**2. Neural network adapting its weights usign fuzzy logic**

The adaptation in a neural net means that it adapts its weights with a law action, seeking the convergence to the output desired answer. The difference between the desired ( ( ) *<sup>j</sup> d k* ) and actual responses ( ( ) *<sup>j</sup> y k* ) is known as convergence *error* ( ( ) *<sup>j</sup> e k* ) and is defined as (3) and is shown in figure 6.

$$d\_j e\_j(k) = d\_j(k) - y\_j(k). \tag{3}$$

The law action could be a sliding mode, proportional gain in its weight and other non-linear models that allows the neural net system converging to the desired answer with respect to the input set.

Fig. 6. Neuronal Weights Adjustment using a law action.

Neuro-Fuzzy Digital Filter 79

and

The Neuro-Fuzzy Digital Filter considers the concepts described in (Ash, 1970) (Abraham, 1991) (Ali, 2003) (Gustafsson, 2000), (Mamdani, 1974) (Morales, 2002) (Medel et.al 2008),

c. Convergence answer considering the stochastic error *e(k) and the its probability bounded* 

d. In a metric sense, *weights distribution* is transformed in Fuzzy Inferences after the law

e. *Rule Base allows the interpretation* of the stochastic weights bounded by distribution

f. *Inference Mechanism* as an expert consequence of the rule base, selects the membership

*connector selecting the dendrite value corresponding to the knowledge base (Yamakawa, 1989).* g. *Activation Function is the stage where the* answer filter is transformed into a natural

h. Neuro-Fuzzy Digital Filter process has a *Natural Actualization* obtaining the linguistic values and actualizes its weights dynamically based on a distribution error and

2 [0,1)

The functional error *J k* has an exponential convergence if the weights set into the Neuro-

The Neuro-Fuzzy Digital Filter needs the knowledge base in order to select the corresponding answer weights in accordance with the desired signal and law action. Firstly, the filtering process uses the neuronal net and requires the adaptation process for nonstationary answer conditions, and the fuzzy rules adjust to the adaptation process guarantying the convergence rate (Takagi and Sugeno, 1986). The filter mechanism makes a selection of the correct variable weights into its data form and selects the correct level into three fuzzy regions. Then, the rules mechanism selects the weights 1, : 1, *ij j m*

adjusting the filter gain, giving the correct answer *y*ˆ*jk* as the filter output (Rajem y Gopal,

2006) (Medel et.al 2008), with MISO (Multi Inputs Single Output) properties.

*e k*

b. Adaptive weights considering the law action and Fuzzy Logic inferences

function accomplishing the actions using the logic *IF* connector.

function described as an adaptive weight 1,

answer approximating to minimal convergence error region.

observing the second probability moment basic law action (5).

<sup>2</sup> <sup>2</sup>

*J k mm R*

in where

: 1, *ij <sup>j</sup> m Z*

*wki nZ*

<sup>1</sup> ( ) 1 ( 1) , . *jj j J k ek k Jk R k Z <sup>k</sup>* (5)

*J J Jd k* min min, inf : min ( ( ), ( )), 1, , *j jj y*ˆ *k j ss Z* (6)

, if 0 () 1 *e k <sup>j</sup>* and (6).

 

is described as a positive value in

 using logic *THEN* 

*wki n*

(*k* is index

 [,] 

The error ( ) *<sup>j</sup> e k* has a interval limit 0,

accordance with inf{ ( ) 0, sup{ ( ) } *j j e k*

interval) (Margaliot and Langholz, 2000) (Morales, 2002).

having the elements needed in its basic description:

action applied in the dendrites stage inputs.

Fuzzy Filter allow that [0,1) lim ( ) , *<sup>j</sup> <sup>k</sup>*

a. Back propagation neural net scheme.

*moments*

The adaptive back*-propagation* procedure is described in (4)

$$
\mu\_{i\_j}^{\cdot}(k) = \mu\_{i\_j}(k) - L\_j(k) \tag{4}
$$

Where corresponds to law action considered by neural net designer.

Now applying the concept considered above with respect to neural net adjusting weights using fuzzy logic concepts gives a great advantage over traditional concepts such as the forgetting factor and sliding modes in other action laws.

The neural net that has adaptive weights is known as a digital identification filter, therefore, the neural net in where the adaptation process considered fuzzy inferences is known as a Neuro-Fuzzy Digital Filter.

A fuzzy neural net classifies, searches and associates information (Huang, Zhu and Siew, 2006) giving a specific answer value in accordance with a reference desired signal process, constructing a control volume described as <sup>2</sup> {( ( ), ( ))} ˆ *T ykyk R N jj* where a variant

scheme has the form <sup>2</sup> 1 <sup>ˆ</sup> : {( ( ), ( )), ( )} <sup>ˆ</sup> *<sup>N</sup> N NN j j <sup>k</sup> T Y Y T ykyk k R* inside the membership intervals delimited by a Knowledge Base (KB) (Schneider y Kandel, 1996), with dynamical and bounded moments. The responses set into the KB represents all the possible correct filter responses (Gustafsson, 2000) (Margaliot and Langholz, 2000) (Zadeh, 1965) in accordance with an objective law, previously defined by the actual natural reference process in a distribution sense. The filtering mechanism adjusts the neural weights selecting the best answer from the KB when the state changes, to use fuzzy rules (*if-then*). The neuro-fuzzy filter is based on the back-propagation algorithm, because its weights have a dynamic actualization (Ali, 2003) (Amble, 1987) (Haykin, 1996) with different levels for each interval iteration (Huang, Zhu and Siew, 2006), using the error described as *ek R* ( ) defined in (3) and considering its distribution function (García, Medel y Guevara, 2008) (Marcek, 2008)); this filter is shown in. Figure 7, integrating the fuzzy logic convenient actions into neural net structure using adaptive weights (Passino, 1998, and Medel 2008).

