小波神经网络若干关键问题研究
|
摘要
基于小波神经网络的理论性分析与应用研究是本论文的主要内容。论文以小波神经网络为研究对象,提出了一类新的加权小波基,分析证明了加权小波基的诸多良好特性;对于常见小波神经网络的一致逼近特性、S型函数组合小波神经网络的鲁棒性分析、多模型小波神经网络的故障检测等问题给出了详细的论证;最后,针对歼击机的常见故障问题,分别给出了应用小波神经网络和BP神经网络的故障诊断结果,实现了小波神经网络对飞机系统的故障诊断。
首先,论文提出了一种新的小波合成方法,不是采用常见的Fourier变换,而是利用细分方案和递推平均插值方法构造了一类新的具有加权性质的小波,同时证明了它的常用性质:紧支撑性,光滑性,消失矩,正交性。表明了加权小波系数的衰减性与一般正交小波的系数衰减性相同,但计算复杂度更低。仿真结果显示,当加权函数跳跃性很大时,尺度函数和加权小波具有非常好的光滑性,而且加权小波的逼近收敛率也快于一般常见小波。并用此加权小波基和常见小波在逼近方面进行了对比分析,得到了加权小波具有更快的收敛速度和更准确的逼近能力。
其次,论文利用小波函数的紧支撑和可积性质,结合实变函数和泛函分析中Hilbert空间,Lebesgue划分、算子理论和相对紧集的性质特点,从理论上严格论证了小波神经网络的逼近特性,以此来扩充小波神经网络的应用范围,为其实用性提供充分的理论指导,并由仿真实例表明小波神经网络逼近非线性函数是一致逼近器。
接着对S型函数组合小波神经网络的扰动敏感性和鲁棒性进行了分析讨论,利用概率统计和数理分析的方法,针对弱扰动情形下S型函数组合小波神经网络的鲁棒性能给出了其保持不变的条件,而对敏感扰动下S型函数组合小波神经网络,得到了系统鲁棒度的检验公式。从理论上证明了S型函数组合小波神经网络中隐层小波元数目的增多,会降低小波神经网络系统鲁棒性的结论,以解析的方式验证了小波神经网络隐层神经元数目和网络系统鲁棒性之间很直观的看法假设。
然后讨论了小波神经网络的系统辨识特性,利用稳定性分析的基本方法证明了其逼近非线性函数的估计误差渐近收敛的性质,同时也得到了小波神经网络辨识器参数趋于理想辨识器参数的结论;并将小波神经网络作为系统辨识器,对多模型系统的故障检测问题进行了分析,得到了较好的结果。
最后,对于歼击机的常见故障,用小波神经网络和BP神经网络进行了诊断分析,多组仿真示例验证了小波神经网络进行歼击机故障诊断的可行性及有效性。
The theoretic analysis and study of wavelet neural networks are main contents of this thesis. Wavelet neural network is the key object in this article, a novel weighted wavelet base is presented, and its favorablecharacters are analyzed and proved. The uniform approximation of normal wavelet neural network and the robust analysis of wavelet neural networks of the combination of Sigmoid function are detailedly introduction; Multiple model failure detection based on wavelet neural network is demonstrated detailedly; At last, the failure diagnosis results of aerocraft is present seperately by employing wavelet neural network and BP neural network, and the fault diagnosis of areocraft system by wavelet neural network is achieved.
At first, a new weighted wavelet is constructed by utilizing the subsection scheme and recursive average interpolated technique, without employing familiar Fourier transform. The weighted wavelet are proved to have such normal characteristics: compactly supported, smooth, vanishing moment, and orthogonal. Though the decay of weighted wavelet coefficient is the same as the common wavelet, the computable complication is observably reduced. The given simulation illustrates that the weighted wavelet and scaling functions are sufficiently smooth even though the weighted function has a large jump, and the approximate convergent rate of weighted wavelet base is faster than that of common wavelet. Compared with approximation rate, the weighted wavelet has faster convergent and more exact approximation.
Next, using the integral characteristics and compact support of wavelet function, the traits of Lebesgue partition, the continuity of the operator theory and the topology structure of the relatively compact set in the Hilbert space are applied to the proof of the approximation of the wavelet neural networks. The wavelet neural networks universally approximating the nonlinear function in compact set with arbitrary precision has been proved in the abstractin theory. It extends the application of wavelet network, and supplies sufficient guidance.The simulation shows that the wavelet neural network can uniformly approximate the nonlinear function. In the following chapter, the disturbing sensitivity and robust of combination of sigmoid function wavelet neural network is discussed by making use of probability statistics and symbolic logic. If the faint perturbation appears, the robust invariability condition of wavelet neural network is obtained, and if the sensitive disturbation appreas, the sensitivity formula and robust formula are given. It proved theoretically that to a certain extent, the more hidden wavelet nerve cells, the less robust of wavelet neural network. Consequently, it validates analytically the intuitionistic hypothesis of the relationship between hidden layer wavelet nerve cells numbers and the robust of wavelet neural network.
Then, the characteristic of wavelet neural network in system identification is introduced.Tthe state estimation error is proved to converge to zero asymptotically with the basic conception of stability analysis, in the mean time, parameters of the identifier can converge to the ideal identifier values. The multiple model failure detection using wavelet neural networks identifiers is also analyzed.The simulation result shows this detection method is effective and feasible.
Finally, the familiar failure of areocraft is detected seperately by normal wavelet neural network and BP neural network. The simulation result shows this fault diagnosis method of wavelet neural network is effective and feasible.
首先,论文提出了一种新的小波合成方法,不是采用常见的Fourier变换,而是利用细分方案和递推平均插值方法构造了一类新的具有加权性质的小波,同时证明了它的常用性质:紧支撑性,光滑性,消失矩,正交性。表明了加权小波系数的衰减性与一般正交小波的系数衰减性相同,但计算复杂度更低。仿真结果显示,当加权函数跳跃性很大时,尺度函数和加权小波具有非常好的光滑性,而且加权小波的逼近收敛率也快于一般常见小波。并用此加权小波基和常见小波在逼近方面进行了对比分析,得到了加权小波具有更快的收敛速度和更准确的逼近能力。
其次,论文利用小波函数的紧支撑和可积性质,结合实变函数和泛函分析中Hilbert空间,Lebesgue划分、算子理论和相对紧集的性质特点,从理论上严格论证了小波神经网络的逼近特性,以此来扩充小波神经网络的应用范围,为其实用性提供充分的理论指导,并由仿真实例表明小波神经网络逼近非线性函数是一致逼近器。
接着对S型函数组合小波神经网络的扰动敏感性和鲁棒性进行了分析讨论,利用概率统计和数理分析的方法,针对弱扰动情形下S型函数组合小波神经网络的鲁棒性能给出了其保持不变的条件,而对敏感扰动下S型函数组合小波神经网络,得到了系统鲁棒度的检验公式。从理论上证明了S型函数组合小波神经网络中隐层小波元数目的增多,会降低小波神经网络系统鲁棒性的结论,以解析的方式验证了小波神经网络隐层神经元数目和网络系统鲁棒性之间很直观的看法假设。
然后讨论了小波神经网络的系统辨识特性,利用稳定性分析的基本方法证明了其逼近非线性函数的估计误差渐近收敛的性质,同时也得到了小波神经网络辨识器参数趋于理想辨识器参数的结论;并将小波神经网络作为系统辨识器,对多模型系统的故障检测问题进行了分析,得到了较好的结果。
最后,对于歼击机的常见故障,用小波神经网络和BP神经网络进行了诊断分析,多组仿真示例验证了小波神经网络进行歼击机故障诊断的可行性及有效性。
The theoretic analysis and study of wavelet neural networks are main contents of this thesis. Wavelet neural network is the key object in this article, a novel weighted wavelet base is presented, and its favorablecharacters are analyzed and proved. The uniform approximation of normal wavelet neural network and the robust analysis of wavelet neural networks of the combination of Sigmoid function are detailedly introduction; Multiple model failure detection based on wavelet neural network is demonstrated detailedly; At last, the failure diagnosis results of aerocraft is present seperately by employing wavelet neural network and BP neural network, and the fault diagnosis of areocraft system by wavelet neural network is achieved.
