This article investigates the problem of fault-tolerant control for stochastic high-order fully actuated systems (FASs) with actuator faults. Different from the majority of existing studies focusing on deterministic high-order FASs, this work introduces stochastic disturbances into the systems. Employing the generalized martingale technique, a novel fault-tolerant equivalent controller is formulated. Additionally, an adaptive compensation law is constructed to address time-varying faults promptly. The designed preclosed-loop strategy advocates the advantage of the FAS methodology and guarantees performance by ensuring that the tracking error complies with the user-defined probabilistic ultimate bound. Finally, a numerical case and a practical example of a rotary steerable drilling platform are exploited to demonstrate the effectiveness of the proposed method.

Although quality-related process monitoring has achieved the great progress, scarce works consider the detection of quality-related incipient faults. Partial least square (PLS) and its variants only focus on faults with larger magnitudes. In this article, a deep quality monitoring network (DQMNet) for quality-related incipient fault detection is developed. DQMNet includes the feature input layer, feature extraction layers, and the output layer. In the feature input layer, collected variables are divided according to quality variables, and then, features are extracted, respectively, through base detectors. For the feature extraction layers, singular values (SVs) of sliding-window patches and principal component analysis (PCA) are adopted to mine the hidden information layer by layer. For the output layer, statistics are constructed from quality-related/unrelated feature matrix through Bayesian inference. The superiority of DQMNet is demonstrated by a numerical simulation and the benchmark data of Tennessee Eastman process (TEP).

Aimed at sequential dynamic modes, a novel multimodal weighted canonical correlation analysis using an attention (MWCCA-A) mechanism is introduced to derive a single model for process monitoring, by integrating two ideas of replay and regularization in continual learning. Under the assumption that data are received sequentially, subsets of data from past modes with dynamic features are selected and stored as replay data, which are utilized together with the current mode data for continual model parameter estimation. The weighted canonical correlation analysis (WCCA) is introduced to achieve appropriate weightings of past modes' replay data so that the latent variables are extracted by maximizing the weighted correlation with its prediction via the attention mechanism. Specifically, replay data weightings are obtained via the probability density estimation from each mode. This is also beneficial in overcoming data imbalance among multiple modes and consolidating the significant features of past modes further. Alternatively, the proposed model also regularizes parameters based on its previous modes' importance, which is measured by synaptic intelligence (SI). Meanwhile, the objective is decoupled into a regularization-related part and a replay-related part, to overcome the potentially unstable optimization trajectory of SI-based continual learning. In comparison with several multimode monitoring methods, the effectiveness of the proposed MWCCA-A approach is demonstrated by a continuous stirred tank heater (CSTH), Tennessee Eastman process (TEP), and a practical coal pulverizing system.

A novel continual learning-based probabilistic slow feature analysis algorithm is introduced for monitoring multimode nonstationary processes. Multimode slow features are extracted and an elastic weight consolidation (EWC) is adopted for sequential modes. EWC was originally introduced in the setting of machine learning of sequential multi-tasks with the aim of avoiding catastrophic forgetting issue, which equally poses as a major challenge in multimode nonstationary process monitoring. When a new mode arrives, a small set of data are collected for continual learning by the proposed algorithm. A regularization term is introduced to prevent new data from significantly interfering with the learned knowledge, where the parameter importance measures are estimated. The proposed method is referred to as PSFA–EWC, which is updated continually and is capable of achieving excellent performance. PSFA–EWC furnishes backward and forward transfer ability by a single model. The significant features of previous modes are retained while consolidating new information, which may contribute to learning new relevant modes. The effectiveness of the proposed method is demonstrated via a continuous stirred tank heater and a practical coal pulverizing system. Note to Practitioners—Since industrial systems operate in varying modes and data are nonstationary within each mode, multimode nonstationary process monitoring is increasingly important. Traditional multimode monitoring methods generally need complete data from all possible modes and may need to be retrained from scratch when a new mode arrives, which require expensive computation and storage resources. Besides, it is difficult to distinguish real faults from normal variations in multimode nonstationary processes. This paper proposes a novel continual learning-based probabilistic slow feature analysis, where elastic weight consolidation is employed to consolidate the previously learned knowledge while extracting multimode slow features. The monitoring model is updated sequentially and provides backward as well as forward transfer learning ability for successive modes. It is able to separate real faults from normal dynamics, which is beneficial to identifying a new mode for multimode nonstationary processes. In addition, the proposed approach delivers excellent model interpretability and deals with missing data as well as uncertainty. In industrial applications, such as power plants and intelligent manufacturing processes, the proposed method can provide excellent monitoring performance.

