Interval type-2 fuzzy inverse controller design in nonlinear IMC structure

Engineering Applications of Artificial Intelligence 24 (2011) 996–1005

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Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai

Interval type-2 fuzzy inverse controller design in nonlinear IMC structure Tufan Kumbasar n, Ibrahim Eksin, Mujde Guzelkaya, Engin Yesil Istanbul Technical University, Faculty of Electrical and Electronics Engineering, Control Engineering Department, Maslak TR-34469, Istanbul, Turkey

a r t i c l e i n f o

a b s t r a c t

Article history: Received 23 February 2011 Accepted 30 April 2011 Available online 23 May 2011

In the recent years it has been demonstrated that type-2 fuzzy logic systems are more effective in modeling and control of complex nonlinear systems compared to type-1 fuzzy logic systems. An inverse controller based on type-2 fuzzy model can be proposed since inverse model controllers provide an efficient way to control nonlinear processes. Even though various fuzzy inversion methods have been devised for type-1 fuzzy logic systems up to now, there does not exist any method for type-2 fuzzy logic systems. In this study, a systematic method has been proposed to form the inverse of the interval type2 Takagi–Sugeno fuzzy model based on a pure analytical method. The calculation of inverse model is done based on simple manipulations of the antecedent and consequence parts of the fuzzy model. Moreover, the type-2 fuzzy model and its inverse as the primary controller are embedded into a nonlinear internal model control structure to provide an effective and robust control performance. Finally, the proposed control scheme has been implemented on an experimental pH neutralization process where the beneficial sides are shown clearly. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Type-2 fuzzy models Model inversion Inverse control Nonlinear internal model control pH neutralization process

1. Introduction Fuzzy systems are very effective in modeling and control of complex nonlinear systems whose complete mathematical models are not available. It is a very well-known fact that fuzzy models can smoothly integrate a priori knowledge with information obtained from process data (Babuska, 1998; Abonyi, 2003; ¨ Celikyilmaz and Tursken, 2009). Since most of the systems and processes are characterized by uncertainties and nonlinearities, various fuzzy modeling and control strategies have been successfully implemented in many problems (Eksin et al., 2001; Babuska ¨ et al., 2002; Guzelkaya et al., 2003; Precup and Preitl, 2004; Fuente et al., 2006; Genc et al., 2009; Ahn and Truong, 2009; Chen et al., 2009). The concept of type-2 fuzzy sets is an extension and generalization of ordinary (type-1) fuzzy sets and it was first introduced by Zadeh (1975). Fuzzy logic systems that are described with at least one type-2 fuzzy set are called type-2 fuzzy logic systems (Mendel, 2000). Lately, it has been shown that type-2 fuzzy sets are more suitable in circumstances where it is difficult to determine the accurate membership function for a fuzzy set (Karnik et al., 1999; Barkati et al., 2008; Lin et al., 2005). It has also been shown that type-2 fuzzy sets are much more powerful to handle uncertainties and nonlinearities directly (Liang and Mendel, 2000; Hagras, 2007). The major problem with type-2

n

Corresponding author. Tel.: þ90 212 2856664; fax: þ90 212 2852920. E-mail address: [email protected] (T. Kumbasar).

0952-1976/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.engappai.2011.04.016

fuzzy logic system is that the computations are much more complicated compared to type-1 fuzzy logic systems. Therefore, Liang and Mendel (2000) and Mendel (2000) proposed a special type of type-2 fuzzy sets called interval type-2 fuzzy sets in which the output of interval type-2 fuzzy sets is uncertain within an interval. In order to obtain a crisp quantity as an output that can be used for real time application purposes, a type-reduction mechanism which maps the type-2 fuzzy set into a type-1 fuzzy set is needed and afterwards routine defuzzification procedure is accomplished (Liang and Mendel, 2000; Wu and Mendel, 2002; Wu and Tan, 2006a). The interval type-2 fuzzy logic systems have been successfully implemented in controller design. The beneficial sides of type-2 fuzzy controllers have been demonstrated in several control applications such as liquid-level process control (Wu and Tan, 2004; Wu and Tan, 2006b); autonomous mobile robots (Martinez et al., 2009; Juang and Hsu, 2009); plants control (Castillo et al., 2005); marine diesel engine control (Lynch et al., 2006); bioreactor control (Galluzzo et al., 2008); and pH control (Liao et al., 2009). Moreover, control structures based on ordinary fuzzy logic systems have been generalized to type-2 fuzzy systems (Kheireddine et al., 2007; Lin et al., 2009). Furthermore, Lin (2010) introduced a state observer based indirect adaptive internal type-2 fuzzy controller. In addition to these approaches, Liao et al. (2009) suggested type-2 fuzzy model based model predictive control structures. In the fuzzy logic literature, there exist various ordinary fuzzy model inversion methods. Different type-1 fuzzy model inversion techniques for certain fuzzy models have been suggested that can

