many non-linear identification methods, such as neural networks and fuzzy models (Narendra, ... Recently, combinations of a priori knowledge with black-box modelling techniques are ... It has been shown that non-linear effect encountered in most chemical processes, distillation ... rules and data gathered from the process.
Hybrid Fuzzy Convolution Modelling and Identification of Chemical Process Systems János Abonyi, Lajos Nagy, Ferenc Szeifert Department of Chemical Engineering Cybernetics University of Veszprém, Veszprém, H-8201, POB. 158, Hungary
Abstract This paper looks at a new method of modelling non-linear dynamic processes, using grid-type Sugeno fuzzy models and a priori knowledge. The proposed hybrid fuzzy convolution dynamic model consists of a non-linear fuzzy steady-state static, and a gain-independent impulse response model-based dynamic part. The modelling of non-linear pH processes is chosen as a realistic case study for the demonstration of the proposed modelling approach. The off-line identified hybrid fuzzy convolution model is shown to be capable of modelling the non-linear process and providing better multi-step prediction than the conventional grid-type Sugeno fuzzy model.
1. Introduction The area of fuzzy-of modelling dynamical systems has developed significantly in recent years. Fuzzy models can be used as a means of both capturing human expert knowledge and dealing with uncertainty. They can be initialised by expert knowledge and can be adapted by the use of process data. This ability can be considered as direct knowledge transfer, which is the main advantage of fuzzy inference systems over classical learning systems and neural networks (Takagi and Sugeno, 1985). The Non-linear AutoRegressive with eXogenous input NARX model is frequently used with many non-linear identification methods, such as neural networks and fuzzy models (Narendra, 1990; Bhat, 1990a; Babuska, 1997). As all system identification strategies, the NARX modelling has several weaknesses. Problems associated with system identification in high dimensions are explained by the course of dimensionality. There are many manifestations of this curse and the most relevant to fuzzy modelling has presented by (Bossely, 1997, Brown 1995): •
Model size It can be proven that the number of the parameters of the fuzzy model is an exponential function of the input dimension. For example the grid-type fuzzy model of a second order process with 7 membership functions on each input variable has 7 4 rules. With growing the number of the rules, the costs of implementation and the costs of calculating the output grow exponentially, making this type of neuro-fuzzy modelling in high dimension infeasible. Also, the interpretation of a fuzzy model in highdimensions is limited.
•
Sample size Arbitrarily, consider n s 1 = 100 as a sufficiently dense sample for an univariate modelling problem. Then for an N dimensional problem n s , N = (n s , 1 ) N sample s are required to give the same sample density. In high dimensions all feasible training samples sparsely populate the input space, providing poor representation of the systems.
Due to the exponentially increasing memory and information requirements, the uses of these type NARX fuzzy models on complex, high order dynamical processes is impractical.
One of the ways of overcoming the course of dimensionality is to try to exploit structural information which is known (either a priori or is discovered during the training process) about the process in order to build parsimonious neuro-fuzzy systems (Bossely, 1997). This approach shows it is highly desirable to incorporate the whole body of a priori knowledge into the construction of dynamic fuzzy models. Recently, combinations of a priori knowledge with black-box modelling techniques are gaining considerable interest. This kind of modelling approach is usually denoted as hybridmodelling or semi-physical modelling (Schubert, 1994; Thompson, 1994; Psichogios, 1992). In this case, the hybrid process model has an internal structure that clearly determines the interaction among process variables and process parameters from physical considerations. It has been shown that non-linear effect encountered in most chemical processes, distillation columns, pH neutralisation processes, heat-exchangers, etc. can be effectively modelled as the combination
of
a
non-linear
static
element
and
a
gain-independent
dynamic
part
(Eskinat,1991; Pottman,1998). Probably the best-known member of this class of models is the Hammerstein and Wiener model (Duwaisht, 1997, 1998). These "next-step-beyond-linear-models" have been shown to be suitable for gray-box modelling where it is assumed that the steady-state behaviour of the process is known a priori (Pottman, 1998). The disadvantage of this approach is that – as a process gets more complex in its physical description – a first-principles steady-state model tends to become increasingly complicated and computationally intensive, often requiring non-linear equation-solving techniques and iterative numerical searches to obtain a solution. Moreover, first-principles steady-state models can be rarely obtained. Therefore, in this study, the steady-state fuzzy model is identified with the help of linguistic rules and data gathered from the process. The identification of the dynamic part of a block-oriented model is also a challenging task (Duwaish, 1997). In this study an impulse response model (IRM) is applied to represent the dynamic behaviour of the process. In practice, the identification of the parameters of the IRM may be troublesome due to their large number (Richalet, 1978). In some cases this problem can be simplified if the modeller has a priori knowledge about the dynamic behaviour of the
system. This is especially true for some chemical processes, when the impulse response model relates to the resident time distribution of the operating unit (Mohilla, 1982). Consequently, this paper looks at a new modelling method, which models the complex nonlinear process as the combination of a steady-state fuzzy and an a priori knowledge based non-linear dynamic part. The idea of the approach is to train the steady-state and the dynamic fuzzy models separately and sequentially. A simulation example will be provided for a pH process to demonstrate the application of the method. The paper is organised as follows: The description of the proposed modelling approach is in section 2. In section 3 the techniques for the identification of the hybrid fuzzy convolution model is presented. Section 4 offers one case study based on the simulation of a non-linear pH process. Finally, some conclusions are drawn.