In recent years, high-order fully actuated (HOFA) systems, founded by Prof. GR Duan, have recorded rapid progress for deterministic systems. However, the control issue of stochastic fully actuated systems is still an open problem. This study develops a novel stochastic HOFA system model that complements the existing HOFA methodology. Notably, stochastic signals can be considered in the proposed model, different from the case in the deterministic model. By adopting a high-order operator, equivalent control and stabilization control laws are realized to guarantee the global asymptotic stability in probability of the closed-loop system. For the system with sensor gain faults, an observer-based fault-tolerant control law is designed. Finally, the simulation results validate the effectiveness of the proposed control schemes.

Effective process monitoring is both a prerequisite and a guarantee for high system reliability. In modern industrial processes, binary variables may appear together with continuous variables, making process monitoring more intractable. Recently, a model named hybrid variable monitoring (HVM) has been proposed to conduct anomaly detection with both continuous and binary variables. Although the performance of HVM has been significantly improved after using the information of binary variables, it assumes that every continuous variable obeys a single Gaussian distribution and each binary variable obeys a single Bernoulli distribution. It is difficult for practical processes to satisfy such strict assumptions. To overcome this problem, this study proposes an improved algorithm called HVM mixture model (HVMMM). The HVMMM contains multiple components with the assumption of an HVM for every component. Compared with the HVM, the HVMMM is suitable for more general situations and has a more accurate characterization of the data features. Subsequently, the expectation-maximization (EM) algorithm is adopted for parameter learning for multiple components. The mathematical expressions of the parameters are derived in detail. In addition, the improvement on the monitoring performance caused by multiple components is analyzed. Finally, a numerical example and a practical case are used to demonstrate the effectiveness and efficiency of HVMMM. After multiple components are considered, the fault detection rate increases by 5.49% in the numerical example and the false alarm rate reduces by 1.6% in the practical case.

The study of identifying human subgroups from videos is a significant topic, which has received a lot of attention in multiple disciplines. So far, however, there has been little consideration about combining it with relevant conceptions in network science. Therefore, this article proposes a novel method for the automatic identification of human subgroups in dynamic pedestrian flows. The spatial proximity and temporal continuity are combined to calculate the interaction intensity between pedestrians, by which a time-dependent pedestrian flow network is constructed. Based on the objective function of weighted partition density, the optimal threshold is used to determine community structures that correspond to human subgroups in frame images. Numerical experiments demonstrate that our method achieves high identification accuracy under various evaluation datasets, and exhibits better performance than existing methods in terms of different crowd densities, various numbers of subgroup members, and certain levels of trajectory noise. Furthermore, this work provides valuable implications for the understanding of subgroup behaviors and the modeling of subgroup movements.