T. Kumbasar et al. / Engineering Applications of Artificial Intelligence 24 (2011) 996–1005

only be applied under certain limitations. Babuska (1998), stated that the exact inverse of a fuzzy model can be obtained under some certain invertibility conditions. In addition, in Babuska (1998) the inverse Takagi–Sugeno fuzzy model controller has been obtained through the principle of rule-by-rule inversion by permutation of the antecedent and consequence parts. Moreover, Boukezzoula et al. (2003) and Boukezzoula et al. (2007) proposed an inversion method in which the fuzzy model is decomposed into fuzzy meshes and the inverse of the global fuzzy system is obtained through inversion of each fuzzy mesh. In Abonyi et al. (1999), the inverse model is obtained directly through mapping the output and input data of the process via fuzzy logic approach. Furthermore, iterative fuzzy model inversion methods are also proposed by researchers which do not require any invertibility conditions to be satisfied (Varkonyi-Koczy et al., 1998; Kumbasar, et al., 2008b). In this study, an inversion method for interval type-2 Takagi– Sugeno fuzzy logic system has been proposed. The inverse type-2 interval TS fuzzy model has been obtained through the principle of rule-by-rule inversion by permutation of the antecedent and consequence parts. The calculations are simple and straightforward. The proposed methodology is a generalization of the type-1 TS fuzzy model inversion proposed by Babuska (1998). It is important to note here that the developed inversion strategy is based on a pure analytic method. However, the proposed method is still restricted to SISO processes. Internal Model Control (IMC) is one of the mostly used control strategy in process control area because of simplicity in design and disturbance rejection properties (Garcia and Morari, 1982). Due to this design simplicity and effectiveness of the IMC, various strategies have been developed to design PID controllers based on this principle (Rivera et al., 1986; Morari and Zafiriou, 1989; Horn et al., 1996; Lee et al., 1998; Skogestad, 2003). Moreover, the internal model control principle has been extended to design other approaches for nonlinear systems (Economou et al., 1986). In this context, several Nonlinear Internal Model Control (NIMC) schemes have been proposed since it is robust against disturbances and model mismatches (Economou et al., 1986; Brown et al., 1997; Narayanan et al., 1997; Rivals and Personnaz, 2000). In the NIMC scheme, the most difficult task is to find a process model which will represent the nonlinear and/or uncertain systems especially for practical chemical processes. Economou et al. (1986) have demonstrated that the main controller can directly be obtained by model inversion in their scheme. However, even if the model matches the process perfectly, the inverse of the nonlinear model may not be obtained analytically. The inversion of nonlinear models is generally obtained by numerical methods in spite of fact that sufficient and necessary conditions are not guaranteed (Economou et al., 1986). In order to overcome this problem, researchers proposed NIMC structures where the main controller design is based on type-1 fuzzy model inversion (Boukezzoula et al., 2003; Kumbasar et al., 2008a, 2011; Oblak et al., 2010) Finally, in this study, the proposed type-2 TS fuzzy model inverse controller is embedded into the nonlinear internal model control structure and Type-2 Fuzzy Inverse Nonlinear Internal Model Controller (T2-FI-NIMC) structure has been developed. If the inverse type-2 fuzzy model is calculated exactly then an offset-free control performance is guaranteed since the type-2 fuzzy models are powerful tools in representing highly nonlinear and uncertain processes. Moreover, the model mismatches and disturbances will be compensated due to the nonlinear internal model control structure. The effectiveness of the proposed method is demonstrated on a pH neutralization experimental set-up which exhibits a highly nonlinear characteristic. The performance of the proposed

997

method T2-FI-NIMC is compared with the type-1 TS Fuzzy Inverse Internal Model Controller (T1-FI-NIMC) structure where the main controller is constructed based on the TS fuzzy model inversion method developed by Babuska (1998). It has been illustrated that the proposed T2-FI-NIMC scheme ameliorates the overall performance of the process much better in every sense compared to the T1-FI-NIMC. This paper is organized in five sections excluding the conclusion section. In Section 1, the type-2 fuzzy model structure is briefly explained. In Section 2, an existing fuzzy model inversion method for type-1 TS fuzzy logic systems is generalized to type-2 fuzzy logic systems and the newly obtained Type-2 fuzzy model based Fuzzy Inverse Controller (T2-FIC) is discussed. In Section 3, the newly proposed T2-FIC is embedded into the nonlinear internal model control structure and Type-2 Fuzzy Inverse Nonlinear Internal Model Controller (T2-FI-NIMC) structure is introduced. The description of pH neutralization process (G.U.N.T pH value Control Trainer RT-552) is briefly given in Section 4. In Section 5, the type-1 and type-2 fuzzy modeling approaches and the comparative performances of these models are given and discussed. Next, the T1-FI-NIMC and the proposed T2-FI-NIMC control schemes have been implemented and tested on the pH neutralization process experimentally. Finally, the servo and disturbance rejection performances of the T1-FI-NIMC and the T2-FI-NIMC structures have been compared based on the same pH neutralization process within this section. The results have been evaluated and discussed in the conclusion section.

