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Computers and Chemical Engineering 32 (2008) 1373–1384

A path constrained method for integration of process design and control

Jigar Patel, Korkut Uygun, Yinlun Huang ∗Department of Chemical Engineering and Materials Science, Wayne State University, Detroit, MI 48202, United States

Received 16 May 2006; received in revised form 12 March 2007; accepted 13 June 2007Available online 19 June 2007

bstract

Integration of process design and control (IPDC) has been the holy grail of process systems engineering since the introduction of heat and massntegration. A proper combination of these separate yet connected tasks carries the promise of achieving superior designs that cannot be realized withonventional procedures. In this work, a bi-level dynamic optimization approach is introduced for achieving IPDC in its true sense. The principaldea proposed here is to utilize an optimal controller (a modified linear quadratic regulator) to practically evaluate the best achievable controlerformance for each candidate design during process design. The evaluation of complete, closed-loop system dynamics can then be meshed with

superstructure-based process design algorithm, thus enabling considering both cost and controllability in design of a process. The practicalityf the introduced approach enables a solution of this complex dynamic optimization problem within reasonable computational requirements, asemonstrated in an evaporator case study.

2007 Elsevier Ltd. All rights reserved.

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eywords: Integration of process design and control; Bi-level optimization; Lin

. Introduction

Process integration has been an ideal method for reducingnergy and raw materials consumption of process industriesEl-Halwagi, 1997; El-Halwagi & Manousiouthakis, 1989; Ho

Keller, 1987; Linnhoff et al., 1982; Rossiter, Spriggs, &lee, 1993). As the level of integration in process plantsid increase, industrial practice made it clear that processontrollability should be considered during process synthesisElliott & Luyben, 1995; Fisher, Doherty, & Douglas, 1988;

cAvoy, 1987; Morari, 1992; Perkins & Walsh, 1994; Sheffield,992). This realization led to the introduction of another typef integration—the integration of process design and controlIPDC).

Process controllability problems of a plant usually haveheir sources originated in the early stages of process synthe-is and design. Naturally, they should be prevented in the sametage. Over the past decades, a primary focus has been on pro-

ess alternative screening, mostly based on steady-state models;hese methods include system flexibility (Floudas & Grossmann,986; Swaney & Grossmann, 1985a,b), disturbance propagation

∗ Corresponding author. Tel.: +1 313 577 3771; fax: +1 313 577 3810.E-mail address: yhuang@wayne.edu (Y. Huang).

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098-1354/$ – see front matter © 2007 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2007.06.005

uadratic regulator

nd rejection (Huang & Fan, 1992; Yan, Yang, & Huang, 2001;ang, Gong, & Huang, 1996; Yang, Lou, & Huang, 2005). Therere also a number of dynamic indices, such as relative gain arrayBristol, 1966; Shinskey, 1995), the resiliency index (Morari,983) and condition number (Grosdidier, Morari, & Holt, 1985;orari & Zafiriou, 1989).While clearly desirable for IPDC purposes, advances in meth-

ds for incorporating dynamic models into the design problemave been slow. Luyben and Floudas (1994a,b) and Walsh anderkins (1996) presented two of the earlier efforts in this direc-

ion; operability (Georgakis, Uzturk, Subramanian, & Vinson,003; Subramanian & Georgakis, 2005) and the dynamic flex-bility index (Dimitriadis & Pistikopoulos, 1995) are also wellorth mentioning.

.1. Current state of IPDC and motivation

As summarized above, for most of the existing algorithmsontrollability is evaluated utilizing controllability indices.owever, the available indices are all indirect indicators of

chievable controllability because closed-loop system dynamic

erformance is measured indirectly. These indices are unable toheck if the system violates any operational constraints duringhe transition process, where the effects of incoming distur-ances to the system are most significant. Therefore, what is

1374 J. Patel et al. / Computers and Chemical

Nomenclature

A, B linearized system matricesd design variablesf system modelfn, gn, hn steady-state system, equality, and inequality

constraints for period ng inequality constraintsh equality constraintshd, gd path constraints (applied during disturbance

rejection)K optimal controller gain matrixQ, R weight matrices in the linear quadratic regulator

problemS solution of the Riccati equationt timeu control (manipulated) variablesus

min, usmax lower and upper bounds for control variables

at steady-stateud

min, udmax lower and upper bounds for control variables,

dynamicx system variables (also referred to as state vari-

ables)x time derivative of system variablesxn steady-state values of x at operation period n (n = 0

indicates nominal)xs

min, xsmax lower and upper bounds for system variables

at steady-statexd

min, xdmax lower and upper bounds for system variables,

dynamicx, u deviation variables (from nominal values)

Greek symbols� disturbances�n steady-state disturbance for period n

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�d dynamic disturbance profile

ecessary is the development of an IPDC methodology thatxamines system closed-loop dynamic performance via solvingystem dynamic equations in the process structure determina-ion and design stage. This is a necessity in order to claimomplete integration of design and control.

