2. Robust Adaptive Self-structuring Fuzzy Control Design for Nonaffine
2.2 Self-structuring Fuzzy System
2.2.2 Structure Learning Algorithm
2.2.2 Structure Learning Algorithm
The developed self-structuring algorithm consists of two parts: growing and pruning of fuzzy rules. Effective membership functions in the input spaces can be generated and ineffective fuzzy rules can be pruned automatically by the self-structuring algorithm, and thus a concise rule base can be obtained. In order to construct the fuzzy rule base, every input space S(Xj) is partitioned into several overlapping clusters to construct the fuzzy sets of Xj. It can happen that for some incoming Xj, the degree of belongings to all its fuzzy sets are quite small, i.e., Fjij(Xj), ij =1 L,2, ,Nj are quite small, as depicted in Fig. 2-1(a). This means that the input space S(Xj) is not properly clustered. Hence, the fundamental concept of the growing of fuzz rules is developed to adjust the inappropriate clustering. Initially, create one initial fuzzy rule with the given initial state as
,1
The SFS will start operating from this single rule. Define the growing criterion as
g j < Θ
µmax , j=1, 2, …, m (2-16)
jth state variable
jth state variable at time t
Fig. 2-1 (a) Improper fuzzy clustering of input variable Xj; (b) Newly created membership function initial mean and standard deviation are
)
q
Nj
j +1 =
σ (2-18) where q>0 can be arbitrarily chosen, and it will be tuned by the adaptive law introduced in later section. The created membership function is shown in Fig. 2-1(b). For the case that one new membership function is created at some time, N1×K×Nj−1×Nj+1×K×Nm new fuzzy rules will be generated according to the new membership function as:
1
Assume that the growing criterion for X1 is satisfied at time t. Then, a new membership function
is created, and two rules are grown according to the new membership function as
1
Rule : IF X2, 1 is F and X12 2 is F THEN y is 21 α2,1
Rule : IF X1 is 2,2 F12 and X2 is F THEN y is 22 α2,2 (2-21) A self-structuring FS with only rule generation algorithm may suffer from the computational load or learning failure caused by an overly large rule base which includes both effective and redundant fuzzy rules. In the following, the strategy to prune redundant rules is developed to solve this problem. Recall that there are n existing fuzzy rules, and then express (2-12) as collections of the singleton consequents and the fuzzy basis functions of the rest of fuzzy rules, respectively. Thus, the contribution made by kth rule on the output y can be defined as
follows:
∑
== n
k k
k k
y C y
1
, k=1, 2, …, n (2-23)
where yk =αkξk. Now, we are ready to introduce the significance index which can help us to decide whether or not to prune a fuzzy rule. Significance index is a measurement of the importance of every fuzzy rule. Sk, which represents the significance index of the kth fuzzy rule, is updated as follows:
⎩⎨
⎧
≥
= <
if ,
if ,
β β τ
k rc
k
k rc
k
k S C
C
S S , k=1, 2, …, n (2-24)
whereS is the most recent Skrc k, 1)τ∈(0 , is a decay constant, and β∈(0 ,1) is a given constant. All Sk, k=1, 2, …, n, are initialized from ones. According to (2-18), if the contribution Ck is equal or larger than β, Sk keeps invariant; if Ck is smaller than β, Sk will be attenuated. An invariant significance implies that the associated rule is still important and should be remained; a decaying significance index implies that the associated rule is becoming less and less important and thus should be pruned. The selection of τ will affect the rate of pruning the fuzzy rules. The smaller the τ is (or the larger the β is), the faster the significance index Sk decays, and thus the faster the ineffective fuzzy rules will be pruned.
The pruning criterion of the kth fuzzy rule is defined as follows based on this knowledge Sk <Θp, k=1, 2, …, n (2-25) where )Θp ∈(0 ,1 is a selected threshold. If the pruning criterion is satisfied for Sk, the associated kth rule is pruned.
Remark 2-1: It is a difficult task to determine the initial values of the singleton consequents of the newly generated fuzzy rules. Because an SFS is in general equipped with a parameter learning algorithm to automatically tune the parameters of the fuzzy rules, the initial values of the singleton consequents can simply set as zeros. However, from (2-10), we can see that this will cause abrupt variation of the fuzzy output y and may deteriorate the performance of the SFS for a short period. This phenomenon can be observed in Fig. 2-5(b). To fix this drawback, we maintain the approximation property of the SFS at the instant that new rules are generated.
Assume that at some time tg, an SFS has n fuzzy rules and the last h rules are just newly generated. Define yp as the “pseudo fuzzy output” of the original n-h rules if h new rules were not generated at tg. The initial consequents of those new rules are chosen so that y(tg)= yp.
Thus, we have
p h
n
k k k
n h n
k k
new
g y
t
y =
∑
+∑
− == +
−
= 1 1
)
( α ξ α ξ (2-26) where αn−h+1 =αn−h+2 =L=αn =αnew. From (2-26), we can easily obtain
∑
∑
+
−
=
−
=
−
= n
h n
k k
h n
k k k
p new
y
1 1
ξ ξ α
α (2-27)
In this way, not only the bad effect caused by the abrupt variation can be mitigated, but also the future performance of the SFS can be improved by the h new rules.
Remark 2-2: While controlling, a membership function is possible to be pruned if all fuzzy rules associated with this membership function are pruned sequentially.
Remark 2-3: In the implementations of practical systems, if computational burden is the issue having highest priority, the threshold Θ can be chosen large enough so that more fuzzy p rules are pruned. Hence, the computational burden will be substantially reduced at the expense of less favorable system performance.
Fig. 2-2 shows the flowchart to summarize the self-structuring algorithm for the SFS.
The growing and pruning effects during the control period will be illustrated in later sections with excellent result.
p
Sk <Θ
mzx
µj
ξ α ˆˆT ufc =
σ c αˆ ,ˆ ,ˆ
n k Sk , =1 ,2,
m j
j , 1 ,2,
max =
µ
Fig. 2-2 The flowchart of the self-structuring algorithm for the SFS