Compound fault diagnosis of rolling mill main drive system based on UIO-TROA-CNN

Abstract

To address compound faults—including actuator faults, sensor faults, and unknown disturbances—in the main drive system (MDS) of a rolling mill, a fault diagnosis model is established based on the d-q model of a three-phase AC motor. To overcome the limitation of conventional model-based fault detection (MFD) in isolating faults under compound fault scenarios, a novel fault diagnosis framework is proposed by integrating model residuals with deep learning. An Unknown Input Observer (UIO) is designed to detect system faults, and its convergence is rigorously proven using Lyapunov theory and linear matrix inequalities (LMIs). The coupled residual signals generated by the observer are segmented into sequential subsequences and processed by a Convolutional Neural Network (CNN) for feature extraction and classification. To account for the temporal and dynamic nature of the residuals, a Tyrannosaurus Rex Optimization Algorithm (TROA) is adopted to optimize the CNN hyperparameters. Numerical simulations on a rolling mill demonstrate that the proposed UIO achieves superior state estimation performance, with a 16.48% reduction in the Root Mean Square Error (RMSE) of angular velocity difference of the motor compared to the Sliding Mode Observer (SMO). Furthermore, the proposed TROA-CNN outperforms Bayesian Optimization (Bayes)-CNN, Whale Optimization Algorithm (WOA)-CNN, and Grey Wolf Optimization (GWO)-CNN in terms of fault classification accuracy (99.76%) and noise robustness (96.27% under 5% noise). In scalability tests where the number of fault types or dataset size are doubled, the inference latency increases by approximately 10%, and the training time rises by about 40%. These results demonstrate that the UIO-TROA-CNN achieves high accuracy, strong robustness, and excellent scalability, making it well-suited for fault diagnosis in industrial environments with high noise and complex fault types.

Keywords

Fault diagnosis unknown input observer convolutional neural networks T-Rex Optimization Algorithm rolling mill main drive system

Introduction

With the development of metallurgical industry, rolling mill main drive system (MDS) tends to be complicated, and AC motor vector control system is widely used. As the core of rolling equipment, the fault of this system will lead to control failure or even stop production.

In complex industrial systems, compound faults denote the simultaneous presence of two or more interrelated and mutually interacting fault modes. Owing to the concurrent and coupled nature of these fault mechanisms, the number of potential compound fault scenarios grows exponentially. This leads to highly diverse and intricate fault signatures, which are difficult to extract and accurately diagnose (Shao et al., 2020). These characteristics make compound fault diagnosis a significant challenge for maintaining and ensuring the reliability of modern industrial equipment.

Traditional fault diagnosis approaches primarily encompass three categories: model-based methods (Mhamdi et al., 2013), experience-based qualitative methods (Chatti et al., 2014), and signal analysis-based techniques for compound fault diagnosis (Zhao et al., 2020). Model-based fault detection relies on accurate physical modeling of the system. For instance, Luenberger (1964) introduced the state observer theory, which has been widely utilized for residual generation and fault detection. Subsequently, Francis and Glover (1983) developed the H∞ control theory, which was later adapted for observer design to improve system robustness against uncertainty and measurement noise. Patton and Frank (1989) proposed a fault diagnosis framework based on multiple observers; however, their method is limited to the detection of multiple single faults and fails to detect or decouple compound faults. Experience-driven methods rely on limited prior expert knowledge for diagnosis. Signal analysis-based techniques, on the other hand, attempt to decompose complex signals to extract fault features. However, both approaches exhibit inherent limitations in dynamic and uncertain industrial environments. They rely on prior knowledge and cannot easily adapt to new or coupled fault patterns.

In contrast to traditional methods, deep learning techniques eliminate the need for extensive prior knowledge. They automatically learn representative features and latent patterns directly from large-scale data, significantly improving the efficiency and accuracy of fault diagnosis (Khan and Yairi, 2018; Shao et al., 2017). With their powerful feature extraction capabilities, deep learning models are well-suited for capturing salient information from complex signals and effectively handling multi-dimensional, multi-source, and compound faults. For example, Ghorbanali and Sohrabi (2024) proposed an intelligent compound fault decoupling method based on Capsule Networks (CapsNet) and ensemble learning, achieving effective separation of fault components. Wang et al. (2021) introduced a deep normalized Convolutional Neural Network (CNN) for classifying mechanical imbalance faults, demonstrating robust performance in feature discrimination. However, the practical deployment of these methods in industrial scenarios remains challenging. This is primarily due to the difficulty of acquiring and labeling compound fault samples, as well as the intricate and entangled nature of information in such environments.

Compound faults generally involve interactions among multiple types of faults. In practical systems, faults rarely occur in isolation; instead, concurrent actuator and sensor faults are frequently observed, often accompanied by unknown external disturbances. Gao et al. (2018) addressed the problem of active fault-tolerant control for rigid spacecraft under multiple actuator faults and saturation, but did not consider simultaneous sensor faults. Qin et al. (2017) developed an active fault-tolerant control strategy for quadrotors affected by speed sensor faults, without addressing actuator faults. Yuan et al. (2020) proposed a hybrid model-based fault diagnosis framework that simultaneously considered actuator and sensor faults, though the fault types addressed were relatively limited. Raoufi and Marquezz (2010) designed an observer capable of simultaneously estimating two types of faults for a class of nonlinear systems; however, the influence of external disturbances was not taken into account.

Unknown Input Observer (UIO) improves fault detection by decoupling unknown inputs such as external disturbances and model uncertainties. It is applicable to both single- and multi-input systems, including compound fault scenarios. However, when fault features are strongly coupled, UIO alone may fail to accurately localize fault sources. The residual signals generated by UIO contain valuable fault information, which can be further leveraged by deep learning methods to extract hidden features and enhance fault classification accuracy.

Kim (2014) applied CNN to sentence classification in natural language processing, demonstrating improved feature extraction performance. Serizawa and Fujita (2020) introduced a linearly decreasing weight particle swarm optimization (LDW-PSO) method for tuning CNN hyperparameters. Zhang et al. (2024a) proposed a UIO-DBO-SVM-based scheme for diagnosing actuator-related compound faults. Nevertheless, these approaches are limited in their ability to process time-dependent residual signals and capture dynamic behaviors of the system. In contrast, the Tyrannosaurus Rex Optimization Algorithm (TROA) proposed by Sahu et al. (2023) incorporates a dynamic adjustment mechanism that is well-suited for modeling the temporal dependencies and dynamic characteristics of complex industrial systems.

According to the above literature review and analysis, this study focuses on compound fault diagnosis for the MDS of a rolling mill, considering the simultaneous presence of actuator faults, sensor faults, and unknown disturbances. Inspired by the work (Zhang et al., 2024a), a hybrid fault diagnosis framework is proposed by integrating model-based observer techniques with deep learning methods. Specifically, a novel UIO-TROA-CNN scheme is developed, as illustrated in Figure 1(a). In the first stage, a UIO is designed to detect system faults, with its convergence and robustness validated using Lyapunov theory and linear matrix inequality (LMI). Once a fault is detected, the residual signals generated by the UIO are divided into subsequences, each representing a distinct fault coupling pattern. These are then used as inputs to the CNN, whose hyperparameters are optimized using the TROA to improve classification accuracy.

