Fuzzy Clustering for System Identification
4.1 Data-Driven Modelling of Dynamical Systems
4.1.1 TS Fuzzy Models of SISO and MIMO Systems
Fuzzy models of dynamical single single output (SISO) and multiple input-multiple output (MIMO) systems are usually based on Nonlinear AutoRegressive with eXogenous input (NARX) model structure . These models can be represented by the following nonlinear vector function:
y(k+ 1) =f(y(k), . . . ,y(k−na+ 1),u(k−nd), . . . ,u(k−nb−nd+ 1)) (4.16) where,f represents the nonlinear model,y=)
y1, . . . , yny
*T
is anny-dimensional output vector, u= [u1, . . . , unu]T is an nu-dimensional input vector, na and nb
are maximum lags considered for the outputs and inputs, respectively, andnd is the minimum discrete dead time.
While it may not be possible to find a model that is universally applicable to describe the unknownf(.) system, it would certainly be worthwhile to build local linear models for specific operating points of the process. The modelling framework that is based on combining a number of local models, where each local model has a predefined operating region in which the local model is valid is calledoperating regime based model[197]. In case of MIMO NARX models, the operating regime is formulated as:
y(k+ 1) = c i=1
βi(z(k))
⎛
⎝
na
j=1
Aijy(k−j+ 1) +
nb
j=1
Biju(k−j−nd+ 1) +ci
⎞
⎠ (4.17)
where theβi(z(k)) function describes the operating regime of the i = 1, . . . , cth local linear ARX model, wherez= [z1, . . . , znz]T is a “scheduling” vector, which is usually a subset of the previous process inputs and outputs,
z(k) =7
y1(k), . . . , yny(k−na+ 1), u1(k−nd), . . . , unu(k−nb−nd+ 1)8 . The local models are defined by the θi = {Aij,Bj,ci} parameter set. As na
andnb denote the maximum lags considered for the previous outputs and inputs, andnd is the minimum discrete dead time, the lags considered for the separate input-output channels can be handled by zeroing the appropriate elements of the Aij and Bij matrices. If there is no a priori knowledge about the order of the nonlinear system, the model orders and the time delays can be directly estimated from input-output data. The next chapter of this book will deal with this problem and give a detailed overview.
The main advantage of this framework is its transparency, because theβi(z(k)) operating regimes of the local models can be represented by fuzzy sets [31]. This representation is appealing, since many systems change behaviors smoothly as a function of the operating point, and the soft transition between the regimes introduced by the fuzzy set representation captures this feature in an elegant fashion. Hence, the entire global model can be conveniently represented by Takagi–
Sugeno fuzzy rules [257]:
Ri : Ifz1 isAi,1and . . . andznz isAi,n then (4.18) yi(k+ 1) =
na
j=1
Aijy(k−j+ 1) +
nb
j=1
Biju(k−j−nd+ 1) +ci,[wi]
whereAi,j(zj) is the ith antecedent fuzzy set for thejth input andwi= [0,1] is the weight of the rule that represents the desired impact of the rule. The value of wi is often chosen by the designer of the fuzzy system based on his or her belief in the goodness and accuracy of theith rule. When such knowledge is not available wi is set aswi = 1,∀i.
The one-step-ahead prediction of the MIMO fuzzy model,y(k+ 1), is inferred by computing the weighted average of the output of the consequent multivariable models,
y(k+ 1) = c i=1
βi(z(k))yi(k+ 1) (4.19) wherecis the number of the rules, andβiis the weight of theith rule,
βi(z(k)) = wi'n
j=1Ai,j(zj) c
iwi'n
j=1Ai,j(zj). (4.20)
To represent theAi,j(zj) fuzzy set, in this chapter Gaussian membership function is used as in the previous chapters
Ai,j(zj) = exp
wherevi,j represents the center andσi,j2 the variance of the Gaussian function.
The presented fuzzy model can be seen as a multivariable linear parameter varying system model (LPV), where at the z operating point, the fuzzy model represents the following LTI model
y(k+1) =
Example 4.1 (Identification of a TS fuzzy model for a SISO Nonlinear System).
The system under study is a second-order nonlinear system
y(k) =f(y(k−1), y(k−2)) +u(k) (4.24) where
f(y(k−1), y(k−2)) = y(k−1)y(k−2) [y(k−1)−0.5]
1 +y2(k−1) +y2(k−2) . (4.25) We approximate the nonlinear componentf of the plant with a fuzzy model. Fol-lowing the approach in[293], 400 simulated data points were generated from the plant model: 200samples of the identification data were obtained with a random input signalu(k) uniformly distributed in [−1.5,1.5], followed by 200samples of evaluation data obtained using a sinusoidal input signalu(k) = sin(2πk/25), k= 1001, . . . ,1200. The simulated data are shown in Figure4.4. The input of the model iszk = [y(k−1), y(k−2)].
Table4.1compares the performance(mean squared error)of the models identi-fied with these techniques. The nomenclature can be found in Example 3.3 (Section 3.3.3).
From the prediction surface and the operating regimes of the local linear models of the fuzzy model (Figure 4.5 and Figure 4.6), one can see that the presented EM-NI method results in almost optimal antecedent and consequent parameters.
0 50 100 150 200 250 300 350 400
−2
−1 0 1 2
k
y(k)
0 50 100 150 200 250 300 350 400
−1.5
−1
−0.5 0 0.5 1 1.5
k
u(k)
Figure 4.4: Simulated output of the plant and the corresponding input signal.
Table 4.1: Fuzzy models of the nonlinear dynamic plant.
4 rules 12 rules
Method Training set Test set Training set Test set GG-TLS 4.6 10−3 2.1 10−3 3.7 10−4 2.9 10−4 GG-LS 4.6 10−3 2.0 10−3 3.7 10−4 2.9 10−4 EM-TI 4.6 10−3 2.0 10−3 2.4 10−4 4.1 10−4 EM-NI 4.4 10−3 3.5 10−3 3.4 10−4 2.3 10−4
We compare our results with those obtained by the optimal rule selection ap-proach proposed by Yen and Wang[293]. Their method uses various information criteria to successively select rules from an initial set of36rules in order to obtain a compact and accurate model. The initial rule base was obtained by partitioning each of the two inputs into six equally distributed fuzzy sets. The rules were selected in an order determined by an orthogonal transform. When linear rule consequents were used, the optimal fuzzy model with24rules achieved the mean squared error of2.0 10−6 on the training data and 6.4 10−4on the evaluation data.
Based on this comparison, we can conclude that the presented modelling ap-proach is capable of obtaining good accuracy, while using fewer rules than other approaches presented in the literature.
−2
−1
0
1
2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−1.5
−1
−0.5 0 0.5 1
y(k−1) y(k−2)
f(y(k−1),y(k−2))
Figure 4.5: Surface plot of the TS model. Available data samples are shown as black dots.
−2
−1
0
1
2 −2
−1.5
−1
−0.5 0
0.5 1
1.5 2
0 0.2 0.4 0.6 0.8 1
y(k−2) y(k−1)
φi((y(k−1),y(k−2))
Figure 4.6: Operating regimes obtained.