**3. Weight properties**

A neuto-fuzzy filter has a weight set 1, : 1, *ij <sup>j</sup> <sup>m</sup> wki n* , where the knowledge base in each layer accomplishes the condition , 1 () 1 *n i j i w k* without losing the transition function basic properties (Medel, 2008):


$$aw\_j(k - k\_0) = \ln \Phi\_j(k) \left(\ln \phi\_j(k\_0)(k - k\_0)\right)^{-1},\tag{7}$$

iii. The velocity changes are limited inside the transition function (8).

$$\ln \text{\(\Phi\_{/}(k)\)} \le \ln \Phi\_{/}(k\_0)(k - k\_0)^T,\\ \ln \Phi\_{/}(k\_i) \le \ln \text{\(\Phi\_{/}(k\_i - 1)(k\_i - (k\_i - 1))\)}^T \tag{8}$$

The transition functions sum is bounded in each layer 1 0 / ( )/ 1 *n ij i k* . In accordance with the value of 0 ln ( ) *<sup>j</sup> k* , the weights are bounded considering (9).

$$w\_j(k - k\_0) \le \ln \Phi\_j(k\_0) \tag{9}$$

The identifier described as (10) considered (6).

$$
\hat{\mathfrak{x}}\_{j}(k) = \mathfrak{w}\_{ij}(k)(k - k\_0)\hat{\mathfrak{x}}\_{j}(k - 1) + K\_{j}(k)\hat{\mathfrak{w}}\_{j}(k) \tag{10}
$$

Where ( ) *K k <sup>j</sup>* is the function gain and is a functional identification error, defined by the second probability moment (5), ˆ ( ) *w k <sup>j</sup>* represents generalized perturbations with <sup>2</sup> ˆ () ( , ) *wk N <sup>j</sup>* .
