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8/18/2019 A Gradient Optimization Scheme for Solution Purification Process 2015 Control Engineering Practice
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A gradient optimization scheme for solution purication process
Bei Sun a,c, Weihua Gui a,b,n, Yalin Wang a,b, Chunhua Yang a,b, Mingfang He a,d
a School of Information Science and Engineering, Central South University, 410083, Chinab Institute of Control Engineering, Central South University, 410083, Chinac Department of Electrical and Computer Engineering, Polytechnic School of Engineering, New York University, 11201, United Statesd Department of Computer Science and Engineering, Polytechnic School of Engineering, New York University, 11201, United States
a r t i c l e i n f o
Article history:
Received 1 June 2014Received in revised form
12 July 2015
Accepted 13 July 2015Available online 12 August 2015
Keywords:
Solution purication
Gradient optimization
Additive utilization ef ciency
Impurity removal ratio
Robust adaptive control
Oxidation reduction potential
a b s t r a c t
This paper presents a two-layer control scheme to address the dif culties in the modeling and control of
solution purication process. Two concepts are extracted from the characteristics of solution puri cationprocess: additive utilization ef ciency (AUE) and impurity removal ratio (IRR). The idea of gradient
optimization of solution purication process, which transforms the economical optimization problem of
solution purication process into nding an optimal decline gradient of the impurity ion concentration
along the reactors, is proposed. A robust adaptive controller is designed to track the optimized impurity
ion concentration in the presence of process uncertainties, disturbance and saturation. Oxidation re-
duction potential (ORP), which is a signicant parameter of solution purication process, is also used in
the scheme. The ability of the gradient optimization scheme is illustrated with a simulated case study of a
cobalt removal process.
& 2015 Published by Elsevier Ltd.
1. Introduction
Solution purication, which belongs to the more general area of separation science and technology, is a key step in hydro-
metallurgy (Flett, 1992). As a widely used approach to obtain pure
metals from their raw ores, hydrometallurgy involves phase and
status transforms of metal elements. Typically, a hydrometallurgy
process is composed of leaching, purication and electrowinning.
The raw ore is rst treated in the leaching process, in which the
valuable metal in the solid state ore is extracted and converted
into soluble salts in liquid solution. As leaching process is not
completely selective, pregnant leaching solution inevitably con-
tains undesired impurity ions. The presence of these impurity ions
would decrease the current ef ciency in the subsequent electro-
winning process in which pure metal is recovered, resulting in
energy waste and downgrade of product quality (Bøckman &
Østvold, 2000). Therefore, the pregnant leaching solution is re-
quired to be puried to a certain degree prior to nal metal
winning.
Owing to the heterogeneous property of raw ores, the types of
impurity ions in the leaching solution are not unique. Conse-
quently, a solution purication process is composed of several sub-
steps designed to remove different impurities. For example,
solution purication of zinc hydrometallurgy consists of copper
removal, cobalt removal and cadmium removal (Fig. 1). These
impurity ions possess different physical and chemical properties.
The technologies and reaction conditions adopted in the sub-steps
are not the same. However, these different sub-steps do share
some common features. The reactions conducted to remove dif-
ferent impurities are essentially oxidation reduction reactions and
require the use of additive and in some occasions catalyst (Sun,
Gui, Wu, Wang, & Yang, 2013).
The dosage of additive is crucial to both purication perfor-
mance and production cost. An excessive amount of additive is a
waste of costly material, while an insuf cient amount fails to re-
move the impurity adequately (Kim, Kim, Park, Song, & Jung,
2007). However, due to the inherent complexity of purication
reaction, uctuation of previous leaching process, stochastic dis-
turbance and interactions between the intermediate sub-steps
inside purication process, it is dif cult for the human operators to
adjust additive dosage precisely in order to achieve economical
and stable operation. As a consequence, some operators prefer to
use an excessive amount of additive to achieve the required pur-
ication performance. More seriously, a large process uctuation
may even cause the failure to meet the desired purication degree.
The existence of these problems has attracted the attention of
researchers from both metallurgy and control community. To the
author's best knowledge, the study on solution purication pro-
cess begins from 1871 (Bøckman & Østvold, 2000). After that, the
research on solution purication process has passed through two
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/conengprac
Control Engineering Practice
http://dx.doi.org/10.1016/j.conengprac.2015.07.008
0967-0661/& 2015 Published by Elsevier Ltd.
n Corresponding author at: School of Information Science and Engineering,
Central South University, 410083, China. Fax: þ 86 731 88876677.
E-mail address: [email protected] (W. Gui).
Control Engineering Practice 44 (2015) 89–103
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stages of rapid development. The rst stage of rapid development
is driven by the mechanism study of solution purication process
(Boyanov, Konareva, & Kolev, 2004; Bøckman & Østvold, 2000;Børve & Østvold, 1994; Dib & Makhlou, 2007, 2006; Dreher,
Nelson, Demopoulos, & Filippou, 2001; Fugleberg & Jarvinen, 1993;
Lew, 1994; Nelson, Wang, Demopoulos, & Houlachi, 2000; Polcaro,
Palmas, & Dernini, 1995; Sun et al., 2013; Tozawa, Nishimura,
Akahori, & Malaga, 1992; van der Pas & Dreisinger, 1996; Yang, Xie,
Zeng, Wang, & Li, 2006). It is found that the impurity removal
reaction is rst order solid-liquid phase kinetics/electrode reaction
with its reaction rate affected by dosage and particle size of ad-
ditive, temperature, pH and type of catalyst, etc. The second stage
of rapid development is caused by the application of control the-
ory to solution purication process which is usually conducted in
CSTRs (Continuous Stirred Tank Reactors). Wu (2001) studied the
use of a LMI (Linear Matrix Inequality) based robust model pre-
dictive controller to an industrial CSTR (Continuous Stirred Tank
Reactor) problem with explicit input and output constraints.
Knapp, Budman, and Broderick (2001) applied to a CSTR process
an adaptive algorithm which uses a neural network representation
to learn the process on-line. Antonelli and Astol (2003) studied
the output feedback regulation of endothermic and exothermic
chemical reactors in the presence of control bounds. Yu, Chang,
and Yu (2007) proposed a stable self-turning PID (Proportional
Integral Derivative) control scheme for multivariable nonlinear
systems with unknown dynamics and applied the scheme to a
simulated CSTR process. Di Ciccio, Bottini, Pepe, and Foscolo (2011)
built a nonlinear feedback control law for a CSTR with recycle by
using tools of differential geometry and observer theory. Hoang,
Couenne, Jallut, and Le-Gorrec (2012) developed nonlinear control
laws for isothermal CSTR based on the Lyapunov method. The
above results mainly focus on the control of a single reactor. Wang,Gui, Teo, Loxton, and Yang (2012) and Li, Gui, Teo, Zhu, and Chai
(2012) studied the use of controlparametrization method to
minimize the zinc dust consumption of a zinc sulphate electrolyte
purication process composed of multiple reactors.
