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Abstract

Driven by the need for remaining useful life prediction of degraded motor systems with feedback controllers, a real-time updated Wiener stochastic process is adopted to model the performance degradation of motor systems. First, a closed-loop performance index of the motor system is derived incorporating the multiple slow time-varying characteristic of motor parameters. On this basis, the drift coefficient and diffusion coefficient of the Wiener degradation model are updated to obtain the prior maximum likelihood function with available historical data. It is followed by the iterative optimization of nonlinear feature parameter in the Wiener degradation model with taking the prior maximum likelihood function as the cost equation. The effectiveness of proposed remaining useful life prediction architecture for closed-loop motor systems is demonstrated by motor systems.

Keywords

Remaining useful life closed-loop motor system performance degradation monitoring Wiener process extended state observer

Introduction

Motors are widely used in modern industrial control systems, as well as key drivers to ensure safe operation in industrial processes.^1–5 But physical structures of motors are prone to damage due to the fact that most of the equipment often operate in harsh working environments. It is necessary to monitor the operational status of motor systems to prevent serious degradation of the performance.⁶ However, the system error will fluctuate in a small allowable range owing to the feedback controller during the early even the middle stage of the motor performance degradation.⁷ Once the obvious change is detected at the later stage of degradation, only extreme ways such as shutdown and replacement can be adopted. In a ward, it is of importance to realize the remaining useful life (RUL) prediction of motor systems in the early stage of degradation for the purpose that subsequent predictive maintenance policy can be taken in time to reduce the economic loss.

The main idea of the RUL prediction is to determine the distribution or expectation of the RUL according to the effective information such as failure mechanism and failure data.⁸ At present, a lot of investigations aim at the degradation of mechanical components in open-loop systems, for instance, Li-ion batteries,⁹ rolling bearings,^10,11 and so on. However, the impact of degraded components can be partially compensated by feedback controller in closed-loop systems. Nguyen et al.^12,13 described the performance degradation process of the closed-loop system as a stochastic shock process with the assumption that the degradation of the actuator can be directly measured. But the performance of systems with feedback controllers cannot generally be reflected in a measurable physical quantity. Considering the above problem, Si et al.¹⁴ put forward a prediction algorithm to predict one degradation parameter which can indicate the degradation of the closed-loop system performance to obtain the RUL prediction by simulating future states of the system. Nevertheless, there are always multiple slow time-varying parameters during the operation of motor systems in practice.^15,16 The prediction result will be inaccurate if the RUL is calculated through fitting of the degradation path by only one parameter. Therefore, it is necessary to calculate a comprehensive performance index which can accurately characterize the degradation of the closed-loop motor system with multiple slow time-varying parameters.

At present, data-driven methods, especially Wiener stochastic process, are widely used to predict the RUL due to its probabilistic framework with the inverse Gaussian distribution. In recent research, it is worth noting that a nonlinear-drifted diffusion process with a constant threshold was transformed to a linear model with a corresponding variable threshold for the purpose of obtaining the probability density function (PDF) of the RUL.¹⁷ In addition, multiple-dimensional search algorithm with the full life cycle data was adopted to estimate parameters offline, which requires considerable time inevitably if suitable initial values were not set in advance.¹⁸ Furthermore, it is difficult to obtain the full life cycle data for majority of industrial processes. Compared with the offline training of model parameters, an RUL algorithm was proposed to online estimate parameters of the degradation model in combination with the Kalman filter.¹⁹ But the posterior estimation of parameters may be affected by the two-stage degradation process which needs to re-estimate the rapidly changing degradation rate. In view of this degradation characteristic, the powerful random global search capability of particle swarm optimization (PSO) algorithm is very helpful for the multi-stage degradation modeling.²⁰ However, estimating all parameters simultaneously is time-consuming which will result in delayed prediction. Therefore, how to online estimate parameters of the degradation model accurately and quickly when there are only the historical data to date is a practical issue.

In this article, an iteratively updated RUL prediction algorithm for closed-loop motor systems with multiple slow time-varying parameters is proposed. For the purpose of constructing a performance index, an extended state observer (ESO) is designed to estimate the augmented degradation state related to degradation when multiple parameters are slowly time-varying. Due to the lack of the full life cycle information, PSO algorithm is taken into account to online optimize the parameter reflecting nonlinear characteristic only by the historical performance index to date, and then expectation maximization (EM) algorithm and Bayes theorem are integrated to update other parameters. The contributions are as follows:

A comprehensive performance index of the degraded closed-loop motor system is constructed in the presence of multiple slow time-varying parameters, based on which the degradation model can be established closed to the actual degradation trajectory.

A two-step fusion algorithm for parameters of the nonlinear Wiener degradation model is designed to iteratively update the PDF only by the available historical data, which contributes to promoting the accuracy of RUL prediction.

This article is organized as follows. In section “Problem formulation,” the problem formulation is introduced. Section “Construction of the performance index” is dedicated to the design of ESO and approximates the calculation of the performance index. Section “Prediction of the performance index” deals with the iteratively updated process of the nonlinear degradation model, and then the PDF of the RUL is obtained. The architecture is summarized in section “RUL prediction architecture for the closed-loop motor system.” Finally, case studies on motor benchmarks are presented, and the validity of proposed approach is discussed in section “Case study on DC motor.”

Problem formulation

In this article, a simplified continuous-time state space expression of the direct current (DC) motor composed of the mechanical motion equation and the voltage equation expressed by equation (1) is considered.^21,22 To begin with, the schematic diagram of the DC motor is shown in Figure 1

{\begin{matrix} C_{m} I_{d} (t) = J \frac{dn (t)}{dt} + T_{L} (t) + ω (t) \\ U_{d} (t) = R I_{d} (t) + L \frac{d I_{d} (t)}{dt} + C_{e} n (t) + υ (t) \end{matrix}

(1)

Figure 1.

