Nonlinear Model Predictive Control of A

7/24/2019 Nonlinear Model Predictive Control of A

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Compurers them. Engng,

Vol. 18, No.

2, PP. 83-102, 1994

0098-I 354/94 6.00

+ 0.00

Printed in Great Britain. All rights reserved Copyright 0 1994 Pergamon Press

Ltd

NONLINEAR MODEL PREDICTIVE CONTROL OF A

FIXED-BED WATER-GAS SHIFT REACTOR:

AN EXPERIMENTAL STUDY

G .

T.

WRIGHT

and T. F.

EDGAR

Department of Chemical Engineering, The University of Texas at Austin, Austin, TX 78712, U.S.A.

Rec ei v e d 16 No vember f 9 9 . 7 ; i n a l r e v i si o n r e c ei v e d 2 J u n e 1 993 ; r e c ei v e d f o r p u b l i c a t i o n I 6 J u ne 199 3 )

Abstract-This paper describes new results on the experimental application of nonlinear model-predictive

control (NMPC) to a fixed-bed water-gas shift (WGS) reactor. The development and experimental

validation of an appropriate first-principles WGS reactor model,

and how it impacts controller

performance is discussed. The implementation of NMPC is computationally intense, requiring that a large

nonlinear program (NLP) be

solved at each sampling period. The significant computational burden

dictates that a relatively slow sam pling rate be used. Infrequent sampling, however, diminishes disturbance

rejection capabilities. To combat this problem, NM PC was implemented

in a master-slave cascade

configuration where a low-level liner controller, having a significantly faster sampling rate, was employed.

The control study was performed using a PC-based distributed control system (DCS) One of the three

processors was dedicated to NMPC calculations. A complete and rigorous implementation strategy is

described in the paper, and the performance of NMPC for set-point tracking of this nonlinear process

is shown to be superior to adaptive or linear control. We also illustrate the ease with which NMPC

accommodated feedforward control.

1. INTRODUCI-ION

The sev erity of the nonlinearities in chemical pro-

cesses influences the selection of control algorithms

for successful control of a process. The linearization

of nonlinear physical m odels about a nominal operat-

ing point is a standard approach for control system

design. This approach is adequate when the process

nonlinearities are mild or when plant operation

is constrained to a narrow region, but for highly

nonlinear systems such as nonisothermal chemical

reactors, linear controllers designed in this way

may perform poorly. The wide range of operating

conditions encountered in start-up or shut-down

of continuous processs and trajectory tracking

of batch processes also prove difficult for linear

control.

One technique that attempts to compensate for the

inadequacies associated with linear controllers uses

an adaptive feedback control law based upon current

and past operating conditions. In general, an adap-

tive controller can be thought of as a nonlinear

control technique that has two distinctly different

types of time-dependent variables operating on very

different time scales. Presum ably, the parameters of

the process m odel vary slowly with time, whereas the

process states (e.g. temperature or concentration)

change at a much faster rate. While many successful

applications of adaptive control have been reported

in the literature, successful control with an adaptive

algorithm may be difficult to achieve. Because of this

deficiency, there appears to be a need for further

improvemen ts in nonlinear controllers, which is the

motivation for this research.

In recent years, model-based control strategies

employing differential geometric theory have been

used to effect linearization via state or output feed-

back. Global feedback linearization involves trans-

forming the states of a nonlinear system into an

eqivalent system such that, under state or output

feedback, the resulting dynamic system becomes ex-

actly linear in a specified manner. Brockett (1978) was

the first to employ this methodology, which was

further advanced by Hunt ef al. (1983). While this

technique is intuitively appealing, the necessary trans-

formations required for implementation often fail to

exist, or do not yield the desired level of robustness

to model error. H ence, we seek a more broadly

applicable nonlinear control strategy.

Recent improveme nts in computer capabilities

permit the use of rigorous models for real-time

optimization and control. A control strategy which

takes advantage of the increased power and speed

of computers is nonlinear model-predictive control

(NMPC). The predictive control strategy involves

a repeated optimization of an open-loop performance

objective over a finite horizon extending from the

current time into the future as done for linear model-

predictive control (Cutler and Ramaker, 1980; Clarke

et al., 1987; Eaton and Rawlings, 1991). Ra wlings

83


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and Muske (1991) have developed a nominally stabi-

Iron oxide, copper-zinc oxide and cobah-

lizing constrained model predictive controller using

molybdenum catalysts are among the catalysts used

an infinite prediction horizon. Linear models with

to promote shift conversion. Iron oxide catalysts have

equality and inequality constraints permit the optim-

been studied in much greater detail than the others.

ization problem to be solved directly using quadratic

They are classified as high-temperature shift catalysts

programming (QP). NMPC extends MPC to nonlin- with maximum operating temperatures of approx.

ear systems and incorporates constraints in an ex-

500C. Since our principal goal w as not to study the

plicit manner. WGS reaction, but to use it as a model chemistry to

The primary objective of this research was to better implement nonlinear control strategies, a well-

develop an advanced nonlinear control strategy for a

understood, high-temperature, iron-oxide shift cata-

fixed-bed reactor and to apply the strategy exper- lyst (United Catalyst Cl2-3-05) was used. Previous

imentally. Section 2 describes the construction of a work on a similar reactor by Bell and Edgar (1991)

laboratory scale fixed-bed water-gas shift reactor

used a sulfur-tolerant cobalt-molybdenum catalyst

with computer facilities capable of acquiring data and that is much m ore difficult to characterize.

implementing advanced control. In Sections 3, 4 and The experimental facility for the WGS reactor

5, a first-principles model for the WGS reactor suit- (Fig. 1) consisted of five parts: the dry gas feed

able for use in real-time, model-based control strat- processing system, the steam generator, the wet gas

egies is developed. A solution technique is adopted,

feed processing system, the fixed-bed reactor an d the

model parameters are estimated and the reactor effluent gas processing system.

model is validated. Section 6 is devoted to NMPC

development and implementation issues, and in Sec-

2. I.

Dry gas feed pr ocess i ng

tion 7, NMPC is evaluated experimentally and com- Carbon monoxide, carbon dioxide, hydrogen and

pared to more traditional control strategies.

nitrogen were supplied to the reactor from pressure-

regulated cylinders via mass flow controllers (MFC).

2. THE EXPERIMENTAL FACILITY AND OPERATING

Compressed air was obtained from building header.

PROCEDURE

MFC isolation valves in conjunction with manual

The water-gas shift reaction (WCS) arises in the

bypass valves were used to direct the supply gases as

production of ammonia, hydrogen and organic

desired.

chemicals. The reaction is reversible and mildly

exothermic:

2.2. S t eam gene r a t o r

The steam generating system w as similar to a

CO(g) + H@(g) - CQl(g) + H,(g)

generator used by Bell and Edgar (1991). The steam

AH ,, = -9.8

kcal/mol.

generator was designed to vaporize a known quantity

of water and to superheat it enough to avoid conden-

In typical industrial applications the dry process

sation prior to entering the reactor. Deionized water,

feed contains carbon monoxide, carbon dioxide,

obtained from a building tap, was stored in 20 I

hydrogen, small quantities of hydrocarbons and

polyethylene reservoir. A Chem-Tech diaphragm

sulfur impurities. A nominal dry-gas reactor feed

pump was used to regulate flow from the reservoir.

composition assuming insignificant quantities of

The deionized water was vaporized in two-parallel

hydrocarbon, is 40 CO, 40 CO, and 20 Hr.

