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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
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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
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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)
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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
G. T. WRIGHT and T. F. EDGAR
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
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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
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90
G. T. WRIGHT and T. F. EDGAR
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.
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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).
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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)]
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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.
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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
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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.
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96
G. T. WRIGHT and T. F. EDGAR
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.
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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.
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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,
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330
325
310
Nonlinear model predictive control
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.
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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
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Nonlinear model predictive control
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
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102
G. T. WRIGHT and T. F. EDGAR
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 ) .
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