Upper Bound Of Convergence Time Research Articles

Distributed predefined-time economic dispatch based on event-triggered strategy for microgrids under directed graphs

The proliferation of new energy solutions and the enhanced accessibility of distributed power sources have significantly increased the complexity and variability of microgrid structures. Due to inherent limitations in its operational mode, the traditional centralized optimization method falls short in effectively addressing the economic dispatch problem in contemporary microgrids. In this paper, a distributed optimization method is devised to address the economic dispatch problem of microgrids under directed graphs by leveraging the consensus algorithm of multi-agent systems. We define the incremental cost of each microgrid unit as the consensus variable, achieving convergence through a pioneering combination of event-triggered strategy and predefined-time control theory. This approach not only optimizes resource use but also ensures timely and efficient solution convergence, addressing critical gaps in existing methodologies. This approach enables control over the upper bound of convergence time for optimal solutions by directly adjusting time parameters in the controller under the directed graph. Consequently, it achieves coordinated control of the power output in microgrids within a predefined time while reducing the communication resources between nodes with event-triggered mechanism. The effectiveness of the proposed algorithm is verified through simulation and analysis.

Exponential control-based fixed/preassigned-time synchronization of output-coupled spatiotemporal networks with directed topology

This paper mainly explores the fixed-time and preassigned-time output synchronization of spatiotemporal networks with directed topology. Firstly, a kind of exponential-type control scheme associated with output feedback is developed. Subsequently, several sufficient criteria are established to ensure fixed-time output synchronization by utilizing the Lyapunov method and Taylor expansion. Moreover, the upper bound of convergence time is evaluated based on fixed-time stability theory. Furthermore, the preassigned-time output synchronization criteria are proposed by improving the fixed-time output controller, which incorporates a specified time parameter. Unlike previous forms, there are no stringent requirements imposed on the output matrix, whether semi-positive, non-singular or full rank in terms of columns. Finally, corresponding simulations are given to confirm the theoretical analysis.

Predefined-time stabilization of Lorenz system with applications for stabilizing and synchronizing chaotic finance systems

This article investigates the predefined-time stabilization (PtS) of the canonical Lorenz system at first, and then applies the derived results into the chaotic finance systems (CFSs) so as to realize the stabilization and synchronization, respectively. Compared with the traditional finite-/fixed-time stability analysis, the upper-bound of convergence time (UbCT) in this investigation can be set beforehand in need, which is an explicit constant regardless of initial values, system dimension, and controlling parameters. Moreover, the designed control schemes are non-chattering, which do not contain the conventional discontinuous signum and absolute value functions anymore. Via adopting the second Lyapunov method, the sufficient conditions are obtained successively for guaranteeing the realization of PtS for Lorenz system, CFS, as well as the predefined-time synchronization between two CFSs. The numerical experiments are finally arranged to manifest the correctness and effectiveness of the theoretical fruits, in which some comparison and perturbation analysis are made.

A novel fixed‐time ZNN model and application to robot trajectory tracking

AbstractIn order to achieve robot trajectory tracking in fixed‐time, a novel fixed‐time zeroing neural network model is designed. Initially, the inverse kinematic model of robot trajectory tracking is translated into a time‐varying quadratic programs problem. Subsequently, a novel fixed‐time zeroing neural network is proposed for solving the time‐varying quadratic programs problem. Furthermore, the fixed‐time stability of this model is rigorously established, and an upper bound of convergence time, irrespective of the initial point, is estimated. Finally, numerical simulation results underscore the efficacy of the proposed methodologies.

