Fuzzy modeling with multivariate membership

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 31, NO. 5, OCTOBER 2001

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Fuzzy Modeling With Multivariate Membership Functions: Gray-Box Identification and Control Design Janos Abonyi, Robert Babuˇska, and Ferenc Szeifert

Abstract—A novel framework for fuzzy modeling and model-based control design is described. The fuzzy model is of the Takagi–Sugeno (TS) type with constant consequents. It uses multivariate antecedent membership functions obtained by Delaunay triangulation of their characteristic points. The number and position of these points are determined by an iterative insertion algorithm. Constrained optimization is used to estimate the consequent parameters, where the constraints are based on control-relevant a priori knowledge about the modeled process. Finally, methods for control design through linearization and inversion of this model are developed. The proposed techniques are demonstrated by means of two benchmark examples: identification of the well-known Box–Jenkins gas furnace and inverse model-based control of a pH process. The obtained results are compared with results from the literature. Index Terms—A priori knowledge, Delaunay triangulation, fuzzy modeling, gray-box identification, inverse control, model-based control.

I. INTRODUCTION

F

UZZY modeling and identification are effective tools for the approximation of uncertain nonlinear systems because they can effectively combine expert knowledge and measured data. Fuzzy models use if–then rules to describe the process through a collection of locally valid relationships. The antecedents (if-parts) of the rules divide the input space into several fuzzy subspaces, while the consequents (then-parts) describe the local behavior of the system in these fuzzy subspaces. Most fuzzy models proposed in the literature use one-dimensional (1-D) (univariate) fuzzy sets, such as triangular or trapezoidal ones, and partition multidimensional input spaces by grid- or tree-type Cartesian products of these univariate membership functions. The advantages of this approach are the simple and transparent representation of the membership functions and the straightforward application of the model in control [1]. The identification of these models is, however,

Manuscript received June 7, 2000; revised June 26, 2001. This work was supported by the Hungarian Ministry of Culture and Education (FKFP-0023/2000, FKFP-0073/2001) and the Hungarian Science Foundation (TO23157). The work of J. Abonyi was supported by the Janos Bolyai Research Fellowship of the Hungarian Academy of Science and the Dutch Senter Project BTS98083. This paper was recommended by Associate Editor N. Pal. J. Abonyi and F. Szeifert are with the Department of Process Engineering, University of Veszprem, Veszprem H-8201, Hungary. R. Babuˇska is with the Faculty ITS, Systems and Control Engineering Group, Delft University of Technology, 2600 GA Delft, The Netherlands. Publisher Item Identifier S 1083-4419(01)08635-6.

involved. When the model is obtained by grid-type partitioning of its input space, the number of rules grows exponentially with the number of input variables, which leads to an unnecessarily complex model (curse of dimensionality). When a tree-type partition is used, the number of fuzzy rules does not grow exponentially, but tree-building procedures, which are usually based on some greedy strategy, often yield suboptimal results in terms of model accuracy and complexity [2]. Fuzzy clustering can be effectively used to initialize fuzzy systems that have a scatter-type rule-base [3]. However, the transformation of the obtained clusters into univariate membership functions is not straightforward [1]. An alternative is to define the antecedent fuzzy sets directly in the product space of the input variables [4]. In this way, the number of rules can be significantly reduced. In this paper, piecewise linear multivariate fuzzy sets are used, whose membership functions are obtained through Delaunay triangulation. Delaunay triangulation-based approximation of an unknown function from scattered data has proven to be an effective tool for classification, modeling, and control [5], [6]. The main advantage of the proposed method is that it yields compact and easily manageable fuzzy models [7]. A multivariate zero-order Takagi–Sugeno (TS) fuzzy model is used. This model is often preferred to higher order TS models, which are, on the one hand, better function approximators [8], but, on the other hand, are prone to yield rules that cannot be interpreted as valid local descriptions of the underlying system [9], [10]. The outcome of the system-identification exercise then depends on the inference method, the parameter estimation criterion, the overlap of the antecedent membership functions, etc. In this sense, zero-order TS models are much less sensitive, as there is only one consequent parameter to be estimated in each rule. Also from the interpretation point of view, the user may simply prefer a zero-order TS model (close to a “linguistic” interpretation) to a higher order one (interpretation in terms of local functional descriptions). Finally, first-order and higher order TS fuzzy models cannot be inverted analytically in general, while for zero-order TS fuzzy models, the inversion is quite straightforward. The contribution of this paper is twofold. • A constrained identification method has been developed for fuzzy models with multivariable membership functions and constant consequents. This method enables the user to incorporate control-relevant a priori knowledge (range for the stationary gain, stability, etc.) into data-driven identification. The a priori knowledge is

