TAI - 8:30APPLICATION OF SINGULAR VALUE DECOMPOSiON TO THE DESiGN,

ANALYSIS, AND CONTROL OF INDUSTRIAL PROCESSES

Dr. Charlie MooreHalliburton Professor of Chemical Engineering andThe Measurement and Control Engineering Center

College of EngineeringUniversity of TennesseeKnoxville, TN 37996

ABSTRACT SINGULAR VALUE DECOMPOSITION

Singular Value Decomposition (SVD) is proving tobe a very useful tool in modern linear system theory.It is also finding an important role in analysis anddesign of control systems for those real industrialprocesses which tend to not be so well behaved asassumed by the theory. This paper will describe howthe concepts of SVD analysis can be used to helpdetermine realistic answers to the many practicalquestions that must be addressed by the controlengineer if he or she is going to tackle realmultivariable control problems. Such questions as:What is the effect of process design alternatives andoperating conditions on the control problem? Thepaper will also show how SVD can be used to designsimple but effective multivariable control systems.

INTRODUCTO

The analysis and design of multivariable controlsystems has received considerable attention in thetheoretical as well as applied literature in recentyears. It is an interesting problem for thetheoreticians and, with the advent of integrated plantwide computer based control systems, it is anincreasingly tempting problem for practitioners totackle. This paper will present an interestingtheoretical tool which has proven to be very helpfulfor the real world design problem. The tool is basedon the numerical concept of Singular ValueDecomposition (SVD) and provides quantitativeinformation about sensor placement, physicalcontrollability, controller pairing and also can be useddirectly as a decoupling control strategy. It can beapplied to an existing process unit or it can be appliedduring the design phase to help screen out inadvertentmultivariable control problems before they becomefixed in concrete and steel.

The concepts and tools presented in this paperare based not only on public domain research done bythis author and others but are also based on a numberof consulting experiences in which the author has hadthe opportunity to apply (or see applied) thesetechniques to a number of full scale industrialsystems. Unfortunately, these industrial experiencescannot be directly documented for publication;however, they have all been quite positive indemonstrating that the techniques do in fact work inthe real world.

The basis of the procedure is Singular ValueDecomposition which is a numerical algorithmdeveloped to minimize computational errors involvinglarge matrix operations. The singular valuedecomposition of a matrix K results in threecomponent matrices as follows:

K = UEVT

Where:

K is an n x m matrixU is an n x n orthonormal matrix called the

"left singular vector"V is an m x m orthonormal matrix called the

"tright singular vector"E is an n x m diagonal matrix of scalers

called the "singular values" and areorganized such that 11>o2>a3---c m>O

Both U and V are ideally conditioned matriceswhere each column vector has unit length and isorthogonal to all other column vectors in the matrixset. In terms of matrix operations both U and Vrepresent a simple coordinate rotation.

SVD is designed to determine the rank and thecondition of a matrix and to map geometrically thestrengths and weaknesses of a set of equations. Thecomputation of the SVD of a matrix has been studiedby numerical analysts for some time and can beperformed with great accuracy. It is discussed inmany textbooks (7,13,20,25) and software is readilyavailable through a number of standard numericalpackages (29,30,31,32).

SVD - A Physical Interpretation

The attractive aspect of SVD in terms of processcontrol is that when applied to a matrix whichdescribes the steady-state characteristics of amultivariable process the singular vectors and thesingular values all have a very strong physicalinterpretation. This physical interpretation canprovide a clear insight into potential control problemsand the physical nature of those problems. It can alsoprovide a physically meaningful structure in which todesign multivariable control strategies. The followingis a general interpretation of the elements in asingular value analysis. The physical concepts will bedeveloped in more detail as specific applications arediscussed.

