Barrier Lyapunov Function Research Articles

Adaptive neural network iterative learning control of long-stroke hybrid robots with initial errors and full state constraints

Research on initial errors and constraint restrictions is one of the main research directions in the field of control of uncertain robotic systems. An adaptive iterative learning control (AILC) method based on radial basis function (RBF) neural network is proposed to address the trajectory tracking problem of the long-stroke hybrid robot system with random initial errors and full state constraints. The RBF neural network is used to approximate the unknown nonlinear terms, and the network weights are updated using an iterative learning law that incorporates a projection mechanism. Additionally, a robust learning strategy is used to compensate for both the approximation error of the neural network and the external disturbances that vary with each iteration. To relax the requirement of traditional iterative learning control (ILC) for identical initial condition, an equivalent error function is constructed based on the time-varying boundary layer. The tangent-type barrier Lyapunov function (BLF) is designed to ensure that the joint position and speed of the robot system are bounded within a predetermined range. Through stability analysis based on barrier composite energy function (BCEF), it can be proved that the boundedness of all signals in the closed-loop system and the tracking error of the robot system will converge to an adjustable residual set asymptotically. Finally, through simulation experiments conducted on the MATLAB platform, the results demonstrate that the method overcomes the random initial errors of the system effectively, ensures that the system satisfies the full-state constraints, and realizes high-precision trajectory tracking.

Backstepping based neural [formula omitted] optimal tracking control for nonlinear state constrained systems with input delay and disturbances

This paper focuses on the problem of H∞ optimal tracking control for a class of nonlinear state constrained systems with input delay and disturbances. With the aid of Pade approximation, an auxiliary variable is devised to eliminate the effects of input delay. Combining barrier Lyapunov functions (BLFs) with backstepping design technique, a feedforward adaptive controller is designed to transform the tracking control problem of nonlinear state constrained system into an equivalent H∞ control problem of input-affine error system without state constraints, wherein neural networks (NNs) are employed to approximate unknown system dynamics. Then based on single-network adaptive dynamic programming (ADP), an H∞ optimal feedback controller is developed by utilizing a single critic network to learn the Nash equilibrium related to Hamilton–Jacobi–Isaacs (HJI) equation. Therefore, the whole tracking controller can be constructed by integrating the feedforward adaptive controller with the optimal feedback controller. Moreover, it is proven by Lyapunov’s theory that all signals within the closed-loop system are uniformly ultimately bounded (UUB), and the tracking error converges to a small neighborhood of the origin without violating any state constraints. Two simulation examples are also presented to validate the effectiveness of the proposed approach.

Prescribed Performance Fault-Tolerant Attitude Tracking Control for UAV with Actuator Faults

This paper proposes a prescribed performance fault-tolerant control based on a fixed-time extended state observer (FXTESO) for a carrier-based unmanned aerial vehicle (UAV). First, the attitude motion model of the UAV is introduced. Secondly, the proposed FXTESO is designed to estimate the total disturbances including coupling, actuator faults and external disturbances. By using the barrier Lyapunov function (BLF), it is proved that under prescribed performance control (PPC), the attitude tracking error is stable within the prescribed range. The simulation results for tracking the desired attitude angle show that the average overshoot and stabilization time of PPC-FXTESO is 0.00455rad and 6.2s. Comparatively, the average overshoots of BSC-ESO and BSC-FTESO are 0.035rad and 0.027rad, with stabilization times of 14.97s and 12.56s, respectively. Therefore, the control scheme proposed in this paper outperforms other control schemes.

Guaranteed Performance Event-Triggered Adaptive Consensus Control for Multiagent Systems under Time-Varying Actuator Faults

A guaranteed performance event-triggered adaptive consensus control is established for uncertain multiagent systems under time-varying actuator faults. To eliminate the impact caused by actuator faults, an adaptive neural network compensation strategy is developed. Simultaneously, by implementing the asymmetric barrier Lyapunov function and transform function, a prescribed time consensus control with guaranteed performance, is constructed. Furthermore, to reduce the frequency of information transmission, an adjustable switching event-triggered control (ASETC) is proposed by using a modified hyperbolic tangent function. It combines the advantage of the relative threshold strategies and the characteristics of the hyperbolic tangent function, giving better flexibility in saving network resources and guaranteeing system performance. By applying the constructed control method, systems with prescribed performance consensus in a prescribed time are achievable while limited network resources and unknown time-varying faults are present. Some simulation examples implemented in MATLAB (R2022a) are given to demonstrate the above results.