2. Hybrid Fuzzy Convolution Dynamic Modelling In this paper, the complex dynamic system is modelled as a combination of a steady-state fuzzy and an a priori knowledge based gain independent dynamic part. For open loop stable processes, the most obvious representation can be given by using a gain-scaled discrete convolution model, which will be discussed in this section. The proposed block-oriented hybrid fuzzy convolution model can be described as: y m (k + 1) = y s + K (u s , x 2 ,..., x N ) ⋅ 144424443 steady − state part
Hm
∑g
((
)
)
i ⋅ u k − i + 1 − us = 1 i 1444424444 3
(1)
dynamic part
where the convolution has to handle the gain independent g i impulse response model and previous input values, u (k − i + 1) , over the H m model horizon. The convolution is multiplied by the K (u s , x 2 ,..., x N ) steady-state gain,
K=
∂f s (u s , x 2 ,..., x N ) ∂u s
(2)
which depends on u s steady-state input and other operating parameters, x 2 ,…,x N ,
having
effects on the steady-state output y s = f s (u s , x 2 ,..., x N ) . According to the choice of the reference point – y s or u s – the convolution model can be applied in several ways. E.g. if the reference point is chosen as u s =
Hm
∑g
l
⋅ u (k − l + 1) the
l =1
Hm proposed model is a Wiener model: y m (k + 1) = y s = f s g i ⋅ u (k − i + 1), x 2 ,..., x N . i =1
∑
This can be seen as a parallel model [17] application of the hybrid convolution model. If the steady-state non-linearity is invertable, u s = f
−1
( y s , x 2 ,..., x N ) ,
it is possible to apply the model
in series-parallel mode by choosing the reference as y s = y (k ) , where y (k ) is the current measured process output. 2.1 The Steady-state part, Fuzzy Model Steady-state modelling basically divides into two main classes: phenomenological and empirical. Phenomenological models, also known as first-principles or a priori models, are based on a mathematical understanding of the underlying physics and chemistry of the process phenomena. Empirical models, on the other hand, are built primarily by regressing available data to fit a suitable mathematical expression. So, they are also known as regression models. While it is desirable to develop phenomenological models for our processes, often there is no sufficient understanding of the underlying physics and chemistry. The advantage of firstprinciple models is that it is usually valid over a greater range of operating conditions than a model developed from empirical data. The disadvantage is that, as a process gets more complex in its physical description, a first-principles model tends to become increasingly complicated and computationally intensive, often requiring non-linear equation-solving techniques and iterative numerical searches to obtain a solution. Empirical models are easier to develop, because they do not need in-depth understanding of the underlying process phenomena. The resultant models are often simple and do not need the computational intensity of the phenomenological models (Ramchandran, 1998).