1083–4419/01$10.00 © 2001 IEEE

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transformed into linear inequalities on the parameter set of the fuzzy model, similarly to [11]–[13]. This improves the robustness of the identification algorithm as it alleviates the common problem of insufficient information content of the identification data. • For a control design based on the obtained fuzzy model, methods are developed for linearization and model inversion. The paper is organized as follows. Section II presents the structure of the fuzzy model along with the methods for its linearization, inversion and identification. In Section III, the application of the presented technique to the modeling and control of dynamic systems is addressed. An application example—the identification of the Box–Jenkins furnace—is also given in this section. A detailed case study is given in Section IV to illustrate the use of a priori knowledge and its effect on the modeling and control performance. The process under study is the pH neutralization process. Conclusions are given in Section V. II. STRUCTURE

OF THE

FUZZY MODEL,

ITS

ANALYSIS,

AND IDENTIFICATION

The purpose of this section is to describe the structure of the fuzzy model and to present methods for its linearization, inversion, and identification. A. Structure of the Model Fuzzy modeling can be regarded as the generation of a non, linear mapping between the input and output spaces, reprewhere denotes the model output and sents the -dimensional input (regression) vector. In zero-order TS fuzzy models (also called singleton models), this nonlinear mapping is represented by a set of fuzzy rules in the following form: is

(1)

is the rule index and is a fuzzy set where defined on . The output of the fuzzy model is computed as the weighted average of the rule consequents

(2)

In the proposed framework, the partitioning of the input space is realized by means of piecewise linear multivariable fuzzy sets defined through their characteristic points (also called nodes) , with , where is the number of nodes. The fuzzy sets are uniquely defined by the membership degrees at these characteristic points, which are constrained such that they sum up to one

The membership functions for these point-wise defined fuzzy sets are obtained by linear interpolation of the membership degrees (described later). This interpolation and the constraint in (3) ensure that the sum of the membership functions over all the rules is one for the entire antecedent domain

Consequently, the output of the fuzzy model (2) can be written as (4) and Here, are the vectors containing the antecedent membership degrees and the consequent parameters, respectively. are defined by linearly inThe membership functions terpolating the membership degrees of the characteristic points. In order to get an efficient interpolation, we partition the input space by Delaunay triangulation of the characteristic points and interpolate by using barycentric coordinates of the obtained simplices. In the following, some definitions are given that are necessary for understanding this method. An -polytope is the smallest convex set containing a finite set of points in . An -simplex is a polytope containing linearly independent points in . The Delaunay triangulation of a set of points is a set of -simplices such that the circumsphere of each simplex does not contain other points in its interior. For example, in two dimensions, three points form a simplex obtained by Delaunay triangulation if and only if the circle that is determined by these points does not contain any other point of that set. Consequently, the bounding spheres of the simplices are as small as possible, and the obtained triangles are as equilateral as possible. It has been proven that among all mappings with a given bound on their second derivative, the piecewise linear approximation based on the Delaunay triangulation (also called Delaunay tessellation) has the smallest worst-case error of all triangulations [14]. This may seem to imply that a large number of simplices must be tested to construct the Delaunay triangulation, which would be impractical for large numbers of points. Fortunately, efficient algorithms have been developed in the area of computational geometry [15] that are capable of forming Delaunay triangulation of points in and -dimensional space in time. Neural networks can also be used for such purposes [16]. The simplices are represented by the connectivity matrix , whose th row contains the indices of the points that form simplex conv For , the membership function the barycentric coordinates

with (3)

is defined through of (5)

ABONYI et al.: FUZZY MODELING WITH MULTIVARIATE MEMBERSHIP FUNCTIONS

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Fig. 1. Example of a piecewise linear membership function.

where

is the extended regression vector, , and

The Delaunay triangulation of these points is represented by the connectivity matrix

The barycentric coordinates are also used to determine in which , all the barycentric coorsimplex the observation is. If , . dinates of are positive: To obtain the interpolation formula for the membership functhe vector of tion, denote by at the characteristic points membership degrees of fuzzy set forming simplex . By arranging these vectors into a matrix , one can determine the vector of membership degrees of for all the membership functions