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K The Steady-State Gain Matrix consists of thephysically scaled steady-state sensitivity of eachprocess sensor to changes in each of the manipulatedvariable. Each element of K is determinedexperimentally or by a detailed simulation and isdefined as follows: kij = 6[sensor i]/6[manipulator j].It is very important that the elements of K be scaledto be representative of what the control system willactually see. A good basis for physical-sealing is interms of percent changes such that the units of kij beapproximately:

% of sensor span% of range of manipulator

U = UL*U2:.....U:n The Left Singular Vectors providethe most appropriate coordinate system for viewingthe process sensors. This coordinate system is suchthat the first singular vector (Ul) indicates the easiestdirection in which the system can be changed. Thesecond singular vector (U2) is the next easiestdirection. The third is the next easiest, and so on....

V = VbV2:.'.Yinm The Right Singular Vectors providethe most appropriate coordinate system for viewingthe manipulated variables. This coordinate system issuch that the first singular vector (VI) indicates thecombination of control actions which has the mosteffect on the system. The second singular vector (V2)is the combination which has the next strongesteffect, and so on...

E = dia(a,cv,2s...,an The Singular Values provide theideal decoupled gain of the open loop process. Theratio of the largest singular value to the smallest0l/Gm is the Condition Number of the gain matrixand is a measure of the difficulty of the decoupledmultivariable control problem. Large conditionnumbers indicate that the degrees-of-freedom aresuch that the number of control objectives need to bereduced.

INDEX OF CONTROLLABILITY

One important aspect of the physical signif-cance of the Singular value analysis can be seen in theCondition Number. The condition number is the ratioof the largest to the smallest singular value and isused in numerical computation to gauge the"condition" of a set of equations.

Condition Number = cma/min

The larger the condition number, the moredifficult it is to compute a matrix inverse which isfree of computational errors. In terms of the processcontrol problem, a large condition number indicatesthat it will be impractical, if not impossible, to satisfythe entire set of control objectives (regardless of thecontrol strategy used). In the physical sense, thecondition number represents the ratio of the maximumand minimum open-loop, decoupled gains of thesystem. A large condition number indicates that therelative sensitivity of the system in one multivariabledirection is very weak.

Consider for example a counter-current tunneldryer. The dryer has an external preheater on theinlet air and three internal heaters distributed in the

air space inside the dryer. All four heaters have aseparate steam control valve. In terms of the sensors,there are eight ports from which to measure themoisture content of the solid and/or the temperatureof the air.

The gain matrix which relates the response ofmoisture content at each of the ports to each heater isas follows:

0.004710.017110.03944

K = 0.057950.065090.060210.048340.03452

0.005220.019030.043150.063150.070910.063340. 046760.02976

0.007130. 026060.050720.070740. 060380. 048020. 031570.01858

0. 0 10660. 023520.027240. 027200.022540. 015990. 010060. 00528

A SVD analysis yields the following Singular Values:

a1 = 0.23133E-0002 = 0.32938E-0103 = 0.11256E-0104 c 0.38279E-02

From these singular values a condition numbercan be calculated for a full 4 x 4 multivariable controlstrategy and can also be calculated for the simplercases.

Dimension4 x 43 x 32 x 2

Condition Number60.4320.557.02

Note, these condition numbers show, on a relativebasis, how much more difficult it will be to controlthree and four moisture locations than it would be tocontrol only two.

It is important to maintain a physicalunderstanding of singular values and the resultingcondition numnbers. The singular values are a directmeasure of the decoupled pins of the process. If themultivariable interactions were removed by an ideal"steady-state" decoupler, the singular values would bethe open loop gains of the noninteracting system.Note in this example the gain of the strongestnoninteracting loop is 0.23 and the pin of the weakestloop is 0.0038. A weak steady-state gain is a veryserious practical problem even in the case of singleloop control. It requires very large controller gainsand results in excessively large controler actions.The typical presence of constraints in the manipulatedvariable and noise in the sensor variable make itdifficult even for the single-input single-outputproblem to be solved with feedback control. In thecontext of multivariable control, the additionalcomplications presented by hidden loops, controllerinteractions, and unusual dynamics make the problemof weak steady-state gains even more severe.