Based on these considerations, the steady-state behaviour of the system, y s = f s (u s , x 2 ,..., x N ) , is described by a fuzzy model, which can be formulated with a set of rules as follows: ri1 ,...,iN : if x1 is A1,i1 and ... and x N is AN ,iN then y s = d i1 ,...,iN
(3)
where x 1 = u s is the steady-state input and x 2 ,…,x N denotes the operating parameters having effects on the steady-state output, y s . The detailed description of the structure of the applied zero-order Takagi-Sugeno fuzzy model is described in the appendix. For modelling and control purpose, the mapping between u s = f s −1 ( y s , x2 ,..., x N ) has to be determined. This means the problem of the inversion of the fuzzy model. In this study the inversion method proposed by (Babuska, 1997) is applied. The fuzzy model of the steady-state behaviour of the process can be described using these rules as well: ri1 ,...,i : if x 2 is A2,i2 and ... and x N is AN ,iN then if u s is Ai1 then y s = d i1 ,...,iN N
(4)
Based on this form, at given arbitrary operating parameters, the model can be simplified to a SISO fuzzy model with using a partial deffuzzification method: ~ if u s is Ai1 then y s = d i1
~ ri1 :
At given observations, x j ∈ a j , m j , a j , m j +1 ,
(5)
the rule consequences of the simplified fuzzy
model are calculated as: m 2 +1
~ di = 1
m N +1
N Al , il (xl )d i ,..., i 1 N iN =mN l =2
∑ ∑ ∏ ...
i2 = m2
(6)
Based on this simplified form the rules of the inverse fuzzy model are: ~ r −1i1 :
~ then u = a if y s is D i1 s 1, i1
(7)
~ is the antecedent membership function on the domain of the output of the fuzzy where D i1
model:
~ y s − d i −1 ~ 1 Di1 ( y s ) = ~ ~ , d i − d i −1 1 1 ~ d i +1 − y s ~ Di1 ( y s ) = ~ 1 ~ , d i +1 − d i 1
~ ~ d i −1 ≤ y s < d i 1
1
(8) ~ ~ d i ≤ y s < d i +1 1
1
1
Based on this consideration, the output of the inverse fuzzy model can be expressed as: i1 = N1
∑ D~
us =
i1 i1 =1 i1 = N1
∑ i1 =1
( y s )a1,i
1
(9)
~ (y ) D i1 s
Above the relation y s = f s (u s , x 2 ,..., x N ) and u s = f s −1 ( y s , x 2 ,..., x N ) have been determined. This reflects the values of the steady-state input-output pairs on the whole operating interval at a given value of process parameters (x 2 ,…,x N ). The next problem of the model’s applicability consists in evaluation of the steady-state gain: K=
∂f s (u s , x2 ,..., x N ) ∂u s
(10)
Fortunately the partial derivatives of the fuzzy model (B-spline model) can be calculated by (Cox M.G., 1982): m +1
K=
1 ∂y s = ... ∂u s i = m i
Γi −1 (u s ) Γi1 (u s ) 1 − a1,i1 − a1,i1 −1 a1,i1 +1 − a1,i1 N = m N mN +1
∑ ∑
1
1, Γi1 = 0,
1
[ ∉ [a
⋅
N
∏ j =2
A j ,i j x j ⋅ d i1 ,...,iN
) 1,i , a1,i +1 )
if u s ∈ a1,i1 , a1,i1 +1 if u s
1
( )
(11)
(12)
1
In computing the gain, K, strictly speaking, the derivative does not exists in the points a1,i1 so at u s = a1,i1 the derivative is taken from the left side of a1,i1 .
2.2 The Dynamic Part (The Impulse Response Model) The identification of the dynamic part of a block-oriented model is a challenging task. In this study an impulse response model (IRM) is applied to represent the dynamic behaviour of the process. In practice, the identification of the parameters of the IRM may be troublesome due
to their large number. In some cases this problem can be simplified if the modeller has a priori knowledge about the dynamic behaviour of the system. This is especially true for some chemical processes, in which the impulse response model relates to the resident time distribution of the operating unit [10]. The residence time of a cascade consisting of continuous perfectly mixed operating units is often used to approximate the behaviour of a partially known process, e.g. a distributed parameter system in chemical engineering practice. The density function of the residence time distribution of this "general" process can be described as: n −1
tc nc ⋅ n t τ ϕ (t ) = c ⋅ ⋅ exp − nc ⋅ (nc − 1) ! τ τ
(13)
where n c is the number of the elements of the cascade and τ is the residence time. In most cases the parameters n c and τ can be easily determined or can be fitted based on the measured impulse-response models because the modelled discrete impulse response model can be calculated as follows: gi =
ϕ (i ⋅ ∆t ) Hm
∑ ϕ (i ⋅ ∆t )
,
i = 1, K , H m
(14)
i =1
where ∆t denotes the sampling time, i the ith discrete time-step, and H m is the model horizon. 2.3 The Hybrid Model, Implementation The fuzzy convolution model can be implemented through the following steps: 1. calculation of the impulse response model from equation (13) and (14), 2. choosing of the value of the reference points u s or y s , and computing the value of y s from the fuzzy model by equation (3) or the value of u s considering the inversion of the fuzzy model by equation (9), 3. calculation of the value of the steady-state gain by equation (11).
