dynamic matrix control_a computer control algorithm

8/10/2019 Dynamic Matrix Control_A Computer Control Algorithm

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TM

DYN MIC M TRIX CONTROL-

COMPUTER CONTROL LGORITHM

PRESENTED AT:

THE NATIONAL MEETING OFTHE AMERICAN INSTITUTE OF CHEMICAL ENGINEERS

HOUSTON TEXASPRIL 1979

BY:

DR. CHARLES R. CUTLER[DYNAMIC MATRIXCONTROL CORPORATION

DR. B.L. RAMAKER[SHELL OIL COMPANY]

[DMC]TM


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DYNAMIC HA1lUX a:MROL - A a :HVIm cnnROL ALOORTIlI1

presented by

C_ R. C11l1D and B. L. RAHAKER

a t the 86th NATIONAL

AMERICAN INS1T1UJE OF OIEHIoo mGINEDS

JaSTRACT

The Dynamie 'iatri x Control (011C) A19orfthmis control technology that has been usedsuccessfu11y in process ~ u t r appH cat ionsI Shell for the last six years. The generaldeweloplllent of the Dr-te Algoritha to incorporate'"dforward and multivariable control is coveredi l this paper. The Dr1t Algorithn evolved frOlll technique of represent i ng process dynaoi es.Ith a set of numerical coeffic:tents. The_meric.l technique, in conjunction with alellSt square fon:ulation to f.'Iinlmize the. inteV.l of the error/tin! curve, make i t possibleto solve c o ~ l e x control probler.lS on a digitalt ~ t r which are not solvable with traditional'ID control concepts. The incorporation of theNlCess dynamics into the synthesis of thedesfgn of the ONe, make i t possfble to naintain1ft awareness of deadt ime and unusual dynar:tlcMbavior.

TllAMlC MATRIX CONTROL OEVELOPIHT

The Dynamic Hatrix Control (Drle) AlgorHhn's a control technology that has been usedsutcessfully In process cerevt er applicationsIII Shell for the last six years. The general~ y e l o p m e n t of the ONC Algoritho to Incorperate.ftedforward and multivariable control wi 11 betoyered In this paper. A subsequent paper wi 11e ~ a n d the scope of the algorlth" to address~ e problem of constrained multivariable control. The need for the expanded algorithn~ o 1 v e d from the application to a catalytic~ ~ a c k l n g unit of a non-linear steady state opti~ l a t l o n which characteristically drives the~ s s to a number of constraints.

. The D 1C Algorithr.l evolved from a techniqucf representing process dynaoi cs with a set

of nUllll!rical coefficients. The nUr.!Cricaltechnique, in conjunct ion with a lea st squarelonulation to minioize the integral of theerror/tfme curve, r.l4ke i t possible to solve

~ l e x control probler.lS on a digital conputer- 'ch are not solvable with traditional PlD

amtrol concepts. The incorporation of tne:rocess dynamics into the synthesis of the~ s l g n of the oro\(. make t possible to r.I4 i ntai np awareness of deadtime and unusual d y n ~ i c ~ h a v i o rAn awareness of deadtime alone:revents the controller frOr.l o v e r c o ~ e n s t i n g .tllch can only be obta i ned in the PI0 contro 11 er~ suppress1ng the integral action with the .;orrespondlng degradation of the control.~ X i l l e s of unusual dynamic response to a step:h.nge are illustrated fn Figure 1. The firstcurve is characteristic of a system which is out

COLUHH

. . J

>W. . J

STEP CKM'GE INFEED TO COLUMN

0"'----'---------FURNACE

CHANCE DUE TOSOOT BLOWER

oTIME

FlC. 1 UNUSUAL DYNAMIC BEHAVIOR

of naterial balance. The second curve is the

response of a furnace tranSfer temperature toa soot blowing operation on a large preheater.:Iote in both illustrations, the response curvecannot be adequately described by a first or~ e c o n d order differential equation typicallyused in control analysis.

