Fixed-time Stability Research Articles

Fixed-time adaptive dynamic event-triggered control of flexible-joint robots with prescribed performance and time delays

In this study, a dynamic event-triggered control strategy was proposed for n-link flexible-joint robots with prescribed tracking performance and time delays. First, an adaptive fixed-time filter was designed to prevent “differential explosion”, and a given-time prescribed performance method was introduced. Then, an auxiliary system and Lyapunov–Krasovskii functionals were designed to compensate for input and full-state delays. After that, neural networks were introduced to handle the unknown dynamics and a dynamic event-triggered controller was designed. The closed-loop system was demonstrated fixed-time stability without Zeno behaviors. Finally, simulations were presented to confirm the effectiveness of the proposed scheme.

Boundary Output Feedback for Fixed-time Stabilization of Distributed Parameter Systems With Time and Space Dependent Reactivity

Boundary Output Feedback for Fixed-time Stabilization of Distributed Parameter Systems With Time and Space Dependent Reactivity

Fixed-time sliding mode control for hypersonic morphing vehicles via event-triggering mechanism

The event-triggered fixed-time control technique for hypersonic morphing vehicles is discussed in this paper. For simplification, the control-oriented model with morphing-span wings is established. Strong robustness and adaption are demanded since the aerodynamic characteristic change caused by the span morphing is significant. To ensure convergence, the multivariable sliding mode manifold is developed based on fixed-time control technology. The switching dynamic event-triggering mechanism is studied to save the resource of the flight control system. The proposed strategy guarantees a direct control input design. The Lyapunov theory is employed to reveal the fixed-time stability of the closed-loop system. Three simulation cases are introduced to examine the detailed performance.

Consistent discretization of homogeneous finite/fixed-time controllers for LTI systems

The paper proposes a discretization (sampled-time implementation) algorithm for a class of homogeneous controllers for linear time-invariant (LTI) systems preserving the finite-time and nearly fixed-time stability properties. The sampling period is assumed to be constant. Both single-input and multiple-input cases are considered. The robustness (Input-to-State Stability) of the obtained sampled-time control system is studied as well. Theoretical results are supported by numerical simulations.

Finite/nearly fixed-time stability of nonlinear impulsive systems with destabilizing impulses and its application to neural networks

This paper investigated the finite-time stability (FTS) and nearly fixed-time stability (nFxTS) of nonlinear impulsive systems with destabilizing impulses. The concept of nFxTS is different from FxTS, as the former may be just asymptotically stable but the latter must be FTS. A Lyapunov inequality with linear terms has been used to derive sufficient conditions based on the dwell time (DT) and average dwell time (ADT) properties of impulsive sequences to ensure FTS/FxTS of the system. Theorems are constructed based on the initial state of the system, which belongs to either δ-neighborhood(nbh) or outside δ-nbh of the origin. When the system starts within δ−nbh of the origin then FTS can be achieved by the system following conditions on DT and ADT of impulsive sequences. The same is true for nFxTS of the system when it starts outside δ-nbh of the origin. We investigated the condition on DT for which nFxTS is not possible, but FTS is possible only if δ>0 is bounded. The main results of this paper are applied to the synchronization problem of impulsive neural networks with destabilizing impulses. Finally, a numerical example of an impulsive neural network is given to illustrate the validity of the proposed theoretical results.

Fixed-time stabilization of periodic linear systems and its application to the elliptical spacecraft rendezvous

A smooth periodic delayed feedback (SPDF) control scheme is proposed for the fixed-time stabilization problem of linear periodic systems subject to input delay. By investigating the monodromy matrix of the periodic system, it is proved that the SPDF controller can achieve the fixed-time stabilization of linear periodic systems with arbitrarily long yet bounded input delays under the condition that the original system is uniformly completely controllable. The proposed controller is continuously differentiable and smooth. The SPDF control scheme is then applied to the elliptical spacecraft rendezvous problem. The effectiveness of the established method is verified on numerical simulations.

