A novel predefined-time neurodynamic approach for mixed variational inequality problems and applications.

In this paper, we propose a novel neurodynamic approach with predefined-time stability that offers a solution to address mixed variational inequality problems. Our approach introduces an adjustable time parameter, thereby enhancing flexibility and applicability compared to conventional fixed-time stability methods. By satisfying certain conditions, the proposed approach is capable of converging to a unique solution within a predefined-time, which sets it apart from fixed-time stability and finite-time stability approaches. Furthermore, our approach can be extended to address a wide range of mathematical optimization problems, including variational inequalities, nonlinear complementarity problems, sparse signal recovery problems, and nash equilibria seeking problems in noncooperative games. We provide numerical simulations to validate the theoretical derivation and showcase the effectiveness and feasibility of our proposed method.

Fixed-time terminal sliding mode control for uncertain robot manipulators.

This paper proposes a fixed-time tracking control for robot manipulators in the presence of parametric uncertainties and disturbances. An auxiliary function is first proposed for constructing a fixed-time sliding manifold. Benefited from this fixed-time sliding manifold, a singularity-free robust control is proposed to evade the effects of algebraic loop problem of the commonly-used sliding mode controls (SMC). The key advantages of the proposed approach are: (i) exact fixed-time stability featuring the convergence time does not relate to the initial conditions and is acquired in advance; (ii) the singularity and algebraic loop problems are eliminated completely; (iii) a simple and intuitive control structure is used for easy implementation of trajectory tracking control for uncertain robot manipulators with faster transient and higher steady-state precision. Simulations and experimental comparisons validate the improved tracking performance of the proposed approach.

A model-free terminal sliding mode control for robots: Achieving fixed-time prescribed performance and convergence.

This paper introduces a new control strategy for robot manipulators, specifically designed to tackle the challenges associated with traditional model-based sliding mode (SM) controller design. These challenges include the need for accurately computed system models, knowledge of disturbance upper bounds, fixed-time convergence, prescribed performance, and the generation of chattering. To overcome these obstacles, we propose the incorporation of a neural network (NN) that effectively addresses these issues by removing the constraint of a precise system model. Additionally, we introduce a novel fixed-time prescribed performance control (PPC) to enhance response performance and position-tracking accuracy, while effectively limiting overshoot and maintaining steady-state error within the predefined range. To expedite the convergence of the SM surface to its equilibrium point, we introduce a faster terminal sliding mode (TSM) surface and a novel fixed-time reaching control algorithm (RCA) with adaptable factors. By integrating these approaches, we develop a novel control strategy that successfully achieves the desired goals for robot manipulators. The effectiveness and stability of the proposed approach are validated through extensive simulations on a 3-DOF SAMSUNG FARA-AT2 robot manipulator, utilizing both Lyapunov criteria and performance evaluations. The results demonstrate improved convergence rate and tracking accuracy, reduced chattering, and enhanced controller robustness.

Adaptive fixed-time sliding mode control for spacecraft reorientation with attitude pointing constraints and disturbance rejection.

Spacecraft reorientation with attitude pointing constraints and the uncertainty of inertia and external disturbance is investigated in this paper. By introducing the potential function into the design of non-singular fixed-time sliding mode surface, the proposed controller can achieve fixed-time convergence and the convergence time of attitude error can be predetermined by selecting appropriate parameters. Meanwhile, the attitude pointing constraints can be satisfied all the time. The designed sliding surface and potential function have two equilibrium points, which guarantees the unwinding-free performance. Furthermore, an adaptive sliding mode control scheme is developed to handle the system lumped disturbance. Rigorous Lyapunov analyses are employed to ensure practical fixed-time closed-loop stability in the presence of system disturbance uncertainties and attitude pointing constraints. Therefore, the fixed-time stability, the feasibility of attitude pointing constraints and disturbance rejection are achieved simultaneously with the proposed controller. Numerical simulations are provided to demonstrate the effectiveness and superiority of the proposed method.

Fixed-time periodic stabilization of discontinuous reaction-diffusion Cohen-Grossberg neural networks.

This paper aims to study the fixed-time stabilization of a class of delayed discontinuous reaction-diffusion Cohen-Grossberg neural networks. Firstly, by providing some relaxed conditions containing indefinite functions and based on inequality techniques, a new fixed-time stability lemma is given, which can improve the traditional ones. Secondly, based on state-dependent switching laws, the periodic wave solution of the formulated networks is transformed into the periodic solution of ordinary differential system. By utilizing differential inclusions theory and coincidence theorem, the existence of periodic solutions is obtained. Thirdly, based on the new fixed-time stability lemma, the periodic solutions are stabilized at zero in a fixed-time, which is a new topic on reaction-diffusion networks. Moreover, the established criteria are all delay-dependent, which are less conservative than the previous delay-independent ones for ensuring the stabilization of delayed reaction-diffusion networks. Finally, two examples give numerical explanations of the proposed results and highlight the influence of delays.