In parallel to the argument above, a recent trend in the fieldf IPDC is to involve dynamic simulation of the process withinhe design procedure (Bansal, Perkins, Pistikopoulos, Ross, &an schijndel, 2000; Kookos & Perkins, 2001; Sakizlis, Perkins,

Pistikopoulos, 2003), in order to ensure that dynamic con-traints and closed-loop controllability measures are satisfied.he simulation of system dynamics in the presence of controllersan, therefore, be performed, which achieves a level of integra-ion completely unattainable by usual IPDC techniques. All theystem variables (i.e., state and manipulated variables) can be

bserved at the dynamic level by this approach.

The main obstacle with this approach is the size of controlystem design problem that has to be augmented to the already

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Engineering 32 (2008) 1373–1384

hallenging process design problem. A major obstacle in IPDC ishat without a control system, it is not possible to evaluate systemynamic performance, i.e., evaluate the input–output controlla-ility (i.e., the ability to satisfy all the steady-state and dynamiconstraints under incoming disturbances and set-point changesoth at static and dynamic levels). However, due to the fact thatossible control configurations, including cascade loops, signalransformation techniques and combinations of control devicesrow ad infinitum, control system design is more an art thancience. The number of options available for selecting controlystem design increases the problem size drastically and thusill not be manageable in process synthesis stage. A usual prac-

ice is to enable a limited number of PI loops with a minimalumber of observed manipulated variable pairing options, andse the observed dynamic performance as an underestimate ofhe achievable control performance (e.g., Kookos & Perkins,001).

As a better method for sidestepping the “curse of dimension-lity” problem involved in controller synthesis decisions, and yetreserving the quality of control performance estimates, effortsave been made to incorporate centralized, on-line optimization-ased controllers in process design (Brengel & Seider, 1992;oeblein & Perkins, 1999; Swartz, Perkins, & Pistikopoulos,000), as well as control structure selection (Zhu & Henson,002). One such approach utilizes model predictive controlMPC). The design problem is reduced in size significantly bysing MPC as control scheme, as there is no control structureo be designed. However, MPC requires the solution of an opti-

ization problem at frequent control steps, and therefore stillisplays high computational requirements. This issue has ren-ered IPDC with MPC practicable for problems of only smallizes (Uygun, 2004).

The latest iteration of modern process design methodseeks to remove this final obstacle by finding a computation-lly feasible centralized control scheme with provable controlerformance. The basic idea of these studies is to derive aimple state-feedback controller, which should demonstratehe control performance similar to that of a centralized con-roller with online-optimization (such as MPC). The works ofohansen, Petersen, and Slupphaug (2002) and Grancharovand Johansen (2002) are notable in terms of developing suchxplicit model-based controllers. But these approaches are basedn trading control performance for computational advantage,hich usually leads to significantly sub-optimal control. On

he other hand, the use of model-based parametric controllersBemporad, Morari, Dua, & Pistikopoulos, 2002; Pistikopoulos,ua, Bozinis, Bemporad, & Morari, 2002; Sakizlis et al., 2003;akizlis, Perkins, & Pistikopoulos, 2005) results in a switchingroportional state-feedback control law that is mathematicallyroven to have optimal control performances. However, the con-rol design (i.e., design of parametric controllers) is still anterative procedure, which has to be augmented to the designroblem (i.e., repeated for each candidate design during the

e prohibitive in computational cost in some circumstances.nother issue is the handling of candidate designs (during iter-

tions of solving the design problem) that are infeasible with

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J. Patel et al. / Computers and Chem

espect to the (dynamic) path constraints; the methods devel-ped for implementation of parametric controllers for IPDC doot seem to address such cases. Even if it is addressed througharious exception handling procedures (such as selective con-traint removal, as is the case with industrial MPC methods), thisssue may render the parametric controller design step exceed-ngly cumbersome for the already computationally expensivePDC problems.

This work introduces an alternative off-line centralizedontroller-based IPDC approach, based on optimal control. Theroposed method enables quick identification of an optimaltate-feedback control law for each candidate design during therocess synthesis stage, based on a modified linear quadraticegulator formulation. Further, the control law developed is aroportional-integral state-feedback controller, hence promisingore robust control and increased control objective formulationexibility as compared to the parametric controllers. The mesh-

ng of the proposed approach with a design algorithm enablesractical consideration of both cost and control performanceuring design process.

. IPDC with modified linear quadratic regulatorIPDC-mLQR)

In this work, an optimal control approach by employing aodified linear quadratic regulator (mLQR) is investigated. The

asic issue here is how to solve a dynamic optimization problemhere the closed-loop control performance is evaluated for eachesign alternative. The approach is presented starting with aormulation of the IPDC problem.

.1. A general IPDC formulation

IPDC in this work is treated as a dynamic optimization prob-em where control-related dynamic properties are consideredimultaneously with cost-related static properties in order toesign a cost effective, highly controllable process. A generalormulation for IPDC problem can be presented below.

minw.r.t.d,xn,un,x(t),u(t)

I

= Φ(d , xn, un)

+λ1

2

∫ ∞

0(xT · Q · x + uT · Rα · u + ˙uT · R� · ˙u) dt (1)

.t.