In summary, the main contributions of this paper are as follows:

By integrating model-based and deep learning-based fault diagnosis techniques, this paper proposes a compound fault diagnosis scheme based on UIO-TROA-CNN. The proposed method combines the physical interpretability of model-based observers with the powerful nonlinear feature extraction capability of deep learning, enabling accurate detection and classification of compound faults in complex systems.

To address the complexity of compound faults in rolling mills, this study introduces an augmented-dimensional system in which sensor faults are incorporated into the system states. By decoupling actuator and sensor faults, the compound fault problem is effectively transformed into multiple independent single-fault scenarios, thereby reducing the diagnostic complexity.

Considering the complexity and nonlinear separability of residual data, the TROA is employed to optimize the hyperparameters of the CNN model, thereby improving the classification performance of the fault diagnosis framework.

System modeling

Rolling mill MDS is a complex system composed of motor, elastic connecting shaft, reducer, roll and various couplings, but the motor and roll are components with large inertia, while the coupling shaft belongs to the lighter elastic components. In order to facilitate the analysis, the MDS of rolling mill is generally idealized as a two-mass spring system composed of motor and roll (Zhang and Li, 2023) by using the concentrated mass method without being divorced from reality, and its mechanical model is shown in Figure 1(b).

The MDS of a rolling mill is composed of two equations, one based on roll parameters and the other based on AC motor, as shown below

Jm \overset{\cdot\cdot}{θ m} + Cm \overset{\cdot}{θ m} + K (θ m - θ L) = Tm

(1)

JL \overset{\cdot\cdot}{θ L} + CL \overset{\cdot}{θ L} + K (θ L - θ m) = TL + Tf

(2)

In the above formula, Jm and JL are the moment of inertia of the motor and the load, respectively; Tm, TL are the electromagnetic torque of the motor and the load resistance moment; Cm, CL is the damping coefficient of motor and load; K is the damping coefficient of elastic axis; Tf is the friction torque between the roll and the rolled piece, and $Tf = μ PaR$ , where Pa is the rolling force, R is the roll radius, and the mixed friction factor μ between the rolls is the nonlinear function of the parameter a, b, c and the relative speed v between the roll and the rolled piece, namely, $μ = a - bv + c v^{2}$ , and $v = R \overset{\cdot}{θ L}$ .

The AC induction motor serves as the driving component of the system and exhibits characteristics such as nonlinearity, strong coupling, and multi-variable interactions. Drawing upon the work (Zhang et al., 2024b), this study introduces rotational speed $ω_{1}$ as an additional control variable. This strategy not only eliminates the relative motion between the stator and rotor but also simplifies the mathematical modeling of the motor, thereby facilitating the design and implementation of the control strategy. The resulting model is transformed into the $d - q$ coordinate system and is expressed as follows

{\begin{matrix} \frac{d ψ_{rd}}{dt} = - \frac{1}{T_{r}} ψ_{rd} + (ω_{1} - ω_{m}) ψ_{rq} + \frac{L_{m}}{T_{r}} i_{sd} \\ \frac{d ψ_{rq}}{dt} = - \frac{1}{T_{r}} ψ_{rq} - (ω_{1} - ω_{m}) ψ_{rd} + \frac{L_{m}}{T_{r}} i_{sq} \\ \frac{disd}{dt} = \frac{L_{m}}{σ L_{s} L_{r} T_{r}} ψ_{rd} + \frac{L_{m}}{σ L_{s} L_{r}} ω_{m} ψ_{rq} - \frac{R_{s} {L_{r}}^{2} + R_{r} {L_{m}}^{2}}{σ L_{s} {L_{r}}^{2}} isd + ω_{1} i_{sq} + \frac{u_{sd}}{σ L_{s}} \\ \frac{disq}{dt} = \frac{L_{m}}{σ L_{s} L_{r} T_{r}} ψ_{rq} - \frac{L_{m}}{σ L_{s} L_{r}} ω_{m} ψ_{rd} - \frac{R_{s} {L_{r}}^{2} + R_{r} {L_{m}}^{2}}{σ L_{s} {L_{r}}^{2}} isq - ω_{1} i_{sd} + \frac{u_{sq}}{σ L_{s}} \end{matrix}

(3)

In the above formula, $ψ_{rd}$ and $ψ_{rq}$ are the motor rotor d axis and q axis flux linkage, respectively; $u_{sd}$ and $u_{sq}$ are the motor stator d shaft and q shaft input voltages, respectively; isd and isq are the motor stator d shaft and q shaft currents, respectively; $ω_{m}$ is the angular velocity of the motor rotor; $R_{s}$ is the stator winding resistance; $R_{r}$ is the rotor winding resistance; $L_{m}$ is the mutual inductance between stator and rotor phase windings; $L_{r}$ is the self-inductance of the rotor phase winding; $L_{s}$ is the self-inductance of stator phase windings; $σ = 1 - {L_{m}}^{2} / L_{s} L_{r}$ is the flux leakage coefficient of the motor; $T_{r} = L_{r} / R_{r}$ is the rotor time constant; The electromagnetic torque of the AC motor is defined as $T_{m} = n_{p} L_{m} (i_{sq} ψ_{rd} - i_{sd} ψ_{rq}) / L_{r}$ , where $n_{p}$ is the number of motor poles and $ω_{1}$ is the rotation speed of the rotating coordinate system.

By combining the AC motor model in the $d - q$ coordinate system with the mechanical model of the rolling mill, an electromechanical coupling model is established. The state vector $x_{p} (t) = [ψ_{rd} ψ_{rq} i_{sd} i_{sq} \overset{\cdot}{θ_{m}} \overset{\cdot}{θ_{L}} θ_{L} - θ_{m}]^{T}$ is defined accordingly, where $\overset{\cdot}{θ m}$ and $\overset{\cdot}{θ L}$ represent the angular velocities of the motor and the load, respectively. The control input vector $u (t) = [u_{sd} u_{sq} T_{L}]^{T}$ is also defined.

Therefore, the MDS fault model of rolling mill considering external disturbance and nonlinear factors of the system is as follows

{\begin{matrix} \overset{\cdot}{x_{p}} = A_{p} x_{p} + g_{p} (x_{p}) + F_{a} f_{a} + Bpu + Dd \\ y_{p} = C_{p} x_{p} + F_{s} f_{s} \end{matrix}

(4)