Full state‐constrained integrated guidance and control for aerial interceptors with tunnel prescribed performance using integral barrier Lyapunov function

AbstractA novel full state‐constrained integrated guidance and control (IGC) schema for aerial interceptors under input saturation with tunnel‐prescribed performance is proposed. To achieve an appointed‐time engagement with less overshoot phenomenon, the tunnel‐prescribed performance control (TPPC) technique is applied to generate virtual guidance laws, where the tracking of desired line of sight (LOS) angles possesses predefined‐time convergent property. For the sake of the flight safety of interceptors in a practical flight environment, the physical limitations on the flight angles and the angular rates of attitude systems are taken into consideration. The integral barrier Lyapunov function (iBLF) technique is adopted to ensure all the attitude states within a user‐defined domain. The nominal virtual commands are generated adaptively by the backstepping method combined with some novel dynamic surfaces. To further eliminate the possible input saturation, a novel one‐order auxiliary system is constructed to compensate for the discrepancy of the virtual input commands. All the signals in the closed‐loop system are strictly proven to be bounded by Lyapunov theory. The simulation results show the effectiveness of the proposed IGC schema by comparison with other IGC laws.

Adaptive neural quantized control for full-state constrained Markov jumping nonlinear systems with incomplete transition probabilities and unknown control directions

This article devises an adaptive quantized control scheme for full-state constrained Markov jumping nonlinear systems with incomplete transition probabilities and unknown control directions. First, the barrier Lyapunov functions were utilized to achieve full-state constraints on the investigated system, while the Nussbaum function has been adopted to overcome the issue of unknown control directions. Then, by means of command filtered backstepping control technique, an adaptive quantized controller is developed, where an improved error compensation mechanism was established to eliminate the effect of filter error, and a hysteresis quantizer was employed to diminish the transmission rate. Furthermore, it is demonstrated that the designed controller assures that all signals of the closed-loop system are bounded in the mean square sense. Finally, two illustrative examples were provided to validate the effectiveness of the proposed control method.

Fixed-Time Adaptive Event-Triggered Guaranteed Performance Tracking Control of Nonholonomic Mobile Robots under Asymmetric State Constraints

A fixed-time adaptive guaranteed performance tracking control is investigated for a category of nonholonomic mobile robots (NMRs) under asymmetric state constraints. For the sake of favorable transient and steady-state properties of the system, a prescribed performance function (PPF) is introduced and a transform function is further constructed. Based on the backstepping technique, an asymmetric barrier Lyapunov function is formulated to ensure the tracking errors converge within a human-specified time. On the foundation of this, the occupation of communication channel is effectively reduced by assigning an event-triggered mechanism (ETM) with relative threshold to the process of controller design. By utilizing the proposed control strategy, the NMR is capable of implementing the enemy dislodging mission while the enemy can always be caught by the NMR and the collision would never be presented. Finally, two simulation experiments are given to verify the effectiveness of the proposed scheme.

Observer-based fuzzy tracking control of nonlinear systems with intermittent output constraints

This paper pays attention to the output tracking control problem of nonlinear systems where the output is uniquely measurable and subject to an intermittent constraint. A multiplicative transformation based on the shifting-replaying function is introduced together with barrier Lyapunov function to solve the intermittent output constraint. A state observer independent of the matching condition of control gains is designed via the non-uniform state transformation to relax the assumption of full-state measurability. By virtue of the estimated states, we propose an output feedback controller that only involves one fuzzy adaptive parameter to restrain the effect of unknown system nonlinearities. Through Lyapunov stability analysis, it is proved that the proposed controller can make all signals in the closed-loop system remain semi-globally uniformly ultimately bounded. Two simulation examples are provided to verify the effectiveness of the proposed method, and also the superiority over the existing ones.

