Robust Adaptive Neural Network-Based Backstepping Tracking for Second-Order Euler-Lagrange Systems with Unknown Parameters

Actor–Critic Predefined-Time Adaptive Tracking Control for Partially Unknown Euler–Lagrange Systems

This paper studies predefined-time trajectory tracking for Euler–Lagrange systems with composite uncertainties encompassing partially known dynamics, parametric variations, and bounded disturbances. A two-step backstepping architecture is developed. In Step 1, a predefined-time virtual control law is constructed for the position subsystem. In Step 2, an energy-based torque controller is designed for the velocity subsystem using nominal model compensation, adaptive parameter estimation, actor neural network approximation of residual dynamics, a critic network for performance-oriented learning, and a continuous robust term. A rigorous Lyapunov analysis shows that all closed-loop signals are uniformly ultimately bounded and that the tracking errors converge to a computable residual set within a predefined time for all initial conditions contained in the selected compact set. Comparative simulations on a 2-DOF planar manipulator demonstrate that the proposed method provides faster convergence and improved steady-state tracking accuracy than both a PID baseline and a classical Slotine–Li adaptive baseline, while respecting the predefined-time bound.

Global Fixed‐Time Exact Tracking for Uncertain High‐Order Nonlinear Systems With Quantized Input

This article investigates the problem of global fixed‐time (FT) exact tracking control for high‐order nonlinear systems (HONSs) characterized by input quantization and external disturbances. Most existing approaches for handling unknown nonlinearities rely on radial basis function neural networks (RBFNNs) or fuzzy logic systems (FLS), which typically ensure only ultimately semi‐globally bounded stability. To address this limitation, an FT output tracking control strategy with prescribed performance is developed by integrating the barrier Lyapunov function (BLF) technique with an auxiliary power integrator and a bounded estimation method. The proposed scheme guarantees that the tracking error converges to zero within an FT, while ensuring the global boundedness of all closed‐loop signals and the strict satisfaction of state constraints. Moreover, the control design does not require function approximation, parameter identification, or command filtering, and effectively avoids the singularity problem. The effectiveness of the proposed method is validated through two numerical examples.

ADAPTIVE ROBUST CONTROL OF SECOND-ORDER NONLINEAR SYSTEMS WITH UNDETERMINED PARAMETERS BASED ON THE BACKSTEPPING METHOD

The article investigates the control problem for a class of nonlinear strict-feedback systems with uncertain parameters and external disturbances. The main objective of the study is to develop an adaptive control algorithm that ensures system robustness and guarantees semi-global uniform boundedness of all signals. To compensate for unknown dynamic functions, the recursive synthesis method, namely backstepping, is combined with adaptive approximation laws. The semi-global stability of the closed-loop system is analytically proven using the Lyapunov function method, and it is shown that the tracking error converges to a bounded neighborhood of zero. As a result of the study, the control problem is formulated for second-order nonlinear strict-feedback systems with uncertain parameters and bounded external disturbances, and it is constructively demonstrated that this problem can be solved by means of adaptive robust control. In particular, the existence of a control law ensuring tracking of a given reference trajectory is established on the basis of the backstepping method, and a step-by-step synthesis procedure for its construction is proposed: first, a virtual control is designed, and then adaptive control laws are defined. Using the Lyapunov function method, the stability of the closed-loop system, the uniform boundedness of all signals, and the convergence of the tracking error to a bounded neighborhood of zero are proven. Thus, the work proposes not only a specific control algorithm, but also a theoretical and constructive approach that substantiates the solvability of the control problem for a class of uncertain nonlinear systems. To verify the proposed method in practice, numerical simulation was carried out for the dynamics of a single-link robotic manipulator. The results showed that the proposed adaptive backstepping algorithm preserves the bounded motion mode of the system under sudden changes in the load parameter and in the presence of external disturbances. A numerical comparison was performed using the MSE, maximum error, and settling time criteria, and it was found that the control performance depends on the choice of algorithm parameters. The proposed method can be applied to the control of mechatronic and robotic systems with parametric uncertainty; however, additional tuning of the control gains is required before practical implementation.

Lyapunov-Based Stability Analysis of Adaptive Neural-Network Controllers for Nonlinear Perturbed Systems

A Lyapunov-based framework for stability analysis and synthesis of adaptive neural-network (NN) controllers for a class of uncertain second-order nonlinear systems (SNS) with bounded external perturbations and unmodelled dynamics is presented. Online learning is employed for the reconstruction of the plant nonlinearity with the use of a radial-basis-function (RBF) network whose weights are adapted using a direct adaptation law deduced from a single composite Lyapunov function. The proposed controller couples the weight update to a persistent robustifying action, while the closed-loop stability is guaranteed throughout the learning transient, in contrast to schemes that guarantee stability after learning has converged. Using a composite Lyapunov function in the filtered tracking error and the weight-estimation error, we prove that all closed-loop signals are uniformly ultimately bounded (UUB) and that the tracking error converges to an explicitly characterized residual set whose radius is governed by the network reconstruction accuracy, the disturbance bound and the design gains. A σ-modification ensures parameter boundedness without persistency of excitation, and a robustness theorem shows that bounded parametric perturbations of the plant preserve stability and enlarge the ultimate bound only gradually (a graceful degradation, rather than a loss of the guarantee). The open-loop plant (a forced double-well Duffing oscillator) is characterized by means of equilibrium and Jacobian analyses. A bifurcation diagram and the largest Lyapunov exponent are presented, which show a chaotic regime (with λ1≈0.17). Numerical experiments indicate that the proposed controller is able to suppress the chaotic motion with a small value of the ultimate bound, and maintain a smooth reference motion with a small and constant RMS error of order 10−3, which is approximately 26 times less than the RMS error obtained with a tuned fixed-gain baseline, and the theoretical dependence of the ultimate bound on the disturbance and the design gains is confirmed by sensitivity sweeps.

Trajectory-tracking control of the UR10 manipulator based on radial basis function neural network and super-twisting sliding mode

Abstract. This paper proposes a composite control strategy combining a radial basis function (RBF) neural network and super-twisting sliding-mode control for high-precision trajectory tracking of the UR10 manipulator under parameter variations, nonlinear friction, and external disturbances. An RBF network is employed to approximate the lumped unknown nonlinear dynamics online, thereby reducing the equivalent disturbance upper bounds. Simultaneously, a super-twisting robust control term is introduced to compensate for approximation residuals and remaining disturbances, which ensures system robustness while effectively suppressing conventional sliding-mode chattering. Furthermore, a projection-based adaptive law is designed to guarantee the strict boundedness of the network weights. Based on Lyapunov theory, the finite-time convergence of the sliding variable and tracking error is proven. Simulations on a 6-degree-of-freedom UR10 manipulator – incorporating mass/inertia perturbations and Coulomb-viscous friction – demonstrate that the proposed method achieves high-precision tracking with small steady-state errors and smooth, chattering-free control torques, verifying its effectiveness and engineering applicability.

Neural Network-Based Adaptive Practical Prescribed-Time Control for a Helicopter System with State Constraints