**
**Implementation Trial of Input-output Linearizing Control on an
Industrial Evaporation Simulator

Kam, K. M.^{1}, Tadé, M. O.^{1} and Le Page, G. P.^{2}

1 School of Chemical Engineering, Curtin University of Technology,
GPO Box U1987, PERTH, Western Australia 6845, Australia

^{
2 Alcoa of Australia Ltd, P. O. Box 161, KWINANA, Western
Australia 6167, Australia
Abstract
An evaporation simulator is being used as a real time simulation
platform for the evaporation stage of the liquor burning process associated with the Bayer
process for alumina production at Alcoa’s Wagerup refinery. The simulator consists of
a falling film effect, three forced circulation effects and one super-concentrator effect.
It is currently being used for on-site operator training, as well as for performing
control studies and controller tunings for the evaporation section prior to actual plant
implementation.
This paper is devoted to the implementation trial of multi-input
multi-output (MIMO) globally linearizing control (GLC) structure of Kravaris and Soroush
(1990) on the simulator. Implementation issues for the nonlinear control trial, such as
handling of plant input’s hard constraints, linking the conventional industrial PI
controllers and the nonlinear control algorithms, initialisation of industrial PI
controllers during start-up of the nonlinear control algorithm are discussed.
Key words:
globally linearizing control; input-output linearization; liquor
burning process; evaporation simulator; implementation trial
Implementation Trial of Input-output Linearizing Control on an
Industrial Evaporation Simulator
Kam, K. M., Tadé, M. O. and Le Page, G. P.
1.0 Introduction
Nonlinear control using input-output linearization technique (Isidori,
1995; Slotine and Li, 1991) for chemical process control has received significant
attention in recent years. In particular, numerous studies have been carried out, both by
simulation and experimental, on the evaporator due to its importance in the chemical
process industries. Montano et al (1991) demonstrated the globally linearizing
control (GLC) of Kravaris and Chung (1987) on a simulated double-effect evaporator. To et
al (1995) and To et al (1997) applied the multi-input multi-output (MIMO) GLC
structure of Kravaris and Soroush (1990) on an industrial single-effect evaporator by
simulation and plant implementation, respectively. The application of MIMO GLC structure
on a simulated industrial triple-effect evaporator was shown by To (1996). Kam et al
(1998a) demonstrated the MIMO GLC structure on a simulated five-effect evaporation system
of an alumina refinery. Simulation and implementation results obtained by the various
researchers indicate that the control performance of input-output linearizing control was
superior to those of linear control using single-input single-output (SISO)
proportional-integral (PI) controllers.
This paper is devoted to the implementation trial of MIMO GLC structure
on a simulator of an industrial five-effect evaporation stage of the liquor burning
process associated with the Bayer process for alumina production at Alcoa’s Wagerup
alumina refinery. The implementation trial on the simulator was carried out to ascertain
the control performance and robustness of the nonlinear control structure on the
evaporation system prior to its implementation on site. The design and implementation
procedures for the nonlinear control structure are discussed.
2.0 THE EVAPORATION SIMULATOR
The evaporation simulator is created by a group of connected basic
unit operation modules from SACDA’s module library. The simulator consists of a
falling film effect, 3 forced circulation effect and a super concentrator. Each effect is
modelled by a combination of pressurised vessel modules representing flash tank volumes,
the shell side of the heater and flash pots. Heater modules and thermal capacitance
modules are used to model the heat exchanges between tubes and shells of heaters.
The simulator was developed to have a one-to-one correspondence to the
five-effect evaporation stage of the liquor burning process associated with the Bayer
process for alumina production at Alcoa’s Wagerup alumina refinery. Similar to the
evaporation system on site, the simulator is connected to a Honeywell distributed control
system (DCS). It has been used extensively for operator training and, studies of control
techniques and tuning of controllers on the evaporator.
