# Fault Diagnosis of an Analog Circuit Based on Hierarchical DVS

^{*}

## Abstract

**:**

## 1. Introduction

## 2. IDVS–CNN Algorithm Principle

#### 2.1. IDVS Algorithm

#### 2.1.1. Hierarchical DVS Model

_{r}is the kernel function of the r-order term, u is the input term, and the r-order term has r accumulation layers. Based on the hierarchical structure shown in Figure 1, the memory parameters are merged to realize the HD of the DVS. The IDVS model with only basic coefficients is obtained [18]:

#### 2.1.2. Complexity of the Hierarchical DVS

_{DVS}multiplication [18]:

_{r}, the IDVS directly provides the parameter calculation, i.e., Equation (4), of each order and the relationship, i.e., Equation (5), between the parameters of each order. To calculate the parameters of each order, we only need to carry out the continuous multiplication and single-layer accumulation operation on the foundation coefficients [a

_{i}, b

_{i}, c

_{i}, d

_{i}] of each layer. However, the DVS does not have an explicit parameter calculation equation, so the input matrix needs to be constructed during calculation. The matrix needs to be reconstructed every time an order is added, and the dimension of the matrix increases as the order of the model increases, which is not conducive to the implementation of the calculation program.

#### 2.1.3. LM Optimization of the Hierarchical DVS

^{T}, where a = [a

_{1}a

_{2}…a

_{M}]

^{T}, b = [b

_{1}b

_{2}…b

_{M}]

^{T}, c = [c

_{1}c

_{2}…c

_{M}]

^{T}, and d = [d

_{1}d

_{2}…d

_{M}]

^{T}.

_{opt}, we introduced the LM optimization strategy. The LM algorithm searches for the best coefficient through the following iteration:

_{i}= a

_{i}+c

_{i}u(k-i), i ≥ 2, and μ

_{1}= [u(k) y

_{1}(k) y

_{2}(k)…y

_{M-1}(k)].

_{2}= [u(k-1) u(k-2)…u(k-M)], y (k) represents the total response of the IDVS, and y

_{i}(k) represents the response of the IDVS in the i-th stage.

- (i)
- Initialize the adjustment factor λ, coefficient θ
_{1}^{(0)}, input matrices μ_{1}and μ_{2}, and Jacobian matrix J. - (ii)
- Calculate the trial iteration step size step_try using Equation (19).$$step={(J{(\theta )}^{T}J(\theta )+\lambda I)}^{-1}J{(\theta )}^{T}(\widehat{y}(\theta )-y)$$
- (iii)
- Calculate the trial coefficient θ_try = θ-step_try.
- (iv)
- Put θ_try substituting Equations (2) and (3) to obtain y_hat, and calculate the chi-square error χ
^{2}_try.$${\chi}^{2}=\frac{1}{2}{\displaystyle \sum _{k=1}^{q}{(\widehat{y}(k)-y(k))}^{2}}$$ - (v)
- Calculate the judgment factor ρ using Equation (21).$$\rho =\frac{{\chi}^{2}(\theta )-{\chi}^{2}(\theta +step)}{ste{p}^{T}({\lambda}_{i}step+J{(\theta )}^{T}(\widehat{y}(\theta )-y))}$$
- (vi)
- If ρ is greater than 0.01, update the adjustment factor using Equation (22) and accept θ_try, χ
^{2}_try and update the iteration matrix J; otherwise, update the adjustment factor using Equation (23).$${\lambda}_{i+1}={\lambda}_{i}\mathrm{max}\left[1/3,1-{\left(2{\rho}_{i}-1\right)}^{3}\right];{\upsilon}_{i}=2$$$${\lambda}_{i+1}={\lambda}_{i}{\upsilon}_{i};{\upsilon}_{i+1}=2{\upsilon}_{i}$$ - (vii)
- Repeat steps ii–vi until the mean squared error (MSE) meets the set value or the iteration number reaches the upper limit; finally, the optimal parameters θ
_{opt}are obtained.

#### 2.1.4. Feature Extraction

_{opt}was obtained using the method described in Section 2.1.3. We then substituted this into Equations (4) and (5) to obtain the first-order kernel h

_{1}and the rest kernel h

_{i}(i = 2, …, r). [h

_{1}, h

_{2}, …, h

_{r}] was considered the eigenvector of the circuit.

