Nonlinear inequalities are widely used in science and engineering areas, attracting the attention of many researchers. In this article, a novel jump-gain integral recurrent (JGIR) neural network is proposed to solve noise-disturbed time-variant nonlinear inequality problems. To do so, an integral error function is first designed. Then, a neural dynamic method is adopted and the corresponding dynamic differential equation is obtained. Third, a jump gain is exploited and applied to the dynamic differential equation. Fourth, the derivatives of errors are substituted into the jump-gain dynamic differential equation, and the corresponding JGIR neural network is set up. Global convergence and robustness theorems are proposed and proved theoretically. Computer simulations verify that the proposed JGIR neural network can solve noise-disturbed time-variant nonlinear inequality problems effectively. Compared with some advanced methods, such as modified zeroing neural network (ZNN), noise-tolerant ZNN, and varying-parameter convergent-differential neural network, the proposed JGIR method has smaller computational errors, faster convergence speed, and no overshoot when disturbance exists. In addition, physical experiments on manipulator control have verified the effectiveness and superiority of the proposed JGIR neural network.

Time-varying quadratic programming (TV-QP) is widely used in artificial intelligence, robotics, and many other fields. To solve this important problem, a novel discrete error redefinition neural network (D-ERNN) is proposed. By redefining the error monitoring function and discretization, the proposed neural network is superior to some traditional neural networks in terms of convergence speed, robustness, and overshoot. Compared with the continuous ERNN, the proposed discrete neural network is more suitable for computer implementation. Unlike continuous neural networks, this article also analyzes and proves how to select the parameters and step size of the proposed neural networks to ensure the reliability of the network. Moreover, how to achieve the discretization of the ERNN is presented and discussed. The convergence of the proposed neural network without disturbance is proven, and bounded time-varying disturbances can be resisted in theory. Furthermore, the comparison results with other related neural networks show that the proposed D-ERNN has a faster convergence speed, better antidisturbance ability, and lower overshoot.

In order to make redundant robot manipulators (RRMs) track the complex time-varying trajectory, the motion-planning problem of RRMs can be converted into a constrained time-varying quadratic programming (TVQP) problem. By using a new punishment mechanism-combined recurrent neural network (PMRNN) proposed in this article with reference to the varying-gain neural-dynamic design (VG-NDD) formula, the TVQP problem-based motion-planning scheme can be solved and the optimal angles and velocities of joints of RRMs can also be obtained in the working space. Then, the convergence performance of the PMRNN model in solving the TVQP problem is analyzed theoretically in detail. This novel method has been substantiated to have a faster calculation speed and better accuracy than the traditional method. In addition, the PMRNN model has also been successfully applied to an actual RRM to complete an end-effector trajectory tracking task.

In this article, a novel approach of utilizing an integral dynamic learning network (IDLN) is presented for addressing a general time-varying Lyapunov matrix equation (TVLME). First, a cost function is defined by designing a variable unbounded vector/matrix-type error function. The goal is to make the cost function approximate to zero. Second, an integral neural dynamic equation with an odd activation function that is monotonically increasing is designed and applied to guarantee that the error function can converge to zero. Third, a novel IDLN with a recurrent topological structure is exploited to find the time-varying theoretical solution. The proposed IDLN with strong robustness to bounded noise with unknown amplitude regardless of the value of hyperparameters, can be implemented through electronic circuits as a method of parallel computing and achieve global convergence from any initial state. In addition, for better convergence rates, the novel linear-arcsine-type and softsign-arcsine-type activation functions are designed and utilized to the proposed IDLN. The effectiveness, stability, and practicability of the proposed IDLN are verified by comparative computer simulations and the application to the analysis of voltage stability in a single-machine infinite bus system.

As for time-varying tracking control problems, many neural-dynamics-based control methods have been proposed because of their high efficiency. The varying-parameter convergent neural dynamics design method with the characteristic of super-exponential convergence has been applied to design controllers for unmanned aerial vehicles. Although the varying-parameter convergent neural dynamics controller has a fast convergence speed, it still needs long enough time to achieve tracking theoretically. By combining the finite-time activation function with the varying-parameter convergent neural dynamics design method, a finite-time-gain-adjustment design method is proposed and proved theoretically in this paper. This controller can make state variables of the system converge to their time-varying targets in finite time. Compared with some traditional methods, contrastive experiments and application to a multi-rotor unmanned aerial vehicle system illustrate that the proposed controller has finite convergence time, better anti-noise performance, and faster convergence speed, which enable the multi-rotor unmanned aerial vehicles to track time-varying targets more quickly and accurately, so as to achieve more complex and efficient control tasks.