Fig. 7. Neuro-fuzzy Digital Filter Process

Now applying the concept considered above with respect to neural net adjusting weights using fuzzy logic concepts gives a great advantage over traditional concepts such as the

The neural net that has adaptive weights is known as a digital identification filter, therefore, the neural net in where the adaptation process considered fuzzy inferences is known as a

A fuzzy neural net classifies, searches and associates information (Huang, Zhu and Siew, 2006) giving a specific answer value in accordance with a reference desired signal process, constructing a control volume described as <sup>2</sup> {( ( ), ( ))} ˆ *T ykyk R N jj* where a variant

membership intervals delimited by a Knowledge Base (KB) (Schneider y Kandel, 1996), with dynamical and bounded moments. The responses set into the KB represents all the possible correct filter responses (Gustafsson, 2000) (Margaliot and Langholz, 2000) (Zadeh, 1965) in accordance with an objective law, previously defined by the actual natural reference process in a distribution sense. The filtering mechanism adjusts the neural weights selecting the best answer from the KB when the state changes, to use fuzzy rules (*if-then*). The neuro-fuzzy filter is based on the back-propagation algorithm, because its weights have a dynamic actualization (Ali, 2003) (Amble, 1987) (Haykin, 1996) with different levels for each interval iteration (Huang, Zhu and Siew, 2006), using the error described as *ek R* ( ) defined in (3) and considering its distribution function (García, Medel y Guevara, 2008) (Marcek, 2008)); this filter is shown in. Figure 7, integrating the fuzzy logic convenient actions into neural net structure using adaptive weights (Passino,

<sup>ˆ</sup> : {( ( ), ( )), ( )} <sup>ˆ</sup> *<sup>N</sup> N NN j j <sup>k</sup> T Y Y T ykyk k R*

() () () *u k u k Lk ij ij j* (4)

1

 inside the

The adaptive back*-propagation* procedure is described in (4)

forgetting factor and sliding modes in other action laws.

Neuro-Fuzzy Digital Filter.

1998, and Medel 2008).

Fig. 7. Neuro-fuzzy Digital Filter Process

'

scheme has the form <sup>2</sup>

Where corresponds to law action considered by neural net designer.

The error ( ) *<sup>j</sup> e k* has a interval limit 0, and is described as a positive value in accordance with inf{ ( ) 0, sup{ ( ) } *j j e k e k* in where [,] (*k* is index interval) (Margaliot and Langholz, 2000) (Morales, 2002).

The Neuro-Fuzzy Digital Filter considers the concepts described in (Ash, 1970) (Abraham, 1991) (Ali, 2003) (Gustafsson, 2000), (Mamdani, 1974) (Morales, 2002) (Medel et.al 2008), having the elements needed in its basic description:


*connector selecting the dendrite value corresponding to the knowledge base (Yamakawa, 1989).*


$$J\_j(k) = \frac{1}{k^2} \left[ e\_j(k)^2 + \left(k - 1\right)^2 J\_j(k - 1) \right] \in R\_{[0, 1)} \quad k \in \mathbb{Z}\_+. \tag{5}$$

The functional error *J k* has an exponential convergence if the weights set into the Neuro-Fuzzy Filter allow that [0,1) lim ( ) , *<sup>j</sup> <sup>k</sup> J k mm R* , if 0 () 1 *e k <sup>j</sup>* and (6).

$$J\_{\min} = \inf \left\{ J\_{\min,j} := \min J(d\_j(k), \hat{y}\_j(k))\_\prime \text{ } j = \overline{1, s}\_\prime \text{ s} \in \mathbb{Z}\_+ \right\} \tag{6}$$

The Neuro-Fuzzy Digital Filter needs the knowledge base in order to select the corresponding answer weights in accordance with the desired signal and law action. Firstly, the filtering process uses the neuronal net and requires the adaptation process for nonstationary answer conditions, and the fuzzy rules adjust to the adaptation process guarantying the convergence rate (Takagi and Sugeno, 1986). The filter mechanism makes a selection of the correct variable weights into its data form and selects the correct level into three fuzzy regions. Then, the rules mechanism selects the weights 1, : 1, *ij j m wki n* adjusting the filter gain, giving the correct answer *y*ˆ*jk* as the filter output (Rajem y Gopal, 2006) (Medel et.al 2008), with MISO (Multi Inputs Single Output) properties.

Neuro-Fuzzy Digital Filter 81

( 1, 1) *xk yk wk vk R ak R* ( ), ( ), ( ), ( ) , ( )

*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 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* 

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

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

*the reference vector desired system signal and, v(k) is the output vector noise.* 

*y*() () () *k xk vk* (12)

And the output described as (12).

( 1, 1) *ak R* ˆ .

technique.

Fig. 9. A membership weights function.
