Because the max function is non-continuous and differentiable, the partial derivative of the above equation is problematic. Here, we introduce an approximation function to overcome this problem.

We define a generalized p mean for the data ${\left(\stackrel{\leftharpoonup }{a},\stackrel{\rightharpoonup }{a}\right)}^{\prime }=[{\left({\stackrel{\leftharpoonup }{a}}_1,{\stackrel{\rightharpoonup }{a}}_1\right)}^{\prime }{\left({\stackrel{\leftharpoonup }{a}}_2,{\stackrel{\rightharpoonup }{a}}_2\right)}^{\prime },$$\dots ,{\left({\stackrel{\leftharpoonup }{a}}_N,{\stackrel{\rightharpoonup }{a}}_N\right)}^{\prime }]:$

$M_p{\left(\stackrel{\leftharpoonup }{a},\stackrel{\rightharpoonup }{a}\right)}^{\prime }$ is a good approximation function [53] for $\underset{i}{\max }{\left({\stackrel{\leftharpoonup }{a}}_i,{\stackrel{\rightharpoonup }{a}}_i\right)}^{\prime }\text{and}\underset{i}{\min }{\left({\stackrel{\leftharpoonup }{a}}_i,{\stackrel{\rightharpoonup }{a}}_i\right)}^{\prime },$ and has the following properties.

Lemma 1.2 [53]

(1)$M_0{\left(\stackrel{\leftharpoonup }{a},\stackrel{\rightharpoonup }{a}\right)}^{\prime }=\underset{p\to 0}{\lim }{M}_p{\left(\stackrel{\leftharpoonup }{a},\stackrel{\rightharpoonup }{a}\right)}^{\prime }={\left[{\left({\stackrel{\leftharpoonup }{a}}_1,{\stackrel{\rightharpoonup }{a}}_1\right)}^{\prime }\cdots {\left({\stackrel{\leftharpoonup }{a}}_N,{\stackrel{\rightharpoonup }{a}}_N\right)}^{\prime }\right]}^{1/N}$

(2)$p<q\Rightarrow M_p{\left(\stackrel{\leftharpoonup }{a},\stackrel{\rightharpoonup }{a}\right)}^{\prime }<M_q{\left(\stackrel{\leftharpoonup }{a},\stackrel{\rightharpoonup }{a}\right)}^{\prime }$

(3)$\underset{p\to \infty }{\lim }{M}_p{\left(\stackrel{\leftharpoonup }{a},\stackrel{\rightharpoonup }{a}\right)}^{\prime }=\max \left[{\left({\stackrel{\leftharpoonup }{a}}_1,{\stackrel{\rightharpoonup }{a}}_1\right)}^{\prime },\cdots ,{\left({\stackrel{\leftharpoonup }{a}}_N,{\stackrel{\rightharpoonup }{a}}_N\right)}^{\prime }\right]$

(4)$\underset{p\to -\infty }{\lim }{M}_p{\left(\stackrel{\leftharpoonup }{a},\stackrel{\rightharpoonup }{a}\right)}^{\prime }=\min \left[{\left({\stackrel{\leftharpoonup }{a}}_1,{\stackrel{\rightharpoonup }{a}}_1\right)}^{\prime },\cdots ,{\left({\stackrel{\leftharpoonup }{a}}_N,{\stackrel{\rightharpoonup }{a}}_N\right)}^{\prime }\right]$

(5)For each variable ${\left({\stackrel{\leftharpoonup }{a}}_i,{\stackrel{\rightharpoonup }{a}}_i\right)}^{\prime },M_0{\left(\stackrel{\leftharpoonup }{a},\stackrel{\rightharpoonup }{a}\right)}^{\prime }$ is continuously differentiable. The partial derivative of $M_p{\left(\stackrel{\leftharpoonup }{a},\stackrel{\rightharpoonup }{a}\right)}^{\prime }\text{with respect to }{\left({\stackrel{\leftharpoonup }{a}}_i,{\stackrel{\rightharpoonup }{a}}_i\right)}^{\prime }$ is

Supposing p > 0 and approximating $\underset{i=1}{\overset{N}{\max }}{\left({\stackrel{\leftharpoonup }{a}}_i,{\stackrel{\rightharpoonup }{a}}_i\right)}^{\prime }\text{with}{M}_p{\left(\stackrel{\leftharpoonup }{a},\stackrel{\rightharpoonup }{a}\right)}^{\prime }.$ (1.39) can be written as

In summary, DFRL can be used to generate or modify dynamic fuzzy rules Rl in a DFMLS. The algorithm can be summarised in Algorithm 1.4.

1.6Summary

In this chapter, we have presented a DFMLM and its related algorithms based on DFSs. We have discussed a DFMLM and DFML algorithm, DFML geometric model, DFMLS parameter learning algorithm and MLE algorithm, DFMLS process control model, and a dynamic fuzzy relation learning algorithm.

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