3.0 MIMO GLOBALLY LINEARIZING CONTROl
Consider a square minimum-phase MIMO nonlinear system with
structurally non-singular characteristic matrix, C(x) as shown (Isidori,
1995; Slotine and Li, 1991),
( 1)
where f, g1, g2, …, gm
are smooth vector fields on Rn, and h1, h2,
…, hm(x) are scalar fields on Rn, x
is n´ 1 dimensional state vector, and u = [u1
u2 … um]T and y = [y1
y2 … ym]T are m-dimensional
input and output vector, respectively.
The modified MIMO GLC structure of Kravaris and Soroush (1990) consist
of state feedback control laws and m external SISO PI controllers to achieve an
overall desired closed-loop performance of the outputs, yi’s. The
state feedback control laws take the form,
( 2)
where v is m´ 1 dimensional
vector of external inputs, and A(x) and B(x) are m´ m matrix and m´ 1 vector
given as,
( 3)
( 4)
where {b ij} are the
design parameters of the state feedback control laws. If the process model is perfect, the
state-feedback control laws yield a linear input-output map of the MIMO nonlinear system
in Equation (1) of the form,
( 5)
The input-output sub-system in Equation (5) is decoupled since the
output, yi is only affected by only one external input, vi.
An external SISO PI controller, as shown in Equation (6), is used to
regulate each input-output pair, (vi-yi) of the
sub-system in Equation (5),
( 6)
where ts is the sampling or scan rate for the PI
controller. The overall closed-lop transfer function (CLTF) for ith
output is given as (Kam and Tadé, 1998a),
( 7)
and the characteristic equation for the CLTF is approximated as,
( 8)
The design parameters, {b ij}
and the PI settings are selected to ensure that all poles of the overall CLTF are on the
left-half of the s-plane (ie. all roots of Equation (8) have negative real parts), so that
the closed-loop feedback control system is stable.
4.0 MIMO GLC OF SIMULATOR
The modified MIMO GLC structure was applied to the first four
evaporator effects of the simulator. The state feedback control laws for the simulator
were formulated from the dynamic equations of the first four effects of evaporator model
M2 of Kam et al (1998b) using Maple (Redfern, 1996) procedures io of Kam et
al (1998c). The equations for the state feedback control laws are not given here due
to space limitations and can be found in Kam and Tadé (1998d).
The state variables were the liquor levels, densities and temperatures
of the flash tanks. The controlled variables were the liquor levels of the flash tanks and
the liquor density of flash tank #4 while the manipulated variables were the liquor
product flow-rates from the flash tanks and steam flow-rate to heater #4. The state,
manipulated and controlled variables of the simulator are summarised in Table 1.
Table 1: State, manipulated and controlled
variables for the simulator
States
Manipulated
Controlled
x1
h1
u1
QP1
y1
h1
x2
h2
u2
QP2
y2
h2
x3
h3
u3
QP3
y3
h3
x4
h4
u4
QP4
y4
h4
x5
r
4
u5
y5
r
4
x6
T1
x7
T2
x8
T3
x9
T4
x10
r
1
x11
r
2
x12
r
3
4.1 Implementation of State Feedback Control Laws and Tuning of PI
Controllers
The state feedback control laws (eg. Equation (2)) for the simulator
were implemented as a set of algebraic equations with the industrial PI controllers (eg.
Equation (6)) in cascade arrangement as shown in Figure 1.
Figure 1
Implementation of State Feedback Control Laws
All equations for the state feedback control laws were implemented into
the DCS using Honeywell control language (CL) code. The following digital PI controllers
and the stat feedback control laws were implemented,
( 9)
( 10)
Due to the cascade arrangement in Figure 1, the outputs
of the industrial PI controllers and the state feedback control laws are referred as the primary
inputs and secondary inputs, respectively. The implementation algorithm for the
nonlinear control structure in DCS is given in Figure 2.
