非线性模拟电路故障诊断的Volterra模型及特征提取研究
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摘要
伴随着电子产业和电子技术的迅猛发展，非线性模拟电路故障诊断问题已经成为电路故障诊断问题中的一大瓶颈。由于非线性模拟电路非常复杂，应用线性模拟电路的故障诊断方法对其进行分析，诊断效果并不理想。利用Volterra级数可以比较准确地建立常见的非线性模拟电路的故障模型，国内外已有部分学者运用这一模型进行了非线性模拟电路的故障诊断，取得了一定的成果。然而，由于Volterra级数存在维数灾难问题，实际使用时只能取其低阶有限项，使得待诊断电路的故障特征难于提取，降低了故障诊断的效果。本文针对非线性模拟电路的故障特征提取问题，致力于在使用低阶Volterra级数的条件下，提高对弱非线性电路的软硬故障的检测和定位能力；从全带Volterra级数、子带Volterra级数、分数阶Volterra级数、非参数型Volterra级数中，对非线性模拟电路进行故障特征的提取。本论文的主要工作和研究成果如下：
（1）针对非线性模拟电路故障诊断的难题，对从全带Volterra级数中提取电路故障特征的方法进行了较为深入的研究，并结合隐马尔科夫模型进行故障诊断以验证该故障特征的有效性。首先对全带Volterra级数和隐马尔科夫模型进行介绍，在此基础上推导出比较准确的非线性模拟电路的全带Volterra级数故障模型，并从这个模型中提取出非线性模拟电路的故障特征。然后进行了计算复杂度的详细分析。最后通过仿真实验对该方法与其他故障诊断方法在故障识别能力和诊断所需要的开销两方面进行了比较；
（2）针对从低阶全带Volterra级数中提取的故障特征在软故障定位能力上的局限性，对从子带Volterra级数中提取非线性电路故障特征的方法进行了较为深入的研究，对其原理、数学模型和具体实施步骤进行了详细的讨论。首先对小波变换进行介绍，在此基础上推导出比较准确的非线性模拟电路的子带Volterra级数模型，从这个模型中提取出非线性模拟电路的故障特征。然后进行了计算复杂度的详细分析。最后通过仿真实验对该方法与其他故障诊断方法在故障识别能力和诊断所需要的开销两方面进行了比较；
（3）为进一步提高对非线性模拟电路的故障诊断能力，对从分数阶Volterra级数和分数阶相关函数中提取故障特征的方法进行了较为深入的研究，对其原理、数学模型和具体实施步骤进行了详细的讨论。首先对分数阶变换和分数阶相关分析方法进行介绍，在此基础上推导出比较准确的非线性模拟电路的分数阶Volterra级数模型和分数阶Volterra相关函数模型，从这些模型中，提取出非线性模拟电路的故障特征。然后进行了计算复杂度的详细分析。最后通过仿真实验对该方法与其他故障诊断方法在故障识别能力和诊断所需要的开销两方面进行了比较；
（4）针对参数型Volterra级数计算复杂度高的问题，对非参数型Volterra级数进行了比较深入的研究。首先应用最优搜索理论得到非线性系统的优化激励信号，然后根据非线性系统的特点和优化的激励信号，推导出比较准确的非线性模拟电路的非参数型Volterra级数模型，从这个模型中，提取出非线性模拟电路的故障特征，然后进行了计算复杂度的分析。最后通过仿真实验对该方法与其他故障诊断方法在故障识别能力和诊断所需要的开销两方面进行了比较。
With the rapid development of electronic industries and technologies, the problemto diagnose faults in nonlinear analog circuits has become the bottleneck in faultdiagnosis of circuits. Since the nonlinear analog circuits are very complicated, if thediagnosis methods of linear circuits are used to analyze the nonlinear analog circuits, thediagnosis results are not satisfactory. Using the Volterra series can establish the faultmodels of common nonlinear analog circuits accurately. Some domestic and foreignresearchers have made use of the models to solve the problems of fault diagnosis ofnonlinear analog circuits and obtained certain achievements. However, since thedimensional disaster problem exists in the Volterra series, only limited lower-orderitems can be used in application, which makes the fault features be difficult to extractand the diagnosis results be decreased. This dissertation aims at the problem ofextracting fault features from nonlinear analog circuits and tries to use lower-orderitems of the Volterra series to improve the capability of detecting and locating soft andhard faults in weak nonlinear analog circuits. From the fullband Volterra series, subbandVolterra series, fractional Volterra series, non-parametric Volterra series, the faultfeatures of nonlinear analog circuits are extracted. The main works and contributions ofthe dissertation are as follows:
(1) Considering the problem of fault diagnosis of nonlinear analog circuits, theapproach that extracts the fault features from the fullband Volterra series is studieddeeply. Combined with the hidden Markov model (HMM), the fault features are used todiagnose faults to verify the effectivenss of the fault features. Firstly, the fullbandVolterra series and the HMM are described. Then, the fullband Volterra series faultmodels of nonlinear analog circuits are derived accurately. From the model, the faultfeatures are extracted. Detailed analysis about computational complexity is made.Finally, the experiment is implemented to compare with other fault diagnosis methodsin the fault recognition capability and diagnosis cost.
