This article develops an event-triggered safe critic learning control (ESCLC) algorithm for nonlinear systems subject to asymmetric state constraints by integrating a safe critic learning control (SCLC) framework with an event-triggering mechanism. The SCLC algorithm innovatively incorporates control barrier functions into the safe value function design, addressing the challenge of deriving optimal control policies that guarantee system safety. Convergence of the SCLC algorithm is rigorously established within the value iteration framework, along with a criterion for assessing the admissibility of control policies. To enhance the application value of the algorithm in resource-constrained scenarios, an event-triggering mechanism is incorporated into the SCLC framework, yielding the ESCLC algorithm. The resulting closed-loop system under the ESCLC algorithm is proved to be asymptotically stable, and an upper bound on the actual value function is derived to ensure bounded performance degradation. In addition, a policy improvement method based on particle swarm optimization is designed that eliminates dependence on the system control matrix. Finally, the effectiveness of the ESCLC algorithm is verified through simulation experiments on a torsion pendulum system and a ball-and-beam system.

Accurate detection of sensor drift fault in wastewater treatment process (WWTP) is essential for maintaining normal system operation and making correct decisions. However, since the WWTP is influenced by numerous internal and external factors, the data acquired from the actual WWTP is always multi-distributed, thus bringing difficulties to the accurate detection of sensor drift fault for the slow gradual change. To address this problem, a PCA-based sensor drift fault detection method with distribution adaptation (DAPCA) is proposed in this study. It presents a novel PCA-based fault detection method including a temporal WaveCluster for adaptive clustering for multi-distributed data, and a robust PCA-based fault detection with a smoothing mechanism using a combined index. Firstly, an improved WaveCluster algorithm is designed to cluster the multi-distributed data adaptively by considering both the spatial and temporal characteristics. Secondly, a robust PCA algorithm is presented that incorporates a smoothing mechanism to increase its robustness to noise interference. Thirdly, to strike a balance between traditional statistical indexes, a combined index is introduced with adaptive thresholds for multi-distributed data, thus enhancing the overall detection accuracy. To assess the performance of DAPCA, it is tested on both benchmark and real datasets. The results show that it attains the superior detection accuracy with higher F1-scores and lower false alarm rates than comparative methods. Furthermore, DAPCA is demonstrated to be more robust to various types of noises, significantly reducing the false alarms caused by the noise. Note to Practitioners—In the context of wastewater treatment process (WWTP), the inherent exposure of sensors to harsh environmental conditions renders them prone to drift fault. Furthermore, the complex operational dynamics of WWTP contribute to the emergence of a multi-distribution of data, thereby exacerbating the challenges associated with accurate detection of drift fault. Motivated by this, the present paper proposes a PCA-based sensor drift fault detection method with distribution adaptation (DAPCA) in WWTP, which prevents the degradation of detection accuracy caused by changes in data distribution. It presents a novel PCA-based fault detection method including a temporal WaveCluster for adaptive clustering for multi-distributed data, and a robust PCA-based fault detection with a smoothing mechanism using a combined index. Consequently, the effectiveness of the proposed DAPCA is validated via comparisons to other models, which performs a superior detection accuracy with higher F1-scores and lower false alarm rates. Furthermore, DAPCA is demonstrated to be more robust to many types of noises, significantly reducing the false alarms caused by the noise. In conclusion, for multi-distributed data, DAPCA is able to accurately detect sensor drift fault in WWTP, and can be further extended for sensor drift fault detection in other industrial processes.

It is well-documented that cross-layer connections in feedforward small-world neural networks (FSWNNs) enhance the efficient transmission for gradients, thus improving its generalization ability with a fast learning. However, the merits of long-distance cross-layer connections are not fully utilized due to the random rewiring. In this study, aiming to further improve the learning efficiency, a fast FSWNN (FFSWNN) is proposed by taking into account the positive effects of long-distance cross-layer connections, and applied to nonlinear system modeling. First, a novel rewiring rule by giving priority to long-distance cross-layer connections is proposed to increase the gradient transmission efficiency when constructing FFSWNN. Second, an improved ridge regression method is put forward to determine the initial weights with high activation for the sigmoidal neurons in FFSWNN. Finally, to further improve the learning efficiency, an asynchronous learning algorithm is designed to train FFSWNN, with the weights connected to the output layer updated by the ridge regression method and other weights by the gradient descent method. Several experiments are conducted on four benchmark datasets from the University of California Irvine (UCI) machine learning repository and two datasets from real-life problems to evaluate the performance of FFSWNN on nonlinear system modeling. The results show that FFSWNN has significantly faster convergence speed and higher modeling accuracy than the comparative models, and the positive effects of the novel rewiring rule, the improved weight initialization, and the asynchronous learning algorithm on learning efficiency are demonstrated.

