Condition Monitoring of Marine Diesel Lubrication System Based on an Optimized Random Singular Value Decomposition Model

Abstract

As modern marine diesel engine systems become increasingly complex, effective condition monitoring methods are essential for ensuring optimal performance and preventing anomalies. This paper proposes a data-driven condition monitoring approach specifically designed for the lubrication system of marine diesel engines. Unlike traditional methods, the proposed approach eliminates the need for explicit modeling and leverages a novel optimization algorithm for data denoising. Additionally, a new noise-resistant monitoring index is introduced to enhance monitoring reliability. The paper is structured into two main sections for validation. The first section addresses advanced data preprocessing, where the Improved Sparrow Search Algorithm (ISSA) is employed to optimize the parameters of Random Singular Value Decomposition (RSVD). This step effectively minimizes noise, reduces manual intervention, and handles high-dimensional data. The second section focuses on analyzing the data characteristics using the Random Matrix Theory (RMT) and establishing novel condition monitoring indicators to achieve more reliable monitoring outcomes. The proposed methodology captures the intricate relationships among key variables within the system, providing a more robust framework for condition monitoring. Applied to a marine diesel engine lubrication system, the method demonstrates significant improvements in noise immunity and monitoring reliability. Comparative analyses of condition monitoring models before and after denoising reveal that the relative error of the proposed monitoring index under varying noise amplitudes is within 1%, substantially lower than that of other indices. Furthermore, the monitoring accuracy is improved by 4.95% when the proposed index is employed for system condition monitoring.

1. Introduction

With the continuous advancement of intelligent technologies in modern ships, the complexity and demand for maintenance in marine systems have significantly increased. Among these systems, monitoring data from marine diesel engines have become increasingly diverse and precise, enabling more effective condition analysis. As a critical subsystem of diesel engines, the lubrication system plays a pivotal role in ensuring stable operation. It supplies sufficient and high-quality lubricating oil to moving components, transforms dry friction into liquid friction, and reduces wear. Furthermore, it dissipates heat generated by friction, removes wear debris, and provides auxiliary functions such as corrosion prevention, noise reduction, and sealing [1]. Leveraging diverse monitoring data to assess the operational status of the lubrication system has become a prominent focus in research on accurate condition monitoring and predictive maintenance.

Current research on diesel engine lubrication systems primarily focuses on three key areas: the impact of lubricating oil on the wear rate of diesel engine mechanical structures, the influence of impurities in the oil on lubrication efficiency, and the effects of parameters such as temperature and pressure within the lubrication system on its overall performance. For example, ref. [2] developed a lubrication model for the piston ring pack in two-stroke marine diesel engines, focusing on an independent oil-feed system. It analyzed the effects of oil-feed parameters on lubrication performance and optimized the characteristics using orthogonal analysis. Similarly, ref. [3] investigated the effects of lubricating oil on the axial motion and crankshaft bearing lubrication efficiency of marine diesel engines by examining parameters such as oil film pressure and the friction power of the oil film formed on mechanical components. Furthermore, ref. [4] evaluated the reliability of lubricating oil by analyzing its chemical properties, including viscosity and alkalinity, and proposed methods for monitoring and maintaining the state of crankcase lubricating oil.

While these studies have advanced the localized monitoring of lubrication system performance, they have notable limitations in comprehensively assessing the system’s overall performance. As highlighted in [5], the pressure parameter of lubricating oil serves as a critical indicator for fault diagnosis in lubrication systems. This parameter reflects the operation of the entire lubrication system, underscoring the need for further research into developing a comprehensive state monitoring model for marine diesel engine lubrication systems from this perspective.

Achieving comprehensive condition monitoring of diesel engine lubrication systems requires the acquisition of parameters from key nodes distributed throughout the entire lubrication system. The integration of these parameters enables the construction of a large-scale condition monitoring data space. As a result, condition monitoring methods need to effectively handle high-dimensional data. While increasing the dimensionality of data unavoidably adds computational complexity, high-dimensional data offer unique spatiotemporal correlations that enhance representational capacity. These data encapsulate richer statistical information and demonstrate greater robustness to issues such as data loss or anomalies. Furthermore, their statistical properties are often more stable when compared to those of low-dimensional data [6].

In current research, condition monitoring methods for high-dimensional data can generally be classified into two categories: random matrix theory (RMT)-based approaches rooted in high-dimensional statistics, and deep learning-based approaches driven by artificial intelligence (AI). Both methods effectively analyze high-dimensional data, but each has distinct strengths. RMT-based methods excel in mathematical analytical capabilities and require minimal dependence on specific models, making them well-suited for theoretical analysis. On the other hand, AI-based methods leverage data modeling and are particularly advantageous for complex data pattern recognition and predictive analysis.

When utilizing high-dimensional data to conduct the condition monitoring of marine diesel engine lubrication systems, the system’s complexity often makes it challenging to achieve the desired outcomes through conventional modeling approaches. To overcome this dependence on models, RMT can be employed to integrate various types of data into high-dimensional matrices without requiring explicit system modeling [7]. Moreover, RMT allows for the retention of spatio-temporal correlations among the original data while analyzing matrix characteristics and data distributions from probabilistic and statistical perspectives [8]. This provides a foundation for selecting appropriate indicators for monitoring system conditions.