Different from the above research, this paper develops a control
scheme for solution purication process based on its character-
istics. The concepts of Additive Utilization Ef ciency (AUE) and
Impurity Removal Ratio (IRR) are proposed based on an analysis of
solution purication process. By using these two concepts, the
control of solution purication process is decomposed into two
problems, i.e., estimated additive dosage optimization and robust
adaptive tracking control of the optimized operating point. Cor-
respondingly, the proposed control scheme is composed of two
layers. The upper layer works on a slow time scale. The additivedosage optimization, which has an economic objective function
subject to constraints on purication performance and process
stability, is transformed into nding an optimal decline gradient of
impurity ion concentration along the reactors. On contrast, the
lower layer works on a fast time scale. A robust adaptive controller,
in which Oxidation Reduction Potential (ORP) plays a central role,
is designed to track the optimized impurity ion concentrations in
the presence of model uncertainties, disturbance and saturation.
The rest of this paper is organized as following. In Section 2, an
analysis of solution purication process is conducted. The pro-
blems arising in the control of solution purication are pointed
out. The two-layer control scheme is introduced in detail in Sec-
tion 3. The ability of the scheme is tested and discussed in Section
4. The concluding remarks are given in Section 5.
2. Process analysis
Solution purication process is a continuous process composed
of N ( N N 1, Z≥ ∈ ) consecutive reactors and a thickener in which
the liquid–solid separation takes place (Fig. 2 shows a solution
purication process composed of four reactors and a thickener).
Consider the reaction described by Eq. (1), along the reactors,
impurity ion B is gradually reduced by reaction with additive A
under specic reaction conditions and the assistance of catalyst.
The overow of the thickener is delivered to subsequent process.
The underow which contains crystal nucleus benecial to im-
purity removal is recycled to promote cementation
mB nA nA mB 1n m+ = + ( )+ +
The technical index and the economical index of solution pur-
ication process are the impurity ion concentration of the puried
solution which reects the purication performance, and additive
consumption which relates to the production cost, respectively.
2.1. Process model
The reaction kinetics of Eq. (1) can be described by a rst-order
kinetic equation
dc
dt kA c
2s= −
( )
in which, c is the concentration of impurity ion, k is the reaction
rate, As is the reaction surface area available for impurity removalin unit volume of the reactor.
Consider the process described in Fig. 2, assume that the uid
in each reactor is perfectly mixed, and the contents are uniform
throughout the reactor volume. Then according to the mass bal-
ance principle, the dynamics of the process can be described by
following equations:
dc
dt
F
V c
F F
V c k A c
dc
dt
F F
V c
F F
V c k A c
i 2, 3, 4 3
in in us
i in ui
in ui i si i
10 1 1 1 1
1
= − +
−
= +
− +
−
= ( )
−
in which V is the volume of the reactor, F in is the ow rate of the
impure input solution from previous stage, F u is the ow rate of the recycled underow solution, c 0 is the impurity ion con-
centration of the input solution, c j is the ef uent impurity ion
concentration of the jth reactor ( j 1, 2, 3, 4= ), k j and As j are the
reaction rate and reaction surface area in unit volume of the jth
reactor ( j 1, 2, 3, 4= ), respectively.
2.2. Role of ORP
Purication process is essentially an oxidation–reduction re-
action and also an electrode reaction composed of many parallel
electrode reactions. According to the independence principle of
parallel electrode reactions (Antropov & Beknazarov, 1972), the
electrode reactions are independent of each other. Their unique
shared characteristic is the electrode potential, which is also called
Fig. 1. Solution purication process in zinc hydrometallurgy.
Fig. 2. Solution purication process.
B. Sun et al. / Control Engineering Practice 44 (2015) 89–10390
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mixed potential and determines the rate of all the electrode re-
actions by affecting the electron transfer rate between oxidant and
reductant.
ORP, which represents the comparative oxidability or re-
ducibility of the solution and the extent of oxidation reduction
reaction, is highly related to the mixed potential of electrode re-
actions. The relation between ORP and reaction rate has been
studied in Sun et al. (2013). It is found that in a certain range, a
more negative ORP represents a faster reaction rate and vice versa
⎛
⎝⎜
⎞
⎠⎟k A
E F e e
RT exp
2
4 f
e mix eq
c
γ = −
+ ( − )
( )
e pe q 5mix orp= + ( )
where k is the reaction rate, emix is the mixed potential, eorp is ORP
of the solution, A f is the frequency factor, E e is the standard acti-
vation energy, F is the Faraday constant, R is the ideal gas constant,
T c is the reaction temperature, eeq is the equilibrium potential, γ isthe inuence factor of electrode potential variation to cathode
activation energy, p and q are the linear approximation parameters
to be identied.
Due to its signicance, in some plants (Fugleberg & Jarvinen,
1993; Sun et al., 2014), the reaction rate is controlled by adjusting
the setting value of ORP which can be monitored online con-
tinuously and controlled by changing the dosage of additive
(Fig. 3). Thus, from the above discussion, ORP is an external re-
presentation of the mixed potential, and an intermediate variable
between reaction rate and additive dosage.
2.3. Control dif culties arising in solution puri cation process
Similar with most industry processes, purication process in-
teracts with both internal and external environments (Fig. 4). Eq.
(3), which describes only the ideal reaction kinetics, is not a suf-
cient description of the overall dynamics.Specically, the dynamics of solution purication is inuenced
by external and internal environments mainly from the following
aspects:
1. The raw ore contains intricately mixed minerals with randomly
varying properties. And the operations taken in the preceding
processes of solution purication may not always leading to
satisfying results. As a result, the characteristics of the input
solution, which include the impurity ion concentration and
types of elements in the solution, are time varying.
2. Under some circumstances, the usually xed conguration of
the reaction conditions needs to be adjusted, such as the ow
rate of underow, type of catalyst, stirring rate, and reaction
temperature.3. In a certain range, a more negative ORP will result in a faster
reaction rate. However, there exists a point beyond which the
effect of ORP on the reaction rate is saturated or even reversed.
Because if ORP is very negative, it is likely to generate more
metal subsulfate which will attach to the surface of additive and
reduce its activity;
Correspondingly, the control dif culties arising in solution
purication include:
1. Model uncertainties: The impurity ion concentration and type of
elements in the input solution are time varying. Also the reac-tion conditions are not consistent. Thus, the parameters in the
process model are not constant. As illustrated by Fig. 5, under
different conditions, the same setting value of ORP would result
in different reaction rates.
2. Disturbance: Equipment failure, external excitation and un-
reasonable operations would bring disturbance to solution
purication process.
3. Saturation: The relation between ORP and the reaction rate ex-
hibits a ‘saturation-like’ phenomenon. In addition, subjects to
the physically constraints, for each reactor, the setting value of
ORP is bounded in an appropriate range (Fig. 6).
The control objective of solution purication process is to re-
duce the impurity ion concentration to a predened acceptablerange using the smallest amount of additive dosage and keep the
process stable. In order to get rid of the control dif culties and
achieve the control objective, a two-layer gradient optimization
scheme is proposed in next section.
3. Gradient optimization scheme
For almost every industrial process, the realization of control
objective relies on a subtly designed control scheme. In this sec-
tion, based on two intuitional concept derived from the char-
acteristics of solution purication process, a two-layer gradient
optimization scheme is developed. The control problem is for-
mulated and transformed into a constrained optimizationFig. 3. ORP-additive dosage controller.
