MULTIVARIABLE CONTROL

CD May 1998 • InTech

Nonlinear multivariablecontrol made easyBy Dan P. Dumdie

Forget complexcontrol theory andmathematics. Here isa simple systematicapproach fordesigning controlsystems.

Hot well

Figure 1. A nonlinear process

Most control loops in the process industriesare nonlinear-their control characteristics andtuning requirements change as operating condi-tions change. When these changes are significantduring normal operation, some form of adaptivecontrol tuning is generally required. In addition,many operations experience some type of controlloop interaction. When the process variable ofone loop severely interacts with or disturbs theprocess variables of other loops, multivariablecontrol can decouple the interaction.

Nonlinear multivariable control can solveboth problems but is generally considered anadvanced topic understood only through com-plex control theory and mathematics. This articlepresents a simple systematic approach for design-ing control systems for nonlinear multivariableprocesses. This method can produce excellentresults and is easy to implement using digitalhardware. It is also easily understood by controlengineers and instrument technicians as it doesnot require the theory and complex mathematics

Where: n is temperature transmitterTC is temperature controller

TwSet point

IFw,Tw

used in more contemporary methods like modelpredictive control.

A design for nonlinear process controlThe simple operation of blending together

two process streams for temperature control isshown in Figure 1. A large change in the operat-ing conditions of this nonlinear process can sig-nificantly affect control loop performance. Forexample, a loop well tuned for a total load of1,000 gallons per minute (gpm) will becomeunstable at flows less than 500 gpm and slow torespond at very high loads. Likewise, changes inthe cold or hot well temperatures will also impactloop behavior. Therefore, the control shown inFigure 1 is effective only over a limited range ofoperation from where it was tuned.

Carefully designed feedforward control can beadded in Figure 1 to linearize the process andmake it controllable at virtually all operating con-ditions without changing control tuning. This isequivalent to steady-state adaptive gain. The first

MULTIVARIABLE CONTROL

InTech • May 1998 CDstep in this feedforward design is to do a processmaterial and energy balance, as shown inEquations 1 and 2.

FcTc + FHTH = FwTw

whereFeTeFHTHFwTw =

cold well flowcold well temperaturehot well flowhot well temperaturewarm (mixed) stream flowwarm (mixed) stream temperature

These equations are then combined andsolved for the manipulated variable, as shown inEquation 3.

(TH- Tw)Fc = n, ( )TH - Tc

Equation 3 becomes the feedforward controlalgorithm, as shown in Figure 2. It computes a setpoint for the cold well flow that will provide thedesired warm stream temperature (Tw) at steady-state conditions.

The final step in this nonlinear design is toselect a feedback trim variable from those used inEquation 3 to use for the feedforward control setpoint. The appropriate selection will linearize theprocess over its entire range of operation. OnlyTw, the controlled variable, can satisfy this crite-rion. Therefore, the temperature controllershown in Figure 2 manipulates the same variableit controls and uniquely sees the process as being

.-••••••••

(1)

linear and as having a unity gain. Moreover, anychange in manipulated variable will produce anidentical change in the controlled variable.Advantages of this design include normalizedcontrol tuning and adaptive gain.

(2)A design for nonlinear multivariable control

If this process must handle changing demandsfrom multiple downstream loads, it may be nec-essary to add pressure control to Figure 2. Thiscould be done as shown in Figure 3; however, itis unclear which valve should be manipulated fortemperature control and which should be manip-ulated for pressure. This decision depends on theprocess operating conditions. Since the twopumps run head to head, both the cold and hotwell valves will affect temperature and pressure.Therefore, the process is both nonlinear andinteractive (i.e., multivariable) in nature.

(3)

Hot well

Where: FT is flow transmitterFC is flow controllerFF is feedforwardFB is feedback

Feedforward controller

Where: PT is pressure transmitterPC is pressure controller

TwSetpoint

Figure 3. A nonlinear multivariableprocess

Figure 2. Nonlinear single-variable process control

MULTIVARIABLE CONTROL

• May 1998 • In Tech

For further reading

This articleis part 2 of anongoingseries.Part 1, by DanDumdie,was tided "Desiregoodprocesscontrol?Trya systemsapproach,"whichappearedinthe September1996issueofInTech, pp. 65-69.