Adaptive Practical Predefined-Time Control for Uncertain Teleoperation Systems With Input Saturation and Output Error Constraints

In this paper, the synchronization tracking control issue is investigated for uncertain teleoperation systems with input saturation and output error constraints. To this end, an adaptive practical predefined-time control scheme integrating backstepping recursive design, predefined time theory, prescribed performance function, and the anti-saturation auxiliary system, is developed for the first time. A fixed-time extended state observer is utilized to eliminate the negative influence of the lumped uncertainty, and the anti-saturation auxiliary system is constructed to address the input saturation. In contrast to finite/fixed-time controllers, the upper bound of convergence time can be obtained in advance by adjusting a single control parameter. The results indicate that the error signals can converge into a small region of the zero domain within a user-defined time and that the output error never violates the prescribed performance boundary. Simulations and experiments are performed on a teleoperation platform made up of two Phantom Touch robots to verify the effectiveness and practicality of the developed controller.

Reinforcement Learning-Based Predefined-Time Tracking Control for Nonlinear Systems Under Identifier-Critic-Actor Structure.

A novel reinforcement learning-based predefined-time tracking control scheme with prescribed performance is presented in this article for nonlinear systems in the presence of external disturbances. First, by employing the backstepping strategy, an adaptive optimized controller is developed under the identifier-critic-actor framework. Therein, the unknown nonlinear dynamics and the system control behavior can be learned effectively through neural networks. Moreover, aiming at obtaining the preset tracking performance, the prescribed performance control is integrated with the predefined-time control. In contrast to previous studies, the proposed scheme can not only constrain the tracking error rapidly to a prearranged vicinity of origin, but also ensure that the upper bound of convergence time can be adjusted in advance via a separate control parameter. In terms of the predefined-time stability theory, the boundedness of all system states can be proven within a predefined time. Finally, the availability and improved performances of the proposed control scheme are demonstrated by a numerical example and a single-link manipulator example.

Efficient Predefined-Time Adaptive Neural Networks for Computing Time-Varying Tensor Moore-Penrose Inverse.

This article proposes predefined-time adaptive neural network (PTANN) and event-triggered PTANN (ET-PTANN) models to efficiently compute the time-varying tensor Moore-Penrose (MP) inverse. The PTANN model incorporates a novel adaptive parameter and activation function, enabling it to achieve strongly predefined-time convergence. Unlike traditional time-varying parameters that increase over time, the adaptive parameter is proportional to the error norm, thereby better allocating computational resources and improving efficiency. To further enhance efficiency, the ET-PTANN model combines an event trigger with the evolution formula, resulting in the adjustment of step size and reduction of computation frequency compared to the PTANN model. By conducting mathematical derivations, the article derives the upper bound of convergence time for the proposed neural network models and determines the minimum execution interval for the event trigger. A simulation example demonstrates that the PTANN and ET-PTANN models outperform other related neural network models in terms of computational efficiency and convergence rate. Finally, the practicality of the PTANN and ET-PTANN models is demonstrated through their application for mobile sound source localization.

Predefined-Time Adaptive Fault-Tolerant Control for Switched Odd-Rational-Power Multi-Agent Systems

This paper studies the distributed adaptive neural network (NN) predefined-time (PT) fault-tolerant consensus tracking control for unknown non-strict feedback nonlinear multi-agent systems (MASs) with time-varying full state constraints (TVFSCs), arbitrary switching behavior functions (possibly asynchronous), actuator faults and high powers. Compared with existing results, the convergence time estimation bound in this paper is less conservative due to the upper bound of convergence time presents as a tuning parameter. This paper avoids the control gain of each virtual controller to appear in the next virtual controller by using separable function technique, so it can reduce the control action complexity. Besides, this paper does not use any filters or piecewise continuous function in the controller design process, which can effectively avoid the potential singularity problem. An adaptive NN fault-tolerant consensus tracking controller is designed via backstepping technique and tangent barrier Lyapunov functions (BLF-Tans), which can ensure that all the closed-loop signals are bounded within a PT and the TVFSC bounds are not violated. Finally, numerical simulation and practical example further verify the effectiveness of the presented control algorithm. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Many practical (underactuated, weakly coupled and unstable mechanical systems) control systems can be modeled as uncertain high-order MAS with high powers. Due to the influence of environment and other factors, the values of some variables in the practical systems are limited and cannot be arbitrarily selected. In addition, the actual systems may fail due to human factors and in most of the relevant literatures, it is often assumed that the powers are equal to 1 and only achieve the controlled systems asymptotic stability as time approaches infinity. Therefore, in this paper, an adaptive PT fault-tolerant control scheme is developed for MASs with high powers, time-varying full state constraints and actuator faults. The effectiveness of the control scheme is illustrated based on simulation study.