For instance, ref. [9] proposes a perimeter security intrusion detection method using an ultra-weak fiber Bragg grating array combined with high-dimensional random matrix analysis. Optical interference signals are processed into random matrices, and statistical characteristics like the Marcenko-Pastur law(M-P Law) and the Single Ring Theorem are used to detect events efficiently. Similarly, ref. [10] applied RMT to the condition monitoring of shafting systems, introducing an RMT-based eigenvalue difference metric as an indicator for monitoring the shafting operational state.

While RMT has primarily been applied in power grid systems, where parameters such as voltage, current, and power are collected for condition assessment, the unique mechanical structures in lubrication systems introduce significant challenges, particularly due to noise. In systems like marine diesel engines, which are heavily affected by noise pollution, it is crucial to consider the impact of noise on the effectiveness of condition monitoring. This necessitates further research into robust techniques for noise handling within the RMT framework to ensure accurate assessments in noisy environments.

Among common high-dimensional matrix denoising methods such as singular value decomposition (SVD), principal component analysis (PCA), and sparse coding (SC), SVD stands out for its broader applicability, better mathematical interpretability, and lower dependence on specific models. Consequently, many studies have adopted SVD for denoising tasks. For example, ref. [11] proposes a novel unsupervised criterion based on SVD-entropy for feature selection, which evaluates the contribution to the entropy (CE) of each feature. The criterion calculates the clustering success rate in the selected feature space using a leave-one-out approach, providing a robust evaluation of feature relevance. However, this approach performs poorly when dealing with high-dimensional data and imposes high hardware requirements for processing massive datasets, limiting its practical applicability. The emergence of Randomised Singular Value Decomposition (RSVD) is a good solution to the trouble caused by high-dimensional data; RSVD can be used to compute the low-rank approximation of matrices [12], in addition its stochastic projection algorithm can be used for dimensionality reduction [13]. For example, ref. [14] presents a model-free voltage stability assessment method using RSVD and Phasor Measurement Units (PMUs). It identifies weak buses, ensures data linearity, and performs efficient online stability analysis. The method is validated on IEEE 14-bus and 118-bus systems, with real-time feasibility confirmed through simulation. Nevertheless, RSVD relies on manually set oversampling parameters, which directly influence its precision.

This study seeks to mitigate the impact of noise on condition monitoring of lubrication systems by employing an Improved Sparrow Search Algorithm (ISSA) [15] to optimize the oversampling parameters of Randomized Singular Value Decomposition (RSVD), thereby enabling the preliminary denoising of high-dimensional random matrices. Leveraging the Marchenko–Pastur (M-P) Law and the Single Ring Theorem from Random Matrix Theory (RMT), a qualitative analysis of the monitoring data from lubrication systems is conducted. Additionally, a novel linear eigenvalue statistic (LES) with superior denoising performance is proposed to replace the traditionally utilized maximum and minimum eigenvalue statistics in RMT, providing an enhanced condition monitoring indicator for lubrication systems. The proposed methodology is applied to a specific type of diesel engine lubrication system, with experimental results demonstrating the effectiveness of the proposed state monitoring approach.

The remainder of the paper is organized as follows: Section 2 outlines the principles of ISSA optimization of RSVD, along with an introduction to the M-P Law and SRT in the context of RMT. Section 3 details the construction of the proposed LES with anti-noise characteristics in a noisy environment, supported by mathematical formulations. Section 4 presents a validation of the proposed method through an analysis of abnormal data from the sliding oil pump in the lubrication system of a specific diesel engine. Finally, Section 5 concludes the study and provides a discussion of the potential directions for future research.

2. Materials and Methods

This study proposes a method to optimize the RSVD parameter settings using the ISSA to address the issue of low accuracy caused by manual parameter tuning in RSVD. A data-driven RMT approach is employed to assess the condition of the lubrication system, and a noise-resistant metric, FE, is established during the quantitative analysis process as a system condition evaluation indicator. The research process is as follows in Figure 1.

2.1. Improved Sparrow Search Algorithm for Optimizing RSVD

2.1.1. RSVD Algorithm

The study in [16] provides an in-depth analysis of the application of sparse sketch matrices and random projection techniques for SVD. The random projection method effectively mitigates the computational challenges associated with SVD on high-dimensional data by reducing its dimensionality, while simultaneously preserving the intrinsic characteristics of the original dataset. The random projection is shown in Equation (1).

(1)$\tilde{H}=H\mathsf{\Psi }$

where $\mathsf{\Psi }$ is a random matrix of dimension $N\times P$, the matrix elements obey the random Gaussian distribution of $N(0,1)$, P satisfies $K\ll P\ll N$, K is the estimated number of signal components, that is, the oversampling parameter to be artificially set, and N is the number of columns of H.

In the context of RMT theory, it is considered that the singular values in the diagonal matrix $\mathsf{\Lambda }$ obtained from the SVD decomposition that satisfy Equation (2) characterize the noise, and Equation (2) is added as a threshold to the RSVD process, and the singular values that satisfy the threshold are set to zero to form the new diagonal matrix $\mathsf{\Lambda }$ and reconstruct the matrix after initial denoising.