In terms of the overall controllability of aprocess there are considerations other than justcondition number. The magnitude of the largestsingular value is also important. Very small singularvalues could well indicate that in spite of a goodcondition number the system is simply not sensitiveenough to control. Very large singular values are also

A_A4

an indication of a practical control problem. In singleloop control large process gain requires very smallcontroller gains. This results in small controlleroutputs which can become lost in the resolution of themanipulator. The symptomatic behavior of suchpoorly scaled systems are cyclic responses which neversettle down to a reasonable steady-state. Such loopperformance can be troublesome in a single loopcontrol but it is a disaster in a multivariable system.

and the sidedraw rate can be manipulated to controltemperatures at selected trays in the column.

The 3x3 SVD analysis for sensors at 41, 34, and28 is as follows:

-29.58 -26.87 1.31K = -32.11 -12.27 19.77

-34.98 -10.83 0.62

CONTROLLER PAIRING

Another simple analysls which can be understoodfrom this physical interpretation is controUler pairing.The pairing which will yield the least open loopmultivariable interaction is one in which the sensorassociated with the largest vector component ofcolumn Ul is paired with the manipulated variableassociated with the largest vector component ofcolumn VI. The largest vector component of U2 is alsopaired with the largest vector component of V2.

To illustrate this procedure consider the classicsimple example of a continuous mixing system inwhich the flow rates of the hot and cold water (Fl andF2) are manipulated to control the total flow andmixed temperature (Tm and Fm). The linearizedsteady state behavior of the system about someoperating conditions can be described by the followingequation:

,&Tm.= K AFIAFm AP2

where:

ATm m change in the mixed temperature frombase condition.

AFm = change in the total flow from the basecondition.

AFI = change in the hot water flow from the basecondition.

AF2 = change in the cold water flow from thebase condition.

For a base case of FP=10 gpm, F2=20 gpm, Th-1000f,To=650f, the gain matrix is as follows:

K = 0.77781. 0000

-0.38891.0000

which decomposes to:

U = 0.27580.9612

V = 0.80910.5877

z = 1.45310

-0.96120.2758

-0.58770.8091

00. 8029

For this base case the U and V matrices indicatethat Pm should be controlled with Fl and Tm should becontrolled with F2. For slightly more complexexample consider a 42 stage Xylene-Benzene-Toluenedistillation column (17). The column represents a 3x3system in which reflux ratio, steam to the reboiler,

-0.595U = -0.583

-0.553

0.869V = 0.454

-0.197

64.10

-0.627 -0.5020.767 -0.669

-0. 134 0.823

-0.084 -0.4880.527 0. 7180.846 -0.495

16.86

(T28)(T34)(T41)

(R)(Q)(S)

10.94

Matching the largest component of each of the lefthand vectors with the largest component of each ofthe right hand vectors yields the following controllerpairing:

R T28S T34Q +T41

It should be noted that in most cases this pairingprocedure gives results consistent with Bristol'sRelative Gain Array (2) and other more traditionalinteraction analyses. From the author's experience theonly cases in which SVD yields controller pairingdifferent from an RGA analysis are for systems whichhave a high condition number. High conditions indicatethat successful multivariable control is highly unlikelyregardless of controller pairing (or decoupling or anyother multivariable strategies.) It should beemphasized that controller pairing should not be donewithout some reference to the condition of themultivariable system.

SENSOR LOCATION

For many multivariable processes there is adecision to make about sensor location which can becritical to the success or failure of a multivariablecontrol system. SVD provides a systematic procedurefor placing sensors in such a way as to both maximizesensor sensitivity and to minimize sensor interaction.

A distillation column is a good example of aprocess with many possible sensor locations. In theEthanol-Water column shown in Figure 1 there are 50trays on which the temperature (or composition) couldbe measured. The basic question is which twolocations are best for the multivariable control of thecolumn. To demonstrate how SVD can be applied tothis problem consider the column operating with a DQscheme (distillate rate and steam rate are themanipulated variables). At the normal operatingconditions the steady state gain matrix is shown inFigure 2. Note that, with no specification on thelocation of the control sensors, the steady-st.te gainmatrix is a 50x2 array of constants which dese tbes

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the temperature sensitivity on each tray to each ofthe two manipulated variables. The decomposition ofthe gain matrix (shown in Figure 3) results in a 50x2 Umatrix, a 2x2 V matrix and it has 2 singular values.

