Fixed-time Stability Research Articles

Walking position commanded NAO robot using nonlinear disturbance observer-based fixed-time terminal sliding mode

The walking stability of a humanoid robot is a fundamental problem due to the complex nonlinear dynamic model of the robot’s legs. This work introduces the performance tracking control for the humanoid NAO robot by using a Nonlinear Disturbance Observer (NDO)-based Fixed-time Terminal Sliding Mode (FTSM). The influence of uncertain external disturbance is considered while implementing the control strategy to improve the walking motion of the NAO robot. An NDO is adapted to estimate the uncertainties and external disturbances. A novel FTSM surface is proposed to drive the tracking errors to zero in fixed-time. The designed NDO-based FTSM control law achieves robustness while reducing the chattering phenomenon. The Lyapunov’s stability theory is used to establish the fixed-time stability of the sliding surface and system states under the proposed control method. To validate the performance of the proposed NDO-based FTSM control, a real-time experiment was conducted on a humanoid NAO robot to demonstrate the improved tracking performance in the presence of the uncertain perturbation effect. The effectiveness of the proposed controller design is validated on a flat, upward inclined surface, and compared to another controller.

Fixed-Time Stabilization of a Class of Stochastic Nonlinear Systems

This paper investigates an improved fixed-time stability theory together with a state feedback controller for a class of nonlinear stochastic systems. First, a delicate transformation is performed, and next, a Gamma function is utilized to directly derive the value of the integral function, which ultimately yields a fixed-time stabilization theorem with a higher precision upper bound for the settling time. Unlike the existing estimation process of amplifying twice, we only performed one amplification, which weakens the effect of amplification. Then, a state feedback controller is constructed for stochastic systems by the method of adding a power integrator. Utilizing the proposed stochastic fixed-time stability theory, simulations show that the intended controller ensures that the trivial solution of the suggested system is fixed-time stable in probability. The results of the simulation demonstrate that the suggested control scheme is meaningful.

Adaptive Distributed Fixed-Time Cooperative Three-Dimensional Guidance Law for Multimissiles against Manoeuvring Target

The problem of cooperative interception of the manoeuvring target is investigated in this paper. Firstly, in light of fast fixed-time consensus theory, time-to-go, and undirected topologies, adaptive cooperative guidance along the line-of-sight (LOS) direction is proposed to guarantee impact time synchronization. Next, novel nonsingular terminal sliding mode (NTSM) is designed to establish adaptive fixed-time guidance law for steering LOS angular rates to the origin or its small neighbourhood. Without the knowledge of target manoeuvre, the proposed cooperative guidance law can be provided by lateral and longitudinal accelerations of each missile, while more reasonable and rigorous analysis of fixed-time stability is carried out through the Lyapunov theory. Within the specified time, both control tasks of simultaneous attack and the desired impact angles can be completed before the final time of the guidance process. Finally, numerical simulations demonstrate the feasibility and effectiveness of the proposed scheme.

Self-triggered fixed-time bipartite fault-tolerant consensus for nonlinear multiagent systems with function constraints on states

This article presents a self-triggered bipartite fault-tolerant consensus control scheme in a fixed-time stability sense for multiagent systems (MASs) with function constraints on states. Neural networks are introduced to identify unknown nonlinearities, thereby relaxing constraints on unknown functions. Then, based on time-varying barrier Lyapunov functions including time and previous states, all states of the considered MASs are constrained to given ranges. In addition, to improve the utilization of system transmission resources, a self-triggered control method in which the next triggering time can be computed according to the current information of the controller is proposed. By applying the Lyapunov theory, the designed self-triggered controller can ensure that all signals containing consensus tracking errors are fixed-time bounded under a given communication topology, and the Zeno phenomenon is successfully avoided. Finally, a practical example is provided to testify the feasibility of the developed control scheme.

Adaptive Fixed-Time Neural Control for Uncertain Nonlinear Multiagent Systems.

In this article, we consider the problem of adaptive fixed-time tracking control for a class of multiagent systems (MASs) with mismatched uncertainty. Unlike the existing methodologies that only implement the practical finite-/fixed-time stability for MASs, a newly adaptive consensus control criterion is developed to reach fixed-time stability, where the controller design includes a series of newly Lyavonov functions and modified tuning functions. Radial basis function neural networks are employed to deal with the unknown functions in each agent, and the direct adaptive strategy solves the obstacle of "explosion of complexity." Under the performance-oriented controller, the error of the MASs converges to a predetermined interval within a fixed time. Two simulations illustrate the results obtained.