Figure 2
Algorithm of Nonlinear Control for Simulator
Design parameters for the state feedback control laws and tuning
parameters for the PI controllers used for the nonlinear control trial are given in Table
2. The design parameters b i0’s were
selected so that the outputs of PI controllers, vi’s in Equation
(6) have the same reference points (ie. same steady state values) as the manipulated
variables, ui’s while b i1’s
were selected to improve the condition of the decoupling matrix, A(x) in
Equation (3) (Kam and Tadé, 1998a). The industrial PI tuning parameters were selected
based on the results of linear control studies that were performed on the simulator.
Table 2
Design parameters and tuning parameters for the simulator
implementation
Design
parameters
PI settings
b
i0
b
i1
KCi
= Kii (%/%)
t
Ii (min)
y1 (h1)
20.5
1
5
75
y2 (h2)
11.2
1
5
75
y3 (h3)
9.8
1
5
75
y4 (h4)
8.7
1
5
75
y5 (r 4)
6.8
10
1
25
4.2 Anti-reset Windup for PI Controller
Anti-reset windup for the integral action of the industrial PI
controllers were done by imposing the constraints of the secondary inputs to the primary
inputs directly,
( 11)
This was possible since the steady state values of the primary
inputs and the secondary inputs were the same due to the selection of design
parameters, b i0’s as discussed
previously. In order to impose upper limits and lower limits on the secondary inputs,
the following saturation law was used for each secondary inputs (ie. manipulated
variables),
( 12)
4.3 Initialisation of PI Controller
The PI controllers in Equation (6) were coded in the velocity form and
need to be initialised once the nonlinear control algorithms were started up.
Initialisation of the PI controllers were done by back transforming the current state and
manipulated variables to the external inputs during start-up by using the inverse of the
state feedback control law,
( 13)
where x0 and u0 are the state and
input vector of the simulator during start-up of the nonlinear control algorithms.
Values of the external inputs, v’s and the manipulated
inputs, u’s during the initialisation of the nonlinear control algorithms are
given in Table 3. Note that the values are expressed in term of the % of the actual values
(in engineering units) to their respective ranges, ie. ui*100%/(ui(max)-
ui(min)). Since the design parameters, b i0’s
were selected such that the primary and secondary inputs have the same
steady state values, the start up values for the manipulated and external inputs should
have been the same. The inconsistencies in their values were due to model uncertainties in
model M2 that was used to generate the state feedback control laws. Consequently, the
implementation tests also provided an indication of the robustness of the MIMO GLC
structure against modelling errors on the five-effect evaporator.
Table 3
External and Manipulated Inputs at Start-up
External inputs
(%)
Manipulated
inputs (%)
v1
50.2
u1
55.8
v2
39.6
u2
46.4
v3
35.5
u3
40.0
v4
34.0
u4
35.6
v5
63.0
u5
54.5
6.0 CONCLUSIONS
Implementation of multi-input multi-output (MIMO) linearizing control
(GLC) structure was carried out on an industrial five-effect evaporation simulator. Detail
design and implementation procedures for the nonlinear control structure were discussed.
It was shown that the MIMO GLC structure can be implemented in cascade arrangement to the
conventional SISO PI controllers.
ACKNOWLEDGMENTS
The authors are grateful to Alcoa of Australia Ltd. for allowing the
simulator to be used for the implementation trial and for allowing this paper to be
presented. The first and second authors would like to thank Mr Graham P. Le Page for his
tremendous effort and time in making the implementation trial a success. The support of an
Australian Research Council (ARC) grant is also acknowledged.
NOMENCLATURE
Abbreviations
CL control language
CLTF close-loop transfer function
FT flash tank
GLC globally linearizing control
ITAE integral time weighted absolute error
MIMO multi-input multi-output
PI proportional-integral
SISO single-input single-output
Symbols
h liquor level
KC proportional gain
KI integral gain
steam flow-rate
QP liquor product flow-rate
T liquor temperature
ts sampling time or scan rate
ui ith manipulated input/secondary
input
vi ith external input/primary input
yi ith controlled output
Greek symbols
b design parameters
r liquor density
t I integral time
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