Sliding-mode observer-based fault diagnosis and fault-tolerant control of the main drive system of rolling mill

Abstract

In order to address the problem that the main drive system of rolling mill is easily affected by the impact of biting steel, and considering the nonlinear friction damping and the external perturbations of the main drive system of rolling mill during the rolling process, a fault model of the main drive system of rolling mill is established, and a fault diagnosis and fault tolerance control method of the main drive system of rolling mill based on the nonlinear sliding-mode observer is proposed. In order to suppress the influence of external perturbations on fault diagnosis, a nonlinear sliding-mode observer is constructed for fault diagnosis and fault reconfiguration of the system, and the robustness of the observer to fault reconfiguration is improved by using the sliding-mode control rate $v$ , and the stability of the designed nonlinear sliding-mode observer is proved using Lyapunov’s stability theorem. In order to ensure that the system can operate normally even after a fault occurs, a reference model is designed, and a new controller is redesigned for fault-tolerant control of the system by adding a fault compensation term to the original control scheme using fault estimation information. Through the simulation study of the main drive system of stand F4 of 2030 mm cold rolling mill, it is verified that the observer can accurately track the system state with an angular velocity error of 2.45% and detect and estimate the main drive system failure of rolling mill with an estimation error of no more than 0.04% after a fault occurs; the fault-tolerant control of the main drive system of rolling mill is carried out by using the fault information to restore the system to its normal state, and the angular velocity error is 1.89%.

Keywords

Fault diagnosis nonlinear sliding-mode observer main drive system of rolling mill fault-tolerant control linear matrix inequality

Introduction

In the past few years, with the rapid developments in the steel industry, the rolling speed of mills has increased substantially, and the vibration problem of mills has gradually become a technical problem plaguing the steel industry. Vibration in rolling mills takes many forms, including torsional vibration of the main drive system of rolling mill (MDSORM; Peng et al., 2021), vertical vibration of the rolling system stand, rolls and mill parts, and horizontal vibration of the rolls and mill parts in the rolling system (Liu et al., 2017). The MDSORM is the core of the strip mill, and its stability is closely related to the quality of steel. Once the MDSORM fails due to vibration, it can cause a huge economic loss to the enterprise. Therefore, the industry’s requirements for the stability of the MDSORM have increased substantially, and it is important to study the fault diagnosis (FD) and fault-tolerant control (FTC) of the MDSORM (Lu et al., 2020; Qian et al., 2021; Zhu et al., 2020).

Among the model-based FD, the observer-based FD method has gained the attention of many experts and scholars. Zhong et al. (2018) considered linear discrete time-varying systems and classified model-based linear discrete time-varying FD techniques as observer-based methods, parity space-based methods, and parameter estimation methods. In the work of Zhirabok et al. (2020), a sliding-mode observer-based method is used for sensor faults in linear dynamic systems under perturbations to resolve the FD problem in the presence of system perturbations. Nevertheless, in the actual production processes, the systematic usually contains nonlinear factors. In the work of Gao et al. (2019), for nonlinear systems satisfying the Lipschitz condition, a method that can eliminate parameter ingestion and measurement noise in the system output is proposed, while a new estimation method is proposed to solve the effects caused by fault bias and to realize the fault estimation problem for nonlinear systems. In the work of Baldi et al. (2019), an adaptation observer for the detection and estimation of sensor and actuator faults was designed considering the nonlinearities factor and for the defects in satellite attitude, body angular rate, flywheel rotation rate, and reaction wheel motor control torque. In contrast, Nemati et al. (2019) considered uncertain inputs and external perturbations to establish a nonlinear system model, designed a nonlinear observer to eliminate the effects of uncertainty and a nonlinear robust position input observer to isolate actuator faults, and finally used a generalized observer to estimate both faults and states. For time-varying systems, Zemzemi et al. (2019) considered nonlinear time-varying uncertain systems that satisfy the Lipschitz condition, proposed a mixed method that models sensor faults as actuator faults based on an integral observer and sliding-mode theory, and proposed an improved design method to improve the fault estimation convergence speed and realize the state estimation and sensor fault reconfiguration. Through the above literature analysis, it has been concluded that observer-based FD techniques have matured, and in order to guarantee the normal operation of the system even after a fault occurs, FTC research is still needed.

