Fixed-time Stability Research Articles

Robust attitude control for hypersonic reentry vehicle via composite fixed-time stable control method

A novel adaptive nonsingular fixed-time converged terminal sliding mode control method is proposed for a standard nonlinear system suffering from uncertainties and disturbance. The obtained results are applied to attitude maneuver controller design for hypersonic reentry vehicles (HRV). First, an adaptive disturbance observer with fixed-time stability is introduced to cope with the total disturbance consisting of uncertainties and disturbance. Considering the singularity issue inherent in classical fixed-time converged terminal sliding mode, an improved nonsingular fixed-time terminal sliding mode(NFxTSM) is designed by implementing the switching function, ensuring faster convergence and singularity-free. The adaptive technique is also incorporated with the controller design to enhance the robustness of the NFxTSM. Then, considering the inherent time-scale separation feature, the composite attitude maneuver controller is further designed for HRV based on the backstepping technique. The outer loop generates the desired angle rate command, and the inner loop is designed to track the outer loop command. Finally, the numerical simulations are established to verify the effectiveness of the proposed controller.

Fixed-time synchronization of coupled neural networks under Hölder continuous nonlinear activation function

This paper investigates the fixed-time stability of dynamical systems and its application on the fixed-time synchronization (FTSYN) of the coupled neural networks (NNs). Firstly, a new theorem on fixed-time stability is established, along with an estimation of the settling time. Based on the new theorem and Lyapunov direct method, two different sufficient criteria are derived to ensure that the NNs in presence of Hölder continuity can achieve synchronization within fixed-time based on 1-norm and 2-norm, separately. Finally, two numerical simulations are conducted to demonstrate the validity and accuracy of the main findings.

Fixed-time adaptive control of quadrotor suspension system with unknown payload mass

This paper investigates the control challenges of quadrotor suspension systems with unknown payload masses. To address the unknown external disturbances, uncertainties in system models, and unknown payloads, fixed-time stable adaptive laws are developed to estimate and compensate for these unknown factors. Additionally, a novel control structure based on a nonlinear backstepping controller is proposed for the quadrotor suspension system, achieving stable control of the quadrotor and reducing the sway of the suspended payload. The fixed-time stability of the designed closed-loop system is analytically proven using Lyapunov functions, ensuring maximum stability time. Simulink simulations validate the proposed control approach, demonstrating accurate estimation of unknown masses and the effectiveness and superiority of the controller.

The Gaussian error function for a new theorem on fixed-time stability with applications in synchronization of chaotic Lorenz systems

The Gaussian error function for a new theorem on fixed-time stability with applications in synchronization of chaotic Lorenz systems

Fixed-time Lyapunov criteria of stochastic impulsive time-delay systems and its application to synchronization of Chua’s circuit networks

In this paper, we investigate the fixed-time stability and synchronization issues of stochastic impulsive delay complex networks (SICNs) with delayed impulses. Firstly, a new fixed-time stability criterion, which provides a direct connection between impulsive strength, impulsive instants, delays and the stability time, is established for stochastic delay systems with delayed impulsive. In this criterion, two types of delays: system delay and impulsive delay are considered, and the impulsive intensity at each impulse instant can be different, (facilitate or disrupt stability). It is the first time to consider the fixed-time stability issues of this case. Secondly, based on this criterion, a novel feedback controller and an adaptive controller are designed to study the fixed-time synchronization of SICNs with delayed impulses; Thirdly, by utilizing Lyapunov functional theory, stochastic analysis technique, matrix inequality analysis techniques and proposed stability criterion, the fixed-time synchronization conditions of SICNs, are derived in the form of linear matrix inequalities (LMIs). Finally, a numerical example and an application example, Chua’s circuit networks, are provided to illustrate the validity of the theoretical analysis.