At first, a new weighted wavelet is constructed by utilizing the subsection scheme and recursive average interpolated technique, without employing familiar Fourier transform. The weighted wavelet are proved to have such normal characteristics: compactly supported, smooth, vanishing moment, and orthogonal. Though the decay of weighted wavelet coefficient is the same as the common wavelet, the computable complication is observably reduced. The given simulation illustrates that the weighted wavelet and scaling functions are sufficiently smooth even though the weighted function has a large jump, and the approximate convergent rate of weighted wavelet base is faster than that of common wavelet. Compared with approximation rate, the weighted wavelet has faster convergent and more exact approximation.
Next, using the integral characteristics and compact support of wavelet function, the traits of Lebesgue partition, the continuity of the operator theory and the topology structure of the relatively compact set in the Hilbert space are applied to the proof of the approximation of the wavelet neural networks. The wavelet neural networks universally approximating the nonlinear function in compact set with arbitrary precision has been proved in the abstractin theory. It extends the application of wavelet network, and supplies sufficient guidance.The simulation shows that the wavelet neural network can uniformly approximate the nonlinear function. In the following chapter, the disturbing sensitivity and robust of combination of sigmoid function wavelet neural network is discussed by making use of probability statistics and symbolic logic. If the faint perturbation appears, the robust invariability condition of wavelet neural network is obtained, and if the sensitive disturbation appreas, the sensitivity formula and robust formula are given. It proved theoretically that to a certain extent, the more hidden wavelet nerve cells, the less robust of wavelet neural network. Consequently, it validates analytically the intuitionistic hypothesis of the relationship between hidden layer wavelet nerve cells numbers and the robust of wavelet neural network.
Then, the characteristic of wavelet neural network in system identification is introduced.Tthe state estimation error is proved to converge to zero asymptotically with the basic conception of stability analysis, in the mean time, parameters of the identifier can converge to the ideal identifier values. The multiple model failure detection using wavelet neural networks identifiers is also analyzed.The simulation result shows this detection method is effective and feasible.
Finally, the familiar failure of areocraft is detected seperately by normal wavelet neural network and BP neural network. The simulation result shows this fault diagnosis method of wavelet neural network is effective and feasible.
引文
[1] Grossmann.A, Morlet.J., Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM. J. Math. Anal., 1984, 15:723-726.
[2] Grossmann.A, Morlet.J,Paul.T.,Transforms associated to square integrable group representions,I:General results. J.Math.Phys., 1985,26(16):2473-2479.
[3] Daubechies.I., Orthonormal bases of compactly supported wavelets. Comm.Pure. Appl.Math.,1988,41 :909-996.
[4] Daubechies I., Ten lecture on wavelets. Philadelphia:SIAM Publ 1992.
[5] Daubechies. I., Han, B., Ron. A, Shen. Z., Framelets: MRA-based Constructions of Wavelet Frames. Appl. Comput. Harmon. Anal., 2003, 14(1): 1-46.
[6] Heng-Tung Hsu,hui-wen Yao, Kauthar Azaki,Ali E Atia., Synthesis of coupled-tesonators Group-delay equalizers.,IEEE Trans.Microwave theory tech., 2002, 50:1960-1967.
[7] Antonini M, et al., Image coding using wavelet transforms. IEEE Trans. on Image Processing, 1992, 1(2): 205-220.
[8] Battle. G., A block spin construction of ondelettes, Part I: Lemarie function. Commun., Math. Phys.,1987,110(3): 601-615.
[9] Rabideau.D.J., Nonlinear synthesis wideband waveforms. IEEE Radar Conference, 2002,Piscataway, IEEE press:212-219.
[10] Donoho.D.L., De-noising by soft thresholding. IEEE Trans.information theory., 1995,41(3):613-627.
[11] Hsung.T.C, Lin.D.P.K,Siu.W.C., Denoising by singularity detection. IEEE Trans on Signal processing., 1999, 47(11): 3139-314.
[12] Mallat.S, Wen Liangf Hwang., Singularity detection and processing with wavelet. IEEE trans.on Information Theory.,1992,38(2):617-643.
[13] Cannon.M, E.Slotine JJ., Space frequency localized basis function networks for nonlinear system estimation and control. Neural Computing., 1995,9:3–342.
[14] Wen-Jyi Hwang, Derin.H., Multiresolution multiresource progressive image transmission. IEEE Transactions on Image Processing.,1995,4(8): 1128-1140.
[15] Cohen.A, I. Daubechies,JC Feauvean., Biorthogonal Bases of. Compactly supported wavelets. Commun. Pure Appl. Math., 1992, 45: 485-560.
[16] Orzel.C, A. K. Tuchman, M. L. Fenselau, M. Yasuda, M. A. Kasevich., Squeezed States in a Bose-Einstein Condensate. Science., 2001, 291: 2386-2389.
[17] Harr.A., Zur theorie der orthogonalen funktionensysteme. Math. Ann.,1910,69: 331-371.
[18] Morlet.J,Arens.G,Fourgeau.E,Giard.D., Wave propagation and sampling theory and complex waves. Geophysics.,1982,47(2):222-236.
[19] Meyer.Y., Principed’incertitude,bases hilbertiennes et algebres d’operateurs. Bourbaki Seminar.,1985-1986,(662).
[20] Meyer.Y., Wavelets with compact support. Zygmund Lecture., 1987.
[21] Mallat.S., Multiresolution representations and wavelets, Ph.D.Thesis, University of Pennsylvania, Philadelphia,PA,1988.
