PlantWIde McAvoy

8/6/2019 PlantWIde McAvoy

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PROCESS DESIGN AND CONTROL

P l a n t w i d e C o n t r o l S y s t e m D e s i g n : Me t h o d o l o g y a n d A p p li c a t i o n t o a

Vi n y l A c e t a t e P r o c e s s

R o n g C h e n a n d T h o m a s M c Av o y *

Department of Chemical Engineering, Institute for Systems Research, University of Maryland,

College Park, Maryland 20742

A new approach to the design of plant wide control systems that is based on a l inear dynamicp r oce s s m od e l a n d ou t p u t op t im a l con t r o l i s p r e se n t e d. Th e a p p r oa ch a l so m a k e s u s e of engineering judgment in eliminating and evaluating candidate architectures. The design of aplant wide ar chitectur e is split into four stages, and r esults from one stage are u sed as th e inputto the next. During th e design process, transient responses a re easily calculated, and they ar eused t o compare candidate architectures to one an other so that those with poor performan cecan be eliminated. The methodology is applied successfully to a model of a vinyl acetate processt h a t h a s 2 6 m a n i pu l a t ed v a r ia b le s a n d 4 3 m e a s u r e m e n t s . Th e m e t h od ol og y p r e s en t e d i s

facilitated t hr ough a user-friendly softwar e package tha t m akes u se of the best cur rent ly availablealgorithms for solving output optimal control problems. A detailed discussion of the variousalgorithms used in the package is presented.

1 . I n t r o d u c t i o n

Plan twide control is an approach th at guides controlsystem design for an entire plant. Plantwide controlt r i es t o a ns wer s om e bas ic ques t ions t hat a cont r olengineer regularly meets in practice:1 which variablesshould be controlled, measured, and manipulated, andhow should these va riables be linked together? The firstcomprehensive discussion of plantwide control wasprovided by Buckley,2 w h o p r es en t e d a n u m be r of

engineering insights into material balance control,production rate control, inventory control, recycle use,impurity purging, and predictive optimization. In 1973,Foss1 stated that the central issue to be solved by newth eories of chem ical p rocess cont rol is how to det ermin ea control system structure. He urged the developmentof new theories to carry out plantwide control design inan efficient a nd organ ized ma nn er. Although th e cont rolobjectives that plantwide control should address wereidentified years ago, there is no agreed-upon systematicmethod for generating control str ucture alternatives.Some of the reasons for the slow progress include thes i ze of t he pr obl em , t he l ack of s t at i c and dynam i cmodels, and the difficulties in tailoring heuristics for aspecific plantwide design.

C ur r ent appr oaches t o plant w ide cont r ol can beloosely categorized as ma th ema tically orient ed (optimi-zation-based) appr oaches, process-oriented (heuristic-based) approaches, a nd a combination of th ese twoapproaches (hybrid approaches).3 Mathematically ori-ented approaches try to identify control structure can-didates by using process models (either static or dy-namic) and quantitative methods from modern controland nonlinear optimization. Process-oriented approaches

use heuristics that are developed from engineeringexperience and process insight. Many authors believethat hybrid approaches are more promising, because thehybrid approaches can facilitate the design task in anefficient way by au tomatically generat ing an d evaluat -ing contr ol stru cture candidat es. In most of the currentapproaches, a hierarchical design procedure that de-composes the plantwide control problem into severalstages and solves them in sequence is used by research-ers, as good scalability is reta ined.

To constru ct a plan twide cont rol system, the simplestapproach is to design a control system for each unitindividually, without considering int eractions. A nu m-ber of case studies show that this approach does notwork, because a control system that is feasible and/oroptimized in controll ing a single unit might not befeasible and/or optimized a fter the unit becomes anintegrated part of the whole plant. For example, in theTennessee Ea stma n (TE) process,4 there ar e three ma jorunits (the reactor, the separator, and the stripper). Fromthe point of view of single-unit operations, the threepressur es in th ese units sh ould be controlled for t he sa feoperation of the individual processes. From the pointof view of the overall process, however, only one of the

t hr ee pr ess ur es needs t o be cont r ol led becaus e t hepr ess ur es a r e hi ghl y cor r elat ed and at t em pt ing t ocont r ol al l t hr ee r es ult s i n s ever e i nt er act i ons . I nanother case, a simple process involving a reactor, aseparator, and a l iquid recycle stream was presentedby Luyben.5 He observed a snowball effect tha t causeda shut-down of the process for a certain type of plant-wide control stru cture, which was obtained by designingcontrol systems for the units separately.

In recent years, the plantwide perspective on design-ing control systems for a chemical process has receivedincreasing at tent ion from people in both a cademia an d

* To whom correspondence should be addressed. Tel.: (301)405-1939. Fa x: (301) 314-9920. E-ma il: [email protected] d.edu.

4753Ind. Eng. Chem. Res. 2003, 42 , 4753-4771

10.1021/ie030202e CCC: $25.00 2003 American Chem ical SocietyPublished on Web 09/06/2003


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industry. There are two key issues in the plantwidecontrol design area: (1) how to qualitatively an d/orquan titatively captu re pr ocess intera ctions and explainthem and (2) in the face of process interactions, how tosystemat ically define a cont rol structu re d esign probleman d pr actically solve it. For the first k ey issue, informa-tion about process interactions can be extracted from aprocess model, which ranges from a simple qualitativemodel, e.g., a process flowsheet with steady-state data,to a complicat ed qua nt itat ive one, e.g., a first-principlesnonlinear dynamic model. When a process m odel isavailable, the issue of how to extract process informationand use the information in control structure design isvery important. For example, the relative gain array(RGA) is a popular tool for representing process interac-tions, a nd several RGA-based r ules ha ve been developedfor loop pairin gs design.6 The second key issu e involvesselecting controlled variables (measurements), manipu-lated varia bles, control configurat ions (decentr alized orcentralized control structures), and control laws.7 F orthis issue, a hierar chical design pr ocedure is generallyused to decompose the design problem into severalstages. The reason for such an approach is that, forchemical process plantwide control design, it is very

difficult to find a global optimum solution to all of thecont rol objectives to be a chieved. A hierar chical d esignprocedure can provide a systematic and practical wayto locate satisfactory solutions in a reduced search space.

The basis of this paper involves taking advantage ofa linear dynam ic process model in designing a plantwidecontrol system. Increasingly dynamic models are avail-able for processes, even in the design stage, and ourobjective is to investigate how much plantwide controldesign can benefit when a l inear process model isavailable. Nonlinear models can be linearized aroundan operating point to generate the models used here.The goal of this paper is to develop a plantwide controldesign methodology in which a linear time-invariant(LTI) state space model is u sed t o provide informa tion

about pr ocess dynam ics a nd interactions. The origina lidea of using a n LTI model and optimal contr ol theoryin control structure design came from Schnelle. 8 Th eorigina l idea of the hierar chical design procedure usedhere came from McAvoy and co-workers previouswork.3,9

Several guidelines are applied in developing ourplant wide cont rol design m eth odology. These guidelinesrepresent our design philosophy:

(1) The methodology should not require an experi-enced control engineer to use i t , and only a l imitedamount of engineering judgment should be involved indesigning a plantwide contr ol system.

(2) The meth odology sh ould easily extract process

informat ion from an LTI model and explain i t in asimple mann er.

(3) The methodology should have good scalability.When the size of the design problem (i.e., the numberof the stat es in t he LTI m odel) increases, th e comput a-tion load should not increase dram atically.

(4) The m ethodology should be implemented u sing ahighly automated computer-aided design tool. After acontrol structure is selected, controllers can be easilytuned, and control performance can be visually evalu-ated.

In this paper, an optimal control-based plantwidecontrol design methodology that complies with theseguidelines is presented. The methodology uses a hier-

archical design procedure involving optimal static out-put feedback (OSOF) controller d esign. After th e deta ilsof t he m et hodol ogy ar e explai ned, t he appr oach i sapplied successfully to a model of a vinyl acetate processthat has 246 states, 26 man ipulated variables, and 43measurements.

