model predictive control of a nonlinear large-scale...

Model Predictive Control of a Nonlinear Large-Scale Process NetworkUsed in the Production of Vinyl AcetateTungSheng Tu, Matthew Ellis, and Panagiotis D. Christofides*

Department of Chemical and Biomolecular Engineering, University of California, Los Angeles, California 90095, United States

ABSTRACT: In this work, we focus on the development and application of two Lyapunov-based model predictive control(LMPC) schemes to a large-scale nonlinear chemical process network used in the production of vinyl acetate. The nonlineardynamic model of the process consists of 179 state variables and 13 control (manipulated) inputs and features a cooled plug-flowreactor, an eight-stage gas−liquid absorber, and both gas and liquid recycle streams. The two control schemes considered arean LMPC scheme which is formulated with a convectional quadratic cost function and a Lyapunov-based economic modelpredictive control (LEMPC) scheme which is formulated with an economic (nonquadratic) cost measure. The economic costmeasure for the entire process network accounts for the reaction selectivity and the product separation quality. In the LMPC andLEMPC control schemes, five inputs, directly affecting the economic cost, are regulated with LMPC/LEMPC and the remainingeight inputs are computed by proportional−integral controllers. Simulations are carried out to study the economic performanceof the closed-loop system under LMPC and under LEMPC formulated with the proposed economic measure. A thoroughcomparison of the two control schemes is provided.

■ INTRODUCTION

Vinyl acetate is mostly used in manufacturing polyvinyl acetateand other vinyl acetate copolymers. Polyvinyl acetate is thefundamental ingredient for polyvinyl alcohol and polyvinylacetate resins. Three raw materials, ethylene (C2H4), oxygen(O2), acetic acid (HAc), react to form the desired product vinylacetate (VAc) as well as two byproducts: carbon dioxide (CO2)and water (H2O). An inert component, ethane (C2H6), entersthe process with the ethylene feed stream. The exothermic andirreversible gas phase chemical reactions are

+ + → +

+ → +

C H HAc12

O VAc H O

C H 3O 2CO 2H O

r

r

2 4 2 2

2 4 2 2 2

1

2(1)

where the heat of the ith reaction (hrxn,i, i = 1, 2) is hrxn,1 =−42 100 kcal/kmol and hrxn,2 = −316 000 kcal/kmol,respectively.The fundamental challenge of controlling a vinyl acetate

process network is operating a highly nonlinear coupled processat an economically optimal steady-state. Luyben and Tyreus1,2

presented a detailed process network design for manufacture ofvinyl acetate monomer and demonstrated that plantwide controlof the process can be accomplished by using a conventionalproportional−integral (PI) control scheme. Chen et al.3

developed a nonlinear dynamic model of a vinyl acetate processin MATLAB based on Luyben and Tyreus’ work, proposed adifferent control structure using combination of proportional(P) control and PI control loops, and studied the dynamics ofthe closed-loop system under several set-point changes anddisturbances. Oslen et al.4 proposed modified control structures,specifically focused on improving the liquid inventory systemcontrol and the controllability of the azeotropic distillationcolumn using a model predictive controller and a static ratiocontroller. Subsequently, Luyben5 modified and optimized a

vinyl acetate process flowsheet using the original unit operationspresented in the previous work1 from a steady-state economicpoint of view using Aspen Dynamics.Optimizing chemical processes from an economic perspec-

tive is an issue of primary importance in industry. Theeconomic optimization of chemical processes is usually realizedvia a two-layer real-time optimization (RTO) system.6 In anRTO system, the upper layer utilizes a steady-state processmodel to compute economically optimal process operationset-points while feedback control systems are used to force thesystem to track the set-points in the lower layer. MPC is usuallyadopted in the lower layer due to its ability of taking advantageof a dynamic model of the process to predict its future evolu-tion along a given prediction horizon. MPC solves an onlineoptimization problem to compute optimal control inputs byoptimizing a quadratic cost function involving penalties on thedeviation of the state and controlled variables from a desiredsteady-state while taking state and input constraints intoaccount.7,8 Lyapunov-based MPC (LMPC) provides an explicitcharacterization of the stability region through utilization of apre-existing Lyapunov-based controller as an auxiliary con-troller.9−11 Recently, a significant number of efforts have beendevoted to integrating MPC and economic optimization ofchemical processes in order to manipulate process capacitiesin response to fast-changing global market demand.12−14

Furthermore, the development of MPC with a generaleconomic cost function has been studied in refs 15−19. Forexample, two economically oriented nonlinear MPC formula-tions using Lyapunov techniques to guarantee nominal stability

Special Issue: John MacGregor Festschrift

Received: February 25, 2013Revised: March 7, 2013Accepted: March 8, 2013Published: March 8, 2013

Article

pubs.acs.org/IECR

© 2013 American Chemical Society 12463 dx.doi.org/10.1021/ie400614t | Ind. Eng. Chem. Res. 2013, 52, 12463−12481

pubs.acs.org/IECR

of the closed-loop system for cyclic processes were studied inref 18. In refs 15 and 19, MPC schemes using an economic-based cost function with established stability properties via aproper Lyapunov function were proposed, in which anincorporated terminal constraint ensures that the closed-loopsystem is driven to a steady-state at the end of the predictionhorizon. Even though a rigorous stability analysis is discussedin ref 15, it is difficult, in general, to characterize, a priori, theset of initial conditions starting from where feasibility andclosed-loop stability (both boundedness and convergence to apotentially economically optimal steady-state) of the proposedMPC are guaranteed. In contrast, a design of economic modelpredictive control (EMPC) using Lyapunov-based techniques(LEMPC) is able to optimize closed-loop performance withguaranteed stability with respect to general economicconsiderations for nonlinear systems.17

Economic model predictive control has been applied toseveral applications.20−22 In ref 22, energy costs in a com-mercial building were effectively reduced via an economicmodel predictive control technique with a shrinking predictionhorizon; likewise, maximizing revenue from a dispatch-capableintegrated gasification combined cycle (IGCC) was studiedby using infinite-horizon economic model predictive control inref 20.Even though plantwide control designs using conventional

control techniques on a vinyl acetate process have beenextensively studied, application of model predictive control to avinyl acetate process network has not been addressed in theliterature. Motivated by this, we apply two model predictivecontrol (MPC) schemes to a large-scale vinyl acetate processnetwork: LMPC formulated with a conventional quadratic costand LEMPC formulated with an economic measure whichaccounts for the reaction selectivity in the plug-flow reactorand vinyl acetate separation quality in the separator and theabsorber. Closed-loop simulations of the vinyl acetate processnetwork with both MPC schemes are carried out to study theclosed-loop economic performance as well as compare the two

MPC schemes. In the following sections, we introduce theprocess model which consists of 179 nonlinear ordinarydifferential equations and 13 manipulated inputs featuring acooled plug-flow reactor, an eight-stage gas−liquid absorber, andboth gas and liquid streams. Next, we formulate the two controlschemes for the vinyl acetate process network where the MPCschemes are used to regulate five manipulated inputs and a setof PI controllers are used to regulate the remaining manipula-ted inputs to maintain closed-loop stability. Lastly, we presentclosed-loop simulation results and provide a thorough discussionon the comparison between the two control schemes.

■ NONLINEAR VINYL ACETATE PROCESS NETWORK

In this section, we describe the vinyl acetate process networkand introduce the nonlinear dynamics of operation units in theprocess network. The variables used in the model and theirdefinitions are listed in the Nomenclature.

Process Description. The vinyl acetate process networkflow diagram is shown in Figure 1. The process network is com-posed of several operation units including a vaporizer (VAP), aplug-flow reactor (RCT), a heat exchanger (HX), a separator(SEP), a compressor (COM), an absorber (ABS), and an aceticacid hold-up tank (TK). In addition to those main operationunits, there are a total two heaters (H1 and H2) and threecoolers (C3, C4, and C5) prior to the main operation units asshown in Figure 1. Using first principles, a nonlinear processmodel is obtained for each of these operation units. The carbondioxide removal unit and the azeotropic distillation in Figure 1are assumed to be component splitters with fixed CO2 removalefficiency and VAc product recovery ratios, respectively.An ethylene gas feed stream (S1) enters the process and is

mixed with a preheated gas recycle stream (S34). The resultingmixed gas stream (S2) enters the vaporizer along with theacetic acid liquid recycle stream (S3) from the acetic acid hold-up tank. The exit gas stream (S4) from the vaporizer is furtherheated to a desired reactor inlet temperature. To keep theoxygen composition in the gas recycle loop below the explosive

Figure 1. Process flow diagram of the vinyl acetate process network.

