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Interval Type-2 Fuzzy Model Based on Inverse Controller Design for the Outlet Temperature Control System of Ethylene Cracking Furnace Taoyan Zhao 1, *, Ping Li 2, *, Jiangtao Cao 2 and Tian Li 3 1 2 3

*

School of Automation, Northwestern Polytechnical University, Xi’an 710129, China School of Information and Control Engineering, Liaoning Shihua University, Fushun 113001, China; [email protected] Sinopec Tianjin Natural Gas Pipe Company, Tianjin 300457, China; [email protected] Correspondence: [email protected] (T.Z.); [email protected] (P.L.)

Received: 30 August 2017; Accepted: 20 September 2017; Published: 22 September 2017

Abstract: Multivariable coupling, nonlinear and large time delays exist in the coil outlet temperature (COT) control system of the ethylene cracking furnace, which make it hard to achieve accurate control over the COT of the furnace in actual production. To solve these problems, an inverse controller based on an interval type-2 fuzzy model control strategy is introduced. In this paper, the proposed control scheme is divided into two parts: one is the approach structure part of the interval type-2 fuzzy model (IT2-FM), which is utilized to approach the process output. The other is the interval type-2 fuzzy model inverse controller (IT2-FMIC) part, which is utilized to control the output process to achieve the target value. In addition, on the cyber-physical system platform, the actual industrial data are used to test and obtain the mathematical model of the COT control system of the ethylene cracking furnace. Finally, the proposed inverse controller based on the IT2-FM control scheme has been implemented on the COT control system of the ethylene cracking furnace, and the simulation results show that the proposed method is feasible. Keywords: interval type-2 fuzzy model; inverse controller; cyber-physical system; ethylene cracking furnace; COT control system

1. Introduction As an important and common raw material in the petrochemical industry, ethylene has a peculiar position in the development of the economy and society. The cracking furnace is the key unit of an ethylene plant, and its control performance directly affects the ethylene yield and operational stabilization of the related following work section. The ethylene cracking furnace mainly consists of a convection segment and a radiation segment. The raw oil is preheated and vaporized in the convection segment, then mixed with dilution steam into the radiation segment. The radiation segment is the reaction area and divided into eight zones; each zone can be heated independently to track the target temperature curve. The schematic diagram of the ethylene cracking furnace is shown in Figure 1. The cracking depth value is the most important control objective for an ethylene cracking furnace. However, it is difficult to obtain the accurate the cracking depth value online in the actual production process. So far, the average of the coil outlet temperature (COT) is usually used as the index to control the running of the ethylene cracking furnace [1]. However, the cracking process is a complex of chemical changes with many issues that are hard to solve by a conventional control scheme, such as the large deviation of the furnace tube outlet temperature, nonlinear performance with the process of the coking furnace tube and the multi-input and multi-output behavior [2]. Thus, it is necessary

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to adopt an advanced process control scheme to solve these challenges, which is also the inevitable it is necessary to adopt an advanced process control scheme to solve these challenges, which is also course of the current petrochemical industry. the inevitable course of the current petrochemical industry. Raw material Dilution steam Convection section

Product

Radiation section

Fuel gas

Figure1.1.Structure Structurediagram diagramofofthe theethylene ethylenecracking crackingfurnace. furnace. Figure