2. Type-2 fuzzy model structure Type-2 fuzzy sets are generalized forms of those of type-1 fuzzy sets (Zadeh, 1975). Mathematically, type-2 fuzzy sets are not easy to describe like type-1 fuzzy sets. A type-2 fuzzy set, e is characterized by a type-2 membership function denoted as A, meðx,uÞ, where xAX and uAJx D[0,1], i.e.: A

e ¼ f ððx,uÞ, m ðx,uÞÞ 98x A X,8u A Jx D ½0,1g A e A

ð1Þ

e in which 0 r meðx,uÞ r 1. For a continuous universe of discourse, A A can be expressed as Z Z e¼ meðx,uÞ=ðx,uÞ, Jx D ½0,1 ð2Þ A A

x A X u A Jx

RR

where denotes union over all admissible x and u. Jx is referred to as the primary membership of x, and meðx,uÞ is a type-1 fuzzy A set known as the secondary set. The uncertainty in the primary e is defined by a region named membership of a type-2 fuzzy set A footprint of uncertainty (FOU) which is also the union of all primary memberships. When me ðx,uÞ ¼ 1 for 8uAJx D [0,1], an A interval type-2 fuzzy set is obtained. An example of a Gaussian type-2 fuzzy set is given in Fig. 1. It can be described in terms of an upper membership function me ðxÞ and a lower membership A function me ðxÞ. The center of the lower and upper membership A function is named as c, while distributions of the lower and upper membership functions are named as s and s , respectively. A gaussian type-2 membership function with fixed center and uncertain distribution can be expressed as 2

meðxÞ ¼ eðxcÞ A

=2s2

,

s A ½s, s 

ð3Þ

The primary membership Jx is also given in Fig. 1, and its associated possible secondary memberships, that are Gaussian and interval, are presented in Fig. 2a and b, respectively. When the interval secondary membership function that is illustrated in Fig. 2b is taken, an interval type-2 fuzzy set is obtained (Mendel, 2000).

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T. Kumbasar et al. / Engineering Applications of Artificial Intelligence 24 (2011) 996–1005

A fuzzy logic system described with at least one type-2 fuzzy set is called a type-2 fuzzy logic system (T2FLS). A block diagram of a T2FLS, that is a special fuzzy logic system, is given in Fig. 3. Similar to a type-1 FLS, a type-2 FLS includes type-2 fuzzifier, rule-base, inference engine, and substitutes the defuzzifier by the output processor which consists of type reduction and defuzzification mechanisms. A T2FLS is characterized by IF–THEN rules where its antecedent and/or consequent fuzzy sets are defined by type-2 fuzzy sets. The type-2 fuzzy set outputs are then processed by the typereducer which combines the output sets and then perform typereduction calculation, which leads to type-1 fuzzy sets called the type-reduced set. The defuzzifier can then defuzzify the typereduced type-1 fuzzy outputs to produce crisp outputs (Mendel, 2000). A type-2 fuzzy rule can be defined as following:

j

j

j

j ¼ 1,2,. . .,N

ð4Þ

n

The firing sets vary the consequent set and inherit the uncertainties of the antecedents’ consequent set. The total firing set f j for each rule is given by f j ¼ m ðx1 Þ \ m ðx2 Þ \    \ m ðxn Þ 1 2 n f j ¼ m1 ðx1 Þ \ m2 ðx2 Þ \    \ mn ðxn Þ

ð6Þ

yj ¼ ½yjl , yjr 

ð7Þ

e ¼ ½yl ,yr  becomes the extended output of the Type-2 Then, y Takagi–Sugeno Fuzzy Logic System (T2-TS-FLS). It inherits the uncertainty of the output of T2-TS-FLS due to antecedent or consequent parameter uncertainties. In this study, the center-of sets type reduction proposed by Karnik et al. (1999), which is used most commonly for type reduction/defuzzification, is used. In this method, yl and yr are calculated iteratively and as the initialization step of this calculation, yjr and yjl are firstly reordered 1 2 N such that y1l ry2l r    r yN l and yr r yr r    r yr , respectively. ðLÞ Defining yðRÞ r and yl , for 0 rL, R rN, as P P j j yðLÞ ¼ Lj ¼ 1 f j yjl þ N j ¼ L þ 1 f yl l P PL L j j j¼1f þ j ¼ Lþ1 f

1 μ ( x)

σ

σ μ ( x)

0

ð5Þ

^ e n ¼ ½m , mn  m

The consequent y of the rule fj is also an interval set

f0 þ p f f f THEN z is yj ¼ p n xn , 1 x1 þ p 2 x2 þ    þ p

Jx’

e 1 ¼ ½m , m1  m 1 e 2 ¼ ½m , m2  m 2

j

e 1 and x2 is A e 2 and. . .and xn is A en, rj :IF x1 is A j

j j j f f f fn j are consequent type-2 fuzzy sets, where p 0 ,p 1 ,p 2 ,. . ., p e1,A e 2 ,. . ., A e n are type-2 fuzzy sets on the universe of discourses A x1,x2y,xn, while yj is the consequent of the jth IF–THEN rule, N is the total number of rules

ð8Þ

x'

c

PR PN j j j j yðRÞ r ¼ j ¼ 1 f yl þ j ¼ R þ 1 f yl P PL N j j j¼1f þ j ¼ Lþ1 f

X

Fig. 1. Illustration of type-2 fuzzy gaussian membership function.