This paper introduces a novel sparse dynamic inner principal component analysis (SDiPCA) based monitoring for multimode dynamic processes. Different from traditional multimode monitoring algorithms, a model is updated for sequential modes by memorizing the significant features of existing modes. By adopting the concept of intelligent synapses in continual learning, a loss of quadratic term is introduced to penalize the changes of mode–relevant parameters, where modified synaptic intelligence (MSI) is proposed to estimate the parameter importance. Thus, the proposed algorithm is referred to as SDiPCA–MSI. When a new mode arrives, a set of normal samples should be collected. The previous significant features are consolidated without explicitly storing training samples, while extracting new information from the current mode. Consequently, SDiPCA–MSI can provide outstanding performance for successive modes. Characteristics of the proposed approach are discussed, including the computational complexity, advantages and potential limitations. Compared with several state-of-the-art monitoring methods, the effectiveness and superiorities of the proposed method are demonstrated by a continuous stirred tank heater case and a practical industrial system. Note to Practitioners—Multimode process monitoring is increasingly significant as industrial systems generally operate in varying operating conditions. However, most researches focus on multiple local monitoring models for complex multimode processes and assume that data of all possible modes are available and stored before learning. When similar or new modes arrive, local models are rebuilt corresponding to each mode and the model’s capacity would increase with the continuous emergence of modes. Adaptive methods are a branch of multimode monitoring algorithms, but they strive to extract information of the current mode to ensure the monitoring performance while forgetting the previously learned knowledge gradually. This paper proposes a novel sparse dynamic inner principal component analysis with continual learning ability for multimode dynamic process monitoring, where modified synaptic intelligence is developed to measure the parameter importance accurately. It requires limited computation and storage resources for successive modes, which is convenient for practical applications. Similar to current multimode process monitoring algorithms, a set of data should be collected before learning a new mode, which may bring difficulties to real–time monitoring. For industrial systems, such as large–scale power plants and chemical systems, the proposed method has outstanding ability to monitor successive dynamic modes.

The study of crowd evacuation has received considerable attention as the frequent occurrence of crowd disasters in public places. Notably, the increasing proportion of disabled pedestrians makes vulnerable crowds an indispensable part of the evacuation process. However, most previous research neglects to introduce the motion characteristics of disabled pedestrians into the modeling of crowd evacuation. Therefore, we develop an extended model to simulate the evacuation process involving nondisabled, visual-disabled, acoustic-disabled, and physical-disabled pedestrians. Numerical simulations indicate that this model achieves a more realistic mixed crowd evacuation in the library scene and reproduces the escape movement of multitype disabled pedestrians. Moreover, several management strategies are provided to guide the evacuation of disabled pedestrians, and the appropriate strategy can be determined by comprehensively considering multiple factors such as efficiency, safety, and cost.

This study investigates nonstationary process monitoring under frequently varying modes, where new modes are allowed to emerge constantly. However, in current multimode process monitoring methods, generally, data are required from all possible modes and mode identification is realized by prior knowledge for multimode nonstationary processes. In contrast, recursive methods update a monitoring model based on the successive data. However, they forget the learned knowledge gracefully and fail to track drastic variations. Aimed at nonstationary data in each mode, this article proposes an adaptive cointegration analysis (CA) to distinguish real faults from normal variations, which updates a model once a normal sample is encountered and adapts to the gradual change in the cointegration relationship. Then, a modified recursive principal component analysis (RPCA) with continual learning ability is developed to deal with the remaining dynamic information, wherein elastic weight consolidation is adopted to consolidate the previously learned knowledge when a new mode appears. The preserved information is beneficial for establishing a more accurate model than traditional RPCA and avoiding drastic performance degradation for future similar modes. In addition, novel statistics are proposed with prior knowledge and thresholds are calculated by recursive kernel density estimation to enhance the performance. An in-depth comparison with recursive CA and recursive slow feature analysis is conducted to emphasize the superiority, in terms of the algorithm accuracy, memory properties, and computational complexity. Compared with state-of-the-art recursive algorithms, the effectiveness of the proposed method is shown by studying on a numerical case and a practical industrial system.

Multiple operating modes have become a key factor affecting the monitoring performance of practical industrial processes. The monitoring of multiple modes with hybrid variables (containing continuous and binary variables) is more intractable. In addition, the label information of the training data may be unavailable, and new modes may arrive or collected modes may disappear in continuous running of the system owing to the influence of production strategies, materials, loads, etc. Therefore, this article proposes an adjustable multimode monitoring with hybrid variables (AMMHV) model. In AMMHV, the expectation maximization algorithm is utilized for parameter estimation when the label information is unknown. AMMHV can not only effectively conduct the multimode process monitoring of hybrid variables without the label information of training samples, but also be updated without retraining when operation modes change. The incremental learning strategy is adopted to extend the model to give it outstanding monitoring performance for new arriving modes. If the originally collected modes no longer appear during operation, AMMHV can improve the monitoring accuracy of the remaining modes by condensing redundant irrelevant information. Finally, the superiority of AMMHV is fully demonstrated first on a numerical example and then on a process of a thermal power plant.

