a gradient optimization scheme for solution purification process 2015 control engineering practice

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.



5/15

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.



6/15

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

= − − ̇

+ ( )

^

+ ( )

^

( )



7/15

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:



8/15

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



9/15

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.



10/15

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


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


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


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.



11/15

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%


time (h)

A U E

Practical value

RBFNN

4 8 12 16 20 240%

20%

40%

60%

80%


time (h)

A U E

Practical value

RBFNN

4 8 12 16 20 240%

20%

40%

60%

80%


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.



12/15

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


time (h)

O R P ( m

v )

4 8 12 16 20 24

−660

−650

−640

−630

−620

−610

−600


time (h)

O R P ( m

v )

4 8 12 16 20 24

−660

−650

−640

−630

−620

−610

−600


time (h)

O R P ( m

v )

Fig. 21. ORP setting value.



13/15

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


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


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


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.



14/15

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


Control result

Reference value

4 8 12 16 20 240

2

4

6

8

10

time (h)

m g / L


Control result

Reference value

4 8 12 16 20 240

2

4

6

8

10

time (h)

m g / L


Control result

Reference value

Fig. 24. Cobalt ion concentration tracking performance.


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