$f_{a}$ denotes the actuator fault function, and $F_{a}$ is the corresponding actuator fault distribution matrix; $f_{s}$ denotes the sensor fault function, and $F_{s}$ is the corresponding sensor fault distribution matrix; d denotes the external disturbance acting on the system, and D is the disturbance distribution matrix; $g_{p} (x_{p})$ is the nonlinear term in the system dynamics; $C_{p}$ is the output distribution matrix; and $y_{p}$ is the system output variable

\begin{matrix} A_{p} = \\ [\begin{matrix} - \frac{1}{T_{r}} & ω_{1} & \frac{L_{m}}{T_{r}} & 0 & 0 & 0 & 0 \\ - ω_{1} & - \frac{1}{T_{r}} & 0 & \frac{L_{m}}{T_{r}} & 0 & 0 & 0 \\ \frac{L_{m}}{σ L_{s} L_{r} T_{r}} & 0 & - \frac{R_{s} {L_{r}}^{2} + R_{r} {L_{m}}^{2}}{σ L_{s} {L_{r}}^{2}} & ω_{1} & 0 & 0 & 0 \\ 0 & \frac{L_{m}}{σ L_{s} L_{r} T_{r}} & - ω_{1} & - \frac{R_{s} {L_{r}}^{2} + R_{r} {L_{m}}^{2}}{σ L_{s} {L_{r}}^{2}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & - \frac{C_{m}}{J_{m}} & 0 & \frac{K}{J_{m}} \\ 0 & 0 & 0 & 0 & 0 & - \frac{C_{L} + b P_{a} R^{2}}{J_{L}} & - \frac{K}{J_{L}} \\ 0 & 0 & 0 & 0 & - 1 & 1 & 0 \end{matrix}] \end{matrix}

\begin{matrix} B_{p} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ \frac{1}{σ L_{s}} & 0 & 0 \\ 0 & \frac{1}{σ L_{s}} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & \frac{1}{J_{L}} \\ 0 & 0 & 0 \end{matrix}] g_{p} (x_{p}) = [\begin{matrix} - ω_{m} ψ_{rq} \\ ω_{m} ψ_{rd} \\ \frac{L_{m}}{σ L_{s} L_{r}} ω_{m} ψ_{rq} \\ - \frac{L_{m}}{σ L_{s} L_{r}} ω_{m} ψ_{rd} \\ \frac{n_{p} L_{m}}{J_{m} L_{r}} i_{sq} ψ_{rd} - \frac{n_{p} L_{m}}{J_{m} L_{r}} i_{sd} ψ_{rq} \\ \frac{a P_{a} R + c R^{3} P_{a} {ω_{L}}^{2}}{JL} \\ 0 \end{matrix}] \\ C_{p} = [\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] . \end{matrix}

Fault detection

Assumption 1. $g (x)$ is locally Lipschitz on a domain (open and connected set) $D \subset R$ if each point of $D$ has a neighborhood $D_{0}$ such that $g (x)$ satisfies $∥ g (\hat{x}) - g (x) ∥ \leq θ ∥ \hat{x} - x ∥$ , $\forall x$ , $x \subset D_{0}$ , where θ is a Lipschitz constant.

Assumption 2. The nonlinear term $g (x)$ in the system is bounded and there exists a positive scalar ρ satisfying inequality: $g (x) \leq ρ$ .

Lemma 1. Let X, Y and $F (t)$ be real matrices of appropriate dimensions. Then for any $λ 0$ and $F (t)^{T} F (t) \leq I$ (Hosseini et al., 2020) we have

XF (t) Y + Y^{T} F^{T} (t) X^{T} \leq λ^{- 1} X X^{T} + λ Y^{T} Y

(5)

Lemma 2. Schur’s complement Lemma for three matrices $S_{1}$ , $S_{2}$ , and $S_{3}$ with appropriate dimensions, and $S_{1} = {S_{1}}^{T}$ , $S_{2} = {S_{2}}^{T}$ , then $S_{3} 0$ and $S_{1} - S_{2} {S_{3}}^{- 1} {S_{2}}^{T} 0$ are equivalent to the following inequality

[\begin{matrix} S_{1} & S_{2} \\ {S_{2}}^{T} & S_{3} \end{matrix}] 0

(6)

Lemma 3. If the nonlinear function g satisfies the Assumption 1 and Lemma 1, then the following inequality holds

2 {e_{x}}^{T} PS (g (E \hat{x}) - g (Ex)) \leq λ^{- 1} {e_{x}}^{T} PP e_{x} + λ θ^{2} ∥ S ∥ ∥ E ∥ {e_{x}}^{T} e_{x}

(7)

In order to realize the fault diagnosis of sensor faults, it is necessary to merge $x_{p} \in R^{n}$ and $f_{s} \in R^{h}$ in the system and define them as new state variables. If $x = [{\begin{matrix} {x_{p}}^{T} & f_{s} \end{matrix}}^{T}]^{T}$ is set, the original system becomes the following singular system form

{\begin{matrix} E \overset{\cdot}{x} = Mx + g (Ex) + F_{a} f_{a} + Bu + Dd \\ y = Hx \end{matrix}

(8)

where $E = [\begin{matrix} I_{n} & 0 \end{matrix}]$ , $M = [\begin{matrix} A_{P} & 0 \end{matrix}]$ , $H = [\begin{matrix} C_{P} & F_{s} \end{matrix}]$ , $B = B_{p}$ .

The following adaptive observer is designed for the system

{\begin{cases} \overset{\cdot}{z} = N z + L y + S g (E \hat{x}) + S B u + S F_{a} f_{a} \\ \hat{x} = z + T y \\ \hat{y} = H \hat{x} \\ {\hat{f}}_{s} = [\begin{matrix} 0 & I \end{matrix}] \hat{x} \end{cases} .

(9)

z denotes the intermediate state of the observer; $\hat{y}$ denotes the estimated output variable; $\hat{x}$ denotes the estimated state variable; ${\hat{f}}_{s}$ denotes the estimated sensor fault. The observer gain matrices N, L, S, and T are to be designed such that the following conditions are satisfied

{\begin{matrix} SE + TH = I \\ N = SM - JH \\ L = J + NT \end{matrix}

(10)

The state error is defined as $e_{x} = \hat{x} - x$ , which can be obtained as follows

e_{x} = \hat{x} - x = z + THx - x = z + (TH - I) x

(11)

According to equation (10), the following can be obtained

e_{x} = z - SEx

(12)

Taking the derivative with respect to $e_{x}$ and combining it with equation (10), we obtain

\begin{matrix} {\overset{\cdot}{e}}_{x} & = \overset{\cdot}{z} - SE \overset{\cdot}{x} \\ = Nz + Ly + Sg (E \hat{x}) + SBu + S F_{a} f_{a} \\ - SMx - Sg (Ex) - S F_{a} f_{a} - SBu - SDd \\ = Nz + (LH - SM) x + S (g (E \hat{x}) - g (Ex)) - SDd \\ = N (\hat{x} - x + x - Ty) + (LH - SM) x + S (g (E \hat{x}) - g (Ex)) - SDd \\ = N e_{x} + (N - NTH + LH - SM) x + S (g (E \hat{x}) - g (Ex)) - SDd \\ = N e_{x} + S (g (E \hat{x}) - g (Ex)) - SDd \end{matrix}

(13)

Let $A = SM$ , $Q = JH$ , $\bar{D} = - SD$ , obtain

{\overset{\cdot}{e}}_{x} = (A - Q) e_{x} + \bar{D} d + S (g (E \hat{x}) - g (Ex))

(14)

Theorem 1. For system (8) satisfying Assumptions 1 and 2, there exist positive definite matrices P, Y and constants δ, λand θ such that the following LMI holds

\bar{Π} = [\begin{matrix} Δ_{1} & P \bar{D} & P & I \\ * & - δ^{2} I & 0 & 0 \\ * & * & - λ I & 0 \\ * & * & * & - I \end{matrix}] 0

(15)

where $Δ_{1} = P A + A^{T} P - Y H - H^{T} Y^{T} + λ θ^{2} ∥ S ∥ ‖ E ‖$ , then the error dynamic system is robust asymptotically stable, and where * denotes the symmetric entries of the symmetric matrix.