Industrial processes generally operate under multiple modes and a global monitoring approach, built upon combining local models that are aimed at each mode, requires complete data from all potential modes to be available. However, practical data are generated and collected in a steady stream, which makes it difficult if not impossible to process. This article proposes an efficient sparse dynamic inner principal component analysis algorithm for multimode nonlinear dynamic process monitoring, which aims to build a single monitoring model with continual learning ability for successive modes. To reduce the storage and computational costs, only a few representative data from each mode are selected based on cosine similarity and replayed for retraining when a new mode arrives, which are sufficient to reflect the operating condition of each mode. Inspired by replay continual learning, data from all existing modes are preprocessed by their own statistics and then regarded as a whole dataset, followed by building a single multimode monitoring model. The multimode dynamic latent variables are extracted from data in raw format, via a vector autoregressive model. Therefore, the proposed method is not constrained by the mode similarity, which makes it appropriate for diverse modes and convenient for long-term monitoring tasks. Besides, the proposed method can deal with nonlinearity and a regularization term is added to avoid the potential overfitting issue. Compared with state-of-the-art multimode monitoring methods, the effectiveness of the proposed approach is demonstrated by a continuous stirred tank heater and a practical industrial system.

This article proposes a novel sparse principal component analysis algorithm with self-learning ability for multimode process monitoring, where the successive modes are learned in a sequential fashion. Different from traditional multimode monitoring methods, a small set of data are collected when a novel mode arrives. The proposed method remembers the learned knowledge by selectively slowing down the changes of parameters important for the previous modes, where the importance measure is estimated by synaptic intelligence. The sufficient condition of fault detectability is proved to provide a comprehensive understanding of the proposed method. Besides, the computation and storage resources are saved in the long run, because it is not necessary to retrain the model from scratch frequently and data are discarded once they have been learned. More importantly, the model furnishes excellent interpretability and the catastrophic forgetting problem is further alleviated owing to the sparsity of parameters. In addition, the hyperparameters are discussed to understand the proposed method comprehensively and the computational complexity is analyzed. Compared with several state-of-the-art approaches, a numerical case, and a practical pulverizing system are adopted to illustrate the effectiveness of the proposed algorithm.

With the increasing complexity and scale of modern industrial processes, there widely exist two-valued variables (TVs), such as status monitoring and numerical range variables. However, the traditional process monitoring approaches (such as principal component analysis and partial least square) are strongly based on continuous variables (CVs), thus they totally ignore the useful merit inherent in TVs. Recently, both CVs and TVs are used in combination for monitoring industrial processes for the first time. The mixed hidden naive Bayesian model (MHNBM) and feature-weighted mixed naive Bayes model (FWMNBM) have been proposed to enhance the monitoring performance by simultaneously and efficiently exploiting the valuable information of TVs and CVs. Nevertheless, both models are not suitable for nonstationary processes, which are consistent with the real property of many practical cases. Therefore, this article mainly proposes a novel self-learning FWMNBM (SL-FWMNBM) for nonstationary process monitoring. SL-FWMNBM constantly updates the model parameters in real time at the online detection stage to overcome the changes in the statistical characteristics of monitoring variables. It has the ability to mine process information carried by newly sampled data through self-learning, which is the main difference between SL-FWMNBM and the above two methods. The effectiveness of SL-FWMNBM is demonstrated through a simulation and an actual vibration case of the Zhoushan thermal power plant, China.