3 m sections of 16 in., 3 16 stainless-steel. The parallel

The steam to dry-gas ratio m ay vary, but typically

tubing wa s wrapped around a 750 W cartridge heater.

satisfies the condition that the steam to CO ratio

This assembly was housed in a 30.5 cm long, 1.25 in.

exceeds four depending upon effluent composition

dia. tube. The tubing w as packed with magnesium

goals.

oxide, an electrical insulator with high thermal con-

The W GS reaction is run in either a single adia-

ductivity. A K-type thermocouple was also placed in

batic fixed-bed reactor or in multiple reactors in series the assembly to monitor the cartridge heater tempera-

when high conversion of carbon monoxide is re-

ture. The heater temperature was varied by manipu-

quired. Steam quench streams are often located be-

lating the power input to the generator. The exterior

tween beds in a multi-reactor configuration. In either

of the steam generator was insulated with 2 in. min-

configuration bed behavior is primarily influenced by eral wool insulation.

varying the reactor inlet temperature or the steam to

dry-gas ratio. Disturbances may include fluctuations

2.3. Wet -gas feed pr ocessin g

in the dry-gas feed composition and flow, and up-

After mixing the dry gases with the steam, the wet

stream temperature variations that cannot be ade-

gas mixture was heated to a temperature of approx.

quately rejected.

165C. This temperature was maintained using PID

84

G. T. WRIGHT and T. F. EDGAR


3/20

Nonlinear model predictive control

k

W8tW

Fig. 1. Schematic of the experimental facility for the water-gas shift reactor.

control to mitigate upstream temperature fluctu-

ations caused by the steam generator. Approximately

0.75 m of 0.25 in. tubing ran from the steam/dry-gas

mixing tee to the reactor inlet. A heat exchanger was

constructed by wrapping this tubing with 3.65 m of

ceramic fiber-insulated, 1.9 n/f nichrom e wire. Tem-

perature at the reactor inlet was controlied by varying

power to the resistance heater. This assembly com-

prised the feed preheater.

2.4.

The r e a c t o r

The body of the reactor w as positioned vertically

and was constructed using 3.175 cm (1.25 in.), 321

stainless-steel with a wall thickness of 0.089 cm

(0.035 in.). The total reactor length w as 99.06 cm

(39 in.). A pair of temperature measurem ents were

taken axially at 11 equally-spaced points along the

reactor. Each pair of measurem ents consisted of a

centerline-bed temperature and a corresponding wall

temperature. The first pair of thermocouples were

located 11.43 cm (4.5 in.) from the top of the reactor

and the last at 72.39 cm (28.5 in.). In order to mini-

mize heat loss, two independent resistance heaters

were wrapped tightly around the reactor. T he first

heater extended from the inlet header to the fourth

thermocouple pair. The second continued from there

and extended to the last thermocouple pair. The

reactor was then insulated with several layers of thick

Cerablanket Insulation.

The feed gases and the reactor inlet section were

heated u sing a 300 W cartridge heater, housed in a

thermo-well assembly located at the top of the reac-

tor. Power to the cartridge heater was supplied in a

mann er analogous to the steam generator heater. The

volume su rrounding the thermo-well and extending

to the first pair of thermocouples was packed with

Pyrex glass beads. Th ese facilitated flow distribution

prior to gas entry into the active bed. They also

served as catalyst su pport when loading the reactor

bed.

The active bed of the reactor w as 36.6 cm long and

extended from the first thermocouple pair through

the seventh. The active bed was packed with iron-

oxide catalyst pellets (0.3175 cm/O.125 in.), yielding a

reactor diameter to particle diameter ratio of approx.

9.4. T he void fraction in the catalyst bed was 0.35.

The exit section of the reactor was packed with

kaolin clay spheres. They supported the catalyst bed

during norm al operation and rested upon a perfo-

rated plate located at the reactor exit. Four thermo-

couple pairs extended into the reactor exit section.

Some were used to examine the validity of model

boundary conditions discussed later. The total exit

section was 51.05 cm (20.1 in.) in length.


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86

G.

T.

WRIGHT and

T. F.

EDGAR

2.5 . Ef l uent gus pr ocessin g

Upon exiting the reactor, the product gases passed

through a series of units designed to eliminate the

water vapor content of the gas and to determine the

effluent dry-gas composition. Infrared analyzers were

used to determine effluent CO and CO2 compositions.

This information coupled with inlet mass flowrates

for each species was sufficient to completely deter-

mine the exit composition by material balance.

2.6.

D i g i t a l equ i pmen t

A global bus architecture, distributed control

system DC S employing two 33 MHz Intel 32-bit

80386/80387 IBM-PC compatible computers and one

16 MHz Intel 16-bit 80386/80387 IBM-PC compat-

ible was used in this research. This architecture

functionally divides the tasks traditionally im-

plemented by a single host computer among several

autonomous computers. The 16 MHz machine w as

used for primary data collection and storage, for

trending and as an operator interface. The 33 MHz

machines were used as platforms for highly intensive

numerical computations such as the optimizations

necessary of NMPC calculations. Ethernet adaptor-

boards were installed in each of the computers to

provide for communication. Thin coaxial Ethernet

cable was used to wire the computers together. The

software used to drive the DCS for the WGS reaction

facility was Intellutions FIX DMACS, a commer-

cially available software package available on several

platforms. DMAC S has an open architecture allow-

ing easy database access for user-written code. Sev-

eral mechanisms were provided for executing user

tasks including event-based execution, fixed interval

execution, specific time execution and continuous

execution. The second of these techniques was used

predominantly and was implemented using a sched-

uler which was designed to spawn independent pro-

grams as user-specified intervals.

For data acquisition and control, digital and anlog

Optomux stations were employed. Each Optomux

station consisted of a brain board and a mounting

rack. The Optomux brain board is a controller which

operates as a slave device to a host computer. The

brain boards are, in essence, intelligent multiplexers,

capable of accomplishing most tasks independent of

the host computer system. The series of Optomux

stations communicated with the host computer over

an RS-422/485 serial link. The Optomux brain boards

were configured for multidrop operation.

2.7. Reac to r ope ra t i on

The procedures used to treat the catalysts were

those suggested by the catalyst manu facturer, United

Catalysts, Inc. (UCI). Activation of the UC1 iron

oxide Cl2 catalyst basically entailed employing a gas

medium to reduce the catalyst from its oxidized state.

The first step in the activation procedure was to

prepare the catalyst by heating it from ambient to

150C using a 2: 3 nitrogen to steam gas mixture.

Steam flowrate was typically 6-8 SLM. Upon reach-

ing this temperature activation was initiated by intro-

ducing a wet process gas mixture with a 1:2 dry-gas

to steam ratio. The dry-gas mixture consisted of a

2: 2: I mixture of COI, Hz, CO. Steam flowrate was

maintained at 68 SLM. The wet process gas was

used to raise the bed temperature to reaction con-

ditions (300C). Special care was exercised through-

out the heating and reducing phases to limit the rate

of temperature increase in the bed to 65C. Th e inlet

temperature was adjusted downward as reaction

slowly proceeded to make certain that catalyst bed

temperatures never exceeded approx. 465C. When

the temperatures ultimately stabilized, the reducing

procedure was complete. The entire process typically

took from 15-20 h.

Normal operation of the WGS reaction facility is

categorized by start-up, data collection and control

and shut-down. Open- and closed-loop experiments

were performed at inlet temperatures ranging from

270 to 300C. Total gas flowrates ranged from 10 to

16 SLM. Steady-state experiments were performed by

maintaining the active-bed inlet temperature at a

desired value using PID control. This proved to be an

effective method for rejecting upstream disturbances

and replicating experiments. Closed-loop experiments

were similarly performed using a cascade control

scheme. An in depth discussion of this is given later

in the text.