DC motor structure and schematic chart.

where $U_{d} (t)$ , $I_{d} (t)$ , and $n (t)$ denote the armature voltage, armature current, and motor speed, respectively. $C_{m}$ , $C_{e}$ , and $J$ denote the torque coefficient, electromotive force coefficient at rated flux, and rotary inertia coefficient, respectively. $R$ and $L$ denote the armature resistance and armature inductance coefficients, respectively. $T_{L} (t)$ denotes the unknown plant load torque with ignoring viscous friction torque and elastic torque. $ω (t)$ and $υ (t)$ denote the white noise in the mechanical motion equation and the voltage equation, respectively.

Proportional–integral–derivative (PID) controller is used due to its widely existence in practical industrial control systems. The measurement of the degraded system will not change significantly because of the feedback controller, so that the RUL prediction algorithm for static systems without the closed-loop controller is not applicable to closed-loop control systems. The PDF of the RUL can be obtained indirectly by predicting the degradation trajectory of the parameter if only one parameter changes. In practice, the degradation of motor performance is often reflected in multiple slow time-varying electrical parameters, such as the armature resistance $R$ and the armature inductance $L$ . Let $R_{0}$ and $L_{0}$ denote the initial values of the armature resistance and the armature inductance, and $Δ R$ and $Δ L$ denote the slow changes in resistance and inductance due to system degradation, respectively. At this time, the DC motor system model can be expressed as

{\begin{matrix} C_{m} I_{d} (t) = J \frac{dn (t)}{dt} + T_{L} (t) + ω (t) \\ \begin{matrix} U_{d} (t) = ((R_{0} + Δ R) I_{d} (t) + (L_{0} + Δ L) \frac{d I_{d} (t)}{dt} \\ + C_{e} n (t) + υ (t)) \end{matrix} \end{matrix}

(2)

In this event, it is not available to estimate and predict all time-varying parameters. One of the reasons is that it is unable to obtain the actual number of time-varying parameters and corresponding thresholds. The other is that the weight of each time-varying parameter in judging the degree of system performance degradation cannot be obtained. In this issue, it is primary to clarify exactly the architecture of the performance degradation prediction algorithm for the closed-loop motor system. It is followed by the construction of the index to describe the performance degradation of the closed-loop motor system under the condition of the unknown load torque and the slow time-varying parameters. On the basis of the index, the ultimate objective of this article is to compute the PDF of the RUL based on the establishing iterative degradation model of the performance index due to the inability to obtain full life cycle data.

Construction of the performance index

In this section, an ESO algorithm is designed to estimate the degradation information, and then a performance index is established for the subsequent prediction process.

To facilitate the calculation in the prediction process, we approximately discretize the system model (1) with multiple slow time-varying parameters by the Euler discretization method. We rewrite equation (1) as follows

{\begin{matrix} n (k + 1) = n (k) + Ts (\frac{C_{m} I_{d} (k)}{J} - \frac{T_{L} (k)}{J} - \frac{ω (k)}{J}) \\ \begin{matrix} I_{d} (k + 1) = I_{d} (k) + Ts (\frac{U_{d} (k)}{(L_{0} + Δ L)} - (\frac{(R_{0} + Δ R)}{(L_{0} + Δ L)} \\ * I_{d} (k)) - \frac{C_{e} n (k)}{(L_{0} + Δ L)} - \frac{υ (k)}{(L_{0} + Δ L)}) \end{matrix} \end{matrix}

(3)

where $Ts$ denotes the time parameter discretization.

We rearrange equation (3) as

{\begin{matrix} \frac{n (k + 1) - n (k)}{Ts} = \frac{C_{m} I_{d} (k)}{J} - \frac{f_{n}}{J} \\ \frac{I_{d} (k + 1) - I_{d} (k)}{Ts} = \frac{U_{d} (k)}{L_{0}} + f_{I_{d}} \end{matrix}

(4)

with

{\begin{matrix} f_{n} = T_{L} (k) + ω (k) \\ \begin{matrix} f_{I_{d}} = (- \frac{(I_{d} (k + 1) - I_{d} (k)) Δ L}{L_{0} Ts} - \frac{(R_{0} + Δ R) I_{d} (k)}{L_{0}} \\ - \frac{C_{e} n (k)}{L_{0}} - \frac{υ (k)}{L_{0}}) \end{matrix} \end{matrix}

(5)

where $f_{n}$ and $f_{I_{d}}$ represent the total changes of system states and parameters except for control input.

ESO is often taken into account to estimate unknown terms and disturbances in motor systems.^23,24 An ESO is designed for the mechanical motion equation and the voltage equation as follows

{\begin{matrix} e_{n} (k) = \hat{n} (k) - n (k) \\ {\hat{f}}_{n} (k) = {\hat{f}}_{n} (k - 1) - Ts β_{1} e_{n} (k) \\ \hat{n} (k + 1) = (\hat{n} (k) + Ts ({\hat{f}}_{n} (k) / J \\ + C_{m} I_{d} (k) / J) - β_{2} e_{n} (k)) \end{matrix}

(6)

{\begin{matrix} e_{I_{d}} (k) = {\hat{I}}_{d} (k) - I_{d} (k) \\ {\hat{f}}_{I_{d}} (k) = {\hat{f}}_{I_{d}} (k - 1) - Ts β_{3} e_{I_{d}} (k) \\ {\hat{I}}_{d} (k + 1) = ({\hat{I}}_{d} (k) + Ts ({\hat{f}}_{I_{d}} (k) \\ + U_{d} (k) / L_{0}) - β_{4} e_{I_{d}} (k)) \end{matrix}