Then, y1 ðxÞ is the minimum of all yðLÞ ðxÞ, and y1 ðxÞ is the maximum l of all yðRÞ r ðxÞ, i.e.   P P  j j yl ðxÞ ¼ min yLl ðxÞ ¼ ylL ðxÞ ðxÞ ¼ Lj ¼ðxÞ1 f j yjl þ N j ¼ L ðxÞ þ 1 f yl 0rLrN ð10Þ PR ðxÞ j PN f þ j ¼ L ðxÞ þ 1 f j j¼1

1

1

ð9Þ

where ðxÞg L ðxÞ ¼ arg min fyðLÞ l

ð11Þ

0rLrN

and yr ðxÞ ¼ max

0 μ ( x)

μ ( x)

1

u

0 μ ( x)

μ ( x) 1 u

0rRrN

 R  P  P  j j yr ðxÞ ¼ yrðR ðxÞÞ ðxÞ ¼ Rj ¼ðxÞ1 f j yjr þ N j ¼ R ðxÞ þ 1 f yr PR ðxÞ j PN f þ j ¼ R ðxÞ þ 1 f j j¼1 ð12Þ

Fig. 2. Illustration of secondary (a) gaussian and (b) interval membership functions.

RULES

OUTPUT PROCESSING DEFUZZIFIER

Crisp Input FUZZIFIER

TYPE-REDUCER

Type-2 Input Fuzzy Sets

INFERENCE

Type-2 Output Fuzzy Sets

Fig. 3. Block diagram of type-2 fuzzy logic system.

Crisp Output

T. Kumbasar et al. / Engineering Applications of Artificial Intelligence 24 (2011) 996–1005

discrete model:

where 

R ðxÞ ¼ arg max

0rRrN

fyRr ðxÞg

ð13Þ

The iterative Karnik and Mendel algorithm is used to obtain the R*(x) and L*(x) points. Then, output of the T2-TS-FLS can be obtained by using the average value of yl(x) and yr(x). Thus, the crisp output of T2-TS-FLS is calculated using (14) y ¼

999

yr þ yl 2

ð14Þ

 xðkÞ þ p  uðkÞ f þ p f f yðk þ 1Þ ¼ p 0 1 2

ð20Þ

   f f f can be calculated using the following where p 0 , p 1 and p 2 relations: PN e ej PN e ej f Up0 f Up1 j ¼ 1 j j ¼ 1 j  ¼ f ¼ P f p , p PN e N 0 1 e f f j ¼ 1 j j ¼ 1 j ð21Þ PN e ej f Up 2 j ¼ 1 j  ¼ f p P N 2 e fj j ¼ 1

3. Interval type-2 Takagi–Sugeno fuzzy model inversion Let us consider a fuzzy logic system without time delay, characterized with a state vector x(k). The fuzzy model can be expressed as xðk þ1Þ ¼ f ðxðkÞ,uðkÞÞ

ð15Þ

where u(k) is the current input, x(k þ1) is the predicted state and f represents the fuzzy mapping. The objective of inverse model is to calculate the input signal such that the system output at the next sampling time will eventually become equal to the desired reference signal denoted as sdes(kþ1). This objective can be formally stated as follows:

Since, here, the type-2 fuzzy logic systems are considered to be of    f f f interval type, the calculated p 0 , p 1 and p 2 type-2 consequent h ij f ¼ p0 p0 ). sets possess upper and lower bounds (e.g. p 0 0

Consequently, (20) can be reformulated for left and right values of the type-2 fuzzy model as follows: yr ðkþ 1Þ ¼ p0 þ p1 xðkÞ þ p2 ur ðkÞ

ð22Þ

yl ðk þ1Þ ¼ p0 þ p1 xðkÞ þ p2 ul ðkÞ

ð23Þ

Since the objective of inverse model is to force the systems output to reach the desired input signal value (sdes(kþ1)) at the next sampling time then naturally (16) can be stated as

ð16Þ

ur ðkÞ ¼ f 1 ðxðkÞ,

sdes ðk þ 1ÞÞ

ð24Þ

Generally, it is difficult, to find inverse fuzzy function in an analytical form, only some special forms can be calculated such as type-1 TS fuzzy model (Babuska, 1998). In this strategy, local linearization of type-1 TS fuzzy model is performed so as to obtain a local model at a certain operating point. Then, the inverse fuzzy model signal is calculated directly using this local model via simple calculations for type-1 TS fuzzy model. This inversion strategy is based on simple manipulation of the antecedent and consequence parts of the fuzzy model. This simple and straightforward methodology can be generalized to the inversion of interval type-2 TS fuzzy models. In this context, let us assume that a type-2 fuzzy model of a SISO process is available and its rules are given as

ul ðkÞ ¼ f 1 ðxðkÞ,

sdes ðk þ 1ÞÞ

ð25Þ

uðkÞ ¼ f 1 ðxðkÞ, sdes ðk þ 1ÞÞ

e 1 and yðk1Þ is A e 2 and. . . and yðkny þ 1Þ is A en rj : IF yðkÞ is A y e 1 and. . . uðknd nu þ 2Þ is B en , and uðknd þ 1Þ is B u j

