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In this paper, we study the finite-time convergence of the time-varying dynamical systems for solving convex and nonconvex optimization problems in different scenarios. We first show the asymptotic convergence of the trajectories of dynamical systems while only requiring convexity of the objective function. Under the Kurdyka–Łojasiewicz (KL) exponent of the objective function, we establish the finite-time convergence of the trajectories to the optima from any initial point. Making use of the Moreau envelope, we adapt our finite-time convergent algorithm to solve weakly convex nonsmooth optimization problems. In addition, we unify and extend the contemporary results on the KL exponent of the Moreau envelope of weakly convex functions. A dynamical system is also introduced to find a fixed point of a nonexpansive operator in finite time and fixed time under additional regularity properties. We then apply it to address the composite optimization problems with finite-time and fixed-time convergence.

Zeroing neural dynamic (ZND) model is widely deployed for time‐variant non‐linear equations (TVNE). Various element‐wise non‐linear activation functions and integration operations are investigated to enhance the convergence performance and robustness in most proposed ZND models for solving TVNE, leading to a huge cost of hardware implementation and model complexity. To overcome these problems, the authors develop a new norm‐based ZND (NBZND) model with strong robustness for solving TVNE, not applying element‐wise non‐linear activated functions but introducing a two‐norm operation to achieve finite‐time convergence. Moreover, the authors develop a discrete‐time NBZND model for the potential deployment of the model on digital computers. Rigorous theoretical analysis for the NBZND is provided. Simulation results substantiate the advantages of the NBZND model for solving TVNE.

This paper considers the L 1 -minimization problem for sparse signal and image reconstruction by using projection neural networks (PNNs). Firstly, a new finite-time converging projection neural network (FtPNN) is presented. Building upon FtPNN, a new fixed-time converging PNN (FxtPNN) is designed. Under the condition that the projection matrix satisfies the Restricted Isometry Property (RIP), the stability in the sense of Lyapunov and the finite-time convergence property of the proposed FtPNN are proved; then, it is proven that the proposed FxtPNN is stable and converges to the optimum solution regardless of the initial values in fixed time. Finally, simulation examples with signal and image reconstruction are carried out to show the effectiveness of our proposed two neural networks, namely FtPNN and FxtPNN.

This paper presents an adaptive multi‐agent coordination (AMAC) strategy suitable for complex scenarios, which only requires information exchange between neighbouring robots. Unlike traditional multi‐agent coordination methods that are solved by neural dynamics, the proposed strategy displays greater flexibility, adaptability and scalability. Furthermore, the proposed AMAC strategy is reconstructed as a time‐varying complex‐valued matrix equation. By introducing a dynamic error function, a fixed‐time convergent zeroing neural network (FTCZNN) model is designed for the online solution of the AMAC strategy, with its convergence time upper bound derived theoretically. Finally, the effectiveness and applicability of the coordination control method are demonstrated by numerical simulations and physical experiments. Numerical results indicate that this method can reduce the formation error to the order of within 1.8 s.

This article investigates a novel fast terminal sliding mode control approach combined with global sliding surface structure for the robust tracking control of nonlinear second‐order systems with time‐varying uncertainties. The suggested control technique is formulated based on the Lyapunov stability theory and guarantees the existence of the sliding mode around the sliding surface in a finite time. Using the new form of switching surface, the reaching phase elimination and the robustness improvement of the whole system are satisfied. Simulation results demonstrate the efficiency of the proposed technique. © 2014 Wiley Periodicals, Inc. Complexity 21: 239–244, 2015

