# Identification Strategy Design with the Solution of Wavelet Singular Spectral Entropy Algorithm for the Aerodynamic System Instability

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Definition of Information Entropy Algorithm

_{1}, a

_{2},⋯, a

_{m}} be the m dimensional sampled random signal, in which each a

_{j}(j = 1, 2,⋯, m) can treated as a state mode. Thus, it may contain m sorts of possible state modes in set A. Appropriately, define the dataset P = {p

_{1}, p

_{2},⋯, p

_{m}}, where p

_{j}(j = 1, 2,⋯, m) denotes the variable of probability distribution corresponding to these state modes. The probability distribution value p

_{j}(j = 1, 2,⋯, m) must satisfy the condition with $0\le {p}_{j}\le 1\cup {\displaystyle \mathit{\sum}_{i=1}^{n}{p}_{j}}$. Then the information contained in the possible state patterns in set A is noted as, I(a

_{i}) = −ln p

_{j}, j = 1, 2, …, m.

## 3. Methodology of Wavelet Singular Spectral Entropy Algorithm

#### 3.1. The Wavelet Transform

_{i}and the high-frequency part D

_{i}. If required for a further solution, a further effort can be made to decompose the low-frequency part of the previous into the low and high parts. With the procedure repeated, the wavelet coefficient reconstruction matrix needed can be constituted by the low-frequency parts and the high-frequency parts. Then, the decomposition relation of the system signal is obtained with x(n) = D

_{1}+ D

_{2}+ … + D

_{k}+ A

_{k}. The m-order wavelet coefficient reconstruction matrix is obtained under the k-scale.

#### 3.2. Singular Value Decomposition

^{T}is the transpose of the h × m orthogonal matrix. The elements λ

_{j}(λ

_{j}≥ 0, j = 1, 2,⋯, h) in the main diagonal of the matrix Σ are called the singular eigenvalues of the matrix W. This kind of solution is named the singular value decomposition of the matrix.

#### 3.3. Wavelet Singular Spectral Entropy

- (1)
- The processing of the signal sequence x(1), x(2), …, x(n) is conducted by wavelet transform. The wavelet basis function is selected for scale decomposition to obtain the wavelet coefficient reconstruction matrix W
_{(k+1),n}; - (2)
- The singular eigenvalue matrix Σ is achieved by the use of singular value decomposition theory with the wavelet coefficient reconstruction matrix. Then the elements λ
_{j}(λ_{j}≥ 0, j = 1, 2, …, h) are obtained on the main diagonal of matrix Σ; - (3)
- Normalize the singular eigenvalues obtained in step (2), which is defined as Equation (7):$${p}_{m}=\frac{{\lambda}_{m}}{{\displaystyle \mathit{\sum}_{i=1}^{r}{\lambda}_{i}}}\left(r\le h\right),$$
- (4)
- Finally, the wavelet singular spectral entropy of the discrete sequence x(1), x(2), …, x(n) is calculated, as shown in Equation (8):$$WS{E}_{r}={\displaystyle \mathit{\sum}_{m=1}^{r}{p}_{m}}\mathrm{ln}{p}_{m}\left(r\le h\right),$$
_{m}satisfies the condition of $0\le {p}_{m}\le 1$, $\mathit{\sum}_{m=1}^{r}{p}_{m}}=1$.

## 4. Data Acquisition Introduction in Experiment

^{5}in the rotor tip chord. A detailed description of the test rig is introduced in the literature [27]. The relevant geometric parameters of the research object are listed in Table 1 for a brief introduction.

## 5. Analysis on the Spatial Mode of Flow Field

#### 5.1. Feature Extraction of Spatial Mode in Time-Domain

_{j}indicates the flow coefficient computed from the signal of the j-th dynamic pressure sensor.