[22] Mallat.S., A theory of multiresolution signal decomposition: the wavelet representation. IEEE trans.Pattern Anal.Machine Intell., 1989,11: 674-693.
[23] Mallat.S., Multiresolution approximation and wavelet orthonormal bases of L2 (R). Trans. Amer.Math.Soc., 1989, 315: 69-87.
[24] Daubechies.I,Grossman.A,Meyer.Y., Painless nonorthogonal expansions. J.Math. Phys., 1986, 27: 1271-1283.
[25] Cohen.A.,Ondelettes,analyses multiresolutions et filters miroirsen quadrature. Ann.Inst.henri Poinccare.,1990,7(5):439-459.
[26] Chui.C,Wang.J.Z., A cardinal spline approach to wavelets. Proc. Amer.Math. Soc., 1991, 113:785-793.
[27] Chui.C,Wang. J.Z., A general framework of compactly supported splines and wavelets. J. Approx. Theory., 1992,71(3):263-304.
[28] Coifman, R, Meyer, Y, Quake, S, Wickerhauser. M. V., Signal processing and compression with wavelet packets, Progress in Wavelet Analysis and Applications (eds. Y. Meyer and S. Roques), Proceedings of the International Conference "Wavelets and Applications", Toulouse, France, 1992, 77-93.
[29] Sweldens W., The lifting scheme: A custom-design construction of biorthogonal wavelets. Applied and Computational Harmonic Analysis., 1996,3(2):186-200.
[30] Sweldens W., The lifting scheme: A construction of second generation wavelets. SIAM Journal of Mathematical Analysis., 1998, 29(2):511-546.
[31] Zhang Q H, Benveniate.A., Wavelet networks. IEEE Trans Neural Networks., 1992,3(6):889-898.
[32] Szu H.H,etal., Neural Network adaptive Wavelets for Signal representation and classification. Optical Engineering.,1992,31(9)2907-1016.
[33] Chen.J,BrunsD.D.,Wavelet neural network development for system identification using a systematic design synthesis.Ind.Eng.Chem.Res.,1995,34:4420-4435.
[34] Pati.C,Krishnaprassad S., Analysis and synthesis of feedforward neural networks using discrete affine wavelet transformations. IEEE Trans on Neural networks.,1993,4(1):73-85.
[35] Bahavik R B,Stephanopoulos G. Wavenet: a mutiresolusion, hierarchical neural network with localized learning.AIChE.,1993,39(1):57-81.
[36] Zhang Jun. Walter Gilber G, Wavelet Neural Network for Function Learning. IEEE Trans on Signal Processing., 1995, 43(6): 1485-1497.
[37] Zhang Q. Using wavelet network in nonparametric estimation. IEEE Transactionson Neural Networks.,1997,8(2):227-236
[38] JIAO L C, PAN J,FANG Y W., Multiwavelet neural network and its approximation properties. IEEE Transactions on Neural Networks.,2001, 12(5):1060-1066.
[39] EricA.Rying,GriffL.Bilbro,Jye-Chyi Lu., Focused Local Learning With Wavelet Neural Networks. IEEE Transactions on Neural Networks.,2002,13(2):304-319.
[40] Rong-JongWai, Han-HsiangChang., Backstepping Wavelet Neural Network Control for Indirect Field-Oriented Induction Motor Drive. IEEE Transactions on Neural Networks., 2004,15(2):367-382.
[41] Abdulhamit Subasia,Ahmet Alkana,Etem Koklukayab,et.al.,Wavelet neural network classification of EEG signals by using AR model with MLE preprocessing. Neural Networks.,2005,18:985–997.
[42] StephenA.Billings,Hua-LiangWei., A New Class Of Wavelet Networks For Nonlinear System Identification. IEEE Transactions on Neural Networks., 2005,16(4):862-874.
[43] Iyengar. S.Sitharama, Cho. E.C, Phoha. Vir V., Foundations of Wavelet Networks and Applications.,Taylor & Francis Ltd,2002.
[44] 丁宇新,沈学勤等.,基于能量密度的小波神经网路.计算机学报,1997,20(9): :832-838.
[45] 何振亚,李文化,蔚承建.,用于信号逼近的自适应时延小波神经网络.电子科学学刊,1998,20(5):604-610.
[46] 李益国,沈炯,吕震中.,一种新的小波神经网络结构优化设计方法.控制理论与应用.,2003,20(3):329-337.
[47] 李换琴,万百五.,基于关键输入和多输入层高维小波网络.自动化学报.,2004, 30(6):939-943.
[48] 高协平,周四望.,正交平衡对称的区间多小波研究.中国科学 E 辑.,2005,35(4): 385-404.
[49] 李俊民,李靖,薄立军.,一类非线性动态系统的自适应模糊小波神经网络控制.工程数学学报.,2006,23(1):37-44.
[50] AntoniziM,BarlandM,MathieuP,DaubechiesI. Image Coding Using Vector Auantization in the Wavelet Transform Domain.In Karlsson &VetteliEds. Pro cof ICASSP’90,NewMexico:1990.2297-2300.
[51] Vetterli M., Wavelet, Filter Banks. Theory and Design. IEEE Trans. Signal Processing.,1992,40(9):2207-2232.
[52] Gurski G C., Optimal Linear Filters for Pyramidal Decomposition. IEEE Trans. On Signal Processing.,1996,41(5):1790-1808.
[53] Daubechies Ingrid. Two recent results on wavelets: wavelet bases for the interval, and biothogonal wavelets diagonalizing the derivative operator. In: Larry L Schumaker, Glenn Webb Eds.Recent Advances in Wavelet Analysis,Academic Press, Incorporeted,1993.
[54] Chui C K,Montefusco L,Puccio L., Wavelets theory,algorithms,and applications. Academic Press Inc.California,1994.
[55] Williams John R, Amaratunga Kevin., Introduction to wavelets in engineering. International Journal for Numerical Method in Engineering.,1994,37(14):2365-2388.
[56] Quak Ewald, Weyrich Norman. Decomposition and reconstruction algorithms for spline wavelets on a bounded interval. Applied and Computational Harmonic Analysis.,1994,1:217-231.
[57] Quak Ewald, Weyrich Norman. Wavelets on the interval. In: Singh S P Ed. Approximation Theory Wavelets and Applications., 1995.
[58] Chen W H, Wu C W. A spline wavelet element method for frame structure vibration. Computational Mechanics.,1995,16(1):11-12.
[59] Chen Mingquayer, Hwang Chyi, Shih Yenping., A wavelet-Galerkin method for solving population balance equations. Computers Chem Engng,1996,20(2):131-145.
[60] Chen Mingquayer, Hwang Chyi, Shin Yehping., The computation of wavelet-Galerkin approximation on a bounded interval. International Journal for Numerical Methods in Engineering,1996,39:2921～2944.
[61] Chen H L., Wavelets form trigonometric spline approach.J.approx.Theory & its Appl.,1996,12(2):99-110.
