Adaptive Stability-Constrained Neural Differential Equations for Controlled Dynamical Systems with Unknown Inputs

Learning Stable Controlled Dynamical Systems via Input-Contraction Neural Differential Models

This paper derives sufficient conditions for input-dependent contraction and formally establish an input-to-state contraction property under bounded external excitations, and develops a novel deep learning framework that seamlessly incorporates time-varying control inputs while ensuring incremental exponential convergence via input-dependent contraction regularization.

A matched-integrator evaluation of Hamiltonian neural networks on pendulum and Kepler dynamics

Hamiltonian Neural Networks (HNNs) parameterize conservative dynamics through a learned scalar Hamiltonian, providing an architectural prior that is absent from generic vector-field neural networks. We evaluate this prior under a controlled protocol in which an HNN and a parameter-matched feedforward baseline are trained on the same RK4-generated trajectories, use the same central-difference derivative targets and optimization settings, and are integrated at inference with the same RK4 scheme. Results are reported over five independent training seeds. On the nonlinear pendulum, the HNN reduces mean energy drift by 42-fold and mean trajectory MSE by 15.8-fold at T = 100, approximately 16 pendulum periods. Its energy drift also remains bounded and exhibits substantially lower seed-to-seed variability than the standard-network baseline. An energy-stratified analysis shows that the difference becomes more pronounced as trajectories explore more nonlinear regions of phase space. As an additional diagnostic, we examine an explicit St\"ormer--Verlet-style rollout of the learned HNN. Because the learned Hamiltonian is not constrained to the separable form H(q,p) = T(p) + V(q), the standard symplecticity guarantee of velocity Verlet does not directly apply. We further apply the same matched-integrator protocol to the three-dimensional Kepler two-body problem. The HNN again exhibits lower trajectory, energy, and angular-momentum drift than the parameter-matched baseline. These experiments provide a controlled study of how Hamiltonian parameterization affects long-horizon prediction and physical consistency across two conservative dynamical systems.

Iterative State- and Control-Dependent Model Predictive Control: A Jacobian-Free Formulation for Constrained Nonlinear Systems

This paper presents an iterative model predictive control algorithm that stabilizes constrained nonlinear systems without evaluating a single plant derivative. By factoring the exact nonlinear dynamics into a pseudo-linear form using state- and control-dependent coefficients (SCDCs), we replace the standard nonconvex optimization with a sequence of constrained linear-quadratic programs. Refreezing the coefficient matrices along the previously predicted trajectory drives the iteration. Near the origin, we prove this sequence contracts to a unique fixed point. We explicitly bound the number of iterations required to reach any stopping tolerance, and we quantify the distance from the fixed point to a true Karush-Kuhn-Tucker point, showing this optimality gap vanishes quadratically as the state approaches the origin. Inflating the discrete algebraic Riccati equation generates terminal ingredients that guarantee recursive feasibility and asymptotic stability, even when the solver terminates early. We adapt the terminal penalty online, proving it remains uniformly bounded, and we secure output feedback through the block-observable canonical form, which extracts the exact system state directly from past inputs and outputs. Retaining the block-banded structure of the subproblem forces the computational cost to scale linearly with the horizon length $\ell$. This $O(\ell)$ complexity matches the iterative linear quadratic regulator (iLQR) but sharply undercuts the $O(\ell^3)$ scaling of dense sequential quadratic programming (SQP). Numerical studies on a saturated quadrotor, a nonholonomic integrator, and a nonminimum-phase plant illustrate the theoretical bounds and map how the algorithm compares with iLQR, SQP, and linear-parameter-varying MPC.

Neural Learning Control for State Constrained Strict-Feedback Systems: A State Predictor Method

This paper concentrates on the neural learning control (NLC) problem for strict-feedback nonlinear systems (SFNSs) despite the presence of time-varying state constraints. A constraint-free system is obtained from the original constrained system through the application of a nonlinear transformed function (NTF). Based on the transformed constraint-free system, a state predictor is constructed for each subsystem. Then, a decoupled neural network weight updating law is developed based on the prediction error instead of the popular used tracking error, which facilitates the convergence of neural weights. Furthermore, by combining the first-order filter and backstepping method, an adaptive neural controller is established to assure the predetermined state constraints. Moreover, different from conventional system decomposition strategy, an effective lemma is given to address the difficulty in verifying the recurrent property of neural inputs, thereby promoting exponential convergence of neural weights. In virtue of the converged neural weights, i.e., learned knowledge, an NLC scheme is constructed, which can not only guarantee the predetermined state constraints, but also upgrade control performance and moderate online computational burden. Finally, simulations are conducted to exhibit the validity and correctness of the presented method. Note to Practitioners—In this paper, we investigate the NLC problem for a kind of SFNSs with time-varying state constraints, and the considered system models have been diffusely applied in the engineering field, such as marine surface vessels, robotic manipulators and so on. Note that two scenarios are widespread in practical applications: 1) system states need to satisfy predetermined constraint conditions; 2) neural weights cannot converge exponentially. Therefore, an NTF is introduced to settle the state constraints problem. Subsequently, an effective neural network weight updating law based on the prediction error, along with a supporting lemma, is developed to facilitate the weights convergence. On that basis, an NLC strategy is proposed by combining the converged neural weights. Simulations show that the developed strategy can efficaciously upgrade control performance and save online computational resources, which has good application prospect.

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

This paper proposes a robust adaptive tracking control scheme for a class of second-order Euler–Lagrange systems with completely unknown parameters and nonlinear dynamics. System uncertainties, including unmodeled dynamics, parametric variations, and external disturbances, are formulated as a time-varying lumped perturbation. Radial Basis Function Neural Networks (RBFNNs) approximate the unknown state-dependent nonlinear component within the perturbation bound, while adaptive laws estimate the unknown bounding constants of input-dependent terms and disturbances. By integrating backstepping with a $\sigma$-modification mechanism, a continuous adaptive control law is developed that eliminates chattering typically caused by discontinuous robust terms. Lyapunov analysis proves that all closed-loop signals are uniformly ultimately bounded, achieving asymptotic trajectory tracking with smooth control inputs. Simulations on an underactuated Unmanned Surface Vehicle (USV) under complete model uncertainty and environmental disturbances validate the effectiveness and superiority of the proposed method.

Koopman-based model predictive control for nonlinear systems with bounded model uncertainty