Consider, for instance, simplex . The matrix becomes

defined by the second row of

and the barycentric coordinates given by (5) are (6) Inserting (6) into (4) gives the output of the fuzzy model

(7) is the vector of parameters of the local where linear model valid in the simplex that contains the current observation. Example (Two-Dimensional Membership Function): The , depicted in two-dimensional (2-D) membership function Fig. 1, is defined by the following five characteristic points:

This equation can be used to evaluate whether or not an obserthe vation is in simplex . For instance, for are all posibarycentric coordinates , however, tive and this point is thus in . Point and is thus has the barycentric coordinates outside of simplex . The membership degree is computed by using (6) with the characteristic membership degrees

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B. Visualization of the Membership Functions Although the membership functions are defined in a multidimensional space, the fuzzy model is transparent and can easily be analyzed and interpreted by mapping the membership functions into a 1-D or 2-D subspace of the input variables. The rules of the original fuzzy model (1) can be also formulated as is

(8)

This is useful, if some a priori knowledge is available that can be transformed into linear inequalities on the parameters [13]. The generation of and is straightforward when upper and lower bounds on or are given. A more interesting situation is the use of constraints on the parameters of the local linear models. From (7), the parameter vector of the local linear model valid in the th simplex is given . Hence, constraints by based on a priori knowledge on

where the original input vector is divided into two parts: . For a given , the model can be simplified to is

(9)

where the mapping of the membership function onto the lowis calculated by dimensional

can be transformed into (14) by choosing , . Constraints for all the simplices are simply stacked and on top of each other

(10) This formula directly follows from (6), where a part of the vector is fixed and the remaining elements are varied to get the membership function. The identification of a fuzzy model using input-output data is divided into two tasks: antecedent identification (fuzzy partition of the input space) and consequent parameter estimation. In the remainder of this section, first unconstrained and then constrained consequent parameter estimation is presented, followed by a method for incremental construction of the antecedent partition.

Here, denotes the number of simplices. For example, a priori knowledge on the monotonicity of the nonlinear function to be represented by the fuzzy model is transformed into inequality equal to the following constraints by choosing matrix .. .

C. Estimation of the Consequent Parameters The output of the model is linear in the consequent parameters [see (7)]. Hence, if there are no constraints, the parameters can be estimated by standard least-squares techniques. By arranging output data samples into a vector and the the available corresponding membership degrees into the following matrix:

.. .

.. .

.. .

..

.

.. . .. .

(15)

for each . An example is presented in

and Section IV.

D. Identification of the Antecedent Partition .. .

.. .

..

.

.. .

consequent parameter estimation can be formulated as the following regression problem: (11) where is a vector of zero-mean modeling errors. If there are no constraints on , the least-squares solution of (11) is (12) In the constrained case, quadratic programming (QP) can be used

To identify the complete model, one has to determine the number and positions of the characteristic points of the membership functions. This can be done in an incremental way, as proposed in [5] or [17]. The construction procedure starts with a small number of nodes and additional nodes are inserted according to the modeling performance of the individual simplices. This approach has also been successfully used in the identification of polynomial models [18], radial basis functions and also for TS fuzzy models with linear consequents [3], [19], [20]. With this method, the modeling accuracy for the training data always improves as the number of rules increases. In order to avoid overfitting, one can use the cross-validation regularity criterion [19] or other structure-selection criteria. One of the most frequently applied criteria is the Bayesian information criterion (BIC) [21]

(13) BIC with

,

(16)

, and the constraints on where (14)

denotes the mean-square modeling error . This criterion is a sum of two


759

terms, the first penalizing prediction errors and the second penalizing model complexity. An initial model is obtained by placing nodes at the vertices of the embedding polytope of the data. The membership functions are then generated by using the following iterative algorithm. Step 1) Compute the Delaunay triangulation of the given set of nodes. Step 2) Identify the consequent parameters of the model by using (12) or (13). Step 3) Compute the model response and the BIC (16) for the training data. If this criterion is bigger than it was in the previous step, stop the construction procedure and use the model obtained in the previous step. Step 4) Compute the modeling error of each simplex (17) denotes the square of where the modeling error at the th data point. Step 5) Select the simplex with the largest , set and insert an additional node at the weighted center of this simplex

(18) This node becomes the core of the new membership function (simultaneously a new rule is created: ). The membership degrees at the remaining , characteristic points are zero, i.e., . Go to Step 1). III. APPLICATION

TO THE MODELING AND CONTROL DYNAMIC SYSTEMS

OF

The remaining part of this paper is focused on the control-relevant properties of the model structure proposed and analyzed in the previous section. A. Modeling of Dynamic Systems Fuzzy models are often used to represent nonlinear dynamic systems with the nonlinear autoregressive with exogenous input (NARX) structure, which establishes a nonlinear relation between the past inputs and outputs and the predicted output

(19) and denote the maximum lags considered for the Here, is the discrete output, and input terms, respectively, dead time, and represents the mapping of the fuzzy model. The rules of the considered fuzzy model (1) that represents a dynamic process are thus in the following form: is with

.