Fixed-Time Controller for Altitude/Yaw Control of Mini-Drones: Real-Time Implementation with Uncertainties

Gradually, it has become easier to use aerial transportation systems in practical applications. However, due to the fixed-length wire, recent studies on load-suspended transportation systems have revealed some practical constraints, especially when using quadrotor unmanned aerial vehicles (UAVs). By actively adjusting the distance between the quadrotor and the payload, it becomes possible to carry out a variety of challenging tasks, including traversing confined spaces, collecting samples from offshore locations, and even landing a payload on a movable platform. Thus, mass variable aerial transportation systems should be equipped with trajectory tracking control mechanisms to accomplish these tasks. Due to the above-mentioned reasons, the present paper addresses the problem of the altitude/yaw tracking control of a mini-quadrotor subject to mass uncertainties. The main objective of this paper is to design a fixed-time stable controller for the perturbed altitude/yaw motions, based on recent results using the fixed-time stability approach. For comparison reasons, other quadrotor motion controllers such as dual proportional integral derivative (PID) loops were considered. To show its effectiveness, the proposed fixed-time controller was validated on a real mini-quadrotor under different scenarios and has shown good performance in terms of stability and trajectory tracking.

Finite/fixed-time stabilization of a chain of integrators with input delay via PDE-based nonlinear backstepping approach

In this paper, we present a general approach to studying the problem of finite-time and fixed-time stabilization of a chain of integrators with input delay. To accomplish this, we first reformulate the chain of integrators with input delay as a cascade ODE–PDE system (i.e., a cascade of a linear transport partial differential equation (PDE) with the chain of integrators) where the transport equation models the effect of the delay on the input. Next, we use a nonlinear infinite-dimensional backstepping transformation to convert the cascade system to a suitable target system that is chosen to be finite-time or fixed-time stable. We perform the stability analysis on the target system by means of classical non-asymptotic concepts and tools such as the linear homogeneity and “generalized KL” functions. Then, we use the inverse transformation to transfer back the stability property to the closed-loop system. Finally, we give some characterizations of finite/fixed time predictor-based controllers followed by numerical simulations.

Event-triggered fixed-time tracking of state-constrained surface ships under actuation saturation with prescribed control performance

This paper investigates the event-trigger-based fixed-time tracking control of a fully-actuated marine surface vehicle subject to full-state constraints, lumped disturbances, model uncertainties, and input saturation. The definition of the constrainedly practically fixed-time stability (CPFTS) is given first, and a sufficient condition for CPFTS is subsequently provided. To handle the input saturation, an auxiliary system is designed elaborately. Then applying the barrier Lyapunov function technology and blending an error transformation into backstepping scheme, the model-based adaptive controller and neural-network-based (NN-based) adaptive controller are proposed, respectively. Under these two controllers, the resulting closed-loop system signals are CPFTS and the settling time is not only independent of the system initial conditions but also assigned by users previously. Besides, it is proved that the tracking errors can converge to an arbitrarily small preset compact set within the preassigned time at a prescribed decay rate whilst all system states preserve the time-varying constraints.

Improved Fixed-Time Stabilization of Fuzzy Neural Networks With Distributed Delay via Adaptive Sliding Mode Control

This paper investigates fixed-time stabilization of fuzzy neural networks with distributed delay by designing an adaptive sliding mode controller. Firstly, according to stability theory and related inequalities, a new fixed-time stability theorem is put forward, and the settling time is given. In order to stabilize the system, a new integral sliding mode surface is designed, and the corresponding sliding mode control strategy and adaptive sliding mode control strategy are established. Some criteria that can be obtained, and it is shown that the neuronal states of neural networks will arrive at the sliding surface in a fixed time, and then approach zero along the sliding surface. Compared with existing sliding mode control techniques, this work extends the previous related results by choosing different parameters of the controller and the sliding mode manifold to gain various protocols. Finally, two examples are provided to verify the validity of the theorems in this work.

