separates points on U.
Finally, Y is proven to vanish at no point of U. By Eq.(A.1), u(j3)(x) is constant and
does not equal zero. That is, for all x∈ℜN, u(j3)(x)>0. If u(j3)(x)>0,(j=1,2,...,R), then
> 0
y
for anyx ∈ ℜ
N. That is, anyy ∈
Y with u(j3)(x)>0 can serve as the required f.In summary, the FLNFN model is a universal approximator, using the Stone-Weierstrass theorem and the fact that Y is a continuous real set on U proves the theorem.
B. Proof of Convergence Theorem
Theorem B1: Let
η
w be the learning rate parameter of the FLNFN weight, and letmax
in the FLNFN model.
Proof of Theorem B1: Since
∑
=u
φ
, the following result holds;R k
Pw( ) ≤ . (B.2)
Then, a discrete Lyapunov function is selected as The change in the Lyapunov function is obtained as
[
( 1) ( )]
The error difference can be represented as [23]
kj layer, respectively. Equations (2.17) and (B.5) yield
) output error between the reference model and actual plant converges to zero as t→∞. This fact completes the proof of the theorem.
The following lemmas [25] are used to prove Theorem 2.
Lemma B1: Let g(h)=hexp(−h2), then g(h) <1,∀h∈ℜ.
Lemma B2: Let f(h)=h2exp(−h2), then f(h)<1,∀h∈ℜ.
Theorem B2: Let
η
m andη
σ be the learning rate parameters of the mean and standarddeviation of the Gaussian function for the FLNFN; let Pmmax be defined as
Proof of Theorem B2: According to Lemma B1,
[
(xi−mij)/σ
ij]
exp{
−[
(xi−mij)/σ
ij]
2}
<1. The upper bounds on Pm(k) can be derived asThe error difference can also be represented as [23]
ij
where ∆mij represents the change of the mean of the Gaussian function in the membership
function layer. Equation (2.18) and (B.11) yield and ∆V <0 given by Eq.(B.3) and Eq.(B.4), is guaranteed. The output error between the reference model and actual plant converges to zero as
t → ∞
.According to Lemma B2,
[
(xi−mij)/σ
ij]
2exp{
−[
(xi−mij)/σ
ij]
2}
<1. The upper boundsThe error difference can be represented as
ij
where ∆
σ
ij represents the change of the variance of the Gaussian function in themembership function layer. Equation (2.19) and (B.17) yield ) reference model and actual plant converges to zero as t→∞. This fact completes the proof of the theorem.
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Vita
博士候選人學經歷資料
姓名：陳政宏 (Cheng-Hung Chen) 性別：男
生日：民國 68 年 2 月 23 日 出生地：高雄市
論文題目：
中文：以函數鏈結為基礎之類神經模糊網路及其應用
英文：A Functional-Link-Based Neuro-Fuzzy Network and Its Applications
學歷：
民國 91 年 6 月，朝陽科技大學資訊工程系畢業
民國 93 年 6 月，朝陽科技大學資訊工程系碩士班畢業
民國 97 年 7 月，國立交通大學電機與控制工程學系博士班，提博士論
文口試
Publication List
著作目錄
姓名：陳政宏(Cheng-Hung Chen) 已刊登或被接受之期刊論文：
[1] Cheng-Hung Chen, Cheng-Jian Lin, and Chin-Teng Lin, “A Functional-Link-Based Neuro-Fuzzy Network for Nonlinear System Control,” accepted to appear in IEEE Trans.
on Fuzzy Systems, 2008. (2.8 點)
[2] Cheng-Jian Lin, Cheng-Hung Chen, and Chin-Teng Lin, “A Hybrid of Cooperative Particle Swarm Optimization and Cultural Algorithm for Neural Fuzzy Networks and Its Prediction Applications,” accepted to appear in IEEE Trans. on Systems, Man, and Cybernetics, Part C: Applications and Reviews, 2008. (2 點)
[3] Cheng-Hung Chen, Cheng-Jian Lin, and Chin-Teng Lin, “Using an Efficient Immune Symbiotic Evolution Learning for Compensatory Neuro-Fuzzy Controller,” accepted to appear in IEEE Trans. on Fuzzy Systems, 2008. (2.8 點)
[4] Cheng-Hung Chen, Cheng-Jian Lin, and Chin-Teng Lin, “An Efficient Quantum Neuro-Fuzzy Classifier Based on Fuzzy Entropy and Compensatory Operation,” Soft Computing, Vol. 12, No. 6, pp. 567-583, Apr. 2008. (1.4 點)
研討會論文：
[1] Cheng-Hung Chen, Chin-Teng Lin, and Cheng-Jian Lin, “A Novel Recurrent Neuro-Fuzzy System and Its Applications,” Cross-Strait Workshop on Controls, Taipei, Taiwan, R.O.C., pp. 69-74, Nov. 22-26, 2007.
[2] Cheng-Hung Chen, Chin-Teng Lin, and Cheng-Jian Lin, “A Functional-Link-Based Fuzzy Neural Network for Temperature Control,” The First IEEE Symposium on Foundations of Computational Intelligence, Honolulu, Hawaii, USA, pp. 53-58, Apr. 1-5, 2007.
[3] Cheng-Hung Chen, Cheng-Jian Lin, and Chin-Teng Lin, “A Recurrent Functional-Link- Based Neural Fuzzy System and Its Applications,” The First IEEE Symposium on Computational Intelligence in Image and Signal Processing, Honolulu, Hawaii, USA, pp.
415-420, Apr. 1-5, 2007.
[4] Cheng-Hung Chen, Cheng-Jian Lin, and Chin-Teng Lin, “A Self-Constructing Compensatory Neural Fuzzy System for Nonlinear System Control,” The 14th National Conference on Fuzzy Theory and Its Applications, Kaohsiung, Taiwan, R.O.C., Dec.
14-15, 2006.
[5] Cheng-Hung Chen, Chin-Teng Lin, and Cheng-Jian Lin, “Identification and Prediction
Using Recurrent Compensatory Neuro-Fuzzy Systems,” The 14th National Conference on Fuzzy Theory and Its Applications, Kaohsiung, Taiwan, R.O.C., Dec. 14-15, 2006.
[6] Cheng-Hung Chen, Chin-Teng Lin, and Cheng-Jian Lin, “A Novel Neuro-Fuzzy Inference System for Skin Color Detection,” The 19th IPPR Conference on Computer Vision, Graphics and Image Processing, Taoyuan, Taiwan, R.O.C., Aug. 13-15, 2006.
[7] Chin-Teng Lin and Cheng-Hung Chen, “An Entropy-Based Neuro-Fuzzy Inference System for Classification Applications,” The First Taiwan Software Engineering Conference, Taipei, Taiwan, R.O.C., pp. 189-194, June 3-4, 2005.
[8] Cheng-Hung Chen and Chin-Teng Lin, 2005, “Identification of Chaotic System Using Recurrent Compensatory Neuro-Fuzzy Systems,” IEEE Int’l Workshop on Cellular Neural Networks and their Applications, Hsinchu, Taiwan, R.O.C., pp. 15-18, May 28-30, 2005.