3. DYNAMIC MODELING OF THE FIXED-BED WATER GAS

SHIFT REACTOR

When modeling fixed-bed catalytic reactors, con-

sideration must be given to a multitude of phenom-

ena ranging from fluid how to intra-particle and

interphase mass and energy transport. These con-

siderations lead to complex partial differential

equation models, even for simple reaction schemes.

The level of model detail is ultimately constrained by

the availability of reliable physical property infor-

mation for reactants and products, accurate rate

expressions and knowledge of catalyst characteristics,

among other things.

How the model is to be used is another key

constraint in the modeling effort. A rigorous fixed-

bed reactor model would prove to be not only

complex, but also unsuitable for on-line app lications

such as optimization and nonlinear control. What


5/20

Nonhnear

mode l predictivecontrol 87

follows is a discussion of the assumptions used to behave ideally. In addition, pressure drop across the

develop a sufficiently simple, yet accurate, model for reactor was small, obviating the need for equations

real-time implementation.

describing this effect. The residence time

for the

The WG S reactor constructed for this study was process gas was on the order of 1 s. This was small

designed to operate under nearly a diabatic con- compared to the time-scale for changes in catalyst-

ditions. This wa s achieved by adding guard heaters to

bed temperatures. Therefore, the quasi-steady state

the reactor to minimize radial heat loss. The reactor

assumption that concentration ch anges are instan-

was also insulated w ith several inches of fiberglas taneous relative to temperature changes was adopted.

insulation. While these efforts did not eliminate heat Dispersion is negligible when the ratio of reaction

loss entirely, an adiabatic assum ption was employed length to particle diameter exceeds 100 (Carberry and

for model simplification. Wendel, 1963, Rase, 1977). It was included here for

Although fixed-bed reactors are heterogeneous sys-

numerical conditioning as suggested by Windes

terns with both fluid and solid phases, it is often

(1986). The final model co nsisted of two partial

reasonable to assume that the mass within a volume

differential equations with axial and tempora l d imen-

element can be characterized by a single bulk tem- sions. Danckwerts (1953) boundary conditions were

perature, pressure and composition. The pseudo -

used at the reactor inlet for the catalyst bed balances,

homo geneous assumption is a valid approximation

and zero gradient boundary conditions were applied

provided comp osition and temperature gradients be-

at the exit. The dimension less model is presented

tween the fluid and solid phases are small. This

below. Model symbo ls a re defined in the Nome ncla-

situation prevails when reaction resistance is large

ture.

relative to mass and heat transfer resistance. Windes

e? nf. (I 989) compared one- and tw o-phase models for

3.1. Carbon monox i de ba l ance

the oxidation of methanol. They concluded that

qualitatively these mode ls compare favorably und er

most circumstances.

They further concluded that

0= - +i [+]-Da (-i ,,) (1 )

pe,

even if the pseudo-hom ogeneous assump tion were

3.2. Cata l ys t bed energy ba lan ce

not strictly applicable,

the one- and two-phase

z-

models compare well quantitatively with some par-

Le

af a?=

ameter adjustment. Bell and Edgar (1991) employed

x=z+& =

I 1

df2

this assumption in a WGS reactor mode l for a system

similar to the one constructed for this research. Their

- %w@ - fw ) + fit-fco).

(2)

results confirmed that assuming homo geneity is a

The WGS reactor model was nondimensiona lized

practical simplification yielding good results for the

using the following definitions:

WGS reaction. The work of Ampay a and Rinker

(1977) and Bonvin (1980) further supported this

conclusion.

f= ,

rsf

The spatial dimensionality of a fixed-bed reactor

model may p rofoundly affect the models capacity for

i=Z

L

accurate prediction. For small diameter reactors run-

ning under adiabatic conditions radial gradients can

often be ignored, but for nonadiabatic exothermic

=I.

r f

reactions where radial gradients can be large, failure

The reference time t,,

was chosen to be the resi-

to mode l the radial dimension may render the model

useless. As stated earlier, the reactor in this research

dence time based upon the initial gas velocity L/v,.

was designed to minimize heat loss and thus large

The reference temperature was taken to be the reactor

radial gradients as done by Be ll and Edgar (1991). A

inlet temperature

TO

These variable definitions led to

1-D model was therefore developed, which also mini-

the dimensionless group s given in Table 1.

mized the number of states upon discretization of the

The dimensionless boundary conditions were:

distributed parameter system.

The simplifications mentioned thus far had the

greatest impact upon the size of the model a nd were

implemented primarily for this reason. Other assump-

tions described below were based solely upon physical

considerations. Because the reactor was operated at

low pressures, the process gases were assumed to

a f

i =o:

- =

Pe,(? - fO ),

ai

3Y,

c - = Pe,(Yco - YO),

a2

K?Yco

i= 1:

-O_

ai- di

(3)

(4)

(5)


6/20

The zero gradient boundary condition employed at

the reactor exit for the above equations is a numerical

approximation to experimentally observed behavior.

However, this is a comm on assumption even when it

cannot be experimentally verified. As the reactive

gases move from the active catalyst bed to the inert

support, reaction ceases. Thus, the effluent concen-

tration becomes fixed, and for adiabatic systems so

does the temperature. The initial conditions were

similarly nondimensionalized and resulted in two

dimensionless profiles, one for each equation.

A num ber of investigators including Moe (1962),

Ampaya and Rinker (1977), L ee (1980), Bo nvin

(1980), Newsom e (1980), and Bell and Edgar (1991)

have studied a n array of shift catalysts. Many rate

expressions for the reaction h ave been proposed, but

in this research special consideration was given to

expressions that account for reaction equilibrium

effects. The following second order rate expressions

provided by United Catalysts, Inc. was employed.

(-rco) = kt&J.Y rilo - JJc,,-YH* I&l(~)l

This expression has the advantage of directly incor-

porating the effects of steam. This is especially useful

if the flow of steam to the reactor is adjustable. The

rate constant k was assumed to have an Arrhenius

temperature dependence.

4.

SOLUTION TECHNIQUES

The model equations developed in the previous

section were solved n umerically using a Galerkin

finite element technique. A linear combination of

piecewise-simple polynomials with respect to some

suitably chosen partition of the spatial domain con-

stitute a finite-dimensional approximation to the true

model solution. Th is approximation ultimately led to

a differential-algebraic equation (DAE) set, which

was integrated using an implicit, predictor-corrector

integration scheme. T welve piecewise linear elements

were used to spatially discretize the WGS reactor

88


model, resulting in 12 differential and 14 algebraic

states. DAEs are notoriously difficult to initialize for

integration since finding a consistent set of initial

conditions is nontrivial. In this research, consistent

initial conditions were determined in a way that

permitted both steady-state and dynamic start-up.

The technique is outlined using an explicit DAE, but

implicit DAEs may be initialized sim ilarly:

k = f(x, Y), (6)

0 = g(x, Y). (7)

The differential states of the DAE, x were given

values xt,, determined by some initial (perhaps arbi-

trary) profile. y0

was then determined so that

equation (7) was satisfied. Having determined both

the differential and algebraic states, the derivatives

were determined to satisfy equation (6). This, of

course, permitted dynamic start-up. Equally import-

ant, the same technique was employed for steady-

state start-up. How best to choose x0 was addressed

as follows.