How to detect incipient faults has been an important problem in the field of fault detection. Although many types of machine and deep learning methods have been proposed, their performance is not as good as expected. In this article, a novel feature ensemble net (FENet) was developed, particularly for faults 3, 9, and 15 in the Tennessee Eastman process (TEP), which are notoriously difficult to detect. For the input feature layer, features extracted by the basic detectors are integrated to expand the detection ability of FENet. For the hidden feature transformer layers, with sliding-window patches and principal component analysis (PCA), the previous feature matrix is transformed. The sliding-window patches can be used to generate singular values, whereas the patches in the well-known convolution technique can only be vectorized, primarily for performing PCA in PCA-based networks. This enhances the sensitivity of the FENet to incipient faults. For the output feature layer, all feature matrices in the last hidden layer are completely stacked into a large feature matrix. The sliding technique is performed at the decision layer, and a detection index is designed with normalized singular values. The superiority of FENet can be completely verified by a continuous stirred tank heater and TEP. As compared with deep PCA, PCA-based monitoring network, and typical ensemble strategies, such as averaging, voting, stacking, and Bayesian inference, FENet can effectively detect Faults 3, 9, and 15 in TEP.

Practical industrial processes usually have nonstationary properties, which make the monitoring more challenging because the fault information may be buried by nonstationary trends. For nonstationary processes, many methods have been proposed for fault detection based on continuous variables. However, binary variables may appear together with continuous variables in modern industrial processes. To address the issue of process monitoring with hybrid variables and nonstationarity, a model named recursive hybrid variable monitoring (RHVM) is proposed in this paper. For RHVM, recursive strategy is utilized to suppress nonstationary trend and to reveal fault information. In addition, RHVM has the ability of model self-updating with arriving samples. The closed-form updates of required parameters are derived in detail and the improvement of performance is analyzed. At last, the superiority of the proposed model is demonstrated by a simulation example and a practical nonstationary process of a power plant.

Actual industrial processes often show nonstationary characteristics, so nonstationary process monitoring is significant to ensure the safety and reliability of industrial processes. However, existing monitoring methods for nonstationary processes usually ignore process uncertainties, caused by random noises and unknown disturbances. It is worth noting that process uncertainties may degrade the monitoring performance for incipient faults, and result in over-fitting of model parameters. To address the problem of monitoring nonstationary industrial processes with uncertainty, a novel algorithm called probabilistic stationary subspace analysis (PSSA) is proposed in this article. PSSA explicitly models process uncertainties, and distinguishes actual process variations from the uncertainty. In view of the coupling between model parameters, the expectation maximization algorithm is used to estimate the parameters of PSSA, and the closed-form updates are derived in detail. Based on PSSA, two detection statistics are designed for process monitoring. Finally, the effective performance of the proposed method is demonstrated by three case studies, including a numerical example, a closed-loop continuous stirred tank reactor, and a real power plant at Zhejiang Provincial Energy Group of China.

As one of the most effective models for human collective motion, the social force model (SFM) simulates the dynamics of crowd evacuation from a microscopic perspective. However, it treats pedestrians as the homogeneous rigid particles, whereas pedestrians are diverse and heterogeneous in real life. Therefore, this paper develops a pedestrian heterogeneity-based social force model (PHSFM) by introducing physique and mentality coefficients into the SFM to quantify physiology and psychology attributes of pedestrians, respectively. These two coefficients can affect the self-driven force by changing the desired speed, thus characterizing the pedestrian heterogeneity more realistically. Simulation experiments demonstrate that the PHSFM designs a more general and accurate theoretical framework for the expression of pedestrian heterogeneity, which realizes special behavior patterns caused by individual diversity. Furthermore, our model provides effective guidelines for the management of crowds in potential research fields such as transportation, architectural science and safety science.

The significant advancement in data-driven fault detection has been made, but incipient faults such as faults 3, 9, and 15 in Tennessee Eastern process (TEP) still remain difficult for the current approaches. In this article, a powerful principal component analysis (PCA)-based ensemble detector (PCAED) is developed for detecting incipient faults. To begin with, multiple PCA-based detectors are designed based on bootstrap sampling in the training dataset. It can generate two matrices according to principal component and residual subspaces. Then, two sensitive detection indices are developed using maximal singular values of one-step sliding windows along the rows of the above two matrices. With this kind of detection index, PCAED can effectively detect incipient faults, specially faults 3, 9, and 15 in TEP, which cannot be detected by an individual PCA detector. Simulations of TEP and a practical coal pulverizing system fully verify the effectiveness of PCAED. Faults can be successfully detected at the incipient stage, which is very helpful to avoid possible economic or human loss.