˙ = f(x(t), u(t), d, �(t), t) (2)

n(d, xn, un) = 0 (3)

smin ≤ xn ≤ xs

max (4)

smin ≤ un ≤ us

max (5)

n n n n

(d, x , u , � ) = 0 (6)

n(d, xn, un, �n) ≤ 0 (7)

dmin ≤ x(t) ≤ xd

max (8)

stst

Engineering 32 (2008) 1373–1384 1375

dmin ≤ u(t) ≤ ud

max (9)

d(d, x(t), u(t), �(t), t) = 0 (10)

d(d, x(t), u(t), �(t), t) ≤ 0 (11)

here

˜(t) = x(t) − xn|n=0 (12)

˜ (t) = u(t) − un|n=0 (13)

|t=t0 = xn|n=0 (14)

|t=t0 = un|n=0 (15)

(t) = �d(t) (16)

here x is the vector of state variables; u the vector of controlariables, � the vector of disturbances, and d is the vector ofesign variables. Superscripts d and s denote dynamic and staticf relevant variables, respectively. The symbols used in this workre outlined in ‘Nomenclature’.

In the objective function, the first part, Φ, represents theesign cost (e.g., total annualized cost (TAC)), and the secondart represents the equivalent control cost reflected by dynamicerformance criterion; an integral square error form has beensed for compatibility with the linear quadratic regulator (LQR)ormulation, as it will be discussed later in the text. It has beenoted that the integral square error form also has an indirectorrelation to the cost (Chintapalli & Douglas, 1975; Shunta,995), although the ˙u term, which leads to a smoother integralontrol action, does not have an economic significance.

The system dynamics is described by a set of differentialquations given in Eq. (2). It should be underlined that the for-ulation above imposes not only dynamic constraints but also

teady-state feasibility constraints in a multi-period formula-ion given by Eqs. (3)–(7). Superscript n is used on steady-stateontrol and state variables. The system equations, fn, and theteady-state constraints, hn and gn, correspond to the particularperation period (note that all variables with superscript n aretatic). Period marked by index n = 0 identifies nominal condi-ions. For instance, Eq. (3) with n equal to 0 requires the systemo be at steady-state when there is no disturbance applied to theystem. The other operation periods correspond to the verticesf the disturbance space, defined by �n as it will be explainedelow. Note that the problem considers designing a process thatan handle a finite number of disturbances (i.e., a design that canope with the worst case scenarios), rather than a dynamic flex-bility index type of study (Dimitriadis & Pistikopoulos, 1995).

typical �n vector, therefore, would be a combination of posi-ive and negative deviations in each disturbance variable, henceeading to a total number of 2dim(�), equal to the number of cor-ers of the hyper-rectangular disturbance space. In general, theynamic disturbance profile, �d, is formed by combinations of

tep disturbances with the values in �n. It is important to ensurehat each combination of disturbance lasts long enough for theystem to settle to the new post-disturbance steady-state in ordero fairly measure the effects of each disturbance on the system.

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376 J. Patel et al. / Computers and Chem

Eqs. (4) and (5) represent the steady-state bounds on systemnd control variables, respectively. Eqs. (8) and (9) are, respec-ively, the dynamic bounds on system and control variables. Eqs.6), (7), (10) and (11) signify other possible static and dynamicpath) constraints.

.2. The IPDC-mLQR methodology—a bi-leveleformulation

There are two alternative solution methods for a dynamicptimization problem, simultaneous and sequential (usually, theormer is used in an infeasible-path solution method, whilehe latter is employed in a feasible-path solution approach). Aimultaneous approach discretizes the dynamic equations so thathe problem is transformed to a conventional static optimiza-ion problem. The system model equations are then imposeds a number of algebraic equality constraints, and system vari-bles are introduced as additional decision variables (Bequette,991; Chachuat, Singer, & Barton, 2005). The problem withhis method is that it produces an exponentially amplified num-er of decision variables and equality constraints. A sequentialpproach, on the other hand, uses an integrator to solve theifferential equations in an inner subroutine, hence creatingpseudo-static correlation between the design decisions and

ynamic properties such as control performance. This results insmall number of black-box functions that have to be evaluateduring the optimization which does not increase the dimension-lity of the problem. However, since the inner routine typicallynvolves numerical solution techniques, it produces very non-inear correlations, which can create various issues in problemolving.

The IPDC problem here is formulated as a bi-level optimiza-ion problem, which is then solved using a two-stage sequential,easible-path method that keeps the problem size manageable.his new problem consists of a main optimization step with

espect to static (design) variables, subject to the solution ofynamic optimization with respect to the dynamic (control)ariables, which is delineated below:

min,xn,un

I = Φ(d, xn, un)

+λ1

2

∫ ∞

0(xT · Q · x + uT · Rα · u + ˙uT · R� · ˙u) dt

(17)

.t.

n(d, xn, un) = 0 (18)

smin ≤ xn ≤ xs

max (19)

smin ≤ un ≤ us

max (20)

n(d, xn, un, �n) = 0 (21)

n n n n

(d, x , u , � ) ≤ 0 (22)

dmin ≤ x(t) ≤ xd

max (23)

dmin ≤ u(t) ≤ ud

max (24)

bwmb

Engineering 32 (2008) 1373–1384

d(d, x(t), u(t), �(t), t) = 0 (25)

d(d, x(t), u(t), �(t), t) ≤ 0 (26)

in˙u(t)

J = 1

2

∫ ∞

0(xT · Q · x + uT · R� · u + ˙uT · R� · ˙u)dt (27)

.t.

