Design and Application of Rough Controller in Three-tank System

Pan Aixian Department of Automatization

Qingdao Technological University Qingdao, China

e-mail: paxg@163.com

Gao Yun School of Electrical and Control Engineering Xi’an University of Science & Technology

Xi’an, China e-mail: xkdgaoyun@xust.edu.cn

Abstract—Rough set theory is an important soft computing method. Some useful knowledge can be extracted from original data with it. In this paper, a design method for rough controller is proposed based on rough set theory, and it be used to control the water-levels of a three-tank system. The key element to design rough controller is how to extract rule sets from human control behavior data according to rough set theory algorithm. The three-tank system consists of two variable frequency controllers, two ac-motors, two pumps and three tanks. The rough controller design steps include sampling control behavior data from a human operator for controlling the levels of the tanks, constructing the primal decision information tables using these data, extracting the control rules from these tables, completizating the rule sets, designing the controller and checking rules. Finally, the rough controllers are used to control the actual system. The experimental results indicate that design method of the rough controller is feasible and control effect is satisfying.

Keywords-rough set theory; three-tank; control rule; rough controller

I. INTRODUCTION Human operators are able to control some complex

objects controlled, while they do not need to know their quantitative models. One of the intelligent control methods is how let machine to imitate the control-behavior of the human beings. The aim is that the control actions of the human are used to synthesize an automatic control system. This paradigm is called learning from examples [1]. The RST is a very useful soft computing technique to achieve this idea.

Rough Set Theory (RST) is a new mathematical approach in data analysis and mining [2, 3]. Developing over 20 years, it has led to many interesting applications and extensions [4], including engineering aspects, e.g. inverted pendulum control [1, 5] and system modeling [6]. In RST, the knowledge is presented with a table (is called information table or decision table). The data of the human’s control behavior to a process is used to construct a decision table. The useful knowledge or rules can be extracted from the data using the rough set algorithm. The human’s control behavior can be represented by these rules. The machine possessed these rules can be used to control the system as human operators approximately.

This paper gives a design method for Rough Controller (RC), the data come from the sampling to control behavior of a human operator for controlling the levels of a three-tank

system, the original Decision Information Tables (DITs) are constructed using these data, the control rule sets of the behavior are extracted from these tables according to RST, the rule sets are completizated and as the decision rule set of the controller, the RC are designed, and applying it to the system. The experimental results indicate that design method of the RC is feasible and control effects are satisfying.

II. ROUGH SET THEORY

A. Indiscernibility Relation In RST, a finite set combined by all investigated objects

is called universe, denoted as U. Another finite set of available information about objects is called attribute, denoted as A. The pair ),( AUS � is called an information system or an attribute-value table.

With every attribute , its values is . Each attribute a determines a function . With every subset of attributes R of A an indiscernibility relation on U denoted and defined thus:

Aa� aV

aa VUf �:

)(RI }.),,(),(:),{()( RaayfaxfUUyxRI ������ The indiscernibility relation is also called an

equivalence relation. The family of all equivalence classes of the relation is denoted as , in short . Equivalence classes of the relation (or blocks of the partition are refereed to R-elementary sets or elementary knowledge about reality. An equivalence class of

containing x can be denoted by .

)(RI

)(RI )(/ RIU RU /)(RI

RU /

RU / )(xR

B. Lower and Upper Approximations With any subset and any subset of attributes

, the indiscernibility relation can be used to define two operations on sets,

UX �AR �

})(:{)(* XxRUxXR ��� (1a) })(:{)(* ��� XxRUxXR � (1b)

where and are called the R-lower and R-upper approximation of X, respectively. The set

)(* XR )(* XR

)()()( ** XRXRXBNR � (2)

is referred to as R-boundary region of X. If �XBNR �)( , the set X is crisp with respect to R, otherwise it is rough with respect to R.

C. Reduction of Attributes

2010 International Conference on Computing, Control and Industrial Engineering

978-0-7695-4026-9/10 $26.00 © 2010 IEEE

DOI 10.1109/CCIE.2010.155

145

2010 International Conference on Computing, Control and Industrial Engineering

978-0-7695-4026-9/10 $26.00 © 2010 IEEE

DOI 10.1109/CCIE.2010.155

144

In an information system (or a DIT), if removing some attributes, the classify ability of residual system is same as primary, the removed attributes are superfluous. The operation of removing superfluous attributes is called reduction.