[62] P. Auscher and P. Tchamitchian. Bases d’ondelettes sur les courbes cordearc noyau de Cauchy et espaces de Hardy associes. Rev .mat. Iberoamericana 5:139-170, 1989.
[63] P. Auscher and P. Tchamitchian. Conjecture de kato sur les ouverts de R. Rev .mat.Iberoamericana. 8(2):149-199, 1992.
[64] Y. Meyer. Ondelettes et Operateurs,I: Ondelettes, II: Operateurs de Calderon-Zygmund, III: (with R.Coifman), Operateurs multilineaires. Wavelets and Operators,Cambridge University Press, 1993.
[65] P. Tchamitchian., Ondelettes et integrale de Chauchy sur les courbes lipchitziennes. Ann. Math., 1989,641-649.
[66] W. Sweldens, R. Piessens. Calculation of the wavelet decomposition using quadrature formulae. Number 1 in Series in Approximations and Decompositions., World Scientific Singapore.,1993:139-160.
[67] DYN, N., Subdivision schemes in computer aided geometric design. In Advances in Numerical Analysis II, Wavelets, Subdivision and Radial Functions, W. Light, Ed., Oxford University Press, New York, NY, 1991,36-104.
[68] D. L. Donoho., Smooth wavelet decompositions with blocky coefficient kernels. Recent Advances in Wavelet Analysis. Academic Press, New York: 1993.259-308.
[69] D. L. Donoho., Unconditional bases are optimal bases for data compression and for statistical estimation. Appl.Comput.Harmon.Anal.,1993,1(1):100-115.
[70] Hong C,Elangovan S.A B-spline wavelet based fault classification scheme for high speed protection relaying[J].Electric mathines and power systems,2000,28(2):313-324.
[71] Mikhail Toupikov, Guang-wen pan and Barry K.Gilbert. Weighted wavelet expansion in the method of moments. IEEE Trans. on Magnetics, 1999,35(3): 1550-1553,
[72] Ganesh Arulampalam,Abdesselam Bouzerdoum., A generalized feedforward neural network architecture for classification and regression. Neural Networks., 2003,16:561-568.
[73] Guang-BinHuang ,P.Saratchandran, NarasimhanSundararajan., A Generalized Growing and Pruning RBF(GGAP-RBF) Neural Network for Function Approximation. IEEE Transactions on Neural Networks.,2005,16(1):57-67.
[74] Funahashi K. On the approximate realization of continuous mappings by neural networks. Neural networks., 1989(2):183-192.
[75] Cybenko G. Approximation by superpositions of a sigmoidal function. Mathematics of control, signals and systems., 1989(2):303-314.
[76] Hornik K. Approximation capabilities of multiplayer feedforward networks, Neural networks, 1991,4:251-257.
[77] Web A.R, Shannon.S., Shape adaptive radial basis functions. IEEE trans. on Neural Networks.,1998, 9(6):1155-1166.
[78] Krishnaswami Alladi., A fundamental invariant in the theory of partitions. Topics in number Theory(University Park,PA,1997) 101-113,Math, Appl.467,Kluwer Acad.Publ,Dordrecht,1999.
[79] R.Henstock., The general theory of integration. Oxford,Clarendon press.,1991.
[80] Bartsch, R, Dencker, P, Poppe, H., Ascoli-Arzela-Theory based on continuous convergence in an (almost) non-Hausdorff setting, categorical topology, Dordrecht, 1996,221-240.
[81] Xu lei,et al.,Robust principle component analysis by self-organizing rules based on statistical physics approach. IEEE Trans. Neural Networks., 1995,6(1): 131-143.
[82] Ying H. Sufficient conditions on uniform approximation of multivariate functions by general Takagi-Sugeno Fuzzy systems with linear rule consequent. IEEE trans.on Systems,Man and Cybernetics.,1998,28(4):515-520.
[83] Sanger T D.,Optimal unsupervised learning in a single –layer feedforward neural network. Neural networks., 1989,1:459-473.
[84] 杨行峻,郑君里., 人工神经网络与盲信号处理,北京:清华大学出版社,2002.
[85] Scott D W. Mulitivariate density estimation. New York:Wiley, 1992.
[86] Walter Gilbert G.,Approximaiton of deta function by wavelets. J.Approx.Theory., 1992,71:329-343.
[87] Huang J N,et al.,Regression modeling in back-propagation and projectionpursuit learning. IEEE Trans.on Neural Networks.,1994,5(3)342-353.
[88] Delyon B,et al., Accuracy analysis for wavelet approximations. IEEE Trans.on Neural networks.,1995,6(2):332-347.
[89] E.J.Davison,E.M.Kurak.,a computational method for determining quadratic lyapunov function for nonlinear systems. Automatica.,1971,7(5):627-636.
[90] G.Zames.,Feedback and optimal sensitiviety:model reference transformations, multiplicative semi-norms and approximate ninverse.IEEE trans on Automatic Control.,1981,26(2):301-320.
[91] A.A. Stoorvogal.,the H ∞control problem.Prentice-hall,1992.
[92] L.Ghaoui,S.Niculescu (Ed.)advances in linear matrix inequality methods in control,SIAM press,2000.
[93] Sam.S.Z, Wang G J.,Robust adaptive tracking for time-varying uncertain nonlinear systems with unknown control coefficients.IEEE Trans On Automatic Control.,2003,48(8):1463-1469.
[94] Fitch. A.J, Kadyrov, A, Christmas, W.J, Kittler, J., Fast robust correlation. IEEE Trans on Image processing., 2005, 14(8):1063 – 1073.
[95] Dong-Mei Li., Identification of chaotic systems with large noise based on regularized feedforward neural networks. Proceedings of 4th International Conference on machine learning and Cybernetics., 2005, 7: 4060-4063.
[96] Nian Cai,Jian Cheng, Jie Yang., Applying a Wavelet Neural Network to Impulse Noise Removal ICNN&B '05. International Conference on Neural netoworks and Brain,2005,2:781-783.
[97] Piche. S., Robustness of feedforward neural networks. International Joint Conference on neural networks.,1992,2:346-351.
[98] C.M.Bishop., Neural networks for Pattern Recognization .London,U.K. Clarendon,1995.
[99] Mizutani, E., Powell's dogleg trust-region steps with the quasi-Newton augmented Hessian for neural nonlinear least-squares learning. IJCNN '99. International Joint Conference on Neural networks.,1999,2:1239-1244.
[100] J.Larsen., A generalization error estimate for nonlinear system. In Proc. IEEE Signal Processing workshop.,1992,29-38.
[101] N.Murata,S. Yoshizawa, S.Amari., network information criterion-determining the number of hidden units for an artificial neural network model. IEEE Trans Neural networks.,1994,5:865-872.
[102] A. Poncet., Asymptotic probability of the generalization error. In Proc. IEEE NICROSP., 1996:66-74.