Recently, Recently,the theinterval intervaltype-2 type-2fuzzy fuzzylogic logicsystem system(IT2-FLS) (IT2-FLS)has hasbeen beenwidely widelyapplied appliedinina avariety variety ofoffields, fields,such suchasasthe thepH pHneutralization neutralizationprocess process[3,4], [3,4],the thenonlinear nonlinearcontinuous continuousbioreactor bioreactorwith with bifurcation [5], the mobile robot [6,7], the coupled-tank system [8], the temperature control [9], bifurcation [5], the mobile robot [6,7], the coupled-tank system [8], the temperature controlthe [9], double-inverted pendulum [10], [10], aerospace [11],[11], the the classification the double-inverted pendulum aerospace classificationand andcurve curve identification identification datasets control [13], [13],the thetravelling travellingsalesman salesmanproblem problem [14], power management datasets[12], [12],airplane airplane flight flight control [14], power management and and electrical control [15], benchmark control problems [16], and so on. The IT2-FLS extends the electrical control [15], benchmark control problems [16], and so on. The IT2-FLS extends the reasoning reasoning design freedom of thebecause fuzzy of system because dimension of the expanding dimension and designand freedom of the fuzzy system the expanding performance provided performance provided by the footprint of uncertainty [17]. In [18–20], has beenisshown that by the footprint of uncertainty (FOU) [17]. In [18–20],(FOU) it has been shown thatitIT2-FLS superior for IT2-FLS is uncertainties superior for handling uncertainties and nonlinearities compared withlogic the conventional handling and nonlinearities compared with the conventional fuzzy system. fuzzy On logic system. the other hand, the high computational cost of the type-reduction process of an IT2-FLS is On the other hand, high computational of the type-reduction process an IT2-FLS is a a major bottleneck [21],the which may hinder itscost application in the real world. Atofpresent, scholars major bottleneck hinder its algorithm application inand the the realWu–Mendel world. At uncertain present, boundary scholars mainly utilize the[21], K-M which iterativemay type-reduction [22] mainly utilize the K-M iterative type-reduction algorithm [22] and The the former Wu–Mendel uncertain type-reduction algorithm [23] in the interval type-2 fuzzy controllers. can offer an exact boundary type-reduction algorithm [23] in the interval type-2 fuzzy controllers. The former can offer computation of the type-reduction sets, but the high computational cost of the iterative computational an exact in computation of the type-reduction sets,the but the high computational cost of theforiterative process type-reduction means that it will limit application of the IT2-FLS, especially the high computational process in meansis that it will limit the application of the which IT2-FLS, real-time requirements of type-reduction the system; the latter an alternative type-reduction algorithm, has especially forexpressions the high by real-time requirements of the the sets. latterThus, is an alternative closed-form calculating the inner-bound and system; outer-bound the Wu–Mendel type-reduction has closed-form expressions by calculating theand inner-bound and type-reductionalgorithm, algorithm iswhich usually faster than the K-M type-reduction algorithm has the potential outer-bound sets. Thus, the Wu–Mendel type-reduction algorithm is usually faster than the K-M to reduce the computational burden of the IT2-FLS. In this paper, the Wu–Mendel type-reduction type-reduction algorithm andproposed has the potential algorithm is adopted in the controller.to reduce the computational burden of the IT2-FLS. In thisInternal paper, the Wu–Mendel type-reduction algorithm is adopted in the model control was put forward by Garcia and Morari inproposed 1982 [24],controller. which is one of Internal model control was put forward by Garcia and Morari in 1982 [24], which is one of theis the most commonly-used control structures in the process control field. Its main characteristic most commonly-used control structures in control the process control field. Its main characteristic is the the simple structure and design, excellent performance and disturbance rejection capability, simple structure and design, excellentsystem. controlTherefore, performance disturbance rejection especially for a nonlinear large-delay manyand scholars have had a strongcapability, interest in especially for a nonlinear large-delay Therefore, many scholars have hadHowever, a strong interest in internal model control and also havesystem. achieved many research results [25–28]. there is the internal control and also have model achieved research [25–28]. However, is the thorny model problem of finding a process thatmany represents theresults actual nonlinear system inthere the internal thorny of finding a processscholars model mostly that represents actual nonlinear system intothe model problem control structure. In general, adopt the the mathematical model approach the internal structure. general, are scholars mostly adopt and the itmathematical model process model model, control but process objects In of industry generally uncertain, is sometimes difficult approach the process model model,especially but process of industry are generally is to build atomathematical for objects the actual petrochemical process. uncertain, In order to and solveitthis sometimes difficult to build a mathematical model especially for the actual petrochemical process. In problem, some scholars use the main controllers based on the fuzzy model in the internal model order to solve this problem, some scholars use results the main based on the fuzzy In model in the control structure, which have achieved good in controllers practical applications [29–31]. [4,17,32,33], internal model control structure, which have achieved good results in practical applications [29–31]. In [4,17,32,33], some inverse controllers based on a type-2 fuzzy model control design strategy are developed, which have been successfully applied to the pH neutralization process and the spherical

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tank process. In controllers [34], a newbased iterative technique based the big bang some inverse on atype-2 type-2fuzzy fuzzymodel modelinversion control design strategy areon developed, which bighave crunch (BB-CC) is introduced, which is feasible through the simulation study of the control been successfully applied to the pH neutralization process and the spherical tank level process. In [34], of aa spherical tank.type-2 fuzzy model inversion technique based on the big bang big crunch (BB-CC) is new iterative In this article, a is novel inverse controller design scheme on IT2-fuzzy (FM)tank. has introduced, which feasible through the simulation study ofbased the level control of amodel spherical been developed. Thea proposed control scheme is divided parts: onemodel is the(FM) approach In this article, novel inverse controller design scheme into basedtwo on IT2-fuzzy has been structure part of IT2-FM, which is utilized to approach the complex nonlinear process output by developed. The proposed control scheme is divided into two parts: one is the approach structure exploiting the advantages of IT2-FM in dealing with nonlinear and uncertain problems. The other part of IT2-FM, which is utilized to approach the complex nonlinear process output by exploiting the part is the interval type-2 inverse controller (IT2-FMIC) design, which is is utilized to advantages of IT2-FM in fuzzy dealingmodel with nonlinear and uncertain problems. The other part the interval control the output process to achieve the target value by exploiting the advantages of IT2-FLS and type-2 fuzzy model inverse controller (IT2-FMIC) design, which is utilized to control the output process thetointernal structure. Finally, the proposed IT2-FM structure and IT2-FMIC are achievemodel the target value by exploiting the advantages of approach IT2-FLS and the internal model structure. applied tothe theproposed COT control of structure the ethylene cracking furnace. The performance of system the Finally, IT2-FMsystem approach and IT2-FMIC are applied to the COT control proposed approach structures of IT2-FM and IT2-FMIC are compared with the corresponding type-1 of the ethylene cracking furnace. The performance of the proposed approach structures of IT2-FM and method. Theare comparison show that the proposed methodThe cancomparison obtain better approach andthe IT2-FMIC comparedresults with the corresponding type-1 method. results show that control performance. proposed method can obtain better approach and control performance. This paper is organized intointo six six sections. In Section 2, the architecture of IT2-FM is introduced. This paper is organized sections. In Section 2, the architecture of IT2-FM is introduced. TheThe approach strategy of IT2-FM is presented in Section 3. Then, in Section 4, the newly proposed approach strategy of IT2-FM is presented in Section 3. Then, in Section 4, the newly proposed inverse controller based on IT2-FM is explained in detail. In Section 5, the proposed approach inverse controller based on IT2-FM is explained in detail. In Section 5, the proposed approach structures structures of and IT2-FM and IT2-FMIC are implemented and the outletoftemperature the of IT2-FM IT2-FMIC are implemented and tested on thetested outleton temperature the ethyleneof cracking ethylene cracking Finally,isthe conclusion is given. furnace. Finally,furnace. the conclusion given. 2. Interval Type-2 Fuzzy SetSet and Model 2. Interval Type-2 Fuzzy and Model TheThe type-2 fuzzy setset (T2-FS) was first introduced type-2 fuzzy (T2-FS) was first introducedbybyZadeh, Zadeh,which whichcan canbe beseen seenas asan an extension extension of of the the traditional traditionalfuzzy fuzzysetset [35,36]. Compared with the traditional fuzzy set, the membership [35,36]. Compared with the traditional fuzzy set, the membership function function ofisT2-FS is one rather thanvalue. a precise value. The MF of general T2-FSofisthe (MF) of(MF) T2-FS one fuzzy setfuzzy ratherset than a precise The MF of general T2-FS is composed composed of the main MF and the secondary MF. Interval type-2 fuzzy set (IT2-FS) as a special case as: main MF and the secondary MF. Interval type-2 fuzzy set (IT2-FS) as a special case is characterized is characterized as: the secondary MF degree of T2-FS is one, so its related computing and operation the secondary MF degree of T2-FS is one, so its related computing and operation are simpler than the arecorresponding simpler than the corresponding type-2 counterpart. of an in Figure type-2 counterpart. An illustration ofAn an illustration IT2-FS is show inIT2-FS Figureis2.show Observe that the 2. Observe that the IT2-FS is bounded from above and below by two type-1 fuzzy sets, which areMF IT2-FS is bounded from above and below by two type-1 fuzzy sets, which are called the upper called theand upper (UMF) and the lower MFThe (LMF), respectively. The footprint of uncertainty (UMF) the MF lower MF (LMF), respectively. footprint of uncertainty (FOU) consists of the area e) area e).UMF ( A~) and LMF ( A~) . (FOU) consists of (the between between UMF A and LMF (A u 1