#### 5.2. Feature Analysis of Spatial Mode with Time-Frequency Information Extraction

## 6. Identification Strategy Design with Recognition Algorithms

#### 6.1. Parameter Selection on Wavelet Singular Spectral Entropy

#### 6.2. Wavelet Singular Mixed Entropy Identification Algorithm

## 7. Discussion

## 8. Conclusions

- (1)
- On the basis of time-frequency analysis of the first-order spatial mode, it is concluded that the essential characteristic of the system is well reflected during the gradual process of stall development. The information of spatial modal amplitude can be used as a key parameter for the identification of the occurrence of compressor system instability. By extending the information entropy algorithm, the wavelet singular spectral mixed entropy algorithm is proposed to integrate wavelet transform analysis into the time-frequency localization with singular value decomposition. The results show that the wavelet singular algorithm is very sensitive to small disturbances during the rise of amplitude;
- (2)
- The WSE represents the complexity of the spatial mode of the flow field in the time-frequency domain framework. Combined with the description of spatial mode, the recognition capacity of the WSE is displayed well for the detection of stall inception in a compressor. Compared with the single information entropy algorithm, the wavelet singular mixed entropy algorithm can better extract the uncertain information contained in the system during the instability process. The wavelet singular spectral indicates an obvious mutation when the system approaches the instability boundary;
- (3)
- In the data processing of instability signals under different speeds, the wavelet singular spectral entropy algorithm shows the ability for early warning of stall inception. According to the threshold set for identification, the wavelet singular mixed entropy algorithm can detect the stall precursor about 23~96 r in advance, which verifies its validity and feasibility for the identification of system instability. Compared to the single information entropy algorithm and related work, the hybrid entropy algorithm is able to shift to an earlier precursor identification by about 11~82 r.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

SVD | Singular Value Decomposition |

WSE | Wave Singular Spectral Entropy |

IEn | Information Entropy |

WT | Wavelet Transform |

DFT | Discrete Fourier Transform |

REV | Revolution |

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**Figure 2.**Amplitudes of spatial mode with pre-sampling and post-sampling: (

**a**) Amplitudes at 2500 rpm; (

**b**) Amplitudes at 3000 rpm.

**Figure 3.**Time-frequency diagram of the first-order spatial mode: (

**a**) Diagram at 2500 rpm; (

**b**) Diagram at 3000 rpm.

**Figure 4.**The effect of sliding window width selection on wavelet singular spectral entropy: (

**a**) Wavelet singular spectral entropy at 2500 rpm; (

**b**) Wavelet singular spectral entropy at 3000 rpm.

**Figure 5.**Variation of singular value in sliding window: (

**a**) Singular value at 2500 rpm; (

**b**) Singular value at 3000 rpm.

**Figure 6.**Wavelet singular spectral entropy of the spatial mode at 2500 rpm: (

**a**) First-order space mode; (

**b**) Wavelet singular spectral entropy.

**Figure 7.**Wavelet singular spectral entropy of the spatial mode at 3000 rpm: (

**a**) First-order space mode; (

**b**) Wavelet singular spectral entropy.

**Figure 8.**Information entropy of the spatial mode at 2500 rpm: (

**a**) First-order space mode; (

**b**) Information entropy.

**Figure 9.**Information entropy of the spatial mode at 3000 rpm: (

**a**) First-order space mode; (

**b**) Information entropy.

Parameter | Value |
---|---|

Design speed n (rpm) | 3000 |

Outer diameter D (mm) | 450 |

Blade height h (mm) | 56 |

Tip speed (m/s) | 70.7 |

Hub-tip ratio | 0.75 |

Rotor blade number | 19 |

Stator blade number | 13 |

Speed | 2500 rpm | 3000 rpm |
---|---|---|

Stall revolution | 998.75 r | 1732 r |

IEn pre-stall revolution | 986.67 r | 1718 r |

WSE pre-stall revolution | 975 r | 1636 r |

REV before stall in IEn | 12.08 r | 14 r |

REV before stall in WSE | 23.75 r | 96 r |

Work | Approach | Method | Pre-Stall REV at 2500 rpm | Pre-Stall REV at 3000 rpm |
---|---|---|---|---|

[12] | Sample Entropy | Time Domain Analysis | 12.083 r | 60 r |

Global-Sample Entropy | 12.083 r | 68 r | ||

Proposed Work | Wavelet Singular Entropy | Frequency-Time Domain Analysis | 23.75 r | 96 r |

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**MDPI and ACS Style**

Zhang, M.; Kong, P.; Hou, A.; Xia, A.; Tuo, W.; Lv, Y.
Identification Strategy Design with the Solution of Wavelet Singular Spectral Entropy Algorithm for the Aerodynamic System Instability. *Aerospace* **2022**, *9*, 320.
https://doi.org/10.3390/aerospace9060320

**AMA Style**

Zhang M, Kong P, Hou A, Xia A, Tuo W, Lv Y.
Identification Strategy Design with the Solution of Wavelet Singular Spectral Entropy Algorithm for the Aerodynamic System Instability. *Aerospace*. 2022; 9(6):320.
https://doi.org/10.3390/aerospace9060320

**Chicago/Turabian Style**

Zhang, Mingming, Pan Kong, Anping Hou, Aiguo Xia, Wei Tuo, and Yongzhao Lv.
2022. "Identification Strategy Design with the Solution of Wavelet Singular Spectral Entropy Algorithm for the Aerodynamic System Instability" *Aerospace* 9, no. 6: 320.
https://doi.org/10.3390/aerospace9060320