In terms of the problem of selecting sensorlocation, consider again the properties of the leftsingular vectors (the U matrix) and the physicalsignificance for the case where the process has moresensors than manipulated variables. Ul and U2 definevectors in sensor space which are orthogonal. Theprojection of a particular temperature profile ontoeach of these vectors defines two "weighted average"steady state temperatures which are both sensitiveand noninteracting.

Ta El UTb

or

TIT2T3

OVERAALL VS SPECIFIC SYD ANALYSIS

It should be noted that in the analysis ofprocesses which have a choice of sensor locations thatthere are at least two separate SVD analysis ofsignificance. One is an Overall SVD Analyss based ona complete system with all possible sensor locationsincluded. The other is of the Speeific SVD Analysiswhich is based only on those sensors which will bedirectly used in the control strategy. (There are otherpossible reduced systems which might need to beanalyzed, for example in distillation control it is notuncommon to base a control strategy on thetemperature difference between two trays rather thana single temperature).

The ethanol-water column provides a goodexample of the difference between an overall analysisand a specific analysis. In the table below the fullcondition number and the 2x2 condition number arelisted for a number of different first level controlschemes.Tn

Ta = U11*T1 + U12*T2 + U13*T3 + .........+ Uln*Tn

Tb = U21*T1 + U22*T2 + U23*T3 + .....+ U2n*Tn

If the column were instrumented such that allthese temperatures were measured the abovecalculation could be made and actually used as twosensitive but independent temperatures on which tobase the column control. This will be discussed as apossibility later in the paper, but for now consider the

utility of the concept in selecting the two good sensorslocations. One obvious selection procedure would beto select the sensors which correspond to the mostsensitive term in Ul and the most sensitive term in U2(the temperatures with the highest weighting factorsin the calculation of the independent averages). Thiswould yield two sensitive temperatures which, if theyrepresent the full vector with any accuracy, will befull of information and will have a low level ofinteraction.

Experience has shown that this procedurenormally works well however, there are cases wheresensor sensitivity shocjld be sacrificed for the sake ofreduced sensor interaction. The trade off betweensensitivity and interaction is easier to see graphicallyby plotting the coefficients of Ul and U2 verses trayposition. Figure 4 shows the Ul and U2 plots for theethanol-water example. Note that trays 8 and 17 arethe most sensitive locations; however, selecting trays7 and 18 would perhaps give less interaction (themagnitude of the other vector component would besmaller).

A less confusing index on which can be applied to2x2 systems to make these decisions is a plot ofABS(Ul) - ABS(U2). This combined index shows in asingle plot both sensitivity and interaction. Figure 5shows the combined index for this column. Note it ismuch clearer that trays 7 and 17 are a good choice forsensor locations.

First LevelSchemeDRQRDQLQLB

Condition Numbers50x2 2x2

60.8 62.6112.3 112.641.9 42.2

369.2 324.7691.9 687.9

As a rule, the overall analysis and specificanalysis may not be close as indicated in this example.The specific analysis can be considerably worse thanthe overall or it can actually be better behaved. Fromthe point-of-view of a detailed control system designthe most meaningful is the specific analysis whichfocuses on the sensors and valves which the controlsystem will use.

SELECTING PROPER MANIPULATED VARILABLES

The table above describing the condition of theethanol-water column for a number of possible firstlevel control strategies illustrates yet another analysisaspect of the SVD tool. Distillation columns arerepresentative of many processes in which thedimensions of the multivariable problem can (andshould) be reduced by a formal recognition that part ofthe control system must be devoted to the maintainingproper material and/or energy balances. Typically ona distillation column there are four control handles:Distillate flow rate, D; Reflux flow rate, L; Bottomsflow rate, B; and the heat rate into the reboiler, Q.However, material balance usually dictates two ofthese handles must be used to control the level in theaccumulator tank and the level in the base of thecolumn. That leaves only two variables which can bemanipulated by any multivariable control scheme.(Note that R in the above example represents thereflux ratio, L/D. This, and other meaningful ratios,can be used to replace the direct manipulation of thefirst level flows). Such physical reasoning simplifiesthe control problem considerably, but it does leave thecontrol engineer with a decision to be made about the"best" of the possible first level schemes.