Fig. 2. BIC criterion as a function the number of rules during the construction procedure.

B. Example: Identification of the Box–Jenkins Gas Furnace The well-known Box–Jenkins benchmark [22] is used to illustrate the proposed fuzzy modeling approach and to compare its effectiveness with that of other methods. The data set consists pairs of input–output samples taken from a furnace of with a sampling time of 9 s. The process input is the methane flow rate and the output is the percentage of CO in the off-gas. A number of researchers concluded that an appropriate structure of a dynamic model for this system is (20) Fig. 2 shows the BIC (16) as a function of the number of rules during the incremental construction procedure. To illustrate the properties of the criterion, the construction algorithm has not been stopped after the increase of the BIC. The minimum is attained for seven membership functions. The input–output map of this fuzzy model (Fig. 3) illustrates the effective partitioning of the input space. The approximation power of the model can be appreciated if one compares the achieved modeling performance [mean-square error (MSE)] with other published results (see Table I). The model in [20] is also of the singleton type, with a structure identical to that of the model presented here. The construction procedure, however, was different. The consequent parameters were identified by the least-squares method, while the antecedent partition was obtained as a Cartesian product of univariate triangular membership functions optimized by gradientdescent techniques. It should be noted that it is recommended to normalize the input domain prior to constructing the model. This is because the Delaunay triangulation yields triangles that are as equilateral as possible. If the domain of is deformed, , the resulting model is less accurate. For instance, without normalization, the best achieved modeling performance is 0.134 (0.124 for the normalized domain).

760


Fig. 3. Input–output map of the fuzzy model.

TABLE I COMPARISON OF THE PERFORMANCE AND COMPLEXITY WITH OTHER MODELS

To illustrate how multivariate membership functions can be identified and interpreted when the input dimension is greater than two, the following model has been also identified: is Fig. 4 shows the BIC (16) as a function the number of rules during the incremental construction procedure for this case. The minimum is attained for 23 membership functions. As this model cannot be interpreted directly, the proposed visualization method has been applied. For example, if is fixed to the mean value of the control signal (0.059), the membership functions are obtained for as depicted in Fig. 5. As the computational complexity of the Delaunay triangulapoints in an -dimensional space is , tion of the construction of this higher order model requires more computational effort than that of the 2-D one. In MATLAB, however, the CPU time needed to compute the triangulation increases

Fig. 4. BIC criterion as a function the number of rules during the construction procedure for the higher order model.


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which is inside the simplex that was used to calculate it; see (5). The inverse model is a part of the IMC scheme [27], where the process model is placed in parallel with the controlled system, and the difference between the output of the process and the model is used by the controller to compensate for the effects of modeling errors and unmeasured disturbances (Fig. 6). This algorithm is based on the assumption that a minimum-phase process can adequately be described by (19) and (the time delay), i.e. that its relative degree is

while

Fig. 5. Membership functions obtained for u(k mean value of the input u.

0 3) and u(k 0 4) set to the

much more slowly than the above theoretical relationship suggests. For this example, the model building and evaluation procedure for 25 rules took 21.75 s for the higher order model (versus 8.12 s for the first-order one). The triangulation was computed by using the Spatial and Geometric Analysis Toolbox [26].

for . By successively iterating (19), it is possible to find an input–output map

(22) is used as an step-ahead predictor. Note In this way, that the input dimension of this model is greater than the input dimension of the original NARX model. For open-loop stable processes, the inverse of the plant is