˙ = f(x(t), u(t), d, �(t), t) (28)

The notation here is the same as in the general IPDC prob-em. Eqs. (12)–(16) also apply and are not repeated for sake ofrevity. It can be shown that this bi-level formulation is equiva-ent to the general IPDC problem formulation in Eqs. (1)–(16),ith the single exception that the control action is assumed

hrough the manipulation of rate of change of the control vari-bles, ˙u, rather than direct manipulation, u. The reason of thishange is to incorporate integral action into the control solution,hich will be discussed in Section 2.3. A major advantage of

his formulation is that the control and design problems are com-artmentalized, which enables focusing on the more problematicontrol part more efficiently. The pseudo-static design problemoes not include the dynamic variables as decision variables,nd enables the evaluation of the dynamic functions (constraintsnd the second part of the objective) as if they are static func-ions of the decision variables. The dynamic part, on the otherand, is specifically formulated so as to be solvable separatelynd analytically. In this way, computational requirements on thisost demanding part of the IPDC problem can be significantly

educed.

.3. Dynamic optimization via modified linear quadraticegulator

In the inner, dynamic optimization, the objective is to mini-ize the integral of squared deviations from the set points. Eq.

28) is a system model, with its nominal conditions in Eqs. (14)nd (15). This sub-problem is with reduced size and only theontrol variables appear as decision variables.

Different control schemes, such as MPC and parametricontrollers, can and have been used to solve such dynamic opti-ization problems (e.g., Uygun, 2004). Non-linear constrainedPC is a general and flexible approach, but the high compu-

ational requirements and possible convergence problems makehe approach limited.

The essential idea in this work is to use the LQR to solvehis problem, with the mathematical elegancy of optimal con-rol, to reduce the computational load. As Lewis stated (1992),Since naturally occurring systems exhibit optimality in theirotion, it makes sense to design man-made control systems in

n optimal fashion”. The LQR (as well as the linear quadraticaussian regulator) was introduced by Kalman (1960a,b) tohom the development of many modern control concepts can

e attributed. Kalman studied the LQR problem where the goalas to minimize a quadratic control performance function (i.e.,inimization of the integral square error) for a system that can

e described by a linear model. A standard LQR problem can

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J. Patel et al. / Computers and Chem

e mathematically defined as follows:

inu(t)

J = 1

2

∫(xT · Q · x + uT · R · u) dt (29)

˙ = Ax(t) + Bu(t) (30)

The objective function J penalizes the squared input and stateeviations from the origin, and includes separate state and inputeight matrices Q and R to allow for tuning trade-offs. Note thatere we assume a linear/linearized model, and both the state andontrol variables are taken as the deviation variables; appropriateethods to obtain linearized models will be discussed in the

ollowing sections. The solution of the LQR problem yields atate-feedback proportional controller (Bryson and Ho, 1975),ith a gain matrix K computed from the solution of the algebraiciccati equation (ARE) specified in the following equation:

T · S + S · A + Q − S · B · R−1 · BT · S = 0 (31)

˜ = −K · x(t) (32)

= R−1 · BT · S (33)

It should be noted that the ARE solution uses a steady-stateain matrix, and is sub-optimal with respect to the full (time-ependent) Riccati equation. This sub-optimality is generallyegligible, and very often the ARE is preferred since its solutions much simpler than that from a full Riccati equation.

The traditional LQR formulation produces a proportional-nly state-feedback controller, similar to the parametricontrollers (Sakizlis et al., 2004). Although a state-feedback con-roller performs (with a proper tuning), in general, better thancollection of traditional single-input single-output controllers,proportional-only controller may lead to either steady-state

ffsets or overly aggressive control variable movements, andt is not possible with this formulation to tune for a smootherontroller profile. To overcome this shortcoming, it is desirableo reformulate the optimal feedback control problem so as tollow a smoother integral control action. The integral action cane included by introducing a penalty term for the manipulatedariables movement. This leads to a modified LQR (i.e., mLQR)s follows:

in˙u(t)

J = 1

2

∫(xT · Q · x + uT · R� · u + ˙uT · R� · ˙u) dt (34)

his is exactly the same as Eq. (27). This alternative objec-ive formulation can be handled by some simple modifications,hich yields a proportional integral (PI) control law as statedelow (see Appendix A for derivation and calculation of gainatrices K1 and K2):

˜ (t) = −K2 · x(t) − K1 ·∫ t

0x(t) dt (35)

The solution of the mLQR problem gives an optimally tunedtate-feedback PI control scheme. A major advantage of this

pproach is that an explicit control law is derived. Therefore,iven a candidate design (i.e., matrices A and B) and an objec-ive function, the ARE can be solved to calculate the optimalain matrices. The solution of the ARE, while not trivial, is very

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Engineering 32 (2008) 1373–1384 1377

uick and fairly reliable. Once the gain matrices are evaluated,he differential equations describing the state and control vari-bles can be integrated simultaneously to yield a simulation withhe optimal mLQR controller, thus enabling evaluation of anyynamic performance measures desired.