[103] C.Alippi., FPE-based criteria to dimension feedforward neural topologies. IEEE Trans Circuits Syst. I,1999, vol 46:962-973.
[104] S.Amari., Statistical and information-geometrical aspects of neural learning.Computational Intelligence: A Dynamic Perspective.,New York: IEEE Press 1995:71-82.
[105] Kreinovich. V, Sirisaengtaksin. O, Cabrera. S.,Wavelet neural networks are asymptotically optimal approximators for functions of one variable.IEEE International Conference on Neural networks.,1994,1:299 -304.
[106] John.J.Dazzo,Constantine H.Houpis,Stuart N.Sheldon.,Linear control system analysis and design with matlab.Marcel Dekker Publisher.,New York,2003.
[107] Xieping Gao,Fen Xiao.,Multiwavelet networks for prediction of chaotic time series. IEEE International Conference on system,man and Cybernetics.,2004,4: 3328-3332.
[108] He.Y, Tan. Y, Sun. Y., Wavelet neural network approach for fault diagnosis of analogue circuits. IEE Proceedings Circuits,Devices and systems.,2004, 151(4):379-384.
[109] I.H.Jeon., Weyl’s heorem and quasi-similarity. Integral Equations Operator Theory.2001,39:214-221.
[110] K.S.Narendra ,J.Balekrishnan., Adaptive control using multiple models. IEEE Trans.Automat .Contr.,1997,42(2):171-187.
[111] K.S.Narendra, Driollet O A, Feilevr M,et al., adaptive control using multiple models ,switching and tuning. Int J of adaptive control and signal process, 2003,17(2):87-102.
[112] P.Dedeoglu G.,Roumeliotis S.I, Sukhatme G.S., Fault Detection and Identification in a Mobile Robot Using Multiple Model Estimation and Neural Network. Proc. IEEE International Conference on Robotics and Automation, San Francisco, California, 2000:2302-2309.
[113] Sastry.Shankar, Bodson.Marc., Adaptive Control:stability, convergence and robustness. Prentice-Hall: Englewood Cliffs, NJ, 1989.
[114] M.M.Polycarpou, Stable adaptive neural control scheme for nonlinear systems. IEEE trans.Automat .Contr., 1996,41(3):447-451.
[115] Narendra.K.S,Annaswamy.A.M,Stable Adaptive Systems, New Jersey: Prentice Hall,1989.
[116] M. Msaad, I. D. Landau, R. Lozano, Yoan D. Landau, M. M'Sand., Adaptive Control.Springer Verlag,London, 1998.
[117] Vidyasagar M, Nonlinear systems analysis (Second Edition). society of industrial and applied mathematics publisher., (SIAM Classics Series), Philadelphia, 2002.
[118] D.W.Lane , P.S.Maybeck., Multiple model adaptive estimation a lied to the lambda URV for failure detection and identification. Proc of 33rd Conference on Decision and Control., Lake Buena Vista FL,1994:678-683.
[119] Erdol N, Basbug F,wavelet transform based adaptive filters:analysis and new results, IEEE trans.on Signal Processing,1996,44(9):2163-2169.
[120] JilKovv P,L X R.,adaptation of transition probability matrix for multiple model estimators. Proc of 4th int.conference on Information fusion Montreal Canada,ISIF press,2001,2:1-3.
[121] Doucet A, Ristic B, recursive state estimation for multiple switching models with unknown transition probabilities.IEEE trans.on Aerospace and Eletronic systems,2002,38(3):1098-1104.
[122] Chen L.J, K.S.Narendra.,nonlinear adaptive control using neural networks and Multiple model.Automatica.,2001,37:1245-1255.
[123] Nasser Sadati,Rasoul Ghadami, Mahdi Bagherpour., Adaptive Neural Network Multiple Models Sliding Mode Control of Robotic Manipulators Using Soft Switching.IEEE computer of science ,431-438, Proceeding of the 17th IEEE International Conference on Tools with Artificial Intelligence (ICTAI'05).2005.
[124] Leonid I. Perlovsky Neural Networks and Intellect : Using Model-Based Concepts Oxford University Press, New York,USA,2000.
[125] Silvio Simani, Cesare Fantuzzi and Ron J. Patton, Model-based fault diagnosis in dynamic systems using identification techniques, Springer: London, 2003.
[126] Andrew D. Back, Tianping Chen., Universal Approximation of Multiple Nonlinear Operators by Neural Networks. Neural Computation., 2002,14: 2561-2566.
[127] Boeing Company., Airplane Maintenance Manual.Seattle,Boeing,2000,55-72.
[128] Boeing Company., Fault Isolation Manual.Seattle,Boeing,2000,243-298.
[129] Airbus Industrie., Trouble Shooting Manual.2001.
[130] 任晓华.,洛克希德马丁公司的 F-22 战斗机装配生产线.航空制造技术,2006, (8):36-38.
[131] 陈文.,F-35 制造上的若干特点.航空维修与工程.2006, (3):24-26.
[132] http://www.scitom.com.cn/museums/aviation/plane/pln004_15.html.
[133] 唐永哲.,基于神经网络的飞机故障诊断.航空维修与工程,2001,(3):30-31.
[134] 潘茂庆,惠克翔.,基于神经网络的飞机故障诊断专家系统.航空维修与工程,2002,(5):34-35.
[2] Grossmann.A, Morlet.J,Paul.T.,Transforms associated to square integrable group representions,I:General results. J.Math.Phys., 1985,26(16):2473-2479.
[3] Daubechies.I., Orthonormal bases of compactly supported wavelets. Comm.Pure. Appl.Math.,1988,41 :909-996.
[4] Daubechies I., Ten lecture on wavelets. Philadelphia:SIAM Publ 1992.
[5] Daubechies. I., Han, B., Ron. A, Shen. Z., Framelets: MRA-based Constructions of Wavelet Frames. Appl. Comput. Harmon. Anal., 2003, 14(1): 1-46.
[6] Heng-Tung Hsu,hui-wen Yao, Kauthar Azaki,Ali E Atia., Synthesis of coupled-tesonators Group-delay equalizers.,IEEE Trans.Microwave theory tech., 2002, 50:1960-1967.
[7] Antonini M, et al., Image coding using wavelet transforms. IEEE Trans. on Image Processing, 1992, 1(2): 205-220.
[8] Battle. G., A block spin construction of ondelettes, Part I: Lemarie function. Commun., Math. Phys.,1987,110(3): 601-615.
[9] Rabideau.D.J., Nonlinear synthesis wideband waveforms. IEEE Radar Conference, 2002,Piscataway, IEEE press:212-219.
[10] Donoho.D.L., De-noising by soft thresholding. IEEE Trans.information theory., 1995,41(3):613-627.
[11] Hsung.T.C, Lin.D.P.K,Siu.W.C., Denoising by singularity detection. IEEE Trans on Signal processing., 1999, 47(11): 3139-314.
