Fixed-time Stability Research Articles

Finite time stability of time-varying stochastic nonlinear systems with random impulses

ABSTRACT This article studies the finite time stability (FNTS) of time-varying nonlinear stochastic systems with random impulses. We provide sufficient conditions for the FNTS or even fixed time stability (FXTS) for time-varying stochastic systems under two cases: (1) The impulse time is deterministic; (2) The impulse time is random. For case (1), we use the reverse average dwell time to investigate FNTS, but for case (2), we employ the Poisson process theory to solve the stability problem. In addition, the stability criteria capture the stabilising effect of stochastic noise in the FNTS problem. Finally, the correctness of the theoretical results is verified with two numerical examples. As far as the author knows, no one has studied the FNTS of time-varying stochastic systems with random impulses.

Adaptive Control Design for Euler–Lagrange Systems Using Fixed-Time Fractional Integral Sliding Mode Scheme

This paper presents an adaptive fixed-time fractional integral control for externally disturbed Euler–Lagrange systems. In the first step of the control design, the approach of fractional-order fixed-time integral nonsingular terminal sliding mode control (FoIFxTSM) is introduced. This scheme combines the benefits of fractional calculus with integral sliding mode control, resulting in fast convergence, smooth nonsingular control inputs, and fixed-time stability. By integrating an adaptive scheme, the proposed method is used to control the dynamical system in the presence of uncertainty and external perturbations. The findings of the fixed-time stability using Lyapunov analyses are provided for the closed-loop system. The simulation results are compared with the adaptive fractional-order sliding mode control scheme, and they show the better tracking and convergence performance of the proposed method.

Fixed-time controller of visual quadrotor for tracking a moving target

In order to eliminate the dependence of finite-time control on the initial state of the system in the image-based visual servoing (IBVS) of a quadrotor UAV (QUAV) for the moving target, we propose a control scheme for the moving target of a QUAV based on fixed-time stability. We select the image moments of the virtual camera plane as features and establish an image moment dynamics model that includes target motion parameters based on them. Then, we use the backstepping method to design the fixed-time controller for the system. The design steps of the controller mainly consist of two parts. Firstly, we designed a fixed-time stable linear velocity observer to address the unmeasurable linear velocity and unmeasurable monocular camera depth issues of QUAV. Then, using a high-order tracking differentiator to estimate the target linear velocity as a feedforward, combined with the fixed-time linear velocity observer estimated QUAV linear velocity, we designed a fixed-time controller for the system using the backstepping method. We demonstrated the fixed-time stability of the controller using the Lyapunov theory. The effectiveness and robustness of the proposed method are demonstrated by numerical simulation. At the same time, the comparative simulation shows that the method can guarantee the rate of convergence of the system and has high control accuracy.

Adaptive neural network fixed-time sliding mode control for trajectory tracking of underwater vehicle

The effectiveness of Autonomous Underwater Vehicles (AUVs) in diverse underwater tasks is heavily reliant on their ability to perform accurate trajectory tracking. However, due to uncertainties in AUVs modeling and the complex underwater environment disturbances, designing effective trajectory tracking controllers and disturbance observers for AUVs is still a major challenge. To address these uncertainties and enable faster convergence of tracking errors, a trajectory-tracking controller based on fixed-time sliding mode control (FTSMC) and a Radial Basis function neural network (RBFNN) observer are used in this paper. In most cases, the AUV platform carries limited computational resources. In most cases, AUV platforms carry limited computational resources, which restricts the practical use of online neural network methods, and it is particularly important to reduce the complexity of computational neural networks and enhance the real-time performance of the observer. Therefore, we adopted a fast online weight update strategy based on a single parameter. Considering actuator faults and input saturation, passive fault-tolerant control (PFTC) is used in this scheme to further reduce the computational burden. Furthermore, the Lyapunov method is used to demonstrate the fixed-time stability of the individual signals of the system. Finally, simulation results and theoretical analysis demonstrate the superiority and effectiveness of the proposed method.

Fixed-time stability super-twisting sliding mode control with mismatched disturbances

In practice, many systems are affected by mismatched disturbances, which do not satisfy the match condition. This study focuses on a variable gain super-twisting sliding mode controller that achieves global robust fixed-time stabilization of second-order systems with matched and mismatched disturbances without disturbances observers. The proposed controller and the sliding variables adopt a state-dependent variable exponent coefficient, which improves the robustness properties of a sliding surface and switching control law. At the same time, the control strategy alleviates the chattering phenomenon by using the variable-gain super-twisting algorithm. The settling time is calculated a priori by the Lyapunov method. Simulation results show that, compared with recently published controllers, our proposed control strategy can suppress the mismatched disturbance more rapidly, simply, and effectively.