2 . Ov e r v i e w o f O p t i m a l C o n t r o l -B a s e dP l a n t w i d e C o n tr o l D e s i g n Me t h o d o l o g y

The optimal contr ol-based plantwide contr ol designmethodology is a design procedure that identifies fea-sible contr ol stru ctures for a n en tire chemical plan t. Thecontrol structures are determined by a combination ofmathematical analysis and engineering judgment. Thereare three major chara cteristics of t he methodology:First, the meth odology extracts process informa tionfrom a l inear t ime-invariant (LTI) state space modeltha t is developed a t a local operat ing point. Because adynamic model gives more process insight tha n a st aticmodel does, dynamic models should be used in plant-wide control design whenever available. The processgain m at r i x i s t he s im plest pr ocess m odel, and i tconta ins st atic informa tion about process inter actions.

Several popular tools, e.g., RGA and singular valuedecomposition (SVD),3 are available for designing de-centralized control structures. The focus of this paperis to find a simple form, analogous to a process gainm at r i x, t o r epr es ent dynam i c i nfor m at i on about aprocess an d t o develop tools to extra ct an d explain t heinformation. In t he n ext section, a m ulti-input -multi-output optimal static output feedback controller isintroduced as the simple form that meets this require-ment for plantwide control design. Second, the optimalcontrol-based methodology is a hierarchical designprocedure, consisting of four stages that are extendedfrom an original design procedure used by McAvoy andco-workers.3,9 In stage 1, the process model is scaled,and the contr olled variables related principally to safe

process operation ar e identified. In st age 2, decentra l-ized control structures are determined for the stage 1variables. In sta ge 3, either cent ra lized or decentr alizedcont r ol s t r uct ur es ar e gener at ed for t he cont r ol of product rate and quality. In stage 4, either centralizedor decentralized control structures are designed forother contr olled variables relat ed t o component inven-tory and unit operation control. Third, th e optimalcontrol-based methodology is implemented using ahighly automated computer-aided toolkit . The coderequires a limited am ount of inter action from a u ser whois not necessarily an experienced control engineer. Someengineering judgment is required to determine designparameters and evaluate control structure candidates

that are generated in each design stage. Because theplantwide control design problem is quite computation-ally intensive, special consideration is t aken in develop-ing th e numerical algorithms used to provide goodscalability.

3 . B a s i c O p t i m a l S t a t i c O u t p u t F e e d b a c kC o n t r o l P r o b l e m

Given an LTI state space model, an optimal staticoutput feedback (OSOF) controller is designed to sta-bil ize the system and bring the states from arbitraryinitial values t o zero, following a tra jectory th at mini-mizes a linear quadratic objective function (LQR). Thebasic formulation of the OSOF LQR design problem is

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presented by Lewis as follows:10

where x represents the states, u r epr es ent s t he m a-nipulated var iables, y represents th e measurements, Qan d R ar e w ei ght m at r i ces , and gij i s a w ei ght onelement kij in K. In most cases where eq 3 is solved todevelop a plantwide control system, the gijs are zero.However, as discussed below, in some cases, a multiloopsingle-input -single-output (SISO) structure is used.Then, the gijs a re u sed t o force the off-diagona l element s

of K to be zero so that the resulting controller has adiagonal structure. To make element kij small, a largevalue for the corresponding weight, gij, should be used.

The design equations for K and two auxiliary matri-ces, P a nd S , that result from the first-order necessaryconditions for optimality are given by Lewis 10 as pre-sented in eq 4

where g*K is a m atrix with elements gij*kij. In solvingeq 4, R should be positive definite, and Q should bepositive semidefinite to ensure that CQC is positivesemidefinite. P will be positive definite (or positivesemidefinite) as long as A C i s s t abl e and (CKRKC +CQC) is positive definite (or positive semidefinite). Sis positive definite (or positive semidefinite) as long as

A C i s s t able and X is positive definite (or positivesemidefinite). It can be noted that including nonzero gijin the LQR design calculation dramatically reduces thecomputation speed when a SISO controller is calculated.An alternative approach to tuning SISO controllers isdiscussed lat er. Becau se th ere is n o explicit a nalyticalsolution for the OSOF controller K, num erical optimiza-t i on r out i nes ar e us ed t o s ol ve t he t hr ee couplednonlinear matrix eqs 4a -c simultaneously and obtaint he K matrix that minimizes J. As discussed later, Kcan be used t o gain insight int o the dyna mic inter actionsthat occur in a process.

The OSOF solution depends on how x, y, a n d u a r escaled. The following scaling guidelines are recom-mended. Whether the states are scaled or not dependson whether they can be compared to one another. If s tates have physical meanings, e.g. obtained from afirst-principles model, they should be scaled by eithertheir steady-state values or the ranges of their desiredmovement . If states do not ha ve physical m eanin gs, e.g.,t he s t at e s pace m odel i s conver t ed fr om a t r ans fer

function model, th ey are left unscaled. Manipulatedvariables (MVs) can be either valve opening per cent agesor set points of inner cascade controllers, and the MVsshould always be scaled. If cascade cont rollers a re u sed,they need to be proportional-only controllers so that theycan be incorporat ed into eq 1. One scaling meth od is touse the ranges of allowable movement. Another scalingmethod is to use physical valve ranges. The measure-ments can be scaled by either the physical ranges of their transmitters or the ranges of their desired move-ment. When the model is scaled, Q a n d R can be chosenas ident ity mat rices. If it is desired to put more or lessweight on measurements and/or manipulated variables,the diagonal elements ofQ a n d R can be adjusted.

As eq 4e shows, the OSOF controller K depends onthe init ial s tates, x0. In some cases, th is dependence isnot desirable because x0 m i ght not be know n. T hisproblem can be sidestepped by min imizing th e expectedvalue ofJ, as discussed by Levine a nd Atha ns.11 In thiscase, eq 4e becomes

where X is the initial a utocorrelation of the stat es. It isusual to assume that the init ial s tates are uniformlydistributed on the unit sphere, and as a result , X ) I,the identity matrix. I t is possible to design controlstructures for specific set-point tracking and/or distur-ban ce rejection pu rposes, where X * I.12 To simplify t hepresentation, designs based on eq 5 are considered inthis paper.

4 . N u m e r i c a l a n d O t h e r C o n s i d e r a t i o n s f o r t h eO S OF P r o b l e m

Several issues need t o be considered when a plant-wide control design approach based on an OSOF con-troller is used. One question involves whether a system

can be stabilized by sta tic outpu t feedback (SOF). Theproblem of the existen ce of a st abilizing SOF cont rollerin multivariable cases is s t i ll open,13 as no t es t abl enecessary and sufficient conditions exist to test thestability of an ar bitrar y system u sing an SOF controller.G iven eq 1, an engi neer does not know w het her asta bilizing SOF exists unt il it is foun d. In our a pproach,the even parity-interlacing property necessary condi-tion 14 is used to check whether a given system can bestabilized by sta tic outpu t feedback. If the system doesnot violate t his necessary condition, it is assum ed th atan SOF controller that stabilizes the system exists.

Num erical a lgorith ms for designing OSOF cont rollerscan be r oughly divided into t wo broad categories. Thefirst category includes all algorithms that iterativelycalculate a solution tha t satisfies th e first-order neces-sary conditions for optimality.15 Although global con-vergence is obtained under certain conditions, globaloptimality is not guarant eed, and most of these algo-rith ms requ ire an initial stabilizing SOF controller. Thesecond category comprises methods using linear matrixinequalities (LMIs).16 These methods do not need aninitial sta bilizing controller an d can conditionally reachglobal optimality, but global convergence is not guar-anteed. In our approach, Moerder and Calises algo-rithm 17 is implemented for i ts conditionally globalconvergence, simplicity, an d efficiency. Four majorcom put at i onal i ss ues w it h t hi s al gor i t hm ar e s um -marized in Table 1.