(2) Considering the problem of soft fault location using the fullband Volterraseries to extract fault features, the approach of extracting fault features from the subband Volterra series is studied deeply. Detailed discussions are made in theprinciples, mathematical models and practical executive steps. Firstly, the wavelettransform (WT) are described. Then, the subband Volterra series models of nonlinearcircuits are derived accurately. From the model, the fault features are extracted. Detailedanalysis about computational complexity is made. Finally, the experiments areimplemented to compare with other fault diagnosis methods in the fault recognitioncapability and diagnosis cost.
(3) To improve the fault diagnosis capability of nonlinear analog circuits further,the approach from the fractional Volterra series and the approach from the fractionalcorrelation analysis to extract fault features are studied deeply. Detailed discussions aremade about the principles, mathematical models and practical executive steps. Firstly,the fractional transform and the fractional correlation are described. Then, the fractionalVolterra series fault models and the model of the fractional Volterra series correlationfunction are derived accurately. From the models, the fault features are extracted.Detailed analysis about computational complexity is made. Finally, the experiments areimplemented to compare with other fault diagnosis methods in the fault recognitioncapability and diagnosis cost.
(4) Considering the problem of the computational complexity using parametricVolterra series, the method of non-parametric Volterra series is studied deeply. Firstly,the optimized excited signals are obtained by the theory of optimal search. Based on thefeatures of nonlinear systems and the optimized excited signals, the non-parametricVolterra series fault models of nonlinear analog circuits are derived accurately. From themodel, the fault features are extracted. Then, detailed analysis about computationalcomplexity is made. Finally, the experiments are implemented to compare the methodwith other fault diagnosis methods in the fault recognition capability and diagnosis cost.
（1）针对非线性模拟电路故障诊断的难题，对从全带Volterra级数中提取电路故障特征的方法进行了较为深入的研究，并结合隐马尔科夫模型进行故障诊断以验证该故障特征的有效性。首先对全带Volterra级数和隐马尔科夫模型进行介绍，在此基础上推导出比较准确的非线性模拟电路的全带Volterra级数故障模型，并从这个模型中提取出非线性模拟电路的故障特征。然后进行了计算复杂度的详细分析。最后通过仿真实验对该方法与其他故障诊断方法在故障识别能力和诊断所需要的开销两方面进行了比较；
（2）针对从低阶全带Volterra级数中提取的故障特征在软故障定位能力上的局限性，对从子带Volterra级数中提取非线性电路故障特征的方法进行了较为深入的研究，对其原理、数学模型和具体实施步骤进行了详细的讨论。首先对小波变换进行介绍，在此基础上推导出比较准确的非线性模拟电路的子带Volterra级数模型，从这个模型中提取出非线性模拟电路的故障特征。然后进行了计算复杂度的详细分析。最后通过仿真实验对该方法与其他故障诊断方法在故障识别能力和诊断所需要的开销两方面进行了比较；
（3）为进一步提高对非线性模拟电路的故障诊断能力，对从分数阶Volterra级数和分数阶相关函数中提取故障特征的方法进行了较为深入的研究，对其原理、数学模型和具体实施步骤进行了详细的讨论。首先对分数阶变换和分数阶相关分析方法进行介绍，在此基础上推导出比较准确的非线性模拟电路的分数阶Volterra级数模型和分数阶Volterra相关函数模型，从这些模型中，提取出非线性模拟电路的故障特征。然后进行了计算复杂度的详细分析。最后通过仿真实验对该方法与其他故障诊断方法在故障识别能力和诊断所需要的开销两方面进行了比较；
（4）针对参数型Volterra级数计算复杂度高的问题，对非参数型Volterra级数进行了比较深入的研究。首先应用最优搜索理论得到非线性系统的优化激励信号，然后根据非线性系统的特点和优化的激励信号，推导出比较准确的非线性模拟电路的非参数型Volterra级数模型，从这个模型中，提取出非线性模拟电路的故障特征，然后进行了计算复杂度的分析。最后通过仿真实验对该方法与其他故障诊断方法在故障识别能力和诊断所需要的开销两方面进行了比较。
With the rapid development of electronic industries and technologies, the problemto diagnose faults in nonlinear analog circuits has become the bottleneck in faultdiagnosis of circuits. Since the nonlinear analog circuits are very complicated, if thediagnosis methods of linear circuits are used to analyze the nonlinear analog circuits, thediagnosis results are not satisfactory. Using the Volterra series can establish the faultmodels of common nonlinear analog circuits accurately. Some domestic and foreignresearchers have made use of the models to solve the problems of fault diagnosis ofnonlinear analog