C. Model Inversion and its Application to Control An inverse of a dynamic process model is often used as a feedforward controller which computes the control input such that a process output is obtained that is close to the desired reference. An internal model control (IMC) scheme or another kind of feedback mechanism is then used to compensate for disturbances and effects of unmodeled dynamics. The aim of this section is to show how the proposed model can be applied in this control scheme. Readers interested in stability of nonlinear IMC control may refer to [27]. Additional stability results readily applicable to the fuzzy model class under consideration are given, for instance, in [28] and [29], among others. Ullrich and Brown [30] proposed to invert a Delaunay network by means of nonlinear optimization. This approach is very general and can be applied to any nonlinear model [31]. The piecewise linear nature of the Delaunay network model, however, enables us to use a much more efficient analytical inversion technique. denote the fuzzy model, Let where the input vector is decomposed into the vector and the scalar . Similarly, the parameter vector is partitioned into and . It is easy to see that the inverse model is given by

(23) denotes the inverse of the nonlinear system generwhere ated by the fuzzy model inversion (21). Under ideal circumstances (no model–plant mismatch), this control law will steer the output of the plant to the reference. D. Linearization of the Dynamic Model The possibility to effectively linearize a nonlinear dynamic model is extremely important for model-based control applications. Instantaneous on-line linearization around the current operating point can be used, for instance, in model predictive control to avoid nonconvex optimization [32], [33]. A schematic diagram of such a control scheme is shown in Fig. 7. Although the controller is not adaptive in the sense that it has not been designed specifically for time-varying processes, this concept is closely related to the indirect self-tuning regulator [34]. The main difference is that the linear model is extracted at each sampling instant from the nonlinear fuzzy model rather than recursively estimated from observed data. The linearization technique yields an approximate linear time-invariant (LTI) ARX model

(21) As it is not known beforehand which simplex should be inverted, (21) is applied to all simplices and the resulting is chosen for

(24)

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Fig. 6. Internal model control scheme.

Fig. 7.

Control architecture based on instantaneous linearization.

with

Fig. 8.

Considered continuous stirred tank reactor.

IV. FUZZY MODELING OF A

MODEL-BASED CONTROL pH PROCESS

AND

The control of the pH (the concentration of hydrogen ions) in a continuous stirred-tank reactor (CSTR) is a well-known benchmark problem with nonlinear dynamics. Three fuzzy models were constructed for this process by using dynamic data and various pieces of a priori knowledge. A detailed analysis of their dynamic properties is given and their performances are compared both in open-loop simulation and in closed-loop control (IMC). A. Process Description and the parameter vector

The linearization method is also useful for the analysis of the dynamic behavior of the model. For instance, by using the Lyapunov indirect method, the local stability of the nonlinear model and its inverse can be analyzed [35]. Stability analysis of piecewise linear systems has also been extensively studied in the fuzzy-control community [28]. From the modeling and identification point of view, qualitative information about the local dynamics of the process, like local stability, is also useful. It can be incorporated into the identification procedure as a priori knowledge [13]. An example of this “gray-box” identification is given in the next section.

The CSTR depicted in Fig. 8 has two input streams: sodium hydroxide and acetic acid. The dynamic model for the pH in the tank is given in [36]. The training and validation data were in the range of generated by randomly manipulating 513–525 l/min. The sampling period is 12 s, both for data collection and for control. B. Fuzzy Modeling As the process can be modeled as a first-order dynamic system [36], the fuzzy model consists of the following rules: pH

is

pH

Three different models were constructed. • Model 1: no a priori knowledge was used, the model was constructed using the available input–output data only.


Fig. 9.

Fig. 10.

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0 1 0), steady-state of the model (0 0), steady-state of the process (—).

Input partition of Model 1, stable simplex (—), unstable simplex (



• Model 2: a priori knowledge on the invertibility of the model was assumed [see (15)]. If a given simplex contained steady-state input–output data, stability was also assumed [13]. • Model 3: the previous case was extended by including constraints based on a priori knowledge about the minimum and maximum steady-state gain of simplices that contained steady-state data.

Generally, this sort of a priori knowledge can be extracted from available steady-state data or from a steady-state mathematical or empirical model of the process [37]. Such information has been commonly used in the identification of nonlinear blockoriented models [38]. In our example, the steady-state map of the process is readily available, as it is identical to the titration curve. The steady-state gain can be estimated from the derivative of this curve. As the steady-state information is only used to

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Fig. 11.


Fig. 12.

Model validation experiment: (—) process,

0 1 0 model.

decide whether a simplex contains a steady-state operating point and what the bounds of the steady-state gain are, it is sufficient to have only a few steady-state input–output data pairs. The initial fuzzy model contained eight rules that covered the whole operating domain of the system. The cores of the fuzzy sets were manually arranged at the edges of the domain. For the sake of a fair comparison, the number of rules in the final model

was set to 20 for all three models. The resulting input partitions are shown in Figs. 9–11 along with the steady-state behavior of the process and the obtained model. As each simplex defines a local linear dynamic model given by (24), it is possible to extract or incorporate knowledge about the dynamic properties of these local models. As an example of knowledge extraction, the stability of the local models is eval-


Fig. 13. output.