Pr ocess Model

{x ) A x + B u (a )

y ) Cx (b)

x(0) ) x0 (c)

(1)

Output Feedback

u ) -Ky (2)

General Objective Function

m inK

J )1

20

(y

TQy + u

TR u ) dt +

1

2

i

j

gijkij2

(3)

{A C

TP + PA C + C

TK

TRKC+ C

TQC ) 0 (a )

A CS + S A CT

+ X ) 0 (b)

RKCSCT

- BT

PS CT

+ g*K ) 0 (c)

A C ) A - BKC (d)

X ) x(0) x(0)T (e)

(4)

X ) E[x(0) x(0)T

] (5)

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I s s u e 1 : Al g o ri t h m S e l e c t i o n . T h e r e a r e t h r e egroups of numerical OSOF LQR design algorithms thatiteratively solve for the necessary optimal conditions,15

descent Anderson -Moore-like methods (e.g., Moerderand Calises algorithm), Levine-Athans-like methods(e.g., Toivonens algorithm 18), an d Newton-like m eth ods(e.g., Toivonen and Makilas algorithm 19). Ba sically,descent Anderson -Moore-like m ethods run faster th anLevine-Athans-like methods. For example, from ourexperience with the algorithms, represented in Table2, it is obvious tha t Moerder an d Calises algorithm r un smuch faster t ha n Toivonens algorithm. Descent Ander-son -Moore-like methods are also less complex to imple-ment than Newton-like methods. Therefore, Moerderand Calises algorithm is the defau lt algorithm used incalculating K in our a pproach. It should be pointed out

that, as the size of a design problem gets larger, e.g.,when hu ndreds of stat es are present, the run ning timesof all of these algorithms grow rapidly. Therefore, modelorder reduction might be necessary before an OSOFcalculation is executed.

I s s u e 2 : C o n v e r ge n c e P r o p e r ti e s . Accordin g to t heconcl usi ons given i n r efs 13 and 15, M oer der andCalises a lgorith m converges to a local optimum . If theset of stabilizing static output feedback gains is convexand the solution to eq 4 for K, P, and S is unique, thenthe global optimum is obtained. However, these twosufficient conditions are not t estable, an d t herefore, itis necessary to compa re different solut ions t o determinethe global optimum. F or all of the cases we ha ve studiedso far for th e Tennessee Ea stma n process,4 Moerder andCalises algorithm always gives the sam e solut ion wh enstarting from a number of different initial stabilizinggains.

I s s u e 3 : S u f fi c i e n t C o n d i t i o n s f o r G l o ba l C o n -v e r g e n c e . All five sufficient conditions for global con-ver gence s houl d be s at is fi ed t o guar ant ee t hat t healgorithm converges to a stationary point of the objectivefunction. In practice, there are cases in which A is not

stable and (QC,A ) is not detectable, and as a result ,the fifth condition is not satisfied. For example, theTennessee Eastma n pr ocess4 is open-loop unst able, an das stated by Lewis,10 this detectability condition basi-cally means th at all unsta ble states should be weightedin the objective function. From the point of view of

num erical algorithms, the detectability condition guar-ant ees t hat t he L yapunov m at r i x, P, i n e q 4 a h a s aunique positive semidefinite solution. F ortuna tely, thedetectability condition is not compulsory,20 which means

t hat , even w hen (QC,A) is not detectable, eq 4 canstill be solved, and the algorithm can st ill converge.

Two methods are implemented in our approach tohandle situations in which the detectability conditionis not satisfied. The first method is based on executingthe algorithm without checking t he detectabili ty of

(QC,A) . If a solution to eq 4a cannot be found or isnot unique, the LQR calculation is aborted. The algo-rithm is coded in MATLAB, and a standard routine,lyap(), is used to solve for P. When a solution cannot befound or i s not un i que, t hi s r out ine r et ur ns w i t h afailure signal that will stop the algorithm, and thenan alternative method is used to calculate the OSOFcontroller.

The alternative method removes th e detectabili tycondition by altering the objective function. In thismethod, the objective function is changed as follows 10

where a time-varying weighting t places a hea vy penaltyon errors that occur late in the response. As pointed outby Lewis,10 when eq 6 is used, the detectability conditionis no longer necessary. The price one pays is that thenumber of design equations, given in eq 7, is increasedand, a s a result , the computat ion t ime increases.

For example, using the scaled state space model inthe design of the Tenn essee Eastma n pr ocess,4 both thebasic and altern ative Moerder and Calises a lgorithm swere tested. The alternative method ran much moreslowly than the default method, because additionalnonl i near m at r i x equat i ons had t o be . T he r es ul t s ,w hich w er e cal cul at ed on a Pent ium I I I 550-M H zpersonal compu ter, ar e shown in Table 2. Therefore, th ealtern ative method is not recommen ded unt il the defau ltmethod fails. The same conditions on R , Q, P, a n d S ,discussed earlier after eq 4, apply to eq 7 as well.

I s s u e 4 : C a l c u l a t i n g a n I n i t i a l S t a b i l i z i n g S O FC o n t r o l l e r K. T hr ee m et hods have been s t udi ed t ocalculate an initial guess for K that makes an unstableopen-loop process model asymptotically stable. The firstmethod is based on a ran dom search. Random numbers,ran ging between (R, are genera ted for t he elements ofK until A - BKC is asymptotically stable. R is a designpar am et er , and i t s val ue i s gi ven by us er s , w i t h adefault value of 1.0. This m ethod is s low for largeproblems. The second method is based on optimization.A nonlinear optimization routine is used to f ind a Kmatrix th at can minimize the maximum eigenvalue of

T a bl e 1 . C o m p u t a ti o n a l I s s u e s w i t h M o e rd e r a n d

C alises A lgorit h m

issu es Moer der a nd Ca lises a lgor it hm

(1) numerical algorithmselection

faster than the Levine -Athans-likemethods18

simpler than the Newton-like methods19

(2) sufficient condit ions a K exists such that A - BKC isasymptotically stable

for global convergence C has full row rankR is positive definiteCTQC is positive semidefinite

(QC,A) is detectable when A is not stable(3) convergence propert ies local optim um(4 ) ca lcu la t in g a n r a n dom s ele ct ion

initial stabilizing K minimize the maximum of theeigenvalues ofA - BKC

Pet kovski and Rakics meth od21

T a b l e 2 . R u n n i n g T i m e C o m p a r i s o n o f N u m e r i c a l L QRA l g o r i t h m s o n t h e T e n n e s s e e E a s t m a n M o d e l

algorithm

M oe r de r a n d C a li se T oi von e n

defa u lt m eth od 10-100 s 1000-2000 sa lt er na t ive m et h od 1000-2000 s 8000-12 000 s

m inK

J )1

2

E

[0

(ty

TQy + u

TR u ) dt +

1

2

i

j

gijkij2

](6)

{

A CT

P0 + P0A C + CT

QC ) 0 (a )

A CT

P1 + P1A C + P0 + CT

KTRKC) 0 (b)

A CS

1 + S 1A CT

+ X ) 0 (c)

AC

S0

+ S0

AC

T+ S

1

) 0 (d)

RKCS 1CT

- BT

(P0S0 + P1S 1)CT

+ g*K ) 0 (e)

A C ) A - BKC (f)

X ) E[x(0) x(0)T

] ) I (g)

(7)



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A - BKCin a fixed number of iterations. The optimiza-tion routine is repeated until A -BKCis a symptoticallysta ble. This meth od is also slow for large pr oblems. Thethird meth od is called P etkovski a nd Rakics m ethod21

in which a minimum error excitation criterion is usedt o gener at e an i nit i al gues s t h at can s t abil ize t hesystem. This method requires the solution of only oneR iccat i equat i on and one L yapunov equat i on, andtherefore, i t is fast for large problems. H owever, a

neces sar y condi t ion for us ing t hi s m et hod i s t hat(QC,A) should be detectable when A in not stable.Therefore, Petkovski and Rakics method is recom-

mended, except when (QC,A) is not detectable and Ais not stable. The OSOF calculation procedure is out-l ined in Appendix A, using a generic procedure lan-guage. The details of the algorithms and a l ist ing of software used to implement them are available in ref 12.