For the WGS reactor model, the differential states

corresponded to spatially distributed centerline-bed

temperatures, which were measurable. Spatially dis-

tributed, radially averaged bed compositions com-

prised the algebraic states, but only the composition

at the reactor exit was measurable. The seven equally-

spaced temperature m easuremen ts collected axially

along the reactor w ere used to determine approxi-

mate model temperatures at the spatial nodes via

linear interpolation. These w ere used to initialize the

model for experimental applications.

The nonlinear algebraic equation sets that arise

when implementing the above procedures were solved

using HYBRD, a subroutine from the MINPACK

libraries (More et al., 1980). Caracotsios DASAC

(1986) was used to integrate the model in time and to

evaluate state and output sensitivities with respect

to the sequence of manipulated variable moves.

DASAC is based upon the predictor-corrector inte-

gration algorithm, DASSL, developed by Petzold

(1983).

5. PARAMETER FSTIMATION AND MODEL

VERIFICATION

For the WGS reactor model, the parameter set of

to be fitted consisted of the following parameters:

E,,

activation energy.

A, pre-exponential factor.

PY

heat capacity of solid medium.

These parameters were chosen because they were

poorly known and could not accurrately be deter-

mined a ~riori. The first two of these parameters are


7/20

Nonlinear

model

predictive control

89

clearly kinetic rate parameters and the last was used

primarily for fitting reactor dynamics since it appears

only in the Lewis number which is the coefficient for

the time derivative of dimensionless temperature. The

need to estimate heat transfer parameters was re-

moved by virtue of the adiabatic a ssumption which

eliminated the reactor wall energy balance. This

assump tion proved to be quite reasonable since heat

losses to the surrounding were small relative to the

heat generated by reaction.

Seven temperature measurements located axially

along the centerline of the active bed were used for

parameter estimation. The m easured effluent CO

composition was also employed initially, but sub-

sequent studies proved the parameters to be insensi-

tive to this value when used with the multiple

temperature measurements.

The model states were most sensitive to the rate

parameters on the interior of the active bed for

high-temperature operation and at the exit for low-

temperature operation. The state sensitivities with

respect to the activation energy and the pre-exponen-

tial constants varied roughly in proportion to one

another. Linearly dependent sensitivities lead to vir-

tually dependent first-order necessary conditions for

the parameter estimation problem and an ill-posed

estimation problem . For this reason the centering

technique described by Bates and Watts (1988), which

improves the conditioning of the estimation problem ,

was employed. The Arrhenius expression:

was rewritten as

,

where

A=Aexp(- ) .

The mean temperature T , was chose n to lie within

the range of observed bed temperatures. The primary

effect of centering was to reduce the collinear depen-

dence between the sensitivities. What was not clear,

however, was how best to choose T , for optimal

conditioning of the estimation problem. The par-

ameter estimation problem was solved for several

values of

T,,,

. The value that was ultimately employed

yielded the smallest 2 - d intervals for the parameter

estimates as determined by GREG (Caracotsios,

1986), a parameter estimation package developed by

Caracotsios.

A weighted least squares cost function was used as

the measu re of plant-model misma tch in this re-

search:

The cost function is equivalent to the maximum

likelihood estimator when the measurement errors

are uncorrelated and normally distributed and their

variances are constant. Since these assumptions do

not apply to our data, we are content to interpret the

results simply as weighted least squares estimates.

Because each temperature measurement was assumed

to be similarly accurate, the weights u, were each

given a value of unity. For mea sureme nts of varying

qualities, how ever, the weights can be- adjusted to

reflect measu rement confidence. The weights can also

be used as scaling factors for measurements of differ-

ent magnitudes. This was unnecessary here since the

equations and the data were scaled via nondimen-

sionalization.

The parameters were estimated using eight steady-

state data sets and three dynamic data sets. The

dynamic experiments consisted of perturbations to

the active bed inlet temperature usually in the form

of first-order exponentially filtered steps. Flowrates

ranged from approx. 9.6 to 12.0 SLM, and the inlet

conditions varied as indicated in Table 2 for the

steady-state experiments. Experiments rb1028a,b,

and c were performed to verify reproducibility o f

results from

several months earlier. Parameter

estimates obtained from steady-state data for the

activation energy and the pre-exponential constant

are given in Table 3. A lso listed are the 2 - cr

intervals associated with each parameter estimate.

The 2 - D interval is a simple measure of the quality

of the parameter estimate-small values relative to

Table 2. Operating conditions for steady-state experiments

Inlet

Inlet mol fraction

tcmperaturc

Experiment ID C

co

H>O CO1

HZ

Total

flowrate

SLM

rbO7231a

rbO723lb

rbO7241a

rbO724lb

rbO724lc

rb10281a

rbl028lb

rb10281c

285.00 0.154 0.534

0.156 0.156 9.6 I

280.00 0.154 0.534

0.156 0.156 9.6 I

28 I .24 0.167 0.588

0.167 0.084 11.95

285.00 0.167 0.588

0.167 0.084 11.95

290.00 0.167 0.588

0.167 0.084 11.95

281.24 0.165 0.585

0.166 0.084 I i.97

285. I7 0.165 0.585

0.166 0.084 11.97

290.00 0.165

0.585

0.166

0.084 Il.97


8/20

90


Table 3. Optimal parameter estimates from steady-state experiments

Parameter

A

E.

T, = 297

SE of residuals

Estimated 2-u

value

Interval

1.055 x 10-s 4.258 x lO-7

2.783 x IO+ 1.189 x lo+

7.199X 10-3

the estimates are preferred. These intervals are strictly

valid only if the parameter estimates are independent

and normally distributed. Figure 2 illustrates the

good experimental data/model agreeme nt for three of

the eight steady-state experiments. Th e sam ple given

represents low; m edium- and high-temperature oper-

ation.

As stated earlier the heat capacity of the solid

phase was estimated to obtain a good dynamic fit.

The rate parameters were also re-estimated using the

steady-state parameter estimates as initial guesses.

Table 4 lists the parameter estima tes obtained w hen

the dynamic experimental data were employed. Be-

cause the rate parameters varied only slightly from

the values obtained using steady-state data and since

all parameters were well determined, we may con-

clude that the good dynamic fit illustrated in Fig. 3

for the seven equally spaced axial centerline bed

temperature measurements was primarily achieved

via the solid heat capacity estimate. Note that the

dynamic data led to sma ller 2 - o intervals for the

kinetic parameters. Figure 3 shows experiment

rb7241d. The rem aining dynamic experim ents be-

haved similarly. We may also conclude that the mode l

was valid over the nonlinear operating space given by

the conditions in Table 2.

6. CONTROLLER DEVELOPMENT

There are two ways of performing model-predictive

control calculations. T he first method is sequential

and employs separate algorithms to solve the differ-

ential equations, and carry out the optimization.

First, a manipulated variable profile is guessed, and

the differential equations are solved numerically to

obtain an open-loop variable profile. Based upon the

numerical solution, the objective function is evalu-

ated. The gradient of the objective function w ith

respect to the manipulated variable is determined

either by finite differencing or by solving sensitivity

equations. Finally, the control profile is updated

using some optimization algorithm, and the process

repeated until the optimal profiles are obtained. This

constitutes a sequential solution and optimization

strategy, and recent versions of this strategy have

been reported by: Asselmeyer (1985), Morshedi

(1986), Jang et al. (1987), Kiparissides and Georgiou

(1987) and Peterson er al. (1989). The av ailability of

accurate and efficient integration and optimization

packages permits implementation o f this method

with little programm ing effort. However, constraint

handling is poorer than in an alternative method

which uses a simultaneous solution and optimization

strategy.