Fixed-time fault-tolerant attitude control for rigid spacecraft with torque saturation.

This paper investigates the fixed-time attitude control problem for spacecraft under input saturation, actuator faults, and system uncertainties. Three novel saturated fixed-time nonsingular terminal sliding mode surfaces (NTSMSs) are designed, which can keep the system states fixed-time stable after the establishment of their sliding manifolds. Two of them are time-varying and firstly designed. Each of the two NTSMSs has an adjustment parameter that is adjusted dynamically and used to handle saturation and cancel the attitude dynamics. According to other related predesigned parameters, a conservative lower bound of this parameter is obtained. A saturated control scheme is then designed in conjunction with a newly proposed saturated reaching law. A modification strategy is carried out to facilitate the engineering applications of our methods. The fixed-time stability of the closed-loop systems is validated by Lyapunov stable theory. Simulation results validate the effectiveness and superiority of the proposed control scheme.

Fixed-time synchronization of delayed memristive neural networks with impulsive effects via novel fixed-time stability theorem.

In this study, the fixed-time synchronization (FXTS) of delayed memristive neural networks (MNNs) with hybrid impulsive effects is explored. To investigate the FXTS mechanism, we first propose a novel theorem about the fixed-time stability (FTS) of impulsive dynamical systems, where the coefficients are extended to functions and the derivatives of Lyapunov function (LF) are allowed to be indefinite. After that, we obtain some new sufficient conditions for achieving FXTS of the system within a settling-time using three different controllers. At last, to verify the correctness and effectiveness of our results, a numerical simulation was conducted. Significantly, the impulse strength studied in this paper can take different values at different points, so it can be regarded as a time-varying function, unlike those in previous studies (the impulse strength takes the same value at different points). Hence, the mechanisms in this article are of more practical applicability.

Fixed-time regulation of spacecraft orbit and attitude coordination with optimal actuation allocation using dual quaternion.

On-orbit service spacecraft with redundant actuators need to overcome orbital and attitude coupling when performing proximity maneuvers. In addition, transient/steady-state performance is required to fulfill the user-defined requirements. To these ends, this paper introduces a fixed-time tracking regulation and actuation allocation scheme for redundantly actuated spacecraft. The coupling effect of translational and rotational motions is described by dual quaternion. Based on this, we propose a non-singular fast terminal sliding mode controller to guarantee fixed-time tracking performance in the presence of external disturbances and system uncertainties, where the settling time is only dependent on user-defined control parameters rather than initial values. The unwinding problem caused by the redundancy of dual quaternion is handled by a novel attitude error function. Moreover, optimal quadratic programming is incorporated into null space pseudo-inverse control allocation that ensures the actuation smoothness and never violates the maximum output capability of each actuator. Numerical simulations on a spacecraft platform with symmetric thruster configuration demonstrate the validity of the proposed approach.

A fixed-time distributed extended state observer for uncertain second-order nonlinear system.

This paper investigates the fixed-time distributed estimation problem for a class of second-order nonlinear systems with uncertain input, unknown nonlinearity and matched perturbation. A fixed-time distributed extended state observer (FxTDESO) consisting of a group of local observer nodes under directed communication topology is proposed, and each node can reconstruct both the full state and unknown dynamics of the system. To achieve fixed-time stability, a Lyapunov function is elaborated, and based on this, sufficient conditions for the existence of the FxTDESO are established. Under time-invariant and time-varying disturbance, the observation errors can converge to the origin and a small region of the origin within a fixed time, respectively, where the upper bound of the settling time (UBST) is irrelevant to the initial conditions. Compared to the existing fixed-time distributed observers, the proposed observer can reconstruct both the unknown states and uncertain dynamics, and only the output of the leader and 1-dimensional output estimates from the neighboring nodes are needed in the observer design which effectively reduces the communication load. The paper also extends previous finite-time distributed extended state observer to the case of time-variant disturbance and eliminates the complex linear matrix equation assumption that guarantees the finite-time stability. Furthermore, the FxTDESO design for a class of high-order nonlinear systems is also discussed. Finally, simulation examples are conducted to demonstrate the effectiveness of the proposed observer.

Comprehensive analysis of fixed-time stability and energy cost for delay neural networks.