(7)

where $β_{1}, β_{2}, β_{3}, β_{4}$ denote the discrete ESO gains; closed-loop transfer functions on $z$ domain are respectively

\begin{matrix} G_{n} = \frac{β_{2} Jz + β_{1} Ts - J β_{2}}{J {(z - 1)}^{2} + β_{2} Jz + β_{1} Ts - β_{2} J} \\ G_{I_{d}} = \frac{β_{4} z + β_{3} Ts - β_{4}}{{(z - 1)}^{2} + β_{4} z + β_{3} Ts - β_{4}} \end{matrix}

(8)

By calculating the pole position, the discrete ESO stability condition is²⁵

{\begin{matrix} β_{1} < J (4 + β_{2}^{2}) / 4 Ts \\ 0 < β_{2} < 4 \\ β_{3} < (4 + β_{4}^{2}) / 4 Ts \\ 0 < β_{4} < 4 \end{matrix}

(9)

When the discrete ESO gains satisfy equation (9), the estimated ESO converges to the true value. The estimated load and the performance index can be obtained, respectively, from equations (6) and (7). To be noted, this article only studies the method of establishing performance index when multiple parameters in the voltage equation degrade, which often occurs in practice. Similarly, the degradation in the mechanical equation can be transformed into a similar performance index through the state variable of the current. The performance index is derived as equation (10)

\begin{matrix} γ (k) = ((1 - β_{3}) e_{I_{d}} (k) / Ts + {\hat{f}}_{n} (k) + U_{d} (k) / L_{0} \\ + R_{0} I_{d} (k) + C_{e} n (k)) / U_{d} (k) \end{matrix}

(10)

where $γ \in (0, 1)$ . There is no doubt that $γ = 1$ when the motor is not degraded. The threshold of $γ$ is the value when the system error or the control input signal exceeds the specified limit. Meanwhile, the impact of noise in the performance index will be considered as a random process in the degradation model in the next section.

Prediction of the performance index

In this section, the PDF of the RUL will be obtained by a nonlinear Wiener process. It is followed by iteratively updated parameters of degradation model, which can be derived by integrating PSO algorithm, EM algorithm, and Bayes theorem.

PDF of RUL based on a nonlinear Wiener process

The nonlinear Wiener degradation process for the performance index can be modeled as¹⁹

\hat{γ} (i) = γ (i - 1) + \sum_{i - 1}^{i} μ (i, θ) + σ_{B} Δ B (i)

(11)

where $i$ denotes the specific dispersed time point, and it is different from $k$ which denotes the number of iterations. $i = 1 * T_{W}, 2 * T_{W}, . . ., q * T_{W}$ . $T_{W}$ is generally equal to the integer multiple of the time parameter of discretization. $q$ denotes the number of available performance indicator, $\hat{γ} (i)$ is the prediction of the performance index, and $σ_{B}$ denotes the diffusion coefficient. $Δ B (i) = B (i) - B (i - 1)$ denotes the standard Brownian motion with $B (i)$ . $θ$ in the nonlinear drift term $μ (i, θ)$ represents the parameter set.

We choose the power function $μ (i, θ) = a i^{b}$ which can conform to the majority of degradation path.¹⁴ For the sake of considering the noise effect, $a ~ N (μ_{a}, σ_{a})$ . $b$ is fixed for charactering the nonlinearity of degradation trajectory. When the motor system runs to time $i$ and the RUL for the motor system is $t_{RUL}$ , the PDF of the RUL is described as equation (12)

\begin{matrix} \begin{matrix} \begin{matrix} (t_{RUL} | Θ) = (ζ - γ (i) - ({(t_{RUL} + i)}^{b} - i^{b} \\ {- b t_{RUL} {(t_{RUL} + i)}^{b - 1})}^{b - 1} \end{matrix} \\ * \frac{σ_{a}^{2} (ω - γ (i)) ({(t_{RUL} + i)}^{b} - i^{b}) + μ_{a} σ_{B}^{2} t_{RUL}}{σ_{a}^{2} {({(t_{RUL} + i)}^{b} - i^{b})}^{2} + σ_{B}^{2} t_{RUL}}) \end{matrix} \\ \begin{matrix} * \exp [- \frac{{(ω - γ (i) - μ_{a} ({(t_{RUL} + i)}^{b} - i^{b}))}^{2}}{2 {(σ_{a}^{2} ({(t_{RUL} + i)}^{b} - i^{b}))}^{2} + σ_{B}^{2} t_{RUL}}] \\ * \frac{1}{\sqrt{2 π t_{RUL}^{2} (σ_{a}^{2} {({(t_{RUL} + i)}^{b} - i^{b})}^{2} + σ_{B}^{2} t_{RUL})}} \end{matrix} \end{matrix}

(12)

$Θ = {(μ_{a}, σ_{a}, b, σ_{B})}^{T}$ denotes the parameter set of the degradation model to be determined. $ζ$ denote the threshold of the performance index. It is followed by the maximum likelihood estimation (MLE) approach to estimate $Θ$ .

Iterative update of the nonlinear Wiener degradation model

It is necessary to iteratively update the degradation model in real-time due to the inability to obtain the full life cycle data. Meanwhile, parameter estimation methods based on traditional search algorithm will lead to low accuracy and time consumption in the nonlinear degradation model. In order to obtain parameters of degradation model $Θ = {(μ_{a}, σ_{a}, b, σ_{B})}^{T}$ by MLE, the drift coefficient and diffusion coefficient $Θ_{1} = {(μ_{a}, σ_{a}, σ_{B})}^{T}$ are estimated first, and the point is that the optimization algorithm is then integrated to estimate the parameter $Θ_{2} = b$ which describes nonlinear characteristics in this section.