Fig. 4. Internal and external environments of purication process.
−570−560−550−540−530−520−510−5000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6x 10
Reaction rate and ORP under different conditions
R e a c t i o n r a t e ( s
)
ORP (mv)
Condition 1
Condition 2
Condition 3
Same ORP result in different reaction rate
Fig. 5. Relation between ORP and reaction rate under different conditions.
B. Sun et al. / Control Engineering Practice 44 (2015) 89–103 91
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problem, or in other words, nding an optimal decline gradient of
impurity ion concentration along the reactors. The control dif -
culties, which include model uncertainties, disturbance and sa-
turation, are handled by a robust adaptive controller.
3.1. Concept design
3.1.1. Additive utilization ef ciency
The amount of additive fed into the reactor affects the pur-
ication effect and production cost. A major problem in deciding
the additive dosage is that not all the additive added involve in
impurity removal (Sun, Gui, Wang, & Yang, 2014). According to the
mass balance principle, the proper amount of additive to be fed
into the reactor depends on its ef ciency in removing the impurity
(Kim et al., 2007).Concept 1: Consider a certain period of time, Additive Utiliza-
tion Ef ciency (AUE) is the ratio of additive practically involved in
impurity removal when a certain amount of additive is added intothe reactor
z
z 6
real
all
μ =( )
where z all is the amount of additive added, z real is the amount of
additive involved in impurity removal.
Consider a block of solution with volume V , for reactor i
(i N 1, 2, ,= … ), denote μi as its AUE, c i 1− and c i as the impurity ionconcentration before and after its retention in the reactor, assume
that μi is constant during the retention, then the required additivedosage is
z M
M V c c
7i i
A
Bi i
11 μ= ( − )
( )−
−
where M A and M B are the atomic weight of additive and impurity,
respectively.
3.1.2. Impurity removal ratio
Concept 2: Consider a block of leaching solution with volume V
and impurity ion concentration c 0, its retention in solution pur-
ication process is essentially a gradually decline process of the
impurity ion concentration along the reactors. For reactor i
( i N 1, 2, ,= … ), the Impurity Removal Ratio (IRR) is the ratio of
impurity ion removed in it
c c
c 8i
i i1
0
λ = −
( )
−
IRR of each reactor needs to be arranged in order to achieve the
required purication performance.
3.2. Basic idea of gradient optimization
Using the two concepts, for a block of leaching solution with
volume V and impurity ion concentration c 0, assume that AUE of
each reactor is constant, then during its retention in a solution
purication process with N consecutive reactors, the total additive
dosage can be formulated as
z z M
M Vc z
9w
i
N
i A
B i
N
i i w
i
N
i i
1
0
1
10
1
1∑ ∑ ∑ μ λ μ λ= = =( )= =
−
=
−
where z Vc wM
M 0 0 A
B= is the ideal additive dosage needed to remove
all the impurity ion.
Eq. (10) indicates that AUE can not only be used to estimate
additive dosage of a single reactor, but also optimize additive
consumption of purication process composed of multiple re-
actors. If more IRR is assigned to the reactor with larger AUE, fewer
IRR is assigned to the reactor with smaller AUE, the overall ad-
ditive consumption could be optimized. Moreover, IRR of a reactor
reects its internal reaction state. A stable IRR indicates a stable
reaction status. So limiting IRR of each reactor in suitable ranges is
benecial to the stability of purication process.
Thus for a block of leaching solution with volume V and im-
purity ion concentration c 0, assume that AUE of each reactor are
constant, then the problem of optimizing the additive consump-
tion can be formulated as an allocation problem of IRR according
to AUE while considering the stability and purication require-
ment:
⎛
⎝⎜⎜
⎞
⎠⎟⎟
z z
c c c
i N
min , , ,
st. 0 1
, 1, 2, , 10
w N w
i
N
i i
N
i
N
i dp
mini
i maxi
1 2 0
1
1
0
1
∑
∑
λ λ λ μ λ
λ
λ λ λ
( … ) =
< = − ≤
≤ ≤ = … ( )
=
−
=
where c dp is the desired ef uent impurity ion concentration, mini λ
and maxi λ are predened lower and upper bounds of IRR of reactor
i.
3.3. Two layer control scheme
Solving Eq. (10), which would obtain the optimized IRR of each
reactor, is equal to nd the optimized setting values of the ef uent
impurity concentration of each reactor, or in other words, nd a
best decline gradient of impurity ion concentration along the re-
actors (Fig. 7). However, due to the control dif culties discussed in
Section 2, solving Eq. (10) itself is not suf cient to achieve the
required purication performance. A robust adaptive controller,
which is capable of driving the ef uent impurity ion concentrationof each reactor to follow the optimized value, is required to be
designed. Thus, a complete control scheme of solution purication
process should include two layers (Fig. 8). The upper layer, which
works on a slow time scale, solves the estimated economical op-
timization problem. The lower layer, which works on a fast time
scale, handles the model uncertainties, disturbance and saturation.
3.4. Upper layer: estimated economical optimization
The main task of the upper layer is to calculate the best decline
gradient of impurity ion concentration according to the impurity
ion concentration in the leaching solution c 0 and AUE of the re-
actors μi ( i N 1, 2, ,= … ). However, for each reactor, AUE is related
to the reaction status and changes with time. The impurity ion
−620−600−580−560−540−520−5000
0.5
1
1.5
2x 10
−3 Bounded input and input saturation
R e a c t i o n r a t e ( s − 1 )
ORP (mv)
Fig. 6. Bounded input and input saturation.
B. Sun et al. / Control Engineering Practice 44 (2015) 89–10392
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concentration of the leaching solution is also not always constant.
Owing to the time varying property of c 0 and μi (i N 1, 2, ,= … ), Eq.
(10) needs to be solved periodically. Selection of the optimizationperiod is related to the time-variation characteristic of the target
process. If c 0 or μi ( i N 1, 2, ,= … ) changes drastically and fre-quently, then a short optimization period is required. If c 0 and μ i( i N 1, 2, ,= … ) are stable with small variations, then a long opti-
mization period is preferred.
Besides the selection of optimization period, the estimation of
AUE is another important issue. There exist various kinds of
methods to estimate the value of AUE (Kim et al., 2007), such as
regression method and Box– Jenkins method. In this paper, Radial
Basis Function Neural Network (RBFNN) (Moody & Darken, 1989),
which has been applied successfully in many engineering pro-
blems (Han, Qiao, & Chen, 2012; Iliyas, Elshafei, Habib, & Adeniran,
2013; Seshagiri & Khalil, 2000), is selected to estimate AUE.
RBF neural network is a feedforward neural network composedof three layers including one input layer, one hidden layer and one
output layer. Fig. 9 shows a RBF neural network with n input, m
hidden node and a single output.