FTC can be classified into two main categories, passive FTC (Li et al., 2019) and active FTC (Abbaspour et al., 2020; Boem et al., 2019), depending on whether FD information is utilized or not. Among the passive FTC methods, an adaptive passive FTC method based on sliding-mode control suggests in the work of Nasiri et al. (2019) for the control of affine class actuator faults in uncertain multi-input multi-output nonlinear systems. In the work of Chen et al. (2019), an autonomous distributed-drive electric vehicle’s passive fault-tolerant path-following control method is proposed for vehicle steering system faults, using a reduced-order Kalman filter to evaluate the vehicle sideslip angle and steering system faults. Although passive FTC methods can maintain a certain system performance, the passive FTC has limited adaptive capability to fault tolerance when faults occur, while the active FTC can proactively handle the occurring faults and can increase the control system performance to a greater extent. Therefore, active FTC has gained a wide range of scholarly attention. In the active FTC method, in the work of Gao et al. (2018), a nonlinear attitude system model for satellites with multiple actuator faults is developed for satellite attitude system with external perturbations and multiple actuator faults, a fault detection and fault estimation module is designed to detect the time of actuator fault and estimate the fault value, and then a fault-tolerant controller based on terminal sliding mode is designed using backstepping control technique to achieve FTC of the satellite attitude system. In contrast, a new adaptive sliding-mode control method is presented in the work of Wang et al. (2020) which guarantees the system tracking performance without causing control jitter, while a model-based fault estimation scheme was proposed for the uncertainty in the system and combined with the proposed adaptive sliding-mode control for FTC of the quadrotor helicopter system. Wang et al. (2021) proposed an adaptive interval observer for fault detections in an uncertain system, which uses adaptive parameters instead of uncertainty bounds, to improve the sensitivity of the observer to faults and activate a reconfigured controller for FTC of the system when a fault isolation module determines the location of the fault. In contrast, Qin et al. (2017) proposed a two-level control scheme consisting of outer-loop PD control lemma and inner-loop PID control lemma, while designing a residual generation based on Luenberger observer to achieve FD and a new incremental generalized variable observer to achieve fault estimation, which combines fault estimation with the outer-loop PD control lemma to achieve fault-tolerant quadrature sensor fault control. Wang and Shen (2018) constructed an augmented system to convert sensor faults to actuator faults, collected fault information through equivalent output control, gave an algorithm for the simultaneous reconfiguration of sensor faults and actuator faults, and proposed a robotic sliding-mode observer to implement multi-fault reconfiguration and achieved FTC of simultaneous sensor and actuator faults of the system. Liu et al. (2018) addressed the problem that induction motor sensors are prone to faults and achieve FTC of induction motor drives smoothly using active direct torque control, indirect field orientation control, and volt/hertz controllers in place of each other in the event of sensor failure. In contrast, Ebadpour et al. (2021) addressed the problem that the Hall sensor of a brushless DC motor is prone to failure and proposed an improved fast FTC algorithm that can quickly identify and compensate for Hall sensor failures, realizing FTC of brushless DC motor. In the work of Zhang et al. (2021), an observer based on the super twist algorithm is proposed and combined with an established extended finite state machine to establish a real-time FD strategy for Hall sensors for the problem that Hall sensors of permanent magnet brushless DC motor are prone to faults. The Hall signal reconfiguration is used to generate a compensated Hall signal after the fault is diagnosed to achieve FTC under Hall sensor faults.