[12] Mallat.S, Wen Liangf Hwang., Singularity detection and processing with wavelet. IEEE trans.on Information Theory.,1992,38(2):617-643.
[13] Cannon.M, E.Slotine JJ., Space frequency localized basis function networks for nonlinear system estimation and control. Neural Computing., 1995,9:3–342.
[14] Wen-Jyi Hwang, Derin.H., Multiresolution multiresource progressive image transmission. IEEE Transactions on Image Processing.,1995,4(8): 1128-1140.
[15] Cohen.A, I. Daubechies,JC Feauvean., Biorthogonal Bases of. Compactly supported wavelets. Commun. Pure Appl. Math., 1992, 45: 485-560.
[16] Orzel.C, A. K. Tuchman, M. L. Fenselau, M. Yasuda, M. A. Kasevich., Squeezed States in a Bose-Einstein Condensate. Science., 2001, 291: 2386-2389.
[17] Harr.A., Zur theorie der orthogonalen funktionensysteme. Math. Ann.,1910,69: 331-371.
[18] Morlet.J,Arens.G,Fourgeau.E,Giard.D., Wave propagation and sampling theory and complex waves. Geophysics.,1982,47(2):222-236.
[19] Meyer.Y., Principed’incertitude,bases hilbertiennes et algebres d’operateurs. Bourbaki Seminar.,1985-1986,(662).
[20] Meyer.Y., Wavelets with compact support. Zygmund Lecture., 1987.
[21] Mallat.S., Multiresolution representations and wavelets, Ph.D.Thesis, University of Pennsylvania, Philadelphia,PA,1988.
[22] Mallat.S., A theory of multiresolution signal decomposition: the wavelet representation. IEEE trans.Pattern Anal.Machine Intell., 1989,11: 674-693.
[23] Mallat.S., Multiresolution approximation and wavelet orthonormal bases of L2 (R). Trans. Amer.Math.Soc., 1989, 315: 69-87.
[24] Daubechies.I,Grossman.A,Meyer.Y., Painless nonorthogonal expansions. J.Math. Phys., 1986, 27: 1271-1283.
[25] Cohen.A.,Ondelettes,analyses multiresolutions et filters miroirsen quadrature. Ann.Inst.henri Poinccare.,1990,7(5):439-459.
[26] Chui.C,Wang.J.Z., A cardinal spline approach to wavelets. Proc. Amer.Math. Soc., 1991, 113:785-793.
[27] Chui.C,Wang. J.Z., A general framework of compactly supported splines and wavelets. J. Approx. Theory., 1992,71(3):263-304.
[28] Coifman, R, Meyer, Y, Quake, S, Wickerhauser. M. V., Signal processing and compression with wavelet packets, Progress in Wavelet Analysis and Applications (eds. Y. Meyer and S. Roques), Proceedings of the International Conference "Wavelets and Applications", Toulouse, France, 1992, 77-93.
[29] Sweldens W., The lifting scheme: A custom-design construction of biorthogonal wavelets. Applied and Computational Harmonic Analysis., 1996,3(2):186-200.
[30] Sweldens W., The lifting scheme: A construction of second generation wavelets. SIAM Journal of Mathematical Analysis., 1998, 29(2):511-546.
[31] Zhang Q H, Benveniate.A., Wavelet networks. IEEE Trans Neural Networks., 1992,3(6):889-898.
[32] Szu H.H,etal., Neural Network adaptive Wavelets for Signal representation and classification. Optical Engineering.,1992,31(9)2907-1016.
[33] Chen.J,BrunsD.D.,Wavelet neural network development for system identification using a systematic design synthesis.Ind.Eng.Chem.Res.,1995,34:4420-4435.
[34] Pati.C,Krishnaprassad S., Analysis and synthesis of feedforward neural networks using discrete affine wavelet transformations. IEEE Trans on Neural networks.,1993,4(1):73-85.
[35] Bahavik R B,Stephanopoulos G. Wavenet: a mutiresolusion, hierarchical neural network with localized learning.AIChE.,1993,39(1):57-81.
[36] Zhang Jun. Walter Gilber G, Wavelet Neural Network for Function Learning. IEEE Trans on Signal Processing., 1995, 43(6): 1485-1497.
[37] Zhang Q. Using wavelet network in nonparametric estimation. IEEE Transactionson Neural Networks.,1997,8(2):227-236
[38] JIAO L C, PAN J,FANG Y W., Multiwavelet neural network and its approximation properties. IEEE Transactions on Neural Networks.,2001, 12(5):1060-1066.
[39] EricA.Rying,GriffL.Bilbro,Jye-Chyi Lu., Focused Local Learning With Wavelet Neural Networks. IEEE Transactions on Neural Networks.,2002,13(2):304-319.
[40] Rong-JongWai, Han-HsiangChang., Backstepping Wavelet Neural Network Control for Indirect Field-Oriented Induction Motor Drive. IEEE Transactions on Neural Networks., 2004,15(2):367-382.
[41] Abdulhamit Subasia,Ahmet Alkana,Etem Koklukayab,et.al.,Wavelet neural network classification of EEG signals by using AR model with MLE preprocessing. Neural Networks.,2005,18:985–997.
[42] StephenA.Billings,Hua-LiangWei., A New Class Of Wavelet Networks For Nonlinear System Identification. IEEE Transactions on Neural Networks., 2005,16(4):862-874.
[43] Iyengar. S.Sitharama, Cho. E.C, Phoha. Vir V., Foundations of Wavelet Networks and Applications.,Taylor & Francis Ltd,2002.
[44] 丁宇新,沈学勤等.,基于能量密度的小波神经网路.计算机学报,1997,20(9): :832-838.
[45] 何振亚,李文化,蔚承建.,用于信号逼近的自适应时延小波神经网络.电子科学学刊,1998,20(5):604-610.
[46] 李益国,沈炯,吕震中.,一种新的小波神经网络结构优化设计方法.控制理论与应用.,2003,20(3):329-337.
[47] 李换琴,万百五.,基于关键输入和多输入层高维小波网络.自动化学报.,2004, 30(6):939-943.
[48] 高协平,周四望.,正交平衡对称的区间多小波研究.中国科学 E 辑.,2005,35(4): 385-404.
[49] 李俊民,李靖,薄立军.,一类非线性动态系统的自适应模糊小波神经网络控制.工程数学学报.,2006,23(1):37-44.
[50] AntoniziM,BarlandM,MathieuP,DaubechiesI. Image Coding Using Vector Auantization in the Wavelet Transform Domain.In Karlsson &VetteliEds. Pro cof ICASSP’90,NewMexico:1990.2297-2300.
[51] Vetterli M., Wavelet, Filter Banks. Theory and Design. IEEE Trans. Signal Processing.,1992,40(9):2207-2232.
[52] Gurski G C., Optimal Linear Filters for Pyramidal Decomposition. IEEE Trans. On Signal Processing.,1996,41(5):1790-1808.