Based on the choice of the reference point the hybrid fuzzy convolution model can be applied in series-parallel (one-step ahead prediction) and parallel model (multi-step ahead prediction) mode. At series-parallel mode, in the second step, the y s =y(k) is chosen as reference point, while at "free run" modelling the chosen reference point, u s , is based on the previous process inputs, u(k),…u(k-H m +1).
us =
Hm
∑g
l
⋅ u (k − l + 1)
(15)
l =1
In this case the fuzzy model has a special a priori knowledge based filtered regressor (Wahlberg, 1994).
3. Identification of the Fuzzy Convolution Process Model In this section the identification of the hybrid fuzzy convolution model is discussed. The novelty of the presented method is to train the steady-state fuzzy model and the dynamic model independently (Duwaish, 1996). The gain-independent dynamical model is identified based on the measured impulse response of the process. Based on a priori knowledge, the density function of the residence time distribution of a cascade consisting of perfectly stirred units is used for the approximation of the normalised measured impulse response. Therefore, the identification of the dynamic model means the identification of the value of n c and τ in equation (13). The most obvious way to identify the steady-state fuzzy model is based on measured steadystate input-output data pair. If there are enough training data, this means a standard fuzzy model identification problem (Babuska,1998). The remainder part of this section is intended to describe a new method, in which the fuzzy model is identified based on transient data and the previously identified dynamic model of the process. Assuming the given training data set has n t entries, the mean square error (MSE) error criterion is defined by:
MSE =
1 nt
∑ (y(k + 1) nt
− y m (k + 1)t
t
)
2
(16)
t =1
where y (k + 1)t is the t t h reference for the hybrid fuzzy convolution model output, y m (k + 1)t .
{x , x ,..., x }, where t 1
At given training data
t 2
t N
{
x1t = u st ; and u (k ) t ,..., u (k − H m ) t
} the output of the
fuzzy convolution model can be expressed as follows:
ymt (k + 1) =
∑ ...∑ ∏ A (x )d M1
i1 =1
MN
N
i N =1 i = 2
i ,il
t i
i1 ,..,i N
m1 +1
+
m N +1
∑ ... ∑
i1 = m1 i N = m N
( )
( )
Γ u t Γi1 ust i1 −1 s − a1,i1 − a1,i1 −1 a1,i1 +1 − a1,i1
⋅
∏ A (x )⋅ d N
i=2
t i
i ,il
i1 ,...,i N
Hm ⋅ g j ⋅ u t (k − i + 1) − ust j =1
∑ (
)
(17) Introducing matrix notation this can be written as y = W ⋅d
(18)
where
[
y = y (k + 1)1 , L , y (k + 1) nt
[
d = d1,...,1 L d M1 ,..., M N
]
T
(19)
]
T
(20)
w11,...,1 L w1M ,..., M 1 N W= M O M nt w nt L w M 1 ,..., M N 1,...,1
(21)
where the elements of the W matrix are:
wit1 ,...,iN
∏A ( ) N
=
j =1
j ,i j
x tj
( )
( )
Γi −1 u st Γi1 u st 1 + − a1,i − a1,i −1 a1,i +1 − a1,i 1 1 1 1
⋅
Hm
∏ A ( )⋅ ∑ g ⋅ (u(k − l + 1) − u ) N
j =2
j ,i j
x tj
t s
l
(22)
l =1
where the reference operating point, u s , is determined as u s =
Hm
∑g
l
⋅ u (k − l + 1) .
l =1
The identification of the rule consequent parameters is a standard linear least-squares estimation problem, and the best solution for d, which minimises W ⋅ d − y
2
.
As it was mentioned in the previous section the steady-state non-linearity is assumed to be invertabe. In the case when the a priori knowledge says, that the process gain is positive in
the whole operating range the following relation has to be held: d i1 ,i2 ,K,iN < d i1 , +1,i2 ,K,iN . This knowledge can be represented by constraints like: d i1 ,i2 ,K,iN − d i1 , +1,i2 ,K,iN < 0 .