To realize the robust control of multirotor unmanned aerial vehicle (UAV) systems, adaptive multilayer neural dynamics (AMND) controllers are proposed and analyzed. The proposed AMND controllers with the strong anti-perturbation property can drive multirotor UAVs to track time-varying tasks and deal with parameter uncertainty problems. First, the design method of the general multilayer neural dynamics (MLND) controllers is introduced and analyzed. Second, based on the design method, the attitude angles, height, and position controllers of a UAV system are designed. Third, according to the adaptive control theory, a novel AMND controller is designed, which can self-tune the parameters of the UAV. Finally, the proposed AMND method applies to a real-world hexrotor UAV system to illustrate its reliability. Mathematical analysis, computer simulations, and experiments verify the reliability, stability, and effectiveness of the proposed controllers which are used to track time-varying tasks.

To solve a general time-varying Sylvester equation, a novel integral recurrent neural network (IRNN) is designed and analyzed. This kind of recurrent neural networks is based on an error-integral design equation and does not need training in advance. The IRNN can achieve global convergence performance and strong robustness if odd-monotonically increasing activation functions [i.e., the linear, bipolar-sigmoid, power, or sigmoid-power activation functions (SP-AFs)] are applied. Specifically, if linear or bipolar-sigmoid activation functions are applied, the IRNN possess exponential convergence performance. The IRNN has finite-time convergence property by using power activation function. To obtain faster convergence performance and finite-time convergence property, an SP-AF is designed. Furthermore, by using the discretization method, the discrete IRNN model and its convergence analysis are also presented. Practical application to robot manipulator and computer simulation results with using different activation functions and design parameters have verified the effectiveness, stability, and reliability of the proposed IRNN.

Because of the simple structure and strong flexibility, multirotor unmanned aerial vehicles (UAVs) have attracted considerable attention among scientific researches and engineering fields during the past decades. In this paper, a novel adaptive multilayer neural dynamic (AMND)-based controllers design method is proposed for designing the attitude angle (the roll angle ϕ , the pitch angle θ , and the yaw angle ψ ), height ( z ), and position ( x and y ) controllers of a general multirotor UAV model. Global convergence and strong robustness of the proposed AMND-based method and controllers are analyzed and proved theoretically. By incorporating the adaptive control method into the general multilayer neural dynamic-based controllers design method, multirotor UAVs with unknown disturbances can complete time-varying trajectory tracking tasks. AMND-based controllers with the self-tuning rates can estimate the unknown disturbances and solve the model uncertainty problems. Both the theoretical theorems and simulation results illustrate that the proposed design method and its controllers with strong anti-interference property can achieve the time-varying trajectory tracking control stably, reliably, and effectively. Moreover, a practical experiment by using a mini multirotor UAV illustrates the practicability of the AMND-based method.

As an increasing number of unmanned aerial vehicles (UAVs) have been widely applied in many aspects, controllers with higher performance are preferred. In this paper, a new varying-parameter adaptive multi-layer neural dynamic based controller (termed as VP-AMND controller) design method is proposed and applied to controllers of multi-rotor UAVs. First, a varying-parameter convergent neural dynamic (VP-CND) based controller is proposed and its convergence and robustness are theoretically proven. Second, by incorporating the adaptive control method into the VP-CND controller, the VP-AMND controller design method is proposed, of which the global stability, fast convergence speed and strong robustness can be guaranteed. Different from traditional triple zeroing dynamic (TZD) and VP-CND controllers, the proposed VP-AMND controller with self-tuning rates can estimate the unknown disturbances and enhance the stability of the system in the face of uncertainty. Third, computer simulation results verify that the multi-rotor UAVs with VP-AMND controllers can track time-varying trajectories quickly and solve the parameter uncertainty and disturbances problems effectively.

Collision between dual robot manipulators during working process will lead to task failure and even robot damage. To avoid mutual collision of dual robot manipulators while doing collaboration tasks, a novel recurrent neural network (RNN)-based mutual-collision-avoidance (MCA) scheme for solving the motion planning problem of dual manipulators is proposed and exploited. Because of the high accuracy and low computation complexity, the linear variational inequality-based primal-dual neural network is used to solve the proposed scheme. The proposed scheme is applied to the collaboration trajectory tracking and cup-stacking tasks, and shows its effectiveness for avoiding collision between the dual robot manipulators. Through network iteration and online learning, the dual robot manipulators will learn the ability of MCA. Moreover, a line-segment-based distance measure algorithm is proposed to calculate the minimum distance between the dual manipulators. If the computed minimum distance is less than the first safe-related distance threshold, a speed brake operation is executed and guarantees that the robot cannot exceed the second safe-related distance threshold. Furthermore, the proposed MCA strategy is formulated as a standard quadratic programming problem, which is further solved by an RNN. Computer simulations and a real dual robot experiment further verify the effectiveness, accuracy, and physical realizability of the RNN-based MCA scheme when manipulators cooperatively execute the end-effector tasks.