~ UMF ( A )

~ UMF ( A )

~ FOU (A) ~ LMF ( A)

x

Figure 2. Illustration of interval type-2 (IT2)-membership function (MF). UMF, upper MF; LMF, lower Figure 2. Illustration type-2 (IT2)-membership function (MF). UMF, upper MF; LMF, MF; FOU, footprintof of interval uncertainty. lower MF; FOU, footprint of uncertainty.

The general structure of IT2-FM is shown in Figure 3. Unlike the conventional fuzzy system, The general structure of IT2-FM is shown in Figure 3. Unlike the conventional fuzzy system, the the output of the IT2-FM is an IT2-FS, which needs to be made into the type-1 counterpart by the output of the IT2-FM is an IT2-FS, which needs to be made into the type-1 counterpart by the type-reduction process. type-reduction process.

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OUTPUT PROCESSING

RULES

Crisp Output DEFUZZIFIER

Crisp Input FUZZIFIER

TYPEREDUCER INFERENCE

Type- 2 Input Fuzzy Sets

Type- 2 Output Fuzzy Sets

Figure Figure 3. 3. Structure diagram diagram of of the the IT2-fuzzy IT2-fuzzy model model (FM). (FM).

An IT2-FM IT2-FM rule rule can can be be expressed expressed as as follows: follows: An ~ ~ ~ j r j :xIFisxA A1 and A2 and THEN yy is eis2 and eAnnTHEN 1eis r j : IF x2 isx2A · · · xnxnisisA is C C j ,, j j= 11,, 22 · ·, ·N, N 1 1 and ~ ~

(1) (1)

~

where A antecedent MFs of the IT2-FM, while C j j is the consequent MF and usually eA11,, A e22,, eare where . .,.A,nA A n are antecedent MFs of the IT2-FM, while C is the consequent MF and usually used with with centroid centroid representation, representation,and andN N isis the thetotal totalnumber numberof offuzzy fuzzyrules. rules. used ~ ~ ~  [ A e ,  Ae 1 ]1 ] e µ 1 1= [ µ A11 , µ A ~ ~ e e µ 2~2=[[µA A22,, µ A Ae22]] (2) (2) .. ..  .  . ~e e~n , µ A en~ =[[µA µ A ,  A n]] n

n

n

where µ and µ j are the lower and upper firing intervals, respectively. The total of the firing intervals where j and  j are the lower and upper firing intervals, respectively. The total of the firing of IT2-FMj is described as: j intervals of IT2-FM is described as: fj = [fj f ] (3) j f j = µ ( xf 1j) ∩  [µf (jx2f) ∩] µ ( xn ) 1

j

n

2

f = µ1 ( x1 ) ∩ µ2 ( x2 ) ∩ µ n ( x n ) f j   ( x1)   ( x2 )   ( xn ) 1 n set, defined as: The consequent C j of the rule of the IT2-FM is 2the interval f

j

 1( x1)   2j( x 2j )   n ( x n )

C j = [ c l , cr ]

(3) (4) (4) (5)