When the second or simultaneous approach is

adopted, the model differential equations are dis-

cretized,

and along with the algebraic model

equations are included as constraints in a nonlinear

programming (NLP) problem. The optimization

of the objective function is performed such that

460

440

420

E

3

380 t

z

P

360

p1

340

t

model - rb7241a -

model - rb724lb ----.

model - rb7241c -----

rb724la

rb7241b +

rb7241c D

320

I

I .

0 0.2 0.4 0.6

0.8

1

Normalized Axial Length

Fig. 2. Steady -statemodel predictions

nd experimental observations

for experiments b724

a,

rb724

b

rb724 lc.


9/20

Nonlinear model predictive

control 91

Table 4. Optimal parameter estimates from dynamic experiments

timization. The num ber of NLP con straints arising

Estimated 2-a

Parameter value Interval

from sequential optim ization and solution is indepen-

A 1.070 x 10-S 3.210 x IO-

dent of the prediction horizon.

6

2.734 x 1O+4 9.953 x IOf

Cp* 4.480 x 10-l 6.123 x IO-

Having selected a sequential optimization and sol-

7- =297C

ution strategy, we opted for a feasible path approach

m

SE of residuals 7.282 x 1O-3

vs

an infeasible path strategy. Infeasible path strat-

egies do not require that the constraints be satisfied

(the model equations be solved) at each iteration, but

the discretized model differential equations are sat-

find the optimum and satisfy the constraints simul-

isfied and other constraints on the states and manip- taneously. Feasible path strategies, on the contrary,

ulated variables are met. Key results em ploying satisfy the constraints at each iteration while seeking

this method have been reported by Hertzberg and

the optimum. The sequential optimization and sol-

Asbjornsen (1977), Biegler (1984), Cuthrell and ution strategy is necessarily a feasible path technique

Biegler (1987), Renfro et al. (1987), P atwardhan et al. (at least in terms of the model equations), but the

(1988, 1989, 1990, 1991) and Eaton and Rawlings simultaneous optimization strategy can employ either

( 1990). a feasible or infeasible path solution strategy. While

In this work, a sequential optimization and sol- infeasible path strategies appear to be computation-

ution strategy was employed. We justified this choice

ally more efficient, the feasible path techn ique offers

for experimental application based predominantly

several advantages. The first of these is that if the

upon the dimensionality of the NLP which aro se optimizer sh ould fail, the controller need only com-

for the two strategies. The num ber of constraint pare the values of the objective function at the

equations arising from discretization in the simul-

beginning of the optimization to the value at failure.

taneous optimization and solution strategy varies

If this value improves, the controller output can be

directly with the prediction horizon (PH), an integer

implemented. When infeasible path strategies fail,

multiple of the sampling time. If the model order is

there is no direct recourse since the model is usually

sufficiently large, the computational burden of a large infeasible. Another advantage of feasible path tech-

prediction horizon becomes more intense than that

niques is that one enjoys the luxury of intentionally

associated with the integrations required in the

solving the NLP suboptim ally. This becomes import-

sequential strategy. Moreover, the relative increase in

ant for real-time implementation since the solution of

the computation time required wh en the prediction

each NLP must not exceed som e fixed time, usually

horizon is extended is smaller for the sequential

the sampling interval. If the time limit is approached

strategy than for the simultaneous approach to op-

in the course of solving the NLP, optimization can be

460

440

420

400

z

-

t

380

2

::

360

H

d

340

320

300

280

Experimental Data -

Model Data ----.

0

50 100 150 200 250 300 350

400 450

Time minutes)

Fig. 3. Dynamic model predictions and experimental observations for experiment rb7241d (see Table 2,

experiment number 3 for conditions).


10/20

92

G. T. WRIGHTand T. F. EDGAR

halted and the solution from the last complete iter-

ation can be implemented.

6.1. Sequenti&

so l u t i o n an d op t im i z a t i o n st r a t e gy

The nonlinear model predictive controller im-

plemented in this research was formulated as the

following NLP:

mip @[x(ri), u(zi)] i = 1,2,. _ , PH,

subject to satisfying:

1.

2.

3.

4.

5.

6.

Model differential and algebraic equations:

E(t)* = f]x(O, uU)l> 8)

where x(2) E W,

u(t) E 4t, E(t) E W X ,

rank[E(t)]


11/20

Nonlinear model

sequence of piecewise constant inputs are denoted by

x,,. In order to determine the gradient of the objective

function with respect to v, the sensitivity equ ations

for the states of the DAE must be determined.

Let W represent the sensitivity matrix of the state

vector with respect to u,:

w, = x*,.

Then the dynamic evolution of W, is determined by:

E(t)W, = J(x, v)W, + B,(x, v),

i = 1, . . . , CH,

(11)

where

Bj(Xv VII =

f (x. q 1,

zj-, < t -c t,,

0

3

otherwise,

and J(x, v) is the Jacobian of f(x, v). The sensitivity

equations are subject to the initial conditions:

W,(r,) = 0.

7. EXPERIMENTAL CONTRO L STUDIES

As a consequenc e of the computational burden

associated with NMP C, a slow sampling rate was

required to accomm odate the relatively long compu-

tation times evolving from the solution process.

Because this factor also diminished the disturbance

rejection capabilities of NMPC , NMPC was im-

plemented in a master-slave cascade control

configuration where a low-level linear controller was

a significantly faster sampling rate was employed.

As described previously, the primary input to the

reactor was power to the inlet feed heater. W hile

varying the power level affected all bed temperatures

and compositions, we focused specifically upon its

impact on the active bed inlet temperature. Power

was never determined directly as the NMP C output.

Instead, NMPC determined a target value for the

active bed inlet temperature that would presumably

lead to the desired bed behavior. Recall that the

active bed inlet temperature constituted the inlet

boundary condition for the reactor energy balance.

This relationship was used to construct the NMPC

control strategy for the WGS reactor illustrated in

Fig. 4.

7.1. The WGS r eac t o r i n l e t t empe r a t u r e f oop

The open-loop reactor inlet temperature behavior

was influenced predominantly be feed-gas flowrate.

The inlet behavior was quite linear for flowrates

ranging from 9 to 13 SLM. Time constan ts varied

from 10.3 to 14.3 min and the dead time was approx.

3 min. The static gain w as 2.8 with a maximum

variation of 15%. Since the reactor inlet section w as

predictive control

__......................

93

Feed

R

e

a

:

0

a

-r ----------

*

Fig. 4. Cascade control con figuration for implementing

model-based control strategies.

packed with inert Pyrex glass beads, no reaction

occurred in this region. Therefore, heating and cool-

ing of the reactor inlet was virtually independent of

feed composition. However, when the reactor feed

stream was composed of a reactive gas mixture (e.g.

the gas mixture contained CO), a portion of the heat

generated from reaction in the active bed diffused

upstream to the inlet, marginally impacting reactor

inlet behavior. Because the static gain, time constant

and time delay varied little for the flowrates of

interest, PID control was used to close the loop.

In light of the master-slave, cascade control

configuration, it was imperative that the PID con-

troller (the slave) effectively track the reactor inlet

temperature target value computed by a model-based

controller (the master). The PID controller was

tuned, therefore, using an ITAE tuning rule for

set-point tracking. Figure 5 is typical of the closed-

loop response, which h ad the appearance of a first-

order plus dead-time transfer function step response.

The closed-loop time constant for the flowrates of

interest was approx. 6 min and the dead-time was

3 min.