Disturbance observer-based control with hysteresis identification in piezoelectric deformable mirrors

Piezoelectric deformable mirrors are complex systems with a reflective face-sheet and underlying actuators. The inherent hysteresis phenomenon in piezoelectric materials introduces nonlinearity and delay, rendering conventional control methods inappropriate for deformable mirror actuation. This study presents a disturbance observer-based controller with the generalised Bouc–Wen hysteresis parameter identification algorithm. Analytical proofs establish fixed-time stability for the observer and controller. A key feature of the proposed disturbance observer is the convergence of the estimated hysteresis to the actual value after a prescribed settling time, which facilitates precise hysteresis parameter identification. This is based on a data clustering technique, unlike most previous works. The proposed controller has the capability of hysteresis compensation and reference tracking in the piezoelectric deformable mirror actuators. Numerical simulation results are present to evaluate the tracking performance of the controller in the presence of random wavefront, the convergence of disturbance to the actual value in the fixed time, and identification of hysteresis parameters.

Adaptive fixed-time fuzzy control for delayed recycling continuous stirred tank reactor with asymmetric time-varying full-state constraints

Recently, there are some results concerning stability and finite-time stability control for continuous stirred tank reactors (CSTR). However, when considering the control of time-delay CSTR systems or fixed-time stability of CSTR systems, there are relatively few articles involved. In this paper, we first consider the CSTR system that includes both of the above factors simultaneously, that is, the adaptive fixed-time fuzzy control design problem for a CSTR with a delayed recycle stream and asymmetric time-varying full-state constraints. Firstly, based on existing works, we present an improved practical fixed-time stability criterion, which can be applied to more complex nonlinear systems. Then, we construct time-varying asymmetric integral barrier Lyapunov functions to handle asymmetric time-varying full-state constraints. Next, fuzzy logic systems are used to approximate the unknown functions of the system, and a sliding mode differentiator is introduced to avoid the “explosion of complexity” of the time derivative of the virtual controller. Furthermore, using the improved practical fixed-time stability criterion, we present the adaptive fixed-time fuzzy controller design processes that can ensure tracking performance in a fixed-time, and states of the system never violate their constraint bounds. Finally, a simulation example is provided to verify the effectiveness of the proposed control strategy.

High-Order Disturbance Observer-Based Fuzzy Fixed-Time Safe Tracking Control for Uncertain Unmanned Helicopter with Partial State Constraints and Multisource Disturbances

In the real-world operation of unmanned helicopters, various state constraints, system uncertainties and multisource disturbances pose considerable risks to their safe fight. This paper focuses on anti-disturbance adaptive safety fixed-time control design for the uncertain unmanned helicopter subject to partial state constraints and multiple disturbances. Firstly, a developed safety protection algorithm is integrated with the fixed-time stability theory, which assures the tracking performance and guarantees that the partial states are always constrained within the time-varying safe range. Then, the compensation mechanism is developed to weaken the adverse impact induced by the filter errors. Simultaneously, the influence of the multisource disturbances on the system stability are weakened through the Ito^ differential equation and high-order disturbance observer. Further, the fuzzy logic system is constructed to approximate the system uncertainties caused by the sensor measurement errors and complex aerodynamic characteristics. Stability analysis proves that the controlled unmanned helicopter is semi-globally fixed-time stable in probability, and the state errors converge to a desired region of the origin. Finally, simulations are provided to illustrate the performance of the proposed scheme.

Fixed-Time Control and Estimation of Discontinuous Fuzzy Neural Networks: Novel Lyapunov Method of Fixed-Time Stability.

This article considers the fixed-time stability (FXTS) issue for discontinuous system described by differential equation (DE) with time-varying parameters. Using the tool of differential inclusion (DI), several improved FXTS criteria and estimation formulas of settling time (SET) are derived by employing a relaxed Lyapunov method. The special novelty of the developed Lyapunov method is that the derivative of Lyapunov function possesses indefiniteness. As an important application, the established FXTS theorems are utilized to deal with fixed-time (FXT) synchronization issue for fuzzy neural networks (FNNs) possessing state discontinuity, where the fuzzy operation is used in the synaptic law computing, and one typical time-varying switching control protocol is designed. Moreover, a series of estimations of SET for FXT synchronization are given out. Finally, the simulation examples are present to substantiate the validity of the obtained results.

Internal/Boundary Control-Based Fixed-Time Synchronization for Spatiotemporal Networks.