j

j

f f f THEN yj ðk þ 1Þ is wj ¼ p 0 þp 1 xðkÞ þ p 2 uðkÞ, j ¼ 1,2,. . .,N j

ð17Þ

j e 1 . . .A e and B e 1 . . .B e j and are type-2 fuzzy sets, p f0 , p f f where A 1 and p 2 j consequent type-2 fuzzy sets, nd transport delay, ny past outputs, nu past inputs. For the sake of simplicity, it is assumed that ny ¼nu ¼N and nd ¼1 (assuming no transport delay). Thus, the defined fuzzy sets could then be represented as one multi-dimensional state type-2 e1      A eN  B e ¼A e1     B e N . Substituting B e for B e1 , fuzzy set X the rule base structure can then be redefined as follows:

e and uðkÞ is B, e THEN yðk þ1Þ is wj ðk þ 1Þ rj : IF xðkÞ is X f f f ¼ pj0 þ pj1 xðkÞ þ pj2 uðkÞ

j

ð18Þ

Therefore, the type-2 fuzzy model output y(kþ1) can be calculated as the weighted average of the linear consequents in the individual rules: PN yðkþ 1Þ ¼

f f j j j e f j ¼ 1 fj ðp0 þ p1 xðkÞ þ p2 uðkÞÞ PN e j ¼ 1 fj

ð19Þ

j j j f f f If the parameters p 0 ,p 1 and p 2 are assumed to be frozen at a certain operating point (x(k),u(k)), then type-2 fuzzy model can be represented as the following uncertain linear time invariant

Thus, the inverse type-2 TS fuzzy model law can be formulated as ur ðkÞ ¼ ðsdes ðk þ1Þp0 p1 xðkÞÞ=p2

ð26Þ

ul ðkÞ ¼ ðsdes ðk þ1Þp0 p1 xðkÞÞ=p2

ð27Þ

where p0 ,p1 ,p2 ,p0 ,p1 and p2 are updated at each sampling time. Thus, the crisp output of inverse interval TS-T2FLS is uðkÞ ¼

ul ðkÞ þur ðkÞ 2

ð28Þ

If there exists any time delay in the system then the method should be applied after nd steps of prediction have been performed, similar to what has to be done for the case of the type-1 TS fuzzy model inversion (Babuska, 1998). The proposed interval type-2 fuzzy model inversion algorithm is presented in Table 1. The inversion method has been embedded in the original Karnik–Mendel type reduction/defuzzification algorithm. 3.1. Illustrative example To illustrate the effectiveness of the proposed inversion method, a simple example with two input and one output interval type-2 Takagi–Sugeno fuzzy model is presented. The inputs are represented by three Gaussian shape type-2 membership functions which are illustrated in Fig. 4. The output is represented by type-2 linear function sets. The consequent parameters ff f pj0 , pj1 and pj2 of the interval type-2 fuzzy input sets are tabulated in Table 2. Since the inputs and outputs are defined with type-2 fuzzy sets; this is the most general type-2 Takagi–Sugeno fuzzy system (Mendel, 2000). The rule base of the model consists of three Takagi–Sugeno type rules as following: e and uðkÞ is B e j , THEN yðk þ1Þ is wj rj : IF yðkÞ is A j f f f ¼ pj0 þ pj1 yðkÞ þ pj2 uðkÞ, j ¼ 1,2,3

ð29Þ

In this illustrative study, the proposed inversion method has been tested whether it can force the system output to the desired

1000

T. Kumbasar et al. / Engineering Applications of Artificial Intelligence 24 (2011) 996–1005

Table 1 Calculation of the left and right inverse type-2 fuzzy model signals. Step Calculation of the left inverse type-2 fuzzy model signal 1.

Sort wj (j¼1,2,y,N) in increasing order such that w1 r w2 r    r wN . fj ,pj0 ,pj1 ,pj2

Match the corresponding weights corresponds to the renumbered wj ) 2.

Calculation of the right inverse type-2 fuzzy model signal Sort wj (i ¼1,2,y,N) in increasing order such that w1 r w2 r    r wN . Match the corresponding weights fj ,pj0 ,pj1 ,pj2 (with their index corresponds to the renumbered wj )

(with their index

Initialize fj by setting fj þ fj

Initialize fj by setting j ¼ 1,2,. . .,N PN fj wj Compute y ¼ Pj ¼N 1

Find the switch point L(1r Lr N  1) such that wl r yr wl þ 1 ( wj j r L Set wj ¼ wj j 4L PN fj wj and compute y0 ¼ Pj ¼N 1

Find the switch point R(1r Rr N  1) such that wr r yr wr þ 1 ( wj jr R Set wj ¼ wj j4 R PN fj wj and compute y0 ¼ Pj ¼N 1

Check if y¼ y0 . If not go to step 3 and set y0 ¼y. If yes stop and set y¼ y0 PN PN f Upj0 f Upj1 j ¼ 1 j j ¼ 1 j p0 ¼ P p1 ¼ P N N f f j j ¼ 1 j PN j ¼ 1 j j ¼ 1 fj Up2  p2 ¼ PN j ¼ 1 fj

Check if y¼ y0 . If not go to step 3 and set y0 ¼ y. If yes stop and set y ¼y0 PN PN f Upj0 f Upj1 j ¼ 1 j j ¼ 1 j p0 ¼ P , p1 ¼ P N N

ul ðkÞ ¼ ðsdes ðkþ 1Þp0 p1 yðkÞÞ=p2

ur ðkÞ ¼ ðsdes ðkþ 1Þp0 p1 yðkÞÞ=p2

2

,

fj ¼

4.

f j ¼ 1 j

PN f Upj2 j ¼ 1 j p2 ¼ P N

f j ¼ 1 j

f j ¼ 1 j

1

1

0.8

0.8 Membership Grade

Membership Grade

,

f j ¼ 1 j

f j ¼ 1 j

5.