Practical fixed-time output feedback trajectory tracking control for marine surface vessels with unknown disturbances

This paper proposes a fixed-time output feedback trajectory tracking control scheme based on an extended state observer for marine surface vehicles (MSVs) with uncertain dynamics and unknown disturbances. The scheme utilizes a high-gain fixed-time extended state observer (HGFxESO) to estimate the reconstructed unmeasured velocities and lumped disturbances. The upper bound of the settling time of the HGFxESO is only determined by the design parameters, and the estimation errors can converge to a bounded set within a fixed time. Moreover, a novel fixed-time control law is proposed based on the backstepping technique and “adding a power integrator” (API) technique. The designed continuous control law ensures the trajectory tracking errors of MSVs converge to a neighborhood around the origin within a fixed time. The fixed-time stability of the MSVs trajectory tracking control system is proved using Lyapunov theory. Finally, simulation results are provided to demonstrate the effectiveness of the proposed control scheme.

Practical finite-/fixed-time improved command-filtered backstepping control for nonlinear systems via immersion and invariance

This paper presents a practical finite-/fixed-time adaptive tracking control scheme based on improved command filtered backstepping (ICFBS) and immersion and invariance (I&I) for a class of single-input single-output (SISO) strict-feedback nonlinear systems with uncertainty parameterization. The conventional command-filtered backstepping method (CFBS) is improved by introducing any bounded function such that ICFBS relaxes the boundedness assumption of the control direction function and elevates the tunable space of the design scheme, with almost no increase in computational complexity. The higher-power composite term of a parameter estimation is designed in the parameter update law based on I&I and the inequality constructed by us proves that this form of the adaptive update law can be augmented to suit for not only uniform ultimate bounded (UUB) and practical finite-time stability but also practical fixed-time stability of the whole closed-loop system. The error compensation system is also extended to realize the practical finite-/fixed-time convergence of the error compensation signals. The proposed controller guarantees practical finite-/fixed-time stability of the closed-loop system. Therefore, our work solves two unsolved problems: both practical fixed-time and practical finite-/fixed-time ICFBS and I&I adaptive tracking control. Finally, the proposed control scheme is validated by a practical electromechanical system and a nonlinear system with unbounded control direction functions.

Fixed-Time Convergent Guidance Law with Angle Constraint and Autopilot Lag Compensation Using Partial-State Feedback

In this paper, by accounting for the angle constraint (AC) and autopilot lag compensation (ALC), a novel fixed-time convergent guidance law is developed based on a fixed-time state observer and bi-limit homogeneous technique. The newly proposed guidance law exhibits three attractive features: (1) unlike existing guidance laws with AC and ALC which can only guarantee asymptotic stability or finite-time stability, the newly proposed guidance scheme can achieve fixed-time stability. Thus, the newly proposed scheme can drive the guidance error to zero within bounded time which is independent of the initial system conditions. (2) To compensate for autopilot lag, existing guidance schemes need the unmeasurable second derivative of the range along line-of-sight (LOS) and second derivative of LOS angle or the derivative of missile’s acceleration. Without using these unmeasurable states, the newly proposed guidance law still can guarantee the fixed-time stability. (3) By using the bi-limit homogeneous technique to construct an integral sliding-mode surface, the proposed scheme eliminates the singular problem without using the commonly-used approximate method in recent fixed-time convergent guidance schemes. Finally, the simulation results demonstrate the effectiveness of the proposed scheme.

Adaptive fixed-time inverse optimal consensus of multi-agent systems with limited-time interval state constraints

This article presents a study on fixed-time adaptive fuzzy inverse optimal consensus for multi-agent systems (MASs) subject to limited-time interval state constraints. To handle the constraints that occur within a limited-time interval during the system's operating cycle, we introduce a shifting function to shift the boundary and a state-dependent transformation function to transform the constrained MASs into non-constrained ones. We then propose an adaptive fixed-time inverse optimal consensus controller that achieves the convergence of consensus error within a fixed-time and inverse optimality without directly learning the solution of the Hamilton-Jacobi-Bellman (HJB) equations. To establish the criteria for fixed-time stabilization, we design an auxiliary controller instead of the auxiliary system and introduce two Lyapunov functions to prove the inverse optimality and fixed-time stabilization. The developed consensus scheme ensures that the synchronization error asymptotically converges to a user-predefined region within a fixed settling time. We validate the effectiveness of the developed scheme through a simulation example.