[53] Daubechies Ingrid. Two recent results on wavelets: wavelet bases for the interval, and biothogonal wavelets diagonalizing the derivative operator. In: Larry L Schumaker, Glenn Webb Eds.Recent Advances in Wavelet Analysis,Academic Press, Incorporeted,1993.
[54] Chui C K,Montefusco L,Puccio L., Wavelets theory,algorithms,and applications. Academic Press Inc.California,1994.
[55] Williams John R, Amaratunga Kevin., Introduction to wavelets in engineering. International Journal for Numerical Method in Engineering.,1994,37(14):2365-2388.
[56] Quak Ewald, Weyrich Norman. Decomposition and reconstruction algorithms for spline wavelets on a bounded interval. Applied and Computational Harmonic Analysis.,1994,1:217-231.
[57] Quak Ewald, Weyrich Norman. Wavelets on the interval. In: Singh S P Ed. Approximation Theory Wavelets and Applications., 1995.
[58] Chen W H, Wu C W. A spline wavelet element method for frame structure vibration. Computational Mechanics.,1995,16(1):11-12.
[59] Chen Mingquayer, Hwang Chyi, Shih Yenping., A wavelet-Galerkin method for solving population balance equations. Computers Chem Engng,1996,20(2):131-145.
[60] Chen Mingquayer, Hwang Chyi, Shin Yehping., The computation of wavelet-Galerkin approximation on a bounded interval. International Journal for Numerical Methods in Engineering,1996,39:2921～2944.
[61] Chen H L., Wavelets form trigonometric spline approach.J.approx.Theory & its Appl.,1996,12(2):99-110.
[62] P. Auscher and P. Tchamitchian. Bases d’ondelettes sur les courbes cordearc noyau de Cauchy et espaces de Hardy associes. Rev .mat. Iberoamericana 5:139-170, 1989.
[63] P. Auscher and P. Tchamitchian. Conjecture de kato sur les ouverts de R. Rev .mat.Iberoamericana. 8(2):149-199, 1992.
[64] Y. Meyer. Ondelettes et Operateurs,I: Ondelettes, II: Operateurs de Calderon-Zygmund, III: (with R.Coifman), Operateurs multilineaires. Wavelets and Operators,Cambridge University Press, 1993.
[65] P. Tchamitchian., Ondelettes et integrale de Chauchy sur les courbes lipchitziennes. Ann. Math., 1989,641-649.
[66] W. Sweldens, R. Piessens. Calculation of the wavelet decomposition using quadrature formulae. Number 1 in Series in Approximations and Decompositions., World Scientific Singapore.,1993:139-160.
[67] DYN, N., Subdivision schemes in computer aided geometric design. In Advances in Numerical Analysis II, Wavelets, Subdivision and Radial Functions, W. Light, Ed., Oxford University Press, New York, NY, 1991,36-104.
[68] D. L. Donoho., Smooth wavelet decompositions with blocky coefficient kernels. Recent Advances in Wavelet Analysis. Academic Press, New York: 1993.259-308.
[69] D. L. Donoho., Unconditional bases are optimal bases for data compression and for statistical estimation. Appl.Comput.Harmon.Anal.,1993,1(1):100-115.
[70] Hong C,Elangovan S.A B-spline wavelet based fault classification scheme for high speed protection relaying[J].Electric mathines and power systems,2000,28(2):313-324.
[71] Mikhail Toupikov, Guang-wen pan and Barry K.Gilbert. Weighted wavelet expansion in the method of moments. IEEE Trans. on Magnetics, 1999,35(3): 1550-1553,
[72] Ganesh Arulampalam,Abdesselam Bouzerdoum., A generalized feedforward neural network architecture for classification and regression. Neural Networks., 2003,16:561-568.
[73] Guang-BinHuang ,P.Saratchandran, NarasimhanSundararajan., A Generalized Growing and Pruning RBF(GGAP-RBF) Neural Network for Function Approximation. IEEE Transactions on Neural Networks.,2005,16(1):57-67.
[74] Funahashi K. On the approximate realization of continuous mappings by neural networks. Neural networks., 1989(2):183-192.
[75] Cybenko G. Approximation by superpositions of a sigmoidal function. Mathematics of control, signals and systems., 1989(2):303-314.
[76] Hornik K. Approximation capabilities of multiplayer feedforward networks, Neural networks, 1991,4:251-257.
[77] Web A.R, Shannon.S., Shape adaptive radial basis functions. IEEE trans. on Neural Networks.,1998, 9(6):1155-1166.
[78] Krishnaswami Alladi., A fundamental invariant in the theory of partitions. Topics in number Theory(University Park,PA,1997) 101-113,Math, Appl.467,Kluwer Acad.Publ,Dordrecht,1999.
[79] R.Henstock., The general theory of integration. Oxford,Clarendon press.,1991.
[80] Bartsch, R, Dencker, P, Poppe, H., Ascoli-Arzela-Theory based on continuous convergence in an (almost) non-Hausdorff setting, categorical topology, Dordrecht, 1996,221-240.
[81] Xu lei,et al.,Robust principle component analysis by self-organizing rules based on statistical physics approach. IEEE Trans. Neural Networks., 1995,6(1): 131-143.
[82] Ying H. Sufficient conditions on uniform approximation of multivariate functions by general Takagi-Sugeno Fuzzy systems with linear rule consequent. IEEE trans.on Systems,Man and Cybernetics.,1998,28(4):515-520.
[83] Sanger T D.,Optimal unsupervised learning in a single –layer feedforward neural network. Neural networks., 1989,1:459-473.
[84] 杨行峻,郑君里., 人工神经网络与盲信号处理,北京:清华大学出版社,2002.
[85] Scott D W. Mulitivariate density estimation. New York:Wiley, 1992.
[86] Walter Gilbert G.,Approximaiton of deta function by wavelets. J.Approx.Theory., 1992,71:329-343.
[87] Huang J N,et al.,Regression modeling in back-propagation and projectionpursuit learning. IEEE Trans.on Neural Networks.,1994,5(3)342-353.
[88] Delyon B,et al., Accuracy analysis for wavelet approximations. IEEE Trans.on Neural networks.,1995,6(2):332-347.
[89] E.J.Davison,E.M.Kurak.,a computational method for determining quadratic lyapunov function for nonlinear systems. Automatica.,1971,7(5):627-636.
[90] G.Zames.,Feedback and optimal sensitiviety:model reference transformations, multiplicative semi-norms and approximate ninverse.IEEE trans on Automatic Control.,1981,26(2):301-320.
[91] A.A. Stoorvogal.,the H ∞control problem.Prentice-hall,1992.
[92] L.Ghaoui,S.Niculescu (Ed.)advances in linear matrix inequality methods in control,SIAM press,2000.
[93] Sam.S.Z, Wang G J.,Robust adaptive tracking for time-varying uncertain nonlinear systems with unknown control coefficients.IEEE Trans On Automatic Control.,2003,48(8):1463-1469.