.4. Implementation

The solution of the mLQR problem yields an optimally tunedtate-feedback PI control scheme, and simulation with this con-roller enables evaluation of the entire set of path constraintss well as the control performance. The performance of theptimal controller is the best achievable performance for theefined problem, and thus is a fair method for comparing thectual control performances that will be displayed by differ-nt candidate designs. The centralized control scheme enablesvoiding the complexity of control system design. However,ince the mLQR scheme does not require a large number ofn-line optimizations during simulation, the computational effi-iency is vastly improved as compared to MPC, rendering thentroduced IPDC-mLQR method a practical solution for IPDCroblems.

For implementation, the modified LQR is used as the solu-ion of the dynamic optimization. The main optimization thencquires the evaluation of the dynamic performance, combineshis information with the static criteria such as cost and flexibil-ty constraints, and produces a new candidate by adjusting theector of design variables. The new candidate is then evaluatedn the dynamic optimization part (with a new optimal controllerased on the new design), until the iterations converge on aeasible design that cannot be improved further. Fig. 1 depictshe proposed sequential optimization algorithm. Note that thistrategy is sequential since the dynamic optimization is solvedith an integration routine as opposed to a discretization of the

ystem equations. The sequential nature of the evaluation algo-ithm does not change the fact that the methodology evaluates aesign via simultaneously considering cost and controllability.lso note that the focus of this work is on finding a practi-

al method for the solution of control problem, and the designroblem is solved with a fairly generic method. Developmentf IPDC methods taking further advantage of the basic ideasroposed here could prove a profitable direction of research.mplementation of global mixed-integer dynamic optimizationechniques (Chachuat et al., 2005), for instance, appears to behe next logical step.

It should be noted that a linear system model is required forhe mLQR setup. However, this requirement can be satisfied for

ost non-linear systems by linearization of the model aroundhe nominal conditions. Depending on the non-linearity of the

odel, the linearization can be performed either once at thetart of the simulation (i.e., once for each candidate design), orultiple times at a desired frequency during the simulation in

n adaptive-control like fashion. The linearization approach pro-

uces satisfactory results for most cases (Morari and Lee, 1999).ne other attractive alternative is to construct multiple linearodels to approximate the actual system model at the start of

he simulation, and corresponding optimal controllers, and then

1378 J. Patel et al. / Computers and Chemical Engineering 32 (2008) 1373–1384

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witching between these models and controllers as necessaryuring simulation (Banerjee and Arkun, 1998).

The LQR formulation does not allow for the inclusion ofnequality constraints in the dynamic optimization. Accordingly,he path constraints (Eqs. (18)–(21)) are imposed during theain optimization routine. Ideally, these constraints should be

mposed within the dynamic optimization. However, as the con-rol problem is implemented within the design problem, it isossible to encounter designs that cannot satisfy these pathonstraints (i.e., designs that are completely infeasible). Theolution algorithm should be able to continue iterations in suchases, which means that these constraints have to be formu-ated either as soft-constraints (i.e., violations penalized in thebjective) or be hierarchically dropped if impossible to meethem. Either case is very sophisticated for direct implementa-ion within an IPDC algorithm. Hence, the algorithm employedere does not enforce the constraints in the dynamic optimiza-ion (i.e., the constraints are not considered during controlleresign). The final designs will adhere to these constraints ashey are evaluated and enforced in the main optimization. Ithould be noted however that omission of these constraints dur-ng controller design will likely result in sub-optimality; this isccepted as a trade-off to avoid to the exponential increase inroblem complexity and computational burden.

. Case study—integrated design and control of anvaporation process

To test the effectiveness of the developed method, an evapora-

ion process, which was investigated by Newell and Lee (1989),nd Kookos and Perkins (2001) is studied. This process removesvolatile liquid from a non-volatile solute through evapora-

ion with steam, producing a concentrated product stream as

Q

Q

gy for IPDC-mLQR.

he bottom product. Fig. 2 depicts the process. It uses two heatxchangers as the evaporator and the condenser, and a pump forecycling part of the condensate stream. The flow rate of coolantnd the inlet pressure of steam are used as manipulated variableso adjust product composition and operating pressure.

.1. System model

The evaporator system is modeled by a set of differential andlgebraic equations, as given by Kookos and Perkins (2001) andeproduced here:

dC2

dt= F1C1 − F2C2 (36)

dP2

dt= F4 − F5 (37)

number algebraic equations are derived based on mass andnergy balances:

1 − F4 − F2 = 0 (38)

1CpT1 − F4(λ + CpT4) − F2CpT2 + Q100 = 0 (39)

.5616P2 + 0.3126C2 + 48.43 − T2 = 0 (40)

.5070P2 + 55 − T4 = 0 (41)

.1538P100 + 90 − T100 = 0 (42)

− UA (T − T ) = 0 (43)

100 1 100 2

100 − F100λs = 0 (44)

200 − F200Cp(T201 − T200) = 0 (45)

J. Patel et al. / Computers and Chemical Engineering 32 (2008) 1373–1384 1379

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200 − UA2

(T4 − (T201 + T200)

2

)= 0 (46)

5λ − Q200 = 0 (47)

here are a number of inequality constraints in operation:

5 − C2 ≤ 0 (48)

2 − 30 ≤ 0 (49)

0 − P2 ≤ 0 (50)

2 − 80 ≤ 0 (51)

100 − 400 ≤ 0 (52)

200 − 600 ≤ 0 (53)

here F1 is the feed flow rate (kg/min), F2 the product flow ratekg/min), F3 the circulation flow rate (kg/min), F4 the vaporow rate (kg/min), F5 the condensate flow rate (kg/min), F100

he steam flow rate (kg/min), F200 the cooling water flow ratekg/min), T1 the feed temperature (◦C), T2 the product tem-erature (◦C), T3 the vapor temperature (◦C), T100 the steamemperature (◦C), T200 the cooling water inlet temperature (◦C),201 the cooling water outlet temperature (◦C), C1 the feed com-osition (%), C2 the product composition (%), P2 the operatingressure (kPa), P100 the steam pressure (kPa), A1 and A2 are heatransfer areas of evaporator and condenser, respectively (m2).

The parameters used in the solution of system model are:1 = 40, T200 = 25, M = 20, C = 4, Cp = 0.07, λ = 38.5, λs = 36.6,

A1 = 9.6, and UA2 = 6.84. Eq. (36) represents the mass balancef the system, Eqs. (37)–(38) describes the interrelation of pres-ure, temperature and composition, and Eqs. (39) and (43)–(47)epresent the energy balance of the system.

btrp

or system.

The feed flow rate (F1) and feed composition (C1) are thenlet disturbances in the system having following description:

1 = 10 ± 0.5 (54)

1 = 5 ± 0.25 (55)

It should be noted that the algebraic Eqs. (38)–(47) areinear and well defined. This has allowed the algebraic sub-titution of these variables, when the differential Eqs. (36)nd (37) were being solved. This eliminates the need for aifferential-algebraic solver. In the original design, the flow ratef coolant, and the inlet pressure of steam are used as manip-lated variables to adjust product composition and operatingressure.

.2. Problem description

The design objective is to minimize the TAC, including thetility cost of steam and cooling water and the equipment cost.o ensure the choice of manipulated variables is not trivial, theosts of measuring devices and control valves are also includedn the design cost, therefore reflecting the drawback in usinghe additional manipulated variables. The cost functions andarameters used in calculations are listed in Table 1.

It should be noted that the design is required to handle aulti-period operation both at steady-state and dynamically.he first period corresponds to the nominal case with no dis-

urbances. The other four periods represent the four verticesf the disturbance space with ±5% disturbances in the feedow rate and inlet composition. The system is required to beapable of functioning with these disturbances at steady-state

y virtue of Eqs. (18)–(22) in the problem formulation. Also,he system should be able to satisfy the path constraints whichequire that the concentration remains between 25 and 30%, theressure remains within 40–80 kPa, the coolant flow rate can-

1380 J. Patel et al. / Computers and Chemical Engineering 32 (2008) 1373–1384

Table 1Objective and cost functions used in the case study

Objective function I = utility cost + controller cost + equipment cost

Utility cost ($/year) = steam + cooling waterUtility Utility flow rate × cost factor × total hours of

operation per yearSteam F100 × 0.6 × 24 × 320Cooling water F200 × 0.00061 × 24 × 320

Controller cost ($/year) = steam controller + water controllerController Controller cost × depreciation factorSteam controller 6000/2Cooling water controller 4000/2

Equipment cost ($/year) = evaporator + condenser + shell + recycle pumpTotal equipment Equipment installed cost × depreciation factor

Installation Inflation factor × 101.3area0.65 × (2.29 + correction factor)

Inflation factor Index value at present/index value at base yearCorrection factor (design factor + pressure factor) × material

factor

Evaporator 394.3/119 × 101.3 × A0.651 × (2.29 + 2.81 ×

1.35)/10Condenser 394.3/119 × 101.3 × A20.65 × (2.29 + 1 ×

0.85)/10

n4gsmdpt

dmatcwaspw

3

tccTbtmatwaaFtdtr

pa

TR

DNNTD

MC

Shell 45,000/10Recycle pump 10,000/5

ot exceed 600 kg/min, and the steam pressure cannot exceed00 kPa. These constraints correspond to Eqs. (23)–(26) in theeneral formulation. The same operational limits are imposedtatically and dynamically; hence the constraints listed should beet by the systems at all times. Note that it is possible to specify

ifferent limits for steady-state and dynamic behavior, but thisath has been avoided in order not to unnecessarily complicatehe case study.

The inclusion of the controller selection decisions in theesign problem results in a mixed-integer non-linear program-ing (MINLP) problem. Since the number of combinations

re only four (Case I—no controller, Case II—pressure con-rol only, Case III—flow control only, and Case IV—with bothontrollers), rather than employing an MINLP solver, each caseas solved separately with an NLP solver, which enables better

nalysis of this study problem. All the four cases were solvedeparately using fmincon non-linear constrained optimizationrogramming solver in MATLAB. The total time requirementas less than 10 min (on a Pentium 4 desktop PC).

dpct

able 2esults

Original design (both controllers exist) Optim

ecisions UA1 = 9.60, UA2 = 6.84 UA1 =ominal control values P100 = 193.43, F200 = 207.36 P100 =ominal state values C2 = 25.00, P2 = 50.57 C2 = 2otal annualized cost ($/year) 64,136 61,79ynamic constraints 30 > C2 > 25, 80 > P2 > 40,

600 > P100 > 101.33, 400 > F200 > 030 > C600 >

anipulated variable bounds 600 > P100 > 101.33, 400 > F200 > 0 600 >onstraint adherence Violates dynamic constraints (minimum

composition)Confo

Fig. 3. Disturbance profiles in dynamic simulation.