Reinforcement Learning-Based Fixed-Time Trajectory Tracking Control for Uncertain Robotic Manipulators With Input Saturation.

A fixed-time trajectory tracking control method for uncertain robotic manipulators with input saturation based on reinforcement learning (RL) is studied. The designed RL control algorithm is implemented by a radial basis function (RBF) neural network (NN), in which the actor NN is used to generate the control strategy and the critic NN is used to evaluate the execution cost. A new nonsingular fast terminal sliding mode technique is used to ensure the convergence of tracking error in fixed time, and the upper bound of convergence time is estimated. To solve the saturation problem of an actuator, a nonlinear antiwindup compensator is designed to compensate for the saturation effect of the joint torque actuator in real time. Finally, the stability of the closed-loop system based on the Lyapunov candidate is analyzed, and the timing convergence of the closed-loop system is proven. Simulation and experimental results show the effectiveness and superiority of the proposed control law.

A Segmented Variable-Parameter ZNN for Dynamic Quadratic Minimization With Improved Convergence and Robustness.

As a category of the recurrent neural network (RNN), zeroing neural network (ZNN) can effectively handle time-variant optimization issues. Compared with the fixed-parameter ZNN that needs to be adjusted frequently to achieve good performance, the conventional variable-parameter ZNN (VPZNN) does not require frequent adjustment, but its variable parameter will tend to infinity as time grows. Besides, the existing noise-tolerant ZNN model is not good enough to deal with time-varying noise. Therefore, a new-type segmented VPZNN (SVPZNN) for handling the dynamic quadratic minimization issue (DQMI) is presented in this work. Unlike the previous ZNNs, the SVPZNN includes an integral term and a nonlinear activation function, in addition to two specially constructed time-varying piecewise parameters. This structure keeps the time-varying parameters stable and makes the model have strong noise tolerance capability. Besides, theoretical analysis on SVPZNN is proposed to determine the upper bound of convergence time in the absence or presence of noise interference. Numerical simulations verify that SVPZNN has shorter convergence time and better robustness than existing ZNN models when handling DQMI.

Fixed-time fully distributed observer-based bipartite consensus tracking for nonlinear heterogeneous multiagent systems

This paper focuses on the problem of the fully distributed fixed-time bipartite output consensus tracking for nonlinear heterogeneous multiagent systems (MASs) via both state-feedback and output-feedback methods under the switching topology. The adaptive fully distributed state observers with quantization information are designed to eliminate the dependence on the Laplacian matrix. For MASs with unknown model matrix, a novel fixed-time observer-based regulator equation is employed, which avoids repeatedly getting the unnecessary solution of universal one in the time-varying cooperation-competition communication topology. In this case, both fixed-time state-feedback and output-feedback controllers are constructed such that the output consensus tracking is achieved regardless the state value is available or unavailable. Besides, the upper bound of convergence time can be adjusted only by parameters without initial states. Lyapunov functions are established to derive conditions of achieving consensus tracking by mathematical analysis. Eventually, the effectiveness of the proposed control strategy is manifested by simulation results.

Bi‐homogeneous observers for uncertain 1‐DOF mechanical systems

A global, bi-homogeneous, observer for a class of 1-DOF mechanical systems with Coriolis and centrifugal forces, dry and viscous frictions and non-vanishing bounded uncertainties/perturbations is proposed, ensuring predefined upper bound of convergence time. The options for convergence acceleration are discussed in a simulation example.

