax] ) [-45,20]. For these values, it was verified that the operating steady state, Z(σ,t) ) 0, is an unstable one (see Figure 1). The control objective is to stabilize the state profile at an unstable zero steady state by manipulating u(t), subject to the input and state constraints. We first formulate the PDE of eq 35 into the infinite-dimensional equation of the form of eq 12 by formulating the operator A ) bh(d2/dσ2) + cj with domain

D(A) ) {φ(σ) ∈ L2(0,1): φ(σ), φ′ (σ) are absolute constraints, Aφ(σ) ∈ L2(0,1) and φ′(0) ) 0 ) φ′(1)} (36)

(35) (35b) (35c,d,e)

The eigenspectrum and associated eigenfunctions of the symmetric operator A are given by

λκ ) cj - bhκ2π2

(for κ g 0), φ0 ) 1

φκ(σ) ) x2 cos(κπσ),

(for κ ) 1, ..., ∞)

(37a,b) (37c)

(35f)

(for w ) 1, 2, 3) (35g)

The following actuator/sensor position parametrized infinitedimensional equation modal representation is obtained through Galerkin’s method and it is given in the following form:

112


z˘ (t) ) Az(t) + B(pai)u(t), z(0) ) a0 Z h pai(t) ) C(pai)z(t)

(38a,b) (38c)

where

z(t) ) [z0(t) z1(t) ‚‚‚ zl(t) ] In the ensuing calculations, eq 39 is approximated using a finitedimensional approximation with 20 modes (further increases in the number of equations led to identical numerical results). The matrix A is a diagonal matrix, given by A ) diag{λκ}, for κ ) 0, 1, ..., l. Bpai is a 20 × pai matrix, whose (i,pai)th element is given by Bipai ) (bpai(σ),φi(σ)), and Cpai ) BpTai by the collocated nature of the actuators and sensors. A discrete finitedimensional model, given by eqs 22 and 23, is obtained by a standard transformation applied with the sampling time δ (defined as 1/(2 max|{σ(A)}|) ) 7.0234 × 10-4, where A ˜ ) eAδ, B ˜ ) δeBδB, C ˜ ) C).36 Slow modes are given as

zs(k) ) [z0(k) z1(k) z2(k) ]T and fast modes are given as

Figure 2. Closed-loop state profile under the implementation of the optimal activation actuator policy described by eqs 39 and 40, with the input and state constraints that account for slow modes in the constraints.

zf(k) ) [z3(k) ‚‚‚ zl(k) ]T Using these projections, the state constraints are expressed by eq 25. We now proceed with the design and implementation of the two optimal switching predictive control policies presented in the previous section. These two control policies are implemented under the assumption that the entire PDE state is known, because exact evolution of the zs and zf modal states is required in the algorithm realization. The appearance of sensors in the performance functional illustrates that low-dimensional representation of the PDE statesthat is, Z h (t) ) Cszs(t)scan be used efficiently in such a constructed MMPC law. Under the first scenario, we use the zs subsystem in eq 22 as the basis for the predictive controller design (the zf subsystem is neglected). For this case, we consider an actuator/sensor activation policy that considers predictive control law as given by eqs 30 and 31 and an optimal switching policy: N-1

arg min min pai

u

T [zs(k + ι|k)TC ˜ sp ∑ ι)0

ai

QC ˜ spaizs(k + ι|k) +

R(σw1,t) )

∫01 rw(σw1)Z(σ,t) dσ

is being violated at the lower constraint at σw1 ) 0.64 for some time. Violation of the state constraint ensues from the neglect of the contribution of the zf modal states in the PDE state in the predictive controller formulation. To account for the evolution of fast states in the optimal switching policy formulation, we consider the following predictive control formulation with the objective function and constraints: N-1

u(k + ι|k)TRu(k + ι|k)] + zs(k + N|k)TQpaizs(k + N|k) (39) zs(ι + 1|k) ) A˜szs(ι|k) + B ˜ spaiu(ι|k)

predictive policy given by eqs 39 and 40, which stabilizes the PDE state at the unstable zero steady state, starting from the initial condition Z(σ,0) ) 0.65 sin(πσ). In all considered simulation studies, controller parameters Q, R, T, and the initial condition are kept identical. By examining the solid line shown in Figure 4 (presented later in this paper), we can see that the state constraint R(σw1,t), which is given as