(23)
In order to be able to apply these constraints in the identification algorithm standard quadratic programming (QP) should be used instead of linear least-squares optimisation. Therefore, the optimisation problem should be formulated as: 1 min d T ⋅ H ⋅ d + c T ⋅ d d 2
(24)
(
)
(
)
by choosing H and c as, H = 2 ⋅ W T W and c = −2 ⋅ W T ⋅ y and satisfying the constrains on d: Λ ⋅ d ≤ ω , where the elements of ω and Λ can be generated straightforward way based on
equation (23).
4. Simulation example: application to pH process
4.1 The Modelling Problem The modelling of pH (the concentration of hydrogen ions) in a continuous stirred tank reactor (CSTR) is a well-known problem that presents difficulties due to large variation in process dynamics. The CSTR is shown schematically in Figure 1. The CSTR has two input streams, one containing sodium hydroxide and the other acetic acid. By writing material balances on [Na + ] and total acetate [HAC+AC - ] and assuming that acidbase equilibrium and electroneutrality relationships hold one we get: Total acetate balance:
[
]
FHAC [HAC ]in − (FHAC + FNaOH ) HAC + Ac − = V
Sodium ion balance:
[
d HAC + Ac − dt
]
(25)
[ ]
FNaOH [NaOH ]in − (FHAC + FNaOH ) Na + = V
[ ]
d Na + dt
(26)
HAC equilibrium:
[Ac ] [H ] = K −
+
[HAC ]
(27)
a
Water equilibrium:
[H ] [OH ] = K +
−
(28)
w
Electroneutrality:
[Na ] + [H ] = [OH ] + [Ac ] +
+
−
−
(29)
Equations (25) through equation (29) are a set of five independent equations which completely describe the dynamic behaviour of the strirred tank reactor. The pH can be calculated from equations (27-29) as follows:
[H ] + [H ] ⋅ ( K + [Na ] )+ [H ]⋅ ( [Na ]⋅ K − [HAC + Ac ]⋅ K + 3
+ 2
a
+
+
+
a
−
a
)
− K w − K w ⋅ K a = 0 (30)
[ ]
pH = − lg H +
(31)
The parameters used in our simulation study are the same as in (Bhat, 1990a) and are given in Table 1.
4.2 The Fuzzy Convolution Process Model The fuzzy model describing the steady-state behaviour of the pH system can be formulated with a set of rules as follows: ri1 : if FNaOH s is A1,i1 then pH s = d i1
(32)
where FNaOH s and pH s denotes the steady-state process input and output respectively. For a good modelling performance 8 antecedent fuzzy sets (Figure 2) on the input universe were utilised.
As it was mentioned, the dynamic part of the model is based on a priori knowledge about the process dynamics represented by the impulse response model formulated resident time distribution. Because for the pH reactor n c = 1, the density function of the residence time distribution can be expressed as below:
1
t
ϕ (t ) = ⋅ exp − τ τ where τ =
(33)
V is the residence time in the reactor. FNaOH + FHAC
In this study the volume of the reactor, V, is assumed to be unknown and the non-linear leastsquares optimisation (Garce,1994) is used to estimate its value by fitting equation (14) and (33) to the normalised impulse response generated from the process. As Figure 3. shows the fitted function gives a good approximation of the measured normalised impulse response when the estimated reactor volume is V = 1120 .
4.3 The Conventional Fuzzy Model In order to show the advantages and disadvantages of the proposed method the presented modelling approach is compared to conventional fuzzy modelling. The studied pH system can be correctly identified as a first-order system (McAvoy, 1972). Therefore, the rules of the fuzzy model representing the series-parallel NARX model (Narendra,1990) of the process can be formulated as: ri1 ,i2 : if pH (k ) is A1,i1 and FNaOH (k ) is A2,i2 then pH m (k + 1) = d i1 ,i2
(34)
In the parallel model, (NOE) structure (Narendra,1990), the past outputs of the non-linear model are used instead of the plant outputs, and therefore, there is a feedback around the model. This can be described by the following rules: ri1 ,i2 : if pH m (k ) is A1,i1 and FNaOH (k ) is A2,i2 then pH m (k + 1) = d i1 ,i2
(35)
In this application mode, multistep ahead prediction can be made, using former predictions of the system output. The fuzzy model has identical structure with the previous series-parallel (NARX) structure, and only differs in the input vector.