Let , if and P is independent, the subset P is a reduct of R, otherwise subset

RRP ��)( � )()( RIPI �R� is

superfluous. D. Dependency of Attributes Let an information system , and

, thus we have a sub-system . In RST, the degree of dependency of a set

of attributes D on a set of attributes C can be represented by a measure, it is

),( AUS � AR �DCR �� �DC ��

),,,(1 fVRUS �

)())(()(

UcardDPOScardDk C

C �� � (3)

where is called a positive

region of the partition with respect to C. It is the set of all elements of U that uniquely classified to blocks of the partition by means of C. Obviously,

)()( */

XCDPOSDUX

C�

� �

DU /

DU / 10 k . E. Significance of Attributes The significance of an attribute can be evaluated by

measuring the effect of removing an attribute from an information table on classification defined by the table. Let C and D be sets of condition and decision attributes, respectively, (thus, the information table is called decision table). Let , the significance of the attribute a with respect to the attribute D is defined as

Ca�

)()(

1)(

)()()( }{}{

DD

DDD

aC

aC

C

aCCCD �

��

��� �

� (4)

Obviously, 1)(0 aCD� . The more important attribute a is, the greater the number )(aCD� is.

F. Decision Rules Let and � � be logical formulas representing the sets

of the conditions and the decisions, respectively, the certainty factor of decision rule is defined as �� �

)()(),(

S

SS

�card��card�� �

�� (5)

where and denote the meaning of � and S� S� � in the system . Of course, 1S 1),(0 ��� ; if 1),( ���� , the rule is deterministic, otherwise it is non-deterministic. The factor

�� �),( ��� , the larger the value, the

greater the creditability of rule . �� �

III. EXTRACTING RULE SET OF THE RC

A. Getting the DITs of human control behavior The system controlled consists of two Variable

Frequency Controllers (VFC, SIEMENS-MM12), two ac motor (380V, 0.33KW) pumps, and a three-tank. The block diagram is shown in Fig. 1.

In Fig. 1, and are the control voltages (input, 2~5V); and (Hz) are the frequencies of output voltage of the VFC1 and VFC2; and (cm

1x 2x

1f 2f

1iQ 2iQ 3/s) are the output flow rate of pump1 and pump2; , and are the liquid levels (output 1~5V, height range 0~0.4m) of the water tank 1, 2 and 3.

1h 2h 3h

Figure 2 shows the three-tank, they are cylinders with cross section A (=3.14×(0.094)2=0.0277m2).

In Fig. 2, and are the flow rate into the tank1 and tank2, 20 (cm

1iQ 2iQQ 3/s) is the flow rate (assumed to be constant)

out of the tank. The control block diagram of a human operator is shown

in Fig. 3, the control objectives are to track or maintain the liquid levels of the tank1 and tank2 at given values ( 1 2�h 95. V, 0.2m; V15.02 �h , 0.1m) through regulating the input voltages of VFC1 and VFC2 according to the error signals.

x1 VFC1Ac motor

pump1

Three

tank

h1Qi1 f1

x2 VFC2 Ac motor pump 2

Qi2 f2

h3

h2

Figure 1. Block diagram of the system controlled

Qi1

tank1h1

h3 Q12

tank3

Q20

A20A12

tank2

Figure 2. Layout of the three-tank

Figure 3. Control block diagram of human operator

In Fig. 3, and are the given outputs of the tank1 and tank2, and are the actual outputs (the liquid-levels)

1g 2g

1h 2h

Ac motor Pump2

Three

tankVFC2

Controlled system Opreator

Qi2 h2h3

h1

Ac motor Pump1 VFC1

Qi1

x2

g1 e1 x1 f1

f2 e2g2

pump1 pump2

Qi2

h2Q32

A32

146145

of the system, and are two errors, and are the regulation outputs.

111 hge � 222 hge � 1x

2xThe control behavior of the operator is obtained by

sampling , , and . The data sampling system consists of Advantech IPC-610 Industrial PC and PCL-812PG data sampling card.