Proof: Define the Lyapunov function as $V = {e_{x}}^{T} P e_{x}$ and take the derivative of V along (14) as follows

\begin{matrix} \overset{\cdot}{V} = {e_{x}}^{T} P {\overset{\cdot}{e}}_{x} + \overset{\cdot}{e_{x}^{T}} P e_{x} \\ = {e_{x}}^{T} P [(A - Q) e_{x} + \bar{D} d + S (g (E \hat{x}) - g (Ex))] \\ + [(A - Q) e_{x} + \bar{D} d + S (g (E \hat{x}) - g (Ex))]^{T} P e_{x} \\ = {e_{x}}^{T} [P (A - Q) + (A - Q)^{T} P] e_{x} \\ + {e_{x}}^{T} P \bar{D} d + d \overset{T}{\bar{D}} P e_{x} + 2 {e_{x}}^{T} PS (g (E \hat{x}) - g (Ex)) \end{matrix}

(16)

According to Lemma 3, the following follows

\begin{matrix} \overset{\cdot}{V} \leq {e_{x}}^{T} [P (A - Q) + (A - Q)^{T} P \\ + λ^{- 1} PP + λ θ^{2} ∥ S ∥ ∥ E ∥] e_{x} + {e_{x}}^{T} P \bar{D} d + d^{T} {\bar{D}}^{T} P e_{x} \end{matrix}

(17)

In order to suppress the influence of unknown input interference d on the system, the performance index $H \infty$ is designed such that

∥ e_{x} ∥ \leq δ ∥ d ∥

(18)

where $E = [\begin{matrix} I_{n} & 0 \end{matrix}]$ .

Redefine $\bar{V}$ , let $Δ = λ θ^{2} ∥ S ∥ ∥ E ∥$ , then we have

\begin{matrix} \bar{V} \leq \overset{\cdot}{V} + {e_{x}}^{T} e_{x} - δ^{2} d^{T} d \\ = {e_{x}}^{T} [P (A - Q) + (A - Q)^{T} P + λ^{- 1} PP + I + Δ] e_{x} \\ + {e_{x}}^{T} P \bar{D} d + d^{T} {\bar{D}}^{T} P e_{x} - δ^{2} d^{T} d \\ = ε^{T} {\bar{Π}}_{1} ε \end{matrix}

(19)

where $ε = {[\begin{matrix} e^{T} & d^{T} \end{matrix}]}^{T}$ , ${\bar{Π}}_{1} = [\begin{matrix} P (A - Q) + {(A - Q)}^{T} P + λ^{- 1} PP + I + Δ & P \bar{D} \\ * & - δ^{2} I \end{matrix}]$ .

The derivation of $\underset{1}{\bar{Π}}$ using Lemma 2 yields ${\bar{Π}}_{2}$

{\bar{Π}}_{2} = [\begin{matrix} P (A - Q) + {(A - Q)}^{T} P + Δ & P \bar{D} & P & I \\ * & - δ^{2} I & 0 & 0 \\ * & * & - λ I & 0 \\ * & * & * & - I \end{matrix}]

(20)

Let $PJ = Y$ , compute to know $P (A - Q) = PA - YH$ , and thus ${\prod^{¯}}_{2} < 0$ becomes ${\bar{Π}}_{2} < 0$ , so that when $\prod^{¯} < 0 \to \bar{V} < 0 \to \overset{\cdot}{V} < 0$ , the system is asymptotically stable at this time.

Fault classification

T-Rex Optimization Algorithm

The TROA, proposed by Sahu et al. (2023), is a recently developed nature-inspired metaheuristic. TROA mimics the behavioral characteristics of Tyrannosaurus Rex during foraging, hunting, and interspecies competition. By incorporating probabilistic strategies, the algorithm effectively balances global exploration and local exploitation. It features fast convergence and strong robustness, particularly in solving high-dimensional optimization problems and addressing challenges in nonlinear control applications.

The implementation steps of the method are as follows:

1. Initialize the T-Rex population and related parameters. Randomly generate the initial positions and fitness values of individuals, and record the global best solution

\begin{matrix} X_{i} = LB + rand () \cdot (UB - LB) \\ J_{i} = f (X_{i}), i = 1, 2, \cdot\cdot ., N \\ X_{best} = \arg \min_{X_{i}} f (X_{i}) \end{matrix}

(21)

$X_{i}$ represents the initial position of the i th Tyrannosaurus Rex individual, which represents its current state in the search space. LB and UB are the lower and upper boundaries of the search space, respectively. $J_{i}$ is the value of the fitness function. $X_{best}$ denotes the current global best solution; $rand ()$ is a uniformly distributed random number.

2. Simulate the hunting behavior of T-Rex and update its position accordingly. Based on the foraging strategy, the position of each individual is dynamically adjusted. The hunting mechanism is regulated by a probabilistic factor ES, which is defined as follows

ES = N (0, 1 (1 - \frac{t}{T_{\max}})

(22)

where $N (0, 1)$ is a standard normal distributed random variable, t is the current iteration number, and $T_{\max}$ is the maximum iteration number.

When the hunt is successful (global search), $rand < ES$ , the T-Rex will explore new prey in a large area and update its position by

X_{i}^{t + 1} = {X_{i}}^{t} + sr \cdot (tr \cdot V_{i} - X_{best} \cdot rand () \cdot pr)

(23)

When the hunt fails (local exploitation), $rand \geq ES$ , the T-Rex randomly moves to a new location

X_{i}^{t + 1} = {X_{i}}^{t} . rand ()

(24)

Here, sr denotes the probability of a successful hunt; tr denotes the T-Rex velocity factor; pr denotes the prey’s escape velocity factor; $V_{i}$ denotes the velocity vector of the T-Rex

3. After each position update, evaluate the fitness ${J_{i}}^{t + 1} = f ({X_{i}}^{t + 1})$ of the new position. Then, update the individual position and the global best solution according to the following rules

\begin{matrix} X_{i}^{t + 1} = {\begin{cases} {X_{i}}^{t + 1}, i f f ({X_{i}}^{t + 1}) < f ({X_{i}}^{t}) \\ X_{i}^{t}, otherwise \end{cases} \\ X_{best} = \arg \min_{X_{i}} f (X_{i}) \end{matrix}

(25)

4. Repeat steps 2 and 3 until the maximum number of iterations is reached or the change of the current optimal fitness value is less than the threshold. Finaly, output the global best position $X_{best}$ and the corresponding fitness value $f (X_{best})$ .

TROA-optimized CNN

In the fault classification task, residual data are first normalized using min–max scaling, where training samples are linearly mapped to the interval [0,1]. Considering the MDS of the rolling mill, ten state variables are selected, and a corresponding set of residual signals is collected for each. After normalization, these ten sets of residual signals jointly form a complete residual dataset, which serves as the input to the CNN. The structure of the CNN and its associated parameters are illustrated in Figure 2, and the fault configurations are summarized in Table 2.