From the modeling side, a major concern related tothe computation load is the order of the model, whichis equal to th e nu mber of state var iables. If one wishesto speed up calculations, i t is recommended that themodel order be redu ced. The m ajor cha ra cteristics of theprocess dynam ics and intera ctions, as well as the inpu ts(MVs) and outputs (measurements), should be retained.In t he plan twide control design approach, the r elation-ships among the inputs a nd output s are major concerns.Sta tes a re u sed only for the pur pose of controller design,and i t does not m at t er w het her t hey r epr es ent r ealprocess variables or not. As a result, some model erroris introduced by model reduction techniques. Thisinaccuracy is the price one m ust pay for retaining anefficient design algorithm. If one uses a reduced-ordermodel, then there is a possibil i ty that the resultingplant wide cont rol design could be different from th e onethat would result from using the full model.

Currently, several model reduction methods withsoftwar e ar e available. The m odel reduction method th atwe have implemen ted in our meth odology is th e balan ceand truncate approximation method without coprimefactorization22 an d t he related MATLAB software pro-gram, sysred(), provided in the SLICOT package.23,24 Auser can specify either a desired order or a tolerancefor the model error. In the latter case, the model orderis automatically determined by the number of Hankel-singular values greater than the tolerance. An impor-t a n t a s pe ct of t h i s s oft w ar e is t h a t it ca n h a n d leuns t able s ys t em s becaus e al l un s t able s t at es, andi nt egr at or s ar e r et ained i n t he r educed m odel. T heTennessee E astma n process4 is used to test the feasibil-

ity of the m odel reduction meth od. The original m odelconta ins 50 stat es, and t he redu ced-order model ha s 23states. Two OSOF controllers with 5 measurements and12 m ani pul at ed var i abl es ar e gener at ed f or t he t w omodels. The contr ol structures developed from th e t wodifferent OSOF controllers are essentially the same.However, th e computat ion t ime related to generatingthe OSOF controller for the reduced-order model isroughly 50 times less. The model reduction method isimplemented a s an option in our plantwide controldesign a pproach. In the a pplication t o the vinyl acetat eprocess discussed below, model redu ction wa s not u sed,becaus e t he com put at i on s peed i s accept abl e on aPentium III 1-GHz personal computer.

5 . U s i n g t h e O S O F S o l u t i o n f o r P l a n t w i d eC o n t r o l D e s i g n

a . D e s i g n i n g S I S O Co n t r o l l e r s . In part of a plant-w ide cont r ol desi gn, i t can be desi r abl e t o have amultiloop SISO control structure. In stage 1, discussedbelow, such a decentr alized control architectur e is used.The OSOF contr oller K conta ins informa tion aboutprocess dynamics and interactions that can be used indesigning a SISO architecture. After K is obtained, th e

next qu estion is how to extract th e informa tion from Kand use it for SISO control structure design. Becausethe process model has been scaled, K is dimensionless,a n d t h e m a gn it u d es of t h e e le m en t s in K ca n b ecompar ed to one a nother. To extract information aboutprocess dynamics, the simplest metric is the absolutevalue of each element in the OSOF controller. Generally,an element with absolute value close to zero indicatesa weak relation between the manipulated variable andthe measurement. The following general rules can beused for control structure design: (1) If a row of theOSOF contr oller contains only small elements, e.g.,absolute values less than 0.1, then the correspondingmanipulated variable should not be included in thecontrol structure design. (2) If a column of the OSOFcontroller contains only small elements (e.g., absolutevalues less than 0.1), then the corresponding measu re-ment should not be included in the control structuredesign. (3) For decentralized control structures, if anelement of the OSOF controller is small (e.g., absolutevalues less than 0.1), then the corresponding pairingshould not be used.

In an earlier paper,25 two approaches to defining adynamic relative gain array based on the OSOF control-ler K were proposed. In essence, these approaches tryto extract the information about process interactionsthat is contained in K. One of the two approaches is asensit ivity approach, which is used in our plantwidedesign meth odology. The second approach is based on

K itself, an d t wo simple plus one industr ial example ofthe use of the second approach are given in the earlierpaper.25 The motivation for using the sensitivity matrixcomes from t he r elative gain arr ay,26 which is frequentlyused t o detect pr ocess intera ctions u sing a st at ic processgain m odel. In our design p hilosophy, an OSOF cont rol-ler is ana logous t o a process gain ma trix, and a sensitiv-ity matrix is analogous to an RGA. Mathematically, thesensitivity matrix S is given by

To calculate the sensit ivity matrix, one proceeds asfollows. F irst , a base-case OSOF design problem issolved with the R matrix equal to R 0 (e.g., an identitymatrix). Then, the same problem is re-solved with eachof the manipulated variables empha sized. F irst , al ldiagonal entries in the R matrix are multiplied by 100except for t he ent ry for t he ma nipulated var iable beingempha sized (e.g., for the ith manipulated variable u i,

R equals R i in which R kk ) 10 0R 0kk, k k * i, and R kk )R 0k k, k k ) i). Then, the OSOF design problem with R iis solved. The sensit ivity matrix is calculated as theratio of the gains for u i from th e base case divided intothe gains when u i is emphasized. The terms in eq 8 givet he gai n of u i t o yj dur i ng a t r ans i ent i n w hi ch t heprocess is controlled using an optimal output propor-

S ) matrix[ij] ) matrix[(Kij)R )R 0

(Kij)R )R i] (8)



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tional controller. The numerator gives the gain of themanipulated variable, u i, to a change in measurement

yj for the case in which the full base-case optimalcont roller is bringing th e system ba ck to the origin. Thedenominator gives th e sam e gain for t he case in whichcontrol is achieved primarily using only u i because theother man ipulated variables are heavily penalized. Theij ratio given by eq 8 is scale-independent, and i tsdetermination requires the solution of a number of OSOF design problems, one problem for th e ba se caseand one for each manipulated variable.

The following int erpreta tion ofij is proposed. Ifijis close to 1.0, then the optimal gain between u i a n d yjremains the same regardless of whether the remainingmanipulated variables are changing aggressively ormoving very l i t t le because of the penalty on them. If ij 1.0, then one can int erpret th is result to mean th atthe m an ipulated variable under consideration does notinteract with the other manipulated variables insofaras yj is concern ed. Thus, pa iring yj with u i would qualifyfor a multi loop control stru cture. If a manipulatedvariable h as a negative value of ij, then its behaviorswitches sign depending on how aggressively the otherm ani pul at ed var iables ar e m oving. Such a pair i ng

should be avoided as intera ctions are likely to be high.Large or sma ll values ofij also indicate large chan gesin the behavior of a manipulated variable, which, inturn, indicate that interactions are high. To use eq 8for pairing a multiloop control system, one would choosepairings in which the corresponding element in thesensitivity matrix is close to 1 (i.e., between 0.2 and 5).For such pa irings, the coupling between u i a nd yj shouldnot change much regardless of whether other manipu-lated var iables are m oving. As a result, if a row of thesensitivity matrix does not contain any element in therecommended range, th en the corresponding ma nipu-lated variables should not be included in the controlstructure design. If a column of the sensitivity matrix

does not cont ai n any elem ent i n t he r ecom m endedran ge, then the corresponding measur ement should notbe included in the control structure design. This pairingrule is a heuristic because, at present, a theoreticallybased pairing rule has not been developed. A differencebetween the sensitivity mat rix and th e RGA is that theelements of the sensitivity mat rix do not sum to 1.0 evenfor a square system. Using the OSOF controller and thesensitivity matrix, loop-pairing structures can be de-termined for decentr alized contr ol. Compar ed with thetraditional RGA method, this approach has two majoradvan tages: (1) process dynamics are included in thedesign as well as analysis of process interactions and(2) as discussed by Chen,12 process disturban ces an d set-point changes can be handled directly.

b . P l a n t w i d e C o n t r o l D e s i g n M e t h o d o l o g y . Inthis section, the optimal cont rol-based plant wide contr oldesign m eth odology is described in d eta il. As discussedabove, a hierarchical design procedure that containsfour design stages is used. In this methodology, a nOSOF controller is designed on the basis of a set of preselected measur ements an d a set of the m anipulatedvariables that a re available in a pa rticular stage. Then,control stru cture candidates ar e determined u sing bothmathematical analysis and engineering judgment. Foreach contr ol structure candidate, a corresponding cen-tralized controller or decentralized controller is auto-matically tun ed, and process tran sients are generatedon the basis of the linearized model so that a user can

compare the control performances of different plantwidear chi t ect ur es . I n t he r em ainder of t hi s s ect i on, t hehierarchical design procedure is presented, and then thedetails of the calculation a re discussed.