To obtain the optimal solution to the time-varying quadratic programming (TVQP) problem with equality and multitype inequality constraints, a penalty strategy combined varying-parameter recurrent neural network (PS-VP-RNN) for solving TVQP problems is proposed and analyzed. By using a novel penalty function designed in this article, the inequality constraint of the TVQP can be transformed into a penalty term that is added into the objective function of TVQP problems. Then, based on the design method of VP-RNN, a PS-VP-RNN is designed and analyzed for solving the TVQP with penalty term. One of the greatest advantages of PS-VP-RNN is that it cannot only solve the TVQP with equality constraints but can also solve the TVQP with inequality and bounded constraints. The global convergence theorem of PS-VP-RNN is presented and proved. Finally, three numerical simulation experiments with different forms of inequality and bounded constraints verify the effectiveness and accuracy of PS-VP-RNN in solving the TVQP problems.

To solve the disturbed time-varying inversion problem, an exponential-type anti-noise varying-gain network (EAVGN) is proposed and analyzed. To do so, a vector-based error function is first defined. By using the varying-gain neural dynamic design method, an EAVGN model is then formulated. Furthermore, the differentiation error and the model-implementation error are considered into the model, and the perturbed EAVGN model is obtained. For better illustrations, comparisons between the EAVGN and the conventional fixed-parameter recurrent neural network (FP-RNN) are conducted to illustrate the advantages of the proposed EAVGN. Mathematical proof demonstrates that the proposed EAVGN has much better anti-noise properties than FP-RNN. On one hand, the residual error of EAVGN can be reduced to zero in any case, but that of FP-RNN is large and cannot be convergent, in particular when the bound of Frobenius norm of the exact solution is large or the noise is large. On the other hand, the bound of the residual error of EAVGN is always smaller than that of FP-RNN. Simulation results verify that when different types of noises exist, the proposed EAVGN owns better anti-noise property compared with the state-of-the-art methods. In addition, a practical application is presented to illustrate the implementation process and the practical benefits of the EAVGN.

Varying-parameter recurrent neural network, being a special kind of neural-dynamic methodology, has revealed powerful abilities to handle various time-varying problems, such as quadratic minimization (QM) and quadratic programming (QP) problems. In this paper, a novel power-type varying-parameter recurrent neural network (PT-VP-RNN) is proposed to solve the perturbed time-varying QM and QP problems. First, based on the generalization of time-varying QM and QP problems, the design process of the PT-VP-RNN is presented in detail. Second, the robustness performance of the proposed PT-VP-RNN is theoretically analyzed and proved. What is more, two numerical examples are simulated to illustrate the robustness convergence performance of PT-VP-RNN even in a large disturbance condition. Finally, two practical application examples (i.e., a robot tracking example and a venture investment example) further verify the effectiveness, accuracy, and widespread applicability of the proposed PT-VP-RNN.

A novel recurrent neural network, which is named as complex varying-parameter convergent-differential neural network (CVP-CDNN), is proposed in this paper for solving the time-varying complex Sylvester equation. Two kinds of CVP-CDNNs (i.e., CVP-CDNN Type I and Type II) are illustrated and proved to be effective. The proposed CVP-CDNNs can achieve super-exponential performance if the linear activation function is used. Some activation functions are considered for searching the better performance of the CVP-CDNN and the finite time convergence property of the CVP-CDNN with sign-bi-power activation function is testified. The convergence time of the CVP-CDNN with sign-bi-power activation function is shorter than complex fixed-parameter convergent-differential neural network (CFP-CDNN). Moreover, compared with traditional CFP-CDNN, better convergence performances of novel CVP-CDNN are verified by computer simulation comparisons.

Many practical problems can be solved by being formulated as time-varying quadratic programing (TVQP) problems. In this paper, a novel power-type varying-parameter recurrent neural network (VPNN) is proposed and analyzed to effectively solve the resulting TVQP problems, as well as the original practical problems. For a clear understanding, we introduce this model from three aspects: design, analysis, and applications. Specifically, the reason why and the method we use to design this neural network model for solving online TVQP problems subject to time-varying linear equality/inequality are described in detail. The theoretical analysis confirms that when activated by six commonly used activation functions, VPNN achieves a superexponential convergence rate. In contrast to the traditional zeroing neural network with fixed design parameters, the proposed VPNN has better convergence performance. Comparative simulations with state-of-the-art methods confirm the advantages of VPNN. Furthermore, the application of VPNN to a robot motion planning problem verifies the feasibility, applicability, and efficiency of the proposed method.

Solving Sylvester equation is a common algebraic problem in mathematics and control theory. Different from the traditional fixed-parameter recurrent neural networks, such as gradient-based recurrent neural networks or Zhang neural networks, a novel varying-parameter recurrent neural network, [called varying-parameter convergent-differential neural network (VP-CDNN)] is proposed in this paper for obtaining the online solution to the time-varying Sylvester equation. With time passing by, this kind of new varying-parameter neural network can achieve super-exponential performance. Computer simulation comparisons between the fixed-parameter neural networks and the proposed VP-CDNN via using different kinds of activation functions demonstrate that the proposed VP-CDNN has better convergence and robustness properties.