TROA is integrated with CNN to optimize key hyperparameters, including the learning rate α and the regularization parameter β. This optimization enhances the classification performance of CNN, particularly when handling complex and nonlinearly separable residual datasets. The flowchart illustrating the TROA-based CNN hyperparameter optimization process is shown in Figure 3, and the main steps are summarized as follows:

The parameters of TROA are first initialized, including the population size, maximum number of iterations, and the boundaries of the search space. A population of T-Rex individuals is then randomly generated, and the initial positions of each individual—corresponding to candidate values for α and β are assigned accordingly.

Each individual’s parameter pair (α, β) is applied to configure the CNN, which is trained on the training and validation sets. The fitness value is then computed based on classification accuracy.

Based on the behavioral rules of TROA, each individual’s position is dynamically updated. The corresponding parameters (α, β) are reapplied to the CNN to recompute the fitness. If the new fitness improves upon the previous value, the position is updated; otherwise, the original position is retained.

Among all individuals, the current best parameter pair (α, β) is selected as the global optimal solution.

The iteration terminates either when the maximum number of iterations is reached or when the change in the global best fitness falls below a predefined threshold. The final optimized parameters, α and β, are then output as the optimal configuration for the CNN.

Experimental simulation

Experimental parameter setting

To verify the correctness and reliability of the proposed method, the parameters (Zhang and Tong, 2006) of MDS of a cold strip mill are shown in Table 1.

Table 1.

List of system model parameters.

Symbol	Value	Units	Symbol	Value	Units
$T_{L}$	60	$N \cdot m$	$L_{m}$	13.28	mH
$J_{m}$	0.1766	$kg \cdot m^{2}$	$L_{s}$	13.65	mH
$J_{L}$	0.1746	$kg \cdot m^{2}$	$L_{r}$	13.95	mH
K	695.567	$\frac{N}{m}$	$R_{s}$	0.0413	Ω
$C_{m}$	3.532	$\frac{N}{(\frac{m}{s})}$	$R_{r}$	0.0407	Ω
$C_{L}$	8.73	$\frac{N}{(\frac{m}{s})}$	$n_{p}$	2	-
R	0.5	m	a	6.25×10-5	-
P	80	kN	b	1.25×10-5	-
$u_{s}$	380	V	c	1.25×10-6	-
f	50	Hz

The initial states of the MDS of the mill and the initial states of the observer are set to be both zero states.

The disturbance distribution matrix is $D = {[\begin{matrix} 0 & 0 & 1 & \begin{matrix} 1 & 0 & 1 1 \end{matrix} \end{matrix}]}^{T}$ ; Actuator fault matrix and sensor fault matrix are as follows

\begin{matrix} F_{a} = {[\begin{matrix} 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \end{matrix}]}^{T}, \\ F_{s} = {[\begin{matrix} 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{matrix}]}^{T}, \\ f_{a} = [\begin{matrix} f_{a 1} \\ f_{a 2} \\ f_{a 3} \end{matrix}], f_{s} = [\begin{matrix} f_{s 1} \\ f_{s 2} \\ f_{s 3} \end{matrix}] \end{matrix}

Through the toolbox module, $u_{sd}$ and $u_{sq}$ are available directly from the AC three-phase voltages $U_{A}$ , $U_{B}$ and $U_{C}$

\begin{matrix} U_{A} (t) = 380 \sin (100 π) \\ U_{B} (t) = 380 \sin (100 π - 2 π / 3) \\ U_{C} (t) = 380 \sin (100 π - 4 π / 3) \end{matrix}

(26)

According to Theorem 1, define $δ = 10.101$ , $λ = 0.009$ , $θ = 14.5979$ . We solve the linear inequalities using the LMI toolbox to obtain matrices P, Y, and J (Appendix A), and subsequently derive matrices N and L from equation (9) (Appendix B).

Six typical independent faults, along with six compound faults formed by their coupling, are designed as summarized in Table 2.

Table 2.

Classification of rolling mill MDS faults.

Faults model	Occurrence time	Data label
normal	$0 s - 4 s$	0
$f_{a 1} f_{a 2}$	$4 s - 8 s$	1
$f_{a 1} f_{a 2} f_{a 3}$	$8 s - 12 s$	2
$f_{s 1} f_{s 2}$	$12 s - 16 s$	3
$f_{s 1} f_{s 2} f_{s 3}$	$16 s - 20 s$	4
$f_{a 1} f_{s 1}$	$20 s - 24 s$	5
$f_{a 1} f_{a 2} f_{s 2}$	$24 s - 28 s$	6

Assuming a rated power of $50 kW$ , with a line voltage of $380 V$ (three-phase), and adopting typical motor parameters with an efficiency of $η = 0.9$ and a power factor of $\cos φ = 0.85$ , the rated current is approximately $99 A$ and the rated torque is about $114 Nm$ .

Actuator fault 1: In the case of an armature short-circuit fault occurring in the MDS motor of the rolling mill, the fault current can reach 3 to 5 times the rated current. The corresponding actuator fault function is defined as

f_{a 1} = {\begin{matrix} 0 & 0 s \leq t < 4 s \\ 5 \times 99 s i n (31.4 t) & 4 s \leq t \leq 12 s \\ 0 & 12 s < t < 20 s \\ 5 \times 99 s i n (31.4 t) & 20 s < t < 28 s \end{matrix}

(27)

Actuator fault 2: A transmission shaft fault in the rolling mill MDS is modeled to introduce periodic fluctuations in the load angular velocity. Considering the nominal motor speed of approximately 42 rad/s, the fault disturbance amplitude is set to 30 rad/s, representing about 70% of the nominal speed, to simulate a severe fault scenario. The fault function is defined as

f_{a 2} = {\begin{matrix} 0 \\ 30 \sin (50 t) \\ 0 \\ 30 \sin (50 t) \end{matrix} \begin{matrix} 0 s \leq t < 4 s \\ 4 s \leq t \leq 12 s \\ 12 s < t < 24 s \\ 24 s < t < 28 s \end{matrix}

(28)

Actuator fault 3: A biting-induced impact fault in the rolling mill MDS is modeled with two components: a ramp term with an amplitude of $60 N \cdot m$ (approximately 50% of the rated torque) to simulate strong impact torque accumulation, and a sinusoidal term with an amplitude of $40 N \cdot m$ (about 33% of the rated torque) to represent torsional vibration. The fault function is defined as

f_{a 3} = {\begin{matrix} 0 & 0 s \leq t < 8 s \\ 60 (t - 8) + 40 s i n (30 t) & 8 s \leq t \leq 12 s \\ 0 & 12 s < t < 28 s \end{matrix}

(29)

Sensor fault 1: A rotational encoder fault in the MDS motor of the rolling mill occurs when interference in the encoder signal prevents the controller from obtaining accurate angular velocity feedback. The actual motor speed is approximately $42 rad / s$ . The corresponding fault function is defined as

f_{s 1} = {\begin{matrix} 0 & 0 s \leq t < 12 s \\ 42 s i n (100 t) & 12 s \leq t \leq 24 s \\ 0 & 24 s < t < 28 s \end{matrix}

(30)

