Description of SFNN - Adaptive Self-structuring Asymmetric Fuzzy Neural-network Control

2. Adaptive Self-structuring Asymmetric Fuzzy Neural-network Control

2.2 Description of SFNN

2 ( ) 1

( L . (2-6) Substituting (2-5) into (2-1) and using (2-3) and (2-6), yields

rb sfnn u u u

s&= ^*− − . (2-7)

2.2 Description of SFNN

Fuzzy logic and NNs are complementary technologies in the design of intelligent systems. FNNs retain the basic properties and functions of NNs with some of their elements being fuzzified. In this approach, a network’s domain knowledge becomes formalized in terms of fuzzy sets, later being applied to enhance the learning of the network and augment its interpretation capabilities. By incorporating fuzzy principles into a NN, more user flexibility is attained and the resultant network or system becomes more robust [4]. FNNs are generally a fuzzy inference system constructed from structure of NN. Learning algorithms are used to adjust the weightings of the fuzzy inference system.

Figure 2-2 shows the configuration of the proposed SFNN which is composed of the input, the membership, the rule, and the output layers. Layer 1 accepts the input variables.

Nodes at layer 2 are term nodes which act as membership functions to represent the terms of the respective linguistic variables. The asymmetric Gaussian membership function constituted by a center, a left-side variance, and a right-side variance is considered. Nodes of layer 3 are regarded as fuzzy rules. The links before layer 3 represent the preconditions of rules and the links after layer 3 represent the consequences. Layer 4 is the output layer, where the node in this layer is the output of the NN. The interactions for those layers are given as follows.

cancel the r-th rule add a new rule

th r<P β

rule generating process

rule pruning process

βk

cancel the r-th rule add a new rule

th r<P β

rule generating process

rule pruning process

βk

Fig. 2-2. The structure of SFNN.

Layer 1 - Input layer: For every node i in this layer, the net input and the net output are represented as

i variables. They mean that output equals input in this layer. This layer of SFNN just executes the transmission work.

Layer 2 - Membership layer: In this layer, each node performs a membership function and acts as a unit of memory. The bell-shaped function is adopted as the membership function.

For the ith input, the corresponding net input and output of the jth node can be expressed as

( ) ( )

² ²

2 2 2 2

ij ij i ij

m net x

− −

= (2-10)

(

) (

)

ij f net net

y ² ² ²

2 = =exp , j=1 ,2,L,M (2-11) where ²m_ij is the mean, ²σ_ij is the variance and M is the total number of membership functions with respect to the respective input node. In this study, the input linguistic variable is the tracking error vector.

Layer 3 - Rule layer: Each node k in this layer is denoted by ∏ which multiplies the incoming signals and outputs the result of the product. For the kth rule node, the operation of the net input and output of this layer is presented as

∏

= _ij _ij

k w x

net ³ ³

3 (2-12)

(

)

k f net net

y ³ ³ ³

3 = = , k =1 ,2,L,N (2-13) where ³x_ij represents the i, th input to the kth node of layer 3, j ³w_ij between the membership and the rule layers are assumed as unity, and N is the total number of fuzzy rules.

Layer 4 - Output layer: The single node o in this layer is labeled as Σ , which computes the overall output as the summation of all incoming signals. It executes the sun-of-weighting defuzzification. The description of the net input and output is expressed as

k k k

o w x

net ⁴ ⁴

4 =∑ (2-14)

(

)

o f net net

y ⁴ ⁴ ⁴

4 = = (2-15) where ⁴w_k is the output action strength of the output associated with the kth rule, ⁴x_k represents the kth input to the node of layer 4, and ⁴y_o is the output of SFNN.

In order to improve the learning capability and flexibility of a NN, asymmetric Gaussian membership functions are adopted, instead of ball-shaped functions described in layer 2.

According to the above description, the output of the SFNN with N existing fuzzy rules can be represented simply as

∑

= ^N

k k k

o w

)

φ (x (2-16) in which w is the output action strength associated with the k-th rule and _k φ_k is the response of the firing weight for an input vector x=[x₁x₂Lx_L]^T and composed of asymmetric Gaussian membership functions defined as [42]

( )

where M is the total number of membership functions with respect to the respective input node; m , _ij σ_ij^l , and σ_ij^r are the mean, left-side variance, and right-side variance of the asymmetric Gaussian function in the j-th term of the i-th input linguistic variable x , _i respectively. However, σ and _ij^l σ may become zero in the training procedure, the _ij^r membership function ζ_ij will not be defined. To avoid this problem, this dissertation considers a membership function form as [44]

( )

where ϖ is a small positive constant. Then, the associated firing strength can be defined as

∏

Thus, the output of the SFNN can be represented in a vector form as ) structure of the FNN should be determined in advance by empiricism. However, it is difficult to consider the balance between the rule number and the desired performance. Therefore, the structure adaptation algorithm which contains the growing and pruning of membership functions and fuzzy rules is proposed in this dissertation. The descriptions are given as follows.