(40)

arg min min pai

u

T [zs(k + ι|k)TC ˜ spa QC ˜ spa zs(k + ι|k) + ∑ ι)0 i

i

h paizs(k + N|k) + u(k + ι|k)TRu(k + ι|k)] + zs(k + N|k)TQ (41)

subject to

umin e u(ι|k) e umax max Zmin w e S˜swzs(ι|k) e Zw

zus(N) ) 0

(for w ) 1, 2, 3)

(for ι ) 0, 1, ..., N - 1)

where Q ) 100, R ) 0.001, and N ) 150. To ensure stability, we also impose a terminal equality constraint of the form zus(N) ) 0 to the optimization problem. The resulting quadratic program is solved using the MATLAB subroutine QuadProg. The quadratic program uses a projection method, which is a variation of the well-known Simplex method for linear programming.43 The control action is then implemented on the 20thorder model of eqs 22 and 23. Figure 2 shows the closed-loop state evolution under the implementation of the optimal model

zs(ι + 1|k) ) A˜szs(ι|k) + B ˜ spaiu(ι|k)

(42a)

zf(ι + 1|k) ) A˜fzf(ι|k) + B ˜ fpaiu(ι|k)

(42b)

subject to

umin e u(ι|k) e umax S˜swzs(ι|k) e Zmax - S˜fwzf(ι|k) g

(for w ) 1, 2, 3)

-S˜swzs(ι|k) e Zmin g + S˜fwzf(ι|k) zus(N) ) 0

(for ι ) 0, 1, ..., N - 1)

Results are shown in Figures 3 and 4, where it is demonstrated that the predictive controller, when designed with optimal


Figure 5. Control inputs under the optimal policy described by (s) eqs 39 and 40 and (‚ ‚ ‚) eqs 41 and 42. Figure 3. Closed-loop state profile under the implementation of the optimal activation actuator policy described by eqs 41 and 42, with the input and state constraints that account for slow and fast modes in constraints.

Figure 6. Optimal switching sequences among actuators under the optimal policy described by (s) eqs 39 and 40 and (‚ ‚ ‚) eqs 41 and 42.

Figure 4. Closed-loop state profiles at zw1 ) 0.64 ((s) under the implementation of eqs 39 and 40 and (‚ ‚ ‚) under the implementation of eqs 41 and 42) and at zw1 ) 0.17 ((- ‚ -) under the implementation of eqs 41 and 42), which undergoes time-varying disturbance.

switching and using eqs 41 and 42, successfully stabilizes the state profile at the zero steady state, and that the state constraints are satisfied for all times. Although, in the presented simulation studies, one of the prespecified locations for the actuator implementation coincides with the location where the PDE state constraints are enforced, this does not depreciate the generality of the proposed optimal switching policy algorithm, because such collocated architecture may initially create an advantage in the actuation authority to the actuator/sensor closest to the constraint, but still it will take both the input and PDE state constraints conditions at other locations into account. The corresponding manipulated input profiles are given in Figure 5. It is demonstrated that, in the case of optimal switching policy (eqs 39 and 40), the actuator position is alternated (see the solid line in Figure 6), while the input constraints are active (see the solid line in Figure 5; the upper and lower input constraints are

