Barrier Lyapunov Function Research Articles

Barrier Lyapunov function-based dual event-triggered prescribed performance path following control for marine surface vessel under input saturation

Concentrated on the marine surface vessel (MSV) with input saturation, model uncertainties and unknown disturbances, this paper proposes the error-constrained line-of-sight guidance-based prescribed performance dual event-triggered path following control (ECLOS-PPDETPFC) strategy which applies the dual relative threshold event-triggered mechanism (ETM) to both guidance and controller of the MSV path following. We apply the prescribed performance function (PPF) as the bounds of asymmetric modified barrier Lyapunov function (AMBLF) to construct the prescribed performance asymmetric modified barrier Lyapunov function (PPAMBLF). In guidance, event-triggered error-constrained line-of-sight (ETECLOS) is constructed by relative threshold ETM and AMBLF to reduce the calculation pressure and keep position errors within the constraint range; In controller, prescribed performance event-triggered path following control (PPETPFC) is based on the relative threshold ETM and PPAMBLF, which makes the heading error meet the prescribed performance requirements, while reducing the loss of actuator. At the same time, the adaptive auxiliary systems are used to compensate the influence of input saturation, radial basis function neural networks (RBFNNs) and adaptive laws are used to approximate the model uncertainties and composite disturbances. According to the Lyapunov stability theory, all signals are semi-globally uniformly ultimately bounded (SGUUB) and the Zeno phenomenon is avoided. Finally, the comparative experiment shows the effectiveness and superiority of the designed control strategy.

Prescribed performance sliding mode control for the bursting oscillation of a fractional-order hydro-turbine governing system based on time-varying tangent barrier Lyapunov function

A dynamic model for a hydro-turbine governing system is established that accounts for the periodic excitation caused by the dynamic transfer coefficient. The dynamic analysis shows that the turbine regulation system produces bursting oscillatory behaviors when its transmission coefficient changes. The bursting oscillation behavior can adversely affect its stable operation. In response to this challenge, a prescribed performance sliding mode controller based on the time-varying tangent barrier Lyapunov function is proposed. Specifically, a time-varying constraint function of the exponential decay type is initially established. This function is then combined with the prescribed performance constraint function and the dynamic sliding surface to construct the time-varying tangent barrier Lyapunov function. In addition, to ensure that the system output exhibits a slight overshoot and fast-tracking speed at the initial time while satisfying the steady-state index, the relevant parameters of the time-varying constraint function are presented. Finally, the stability of the proposed controller is rigorously proven, verifying the effectiveness of the controller under five working conditions. The simulation results demonstrate that the prescribed performance sliding mode control based on the time-varying tangent barrier Lyapunov function can enable the hydro-turbine governing system to quickly break away from the bursting oscillation state. The proposed control strategy achieves faster convergence times for the rotor angle and generator speed (0.3 s) as well as the guide vane opening (0.3 s) than the ordinary sliding mode controller. The proposed strategy is also superior in controlling the steady-state error, which means that the proposed controller has a faster response speed, higher control accuracy, and optimized transient performance of the system than the traditional controller.

Event-triggered control for [formula omitted]-normal switched stochastic nonlinear systems with asymmetric output constraints

This article focuses on the event-triggered control design for a class of p-normal switched stochastic nonlinear systems(SNS) with asymmetric output constraints. Firstly, in order to handle the pre-specified output constraints, a modified asymmetric tan-type common barrier Lyapunov function(CBLF) is put forward through the idea of piece-wise function. Combining the proposed CBLF with adding a power integrator technique, an event-triggered state-feedback controller is designed to ensure the switched SNS states asymptotically converge to zero in probability. Simultaneously, it is proved that the output will be constrained in the pre-specified bound and Zeno-behavior will not happen. Finally, a simulation of two-tank liquid-level system is given to indicate the effectiveness of the proposed method.

Adaptive Neural Network Control of an Uncertain 2-DOF Helicopter With Unknown Backlash-Like Hysteresis and Output Constraints.