Adaptive Neural Control for a Class of Random Fractional-Order Multi-Agent Systems with Markov Jump Parameters and Full State Constraints

Based on an adaptive neural control scheme, this paper investigates the consensus problem of random Markov jump multi-agent systems with full state constraints. Each agent is described by the fractional-order random nonlinear uncertain system driven by random differential equations, where the random noise is the second-order stationary stochastic process. First, in order to deal with the unknown functions with Markov jump parameters, a radial basis function neural network (RBFNN) structure is introduced to achieve approximation. Second, for the purpose of keeping the agents’ states from violating the constraint boundary, the tan-type barrier Lyapunov function is employed. By using the stochastic stability theory and adopting the backstepping technique, a novel adaptive neural control design method is presented. Furthermore, to cope with the differential explosion problem in the design course, the extended state observer (ESO) is developed instead of neural network (NN) approximation or command filtering techniques. Finally, the exponentially noise-to-state stability in the mean square is analyzed rigorously by the Lyapunov method, which guarantees the consensus of the considered multi-agent systems and all the agents’ outputs are bounded in probability. Two simulation examples are provided to verify the effectiveness of the suggested control strategy.

Nonlinear adaptive fault-tolerant control for full-car active suspension with velocity measurement errors and full-state constraints

This work concentrates on the vibration suppression problem for full-car active suspension in the presence of multiple actuator faults, velocity measurement errors, full-state constraints and external disturbances, simultaneously. Herein, a novel nonlinear fault-tolerant control (FTC) strategy is developed by the aid of the online estimation of fault parameters, and it can accommodate multiple actuator faults without the information obtained from costly fault detection and isolation mechanism. Then, the violation of full-state constraints can be prevented by introducing the log-type barrier Lyapunov function (BLF) into the adaptive backstepping method, and the problems related to velocity measurement errors of car body and external disturbances are handled under the robust control framework, so the complexity caused by state constraints, actuator faults and velocity measurement errors can thus be taken into consideration. The stability analysis is subsequently conducted with the assistance of Lyapunov theory, the boundness of all signals is confirmed, and the non-violation of the displacement and velocity constraints for car body motions is guaranteed. In the end, the performance of the proposed controller is evaluated under sinusoidal and random road surfaces. Importantly, the simulation results are presented to indicate the validity and advantages of the developed approach despite the existence of multiple actuator faults, velocity measurement errors and external disturbances.

Backstepping-Based Fuzzy Adaptive Stabilization of Reaction-Diffusion Equation With State Constraints.

This article proposes a stabilization scheme for a cascaded parabolic partial differential equation (PDE)-ordinary differential equation (ODE) system with state constraints. To begin, by employing the fuzzy-logic system (FLS) technique for the unknown nonlinear functions in the ODE subsystem, the influence of the nonlinearities is successfully eliminated. Then, the infinite and finite-dimensional backstepping methods are skillfully applied to the design of control schemes. Specifically, the infinite-dimensional backstepping transformation and its inverse are utilized to obtain a new PDE subsystem, which is convenient for control design. Next, based on the new target PDE-ODE cascade system, a novel first-order filter is subtly introduced into each step of the finite-dimensional backstepping-based controller design procedure to avoid the issue of "explosion of complexity." Furthermore, a novel barrier Lyapunov function (BLF) is constructed to guarantee that the states of the ODE subsystem do not violate the constraints. The controller designed in this article ensures that all signals in the closed-loop system are bounded and the states can converge to zero. Finally, a simulation example verifies that the proposed scheme can achieve satisfactory performance.

Disturbance Observer-Based Adaptive Intelligent Control of Marine Vessel With Position and Heading Constraint Condition Related to Desired Output.

This article studies the adaptive control about the geodetic fixed positions and heading of three-degree-of-freedom dual-propeller vessel. During the navigation of a vessel at sea, due to the unpredictable sea, on the one hand, it is important to ensure that the vessel can smoothly follow the desired geodesic fixed position and heading; on the other hand, when the sailing environment is harsh, it is even more important that the vessel can adapt to the desired geodesic fixed position and heading that change at any time for safe driving. Therefore, this article selects the time-varying function related to the desired geodesic fixed position and heading as the constraint condition, and the constraint condition will change in real time as the expected position and heading change. The design of the control strategy is difficult, and the designed control strategy will be more suitable for complex maritime navigation conditions. First, the article constructs a log-type barrier Lyapunov function. Second, by introducing an unknown external disturbance observer, the external disturbances caused by the environment that may be encountered during the vessel's voyage can be observed. Then, combined with the backstepping algorithm, a neural network (NN) control strategy and adaptive law are designed. Among them, for the uncertain function in the process of designing the control strategy, the NN is used to approximate it. Furthermore, through the Lyapunov stability analysis, it is shown that applying the designed control strategy to the vessel system in this article can ensure that the system is closed-loop stable. The final simulation experiment shows the effectiveness of the designed control strategy.