1x 2x 1h 2h

In sampling process, the period is set 2.5s. In each period, the data sampled is filtered by using arithmetic mean filtering method (N =100), the average value is regarded as the sampling value. Fig. 4 shows the signals of control and outputs obtained through sampling. And their data sets are

)},(,),(,),2(),1({ 11111 nxkxxxX ��� , )},(,),(,),2(),1({ 22222 nxkxxxX ���

)},(,),(,),2(),1({ 11111 nhkhhhH ��� and . )}(,),(,),2(),1({ 22222 nhkhhhH ���In Fig.4, after the outputs of the system have basically

been stabilized, the goal to regulate afresh the inputs is to obtain more changing data, thus more information about control behavior can be gotten.

According to sampling data of and , and , let DIT

1x 1h 2x 2h1 and DIT2 of human control behavior are respectively

),,,,,( 11111 11fVVDCUDIT DC� (6)

and ),,,,,( 22222 22

fVVDCUDIT DC� (7)

0 200 4000

3

5

Discrete time / 2.5s

Con

trol in

put a

nd o

utpu

t / V

h1

x1

(a) Curve of and 1x 1h

0 200 400

0

3

5

Discrete time / 2.5s

Con

trol in

put a

nd o

utpu

t / V

h2+2

x2

(b) Curves of and 2x 2h

Figure 4. Control and output signals obtained through sampling

where )}2(),1(),(),2(),1(),({ 1111111 ���� kekekekekekeC

)}({ 11 kxD � , )()( 111 khgke � , )1()()( 111 �� kekeke , ; },,,{ 1

44112

111 uuuU ��

)}2(,)1(,)(�,)2(,)1(,)({ 2222222 ��� kekekekekekeC)}({ 22 kxD � ,

)()( 222 khgke � , )1()()( 222 �� kekeke , },,,{ 2

44122

212 uuuU �� ,

V65.41 �g ( 0.37m) V46.22 �g ( 0.15m)

B. Discretization of DITs Because the quality of the original DITs are very poor

and induced decision rules are numerous, so they must be discretized [7]. Here, take factitious discretization method.

In (6), the value domains are ]9.3,35.0[)(1 �ke , ]07.0,16.0[)(1 �� ke , and , respectively. ]0.5,0.2[)(1 �kx

The value domain of attributes , )(1 ke )1(1 ke and )2(1 ke are discretized by the cut-point set{ 0.10, 0.05,

0.02, 0.02, 0.05, 0.10, 0.20, 0.50, 1.0}, corresponding discrete codes are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, respectively.

The value domains of attributes , )(1 ke� )1(1 � ke and )2(1 � ke are discretized by the cut-point set{ 0.10, 0.05,

0.03, 0.01, 0.01, 0.03, 0.05}, corresponding discrete codes are 1, 2, 3, 4, 5, 6,7 and 8, respectively.

And the value domain of is discretized by the cut-point set {2.4, 2.7, 3.0, 3.3, 3.6, 3.9, 4.3}, corresponding discrete codes are 1, 2, 3, 4, 5, 6, 7 and 8, respectively.

)(1 kx

In this way, the discretization result of the DIT1 is shown in Table I.

In Table I a1, a2, a3, a4, a5 and a6 are )(1 ke)1(1 ke , )2(1 ke , )(1 ke� )1(1 � ke and )2(1 � ke ,

respectively d1 is decision attribute . )(1 kxIn (7), the value domains are ]0.2,30.0[)(2 �ke ,

]05.0,15.0[)(2 �� ke , and , respectively. ]0.5,0.2[)(2 �kxThe value domain of attributes , )(2 ke )1(2 ke and

)2(2 ke , the discretization intervals and discrete codes and are same. )(1 ke

The value domain of attributes , )(2 ke� )1(2 � ke and )2(2 � ke , the discretization intervals and discrete codes

and )(1 ke� are same.

TABLE I. DISCRETIZATION DIT1

U1 a1 a2 a3 a4 a5 a6 d1

1 9 9 9 2 4 1 6 � � � � � 150 4 4 4 5 5 5 3 � � � � � 300 6 6 6 6 5 7 1 � � � � � 441 3 3 3 5 6 6 3

147146

And the value domain of is discretized by the cut-point set {2.3, 2.6, 2.8, 2.9, 3.1, 3.4, 3.8, 4.2}, corresponding discrete codes are 1, 2, 3, 4, 5, 6, 7, 8 and 9, respectively.

)(2 kx

Thus, the discretization result of the DIT2 is shown in Table II.