The flowsheet of the hierarchical design procedure isshown in Figure 1. The pr ocedur e vertically decomposesa plantwide control design problem into three subprob-lems, according to the priorities of the control objectives.

The three subproblems are (1) controll ing variablesrelated principally to safety issues, (2) controll ingvariables related to production ra te an d product quality,and (3) controlling component balances and unit opera-tions. The output of the current design st age includingcontrollers is the input to the next stage.

( 1) I n p u t . T he i nput of t he hi er ar chi cal desi gnprocedure includes the following information: (1) a statespace linear time-invariant process model as describedby eq 1; (2) process flowsheet and steady-state processdata for state variables, manipulated variables, andmeasurements; (3) operating ranges of the measure-ments and manipulated variables, which are typicallyused in scaling the model; (4) control objectives, whichare used to define controlled variables, involving the

specifications of production rate, product quality, oper-at ing mode chan ges, etc.; (5) process const ra ints, wh ichare also used to define contr olled var iables, involvingboth har d a nd soft constraint s in th e process operat ion(Har d constr aints, e.g. safety related issues, can not beviolated at any time. Soft constraints can be violatedover a short period of time. However, if the violation isnot corrected, operating performance suffers. For ex-ample, a valve might have a constraint on how fre-quent l y i t can m ove. ) and (6) pr ocess i nsi ght andengineering judgment.

(2) Stage 1. Sta ge 1 considers a ll variables associatedwith safe operation, e.g., liquid levels, variables withconstraints, and variables that have a very slow re-sponse so tha t th ey respond in a ma nner t ha t is similarto pure integrating variables. The major tasks of stage1 design involve scaling th e process model and identify-ing measurements for safety and other slow-respondingvar iables t ha t have t o be cont r ol led i n t he s t age 2design. Scaling is required, as th e elements in th e OSOFcontr oller should be dimensionless a nd have values ina r elat i vely s m all r ange t o be com par ed w it h oneanother. The other task in stage 1 is to select a set of man ipulated variables and a set of measur ements to beused in the stage 2 design. In our methodology, onlyman ipulated var iables without frequency constrain ts ontheir movement a re used in the sta ge 2 design, as thesafety-related controlled variables can require a fastresponse. Determining which measurements to use is

F i g u r e 1 . Outline of the hierarchical design procedure.



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comparatively more difficult, and in our approach, itconsists of three operations. First, a set of controlledvariables is identified on the basis of control objectivesand process constraints. For example, if the pressurein a g a s l oop h a s a h igh lim it , t h i s p r es su r e is acont rolled variable. Second, measur ement s ar e assignedto th ese cont rolled var iables. In some cases, a cont rolledvariable can be m easu red a t different locations, e.g., th egas-loop pressure can be measured at any unit in thegas loop. Therefore, it is necessary to select the bestlocation to measu re t he pr ocess variable using en gineer-ing judgment or some numerical tools. In other cases,there are high correlations among some of the measure-ments, and these correlations indicate that it is betternot control these variables at the same t ime. Third, anumerical method based on an eigenvalue analysis ofthe state space model is proposed to identify unstableand slow-responding measurements. The reason forusing the eigenvalue analysis is that the OSOF control-ler must stabilize the plant, a nd consequently, thisrequires that the positive, zero, and small eigenvaluesi n t he s t at e s pace m odel s houl d be det ect ed i n t hemeasurements. If the process is open-loop unstable, itis necessary to determine which m easurement s ar e best

to detect the instability. The following steps are usedin the eigenvalue analysis:(1) Calculate eigenvalues of the A matrix in the scaled

model. Pick out th e eigenvalues with positive, zero, andvery small negative real parts . Posit ive eigenvaluesindicate open-loop instability, a nd zero eigenvaluesindicate integrators. Eigenvalues with very small nega-t i ve r eal par t s i ndi cat e a ver y s luggi sh open-l oopresponse.

(2) If the process has eigenvalues with positive realparts, each unit operation is checked. Zero eigenvaluesare considered below. If a unit has eigenvalues withpositive r eal pa rts, t he instability is eliminat ed locallybefore a plantwide control structure is designed. Thefollowing approach can be used to suggest measure-

ments t ha t can be used to stabilize the system. Assumet hat A* corresponds t o a dynamic model in which onlyt he s t at es i n t he uni t change and t he ot her s t at es i nthe process are zero, i.e., they are at their steady-statevalues. A * is then diagonalized as A* ) V S V-1, whereS is a diagonal matrix of the eigenvalues. Then eq 1becomes equivalent to

where ) V -1x. A ss um e t ha t t he elem ent S jj h a s apos i t i ve r eal par t ; t hen j ) V -1(j,:)x i s t he l i nearcombination of the original stat es th at is unst able. Thefollowing steps are used to stabilize the unit:

(a) The origina l stat e, say xi, which corresponds to th elargest element in the jth row of V-1, contributes themost to the unstable mode.

(b) The original measurement, say yk, which corre-sponds to the largest element in the jth column ofCV,is most appropriate to detect the instability.

(c) The original m anipulat ed var iable, say up, wh ichcorresponds to the largest element in the jth row of V-1B , is most appropriate to control the instability.

By following th ese steps, in most cases, th e insta bilitycaused by positive eigenvalues can be eliminated by aproportional-only controller between yk a n d up. I t canbe noted that this same approach can be used to suggest

measurements and manipulated variables that can beused to control slow-responding modes.

(3) After the positive eigenvalues have been elimi-nated in each unit operation, determine whether anyremaining eigenvalues with posit ive real parts exist .The remaining process instability would be caused bythe interconn ection of th e un it operations in th e process.The eigenvalue analysis method can again be appliedto determine the best measurement using the updated

process model, wh ich would include contr ollers tha tstabilized the individual units.

(4) Identify t he integrators. The simplest way t oidentify them is to check the first-order process gainmatrix, G1, using the Arkun and Downs approach.27

(5) Identify slow responding modes u sing th e eigen-value analysis.

( 3 ) S t a g e 2 . T he goal of t he s t age 2 des i gn i s t ogenerate decentra lized contr ol structure candidates forthe variables tha t a re identified in sta ge 1. Typically aSISO loop is used to control a cri t ical variable in aprocess. As a result, in t he st age 2 design, a multiloopSISO control architecture is used to increase the reli-ability of the plantwide contr ol system. The outputs of

the approach include a set of feasible contr ol structurecandi dat es t hat ar e i m pl em ent ed i n t he s t at e s pacemodel u sing proportional-only contr ollers. These con-trollers a re incorporated into the model for use in laterstages. Transients can be easily calculated to evaluatecont rol structur es. The calculat ions carr ied out in sta ges2-4 are essentially the same. These calculations arediscussed below after a discussion of the design stages.

( 4 ) S t a g e 3 . T he goal of t he s t age 3 des i gn i s t ogenerate control stru cture candidates, which can beeither centr alized or decentra lized, for the product ra teand quality variables tha t a re identified from t he contr olobjectives. Ifn control structure candidates are gener-ated in the stage 1 design, the stage 3 design will be

executed n times, with one of these control structuresbeing implemented. In stage 3, the set points of loopsclosed in sta ge 2 can be used a s ma nipulated va riables.One important question in stage 3 design is how todetermine whether a centralized or a decentralizedcontrol system should be implemented from the pointof view of the pr ocess dyna mics a nd interactions. Th isproblem is simply attacked by using process simulationbased on the linearized model. The following heuristicsare proposed: (1) The tr ansients produced by imple-menting a proportional-only diagonal control (when theOSOF controller contains only diagonal terms) indicatethe performa nce of a totally decentr alized control stru c-ture. (2) The transients produced by implementing amultivariable contr oller (when t he OSOF controller is

a full ma trix) indicate th e performa nce of a multivari-able control structure. These two tra nsients can becompared to estimate the benefit of using a multivari-able contr ol architectur e.