active). However, this computed control action, when applied at the allowable actuators’ positions, does violate state constraints, whereas the one that is computed based on eqs 41 and 42 does not violate state constraints, because it is lower in magnitude for the same corresponding time of the controller usage (see the dotted line in Figure 5). Remark 9: The computational advantages of the proposed approach are demonstrated in the case of the controller synthesis given in eqs 39 and 40, in which low-dimensional PDE state representation is used in the performance functional. This algorithm provides optimal stabilization and input state constraints satisfaction for the “coarse’’ PDE state representation, which, in principle, contains a few spatial modes that capture 60%-80% of the entire PDE state. The computational burden of integration over the horizon length of entire finite-dimensional PDE modal states,

z(t) ) [z0(t) z1(t) ‚‚‚ z19(t) ] to the case when only slow modal states are considered in the optimization functional; that is,

z(t) ) [z0(t) z1(t) z2(t) ]

114


Figure 7. Closed-loop state profile under the implementation of the optimal activation actuator policy described by eqs 41 and 42, with the input and state constraints that account for slow and fast modes in constraints under the influence of a disturbance d(σ,t) (d(σ,t) ) δ(σ - σd) sin(2/πt)) applied at σd ) [0.0409;0.1409] for a period of time t ) [0.1;0.12].

is recognized in the light of the computational time required for the QP program to be constructed. Namely, the matrix H h in the QP problem described by eq 29 is of the size of the horizon length (N × N). However, the construction of this matrix, in the case of the considered full model (A,B,C) will require much more computational time, compared to that where only three modal states are considered (As,Bs,Cs), because of the size of the matrices. This difference is even more emphasized when one considers the large number of actuator/sensors available for the generation of optimal switching policy, because the QP problems described by eq 29 must be constructed for all available actuator/sensor locations. Remark 10: Another important issue is the robustness of the proposed control synthesis, with respect to the unknown disturbances that may appear within the domain (see Figures 4-7; in the affine manner, a time-varying disturbance is added to the PDE in eq 35) and perturb a system so that the state constraints or input constraints satisfaction may not hold. Still, the proposed synthesis successfully resolves the appearance of time-varying disturbances within the domain and it stabilizes the profile in the optimal and smooth manner. However, a control practitioner must be cautious in the case of possible parametric uncertainty in eq 35, which may induce the redistribution of the eigenmodes and change the number of unstable modes in the slow zs subsystem. Nevertheless, to achieve robustness of the proposed model predictive algorithm that satisfies the input and state constraints, the crucial feature is to formulate a precise identification of the modal states evolution, which is not addressed in this work and lies in the scope of future research directions. 5. Summary In summary, a actuator switching policy for spatially distributed control of a parabolic partial differential equation (PDE) with state and input constraints obtained by means of the loworder modal model predictive controller (MMPC) design has been developed. The synthesis of the low-order modal controller benefits from fundamental properties of dissipative systems that