An adaptive neural network (NN) control is proposed for an unknown two-degree of freedom (2-DOF) helicopter system with unknown backlash-like hysteresis and output constraint in this study. A radial basis function NN is adopted to estimate the unknown dynamics model of the helicopter, adaptive variables are employed to eliminate the effect of unknown backlash-like hysteresis present in the system, and a barrier Lyapunov function is designed to deal with the output constraint. Through the Lyapunov stability analysis, the closed-loop system is proven to be semiglobally and uniformly bounded, and the asymptotic attitude adjustment and tracking of the desired set point and trajectory are achieved. Finally, numerical simulation and experiments on a Quanser's experimental platform verify that the control method is appropriate and effective.

Event-Triggered Tracking Control for a Class of Nonlinear Systems: A Dynamic Gain Approach

This note presents a dynamic gain approach to the event-triggered tracking control for a class of strict-feedback nonlinear systems. A novel shifting function is designed to make the initial tracking error arbitrarily bounded, which significantly relaxes the requirement on the initial tracking error. To achieve the objective that the tracking error does not exceed the constraint after the pre-specified finite time, a barrier Lyapunov function and a new reduced-order observer are constructed by using the shifting function and the desired tracking signal. Based on the proposed dynamic gain approach, an event-triggered output feedback tracking controller is designed. It is proved that the developed control scheme guarantees that all the closed-loop signals are bounded and the tracking error constraint is strictly obeyed after a specified finite time. Compared with the celebrated backstepping method, the dynamic gain approach does not involve any virtual control variables, which avoids the complicated iterative procedure. The effectiveness of the proposed approach is verified by an example.

Adaptive neural tracking control for upper limb rehabilitation robot with output constraints

AbstractThe authors investigate the trajectory tracking control problem of an upper limb rehabilitation robot system with unknown dynamics. To address the system's uncertainties and improve the tracking accuracy of the rehabilitation robot, an adaptive neural full‐state feedback control is proposed. The neural network is utilised to approximate the dynamics that are not fully modelled and adapt to the interaction between the upper limb rehabilitation robot and the patient. By incorporating a high‐gain observer, unmeasurable state information is integrated into the output feedback control. Taking into consideration the issue of joint position constraints during the actual rehabilitation training process, an adaptive neural full‐state and output feedback control scheme with output constraint is further designed. From the perspective of safety in human–robot interaction during rehabilitation training, log‐type barrier Lyapunov function is introduced in the output constraint controller to ensure that the output remains within the predefined constraint region. The stability of the closed‐loop system is proved by Lyapunov stability theory. The effectiveness of the proposed control scheme is validated by applying it to an upper limb rehabilitation robot through simulations.

Barrier Lyapunov Function-Based Adaptive Output Feedback Prescribed Performance Controller for Hydraulic Systems With Uncertainties Compensation

In this paper, an adaptive output feedback prescribed performance controller is proposed for a hydraulic system with matched and mismatched uncertainties (including parametric uncertainties and uncertain nonlinearities), which always exist in physical hydraulic systems, and degrade the internal state stability and tracking response performance. By integrating the adaptive control and disturbance observer into the extended state observer, the matched and mismatched uncertainties can be feedforward compensated in the output feedback control. Then, based on the fusion of prescribed performance function and barrier Lyapunov function, the proposed practical method is designed via backstepping method for hydraulic systems, all states are maintained within a preset range by constraining the upper bound of the control errors and the output tracking error is kept within a predetermined boundary that decreases with time. Moreover, the stability of closed-loop system is analyzed by Lyapunov theory. Finally, comparative experimental results are obtained to verify the control performance of the proposed control strategy.

Robust adaptive tracking control for dynamic positioning ships subject to dynamic safety constraints and actuator saturation

This paper proposed a novel robust adaptive trajectory tracking control scheme for dynamic positioning (DP) ships subjected to unknown time-varying disturbances, unknown model parameters, dynamic safety constraints, and actuator saturation. The dynamic safety constraints were guaranteed based on the tan-type barrier Lyapunov function (BLF) technique, which ensured the position and velocity of the DP vessel within the given constraints. Moreover, the radial basis function neural network (RBFNN) was adopted to approximate the unknown parameters of the dynamics model, and an adaptive neural disturbance observer (ANDO) was designed to compensate for lumped disturbances. An anti-windup system was developed to handle the actuator saturation effect. By combining the above techniques and back-stepping method, the final control law was presented, and the semi-global asymptotic convergence was achieved via rigorous theoretical analysis and Lyapunov proof. The results of simulations demonstrated the superiority and robustness of this approach in resolving the trajectory tracking challenges of DP ships. Moreover, the proposed control scheme is highly effective in optimizing the trajectory tracking of DP ships, as it takes into account the multiple constraints that may arise during the process, which is a valuable contribution to the field of DP ship trajectory tracking, providing a more comprehensive and effective solution to this complex problem.