Industrial & Engineering Chemistry Research Article

dx.doi.org/10.1021/ie400614t | Ind. Eng. Chem. Res. 2013, 52, 12463−1248112464

region while providing sufficient oxygen for the reactions,oxygen is fed prior to the reactor (S5). The catalytic packed bedplug-flow reactor gas effluent (S8) enters the heat exchanger asthe hot-side stream. The hot-side effluent from the heatexchanger (S9) is partially condensed through a pressure let-down valve and a cooler. In the separator, the exit liquid stream(S13) consists of mostly heavy components such as VAc, H2O,and HAc; the exit vapor stream (S12) passes through acompressor and a heater before it enters the absorber where theremaining heavy components are further recovered and lightcomponents such as O2, CO2, C2H4, and C2H6 are recycled. Alarge portion of liquid from the base of the absorber isrecirculated through a cooler and fed to the second stage of theabsorber. The liquid effluents from the separator (S13) and theabsorber (S17) are mixed and fed to an azeotropic distillationcolumn in which the overhead stream (S26) contains the finalproduct, vinyl acetate, of the process network. The liquidproduct from the bottom of the distillation column (S24)enters an acetic acid hold-up tank where fresh acetic acid is alsofed. A portion of liquid in the tank (S23) is pumped to the topstage of the absorber to provide final gas scrubbing. Theoverhead gas leaving the absorber (S16) is split and part of thestream (S27) with constant molar flow rate of 6.556 kmol/minenters the carbon dioxide removal system where the carbondioxide removal system is able to remove 75% of the carbondioxide from the stream entering the unit. The vapor streamleaving the CO2 removal unit is split and 0.00318 kmol/min ispurged from the process and the remainder is combined withthe remaining vapor stream (S28) that was not sent to the CO2removal unit. The combined recycle vapor stream (S33) is fedto the heat exchanger as the cold-side stream before mixingwith ethylene feed stream. We define the gas streams that forma loop with the gas recycle stream (S34) as the gas loop of theprocess network.In this process, one key safety constraint which needs to be

taken into account is that the oxygen composition must be keptbelow 8 mol % throughout the gas loop to remain outside theexplosive envelop of ethylene. In addition, several operationconstraints must be satisfied. The peak reactor temperaturemust stay below 200 °C to avoid damage to the reactor catalyst.To avoid condensation in the reactor, the reactor inlettemperature must be greater than 130 °C. Since the process-to-process heat exchanger is designed to handle gas fluids, theheat exchanger hot fluid exit temperature must be greater than130 °C. In addition, the liquid hold-ups must remain within thelimits of 10−90% of allowable working liquid volume in thevessels.Thermodynamics and Physical Data. Ideal gas is

assumed for the vapor phase. A standard Wilson liquid activitycoefficient model is used to model the liquid phase with theWilson parameters for each species pair i−j (aij) given in Table 1.Pure component physical properties including molecular weight(MWi), the liquid specific gravity (SpGi), latent heat (hL,i), liquid

(ai and bi) and vapor (ai and bi) heat capacity parameters ofspecies i, i = VAc, H2O, HAc, O2, CO2, C2H4, C2H6, and molarvolume (vi) are provided in Table 2. The liquid specific gravity isdetermined based on the density of water at 0 °C. The temperaturedependence on the vapor and liquid component constant pressureheat capacity is modeled as a linear dependence, as follows:

= +c a b T( )MWi i i i (2)

= + c a b T( )MWi i i i (3)

The temperature dependence on the vapor and liquid enthalpy ismodeled as follows (i.e., (∂hi/∂T)P = ci):

= + +h a T b T h( 0.5 )MWi i i i i2

L, (4)

= + h a T b T( 0.5 )MWi i i i2

(5)

where hi denotes the vapor enthalpy of species i and hi denotesthe liquid enthalpy of species i. The pressure dependence on heatcapacity and enthalpy is assumed to be negligible. The Antoineequation is used to calculate the component saturated pressure(pi

sat):

=

+ °+

⎛⎝⎜

⎞⎠⎟p

BC T C

Aexp( )i

i

ii

sat

(6)

where the Antoine coefficients, Ai, Bi, and Ci are listed in Table 3.Vaporizer. The vaporizer is assumed to be well mixed, and

vapor−liquid equilibrium is also assumed. Only the dynamics ofthe liquid phase in the vaporizer is taken into account. Vaporpressure and the vapor compositions are found using a bubble-point calculation

γ = P y p xi i i iVAP VAP sat,VAP VAP VAP

(7)

where γiVAP represents the activity coefficient of species i in the

liquid phase of the vaporizer. There are eight state variables inthe vaporizer such as the liquid hold-up, the molar fractions,and the liquid temperature. The dynamic equations of thevaporizer are as follows:

= + −Mt

f f fd

d

VAP

S2 S3 S4 (8)

= − + − − −

xt

f y x f x x f y x

M

dd

( ) ( ) ( )

i

i i i i i i

VAP

S2 ,S2VAP

S3 ,S3VAP

S4 ,S4VAP

VAP

(9)

= − + −

− − +

Tt

f H H f H H

f H H Q M C

dd

[ ( ) ( )

( ) ]/( )

VAP

S2 S2VAP

S3 S3VAP

S4 S4VAP VAP VAP VAP

(10)

where MVAP is the total number of moles in the liquid phase ofthe vaporizer, x iVAP is the mole fraction of species i in the liquidphase of the vaporizer, and TVAP is the temperature of the liquidphase of the vaporizer. The maximum allowable liquid volumein the vaporizer is vmax

VAP = 4 m3.Catalytic Plug-Flow Reactor. A tubular packed-bed

catalytic reactor is used to convert ethylene and acetic acidinto the desired product, vinyl acetate. A combustion reactionalso consumes ethylene in the reactor. The two reactions aregiven in eq 1. The reactor length is 10 m, and the diameter is

Table 1. Wilson Parameters (aij) of Each Species PairExisting in the Liquid Phase of the Vinyl Acetate ProcessNetwork Taken from Reference 2

i

aij (kcal/kmol) VAc H2O HAc

j VAc 0 1384.6 −136.1H2O 2266.4 0 670.7HAc 726.7 230.6 0



3.71 cm. It is assumed that the reactions only take place in thereactor. Plug flow through the reactor is assumed, and thus, thetemperature and concentration gradients are ignored in the radialdirection. In addition, diffusion is negligible in the axial direction.Temperature and concentration gradients from the bulk fluid tothe external surface of the catalyst are negligible because massand heat transfer are assumed to be very fast between phases.The mass flow rate per unit cross-sectional area is assumed to beconstant. Catalyst deactivation is not considered so the catalystactivity is unity. The catalyst heat capacity Λcat, density ρcat, andporosity ε are 0.23 kcal/(kg °C), 385 kg/m3, and 0.8,respectively. The dynamic model of the reactor is

ε ρ ζ ζ∂

∂= −

∂ ∂

+ +c

tc v

zr r

( )( )i i

i i

RCT RCT RCT

cat 1, 1 2, 2 (11)

ε ρ

ρ

+ Λ ∂∂

= − ∂ ∂

− + −

C CT

t

v C C Tz

r h r h q

( )

( )( )

RCT RCTcat cat

RCT

RCT RCT RCT RCT

cat 1 rxn,1 2 rxn,2RCT

(12)

where CRCT is the total vapor concentration, CRCT is the vaporheat capacity, z is the axial coordinate along the length of thereactor (z ∈ [0 m, 10 m]), v is the superficial velocity, and ζ1,iand ζ2,i are the stoichiometric coefficients for the reactions 1 and2, respectively. To approximate the reactor dynamics of eqs 11and 12, the reactor is modeled in ten sections in the axialdirection (l = 1, 2, ..., 10) and the convective mass transfer andtemperature gradients are assumed to have linear dependence inaxial direction in each section:

ε ρ ζ ζ= −

−+ +− −

−

c

t

c v c v

z zr r

d

d

( )

( )( )i l i l l i l l

l li l i l

,RCT

, 1RCT

1RCT

,RCT RCT

1cat 1, 1, 2, 2,

(13)

ε ρ

ρ

+ Λ

= −

−

− + −

−

−

C CT

t

v C C T Tz z

r h r h q

( )d

d

( )( )

( )

l ll

l l l l l

l l

l l l

RCT RCTcat cat

RCT

RCT RCT RCT1

RCT RCT

1

cat 1, rxn,1 2, rxn,2RCT

(14)

Since the Reynold’s number of the packed bed reactor wasestimated to be much greater than 1000, the pressure dropthroughout the reactor was estimated to be less than 5% ofthe inlet pressure of the reactor using the Burke−Plummerequation.23,24 Furthermore, the pressure drop does not have asignificant effect on the reaction rates as confirmed by extensiveopen-loop simulations of the reactor model. As a result, thepressure drop throughout the reactor is not taken into account.The expressions of the chemical reaction rates1 are

= −

×+

+ + +

⎛⎝⎜

⎞⎠⎟r

Tp p p p

p p p

0.1036exp3674(K)

(1 1.7 )

(1 0.583 (1 1.7 ))(1 6.8 )

1

O C H HAc H O

O H O HAc

2 2 4 2

2 2 (15)

= × − +

+ +⎛⎝⎜

⎞⎠⎟r

T

p p

p p1.9365 10 exp

10116(K)

(1 0.68 )

(1 0.76 (1 0.68 ))25 O H O

O H O

2 2

2 2

(16)

where r1 has units of moles of vinyl acetate produced per minuteper gram of catalyst and r2 has units of moles of ethyleneconsumed per minute per gram of catalyst. Heat released perunit volume by the reactions (ql

RCT) in the lth section is removedby water coolant on the shell side of the tubes, and it is calculatedby the following equation:

= −q U T T( )l lRCT RCT RCT

coolantRCT

(17)

where TcoolantRCT is the water coolant temperature on the shell side

which is assumed to be uniform and URCT = 269.84 kcal/(m3 min °C) is the overall heat transfer coefficient per unitvolume. There are 70 state variables in the reactor model such asfluid temperatures and concentrations of components for eachsection in the reactor.