(2)${\sigma }_j<\sqrt{{\lambda }_{max}}$

where $\mathsf{\Lambda }=diag({\sigma }_1,{\sigma }_2,\cdots ,{\sigma }_r),{\sigma }_1⩾{\sigma }_2⩾\cdots ⩾{\sigma }_1>0,{\sigma }_{1=(1,2,\cdots ,r)},{\lambda }_{max}$ is the maximum eigenvalue of the sample covariance matrix of the random matrix.

2.1.2. Improved Sparrow Search Algorithm

This paper introduces ISSA to optimize the oversampling parameters in RSVD, addressing the challenges associated with parameter settings. The Tent chaotic mapping function enhances the sparrow search algorithm by leveraging its uniform distribution, irreducibility, and unpredictability characteristics. Data generated by the Tent mapping function serve as the initial position information for the sparrow population, which helps preserve the diversity of the population’s search and enables sparrow individuals to escape local optima during the search process. This enhancement improves both the convergence speed and global search capability of the sparrow search algorithm [17,18]. The corresponding flowchart is illustrated in Figure 2.

The Tent chaotic mapping function, also referred to as the Tent mapping function, generates uniform chaotic sequences within the interval [0, 1] and features a fast iteration rate [19]. The mathematical representation of the Tent mapping function is provided in Equation (3).

(3)$x_{i+1}=\left\{\begin{array}{cc}2x\hfill & 0\le x_i\le 1/2\hfill \\ 2(1-x_i)\hfill & 1/2\le x_i\le 1\hfill \end{array}\right.$

Since the Tent chaotic sequence iteration may fall into small and unstable cycle points, it is necessary to add a random variable to the initial Tent chaotic mapping function. In addition, the Tent mapping function needs to undergo the Bernoulli shift transformation to enhance the expressive power of the model and improve the convergence speed and stability of the model. The transformed formula is shown in Equation (4).

(4)$x_{i+1}=\left(2x_i\right)mod1+rand(0,1)\times {\displaystyle \frac{1}{N}}$

In the context of the entire algorithm, the finder in the sparrow population is more inclined to search for food, resulting in superior fitness values and positions closer to the optimal solution within the entire solution space. During each iteration of the search process, the finder updates its position based on the equations provided in Equation (5).

(5)$X_{i,j}(t+1)=\left\{\begin{array}{cc}{X}_{i,j}\left(t\right)·exp(-{\displaystyle \frac{i}{\alpha ·t_{max}}})\hfill & if R<ST\hfill \\ X_{i,j}\left(t\right)+Q·L\hfill & if R\ge ST\hfill \end{array}\right.$

where $X_{i,j}$ denotes the location information of the ith sparrow in the jth dimension in the solution space; t is the current number of iterations of the algorithm, and $t_{max}$ denotes the maximum number of iterations; $\alpha$ is a random value of (0, 1]; R is in the range of [0, 1], which denotes the warning value of the sparrow individual in the algorithm that sends out a chirping call when it encounters danger; $ST$ is in the range of [0.5, 1], which denotes the warning value above which an individual sparrow in the algorithm will move to a safe place for foraging; is a random number that satisfies normal distribution; is the dimension, and the elements are all one matrix warning value which exceeds the threshold and will move to a safe place to forage; Q is a random number that satisfies the normal distribution; L is a matrix with dimension $l\times d$ and the elements are all 1.

Once the discoverer completes an exhaustive search for suitable food sources, the followers move to the discoverer’s location. If the foraging attempt is unsuccessful, the individual relocates to an alternative position to resume foraging, thereby improving fitness values and expanding exploration into previously unsearched areas. The formula governing this process is provided in Equation (6).

(6)$X_{i,j}(t+1)=\left\{\begin{array}{cc}Q·exp\left({\displaystyle \frac{X_w-X_{i,j}\left(t\right)}{i^2}}\right)\hfill & \mathit{if} i>n/2\hfill \\ X_{i,j}\left(t\right)+\left|X_{i,j}\left(t\right)-X_p\left(t\right)\right|·A^{*}·L\hfill & \mathit{otherwise}\hfill \end{array}\right.$

where $X_p$ represents the position where the discoverer has the most food, that is, the optimal fitness value; $X_w$ represents the place with the worst fitness value in the current solution space; A is a matrix with dimension $l\times d$ and elements 1 or −1 and satisfies relation $A^{*}=A^T{\left(A{A}^T\right)}^{-1}$.

When a sparrow individual encounters danger during the foraging process, it moves closer to the search circle or to other companions. The method of updating the position of individual sparrows for this process is shown in Equation (7).