Prescribed time convergence and robust zeroing neural network for solving time-varying linear matrix equation

Zeroing neural network offers a new solution method to solve the time-varying linear matrix equation. As an important component, the activation function directly affects the performance of zeroing neural network in solving time-varying linear matrix equation. Focusing on the unification of prescribed time convergence and strong robustness without changing the basic structure of zeroing neural network, a novel activation function is proposed for the first time in this paper. Compared with commonly used activation functions in previous work, the novel activation function has superior performance for zeroing neural network to solve the time-varying linear matrix equation. The first item is global asymptotic convergence, capable of converging from random initial states to the theoretical solution. The second item is the prescribed time convergence, i.e. the upper bound of convergence time is only related to the parameters of the novel activation function and zeroing neural network, which facilitates the prediction of the convergence process. The third item is strong robustness, which ensures that the solution converges in various noisy environments. Theoretical analysis and comparative simulation experiments verify that zeroing neural network with the novel activation function has these performances for both low- or high-dimensional time-varying linear matrix equation.

Bi-Homogeneous Observers for Uncertain 2-DOF Mechanical Systems

A global, bi-homogeneous, observer for a class of 2-DOF mechanical systems with Coriolis and centrifugal forces, dry and viscous frictions and non-vanishing bounded uncertainties/perturbations is proposed, ensuring predefined upper bound of convergence time.

Nonsingular Terminal Sliding Mode-Based Distributed Cooperative Event-Triggered Control for Multiple Surface Vessels Under Actuator Faults With Input Saturation

In this paper, a fixed-time distributed cooperative control under actuator faults with input saturation scheme is designed for multiple surface vessels (MSVs). Firstly, a fixed-time disturbance observer (FxDO) is introduced to observe unknown environmental disturbances and actuator faults (AF). Secondly, to deal with the actuator saturation, an improved nonlinear antiwindup compensator is introduced to compensate for the saturation effect of the actuator. Thirdly, a fixed-time fault-tolerant event-triggered control law with the actuator saturation is applied to cooperative control based on a fixed-time nonsingular terminal sliding mode manifold (FxNTSMM), and their cooperative errors can converge within fixed time. At the same time, the upper bound of convergence time is independent of the initial state. Finally, the stability analysis of the cooperative control system is given to prove that the system can realize the practical fixed time stability, and Zeno phenomenon is avoided by theoretical proof. Simulation results are given to demonstrate the effectiveness of the proposed control scheme.

Fast Nonsingular Fixed-Time Fuzzy Fault-Tolerant Control for HFVs With Guaranteed Time-Varying Flight State Constraints

This article proposes a fuzzy adaptive fault-tolerant control strategy solving the fast fixed-time constrained tracking problem for hypersonic flight vehicles. To handle the actuator faults, a two-step design method is first proposed that only needs to estimate the upper and lower bounds of actuator parameters, under which the adverse influences of actuator faults (e.g., lock-in-place, loss of effectiveness, etc.) have been compensated via an adaptive fashion instead of conventional robust ways. In comparison with the state-of-the-art, a new fixed-time corollary that has proved a smaller upper bound of convergence time under the same condition of conventional fixed-time stability is presented. To avoid singularity issues often encountered in fixed-time designs, a piecewise but differentiable switching control laws whose continuity and differentiability are guaranteed everywhere through an appropriate design is introduced. Such a design not only preserves the continuity and differentiability of virtual and actual control laws, but also ensures the continuity of their time derivatives. Fuzzy logic systems are exploited to tackle continuous unknown dynamics and asymmetric time-varying barrier functions are utilized to confine flight states within some predefined compact sets all the time provided their initial conditions remain therein. This property is of great significance in dealing with the challenge that the operating regions of the flight state variables are asymmetric and time-varying in practice, especially when executing various flight missions. Comparative simulations have been performed to validate the effectiveness of the proposed control scheme in terms of fixed-time convergence, smoothness, and time-varying constraints satisfaction.