The NARX model is frequently used with many non-linear identification methods for dynamic modelling of chemical process systems, because its parameter are easy to estimate because its structure is non-recursive. The identified NARX or series-parallel model is often used and tested in NOE or parallel model mode, when the past outputs of the non-linear model are used instead of the plant outputs. By using this approach, a multistep ahead prediction can be made, using former predictions of the system output. This procedure is often called "free run" modelling. The free run is in a very rigorous test of the predictive power of the used model, because in this way small errors can accumulate to major ones. Based on the previous consideration, the conventional fuzzy model was identified in NARX mode based on transient process data. For good model performance eight and six antecedent fuzzy sets on the input universes were utilised, as shown in Figure 2 and 4.
4.4 Simulation Results The identification of the conventional fuzzy model was worked out by the determination of the consequent parameters using least-squares estimation method (Babuska,1997). The training and the validation database was generated by forcing the F N a O H
with a uniformly
distributed signal in the range of 515-525 l/min (Figure 5), because the conventional use of PRBS signals may cause loss in identifiabilily of the non-linear system (Leontaritis, 1987).The steady-state part of the hybrid convolution model was also identified based on this transient data by using quadratic programming. For the validation of the model accuracy, at first the equilibrium points of the identified models were compared to the steady-state behaviour of the process. In conventional fuzzy modelling, the pair (y*,u*) is an equilibrium point of the dynamical system, when the following holds: y* = f ( y*,..., y*, u*,..., u *)
where y (k + 1), y (k ),..., y (k − na + 1) have been substituted with y* and
(36) u (k ),..., u (k − nb + 1) have
been substituted with u* in the NARX model: . y (k + 1) = f (, (k ),..., y (k − na + 1), u (k ),..., u (k − nb + 1) ) .
In general, the previous equation can be solved using Newton's method, but for a first order system this can easily be solved by the inversion of the fuzzy model. In hybrid fuzzy convolution modelling, the equilibrium data points (y*,u*) of the model can be generated from the steady-state fuzzy model y* = f s (u*, x 2 ,..., x N )
(37)
where y s and u s have been substituted with y* and u*. Figure 6 and 7 compare the real equilibrium points to the conventional and the hybrid fuzzy model's equilibrium points respectively. In these figures, the dashed line corresponds to the real system's equilibrium points, which is in fact the titration curve, and the solid line to the estimated points. The contrast between the two figures shows that the proposed hybrid convolution fuzzy model gave much accurate steady-state modelling performance than the conventional one. The fuzzy models were also compared in their dynamic modelling capability. In the experiments a time step ∆t =0.2 min is used according to (Bhat, 1990b). In order to compare the two models the mean square output error (MSE) performance index was used. Table 2 shows the modelling performance of the proposed hybrid and conventional fuzzy models in series-parallel and parallel structure. As table 2 and the contrast between Figure 9 and 10 shows the proposed hybrid fuzzy convolution model gave better multi-step prediction performance than the conventional one. The good steady-state representation and multi-step prediction capability of the proposed hybrid fuzzy convolution model indicates its usability in building model-based controllers.
6. Conclusions With the combination of a grid-type fuzzy model for the representation of the steady-state behaviour, and a priori knowledge based impulse response model for the representation of the process dynamics the hybrid fuzzy convolution modelling technique has been presented. The advantages of the proposed method are that the new model handles the steady-state part of the system separately from the dynamical. Therefore, the steady-state fuzzy model does not suffer
from the course of dimensionality. The algorithm for the identification of the fuzzy model is based on the quadratic programming in order to generate invertable steady-state non-linearity. The proposed hybrid fuzzy convolution model has better generalisation properties then the conventional grid type fuzzy model. The proposed model could be very useful in model based predictive control, because it is capable of provideing good multi-step ahead prediction.
Appendix: The Structure of the Sugeno Fuzzy Model This paper deals with a Sugeno fuzzy model proposed by Takagi, Sugeno, and Kang (Takagi, 1985) in an effort to develop a systematic approach to generating fuzzy rules from a given input-output data set. The fuzzy model can be formulated with a set of rules as follows: ri1 ,..., i N : if x1 is A1, i1 and ... and x N is AN , i N then y = f i1 ,..., i N ( x1 ,..., x N )
(A1)
where ri1 ,...,iN denotes the fuzzy implication, N is the number of inputs, x = [x1 ,.., x N ]T is an N
( )
vector containing all inputs of the fuzzy controller. A j , i x j is the i j = 1,2,..., M j -th antecedent j
fuzzy set referring to the j - th input, whose membership functions are denoted by the same symbols as the fuzzy values, where M j is the number of the fuzzy sets on the j -th input domain. f i1 ,..., i N ( x ) is a consequent crisp function corresponding to the output of the fuzzy model, y .
