Related Modelling Paradigms

c

i=1

βi(x)

. (3.19)

3.2.2 Related Modelling Paradigms

There are many well-known or developing modelling strategies that can be seen as special case of the previously presented fuzzy model. The remaining part of this section presents the connections with these methods to show the possible interpretations of the TS fuzzy models.

• Operating Regime Based Modelling

As the combination of fuzzy sets partition the input space into a number of fuzzy regions and the consequent functions (local models) describe the system

behavior within a given region, the TS fuzzy model can be seen as multiple model network [197]. The soft transition between the operating regimes is handled by the fuzzy system in elegant fashion [31]. This representation is appealing, since many systems change its behavior smoothly as a function of the operating point.

From (3.15) and (3.19) one can see that the TS fuzzy model is equivalent to the operating regime-based model when the validity function is chosen to be the normalized rule degree of fulfillment:

φi(x) = βi(x) c i=1

βi(x)

. (3.20)

In this chapter, Gaussian membership functions are used to represent the fuzzy sets Ai,j(xj):

with vi,j being the center andσ_i,j² the variance of the Gaussian curve. This choice leads to the following compact formula for (3.18):

βi(x) =wiA_i(x) =wiexp inverse of the matrix containing the variances on its diagonal:

F^xx_i =

When the antecedent of a first-order TS fuzzy model consists of crisp sets or the fuzzy sets are defined by piece-wise linear membership functions, the resulted fuzzy model has piece-wise (linear or quadratic) behavior. Mod-elling techniques based on piece-wise linear models are widely applied for control relevant modelling [127]. Skeppstedt describes the use of local models for modelling and control purposes with hard transfer from one model to the next [248]. Pottman describes a multi model approach where the local models overlap [219]. These models can be effectively used in model based control [94].

When the rule consequent is a crisp number (Singleton) and the rule an-tecedent contains piece-wise linear membership functions, the resulted fuzzy model has piece-wise linear input-output behavior. Piece-wise linear multi-dimensional fuzzy sets can be obtained by Delaunay triangulation of charac-teristic points defined on the input space of the fuzzy model. This technique has already been suggested in the context of complexity reduction of fuzzy systems [242]. Moreover, Delaunay-based multivariable spline approximation from scattered samples of an unknown function has proven to be an effective tool for classification [67]. Recently, Delaunay Networks were introduced to represent interpolating models and controllers [270] and the integration of expert knowledge in these models has been also studied [269]. Fuzzy models based on this concept are pursued in [1].

• B-spline Networks

The presented grid-type fuzzy model makes piece-wise polynomial approx-imation of the nonlinear system to be modeled. Piecewise polynomials are closely related to spline modelling and have been successfully applied to many real-life problems, e.g., in experimental pH control and for the prediction of the viscosity for an industrial polymerization reactor [175]. B-spline basis functions can be regarded as piece-wise polynomials, where B-spline basis functions are defined by a set of knots that represent piecewise polynomial intervals. The order of these local polynomials is defined by the order of the B-splines, denoted by k. A knot vector of a set ofkth order basis functions is defined by

a_i = [ai,1, ai,2, . . . , ai,Mi+k−1]^T (3.24) where Mi is the number of the basis functions defined on the ith variable, and ai,j is thejth knot. The univariate basis functions are calculated using the following recurrence relationship [50]:

A^k_i,j(zi) =

zi−ai,j−k

ai,j−1−ai,j−k

A^k−1_i,j−1(zi) +

ai,j−zi

ai,j−ai,j−k+1

A^k−1_i,j (zi) (3.25) A¹_i,j(zi) =

1, ifzi∈[ai,j−1, ai,j]

0,otherwise (3.26)

whereA^k_i,j(zi) is thejth univariate basis function of orderk. Triangular mem-bership functions are identical to second-order B-spline basis functions (3.12).

Multi-dimensionality of the model is achieved by the means of the ten-sor product of the univariate B-splines. Given a set of B-splines defined over the input variables the multivariate B-spline can be defined as

β_j^k(z) =

&n i=1

A^k_i,j(zi). (3.27)

This multivariate B-spline is identical to the weight of the jth rule.

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

0.2 0.4 0.6 0.8 1

z₂ z1

βl

Figure 3.9: Example of multi(bi)variate B-spline (membership) functions.

Figure 3.9 shows an example of bivariate B-splines that are identical to the rule weights of nine rules in a fuzzy system with three membership functions defined on its two input domains.

• Radial Basis Function Networks

Basis function networks have been used for function approximation and mod-elling in various forms for many years. The original radial basis function methods come from interpolation theory [220], where a basis function is as-sociated with each data point. Basis function nets have also received attention from neural network and the control community [182].

Radial Basis Functions Networks (RBFNs), as proposed in 1989 by Moody and Darken [195], are often considered to be a type of neural net-work in which each unit has only a local effect instead of a global effect as in multi-layer percetron based neural networks (MLPs). In a similar way to MLPs, RBFNs perform function approximation by superimposing a set of Nr Radial Basis Functions (RBFs) as follows:

βj = exp

− n i=1

xi−ai,j

σi,j

2

(3.28)

y =

Nr

j=1βjθj

Nr

j=1βj

(3.29)

where βj : j = 1, . . . , Nr is the firing strength of unit j,Nr is the number of RBFs, xi : i = 1, . . . , n are the inputs,y is the output, and ai,j : i = 1, . . . , n;j = 1, . . . , Nr, σi,j : i = 1, . . . , n;j = 1, . . . , Nr, and θj : j = 1, . . . , Nr are free parameters which respectively determine the position, width, and height of the humps.

RBFNs can be trained in the same way as MLPs, i.e., they are initialized randomly and then minimized by gradient-descent. Alternatively, the position of the centres of the RBFs and their widths can be determined by a clustering algorithm and then the heights can be set by a least-squares type of algorithm [195]. Like MLPs, they have been proven to be universal approximators [112].

Jang has pointed out, under certain constraints, the radial basis func-tion network (RBFN) is funcfunc-tionally equivalent to zero-order TS fuzzy model [22, 129] as

βj = exp

− n i=1

xi−ai,j

σi,j

2

(3.30)

=

&n i=1

exp

−

xi−ai,j

σi,j

2

Ai,j(xi)

.

Hunt has developed a generalized radial basis function network (GBFN) that is similar to the first-order TS fuzzy model [121]. These models are identical to TS fuzzy models under the following conditions.

– The number of the basis function units is equal to the number of the fuzzy if-then rules.

– Both the basis function network and the fuzzy inference system use the same method to calculate their overall outputs.

Thus, the model presented in Section 3.2.1 can be seen as RBFN if q={0} and GBFN if q={0}, with piecewise linear basis functions.

The realization that fuzzy models are very similar to RBFN function approximators also means that methods which have been developed in fuzzy control, such as those analyzed in this work, can be applied to neural control.