Sensor fault 2: A current sensor fault in the MDS motor of the rolling mill directly affects the motor’s current. The rated current of the motor is approximately $99 A$ . The corresponding fault function is defined as

f_{s 2} = {\begin{matrix} 0 & 0 s \leq t < 12 s \\ 99 c o s (25 t) & 12 s \leq t \leq 20 s \\ 0 & 20 s < t < 24 s \\ 99 c o s (25 t) & 24 s < t < 28 s \end{matrix}

(31)

Sensor fault 3: The torque sensor is used to monitor and regulate the torque of the rolling mill. In the event of a fault, the torque is limited to $60 N \cdot m$ with low-frequency fluctuations, causing the torque loop to become unstable. The corresponding fault function is defined as

f_{s 3} = {\begin{matrix} 0 & 0 s \leq t < 16 s \\ 60 s i n (5 t) & 16 s \leq t \leq 20 s \\ 0 & 20 s < t < 28 s \end{matrix}

(32)

Computational complexity analysis shows that each UIO iteration requires 254 multiplications, with matrix multiplication accounting for 93.7% (238 operations). Nonlinear terms contribute the remaining 6.3% (14 operations). Specifically, the 10×10 state transition matrix calculation consumes 100 multiplications, representing the primary bottleneck, while the nonlinear coupling terms require only 14 multiplications.

A dynamic model of the rolling mill MDS was built in Simulink based on equation (8), with parameters listed in Table 1. Compound fault scenarios (Table 2) were introduced, and a UIO designed via equation (9) was employed to estimate system states. The resulting 10-dimensional residuals served as input features. Simulations were run for 28 seconds at 1000 Hz, yielding 6000 compound fault samples, with numerical integration performed using the ode4 (Runge-Kutta) solver. The plotted curves are discrete simulation results where a single-point jump is visually smoothed by interpolation.

The estimated state variables of the MDS are shown in Figure 4. As seen in Figure 4(a) from 4 to 8 seconds, when $f_{a 1} f_{a 2}$ occur, the observer tracks the system state within 0.01 seconds. After 0.05 seconds, the motor’s angular velocity reaches its peak, and the observer’s state estimate closely matches the actual system state with an estimation error not exceeding 0.2%, indicating rapid state tracking and high accuracy. From 8 to 12 seconds in Figure 4(a), when $f_{a 1} f_{a 2} f_{a 3}$ occurs, the motor’s angular velocity undergoes severe oscillations. The system still tracks the state promptly, with an estimation error not exceeding 0.3%. In Figure 4(b), when $f_{a 1} f_{s 1}$ occur simultaneously, slight oscillations in the roller’s angular velocity are observed. The observer continues to track the system state effectively, with the estimation error remaining below 0.01%.

By constructing an augmented-dimensional system, the decoupling of compound sensor faults is directly achieved, with the system state variables $x_{8}$ , $x_{9}$ and $x_{10}$ representing their characteristics. As shown in Figure 4(d), $f_{s 1}$ occurs between 12 and 24 seconds; in Figure 4(e), $f_{s 2}$ occurs between 12 and 20 seconds and again between 24 and 28 seconds; and in Figure 4(f), fault $f_{s 3}$ occurs between 16 and 20 seconds. This process successfully decouples the sensor faults, as well as the sensor-actuator faults, transforming complex compound faults into simpler, individual faults.

To demonstrate the advantages of the proposed fault detection method, we compared it with the SMO-based fault diagnosis method proposed by Gou et al. (2020). In the experiment, the fault type and occurrence time, as shown in Table 2, are selected for simulation, and the same external disturbance signal $d (t) = rand 0.003 \sin (20 π t)$ is applied to the rolling mill MDS. Taking the angular velocity difference of the motor as an example, the simulation results are shown in Figure 5. At 2.9 seconds, the system is fault-free and is only affected by external disturbances. As shown in Figure 5, the UIO-based method exhibits minimal numerical fluctuation, demonstrating strong anti-interference capability. The RMSE observed by the UIO is 0.11096 rad/s, while the RMSE observed by the SMO is 0.13286 rad/s. The observation error of the UIO is approximately 16.48% lower than that of the SMO, indicating that the UIO-based method effectively filters interference signals while maintaining high output accuracy.

The representative residual signals $r_{1}$ , $r_{6}$ , and $r_{10}$ are selected from the 10 groups of state residuals collected for analysis, corresponding to the motor flux, roll angular velocity, and sensor fault 3, as shown in Figure 6. Between 0 and 1 seconds, the residual errors of $r_{1}$ , $r_{6}$ , and $r_{10}$ fluctuate significantly, indicating system instability during the startup phase; however, their maximum value does not exceed 0.04. From 1 to 4 seconds, before any faults occur, the residuals $r_{1}$ , $r_{6}$ , and $r_{10}$ are all close to 0, demonstrating the strong robustness of the proposed UIO. Between 4 and 12 seconds, an actuator fault occurs in the MDS, causing significant fluctuation in $r_{1}$ and $r_{6}$ , while $r_{10}$ shows no fluctuation. Between 16 and 20 seconds, a sensor fault occurs in the MDS, with no fluctuation in $r_{1}$ and $r_{6}$ , but significant fluctuation in $r_{10}$ , demonstrating the good sensitivity and accuracy of the observer. Throughout the operation, the residual signals $r_{1}$ , $r_{6}$ , and $r_{10}$ fluctuate within the range of 0.04, indicating that the designed UIO exhibits good convergence.

Fault classification results based on TROA-CNN

To validate the effectiveness of the proposed UIO-TROA-CNN fault diagnosis method, the residual signals generated by the UIO during high-precision state estimation of the MDS are utilized in combination with the TROA-CNN classifier for fault classification. During the construction of the experimental dataset, multi-fault residual data were collected by controlling the occurrence times of specific faults. The detailed timing configuration of each fault is presented in Table 2.

To evaluate the generalization capability of the proposed model, the residual dataset is randomly shuffled, with 60% allocated for training, 10% for validation, and 30% for testing. An early stopping mechanism is applied by monitoring validation loss every 30 iterations; training is terminated if the validation loss fails to improve after 30 consecutive checks, while preserving the best-performing weights.

The Adam optimizer is used for initial gradient descent, with the search ranges for α and β set to [1e-4,0.1] and [1e-5,0.1], respectively. Noise with a standard deviation of 5% is added to the test set to evaluate the model’s robustness.

To compare the convergence mechanisms and randomness sources of different optimization algorithms, their characteristics are summarized in Table 3.

Table 3.

Comparison of algorithm characteristics.

Algorithm	Convergence Mechanism	Key Parameters	Randomness Source
TROA	Probabilistic factor ES	$sr = 0.8$ $pr = 0.25$ $tr = 0.3$	Dynamic adjustment
WOA	Linear decay $a = 2 - 2 t / T$ helix parameter $b = 1$	Probability threshold $p = 0.5$ Auxiliary factor $a 2 = - 1 - t / T$	Uniform distribution
GWO	Linear decay $a = 2 - 2 t / T$	Social weight $A = 2 arand - a$	Uniform distribution
Bayes	Gaussian process surrogate model	Kernel bandwidth $σ \in [1 e - 4, 1]$ Acquisition function $EI (x)$	Probabilistic sampling

Figure 7(a)–(d) present the confusion matrices obtained using different optimization methods. Figure 7(e) summarizes the comparison of the proposed TROA-CNN model with other algorithms across multiple key metrics. The detailed analysis is as follows:

1. Classification Accuracy: TROA-CNN significantly outperforms the other models in both average accuracy (99.33%) and highest accuracy (99.76%), with improvements of 5.18% and 3.88%, respectively, over the next best model, WOA-CNN. This demonstrates that the algorithm achieves an optimal balance between exploration and exploitation by setting the hunting success rate sr.