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0 1 0) setpoint, (—) process output, (- -) model

Controller based on Model 3 which was constructed by combining data and a priori knowledge, (

uated and depicted in these figures. Unstable simplices are de). noted by dashed-dotted lines ( In Fig. 9, one can see that Model 1, which was constructed without using a priori knowledge, does not correctly represent the dynamic properties of the system as it contains unstable local models in the steady-state region of the process. Moreover, there where the model has is a small region around output multiplicity. Model 2, shown in Fig. 10, does not have these undesired properties. However, the steady-state gain in is too big and the one around pH the region around pH too small. Knowledge about the minimum and maximum steady-state gain has been used in the construction of Model 3. This model gives a good piecewise linear approximation of the steady-state behavior of the real process (Fig. 11). Recurrent simulation with a separate validation data set was used to evaluate the differences in the modeling performance of the three models. The true and the simulated pH are shown in Fig. 12 for Model 2. The two curves can hardly be distinguished from each other. The performance was measured by the MSE. For a comparison, a linear ARX model was identified around the working . Table II shows that the fuzzy point pH 7.35, models are much more accurate than the linear model. This is not surprising, because the linear model only has two parameters and cannot capture the nonlinearity. The differences in the performance of the identified fuzzy models are more interesting. Model 2 is superior to Model 1. This can be explained by the use of a priori knowledge, which compensates for the missing information in the data. By including a priori knowledge about the steady-state gain of the system (Model 3), the modeling accuracy of the model

TABLE II MODELING PERFORMANCE (MSE) OF THE IDENTIFIED MODELS

with 20 rules actually dropped. This is because the freedom available within the given model structure was used to represent the steady-state characteristic. If simultaneous improvement is required in the steady-state behavior and the off-equilibrium dynamic performance, the complexity of the model (number of rules) has to be increased. This is confirmed by identifying a new model with 24 rules (see the right-most column in Table II). C. Model-Based Control In model-based control, the most relevant model validation is the control performance of the model-based controller. We use the IMC scheme of Fig. 6 based on the inverse fuzzy model (21). As three models were identified, three different controllers will be evaluated. For comparison, the linear model was used to design a fourth IMC, which is, for this linear model, functionally equivalent to a PI controller [39]. The performance of these controllers was measured by the integral squared error (ISE) index given by ISE where

(25)

denotes the setpoint (reference) to be followed.

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TABLE III COMPARISON OF THE CONTROL PERFORMANCE OF THE IDENTIFIED MODELS

The best performance, obtained with the controller based on Model 3, is shown in Fig. 13. In Table III, the performances of the three controllers are summarized. It should be noted that the differences are not so small as they might seem. The optimal IMC controller, based on an inverse of the nonlinear process model, has a response which is one sample delayed after the reference and results in an ISE of 5.3. Relative to this value, the IMC based on Model 3 gives an improvement of 18% compared to the linear PI controller. V. CONCLUSION A fuzzy modeling framework has been developed which uses multivariate membership functions obtained by Delaunay triangulation of the antecedent data. As the input–output map of such a model is piecewise linear, local linearization and inversion algorithms are easily derived and the model is computationally cheap to evaluate. The model also lends itself to interpretation and analysis, which enables the user to incorporate a priori knowledge (such as steady-state or stability information) into the estimation of the consequent parameters. The performance of the proposed modeling technique was first demonstrated in the identification of the well-known Box–Jenkins gas furnace benchmark. Better accuracy was achieved than with models which use grid-type partitioning of the input space. The second example is an IMC for a pH process, which is based on the analytic inversion of a fuzzy model. It was shown that fuzzy models constructed by using data combined with a priori knowledge perform better in control than models obtained from data only. This is because a priori knowledge can compensate for the lack of training data and helps balancing the tradeoff between the accuracy of the steady-state representation and the dynamics. REFERENCES [1] R. Babuˇska, Fuzzy Modeling for Control. Norwell, MA: Kluwer, 1998. [2] O. Nelles, A. Fink, R. Babuˇska, and M. Setnes, “Comparison of two construction algorithms for Takagi–Sugeno fuzzy models,” Int. J. Appl. Math. Comput. Sci., vol. 10, no. 4, pp. 835–855, 2000. [3] M. Sugeno and T. Yasukawa, “A fuzzy-logic-based approach to qualitative modeling,” IEEE Trans. Fuzzy Syst., vol. 1, pp. 7–31, Jan. 1993. [4] A. Kroll, “Identification of functional fuzzy models using multidimensional reference sets,” Fuzzy Sets Syst., vol. 80, pp. 149–158, 1996. [5] D. Cubanski and D. Cyganski, “Multivariable classification through adaptive Delaunay-based c spline approximation,” IEEE Trans. Pattern Anal. Machine Intell., vol. 17, pp. 403–417, Mar. 1995. [6] T. Ullrich and H. Tolle, “Delaunay-based local model networks for nonlinear system identification,” in Proc. IASTED Int. Conf. Applied Modeling and Simulation, 1997. [7] M. Setnes, V. Lacrose, and A. Titli, “Complexity reduction methods for fuzzy systems design,” in Fuzzy Logic Control: Advances in Applications, H. Verbruggen and R. Babuˇska, Eds. Singapore: World Scientific, 1999, pp. 185–218.