Fixed-time event-triggered sliding mode cooperative path-following control with prescribed performance for USVs based on lumped disturbance observer

In order to solve the problem of Unmanned Surface Vehicles being unable to accurately cooperative path-following under the parameter uncertainties, actuator faults, and external disturbances, this paper mainly combines the fixed-time and sliding mode theory to design the control strategy. Firstly, a fixed-time disturbance observer is adopted to estimate composite disturbances. Then, a fixed-time heading-surge guidance law that satisfies the fixed-time stability of the position error system is designed, and the synchronization of the formation on the full paths is realized by the fixed-time path parameter consensus. On this basis, a fixed-time dynamic event-triggered sliding mode controller with prescribed performance is proposed to ensure that the velocity error system is stable in fixed time and the tracking error is strictly within the specified range. Due to the application of fixed-time theory in cooperative path-following, the whole closed-loop system is proved to be fixed-time stable. Moreover, the dynamic event-triggered mechanism is introduced to reduce the energy consumption. Finally, the effectiveness and superiority of the proposed control method are verified by simulations.

A fixed-time converging neurodynamic approach with time-varying coefficients for l1-minimization problem

In this paper, we propose a novel neurodynamic network to deal with l1-minimization problem. In the framework of the fixed-time converging neurodynamic network (FxNN), time-varying coefficients are introduced to design the time-varying fixed-time converging neurodynamic network (TFxNN). It is shown that the fixed-time stability of the proposed TFxNN via the Lyapunov stability conditions. It is further shown that the proposed TFxNN trajectories from any initial points quickly converge to the unique equilibrium solution in fixed-time. A distinctive feature of the proposed TFxNN is its flexibility to choose time-varying coefficients to accelerate convergence. Simulation results based on signal and image reconstruction are presented that the proposed neurodynamic network is feasible and effective.

Robust Constant Exponent Coefficient Fixed-Time Control Based on Finite-Time Extended Sliding Mode Observer of Permanent Magnet Synchronous Motors

This paper presents the Robust Constant Exponent Coefficient Fixed-Time Control (CECFSMC), an innovative control technique for precisely regulating the speed of a permanent magnet synchronous motor (PMSM) by utilizing fixed-time stability with constant exponent coefficients to provide not only faster convergence but also in a specific period of time. The effect of chattering is also lessened. To ensure that the designed controller produces the desired performance under bounded disturbances, a finite-time extended sliding-mode observer (ESMO) is also designed to estimate the PMSM velocity while also estimating lumped load disturbances. The considered PMSM is the surface-mounted PMSM. Finally, a numerical simulation with PMSM drive shows good robustness against load disturbances, better convergence, and a reaching time of less than 2 s, thereby demonstrating that the proposed fixed-time constant exponent coefficient offers good performance and is much simpler than the conventional finite-time method.

Intermittent boundary control for robust fixed‐time stabilization of uncertain reaction‐diffusion systems

AbstractAchieving the robust fixed‐time stabilization (FxTS) of uncertain reaction‐diffusion systems (RDSs) utilizing intermittent boundary control is a valuable and unresolved problem. This paper simultaneously considers the influence of the uncertainty of non‐diffusion and diffusion coefficients on RDSs and provides an effective solution for the treatment of the uncertainty of diffusion coefficient in the investigation of FxTS. Then, a fixed‐time control scheme is presented to stabilize uncertain RDSs within a fixed time, which incorporates the merits of intermittent and boundary control and reduces control costs in both the spatial and temporal domains. Besides, some sufficient conditions in the form of linear matrix inequalities are deduced to guarantee the robust FxTS of uncertain RDSs with Neumann and mixed boundary conditions, respectively, which contain and generalize some existing results. Finally, the validity of the intermittent boundary control scheme and the FxTS criteria developed in this paper are verified by numerical simulations.

Adaptive Fixed-Time Stabilization for a Class of Uncertain Nonlinear Systems

This paper studies the problem of fixed-time stabilization for a class of nonlinear systems with parametric uncertainties, where the nonlinear functions are constrained by an unknown linear growth condition. By establishing a new adaptive fixed-time stability analysis criterion, it is proved that the closed-loop system is fixed-time stable and all the states can be regulated to zero. In addition, the singularity problem caused by the backstepping method under fixed-time control can be avoided. Finally, the effectiveness of the control strategy is verified by a simulation example.

Adaptive Fuzzy Nonsingular Fixed-Time Control for a Class of MIMO Nonstrict Feedback System

This paper studies the adaptive fuzzy practical fixed-time control design for MIMO nonlinear systems with non-strict feedback. The extant practical fixed-time stability criterion cannot obtain an accurate settling time upper bound or exists some constraints on parameters that lead to a conservative result. Thus a novel practical fixed-time stability criterion is proved in this paper to resolve the above problem and can obtain the precise value of the settling time upper bound. The control singular issue caused by the fixed-time stability criterion form is avoided by introducing adding power integration method. Compared with other approaches, this method is easy to apply in control design, and it's convenient to prove its effectiveness. And fuzzy logic system is adopted to approximate the unknown nonlinear items. Combining the proposed fixed-time stability criterion, traditional backstepping scheme, and fuzzy logic system, an adaptive practical fixed-time controller is proposed. The simulation of the actual physical system demonstrates the effectiveness of the controller and practical fixed-time criterion.