Dynamic Observer-Based Fast Fixed-Time Filtered Backstepping Controller Design for a Constrained Uncertain Nonlinear System

In this article, a fast fixed-time filter (FFTF)-based-backstepping control system is proposed for a class of uncertain high-order nonlinear systems with full-state constraints. To effectively tackle the lumped uncertainty, a dynamic gain fixed-time disturbance observer (DGFDO) is constructed to estimate the uncertainty, where the observer parameters are dynamically tuned and the estimation error converges to zero in fixed time. In order to solve the full-state constraints problem, the state transformation technique is employed with avoiding “feasibility conditions.” For overcoming the “explosion of complexity” problem of the backstepping method, an FFTF is constructed, where the filter error can be ensured to achieve fixed-time stability. In the light of Lyapunov theory, it is proved that all signals of the closed-loop system are bounded, all states are kept in their constraints, and output error converges to an arbitrarily small neighborhood around zero in fixed time. Finally, simulation results are demonstrated to verify the effectiveness of the theoretical results.

Adaptive Fixed-Time Control for the Postcapture Tethered Spacecraft With Full-State Constraints

This paper studies the fixed-time control problem for the post-capture tethered spacecraft subject to full-state constraints and uncertainties. A <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> smooth adaptive fixed-time controller is constructed via introducing a smooth switch between fractional and quadratic form feedback, such that the undesired chattering and singularity issues are removed. With the aid of universal transformed function (UTF) technique, the proposed controller not only ensures that the constraint boundaries are not violated, but also eliminates the feasibility condition existing in the traditional barrier Lyapunov functions (BLFs)-based control solutions. Furthermore, by incorporating the generalized first-order filter and adaptive laws into control design procedure, the fixed-time stability of the closed-loop system can be guaranteed. The validity of the proposed control scheme is demonstrated by a simulation example.

A Novel Fast Fixed-Time Backstepping Control of DC Microgrids Feeding Constant Power Loads

Nonlinear behavior of constant power loads (CPLs) and their negative incremental impedance, which impose adverse effects on system damping and stability margins, necessitate developing proper strategies for efficient implementation framework of DC microgrid. In this paper, a fixed-time sliding mode disturbance observer (SMDO) is first addressed to provide an estimation of the instantaneous power flow moving along the uncertain CPLs with time-varying nature within a fixed time. It not only expedites the estimation rate but also improves the robustness against physical parameter variation. The fast fixed-time backstepping technique is then developed based on the estimated load power to control the duty cycle of the boost converter such that the entire power grid becomes stable and the desired voltage of the DC bus is tracked within a fixed time irrespective of the initial conditions. A rigorous Lyapunov-based approach is represented to guarantee the fixed-time stability of the proposed scheme. Finally, to validate the merits and implementation feasibility of the proposed methodology, the experimental realization is represented under various operating case studies.

Novel fixed-time stability criteria of nonlinear systems and applications in fuzzy competitive neural network and Chua’s oscillator

Novel fixed-time stability criteria of nonlinear systems and applications in fuzzy competitive neural network and Chua’s oscillator

Adaptive Finite/Fixed Time Control Design for a Class of Nonholonomic Systems with Disturbances

This paper addresses the fixed-time stability analysis of a mobile unicycle-like system (UTMS) with chained shape dynamics (CFD) and subjected to unknown matched uncertainties. To achieve fixed-time stabilization of a nonholonomic (NS) system in CFD, an adaptive nonsingular fast terminal sliding mode control scheme (ANFTSMC) is proposed. To determine the upper bounds of the disturbances, only velocity and position measurements are required. In addition, the control rule uses the Lyapunov theory, which guarantees the stability of the closed-loop system. To emphasize/evaluate the efficacy of the proposed method, simulations are performed in different disturbance situations.

Fixed/Preassigned-Time Stabilization for Complex-Valued Inertial Neural Networks with Distributed Delays: A Non-Separation Approach

This article explores complex-valued inertial neural networks (CVINNs) with distributed delays (DDs). By constructing two new feedback controllers, some novel results on fixed-time stabilization (FTS) and preassigned-time stabilization (PTS) of CVINNs are established. Unlike most of the previous works, FTS and PTS obtained here are explored without dividing the original complex-valued system into two separate real valued subsystems. Eventually, to verify the effectiveness and reliability of the results of this article, we provide several numerical examples. The FTS and PTS of CVINNs are successfully implemented at T = 6, 5.5, and 5, and the settling time is not affected by system parameters and initial values.