The guidance laws for intercepting maneuvering targets in three‐dimensional (3D) space poses considerable challenges owing to various inescapable factors. These factors include the impact angle, input saturation constraints, and uncertainty. To address these challenges, a novel universal operator, denoted as |||·||| is introduced for the first time. This operator, when employed to represent sliding surface vectors, demonstrates a closer alignment with practical scenarios compared to the traditional Euclidean vector norm ||·||. Following this, a novel universal fixed‐time nonsingular terminal sliding surface is introduced in both scalar and vector representations, effectively resolving issues related to singular points and achieving reduced convergence times. Additionally, Furthermore, a new continuous adaptive finite‐time nonsingular terminal sliding mode guidance law (CAFnTNTSMGL) has been formulated. This guidance law incorporates a newly proposed sliding surface, a modified finite‐time super‐twisting algorithm, and a parameter‐adaptive law. The system's stability and its finite convergence time are subsequently demonstrated. Finally, the effectiveness of CAFnTNTSMGL is validated through a comparative analysis of simulation results. CAFnTNTSMGL has the capacity to effectively mitigate the negative effects resulting from the indeterminate upper limit of the overall uncertainty has less intercept time, smaller terminal line‐of‐sight (LOS) angle error, smaller maximum field of view, and smaller total cost of energy. The article introduces novel operator III.III, innovative non‐singular sliding surface, and advanced guidance law. A comparative analysis of theoretical and numerical simulations against four alternative methodologies has yielded favorable outcomes.

This study addresses the finite-time attitude tracking control for rigid spacecraft with external disturbances and inertia uncertainties. A novel adaptive-gain super-twist algorithm (STA) improves the control performance of standard STA, and the dynamically adapted control gains can resolve non-overestimating problem. The presented controllers do not require any knowledge on inertial uncertainties and external disturbances, and are anti-chattering and anti-singularity. The closed-loop spacecraft system under the proposed controllers can provide rapidity, robustness, accuracy and anti-wasting energy simultaneously, which is largely ignored in the existing literatures. The finite-time rigorous convergence, an estimation of the convergence time and accurate expression of convergence region are also provided. Finally, comparison results demonstrate that the presented controllers can achieve higher control performance than existing methods. Furthermore, digital simulations utilising the physical parameters of Uosat-12 verify the effectiveness of the proposed controllers.

Microgrid (MG) technology evolves as a promising solution to deal with the intermittent renewable generations and frequently changing load demand. This paper proposes a fully distributed and bounded secondary control algorithm with flexible convergence time for voltage and frequency restoration. It also enables accurate active power sharing for an islanded MG, compared with the well-known consensus-based distributed control approach. The proposed control scheme achieves accelerated fixed-time convergence. The upper bound on the convergence is established by using the Lyapunov stability theory. The bounded, distributed control approach restores the voltage and frequency in fixed-time while sharing the active power precisely. Further, the proposed controller is adaptive to the communication topology change and supports the plug and play feature of MG. Extensive simulations have been pursued using MATLAB/SimPowerSyetem toolbox considering frequent load perturbation and communication topology change. The obtained results are analysed to verify the performance of the proposed control algorithm. It is observed that the proposed bounded input controller converges faster than the conventional method.

In this paper, a new zeroing neural network (NZNN) with a new activation function (AF) is presented and investigated for solving dynamic Sylvester equation (DSE). The proposed NZNN not only finds the solutions of the DSE in fixed-time but also has better robustness, and its superior effectiveness and robustness are proved by rigorous mathematical analysis. Numerical simulation results of the proposed NZNN, the original zeroing neural network activated by other recently reported AFs and the existing robust nonlinear ZNN (RNZNN) for solving second-order dynamic Sylvester equation and third-order dynamic Sylvester equation are provided for the purpose of comparison. Comparing with the existing ZNN models, the proposed NZNN has better robustness and faster convergence performance for solving DSE in the same noise environment. Moreover, a successful robot manipulator trajectory tracking example in noise-disturbed environment using the proposed NZNN is also applied for illustrating its further practical applications.

A finite-time consensus protocol is proposed for multi-dimensional multi-agent systems, using direction-preserving signum controls. Filippov solutions and nonsmooth analysis techniques are adopted to handle discontinuities. Sufficient and necessary conditions are provided to guarantee infinite-time convergence and boundedness of the solutions. It turns out that the number of agents which have continuous control law plays an essential role in finite-time convergence. In addition, it is shown that the unit balls introduced by ℓp norms, where p∈[1,∞] , are invariant for the closed loop.