Operation safety and efficiency are two main concerns in power plants. It is important to detect the anomalies in power plants, and further judge whether they affect key performance indicators (KPIs), such as the thermal efficiency. These two goals can be achieved by KPI-related nonstationary process monitoring. Although the thermal efficiency cannot be accurately measured online, it can be strongly characterized by some online measurable variables, including the exhaust gas temperature and oxygen content of flue gas. These critical variables closely related to the thermal efficiency are termed as output variables. Inspired from nonstationary common trends between input and output variables in thermal power plants, the output-relevant common trend analysis (OCTA) method is proposed, in this article, to model the input–output relationship. In OCTA, input and output variables are decomposed into nonstationary common trends and stationary residuals, and the model parameters are estimated by solving an optimization problem. It is pointed out that OCTA is a generalized form of partial least squares (PLS). The superior monitoring performance of OCTA is illustrated by case studies on a real power plant in Zhejiang Provincial Energy Group of China. Compared with the other PLS-based recursive algorithms, OCTA can effectively detect the anomalies in power plants and accurately determine whether they have an impact on the thermal efficiency or not.

For some practical industrial systems or components, such as blast furnaces and Li-ion batteries, there are two important factors to model the degradation processes. One is the long-range dependence, which can reflect the non-Markovian nature of the degradation processes. The other factor is the existence of multiple modes, because the operating conditions and external environments inevitably change during the whole lifetime of these systems. In this paper, we first propose a fractional Brownian motion (FBM) based degradation model with long-range dependence and multiple modes, and then consider the prediction of remaining useful life. To identify the multiple modes in the degradation process, we propose a two-step method, including change-points detection and linear segments clustering. In each degradation mode, the degradation rate is assumed to be normally distributed. The means and variances of these distributions can be obtained by the maximum likelihood estimation. To describe the switching between different modes, the continuous-time Markov chain is applied, and its transition rate matrix can be estimated by the historical switching time. An approximation of the first passage time with a predefined threshold can be obtained by a weak convergence theorem and a time-space transformation. A numerical simulation and a practical case of a blast furnace wall are provided to demonstrate the effectiveness of the proposed method.

This study considers the decentralized maintenance of a multistate system (MSS) with low-priority components (LPCs) and high-priority components (HPCs). By introducing imperfect observations of the state, the MSS can be modeled as a partially observable Markov decision process. We propose an (m, N) maintenance policy, where it is considered that the MSS has failed when an HPC fails or when the number of failed LPCs reaches m. In contrast to a centralized maintenance mode, two maintenance teams conduct reliability evaluations and maintenance actions. One team employs the Markov method to predict the trends in the deterioration of the components. The other team estimates the status of the MSS based on the sample data, which are stochastically related to the condition of the system. The different teams may have different maintenance costs and effects, and either maintenance team can be selected based on the system's status. We discuss in detail how to arrange the maintenance teams in order to obtain the lowest expected cost rate with a guarantee of system reliability. Illustrative numerical examples are provided to show the significant cost savings under decentralized maintenance compared with centralized maintenance due to either lower expenditure or shorter time requirements.

Some practical systems such as blast furnaces and turbofan engines have degradation processes with memory effects. The term of memory effects implies that the future states of the degradation processes depend on both the current state and the past states because of the interaction with environments. However, most works generally used a memoryless Markovian process to model the degradation processes. To characterize the memory effects in practical systems, we develop a new type of degradation model, in which the diffusion is represented as a fractional Brownian motion (FBM). FBM is actually a special non-Markovian process with long-term dependencies. Based on the monitored data, a Monte Carlo method is used to predict the remaining useful life (RUL). The unknown parameters in the proposed model can be estimated by the maximum likelihood algorithm, and then the distribution of the RUL is predicted. The effectiveness of the proposed model is fully verified by a numerical example and a practical case study.