Iterative update of drift coefficient and diffusion coefficient $Θ_{1}$

First, we rewrite equation (11) as

Δ ϒ = a Ψ (i) + σ_{B} Δ B (i)

(13)

with $Δ ϒ = {(Δ ϒ_{1}, . . ., Δ ϒ_{q})}^{T}$ , $Δ ϒ_{m} = γ (m) - γ (m - 1)$ , $Ψ = {(Ψ_{1}, . . ., Ψ_{q})}^{T}$ , $Ψ_{m} = Λ_{m} - Λ_{m - 1}$ , $Λ_{m} = {(i_{m})}^{b}$ , and $i_{m} = T_{W} * m$ when $m = 2, 3, . . ., q$ . It is easily obtained that $Δ ϒ$ follows a multivariate normal distribution with the mean $E_{Δ ϒ | a} = a Ψ$ , the variance matrix $cov (Δ ϒ_{m}, Δ ϒ_{l} | a) = σ_{B}^{2} Ω$ which is a positive definite matrix given by

Ω = {\begin{matrix} T_{W}, l = m \\ 0, otherwise \end{matrix}

(14)

with $1 \leq m, l \leq q$ . So that the joint PDF of $Δ ϒ$ can be written as

\begin{matrix} p (Δ ϒ | a, Θ) = {(2 π)}^{- \frac{q}{2}} {(σ_{B}^{2} | Ω |)}^{- \frac{1}{2}} \\ * \exp (- \frac{{(Δ ϒ - a Ψ)}^{T} Ω^{- 1} (Δ ϒ - a Ψ)}{2 σ_{B}^{2}}) \end{matrix}

(15)

According to the property of the normal distribution, the PDF of $a$ is easily obtained that

p (a | Θ) = {(2 π σ_{a}^{2})}^{- \frac{1}{2}} \exp (- \frac{{(a - μ_{a})}^{2}}{2 σ_{a}^{2}})

(16)

It is followed by the log likelihood function involving the parameter set $Θ$ as

\begin{matrix} L (Θ | a, Δ ϒ) \\ = \ln p (Δ ϒ | a, Θ) + \ln p (a | Θ) \\ = - \frac{1}{2} [(m + 1) \ln (2 π) + \ln σ_{a}^{2} \\ + \ln σ_{B}^{2} + \ln | Ω | \\ + \frac{{(Δ ϒ - a Ψ)}^{T} Ω^{- 1} (Δ ϒ - a Ψ)}{σ_{B}^{2}} + \frac{{(a - μ_{a})}^{2}}{σ_{a}^{2}}] \end{matrix}

(17)

It is difficult to maximize the log likelihood function in equation (17) because the distribution of $a$ cannot be determined. The EM algorithm provides an effective way. More details and convergence proofs are given in the literature.^26–28 Before E-step, the Bayes theorem can be employed to determine the posterior distribution that corresponds to the normal distribution. It is assumed that $Θ^{(j)} = (μ_{a}^{(j)}, σ_{a}^{2 (j)}, b^{(j)}, σ_{B}^{2 (j)}) (j = 0, 1, \dots, K)$ represents estimated parameters after the $j th$ EM algorithm iteration, and $K$ denotes the circulation iteration numbers. In the Bayes framework, the logarithmic terms related to $a$ should be considered, then the posterior distribution of $a$ can be updated as follows

\begin{matrix} p (a_{q} | Δ ϒ, Θ^{(j)}) \\ \propto p (Δ ϒ | a_{q}, Θ^{(j)}) p (a_{q} | Θ^{(j)}) \\ \propto \exp (- \frac{1}{2} ((\frac{σ_{a}^{2 (j)} Ψ^{T} Ω^{- 1} Ψ + σ_{B}^{2 (j)}}{σ_{B}^{2 (j)} σ_{a}^{2 (j)}}) a_{q}^{2} \\ - 2 (\frac{σ_{a}^{2 (j)} Ψ^{T} Ω^{- 1} Δ ϒ + μ_{a}^{(j)} σ_{B}^{2 (j)}}{σ_{B}^{2 (j)} σ_{a}^{2 (j)}}) a_{q})) \\ \propto \exp (- \frac{{(a_{q} - \frac{σ_{a}^{2 (j)} Ψ^{T} Ω^{- 1} Δ ϒ + μ_{a}^{(j)} σ_{B}^{2 (j)}}{σ_{a}^{2 (j)} Ψ^{T} Ω^{- 1} Ψ + σ_{B}^{2 (j)}})}^{2}}{\frac{σ_{B}^{2 (j)} σ_{a}^{2 (j)}}{σ_{a}^{2 (j)} Ψ^{T} Ω^{- 1} Ψ + σ_{B}^{2 (j)}}}) \end{matrix}

(18)

In addition, combine the normal distribution characteristics of $a$ like equation (16) when the $q$ th performance index is available. On this basis, the posterior mean and variance are easily available like

\begin{matrix} p (a_{q} | Δ ϒ, Θ^{(j)}) \propto \exp (- \frac{{(a_{q} - μ_{a, Bayes}^{(j)})}^{2}}{σ_{a, Bayes}^{2 (j)}}) \\ μ_{a, Bayes}^{(j)} = \frac{σ_{a}^{2 (j)} Ψ^{T} Ω^{- 1} Δ ϒ + μ_{a}^{(j)} σ_{B}^{2 (j)}}{σ_{a}^{2 (j)} Ψ^{T} Ω^{- 1} Ψ + σ_{B}^{2 (j)}} \\ σ_{a, Bayes}^{2 (j)} = \frac{σ_{B}^{2 (j)} σ_{a}^{2 (j)}}{σ_{a}^{2 (j)} Ψ^{T} Ω^{- 1} Ψ + σ_{B}^{2 (j)}} \end{matrix}

(19)