.3. Results

Feasible results were attained for all the cases, except forhe case employing no controllers. For Case I (employing noontrollers), the operational constraints requirement that theoncentration should not fall below 25% could not be satisfied.hus, an infeasible result was obtained. Case IV (employingoth the controllers) is the optimal design, which results inhe least total annualized cost and a control performance that

atches the required static and dynamic constraints. The totalnnualized cost for the optimal design is US$ 2340 less thenhe original design and control performance is also much betterhen compared to original design. The results of optimization

re summarized in Table 2 (this problem/design is designateds optimal design I). Fig. 3 depicts the disturbance profile, andig. 4 depicts the effect of disturbance on the uncontrolled sys-

em. Figs. 5–8 display the dynamic response of the originalesign (employing the multi-variable proportional-integral con-roller as described by Kookos and Perkins, 2001) and dynamicesponse of the optimal design under mLQR control.

To investigate the sensitivity to the constraints, the designroblem was also solved with relaxed constraints (designateds optimal design II), which produced interesting results. As

escribed above, the operational constraint, which requires thatroduct composition remains between 25 and 30%, is the activeonstraint in the first design problem. For the second problem,he upper bound was relaxed and set to 100%, such that product

al design (both controllers exist) Alternate design (no controller exists)

4.93, UA2 = 8.49 UA1 = 5.30, UA2 = 8.67371.38, F200 = 157.76 P100 = 374.62, F200 = 164.755.62, P2 = 40.45 C2 = 31.09, P2 = 41.92

5 58,812

2 > 25, 80 > P2 > 40,P100 > 101.33, 400 > F200 > 0

100 > C2 > 25, 80 > P2 > 40,600 > P100 > 101.33, 400 > F200 > 0

P100 > 101.33, 400 > F200 > 0 600 > P100 > 101.33, 400 > F200 > 0rms to constraints Conforms to constraints

J. Patel et al. / Computers and Chemical Engineering 32 (2008) 1373–1384 1381

Fig. 4. Effect of disturbance on the uncontrolled system.

Fs

svsasotf

Fs

iptiTrdctaacbdaw

oaT

ig. 5. Dynamic behavior of original design (dotted line represents the con-traint: C2 > 25%).

olute concentration can vary between 25 and 100%. This alteredalue is viable if product exceeding the specifications can still beold or used upstream in the process. All other constraints werepplied without any alteration. All the four cases were solved

eparately using the same methodology employed for solutionf design problem I. Computational time required was similaro the previous case. Feasible results were obtained for all theour cases.

ig. 6. Dynamic behavior of optimal design (dotted line represents the con-traint: C2 > 25%).

losc

Fig. 7. Manipulated variable movements of the original design.

Remarkably, it was observed that the optimal solutionnvolves no controllers, results in the least annualized cost, androvides a poor control performance that nevertheless matcheshe required operational constraints. The result of optimizations summarized in Table 2 under the “Alternate design” column.he result is interesting and counterintuitive in nature, as the

esult of an integration of process design and control study is aesign without controllers. Despite the lack of controllers, theoncentration fluctuates between 25 and 40%, and the opera-ional constraints are not violated (see Fig. 9). This is evidentlychieved by overdesigning the system so that the nominal oper-tion concentration is about 33%. The eliminated cost of theontrollers compensate for the cost of this overdesign. It shoulde noted that, the case with both controllers existent results in aesign with excellent control performance, but one that has annnualized cost of US$ 61,795, significantly more than Case IIith the cost of US$ 58,812.While counterintuitive, Case II demonstrates the true value

f IPDC algorithms, i.e., novel designs that push the bound-ries to create the optimal for the particular problem studied.he objective in the problem here (and in general design prob-

ems) is to minimize the total operational cost, as long as theperational constraints can be satisfied. As demonstrated above,lightly different operational constraints can result in dramati-ally different designs with true IPDC algorithms, whereas the

Fig. 8. Manipulated variable movements of the optimal design.

1382 J. Patel et al. / Computers and Chemical

Fs

mlaemhptpbsactpceht

4

itapfTdeoior

eoas

rtcme

racitcswHt

cgliwtctcf(

tOmtbesec

A

PU

A

m

ig. 9. Dynamic behavior of alternate design (dotted line represents the con-traint: C2 > 25%).

ethods that do not consider actual process dynamics will beimited to conventional control wisdom. It is only the power ofn IPDC method to implement the exact amount of control nec-ssary in a design; a conventional first design and then controlethod would never be able to identify such an optimum and

ave an extra cost. This case in fact parallels the six-sigma princi-les, which eliminates (most) quality control steps by ensuringhat any variation of the process will still yield an acceptableroduct. It should, however, be stated that such cases should note expected to be the norm; this case was generated by carefulelection of control requirements to produce such a solution, asn example of “out of the box” solutions IPDC may produce. Ofourse, if control performance is a highly important criterion,hen the problem could be revised as to attain a better controlerformance. For instance, a limit on the composition variationould be added to the problem, which would clearly require thexistence of controllers. One advantage of the method proposedere is that all such problem variations can be considered withinhe IPDC-mLQR algorithm introduced in this work.