The consequent C j of the rule of the IT2-FM is the interval set, defined as: The center-of-sets is one of the most popular methods for an IT2-FM. Thus, the type-reduced set for the IT2-FM is expressed as: (5) C j  [clj , crj ] YTR = [yl ( x ), yr ( x )] (6) The center-of-sets is one of the most popular methods for an IT2-FM. Thus, the type-reduced set where YTR is an interval type-reduced set, and it is determined by its two end points yl ( x ) and yr ( x ). for the IT2-FM is expressed as: The end-points yl ( x ) and yr ( x ) of the YTR of an IT2-FM for the input x are bounded from below and above by y ( x ) ≤ yl ( x ) ≤ yl ( x ) and ( x[)y≤( x yr ( x ) ≤ y ( x ). The inner-bound sets of YTR are YTRyr (6) l l ), yr ( x)]r represented as: n o ( M)

(0)

yl ( x ) = ), yl ( x ) by its two end points yl (x) and (7) where YTR is an interval type-reduced set,min and yitl is( xdetermined ofor the input x are bounded from yr (x) . The end-points yl (x) and yr (x) of the YnTR ( M of) an IT2-FM (0) y ( x ) = max yr ( x ), yr ( x ) (8) r below and above by yl ( x)  yl ( x)  yl ( x) and y r ( x)  yr ( x)  y r ( x) . The inner-bound sets of YTR are where: M M represented as: (0) yl ( x ) = ∑ f i yil / ∑ f i (9)

yl ( x)  min{yi=l(1M ) ( x),i=y1l(0) ( x)} M

i f yil /

M

i

yl ( x ) = ∑ ∑f y ( x)  max{iy=r(1M ) ( x),i=yr1(0) ( x)} ( M)

r

(7) (10) (8)

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(0)

yr ( x ) = ( M)

yr

M

M

i =1

i =1

∑ f i yri / ∑ f i M

(x) =

∑f

i i yr /

i =1

M

∑f

(11)

i

(12)

i =1

The outer-bound sets of YTR are represented as: 

M

i

M

i

i

(yil

− y1l )

M

i

(ylM

− yil )



∑ f ∑ f  ∑ (f − f )   i =1  i =1 i =1 × y ( x ) = yl ( x ) −   l M i M  M iM i  i ∑ f ∑ f ∑ f (yil − y1l ) + ∑ f (ylM − yil ) 

i =1

i =1

M

i

i =1

i =1

M

i

i

(yri

− y1r )

M

i

(yrM

− yri )



∑ f ∑ f   ∑ (f − f )   i =1 i =1 × y r ( x ) = y ( x ) +  i =1  r M i M   M iM i i M i 1 i ∑ f ∑ f ∑ f ( yr − yr ) + ∑ f ( yr − yr ) i =1

i =1

(13)

i =1

(14)

i =1

The approximation of the type-reduced set is shown as follows:

[yl ( x ), yr ( x )] ≈ [

yr ( x ) + y ( x ) y ( x ) + y l ( x ) r , l ] 2 2

(15)

The precise output of the IT2-FM is defined as: " # 1 yr ( x ) + yr ( x ) y l ( x ) + y l ( x ) y( x ) = + 2 2 2

(16)

3. Approach Structure of the Interval Type-2 Fuzzy Model The choice of process model inversion is crucial for the internal model control structure. However, the choice of process model inversion is difficult for some complex uncertain systems. IT2-FM is a powerful tool that deals with highly nonlinear and uncertain processes. In this paper, the IT2-FM is utilized to approach the complex process output. The structure diagram of the approach strategy of IT2-FM is2017, shown Information 8, 116in Figure 4. 6 of 14 y process(k )

u(k) Process

Interval Type-2 Fuzzy Model

y IT 2  FM ( k )

Figure IT2-FM. Figure 4. 4. Open Open loop loop approach approach structure structure diagram diagram of of IT2-FM.

Let us design a simple IT2-FM with two inputs (u(k), yPROCESS (k ) ) and one output ( yIT 2 FM (k ) ). Let us design a simple IT2-FM with two inputs (u(k), y PROCESS (k )) and one output (y IT2− FM (k )). The u(k) represent the the output output of of the the inverse inverse controller controller to to be be designed designed and and process process PROCESS ((k The u(k) and and yyPROCESS k)) represent yIT ) represents output, respectively. respectively.The The the output the IT2-FM, is toutilized to y IT2 (k)(krepresents the output of theof IT2-FM, which iswhich utilized approach 2 FM − FM the output of the original process. The antecedent MFs for the inputs u(k) and y ( k ) are shown approach the output of the original process. The antecedent MFs for the inputs PROCESS u(k) and yPROCESS (k ) in Figure 5. The total number of the fuzzy rules is 3 × 4 = 12. The rule base and consequents are given are shown in Figure 5. The total number of the fuzzy rules is 3 × 4 = 12. The rule base and in Table 1. The reasoning process of the IT2-FM is described in detail in Section 2. consequents are given in Table 1. The reasoning process of the IT2-FM is described in detail in Section 2. 1

1

0.9

0.9

0.8

0.8

The u(k) and y PROCESS (k ) represent the output of the inverse controller to be designed and process output, respectively. The y IT 2 − FM (k ) represents the output of the IT2-FM, which is utilized to

1

1

0.9

0.9

0.8

0.8

0.7

0.7

Membership Funtions

Membership Funtions

approach the output of the original process. The antecedent MFs for the inputs u(k) and y PROCESS (k ) are shown in Figure 5. The total number of the fuzzy rules is 3 × 4 = 12. The rule base and consequents are given in Table 1. The reasoning process of the IT2-FM is described in detail in Information 2017, 8, 116 6 of 13 Section 2.