Finite-Time and Fixed-Time Synchronization of Delayed Memristive Neural Networks via Adaptive Aperiodically Intermittent Adjustment Strategy.

This article investigates the finite-time and fixed-time synchronization for memristive neural networks (MNNs) with mixed time-varying delays under the adaptive aperiodically intermittent adjustment strategy. Different from previous works, this article first employs the aperiodically intermittent adjustment feedback control and adaptive control to drive the MNNs to achieve synchronization in finite time and fixed time. First of all, according to the theories of set-valued mappings and differential inclusions, the error MNNs is derived, and its finite-time and fixed-time stability problems are discussed by applying the Lyapunov function method and some LMI techniques. Moreover, by meticulously designing an effective aperiodically intermittent adjustment with adaptive updating law, sufficient conditions that guarantee the finite-time and fixed-time synchronization of the drive-response MNNs are obtained, and the settling time is explicitly estimated. Finally, three numerical examples are provided to illustrate the validity of the obtained theoretical results.

Stability analysis of switched stochastic nonlinear functional systems

AbstractThis article is concerned with stochastic finite time stability (SFTS) and stochastic fixed time stability (SFIXTS) of switched stochastic nonlinear functional systems. By using the Lyapunov–Razumikhin approach, stochastic analysis techniques, and stochastic process theory, several novel theorems on SFTS and SFIXTS are established. Concurrently, the estimations of stochastic settling time are given. Different from the previous general results, this article considers the effects of noise, time‐delays, and switches. The SFTS and SFIXTS conditions are established, which are quite different from the traditional exponential stability or almost sure stability. Furthermore, two examples are given to illustrate the effectiveness of the obtained results.

Multiple adaptive fuzzy Nussbaum-type functions design for stochastic nonlinear systems with fixed-time performance

Nussbaum design has been successfully exploited to deal with uncertain systems with a single unknown control coefficient. However, it is challenging for the traditional Nussbaum method to efficiently handle the feedback control design with multiple unknown time-varying control coefficients because the multiple Nussbaum-type gains might mutually cancel out, which makes the Lyapunov-based stability analysis infeasible. To tackle this issue, a novel type of Nussbaum function is designed for the tracking problem of a class of stochastic strict-feedback nonlinear systems. In the presented Nussbaum design, the multiple unknown control coefficients are separately compensated, where the effect of mutual cancellation no longer exists and the Lyapunov stability analysis can be directly applied. Moreover, the proposed Nussbaum design is combined with the fixed-time stability theory, which ensures the settling time is bounded and independent of the initial condition. Within the adaptive fuzzy backstepping framework, it is shown that all signals in the closed-loop system remain bounded through the theoretical discussions. The simulation experiment is conducted to verify the effectiveness of the presented design for the continuous-time stochastic nonlinear dynamical system.

A novel approach to state estimation of HIV infection dynamics using fixed-time fractional order observer

This paper presents a novel approach to designing a fixed-time fractional order observer for estimating the states of the dynamic model of human immunodeficiency virus (HIV) infection. The proposed approach combines output injection terminal sliding mode and RBF neural network strategies to achieve a robust and efficient estimation of the states of the HIV model within a fixed time frame. The main contributions of this work are the introduction of an output injection observer that ensures the stability of the error system along with a novel nonlinear sliding surface that guarantees the fixed-time error convergence to the neighborhood of zero. Moreover, the closed-loop scheme of the observer design is proven to be bounded, and the fixed-time stability of the observer error is obtained using the fractional Lyapunov stability approach. Simulation results show that the proposed fixed-time fractional order observer design provides accurate and efficient estimation of the states of the HIV model.

Fixed-time stabilization of fuzzy neutral-type inertial neural networks with proportional delays

The issue of fixed-time stabilization (FTS) for a class of fuzzy neutral-type inertial neural networks (FNTINNs) with proportional delays (PDs) is discussed in this research. Two feedback controllers were constructed utilizing the fixed-time stability theorem. Two criteria for figuring out the FTS of FNTINNs with PDs were acquired by building suitable Lyapunov functions (LFs). The theoretical conclusions offered could potentially contribute to a more precise calculation of the upper settling-time (ST) when compared to previously conducted research. Two simulation examples are provided to demonstrate the applicability of the stated theoretical conclusions.