Fixed-Time Formation Tracking for Heterogeneous Multiagent Systems Under Actuator Faults and Directed Topologies

This article investigates the problem of fixed-time formation tracking control for a heterogeneous multiagent system composed of unmanned aerial vehicles and unmanned ground vehicles in the presence of actuator faults, model uncertainties, and external disturbances. First, two fixed-time distributed observers are proposed, one of which is applied to detail-balanced directed graph and the other is applied to general directed graph. These two observers remove the requirements of global information and control signal of leader, as well as reduce communication flow. Second, a fixed-time disturbance observer is designed for each follower to estimate lumped uncertainties. Third, a Lyapunov-based formation controller is proposed to guarantee that the fixed-time synchronized formation is achieved for heterogeneous multiagent systems using estimation and backstepping technique. Furthermore, the obtained upper bound of convergence time only depends on the observer gains and controller parameters, which facilitates the adjustment of the settling time offline regardless of initial conditions under different formation requirements. Finally, the effectiveness of the proposed fixed-time protocol is verified using numerical simulations.

A parameter-changing zeroing neural network for solving linear equations with superior fixed-time convergence

Linear equations (LEs) play an essential role in mathematics. Nevertheless, the vast majority of fixed-parameter zeroing neural network (ZNN) models are not fast enough to solve LEs. So, a novel parameter-changing ZNN (PCZNN) is proposed and studied in the quest for resolving LEs more efficiently. In contrast to the conventional ZNN and finite-time ZNN (FTZNN), the tuning parameter of the presented PCZNN is time-varying, resulting in three advantages of the PCZNN model: (1) less time for parameter adjustment, (2) faster convergence rate, and (3) less conservative upper bound on the convergence time. Furthermore, the upper bound of convergence time is derived through analysis, which is independent of its initial value. Simulation comparisons conclude that the PCZNN model is more efficient than the ZNN and FTZNN models in solving LEs. Finally, the synchronization control experiments of chaotic systems by two ZNN models with time-varying parameters are offered to verify the superiority of the PCZNN model.

Distributed Adaptive Neural Fixed-Time Tracking Control of Multiple Uncertain Mechanical Systems With Actuation Dead Zones

This article is mainly concerned with the fixed-time tracking control problem for multiple uncertain mechanical systems with actuation dead zones. As the dead zones and control gains are time varying and completely unknown, the control impact on mechanical system becomes completely unknown, making the achievement of fixed-time tracking control nontrivial. The existing studies on fixed-time tracking control cannot effectively estimate and compensate such dead zones, resulting in the upper bound of convergence time being uncertain and even system becoming unstable. To compensate the dead zones in a fixed time, we introduce a bound estimation method to estimate the impact of dead zones and design two fixed-time adaptive laws. Two cases on system uncertainties are considered. By introducing distributed control method, adaptive control method, neural networks and some skillful treatments, two novel distributed adaptive neural fixed-time tracking control schemes are proposed. It is confirmed that, with the proposed control schemes, all followers which are uncertain mechanical systems with completely unknown dead zones track the leader with sufficient precision in a predefined time.

Bounded Observer-Based Consensus Algorithm for Robust Finite-Time Tracking Control of Multiple Nonholonomic Chained-Form Systems

This article is concerned with the leader–follower consensus tracking problem for multiagent systems with nonholonomic high-order chained-form dynamics subject to external disturbances. A novel distributed and bounded observer is first developed for each follower to estimate the leader information in a finite time. Then, a bounded fast terminal sliding-mode control protocol is constructed for each follower to track the estimated leader's states leading to the fast convergence performance as well as strong robustness. Contrary to some existing finite-time consensus tracking schemes, the control input constraint is taken into account by utilizing hyperbolic tangent saturation function to reduce the risk of the actuator saturation. Moreover, an approximation-based technique is introduced to reduce the conservatism of the upper bound of convergence time. Finally, some simulations for a set of wheeled mobile robots are carried out to demonstrate the efficiency of the proposed control algorithm.