2

f j ¼ 1 j

f j ¼ 1 j

3.

fj þ fj

j ¼ 1,2,. . .,N PN fj wj Compute y ¼ Pj ¼N 1 fj ¼

0.6

0.4

0.2

0.6

0.4

0.2

0 -5

0 MFs of u

5

0 -5

0 MFs of y

5

Fig. 4. Antecedent membership functions of the T2FLS.

The tracking error between the fuzzy model output and reference signal for both cases are illustrated in Fig. 6. It can be concluded that the type-2 fuzzy system inversion is exact since the amplitude of the error is around 10  6 for sdes1 and 10  3 for sdes2.

Table 2 Parameters of the consequent type-2 fuzzy sets. Rule number

po

po

p1

p1

p2

p2

r1 r2 r3

 0.0002 0.0004 0.0003

 0.0001 0.0002 0.0002

0.9953 0.9353 0.9253

0.2429 0.6429 0.7429

0.1147 0.2048 0.1461

0.0171 0.1571 0.0571

sdesired

Inverse Interval Type -2 Fuzzy Model

u

Interval Type -2 Fuzzy Model

s

Fig. 5. Interval type-2 T–S fuzzy model inversion scheme.

reference signal in an open loop fashion as illustrated in Fig. 5. Two different reference signals have been applied, a sinusoidal signal ðsdes1 ¼ sinðtÞÞ and a uniform distributed random number within the interval [  2, 2] (sdes 2).

4. Type-2 fuzzy model based internal model control design The simplest interpretation of a control problem is to determine a control output u(k) that will force the system or the process to converge to a desired set point without any offset. One of the ways of achieving this goal is to set up the inverse model of the system (Economou et al., 1986). Here, the inverse controller is formed using the fuzzy model of the process which is illustrated in Fig. 7. Since the type-2 fuzzy models are more powerful to represent process dynamics compared to type-1 fuzzy models, an inverse type-2 model based controller should naturally provide a more effective control performance than type-1 inverse controller. In Section 3, it has been illustrated that the type-2 fuzzy model

T. Kumbasar et al. / Engineering Applications of Artificial Intelligence 24 (2011) 996–1005

Error

x 10

3

3

1001

Error

x 10

2.5

2

2

1

1.5 0 1 -1

0.5

-2

0 -0.5

-3 0

5

10

15

20

0

5

Time [s]

10

15

20

Time [s]

Fig. 6. Illustration of the tracking error in the case of (a) a sinusoidal reference signal and (b) a uniform distributed random reference signal.

ref

Inverse Fuzzy Model

u(k)

System

illustrated in Fig. 9. This process has two input streams, one containing strong acid (HCl) and the other one strong base (NaOH). For the simplicity the acetic acid stream, Fa, is considered to be constant at its nominal value while the base flow rate is considered as the manipulated variable (Fb). Therefore, the process can be described as the sodium hydroxide stream as input (Fb) and the pH (pH2) as output. The operating conditions and the parameters of the neutralization process are given in Table 3.

y(k)

y (k)

Fuzzy Model Fig. 7. Illustration of type-2 fuzzy model inverse controller (T2FMIC).

5.2. Fuzzy modeling inversion is almost exact, i.e. y(k þ1)¼sdes(kþ1). Therefore, if the type-2 fuzzy model matches the process perfectly, then one could assume that the process output would converge to the desired set point. Even though inverse model controllers may produce perfect control while operating in an open loop fashion, this open loop control would not be sufficient in the case of modeling mismatches or disturbances. Thus, the control of a nonlinear process within a nonlinear internal model control (NIMC) structure, which is illustrated in Fig. 8 is implemented to provide an effective control performance (Economou et al., 1986). The main advantage of this scheme is that the inverse fuzzy model controller has been obtained through simple calculations. If the fuzzy model is a perfect model-match of the process then the NIMC structure is equivalent to an open-loop control configuration. In case of disturbances and model mismatches, the NIMC scheme will be able to compensate the steady state error; that is, the proposed scheme guarantees offset-free performances (Economou et al., 1986). Moreover, a robustness filter in the NIMC scheme is used to filter the noise and to stabilize the control loop by reducing the sensitivity gain (Boukezzoula et al., 2003). The robustness filter is chosen as follows: RðsÞ ¼