[94] Fitch. A.J, Kadyrov, A, Christmas, W.J, Kittler, J., Fast robust correlation. IEEE Trans on Image processing., 2005, 14(8):1063 – 1073.
[95] Dong-Mei Li., Identification of chaotic systems with large noise based on regularized feedforward neural networks. Proceedings of 4th International Conference on machine learning and Cybernetics., 2005, 7: 4060-4063.
[96] Nian Cai,Jian Cheng, Jie Yang., Applying a Wavelet Neural Network to Impulse Noise Removal ICNN&B '05. International Conference on Neural netoworks and Brain,2005,2:781-783.
[97] Piche. S., Robustness of feedforward neural networks. International Joint Conference on neural networks.,1992,2:346-351.
[98] C.M.Bishop., Neural networks for Pattern Recognization .London,U.K. Clarendon,1995.
[99] Mizutani, E., Powell's dogleg trust-region steps with the quasi-Newton augmented Hessian for neural nonlinear least-squares learning. IJCNN '99. International Joint Conference on Neural networks.,1999,2:1239-1244.
[100] J.Larsen., A generalization error estimate for nonlinear system. In Proc. IEEE Signal Processing workshop.,1992,29-38.
[101] N.Murata,S. Yoshizawa, S.Amari., network information criterion-determining the number of hidden units for an artificial neural network model. IEEE Trans Neural networks.,1994,5:865-872.
[102] A. Poncet., Asymptotic probability of the generalization error. In Proc. IEEE NICROSP., 1996:66-74.
[103] C.Alippi., FPE-based criteria to dimension feedforward neural topologies. IEEE Trans Circuits Syst. I,1999, vol 46:962-973.
[104] S.Amari., Statistical and information-geometrical aspects of neural learning.Computational Intelligence: A Dynamic Perspective.,New York: IEEE Press 1995:71-82.
[105] Kreinovich. V, Sirisaengtaksin. O, Cabrera. S.,Wavelet neural networks are asymptotically optimal approximators for functions of one variable.IEEE International Conference on Neural networks.,1994,1:299 -304.
[106] John.J.Dazzo,Constantine H.Houpis,Stuart N.Sheldon.,Linear control system analysis and design with matlab.Marcel Dekker Publisher.,New York,2003.
[107] Xieping Gao,Fen Xiao.,Multiwavelet networks for prediction of chaotic time series. IEEE International Conference on system,man and Cybernetics.,2004,4: 3328-3332.
[108] He.Y, Tan. Y, Sun. Y., Wavelet neural network approach for fault diagnosis of analogue circuits. IEE Proceedings Circuits,Devices and systems.,2004, 151(4):379-384.
[109] I.H.Jeon., Weyl’s heorem and quasi-similarity. Integral Equations Operator Theory.2001,39:214-221.
[110] K.S.Narendra ,J.Balekrishnan., Adaptive control using multiple models. IEEE Trans.Automat .Contr.,1997,42(2):171-187.
[111] K.S.Narendra, Driollet O A, Feilevr M,et al., adaptive control using multiple models ,switching and tuning. Int J of adaptive control and signal process, 2003,17(2):87-102.
[112] P.Dedeoglu G.,Roumeliotis S.I, Sukhatme G.S., Fault Detection and Identification in a Mobile Robot Using Multiple Model Estimation and Neural Network. Proc. IEEE International Conference on Robotics and Automation, San Francisco, California, 2000:2302-2309.
[113] Sastry.Shankar, Bodson.Marc., Adaptive Control:stability, convergence and robustness. Prentice-Hall: Englewood Cliffs, NJ, 1989.
[114] M.M.Polycarpou, Stable adaptive neural control scheme for nonlinear systems. IEEE trans.Automat .Contr., 1996,41(3):447-451.
[115] Narendra.K.S,Annaswamy.A.M,Stable Adaptive Systems, New Jersey: Prentice Hall,1989.
[116] M. Msaad, I. D. Landau, R. Lozano, Yoan D. Landau, M. M'Sand., Adaptive Control.Springer Verlag,London, 1998.
[117] Vidyasagar M, Nonlinear systems analysis (Second Edition). society of industrial and applied mathematics publisher., (SIAM Classics Series), Philadelphia, 2002.
[118] D.W.Lane , P.S.Maybeck., Multiple model adaptive estimation a lied to the lambda URV for failure detection and identification. Proc of 33rd Conference on Decision and Control., Lake Buena Vista FL,1994:678-683.
[119] Erdol N, Basbug F,wavelet transform based adaptive filters:analysis and new results, IEEE trans.on Signal Processing,1996,44(9):2163-2169.
[120] JilKovv P,L X R.,adaptation of transition probability matrix for multiple model estimators. Proc of 4th int.conference on Information fusion Montreal Canada,ISIF press,2001,2:1-3.
[121] Doucet A, Ristic B, recursive state estimation for multiple switching models with unknown transition probabilities.IEEE trans.on Aerospace and Eletronic systems,2002,38(3):1098-1104.
[122] Chen L.J, K.S.Narendra.,nonlinear adaptive control using neural networks and Multiple model.Automatica.,2001,37:1245-1255.
[123] Nasser Sadati,Rasoul Ghadami, Mahdi Bagherpour., Adaptive Neural Network Multiple Models Sliding Mode Control of Robotic Manipulators Using Soft Switching.IEEE computer of science ,431-438, Proceeding of the 17th IEEE International Conference on Tools with Artificial Intelligence (ICTAI'05).2005.
[124] Leonid I. Perlovsky Neural Networks and Intellect : Using Model-Based Concepts Oxford University Press, New York,USA,2000.
[125] Silvio Simani, Cesare Fantuzzi and Ron J. Patton, Model-based fault diagnosis in dynamic systems using identification techniques, Springer: London, 2003.
[126] Andrew D. Back, Tianping Chen., Universal Approximation of Multiple Nonlinear Operators by Neural Networks. Neural Computation., 2002,14: 2561-2566.
[127] Boeing Company., Airplane Maintenance Manual.Seattle,Boeing,2000,55-72.
[128] Boeing Company., Fault Isolation Manual.Seattle,Boeing,2000,243-298.
[129] Airbus Industrie., Trouble Shooting Manual.2001.
[130] 任晓华.,洛克希德马丁公司的 F-22 战斗机装配生产线.航空制造技术,2006, (8):36-38.
[131] 陈文.,F-35 制造上的若干特点.航空维修与工程.2006, (3):24-26.
[132] http://www.scitom.com.cn/museums/aviation/plane/pln004_15.html.
[133] 唐永哲.,基于神经网络的飞机故障诊断.航空维修与工程,2001,(3):30-31.
[134] 潘茂庆,惠克翔.,基于神经网络的飞机故障诊断专家系统.航空维修与工程,2002,(5):34-35.