Through the above literature analysis, FTC has been widely used in DC motor faults, but there is less research on FTC for DC motor load-side faults. Therefore, considering the external perturbations, nonlinear factors, and load-side faults in the MDSORM, a fault mathematical model of the main drive system of the strip mill is established, a nonlinear sliding-mode observer is devised, and a model-referenced FTC method is used to study the feasibility of its use for FD and FTC of the MDSORM. To further optimize the anti-disturbance capability of the observer, the impact of external perturbations on the system is eliminated by using the sliding-mode control rate $v$ in the design process of the observer, which improves the robustness of fault estimation. The simulation study of the MDSORM proves the effectiveness of the proposed method in FD and FTC.

In conclusion, the main contributions of this paper can be summarized as follows:

An FD scheme based on a nonlinear sliding-mode observer for the MDSORM is proposed, and an adaptation law of fault estimation is given for fault reconfiguration.

A model-reference FTC scheme for the MDSORM is proposed.

The structure of the rest of the paper is as follows: Section “The fault system mathematical model” will introduce the fault model of the MDSORM. Section “Observer-based fault diagnosis of the main drive system of rolling mill” will introduce the stability analysis of nonlinear sliding-mode observer, FD criterion, and fault reconfiguration method. Section “Model-referenced fault-tolerant control of the main drive system of rolling mill” will introduce the model-referenced FTC of the MDSORM. Section “Simulation and verification” will simulate the MDSORM under the impact of biting steel, and the effectiveness of the scheme will be verified by adding the model-referenced FTC to make the system operate normally. Finally, conclusions are given in Section “Conclusion.”

The fault system mathematical model

The MDSORM is a complex system of multiple degrees of freedom composed of motor, elastic connecting shaft, reducer, roll, and various coupling. In general, the MDSORM can be simplified into a mechanical model with two moments of inertia connected by a spring, as shown in Figure 1.

In Figure 1, variables $T_{1}$ , $T_{2}$ , and $T_{f}$ denote the electromagnetic torque, load torque, and friction torque, respectively, $θ_{1}$ and $θ_{2}$ denote the angular velocity of the motor and load, respectively, $J_{1}$ and $J_{2}$ denote the rotational inertia of the motor and load, respectively, $C_{1}$ and $C_{2}$ denote the damping coefficients of the motor and load, respectively, and $K$ is the connecting shaft stiffness coefficient.

In consideration of the MDSORM in the rolling process may encounter external perturbations and consider nonlinear factors and the impact of biting steel shock vibration on the MDSORM, the establishment of the MDSORM nonlinear failure model as (Qian et al., 2022; Zhang and Li, 2023)

{\begin{matrix} \overset{\cdot}{x} = Ax + Bu + f (x) + Dd + {Ef}_{a} \\ y = Cx \end{matrix}

(1)

where $A, B, C$ denotes the system matrix; $D$ is the disturbance distribution matrix; $d$ is the matrix of external perturbations to the system; $E$ is the fault distribution matrix; $f_{a}$ is the system fault function; and $f (x)$ is the nonlinear function and satisfies the Lipschitz condition, that is, $| | f (x) - f (\hat{x}) | | \leq γ | | x - \hat{x} | |$ , where $γ$ is the Lipschitz constant and $γ > 0$ .

Define the state variable matrix and input matrix of the MDSORM as $x = [{\overset{\cdot}{θ}}_{1} {\overset{\cdot}{θ}}_{2} θ_{1} - θ_{2}]^{T}$ and $u = [I_{c} T_{2} 0]^{T}$ , where $θ_{1}$ is the motor rotation angular displacement; $θ_{2}$ is the roll rotation angular displacement; $I_{c}$ is the motor rated current; and $T_{2}$ is the load torque at the roll end.

Observer-based fault diagnosis of the main drive system of rolling mill

Assumption 1. The system is observable. Then there exists a feedback matrix $L \in R^{3 \times 3}$ such that all the characteristic roots of $A - LC$ have negative real parts.

Assumption 2. There exist matrices $P \in R^{3 \times 3}$ , $Q \in R^{3 \times 3}$ , and $F \in R^{1 \times 3}$ such that $(A - LC)^{T} P + P (A - LC) = - Q$ and $PD = C^{T} F^{T}$ .

