This paper examines approximation-based fixed-time adaptive tracking control for a class of uncertain nonlinear pure-feedback systems. Novel virtual and actual controllers are designed that resolve the meaninglessness of virtual and actual controllers at the origin and in the negative domain, and the sufficient condition for the system to have semiglobal fixed-time stability is also provided. Radial basis function neural networks are introduced to approximate unknown functions for solving the fixed-time control problem of unknown nonlinear pure-feedback systems, and the mean value theorem is used to solve the problem of nonaffine structure in nonlinear pure-feedback systems. The controllers designed in this paper ensure that all signals in the closed-loop system are semiglobally uniform and ultimately bounded in a fixed time. Two simulation results show that appropriate design parameters can limit the tracking error within a region of the origin in a fixed time.

In this brief, we study the distributed adaptive fixed-time tracking consensus control problem for multiple strict-feedback systems with uncertain nonlinearities under a directed graph topology. It is assumed that the leader’s output is time varying and has been accessed by only a small fraction of followers in a group. The distributed fixed-time tracking consensus control is proposed to design local consensus controllers in order to guarantee the consensus tracking between the followers and the leader and ensure the error convergence time is independent of the systems’ initial state. The function approximation technique using radial basis function neural networks (RBFNNs) is employed to compensate for unknown nonlinear terms induced from the controller design procedure. From the Lyapunov stability theorem and graph theory, it is shown that, by using the proposed fixed-time control strategy, all signals in the closed-loop system and the consensus tracking errors are cooperatively semiglobally uniformly bounded and the errors converge to a neighborhood of the origin within a fixed time. Finally, the effectiveness of the proposed control strategy has been proved by rigorous stability analysis and two simulation examples.

Log interpretation is critical in locating pay zones and evaluating their potential. Conventional log interpretation is done manually. In our work, deep learning methods are utilized to deal with preliminary pay zone classification. In this way, human expertise can be liberated from trivial and repetitive tasks during logging interpretation. In a fluvial depositional environment, the sand distribution varies both vertically and horizontally. Thus, a large dataset covering a large area may lead to a too "averaged" model. In our work, we select a relatively small dataset (e.g., seven wells) to reflect the regional features. Standard deep learning processes are employed. The log data are cleaned, visualized, and preprocessed for the algorithms. A preliminary random forest (RF) model is used to separate the sand (interpretation needed) from the shale (interpretation not needed) facies. In the classification model building and training stages, various types of algorithms are tried and compared, from the simple K-nearest neighbor (KNN) to dense neural network (DNN). To account for the continuity and influence of adjacent depths, a 1D convolutional neural network (CNN) model is tested. With the model, a simple self-training model is developed and discussed. K-fold validation methods are used to fully reflect the model's performance in such relatively small dataset. With the given dataset, common deep learning methods generate only moderate accuracy and are easily overfitted. On the other hand, the CNN outperforms the other approaches due its features for pattern recognition. With special caution, a self-learning approach can also further improve the performance. A comparison of different deep learning approaches in terms of time of computation, accuracy, and stability is established. Even trained from a small dataset, with the CNN model, it is possible to identify the zones of interest automatically and consistently. Due to the size of dataset, a series of techniques is utilized to reduce the impact of overfitting, including balance sampling, drop out, regularization, and early stopping, among others. During the optimization of critical hyperparameters, grid search with Bayesian statistics is used together with K-fold validation.

This paper investigates adaptive fixed-time tracking consensus control problems for multiagent nonlinear pure-feedback systems with performance constraints. Compared with existing results of first/second/high-order multiple agent systems, the studied systems have more complex nonlinear dynamics with each agent being modeled as a high-order pure-feedback form. The mean value theorem is introduced to address the problem of nonaffine structure in nonlinear pure-feedback systems. Meanwhile, radial basis function neural networks (RBFNNs) are employed to approximate unknown functions. Furthermore, a constraint variable is used to guarantee that all local tracking errors are within the prescribed boundaries. It is shown that, by utilizing the proposed consensus control protocol, each tracking consensus error can converge into a neighborhood around zero within designed fixed time, the tracking consensus performance can be ensured during the whole process, and all signals in the investigated systems are bounded. Finally, two simulations are performed and the results demonstrate the effectiveness of the proposed control strategy.

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