( 5 ) S t a g e 4 . T he goal of t he s t age 4 des i gn i s t ogenerate control stru cture candidates, which can beeither centralized or decentralized, for maintainingcomponent balances an d controlling u nit operat ions.Basically, users can employ either heuristics or theoptimal cont rol-based d esign meth od to solve th e designproblem. One question that is par ticularly importan tfor the stage 4 design is how to determine t he var iablesto be used to control component inventories. Controlobjectives and process constraints can provide hints for

{ ) S + (V-1

B )u

y ) (CV)(9)



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unit operation control, but normally, they provide nohelp in identifying uncontrolled chemical components.

Genera lly, the in ventory of a componen t in a chemicalprocess has t o be controlled unless t he in ventory is self-regulating or made self-regulating by closing otherloops. A Downs drill table28 is used for checking com-ponent balances aft er a cont r ol s t r uct ur e has beendesigned. In the stage 4 design, a Downs drill table ismade for ea ch control structure candidat e given by th estage 3 design to identify any uncontrolled chemicalspecies. The measurements for these uncontrolled chemi-cal components should also be determined. After thenumber of control loops for the remaining uncontrolledchemical inventories is identified, the sa me n umber ofman ipulated var iables is consu med. Therefore, after th ecomponent balance loops are closed, the remainingdegrees of freedom can be used for unit operation controlor process optimization.

(6) O utput. At the en d of the st age 4 design, a set ofplantwide contr ol stru ctures is generat ed. They can bemultiloop SISO control (with loop pairings) or multi-variable control. Tra nsients based on the l inearizedm odel a nd t he O SOF cont r ol ler s ar e avail abl e forevaluating contr ol performance. Nonlinear simulation

is strongly recommended for testing control structurecandidat es when a nonlinear p rocess model is available.

c . D e t a i l s o f O S OF C a l c u l a t i o n s f o r S t a g e s 2 -4.The various steps in each design stage are as follows.

(1) Calculate an O SO F C ontroller. In th is step, anOSOF controller is designed for the given manipulatedvariables and m easur ements. The OSOF contr oller canbe a n onsquare system, an d its formu lation is given byeqs 1-3 and 5. U s er s ar e r equi r ed t o det er m i ne t hedesign parameters, which are the weighting matr icesto use in the objective function. The default values oft h e Q a n d R matrices are identity matrices, and thedefault value of gij is zero. Users also need to specifywhich numerical LQR design algorithm to u se ( the

defau lt met hod is Moerder and Calises met hod) andhow to generat e th e initial guess of th e OSOF controller(the default method is the random search method).

( 2 ) Ca l c u l a t e t h e S e n s i t i v i t y M a t ri x . The calcula-tion is performed exactly as described in section 5aabove. I f an O SO F cont r ol ler gener at ed i n s t ep 1contains m manipulated variables, then an additionalm O SO F cal cul at ions ar e r equi r ed t o gener at e t hesensitivity matrix. In each of these calculations, onlyone of the manipulated variables is not heavily penal-ized.

( 3 ) G e n e r a t e D e c e n t r a l i z e d C o n t r o l S t r u c t u r e s .In t his step, decentr alized contr ol stru ctures ar e gener-ated using th e OSOF contr oller, the sensitivity ma trix,and engineering judgment. The following heuristics areapplied in our approach: (1) Only pairings on elementsw i t h abs ol ut e val ues gr eat er t han 0. 2 i n t he O SO Fcontroller are considered. (2) Only pairings on theelements with values between 0.2 and 5 in the sensitiv-ity mat rix are considered. (3) The pairings a ccepted byheuristics 1 and 2 are screened by engineering judg-ment, and those that do not pass the screening are notconsidered in further stages. For example, one wouldnot contr ol a measu rement with a man ipulated variablethat is located physically far from the measurement.

( 4 ) F o r E a c h S t r u c t u r e , G e n e r a t e a F u l l O S O FC ontroller. When a user wan ts to evaluat e the contr olperformance of a multivariable controller, a full OSOFcont roller is designed for th e given model. The contr oller

can be viewed as a simplest multivariable controller,and its performance can be compared with the perfor-mance of diagonal OSOF controllers designed in step 5below.

( 5 ) F o r E a c h S t r u c t u r e , G e n e r a t e a D i a g o n a lO SO F C ontroller. When a multiloop SISO structureis preferred, a diagonal OSOF cont roller is designed forthe given m odel. To avoid using t he gijs in the calcula-tion, a n alternative method is developed to automati-cally tu ne t he p roport iona l-only controllers. The generalidea is to generate a diagonal initial OSOF controllerand keep i t diagonal when updating i t by solving thecoupled design equat ions. We ha ve developed a n algo-rithm based on this idea that gives the same result asus i ng t he gijs but r uns m uch fas t er . T he det ail edalgorithm is given in Appendix B.

( 6 ) S i m u l a t e U s i n g t h e S t a t e S p a c e M o d e l . AfterOSOF controllers are generated, t hey can be easilytested by simulating the l inearized model. StandardMATLAB routines (i.e., initia l an d lsim) can efficientlysimulate the closed-loop system, and tran sients forspecific disturbance rejection and/or set-point trackingcan be easily generated and compared. Judging fromt he t r ans ient s , us er s can det er m i ne w hich cont r ol

structures can be sent to th e next design stage.( 7 ) U p d a t e t h e S t a t e S p a c e M o d e l a n d t h e R e -

m a i n in g D e g re e s o f F r e e d o m . For each feasiblecontrol candidate, the closed-loop state space model isthe process model for the next design stage. The setpoints of any loops closed in an earlier stage can bemanipulated in a later sta ge. The hierarchical designprocedure and the optimal contr ol-based control struc-t ur e des i gn appr oach ar e t he key t echni ques of ourplant wide cont rol design met hodology. The n ext sectiondiscusses the application of the methodology to the vinylacetate process.

6 . A p p l i c a t i o n o f P l a n t w i d e C o n t r o l D e s i g nM e t h o d o l o g y t o t h e V i n y l A c e t a t e P r o c e s s

a . O v e r v i e w . The vinyl acetate (VA) monomer pro-cess was first presented by Luyben and Tyreus as a testproblem for process control technologies.29 A flow dia-gram of the process is shown in Figure 2. In the VApr ocess , t her e ar e 10 bas ic uni t oper at ions , w hichinclude a vaporizer, a catalytic plug-flow reactor, afeed-effluent heat exchanger (FEHE), a separator, agas compressor, an absorber, a carbon dioxide (CO 2)removal system, a gas removal system, a tank for theliquid r ecycle str eam, and an azeotropic distillationcolumn with a decanter. The manipulated variables andmeasurements for the VA process are listed in Tables 3and 4, respectively. There are seven chemical compo-nent s in th e VA process: Eth ylene (C2H 4), pure oxygen(O2), and acetic acid (HAc) are converted into the vinylacetate (VAc) product, and water (H 2O) and carbondioxide (CO2) are byproducts. An inert, ethane (C2H 6),enters with the fresh C2H 4 feed stream. The followingreactions take place

R eader s ar e r efer r ed t o s ect i on 2 i n r ef 29 f or adetailed process description, including the reaction rate

C2H 4 + CH 3COOH +1/2O2 f

CH 2dCHOCOCH 3 + H 2O (10)

C2H 4 + 3O 2 f 2C O2 + 2H 2O (11)



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expressions and the major aspects of each unit opera-tion. The base operation considered h ere, in which th epeak t emperat ure in the reactor is below 162 C, is th es am e as t hat dis cus s ed i n C hapt er 11 of r ef 28. Anonl inear dynam i c m odel for t he VA pr ocess w asdeveloped by Luyben a nd Tyreus in TMODS, which isa DuPont in-house process simulation software sys-tem.29 A heuristic-based plan twide contr ol design met h-odology30 was applied to the VA process, and a decen-tralized plantwide control system was given in ref 28.The contr ol system gives good contr ol performa nce indynamic simulation using t he n onlinear process model.