The most signicant characteristic of RBF neural network is the
use of nonlinear activation function, i.e., radial basis function, in
the hidden layer. The output of a RBF neural network is a weighted
linear combination of the output of hidden neurons (Fig. 9)
y11i
m
i i
1
∑ χ β ψ = ( )( )=
where , , , nT
1 2 χ χ χ χ = [ … ] and y are the input and the output of the
network, β i ( i m1, 2, ,= … ) is the connecting weight between the
ith neuron in the hidden layer and the output layer, ψ i is the
output value of the ith neuron in the hidden layer. The most
commonly used radial basis function is the Gaussian function:
e 12ic /i i
2 2 χ ψ ( ) = ( ) χ σ − ∥ − ∥
where ci χ ∥ − ∥ is the Euclidean distance between χ and ci, i andsi are center and spread of the ith ( i m1, 2, ,= … ) node in the
hidden layer.
The performance of a RBF neural network is determined by the
number of hidden neurons, center and spread of each neuron andthe connecting weight , , , m1 2 β β β β = [ … ]. According to the way the
centers are selected, there exist four types of training method,
including random center selection, self-organized center selection,
supervised center selection and orthogonal least squares learning
algorithm. In this paper, the network is trained using orthogonal
least squares (OLS) method (Chen, Cowan, & Grant, 1991), while
the aim is to minimize the following criterion:
J y dmin13i
P
i i
1
2∑= | − |( )=
where P is the number of training data samples, d and y are the
practical value and the output of the RBF neural network,
respectively.
A complete development procedure of the RBF neural networkfor AUE estimation includes following steps:
Step 1: Select the input variables χ . According to the reaction
mechanism, for the AUE estimation of each reactor, the input
variables should include ow rate and impurity ion concentration
of the inlet solution, ow rate of recycled underow, dosage of
additive and catalyst and ORP.
Step 2: Select the training data. The training data is selected
from routinely collected production data of real plants. The
training data should cover as many production mode as possible.
However, the training data must be selected carefully, the data
from the cases with an excessive additive dosage should be
avoided.
Step 3: Select the centers { i} using OLS, and choose the spread
of each neuron as the closest Euclidean distance between itscenter and centers of other neurons.
Step 4 : Adapt the connecting weights β using the adaptive
gradient descent procedure (Iliyas et al., 2013).
3.5. Lower layer: robust adaptive controller of impurity ion
concentration
The function of the lower layer is to eliminate the effect caused
by the model uncertainties, disturbance and saturation. A con-
troller which forces the impurity ion concentration to track the
desired reference is designed.
3.5.1. Nominal state space model
According to Eqs. (3)–
(5), the nominal state space model of the
0 t1 t2 t3 t40
c0Gradient optimization
I m p u r i t y i o n c o n c e n t r a t i o n
time
Gradient 1Gradient 2
Gradient 3Gradient 4Required technical index
out of predefined range of IRR
within predefined
range of IRR
failed to meet the required purification performance
Fig. 7. Gradient optimization along reactors ( t i means the time when solution
outows from reactor i i N 1, 2, ,( = … )).
Fig. 8. Two layer control scheme.
Fig. 9. Structure of a RBF neural network.
B. Sun et al. / Control Engineering Practice 44 (2015) 89–103 93
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process can be represented as
x s Ax F x g v 140 ̇= + − ( ) ( ) ( )
where x x x c c c x , , , , , ,N T N T 1 2 1 2= [ … ] = [ … ] is the outlet impurityion concentration of the reactors which is considered as the sys-
tem state, v v v e e e v , , , , , ,N T orp orp orp T 1 2 N 1 2= [ … ] = [ … ] is the ORP
setting value of the reactors, c , 0, , 0F V
T 0 0= [ … ] . A is the system
matrix
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥
a
a a
a a
a a
a a
A
0 0 0 0
0 0 0
0 0 0
0 0 0
0 0 0 15
=
− ⋯
− ⋯
− ⋯⋮ ⋱ ⋱ ⋱ ⋱ ⋮
⋯ −
⋯ − ( )
In which, a F F
V
in u= +
. F x ( ) and g v ( ) are functions of state and input
variables, respectively
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
x
x
x
F x
0 0
0 0
0 0 16N
1
2( ) =
⋯
⋯
⋮ ⋮ ⋱ ⋮
⋯ ( )
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
g v
g v
g v
g v
17N N
1 1
2 2( ) =
( )
( )
⋮
( ) ( )
In which,
g v A A e i N , 1, 2, , 18i i si fi E F p v q e RT 2 /ei i i i i eq c ( ) = = … ( )
γ −( + [( + )− ])
A E p q, , , ,i i ei i i i0ϕ γ = { } is the parameter set of reactor i
( i N 1, 2, ,= … ) to be identied.
3.5.2. Model uncertainties, disturbance and saturation
The discussion in Section 2.3 indicates that the dynamics of
purication process is not always consistent. The identied para-
meter sets ϕi ( i N 1, 2, ,= … ) exhibit uncertainties iϕΔ
(i N 1, 2, ,= … ) which are time varying and depend on the internal
and external environments. So function g vi i( ) can be written as
g v g v d t , 19i i i i g i θ ( ) = ¯ ( ) + ( ) ( )
In which, , p c T θ θ θ = [ ] , g vi i¯ ( ) and d t , g i θ ( ) are the deterministic part
and the uncertain part of g vi i( ), respectively.
In Eq. (19), g vi i¯ ( ) is saturated when input vi exceeded certain
value as shown in Fig. 6. There is a sharp corner when the sa-
turation happens, thus backstepping technique cannot be directly
applied. In order to remove this barrier, the saturation is ap-
proximated by a smooth function (Wen, Zhou, Liu, & Su, 2011)
⎧⎨⎪
⎩⎪ g v
a v b v c v v
g v v v
if
if ai i
i i i i i i M
i i i M
2i
i
( ) =+ + ≥
¯ ( ) <
in which the parameter ai, bi, c i and vM i are chosen according to
g vi i¯ ( ) (Fig. 6).
Then g vi i¯ ( ) can be expressed as
g v g v d v 20i i ai i ai i¯ ( ) = ( ) + ( ) ( )
where d vai i( ) is the bounded approximation error
d v g v g v D
21ai i i i a
i
i ai| ( )| = | ¯ ( ) − ( )| ≤( )
Besides model uncertainties d t , g i θ ( ) and approximation error
d vai i( ), external disturbance d t i¯ ( ) also affects the process dynamics,
thus a more comprehensive description of the process is
x f t
x a x ax x g v d v t
x ax ax x g v d v t
, ,
, ,
, , 22
p
a
j j j j a j j j j
0 0
1 0 0 1 1 1 1 1 1
1
φ θ
θ
θ
̇ = ( )
̇ = − − ( ) + ( )
̇ = − − ( ) + ( ) ( )−
in which N 2, 3, ,= … , x0 is the impurity ion concentration of the
input leaching solution, φ is the physical and chemical character-istics of raw ore, f t , , p0 φ θ ( ) is an unknown bounded function,
a F
V 0 = , d v t x d t d v d t , , ,i i i g i ai i iθ θ ( ) = ( ( ) + ( )) +
¯ ( ) (i N 1, 2, 3, ,= … ) is
the synthetical effect of model uncertainties, approximation error
and external disturbance.
Before the development of the robust adaptive tracking con-
troller, following assumptions are made:
Assumption 1. Solution purication process is input-to-state
stable (ISS).