[8] R. Rovatti, “Takagi–Sugeno models as approximators in Sobolev norms: The SISO case,” in Proc. 5th IEEE ICFS, New Orleans, LA, Sept 1996, pp. 1060–1066. [9] J. Yen, L. Wang, and W. Gillespie, “Improving the interpretability of TSK fuzzy models by combining global learning and local learning,” IEEE Trans. Fuzzy Syst., vol. 6, pp. 530–537, Apr. 1998. [10] J. Abonyi and R. Babuˇska, “Local and global identification and interpretation of parameters in Takagi–Sugeno fuzzy models,” in Proc. 9th IEEE ICFS, San Antonio, TX, May 2000, pp. 835–840. [11] H. Tulleken, “Gray-box modeling and identification using physical knowledge and Bayesian techniques,” Automatica, vol. 29, pp. 285–308, 1993. [12] W. Timmons, H. Chizeck, and P. Katona, “Parameter-constrained adaptive control,” Ind. Eng. Chem. Res., vol. 36, pp. 4894–4905, 1997. [13] J. Abonyi, R. Babuˇska, H. Verbruggen, and F. Szeifert, “Incorporating prior knowledge in fuzzy model identification,” Int. J. Syst. Sci., vol. 31, no. 5, pp. 657–667, 2000. [14] S. M. Omohundro, “The Delaunay triangulation and function learning,”, Tech. Rep. 90-001, 1989. [15] D. F. Watson, “Computing the -dimensional Delaunay tessellation, application to Voronoi polytopes,” Comput. J., vol. 24, pp. 167–172, 1981. [16] A. Garga and N. Bose, “A neural network approach to the construction of Delaunay tessellation points in ,” IEEE Trans. Circuits Syst. I, vol. 41, pp. 611–613, 1994. [17] M. Brown and T. Ullrich, “Comparison of node insertion algorithms for Delaunay networks,” in Proc. IMACS 2nd MATHMOD Conf., 1997, pp. 445–780. [18] A. Banerjee and Y. Arkun, “Model predictive control of plant transitions using a new identification technique for interpolating nonlinear models,” J. Process Control, vol. 5–6, pp. 441–457, 1998. [19] M. Sugeno and K. Tanaka, “Successive identification of a fuzzy model and its application to prediction of a complex system,” Fuzzy Sets Syst., vol. 42, pp. 315–334, 1991. [20] J. Abonyi, L. Nagy, and F. Szeifert, “Adaptive fuzzy inference systems and its application in modeling and model based control,” Chemical Engineering Research and Design, Trans IChemE, vol. 77A, pp. 281–290, 1999. [21] K. M. Boosley, “Neurofuzzy modeling approaches in system identification,” Ph.D. dissertation, Univ.Southampton, Southampton, U.K., 1997. [22] G. Box and G. Jenkins, Time Series Analysis, Forecasting and Control. San Francisco, CA: Holden Day, 1970. [23] B. Posthlethwaite, M. Brown, and C. Sing, “A new identification algorithm for fuzzy relational models and its applications in model-based control,” Chemical Eng. Res. Design, Trans IChemE, vol. 75A, pp. 453–458, 1997. [24] W. Pedrycz, “An identification algorithm in fuzzy relational systems,” Fuzzy Sets Syst., vol. 13, pp. 153–167, 1984. [25] C. Xu and Y. Lu, “Fuzzy model identification and self-learning for dynamic systems,” IEEE Trans. Syst., Man, Cybern., vol. SMC-17, pp. 683–689, Sept. 1987. [26] K. Pankratov. (1995) Spatial and geometric analysis toolbox, SaGA ver. 1.3.. [Online]. Available: http://puddle.mit.edu/glenn/kirill/saga.html [27] C. Economou, M. Morari, and B. Palsson, “Internal model control. 5. Extension to nonlinear systems,” Ind. Eng. Chem. Process Des. Dev., vol. 25, pp. 403–411, 1986. [28] H. Kiendl and J. Rüger, “Stability analysis of fuzzy control systems using facet functions,” Fuzzy Sets Syst., vol. 70, pp. 275–285, 1995. [29] M. French and E. Rogers, “Input/output stability theory for direct neurofuzzy controllers,” IEEE Trans. Fuzzy Syst., vol. 6, pp. 331–345, Mar. 1998. [30] T. Ullrich and M. Brown, “Delaunay-based nonlinear internal model control,” in Proc. ICACS’97, pp. 33–41. [31] D. Psichogios and L. Ungar, “Direct and indirect model based control using artificial neural networks,” Ind. Eng. Chem. Process Des. Dev., vol. 30, pp. 2564–2573, 1991. [32] J. Abonyi, L. Nagy, and F. Szeifert, “Fuzzy model based predictive control by instantaneous linearization,” Fuzzy Sets Syst., vol. 120, pp. 109–122, 2001. [33] J. Roubos, S. Mollov, R. Babuˇska, and H. Verbruggen, “Fuzzy model based predictive control by using Takagi–Sugeno fuzzy models,” Int. J. Approx. Reason., vol. 22, no. 1/2, pp. 3–30, 1999. [34] K. Astrom and B. Wittenmark, Adaptive Control. Reading, MA: Addison-Wesley, 1989. [35] E. Hernandez and Y. Arkun, “Study of the control-relevant properties of backpropagation neural network models of nonlinear dynamic systems,” Comput. Chem. Eng., vol. 16, pp. 227–240, 1992.