Assumption 3. There exists a constant $ρ > 0$ such that $| | d | | \leq ρ$ .

Assumption 4. There exists a symmetric positive definite matrix $P_{0}$ such that $G = - P_{0}^{- 1} E^{T} P C^{- 1}$ .

Nonlinear sliding-mode observer

Based on Assumptions 1–4, the following form of sliding-mode observer is designed for the main drive failure system of rolling mill described in equation (1)

{\begin{matrix} \overset{\cdot}{\hat{x}} = A \hat{x} + Bu + \hat{f} (x) - L (\hat{y} - y) + Dv + E {\hat{f}}_{a} \\ \hat{y} = C \hat{x} \end{matrix}

(2)

where $\hat{x}$ and $\hat{y}$ are the estimates of $x$ and $y$ , respectively; $L$ is the gain matrix; ${\hat{f}}_{a}$ is the fault estimate; and $v$ is the sliding-mode control rate function.

Definition: $e_{y} = \hat{y} - y$ is the residual generated by the observer, and let the sliding-mode surface be $e_{y}$ to obtain the expression for the sliding-mode control rate $v$ as

v = - ρ \frac{F e_{y}}{‖ F e_{y} ‖ + δ}

(3)

where $δ$ is a very small scalar constant, and $δ$ is designed to avoid jitter in the system (Spurgeon, 2008).

The sliding-mode control rate $v$ is used to suppress the effect of external perturbations so that the residuals contain only fault information, and the fault estimation algorithm can be designed based on the residuals, whose expression is (Han et al., 2018)

{\dot{\hat{f}}}_{a} = G e_{y}

(4)

where $G \in R^{3 \times 3}$ is the appropriate dimensional matrix to be designed.

Define the status errors as

e = \hat{x} - x

(5)

The fault estimate error is

e_{f_{a}} = {\hat{f}}_{a} - f_{a}

(6)

From equations (1) and (2), we have

\begin{array}{l} \dot{e} = \dot{\hat{x}} - \dot{x} = (A - L C) e + (\hat{f} (x) - f (x)) \\ + D (v - d) + E e_{f_{a}} \end{array}

(7)

In order to achieve fault reconfiguration, the derivation of equation (6) yields

{\overset{\cdot}{e}}_{f_{a}} = {\overset{\cdot}{\hat{f}}}_{a} - {\overset{\cdot}{f}}_{a}

(8)

Stability proof

Theorem 1. Consider the MDSORM and a nonlinear sliding-mode observer if there exist symmetric positive definite matrices $P$ and $Q$ and constants $γ$ such that the following linear matrix inequality holds

[\begin{matrix} A^{T} P + PA + C^{T} Q + QC + I P \\ * - \frac{1}{γ^{2}} I \end{matrix}] < 0

(9)

Then the state estimation error dynamic equation (7) of the nonlinear observer is asymptotically stable with the fault estimation error dynamic equation (8).

Proof. Select Lyapunov function

V = e^{T} Pe + e_{f_{a}}^{T} P_{0} e_{f_{a}}

where $P_{0}$ is the symmetric positive definite matrix.

The derivative is obtained by taking $V$

\begin{array}{l} \dot{V} = 2 e^{T} P \dot{e} + 2 e_{f_{a}}^{T} P_{0} {\dot{e}}_{f_{a}} \\ = 2 e^{T} P (A - L C) e + 2 e^{T} P (\hat{f} (x) - f (x)) \\ + 2 e^{T} P D (v - d) + 2 e^{T} P E e_{f_{a}} + 2 e_{f_{a}}^{T} P_{0} {\dot{e}}_{f_{a}} \end{array}

(10)