Adaptive RBF Neural Network Tracking Control of Stochastic Nonlinear Systems with Actuators and State Constraints

This paper investigates the adaptive neural network (NN) tracking control problem for stochastic nonlinear systems with multiple actuator constraints and full-state constraints. The issue of system full-state constraints is tackled by a generalized barrier Lyapunov function (GBLF), and the output constraints of the system are considered to be in the form of time-varying functions, which are more in line with the needs of real physical systems. The NN approximation technique is utilized to overcome the influence of the uncertainty term on controller design due to randomness. Based on the backstepping technique, a neural adaptive fixed-time tracking control strategy is designed. Under the designed control strategy, the tracking accuracy of the controlled system can reach the expectation in a fixed time. The multi-actuator constraints are converted into a generalized mathematical model to simplify the controller design process. Using the characteristics of the hyperbolic tangent function, a new function called practical virtual control signal is designed using the virtual control signal as the input. Due to the saturation constraint property of the hyperbolic tangent function, it is theoretically ensured that no state of the system exceeds the constraints through to the new form of the virtual controller. Using the adaptive controller constructed in this paper, the controlled system is semi-global fixed-time stabilized in probability (SGFSP). Finally, the effectiveness of the proposed control strategy is further verified by simulation examples.

Adaptive optimized backstepping tracking control for full‐state constrained nonlinear strict‐feedback systems without using barrier Lyapunov function method

AbstractIn this article, the problem of adaptive optimal tracking control is studied for nonlinear strict‐feedback systems. While not directly measurable, the states of these systems are subject to both time‐varying and asymmetric constraints. Bypassing the conventional barrier Lyapunov function method, the constrained system is transformed into its unconstrained counterpart, thereby obviating the need for feasibility conditions. A specially designed reinforcement learning (RL) algorithm, featuring an observer‐critic‐actor architecture, is deployed in an adaptive optimal control scheme to ensure the stabilization of the converted unconstrained system. Within this architecture, the observer estimates the unmeasurable system states, the critic evaluates the control performance, and the actor executes the control actions. Furthermore, enhancements to the RL algorithm lead to relaxed conditions of persistent excitation, and the design methodology for the observer overcomes the restrictions imposed by the Hurwitz equation. The Lyapunov stability theorem is applied for two primary purposes: to ascertain the boundedness of all signals within the closed‐loop system, and to ensure the accuracy of the output signal in tracking the desired reference trajectory. Finally, numerical and practical simulations are provided to corroborate the effectiveness of the proposed control strategy.

Event-Triggered Adaptive Neural Network Control for State-Constrained Pure-Feedback Fractional-Order Nonlinear Systems with Input Delay and Saturation

In this research, the adaptive event-triggered neural network controller design problem is investigated for a class of state-constrained pure-feedback fractional-order nonlinear systems (FONSs) with external disturbances, unknown actuator saturation, and input delay. An auxiliary compensation function based on the integral function of the input signal is presented to handle input delay. The barrier Lyapunov function (BLF) is utilized to deal with state constraints, and the event-triggered strategy is applied to overcome the communication burden from the limited communication resources. By the utilization of a backstepping scheme and radial basis function neural network, an adaptive event-triggered neural state-feedback stabilization controller is constructed, in which the fractional-order dynamic surface filters are employed to reduce the computational burden from the recursive procedure. It is proven that with the fractional-order Lyapunov analysis, all the solutions of the closed-loop system are bounded, and the tracking error can converge to a small interval around the zero, while the state constraint is satisfied and the Zeno behavior can be strictly ruled out. Two examples are finally given to show the effectiveness of the proposed control strategy.