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[36] N. Bhat and T. McAvoy, “Determining model structure for neural models by network stripping,” Comput. Chem. Eng., vol. 16, pp. 271–281, 1992. [37] S. Ramchandran, “Consider steady-state models for process control,” Chem. Eng. Progress, pp. 75–81, Feb. 1998. [38] M. Pottman and R. Pearson, “Block-oriented NARMAX models with output multiplicities,” AIChE J., vol. 44, pp. 131–140, 1998. [39] D. Seborg, T. Edgar, and D. Mellichamp, Process Dynamics and Control. New York: Wiley, 1989.

Janos Abonyi received the M.Eng. and Ph.D. degrees in chemical engineering from the University of Veszprem, Hungary, in 1997 and 2000, respectively. He is currently an Assistant Professor at the Department of Process Engineering at the University of Veszprem. From 1999 to 2000, he was a Research Fellow at the Control Laboratory at Delft University of Technology, The Netherlands. His current research interests include process engineering, data-mining, research and design of fuzzy systems for modeling, identification, and control.

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Robert Babuˇska received the M.Sc. degree in control engineering from the Czech Technical University, Prague, in 1990, and the Ph.D. degree from the Delft University of Technology, The Netherlands, in 1997. Currently, he is an Associate Professor at the Control Laboratory of the Electrical Engineering Department of the Delft University of Technology. He has coauthored more than 30 journal papers and chapters in books and has published a research monograph Fuzzy Modeling for Control (Norwell, MA: Kluwer, 1998). His research interests include the use of fuzzy set techniques and neural networks in nonlinear system identification and control. He is an Associate Editor of Engineering Applications of Artificial Intelligence and Fuzzy Sets and Systems. Dr. Babuˇska is an Associate Editor of the IEEE TRANSACTIONS ON FUZZY SYSTEMS.

Ferenc Szeifert received the M.Eng. and C.Sc. degrees in chemical engineering from the University of Veszprem, Hungary, in 1973 and 1991, respectively. He is a Professor at the Department of Process Engineering at the University of Veszprem. In 1972 and 1983, he has held visiting appointments at Lensoviet University, St. Petersburg, Russia and at the University of Akron, Akron, OH. His research interests include process engineering, predictive control, control of batch processes, and intelligent systems. He is an Associate Editor of the Hungarian Journal of Industrial Chemistry.