(7)$X_{i,j}(t+1)=\left\{\begin{array}{cc}{X}_b\left(t\right)+\beta ·|X_{i,j}\left(t\right)-X_b\left(t\right)|\hfill & if f_i\ne f_b\hfill \\ X_{i,j}\left(t\right)+K·{\displaystyle \frac{|X_{i,j}\left(t\right)-X_w\left(t\right)|}{(f_i-f_w)+\epsilon }}\hfill & if f_i=f_b\hfill \end{array}\right.$

where $X_b$ is the position where the fitness value is optimal in the current iteration global solution space; $\beta$ is a random value that satisfies a normal distribution with mean 0 and variance 1; K indicates the direction of individual sparrow movement; $\epsilon$ is a constant close to zero; $f_i$ is the fitness value of individual sparrow $ith$ in the current iteration; $f_b$ is the optimal fitness value in the global solution space under the current iteration; $f_w$ is the value with the worst fitness value in the global solution space under the current iteration.

The advantages of ISSA are demonstrated by comparing its performance with Particle Swarm Optimization (PSO), Grey Wolf Optimization (GWO) [20], and the original Sparrow Search Algorithm (SSA). This study uses four randomly selected test functions from the 23 public test functions in the CEC2005 benchmark set. Instead of listing all 23 functions, only the four selected functions are summarized in Table 1. The corresponding test results are depicted in Figure 3.

By comparing the convergence speed performance of the four algorithms in the case of the four randomly selected test functions, F1, F3, F7 and F9, it can be found that ISSA has a faster convergence speed and fewer iterations, and this result also provides support for the selection of ISSA for the optimization of the subsequent algorithms.

2.2. Optimizing RSVD Using ISSA

In RSVD, the oversampling parameter determines the number of additional random vectors required to ensure a good approximation and reasonable settings of the oversampling parameter can improve the quality of the low-rank approximation, thus improving the RSVD denoising accuracy. Optimizing the oversampling parameters using ISSA can reduce the unreasonable parameter settings caused by human intervention. The specific optimization steps are as follows:

Step 1: Generate an initial population containing sparrow individuals, each representing a possible oversampling parameter; chaos mapping initialization.

Step 2: RSVD was performed for each individual sparrow and the reconstruction error was calculated.

Step 3: Update the location of individual sparrows according to the sparrow algorithm. Use Tent chaotic mapping to generate chaotic sequences to replace random numbers in traditional SSA to enhance the exploration of the search space.

Step 4: Selection of best-adapted individuals.

Step 5: The follower in the sparrow adjusts its position according to the finder’s position and the chaotic sequence, and the less adapted individual performs a wide search based on the chaotic sequence to avoid falling into a local optimum.

Step 6: Repeat steps 2 to 5 until the best fitness value, which is the optimal oversampling parameter, is obtained.

2.3. Random Matrix Theory

A matrix with random variables as elements is called a random matrix. When the number of rows and columns of the random matrix tends to infinity and the row–column ratio remains constant, the Empirical Spectral Distribution (ESD) function of the random matrix has many excellent properties, such as the M-P Law, Sigle Ring Theorem, and so on. Although theoretically the convergence of random matrices can be satisfied when the matrix dimension tends to infinity, fairly accurate asymptotic convergence results can be observed for matrices of more modest sizes (with dimensions ranging from a few tens to a few hundreds) [21]. This is a prerequisite for RMT to be able to be used for processing practical high-dimensional data. Since RMT is very rich, in this paper, only the M-P Law as well as the Single Ring Theorem from it are chosen for the study.

2.3.1. M-P Law

For an M-dimensional Hermitian matrix Y, with M real eigenvalues, define its empirical spectral distribution function as follows:

(8)$F_M^Y\left(x\right)={\displaystyle \frac{l}{M}}\sum _{{\lambda }^Y=l}^{M}f\left({\lambda }^Y\right)({\lambda }_i^Y\le x)$

where M is the dimension of Y, ${\lambda }_i^Y$ denotes the i-th eigenvalue, $f\left({\lambda }^Y\right)$ denotes the indicator function, and the value of $f\left({\lambda }^Y\right)$ is 1 when the judgment condition is satisfied; otherwise, the value is 0.

However, when using the M-P law for determination, the monitored system data often do not satisfy the properties of the Hermitian matrix. The monitoring data are often $M\times N$ dimensional non-square matrices, for which the matrix needs to be normalized and standardized to obtain a non-square matrix $X_{M\times N}(M<N)$ whose internal elements are independently and identically distributed and satisfy the $\mu \left(x\right)=0,{\sigma }^2\left(x\right)=d$ distribution. The sample covariance matrix is defined as follows:

(9)$S={\displaystyle \frac{d}{N}}X{X}^H$

where d is the variance of matrix X, N is the number of columns of matrix X, and $X^H$ is the conjugate transpose matrix of X. According to the M-P law, the empirical spectral distribution of the sample covariance matrix S satisfies the probability density function:

(10)$P_{\mathrm{MP}}\left(\lambda \right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2\pi c{\sigma }^2\lambda }}\sqrt{({\lambda }_{max}-\lambda )(\lambda -{\lambda }_{min})},\hfill & {\lambda }_{min}\le \lambda \le {\lambda }_{max},\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$