A prerequisite for the existing remaining useful life prediction methods based on stochastic processes is the assumption of independent increments. However, this is in sharp contrast to some practical systems including batteries and blast furnace walls, in which the degradation processes have the property of long-range dependence. Based on the fractional Brownian motion, we adopt a degradation process with long-range dependence to predict the remaining useful life of the above systems. Because the degradation process with long-range dependence is neither a Markovian process nor a semimartingale, the exact analytical first passage time is difficult to derive directly. To address this problem, a weak convergence theorem is first adopted to approximately transform a fractional Brownian motion-based degradation process into a Brownian motion-based one with a time-varying coefficient. Then, with a space-time transformation, the first passage time of the degradation process with long-range dependence can be obtained in a closed form. Unknown parameters in the degradation model can be identified using discrete dyadic wavelet transform and maximum likelihood estimation. Numerical simulations and a practical example of a blast furnace wall are given to verify the effectiveness of the proposed method.

A general (N, T, τ) maintenance model is developed for multicomponent systems with two types of components, namely main and auxiliary components. The main component suffers from evident failures, which are assumed to be found or detected as soon as they occur. Auxiliary components with protective or standby functions are modeled by a k-out-of-n:F subsystem, in which failures are hidden and assumed to be detected and fixed only at inspections. Although the shutdown of a subsystem may not halt the system, it could cause a potential risk to the system or financial losses. In this model, the whole system can be renewed at the Nth failure of the main component or at time T, whichever occurs first. Further, incomplete periodic inspections and the optimal number of repairs before replacement are also considered in opportunistic maintenance. Incomplete periodic inspections can efficiently overcome the drawbacks of existing maintenance based on periodic inspections and opportunistic maintenance. For the case of exponential lifetime distribution, an explicit analytical expression of the maintenance cost rate in a renewal cycle is derived by applying the Laplace transform to recursive equations. By setting parameters N, T, τ to tend to infinity, respectively, special properties are derived and a comparison with several maintenance models is performed. For a given life expectancy of the system (namely T), the existence of optimal parameters N and τ is proven. Numerical examples are presented to show the effectiveness of the proposed model.

The problem of finding the optimal maintenance policy for partially observed systems is considered, where only a limited number of maintenance actions can be performed. The maintenance is assumed to be imperfect in that the system can be only restored to a less deteriorated level rather than to a state as good as new. In addition, the true deterioration state is known just after each replacement action, but the deterioration state at other instants during the system's operation cannot be observed, evolving as a discrete-time Markov chain with a finite state space. In this paper, the described problem can be formulated as a partially observed Markov decision process (POMDP) over the infinite time horizon. To increase the computational efficiency, several key structural properties are developed through minimizing the total expected cost per unit time. The existence of the optimal threshold-type maintenance policy is strictly proved, and the monotonicity of the threshold is obtained. The effectiveness of the optimal policy can be verified by a numerical example.

For a class of multi-sensor dynamic systems subject to latent degradation, the remaining useful life prediction with anticipated performance is mainly considered in this paper. The hidden degradation process is first identified recursively by adopting distributed fusion filtering based on observations from multiple sensors. Then the remaining useful life distribution is predicted on the basis of converged degradation state and parameter updating during the operating process. The uncertainty index is aanalyzed to quantitatively evaluate the benefits of increasing multi-sensor information for predicted remaining useful life, and the sensor selection is also discussed for satisfying the anticipated performance such as variance. Our main results are verified by a numerical example, and a practical case study of the milling machine experiment.