Fixed-time adaptive fuzzy event-triggered control for uncertain nonlinear systems with output constraint and actuator failures

For a category of uncertain nonlinear systems with output constraint and actuator failures, the fixed-time event-triggered control method is investigated. Output constraint and actuator failures are common in practical systems, which may affect the performance of the system. Most of the existing schemes to deal with these problems only guarantee the system's asymptotic stability. It is a challenge to design event-triggered based controllers to achieve fixed-time stability of a system with the above constraints. To handle the above problems, barrier Lyapunov functions (BLFs) and extended fuzzy logic systems (EFLSs) are introduced to the backstepping procedure. Besides, to reduce redundant data transmissions, an event-triggered mechanism (ETM) is designed. On this foundation, a fixed-time event-triggered control method is proposed. Technically, by using the proposed method, the controlled system is practically fixed-time stable, and the system output abides by the corresponding constrained requirements. Meanwhile, the communication resources are saved. Finally, some simulation examples are used to demonstrate the effectiveness of the control method.

Nonsingular Fixed-Time Control of Nonstrict Feedback MIMO Nonlinear System With Asymptotically Convergent Tracking Error

In this paper, the research on fixed-time tracking control problem of multiple-input multiple-output (MIMO) nonlinear systems with non-strict feedback control structure is carried out. By means of the fuzzy logic systems (FLSs), the unknown system nonlinear dynamics are identified, and the “algebraic loop problem” is solved. Then, through the combination of adaptive back-stepping recursive technology and adding power integration technology, a non-singular fixed-time adaptive tracking control scheme is presented. Under the action of the presented control scheme, the system can track the specified reference signal within a fixed time independent of the initial state of the system, and the tracking control error can converge to zero asymptotically. Next, on the basis of the fixed-time Lyapunov stability theory, the practical fixed-time stability of the closed-loop system is theoretically demonstrated. Finally, two simulations verify the validity and practicability of the proposed control scheme.

Neural Adaptive Fixed-Time Control for Nonlinear Systems With Full-State Constraints.

This article aims at this problem of adaptive neural tracking control for state-constrained systems. A general fixed-time stability criterion is first presented, by which an adaptive neural control algorithm is developed. Under the action of the proposed adaptive neural tracking controller, the tracking error converges into a small neighborhood around the origin in fixed time; meanwhile, all system states abide by the corresponding state constraints for all the time. The main difference between the present research and the previous control schemes for state-constrained systems is that this article proposes a novel and feasible approach to ensure that the constructed virtual control signals satisfy the state constraints on the corresponding states viewed as the virtual control inputs. Such an approach guarantees theoretically that all the system states cannot violate their constrained requirements at any time. Finally, two simulation examples provide support to the proposed results.

Fully Distributed Pull-Based Event-Triggered Bipartite Fixed-Time Output Control of Heterogeneous Systems With an Active Leader.

This article deals with the fully distributed pull-based event-triggered bipartite fixed-time output consensus problem of heterogeneous linear multiagent systems (HLMASs) with an active leader, whose information can be merely accessed by a small fraction of followers. First, a class of fully distributed fixed-time observers is proposed for each follower to estimate the leader's system matrices, position, and control input under the signed communication topology, respectively. Then, based on the estimations of leader's system matrices, two adaptive algorithms are given to solve the regulator equations. Furthermore, the fully distributed fixed-time observer-based controllers associated with state feedback and output feedback are, respectively, proposed by employing the pull-based event-triggered mechanism (ETM) where each agent merely updates controller at its own triggering instants. Correspondingly, some sufficient criteria and the rigorous proofs are provided to ensure the implementation of bipartite output consensus in fixed time by using the Lyapunov stability theory and fixed-time stability theory. Moreover, the strictly positive lower bounds of intervals between two adjacent event-triggered times are derived, which means the Zeno behavior is ruled out. Finally, numerical simulations are performed to demonstrate the theoretical analysis.