In Table II, b1, b2, b3, b4, b5 and b6 are , , and

)(2 ke)1(2 ke )2(2 ke )(2 ke� )1(2 � ke )2(2 � ke ,

respectively d2 is decision attribute . )(2 kx

C. Reduction of Discretization DITs In discretization DIT1, determining the significances of

the elements in condition attributes set C1 relative to decision attribute in terms of (4) are listed below: )(1 kx

0570.0)]([ 1 �keCD� , 1654.0)]([ 1 �� keCD� , 0632.0)]1([ 1 �keCD� , 1948.0)]1([ 1 �� keCD� , 0785.0)]2([ 1 �keCD� , 2786.0)]2([ 1 �� keCD� .

Visible, )]([)]2([ 11 keke CDCD �� � , )]1([ 1 keCD� , )]2([ 1 � keCD� )]([ 1 keCD ��� and )]1([ 1 � keCD� .

With respect to the attribute , the)(1 kx )2(1 ke and more important than and , as well

as and , respectively. )2(1 � ke )(1 ke )1(1 ke

)(1 ke� )1(1 � keSimilarly, in discretization DIT2, the significances of C1

relative to as follows: )(1 kx0791.0)]([ 2 �keCD� , 1337.0)]([ 2 �� keCD� ,

0824.0)]1([ 2 �keCD� , 1348.0)]1([ 2 �� keCD� , 0986.0)]2([ 2 �keCD� , 1786.0)]2([ 2 �� keCD� .

With respect to the attribute , the)(2 kx )2(2 ke and more important than and , as well

as and , respectively. )2(2 � ke )(2 ke )1(2 ke

)(2 ke� )1(2 � keRemoving the unimportant attributes, the new DITs can

be obtained from Table I and Table II, they are respectively ),,,,,( 11111 11

fVVDCUTDI DC ���� � (8) and

),,,,,( 22222 22fVVDCUTDI DC ���� � (9)

where ,},,,{ 1441

12

111 uuuU �� )}2(),2({ 111 ��� kekeC ,

},,,{ 2441

22

212 uuuU �� and )}.2(),2({ 222 ��� kekeC

TABLE II. DISCRETIZATION DIT2

U2 b1 b2 b3 b4 b5 b6 d2

1 9 9 9 4 5 1 9 � � � � � 150 4 6 6 2 4 3 8 � � � � � 300 4 3 3 7 6 6 1 � � � � � 441 3 3 3 4 5 5 4

D. Production of Decision Rules

According to (5), the certainty factors are calculated in (8) and (9). Thus two decision tables can be built, the decision table I (DT1) corresponds to the (8), and the DT2 corresponds to the (9), namely

),,,,,( 11111 11fVVDCUDT DC ���� � (10)

and ),,,,,( 22222 22

fVVDCUDT DC ���� � (11) where , . },,,{ 1

8012

111 uuuU ��� },,,{ 2

7022

212 uuuU ���

In DT1, there are 49 decision rules which their certainty factors 5.0),( ���� , the certainty factors of 19 rules is less than 0.5, and 12 items are empty; In DT2, the certainty factors of 45 rules are greater than or equal 0.5, 16 items are less than 0.5, and 9 items are empty. In DT1 and DT2, the decision rules of 5.0),( ���� are retained, the rules of

5.0),( ���� and empty are reconfirmed through experiment, thus, 13 (in DT1) and 11 (in DT2) rules are obtained. But there still are 18 (DT1) and 15 (DT2) uncertain and imperfectness terms, so the decision rule set (decision tables) are incomplete. According to the method in [8], they are factitiously filled, the items are denoted as “*” in Table III and Table IV. Finally, all decision rules are determined, they can be written into the form in Table III and Table IV, and correspond to DT1 and DT2 respectively.

TABLE III. DISCRETIZATION DIT3

b a

8 7 6 5 4 3 2 1

0 1* 1* 1 2 2 2 2* 3* 1 2* 2 2* 3 3 3 4 4* 2 2* 2 3 3 3 4 4 5* 3 2 3 3 3 4 4 5 5 4 3 4 4 4 5 5 5 6 5 4 4 5 5 6 6 6 7 6 5* 5 5 6 6* 6 7 7 7 5 6* 6 6 7 7* 7 8 8 6* 6 7* 7 7* 8 8 8 9 7* 7* 8 8 8 8 8 8

TABLE IV. DISCRETIZATION DIT4

d c

7 6 5 4 3 2 1

0 1* 1 1* 2 2 2* 3* 1 1* 2* 2 3 3 3 4* 2 2 3 3 3 4 4 5* 3 3* 4 4 4 4 5 6* 4 4 4 5 5 5 6 7 5 5 5 6 6 7 7 8 6 6 7 7 8 8 8 9 7 7 8 8 8 8 9 9 8 8* 8 9* 9 9 9* 9 9 9* 9 9 9 9 9 9*

In Table III, a and b are and )2(1 ke � )2(2 ke , respectively.

