The batch process generally covers high nonlinearity and two-directional dynamics: time-wise dynamics, which correspond to inherently time-varying dynamics resulting from the slowly varying underlying driving forces within each batch duration; and batch-wise dynamics, which are associated with different operating modes among different batches. However, most existing dynamic nonlinear monitoring methods cannot extract the slowly varying underlying driving forces of the nonlinear batch process and rarely tackle the batch-wise dynamic characteristics among batch runs. In order to address these issues, a new monitoring scheme based on two-directional dynamic kernel slow feature analysis (TDKSFA) is developed by combining kernel SFA with a global modelling strategy. In the TDKSFA method, kernel SFA is integrated with the ARMAX time series model to mine the nonlinear and timewise dynamic properties within a batch run due to its capability of extracting the slowly varying underlying driving forces. Furthermore, the global modelling strategy is presented to handle the batch-wise dynamics among batches by calculating the total average kernel matrix of all training batches. After the slow features are extracted, Hotelling's T 2 and SPE statistics are built to detect faults. To solve the issue of fault variable nonlinear identification, a novel nonlinear contribution plot inspired by the pseudo-sample variable projection trajectories in the TDKSFA model is further proposed to identify fault variables. Finally, the feasibility and effectiveness of the TDKSFA-based fault diagnosis strategy is demonstrated through a numerical system and the penicillin fermentation process. K E Y W O R D S batch process, fault diagnosis, slow feature analysis, two-directional dynamics 1 | INTRODUCTION Batch processes are becoming increasingly important in the pharmaceutical, semiconductor, polymer, and biochemical industries to manufacture high value-added products. However, it is difficult to establish a batch process monitoring model due to the high nonlinearity in time-wise and the time-varying dynamics both time-wise

In this paper, a class of fractional-order nonlinear systems are considered in the presence of actuator faults. A novel fault tolerant control scheme based on disturbance observer has been presented, where the actuator faults are considered as the system disturbance and can be approximated by the proposed disturbance observer. The developed fault tolerant control guarantees the convergence of the closed-loop system and the output tracking performance. Finally, a simulation example is presented to verify the effectiveness of the new method. K E Y W O R D Sadaptive control, disturbance observer, fault tolerant control, fractional-order system INTRODUCTIONSince actuator fault is inevitable in modern control systems, which may degrade system performance, or even lead to system instability, compensation of actuator faults has significant impact on some critical physical systems. Adaptive fault tolerant control (FTC) has been shown as a desirable tool to this problem and remarkable progress have been made in the area. 1-8 Guang-Hong Yang proposes an adaptive output feedback control to deal with unknown nonaffine nonlinear faults for a class of nonlinear systems. 1 Salman Ijaz studies an active fault tolerant control (FTC) scheme for aircraft with dissimilar redundant actuation system in Reference 2. In Reference 3, fault-tolerant tracking control problem for a class of strict-feedback nonlinear systems subjected to actuator faults and external disturbances is investigated. Due to the global approximation performance of fuzzy logic system and neural networks, the authors in References 4-9 successfully solved the problem by introducing adaptive fuzzy logic control or adaptive neural networks control. Over the past decades, fractional-order systems 10,11 have attracted considerable attention due to the ability of describing long memory and hereditary characteristic of complex phenomenon in some practical systems. It can describe some systems more accurately than the traditional integer order method and has been widely applied in fields in engineering and physics, such as system biology, physics, chemistry, automatic control, materials science, engineering, etc. More and more fractional order systems have been studied, especially in tracking control performance and stability analysis. 12-18 Y. Wei et al. propose a variety of methods for the stabilization of fractional order systems. 12,14,15 Sufficient and necessary conditions for stabilizing singular fractional order systems are given in Reference 16. For a class of continuous-time fractional positive systems, stabilization control is addressed based on disturbance observer in Reference 17. The Lyapunov stability theory is extended to fractional-order nonlinear systems in References 13 and 18 and, where the concept of Lyapunov stability is defined and Lyapunov design method is proposed. As we all know, actuator or sensor faults may also occur in fractional-order systems due to poor working conditions, mechanical fatigue, and other factors. Consequently, how to detect and ...

Recently, the unsupervised extreme learning machine (UELM) technique as a nonlinear data mining approach has been employed to diagnose nonlinear process faults. However, during the dimensionality reduction of process observation data, UELM only aims at mining the local structure feature information of process data and lacks of the ability of preserving the global structure analysis of process data, which would perform unsatisfactorily for monitoring nonlinear process. Furthermore, the choice of the optimal number of hidden layer nodes in UELM method is still a challenging problem. To handle these two tough problems, a new fault detection approach on the basis of global preserving unsupervised kernel extreme learning machine (GUKELM) technique is proposed to monitor nonlinear process effectively. In GUKELM technique, the data global structure preserving framework is naturally incorporated into standard UELM to capture the local structure information as well as to preserve the global structure feature information of nonlinear process observation data during the dimensionality reduction. Meanwhile, to tackle the strong nonlinearity of process observation data, the kernel trick is utilized to successfully solve the challenging issue of setting the optimal number of hidden layer nodes. Based on extracted process data low dimensional feature information from GUKELM method, support vector data description algorithm is adopted to construct the monitoring statistic to detect process fault. Finally, the feasibility and effectiveness of the proposed process monitoring strategy is illustrated through a numerical nonlinear system and the continuous stirred tank reactor process which is a typical nonlinear process.INDEX TERMS Nonlinear process monitoring, unsupervised extreme learning machine, global structure analysis, kernel trick, support vector data description.

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