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 event‐triggered control for high‐order nonlinear uncertain systems with time‐varying full state constraints

AbstractThe fixed time event‐triggered control for high‐order nonlinear uncertain systems with time‐varying full state constraints is investigated in this paper. First, the event‐triggered control (ETC) mechanism is introduced to reduce the data transmission in the communication channel. In consideration of the physical constraints and engineering requirements, time‐varying barrier Lyapunov function (BLF) is deployed to make all the system states confined in the given time‐varying constraints. Then, the radial basis function neural networks (RBF NNs) are used to approximate the unknown nonlinear terms. Further, the fixed time stability strategy is deployed to make the system achieve semiglobal practical fixed time stability (SPFTS) and the convergence time is independent of the initial conditions. Finally, the proposed control scheme is verified by two simulation examples.

Reconfigurable tolerant control of nonlinear Euler–Lagrange systems under actuator fault: A reinforcement learning-based fixed-time approach

This paper presents a novel fixed-time adaptive Fault Tolerant Control (FTC) framework for MIMO nonlinear Euler-Lagrange systems using sliding mode-based strategy, reinforcement learning (RL) and fixed-time disturbance observer. The primary objective is to enhance system reliability in the presence of actuator faults, uncertainties and disturbances. The proposed RL algorithm incorporates an actor-critic neural network (NN), where the actor NN estimates the uncertainty and the critic NN evaluates the performance cost function. Additionally, a fixed-time adaptive observer is designed to estimate the lumped term of faults and disturbances. To achieve high precision trajectory tracking within a fixed-time interval, a nonsingular fast terminal sliding mode scheme is designed. This scheme ensures fixed-time convergence of the tracking error and facilitates disturbance attenuation and fault mitigation, which are key features of the proposed fixed-time secure control strategy. Furthermore, the closed-loop system's fixed-time stability is analyzed using Lyapunov theory. Experimental results demonstrate the effectiveness of the proposed FTC framework in mitigating the adverse effects of faults, uncertainties and disturbances, thereby enhancing system performance and reliability.

Global Fixed-Time Sliding Mode Trajectory Tracking Control Design for the Saturated Uncertain Rigid Manipulator

The global fixed-time sliding mode control strategy is designed for the manipulator to achieve global fixed-time trajectory tracking in response to the uncertainty of the system model, the external disturbances, and the saturation of the manipulator actuator. First, aiming at the lumped disturbance caused by system model uncertainty and external disturbance, the adaptive fixed-time sliding mode disturbance observer (AFSMDO) was introduced to eliminate the negative effects of disturbance. The observer parameters can adaptively change with disturbances by designing the adaptive law, improving the accuracy of disturbance estimation. Secondly, the fixed-time sliding surface was introduced to avoid singularity, and the nonsingular fixed-time sliding mode control (NFSMC) design was put in place to ensure the global convergence of the manipulator system. Finally, the fixed time saturation compensator (FTSC) was created for NFSMC to prevent the negative impact of actuator saturation on the manipulator system, effectively reducing system chatter and improving the response speed of the closed-loop system. The fixed-time stability theory and Lyapunov method were exploited to offer a thorough and rigorous theoretical analysis and stability demonstration for the overall control system. Simulation experiments verify that the designed control scheme has excellent control effects and strong practicability.

Robust cooperative output regulation for heterogeneous nonlinear multi-agent systems with an unknown exosystem subject to jointly connected switching networks

This paper investigates the robust cooperative output regulation problem for heterogeneous lower triangular nonlinear multi-agent systems with an unknown exosystem over jointly connected switching networks. The problem has been studied for the exactly known exosystem over switching networks. However, the existing result for the unknown exosystem is still limited to the static networks. To ensure that all followers acquire the reference trajectory generated by the unknown exosystem through the jointly connected switching networks, by combining a set of auxiliary filtering variables and fixed-time stability theory, an adaptive distributed observer is designed. On the basis of the adaptive distributed observer and the distributed internal model approach, we propose a distributed controller under several standard assumptions to solve the problem. Compared with the similar work subject to the static networks, the controller in this paper is applicable to the more general communication network while weakening the assumptions of the controlled system.