1 Tf s þ 1

ð30Þ

5. Application to a pH neutralization process 5.1. The G.U.N.T. pH value control trainer RT 552 The control of pH in a continuous stirred tank reactors (CSTR) are complex industrial process with dominant nonlinearities. Since the pH process inherits severe nonlinearities, uncertainties and time varying behavior, it is a benchmark process to evaluate control performances. The experimental apparatus considered in this study is the pH Value Control Trainer RT 552, which is

Since both the T1FMIC and T2FMIC strategies are model based, an identification experiment is applied to G.U.N.T. RT-552 to obtain fuzzy models. For that purpose, a multi-signal (three random signal generators with different frequencies and amplitudes) is used as input signal of the process and data set with the sampling time of 2 s has been collected. The regression vector for both the type-1 and type-2 fuzzy model is constructed by considering pH(k) and Fb(k) (Kumbasar et al., in review). The identification of the fuzzy models is handled as an optimization problem. The learning procedure is accomplished in two stages. In the forward pass, the antecedent parameters are fixed while the consequent parameters are adjusted via the recursive least square (RLS) algorithm. In the backward pass, this time the consequent parameters are fixed, and the antecedent parameters of the membership functions are determined via the Big Bang Big Crunch algorithm (Erol and Eksin, 2006). Firstly, the identification of the type-1 TS fuzzy model (T1FM) is accomplished based on this procedure. Then, the identification related to the type-2 fuzzy model (T2FM) has been accomplished as an extension of type-1 FLS and here the same optimal results used for the type-1 fuzzy model are chosen as initial conditions in order to reduce the complexity and to be fair (Kumbasar et al., in review). In order to map the nonlinearities of the process better, the antecedent membership functions of the type-1 fuzzy rules are defined with seven Gaussian combinational membership functions (Gauss2mf) and the consequent parts are defined with seven linear functions with crisp parameters. Similarly, the inputs of the type-2 fuzzy model are represented by seven type-2 Gaussian combinational membership functions, and the consequent parts are defined with seven linear functions with interval parameters to match the nonlinearities of the process enhanced (Kumbasar et al., in review). A uniform random test signal is applied for validation purposes of the type-1 and type-2 fuzzy models obtained. Two performance indices are used to provide a quantitative measure of the model accuracy of the obtained models;

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d(k)

Ref + –

Robustness Filter

Inverse Fuzzy Model

u(k)

y(k)

Process

Fuzzy Model

+

y (k) –

Fig. 8. Inverse fuzzy model based nonlinear IMC scheme.

Fig. 9. (a) A general view of RT552-pH process set and (b) schematic working principle of the RT 552-pH process set.

ii) Total variation (TV) of the control input (Skogestad, 2003), which is defined as

Table 3 Parameters of the pH neutralization process. Symbols

Description

V Fa Fb Ca Cb

Volume of the CSTR Flow rate of the influent stream Flow rate of the titrating stream Concentration of the influent stream Concentration of the titrating stream

Value 0.8 l 1 l/s 0–2.1 ml/s (0–100%) 6.3096  10–4 M 13  10–4 M

namely, root mean square error (RMSE) and the variance accounted for (VAF) (Babuska, 1998). These two indices are often used to assess the quality of a model. For the validation data, the calculated performances values for the T2FM are RMSE ¼0.2634 and VAF ¼94.5942 while the T1FM has correspondingly RMSE¼0.5998 and VAF ¼83.0993 on the same data set (Kumbasar et al., in review). Thus, it can be concluded that T2FM is superior to T1FM in the case of uncertainties and strong nonlinearities are present. Moreover, it has been illustrated in Fig. 10 that the T2FM represents the nonlinear neutralization process much better than the type-1 fuzzy model. 5.3. Performance comparison of the inverse fuzzy controllers The effectiveness and the superiority of the proposed T2-FI-NIMC method over the T1-FI-NIMC is tried to be shown on the real time control application. The time constant (Tf) of the robustness filter of the nonlinear internal model control scheme has been set as 20 s via some empirical tryouts for both the T1-FI-NIMC and the T2-FI-NIMC structures. Moreover, in order to make a fair comparison of the both schemes, two performance measures are considered as follows: i) Integral of time squared error (ITSE), which is defined as Z 1 ITSE ¼ tðrðtÞyðtÞÞ2 dt ð31Þ 0

TV ¼

1 X

9ui þ 1 ui 9

32Þ

i¼1

Since the major difficulty of controlling pH neutralization processes is the nonlinear S-shaped pH titration curve, a servo performance of the T1-FI-NIMC and T2-FI-NIMC structures have been compared. The desired reference trajectories have been chosen as 6, 8, and 7 pH values, respectively. It has been illustrated in Fig. 11 that the T2-FI-NIMC scheme provides better transient state performances than the T1-FI-NIMC scheme under varying set points. Moreover, the ITSE value of T2-FI-NIMC structure is less then T1-FI-NIMC controller structure, as it can be seen in Table 4. Additionally, the T2-FI-NIMC scheme has a low value of TV that shows that it has the smoothest control signal. Secondly, the disturbance rejection performance of the T2-FINIMC and T1-FI-NIMC scheme has been examined. First of all, a reference signal of ‘‘7 pH’’ has been applied. After the process converged to the set point value, a unit step output disturbance has been applied in 50th second. As it can be seen from Fig. 12 the proposed T2-FI-NIMC structure compensated very effectively the output disturbance in a short period of time compared to the T1FI-NIMC structure. Moreover, the control signal of the T2-FI-NIMC scheme is smoother and therefore has a lower TV value. Also, the ITSE value is almost 46% better. Moreover, since the main drawback of type-2 fuzzy models is caused by type reduction/defuzzification inference mechanism, the computational time is examined for both the T1FMIC and T2FMIC. The time needed for experiment runs was measured and compared on a personal computer with an Intel Pentium Dual Core E7500 2.93 MHz processor, 2.99 GB RAM, and software package Matlab/ SIMULINK R2007A. For the servo performance study the total computational time of the T2FMIC is 0.3246 s while for T1FMIC it is 0.2154 s, whereas the average computational times are 8.115 and 5.384 ms for the T2FMIC and T1FMIC, respectively. This means