We have already concluded that given flowrate an d

composition, the relationship between inlet bed tem-

perature and subsequen t bed behavior was well

defined. The relationship governing bed behavior as

a function of power to the inlet heater was poorly

defined and subject to disturbances which we re not

easily measured or modeled. Fortunately, the need to

model the reactor inlet was eliminated by assumin g

that the PID controller consistently generated a

first-order plus dead-time closed-loop response as

described above.


12/20

288

287

286

285

284

48

46

G

T. W RIGHT and T. F. EDGAR

1

44

1

0 20

40 60

80 100

T i m e m i n u te s )

Fig. 5. C lose-loop response of the reactor inlet temperature for a 5C set-point increase and a Aowrate

of 13 SLM.

7.2 . Closed- loop NM PC exper im ents

For all NMPC experiments the control horizon

CH was unity, permitting only one manipulated

variable move over the entire time horizon. The

prediction horizon PH was 24 sampling intervals or

120 min. Aggressive control is generally achieved for

large CH and small PH. Our objective, however, was

not to demonstrate aggressive control, but smooth

consistent control over a broad operation region.

Therefore, the tuning parameters were selected ap-

propriately. Furtherm ore, a larger control horizon

would have made the problem computationally in-

feasible for real-time application on the WGS system

because it would have been accompanied by an

increased number of optimization variables.

Although every attempt was made to make the

NMPC algorithm computationally efficient, the sol-

ution of each NLP required from 3 to 4.5 min. In

light of this, the sampling interval

T ,

for control of

the sixth bed temperature was chosen to be 5 min.

This value complied with established guidelines pre-

sented by Seborg et a l . (1989) and Astrom and

Wittenmark (1990), who suggest that the sampling


13/20

Nonlinear model predictivecontrol

95

rate be less than a tenth of the dominant time

constant or that the ratio of the sampling rate to the

time constant lie between 0.1 an d OS. The dominant

time constant for the system was approx. 55 min and

the

dead time, 30 min. A nominal dry-gas inlet com-

position of 40% CO, 40% CO2 and 20% H2 was used

for all experiments unless otherwise stated. The volu-

metric dry-gas to steam ratio was 0.625, and the total

gas flow was 13 SLM.

The first experiment w as intended simply to

demon strate that NMP C handles set-point tracking

smoothly and efficiently. Figure 6 depicts the closed-

loop response of bed temperature 6 to a 16.5%

step-change in its set-point. For clarity, we reiterate

that the manipulated variable for the NMP C loop

was the inlet temperature set-point, and that the

manipulated variable for the PID

loop was power to

the inlet heater. NMP C was permitted to change the

325

I

I

320

G

.m

t

~______________________________._____~

300

240

288

286

284

282

280

278

276

3

44.0

2

H

42.0

set

-Point -----

Outpuf -

____________I

4

1

set-POi

nt

output

I

1

c.

I

0 50

100 15

200 250

300

Time (minutes)

Fig. 6. NMPC experimentnumbe r I: WGS reactor response o a 16.5C step increase n the set-point

for bed temperaturenumber 6.


14/20

96


inlet temperature set-point by no more than +2.5 C

per control interval. Absolute limits of 270 and 300C

were also enforced.

Although small oscillations of f 1C persisted, it is

apparent that NM PC achieved the desired set-point.

These minor oscillations were the direct result of

small inlet temperature oscillations about the inlet

temperature target value. Since NMP C is model-

based, dead-time compen sation is inherent, provided

360

350

340

330

320

300

290

286

284

42.0

36.0

the model accounts for it. For the WGS reactor, a

temperature variation at the inlet initiates a thermal

wave, which amplifies as it propagates through the

active bed. This phenome non effectively creates a lag

that would be modeled as pure delay in a transfer

function representation of the system. Notice that

NMP C required only three sampling intervals to

determine the inlet temperature that would drive bed

temperature 6 to set-point. Furthermore, once this

I

1

1 1

Set Point

ourput -

Set Point

output -

0 100

200 300

400 500

Time (minutes)

Fig. 7. NMPC experiment number 2: WGS reactor response to a sequence of set-point increases for bed

temperature number 6 which spans the operating space for nominal feed composition and flowrate.


15/20

Nonlinear model predictive ontrol

value had been determined, manipulated variable

changes virtually ceased, despite the initial absence of

response of temperature 6. This example and others

that follow powerfully illustrate the inherent dead-

time compen sation of NMP C.

The second experiment was intended to illustrate

the ease with which NMPC progresses from a

state of virtually no reaction to a state of almost

complete reaction when applied to the WGS reactor.

This example highlights the effective use of NMPC

for plant start-up. Figure 7 shows a sequence of

set-point changes. The first was an 18S C step-

increase from 306.5 to 325C and the second a

25C step-increase to 350C. A velocity constraint

permitted NM PC to manipulate the inlet temperature

set-point by n o m ore than + l.O C per control

interval.

We note at this point that when Ziegler-Nichols

tuning rules are adopted for PID tuning, positive

static gain variations should not exceed approx.

lOO%, and even this value is borderline. While more

advanced PID tuning strategies have been de-

veloped more recently, this rule of thumb still loosely

applies.

As with the previous example delay time was easily

accomm odated as evidenced by the absence of exces-

sive manipulated variable move ment. More signifi-

cant, however, was the successful handling of the

static gain variations. Figure 7 clearly illustrates

that for an 18.5C change in the output, an inlet

temperature change of approx. 5C was required.

For the subsequent 25C output increase, which

occurred at higher CO conversion, an inlet tempera-

ture change of approx. 2.4C was required, a tripling

of the static gain. NMPC inherently recognized these

gain varitions and responded accordingly. For the

same operating conditions, Fig. 8 illustrates the

poor simulated response achieved using a PID con-

troller, tuned with ITAE rules for set-point tracking.

The third and final NMP C experiment dealt

with the disturbance rejection capabilities of NMPC.

First, steady-state was achieved with an output set-

point of 310C. At time equal to 30 min, the dry-gas

flowrate was decreased by 10% to 4.5 SLM. The

dry-gas com position and steam flowrate were not

altered. Since flowrate and composition measure-

ments appeared as parameters in the NMP C model,

an inherent feedforward action caused an immed iate

drop in the inlet temperature set-point (Fig. 9, arrow s

mark the flowrate decrease). This occurred before any

significant response in bed temperature 6, the feed-

back variable. In fact, the output only began to

respond approx. 10 min later.

This disturbance rejection example highlights

a flaw of the NMP C control implementation.

When constructing the controller, it was assumed

that the inlet temperature loop had a perfectly

consistent first-order response for set-point track-

ing. Figure 9 clearly demonstrates that this

assumption was violated in the presence of a dis-

turbance. In the next section we discuss the rami-

fications of this assump tion by examining the

model states with and without inlet temperature

feedback.

97

370

360

350

iz

z 340

2

::

1 330

300 I

1

0 100

200 300 400 500 600

Time (minutes)

Fig. 8. Simulated WGS reactor response to a sequence of set-point increases for bed temperature

numb er 6, wh ich spans the operating space for nominal feed composition and ilowrate.


16/20

98

G T. WRIGHTZUI~ T F EDGAR

set-Polnc ----

output - _

320

318

316

314

312

310

308

306

286 0

265 0

284 0

282 0

281 0

260 0

40 0

39 5

39 0

36 5

36 0

37 5

37 0

36 5

36 0

35 5

35 0

Set-Point ----.

output -

1

1 1

0

so

100 150 200

Time lminucesl

Fig. 9. NMF C experiment number 3: WGS reactor response to a 10% step decrease in the nominal dry-gas

flowrate.