This article is concerned about fixed-time (FT) synchronization of spatiotemporal networks (STNs) with the Robin boundary condition. Above all, a switching-type FT stability theorem and an integral inequality are established, which provide a novel theoretical tool for the rigorous analysis of FT control in STNs. Subsequently, three kinds of nontrivial power-law controllers are developed which are separately acted on the interior, the boundary, and the whole of the spatial domain. Based on these control schemes and Lyapunov-like method, several flexible criteria are obtained to achieve FT synchronization of STNs, and the upper bound of the synchronization time is explicitly estimated. Note that, the derived results here are also perfectly applicable to STNs with Neumann or Dirichlet boundary condition. Several illustrate examples are presented at final to confirm the developed controllers and criteria.

Adaptive neural network based fixed-time attitude tracking control of spacecraft considering input saturation

Aiming at the issues of actuator saturation, inertia uncertainties, and external unknown disturbances in the attitude tracking control process of spacecraft, an adaptive fixed-time attitude control method is proposed, which is based on a radial basis function neural network (RBFNN). Firstly, a spacecraft attitude kinematics and dynamics model is established based on the quaternion method and a Gaussian error function is introduced to constrain the controller amplitude. Secondly, the external unknown disturbances are addressed by a fixed-time disturbance observer, and the controller is designed utilizing the backstepping method. To eliminate the adverse effects caused by actuator saturation, we design an enhanced auxiliary system to improve the stability of the system. Aiming at inertia uncertainties, RBFNN is used to approximate it, and an innovative fixed-time convergence adaptive law with RBFNN weights is devised. Subsequently, based on Lyapunov theory, the fixed time stability of the closed loop system is proven, and an expression for the settling time is given. Finally, simulation analysis validates the effectiveness of the designed controller.

Fixed-Time Backstepping Sliding-Mode Control for Interleaved Boost Converter in DC Microgrids

Interleaved boost converters (IBCs) are commonly used as interface converters for DC microgrids (MGs) due to their high efficiency and low output ripple. However, the MGs system can easily become unstable due to the negative impedance characteristics of constant power load (CPL) and rapid power fluctuations. This paper proposes a fixed-time backstepping sliding-mode controller (FTBSMC) aimed at stabilizing the MGs system and achieving fixed-time tracking of the DC bus voltage. Firstly, the fixed-time disturbance observer (FxTDO) estimates the load disturbance at a fixed time, which improves the fast disturbance resistance of the system. Then, based on the dis-turbance estimation, the FTBSMC is designed, which combines the fast dynamic response of the sliding-mode control with the global stability of the backstepping control, avoiding the singularity problem of the conventional sliding-mode control. In addition, a first-order nonlinear filter is employed to avoid the direct differentiation of conventional backstepping control and at the same time to ensure global fixed-time stability. The fixed-time convergence of the proposed FTBSMC is rigorously demonstrated by using Lyapunov stability analysis. Finally, the FTBSMC proposed is verified by simulation and experiment in terms of faster dynamic response and stronger robustness.

Cluster formation tracking of networked perturbed robotic systems via hierarchical fixed-time neural adaptive approach

This paper investigates the fixed-time cluster formation tracking (CFT) problem for networked perturbed robotic systems (NPRSs) under directed graph information interaction, considering parametric uncertainties, external perturbations, and actuator input deadzone. To address this complex problem, a novel hierarchical fixed-time neural adaptive control algorithm is proposed based on a hierarchical fixed-time framework and a neural adaptive control strategy. The objective of this study is to achieve accurate CFT of NPRSs within a fixed time. Specifically, we design a distributed observer algorithm to estimate the states of the virtual leader within a fixed time accurately. By using these observers, a neural adaptive fixed-time controller is developed in the local control layer to ensure rapid and reliable system performance. Through the use of the Lyapunov argument, we derive sufficient conditions on the control parameters to guarantee the fixed-time stability of NPRSs. The theoretical results are eventually validated through numerical simulations, demonstrating the effectiveness and robustness of the proposed approach.