Fixed time prescribed performance dynamic positioning control design for surface vessels

This paper addresses the problem of fixed time dynamic positioning (DP) control for surface vessels under the influence of external disturbances, while considering input and output constraints. Using the hyperbolic tangent function, a saturation handling method with an auxiliary control system is constructed, which can handle the saturation constraints of both the control input and its rate of change within a same control framework. By incorporating an auxiliary sliding mode variable, a fixed time disturbance observer (FTDO) is developed to estimate external disturbances. This FTDO ensures that the estimation error converges within a fixed time and provides users with the flexibility to determine the desired convergence time. Moreover, a nonlinear function is introduced in the FTDO to reduce the possibility of singular problems in the control system. Combining the developed FTDO, prescribed performance control strategy and backstepping control technique, a novel DP control strategy is proposed, which not only ensures that the positioning error satisfies the output constraint requirements within a fixed time but also guarantees the fixed time stability of the closed-loop system. Finally, to validate the effectiveness of the proposed DP control law, numerical simulation analysis is carried out on a supply vessel.

New Inequality Approaches for Fixed-Time Stability Lemmas and Application to Discontinuous CGNNs With Nondifferentiable Delays

This article proposes fixed-time stability lemmas for the Filippov system via some new inequality approaches. The adopted method no longer needs to integrate the Lyapunov function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$V$</tex-math> </inline-formula> on the two integral intervals, which is quite different from the existing ones. Some new estimations of the settling times are provided. Also, the steepness exponents of the Lyapunov function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$V$</tex-math> </inline-formula> in the previous fixed-time stability lemmas are improved. In order to further study the nondifferentiable delayed neural networks modeled by the Filippov system, a class of discontinuous uncertain Cohen–Grossberg neural networks (CGNNs) with mixed delays is formulated and the distributed delays are nondifferentiable, which is more general. Due to the existence of nondifferentiable distributed delays, the existence of the periodic solutions is proved by Kakutani’s fixed-point theorem before considering the stability. By virtue of the obtained fixed-time stability lemmas and the constructed delay-product-type Lyapunov–Krasovskii functional, the fixed-time stabilization is obtained via a no-chattering controller. Clearly, the designed controller does not contain integral terms and delay terms for dealing with the time delays in the closed-loop system, which is more simplified and practical. Finally, two examples help examine the correctness of the main results.

Robust consensus control of second-order uncertain multiagent systems with velocity and input constraints

In this paper, we investigate the consensus problem of second-order multiagent systems under directed graphs. Simple yet robust consensus algorithms that advance existing achievements in accounting for velocity and input constraints, agent uncertainties, and lack of neighboring velocity measurements are proposed. Furthermore, we show that the proposed method can be applied to the consensus control of uncertain robotic manipulators with velocity and control torque constraints. We rigorously prove that the velocity and control inputs stay within prespecified ranges by tuning design parameters a priori and that asymptotic consensus can be achieved through Lyapunov functions and fixed-time stability. Simulations are performed for symmetric and asymmetric constraints to show the efficiency of the theoretical findings.

Proximate fixed-time fault-tolerant tracking control for robot manipulators with prescribed performance

This paper presents a proximate fixed-time fault-tolerant prescribed performance tracking control for robot manipulators with actuator faults, parametric uncertainties, and bounded disturbances. An exact convergent fixed-time prescribed performance function is first introduced. A proximate fixed-time fault-tolerant prescribed performance control (FTPPC) is proposed within the framework of terminal sliding mode control. Lyapunov stability is employed to prove the position tracking error can be ensured proximate fixed-time stability and prescribed performance with an exact convergence time. Advantages of the proposed FTPPC include proximate fixed-time prescribed performance guarantees featuring faster transient and higher steady-state precision and strong robustness to actuator faults, parametric uncertainties, and bounded disturbances. Simulations and experiments utilizing artificial faults demonstrate the improved performances of the proposed approach.