circuits and obtained certain achievements. However, since thedimensional disaster problem exists in the Volterra series, only limited lower-orderitems can be used in application, which makes the fault features be difficult to extractand the diagnosis results be decreased. This dissertation aims at the problem ofextracting fault features from nonlinear analog circuits and tries to use lower-orderitems of the Volterra series to improve the capability of detecting and locating soft andhard faults in weak nonlinear analog circuits. From the fullband Volterra series, subbandVolterra series, fractional Volterra series, non-parametric Volterra series, the faultfeatures of nonlinear analog circuits are extracted. The main works and contributions ofthe dissertation are as follows:
(1) Considering the problem of fault diagnosis of nonlinear analog circuits, theapproach that extracts the fault features from the fullband Volterra series is studieddeeply. Combined with the hidden Markov model (HMM), the fault features are used todiagnose faults to verify the effectivenss of the fault features. Firstly, the fullbandVolterra series and the HMM are described. Then, the fullband Volterra series faultmodels of nonlinear analog circuits are derived accurately. From the model, the faultfeatures are extracted. Detailed analysis about computational complexity is made.Finally, the experiment is implemented to compare with other fault diagnosis methodsin the fault recognition capability and diagnosis cost.
(2) Considering the problem of soft fault location using the fullband Volterraseries to extract fault features, the approach of extracting fault features from the subband Volterra series is studied deeply. Detailed discussions are made in theprinciples, mathematical models and practical executive steps. Firstly, the wavelettransform (WT) are described. Then, the subband Volterra series models of nonlinearcircuits are derived accurately. From the model, the fault features are extracted. Detailedanalysis about computational complexity is made. Finally, the experiments areimplemented to compare with other fault diagnosis methods in the fault recognitioncapability and diagnosis cost.
(3) To improve the fault diagnosis capability of nonlinear analog circuits further,the approach from the fractional Volterra series and the approach from the fractionalcorrelation analysis to extract fault features are studied deeply. Detailed discussions aremade about the principles, mathematical models and practical executive steps. Firstly,the fractional transform and the fractional correlation are described. Then, the fractionalVolterra series fault models and the model of the fractional Volterra series correlationfunction are derived accurately. From the models, the fault features are extracted.Detailed analysis about computational complexity is made. Finally, the experiments areimplemented to compare with other fault diagnosis methods in the fault recognitioncapability and diagnosis cost.
(4) Considering the problem of the computational complexity using parametricVolterra series, the method of non-parametric Volterra series is studied deeply. Firstly,the optimized excited signals are obtained by the theory of optimal search. Based on thefeatures of nonlinear systems and the optimized excited signals, the non-parametricVolterra series fault models of nonlinear analog circuits are derived accurately. From themodel, the fault features are extracted. Then, detailed analysis about computationalcomplexity is made. Finally, the experiments are implemented to compare the methodwith other fault diagnosis methods in the fault recognition capability and diagnosis cost.
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