In the following, the updated expectation and covariance of the hidden variable $a$ are substituted into the complete log likelihood function to achieve E-step. As a result, equation (17) is derived as

\begin{matrix} \begin{matrix} L (Θ | Θ^{(j)}, Δ ϒ) \\ = E_{a_{q}} L (Θ | a_{q}, Δ ϒ) \end{matrix} \\ = - \frac{1}{2} [(m + 1) \ln (2 π) + \ln σ_{a}^{2} + \ln σ_{B}^{2} + \ln | Ω | \\ \begin{matrix} + \frac{Δ ϒ^{T} Ω^{- 1} Δ ϒ - 2 μ_{a, Bayes}^{(j)} Ψ^{T} Ω^{- 1} Δ ϒ}{σ_{B}^{2}} \\ + \frac{({(μ_{a, Bayes}^{(j)})}^{2} + σ_{a, Bayes}^{2 (j)}) Ψ^{T} Ω^{- 1} Ψ (t)}{σ_{B}^{2}} \end{matrix} \\ + \frac{({(μ_{a, Bayes}^{(j)})}^{2} + σ_{a, Bayes}^{2 (j)}) - 2 μ_{a, Bayes}^{(j)} μ_{a} + μ_{a}^{2}}{σ_{a}^{2}}] \end{matrix}

(20)

The target of M-step is to find $Θ_{1} = {(μ_{a}, σ_{a}, σ_{B})}^{T}$ that maximizes function $\partial L (Θ | Θ^{(j)}, Δ ϒ)$ . Based on MLE, we calculate the partial derivative of equation (20), namely, $\frac{\partial L (Θ | Θ^{(j)}, Δ ϒ)}{\partial Θ_{1}}$ . Thus, $μ_{a}^{(j + 1)}$ $σ_{a}^{2 (j + 1)}$ and $σ_{B}^{2 (j + 1)}$ can be obtained by $\frac{\partial L (Θ | Θ^{(j)}, Δ ϒ)}{\partial μ_{a}} = 0$ , $\frac{\partial L (Θ | Θ^{(j)}, Δ ϒ)}{\partial σ_{a}^{2}} = 0$ , and $\frac{\partial L (Θ | Θ^{(j)}, Δ ϒ)}{\partial σ_{B}^{2}} = 0$ with fixing $b^{(j)}$ as

\begin{matrix} μ_{a}^{(j + 1)} = μ_{a, Bayes}^{(j)} \\ \begin{matrix} σ_{a}^{2 (j + 1)} = ({(μ_{a, Bayes}^{(j)})}^{2} + σ_{a, Bayes}^{2 (j)}) \\ - 2 μ_{a}^{(j + 1)} μ_{a, Bayes}^{(j)} + {(μ_{a}^{(j + 1)})}^{2} \end{matrix} \\ \begin{matrix} σ_{B}^{2 (j + 1)} = Δ ϒ^{T} Ω^{- 1} Δ ϒ - 2 μ_{a, Bayes}^{(j)} Ψ^{T} Ω^{- 1} Δ ϒ \\ + ({(μ_{a, Bayes}^{(j)})}^{2} + σ_{a, Bayes}^{2 (j)}) Ψ^{T} Ω^{- 1} Ψ \end{matrix} \end{matrix}

(21)

Iterative update of nonlinear characteristic parameter $Θ_{2}$

Up to now, $Θ_{1}$ has been updated in the $j th$ iteration, but there is another parameter $b$ representing the nonlinear characteristic of the degradation process that has not been updated in $Ψ$ . Meanwhile, $b$ needs to be timely and accurately optimized when the rate of degradation changes. To solve this issue, PSO algorithm can be applied to find the optimal solution. More details about PSO are given in Shi and Eberhart.²⁹ Substituting equation (21) into equation (20) yields

\begin{array}{l} L (b | μ_{a}^{(j + 1)}, σ_{a}^{2 (j + 1)}, σ_{B}^{2 (j + 1)}, Δ ϒ) \\ \begin{array}{l} = - \frac{1}{2} [Q + \ln | Ω | + \ln ({(μ_{a, B a y e s}^{(j)})}^{2} + σ_{a,, B a y e s}^{2 (j)} \\ - 2 μ_{a}^{(j + 1)} μ_{a,, B a y e s}^{(j)} + {(μ_{a}^{(j + 1)})}^{2}) \end{array} \\ \begin{array}{l} + \ln (Δ ϒ^{T} Ω^{- 1} Δ ϒ - 2 μ_{a, B a y e s}^{(j)} Ψ^{T} Ω^{- 1} Δ ϒ \\ + ({(μ_{a, B a y e s}^{(j)})}^{2} + σ_{a, B a y e s}^{2 (j)}) Ψ^{T} Ω^{- 1} Ψ (t))] \end{array} \end{array}

(22)

where $Q$ denotes a merged constant. To facilitate the writing, equation (22) is shortened to $L (b)$ which is regarded as the objective function of PSO algorithm simultaneously. Generally, equation (23) is used to operate on $d$ particles with $i = 1, . . ., d$ given by

\begin{array}{l} \begin{array}{l} {\vec{V}}_{i}^{j + 1} = {\vec{V}}_{i}^{j} + w * c 1 * r 1 * ({\overset{\leftarrow}{H}}_{i}^{j} - {\vec{X}}_{i}^{j}) \\ + c 2 * r 2 * ({\overset{\leftarrow}{H}}_{g}^{j} - {\vec{X}}_{i}^{j}) \end{array} \\ {\vec{X}}_{i}^{j + 1} = {\vec{X}}_{i}^{j} + {\vec{V}}_{i}^{j + 1} \end{array}

(23)

where ${\vec{H}}_{g}$ denotes the position with the minimum function $L (b)$ when all search positions of all particles are brought into the function $L (b)$ .³⁰

The optimal value of $Θ^{T} = (μ_{a}, σ_{a}, σ_{B}, b)$ is available with the condition of $| Θ^{(j + 1)} - Θ^{(j)} | \leq t h$ where $th$ denotes a predetermined threshold. It is worth noting that the calculation of $| Θ^{(j + 1)} - Θ^{(j)} | \leq t h$ can be regarded as the process of computing two-norm because of vector $Θ^{(j)}$ . So far, the PDF of the RUL can be obtained by bringing all updated parameters into equation (12). Analogously, $Θ$ can be updated when the next performance index is available. The iteratively updated algorithm for parameters of the degradation model proposed in this article is abbreviated as algorithm PBEA. Algorithm PBEA indicates the online updated process for parameters of degradation model in Table 1.