646

The overall condition number is a good basis forsorting through the first level alternatives. Note inthis example, that DQ and DR are by the best firstlevel schemes. This steady-state information isinvaluable in terms of reducing the scope of thefurther dynamic and experimental studies.

IMPACT OF SYD ON BASIC PROCES DESIGN

The SVD analysis can of course be applied to anexisting process as well as to a process which is still

on the drawing boards. There are some majoradvantages to applying the analysis very early beforethe final design is complete. Not only does it providethe control engineer with information which is neededfor that aspect of the design, but it also provides away to factor the question of controllability into theevaluation of various design alternatives.

For example consider the overall analysis of areal (but unnamed) column which was designed "foreconomic reasons" with an internal condensor.

Control SchemeBLBRDQQRLQ

Cond. *1860.1413.199.14.72.24

It is obvious looking at these numbers why thecolumn design was difficult to control. With aninternal condensor the only first level scheme possiblewas DQ. While this was not the worst scheme it isclear that an external condensor would have made itpossible to implement an LQ scheme which would havebeen very wel conditioned indeed.

The same column can be used to make anothervery important point about the importance of an SVDanalysis before the column is built. This column wasdesigned and was operating with a liquid feed. TheSVD analysis indicated the following which would havecertainly been interesting results to have before thedesign was finalized.

SchemeLQDQ

Condition NumberLiquid Feed Vapor Feed

2.24 2.54199. 0 2.96

In the DQ case (which is the control scheme used) thecondition of the feed effects in a major way thecontrollability of the column. With a liquid feed, onlyone product can be controlled. With a vapor feed bothproducts could be controlled.

IMPACT OF SVD ON PROCESS OPERATION

SVD analysis can also be used to factor in thequestion of controllability in the establishment of thetarget operating conditions for some multivariableprocess unit. Consider, for example, part of a studymade on an ethanol-water column shown in Figures 6-11. The effect of feed temperature, sidedraw location,sideraw rate, reflux ratio, and distillate rate areshown on the overall condition number. Note that inevery case there are definitely some preferred ares of

operation and some area which should clearly, fromthe point-of-view of "robust" multivariable control, beavoided.

SVD AS A STRUCTURE FOR MULTIVARIABLE CONTROL

So far this paper has discussed the use of SVD inthe analysis of processes which have more than onevalve and more than one sensor to determine controlproblems and to suggest simple ways to reduce theproblem. It is also a very useful tool in the design ofthe multivariable control strategies for those units.At the simplest level of application the U and Vmatrices can be applied as a steady-state decoupler.Figure 7 shows a simple form of a SVD based controlsystem. This form uses the UT and the V matrices tostructure a "decoupled space" for the single loopcontrollers to operate without steady-stateinteraction. It should be noted that this form isdifferent than a conventional steady-state decouplerwhich consist of only one matrix operation, K-1, whichis typically between the controllers and the valves. Itdoes have two separate computations instead of onebut it offers the advantage that U and V are both wellbehaved. Computing the matrix inverse of K quitefrequently results in excessively large control actionswhich are beyond the physical possibilities of thevalves.

SVD can also be used directly as a "profile"controller in which there are more sensors thanmanipulated variables. For example, in some highlynonideal distillation columns (ie, azeotropic andextractive columns) there may be some distinctoperational advantage to holding the entire columnprofile close to some nominal value. Thenonlinearities may be such that holding a single traytemperature to its setpoint may not be sufficient tokeep the entire column under control. Some suchcolumns even appear to have multiple steady-states(5). The SVD profile controller is shown in Figure 8and is identical in form to the square system shown inFigure 7. The only difference is that the UT matrix isno longer square. The form is very simple yet it canbe shown to yield at steady-state a least-square fitbetween the actual temperature profile and thedesired temperature profile.