Finite-time and fixed-time stability for nonlinear systems via aperiodically intermittent control

Finite-time and fixed-time stability for nonlinear systems via aperiodically intermittent control

Fixed-time second-order sliding mode output feedback trajectory tracking control for an autonomous surface vehicle with model uncertainties and prescribed performance

This paper investigates the fixed-time trajectory tracking control problem of autonomous surface vehicle system with asymmetric performance constraints. The external disturbances, model uncertainties and the difficulty of velocity measurement are all taken into account in the controller design. Asymmetric performance constraints are firstly studied to improve the transient and steady-state performance of the tracking error system. Next, a fixed-time extended state observer is developed to estimate the unknown velocity and external disturbances. Both the fixed-time first-order and the second-order PID-type sliding surfaces are designed. Further, two continuous output feedback controllers are built to achieve high control accuracy and low energy consumption and to achieve fixed-time stability of the closed-loop system. Finally, abundant simulations for CyberShip II demonstrate the effectiveness of the proposed algorithms.

Design of fixed-time sliding mode control using variable exponents

This paper proposes the design of a robust fixed-time sliding mode controller for a second-order nonlinear system with matched and unmatched uncertainties. First, a fixed-time stable scalar dynamics with a variable exponent is introduced. We establish the global fixed-time stability results for the proposed dynamics using Lyapunov analysis. The maximum convergence time has been derived in terms of the design parameters, which is independent of the initial conditions. These results are then extended to solve the robust fixed-time stabilization for uncertain nonlinear second-order system. Specifically, we proceed with the sliding mode control design, where the sliding surface and the control law are motivated by the proposed scalar dynamics. Moreover, the proposed design is free from singularity and guarantees fixed-time robust stabilization. Simulation results with comparative analysis has been included. Further, experimental validation on a single link manipulator is provided to demonstrate the performance of the proposed approach.

Second-order DSC-based adaptive fuzzy tracking control for uncertain unmanned helicopter with fixed-time stability

Second-order DSC-based adaptive fuzzy tracking control for uncertain unmanned helicopter with fixed-time stability

A fast fixed-time backstepping control method for systems with mismatched disturbances

In this paper, a fast fixed-time backstepping control approach is developed for systems with mismatched disturbances. A novel fast fixed-time disturbance observer (FFTDO), which is convergent within fixed time regardless of initial conditions, is designed to efficiently estimate and compensate the disturbances. On the basis of FFTDO, a novel fast fixed-time backstepping control scheme is proposed, which can guarantee fast fixed-time convergence and high steady-state precision. In addition, a novel fixed-time filter is used to circumvent the problem of “explosion and complexity,” and to ensure overall fixed-time convergence. The fixed-time stability of the closed-loop system under the proposed controller is proved by the Lyapunov stability theory. The simulation results are carried out to confirm the feasibility and superiority of the proposed control strategy.

Adaptive model-free disturbance rejection for continuum robots

This paper presents two model-free control strategies for the rejection of unknown disturbances in continuum robots. The strategies utilize a neural network-based approximation technique to estimate the uncertain Jacobian matrix using position measurements. The first strategy is designed for periodic disturbances and employs an adaptive model-free controller in conjunction with an adaptive disturbance observer. The second strategy is designed for robustness against arbitrary disturbances and employs time-varying input and update law gains that grow monotonically, resulting in the achievement of asymptotic, exponential, and prescribed-time reference trajectory tracking. The notion of fixed-time stabilization in prescribed time is particularly noteworthy, as it allows for the predefinition of a terminal time, independent of initial conditions and system parameters. A formal stability analysis is presented for each strategy, and the strategies are both tested experimentally with a concentric tube robot subject to unknown disturbances.

Fixed-time self-triggered fuzzy adaptive control of N-link robotic manipulators

This paper studies the fixed-time fuzzy adaptive self-triggered control of n-link robotic manipulators. To achieve fixed-time stability, a fuzzy adaptive fixed-time control algorithm is proposed by reconstructing the self-triggered controller and using the singularity-free function, which avoids the singularity problem and obtains fixed-time stability. Meanwhile, the self-triggered mechanism reduces the communication resources between the actuator and the controller, improving the system's transmission efficiency. Through the above thoughts, a kind of fixed-time fuzzy adaptive controller based on the self-triggered mechanism is designed, which guarantees that the tracking error of each joint converges to the prescribed range within a fixed time. Finally, the effectiveness of the proposed algorithm is verified by a simulation example of the two-link robotic manipulator.