According to $| | f (x) - f (\hat{x}) | | \leq γ | | x - \hat{x} | |$ , we can further obtain

\begin{matrix} \overset{\cdot}{V} \leq e^{T} ({(A - LC)}^{T} P + P (A - LC)) e + γ^{2} e^{T} PPe + e^{T} e \\ + 2 e^{T} PD (v - d) + 2 e^{T} PE e_{f_{a}} + 2 e_{f_{a}}^{T} P_{0} {\overset{\cdot}{e}}_{f_{a}} \\ \leq e^{T} ({(A - LC)}^{T} P + P (A - LC)) e + γ^{2} e^{T} PPe \\ + e^{T} e + 2 {(FCe)}^{T} (v - d) + 2 e_{f_{a}}^{T} (E^{T} Pe + P_{0} {\overset{\cdot}{e}}_{f_{a}}) \\ \leq e^{T} ({(A - LC)}^{T} P + P (A - LC)) e + γ^{2} e^{T} PPe \\ + e^{T} e - 2 ‖ F e_{y} ‖ (ρ - ‖ d ‖) + 2 e_{f_{a}}^{T} (E^{T} Pe + P_{0} {\overset{\cdot}{e}}_{f_{a}}) \\ \leq e^{T} ({(A - LC)}^{T} P + P (A - LC)) e + γ^{2} e^{T} PPe \\ + e^{T} e + 2 e_{f_{a}}^{T} (E^{T} Pe + P_{0} {\overset{\cdot}{e}}_{f_{a}}) \end{matrix}

(11)

where the observer gain matrix is $L = - P^{- 1} Q$ .

According to the Schur complementary lemma, if

[\begin{matrix} A^{T} P + P A + C^{T} Q + Q C + I & P \\ * & - \frac{1}{γ^{2}} I \end{matrix}] < 0

(12)

And, when ${\overset{\cdot}{e}}_{f_{a}} = - P_{0}^{- 1} E^{T} Pe = G e_{y}$ , it can be guaranteed that $\overset{\cdot}{V} < 0$ , the system error dynamic equation is asymptotically stable, and the proof is over.

Threshold setting

The output estimation error $e_{y}$ is the residual, and the residual is $e_{y} = 0$ when there is no fault in the MDSORM and $e_{y} \neq 0$ when a fault occurs, but, in practice, it is difficult to keep the residual strictly at zero due to the existence of external perturbations in the system. So, a suitable decision system needs to be established to make the decision function $J (e_{y})$ sensitive to faults, and the decision logic is as follows

{\begin{matrix} J (e_{y}) > ϕ \Rightarrow normal \\ J (e_{y}) \leq ϕ \Rightarrow a fault occurs \end{matrix}

Based on the randomness of the residuals, a confidence interval is introduced to determine the threshold value. Let the expectation of the residuals $E (X) = μ$ , the variance $D (X) = σ^{2}$ , $N$ be the number of samples, and $(1 - α)$ be the confidence level

\begin{matrix} μ = \frac{1}{n} \sum_{j = 1}^{n} e_{y} \\ σ^{2} = \frac{1}{n - 1} \sum_{j = 1}^{n} {[e_{y} - μ]}^{2} \end{matrix}

Therefore, the $μ$ confidence interval for $(1 - α)$ is

P (\bar{X} - \frac{σ}{\sqrt{N}} μ_{1 - \frac{α}{2}}, \bar{X} + \frac{σ}{\sqrt{N}} μ_{1 - \frac{α}{2}}) = 1 - α

The confidence level $(1 - α)$ is chosen to be 0.95, and the quantile $(1 - α / 2)$ is 1.96. Therefore, the threshold value for fault detection is $ϕ = \bar{X} + \frac{σ}{\sqrt{N}} μ_{1 - \frac{α}{2}}$

Fault reconfiguration

Theorem 2. For nonlinear systems and observers, if there exist positive definite matrices $P$ and $Q$ such that the linear matrix inequality has a solution, then there is ${\hat{f}}_{a} = \int_{0}^{\infty} G e_{y} dt$ , an asymptotic reconfiguration of the actual fault signal $f_{a}$ .