Table 1.

Algorithm PBEA: update of $Θ = {(μ_{a}, σ_{a}, b, σ_{B})}^{T}$ .

Step 1: initialize parameters of the degradation model

Θ^{T (0)} = (μ_{a}^{(0)}, σ_{a}^{2 (0)}, σ_{B}^{2 (0)}, b^{(0)})

Step 2: complete the calculation of E-step asequations (19) and (20).

Step 3: complete the calculation of M-step as equation (21) to obtain

μ_{a}^{(j + 1)}

σ_{a}^{2 (j + 1)}

, and

σ_{B}^{2 (j + 1)}

Step 4: implement PSO algorithm with the objectivefunction as (22) to obtain

{\vec{H}}_{g}^{(j + 1)}

Step 5: if

| Θ^{(j + 1)} - Θ^{(j)} | \leq th

, obtain

Θ

Step 6: if

| Θ^{(j + 1)} - Θ^{(j)} | > th

, repeat Steps 2–5.

RUL prediction architecture for the closed-loop motor system

Figure 2 shows the closed-loop RUL prediction architecture based on the performance degradation index for the motor system. $y^{*}, u, y$ denote the system reference input, control input, and output, respectively. The closed-loop motor control system is represented within the green curve range. The performance index is constructed within the blue curve. The online updated process of degradation model is described within the red curve. Combining above information, the PDF of the RUL will be calculated.

Figure 2.

Closed-loop performance degradation prediction architecture of motor system.

The aforementioned RUL calculation process of the closed-loop motor system is represented as Algorithm RUL-PBEA in Table 2.

Table 2.

Algorithm RUL-PBEA: RUL prediction algorithmfor the closed-loop motor system.

Step 1: determine initialization parameters, controller parameters, and motor parameters.

Step 2: establish and discretize the mathematical model of DC motor.

Step 3: design ESO to estimate the real-time degradation performance index.

Step 4: construct Wiener degradation model of the degradation performance index.

Step 5: update iteratively parameters of nonlinear Wiener model according to PBEA.

Step 6: calculate PDF of RUL.

Case study on DC motor

In this section, the proposed closed-loop RUL prediction algorithm is verified by case studies on two kinds of motor systems with multiple slow time-varying parameters.

Verification of the accuracy of ESO

Case study on DC motor

The closed-loop control block diagram of DC motor can be exhibited in Figure 3. Motor parameters, controller parameters, and initial values are summarized as follows. $C_{m} = 1.95 N . m / Amp$ , $C_{e} = 1.1 {V / rad /}_{s}$ , $R = 1 Ω$ , $L = 0.5 H$ , $J = 0.01 kg . m^{2}$ , $ω (t) ~ N (0, e - 04)$ , $υ (t) ~ N (0, e - 04)$ , $y^{*} = 1 rad / s$ ; $K_{P} = 100, K_{I} = 60, K_{D} = 1$ ; $n (0) = 0$ , $I_{d} (k) = 0$ , $\hat{n} (0) = 0$ , ${\hat{I}}_{d} (0) = 0$ , ${\hat{f}}_{n} (0) = 0$ , ${\hat{f}}_{I_{d}} (0) = 0$ ; $β_{1} = 3.9$ , $β_{2} = 4.8$ , $β_{3} = 3.5$ , $β_{4} = 150$ . In this case study, the time parameter of discretization is chosen as $Ts = 0.01 s$ , sampling interval of Wiener process is chosen as $T_{W} = 10^{3} * Ts$ , and the full life cycle of the motor is set 100 h.

Figure 3.

Control schematic diagram of DC motor.

To better demonstrate the effectiveness of the proposed RUL algorithm, it is assumed that the degradation rate of resistance and inductance increases at $T = 80 h$ . In this instance, the known turning point of the temperature based on expert knowledge is close to the turning point of degradation performance index. Then, the turning point of the performance index can be calculated through the relationship between the temperature and the performance index. Meanwhile, the thresholds should be set with some margin owing to the presence of noise so that thresholds of the first half and the second half of the performance index are set to $0.5$ and $0.85$ . The corresponding time are $T = 80 h$ and $T = 100 h$ . Therefore, the evolution processes of the resistance $R$ and the inductance $L$ are shown in Figure 4.

Figure 4.

Evolution of motor resistance and motor inductance.

It is worth noting that $γ = 1$ when the motor is not degraded. For the convenience of displaying, the motor performance is adjusted to $| 1 - γ (k) |$ . As shown in Figure 5, the performance degradation index of DC motor represented by red line can effectively track the real performance degradation index represented by blue line. Moreover, it is also supported by the error curve in Figure 6.

Figure 5.

Real and estimated motor performance degradation index of DC motor.

Figure 6.

Error between real and estimated motor performance degradation index of DC motor.