Because TMODS is n ot accessible for public use, a first-principles dynam ic model has been developed an d ma deavailable a nd t he code for the m odel can be downloadedfrom the Internet.31 In total , the model includes 246states, 26 manipulated variables, and 43 measurements.This n ew VA model was originally writt en in MATLABand t hen t r ans lat ed i nt o t he C l anguage t o s peed upexecution. The compiled version of the nonlinear VAmodel run s at approximat ely 80 times real time in bothWindows and UNIX environments.

At present, only one published decentralized plant-wide contr ol system for t he VA process is a vailable.28

Using t he optimal contr ol-based plantwide contr ol de-sign methodology, other feasible control structures canbe developed. The sections below discuss the applicationof the methodology to the VA process.

b . D e v e l o p m e n t o f a L i n e a r i z e d M o d e l . A MIMOLTI sta te spa ce model of the VA process is required, a ndt he s t at e s pace m odel i s obt ained by num er ical lycalculat ing t he first -order Taylor expan sion coefficientsof the nonlinear first-principles model. It is importa ntthat the l inearized m odel {A , B , C, D} be obtained a t

an oper at i ng poi nt t hat s t ar t s exact l y at t he s t eadystat e. Otherwise, some err ors will be introduced into thestate space model. A MATLAB nonlinear equationsolver, fsolve(), is used to force all state derivatives tozero by manipulating the steady-state values of thestat es. For the VA process, the lar gest stat e derivativehas a magnitude of 3 10 -8 at steady state, which isconsidered sufficiently a ccura te. After the stat e spacemodel is determined, the process gain matrix, includingthe derivative of any integrating variables, is calculatedusing Arkun and Downs method.27

c . Co n t r o l S t r u c t u r e D e s i g n f o r t h e V A P r o c e s s .The operating mode studied is the same as that pre-sented in ref 29. According to the discussion in the

F i g u r e 2 . VAC process flow sheet.

T a b l e 3 . S t e a d y - S t a t e V a l u e s o f Ma n i p u l a t e d V a r i a b l e s

MV descr ipt ion st ea dy st a t e r a n ge u nit s

1 fr esh O2 feed 0.523 43 0-2.268 k mol/m in2 fr esh C2H 4 feed 0.835 22 0-7.56 k mol/m in3 fr esh H Ac feed 0.790 03 0-4.536 k mol/m in4 va porizer st ea m du t y 21 877 0-1 433 400 kcal/min5 va porizer va por exit 18.728 0-50 km ol/m in6 va por izer h ea ter du ty 9008.54 0-1 5 0 00 k ca l/m in7 r ea ctor sh ell t em p 135.02 110-150 C8 sepa ra tor liqu id exit 2.7544 0-4.536 k mol/m in9 s ep ar at or pr eh ea ter t em p 36.001 0-80 C10 s epa ra tor va por exit 16.1026 0-30 km ol/m in11 com pr es sor h ea ter d ut y 27 192 0-5 0 0 00 k ca l/m in12 a bsorber liqu id exit 1.2137 0-4.536 k mol/m in1 3 a b sor be r ci rcu la t ion fl ow 1 5. 11 98 0-50 km ol/m in14 cir cu la tion cooler d ut y 10 730 0-3 0 0 00 k ca l/m in15 absorber scr u b flow 0.756 0-7.560 k mol/m in16 scr u b cooler du t y 2018.43 0-5000 k ca l/m in17 CO2 r emova l inlet 6.5531 0-22.68 k mol/m in18 pu r ge 0.003 157 0-0.022 68 kmol/min19 F E H E bypass r a t io 0.313 03 0-120 colu mn r eflu x 4.9849 0-7.56 k mol/m in21 colu mn r eboiler du ty 67 179 0-1 00 0 00 k ca l /m i n22 colu mn con den ser d ut y 60 367 0-1 50 0 00 k ca l /m i n23 colu mn or ga nic exit 0.8290 0-2.4 km ol/m in24 colu mn a queou s exit 0.8361 0-2.4 km ol/m in25 colu mn bot tom exit 2.1584 0-4.536 k mol/m in26 va por izer liqu id in let 2.1924 0-4.536 k mol/m in



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Chapter 11 of ref 28, th e eth ylene (C2H 4) and oxygen(O2) feed streams come from gas headers, and the aceticacid (HAc) feed stream is drawn from a supply tank.No frequency constraint s are specified for flow-rat echanges in t he feed strea ms. The optimal cont rol designmethodology generates decentralized control structuresfor t he cont rolled varia bles in t he st age 2 design. Eitherdecentra lized contr ollers or multivariable contr ollerscan be implement ed in the st age 3 and 4 designs. Here,decentralized control structures are generated in thestage 3 and 4 designs so that a comparison with thearchitecture in Chapter 11 in ref 28 can be made. Noassessment of the benefits of multivariable control is

made here.S t a g e 1 : P r e p a r a t io n . The major task in stage 1 is

to properly scale the process model and identify themeasur ements for the safety a nd slow-responding var i-ables. The measurements are determined according tothe process gain matrix, an eigenvalue analysis, andengineering judgment:

(1) Scale t he stat e spa ce model. The scale factors a reas fol low s: St at es ar e s cal ed by t heir s t eady-s t at evalues, and manipulated variables are scaled by theran ges of allowable movement . For the measu rement s,pressures, temperatures, a nd levels are scaled by th eranges of allowable movement, which are 10 psia, theminimum value of 40 C or the steady-state tempera-

ture, and 50%, respectively. Molar compositions andmass flow rat es are scaled by their steady-state values.

(2) In each unit operation, identify a nd eliminateprocess instabilities associated with positive eigenval-ues. Each unit in the process, as well as the overallprocess, should not have posit ive eigenvalues in i tsmodel. The VA process model does not have any positiveeigenvalues.

(3) Identify integra ting var iables. The simplest way

to identify integrating variables is to check the first-order process gain matrix, G1,27 calculated from thescaled model. This approach is based on singular valuedecomposition, and it involves determining the numberof zero singular values. There are seven singular valuesthat have values less than 1.0 10 -8, indicating t hatthere are seven integrators (0 eigenvalues) in the VAmodel. A column vector is generated by calculating themaximum absolute value of the elements in each rowofG1, and th e measurements with large elements in thisvector are the integrators. In the VA model, the sevenlargest nonzero elements in this vector are 11.72, 5.60,4.93, 2.47, 0.681, 0.555, and 0.312. It can be noted thatthe next lar gest element in the vector is 0.0416, which

is about 8 times sma ller tha n th e seventh element. Themeasurements corresponding to the seven largest ele-m ent s ar e t he decant er or ganic phas e, r eboi ler , a b-sorber, vaporizer, separator, decanter aqueous phase,and HAc tank levels, respectively. These measurementsare expected to indicate integrators because th ey ar emeasu rement s of liquid levels.

(4) Calculate eigenvalues of the scaled model. The 246eigenvalues are sorted by their real parts , and thosewith posit ive, zero, or small negative real pa rts areanalyzed in the stage 1 design. The nine eigenvalueswith the smallest negative real parts are -5.4304 10 -5, -2.3987 10 -5, -6.7229 10 -8, -5.517 10 -10,-4.5862 10 -11, -8.6798 10 -17, -9.8002 10 -20,3.772 10 -12 ,9 and 2.1411 10 -10. Zero eigenvalues,which indicate integrators, a re considered a s t hose forw hi ch t he abs ol ut e val ues of t he r eal par t s of t hei reigenvalues are less than 1 10 -7. In the VA model,seven integrators are identified, in a greement with t heana lysis in step 2 above. Positive eigenvalues indicateopen-loop instability, and in the VA model, there areno posit ive eigenvalues with real parts greater than1 10 -7. There ar e two very small negative eigenvalues(-5.4304 10 -5 and -2.3987 10 -5), and they indicatevery slow process dynamics. The next smallest eigen-val ues ar e about 16 t i m es gr eat er t han t he l ar ger of these two eigenvalues. Because the two small eigenval-ues act as approximate integrators, i t is necessary tocontr ol these two slow modes. It should be n oted tha t a

decision as to which modes need to be controlled has tobe made. If satisfactory dynamic performan ce is notachieved, then one can go back and control additionalmodes. Using the eigenvalue approach discussed above,the measurements t hat have large elements in the CVmatrix, indicating that they contribute to these eigen-modes are the seven levels, the tr ay 5 temperat ure, andthe bottom composition of VAc. Because the tempera -ture measurement is more reliable than the analyzermeasurement, it is selected to control the modes associ-a t e d w i t h t h e -5.4304 10 -5 a n d -2.3987 10 -5

eigenvalues. Controlling this temperature results inboth of these slow modes being controlled.