Adaptive Decentralized Control for Interconnected Time-Delay Uncertain Nonlinear Systems With Different Unknown Control Directions and Deferred Full-State Constraints.

In this article, the problem of adaptive decentralized control is investigated for a class of interconnected time-delay uncertain nonlinear systems with different unknown control directions and deferred asymmetric time-varying (DTV) full-state constraints. By constructing the novel time-varying asymmetric integral barrier Lyapunov function (TVAIBLF), the conservative limitation of constant integral barrier Lyapunov function (IBLF) or symmetric IBLF is reduced and the need on the prior knowledge of control gains is also avoided, while the deferred constraints directly imposed on the states of system are achieved by introducing the shifting function into the controller design. Furthermore, based on the Nussbaum-type functions, a new adaptive decentralized control strategy for interconnected time-delay nonlinear systems with subsystems having different control directions is proposed via backstepping method. And it is proven that the proposed control method can guarantee that all signals in closed-loop system are bounded and the transform errors asymptotically converge to zero. Finally, the effectiveness of the proposed control strategy is illustrated through the simulation results.

Adaptive Event-Triggered Neural Network Fast Finite-Time Control for Uncertain Robotic Systems

A fast convergence adaptive neural network event-triggered control strategy is proposed for the trajectory tracking issue of uncertain robotic systems with output constraints. To cope with the constraints on the system output in the actual industrial field while reducing the burden on communication resources, an adaptive event-triggered mechanism is designed by using logarithm-type barrier Lyapunov functions and an event-triggered mechanism. Meanwhile, the combination of neural networks and fast finite-time stability theory can not only approximate the unknown nonlinear function of the system, but also construct the control law and adaptive law with a fractional exponential power to accelerate the system’s convergence speed. Furthermore, the tracking errors converge quickly to a bounded and adjustable compact set in finite time. Finally, the effectiveness of the strategy is verified by simulation examples.

Adaptive Event-Triggered Fuzzy Control of State-Constrained Stochastic Nonlinear Systems Using IBLFs

In this article, an adaptive tracking control problem is addressed for nonstrict-feedback stochastic nonlinear systems subject to state constraints. Fuzzy logic systems (FLSs) are used to model unknown nonlinearities and avoid the algebraic loop arising from the system structure. Appropriate integral Barrier Lyapunov functions (IBLFs) are chosen so that time-varying full state constraints can be guaranteed directly rather than by transforming the constraint object. In the framework of backstepping technology, the relative threshold strategy is introduced to modify the adaptive control scheme so that the controller can be updated only after the trigger condition has been met, which reduces the update frequency of the controller and the loss of the actuator. Combined with Lyapunov stability theory, it is shown that all closed-loop signals are bounded in probability, in which the states remain within the specified constraints, and there is no Zeno behavior. A series of simulation results are given to reveal the effectiveness of the constructed control scheme.

Prescribed performance control of a robotic manipulator with unknown control gain and assigned settling time

This paper presents a control method for trajectory tracking of a robotic manipulator, subject to practical constraints and uncertainties. The proposed method is established upon an adaptive backstepping procedure incorporating a tangent-type barrier Lyapunov function and it preserves some important metrics of trajectory tracking such as fast and user-defined settling time response and robustness against actuation faults and unknown control gain. The proposed design maintains the system trajectory within a prescribed performance bound and relaxes the assumption of the bounded initial condition. These salient features preserve the system within a safety bound and, consequently, guarantee the system stability and safety. The performance of the proposed control method is validated on a 3-DOF PUMA 560 robotic manipulator benchmark model, with different operation scenarios. The simulation results confirm the effectiveness and robustness of the proposed control method.