Case study on permanent magnet synchronous motor

Permanent magnet synchronous motor (PMSM) is another kind of widely used motor as core power mechanism so that designed ESO should be validated on PMSM for stronger persuasiveness. The closed-loop control block diagram of PMSM can be exhibited in Figure 7. The dynamic mathematical model of PMSM under general assumptions can be arranged as³¹

{\begin{matrix} \overset{\cdot}{ω} = \frac{1}{J} (\frac{3}{2} p φ_{f} i_{q} - T_{L} - B_{f} ω) \\ {\overset{\cdot}{i}}_{q} = \frac{1}{L} (U_{q} - R i_{q} - p ω L i_{d} - p ω φ_{f}) \\ {\overset{\cdot}{i}}_{d} = \frac{1}{L} (U_{dq} - R i_{d} + p ω L i_{q}) \end{matrix}

(24)

where $i_{q}$ and $i_{d}$ denote the current of $q$ -axis and $d$ -axis, $U_{q}$ and $U_{dq}$ denote the voltage of $q$ -axis and $d$ -axis, $B_{f}$ denotes the coefficient of viscous friction, $p$ denotes the number of pole pairs, $ω$ denotes the rotor mechanical angular speed, and $φ_{f}$ denotes the flux linkage. The physical meaning of other symbols is similar to equation (1). Proportional–integral (PI) controllers are used in the speed closed-loop and the current closed-loop. Wherein proportional parameters of three controllers are $K_{P 1} = 80$ , $K_{P 2} = 60$ , and $K_{P 3} = 80$ , and integral time constants are $τ_{1} = 43.2$ , $τ_{2} = 2$ , and $τ_{3} = 10$ , respectively. The gains of ESO are $β_{1} = 3.9$ , $β_{2} = 4.8$ , $β_{3} = 3.5$ , $β_{4} = 150$ $β_{5} = 8$ , and $β_{6} = 170$ . Initial values of each state are set to zero. Similarly, the evolution processes of the resistance $R$ and the inductance $L$ like the trend in Figure 4. The performance degradation index of PMSM represented by red line can effectively track the real performance degradation index represented by blue line in Figure 8. Moreover, it is also supported by the error curve in Figure 9.

Figure 7.

Control schematic diagram of PMSM.

Figure 8.

Real and estimated motor performance degradation index of PMSM.

Figure 9.

Error between real and estimated motor performance degradation index of PMSM.

It can be concluded that designed observer in this article aimed at whether degraded DC motor or degraded PMSM can estimate the performance degradation indexes correctly. The only difference is that the degradation process of PMSM is slower than of DC motor under the same degradation conditions. In order to keep consistent with previous derivation of RUL prediction algorithm, subsequent simulation will focus on the performance degradation index of DC motor. It should be noted that the degradation process of PMSM can also be predicted in the same way like DC motor.

Verification of proposed RUL prediction algorithm

In order to reduce the load of calculation, it is not necessary to calculate the PDF at each time. Eight time points $T = 20 h$ , $T = 30 h$ , $T = 40 h$ , $T = 50 h$ , $T = 84 h$ , $T = 88 h$ , $T = 92 h$ , and $T = 96 h$ are selected to verify the accuracy of the RUL prediction algorithm proposed in this article. The trend of the performance degradation index in the future can be described by $\hat{γ} (t) = γ (t - 1) + \sum_{t - 1}^{t} μ (t, θ) + σ_{B} Δ B (t)$ and $μ (t, θ) = a t^{b}$ , $a ~ N (μ_{a}, σ_{a})$ . The maximum number of iterations at each sampling time is $10^{6}$ . The parameters of PSO algorithm are $c 1 = 1.5$ , $c 2 = 1.5$ , $w = 0.3$ , $P_{n} = 100$ , $[V_{min}, V_{max}] = [- 0.7, 0.7]$ , $[X_{min}, X_{max}] = [- 2, 2]$ , $μ_{a}^{(0)} = 1$ , $b^{(0)} = 1$ , $σ_{a}^{2 (0)} = 0.01$ , and $σ_{B}^{2 (0)} = 0.01$ . For the sake of simplicity, EM algorithm based on the Bayes theorem, the three-dimensional search (TDS) algorithm, and the Simplex algorithm are abbreviated as BEA, TDSA, and SA, respectively.

Figures 10(a), 11(a), and 12(a) show the RUL prediction of RUL-SA, RUL-TDSA, and RUL-BEA before the turning point. Figures 10(b), 11(b), and 12(b) show the RUL prediction of RUL-SA, RUL-TDSA, and RUL-BEA after the turning point. Figure 13(a) and (b) shows the accuracy of the RUL prediction by RUL-PBEA before and after the turning point, respectively. Particularly, MRUL denotes maximum probability RUL in PDF curve.²⁰ It can be seen intuitively that the result obtained by RUL-PBEA is more accurate than RUL-SA, RUL-TDSA, and RUL-BEA. Under the projection angle of view, the closer the red dotted line and the black dotted line are, the higher RUL prediction accuracy achieve. Consequently, whether before or after the turning point, the RUL prediction accuracy of RUL-PBEA proposed in this article is higher than that of algorithm RUL-SA, RUL-TDSA, and RUL-BEA.

Figure 10.

RUL prediction by RUL-SA: (a) PDF after the turning point and (b) PDF after the turning point.

Figure 11.

RUL prediction by RUL-TDSA: (a) PDF after the turning point and (b) PDF after the turning point.

Figure 12.

RUL prediction by RUL-BEA: (a) PDF after the turning point and (b) PDF after the turning point.

Figure 13.

RUL prediction by RUL-PBEA: (a) PDF after the turning point and (b) PDF after the turning point.

To quantitatively obtain the error between the real degradation trajectory and the established degradation model, Akaike information criterion (AIC) and mean-squared error (MSE) were both useful measure.¹⁷ AIC can be expressed as

AIC = 2 (N_{Θ} - (max L_{Θ}))

(25)

where $N_{Θ}$ denotes the number of the degradation model parameters, and $L_{Θ}$ denotes the function in equation (17). Meanwhile, MSE is calculated as

MSE = \int_{0}^{\infty} {(t_{RUL} - T_{RUL})}^{2} f (t_{RUL} | Θ) d t_{RUL}

(26)

where $T_{RUL}$ denotes the real RUL obtained at current time point, and $f (t_{RUL} | Θ)$ denotes the according conditional PDF of RUL by equation (12). Considering the above two criterion, the smaller values represent that the degradation model is closer to the real trajectory; thus, the corresponding iterative updated degradation model can be more accurate.