(11)$\left\{\begin{array}{c}{\lambda }_{min}={\sigma }^2{\left(l-\sqrt{c}\right)}^2\hfill \\ {\lambda }_{max}={\sigma }^2{\left(l+\sqrt{c}\right)}^2\hfill \end{array}\right.$

where ${\sigma }^2$ denotes the variance of the noise matrix Q, c denotes the rank ratio of the matrix H, and $\lambda$ is the eigenvalue of the sample covariance matrix S. The noise matrix Q is derived from the noise accompanying the monitoring data [22], and the real monitoring data should be as follows:

(12)$H=H_{clean}+Q$

2.3.2. Single Ring Theorem

For a matrix $X_{M\times N}(M<N)$ whose elements in each row satisfy an independent homogeneous distribution and obey the $\mu \left(x\right)=0,{\sigma }^2\left(x\right)=d$ distribution, the singular value equivalence matrix $X_U$ is obtained by processing it using the Haryu matrix U:

(13)$X_U=U\sqrt{X{X}^H}$

where U is of the same dimension as matrix X. For multiple singular value equivalent matrices constructed at the same time, the matrix product is used in the calculation as the result of the matrix transformation, and the matrix product is defined as follows:

(14)$Z=\prod _{i=1}^L{X}_{U,i}$

where L is the number of singular value equivalent matrices (the case of =1 is considered in this paper to simplify the calculation). The singular value equivalent matrix obtained by transforming the matrix can be mapped to its eigenvalues in the complex plane, i.e., the real part of the complex eigenvalues of the matrix is extracted as well as the imaginary part of the matrix, which satisfies the probability density function when mapped to the complex plane:

(15)$P_{RL}\left(\lambda \right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\pi c{\sigma }^{2}L}}|\lambda {|}^{{\displaystyle \frac{L}{2}}-2},\hfill & {(1-c)}^{{\displaystyle \frac{L}{2}}}\le |\lambda |\le 1,\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$

As demonstrated by Equation (15), the complex eigenvalues of the equivalence matrix of singular values of the monitoring matrix are distributed within a circle with an inner ring radius and an outer ring radius of 1 when the system is in a normal state [23].

3. Construction of the Fusion Eigenvalue Indicator

The M-P Law and the Single Ring Theorem are commonly employed to determine whether the eigenvalues of corresponding matrices conform to specific probability distributions. However, relying on visual inspection of the resulting plots does not allow for the accurate identification of abnormalities in the current system. This observational and statistical judgment method is, therefore, inadequate for the quantitative analysis and evaluation of system states [24]. To address this limitation, the introduction of an LES is necessary to enable quantitative analysis and provide a more precise evaluation of the system state [25].

As shown in Equation (11), the maximum and minimum eigenvalues are typically used as LES to assess the system state. However, it is evident that these LES are influenced by noise, and even after applying RSVD for matrix denoising, the noise cannot be completely eliminated. To further mitigate the impact of noise on the LES, a new noise-resistant statistic, FE, is proposed as the system state evaluation metric. The mathematical definition of FE is presented in Equation (16).

(16)$\begin{array}{cc}\hfill \mathrm{FE}& ={\displaystyle \frac{{\lambda }_{max}-{\lambda }_{min}}{{\lambda }_{min}}}\hfill \\ \hfill & ={\displaystyle \frac{{\sigma }^2{(l+\sqrt{c})}^2-{\sigma }^2{(l-\sqrt{c})}^2}{{\sigma }^2{(l-\sqrt{c})}^2}}\hfill \\ \hfill & ={\displaystyle \frac{4c}{{\left(l-\sqrt{c}\right)}^2}}\hfill \end{array}$

where ${\lambda }_{max}-{\lambda }_{min}$ reflects the degree of change in the eigenvalues of the current matrix. When the system is stable, the difference should remain within a stable range; however, when the system state is abnormal, the difference will exhibit larger fluctuations. From the perspective of the mathematical formula, dividing the difference by ${\lambda }_{min}$ not only reduces the impact of noise but also further amplifies the trend of eigenvalue changes in the system.

4. Discussion

4.1. Data Preparation

4.1.1. Data Acquisition

This study aims to monitor the operational status of the lubrication system by using pressure data collected from key nodes as the primary parameter. However, RMT can fully exploit the spatiotemporal correlations within data only when processing high-dimensional matrices. The collection of pressure data alone does not sufficiently capture the high-dimensional characteristics of the system. Therefore, this study collects both temperature and pressure data from key nodes of the lubrication system in the marine diesel engine, as illustrated in Figure 4. The main structure of the lubrication system is shown in Figure 5.

Since it is challenging to externally attach sensors for data collection within the internal nodes of the diesel engine frame, the structural diagram indicates that the lubrication oil pump plays a critical role in the lubrication system. It extracts oil from the lubrication tank and pressurizes it for delivery to other components through the lubrication oil pipelines. Therefore, the key nodes for data collection in this study are primarily located at the installation points of valves along the lubrication oil pipelines, as indicated by the star in Figure 6, with a total of 15 sampling points. The relevant parameters of the lubrication oil pump are listed in Table 2.