Many works on preventive maintenance (PM) of systems only consider either a single failure mode, or statistically independent failure modes. Here, we study the maintenance policy for systems with two statistically dependent failure modes (namely maintainable, and nonmaintainable), and resource constraints. Assume (i) the nonmaintainable failures unidirectionally affect the maintainable failure rate; (ii) due to the constrained resource, only a limited number of imperfect PM actions are performed to reduce the maintainable failure; and (iii) the improvement factor due to each imperfect PM is fixed, and the maximal number before replacement is fixed. By combining a Castro model for statistically dependent failure modes with a Zhang-Jardine model for a single failure mode and imperfect PM, we propose a hybrid maintenance model for systems with statistically dependent failure modes and limited imperfect PM. To examine the maintenance policy, assume that both the nonmaintainable failures and the maintainable failures follow the same type of failure rate functions such as the increasing power law failure rate function. We discuss the relation between expected cost rate per unit-time and each decision variable, and then give a solution to a constrained optimization problem provided that the length of the interval between two successive maintenance actions cannot be too small. Numerical simulations for the increasing power law failure rate fully verify the proposed maintenance policy, which can be also extended to other increasing failure rate functions such as an exponential failure rate function.

Sheu and Chang (IEEE Trans. Rel., vol. 58, no. 2, pp. 397-404, 2009) presented an interesting extended periodic imperfect preventive maintenance (EPIPM) model for a system with age-dependent failure type. Many cases studied previously are special cases of the EPIPM model. In the Errata (IEEE Trans. Rel., vol. 60, no. 2, 2011), Sheu and Chang showed that the proposed effective age and the proposed hazard rate function after the PM are incorrect. In this paper, based on the correct failure characteristics (effective age and hazard rate function after PM), the corrects-expected total cost per unit time for the EPIPM model is presented. By assigning three types of failure characteristics for the EPIPM model, we analyse and compare the correspondings-expected total costs per unit time. We find that thes-expected total cost per unit time developed by Sheu and Chang (IEEE Trans. Rel., vol. 58, no. 2, pp. 397-404, 2009) is only one upper bound of the exacts-expected total cost per unit time. In addition, we also give some results on the existence of the optimal solution for the exacts-expected total cost.

Manufacturing systems run in time-varying environmental and operational conditions. For the effective manager to make a long-term preventive maintenance decision, it is necessary to integrate the time-varying environment into preventive maintenance (PM) policies. This paper considers PM for systems running in the time-varying environment, modeled as a two-state homogeneous Markov process, where one state represents a typical condition, and the other represents a severe condition. Environmental conditions affect the hazard rate function through a proportional hazard model. To avoid sudden failures in a system due to either minor failures or catastrophic failures, an extended periodic imperfect preventive maintenance model is carried out, and the maintenance effect is modeled with an age reduction factor, and a hazard improvement factor. We prove the discontinuity of the hazard rate function of the system in a time-varying environment through a Markov additive process. We also give a method to compute the probability density function of failure at any time. Further, the$s$-expected cost rate of the system in the time-varying environment is compared with the$s$-expected cost rates of the system always working in typical, and severe conditions. Finally, numerical examples fully verify our main results.

Many works on condition-based maintenance of repairable systems apply to either a single failure mode, or statistically independent failure modes. Different from these works, this paper considers the problem of predictive maintenance of repairable systems with dependent failure modes, and resource constraints. Assume that (i) a repairable system is subject to two statistically dependent failure modes bidirectionally affecting each other, (ii) imperfect maintenance actions are cooperatively performed on two dependent failure modes by allocating insufficient resources spent for maintenance, and (iii) future maintenance scheduled at the current time depend on both the predicted number of future failures and the minimization of the expected maintenance cost rate defined in the long term. To resolve the above problem, a novel cooperative predictive maintenance model is proposed. Its basis is the incorporation of the hazard-rate function, and effective age. In this model, two failure modes are statistically dependent in such a way that the hazard rate of one failure mode depends on the accumulated number of failures of the other failure mode. The effect of imperfect maintenance is interpreted in terms of how the hazard rate function and the effective age are changed by maintenance actions. The age reduction factor for each failure mode due to maintenance has some deterministic relation to the degree of resources cooperatively allocated to perform maintenance. The decision variables in the maintenance policy, namely the number of maintenance actions to be performed, the interval between successive maintenance actions, and the cooperatively allocated degree of resources, can be recursively updated when new monitored information arrives. This approach relies on both the predicted number of future failures, and the minimization of the expected maintenance cost rate defined in the long term.