(5) Identify other safety variables (hard process

T a b l e 4. M e a s u r e m e n t s a t S t e a d y S t a t e

m ea su r em en t descr ipt ion va lu e u n it s

1 va por izer pr essu r e 128 psia2 va por izer level 0.73 va por izer t em p 119.145 C4 h ea t er exit t em p 150 C5 r ea ct or exit t em p 159.17 C6 r ea ct or exit flow r a t e 18.857 km ol/m in7 F E HE cold exit t em p 97.1 C8 F E HE h ot exit t em p 134 C

9 sepa r at or level 0.510 sepa r at or t em p 40 C11 com pr essor exit t em p 80 C12 a bsor ber pr essu r e 128 psia13 a bsor ber level 0.514 cir cu lat ion cooler exit t em p 25 C15 scr u b cooler exit t em p 25 C16 ga s r ecycle flow r at e 16.5359 k mol/m in17 or ga nic p rod uct flow r a te 0.829 k mol/m in18 deca nt er level (or ga nic) 0.519 d eca nt er level (a qu eou s) 0.520 deca n t er t em p 45.845 C21 colu m n bot t om level 0.522 t r a y 5 t em p 110 C23 H Ac t a n k level 0.52 4 or g a n ic p r od u ct com p os it i on 0 .9 49 7 86 m ol %25 (VAc, H 2O, H Ac) 0.049 862 mol %26 0.000 352 m ol %

2 7 col um n b ot t om com p os it i on 0 .0 00 0 10 m ol %28 (VAc, H 2O, H Ac) 0.093 440 mol %29 0.906 550 m ol %3 0 ga s r ecycle com pos it ion 0 .0 55 6 64 m ol %31 (O2, CO 2, C2H 4, C2H 6, 0.007 304 m ol %32 VAc, H 2O, HAc) 0.681 208 m ol %33 0.249 191 m ol %34 0.001 597 m ol %35 0.000 894 m ol %36 0.004 142 m ol %3 7 r ea ct or fe ed com pos it ion 0 .0 75 0 00 m ol %38 (O2, CO 2, C2H 4, C2H 6, 0.006 273 m ol %39 VAc, H 2O, HAc) 0.585 110 m ol %40 0.214 038 m ol %41 0.001 373 m ol %42 0.008 558 m ol %43 0.109 648 m ol %



11/19

constraints). Luyben and Tyreus gave five safety con-straints in ref 29:

(a) The oxygen composition must not exceed 8 mol %an ywhere in t he gas loop. Therefore, the O 2 compositionin the gas loop is a controlled variable. Because acontinuous and reliable analyzer is installed at the inletof the reactor, this analyzer is used for the O 2 measure-ment.

( b) T he pr es s ur e i n t he gas l oop m us t not exceed140 psia. Therefore, the pressure in the gas loop is acontrolled variable, and it is measured in two locations,the a bsorber pressur e and t he vaporizer pressure. Bothmeasurements are controlled.

(c) The peak reactor temperature must not exceed

200 C. Therefore, the peak reactor temperature is acontrolled variable. Because no temperature sensor isinstalled inside the reactor, the reactor outlet temper-ature, which is close to th e peak temperature, is u sedas t h e m eas ur em ent .

(d) The rea ctor inlet tem perat ur e must exceed 130 C.Because no temperature sensor is installed at the inletof t he r eact or , t he out l et t em per at ur e of t he heat erinstalled on the vaporizer outlet stream is used as themeasur ement for contr ol.

(e) The hot-side exit temperature from the FEHE,which is measured, must be greater than 130 C, andthis temperature is controlled.

(6) Summ arize t he identified contr olled variables tobe used in the stage 2 design. In the base operation,

th ere is one choice for t he contr olled varia bles, and it islisted in Table 5.

S t a g e 2 : D e c e n t r a l iz e d C o n t r o l S t ru c t u r e f o rS a f e t y V a r i a b l e s . For the controlled variables deter-m i ned i n t he s t age 1 desi gn, decent r ali zed cont r ols t r uct ur es ar e gener at ed u s ing al l 26 m ani pulat edvariables. First, an OSOF controller is calculated bysolving eqs 1-3 and 5, and then the sensit ivity matrixis generated. Control structure candidates are deter-mined by analyzing the OSOF controller , using thesensitivity mat rix, and a pplying engineering judgment.Fina lly, proport iona l-only controllers ar e au toma ticallyt uned for each cont r ol s t r uct ur e candi dat e, an d t heclosed-loop systems with good performance in trackings t ep s et -poi nt changes ar e r et ained for t he s t age 3design.

The options used in the OSOF controller are listed inT abl e 6. I n t he s ensi t ivit y m at r i x cal cul at ion, t heweights used in the R matrix are either 1 or 100. TheOSOF controller is a large mat rix, 26 1 4 , a n d aportion of it is given in Table 7 with elements whoseabsolute value is greater than 0.2 shown in boldface.There are several weak manipulated variables presentin t he overall K matrix, the absorber gas inlet coolerdut y (MV11), th e ab sorber circulat ion flow rat e (MV 13),the absorber circulation cooler duty (MV 14), and theabsorber scrub cooler duty (MV16). The sensit ivitymatrix corresponding to the part ofK given in Table 7is shown in Table 8 with acceptable values between 0.2

and 5 shown in boldface. If the elements in the samepositions of Tables 7 and 8 are both in boldface, thecorresponding loop pairing is accepted from the dynamiccontrol point of view. These loop pairings are listed inTable 9, from which hundreds of different pairingstru ctures can be gener ated. In this case, the followingengineering judgment is used to reduce the number.First, parings in which the measured and manipulatedvariables are ph ysically far from one another are ruledout . Second, a pair i ng i n w hich t he m eas ur ed andman ipulated variables involve different pha ses, i.e.,l iquid and gas pha ses, are ruled out. Finally, judgingfrom th e pur pose of the absorber scrub st ream , it is notrecommended th at the absorber scrub stream be usedto contr ol a liquid level unless t here is no other choiceavailable. Nine feasible 14-by-14 multiloop structuresresult from applying the OSOF contr oller ru les, sensi-t ivity rules, and engineering judgment, and they arelisted in Table 10. In each column of Table 10, theindexes of the ma nipulated variables are listed to showthe pairing structures.

From th e differences among the n ine contr ol stru cturecandidat es, i t can be obser ved t hat (1) eit her t hevaporizer heat duty or t he vaporizer liquid inlet str eamcan be used control the vaporizer level; (2) either thefresh HAc stream or the vaporizer liquid inlet streamcan be used control the HAc tank level; and (3) eithert h e C2H 4 feed stream, the separator cooling jacket

T a b l e 5 . C o n t r o l l e d Va r i a b l e s f o r S t a g e 2

n o. m ea su r em en t n o. m ea su r em en t

1 va por izer level 8 t r a y 5 tem p2 sepa r a tor level 9 % O2 in reactor inlet stream3 a bsor ber level 10 va por izer pr essu re4 or ga n ic p ha s e le ve l 1 1 a bs or be r p re ss ur e5 a qu eou s p ha s e le ve l 1 2 h ea t er e xit t e mp er a t ur e6 col um n b as e le ve l 1 3 r ea ct or e xi t t em pe ra t u re7 H Ac t an k level 14 F EH E exit t em per at ur e

T a b l e 6 . O p t i o n s i n t h e O S O F C o n t r o l l e r Ca l c u l a t i o n

option va lu e

m odel sca leda lgor it hm Moer de r a n d Ca lis es m et h od (ba sic)algorithm for initia l

guess ofKrandom guess

Q identity matrixR identity matrixgij 0

X identity matrix

te r m in a tion con d it ion tr a c e(

KT

*

K)

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