0.6 0.5 0.4

0.6 0.5 0.4

0.3

0.3

0.2

0.2 0.1

0.1 0 0

0.5

2.5

2

1.5 u(k)

1

0

3

0

50

100

Yprocess(k)

150

5. Antecedent membershipfunctions functions ofofthe IT2-FM. FigureFigure 5. Antecedent membership the IT2-FM. Table 1. The rule base and consequents of the IT2-FM for an illustrative example.

Table 1. The rule base and consequents of the IT2-FM for an illustrative example. y(k) y(k) e1 A 1 e2 A 1 e3 A 1 e4 A

u(k)

~ B11

~ A11 ~2 A1 ~ A13 ~ A14

ec111 = [0, 30] B

~ B13

2

e, 120 c 3 = [40B 1 ]

3

e[120, 40] c 2 =B

4

= [120 c1 =c [0, 30], 130] c4 = [120, ] ] c 7 = [130 60, 120 c7 = [10 60, 120] c = [40, 120] c10 = [40, 120]

1

~ u(k) B12 5

c = [0, 30 c2 = [20, 40]] cc58= 0, ,30 ]] 150 = [[110 c8 = 11 [110, 150] c = [123, 132] c11 = [123, 132]

cc36 = , 70120 ] ] =[30 [40, 6

cc9 ==[5[,30, 30] 70]

c9 = [5, 30] c12 = [130, 150] c12 = [130, 150]

In order to test the approach structure of IT2-FM, it is examined via simulations on a mathematical model. Many complex processes, especially petrochemical processes, canon be adescribed In order to test the approach structure of IT2-FM, it is examined via simulations mathematical by a first order plus large lag model, as shown in the following: model. Many complex processes, especially petrochemical processes, can be described by a first order plus large lag model, as shown in the following: 37 G (s) = e −35s (17) 2937 s + 1 −35s G (s) = e (17) 29s + 1 Information 2017, 8, 116

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Then, the Then, step the signal is adopted in order to activate IT2-FM. The approach result is illustrated step signal is adopted in order to activate IT2-FM. The approach result is illustrated in in Figure 6.Figure 6. 120

Process output IT2-FM output

100

80

60

40

20

0

0

20

40

60

80

100

Figure 6. IT2-FM output approach process output.

Figure 6. IT2-FM output approach process output. From Figure 6, it can be seen that the approach results of the IT2-FM reach a certain requirement, so the IT2-FM approach scheme is effective.

From Figure 6, it can be seen that the approach results of the IT2-FM reach a certain requirement, so the IT2-FM approach scheme is effective. 4. Inverse Interval Type-2 Fuzzy Controller Design The simplest way to design a model-based controller for a process is to set up its inverse model

4. Inverse Interval Type-2 Controller as the controller [37].Fuzzy Let us design a simpleDesign IT2-FM, which is expressed as follows:

The simplest way to design a model-based model as y ( k ) =controller f ( x( k ), u ( kfor )) a process is to set up its inverse (18) the controller [37]. Let us design a simple IT2-FM, which is expressed as follows: where x(k) is the state vector, u(k) is the current input, y(k) is the output and f denotes the fuzzy mapping. Here, the inverse controller is formed utilizing the IT2-FM, which is shown in Figure 7.

0

20

40

60

80

100

Figure 6. IT2-FM output approach process output.

From Figure 6, it can be seen that the approach results of the IT2-FM reach a certain requirement, so the IT2-FM approach scheme is effective. Information 2017, 8, 116

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4. Inverse Interval Type-2 Fuzzy Controller Design The simplest way to design a model-based controller for a process is to set up its inverse model as the controller [37]. Let us design a simple IT2-FM, which is expressed as follows:

y(k) = f ( x (k ), u(k)) (18) y(k )  f ( x(k ),u(k )) (18) where x(k) is the state vector, u(k) is the current input, y(k) is the output and f denotes the fuzzy where x(k) is the state vector, u(k) is the current input, y(k) is the output and f denotes the fuzzy mapping. Here, the inverse controller is formed utilizing the IT2-FM, which is shown in Figure 7. mapping. Here, the inverse controller is formed utilizing the IT2-FM, which is shown in Figure 7. Inverse Fuzzy Controller

ref y(k)

u(k)

Inverse Fuzzy Model

Process

x(k)

Fuzzy Model

Figure 7. Open loop structure of the IT2-FM inverse controller.

Figure 7. Open loop structure of the IT2-FM inverse controller. Since the IT2-FM has a stronger ability to deal with a dynamic process compared with T1-FM, the control performance the inverse controller based on a IT2-FM is superior to the corresponding Since the IT2-FM has a of stronger ability to deal with dynamic process compared with T1-FM, type-1 counterpart. In Section 3, it has been verified that the approach structure of the IT2-FM is the control performance of the inverse controller based on IT2-FM is superior to the corresponding valid, i.e., y(k) = ref. Thus, if the IT2-FM approaches the process exactly, the process output would type-1 counterpart. Section value. 3, it has verified that the inverse approach structure of on theIT2-FM IT2-FM converge to theInreference Thebeen structure of proposed controller based is is valid, i.e., y(k)shown = ref. in Thus, if the IT2-FM approaches thecontrol process exactly, the process output would converge to Figure 8, which provides an effective performance [4,17,37].

the reference value. The structure of proposed inverse controller based on IT2-FM is shown in Figure 8, which provides an effective control performance [4,17,37]. Information 2017, 8, 116 8 of 14 d(k) Inverse Interval Type- 2 Fuzzy Model

ref

u(k)

y(k) Process

Interval Type - 2 Fuzzy Model

y IT 2 FM (k )

Figure 8. Inverse controller based on the IT2-FM design scheme.