It is also possible to use the concept of SVD todesign multivariable control system which attempt todynamically decouple the process interactions over arange of frequencies other than steady-state.Theoretically, this is a very interesting problem;however, it would not apply to most industrialprocesses. The problem with most real processes isthat the condition of the system deteriorates veryrapidly with frequency and there are also suchrealistic complications as nonlinearities, hard limits onthe manipulated variable and nontrivial levels on noisein the sensors. All these factors tend to cancel thepotential effectiveness of decoupling other than atsteady-state.

647

SUMMARY

This paper presents the numerical concept ofSingular Value Decomposition as an analysis and designtool which can be applied quite effectively tomultivariable processes. The procedure presented inthis paper have been applied to a number of realindustrial process units and have proven to be quiteeffective in determining such practical considerationsas the most appropriate sensor locations, the mosteffective first level strategies, and the strongestcontroller pairing. It can also be used as a basis fordesigning multivariable control strategies which aresimple to implement and effective in terms of controlperformance.

One of the most attractive aspects of theanalysis procedure is that it can be applied to anexisting process or it can be used at the design.phasebefore inadvertent control problems become fixed inconcrete and steel. The analysis presents a systematicway to factor the effect of design and operationalalternatives into the overall design decision process.

SVD does not predict (nor solve) all the controlproblems which can be encountered in the vast worldof industrial multivariable control. It is, however,relatively easy to understand and use and does appearto be quite successful in identifying basic controldifficulties and suggesting simlple ways of reducing oravoiding the difficulties.

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ORIG0NALIGI rL IX

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U rTkIX

D.1967349E-03 0.OOOOOOE+000.1907349E-03 0.00000000+040.180734K-03 0.0000800E+000.1907349E-03 D.4766373E-040.10 734940-03 D.4768372E-040.19073400-03 0.4766372E-040.333796E0-03 0.14300510-030.5814697E-03 0.1436511E-030.39146970-03 0.1907349E-030.4291534-03 0.19073490-030.5722046E-03 0.1907349E-03O.57220460-03 0 .1907349E-030. 76293950-03 0.1907349E-030.&M85090-03 0.23641060-030.9536743E-03 0.33378800-030.1144400E-02 0.30146970-030.1335144E-02 0.3614697E-030.1621246E-02 0.4291534E-030.1907349E-02 0.57220460-030.2336502E-02 0.57220460-430.276565-02 0. 782390-030.3337864E-02 O.883069E-030.40054330-02 0.1001350-020.49S910E0-02 0.1239"7770-20.61035100-02 0.1436511E-020.7501711E-02 0.171614E-020.90367430-02 0.0269063E-020. 121116-;01 0.2670200-020.157353M-01 0.33376E-020.2117107-01 0.43e69020-020.2945370-01 0.61511900-020.45156460-41 0.905990-020.04732-01 0.13585 -010.023497K01 0.1573560-010.2222061E-01 0.39100650-020.230418E0-01 0.3305544E-020.2517700E-01 0.281333SE-020.36062990-01 0.2193451E-020.2641C700-01 0.1621246E-020.261306:8-01 0 .114440K-020.252774E-01 0.57220460-030.23904910-01 0.1907349E-030.22172930-01 -0.1430511E-030.200271SE-01 -0.4768372E-030.175590O-01 -0.57220460-030.152111LE-01 -O.76S3990-30.;1206930-01 -0.76293*50-030.1049042E-01 -0.76293950-030.0392340-02 -O.0675706E-030.524243X-02 -0."61023E-03

1 -3.1248757E-02 -0.254589E-022 -0.1240760E-02 -0.2504351E-023 -0.1248760E-02 -0.25043400-024 -0.12)65200-02 0.15e7473E0-2S -0. 1296520-02 0. 156747X-026 -0.12265200-02 0.150747K-027 -0.2329613E-02 0.78926410-020 -0.26400240-02 0.7266756E-029 -0.2688592E0-02 0.11359180-01