Separator. It is assumed that the superheated hot effluentfrom the heat exchanger becomes saturated vapor through apressure let-down valve. The saturated vapor is partiallycondensed through a cooler and then enters the separator. Atemperature−pressure (TP) flash calculation, applicable whentemperature and pressure are known since the equilibriumconstant (K) mainly depends on temperature and pressure, isimplemented to obtain the flow rates and compositions of thevapor ( f S11 and yS11) and liquid ( fS11 and xS11) streams enteringthe separator. The TP calculation sequence is implementedaccording to the following steps:

1. Guess the fraction of feed vaporized α := 0.84 and thefraction of species i in S11 xi,S11 := xi

SEP.2. Calculate Ki,S11 = [γi,S11(xi,S11, TS11)pi,S11

sat (TS11)]/PSEP.

3. Calculate x i,S11 = yi,S10/[1 + (Ki,S11 − 1)α] and yi,S11 =Ki,S11xi,S11.

Table 2. Physical Properties of the Species in the Vinyl Acetate Process Network Taken from Reference 3

species MW (kg/kmol) SpG hL (kcal/kmol) a (kcal/(kg °C)) b (kcal/(kg °C2)) a (kcal/(kg °C)) b (kcal/(kg °C2)) v (L/kmol)

O2 32.000 0.50 2300 0.30 0 0.218 0.0001 64.178CO2 44.010 1.18 2429 0.60 0 0.230 0 37.4C2H4 28.052 0.57 1260 0.60 0 0.370 0.0007 49.347C2H6 30.068 0.57 1260 0.60 0 0.370 0.0007 52.866VAc 86.088 0.85 8600 0.44 0.0011 0.290 0.0006 101.564H2O 18.008 1.00 10684 0.99 0.0002 0.560 −0.0016 18.01HAc 60.052 0.98 5486 0.46 0.0012 0.520 0.0007 61.445

Table 3. Component Vapor Pressure Antoine Coefficients

i A B C

O2 9.2 0 273CO2 7.937 0 273C2H4 9.497 −313 273C2H6 9.497 −313 273VAc 12.6564 −2984.45 226.66H2O 14.6394 −3984.92 233.426HAc 14.5236 −4457.83 258.45



4. Calculate ψ = ∑i(x i,S11 − yi,S11). If ψ is within anacceptable tolerance, STOP. Else, go to step 5.

5. Update the fraction of feed vaporized: α := α − 0.1ψ andgo to step 2.

The number of moles and the pressure in the vapor phase areassumed to be constant while the change in the vapor tem-perature is assumed to exist because it is coupled with thechange in the vapor mole fraction. Through extensivesimulations, it is found that the change in temperature isinsignificant even if mass accumulation and the pressure changeof the vapor phase in the vessel are considered. Therefore, onecould potentially neglect the change in the vapor temperaturein the separator as well. Heat in the liquid phase is removed bya cooling jacket (Tcoolant

SEP ). There are 16 state variables such asliquid hold-up, molar fractions of components, and temper-atures in the liquid phase and the vapor phase as well as thevapor phase pressure. These variables evolve according to thefollowing dynamic equations:

= − Mt

f fd

d

SEP

S11 S13 (18)

=

−

x

t

f x x

Md

d

( )

( )i i iSEP

S11 S11,SEP

SEP(19)

= − − −

T

t

f H H T T

M C

dd

( ) UA ( )

( )

SEPS11 S11

SEP SEP SEPcoolantSEP

SEP SEP

(20)

=

− y

t

f y y

M

d

d

( )

( )i i iSEP

S11 S11,SEP

SEP (21)

=−

Tt

f H H

M C

dd

( )

( )

SEPS11 S11

SEP

SEP SEP(22)

The maximum allowable liquid volume in the separator isvmaxSEP = 8 m3.Absorber. The purpose of the absorber is to recover the

remaining vinyl acetate from the vapor effluent that leaves theseparator. The gas absorber consists of eight theoretical stages(l = 1, 2, ..., 8) and a liquid hold-up base. It is assumed thatthere is no chemical reaction taking place at each stage and theabsorber pressure which is determined by the exit pressure ofthe compressor is uniform throughout the absorber. A masstransfer rate-based model is implemented to describe the masstransfer and heat transfer between the liquid phase and thevapor phase at each stage; vapor−liquid equilibrium is assumedto exist at the interface in the rate-based model. The liquidhold-up at each stage is assumed to be well mixed.For convective flow, the mass transfer coefficient is a function

of flow rate, temperature, and pressure. The mass transfercoefficient in this model, nevertheless, is assumed to beconstant at each stage. For this reason, the mass transferrate calculated from the model for a component at each stagemight exceed the total mass of the component in the bulkphase. Thus, the maximum mass transfer rate for a componentis assumed to be constrained by half the amount in the bulkphase. The convective mass coefficient (k) is 27.22 kmol/min.The individual component mass transfer rate (Ni,l

ABS) betweentwo phases at each stage is determined by the following

equation:

= | − |

= − <

= >

− − −

−

−

⎧⎨⎪⎩⎪

N k y y f y

n N y y

n N y y

min{ ( ) , 0.5 },

, if

, if

i l i l i l l i l

i l i l i l i l

i l i l i l i l

,ABS

,( 1)ABS

,int

( 1)ABS

,( 1)ABS

,ABS

,ABS

,( 1)ABS

,int

,ABS

,ABS

,( 1)ABS

,int

(23)

The vapor compositions of species i (yi,lint) at the interface of

the lth theoretical stage is obtained using an equilibriumcalculation. Two forms of energy transfer occur between thetwo phases at each stage: conductive heat transfer due totemperature gradients and convective heat transfer due to thetransferring components. Conductive heat transfer (Ql

ABS) andconvective heat transfer (Qtr,l

ABS) from the vapor phase to theliquid phase at lth stage are calculated based on the followingequations:

= − −Q T TUA ( )l l l lABS ABS

( 1)ABS ABS

(24)

∑= =

Q n h T( )li

i l i ltr ,ABS

1

7

,ABS

,ABS

(25)

where the overall heat transfer coefficient UAlABS = 100.8 kcal/

kmol for l = 1, 2 and UAlABS = 50.4 kcal/kmol for l = 3, 4, ..., 8.

The absorber is divided into two sections. The bottom sectioncontains two stages and the base. The top section contains sixstages. The base has an inlet stream from the first stage and anoutlet stream which is split into a circulation stream ( fS20) and astream ( fS17) used to regulate the liquid hold-up of the base.The dynamics of the base are described as follows:

= − − M

tf f f

dd

bABS

1ABS

S17 S20 (26)

=

−

x

t

f x x

M

d

d

( )i i i,bABS

1ABS

,1ABS

,bABS

bABS

(27)

= −

T

t

f H H

M C

dd

( )bABS

1ABS

1ABS

bABS

bABS

bABS

(28)

where the subscript b denotes the base. A cooled circulationstream is fed to the second stage (l = 2) for an initial gasscrubbing. The dynamics of the second stage is described asfollows:

= + + − +M

tf f n f

dd

( )ll l l

ABS

1ABS

S20 net,ABS ABS

(29)

= − + − + −

+ +

x

tf x x f x x n n x

M

d

d( ) ( )

i l

l i l i l i i l i l l i l

l

,ABS

1ABS

,( 1)ABS

,ABS

S20 ,bABS

,ABS

,ABS

net,ABS

,ABS

ABS

(30)

= − + −

+ + −

+ +T

tf H H f H H

Q Q n H M C

dd

[ ( ) ( )

]/( )

ll l l l

l l l l l l

ABS

1ABS

1ABS ABS

S20 S20ABS ABS

tr ,ABS ABS

net,ABS ABS ABS ABS

(31)



where nnet,l is the net amount of material transferring to theliquid phase as shown below:

∑ ==

n nli

i lnet,ABS

1

7

,ABS

(32)

For stages (l = 1, 3, 4, 5, 6, and 7), there are an inlet liquidstream from the above stage and an outlet liquid stream whichleaves the stage. The dynamics of these stages are described bythe following equations:

= + − +M

tf n f

dd

( )ll l l

ABS

1ABS

net,ABS ABS

(33a)

=

− + −

+ +x

t

f x x n n x

M

d

d

( )i l l i l i l i l l i l

l

,ABS

1ABS

,( 1)ABS

,ABS

,ABS

net,ABS

,ABS

ABS

(33b)

= − + + −

+ +T

t

f H H Q Q n H

M C

dd

( )l l l l l l l l

l l

ABS1

ABS1

ABS ABStr ,ABS ABS

net,ABS ABS

ABS ABS

(33c)

A liquid stream ( fS23) from the acetic acid hold-up tank whichis cooled and enters the last stage (l = 8) serves as final gasscrubbing stream. The dynamics of the last stage are describedby the following equations:

= + − M

tf n f

dd

( )ll l

ABS

S23 net,ABS ABS

(34a)

=

− + −

x

t

f x x n n x

M

d

d

( )i l i i l i l l i l

l

,ABS

S23 ,S23 ,ABS

,ABS

net,ABS

,ABS

ABS(34b)

= − + + −

T

t

f H H Q Q n H

M C

dd

( )l l l l l l

l l

ABSS23 S23

ABStr ,ABS ABS

net,ABS ABS

ABS ABS

(34c)

Since only liquid phase dynamics is considered, the mass flowrate, the compositions, and the temperature of an exit vaporstream from each stage are calculated using steady-state mass,component, and energy balances around the vapor phase ateach stage. There are a total of 72 state variables including theliquid hold-up, the compositions, and the liquid temperatureat each stage and at the base. The maximum allowable liquidvolume in the absorber is vmax

ABS = 8.5 m3.Acetic Acid Hold-up Tank. The acetic acid tank is used to

mix the bottom product stream from the azeotrope distillationcolumn and the HAc feed stream. The acetic acid hold-up tankallows for a better control of the liquid recycle loop to preventthe snowballing effect. There are four state variables, which arethe liquid hold-up, molar fractions of VAc and HAc, and theliquid temperature. The dynamic equations are as follows:

= + − − Mt

f f f fd

d( )

TK

S24 S25 S3 S22 (35)

=

− + −

x

t

f x x f x x

Md

d

( ) ( )i i i i iTK

S24 ,S24TK

S25 ,S25TK

TK (36)

= − + −

T

t

f H H f H H

M C

dd

( ) ( )TKS24 S24

TKS25 S25

TK

TK TK (37)

The maximum allowable liquid volume in the tank is vmaxTK =

2.83 m3.Compressor, Heat Exchanger, Heaters, and Coolers.

The dynamics of the compressor, heat exchanger, heaters, andcoolers are assumed to be adequately represented by first-ordersystems of the following form:

τ= −x

tx xd

ds

(38)

where xs is the steady-state computed by the steady-stateenergy balance, x is a state variable, and τ is a time constant.

Compressor. Isentropic compression is assumed to calculatethe outlet temperature and pressure of the compressor. Theactual work input (WS

COM) to the compressor is a manipulatedinput. The mean specific heat capacity25 (Cmean

COM) is used andassumed to be linearly dependent on the log-mean temperatureof the inlet and outlet stream. Two state variables are present inthe compressor: the outlet temperature and the pressure. It isassumed that a 5 min time constant can be used to describe thedynamics of the compressor.

Heat Exchanger. The NTU-Effectiveness method26 basedon the single-tube heat exchanger with counterflow is used tocalculate the steady-state heat exchanger exit temperature. Theoverall heat transfer coefficient (UAHX)3 is correlated accordingto the nominal overall heat transfer coefficient, and the ratio ofthe streamflow rates to the nominal flow rates to the power of0.8, as follows:

= +⎛

⎝⎜⎜⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎞

⎠⎟⎟

f

m

f

mUA

UA2

MW MWHX ref S34 S34

cold,ref

0.8S9 S9

hot,ref

0.8

(39)

where the reference overall heat transfer coefficient is UAref =113.33 kcal/(min °C), the reference cold streamflow rate ismcold,ref = 498.952 kg/min, and the reference hot streamflowrate is mhot,ref = 589.67 kg/min. The exit temperatures for thecold stream and the hot stream are the two state variables in theheat exchanger. A 5 min time constant is introduced to describethe dynamics.

Coolers and Heaters. There are three coolers and twoheaters in the process. A 2 min time constant is used to describetheir dynamics. The exit temperature is a state variable and theheat input is a manipulated input in each of these units.

Nonlinear Dynamic Model of the Vinyl Acetate Process.The nonlinear dynamic model of the vinyl acetate consists of179 ODEs, 351 algebraic equations, and 23 inequality processconstraints. The process constraints include the liquid hold-upvolume in vessels, the oxygen composition in the gas loop, theinlet temperature of the reactor, the temperature in the reactor,the hot-side outlet temperature of the heat exchanger, and thepressure drop in the gas loop. In short, the nonlinear dynamicmodel of the vinyl acetate process network can be described bythe following state-space model:

=x t f x t u t u t( ) ( ( ), ( ), ..., ( ))1 20 (40a)

= g x t u t u t0 ( ( ), ( ), ..., ( ))1 20 (40b)

≤ h x t u t u t0 ( ( ), ( ), ..., ( ))1 20 (40c)

where x(t) ∈ Rnx denotes the vector of state variables of thesystem and ui(t) ∈ R, i = 1, ..., 20 denotes the ith control(manipulated) input, respectively. In the nonlinear model ofeq 40, the vector function f is the right-hand-side of the non-linear dynamic equations, the vector function g is the family of



algebraic equations, and the vector function h is the family ofprocess constraints which may depend on states and inputs.In this process, there are 20 available inputs which are shownin Table 4 along with the available range for each input andsteady-state values us. The steady-state that corresponds tothe steady-state input us is denoted as xs. We note that in ourmodel seven of the inputs are fixed to specific values.

■ MODEL PREDICTIVE CONTROL FORMULATIONSIn this section, we initially present a novel economic measurewhich is used in the formulation of the LEMPC and used toassess the economic performance of the closed-loop systemunder LMPC. Next, we provide a detailed description of thecontroller synthesis. Subsequently, we introduce the formula-tions of the Lyapunov-based model predictive control andLyapunov-based economic model predictive control schemesformulated for the vinyl acetate process network.Notation. The operator |·| is used to denote the Euclidean

norm of a vector. The symbol Ωr is used to denote the setΩr := {x ∈ Rnx: V(x) ≤ r} where V is a scalar function. Thesymbol diag(v) denotes a matrix whose diagonal elements arethe elements of a vector v and all the other elements are zeros.The superscript vT denotes the transpose of a vector v.Economic Measure. In this work, we aim to maximize the

overall vinyl acetate production in the azeotropic distillationoverhead stream (S26) so an economic measure is chosen toaccount for the reaction selectivity in the reactor and theseparation quality in the separator and the absorber as follows:

=

+ + L x u Ac

cA x A x( , ) 1

VAc,10RCT

CO ,10RCT 2 VAc

SEP3 VAc,b

ABS

2 (41)

where L(x,u) is the economic measure, A := [A1 A2 A3] =[30 1000 1000] are the constant weighting coefficients whichare chosen such that each term in the economic measure is ofthe same order of magnitude, cVAc,10

RCT and cCO2,10RCT are the VAc and

CO2 concentrations at the 10th section of the reactor, and xVacSEP

and xVAc,bABS are the vinyl acetate liquid molar fractions in the

separator and the absorber base, respectively. The first term of

the measure describes reaction selectivity of VAc with respectto CO2 in the reactor. The second term and the third termaccount for VAc recovery in the separator and the absorber.Remark 1. In this work, the weighting coef f icients have been

chosen such that each term in the economic cost is signif icant. Inpractice, these coef f icients would be chosen based on processobjectives, operating costs, and material prices.

Controller Synthesis. In the process network, the heater(H1) prior to the reactor affects the inlet stream temperaturewhich, in turn, affects the reaction rates and the reactionselectivity. The cooler (C3) prior to the separator determinesthe quality of flashing streams. The heater (H2) prior to theabsorber and the side coolers (C4) and (C5) account for theVAc recovery. Hence, these five inputs have direct influence onthe economic measure and on closed-loop economic perform-ance. Therefore, we propose a control architecture whichcomprises of LMPC or LEMPC regulating the inputs of theheaters and coolers, a set of eight independent PI controllersregulating eight inputs to maintain closed-loop stability, and seveninputs that are fixed. Figure 2 shows the control architecture.