Proof. From Theorem 1, when equation (11) is less than zero, we get $\overset{\cdot}{V} < 0$ , and the error system $e$ is asymptotically stable with the fault estimation error system $e_{f_{a}}$ . According to equation (6), the fault estimation error system $e_{f_{a}}$ tends to 0 as time tends to infinity, that is, $t \to \infty$ at $e_{f_{a}} \to 0$ . Therefore, ${\hat{f}}_{a} \to f_{a}$ , ${\hat{f}}_{a} = \int_{0}^{\infty} G e_{y} dt$ are asymptotically reconstructed for the fault $f_{a}$ , and the proof is over.

Model-referenced fault-tolerant control of the main drive system of rolling mill

To ensure that the MDSORM works properly even after a fault occurs, a model-reference FTC scheme is used for the FTC of the MDSORM. The reference model of the system is (Xiao and Dong, 2020)

{\begin{matrix} {\overset{\cdot}{x}}_{R} = A_{R} x_{R} + B_{R} u_{R} \\ y_{R} = C_{R} x_{R} \end{matrix}

(13)

where $x_{R}$ is the reference system state variable; $y_{R}$ is the reference system output; $u_{R}$ is the reference system input; $A_{R}$ , $B_{R}$ , and $C_{R}$ are the given parameter matrices; and $A_{R}$ is the Hurwitz matrix.

For the main drive failure system of rolling mill described in equation (1), the design model refers to a fault-tolerant controller of the following form

u_{c} = u_{N} + u_{f}

(14)

When no system fault occurs, define the output error as $e_{N} = y - y_{R}$ , and the feedback controller $u_{N}$ in the absence of a fault is designed as

\begin{matrix} u_{N} = - {(CB)}^{+} (CA C^{+} C_{R} x_{R} + Cf (x)) \\ + {(CB)}^{+} (C_{R} A_{R} x_{R} + C_{R} B_{R} u_{R}) \end{matrix}

(15)

Therefore

\begin{matrix} {\overset{\cdot}{e}}_{N} = C (Ax + B u_{N} + f (x) + Dd) \\ - C_{R} (A_{R} x_{R} + B_{R} u_{R}) \\ = A_{1} e_{N} + D_{1} d \end{matrix}

(16)

where $C^{+}$ denotes the generalized inverse matrix of $C$ , $A_{1} = CA C^{+}$ , and $D_{1} = CD$ .

When a system fault occurs, define an output error of $e_{c} = y - y_{R}$ and design the fault compensation controller as

u_{f} = - B^{+} E {\hat{f}}_{a}

(17)

Therefore

\begin{matrix} {\overset{\cdot}{e}}_{c} = C (Ax + B u_{c} + f (x) + Dd + E f_{a}) \\ - C_{R} (A_{R} x_{R} + B_{R} u_{R}) \\ = A_{1} e_{c} - C (\hat{f} (x) - f (x)) - E_{1} e_{f_{a}} + D_{1} d \end{matrix}

(18)

where $E_{1} = CE$

From the above FD section, it is known that the fault estimation error is bounded by the presence of $| | e_{f_{a}} | | < ρ_{1}$ and $ρ_{1} > 0$ .

Theorem 3. The error dynamic systems (16) and (18) are stable if there exist symmetric positive definite matrices $P_{1}$ , $Q_{1}$ , $P_{2}$ , and $Q_{2}$ and constants $λ > 0$ such that the linear matrix inequalities $A_{1}^{T} P_{1} + P_{1} A_{1} + Q_{1} \leq 0$ and $A_{1}^{T} P_{2} + P_{2} A_{1} + 2 λ P_{2} + Q_{2} \leq 0$ hold.