20 -0.300o70CE-02 0.1073249E-0111 -0.3937351E-02 0.8854236E-0212 -0.393735lE-02 0.89I4236E-0213 -0.5106110E-02 0.634995E-0214 -0.5650257E-02 0.919537E-01215 -0.6578173E-02 0.161209C9E-0116 -0.707469K0-02 0.1770847E-0117 -C.9123489OE-02 0.1520412E-0118 -0.1104436E-01 0.1553943E-0119 -0.13060S1E-01 0.2405635E-0120 -0.1507052E-01 0.1842359E-0121 -0.107130E-01 0.291000K-0122 -0.2271311E-01 0.25826600-0123 -0.2722708E-01 0.3333694E-0124 -0.3370971E-01 0.4127434E-012! -0.4139335E-01 0.4261547E-0126 -0.5135784E-01 0.47757760-0127 -0.6453977E-01 0.4942274E-0128 -0.0197121E-01 0.7011590-0129 -0.1063664E+00 0.796S83E-0130 -0.14300700E-DO 0.945070-0131 -0.2022173E+00 0.1346626E+0032 -0.3047197E400 0.18454200+0033 -0.4619186E+00 0.2671007E+0034 -0.5549154E+00 0.2650490+0035 -0.141"74E+00 0.43775150-0136 -0.1059485E+00 -0.2252375E-0137 -0.16765460+00 -0.8915587E-0138 -0.172X52E+00 -0.1542450E+0039 -0.174577K+00 -0.2077255+0040 -0.17222650+00 -0.24491 E+0041 -0.1665030+00 -0.2375+0042 -0.1572226E+00 -0.2985535E+0043 -D.14502500E+00 -0.303404X+0144 -0.130641E0+00 -0.30387300+0045 -0. 114624K+00 -0.28D1271E+0046 -0. 902431E-01 -0.2651901E+0047 -0.83214700-01 -0.233095+0048 -0.6791750E-01 -0.2032077E+0049 -0 .5427"67-01 -0.167476X30050 -0.2094231E-01 -0.4712465E-01

1 2

1 -.09694957E+00 -0.1512489E+ll2 -0.1512489E+00 0.0984957E+11

SINGULAR VALUES

0.1509E0+000.11519E-01

CONDITIO NWISER 13.10091

FIGURE 1

ETHANOL-WATER COLUMN

LO

11-

FIGURE 2ETHANOL-WATERGAIN MATRIX

I- I - r

O 5 1 O 1 S 20 25 30 35 40 45TRAY NUMBER

FIGURE 4 U VECTOR PLOTS

FIGURE 3ETHANOL- WATERSVD ANALYSIS

50 0 5 1 0 1 5 20 25 30 35 40 45 50TRAY NUMBER

FVUKr 5 SENSOR LOCATION PLOT

649

192 PRCOF

5.2 I-EtOH

1067

101912231415

26

17

is

19202122Z32425262728

30

32

432

3334333S3738

4041424344

4556474649SD

I

A

-

50.0

46.0

42.0

38.0

34.0

30.0

26.0

22.0

18.0

14. 010.0

I6 17 18 19 20SIDEDRAW POSITION (TRAY NUMBER)

21

200.0

190.0160.0 '

120.0

100.0 /8o.060.0 /

20.0 -0.0

0.5 1.0 1.6 2.1 2.6 3.2 3.7 4.2 4.8 5.3SIDEORAW RATE (LBMOL/HR1

FIGURE 6

CONDITION OF ETMANOL-WATER COLUMN VS

OPERATING CONDITIONS

80.0

72.064.0

56.0

48.0

40.032.0

24.0

16.0

B.o0.0

FIGURE 7 SVD DECOUPLER

FIGURE 8

SYD PROFILE CONROLLER

650

R1 +

R2

Xl

X2

5

Lt.

0

7-- VT

2:

z

a

-:

m

_o

zo

w

o_a

eJ

0:Q

z

J-

za

'