4.1.2. Data Processing

The presence of multiple pressure relief valves in the lubrication oil pipelines makes it challenging to accurately simulate changes in the lubrication oil state under overload conditions of the lubrication oil pump. Additionally, as highlighted in [26], significant drops in the pump’s rotational speed below its normal operating range lead to notable variations in pipeline oil pressure. To address this, the study simulates both normal and abnormal operating conditions of the lubrication system by reducing the rotational speed of the lubrication oil pump and collecting pressure and temperature parameters from the pipelines under varying conditions. Typically, the pump speed is considered abnormal when it falls to 10–15% of the rated speed, with speeds below 15% generally requiring immediate maintenance. However, in this study, the abnormal condition is defined as the pump operating at 15–20% below the rated speed to observe more pronounced changes in pipeline oil pressure and temperature.

Given the extended length of the lubrication oil pipelines and the relatively slow oil extraction rate of the pump, the sampling interval is set to 5 s, with data collection spanning a total duration of 21 min. Data are collected from the 15 key nodes previously identified, recording both temperature and pressure parameters at each node. This results in 2 × 250 parameters per node. The collected data are then organized as shown in Table 3, forming the dataset used in this study. Specifically, the dataset comprises 30 × 170 samples under normal operating conditions and 30 × 80 samples under abnormal conditions.

Under actual operating conditions of the diesel engine lubrication system, the dataset presented in Table 3 continues to expand over time, resulting in ever-increasing dimensionality. This growth makes it impractical to directly utilize raw data as matrices for RMT analysis in condition monitoring. To address this challenge, a sliding time window approach is employed to segment the data. The time window width is set to 100, with a sliding step of 1 s, ensuring continuous updates to the parameters within the time window matrix. By analyzing the segmented data within the sliding time window, real-time condition monitoring of the lubrication system is achieved.

For the purposes of this study, all collected data are assumed to be under idealized conditions, excluding the influence of factors such as lubrication oil type, frictional resistance, and valve aperture variations. Based on these assumptions, the pipeline lubrication oil pressure is considered to be directly determined by the power of the lubrication oil pump, with operating conditions influencing oil pressure at each node within the pipeline. Within this framework, the effectiveness of the proposed method for condition monitoring of the lubrication system is validated.

4.2. RMT Analysis

4.2.1. Algorithm Applicability Analysis

To apply the RMT algorithm for condition monitoring of the lubrication system, it is necessary to validate the feasibility of the algorithm on the dataset used in this study. For this purpose, the M-P law (as shown in Equation (10)) is employed to analyze the data within the sliding time window. The ISSA-RSVD algorithm mentioned earlier aims to filter out noise in the matrix by setting a threshold, though this effect is not directly observable. However, it can still be validated by comparing the M-P law plots before and after denoising. Thus, in this section, the RMT applicability analysis and the denoising effectiveness of ISSA-RSVD are combined into a unified analysis, as shown in Figure 7. In Figure 7, the monitoring data within a time window at a specific moment are normalized and subjected to RMT analysis. The eigenvalue distribution of the matrix clearly conforms to the M-P law, demonstrating the feasibility of applying RMT to the lubrication system dataset. Furthermore, by comparing Figure 7a,b, it can be observed that eigenvalues not satisfying the threshold described in Equation (2) are effectively filtered out, verifying the effectiveness of ISSA-RSVD in noise reduction.

4.2.2. Monitoring the Process of Matrix Eigenvalue Changes

Although the M-P distribution shown in Figure 7 provides an intuitive representation of the denoising effectiveness of the ISSA-RSVD algorithm, it fails to visually capture the changes in eigenvalue behavior when the system transitions to an abnormal state. However, this issue can be addressed using the Single Ring Theorem, as described by Equation (14). By analyzing the monitoring matrices under different operating conditions, the results are presented in Figure 8. When the system is in a normal state, the real and imaginary parts of the eigenvalues are distributed within the boundaries of the ring, as shown in Figure 8a. As the system begins to experience abnormalities (e.g., the lubrication oil pump’s rotational speed drops significantly below the normal operating speed), the eigenvalues gradually shift toward the interior of the ring, as shown in Figure 8b. Finally, when the system is in a fully abnormal state (where all monitoring matrix data correspond to abnormal conditions), the eigenvalues cluster and distribute entirely within the inner region of the ring, as shown in Figure 8c. This progression, as visualized in Figure 8, effectively demonstrates the applicability of the Single Ring Theorem for diagnosing system abnormalities and monitoring the state transitions of the lubrication system.

4.3. Analysis of the Application of Eigenvalue Statistical Indicators

The applicability of the algorithm proposed in this paper on the lubricating oil system is verified by using two methods commonly used in RMT in Section 4.2, and the distribution of the eigenvalues during the transition of the system’s normal to abnormal operating conditions is analyzed from the Single Ring Theorem plot, but the time point at which the abnormality occurs cannot be accurately obtained from the change in the image alone. At this point, most of the RMT-based studies use the maximum eigenvalue or the minimum eigenvalue shown in Equation (10) as an index to evaluate the system state, and this operation ignores the effect of noise on the eigenvalues, for this reason, this study constructs a fused eigenvalue with denoising effect by Equation (15).