This paper focuses on comprehensive analysis of fixed-time stability and energy consumed by controller in nonlinear neural networks with time-varying delays. A sufficient condition is provided to assure fixed-time stability by developing a global composite switched controller and employing inequality techniques. Then the specific expression of the upper of energy required for achieving control is deduced. Moreover, the comprehensive analysis of the energy cost and fixed-time stability is investigated utilizing a dual-objective optimization function. It illustrates that adjusting the control parameters can make the system converge to the equilibrium point under better control state. Finally, one numerical example is presented to verify the effectiveness of the provided control scheme.

Tracking Design of an Uncertain Autonomous Underwater Vehicle with Input Saturations by Adaptive Regression Matrix-Based Fixed-Time Control.

In this study, a simplified model of an autonomous underwater vehicle (AUV) with input saturation based on kinematic and dynamic equations was built. Subsequently, a simplified model of the AUV was used to represent its main dynamic features. In terms of trajectory tracking, only the system's structure (i.e., the regression matrix, which is flexible and non-unique) from the nominal model of the transformed system was required to design the proposed adaptive regression matrix-based fixed-time controller (ARM-FTC). A nonlinear auxiliary sliding surface was contained in the control design to shape the system's frequency response. When the operating point was in the neighborhood of the zero auxiliary sliding surface, nonlinear filtering gains were increased to accelerate its tracking ability. Furthermore, the skew-symmetric property condition of the time-derivative of the inertia matrix and the Coriolis and centrifugal force matrices was not necessitated for the controller design. Under an appropriate condition for lumped uncertainties, the fixed-time convergence of the auxiliary sliding surface and the corresponding tracking error is guaranteed to go to zero by the Lyapunov stability theory. Finally, a comparative study was conducted through simulations for the AUV with external disturbance and input saturation among the known parameters, learning parameters reflecting a regression matrix, and another asymptotical robust tracking control scheme. The results validate the fast tracking ability of a desired time-varying trajectory of the proposed control scheme.

Fault-tolerant trajectory tracking control for unmanned surface vehicle with actuator faults based on a fast fixed-time system.

In this paper, an observer-based fixed-time tracking control strategy is presented for unmanned surface vehicles (USVs) with model uncertainties, external disturbances, and actuator faults. Firstly, as the theory foundation, a fast fixed-time stable system that has a shorter settling time than the existing systems is proposed. According to this system and the motion characteristics of an USV, a fast fixed-time disturbance observer is developed to obtain the unknown effects caused by lumped uncertainties. By combining the estimated knowledge and a nonsingular fast fixed-time terminal sliding surface, a robust fast fixed-time trajectory tracking controller is designed for the USV. According to Lyapunov stability theory, the fast fixed-time convergence of the proposed controller is proved. Finally, the simulation results demonstrate the effectiveness of the developed control scheme.

Distributed event-triggered fixed-time formation and trajectory tracking control for multiple stratospheric airships.

In this study, an event-triggered fixed-time multiple stratospheric airship formation trajectory tracking controller is designed, and it is composed of two parts: the airship leader trajectory tracking controller (ALTTC) and the airship follower formation tracking controller (AFFTC). First, based on the framework of backstepping, the fixed-time ALTTC is designed to allow the trajectory tracking error to converge to zero within a fixed time. Subsequently, the event-triggered fixed-time AFFTC is designed to reduce the formation tracking error to zero within a fixed time. Two event-triggering conditions are designed to reduce the transmission times of control inputs and calculation times of control outputs. The fixed-time stability and the trajectory-tracking and formation-tracking performance of event-triggered closed-loop systems are theoretically shown to be ensured, and Zeno behavior is excluded in the proposed asynchronous event-triggering mechanism. Finally, simulations indicate the effectiveness of the proposed controller.

Practical fixed-time trajectory tracking control of constrained wheeled mobile robots with kinematic disturbances.

This paper addresses the problem of practical fixed-time trajectory tracking for wheeled mobile robots (WMRs) subject to kinematic disturbances and input saturation. Firstly, considering the under-actuated characteristics of the WMR systems, the WMR model under kinematic disturbances is transformed into a two-input two-output interference system by using a set of output equations. Then, the tracking error state equation with lumped disturbances in the acceleration-level pseudo-dynamic control (ALPDC) structure is established. The lumped disturbances are estimated by a designed fixed-time extended state observer (FESO) without requiring the differentiability of the first-time derivatives of the kinematic disturbances. Meanwhile, a practical fixed-time output feedback control law is developed for trajectory tracking. By resorting to the Lyapunov stability theorem, the fixed-time stability analysis of the closed-loop WMR system in the presence of input saturation is conducted. Finally, simulation results are presented to show the effectiveness of the proposed approach.

Fixed-time terminal sliding mode tracking protocol design for high-order multiagent systems with directed communication topology.