Table 3 indicates that the suitability of degradation models updated by PBEA is more accurate than those by SA, BEA, and TDSA. The marked values emphasize that MSE and AIC of PBEA is the smallest at each sampling time, which indicate that the degradation model updated by PBEA is more consistent with the actual model. Significantly, iterative processes of parameter $b$ which can clearly describe the nonlinear characteristic of the degraded model by PBEA at different times are shown graphically in Figure 14.

Table 3.

Comparison of updated degradation models by three algorithms at different times.

Runtime	Algorithm	$μ_{a}$	$σ_{a}$ $(\times {10^{-}}^{9})$	$σ_{B}$ $(\times {10^{-}}^{9})$	$b$	MSE	AIC $(\times 10^{4})$
$T = 20 h$	SA	0.66	4.47	64.6	1.3157	6.53 $\times 10^{4}$	−0.02
	TDSA	0.59	9.57	75.5	1.2004	9.42 $\times 10^{3}$	−0.66
	BEA	0.27	0.73	37.4	1.2874	4.76 $\times 10^{3}$	−0.83
	PBEA	0.11	4.92	8.28	1.0606	67.2	−2.82
$T = 30 h$	SA	0.34	5.37	56.8	1.3007	9.63 $\times 10^{3}$	−0.14
	TDSA	0.54	9.21	8.92	1.2254	2.85 $\times 10^{3}$	−0.82
	BEA	0.09	4.38	5.36	1.2398	1.03 $\times 10^{3}$	−1.02
	PBEA	0.11	5.93	6.84	1.0933	55.7	−3.75
$T = 40 h$	SA	0.31	7.62	50.3	1.1957	848	−0.7
	TDSA	0.44	9.29	9.25	1.1865	292	−1.55
	BEA	0.26	7.34	19.4	1.1743	149	−2.61
	PBEA	0.12	2.92	1.25	1.1178	37.0	−4.67
$T = 50 h$	SA	0.28	5.53	40.9	1.1865	364	−1.5
	TDSA	0.21	18.6	8.25	1.1701	83.0	−2.62
	BEA	0.14	3.63	48.8	1.1557	39.1	−3.03
	PBEA	0.15	4.85	8.36	1.1188	26.5	−5.57
$T = 84 h$	SA	0.3	7.64	20.7	1.1754	224	−0.12
	TDSA	0.28	6.22	95.2	1.1695	136	−0.12
	BEA	0.3	14.3	86.9	1.1507	125	−0.13
	PBEA	0.22	3.38	5.66	1.1204	2.86	−0.92
$T = 88 h$	SA	0.29	7.86	18.7	1.1734	190	−0.24
	TDSA	0.18	14.25	9.52	1.1621	105	−0.53
	BEA	0.16	4.29	8.63	1.1486	90.7	−0.69
	PBEA	0.27	49.37	28.4	1.1285	2.06	−1.12
$T = 92 h$	SA	0.25	3.75	47.7	1.1633	70.3	−0.48
	TDSA	0.17	19.25	5.25	1.1542	45.3	−1.08
	BEA	0.18	4.23	83.7	1.1487	40.6	−0.98
	PBEA	0.3	46.38	84.6	1.135	2.53	−1.49
$T = 96 h$	SA	0.42	3.76	35.4	1.1667	58.4	−1.19
	TDSA	0.52	5.25	9.24	1.1722	43.5	−1.49
	BEA	0.41	23.38	8.74	1.1865	35.7	−1.67
	PBEA	0.39	13.27	89.7	1.1635	1.73	−1.86

SA: Simplex algorithm; TDSA: three-dimensional search algorithm; BEA: Bayes theorem.

Figure 14.

Estimation results of parameter $b$ at different times: (a) $b$ at $T = 20 h$ , (b) $b$ at $T = 30 h$ , (c) $b$ at $T = 40 h$ , (d) $b$ at $T = 50 h$ , (e) $b$ at $T = 84 h$ , (f) $b$ at $T = 88 h$ , (g) $b$ at $T = 92 h$ , and (h) $b$ at $T = 96 h$ .

Conclusion

This article is devoted to the closed-loop performance degradation–based RUL prediction architecture of motor systems. To be specific, a closed-loop performance index of the motor system with multiple slow-varying parameters is first designed by ESO. Furthermore, it is important to update iteratively Wiener degeneration model by PSO and EM algorithm based on MLE algorithm for the purpose of obtaining the PDF of RUL. Finally, the effectiveness has been illustrated on the DC motor and PMSM. In order to improve the rationality of performance index and the accuracy of RUL prediction, our future work will dedicate to the integrated framework with performance estimation and prediction for the closed-loop control system.

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 work was supported in part by the National Natural Science Foundation of China (61673053, 61903026), the Natural Science Foundation of Beijing Municipality (4212040), and the JianLong Young Scholars Innovation Foundation (2022-0651).

ORCID iDs

Xia Wu

Kaixiang Peng

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Remaining useful life prediction for motor systems by iteratively updated wiener process based on closed-loop performance degradation monitoring

Abstract

Keywords

Introduction

Problem formulation

Construction of the performance index

Prediction of the performance index

PDF of RUL based on a nonlinear Wiener process

Iterative update of the nonlinear Wiener degradation model

Iterative update of drift coefficient and diffusion coefficient Θ 1

Iterative update of nonlinear characteristic parameter Θ 2

RUL prediction architecture for the closed-loop motor system

Case study on DC motor

Verification of the accuracy of ESO

Case study on DC motor

Case study on permanent magnet synchronous motor

Verification of proposed RUL prediction algorithm

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References

Iterative update of drift coefficient and diffusion coefficient $Θ_{1}$

Iterative update of nonlinear characteristic parameter $Θ_{2}$