Adaptive fixed-time path following cooperative control for autonomous surface vehicles based on path-dependent constraints

This paper introduces a fixed-time adaptive path-following cooperative control approach for autonomous surface vehicles (ASVs), aimed at improving navigation accuracy and safety in practical scenarios. The scheme ensures that ASVs accurately follow the reference path during navigation while satisfying position and heading angle constraints. To address the position constraint challenge, an adapted version of the universal barrier Lyapunov function is employed for path-dependent constraint function. Furthermore, a novel form of path-dependent constraint function is introduced, relying on path parameters instead of direct time variables. A new adaptive backstepping technique is also developed to tackle fixed-time convergence issues, mitigate the complexity surge in controller design, and account for the model’s uncertain dynamics. The study demonstrates that this control scheme enables the ASV’s position and heading errors to converge near the equilibrium point at a fixed time rate while consistently satisfying path-dependent constraints. The efficacy of the proposed scheme is further verified through simulation and experimental studies.

Adaptive fuzzy extended state observer for a class of nonlinear systems with output constraint

Abstract In this study, an adaptive fuzzy extended state observer (AFESO) for single-input–single-output nonlinear affine systems in the presence of external disturbances and output constraints is proposed. In this regard, an extended state observer (ESO) was employed to estimate the unmeasured states and external disturbances simultaneously. To improve the estimation accuracy, the observer gains were adjusted using an adaptation law. To obtain a more comprehensive mathematical analysis and an accurate model for the ESO and to increase the degree of freedom, a Takagi–Sugeno fuzzy system was employed. The proposed AFESO relaxes the limitations of the ESO and improves the system performance as compared with the classical methods in the presence of time-varying disturbances. Next, a command-filtered backstepping controller is designed based on the barrier Lyapunov function method, which guarantees fast convergence of the tracking error as well as satisfying the output constraints of the system. The stability analysis showed that both the estimation error of the AFESO and the tracking error of the controller are bounded, and the tracking error converges to a small neighborhood of the origin. A simulating example of a flexible-joint manipulator shows the effectiveness of the proposed method as compared with the recently proposed method in the literature.

Control barrier function based trajectory generation and tracking control for spacecraft inspection mission under multiple safety constraints

In this paper, the safety control problem for spacecraft inspection mission is investigated, which means that there is no position and attitude obstacle collisions between the inspection spacecraft and the defined forbidden zones. We propose a control strategy that ensures safety, utilizing the control barrier function to address multiple safety constraints with logical relationships among them. Firstly, the position and attitude constraints are defined, and the hybrid logical relationship among different attitude constraints is analyzed. To handle complex position and attitude constraints, the Kreisselmeier-Steinhauser function is introduced to approximate the max/min function, thereby expressing them as continuous and differentiable functions. Subsequently, the control barrier function-based quadratic programming (CBF-QP) problems are solved to generate safe velocity and angular velocity. These problems have explicit solutions and do not require extensive computational resources. The safe position and attitude are then obtained by integrating the safe velocity and angular velocity. Notably, the forbidden regions are expanded by a constant in the construction of CBF-QP problems, which acts as a margin for the tracking controller’s performance. Finally, attitude and position tracking controllers based on the control Lyapunov barrier function (CLBF) are designed respectively, taking into account the lumped disturbance in the system, which means that the attitude and position control are computed independently. Numerical simulation results demonstrate that the proposed control strategy effectively handles complex constraints in inspection missions, ensuring system stability and safety, even under severe conditions.

Adaptive tracking control for stochastic nonlinear state-constrained systems with input delay based on multi-dimensional Taylor network

In this paper, an adaptive tracking control strategy based on multi-dimensional Taylor network (MTN) is proposed for a class of stochastic nonlinear state-constrained systems with input delay. Firstly, to avoid constraint violation, the adaptive backstepping approach and barrier Lyapunov function (BLF) are combined in a unified framework. Then, a suitable auxiliary system with the same order as the considered system is constructed to deal with the effect of input delay. In addition, MTN is used to estimate unknown nonlinear functions during the design of the controller. Furthermore, with the help of the Lyapunov stability theorem, the controller presented in this paper can guarantee that all the closed-loop signals are bounded in probability, the system states remain in the defined compact sets, the output signal can track the reference signal successfully, and the tracking error is bounded by the expected bound. Finally, two examples are given to verify the effectiveness of the designed scheme.

