**4. Results**

The MISO neuro-fuzzy filter, considers the digital filter structure (Hayking, 1996) with the transition matrix bounded by the knowledge base in accordance with the functional error criterion (Ash, 1970). The soft system (statistic in variance sense) considers the evolution times bounded by PC with AMD Sempron 3100+ processor performance at *k* intervals, with an average evolution time of 0.004 sec ± 0.0002 sec.

This chapter uses the first order difference discrete ARMA model (11) representing a reference system with *j=1*.

$$\mathbf{x}(k+1) = a(k)\mathbf{x}(k) + w(k) \tag{11}$$

() 1

<sup>1</sup>

*<sup>j</sup> <sup>j</sup> ji ji i i k k kk k k k k* (8)

, (7)

1 0 / ( )/ 1 *n*

*i*

0 0 ( ) ln ( ) *wk k k j j* (9)

*xk akxk wk* ( 1) ( ) ( ) ( ) (11)

<sup>0</sup> ˆ ˆˆ ( ) ( )( ) ( 1) ( ) ( ) *j ij <sup>j</sup> j j x k w k k k x k K kw k* (10)

*ij*

*k*

. In accordance

1

*i*

i. Each weight has a dynamic transition function with natural restrictions:

1. ln( ( )) *<sup>j</sup> <sup>k</sup>* , 2) ln( ( )) 0 *<sup>j</sup> <sup>k</sup>* , 3) <sup>1</sup> ln( ( )) 1 *<sup>j</sup> k k* . ii. The weight is described using the transition function in (7).

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

with the value of 0 ln ( ) *<sup>j</sup> k* , the weights are bounded considering (9).

The transition functions sum is bounded in each layer

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

an average evolution time of 0.004 sec ± 0.0002 sec.

 <sup>2</sup> ˆ () ( , ) *wk N <sup>j</sup>* .

reference system with *j=1*.

**4. Results** 

*i j*

<sup>0</sup> 0 0 ( ) ln ( ) ln ( )( ) *wk k k k k k j jj*

0 0 ln( ( )) ln ( )( ) , ln ( ) ln( ( 1)( ( 1)) *T T*

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

The MISO neuro-fuzzy filter, considers the digital filter structure (Hayking, 1996) with the transition matrix bounded by the knowledge base in accordance with the functional error criterion (Ash, 1970). The soft system (statistic in variance sense) considers the evolution times bounded by PC with AMD Sempron 3100+ processor performance at *k* intervals, with

This chapter uses the first order difference discrete ARMA model (11) representing a

*w k*

*n*

: 1, *ij <sup>j</sup> <sup>m</sup> wki n* , where the knowledge base in

without losing the transition function

**3. Weight properties** 

basic properties (Medel, 2008):

A neuto-fuzzy filter has a weight set 1,

each layer accomplishes the condition ,

And the output described as (12).

$$
\underline{y}(k) = \underline{x}(k) + \underline{v}(k)\tag{12}
$$

$$
\underline{x}(k), \underline{y}(k), \underline{w}(k), \underline{v}(k) \in R, \; \underline{a}(k) \in R\_{(-1, 1)}
$$

*x(k) is the internal states vector, a(k) is the parameter, w(k) is the vector noise into the system, y(k) is the reference vector desired system signal and, v(k) is the output vector noise.* 

*The different operational levels are described in order to operate the distribution function error. The filter process establishes in the fuzzy region the linguistic descriptors adjusted in its ranges. Figure 8 describes the reference signal and its identification without knowing the internal parameter model*  ( 1, 1) *ak R* ˆ .

Fig. 8. Output signal *Y(k)* and its identification *Y k* ˆ using the nero-fuzzy digital filter technique.

*The fuzzy regions considered the distribution weights after applying the law action.* 

Fig. 9. A membership weights function.

Neuro-Fuzzy Digital Filter 83

Figure 13, shows both overlapping final histogram considering the same time interval.

Fig. 13. Overlapping both final histograms with respect to *Y k* and *Y k* ˆ , respectively.

Fig. 12. Histogram of identification signal as *Y k* ˆ

The histogram identification evolution is shown illustratively in Figure 10, in where each weight is adjusted in neuro-fuzzy filter affecting the identification histogram convergence. The convergence in histogram is associated with the membership weights function, allowing that the identification system tends to the reference system.

Fig. 10. The histogram convergence through the time evolution between the identification with respect to reference signal in base to adaptive weights.

Figures 11 and 12 show the *Y k* ˆ and desired signal *Y(k)* final *h*istograms, respectively.

Fig. 11. Histogram of desired signal described as *Y(k)*

The histogram identification evolution is shown illustratively in Figure 10, in where each weight is adjusted in neuro-fuzzy filter affecting the identification histogram convergence. The convergence in histogram is associated with the membership weights function, allowing

Fig. 10. The histogram convergence through the time evolution between the identification

ˆ and desired signal *Y(k)* final *h*istograms, respectively.

that the identification system tends to the reference system.

with respect to reference signal in base to adaptive weights.

Fig. 11. Histogram of desired signal described as *Y(k)*

Figures 11 and 12 show the *Y k*

Fig. 12. Histogram of identification signal as *Y k* ˆ

Figure 13, shows both overlapping final histogram considering the same time interval.

Fig. 13. Overlapping both final histograms with respect to *Y k* and *Y k* ˆ , respectively.

Neuro-Fuzzy Digital Filter 85

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ISBN.

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654-663.

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Figure 14, shows the evolution functional error described by (5).

Fig. 14. Functional error considered in (5).

The Neuro-Fuzzy Digital Filter time evolution responses was less that the reference process time state change proposed with a value of 0.08 sec, and is delimited by the processor, considered in (10). The convergence time is 0.0862 sec, described in (Medel, 2008).