Assumption 2. Impurity ion concentration can be determined
online.
Assumption 3. Variables, such as impurity ion concentration and
the identied parameter sets φi ( i N 1, 2, ,= … ), are bounded andhave their own physical sense.
3.5.3. Robust adaptive tracking controller
In this section, reactor 1 is used as an example to illustrate the
design of robust adaptive controller with the additive dosage as
the control input.
Dynamics of reactor 1 is approximated and augmented as fol-
lows:
x f t
x a x ax h d t
h
, ,
23
p0 0
1 0 0 1 1 1
1
φ θ
ω
̇ = ( )
̇ = − − + ( )
̇ = ( )
where h x g va1 1 1 1= ( ), ω is an auxiliary control to be determined.
The bound of d t 1( ) is denoted as D1 which is not assumed to beknown. v1 is determined for a given state x1 and h1.
The following change of coordinates is made:
z x x
z h 24
r 1 1 1
2 1 1α
= −
= − ( )
where xr 1 is the optimized setting value of x1, z 1 is the tracking
error of impurity ion concentration, z 2 is due to the new state
variable h1, α 1 is the virtual control law to be designed.The backstepping procedure is developed below:
Step 1: Consider the Lyapunov function
V z 1
2 25 z 1 1
2=( )
The derivative of V z 1 is
V z z
z a x ax z d t x 26
z
r
1 1 1
1 0 0 1 2 1 1 1α
̇ = ̇
= [ − − − + ( ) − ̇ ] ( )
If d t 1( ) and x0 are bounded by D1 and D0. Due to the fact that z d t z D1 1 1 1( ) ≤ | | , z x z D1 0 1 0≤ | |
V z ax z x a z D z D 27 z r 1 1 1 2 1 1 0 1 0 1 1α ̇ ≤ [ − − − − ̇ ] + | | + | | ( )
Design α 1 as
c z ax x a z x z Dsgn sgn 28r 1 1 1 1 1 0 1 0 1 1α
= − − ̇
+ ( )
^
+ ( )
^
( )
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with the adaptive law
x a z 290 0 0 1η ̂ ̇ = | | ( )
where x0^ is the estimation of D0.Thus for the following Lyapunov function:
V V x1
2 30 z 1 1
002
η= + ˜
( )
where x D x0 0 0˜ = − ^ is the estimation error of D0.The following equation is obtained:
V z z x x
c z z z z D
1
31
1 1 1
0
0 0
1 12
1 2 1 1
η ̇ = ̇ + ˜ ̃ ̇
≤ − − + | | ˜ ( )
Step 2: Consider the Lyapunov function
V z 1
2 32 z 2 2
2=( )
with its derivative
V z z z
z c a x a z x c x
x z D
z c a a x ax h d t
a z x c x x z D
z c a a x ax h
z c a D a z x
c x x z D
sgn
sgn
sgn sgn
sgn sgn sgn
sgn 33
z
r
r
r r
r r
2 2 2 2 1
2 1 1 0 1 0 1 1
12
1 1
2 1 0 0 1 1
0 1 0 1 1 12
1 1
2 1 0 0 1 1
2 1 1 0 1 0
1 1 12
1 1
ω α
ω
ω
ω
̇ = ̇ = ( − ̇ )
= { − [( − ) ̇ + ( ) ̂ ̇ − ̇
− + ( ) ̂ ̇ ]}
= { − [( − )( − − + ( ))
+ ( ) ̂ ̇ − ̇ − + ( )
̂ ̇ ]}
≤ { − [( − )( − −
− ( ) ( − ) ) + ( ) ̂ ̇
− ̇ − + ( ) ̂ ̇ ]} ( )
( )
( )
( )
Design
c z z c a a x ax h z c a D a z x c x
x z D
sgn sgn
sgn 34
r
r
2 2 1 1 0 0 1 1
2 1 1 0 1 0 1 1
12
1 1
ω = − + + ( − )( − − )− ( )| − |
^+ ( )
̂ ̇ − ̇
− + ( ) ̂ ̇
( )( )
with the adaptive law
D z z c a 351 1 1 2 1η ̇ = (| | + | ( − )|) ( )
For the following Lyapunov function:
V V V D1
2 36 z 2 1 2
112
η= + + ˜
( )
where D D D1 1 1˜ = − ^ is the estimation error of D1.
The following equation is obtained:
V z z z z x x D D
c z c z
1 1
37
2 1 1 2 2
0
0 0
1
1 1
1 12
2 22
η η ̇ = ̇ + ̇ + ˜ ̃ ̇ + ˜ ̃ ̇
≤ − − ( )
Based on Steps 1 and 2, Theorem 1 can be derived.
Theorem 1. For solution puri cation process described by Eq. (1), it
is global asymptotically stable by applying the control law:
v g h x/ 38a1 11
1 1= ( ) ( )−
with
h1 ω ̇ =
c z z c a a x ax h
z c a D a z x c x
x z D
sgn sgn
sgn
r
r
2 2 1 1 0 0 1 1
2 1 1 0 1 0 1 1
12
1 1
ω = − + + ( − )( − − )
− ( )| − | ^
+ ( ) ̂ ̇ − ̇
− + ( ) ̂ ̇( )
and adaptive laws (29) and (35).
Steps 1 and 2 deduced the reference setting value of ORP. In
order to obtain suitable additive dosage that force the practicalORP to follow its setting value, Steps 3 and 4 are required.
Step 3: Consider the ‘ORP-Additive dosage’ system
v x u x v t , , , 391 1 0 1 1 1 1 1ζ μ σ θ ̇ = ( − ) + ( ) ( )
in which ζ 1 and the dynamics of x v t , , ,1 1 1σ θ ( ) are unknown.Before the development of the control law, an auxiliary ap-
proximation system is designed (Rincón, Erazo, & Angulo, 2012)
v f u c sgv z D sat z
D sgv z 40
v1 1 3 3 3 3
3 3 3η
̂ ̇ = ( ) + ( ) + ^
( )
̂ ̇ = | ( ) | ( )
in which, f u x uv 1 1 0 1 1
ζ μ( ) = ( − ), c 3 is a positive design parameter,
z v v3 1 1= − ^ is the approximation error, D3^
is the estimate of D3,which is the bound of x v t , , ,1 1 1σ θ ( ), function sgv z 3( ) and sat z 3( ) aredened as
⎧
⎨⎪
⎩⎪
sgv z
z c z c
z c
z c z c
if
0 if
if
v v
v
v v
3
3 3
3
3 3
( ) =
− ≥
| | <
+ ≤ −
⎪
⎪⎧⎨⎩
sat z z z c
z c z c
sgn if
/ if
v
v v3
3 3
3 3
( ) =( ) | | ≥
| | <
where c v is a positive design parameter.
Thus, for the following Lyapunov function, where D D D3 3 3˜ = − ^
is the estimation error of D3, Theorem 2 can be obtained
⎧
⎨
⎪⎪
⎩
⎪⎪
V
z c D z c
D z c
z c D z c
/ 2 if
/ 2 if
/ 2 if
v v
v
v v
3
1
2 3 2
32
3 3
32
3 3
1
2 3 2
32
3 3
η
η
η
=
( − ) + ˜ ( ) ≥
˜ ( ) | | <
( + ) + ˜ ( ) ≤ −
Theorem 2. For the ‘ ORP-Additive dosage’ system described by Eq.