Usually f i1 ,..., i N ( x ) is a polynomial in the input variables, but it can be any function as long as it can appropriately describe the output of the system within the region specified by the antecedent of the rule. When
f i1 ,...,iN ( x ) is a first order polynomial, the resulting fuzzy
inference system is called first-order Sugeno fuzzy model. When
f i1 ,..., i N ( x ) is a constant
f i1 ,...,iN ( x ) = d i1 ,...,iN , we have a zero-order Sugeno fuzzy model, which can be viewed as a special
case of the Mamdani fuzzy inference systems, which has been described as semi-fuzzy model because of using crisp values in the consequences of the rules. This paper deals with zeroorder Takagi-Sugeno or product-sum crisp type fuzzy models.
Using fuzzy inference based upon product-sum-gravity at a given input, x, the final output of the fuzzy model, y, is inferred by taking the weighted average of the d i1 ,...,iN ‘s: MN
M1
∑ ∑β ...
y=
i1 =1
i1 ,.., i N i N =1 MN M1
∑ ...∑ β i1 =1
i N =1
di
1 ,.., i N
(A2)
i1 ,.., i N
where the weights, β i1 ,..,iN > 0 , implies the overall truth value of the
i1 ,.., i N
–th rule calculated
as:
β i1 ,..,iN =
N
∏ A (x ) j =1
j ,i j
(A3)
j
The triangular membership functions were employed for each fuzzy linguistic value, where a j ,i j denotes the cores of fuzzy set A j ,i j :
( )
a j , i j = core( A j , i j ) = x j A j , i j x j = 1
(A4)
The cores of the adjacent fuzzy sets determine the support of a set thus ensuring that the fuzzy sets on a universe of discourse always form a fuzzy partition, as well as keeping the sum of the membership functions equal with 1. Constraints such as this are important in modelling or in adaptive fuzzy control since they help to obtain an interpretable and grid-type partitioning rule-base. Thus, the membership functions are defined as follows:
( )
A j ,i j x j =
( )
A j ,i j x j =
x j − a j ,i j −1 a j ,i j − a j ,i j −1 a j ,i j +1 − x j a j ,i j +1 − a j ,i j
, a j ,i j −1 ≤ x j < a j ,i j
(A5) , a j ,i j ≤ x j < a j ,i j +1
As the product operator is used for "and" connective, and the sum operator for aggregation, and the fuzzy-mean for defuzzification method, a complete rule base results in: MN
M1
∑ ∑β ...
i1 =1
i N =1
i1 ,...,iN
=1
(A6)
Therefore, at given observations, x j ∈ a j , m j , a j , m j +1 , (23) can be simplified:
y=
m1 +1
m N +1
N
∑ ∑ ∏ ...
i1 = m1 i N = m N
j =1
A j ,i j x j ⋅ d i1 ,..., i N
( )
(A7)
This type of zero-order Sugeno fuzzy model, can be viewed as a (2. order) B-spline, because the rule weights can be represented by multivariate piecewise linear B-spline membership functions formed by taking the tensor product (A3) of N univarieties B-spline membership functions formulated as (A5).
Acknowledgements The financial support of the Hungarian Science Foundation (OTKA T023157) is greatly appreciated.
References BABUSKA, R., 1997, Fuzzy Modelling and Identification, Ph.D. Thesis, Delft University of Technology, Delft BHAT, N., and MINDERMAN, and P. A., MCAVOY, T. J. and WANG, J., 1990b, Modeling Chemical Process Systems via Neural Computation, IEEE Control Systems Magazine, 1990, April, 24-30. BHAT, N., and MCAVOY, T. J., 1990a, Use of Neural Nets for Dynamic Modelling and Control of Chemical Process Systems, Computers Chem. Eng., 14(4/5), 573-583. BOSSELY K. M., 1997, Neurofuzzy Modelling Approaches in System Identification, Ph.D. thesis, University of Southampton, Southampton BROWN, M., and BOSSLEY K.M., and MILLS, D. J., and HARRIS, C.J., 1995, High Dimensional Neurofuzzy Systems: Overcoming the Course of Dimensionality, Proc. Int. Joint Conf. Of the 4 t h IEEE Int Conf. On Fuzzy Systems and the 2 n d Int. Fuzzy Engineering Symp., Yokohama, Japan, 2139-2146.