7 . 3 . Compa r i son o f p l an t and mode l ou t pu t s f o r

NMZC

Experiment three clearly demonstrated a violation

of the assumption that the closed-loop behavior

of the inlet temperature loop was first-order. The

unexpected response of bed temperature 1 was a

consequence of the 10 step-decrease in the dry-gas

flowrate. Figure 10 compares the output response

(bed temp erature 6) actually experienced in the plant

to the model response. Notice that the model tem-

perature increased slightly d ue to the decreased dry-

gas flowrate (arrows mark the flowrate decrease).

When compared to the actual temperature increase in

the plant, however, the model temperature increase

was small.

The temperature increase in the plant output was

the cumulative effect of increased residence time,


17/20

330

325

310


I

Plant

-

Model vf Inlet Temp. Feedback

__-_.

Model w/o Inlet Temp. Feedback --.--

99

0

50 100

150 200

Time (minutes)

Fig. 10. Comparison of plant and model states with and without inlet temperature feedback when a 10

step decrease in the nominal dry-gas flowrate is applied to the reactor system.

which permitted more reaction, and increased inlet

temperature caused simply be a slower rate of heat

removal. Only the first of these effects w as considered

in the model. Incorportion of the second effect would

have required that the reactor inlet section be mod-

eled. Such a model would only increase the overall

reactor model size while adding inform ation that can

be otherwise accounted for. Figure 10 shows a closed-

loop simulation where the actual reactor inlet tem-

perature is used as an input to the model. In this case,

the model-output time derivative was almost pre-

cisely that experienced by the plant. A sm all, slowly

varying bias ranging from 5 to 10C persisted, but the

feedback mechan ism of NMPC is designed to effec-

tively deal with this phenom enon. We conclude,

therefore, that better plant-model agreement is

achieved if the actual inlet temperature is used as an

input to the model. Inlet temperature feedback to

reset the model boundary conditions would effec-

tively achieve a feedforward control strategy (Wright,

1992).

Figures 11 and 12 illustrate that an inlet tempera-

ture feedback would be much less significant for

set-point tracking. In fact, for experiment two there

was no substantial distinction between the model

output with or without feedback. As with experiment

three, the bias varied slowly here, increasing with high

temperature operation. In experiment one, use of

380

1

1

__----_

--._

370 -

,,~~:___---~-~ ---r..__

../-

-9

360 -

,.;?

:,s

:

u

350 -

E

J

E

340 -

P

8

330 -

Plant

-

Model w/ Inlet Temp. Feedback

_____

Model w/o Inlet Temp. Feedback ----_

l-

300 1 i

0 100

200 300

400 500

Time (minutes)

Fig. 11. Comparison of plant and model states with and without inlet temperature feedback when a

sequence of set-point changes was applied to bed temperature 6.


18/20

G. T. WRIGHT and T. F.

EDGAR

325

Model w/

Inlet Temp. Feedback

__--.

Model w/o Inlet Temp. Feedback -----

300 1 I 1 1

0 50 100 150 200 250 300

Time (minutes)

Fig. 12. Com parison of plant and mode l states with and without inlet temperature eedback when a

16SC step-point ncreasewas appliedto bed tem perature .

inlet temperature as a mo del input had a marginally

greater effect than for experiment two. Even so, the

output behavior in the plant could not be captured

for the time interval ranging from 100 to 175 min.

This example illustrates, however, that NMP C is

robust to plant-model mismatch. It is clear, that in

the absence of repeated disturbances, both techniques

lead to the same model output, but inlet temperature

feedback may significantly affect transient behavior.

7.4.

Compar i son w i th c l osed - l oop GPC

Adaptive GPC was implemented using the control

structure outlined for NMP C. Unlike the NMP C

experiments, how ever bed tempera ture 4 w as taken to

be the controlled variable. Because the computation

time required for adaptive GPC was small, a

sampling time of 2 min, based solely upon the open-

loop dynamics of bed temperature 4, was adopted.

For an inlet temperture of 280C and nominal values

of composition and flowrate, the dead-time was

approx. 14min, the time constant 25 mm , and the

gain 1.5. The gain increased by approx. 120% from

this low reaction state to a state of com plete reaction.

The control experiment described below used a

prediction horizon of 20 sampling intervals, and a

control horizon of unity. A move supression factor of

10 was also employed, and GPC was permitted to

change the set-point by no more than _t 1C per

control interval. The recursive least squares esti-

mation algorithm of Chen and Norton (1987) was

employed for parameter estimation. The model took

the form:

A (4 -ly(t) = q -B(q -)24(t) + ci,

(12)

where A (q -) and B(q - ) are polynomials in the

backward shift operator of orders 1 and 3, respect-

ively.

Figure 13 illustrates a sequence of three 5C step-

increases in the target value for bed temperature 4.

While the close-loop response for the first and second

increments were satisfactory, it is clear that the

response became progressively worse with increasing

operating temperature. The oscillatory behavior was

obtained despite parameter adaptation. In addition,

this control pro blem was less challenging than the

problem to which NMP C was applied. We con-

cluded, therefore, that traditional adaptive control is

not well suited for WG S reactor start-up, since the

linear model, even with parameter adaptation, does

not a dequately reflect the rapidly changing nonlinear

dynamics of the system.

8.

CONCLUSlONS

The primary goal of this research was to develop

an advanced nonlinear control strategy for fixed-bed

catalytic reactors. The control method was applied

experimentally using a laboratory-scale water-gas

shift WGS reactor. The following conclusions may be

drawn from the results of this work.

An adiabatic, pseudo-hom ogeneous WGS reactor

mode l represented the physical system well over the

operating space of interest. The physically reason-

able, simplifying assusmptions that were adopted

proved useful in developing a low-order m odel, suit-

able for implementation in an NM PC framework.

The Galkerin technique on finite elements with piece-

wise linear polynomial approximations proved not to


19/20


306

302

296

286

284

Set Point

output -

46.0

3 50 100 150 200 250 300 350 400

450

Time (minufcr)

Fig. 13. Adaptive GFC experiment: W GS reactor response to a sequence of set-point increases for bed

temperature numbe r 4 for nominal composition and flowrate.

be susceptible to oscillatory behavior over the spatial

domain when 12 nodes were employed for discretiza-

tion. Estimation of dynamic and steady-state

parameters were efkctively decoupled for the pseudo-

homo geneous W GS reactor model. Furthermore, in-

formation-rich dyn amic data yielded good parameter

estimates w ith less experimental effort.

The control experiments demonstrated that absol-

ute plant-model agreeme nt was not imperative for

good control using NMPC . However, temporal first-

derivative information,

consistent with

plant

behavior, wa s crucial to good performance. NMP C

was better suited for feedforward dynam ic co mpen-

sation than linear techniques since the nonlinear

model has an inherent characterization of the feed-

forward mechanism. Feedforward control signifi-

cantly

enhances NMP C performance. Finally,

NMP C was effectively used to start-up the WGS


20/20

102


system.

NMPC

was

superior in this regard to tra

ditional control techniques since broad nonlinear

operating regions were traversed. Adaptive linear

control appears to be unsuitable, since parameter

estimates varied a s rapidly a s the state, making

successful parameter adjustment extremely difficult.

REFERENCES

Ampaya J. P. and R. G. R inker, Autothermal reactors with

internal heat exchange. J. C h e m . E n g n g S c i . 3 2 , 13 2 7

( 1 9 7 7 ) .

Asselmeyer B., Optimal control for nonlinear systems calcu-

lated with small computers.

J . O p t . T h e or y A p p l i c . 4 5 , 5 3 3

( 1 9 8 5 ) .