148147

In Table IV, c and d are and )2(2 ke )2(2 � ke , respectively.

IV. DESIGN AND EXPERIMENTS OF THE RC The block diagram of the RC structure and experiment is

shown in Fig. 5. In Fig. 5, and are the reference inputs (4.65V and

2.46V), and are the outputs of the RC, and , and are the outputs of the system. In the RC, the data handing unit judges the current states of the system, including reference inputs and , system outputs , and processed through filtering, and errors and , etc; the database deposits the running states previously; the rule base stores the rules in Table 3 and Table 4; the reasoning machine takes out the control rules from the rule base according to present and former states, and give out the control outputs applying “if…then” formula; and the output conversion unit transforms the decision codes into the output values (average values of discrete intervals).

1g 2g

1x 2x 1h 2h

3h

1g 2g 1h 2h

3h 1e 2e

The RC designed is used to control the system controlled by the human operator, the experimental results are shown in Figs. 6 and 7.

In Fig.6, for the output , the performance specifications of the RC are the rise time is 43 steps, the overshoot is 0.24V, the setting time is 62 steps, and the performance parameter of the steady-state error

1h

Figure 5. RC and experimental block diagram

0 100 200 300

0

3

5

Discrete time / 2.5s

Erro

r and

out

puts

/ V

h1

h3

x1

e1

Figure 6. Error, controller and system outputs

0 100 200 300

0

3

5

Discrete time / 2.5s

Erro

r and

out

puts

/ V h2

x2

e2

Figure 7. Error, controller and system outputs

� � �� 298105

211 0006.0)()194/1( k keJ .

In Fig.7, for the output , the performance specifications of the RC are the rise time is 42 steps, the overshoot is 0.11V, the setting time is 79 steps, and the performance parameter of the steady-state error

2h

� � �� 298121

212 0003.0)()178/1( k keJ .

The results show that the IHC designed is feasible.

V. CONCLUSIONS This paper gives a design method for RC based on RST.

The design steps include mainly establishing DITs, discretizing them, reducting condition attributes, getting original DTs, completing the DTs, designing RC and checking the rules. And it is used in a three-tank system. The experimental results show that design method of the RC is feasible, and the control performance parameter is satisfying.

D/A

RC

Qi2

h2

h3

h1

Data Handling

Unit

Database

x2 g1

x1 x2

g2

A /

D

Rule Base

D/A

System

Output Conversion

Unit Reasoning Machine

REFERENCES [1] L.Plonka, A.Mrózek. “Rule-based stabilization of the inverted

pendulum”, Computational Intelligence, Vol. 11, No.2, pp.348-356, 1995.

[2] Zdis�aw Pawlak, “Rough set approach to knowledge-based decision support”, European Journal of Operational Research, 99, pp. 48-57, 1997.

[3] Zdis�aw Pawlak, “Rough set theory and its applications to data analysis”, Cybernetics and Systems: An international Journal, 29, pp.661-688, 1998.

[4] Z. Pawlak, A.Skowron. “Rudiments of rough sets”, Information Sciences, 177(2007):3-27.

[5] A.Mrózek, L.P�onka. “Knowledge Representation in Fuzzy and Rough Controllers”, Foundamenta Informaticae, 30, pp. 299-311, 1997.

[6] Y.Cho, K.Lee, J.Yoo, and M. Park, “Autogeneration of fuzzy rules and membership functions for fuzzy modelling using rough set theory”, IEE Proc.-Control Theory Appl., Vol.145, No.5, pp. 437-442, 1998.

[7] R.S�owi�ski, J.Stefanowski, S.Greco, and B.Matarazzo. “Rough set based processing of inconsistent information in decision analysis”, Control and Cybernetics, Vol.29, No.1, pp. 379-404, 2000.

[8] Gao Yun. “The Completization Methods of Linear Subsection Interpolation on Incomplete Decision Table”, Chinese Journal of Scientific Instrument, Vol.27, No.6, pp. 1140-1143, 2006(6).

149148