T. Kumbasar et al. / Engineering Applications of Artificial Intelligence 24 (2011) 996–1005

1003

10 9 8

pH

7 6 5

Process T1FM T2FM

4 3

0

500

1000

1500

2000

2500

Time (s) Fig. 10. Illustration of process output, type-1 and type-2 fuzzy model outputs.

8

pH

7 6 5 Reference T1-FI-NIMC T2-FI-NIMC

4 3

0

100

200

300

400 Time (s)

500

600

700

800

100

Control Signal

80 60 40 20 T1-FI-NIMC T2-FI-NIMC

0 0

100

200

300

400 Time (s)

500

600

700

800

Fig. 11. Illustration of (a) the system outputs for varying reference values and (b) the control signals for varying reference values.

Table 4 Performance comparison of the control schemes. Servo performance

Disturbance rejection

ITSE

TV

ITSE

TV

30,535.2 24,086.4

2652.9 592.9

1114.3 593.8

793.8 170.3

T1FMIC’s, it has been illustrated in the servo and disturbance rejection performance studies that T2FMIC scheme outperforms the T1FMIC scheme.

6. Conclusions T1-FI-NIMC T2-FI-NIMC

almost 50% deceleration due to type reduction/defuzzification inference mechanism of T2FLS structure. Although the total computational time of the T2FMIC is relatively big compared to the

In this study, a systematic method has been presented to find or form the inverse model of interval type-2 Takagi–Sugeno fuzzy systems for identification, measurement, and control applications. The inversion technique developed and generalized for a class of type-2 fuzzy systems consists of simple permutation of the antecedent and consequence parts based on a purely analytical

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T. Kumbasar et al. / Engineering Applications of Artificial Intelligence 24 (2011) 996–1005

8 Reference T1-FI-NIMC T2-FI-NIMC

pH

7.5 7

applications demonstrated that the T2FMIC is superior to T1FMIC since type-2 fuzzy models are more effective to map high nonlinearities and uncertainties. Besides, since the sampling time is 2 s for the pH experimental set-up and the average computational time for computing the inverse control signal for the T2FMIC is 8.115 ms, the real time application of the proposed inverse controller is feasible.

6.5

References 6 0

50

100

150

200

Time (s)

Control Signal

100

T1-FI-NIMC T2-FI-NIMC

80 60 40 20 0 0

50

100

150

200

Time (s) Fig. 12. Illustration of (a) the system output and (b) the control signal for output disturbances.

methodology. The effectiveness and the performance of the method has been shown on an illustrative example in which the output of an interval type-2 Takagi–Sugeno fuzzy system has been forced to follow certain desired reference signals in an open loop fashion. Inspecting the tracking error values between the desired input reference signal and the real system output signal, it can easily be concluded that almost exact inversion of interval type-2 Takagi– Sugeno fuzzy system has been accomplished. However, the proposed method is still restricted to SISO processes. Moreover, a nonlinear internal model control strategy where the main controller is formed as an exact type-2 TS fuzzy model inverse is proposed. In case there is no disturbance and no parameter changes occurring within the system an offset-free control performance is ensured since the inverse type-2 fuzzy model is calculated exactly. The developed fuzzy control strategy can potentially be applied to control a wide class of nonlinear processes but it is still restricted to SISO systems. The proposed control design methodology has been implemented on an experimental pH neutralization process setup and the effectiveness of the proposed T2-FI-NIMC method has been shown by comparing the results obtained via the T1-FI-NIMC method based on integral time squared error (ITSE), and total variation (TV) of the control input performance measures. The transient response performances of both methods are investigated under the reference set point of 6, 8, and 7 pH values in this successive order. The T2-FI-NIMC method provides lower ITSE and TV values compared to the T1-FI-NIMC method. Thus, the transient performance of the T2-FI-NIMC method is much better than the T1FI-NIMC method while it produces a smoother control signal. Finally, the disturbance rejection performance of the both methods has been examined and it has been observed that the proposed T2FI-NIMC method has compensated the output disturbance very effectively in a short period of time compared to the T1-FI-NIMC method. On the other hand, the total computational time of the T2FMIC is almost 50% greater than the T1FMIC’s which is caused because of the type reduction/defuzzification inference of T2FLS structure. However, the results of the real time control

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