(39), the approximation error of v1 by using approximation Eq. (40)
converges to c c ,v v[ − ], with that c v can be de ned arbitrarily small
by the user.
Proof. The derivative of V 3 is
V sgv z z D D1
413 3 3
3
3 3η
̇ = ( ) ̇ + ˜ ̃ ̇( )
As z v v f u x v t v, , ,v3 1 1 1 1 1 1 1σ θ ̇ = ̇ − ̂ ̇ = ( ) + ( ) −
̂ ̇:
V sgv z f u v sgv z x v t D D
sgv z f u v sgv z D D D
, , , 1
1
42
v
v
3 3 1 1 3 1 1 1
3
3 3
3 1 1 3 3
3
3 3
σ θ η
η
̇ = ( )( ( ) − ̂ ̇ ) + ( ) ( ) + ˜ ̃ ̇
≤ ( )( ( ) − ̂ ̇ ) + | ( ) | + ˜ ̃ ̇
( )
Using Eq. (40), the following equation is obtained:
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V c sgv z sgv z z D sgv z D
D D sgv z z D sat z D
c sgv z D sgv z D
sgv z z D sat z D
sgn
1sgn
1
sgn 43
3 3 3 2
3 3 3 3 3
3
3 3 3 3 3 3 3
3 3 2
3
3 3 3 3
3 3 3 3 3
η
ηη
̇ ≤ − ( ) − ( ) ( ) ^
+ | ( ) |
− ˜ ̂ ̇ + ( )[ ( )
^− ( )
^]
≤ − ( ) + ˜ ( | ( ) | − ̂ ̇ )
+ ( )[ ( ) ^
− ( ) ^
] ( )
Notice that sgv z z D sat z Dsgn 03 3 3 3 3( )[ ( ) ^ − ( ) ^ ] = , so
V c sgv z 443 3 3 2 ̇ ≤ − ( ) ( )
According to the Lyapunov stability theorem (Khalil & Grizzle,
2002), the approximation error z 3 asymptotically converges to ϵ,where z z c : v3 3ϵ = { | | ≤ }, the estimation error of D3 asymptotically
converges to zero.□
Step 4 : Dene the tracking error as
z v v
v v v v
z z 45
r
r
5 1 1
1 1 1 1
4 3
= −
= − ^ + ^ −
= − ( )
Consider z v vr 4 1 1= − ^
, its derivative is
z v v
v f u c sgv z D sat z
u x c sgv z D sat z v 46
r
r v
r
4 1 1
1 1 3 3 3 3
1 1 1 1 0 3 3 3 3 1ζ μ ζ
̇ = ̇ − ̂ ̇
= ̇ − ( ) − ( ) − ^
( )
= − − ( ) − ^
( ) + ̇ ( )
If
u x c sgv z D sat z v c z 1
47r 1
1 1
1 0 3 3 3 3 1 4 4ζ μ
ζ = [ + ( ) + ^
( ) − ̇ − ]( )
then
z c z 484 4 4 ̇ = − ( )
which means that z 4 converges asymptotically to zero.
From Step 3, it is known that z 3 asymptotically converges to
z z c : v3 3ϵ = { | | ≤ }. Hence the tracking error z 5 converges to z z c : v5 5ϵ = { | | ≤ } asymptotically.
Remark 1. Steps above provide a robust adaptive controller for
solution purication process. The closed loop system is globally
stable. The tracking and transient performance can be controlled
by adjusting the design parameters. Increasing c 1, c 2, c 3 and c 4, or
decreasing c v can improve the tracking performance while in-
creasing η1, η2, η3 and η4 can decrease the effects of the initialerror estimates on the transient performance (Zhou, Wen, &
Zhang, 2004). However, increasing c 1, c 2, c 3 and c 4 indicates a
higher control input. Thus these design parameters are suggested
to be determined through try and error at the development stage.
4. Experimental study
In order to evaluate the feasibility and ability of the proposed
scheme, an experimental study of a cobalt removal process in a
zinc hydrometallurgy plant (Sun et al., 2014) is investigated in this
section.
4.1. Process description
Cobalt removal is an intermediary step of the solution pur-
ication process in zinc hydrometallurgy (Fig. 1). Due to the im-
purity of zinc concentrate, zinc sulfate solution after leaching
contains not only zinc ion, but also other metal ions, such as
copper, cobalt, nickel, cadmium, indium, gallium, arsenic, anti-
mony, and germanium. The presence of these metal ion impurities
would cause large drops of current ef ciency during electrowin-
ning in which metallic zinc is recovered, resulting in energy waste
and downgrade of product quality. Thus, before electrowinning,
these metal ion impurities need to be removed to an acceptable
level. Among the impurity ions, cobalt ion is dif cult to be re-
moved and has the most detrimental effect to electrowinning, so
the result of cobalt removal is used as an index of solution pur-ication performance.
Cobalt removal process studied in this section is composed of
four consecutive continuous stirred tank reactors and a thickener
(Fig. 2). Spent acid is supplied to provide an acid reaction en-
vironment. The solution is heated to around 80 °C to guarantee
enough reaction impetus. Zinc dust and arsenic trioxide are added
into the reactors to conduct complex chemical and electrochemical
reactions with cobalt ions and residual copper ions from previous
copper removal process.
The main reactions taking place in cobalt removal process in-
clude the following:
Cu Zn Zn Cu 492 2+ = + ( )+ +
As 3Cu 4.5Zn Cu As 4.5Zn 503 2 3 2+ + = + ( )+ + +
As Co 2.5Zn CoAs 2.5Zn 513 2 2+ + = + ( )+ + +
By forming alloys or metal compounds, such as CoAs, cobalt ions is
gradually precipitated. After retention in four consecutive reactors,
zinc sulfate solution ows into the thickener in which liquid–solid
separation takes place. Overow of the thickener is delivered to
subsequent cadmium removal process, while the underow which
contains crystal nucleus benecial to cobalt removal is recycled to
the rst reactor.
According to Section 2.1, the nominal model of cobalt removal
process can be described by
dx
dt
F
V x
F F
V x k A x
dx
dt
F F
V x
F F
V x k A x
i 2, 3, 4
in in us
i in ui
in ui i si i
10 1 1 1 1
1
= − +
−
= +
− +
−
=
−
in which V is the volume of the reactor, F in is the ow rate of the
leaching solution, F u is the ow rate of the underow, x0 is the
impurity ion concentration of the leaching solution, xi is the ef-
uent impurity ion concentration of the ith (i 1, 2, 3, 4= ) reactor,
ki and Asi are the reaction rate and reaction surface area in unit
volume of the ith ( i 1, 2, 3, 4= ) reactor, respectively, and
⎛
⎝⎜⎜
⎞
⎠⎟⎟k A
E F pe q eRT
exp 2
52 f
e orp eq
c
γ = − + [( + ) − ]
( )
A g 53s s β = ( )
where A f is the frequency factor, E e is the standard activation en-
ergy, eeq is the equilibrium potential of cobalt reduction, T c is the
reaction temperature, R is the idea gas constant, F is the Faraday
constant, g s is the precipitant content in unit volume of reactor,
eorp is the oxidation–reduction potential, γ is the inuence factor of electrode potential variation to cathode activation energy, β is therelation factor between surface area in unit volume of reactor and
g s, p and q are the approximation terms between oxidation–re-
duction potential and mixed potential. The details of these
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parameters are summarized in Table 1.