Astrom K. J. and B. Wittenmark, C o m p u t e r C o n t r o l l e d

S v s t em s : T h e or v a n d D e s n . 2nd Edn. Prentice-Hall. New

Jkrsey ( 1 990 ) . _

Bates D. M. and D. G. Watts, N o n l i n e a r R e g r es si o n A n a l y si s

a n d I t s A i c a t i o n s . Wiley, New York (1988).

Bell N. H.-&d T. F. Edgar, Modeling. of a fixed-bed

water-gas shift reactor 1: Steady-state model verification.

J. P r o c e ss C o n f r o l 1 , 2 2 - 3 1 ( 1 9 9 1 ) .

Biegler L., Solution of dynamic optimization problems by

successive quadratic programming and orthogonal collo-

cation.

C o m p u t e r s

them. E ngng 8, 243 (1984).

Bonvin D., Dynam ic modeling and control structures for a

tubular autothermal reactor at an unstable state. Ph.D.

Thesis, University of California at Santa Barbara (1980).

Brockett R. W ., Feedback invariants for nonlinear systems.

In P r o c . 7 t h Z F A C W o r l d C o n g r . , Helsinki (1978).

Caracotsios M., Model parametric sensitivity analysis and

nonlinear parameter estimation-theory and appli-

cations. Ph.D. Thesis, The University of Wisconsin at

Madison (1986).

Newsom e D. S., The water-gas sh ift reaction. Cu f a l . R e v . -

S c i . Engng 21, 275 (1980).

Patwardhan A. A. and T. F. Edgar, Nonlinear model-

predictive control of a packed distillation column. In A m .

C o n t r o l C o n f . , Boston, MA (1991).

Patwardhan A. A ., J. B. Rawlings and T. F. E dgar, Model

predictive control of nonlinear processes in the presence

of constraints. In

A Z C h E J I A n n u a l M e et i n g ,

Washington,

DC (1988).

Patwardhan, A. A., J. B. Rawlings and T. F . Edgar, Model

predictive control of nonlinear processes in the presence

of constraints. In Z F A C J Z E E E S ym p . o n N o n l i n e a r C o n -

t r o l S ys t em s D e s i g n , pp. 45&460 (1989).

Patwardhan A. A., J. B. Rawlings and T. F. Edgar,

N o n l i n e a r m o d e l predictive control. C h e m . E n g n g C o m -

m u n . 8 7 , 1 2 3 ( 1 9 9 0 ) .

C a r b e r r y

J. C.

a n d M . M .

Wendel, A

c o m p u t e r

model of the

fixed bed catalytic reactor: the adiabatic and quasi-adia-

batic cases. A Z C h E J f 9 , 1 2 9 - 1 33 ( 1 9 6 3 ) .

Chen M. J. and J. P. Norton, Estimation technique for

tracking rapid parameter changes. ht. J . C o m r o l 45,

1387-1398 (1987).

Clarke, D. W., C. Mohtadi and P. S. Tuffs, Generalized

predictive control-part i the basic algorithm. A u t o m a t i c

2 3 , 1 3 7 - 1 4 8 ( 1 9 8 7 ) .

Cuthrell J. E. and L . T. Biegler, On the optimization of

different/algebraic process systems. A Z C h E J I 3 3 , 1 2 57

( 1 9 8 7 ) .

Cutler C. R. and B. L. Ram aker, D ynamic matrix control-

a computer control algorithm. In J o i n r A u t o m a t i c C o n t r o l

C o n f . P r o c . P a p e r W P - 5 B San Francisco, CA (1980).

Danckwerts P. V., Continuous flow systems-distribution

of residence times. Chem. Engng Sci. 2, 1-13 (1953).

Eaton J. W. and J. B. Rawlings, Feedback control of

chemical processes using on-line optimization techniques.

C o m p u t e r s t h e m . E n g n g 1 4, 169-479 (1990).

Eaton J. W. and J. B. Rawlings, Model predictive control

of chemical processes. Chem. Engng S c i . (1991).

Hertzberg T. and 0. A. Asbjomsen , Parameter estimation

in nonlinear differential equations. In

C o m p u t e r A p p i i -

c a t i o n s i n t h e A n a l y s i s o D a t a a n d P l a n t s . Science Press,

Princeton (1977).

Peterson T. E., E. Hern andez, Y. Arkun and F. J. Schork,

N o n l i n e a r

predictive

c o n t r o l

of a semi-batch polymeriz-

ation reactor by an extended dmc, Paper No. tp4. In A m .

C o n f r o l . C o n f . , Pittsburg, PA (1989).

Petzold L. R., A description of DASSL: a differential-

algebraic system solver. In S c i en t i f i c C o m p u t i n g (R. S.

Stepleman, Ed.), pp. 65 -68 . North-Holland, Amsterdam.

Rase H. F.,

C h e m i c a l R e a c t or D e si g n

or

P r o c e s s P l a n t s ,

Vol. 1. Wiley, N ew York (1977).

Rawlings J. B. and K. R. Muske, The stability of con-

strained receding horizon control. Submitted to I E E E

T r a n s . A u t o . Contr., Sept. (1991).

Renfro J. G., A. M. Morshedi an d 0. A. Asbjornsen.

Simultaneous optim ization an d solution of systems de-

scribed by differential/algebraic equations. C o m p u t e r s

t h e m . E n g n g 11, SO3 1987).

Seborg D. E., T. F. Edgar and D. A. Mellichamp, P r o c e s s

D y n a m i c s a n d C o n t r o l . Wiley, New York (1989).

U n i t e d C a t a l y s t s C - I 2 O p e r a t i n g I n s t r u c t i o n s .

Windes, L. C., Modeling and control of a packed bed

reactor. Ph.D. Thesis, The University of Wisconsin at

Madison (1986).

Windes, L. C., M. J. Schwedock and W. H. Ray. Steady

state and dynamic modeling of a packed bed reactor for

the partial oxidation of m ethanol to formaldehyde-I,

model development. C h e m . E n g n g C o m m u n . 7 8 , 1 43 .

( 1 9 8 9 ) .

Hunt L. R., R. Su and G. Meyer, Global transformations Wright G. T., Modeling and nonlinear predictive control of

of nonlinear systems. I E E E T r a n s . A u t o m a t . C o n t r . A C -

a fixed-bed water-gas shift reactor. Ph.D. Thesis, The

2 8 , 2 4 ( 1 9 8 3 ) . University of Texas at Austin (1992).

Jang S., B. Joseph and H. Mukai, On-line optim ization of

constrained multivariable processes. A Z C h E J f 33, 26

(1987).

Kiparissides C. and A. Georgiou, Finite-element solution of

nonlinear optimal control problems with a quadratic

performance index. C o m p u t e r s t h e m . E n g n g 11,77 ( 1 9 8 7 ) .

Lasdon L. S. and A. D. Waren, G R G U s er s G u i d e . Techni-

cal Report, University of Texas at Austin (1986).

Lee A. L., Effect of raw product gas composition on shift

catalysts. Technical Report Project 61015, DOE Contact

DE-AC 2I-78ET11330, Institute of Gas Technology.

Chicago (1980).

Moe J. M., Design of water-gas shift reactors. Chem. E n g n g .

P r o c . 5 8 , 33 (1962).

Mor& J. J., B. S. Garbow and K. W. Hillstrom, User G u i d e

f o r M i n p a c k - 1.

Technical Report ANL-80-74, Argonne

National Laboratory (1980).

Morshedi A. M ., Universal dynamic matrix control. In

C h e m i c a l C o n t r o l C o n f . I Z I , S es si o n V I , P a p e r N o 2,

Asilomar, CA (1986).

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