Figure 4. Nonlinear multivariableprocess control

A more robust temperature and pressure con-trol system is available that does not require thisuncertain pairing of control variables and valves.Instead, the system can simultaneously manipu-late both valves in just the right proportion toachieve temperature and pressure control withoutloop interaction. For example, it can increasetemperature at constant pressure by decreasingthe cold well flow and increasing the hot wellflow to precisely achieve the desired temperaturewith no change in total flow.Although this mayseem complex, it is easily done using a designmethod similar to that previously described.

As before, the first step in this design is to doa process material and energy balance. This time,with the addition of pressure control, Equations1and 2 must be expanded to include the appro-priate pressure variables. This is done by replac-ing the cold and the hot well flows with theirrespective valve flow equations, as shown inEquations 4 and 5.Fc = Cv cJpc - Pw (4)

whereCVe =CVH =PePHPw

cold well valvecoefficientof dischargehot well valvecoefficientof dischargecold well valveupstream pressurehot well valveupstream pressurewarm (mixed) stream pressure

As before, these equations are combined andsolved for the manipulated variables CVe andCVH' The mathematics involves two equationsand two unknowns. The solution to these equa-tions and the associated nonlinear multivariablecontrol system are shown in Figure 4.

Feedforward controller

t,~-------_..._...: Pc· .• •• •

Fw(TH-Tw)(TwTc)VPc-p~

(Fw-Cvc~)V'H-PW

l-f:iiI-C:~-ofiijl-tJlI---til~~Lood

WIllIeSHoi well

(5)

Daishowa provides real-life exampleThe Daishowa America Port Angeles mill

makes lightweight directory paper from a varietyof pulps, including recycled newsprint and tele-phone directories. The recycle plant containsprocesses for both pulping and flotation inkremoval operations. Each process has multipledemands for warm water dilution with signifi-cant changes in load. Two large pumps provideall the hot and cold well dilution for each unitoperation in both processes. Temperature con-trol optimizes process chemistry, while pressurecontrol ensures adequate supply and preventsdownstream valve interaction. Temperature andpressure set points for the pulping process maybe different from those for the flotation process.In short, two temperatures and two pressuresmust be controlled using four valves and twopumps in a nonlinear hydraulic process wherepressures, flows, and temperatures all interact.

The controls shown in Figure 4 were appliedto this system. Both loops were tuned with pro-portional gains (i.e., steady-state loop gains) of0.5 and moderate integral settings. The controlsperform equally well with little or no interactionat different process operating conditions, includ-ing significant changes in both hot well tempera-ture and total load.

The temperature and pressure response curvesfor a 34% step increase in load are shown inFigure 5 for the pulping process. Although thepressure initially dips about 7 pounds per squareinch (psi), it quickly recovers to steady state inlessthan a minute. The temperature deviates only2°F but requires more than a minute to com-pletely recover due to its slower process dynam-ics. Set-point changes for both temperature andpressure are shown in Figure 6. The temperaturechange has only a minimal effect on pressure,which is short-lived, while the change in pressureset point has Virtuallyno effect on temperature.

Once again, the feedback controllers manipu-late and control the same variables. The feedfor-ward variable,Tw', is manipulated to control Tw,and Pw' is manipulated to control Pw. The pair-ing of feedback and feedforward variables isstraightforward using this design, and the benefitsof process linearity and unity gain are achieved forboth temperature and pressure control.

Cascaded flow loops cannot be used for mul-tivariable control as was done with the single-variable system shown in Figure 2. This is animportant distinction since the valves must bemanipulated directly by the feedforward algo-rithm to decouple control loop interaction andprovide the desired cold and hot well flows.

1'1111" "I'!:)m :1''111/' 'illl'InTech • May 1998 •

Temperature

o 30

Temperatureset point

Figure 5. Temperature and pressureresponse to 34% increase in load.

70

1r

60~iIlIII __ ;;;;;;::;;Il""=ll!!!!!!!!!!!l-==- __ """~;;;';;';';:"-"""'=-----=="ll00Temperalur. P~ ~_ ~e ,

a80 I

60Time (5)

90 120

Temper"u,.

su,e

(p;g) 30

90

Practical considerationsVirtually every control sysrem available on rhe

marker roday can provide feedforward controlcapabiliries by adding a bias ro rhe control ourpurof a feedback loop. Typically, rhe transmitter sig-nal from a disrurbance variable (e.g., load) is mul-tiplied by a gain (i.e., runing paramerer) andadded ro me feedback control ourpur ro deter-mine rhe valve posirion. This is a carryover fromrhe days of pneumarics, when compurarions weredifficulr, and is nor me correcr way ro do feed-forward control using digiral rechnology. Thisolder approach does nor linearize rhe process,requires feedforward tuning, and can only oper-are over a limired range of process condirions.Furthermore, ir does nor realize rhe benefits ofdigiral feedforward designs.