Proof. Select Lyapunov function

V_{1} = e_{N}^{T} P_{1} e_{N} + e_{c}^{T} P_{2} e_{c}

The derivative is obtained by taking $V_{1}$

\begin{matrix} {\overset{\cdot}{V}}_{1} = 2 e_{N}^{T} P_{1} {\overset{\cdot}{e}}_{N} + 2 e_{c}^{T} P_{2} {\overset{\cdot}{e}}_{c} \\ = e_{N}^{T} (A_{1}^{T} P_{1} + P_{1} A_{1}) e_{N} + 2 e_{N}^{T} P_{1} D_{1} d + e_{c}^{T} (A_{1}^{T} P_{2} + P_{2} A_{1}) e_{c} \\ - 2 e_{c}^{T} P_{2} (\hat{f} - f) - 2 e_{c}^{T} P_{2} E_{1} e_{f_{a}} + 2 e_{c}^{T} P_{2} D_{1} d \\ \leq - λ_{min} (Q_{1}) \cdot ‖ e_{N} ‖^{2} + 2 ρ \cdot ‖ e_{N} ‖ \cdot ‖ P_{1} D_{1} ‖ - λ_{min} (Q_{2}) \cdot ‖ e_{c} ‖^{2} \\ + 2 ‖ e_{c} ‖ \cdot (ρ ‖ P_{2} D_{1} ‖ - ρ_{1} ‖ P_{2} E_{1} ‖) \end{matrix}

(19)

Therefore, when $| | e_{c} | | > (2 ρ | | P_{2} D_{1} | | - 2 ρ_{1} | | P_{2} E_{1} | |) / λ (Q_{2})_{\min}$ and $| | e_{N} | | > 2 ρ \cdot | | P_{1} D_{1} | | / λ (Q_{1})_{\min}$ , ${\overset{\cdot}{V}}_{1} < 0$ , that is, the system is stable after performing FTC as described above.

The FTC design of the MDSORM is shown in Figure 2.

Simulation and verification

In order to verify the feasibility of the proposed method, the main drive system of stand F4 of the 2030 mm cold rolling mill is used as the object of study. The corresponding main drive system equipment parameters are: $T_{1} = φ I_{c}$ , $T_{2} = 1.45 \times 10^{4} N \cdot m$ , $J_{1} = 1552 kg \cdot m^{2}$ , $J_{2} = 1542 kg \cdot m^{2}$ , $I_{c} = 1720 A$ , $C_{1} = 2 \times 10^{5} N / (m \cdot s^{- 1})$ , $C_{2} = 2 \times 10^{6} N / (m \cdot s^{- 1})$ , $φ = 28$ , $K = 5.93 \times 10^{6} N \cdot m / rad$ , $R = 0.4 m$ , $P_{a} = 7807 kN$ , $a = 0.49$ , $b = 0.09$ , and $c = 0.0033$ .

The parameters of the design experiment were $γ = 0.01$ and $P_{0} = 0.001$ .

The system matrix of the MDSORM is

A = [\begin{matrix} - \frac{C_{1}}{J_{1}} & 0 & - \frac{k}{J_{1}} \\ 0 & - \frac{C_{2} + b P_{μ} R^{2}}{J_{2}} & \frac{k}{J_{2}} \\ 1 & - 1 & 0 \end{matrix}]

B = [\begin{matrix} \frac{φ}{J_{1}} & 0 & 0 \\ 0 & \frac{1}{J_{2}} & 0 \\ 0 & 0 & 0 \end{matrix}] C = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]

f (x) = [\begin{matrix} 0 \\ \frac{c P_{μ} R^{3}}{J_{2}} x_{2}^{2} + \frac{a P_{μ} R}{J_{2}} \\ 0 \end{matrix}]

The fault matrix and disturbance matrix are $E = [0 1 0]^{T}$ and $D = [1 1 0]^{T}$ , respectively; external perturbations $d = rand 2.4 \sin (20 π t)$ ; bitten steel shock vibration fault is expressed as

f_{a} = {\begin{matrix} 0 (0 < t < 4 seconds) \\ 10^{5} + 8000 \sin (5 t) \cos (2 t) (t \geq 4 seconds) \end{matrix}

The system matrix in the reference model is

$A_{R} = - I_{3 \times 3}$ , $C_{R} = I_{3 \times 3}$

B_{R} = [\begin{matrix} 0.178 & 0 & 0 \\ 0 & 0.000649 & 0 \\ 0 & 0 & 0 \end{matrix}]

