This article addresses the challenging problem of constrained optimal control for human–machine systems subject to external disturbances and the bounded rationality of the human operator. To this end, a novel game-theoretic framework is proposed. Unlike monolithic game formulations, the framework uniquely disaggregates the control problem by transforming it into a multifaceted game via logarithmic barrier functions (BFs): it models human–machine cooperation as a positive-sum game oriented toward shared objectives, and disturbance rejection as a zero-sum game tailored for robustness enhancement. To capture the nonideal human decision-making, we integrate the level-$k$reasoning framework to model the operator’s bounded cognitive dynamics. The corresponding coupled Hamilton–Jacobi–Isaacs (HJI) equations for this human–machine game are derived, and critically, a rigorous proof of global asymptotic stability (GAS) for the transformed system is provided, establishing a solid theoretical foundation. For online implementation without requiring prior knowledge of the system dynamics, we develop a resource-efficient learning architecture based on the adaptive dynamic programming (ADP) and a novel dynamic event-triggered mechanism (DETM). A key feature of this architecture is a fuzzy logic-based module for dynamic authority allocation, which adaptively adjusts control sharing in real time. Rigorous analysis demonstrates that all signals in the closed-loop system are uniformly ultimately bounded and that Zeno behavior is precluded. Simulation results are presented to validate the effectiveness and superiority of the proposed control strategy.

In this article, for the nonzero-sum (NZS) differential games problem of uncertain nonlinear systems with state constraints, an adaptive robust stabilization scheme based on the control barrier function (CBF) is presented under the influence of random disturbances and control input matrix uncertainty. To deal with the impact of uncertainty on the system, the nominal system of the original system is adopted and the cost functions associated with each player are appropriately chosen to convert the robust regulation problem of multiplayer differential games into an optimal regulation problem. Furthermore, the purpose of combining the cost function relevant to each player with the CBF is to make the system states evolve in the safe area. Different from the classical actor–critic dual neural network (NN), each player only needs a critic NN to approach the corresponding cost function without the restriction of the initial stabilizing control. Combined with the Lyapunov stability theory, under the combined influence of random disturbances and state constraints, the state and critic NN weights of the closed-loop system are guaranteed to be uniformly ultimately bounded (UUB). Finally, two simulation examples are used to verify the validity of the presented scheme.

It is well known that the two-player zero-sum differential game problem of the continuous-time nonlinear system relies on the solution of the Hamilton-Jacobi-Isaacs equation, which is a nonlinear partial differential equation that is difficult or impossible to solve. In this paper, a new model-free adaptive dynamic programming algorithm is developed for solving online the Hamilton-Jacobi-Isaacs equation for continuous-time nonlinear system with the fully unknown knowledge of the system dynamics. First, a simultaneous policy iteration algorithm will be given, which can solve the Hamilton-Jacobi-Isaacs equation in an off-line sense, in which the fully knowledge of the system dynamics is required. Second, based on the simultaneous policy iteration algorithm, a new model-free adaptive dynamic programming algorithm is developed for solving online the Hamilton-Jacobi-Isaacs equation, in which the fully knowledge of the system dynamics is not required. Finally, a numerical example is given to demonstrate the convergence and effectiveness of the proposed scheme.

In this paper, a novel iterative two-stage dual heuristic programming (DHP) is proposed to solve the optimal control problems for a class of discrete-time switched nonlinear systems subject to actuators saturation. First, a novel nonquadratic performance functional is introduced to confront control constraints of the saturating actuator. Then, the iterative two-stage DHP algorithm is developed to solve the Hamilton-Jacobi-Bellman (HJB) equation of the switched system with the saturating actuator. Moreover, the convergence and optimality of the two-stage DHP algorithm are strictly proven. To implement this algorithm efficiently, there are two neural networks used as parametric structure to approximate the costate function and the corresponding control law, respectively. Finally, simulation results are given to verify the effectiveness of the proposed algorithm.

The problem of H∞ state feedback control of affine nonlinear discrete-time systems with unknown dynamics is investigated in this paper. An online adaptive policy learning algorithm (APLA) based on adaptive dynamic programming (ADP) is proposed for learning in real-time the solution to the Hamilton-Jacobi-Isaacs (HJI) equation, which appears in the H∞ control problem. In the proposed algorithm, three neural networks (NNs) are utilized to find suitable approximations of the optimal value function and the saddle point feedback control and disturbance policies. Novel weight updating laws are given to tune the critic, actor, and disturbance NNs simultaneously by using data generated in real-time along the system trajectories. Considering NN approximation errors, we provide the stability analysis of the proposed algorithm with Lyapunov approach. Moreover, the need of the system input dynamics for the proposed algorithm is relaxed by using a NN identification scheme. Finally, simulation examples show the effectiveness of the proposed algorithm.

This paper proposes an on-line near-optimal control scheme based on capabilities of neural networks (NNs), in function approximation, to attain the on-line solution of optimal control problem for nonlinear discrete-time systems. First, to solve the Hamilton-Jacobi-Bellman (HJB) equation forward-in-time appearing in the optimal control problem, two neural networks are used to approximate the cost function and to compute the optimal control policy, respectively. And then, according to the Bellman's optimality principle and the adaptive technology, the on-line weight updating laws for the critic network and action network are derived, respectively. Further, considering NNs approximative errors, the stability analysis of the closed-loop system is demonstrated by Lyapunov theory. At last, a numerical example is provided to demonstrate the effectiveness of the proposed method.