Finite-Time Adaptive Event-Triggered Control for Full States Constrained FONSs with Uncertain Parameters and Disturbances

This article focuses the event-triggered adaptive finite-time control scheme for the states constrained fractional-order nonlinear systems (FONSs) under uncertain parameters and external disturbances. The backstepping scheme is employed to construct the finite-time controller via a series of barrier Lyapunov function (BLF) to solve that all the state constraints are not violated. Different from the trigger condition with fixed value, the event-triggered strategy is applied to overcome the communication burden of controller caused by the limited communication resources. By utilizing fractional-order Lyapunov analysis, all variables in the resulted system are proven to be bounded, and the tracking error converges to the small neighborhood around origin in finite time and without the Zeno behavior. Finally, the effectiveness of the proposed control scheme is verified by the simulation analysis of a bus power system.

Full-state constraints and input backlash–based neural network control of a 2-DOF helicopter system

This paper introduces an adaptive neural network compensatory control approach designed for a 2-degree-of-freedom (2-DOF) helicopter system facing challenges such as input backlash and state constraints. The proposed methodology leverages a radial basis function (RBF) neural network to effectively approximate system uncertainties, mitigating the impact of nonlinear dynamics on control performance. To address the presence of nonlinear input backlash, a compensation technique is introduced to enhance the smoothness of input signals. In addition, for enhanced system safety, a barrier Lyapunov function is integrated to impose restrictions on position and velocity states, resulting in constrained control. Through a rigorous analysis using the Lyapunov direct method, this paper demonstrates the effectiveness of the proposed approach in achieving bounded stability of the system. The validation of the approach is further established through the presentation of simulation and experimental results, showcasing its effectiveness and feasibility in real-world applications.

Predefined-time stabilization of stochastic nonlinear systems with application to UAVs

The paper presents a new Lyapunov-type predefined-time stabilization control algorithm for stochastic high-order nonlinear systems with asymmetric output constraints. In contrast to stochastic finite-time and fixed-time stabilization, the average value of the settling-time function for stochastic predefined-time stabilization control is independent of both the initial value and the control factors. To mitigate the significant uncertainties arising from the asymmetric output constraint, a tan-type barrier Lyapunov function is formulated. Furthermore, by harnessing the previously mentioned barrier Lyapunov function and integrating the power integrator technique, a controller design strategy is formulated based on the backstepping method. The rigorous analysis in this study proves that the designed controller ensures both the attainment of predefined-time convergence of the system states to the origin in probability and the satisfaction of the output constraint. Finally, an example of a roll angle subsystem for quadrotor UAVs and a numerical illustration are presented to corroborate the theoretical analysis.

Adaptive fault-tolerant optimal control for hypersonic vehicles with state constrains based on adaptive dynamic programming

This paper addresses a novel adaptive fault-tolerant control (FTC) design based on adaptive dynamic programming (ADP) technique for hypersonic vehicle (HSV) subject to actuator fault and state constrains. The total control input is constructed by the combination of backstepping-based adaptive FTC and the ADP-based adaptive optimal control to improve the tracking performance and fault-tolerant capacity. By introducing a barrier Lyapunov function (BLF) to deal with the state constraints, a backstepping-based FTC scheme is designed to transform the tracking control problem into an equivalent optimal control problem. Subsequently, an adaptive optimal control strategy is developed by using the ADP technique to provide a supplementary control action. A critic network is constructed to solve the Hamilton–Jacobi–Bellman (HJB) equation online. The convergence properties of the closed-loop system are developed by utilizing the Lyapunov stability theory. Finally, simulation results are carried out to illustrate the effectiveness and efficiency of the proposed adaptive ADP-based FTC strategy.

Vision-based finite-time prescribed performance control for uncooperative UAV target-tracking subject to field of view constraints

This paper presents a vision-based finite-time prescribed performance controller for unmanned aerial vehicle (UAV) tracking of uncooperative aerial targets. The relative states between UAV and target are estimated by an onboard monocular camera. The inability of visual measurements to accurately determine the initial state of the target renders conventional prescribed performance controllers ineffective in such situations. As a result, it becomes essential to address the problem of prescribed performance control under conditions of uncertain initial values By utilizing an auxiliary transforming function, an Asymmetric Barrier Lyapunov Function (ABLF) and a finite-time prescribed performance function, a robust adaptive controller based on backstepping framework is proposed to deal with state constraints under unknown initial tracking conditions. It is proved that, the closed-loop relative position is capable of reaching the prescribed performance bound before the preset transforming time and converging to the prescribed steady-state error before a finite setting time. Simulation examples are provided to illustrated the effectiveness of the proposed tracking algorithm.