According to Theorem 1, the observer gain matrix $L$ and the sliding-mode gain matrix $F$ are obtained as

L = [\begin{matrix} 20.216 & 468.732 & - 245.779 \\ - 92.775 & - 965.822 & 200.916 \\ 0.590 & 4.396 & - 2.799 \end{matrix}]

F = [55.109 296.685 589.450]

The matrix $G$ can be derived from equation (4) as

G = [- 55, 109.147 - 296, 684.720 - 589, 449.885]

The initial state of the MDSORM is set to $[x_{1} x_{2} x_{3}]^{T} = [8 24 0.1]^{T}$ , and the initial state of the observer is set to $[{\hat{x}}_{1} {\hat{x}}_{2} {\hat{x}}_{3}]^{T} = [0 0 0]^{T}$ .

Numerical experiment

In order to verify the estimation effectiveness of the nonlinear sliding-mode observer for the angular velocity of the MDSORM, the angular velocity without fault is compared with its estimated value. Figure 3 shows that the nonlinear sliding-mode observer can quickly track the state value of the system within 0.3 seconds and the angular velocity error is 2.45%, which reflects the effectiveness of the nonlinear sliding-mode observer in estimating the angular velocity of the MDSORM only during the start-up phase.

In order to verify the effectiveness of the presented observer-based FD method, the fault value $f_{a}$ and external perturbations $d$ are set up for simulation. By setting the sliding-mode control rate, the robustness of the observer to external perturbations is enhanced, and the effect of persistent external perturbations on the observer fault detection accuracy is eliminated. As can be seen in Figure 4, the system fails at 3 seconds. The state estimate of the observer increases suddenly and converges quickly to the ideal interval within 0.4 seconds, indicating that the observer has good rapidity. The comparison of fault and fault estimation in Figure 4(d) shows that the proposed FD method can achieve accurate fault estimation with an estimation error of only 0.04% even in the presence of external perturbations.

In order to verify the effectiveness of the proposed model-reference FTC method, a fault-tolerant controller is added to the original system for FTC. As shown in Figure 5, when the MDSORM occurs at 3 seconds with a bitten steel shock impact fault, the system angular velocity increases significantly compared with when no fault occurs, and the torsional vibration of the MDSORM increases. Motor angular velocity and roll angular velocity peak when there is no fault increased by 97.7% and 99.6%, respectively. At 7 seconds, the fault-tolerant system was fault-tolerantly controlled using the fault reconfiguration value design model-reference fault-tolerant controller. After 0.1 seconds, the MDSORM quickly returns to normal, and even if there are external perturbations, the MDSORM can still operate normally, further verifying that the FTC method has good fault tolerance for both external perturbations and bitten steel shock impact faults.

Comparative experiments

In order to verify the superiority of the proposed FTC method, it is compared with the FTC method presented in the work of Li et al. (2012). After the fault occurred, the fault-tolerant controller was added at 7 seconds and the system was restored to the normal state after 0.3 seconds. The results are shown in Figure 6, and the comparison with the model-reference FTC method in Figure 5 shows that there is a delay of 0.2 seconds and the estimated root-mean-square error of the state variables increases by 7.8%. This indicates that the proposed FTC method can restore the system to the normal state in a shorter time with good accuracy.

Conclusion

The proposed FD method based on the nonlinear sliding-mode observer can effectively suppress the influence of external perturbations on fault estimation and fault reconfiguration by using the sliding-mode control rate and can improve the robustness of the nonlinear sliding-mode observer to fault reconstruction. In the case of external perturbations, the proposed model-reference FTC method compensates the fault system with the estimated fault value and effectively improves the fault tolerance performance of the MDSORM.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This project was supported by the Natural Science Foundation of Hebei Province (Grant No. F2018209201).

ORCID iDs

Ruicheng Zhang

Pengfei Li

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