The control objective of cobalt removal process is to minimize
the zinc dust dosage and guarantee that the cobalt ion con-
centration after purication is equal to or smaller than a pre-
dened threshold c 0.5 mg/L index =
z u u u u u
c c
min , , ,
st. 54
w
i
i1 2 3 4
1
4
out index
∑( ) =
≤ ( )
=
in which, z w is the zinc dust consumption of four reactors, ui is the
zinc dust dosage of reactor i, c out is the cobalt ion concentrationafter purication.
The manipulated variables of cobalt removal include dosage of
zinc dust and arsenic trioxide, setting value of ORP, reaction
temperature, ow rate of leaching solution and spent acid. Among
these variables, ow rate of leaching solution, which is related
with preceding leaching process and subsequent electrowinning
process, is decided from the prospective of the running of the
whole zinc hydrometallurgy process. Flow rate of spent acid and
reaction temperature are in most cases kept constant to provide a
suitable reaction environment. Dosage of arsenic trioxide is de-termined according to the stoichiometric relationship indicated by
Eqs. (49)– (51). In the plant, the cobalt removal process is con-
trolled by adjusting dosage of zinc dust or the setting value of ORP,
which can then be controlled by a ‘ORP-zinc dust’ PI controller
automatically (Fig. 3).
However, as discussed in Section 2, cobalt removal is a complex
multiphase reaction inuenced by numerous factors, such as pH,
grain size of zinc dust, type of elements and copper ion con-
centration in the leaching solution. These inuences would affect
the process dynamics by causing variations in the parameters of
the model. For example, if γ is increased, then the process is more
sensitive to the variation of ORP; if β is increased, then more re-
action surface is available and the reaction would be accelerated.Besides the inuence of these factors, cobalt removal process also
encounters saturation. An excessive zinc dust dosage may cause
the generation of basic zinc sulfate which would then block the
surface of zinc dust and thus hinder cobalt removal (Fig. 10 shows
a sample of the solution when excessive zinc dust is added).
4.2. Experiment setting
In this simulation, the practical cobalt removal process is si-
mulated using a kinetic model of normal production mode. The
parameters in the kinetic model are identied from the production
data. The performance of the kinetic model is shown from Figs. 11
to 14.
To test the proposed scheme and mimic the real process, fol-
lowing situations are also considered, including:
1. Model uncertainties: The nominal model adopted in the robust
adaptive controller is different with the kinetic model of normal
production mode.
2. Variation of impurity ion concentration and ow rate of leaching
solution: The cobalt ion concentration in the leaching solution
varies from 22 mg/L to 36 mg/L as shown in Fig. 15. The
variation of the ow rate is shown in Fig. 16.
3. External disturbance: The disturbance shown in Fig. 17 is acted
on the rst reactor.
4.3. Simulation steps
The simulation interval is 24 h, and every optimization step is10 min. At each optimization step, following steps are taken:
Step 1: AUE estimation. AUE of each reactor is estimated by a
RBF neural network with 15 centers and following inputs:
1. Flow rate of inlet solution.
2. Inlet cobalt ion concentration of the reactor.
3. Outlet cobalt ion concentration of the reactor.
4. Copper ion concentration in the leaching solution.
5. Zinc dust dosage.
6. Arsenic trioxide dosage.
7. Oxidation reduction potential.
8. Precipitant content in unit volume of the reactor.The structure
of the RBF neural network discussed in Section 3.4 is adopted
here. The estimation result is shown in Fig. 18.
Step 2: IRR calculation. Calculate the IRR of each reactor ac-
cording to estimated AUE and Eq. (10) (Fig. 19). Then calculate the
setting values of the impurity ion concentration according to IRR
and cobalt ion concentration in the leaching solution (Fig. 20). The
constraints on the IRR of each reactor are shown in Table 2.
Step 3: ORP setting. Calculate the ORP setting values of each
reactor using the controller Eq. (38) (Fig. 21). Table 1
Parameters in kinetic model.
Parameter Physical meaning Unit Value
V Volume of reactor m3 450
g s Precipitant content in unit volume of the
reactorkg/m3
β Relation parameter between surface areaand g sm m /kg2 3·
A0 Frequency factor of cobalt removal reaction 1/s
R Ideal gas constant J/ mol K( · ) 8.314472
T c Reaction temperature K
F Faraday constant /mol 96485
E e Standard activation energy of cobalt re-
moval reactionkJ/mol
emix Mixed potential of present solution V
eorp Oxidation reduction potential of present
solution
V
eeq Equilibrium potential of cobalt ion V
χ Inuence factor of electrode potential
change to cathode activation energy
1
p linear term of linear approximation function 1
q offset term of linear approximation function 1Fig. 10. Hinder effect caused by excessive zinc dust dosage.
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10 20 30 40 50 60 70 803.5
4
4.5
O u t l e t c o b a l t i o n c
o n c e n t r a t i o n ( m g / L )
Kinetic model
Practical value
Fig. 11. Kinetic model performance of reactor 1.
10 20 30 40 50 60 70 800.5
1
1.5
2
2.5
3
test point
O u t l e t c o b a l t i o n c o n c e n t r a t i o n ( m g / L )
Kinetic model
Practical value
Fig. 12. Kinetic model performance of reactor 2.
10 20 30 40 50 60 70 800.2
0.4
0.6
0.8
1
1.2
O u t l e t c o b a l t i o n c o n c e n
t r a t i o n ( m g / L )
test point
Kinetic model
Practical value
Fig. 13. Kinetic model performance of reactor 3.
10 20 30 40 50 60 70 800.2
0.3
0.4
0.5
0.6
0.7
test point
O u t l e t c o b a l t i o n c o n c e n t r a t i o n ( m g / L ) Kinetic model
Practical value
Fig. 14. Kinetic model performance of reactor 4.
0 2 4 6 8 10 12 14 16 18 20 22 2420
25
30
35
40Cobalt ion concentration of leaching solution
time (h)
C o b a l t i o n c o n c e n t r a t i o n ( m g / L )
Cobalt ion concentration
Fig. 15. Impurity ion concentration of input solution.
0 2 4 6 8 10 12 14 16 18 20 22 24200
220
240
260
280
300
320Flow rate of input solution
time (h)
F l o w r a t e ( m 3 / h
)
Flow rate of input solution
Fig. 16. Flow rate of input solution.
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Step 4 : Zinc dust dosage determination. The dosage of additive
is obtained via Eq. (47) and shown in Fig. 22.
4.4. Performance evaluation
The control result is shown in Figs. 23 and 24. It is indicated
that by using the two-layer control scheme:
1. The nal impurity ion concentration always satises the pur-
ication requirement.