#### 2.1.5. Order Selection of the Hierarchical DVS

^{2}_hat is the likelihood function, k = (M+1)!/((r-1)!(M+1-r)!). The lower the BIC value, the better the model of the corresponding order. When the order r increases, k increases, and the likelihood function increases. There is a range of r values to minimize the BIC.

#### 2.2. CNN Algorithm

## 3. IDVS–CNN Diagnostic Procedures

- (i)
- Excitation u (n) is applied to the circuit and u (n) is saved, and the sample set of the circuit response in the normal state and the fault state [normal, fault_1, … fault_N] are collected.
- (ii)
- The IDVS algorithm is used to calculate the characteristic parameter set [h(NF), h(F1), …, h(Fi), …, h(FN)] of the circuit in each state; h (Fi) is the eigenvector of the circuit state I. r and M are set appropriately and the dimension of the parameters through PCA is reduced.
- (iii)
- The feature parameter set is used as the input to the CNN to build a diagnosis model, where the fault code is set as [normal, fault_1, …, fault_N] = [0, 1, 2 …, N], and k is set to the appropriate value.
- (iv)
- After calculating the IDVS characteristic parameter h(Fx) of the circuit response data under test, PCA is applied to reduce the dimension.
- (v)
- The characteristic parameters are input to the CNN model established in step (iii), and the corresponding fault mode code of the circuit under test is obtained.

## 4. Simulation Verification of the Improved Algorithm

#### 4.1. Simulated Hardware Equipment

- CPU: Core i5-2450m 2.5 GHz.
- RAM memory: 4 G DDR3 1600.
- Hard disk: 1 TB.
- Software platform: MATLAB 2014B, PSpice 16.6.

#### 4.2. Fault Mode

#### 4.3. Diagnostic Test Process

#### 4.4. IDVS Order Selection

#### 4.5. LM Optimization of Coefficients

#### 4.6. Calculation of the Characteristic Parameters

_{1}–h

_{4}(Data S6–S9) of orders 1–4 were obtained using the parameter calculations in Equations (4)–(9) of the IDVS (Codes S2–S5), as shown in Figure 7a–d. By expanding the parameters of each order into a vector and combining them into a vector, we obtained the eigenvector [h

_{1}h

_{2}h

_{3}h

_{4}] (Data S10), as shown in Figure 8.

_{1}h

_{2}], and eigenvector [h

_{1}h

_{2}h

_{3}h

_{4}] were compared (Data S11–S13), as shown in Figure 9a–c. In Figure 9a, the primary components of each fault mode overlap considerably. In Figure 9b, the fault modes R1↑, R2↑, R1↓, and R2↑ have distinct principal components, and the capacitive faults C1↑, C2↑, C1↓, and C2↓ still have a high overlap. In Figure 9c, the capacitive faults are further separated.

#### 4.7. Distance Calculation

_{1}h

_{2}h

_{3}h

_{4}] were calculated using (22) of the CNN algorithm. Figure 10 shows a comparison of the eigenvector [h

_{1}h

_{2}h

_{3}h

_{4}] with the original data y_dat and commonly used eigenvector [h

_{1}h

_{2}].

_{1}h

_{2}h

_{3}h

_{4}] was used as the eigenvector, the average between-class distance $\overline{\mathrm{d}3}$_btw of the corresponding mode was 0.5, while the average inter-class distance $\overline{\mathrm{d}3}$_in was approximately 0.16, $\overline{\mathrm{d}3}$_btw > $\overline{\mathrm{d}2}$_btw(0.39) > $\overline{\mathrm{d}1}$_btw(0.22) (Data S14–S16), and $\overline{\mathrm{d}3}$_in ≈ $\overline{\mathrm{d}2}$_in > $\overline{\mathrm{d}1}$_in(0.01) (Data S17–S19). The difference Δ d between the between-class distance and the inter-class distance determines the classification performance of the eigenvector. Because Δ d (3) > Δ d (2) > Δ d (1), the eigenvector [h

_{1}h

_{2}h

_{3}h

_{4}] has a better classification performance.