A summary of which manipulated inputs are controlled by MPCor PI or are fixed along with the available control energy for eachmanipulated input and steady-state values us are given in Table 4.In this work, we assume that the full system state x is

measured at every sampling period (Δ). State measurementsare sent to the PI controllers at synchronous time instantstq = qΔPI, q = 0, 1, ... and sent to the LMPC or LEMPC at

Table 4. Input Constraints and Steady-State Values

manipulated input (u) loop input number range steady-state (us) unit

QH1 MPC 1 0−50000 5078.69 kcal/minQH2 MPC 2 −10000−50000 1461.14 kcal/minQC3 MPC 3 0−30000 15491.57 kcal/minQC4 MPC 4 0−30000 7250.42 kcal/minQC5 MPC 5 0−5000 1881.2 kcal/minf S4 PI 6 8−15 12.113916 kmol/minf S5 PI 7 0−2.268 0.47744 kmol/minfS13 PI 8 0−8 2.73964 kmol/minfS17 PI 9 0−4.536 1.20871 kmol/minfS25 PI 10 0−4.536 0.74435 kmol/minQVAP PI 11 0−143340 16933.247 kcal/minTcoolantRCT PI 12 110−150 133.46 deg C

Wscom PI 13 0−1000 275.64 kcal/min

TcoolantSEP

fixed 14 0−80 37.72 deg Cf S1 fixed 15 0−7.56 0.905 kmol/minfS18 fixed 16 0−50 15.35 kmol/minfS22 fixed 17 0−7.56 0.8125 kmol/minfS3 fixed 18 0−4.536 2.1924 kmol/minf S27 fixed 19 0−22.68 6.556 kmol/minf S30 fixed 20 0−0.02268 0.00318 kmol/min

Figure 2. Control architecture designed for the vinyl acetate processnetwork.



synchronous time instants tk = kΔ, k = 0, 1, .... Taking intoaccount closed-loop stability consideration, the sampling timesfor the LMPC/LEMPC and the set of PI controllers are chosento be ΔPI = Δ and Δ = 10Δ, respectively, where Δ = 0.001 min.The manipulated input vector computed by the PI controllerscan be expressed as follows:

∫η ητ

η η= − + − ′ +

=

u KK

t u

i

( ) ( ) d ,

6, ..., 13

i i i ii

i

t

i i iPI, c,set c,

I, 0

sets,

(42)

where uPI is the vector of manipulated inputs on the set of PIcontrollers given by uPI

T = [uPI,6 uPI,7 ... uPI,13] = [f S4 f S5 fS13 fS17fS25 QVAP Tcoolant

RCT WsCOM], η is the vector of the controlled

outputs given by ηT = [νVAP xO2,1RCT νSEP νABS νTK TVAP T10

RCT

PCOMP], ηset is the set-point vector, Kc is the vector of theproportional gains, Kc = [Kc6 Kc7 ... Kc13] and τI is the vector ofthe integral time constants, τI = [τI6 τI7 ... τI13]. The parametersfor the PI controllers along with the tuning methods used tofind these parameters are listed in the Appendix.Lyapunov-Based Controller. In the LMPC and LEMPC

designs proposed in the literature,9,27 the nonlinear system isassumed to be stabilizable by assuming the existence of aLyapunov-based controller h (x)T = [h1(x) ... hm(x)] that canrender the steady-state of the nonlinear system asymptoticallystable under continuous implementation while satisfying theinput constraints for all the states x inside a given stabilityregion. The Lyapunov-based controller (h (x)) used for thevinyl acetate process network is another set of PI controllers,namely,

= = h x u u u( ) [ ... ]TPIT

PI,1 PI,5 (43)

where uPI uses the formulation of eq 42, uPIT = [QH1 QH2 QC3

QC4 QC5] and the set-point vector (η set)T = [TH1,set TH2,set TC3,set

TC4,set TC5,set]. The input constraints and the steady-state inputvalues are listed in Table 4. Closed-loop simulations of the vinylacetate process network confirmed that the Lyapunov-basedcontroller along with other control inputs either on PI controllersor fixed values can regulate the process to the steady-state.Using converse Lyapunov theorems,28,29 this stabilizability

property implies that there exist a continuously differentiableLyapunov function V(x) for the nominal closed-loop system inRnx. We denote the region Ωρ ⊆ O as the stability region of theclosed-loop system under the Lyapunov-based controller h(x).In the vinyl acetate process network, we use a quadraticLyapunov function of the form

= − −V x x x P x x( ) ( ) ( )sT

s (44)

for the design of the LMPC and LEMPC where P ∈ Rnx × Rnx isa positive definite diagonal matrix and xs is the steady-state. Thediagonal elements of P were chosen based on the magnitudeof the vector field f of eq 40a of the closed-loop system withinputs 1, ..., 13 on PI controllers (i.e., both the controllers uPIand h (x) applied to the process network). Specifically, if themagnitude of the ith element of the vector field f wasconsistently positive or negative, the diagonal element pii of thematrix P was increased. The final P was selected throughextensive simulations such that the PI controllers decreasedthe Lyapunov function over each sampling period when the PIcontrollers are implemented in a sample-and-hold fashion andwhen the process network is initialized at an off-steady-state

initial condition. The parameters for the Lyapunov-based con-trollers are listed in the Appendix.Remark 2. Explicit stabilizing control laws that provide

explicitly def ined regions of attraction for the closed-loop systemhave been developed using Lyapunov techniques for various classesof nonlinear systems; the reader may refer to refs 28 and 30−32 forresults in this area.Remark 3. We note that the Lyapunov-based controller is

typically formulated as a static controller as in refs 9 and 11. Eventhough PI controllers are dynamic controllers, the set of PIcontrollers with little integral action can be considered as aLyapunov-based controller as long as the independent PI controlloops can stabilize the process network.

Lyapunov-Based Model Predictive Control. LMPC iscapable of computing optimal control inputs while accountingfor input and state constraints and ensuring the stability of theclosed-loop system. The LMPC design follows the formulationof our previous works9,11 and takes into account the nonlinearsystem of eq 40 (where, without any loss of generality, xs = 0 isthe stabilizing steady-state), as follows:

∫ τ τ τ τ τ + ∈ Δ

+x Q x u R umin [ ( ) ( ) ( ) ( )] d

u u S t

t

,..., ( )

Tc

Tc

k

k N

1 5

(45a)

= ∼x t f x t u t u ts.t. ( ) ( ( ), ( ), ..., ( ))1 20 (45b)

= ∀ ∈ +g x t u t u t t t t0 ( ( ), ( ), ..., ( )), [ , )k k N1 20(45c)

≤ ∀ ∈ +h x t u t u t t t t0 ( ( ), ( ), ..., ( )), [ , )k k N1 20(45d)

∈ = ∀ ∈ +u t U i t t t( ) , 1, ..., 13, [ , )i i k k N (45e)

= = ∀ ∈ +u t u t i t t t( ) ( ), 6, ..., 13, [ , )i i k k NPI, (45f)

= = ∀ ∈ +u t u i t t t( ) , 14, ..., 20, [ , )i i k k Nfixed

(45g)

=x t x t( ) ( )k k (45h)

∂∂

≤ ∂∂

V xx

f x t u t u t u t

u t u u

V xx

f x t h x t h x t u t

u t u u

( )( ( ), ( ), ..., ( ), ( ),

..., ( ), , ..., )

( )( ( ), ( ( )), ..., ( ( )), ( ),

..., ( ), , ..., )

k k k k

k

k k k k

k

1 5 PI,6

PI,13 14fixed

20fixed

1 5 PI,6

PI,13 14fixed

20fixed

(45i)

where S(Δ) is the family of piecewise constant functions withsampling time Δ, N is the prediction horizon, Qc and Rc arestrictly positive definite weight matrices, x is the predicted statetrajectory of the nominal system with inputs 1, ..., 5 computedby the LMPC, 6, ..., 13 computed by the PI controllers, andfixed inputs ui

fixed for inputs 14, ..., 20, and initial state x(tk). Theweighting matrices Qc and Rc are listed in the Appendix.The optimal solution to this optimization problem is denoted

by ui*(τ|tk), i = 1, ..., 5, which is defined for τ ∈ [tk, tk+N) . TheLMPC is implemented in a receding horizon fashion; namely,the optimization problem of eq 45 is solved along theprediction horizon at each sampling time and ui*(t|tk), i = 1,..., 5 are applied to the closed-loop system for t ∈ [tk, tk+1).Equation 45a defines a quadratic cost index penalizing state andmanipulated input deviation from an operating steady-state



which should be minimized. Equations 45b and 45c are used topredict the future evolution of the nominal model of the systemof eq 40. Equation 45d represents the process constraints thatmust be satisfied along the prediction horizon. Equation 45eaccounts for the constraints on the available control energycomputed by LMPC. Equations 45f and 45g account for themanipulated inputs regulated by the PI controllers and themanipulated inputs that are fixed, respectively. Equation 45hprovides the initial state for the predicted state trajectories whichis a measurement of the actual system state. The constraint ofeq 45i ensures that the time derivative of the Lyapunov functionat the initial sampling time (first move) of LMPC is less than orequal to the time derivative of the Lyapunov function obtainedif the Lyapunov-based controller h(x) is implemented in theclosed-loop system in a sample-and-hold fashion. Through thisconstraint, the LMPC inherits the stability and robustness

properties of the Lyapunov-based controller (the reader mayrefer to refs 9 and 10 for more discussion and analysis on thisissue). The control actions that are applied to the closed-loopsystem under the LMPC are defined as follows:

= * | = ∀ ∈ +u t u t t i t t t( ) ( ), 1, ..., 5, [ , )i i k k k 1

Lyapunov-Based Economic Model Predictive Control.Optimization of a chemical process operation with respect togeneral economic considerations has received extensive attention.Within process control, one control scheme that addresses bothdynamic economic optimization and process control is economicmodel predictive control. In our previous work,17 we introducedan economic model predictive control scheme designed viaLyapunov techniques for nonlinear systems. The formulation ofthe LEMPC applied to the vinyl acetate process network ofeq 40 is:

Figure 3. Trajectory of logarithm of the Lyapunov function V(x) under LMPC.

Figure 4. Trajectories of the manipulated inputs for heaters and coolers under LEMPC.



∫ τ τ τ τ ∈ Δ

+L x u umax ( ( ), ( ), ..., ( )) d

u u S t

t

,..., ( )1 20

k

k N

1 5 (46a)

= ∼x t f x t u t u ts.t. ( ) ( ( ), ( ), ..., ( ))1 20 (46b)

= ∀ ∈ +g x t u t u t t t t0 ( ( ), ( ), ..., ( )), [ , )k k N1 20(46c)

≤ ∀ ∈ +h x t u t u t t t t0 ( ( ), ( ), ..., ( )), [ , )k k N1 20(46d)

∈ = ∀ ∈ +u t U i t t t( ) , 1, ..., 13, [ , )i i k k N (46e)

= = ∀ ∈ +u t u t i t t t( ) ( ), 6, ..., 13, [ , )i i k k NPI, (46f)

= = ∀ ∈ +u t u i t t t( ) , 14, ..., 20, [ , )i i k k Nfixed

(46g)

=x t x t( ) ( )k k (46h)

ρ ρ ≤ ∀ ∈ ≤ ′ ≤ +V x t t t t t t V x t( ( )) , [ , ), if and ( ( ))k k N k k

(46i)

ρ ρ

∂∂

≤ ∂∂

> ′ < ≤

V xx

f x t u t u t u t

u t u u

V xx

f x t h x t h x t u t

u t u u

t t V x t

( )( ( ), ( ), ..., ( ), ( ),

..., ( ), , ..., )

( )( ( ), ( ( )), ..., ( ( )), ( ),

..., ( ), , ..., )

if or ( ( ))

k k k k

k

k k k k

k

k k

1 5 PI,6

PI,13 14fixed

20fixed

1 5 PI,6

PI,13 14fixed

20fixed

(46j)

where S(Δ) is the family of piecewise constant functions withsampling time Δ, N is the prediction horizon, L(x (τ),u1(τ), ...,um(τ)) is the economic measure which defines the objectivefunction, x is the predicted state trajectory of the nominalsystem with inputs 1, ..., 5 computed by the LEMPC, 6, ..., 13computed by the PI controllers, and fixed inputs ui

fixed for inputsi = 14, ..., 20, and initial state x(tk). The optimal solution to thisoptimization problem is denoted by ui*(τ|tk) which is defined forτ ∈ [tk, tk+N).Equation 46a defines the economic cost index which should

be optimized for the process. The constraints of eqs 46b−46hare the same as the constraints of eqs 45b−45h, respectively,given in the LMPC optimization problem of eq 45. Theconstraints of eqs 46i and 46j are used to define the twooperation modes of the LEMPC. The first operation mode isrealized while the constraint of eq 46i is active. During the timeperiod of the first operation mode (e.g., t ≤ t′), the LEMPC

Figure 5. Closed-loop state trajectories under LEMPC of the heaters and coolers.



operates the closed-loop system in a possible time-varyingfashion while maintaining the predicted state along theprediction horizon in a predefined set Ωρ to optimize theeconomic cost function. The second operation mode is realizedwhile the constraint of eq 46j is active. The second operationmode corresponds to operation in which the process is drivenby the LEMPC to a steady-state. We note that Ωρ, which is alevel set of the Lyapunov function V(x), is used to estimate thestability region (e.g., region of attraction) of the closed-loopsystem under the Lyapunov-based controller h (x), Ωρ is asubset of Ωρ that defines the safe set of operation in which theLEMPC, operating in mode 1, may optimize the economicobjective freely, and t′ is the switching time between mode 1and mode 2 (e.g., t′ can be an integer multiple of the samplingtime of the MPC). The switching time t′ may be chosen to bearbitrarily large such that the process network is alwaysoperated under the LEMPC operating in mode 1 or it may bechosen to manage the trade-off between dynamically optimalprocess operation and excessive wear on control actuatorsrequired to operate a process in a time-varying fashion. If thestate leaves the set Ωρ, the operation mode of the LEMPCswitches to mode 2 operation.Even though the LEMPC is formulated with an economic

cost function to address both process control and dynamiceconomic optimization of a nonlinear system, no guarantee canbe made that the closed-loop economic performance underLEMPC operating mode 1 (possibly leading to time-varying

operation) will be better compared to the closed-loop economicperformance under LMPC. This issue will be addressed in thesubsequent section. In contrast, Heidarinejad et al.27 recentlyproposed LEMPC algorithms to ensure improved economicperformance under LEMPC where an auxiliary LMPC was usedto formulate constraints in the LEMPC optimization problemthat guarantee that the closed-loop economic performance withLEMPC be at least as good as LMPC while ensuring theLEMPC uses the same amount of control energy as the LMPC(we refer the reader to ref 27 for detailed results).Remark 4. The region Ωρ is chosen such that the LEMPC

optimization problem of eq 46 remains feasible for any x(t)∈ Ωρ,t ≥ 0. Specif ically, if x(t)∈ Ωρ\Ωρ, the LEMPC operates in mode2 to force the process state to Ωρ. Once the process state hasconverged to the set Ωρ and t ≤ t′, the LEMPC operates in mode 1.The process state may come out of Ωρ over one sampling periodwhile the LEMPC is operating in mode 1, but Ωρ is chosen to besuf f iciently small such that the process state will not come out of Ωρ

before the next sampling period. At the next sampling period, theLEMPC operates in mode 2 to force the process state back to Ωρ.Therefore, the LEMPC optimization problem always remainsfeasible for any x(t)∈ Ωρ; see ref 17 for a detailed proof of thisissue.Remark 5. If the switching time t′ = 0, the LEMPC always

operates in mode 2 and will force the process state to converge to asmall neighborhood of the steady-state.

Figure 6. Transient trajectories of the manipulated inputs for heaters and coolers under LEMPC.



■ APPLICATION OF MPC TO THE VINYL ACETATEPROCESS

In this section, we implement the two model predictive controlstructures: the LMPC of eq 45 and the LEMPC of eq 46 on thevinyl acetate process network. The simulations are conductedusing JAVA programming environment with an Intel Core 2

Quad Q6600 computer. The explicit Euler integration methodis used to integrate the nonlinear dynamic process model ofeq 40a with a fixed integration step size equal to 0.001 min. Theintegration step size is chosen to ensure stable numericalintegration and sufficient accuracy of the numeric integration.The open source interior point optimizer IPOPT33 is used to

Figure 7. Transient closed-loop state trajectories under LEMPC of the heaters and coolers.

Figure 8. State trajectories of the temperature of the cooler prior to the separator (top plot) and state trajectories of the vapor phase vinyl acetatecomposition in the separator (bottom plot) with LMPC (solid line) and with LEMPC (dashed line).



solve the optimization problems. The prediction horizon ofN = 5 is chosen for the LMPC and the LEMPC. The firstoptimized control inputs computed by the optimization problemsare applied to the process every sampling time (Δ) following areceding horizon scheme.In this work, we initially demonstrate that LMPC is capable

of driving the closed-loop system to the unstable steady-state

that corresponds to the steady-state input listed in Table 4 froman initial state. The steady-state and steady-state input denotedas xs, us, respectively, have been chosen to satisfy the process(state and input) constraints. The evolution of the Lyapunovfunction of the closed-loop process network over a 300 minsimulation is shown in Figure 3. Since the magnitude of theLyapunov function is initially large, the Lyapunov function is

Figure 9. Temperature (top plot) and VAc concentration (bottom plot) profiles of the absorber at each stage under LMPC (square) and underLEMPC (circle) at the end of simulation.