2. The impurity ion concentration and oxidation reduction po-
tential of each reactor can track their reference trajectories well.
3. There is no large uctuation and excessive increment in the
additive dosage.
An important phenomenon is that the setting value of AUE in
the rst reactor is kept at 70% throughout the simulation. This
indicates that by using the idea illustrated in Sections 3.1 and 3.2,
most of the cobalt ion was removed in the rst reactor whose AUE
is higher than the rest reactors. The economical effect is that the
total zinc dust dosage is decreased, while the over excessive do-
sage of additive is avoided. A practical application of this idea is in
Sun et al. (2014), in which a decrease in the zinc dust consumption
was observed.
0 2 4 6 8 10 12 14 16 18 20 22 240
2
4
6
8
time (h)
D i s t u r b
a n c e
Disturbance
Disturbance
Fig. 17. Disturbance.
4 8 12 16 20 240%
20%
40%
60%
80%
100%AUE of reactor 1
time (h)
A U E
Practical value
RBFNN
4 8 12 16 20 240%
20%
40%
60%
80%
100%AUE of reactor 2
time (h)
A U E
Practical value
RBFNN
4 8 12 16 20 240%
20%
40%
60%
80%
100%AUE of reactor 3
time (h)
A U E
Practical value
RBFNN
4 8 12 16 20 240%
20%
40%
60%
80%
100%AUE of reactor 4
time (h)
A U E
Practical value
RBFNN
Fig. 18. Additive utilization ef ciency of each reactor.
0 2 4 6 8 10 12 14 16 18 20 22 240%
10%
20%
30%
40%
50%Impurity removal ratio
time (h)
I m p u r i t y r e m
o v a l r a t i o
Reactor 1
Reactor 2Reactor 3
Reactor 4
Fig. 19. Optimized impurity removal ratio of each reactor.
B. Sun et al. / Control Engineering Practice 44 (2015) 89–103 99
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Another important phenomenon is that the uctuation in co-
balt ion concentration is gradually attenuated from reactor 1 to
reactor 4. This is due to the constraints on the IRR of each reactor
and the use of robust adaptive controller. By using the controller,
the zinc dust dosage is neither kept constant nor changed drasti-
cally. As a result, the nal cobalt ion concentration is more stable
compared with the open loop control manner adopted in Sun et al.
(2014).
0 2 4 6 8 10 12 14 16 18 20 22 240
5
10
15
time (h)
O u t l e t i m p u r i t y i o n c o n c e n t r a t i o n ( m g / L )
Setting values of impurity ion concentration of reactors
Reactor1
Reactor2
Reactor3
Reactor4
Fig. 20. Setting value of the impurity ion concentration of each reactor.
Table 2
Constraints on IRR of each reactor.
Reactor Reactor 1 Reactor 2 Reactor 3 Reactor 4
Constraint on IRR [55% 70%] [20% 40%] [2.5% 5%] [1% 3%]
4 8 12 16 20 24
−560
−550
−540
−530
−520
ORP setting value of reactor 1
time (h)
O R P ( m
v )
4 8 12 16 20 24
−620
−610
−600
−590
−580
−570
−560
ORP setting value of reactor 2
time (h)
O R P ( m
v )
4 8 12 16 20 24
−660
−650
−640
−630
−620
−610
−600
ORP setting value of reactor 3
time (h)
O R P ( m
v )
4 8 12 16 20 24
−660
−650
−640
−630
−620
−610
−600
ORP setting value of reactor 4
time (h)
O R P ( m
v )
Fig. 21. ORP setting value.
B. Sun et al. / Control Engineering Practice 44 (2015) 89–103100
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4 8 12 16 20 240.25
0.30
0.35
0.40Additive dosage of reactor 1
time (h)
D
o s a g e ( k g / m 3 )
4 8 12 16 20 240.10
0.15
0.20
0.25
0.30Additive dosage of reactor 2
time (h)
D
o s a g e ( k g / m 3 )
4 8 12 16 20 240
0.02
0.04
0.06
0.08
0.10Additive dosage of reactor 3
time (h)
D
o s a g e ( k g / m 3 )
4 8 12 16 20 240
0.02
0.04
0.06
0.08
0.10Additive dosage of reactor 4
time (h)
D
o s a g e ( k g / m 3 )
Fig. 22. Zinc dust dosage.
4 8 12 16 20 24
−560
−550
−540
−530
−520
time (h)
O R P ( m v )
ORP of reactor 1
Practical ORP
Reference ORP
4 8 12 16 20 24
−620
−610
−600
−590
−580
−570
−560
time (h)
O R P ( m v )
ORP of reactor 2
Practical ORP
Reference ORP
4 8 12 16 20 24
−660
−650
−640
−630
−620
−610
−600
time (h)
O R P ( m v )
ORP of reactor 3
Practical ORP
Reference ORP
4 8 12 16 20 24
−660
−650
−640
−630
−620
−610
−600
time (h)
O R P ( m v )
ORP of reactor 4
Practical ORP
Reference ORP
Fig. 23. ORP tracking performance.
B. Sun et al. / Control Engineering Practice 44 (2015) 89–103 101
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5. Conclusion
This paper proposed an intuitive two layer control scheme for
solution purication process. The idea of gradient optimization
was derived based on two concepts (AUE and IRR) extracted fromthe common characteristics of solution purication process. The
desired trajectories of ef uent impurity ion concentration of each
reactor are tracked by adjusting the setting values of ORP and
controlling the additive dosage. The feasibility of the proposed
scheme is proved and illustrated through a case study. However,
there are still some drawbacks of the proposed scheme. The per-
formance of this scheme is affected by the accuracy of the process
model and selection of the design parameters. It is suggested to
determine these design parameters at the development stage. If
the accuracy of the process model is not suf cient or the user is
lack of experience in tuning the design parameters, the perfor-
mance of the scheme may deteriorate. Thus model free controller
design approach, and more precise and comprehensive process
modeling method still needs to be studied to increase the ability of
the scheme in the future.
Acknowledgments
This research was supported by Innovation-driven Plan in
Central South University (Grant no. 2015cx007), the National
Natural Science Foundation of China (Grant nos. 61174133,
61273185) and Science Fund for Creative Research Groups of the
National Natural Science Foundation of China (Grant no.
61321003). Bei Sun and Mingfang He would like to thank China
Scholarship Council for the nancial support (Nos. 201206370097
and 201306370089). These are gratefully acknowledged.
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4 8 12 16 20 240
2
4
6
8
10
time (h)
m g / L
Impiurity ion concentration of reactor 1
Control result
Reference value
4 8 12 16 20 240
2
4
6
8
10
time (h)
m g / L
Impiurity ion concentration of reactor 2
Control result
Reference value
4 8 12 16 20 240
2
4
6
8
10
time (h)
m g / L
Impiurity ion concentration of reactor 3
Control result
Reference value
4 8 12 16 20 240
2
4
6
8
10
time (h)
m g / L
Impiurity ion concentration of reactor 4
Control result
Reference value
Fig. 24. Cobalt ion concentration tracking performance.
B. Sun et al. / Control Engineering Practice 44 (2015) 89–103102
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