Figure 8. Inverse controller based on the IT2-FM design scheme. 5. Simulation Studies

5. Simulation Studies A cyber-physical system (CPS) is a new complex system with heterogeneous computing units and physical objects, which are highly integrated and interactive in the network environment [38]. In

A cyber-physical system (CPS) is a new complex system with heterogeneous computing units and the control of the ethylene cracking furnace outlet temperature system, the construction of physicalcommunication, objects, whichcomputing are highly integrated and interactive theand network environment and control between the physicalin layer the information layer is[38]. an In the control of the ethylene cracking furnace outlet temperature system, the construction of communication, essential part. The advanced process control system is implemented in an industrial upper computing and control and the information layer is an essential part. computer, which isbetween connectedthe withphysical the objectlayer linking and embedding for process control (OPC) server through thecontrol hub. The physical of data transmission is established between the OPC The advanced process system is link implemented in an industrial upper computer, which is server andthe theobject controllinking station through the internalfor protocol; thus, advanced process control systemthe hub. connected with and embedding process control (OPC) server through upper computer utilizes the corresponding communication configuration, which can achieve the The physical link of data transmission is established between the OPC server and the control station physical link of data transmission with the OPC server. The data exchange between the advanced throughprocess the internal protocol; advanced process controlbysystem computer the control system andthus, the DCS control station is realized the OPCupper standard interface.utilizes The corresponding communication configuration, which can achieve the physical link of data transmission block diagram of the CPS is shown in Figure 9. A large amount of field data is collected and saved in upper computer CPS. The mathematical model of the COT controlcontrol system system of the ethylene with thethe OPC server. The via data exchange between the advanced process and the DCS furnace is obtained usingstandard the step signal as the test signal: control cracking station is realized by thebyOPC interface. The block diagram of the CPS is shown in Figure 9. A large amount of field data is 5collected .2634 and saved in the upper computer via CPS.

G( s) 

146.97 s  1

HUB

Convention Control DCS Operation station

e6.7259s

(19)

Advanced Control System Upper Computer

computer, which is connected with the object linking and embedding for process control (OPC) server through the hub. The physical link of data transmission is established between the OPC server and the control station through the internal protocol; thus, advanced process control system upper computer utilizes the corresponding communication configuration, which can achieve the physical link of data transmission with the OPC server. The data exchange between the advanced Information 2017, 8,control 116 process system and the DCS control station is realized by the OPC standard interface. The 8 of 13 block diagram of the CPS is shown in Figure 9. A large amount of field data is collected and saved in the upper computer via CPS. The mathematical model of the COT control system of the ethylene The mathematical model of the by COT control system ofthe thetest ethylene cracking furnace is obtained using the step signal as signal: cracking furnace is obtained by

using the step signal as the test signal:

5.2634 G( s)  e6.7259s 5.2634 146 . 97 s  1 G (s) = e−6.7259s 146.97s + 1

(19)

(19)

Advanced Control System Upper Computer

HUB

Convention Control DCS Operation station OPC Server

Inner Protocol FCS

Figure 9. The blockdiagram diagram of the system (CPS). Figure 9. The block thecyber-physical cyber-physical system (CPS).

5.1. Performance Comparison of Type-1 and Interval Type-2 Fuzzy Approach Models

5.1. Performance Comparison of Type-1 and Interval Type-2 Fuzzy Approach Models In this part, the effectiveness of the interval type-2 fuzzy approach model for the COT control

In system this part, theethylene effectiveness the interval type-2 fuzzyThe approach forofthe of the cracking of furnace will be demonstrated. approachmodel structure the COT fuzzy control system model of theisethylene cracking furnace willcontroller be demonstrated. The approach structure of the fuzzy a system with two inputs (inverse output, process output) and one output (fuzzy Information 2017, 8, 116 9 of 14 model output), as shown in Figure 4. The antecedent for theprocess inputs are shown and in Figure The (fuzzy model is a system with two inputs (inverse controllerMFs output, output) one 10. output model output), in Figure 4. The antecedent MFs forsets, the which inputsare areillustrated shown inin Figure consequent partas of shown the fuzzy rule is illustrated with 12 interval Table10. 2. The consequent part of the fuzzy rule is illustrated with 12 interval sets, which are illustrated in The total number of rules is 12. The approach process is similar to the third part of the paper. The Table 2. The totaland number rules outputs is 12. The approach is similar torespectively. the third partThe of the paper. process output fuzzyof model are added toprocess the disturbances, approach The process output and fuzzy model outputs are added to the disturbances, respectively. The approach results are illustrated in Figure 11. results are illustrated in Figure 11. 1

0.9

0.9

0.8

0.8

Membership Functions

Membership Functions

1

0.7 0.6 0.5 0.4 0.3

0.7 0.6 0.5 0.4 0.3

0.2

0.2

0.1

0.1

0

0

20

40

60

80 u(k)