On the fixed-time extinction based nonlinear control and systems decomposition: applications to bilinear systems

This paper deals with new criteria of the fixed-time extinction of nonlinear dynamical systems. We show, by comparison principal, that Gronwall-like integral inequalities leading to the extinction of the energy system in fixed-time; this helps us to build fixed-time stabilizing feedback laws for bilinear control systems in Hilbert space. In addition, in order to relax some existing results on the fixed-time stability of systems by Lyapunov function, we prove that the decomposition of the system into two subsystems so that one is finite-time stable and the other is polynomially stable, leads to fixed-time extinction of the whole system. Our results are applied to the double integrator where new fixed-time stabilizing feedbacks, with various aspects, are constructed.

Fixed-time synchronization of complex-valued memristive competitive neural networks based on two novel fixed-time stability theorems

Fixed-time synchronization of complex-valued memristive competitive neural networks based on two novel fixed-time stability theorems

Novel results on event-triggered-based fixed-time synchronization and stabilization of discontinuous neural networks with distributed delays

This article focuses on fixed-time synchronization(FXS) and fixed-time stabilization(FTS) for discontinuous neural networks(DNNs) with distributed delays. An improved fixed-time stable lemma for differential systems is proposed. Then, by using theory of non-smooth analysis and inequality techniques, some novel sufficient results are established to realize FXS and FTS for such DNNs. What is more, unlike feedback controllers used in most of earlier works, FXS and FTS of this article are achieved by constructing new effective event-triggered control(ETC), which conserve control costs and easier to materialize. In the end, examples and comparisons are displayed to corroborate the correctness and advantages of the derived FXS and FTS.

A Model-Free-Based Control Method for Robot Manipulators: Achieving Prescribed Performance and Ensuring Fixed Time Stability

This paper addresses three significant challenges in controlling robot manipulators: improving response time, minimizing steady-state errors and chattering, and enhancing controller robustness. It also focuses on eliminating the need for computing the robot’s dynamic model and unknown functions, as well as achieving global fixed-time convergence and the prescribed performance for the control system. To achieve these objectives, a fixed-time sliding mode function is designed, which uses transformation errors to achieve prescribed control performance, with adjustments made to the maximum overshoot, convergence time, and tracking errors to keep them within predefined bounds. Additionally, a radial basis function neural network (RBFNN) is used to eliminate the need for knowledge of the robot’s dynamical properties and uncertain terms, which also reduces negative chattering. Finally, a novel fixed-time terminal sliding mode control (TSMC) algorithm is developed for robot manipulators without using their dynamical model. The fixed-time stability of the control system is thoroughly demonstrated by applying Lyapunov criteria and conducting simulations on a robot manipulator to showcase its effectiveness.

Fixed-Time Control for a Class of Nonlinear PH-DAE Systems

The globally fixed-time stabilization and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty}$</tex-math> </inline-formula> control for the port-Hamiltonian differential algebraic equation (PH-DAE) is addressed by applying the structural properties of the port-Hamiltonian (PH) systems, and the interconnection and damping assignment passivity-based control (IDA-PBC) technique. For fixed-time control design of the differential algebraic equation systems, the main obstacle lies that the algebraic variables are hidden in the system and their dynamics are ambiguous. We propose an available and effective approach to overcome this obstacle. The system variables are decomposed into differential component and the algebraic component, and the explicit representation of the dynamics of algebraic component is obtained. Fist, we investigate the locally fixed-time stabilization control as well as globally attractive for PH-DAE. Second, we focus on solving the globally fixed-time stabilization and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty}$</tex-math> </inline-formula> control problems by jointly taking advantage of the locally fixed-time stabilization control for the origin and the globally fixed-time attractivity control for a predeterminate region. Finally, in order to address fixed-time stabilization control and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty}$</tex-math> </inline-formula> control for PH-DAE, new controllers are designed. Moreover, the convergence time of systems can be easily estimated after external disturbance vanishes. The efficiency of the theoretical results acquired in present article is verified via two examples with numerical simulation.

Fixed-Time Lyapunov Criteria of Stochastic Nonlinear Systems and Its Generalization

This study investigates the fixed-time stability of stochastic nonlinear systems described by Itô differential equations. The improved fixed-time Lyapunov theorem is given and an important corollary is obtained. It is proved that the upper-bound estimate of the settling time is less conservative. We establish a new definition of practically fixed-time stability in probability (PFxTSp) and propose the corresponding Lyapunov criterion theorem. In addition, different cases of the parameters are discussed to provide a less conservative upper-bound estimate of the settling time. The effectiveness of the proposed stability and controller is demonstrated by two simulation examples.