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AND KARIM, N.M., 1997, A New Method for Identification of
Hammerstein Model, Automatica, 33(10), 1871-1875 DUWAISH, H.AL., KARIM, N.M., AND CHANDRASEKAR, V., 1996, Use of Multilayer Feedforward Neural Networks in Identification and Control of Wiener Model, IEE Proceedings of Control Theory and Applications, 143(3), 255-258. ESKINAT, E., JOHNSON, S. H., AND LUYBEN, W., 1991, Use of Hammerstein Models in Identification of Non-linear Systems, AIChE Journal, 37(2), 255-268 GRACE, A., Optimization Toolbox For Use with MATLAB, MathWorks Inc., 1994 LEONTARITIS, I. J., and BILLINGS, S. A., 1987, Experimental Design and Identifiably for Non-linear Systems, Int. Journal of Systems Science, 18, 189-202. MCAVOY, T. J., and HSU, E., and LOWENTHAL, S., 1972, Dynamics of pH in Controlled Stirred Tank Reactor, Ind. Eng. Chem. Process Des. Develop., 11(1), 68-70. MOHILLA, R., FERENCZ, B., 1982, Chemical Process Dynamics, Vol. 4 in the series of "Fundamental
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Amsterdam. NARENDRA, K.S. AND PARTHASARATHY, K., 1990, Identification and Control of Dynamical Systems, IEEE Transactions on Neural Networks, 1(1), 4-27. POTTMAN, M., AND PEARSON, R.K., 1998, Block-Oriented NARMAX Models with Output Multiplicities, AIChE Journal, 44(1), 131-140 PSICHOGIOS, D.C., and UNGAR, L. H., 1992, A Hybrid Neural Network-First Principles Approach to Process Modeling, AIChE Journal, 38(10), 1499-1511. RAMCHANDRAN, S., 1998, Consider Steady-State Models for Process Control, Chemical Engineering Progress, February 1998, 75-81
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Table 1. Parameters used in the simulation
Parameter
Description
Nominal Value
V
Volume of the tank
1000 l
FHAC
Flow rate of acetic acid
81 l/min
FNaOH
Steady flow rate of NaOH
515 l/min
[NaOH] i n
Inlet concentration of NaOH
0.05 mol/l
[HAC] i n
Inlet concentration of acetic acid
0.32 mol/l
Initial concentration of sodium in the CSTR
0.0432 mol/l
[HAC+Ac ]
Initial concentration of acetate in the CSTR
0.0432 mol/l
Ka
Acid equilibrium constant
+
[Na ] -
1.753·10 - 5
Table 2. Modelling errors MSE values Conventional
Hybrid Fuzzy Convolution
Series-Parallel
0.0064
0.0262
Parallel
0.867
0.0252
FNaOH
FHAC
V pH FNaOH+FHAC
Figure 1. The pH continuous stirred tank reactor
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 515
516
517
518
519
520
521
522
523
524
525
Figure 2. The antecedent membership functions on the F N a o H input domain
Figure 3. The fitted (−) and the measured (--) gain-independent impulse response of the pH process
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
6
6.5
7
7.5
8
8.5
9
9.5
10
10.5
11
Figure 4. The antecedent membership functions on the pH(k) input domain
526
524
F (NaOH) [l/m in]
522
520
518
516
514
0
50
100
150
200
250 300 tim e [m in]
350
400
Figure 5. The applied forcing signal
450
500
11 10.5 10 9.5
pH
9 8.5 8 7.5 7 6.5 515
516
517
518
519 520 521 F (NaOH), [l/min]
522
523
524
525
Figure 6. The "real" titration curve (---) and the equilibrium points calculated by the conventional fuzzy model ()
11 10.5 10 9.5
pH
9 8.5 8 7.5 7 6.5 515
516
517
518
519 520 521 F (NaOH), [l/min]
522
523
524
525
Figure 7. The "real" titration curve (---) and the equilibrium points calculated by the hybrid fuzzy convolution model ()
11 10.5 10 9.5 9
pH
8.5 8 7.5 7 6.5
0
50
100
150
200
250 300 time [min]
350
400
450
500
Figure 8. The free-run modelling performance of the conventional fuzzy model, real system (), conventional fuzzy model (---)
11 10.5 10 9.5
pH
9 8.5 8 7.5 7 6.5
0
50
100
150
200
250 300 tim e [m in]
350
400
450
500
Figure 9. The free-run modelling performance of the hybrid fuzzy convolution model, real system (),hybrid fuzzy convolution model (---)