. Concluding remarks

In this work, an optimal-control-based approach has beenntroduced for achieving integration of process design and con-rol in a practical manner. The principal idea proposed is to utilizen optimal controller (a modified linear quadratic regulator) toractically evaluate the best achievable control performanceor each candidate design during the process synthesis stage.he evaluation of complete and detailed closed-loop systemynamics are then meshed with a process design algorithm, thusnabling consideration of both cost and operability in designf the process. The practicality and computational efficiencyntroduced by the IPDC-mLQR method enables the solutionf this complex dynamic optimization problem within a veryeasonable computational time.

In the IPDC-mLQR method, the use of optimal control

nsures that each candidate design is fully evaluated in termsf its closed-loop control performance, whereas the designlgorithm employed ensures all static and dynamic criteria areimultaneously considered. As a result, the IPDC-mLQR algo-

F

x

Engineering 32 (2008) 1373–1384

ithm is computationally far more effective compared to methodshat involve heavy online-optimization, such as model predictiveontrol. On the other hand, the control performance assessmentethod is not questionable as is the case with the methods

mploying limited control structures.It should be noted, however, that while the IPDC-mLQR algo-

ithm is computationally more practical as compared to IPDCpproaches employing parametric controllers, this advantageomes at the expense of ability to address path constraints dur-ng the mLQR controller design stage. However, in our opinion,he possibility of infeasibility renders full consideration of theseonstraints extremely problematic. The proposed approach con-iders these constraints only within the static design procedure,hich should be a satisfactory approximation for most cases.owever, it should be re-emphasized that this is a premeditated

rade-off between computational cost and accuracy.The proposed IPDC-mLQR algorithm presents a method that

an be utilized to design or improve process plants through inte-ration of process design and control. However, it has certainimitations. One of them is that the algorithm allows check-ng only certain predetermined points in the disturbance space,hich should logically correspond to the vertices of the uncer-

ainty space for the system. While the number of points checkedan be increased, it is not possible to consider the entire uncer-ainty space with the current algorithm. The problem, however,an be resolved by adding an additional layer of iterationsor ensuring dynamic flexibility, as outlined by Sakizlis et al.2004).

A major advantage of the optimal control-based approach ishat a unique control law is identified for each candidate process.ne possible idea that is currently being explored is to createultiple linear approximations to the process model, combine

hem with the mLQR control law identified (which can alsoe expressed in linear form), and then solve the system differ-ntial equations analytically. This approach would result in aingle unified MINLP problem that could be solved much morefficiently, and could also help resolve some of the issues withonstraints in the dynamic optimization.

cknowledgments

This work is in part supported by NSF (CTS 0091398), ACS-RF, and the Institute of Manufacturing Research of Wayne Stateniversity.

ppendix A

Derivation of Eq. (35).The mLQR objective is:

in˙u(t)

J = 1

2

∫(xT · Q · x + uT · R� · u + ˙uT · R� · ˙u) dt (34)

irst, the state equation in Eq. (30) is differentiated, which yields:

¨ (t) = A ˙x(t) + B ˙u(t) (A-1)

ical

T

w

w

w

v

T

m

Ntm

A

w

A

B

Q

R

Tc

v

w(

u

Il

u

R

B

B

B

B

B

B

B

C

C

D

E

E

E

F

F

G

G

G

H

H

J

K

K

K

L

L

L

L

J. Patel et al. / Computers and Chem

his new state equation can be rewritten as:

˙ =[

0 I

0 A

]· w +

[0

B

]· v (A-2)

here

=[

w1

w2

]=

[x(t)˙x(t)

](A-3)

(t) = ˙u(t) (A-4)

herefore, the objective function becomes:

inv(t)

J = 1

2

∫(wT · Q · w + vT · R · v) dt (A-5)

ote that Eq. (A-5) is different from Eq. (29), as an additionalerm for derivative of the control variables also appears in it. The

odified ARE can be written as:

ˆ T · S + S · A + Q − S · B · R−1 · B

T · S = 0 (A-6)

here

ˆ =[

0 I

0 A

](A-7)

ˆ =[

0

B

](A-8)

ˆ =[

Q 0

0 0

](A-9)

ˆ = R (A-10)

he solution of the above stated Riccati equation gives feedbackontrol law:

= −K · w(t) (A-11)

here the gain matrix, K, can be split into gain matrices K1integral term) and K2 (proportional term), and thus,

˙ (t) = −[ K1 K2 ] ·[

x˙x

](A-12)

ntegrating Eq. (A-12) yields a proportional integral (PI) controlaw as stated below:

˜ (t) = −K2 · x(t) − K1 ·∫ t

0x(t) dt (35)

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