are described by linear parabolic PDEs, whose abstract evolutionary equation representation provides the basis for the loworder controller synthesis on the premises of the existence of dominant dynamics, described by a few dominant modes given by the abstract evolutionary equation. An important feature of such structured dynamics is the way the modes are coupled. In particular, for the synthesis of the low-order modal model predictive controller, the diagonal dominant feature of evolutionary operators associated with dissipative parabolic PDEs systems where modes are not coupled through the states but only by way of input injection are the main features utilized in the constructed control law. The dominant dynamics is utilized in the standard model predictive control (MPC) formulation as the low-dimensional actuator/sensor parametrized dynamics that captures the most dominant features of the state evolution, and, therefore, it is used in the construction of the cost functional for each of preset available actuator/sensor locations. Optimization of the actuator/sensor position-dependent cost functionals is performed, with respect to the control input vector and with respect to the actuator/sensor position. Fast modal dynamics evolution is accounted exactly in the modal state constraints construction, to guarantee PDE state constraints satisfaction. The application of a such a constructed actuator guidance policy has been successfully demonstrated by the illustrative example where stabilization, input, and PDE state constraints satisfaction, in the case of the class of distributed parameter systems described by the linear parabolic PDEs, are accomplished. Literature Cited (1) Butkovskii, A. G. Distributed Control Systems; Kranc, G. M., Ed.; Modern Analytic and Computational Methods in Science and Mathematics, Vol. 11; American Elsevier Publishing Co.: New York, 1969. (2) Triggiani, R. On the stabilizability problem in Banach space. J. Math. Anal. Appl. 1975, 52, 383-403. (3) Balas, M. J. Feedback control of linear diffusion processes. Int. J. Control 1979, 29, 523-533. (4) Ray, W. AdVanced Process Control; McGraw-Hill: New York, 1981. (5) Curtain, R. F. Finite dimensional compensators for parabolic distributed systems with unbounded control and observation. SIAM J. Control Optim. 1984, 22, 255-276. (6) Curtain, R. F. On stabilizability of linear spectral systems via state boundary feedback. SIAM J. Control Optim. 1985, 23, 144-152. (7) Curtain, R. F.; Zwart, H. An Introduction to Infinite-Dimensional Linear Systems Theory; Springer-Verlag: New York, 1995. (8) Christofides, P. D. Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes; Birkha¨user: Boston, 2001. (9) Christofides, P. D. Robust control of parabolic PDE systems. Chem. Eng. Sci. 1998, 53, 2949-2965. (10) Lions, P. D. Optimal Control of Systems GoVerned by Partial Differential Equations; Springer-Verlag: New York, 1971. (11) Gibson, P. D. The Riccati integral equations for optimal control problems on Hilbert space. SIAM J. Control Optim. 1971, 17, 537-565. (12) Kubrusly, C. S.; Malebranche, H. Sensors and controllers location in distributed systemssA survey. Automatica 1985, 21, 117-128. (13) Antoniades, C.; Christofides, P. D. Integrating nonlinear output feedback control and optimal actuator/sensor placement for transport reaction processes. Chem. Eng. Sci. 2001, 56, 4517-4535. (14) Waldraff, W.; Dochain, D.; Bourrel, S.; Magnus, A. On the use of observability measures for sensor location in tubular reactor. J. Process Control 1998, 8, 497-505. (15) Alonso, C. S.; Frouzakis, C. E.; Kevrekidis, I. G. Optimal sensor placement for state reconstruction of distributed process systems. AIChE J. 2004, 50, 1438-1452. (16) El-Farra, N. H.; Armaou, A.; Christofides, P. D. Analysis and control of parabolic PDE systems with input constraints. Automatica 2003, 39, 715-725. (17) El-Farra, N. H.; Christofides, P. D. Coordinated feedback and switching for control of spatially-distributed processes. Comput. Chem. Eng. 2004, 28, 111-128.