Adaptive finite-time output tracker design of uncertain nonlinear systems with sandwich structure and actuator saturation constraint

This paper studies the finite-time output tracking problem for a class of nonlinear sandwich systems with sandwiched dead-zone function and input actuator saturation constraint. To achieve the goal, a systematic adaptive backstepping controller is proposed based on the practical finite-time stability criterion which guarantees the output tracking even if the dead zone and saturation constraints occur, simultaneously. In the control approach to confine the system output to a predefined set, the barrier Lyapunov function is utilised. To cope with the problem of ‘explosion of complexity’, a finite-time command filter is also employed. Furthermore, the values of the unknown parameters are estimated by appropriate adaptive laws. The finite-time stability of the closed-loop system is proved based on the Lyapunov approach which ensures the boundedness of all signals. Finally, the efficiency and applicability of the proposed method are confirmed by providing the simulation results for the hydraulic servo press system.

Adaptive backstepping control for multiple Euler–Lagrangian systems with independent dynamic communication

This paper presents a novel backstepping formation control strategy for multiple Euler–Lagrange (EL) systems. To enhance parallel communication capability, the communication channels for systems’ position vectors and filters’ states are independent. Subsequently, the corresponding dynamic event-triggered mechanisms are designed specifically for them to reduce the frequency of interactive communication networks, and a distributed filter is devised to eliminate the non-differentiability problem of event-triggered signals, respectively. Furthermore, we introduce a new barrier Lyapunov function (BLF) to indirectly restrict the velocity vector. The key novelty of this BLF is that the controller gain matrix caused by the designed BLF can be approximated as an identity matrix, when the distance between the velocity component and the designed boundary is substantial. To prevent the designed dynamic event-triggered mechanisms from being triggered infinitely for a period of time, Zeno behavior is excluded. By leveraging Lyapunov stability theory and Barbalat lemma, we demonstrate that all signals of the closed-loop system are bounded and the formation error asymptotically converges to the origin. Finally, simulation results are provided to demonstrate the efficacy of the proposed control schemes.

Prescribed performance formation control for second-order multi-agent systems with connectivity and collision constraints

This paper studies the distributed formation control problem of second-order multi-agent systems (MASs) with limited communication ranges and collision avoidance constraints. A novel connectivity preservation and collision-free distributed control algorithm is proposed by combining prescribed performance control (PPC) and exponential zeroing control barrier Lyapunov functions (EZCBFs). In particular, we impose the time-varying performance constraints on the relative position and velocity errors between the neighboring agents, and then a PPC-based formation control algorithm is developed such that the connectivity of the communication graph can be preserved at all times, and the prescribed transient and steady performance on the relative position and velocity error can be achieved. Subsequently, by introducing the EZCBFs method, an inequality constraint condition on the control input is derived to guarantee the collision-free formation motion. By regarding the PPC-based formation controller as a nominal input, an actual formation control input is given by solving the quadratic programming (QP) problem such that each agent achieves collision-free formation motion while guaranteeing the connectivity and prescribed performance as much as possible. Finally, numerical simulation is carried out to validate the effectiveness of the proposed algorithm.

Adaptive constrained tracking control for dynamical actuator driven linear 2 × 2 hyperbolic PDE systems with nonlinear uncertainties

The paper develops an adaptive constrained tracking control technique for a class of 2 × 2 hyperbolic partial differential equation (PDE) systems with boundary actuator dynamics, which are described by a set of ordinary differential equations (ODEs) in the presence of unknown parametric nonlinearities. Since the control input only appears in the uncertain ODE subsystem rather than directly on the boundary of PDE subsystem, the control task becomes quite difficult and the existing direct boundary control approaches are ineffective. Moreover, in this paper, a more challenging problem is considered such that the controlled output and the states of ODE actuators are constrained. To this end, by utilizing finite and infinite dimensional backstepping techniques, barrier Lyapunov functions (BLFs) and adaptive methods, a novel adaptive tracking control approach is proposed. It is the first time that such a constrained tracking control problem is addressed for the PDE-ODE coupled systems considered in this paper. On the basis of the presented method, the rigorous theoretical proof is provided to show that the PDE controlled output and all the states of the ODE actuator stay within the predefined compact sets. Finally, the results are illustrated via a comparative numerical simulation.