This paper presents a novel fixed-time consensus tracking protocol for high order multi-agent system (MAS) with directed communication topology. A new distributed observer is proposed such that fixed-time leader's state estimation can be achieved, which overcomes the difficulty arising from asymmetry of communication topology. A series of terminal sliding surfaces are constructed and a singularity-free sliding mode fixed-time tracking protocol is developed. It is proved that the proposed tracking protocol achieves fixed-time consensus tracking. Particularly, we can obtain the controller gain from the pre-specified time, which helps to tune the gain in accordance with consensus time requirement. Moreover, a less conservative convergence time bound estimation is attained. Simulation examples demonstrate the effectiveness of the presented scheme.

Fixed-time ESO based fixed-time integral terminal sliding mode controller design for a missile.

This paper studies a novel fixed-time extended state observer based fixed-time integral terminal sliding mode controller for partial integrated guidance and control design. Firstly, a class of arbitrary-order systems with fixed-time stability is proposed by utilizing homogeneous approach, whose upper bound of convergence time is given. Then, an arbitrary-order fixed-time integral terminal sliding mode control is designed based on the proposed arbitrary-order fixed-time stable system, which avoids the singular problem. Subsequently, this paper constructs a new fixed-time extended state observer to further actively compensate for the disturbance caused by unknown target acceleration. Finally, numerical simulations show the effectiveness of the proposed controller.

Practical fractional-order nonsingular terminal sliding mode control of spacecraft.

In this study, a new adaptive fractional-order nonsingular terminal sliding mode (AFONTSM) controller is presented. A novel multi-purpose sliding surface is constructed, with the aim of bringing the reaction wheels in to rest after every attitude stabilization maneuver, utilizing the fractional-order difference of the quaternion error and the reaction wheels angular momentum error. The closed-loop system's practical fixed-time stability is investigated using the Lyapunov theorem under uncertainty and external disturbance. The AFONTSM controller's performance is compared with the existing nonsingular terminal sliding mode (NTSM), full-order NTSM, and fractional-order sliding mode controllers. Finally, the proposed AFONTSM controller's effectiveness is studied in close-to-reality situations through practical experiments on the spacecraft attitude control subsystem simulator under internal/external disturbance and uncertainty; then, the results are compared with previous studies.

Fixed time stability of a class of chaotic systems with disturbances by using sliding mode control.

The main goal of this article is to consider the fixed time control problem of perturbed chaotic systems by virtue of sliding mode control. For this aim, this article presents a novel fixed time stability theorem at first by the Lyapunov tools. Then combining the obtained stability theorem and sliding mode technique, a new sliding mode surface is constructed and some novel controllers are designed appropriately to stabilize the discussed chaotic system. The proposed controllers have two main advantages: (1) The control criteria is robust against the effects of perturbations. (2) The convergence time, which is only dependent on the control parameters regardless of the initial conditions, is bounded by a fixed constant. Finally two typical systems are taken as the numerical examples to verify the validity of the control strategy.

Fixed-time attitude coordination control for spacecraft with external disturbance.

This paper studies the fixed-time stability of attitude coordination control for spacecraft formation flying (SFF) in the presence of some external disturbance. Firstly, to ensure that the states converge to the origin within a fixed time, a novel nonsingular terminal sliding mode surface (NTSMS) is designed. The convergence time is bounded by some predefined constants. Secondly, an attitude synchronization controller is proposed based on the designed NTSMS, which guarantees the fixed-time stability of SFF under an undirected communication topology. Finally, a fixed-time adaptive control law is designed for cases in which the boundary of the external disturbance is unknown. The fixed-time stability is guaranteed by a revised form of the proposed NTSMS. Simulation results show that the proposed controllers provide fixed-time stability and outperform existing finite-time controllers.

Fixed-time trajectory following for quadrotors via output feedback.

A fixed-time trajectory following problem for quadrotors via output feedback is concerned. Based on the inner-outer separation design philosophy, the under-actuated quadrotor is formulated as a hierarchical structure composed by position and attitude dynamics. With an emphasis on removing the demand on unmeasured velocity and eliminating the negative effect of disturbances, fixed-time extended state observers utilizing two kinds of polynomial feedback terms are proposed to simultaneously identify unavailable velocity states and unknown uncertainties with a fixed-time estimation capability. With these observation results, a velocity free fixed-time control protocol is synthesized to enable a satisfied trajectory regulation with a uniform convergence time independent of initial positions, such that a prescribed fixed-time stability and enhanced robustness can be obtained with chattering-free inputs. By virtue of bi-limit homogeneity properties, all error variables of the resultant quadrotor system are demonstrated to be fixed-time convergent. Eventually, the benefits of developed algorithm are illustrated via simulations.