Vector-valued regularized kernel function approximation based fault diagnosis method for analog circuit

A vector-valued regularized kernel function approximation (VVRKFA) based fault diagnosis method for an analog circuit, comprising the following steps:

• (1) extracting a time domain response voltage signal of each node of a tested analog circuit, that is, collecting output signals;
• (2) performing wavelet packet decomposition on the collected output signals, and calculating energy of each node as original sample feature data;
• and averagely dividing the original sample feature data into two parts:
• a training sample set and a test sample set;
• (3) optimizing regularization parameters and kernel parameters of a VVRKFA mathematical model by using Quantum Particle Swarm Optimization (QPSO) on the basis of the training sample set and constructing a VVRKFA-based fault diagnosis model;
• and
• (4) inputting the test sample set into the constructed VVRKFA-based fault diagnosis model to identify a circuit fault class;
• wherein in Step (3), the VVRKFA-based fault diagnosis model is constructed as follows:
• (3.a) determining a type of a kernel function:
• selecting a Gaussian kernel function K(xi,xj)=exp(σ∥xi−xj∥2) as the kernel function of VVRKFA to establish the VVRKFA mathematical model, wherein σ is a width factor of the Gaussian kernel function;
• and
• the VVRKFA mathematical model is as follows:
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• where, Θ∈N×m is a regression coefficient matrix, for the purpose of mapping feature inner product space to label space;
• N is a number of fault classes;
• m is a dimension of a training sample in the training sample set;
• m is the dimension of the sample in a dimension-reduced sample data set after a dimension reduction of the training sample set;
• b∈N×1 is a base vector;
• C is a regularization parameter;
• ξi∈N×1 is a slack variable;
• a matrix B∈m×n is the dimension-reduced sample data set (m≤m) after the dimension reduction of the training sample set;
• n is a total number of all training samples in the training sample set;
• K(.,.) is a kernel function;
• xi is a training sample in the training sample set;
• Yi a is a class label vector of the training sample xi;
• T is a matrix transpose symbol;
• J is a target function name;
• [Θ b]=YPT[CI+PPT]−1, where P=[k(A,BT)e]T∈(m+1)×m);
• A∈m×n is a training sample data set;
• I is a unit matrix, and Y∈N×m is a matrix containing a number m of class label vectors of training samples;
• (3.b) optimizing the VVRKFA mathematical model by using the QPSO algorithm to obtain an optimal regularization parameter and an optimal kernel parameter of the VVRKFA mathematical model;
• (3.c) with the training sample xi in the training sample set as input data, constructing the following vector value mapping function on the basis of the optimal regularization parameter and the optimal kernel parameter obtained in Step (3.b):
• p(xi)=ΘK(xiT,BT)T+b;
• (2)
• where, Θ∈N×{circumflex over (m)} is a regression coefficient matrix, and B∈m×n is a dimension-reduced sample data set after dimension reduction of the training sample set;
• K (.,.) is a kernel function;
• in Equation (2), the kernel function K (.,.) is the optimal kernel function obtained in Step (3.b);
• C is the regularization parameter, and in Equation (2), the regularization parameter C is the optimal kernel parameter obtained in Step (3.b);
• b ∈N×1 is a base vector, and N is the number of fault classes;
• m is the dimension of a training sample in the training sample set;
• m is the dimension of a sample in the dimension-reduced sample data set after the dimension reduction of the training sample set;
• and n is the total number of all training samples in the training sample set;
• and
• (3.d) using the mapping function constructed in Step (3.c) to establish a VVRKFA decision function which is expressed as:
• Class(x)=argmin1≤j≤NdM({circumflex over (p)}(xt),{circumflex over (p)}(j))|{circumflex over (Σ)});
• (3)
• where, xt is a test sample in the test sample set;
• p(j)=(1/nj){circumflex over (Σ)}i=1nj{circumflex over (p)}(xi), is the center point of all training samples of the j-th class fault in the training sample set, dM represents a Mahalanobis distance, xi is a training sample in the training sample set, {circumflex over (Σ)}=Σj=1N(nj−1){circumflex over (Σ)}(j)/(n−N) is an intra-class covariance matrix, and n is the total number of training samples in the training sample set;
• N is the number of fault classes;
• nj is the number of training samples of the j-th class fault in the training sample set;
• {circumflex over (p)}(xi) is a mapped projection of the training sample xi in the feature subspace;
• and {circumflex over (p)}(xi) is a mapped projection of the test sample in the feature subspace;
• wherein the establishment of the decision function indicates the completion of the construction of the VVRKFA-based fault diagnosis model.