100

120

140

160

0

0

100

200

300

400

500

Yprocess (k )

600

700

800

900

Figure Figure 10. 10. Antecedent Antecedent membership membership functions functions of of the the IT2-FM. IT2-FM. Table Table2.2. The The rule rule base baseand andconsequents consequentsof ofthe theIT2-FM. IT2-FM.

y(k) y(k) ~ Ae111 A ~ 212 e1 AA 1 ~A e3 A131

e4

~A1 A14

u(k) u(k) ~2

~1 1B1

1

e1 B

[22.636,32.915] 32.915] = [22.636, cc1 = cc22 == [226.66, 240.02] [226.66, 240.02] c33 = [543.33, 556.66] c = [543.33, 556.66] c4 = [800.00, 812.59]

c 4 = [800.00, 812.59]

1000 900

5

e 21B1 B

[22.636, 32.915] cc5 ==[22.636, 32.915] cc6 6==[226.66, 240.02] [226.66, 240.02] c7 7= [543.33, 556.66] c = [543.33, 556.66] c8 = [800.00, 812.59]

c 8 = [800.00, 812.59]

9

e 31 B

~ B13

[22.636, 32.915] c9c= =[22.636, 32.915] 10 c10 = [226.66, 240.02] c = [226.66, 240.02] c11 11 = [543.33, 556.66] c = [543.33, 556.66] c12 = [800.00, 812.59]

c 12 = [800.00, 812.59]

Process T1FMIC T2FMIC

~ A11 ~ A12 ~ A13

~ Information 2017,A148, 116

c 1 = [22.636, 32.915]

c 5 = [22.636, 32.915]

c 9 = [22.636, 32.915]

c 2 = [226.66, 240.02]

c 6 = [226.66, 240.02]

c 10 = [226.66, 240.02]

c 3 = [543.33, 556.66]

c 7 = [543.33, 556.66]

c 11 = [543.33, 556.66]

c 4 = [800.00, 812.59]

c 8 = [800.00, 812.59]

c 12 = [800.00, 812.59]

1000

Process T1FMIC T2FMIC

900

820

600

815

500

810

300

Process T1FMIC T2FMIC

805

℃

400

Temperature/

700

℃

Temperature/

800

800

200

795

100 0

9 of 13

790 150

0

50

100

150

200

200

250

250 time(s)

300 350 time(s)

300

350

400

400

450

450

500

500

Figure 11. The output comparison of the process, type-1 fuzzy model (T1-FM) and IT2-FM. Figure 11. The output comparison of the process, type-1 fuzzy model (T1-FM) and IT2-FM.

In order to make a fair comparison between T1-FM and IT2-FM, two performance indexes In order to make a fair comparison between T1-FM and IT2-FM, two performance indexes are given. are given. (1)(1) Root Root mean mean square square error error (RMSE), (RMSE), which which is is described described as as the the following following formula: formula: s 1 n RMSE = (y p − y FM )2 n k∑ =1 (2)

(20)

Variance accounted for (VAF) [4], which is described as the following formula: var(y p − y FM ) VAF = 100% × 1 − var(y p ) 

 (21)

where yp represents the process output and y FM represents the fuzzy model output. The VAF index is usually utilized to evaluate the performance of a model, by comparing the real process output with the model output. If the two signals are closer, then the VAF value is bigger. The performance results obtained from Formulas (20) and (21) are shown in Table 3. It can be clearly seen that the IT2-FM has a smaller RMSE value and a higher VAF value compared to the corresponding T1-FM. Therefore, it is seen obviously that IT2-FM has better approximation accuracy than T1-FM. Here, it is illustrated in Figure 11 that the IT2-FM represents the outlet temperature in the ethylene cracking furnace obviously better than the T1-FM. Table 3. Comparison of the fuzzy models. VAF, variance accounted for. -

RMSE

VAF

T1-FM IT2-FM

22.0541 16.2918

0.9816 0.9900

5.2. Control Performance Evaluation of the T1-FMIC and IT2-FMIC Strategies for the Outlet Temperature of the Ethylene Cracking Furnace It is a known fact that the ethylene cracking furnace is a complex industrial process with nonlinearity and time delay. The control goal for the coking furnace is to stabilize the outlet

T1-FM IT2-FM

22.0541 16.2918

0.9816 0.9900

5.2. Control Performance Evaluation of the T1-FMIC and IT2-FMIC Strategies for the Outlet Temperature of the Ethylene Cracking Furnace

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10 of 13

It is a known fact that the ethylene cracking furnace is a complex industrial process with nonlinearity and time delay. The control goal for the coking furnace is to stabilize the outlet temperature, to improve its disturbance rejection capability prolongthe therun runtime timeofofthe thefurnace. temperature, to improve its disturbance rejection capability andand to to prolong furnace. Here, IT2-FMIC the proposed IT2-FMIC to the COT control system of the ethylene Here, the proposed strategy is strategy appliedistoapplied the COT control system of the ethylene cracking cracking furnace. The proposed IT2-FMIC has two inputs (set value of COT, fuzzy model output) furnace. The proposed IT2-FMIC has two inputs (set value of COT, fuzzy model output) and one and one output (control variable); the antecedent MFs for the inputs are shown in Figure 12. The output (control variable); the antecedent MFs for the inputs are shown in Figure 12. The consequent consequent part of the fuzzy rule is illustrated with 16 interval sets, which are illustrated in Table 4. part of the fuzzy rule isofillustrated with 16 interval sets, which aremodel illustrated in is Table 4. The total The total number rules is 16. The performance of the fuzzy inverse controller illustrated numberinofFigure rules 13. is 16. The performance of the fuzzy inverse model controller is illustrated in Figure 13. 1 0.9

Membership Functions

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Information 2017, 8, 116

0

100

200

300

400 500 inputs

600

700

800

900

Figure 12. Antecedent membership functions of the inputs.