In a large marine diesel engine lubricating oil system in normal operating conditions, the typical amplitude of the pressure noise is 0.3~0.5 bar; in the slipper pump failure or pipeline vibration and other abnormal conditions, the pressure noise fluctuations may reach 0.6~1.0 bar or even higher; similarly, in normal operating conditions, the typical amplitude of the temperature noise is ±1~3 °C or so; in the failure of the cooler or when the flow of the pipe is not stable and other abnormal conditions, the amplitude of the temperature noise will exceed ±5 °C. Under abnormal conditions such as cooler failure or unstable flow in the pipeline, the amplitude of temperature noise can exceed ±5 °C. Since it has been stipulated that the data collected in this study are under ideal conditions, the noise amplitude of the collected data is in a steady state. In order to study the effect of different amplitude noise on the FE proposed in this study, Gaussian white noise of different amplitudes will be added additionally to analyze the FE, and the FE without Gaussian white noise will be considered to be free of noise. The Gaussian white noise is added to the time-window matrix of the system in a normal operation state; the noise follows a normal distribution; the mean value of the noise is set to 0, and the standard deviation is 3%, 6%, 9%, 15%, and 20% of the standard deviation of the current monitoring matrix, respectively. In order to verify the anti-interference ability of FE to noise, it is compared with the maximum eigenvalue statistic and the minimum eigenvalue statistic, and the relative errors of the three are calculated under different noise amplitudes, and the experimental results are shown in Table 4.

Table 4 shows that under the noise amplitude of 3%, 6%, 9%, 15% and 20%, although the relative errors of the three eigenvalue statistics increase with the increase in the noise amplitude, the relative errors of the proposed indexes are always lower than those of the original two indexes, and the relative errors are always kept within 1%. When the indicators proposed in this paper are used as the lubricant system condition monitoring indicators, even if the sudden noise is encountered in the actual experimental process, there will be no significant jitter in the data, which will reduce the probability of misjudgment in the system condition monitoring.

4.4. Verification of the Effectiveness of Condition Monitoring

4.4.1. Threshold Setting

In Section 4.1, the data used in this study have been fully described. The dataset is divided using a sliding time window, and since the resulting monitoring matrix is employed for RMT analysis, the matrix is updated every second. Consequently, a monitoring index (FE) is obtained every second. Under ideal conditions, when the lubrication pump operates continuously at its rated speed, the FE should remain stable over this period. Furthermore, since this study focuses solely on analyzing the impact of reduced pump speed on the lubrication system, the pump speed is set to 15% below its rated speed, as described in Section 4.1.2. A total of 30 × 200 sets of data are collected under abnormal conditions. Using the sliding time window method, the parameters within the monitoring matrix are fully updated, and the average value of the 100 obtained monitoring indices (FE) is calculated, which is used as the threshold for the abnormal state of the lubrication system. The calculation method is shown in Equation (17).

(17)$\begin{array}{cc}\hfill \mathrm{Threshold}& =\overline{FE}={\displaystyle \frac{1}{100}}\sum _{i=1}^{100}F{E}_i\hfill \end{array}$

4.4.2. Effectiveness Verification

The threshold obtained from the above method is applied to the lubrication system’s state monitoring, with the monitoring results shown in Figure 9. As observed in Figure 9a, when the lubrication system operates under normal conditions, the FE index should remain around 350, with only minimal deviation. However, a significant anomaly occurs at 71 s, and the FE index exceeds the threshold at 81 s, finally stabilizing around 140 at 92 s. This result is in line with expectations, as the time point for gradually reducing the pump speed in this study occurs after approximately 14 min of data collection, i.e., after the 170th dataset. Given the time window width of 100 and the data updating every second, this time point coincides with the moment when the lubrication pump speed is reduced, further validating the feasibility of the method proposed in this study for lubrication system state monitoring.

The state monitoring curve in Figure 9a is derived from the FE indices after applying ISSA-RSVD to denoise the monitoring matrix. In Figure 9b, an additional state monitoring curve is plotted without using ISSA-RSVD for denoising. It is evident that the FE indices without ISSA-RSVD denoising are consistently higher than those with denoising. This is because the matrix without denoising contains smaller eigenvalues with noise, which while leaving the largest eigenvalue of the monitoring matrix unchanged, results in higher FE indices. Consequently, this impacts the accuracy of state monitoring. In Figure 9b, the FE index without denoising only exceeds the threshold around the 85th second, which is 4 s later than the denoised index. This demonstrates that the ISSA-RSVD-based RMT state monitoring model proposed in this study not only provides excellent denoising performance but also improves state monitoring accuracy by 4.95% compared to conventional RMT models.

Further analysis was conducted to examine the impact of abnormal lubrication pump speed on the pressure and temperature of the lubrication oil within the pipeline. To avoid the attenuation effects that might arise due to the length of the pipeline, which could influence the analysis of the two parameters during lubrication oil flow, data analysis was performed at the monitoring matrix corresponding to the node closest to the lubrication pump at t = 81 s. The results are shown in Figure 10.