Bayes-CNN, which benefits from Bayesian optimization, demonstrates superior parameter tuning efficiency, achieving a training time of 27 seconds. However, its accuracy (90.12%) remains lower than that of TROA-CNN, indicating that traditional optimization algorithms have inherent limitations when handling complex feature classification tasks.

2. The training time (53 seconds) and number of iterations (7140) for the TROA-CNN are higher than those of other models, primarily due to the additional computational overhead introduced by its multi-stage optimization mechanism. WOA-CNN, due to the population iteration characteristics of the Whale Optimization Algorithm, has a higher training time (41 seconds) and number of iterations (5370). However, its accuracy (94.15%) does not reach the optimal value, which validates the inherent limitations of heuristic algorithms in local optima traps. The computational complexity, including both time and space complexity, is evaluated using FLOPs and the number of parameters, consistent with the majority of existing methods. The FLOPs of the proposed network are 7936, and the number of parameters is 2262.

3. Noise Robustness Validation: On the test set with 5% noise, the noise robustness accuracy of the TROA-CNN (96.27%) decreased by only 3.06% compared to clean data (99.33%), which is significantly lower than that of CNN (6.12%) and Bayes-CNN (4.34%). This demonstrates that, through the settings of pr and tr, the TROA-CNN forces the algorithm to consistently escape from suboptimal regions, with the prey escape mechanism effectively enhancing its anti-interference capability. In the horizontal comparison, the noise robustness of TROA-CNN outperforms the next best model, WOA-CNN (90.28%), by 5.99%, indicating that its prey escape mechanism effectively filters noise interference and enhances generalization ability.

4. Scalability Analysis: The hardware platform used in this study is an NVIDIA RTX 4060 Laptop GPU (8GB GDDR6). For the experimental dataset with six fault types and 6000 samples, the model requires 45 seconds for training, and the inference time per sample is only 1.1 ms.

To evaluate the scalability of the proposed TROA-CNN model, a controlled variable experiment was conducted. When the number of fault types increased from 6% to 12 (a 100% increase), the inference time rose slightly to 1.1 ms (a 10% increase), while the training time increased to 70 seconds (a 35% increase). Similarly, when the number of training samples was doubled from 6000 to 12,000, the inference time remained at 1.1 ms (+10%), and the training time increased to 77 seconds (a 41% increase).

These results indicate that TROA-CNN maintains stable inference efficiency while showing predictable growth in training time. It is therefore well-suited for industrial applications that require real-time performance and must accommodate evolving system complexity or expanding data volumes.

Conclusion

To address compound fault diagnosis involving actuator faults, sensor faults, and unknown disturbances in the rolling mill MDS, this study proposes a scheme combining a model-based UIO with a TROA-optimized CNN. Based on experimental simulations, the following conclusions are drawn:

The proposed UIO-TROA-CNN framework effectively decouples and isolates compound faults while overcoming the difficulty of acquiring sufficient compound fault samples. By solving the observer parameter matrices, the UIO generates high-quality residual signals, providing reliable input for CNN-based classification.

Under narrow-band sinusoidal interference with a 10 Hz center frequency and randomized phase, the designed UIO mitigates external disturbance effects. Compared to the traditional SMO method, the proposed approach reduces the RMSE by 16.48%, with a significantly smaller fluctuation range, ensuring more reliable state estimation.

Through multi-stage optimization and adaptive parameter tuning, TROA-CNN achieves high accuracy, strong robustness, and good scalability, making it well-suited for industrial environments with high noise and complex fault types.Future work may focus on further improving computational efficiency to accommodate larger-scale datasets and more intricate fault patterns.

Footnotes

Appendix A

P = [\begin{matrix} 22.04 & 0.68 & - 5.92 & 0.07 & 1.01 \times 10^{- 8} \\ 0.68 & 39.11 & - 1.69 & 0.03 & - 3.32 \times 10^{- 9} \\ - 5.92 & - 1.69 & 53.33 & - 0.40 & 2.62 \times 10^{- 8} \\ 0.07 & 0.03 & - 0.40 & 38.84 & - 2.87 \times 10^{- 9} \\ - 1.01 \times 10^{- 8} & - 3.32 \times 10^{- 9} & 2.62 \times 10^{- 8} & - 2.87 \times 10^{- 9} & 19.81 \\ - 6.05 \times 10^{- 5} & - 9.07 \times 10^{- 6} & 1.37 \times 10^{- 4} & 5.75 \times 10^{- 5} & - 1.77 \times 10^{- 6} \\ - 1.05 \times 10^{- 4} & - 1.23 \times 10^{- 4} & 6.09 \times 10^{- 4} & - 4.77 \times 10^{- 4} & 0 \\ 1.01 \times 10^{- 8} & 3.32 \times 10^{- 9} & 2.62 \times 10^{- 8} & 2.87 \times 10^{- 9} & 19.10 \\ 16.87 & - 0.68 & 5.92 & - 0.07 & 1.01 \times 10^{- 8} \\ 6.05 \times 10^{- 5} & 9.07 \times 10^{- 6} & - 1.37 \times 10^{- 4} & - 5.75 \times 10^{- 5} & 1.77 \times 10^{- 6} \end{matrix} \begin{matrix} - 6.05 \times 10^{- 5} & - 1.05 \times 10^{- 4} & 1.01 \times 10^{- 8} & 16.87 & 6.05 \times 10^{- 5} \\ - 9.07 \times 10^{- 6} & - 1.23 \times 10^{- 4} & 3.32 \times 10^{- 9} & - 0.68 & 9.07 \times 10^{- 6} \\ 1.37 \times 10^{- 4} & 6.09 \times 10^{- 4} & - 2.62 \times 10^{- 8} & 5.92 & - 1.37 \times 10^{- 4} \\ 5.75 \times 10^{- 5} & - 4.77 \times 10^{- 4} & 2.87 \times 10^{- 9} & - 0.07 & - 5.75 \times 10^{- 5} \\ - 1.77 \times 10^{- 6} & 0 & 19.10 & 1.01 \times 10^{- 8} & 1.77 \times 10^{- 6} \\ 19.62 & - 0.01 & 1.77 \times 10^{- 6} & 6.05 \times 10^{- 5} & 19.29 \\ 0 & 38.92 & - 0.01 & 1.05 \times 10^{- 4} & 0 \\ 1.77 \times 10^{- 6} & - 0.01 & 19.81 & - 1.01 \times 10^{- 8} & - 1.77 \times 10^{- 6} \\ 6.05 \times 10^{- 5} & 1.05 \times 10^{- 4} & - 1.01 \times 10^{- 8} & 22.04 & - 6.05 \times 10^{- 5} \\ 19.29 & 0 & - 1.77 \times 10^{- 6} & - 6.05 \times 10^{- 5} & 19.62 \end{matrix}]