Figure 10. Trajectories of the manipulated inputs for heaters and coolers under LMPC.



plotted on a semilogarithm scale in order to observe thedescending trajectory of the Lyapunov function throughout thesimulation.The region Ωρ is approximately calculated by extensive

closed-loop simulations under LMPC by initializing the processnetwork at different initial states. From these simulations, theregion Ωρ is defined as the level set Ωρ with ρ = 5600 which hasbeen taken as the largest level set of the Lyapunov functionwhere the Lyapunov function under LMPC is decreasing overeach sampling period for any initial state starting inside Ωρ.Subsequently, we define the parameter ρ through a series ofclosed-loop simulations under LEMPC operating in mode 1only. The region Ωρ is the subset of the stability region Ωρ usedin the formulation of LEMPC of eq 46 (i.e., the set where theprocess network is allowed to evolve when LEMPC is operatingin mode 1). The procedure for determining ρ is as follows:starting from ρ = 5600, ρ is decreased until the LEMPCmaintains stability of the process network (i.e., boundedness ofthe state inside of Ωρ). Through this procedure, the parameterρ used in the formulation of the LEMPC is ρ = 3275.The LEMPC, operating in mode 1 only, is applied to operate

the process network in a possible time-varying (transient)manner as to optimize the economic cost while maintaining theprocess states inside Ωρ. The Lyapunov-based constraint of theLEMPC (eq 46i) is formulated with the Lyapunov functionbased on the steady-state xs and the parameter ρ = 3275. Theprocess network under LEMPC is initialized at the same state

as the closed-loop system under LMPC demonstrated inFigure 3. The closed-loop input and state trajectories underLEMPC over the entire 300 min simulation are shown inFigures 4 and 5, respectively. The transient input and stateprofiles are shown over the first 10 min of the simulation inFigures 6 and 7, respectively. From Figure 4, the LEMPCoperates the process network in a time-varying manner(continuously changing control actions being computed formanipulated inputs on LEMPC). However, since the inputtrajectories change at a high frequency to maintain the process asclose as possible to an economically optimal steady-state, theobserved behavior of the closed-loop process state is similar tosteady-state operation (Figure 5). The high frequency fluctuations(appearing as chattering in the plots) in the computed inputprofiles by the LEMPC occur because the LEMPC is formulatedwith an economic cost that does not explicitly depend on themanipulated inputs and the LEMPC (operating in mode 1) doesnot include a constraint that imposes convergence to a steady-state (this chattering behavior can be eliminated if LEMPCmode 2 is implemented and this has been verified in the presentcase via simulations).Compared to LMPC that forces the closed-loop process

network to the steady-state xs, LEMPC significantly decreasesthe duty of the cooler C3 which results in the low recovery ofVAc in the separator as shown in Figure 8. Consequently,the majority of VAc from the reactor is recovered in theabsorber. Figure 9 shows the temperature and molar fraction of

Figure 11. Closed-loop state trajectories under LMPC of the heaters and coolers.



VAc profiles of the absorber at each stage at the end ofsimulation under LMPC and LEMPC and points out thatthe evolution of the closed-loop system under LMPC isdifferent from the evolution of the closed-loop system underLEMPC.Since the closed-loop process network under LEMPC evolves

like steady-state operation, we compare the closed-loopeconomic performance of the process network under LEMPCwith the closed-loop economic performance of the processnetwork under LMPC formulated with the operating steady-state of the LEMPC denoted as xs* with corresponding steady-state input us*. The steady-state xs* is defined from the closed-loop simulation of the process network under LEMPC. Sincethe control actions computed by the LEMPC and the cor-responding state profiles have significant fluctuations asdisplayed in Figures 4 and 5, xs* and us* are computed byaveraging out the profiles of the state and inputs over the last10 min of the simulation to determine the operating steady-stateunder LMPC. In this manner, the RTO layer (upper-layereconomic steady-state optimization) for the closed-loopprocess network under LMPC is the LEMPC that providesthe optimal steady-state by averaging the state profiles. Tocompare the closed-loop economic performance of theprocess network under LMPC and LEMPC, we initialize theprocess network at the same initial state and define the averagetotal economic measure (Je) over two 300 min closed-loopsimulations as

∑==

Jt

L t1

( )f k

f

ke0 (47)

where tk = kΔ, k = 0, 1, ..., f.Figures 10 and 11 show the closed-loop input and state

trajectories of the process network under LMPC over the entire300 min simulation; meanwhile, Figures 12 and 13 show theclosed-loop input and state trajectories over the first 10 min.Figure 14 shows the trajectories of the economic measure underLEMPC and LMPC, in which the average economic perform-ance of the closed-loop system under LEMPC (576.6) is notbetter than the average economic performance of the closed-loop system under LMPC (576.8). Indeed, the formulationof LEMPC (operating in mode 1 only) utilized in this workdoes not ensure improved economic performance of LEMPCover LMPC. As observed in Figure 4 and Figure 10, LEMPC,operating in mode 1, allows for unfavorable and impracticaltrajectories of control inputs compared to the trajectories ofcontrol inputs computed by LMPC. In addition, LEMPCrequires significantly larger evaluation time than LMPC (on theorder of days).

When is EMPC Needed? In this work, we applied LMPCand LEMPC to a vinyl acetate process network. The closed-loop economic performance of two simulations under LMPCand LEMPC were similar. This raises an important questionwhen considering the application of EMPC to a chemicalprocess network: When is EMPC needed? With large-scale

Figure 12. Transient trajectories of the manipulated inputs for heaters and coolers under LMPC.



process networks such as the vinyl acetate process network,operating process networks in a time-varying fashion as tooptimize economic measures while meeting process constraintsis difficult because the dynamics of each unit are coupled to thedynamics of the other units through the recycle streams. Whilethe topic of guaranteed closed-loop economic performance hasbeen addressed in ref 27, if we apply the LEMPC introduced inref 27 to this particular large-scale chemical process network,the LEMPC would compute control actions that wouldapproach the auxiliary LMPC. The computational requirementfor solving the LEMPC and LMPC problems in such controlframework would not likely be offset by the little improvementin closed-loop economic performance. Therefore, the bestoperating strategy for large-scale chemical process networkswith coupling of process dynamics would likely be operating atthe economically optimal steady-state. Since process networks

are typically operated continuously for long periods of time, theobjective of the controller formulated for steady-state operationis to drive and regulate the process network to a steady-stateand the effect of the transient state becomes insignificant onclosed-loop economic performance. Hence, the difference ineconomic closed-loop performance of the process networkunder LMPC formulated with a conventional quadratic costand under LEMPC formulated with the economic costfunction, but operating in mode 2, will also be insignificant ifoperation can be maintained at the steady-state (bound on thedisturbances is small).If there exists any time-varying operating strategy that is

better than operating at the economically optimal steady-state,then one would have to increase the prediction horizon of theLEMPC in an attempt to improve closed-loop economicperformance. The computational cost of increasing the horizon

Figure 13. Transient closed-loop state trajectories under LMPC of the heaters and coolers.

Figure 14. Trajectories of the economic measure under LMPC (solid line) and under LEMPC (dashed line).



may make, however, EMPC impractical to implement. Onemay consider the closed-loop economic performance of theprocess network under the other formulations of EMPC in theliterature like, for instance, the EMPC studied in refs 15 and 19where a terminal constraint is used in the formulation instead ofthe Lyapunov-based constraint of eq 46i in LEMPC. However,the Lyapunov-based constraint allows the LEMPC to operatethe system in a time-varying fashion while maintaining the stateinside a subset of the estimated closed-loop stability region asopposed to the EMPC with terminal constraint which allowsthe EMPC to operate the system in a time-varying fashion aslong as the system is forced to the operating steady-state atthe end of the prediction horizon. Therefore, the terminalconstraint is, in some sense, more restrictive in terms of theavailable time-varying operating trajectories. For this reason, itis also unlikely that using EMPC with a terminal constraint willprovide significant economic closed-loop performance improve-ment in this example compared to MPC formulated with aconventional quadratic cost.The main advantage of EMPC is when it operates a process

in a time-varying (transient) fashion to yield better closed-loopeconomic performance. Clearly, some processes do not benefitfrom time-varying operation based on the intrinsic propertiesof the dynamics. However, these systems may still benefitfrom application of EMPC if the economic cost function andconstraints become time-varying (e.g., the weights A1, A2, andA3 of the economic cost become time-varying). Time-varyingeconomic cost and constraints allows LEMPC to fully manipulateprocess capacities to meet the continuously changing marketdemand, feedstock variability and energy cost.