Any system which can be described orapproximated by a systeo of linear differentialequations can utilize the Dynamic Matrix Controltechnique which is based upon the n u ~ r i c l representation of the system dynamics. Twoproperties of linear systems makcs the numeri-.cal representation possible. The first ofthese principles is the preservation of thescale factor. It is illustrated in Figure 2

where the response of .the output variable 0 isshown for a changc in the input varfable I .The solid line represents the response of theoutput variable to a unit change in the inputvariable and the dashed line illustrates theresponse for a two-unit change in the inputvariable. Note the response of the two-unitchange has twice the amplitude of the Dne-unitchange. For a linear system, the response ofthe output variable for any size change in thefnput variable may be obtained by multiplyingthe scalar value of the input variable times
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, , --,

, '/ ' TWO UNIT CH.eJ:GE

HI UfPUT VARIABLE,/.-

,"""

o

rtc. 2 PRESERYATlOIl OF SCALE FACTOR

the unit response curve for the output variable.Further note on Figure 2 that the unit responsecurve can be a p p r o x i ~ t e d by a set of numbersi f the curve i s broken into discrete intervals

of tiMe. The two-unit response curve in Figure2 can be obtained by ~ l t i p y i n g the set ofnumbers for the unlt response by 2. The secondcharacteristic of a linear system is the principle of superposit ion. This princ iple isi l lustrated in Figure 3 where the response Ofthe output variable is shown for a unit change.in two input variables. Also, the responseof the output variable to a s i ~ l t a n e o u s unitchange in both input variables is shown. Theresponse for this curve was obtained by s ~ i r i g the responses for the unit response curves o r

usrr mCREASE IflINPUT VARIABLE I

TIME

UNIT INCREASEINPUT VARIABLE

INW tb J

~ ~ -1;:) a:: 0

UNIT INCREASE IHINPUT VARIABLESla & Ib

TIME

FIGURE 3 PRINCIPLE OF SUPERPOSITION

the input variables. Hathemitically theseconcepts are given by:

6 01 a1 6 11 + b1 6 12 +

6 02 az 6 11 + b2 6 12 +

6 ~ 3 a3 6 11 + b3 6 12 + 1)

6 0i ai 6 11 + bi A 12 +

where the 6 0i are the changes in the outputvariable frGQ i t s init ial value to i ts valueat t i -e interval t and t h e I j are thechanges in the input variables from theirinit ial value at t i-e equal to zero. The aiand bi are the numerical coefficients referredto in the preceding paragraphs. Fi9lr e 4illustrates the response of the outlet t e ~ e r -ature of a preheat furnace to i step change inthe fuel to the furnace and to a step changein the inlet t e ~ e r a t u r e of the feed to thefurnace.

The fuelcoefficients

shown lnfigure 4

are an illustrative e x a ~ l e of the at and theinlet t e ~ e r a t u r e coefficients are i lustrat1veof the b i The input variables can be ~ n i p u -lated control variables or ~ a s u r e d disturbances. For e x a ~ l ein the furnace controlproblem to be described in this paper. thefuel i s a ~ n i p u l a t e d input variable, theinlet feed tecperature is a leISured disturbance. and the controlled variable is thefurnace transfer teqerature.

b.O at INlET

FUEL TEMP.COEF. COEF.

.014 0.0

.086 .240

.214 .340

.414 .465

.600 .540

.736 .590.5

.836 6ZZ

.904 .640

.949 .653

.986 .658

TlHE

FIC. 4 FURNACE RESP.ONSE, FUEL, TEHPERAllJRE

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The change in the controlled variable fro"time equal to zero to s ~ ~ future tine t h t will 6 1 b1 A 12 a1 6 I 'result from changes in the ~ n i p u l t e d ~ u t can be represented the following equations: 1

1(2 )6 01 al A 1 6 02 + b2 A 12 aZ A I 1

1 2 36 03 a3 A 1 1 ~ A 11 a,

A 11

2 36 0i ai A I ai-l A I ai-2 A I1 1 1

where I:lOvement of manipulated variable I , sconsidered for three intervals of t ~ intothe future with the superscripts o ~ 11 representing the t ~ interval. The A II is the

step change in the~ n i p u l t e d

input variablefrom i ts value at the end of the first .tineinterval and the beginning of the second.Similarly A 311 s the etlange in the ~ n i p u l t einput variable from i ts value at the end of thesecond t ~ interval and the beginning cf thethird . Further. note the s ~ set of coefficients is used for each colunn. The coefficients in each column are shifted down onetime interval for each successive column tocorrespond-with .the first time interval whichthe future inputs can impact on the outrut variable.