Y = [\begin{matrix} 1.44 \times 10^{5} & 689.49 & 9.33 \times 10^{3} & - 869.41 & 3.32 \times 10^{- 5} & 2.40 \times 10^{- 3} & 0.41 \\ - 2.82 \times 10^{3} & 2.99 \times 10^{5} & - 9.09 \times 10^{3} & 2.67 \times 10^{4} & 2.53 \times 10^{- 5} & 2.56 \times 10^{- 4} & - 0.66 \\ 2.45 \times 10^{4} & - 9.09 \times 10^{3} & 3.56 \times 10^{5} & - 20.68 & - 9.22 \times 10^{- 5} & - 3.50 \times 10^{- 3} & 1.20 \\ - 189.45 & 2.67 \times 10^{4} & - 20.68 & 2.96 \times 10^{5} & 6.96 \times 10^{- 5} & - 1.70 \times 10^{- 3} & - 1.37 \\ 1.22 \times 10^{- 5} & - 1.48 \times 10^{- 5} & 2.03 \times 10^{- 5} & 4.62 \times 10^{- 5} & 1.48 \times 10^{5} & 2.77 \times 10^{- 4} & - 2.81 \times 10^{3} \\ 0.09 & 0.01 & 0.51 & 0.28 & 7.81 \times 10^{- 4} & 1.49 \times 10^{5} & - 1.34 \times 10^{3} \\ 0.28 & - 0.66 & 1.20 & - 1.37 & - 6.57 & 6.52 & 2.98 \times 10^{5} \\ - 2.45 \times 10^{- 5} & 2.65 \times 10^{- 5} & - 1.44 \times 10^{- 4} & 5.37 \times 10^{- 5} & 1.49 \times 10^{5} & - 7.09 \times 10^{- 4} & 2.81 \times 10^{3} \\ 1.53 \times 10^{5} & - 3.51 \times 10^{3} & 1.52 \times 10^{4} & 679.96 & - 2.09 \times 10^{- 5} & - 2.40 \times 10^{- 3} & - 0.13 \\ - 0.09 & - 0.01 & - 0.52 & - 0.28 & - 3.49 \times 10^{- 4} & 1.49 \times 10^{5} & 1.35 \times 10^{3} \end{matrix}]

J = [\begin{matrix} 3.46 \times 10^{3} & - 747.91 & 9.86 \times 10^{3} & - 348.58 & 2.78 \times 10^{- 6} & 5.24 \times 10^{- 5} & 0.09 \\ - 43.19 & 7.67 \times 10^{3} & - 186.62 & 688.95 & 5.91 \times 10^{- 8} & - 2.01 \times 10^{- 6} & 1.40 \times 10^{- 3} \\ 379.34 & - 80.32 & 8.79 \times 10^{3} & 2.82 & 7.82 \times 10^{- 7} & 4.64 \times 10^{- 5} & - 0.02 \\ 0.50 & 683.28 & 56.05 & 7.62 \times 10^{3} & - 2.5 \times 10^{- 7} & 1.83 \times 10^{- 5} & 0.06 \\ 2.78 \times 10^{- 6} & - 3.93 \times 10^{- 5} & 1.95 \times 10^{- 4} & - 1.2 \times 10^{- 4} & 3.78 \times 10^{3} & 1.93 \times 10^{- 4} & - 3.95 \times 10^{3} \\ - 2.70 \times 10^{- 3} & - 0.10 & 1.37 & - 0.55 & 4.28 \times 10^{- 4} & 3.80 \times 10^{3} & - 3.94 \times 10^{3} \\ - 6.61 \times 10^{- 4} & 0.01 & - 0.05 & 0.05 & - 0.16 & 0.16 & 7.65 \times 10^{3} \\ - 3.10 \times 10^{- 6} & 3.96 \times 10^{- 5} & - 1.99 \times 10^{- 4} & 1.23 \times 10^{- 4} & 3.86 \times 10^{3} & - 2.04 \times 10^{- 4} & 3.95 \times 10^{3} \\ 4.19 \times 10^{3} & 675.30 & - 9.23 \times 10^{3} & 343.71 & - 2.47 \times 10^{- 6} & - 5.09 \times 10^{- 5} & - 0.08 \\ 2.70 \times 10^{- 3} & 0.10 & - 1.37 & 0.55 & - 4.17 \times 10^{- 4} & 3.84 \times 10^{3} & 3.94 \times 10^{3} \end{matrix}]

Appendix B

N = 1000 \times [\begin{matrix} - 3.4699 & 1.0621 & - 9.8656 & 0.3486 & 0 \\ - 0.1139 & - 7.6725 & 0.1866 & - 0.6889 & 0 \\ 0.9984 & 0.0803 & - 8.8318 & 0.1543 & 0 \\ - 0.0005 & 0.6945 & - 0.2131 & - 7.6663 & 0 \\ 0 & 0 & 0 & 0 & - 3.8686 \\ 0 & 0.0001 & 0.0014 & 0.0006 & 0 \\ 0 & 0 & 0.0001 & - 0.0001 & - 0.0003 \\ 0 & 0 & 0 & 0 & - 3.7886 \\ - 4.1874 & - 0.9895 & 9.2337 & - 0.3437 & 0 \\ 0 & - 0.0001 & 0.0014 & - 0.0006 & 0 \end{matrix} \begin{matrix} 0 & - 0.0001 & 0 & - 3.4670 & 0 \\ 0 & 0 & 0 & 0.0432 & 0 \\ 0 & 0 & 0 & - 0.3793 & 0 \\ 0 & - 0.0001 & 0 & - 0.0005 & 0 \\ 0 & 0.0182 & - 3.7886 & 0 & 0 \\ - 3.8486 & - 0.0385 & 0 & 0 & - 3.8086 \\ 0.0003 & - 7.6572 & 0.0002 & 0 & - 0.0002 \\ 0 & - 0.0181 & - 3.8686 & 0 & 0 \\ 0 & 0.0001 & 0 & - 4.1903 & 0 \\ - 3.8087 & 0.0383 & 0 & 0 & - 3.8487 \end{matrix}]

L = 1000 \times [\begin{matrix} 0 & - 0.2169 & 4.9329 & - 0.1743 & 0 & 0 & 0 \\ 0 & 3.8348 & - 0.0933 & 0.3443 & 0 & 0 & 0 \\ 0 & - 0.0402 & 4.3771 & 0.0800 & 0 & 0 & 0 \\ 0 & 1.0305 & - 0.0505 & 3.7944 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - 3.9478 \\ 0 & 0.0001 & 0.0007 & - 0.0003 & 0 & 0 & - 3.9645 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3.8286 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3.9477 \\ 0 & 0.1806 & - 4.6169 & 0.1719 & 0 & 0 & 0 \\ 0 & 0.0001 & - 0.0007 & 0.0003 & 0 & 0 & 3.9646 \end{matrix}]

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publicationof this article: This project was supported by TangshanTalent Support Project (B202302009); TangshanScience and Technology BureauScience and Technology PlanProject (22130213G); Key Scientific ResearchProject of North China Universityof Science and Technology in2023 (ZD-ST-202302-23).

ORCID iDs

Ruicheng Zhang

Weizheng Liang

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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