■ CONCLUSIONS

In this work, two Lyapunov-based MPC schemes are developedand applied to a large-scale chemical process network usedin the production of vinyl acetate. The nonlinear dynamicprocess network consists of 179 nonlinear differential equationsand 13 manipulated inputs and is composed of process unitssuch as a vaporizer, a separator, a plug-flow reactor, a gas−liquid absorber, a process-to-process heat exchanger, andheaters/coolers. Specifically, we propose an economic measureaccounting for the reaction selectivity in the reactor and theseparation quality of vinyl acetate in the separator and theabsorber. Subsequently, a novel control architecture whichcomprises of a LMPC/LEMPC acting to regulate fivemanipulated inputs which directly influence in the economicmeasure and a set of PI controllers acting to regulate the othermanipulated inputs to maintain closed-loop stability is appliedto the process network. The economic performance of theclosed-loop system under LMPC formulated with a conven-tional quadratic cost function associated with the systemsteady-state is compared with the economic performance of theclosed-loop system under LEMPC which explicitly considers aneconomic measure as its objective function through extensivesimulations.

■ APPENDIX: CONTROLLER PARAMETERS

Several closed-loop simulations were completed to tune theproportional−integral controllers. Closed-loop simulationswere initially performed under proportional control loopstuned to provide sufficiently small rise times and no oscillationin the controlled outputs. Subsequently, integral action wasadded to eliminate the residual steady-state errors. For example, Table

5.Weights

inPMatrix

weights

states(x)

1T1ABS ,T2ABS ,T5ABS ,T6ABS ,TCOMP

10x H

AC

VAP,x HA

cSE

P,x

H2O

SEP,x

HAc

SEP,x

HAc,b

ABS,x H 2

O,b

ABS,x

HAc,b

ABS,x

VAc,1

ABS,x

H2O

,1ABS,x

VAc,2

ABS

x H2O

,2ABS,x H 2

O,3

ABS,x

HAc,l

ABS,x

HAc

TK,x C 2H

4

SEP,T

SEP ,

TSE

P ,TVAP,T

lRCT,T

3ABST4ABS ,T7ABS ,T8ABS ,TbABS ,TTK,T

H1 ,TH2 ,TC3 ,TC4 ,TC5 ,TS34,TS9M

lABS ,M

bABS ,M

SEP ,M

VAP ,M

TK,P

SEP ,PC

OMP ,c C

2H4,l

RCT

100

x C2H

4

VAP,x

C2H

VAP ,x C

2H4

SEP,x

VAc,5

ABS,x

VAc,6

ABS,x

C2O

,4ABS,x

C2O

,5ABS,x

C2O

,6ABS,x

C2O

,7ABS,x VA

c,3

ABS,x

VAc,4

ABS,x

H2O

,8ABS,x

b,C2H

4

ABS

,

x C2H

4,l

ABS,y

VAc

SEP,y

H2O

SEP,y

O2

SEP ,y C

O2

SEP,c

O2,3

RCT,c

O2,4

RCT,c

O2,5

RCT,c

VAc,10

RCT,c

H2O

,8RCT,c

H2O

,9RCT,c

H2O

,10

RCT

,cO

2,1RCT,

c O2,2

RCT,c

HAc,l

RCT

1000

x O2

SEP ,x O

2,1ABS ,x O

2,2ABS ,x V

Ac,7

ABS,x

VAc,8

ABS,x O 2

,bABS ,y H

Ac

SEP,x

O2,6

RCT,x

O2,7

RCT,x O 2

,8RCT,x

O2,9

RCT,x O 2

,10

RCT,c

VAc,2

RCT,c

VAc,3

RCT,c

VAc,4

RCT,

c VAc,5RCT,c

VAc,6

RCT,c

VAc,7

RCT,c

VAc,8

RCT

c VAc,9

RCT,c

H2O

,1RCT,c

H2O

,2RCT,c

H2O

,3RCT,c

H2O

,4RCT,c

H2O

,5RCT,c

H2O

,6RCT,c

H2O

,7RCT,c

CO

2,lRCT

10000

x O2

VAP ,x C

O2

VAP ,x V

Ac

VAP ,x C

2H4

SEP,x O 2

,4ABS ,x O

2,4ABS ,x O

2,5ABS ,x O

2,6ABS ,x O

2,7ABS ,x O

2,8ABS ,x b

,CO

2

ABS,x

CO

2,l

ABS,c

VAc,1

RCT

100000

x VAc

TK



the proportional gain and integral time constant for the PIcontroller that regulates oxygen feed flow rate were chosen toavoid excessive overshoot such that the oxygen concentrationin the gas loop is kept outside the explosive region.The proportional gains (Kc), integration time constant (τI),

and set-points (ηset) for PI controllers are

= ·K (10 ) [1 0.1 1 1 1 75 0.3 500]cT 2

τ = [80 0.1 350 200 50 40 4.5 50]IT

η =( ) [2.72 0.01256 4 4.251.415120 158.5128]set T

The proportional gains (Kc), integration time constant (τI), andset-points (η set) for the Lyapunov-based controllers are

= ·K (10 ) [2.5 30 10 50 5]cT 2

τ = [80 220 7.14 90 50 52.6]IT

η =( ) [145 83 47 32.5 40]set T

The weights in positive definite diagonal matrix (Qc) and thepositive definite diagonal matrix (Rc) are

= ·− −Q x(10 ) diag( )c5

s1

= × × × × × ×− − − − − −R diag[1 10 1 10 1 10 1 10 1 10 1 10 ]c11 8 8 11 11 7

where xs is the steady-state vector. The weights in P matrix andthe corresponding states are listed in Table 5.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: [email protected] authors declare no competing financial interest.

■ NOMENCLATURE

Roman Symbolsai = vapor heat capacity/enthalpy coefficient for species i,kcal/(kg °C)ai = liquid heat capacity/enthalpy coefficient for species i,kcal/(kg °C)a ij = Wilson parameters of species pair i−j kcal/kmolAi = Antoine coefficient of species ibi = vapor heat capacity/enthalpy coefficient for species i,kcal/(kg °C2)bi = liquid heat capacity/enthalpy coefficient for species i,kcal/(kg °C2)Bi = Antoine coefficient of species ici = vapor concentration of species i, kmol/m3

ci = liquid heat capacity of species i, kcal/(kmol °C)ci = vapor heat capacity of species i, kcal/(kmol °C)C = total vapor concentration, kmol/m3

C = liquid heat capacity, kcal/(kmol °C)C = vapor heat capacity, kcal/(kmol °C)

Ci = Antoine coefficient of species if = vapor molar flow rate, kmol/minf = liquid molar flow rate, kmol/minhi = vapor enthalpy of species i, kcal/kmolhL,i = latent heat of species i, kcal/kmolhrxn,i = heat of the ith reaction (i = 1, 2), kcal/kmolh i = liquid enthalpy of species i, kcal/kmolH = vapor phase enthalpy, kcal/kmol

H = liquid phase enthalpy, kcal/kmolk = mass transfer coefficient, kmol/minK = equilibrium constantM = total moles in vapor phase, kmolMWi = molecular weight of species i, kg/kmolM = total moles in liquid phase, kmoln = mass transfer rate to the liquid phase, kmol/minNi = mass transfer rate of species i, kmol/minpi = partial pressure of species i, psiapisat = saturated pressure of species i, psiaP = pressure, psiaq = heat rate supplied or removed per unit volume,kcal/(m3 min)Q = heat rate supplied or removed, kcal/minQtr = convective heat transfer rate due to transferringmaterials, kcal/minSpGi = specific gravity of species iT = vapor temperature, °CTcoolant = coolant temperature, °CT = liquid temperature, °CUA = overall heat transfer coefficient, kcal/(min °C)U = overall heat transfer coefficient per unit volume,kcal/(m3 min °C)v = liquid volume, m3

v = superficial velocity, m/minvi = molar volume of species i, L/kmolWs = actual work supplied to the compressor, kcal/kmolxi = liquid mole fraction of species i, kmol/kmolyi = vapor mole fraction of species i, kmol/kmolz = axial coordinate along the length m

Greek Symbolsα = fraction of stream in vapor phaseε = catalyst porosityγi = activity coefficient of species iΛcat = catalyst heat capacity, kcal/(kg °C)ρcat = catalyst density, kg/m3

ζj,i = stoichiometric coefficients for species i and reaction j, j= 1, 2

Subscripts0 = inlet stream of the absorber or the plug-flow reactori = component (i = C2H4, O2, HAc, VAc, CO2, H2O, C2H6)int = vapor−liquid interfacej = stream number (j = S1, S2, ..., S34)l = section number of the plug-flow reactor l = 1, 2, ..., 10 ortheoretical stage number of the absorber l = 1, 2, ..., 8max = maximumnet = net amountb = absorber base

Superscriptsj = vessel number (j = VAP, RCT, HX, SEP, COM, ABS,TK, H1, H2, C3, C4, C5)

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