In the process of the growing of membership functions, the concept which decides

whether to add a new node (membership function) in layer 2 and the associated fuzzy rule in layer 3 will be introduced. The mathematical description of the existing rules can be expressed as a set of clusters. For constructing the initial fuzzy rules of the SFNN, the fuzzy clustering method is used to partition a set of data into a number of overlapping clusters based on the distance in a metric space between the data points and the cluster prototypes. Each cluster in the product space of the input-output data represents a rule. The firing strength of a rule for each incoming data x can be represented as the degree that the incoming data _i belong to the cluster [19]. If the value of firing strength is too small, it indicates that the input value is on the edge of range of the existing membership functions. Under this situation, the output will cause unsatisfactory performance. Therefore, a new membership function and a new fuzzy rule should be generated to improve the performance.

The firing strength from (2-19) is used as the degree measure

k φ

β = , k =1,2,...,N(t) (2-24) where )N(t is the number of the existing fuzzy rules at the time t. Define the maximum degree β_max as

t k N

k β

) (

max 1max

≤

= ≤ . (2-25) If β_max ≤G_th is satisfied, where G_th∈(0,1) is a pre-given threshold, the incoming data is far from the edge of range of the existing membership functions. Hence, a new membership function is generated. The mean and the variance of the new membership function and the weight are selected as follows

i new

i x

m = , (2-26)

i new l

i σ

σ ^, = , (2-27)

i new r

i σ

σ ^, = , (2-28)

wnew (2-29)

where x is the new incoming data and _i σ_i is a pre-specified constant. If the unknown control system dynamics is too complex, we can choose the larger G_th so that many membership functions can be created.

sampling time

Rise and decay curves of the used frequency index

sampling time

Rise and decay curves of the used frequency index

Fig. 2-3. The rise and decay curves of the used frequency index.

Next, to avoid the unrestricted growth of network structure and an overload computation, the pruning algorithm is developed to eliminate irrelevant fuzzy rules. In Ref. [21], a significance index is determined for the importance of the fuzzy rules. The elimination algorithm is derived from the observation that if the significance index fades when the firing weight is smaller than a special threshold value and if the significance index fixes when the firing weight is larger than a special threshold value [21]. In this dissertation, when the r-th firing strength β_r is smaller than the threshold value P , it indicates that the relationship _th becomes weak between the input and the r-th rule. Then, the significant index of r-th fuzzy rules will be decreased. When the r-th firing strength β_r is larger than the threshold value P , it indicates that the incoming inputs fall into the range of the r-th fuzzy rule. Thus, the th

significant index of r-th fuzzy rules should be raised. The rise and decay curves of the used frequency index show in Fig. 2-3. The significance index is determined for the r-th rules can be given as

( )

[ ]

⎪⎩

⎪⎨

⎧

≥

−

⋅

−

= ⋅ +

th r r

r I t I t if P

P if

t t I

I τ β

β τ

, ) ( 1 exp 2 ) (

exp(

) ) (

1 (

1 , )r=1,2,L,N(t (2-30)

where I_r is the significant index of the r-th rule and its initial value is 1, P is the pruning _th

threshold value, and τ₁ and τ₂ are the designed constant. Exponential functions in (2-30) are used to rise or decrease the values of significant index in [0, 1]. If I_r ≤I_th is satisfied, where I is another pre-given threshold, the r-th fuzzy rule will be deleted. For real-time _th implementation, if the computation load is the issue having highest priority, P should be _th chosen large, so that more fuzzy rules can be pruned. This operation will prevent the fuzzy rule, which may be less used but still significant, from being deleted in the training process.

Hence, the computation load would be reduced.

In summary, the flow chart of the structure learning algorithm is shown in Fig. 2-4. The major contributions of the SFNN are: 1) SFNN can be operated directly without spending much time pre-determining membership functions and fuzzy rules; and 2) the computation load can be reduced simultaneously.

start

calculate the tracking error and sliding surface

max ≤Gth?

grow a membership function and fuzzy

rule

calculate the adaptive law and update the parameter

calculate the

prune the r-th membership function and rule

end control ?

End Yes

No calculate the

robust control

apply control law into system