Ind. Eng. Chem. Res., Vol. 47, No. 1, 2008 115 (18) Armaou, A.; Christofides, P. D. Dynamic optimization of dissipative PDE systems using nonlinear order reduction. Chem. Eng. Sci. 2002, 57 (24), 5083-5114. (19) Patwardhan, A. A.; Wright, G. T.; Edgar, T. F. Nonlinear modelpredictive control of distributed-parameter systems. Chem. Eng. Sci. 1992, 47, 721-735. (20) Demetriou, M. A.; Kazantzis, N. A new actuator activation policy for performance enhancment of controlled diffusion processes. Automatica 2004, 40, 415-421. (21) Demetriou, M. A.; Iftime, O. V. Finite horizon optimal control of switched distributed parameter systems with moving actuators. Proc. Am. Control Conf., 2005 2005, 3912-3917. (22) Biegler, L. T.; Ghattas, O.; Heinkenschloss, M.; van Bloemen Waanders, B. Large-Scale PDE-Constrained Optimization; Lecture Notes in Computational Science and Engineering, Vol. 30; Springer: Berlin, 2003. (23) Garcia, C.; Prett, D.; Morari, M. Model predictive control: Theory and practicesa survey. Automatica 1989, 25, 335-348. (24) Rawlings, L. T. Tutorial overview of model predictive control. IEEE Control Syst. Mag. 2000, 20, 38-52. (25) Mayne, L. T.; Rawlings, J. B.; Rao, C. V.; Scokaert, P. O. M. Constrained model predictive control: Stability and optimality. Automatica 2000, 36, 789-814. (26) Dubljevic, S.; Mhaskar, P.; El-Farra, N. H.; Christofides, P. D. Predictive control of transport-reaction processes. Comput. Chem. Eng. 2005, 29, 2335-2345. (27) Dubljevic, S.; El-Farra, N. H.; Mhaskar, P.; Christofides, P. D. Predictive control of parabolic PDEs with state and control constraints. Int. J. Robust Nonlinear Control 2006, 16, 749-772. (28) Dubljevic, S; Christofides, P. D. Predictive output feedback control of parabolic PDEs. Ind. Eng. Chem. Res. 2006, 45 (25), 8421-8429. (29) Dubljevic, S; Christofides, P. D. Predictive control of parabolic PDEs with boundary control actuation. Chem. Eng. Sci. 2006, 61, 62396248. (30) Hoo, K. A.; Zheng, D. System identification and model-based control for distributed parameter systems. Chem. Eng. Sci. 2004, 28 (8), 1361-1375. (31) Demetriou, M. A. Integrated actuator/sensor placement and hybrid controler design of flexible structures under worst case spatiotemporal disturbance variations. J. Intell. Mater. Syst. Struct. 2004, 15, 901-931.

(32) Demetriou, M. A.; Kazantzis, N. Compensation of spatiotemporally varying disturbances in nonlinear transport processes via actuator scheduling. Int. J. Robust Nonlinear Control 2005, 14, 191-197. (33) Iftime, O. V.; Demetriou, M. A. Optimal Control for Switched Distributed Parameter Systems with application to the Guidance of a Moving Actuator. In Proceedings of the 16th IFAC World Congress, Prague, Czech Republic, 2005. (34) Demetriou, M. A.; Kazantzis, N. A new integrated output feedback controller synthesis and collocated actuator/sensor scheduling framework for distributed parameter systems. Comput. Chem. Eng. 2005, 29, 867876. (35) Kato, T. Perturbation Theory for Linear Operators; SpringerVerlag: New York, 1966. (36) Elaydi, S. N. An Introduction to Difference Equations, Second Edition; Springer-Verlag: New York, 1999. (37) Muske, K. R.; Rawlings, J. B. Model predictive control with linear models. AIChE J.1993, 39, 262-287. (38) de Oliveira, N. M. C.; Biegler, L. T. Constraints handling and stability properties of model predictive control. AIChE J. 1994, 40, 11381155. (39) Zheng, A.; Morari, M. Stability of model predictive control with mixed constraints. IEEE Trans. Autom. Control 1995, 40, 1818-1823. (40) Scokaert, P. O. M.; Rawlings, J. B. Feasibility issues in linear model predictive control. AIChE J. 1999, 45, 1649-1659. (41) Demetriou, M. A.; Vayena, P. A. O.; Doumanidis, H. Scanning actuator guidance scheme in a 1-D thermal manufacturing process. IEEE Trans. Control Syst. Technol. 2003, 11, 757-764. (42) Muske, K. R. Linear model predictive control of chemical processes, Ph.D. Dissertation, The University of Texas-Austin, Austin, TX, 1995. (43) Dantzig, G. B.; Orden, A; Wolfe, P. Generalized simplex method for minimizing a linear from under linear inequality constraints. Pac. J. Math. 1955, 5, 183-195.

ReceiVed for reView April 18, 2007 ReVised manuscript receiVed September 20, 2007 Accepted September 24, 2007 IE070546V

Constraints-Driven Optimal Actuation Policies for Diffusion-Reaction

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