As a key water quality parameter in the wastewater treatment process (WWTP), the accurate measurement of total phosphorus (TP) would effectively prevent the effluent water from eutrophication. Although soft measurement models can successfully predict effluent TP, the model prediction is unreliable because outliers will inevitably exist in actual WWTP due to a variety of disturbances. To solve this problem, a novel robust small-world feedforward neural network (RSWFNN) is proposed to improve the robustness of effluent TP prediction. First, the robust Spearman rank correlation analysis is used to determine auxiliary variables intrinsically correlated with the effluent TP. Second, inspired by the fault tolerance of the human brain from its small world property, the small-worldness is introduced to obtain a robust network architecture. Third, the robust learning algorithm using the loss function of regularized M-estimation is proposed to suppress the responses of outliers to improve the robustness of the model. Finally, the corresponding two hyperparameters are determined by an adaptive adjustment strategy, thus ensuring the effectiveness of suppressing outliers. Our experimental results have shown that RSWFNN has stronger robustness and better prediction performance to predict effluent TP than other modeling methods, and the superiority of robustness becomes more obvious with the increase of outlier proportion.

A deep belief network (DBN) is effective to create a powerful generative model by using training data. However, it is difficult to fast determine its optimal structure given specific applications. In this paper, a growing DBN with transfer learning (TL-GDBN) is proposed to automatically decide its structure size, which can accelerate its learning process and improve model accuracy. First, a basic DBN structure with single hidden layer is initialized and then pretrained, and the learned weight parameters are frozen. Second, TL-GDBN uses TL to transfer the knowledge from the learned weight parameters to newly added neurons and hidden layers, which can achieve a growing structure until the stopping criterion for pretraining is satisfied. Third, the weight parameters derived from pretraining of TL-GDBN are further fine-tuned by using layer-by-layer partial least square regression from top to bottom, which can avoid many problems of traditional backpropagation algorithm-based fine-tuning. Moreover, the convergence analysis of the TL-GDBN is presented. Finally, TL-GDBN is tested on two benchmark data sets and a practical wastewater treatment system. The simulation results show that it has better modeling performance, faster learning speed, and more robust structure than existing models. Note to Practitioners-Transfer learning (TL) aims to improve training effectiveness by transferring knowledge from a source domain to target domain. This paper presents a growing deep belief network (DBN) with TL to improve the training effectiveness and determine the optimal model size. Facing a complex process and real-world workflow, DBN tends to require long time for its successful training. The proposed growing DBN with TL (TL-GDBN) accelerates the learning process by instantaneously transferring the knowledge from a source domain to each new deeper or wider substructure. The experimental results show that the proposed TL-GDBN model has a great potential to deal with complex system, especially the systems with high nonlinearity. As a result, it can be readily applicable to some industrial nonlinear systems.

An echo-state network (ESN) is an effective alternative to gradient methods for training recurrent neural network. However, it is difficult to determine the structure (mainly the reservoir) of the ESN to match with the given application. In this paper, a growing ESN (GESN) is proposed to design the size and topology of the reservoir automatically. First, the GESN makes use of the block matrix theory to add hidden units to the existing reservoir group by group, which leads to a GESN with multiple subreservoirs. Second, every subreservoir weight matrix in the GESN is created with a predefined singular value spectrum, which ensures the echo-sate property of the ESN without posterior scaling of the weights. Third, during the growth of the network, the output weights of the GESN are updated in an incremental way. Moreover, the convergence of the GESN is proved. Finally, the GESN is tested on some artificial and real-world time-series benchmarks. Simulation results show that the proposed GESN has better prediction performance and faster leaning speed than some ESNs with fixed sizes and topologies.