As seen in Figure 10a, with the continuous reduction in lubrication pump speed, the pressure of the lubrication oil in the pipeline also significantly decreases. It is generally accepted that the lubrication oil pressure should remain between 50–80 psi, and even under low load or idle conditions, it should be maintained around 50 psi. However, as observed in Figure 10a, the lubrication oil pressure is clearly lower than this range, demonstrating the substantial impact of abnormal pump operation on the lubrication oil pressure. Figure 10b reveals that the effect of excessively low pump speed on lubrication oil temperature is not as pronounced as its effect on pressure. When the pump speed is too low, the flow rate of the lubrication oil decreases, preventing excess heat from being carried away by the flow, resulting in a slight accumulation of heat and a small rise in temperature. Due to the relatively short duration of the test, the temperature change is limited. Nevertheless, this also confirms the influence of abnormal lubrication pump operation on lubrication oil temperature.

5. Conclusions

This study proposes a data-driven approach for state monitoring of the lubrication system in marine diesel engines. The proposed method effectively mitigates the impact of noise in raw data on the monitoring matrix and introduces a novel state monitoring index with enhanced denoising capabilities. By eliminating the reliance on models inherent in traditional monitoring methods, this approach significantly improves monitoring accuracy. The key conclusions of this study are summarized as follows:

• Applying ISSA to RSVD can effectively address the parameter setting issues in RSVD and enhance the reliability of denoising results.

• The newly proposed FE index exhibits superior noise resistance compared to traditional indices. Across environments with varying noise amplitudes, the FE index consistently reduces the influence of noise, ensuring more reliable performance.

• Using the FE index as a state monitoring indicator enables real-time monitoring of the lubrication system’s condition with good results.

Despite the contributions of this study, several limitations remain. For instance, the analysis focused solely on the impact of a significantly reduced lubrication pump speed on the system. While abnormal lubrication pump operation is a critical issue in lubrication systems, other anomalies, such as valve damage within the pipeline, may also significantly affect system performance. This study considered a single abnormal condition and did not explore the potential impact of multiple concurrent abnormalities on state monitoring, which represents a promising direction for future research.

Furthermore, this study achieved state monitoring of the lubrication system but did not address the localization of specific anomalies. Future research could focus on anomaly localization by analyzing the eigenvector elements of the monitoring matrix across different time points to pinpoint the exact source of data anomalies. Once anomaly localization is realized, machine learning or deep learning methods could be employed for fault diagnosis and prediction. The outcomes of fault diagnosis and prediction could then be integrated into an auxiliary decision-making system to enhance maintenance strategies and fault management.

Author Contributions

Data curation, B.D., H.X. and S.L.; Funding acquisition, S.Y. and L.Q.; Methodology, S.Y. and L.Q.; Project administration, S.Y. and L.Q.; Software, S.Y.; Visualization, B.D., H.X. and S.L.; Writing—original draft, B.D. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:

ISSA	Improved Sparrow Search Algorithm
RSVD	Randomized Singular Value Decomposition
FE	Fused Eigenvalue
RMT	Random Matrix Theory
M-P	Marcenko–Pastur Law

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Proposed framework for condition monitoring of lubrication systems.

Figure 2. Flowchart of improved sparrow algorithm.

Figure 3. Comparison of results across four test scenarios.

Figure 4. Marine diesel engines.

Figure 5. Marine diesel engine lubricating oil system structure diagram.

Figure 6. Lube line data acquisition node.

Figure 7. M-P distribution before and after denoising.

Figure 8. Single Ring Theorem distribution of eigenvalues in selected time nodes.

Figure 9. Status monitoring graph.

Figure 10. Temperature and pressure analysis chart.

Table 1

Test Functions.

Function Number	Function Type
F1	$$F_1\left(x\right)=\sum _{i=1}^d{x}_i^2$$
F3	$$F_3\left(x\right)=\sum _{i=1}^d{\left(\sum _{j=1}^i{x}_j\right)}^2$$
F7	$$F_7\left(x\right)=\sum _{i=1}^{d}i·x_i^4+\mathrm{rand}$$
F9	$$F_9\left(x\right)=\sum _{i=1}^d\left[x_i^2-10cos\left(2\pi x_i\right)+10\right]$$

Table 2

Lubricating oil pump-related parameters.

Performance Indicators	Parameter Value
Rated Voltage	380 V
Rated Current	23.9 A
Rated Speed	790 r/min
Continuous Power	11 KW
Power Factor	0.79
Motor Efficiency	88.7%

Table 3

Description of the internal parameters of the dataset.

Number of Matrix Rows	Signal Source	Signal Type	Data Dimension
1	Node 1	Pressure	1 × 250
2	Node 1	Temperature	1 × 250
3	Node 2	Pressure	1 × 250
4	Node 2	Temperature	1 × 250
	⋮	⋮	⋮
29	Node 15	Pressure	1 × 250
30	Node 15	Temperature	1 × 250

Table 4

Relative error of LES in noise of different amplitudes.

LES	3%	6%	9%	15%	20%
${\lambda }_{max}$	6.0852%	12.3568%	18.8089%	32.2326%	43.9967%
${\lambda }_{min}$	5.8823%	11.7647%	19.6078%	32.2404%	43.7158%
FE	0.7801%	−0.1899%	0.1957%	0.1688%	0.1973%

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