FEED FORWARD DM

Feedforward control is accomplished bymoving the measured disturbance input variablesin equation set (1) to the left hand side.For example. if input 12 is a disturbance input.equation set (1) becoDes:

6 1 bl A 12 al A 116 02 bZ A IZ aZ A 11

6 03 b3 A IZ a3 A 116 4 b4 A 12 a4 A 11 (3)6 Os bS A 12 as A 116 06 b A 12 a6 A 11

61 bi A 12 ai A 11

Combining equation set (Z) with set (3)yields the general form of the equations usedto do feedforward predic tive control of avariable 0.

The desired response of the systeGI is determined by subtracting the predicted response ofthe system from the setpoint. which is determined from the past h i s t o ~ of inputs to thesystem. With the desired output response 6 0iknown. the measured input disturbances A 12known. and the numerical o e f f ~ i e n t s ai andbi known. all the infonnation is availableto solve equation set (4) for the set of time

dependent moves in the manipulated inputvariable.

DHC SOLUTION TECHNIQUE

The set of equations is over-detenginedwhich prevents direct solution. but can besolved using a least square criterion. Such asolution produces a projected set of moves inthe manipulated variable that minimizes theerror in the output variable from i ts setpoint.The obvious difficulty with the use of the leastsquare method to calculate the movement of themanipulated variable is the unconstrainednature of the solution. The method withoutconstraint will yield very large changes inthe manipulated variable that would not bephysically realizable.

One technique for sappressing the changein the manipulated variable is to multiplY bya number greater than one. the main diagonalelementsof the square matrix that evolves frOmthe least square formulation. The effectivenessof such a multiplier is illustrated in Figure 5where a square wave change in setpoint was made.The unconstrained least square reduction inthe error resulted in a total change in theabsolute value of the manipulated input variableof 37.57 taken over 10 intervals of time. Witha multiplier of 1.005 the total change was4.91 and with 1.010 the change was 3.42. Themultiplier effectively adds another row to

the original data for the input variable foreach interval in which i t is allowed to move.All elements in the row are zero except for the

. specific input variable which has a coefficientrelated to the size of the multiplier. As canbe seen from figure 5. the suppression of themanipulated input variable moved by an orderof magnitude did not significantly impair thereduction in the projected error.

The matrix of coefficients which describe.the clYnu1cs of the systl lll 1s the basis for

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TU-I

FIG. 5 HOVE SUPPRESSIon FUEL TO F U R r ~ C E

the m1C A l g o r i t ~ For the furnace controlp r o b l ~ the response of the output variablewasconsidered for 30 intervals of t i ~ and themovel:\ent of the fuel gaswas considered for 10intervals. Thirty intervals of t i ~ representsabout 4 l Z time constants for the response ofthe outlet temperature to a change in the fuel.At the tenth time interval. the outlet t e ~ e r - ture has three t i ~ constants to settle f r o ~ the last change in the fuel. This choice oftime intervals results in a Rltrix with10,colunns and 30 rows. To initialize the algori t ~ the measured outlet t e ~ e r a t u r e isstored into the 30e l ~ n t vector that represents predicted values of the output variable.This assumes the system is at steady state,but is not a necessarycriterion. An error isthen calculated from the projected value of theoutput variable and the setpoint for the 30intervals of t i ~ e This vector of errorsbecomes the right hand side for the 10 by 30~ t r i x The least square solution of this set

.of equations yields the best set of fuel~ Y e s to eliminate the projected errors for 30 timeintervals. The projected set of fuel moves isused to calculate the outlet temperature changefor the forthcoming 30 intervals of time, andthe temperature changes are then added intothe 30 element vector for the predicted valueof the dependent variable. The first fuelmove is implemented and the entire vector ofpredicted output variable values is shifted

forward one interval of t i - . . At the start ofthe next interval of time the predicted valueof the output variable is compared with themeasured value. The error in the projectionis used to adjustall 30 values in the predictedoutput variable vector. This a d j u s ~ n t inthe prediction provides the feedback to cocpensate for u ~ s u r edisturbances and errors inthe dynaQic prediction. At the next intervalthe ,set of errors between the setpoint and thepredicted values of the output variable is usedto solve for another set of 1 fuel aoves.The ntne remaining fuel ~ v e s fram the prevtous

SET P O I ~ T CHArlGE

HAIN D I ~ C i m I A L f' UL TI PLIER OF1.005

- - - - - - - HAIII I H ~ ~ O t I A L ~ U L TlPl,lER OF1.000

a::"'2 .....- ; - 1 -

dynamic matrix control_a computer control algorithm

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