There are usually a number of pracrical con-siderarions ro address when going from controltheory ro online operarion. For example, mere isno substitute for a good process design and prop-erly se!ecred, sized, and installed instrumenrs.Equipment sizing can also play an important rolein providing good process control. Even me mosrsophisticated controls cannot compensare for

Pres~r.~-- __ I~poc~~

120 150 1801101.(,)

210

deficiencies in these areas. Furthermore, me sys-rem musr be easy ro use by operarions and easy rounderstand and rroubleshoor by maintenance.The following are some of rhe pracrical consider-ations for implementing me controls shown inFigure 4:• Both the hot and cold well pumps should be

sized ro operare on or near me flar part of meroral head pump curve to meet all loaddemands and ro accommodare feedforwarddecoupling.

• Because operarors musr be able ro stroke thevalves when in manual, rhe feedback controlourpur must be a valve position (%) when inmanual and a feedforward control ser point(engineering unit) when in auromaric.

• Since proper feedforward control requires marboth feedback loops (i.e., remperarure andpressure) are run in auromaric, me control con-figurarion must ensure an auromaric or manu-al mode change by me operaror is rransferredto both loops simulraneously.

• Since the remperarure and pressure loopsmanipulare and control me same variable,bumpless transfer from manual to auromaric is

Figure 6. Temperature andT pressure response to set-em point change

u,70 e

IQ

Behind the byline

Dan P. Dumdie is process andcontrol engineering supervisor atthe Port Angeles mill ofDaishowa America Co. Ltd., adirecrory paper manufacturer inWashington. He has a B.S. inchemical engineering from meUniversity of Washing ron andmore man 20 years of experiencein process control. His careerobjectives have placed specialemphasis on making advancedcontrol theory practical andeasily applied.

MULTIVARIABLE CONTROL

• May 1998 • InTech

easily done by tracking the comrolled variableduring manual operation.

• The comrol configuration must appropriatelyhandle all anomalies in the feedforward calcu-lations, including division by zero (e.g., T H =

Td' calcularion of negative valve coefficients(e.g.,T W' > TH)' and any other exceptions.

• Valve characterizarion to determine theinstalled valve coefficiem as a function of valveposition can be done by collecting online datausing me instruments shown in Figure 4 or byusing the supplier's data sheets.

• The process should be designed to reduce con-

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trol loop deadrime and lag by locating thevalves and temperarure sensor close to thein-line-mixing tee plus using a min-walledrherrnowell, or by placing me sensor directlyimo me process stream.

• Although field instruments are used to provideall of the feedforward variables shown in Figure4, only me variables mat change significandyduring normal operation need be instrument-ed; others can be assumed constant for feedfor-ward computation.

• Dynamic compensation can be added as need-ed for each feedforward variable ro improve thecomrol system's response to load change.

Design benefitsThere are several significam benefits associat-

ed with this feedforward design method. Themost important is a resulr of placing the feed-forward comrol calculations direcdy inside thefeedback loop. This in-loop computation makesthe comrolled variable respond linearly withchanges in the manipulated variable. Further-more, when the feedback comroller manipulatespressure to control pressure or manipulates tern-perarure to comrol temperature, the processresponds with a gain of 1.0. Consequently, agiven increase in the manipulated variable willproduce the same increase in the comrolledvariable. This normalizes comrol tuning bymaking the overall steady-state loop gain equalto the controller's proportional gain. The abilityro directly set this gain adds meaning during thetuning process for both engineering and main-tenance personnel. For example, differem loopswith fast dynamics (e.g., temperature, pressure,and pH) will tune with similar comrol gainsaround 0.5, independem of field instrumentcalibrations and other factors that affect con-vemionalloop tuning.

In addition, all feedback comrollers used withthis feedforward design are reverse acting. Atsteady state, me set point, comrolled, and manip-ulated variables will all be the same. Operatorscan use this as a criterion ro help determine whensomething in the field has changed or failed andit is time to call for maimenance.

This approach provides a comrol system withadaptive gain runing. The design can performwell at virtually any process operacing conditionsand makes decoupling of control interactionstraightforward. It uses simple material and ener-gy balance equations -and does not require theknowledge of complex mathematics and comroltheory. The design has been proven with numer-ous industrial applications and is worthy of con-sideration for any critical process control need. IT