#### 4.8. Comparison of the Diagnostic Effects

#### 4.9. Real Circuits Experiment

_{B1}↑, R

_{B1}↓, R

_{B21}↑, R

_{B21}↓, RF↑, RF↓, RL↑, RL↓) are 87%, 85%, 82%, 88%, 89%, 88%, 89%, and 85%, respectively. This further demonstrates the effectiveness of our method in engineering applications.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Fault diagnosis implementation: Block diagram of the improved discrete Volterra series (IDVS)–condensed nearest neighbor algorithm (CNN).

**Figure 6.**Coefficient optimization comparison. (

**a**) Output estimation of the least-squares (LS) algorithm; (

**b**) output estimation of the Levenberg–Marquardt (LM) algorithm; (

**c**) mean square error (MSE) comparison of the LS and LM algorithms.

**Figure 7.**Comparison of the characteristic parameters of each state. (

**a**) h

_{1}; (

**b**) h

_{2}; (

**c**) h

_{3}; (

**d**) h

_{4}. Note: h

_{4}is the result of five-dimensional data h

_{4}(m

_{1}, m

_{2}, m

_{3}, m

_{4}) expanded into a vector.

**Figure 9.**Principal component comparison of the eigenvectors. (

**a**) Principal component of the raw data; (

**b**) principal components of the DVS eigenvectors (r = 2); (

**c**) principal component of the IDVS eigenvectors (r = 4).

**Figure 10.**Comparison of the inter- and between-class distances. Note: d1, d2, and d3 are the distances of the original signal, feature [h

_{1}h

_{2}], and feature [h

_{1}h

_{2}h

_{3}h

_{4}] of fault modes NF–F8, respectively.

**Figure 11.**Diagnosis results of the BLPF circuit: (

**a**) CNN algorithm; (

**b**) DVS–CNN algorithm; (

**c**) IDVS–CNN algorithm.

**Figure 12.**Integrated operational amplifier (µA741). Note: The left is a physical picture of the circuit, and the right is an equivalent circuit diagram.

**Figure 14.**Continuous-time state variable filter (CTSVF) circuit. (

**a**) Schematic diagram; (

**b**) test board.

CNN | DVS–CNN | IDVS–CNN | |
---|---|---|---|

S1 | 67% | 72% | 94% |

S2 | 62% | 69% | 97% |

S3 | 23% | 15% | 2.4% |

MacroF1 | 0.544 | 0.607 | 0.903 |

MicroF1 | 0.541 | 0.624 | 0.894 |

Time cost | 11.2 s | 15.2 s | 13.5 s |

Method | Mean Accuracy (%) | Mean Time Cost (s) | ||
---|---|---|---|---|

SNR = 30 dB | SNR = 20 dB | SNR = 10 dB | ||

Reference [28] | 88 | 77 | 60 | 13 |

Reference [29] | 87 | 78 | 62 | 12 |

Reference [30] | 91 | 82 | 65 | 15 |

IDVS–CNN | 92 | 85 | 76 | 12 |

NF | R1↑ | R1↓ | R3↑ | R3↓ | C2↑ | C2↓ | |
---|---|---|---|---|---|---|---|

NF | 46 | 1 | - | - | 1 | 1 | 1 |

R1↑ | - | 49 | - | - | - | 1 | - |

R1↓ | 1 | - | 45 | - | - | 2 | 2 |

R3↑ | - | - | - | 45 | - | 3 | 2 |

R3↓ | - | - | - | - | 47 | 2 | 1 |

C2↑ | 1 | - | - | - | 1 | 44 | 4 |

C2↓ | 1 | - | - | - | - | 5 | 44 |

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**MDPI and ACS Style**

Deng, Y.; Zhou, Y.
Fault Diagnosis of an Analog Circuit Based on Hierarchical DVS. *Symmetry* **2020**, *12*, 1901.
https://doi.org/10.3390/sym12111901

**AMA Style**

Deng Y, Zhou Y.
Fault Diagnosis of an Analog Circuit Based on Hierarchical DVS. *Symmetry*. 2020; 12(11):1901.
https://doi.org/10.3390/sym12111901

**Chicago/Turabian Style**

Deng, Yong, and Yuhao Zhou.
2020. "Fault Diagnosis of an Analog Circuit Based on Hierarchical DVS" *Symmetry* 12, no. 11: 1901.
https://doi.org/10.3390/sym12111901