Figure 12. Antecedent membership functions of the inputs.

11 of 14

TableTable 4. The rulerule basebase andand consequents IT2-FMIC. 4. The consequentsofofthe theproposed proposed IT2-FMIC.

y(k) e1 A 1 e2 A 1 e3 A 1 e4 A 1

y(k)

~ B11

e1 ~ A11 B1 c1 = [14.44, 24.59] ~ 2 ]= 5 c1 = A 2[14.44, c24.59 = [14].44, 24.c59

u(k) u(k)

~ B12

e2 21 cB = [91.84, 105.4]

~ B13 e 31 B c 3 = [128 .8, 143.3]

~ B14

e 41 B c 4 = [163.4, 174 .0] 84 c c = [= 128[.163.4, 8, 143.3]174.0]

[91.84, 105.4 ] .4] c3 = 143.3 .4] ] c 6 = [91 .84, 105 [91.84, 105 c 7 [=128.8, 1 ~ 5 6 7 3 9 10 c = 105.4 ] .3] c = 91.84, 105.4 A1 [14.44, c24.59 = [91].84, 105c.4]= [91.84, c = [128 .8, 143 [163.4, 174 .0] ] c11[= c9 =~ 4[91.84, 13 105.4] c10 = [128.8, 143.3] c11 =15 [163.4, 174.0] 14 A1 c = [128.8, 143.3] c = [163.4, 174.0] c = [91.84, 105.4] c13 = [91.84, 105.4] c14 = [128.8, 143.3] c15 = [163.4, 174.0] 1000

Reference T1FMIC T2FMIC

900

830

600

820

500

810

300

Reference T1FMIC T2FMIC

800

℃

400

Temperature/

700

℃

Temperature/

800

790

200

780

100 0

c8= = c12 [163[128.8, .4, 174.0]143.3] c12 = [163.4, 174.0] 16 c = [163.4, 174.0] c16 = [163.4, 174.0]

770 200

0

50

100

150

250

200

250 time(s)

300

350 time(s)

300

350

400

400

450

450

500

500

Figure 13. Tracking curve ofofIT1-FMIC Figure 13. Tracking curve IT1-FMICand and IT2-FMIC. IT2-FMIC. In order to make a fair comparison between the T1-FMIC scheme and the IT2-FMIC scheme, the In order to make a fair comparison between the T1-FMIC scheme and the IT2-FMIC scheme, performance indexesare areconsidered considered as the performance indexes asfollows: follows: (1) Integral of the square error (ISE), which is described as: ∞

ISE =

∫ ((r(t) − y(t)) )dt 2

0

(2) Integral of the absolute error (IAE), which is described as: ∞

(22)

Information 2017, 8, 116

(1)

11 of 13

Integral of the square error (ISE), which is described as:

ISE =

Z∞

((r (t) − y(t))2 )dt

(22)

0

(2)

Integral of the absolute error (IAE), which is described as:

I AE =

Z∞

|r (t) − y(t)|dt

(23)

0

(3)

Integral of the time-weighted absolute error (ITAE), which is defined as:

ITAE =

Z∞

t|r (t) − y(t)|dt

(24)

0

where r(t) and y(t) are the target profile and the process output, respectively. It has been illustrated in Figure 13 that the IT2-FMIC strategy has better control performances compared with the T1-FMIC strategy. Moreover, the performance index value of the IT2-FMIC is superior to the corresponding value of T1-FMIC, as seen in Table 5. Table 5. Regulation performance comparison of the inverse controllers. ISE, integral of the square error; IAE, integral of the absolute error; ITAE, integral of the time-weighted absolute error. T1-MIC IT2-MIC

ISE

IAE 107

1.3544 × 1.3241 × 107

ITAE 104

3.3798 × 3.0134 × 104

2.1988 × 106 1.3272 × 106

6. Conclusions This article is focused on IT2-FMIC design and its application in the outlet temperature control system of the ethylene cracking furnace, which is a typical nonlinear object in petrochemical industry. The proposed IT2-FMIC scheme is divided into two parts: the approach part and the control part. Simulation results reveal that the IT2-FM can accurately approximate the complex nonlinear chemical process, and the proposed IT2-FMIC scheme has better disturbance rejection capability. However, the proposed control strategy is still constrained to a single output process. In future work, extension of the proposed method to a multi-input multi-output process will be a hot issue. Then, the optimization of the inverse IT2-FM and its robustness analysis will also be the focus of research. Acknowledgments: This work is supported by the National Science Foundation of China under Grant No. 61673199 and the Program for Liaoning Excellent Talents in University under Grant LR2015034. Author Contributions: Taoyan Zhao conceived of and designed the experiments. Taoyan Zhao and Tian Li designed the control algorithm. Taoyan Zhao wrote the paper. Jiangtao Cao and Ping Li conducted the work. Conflicts of Interest: The authors declare no conflict of interest.

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