In this article, an online adaptive optimal control algorithm based on adaptive dynamic programming is developed to solve the multiplayer nonzero-sum game (MP-NZSG) for discrete-time unknown nonlinear systems. First, a model-free coupled globalized dual-heuristic dynamic programming (GDHP) structure is designed to solve the MP-NZSG problem, in which there is no model network or identifier. Second, in order to relax the requirement of systems dynamics, an online adaptive learning algorithm is developed to solve the Hamilton-Jacobi equation using the system states of two adjacent time steps. Third, a series of critic networks and action networks are used to approximate value functions and optimal policies for all players. All the neural network (NN) weights are updated online based on real-time system states. Fourth, the uniformly ultimate boundedness analysis of the NN approximation errors is proved based on the Lyapunov approach. Finally, simulation results are given to demonstrate the effectiveness of the developed scheme.

In this paper, a new iterative adaptive dynamic programming (ADP) algorithm is developed to solve optimal impulsive control problems for infinite horizon discrete-time nonlinear systems. Considering the constraint of the impulsive interval, in each iteration, the iterative impulsive value function under each possible impulsive interval is obtained, and then the iterative value function and iterative control law are achieved. A new convergence analysis method is developed which proves an iterative value function to converge to the optimum as the iteration index increases to infinity. The properties of the iterative control law are analyzed, and the detailed implementation of the optimal impulsive control law is presented. Finally, two simulation examples with comparisons are given to show the effectiveness of the developed method.

In this article, off-policy reinforcement learning (RL) algorithm is established to solve the discrete-time N-player nonzero-sum (NZS) games with completely unknown dynamics. The N-coupled generalized algebraic Riccati equations (GARE) are derived, and then policy iteration (PI) algorithm is used to obtain the N-tuple of iterative control and iterative value function. As the system dynamics is necessary in PI algorithm, off-policy RL method is developed for discrete-time N-player NZS games. The off-policy N-coupled Hamilton-Jacobi (HJ) equation is derived based on quadratic value functions. According to the Kronecker product, the N-coupled HJ equation is decomposed into unknown parameter part and the system operation data part, which makes the N-coupled HJ equation solved independent of system dynamics. The least square is used to calculate the iterative value function and N-tuple of iterative control. The existence of Nash equilibrium is proved. The result of the proposed method for discrete-time unknown dynamics NZS games is indicated by the simulation examples.

This paper establishes an approximate optimal critic learning algorithm based on single neural network (NN) policy iteration (PI) aiming at solving for continuous-time (CT) 2-player zero-sum games (ZSGs). In fact, we have to face the problem that the errors will disturb the dynamics and in turn identifying dynamics will generate errors. In order to prevent the effect of errors, in this paper, a single NN-based online PI algorithm is developed for the CT system, which is disturbed nonlinear ZSG. With plenty of online data, the Hamilton-Jacobi-Isaacs equation can be solved without complete dynamics. Then by the least-squares method, we can obtain the NN weights. Moreover, in the process of dealing with the undisturbed system, we find the way that obtains NN weights in this paper is equal to the way that obtains the optimal solution by the Gauss-Newton method. Based on the convergence of the Gauss-Newton method, we can efficiently obtain the optimal controller for the undisturbed system by utilizing online data. After getting the controller of the undisturbed system, it is time to take disturbance into consideration, so that we design a robust control pair to overcome the disturbance. In order to demonstrate the effectiveness of this algorithm, we design a set of simulations. The results verify that we can solve the disturbed nonlinear ZSG by this algorithm.

Taking safety and performance into consideration, the state and control input of the actual engineering system are often constrained. For this kind of problem, this article puts forward an online dual event-triggered (ET) adaptive dynamic programming (ADP) optimal control algorithm for a class of nonlinear systems with constrained state and input. First, the original system is transformed into another system through the barrier function, after that, a suitable value function with a nonquadratic utility function is designed to obtain the optimal control pair. In addition, on the premise of the asymptotic stability of the system, the trigger condition is devised, and the intersampling time analysis is proved that the algorithm can avoid the Zeno phenomenon. What is more, the critic, action, and disturbance neural networks (NNs) are trained to approximate value function and control sequences, subsequently, the approximation error is proved to be uniformly ultimately boundedness (UUB). Finally, two comparative experiments based on the robot arm model are simulated to verify that the algorithm can make control policies update only when the system has the requirement and keep satisfactory control effect, which can effectively decrease the number of data transfers and reduce the calculation burden.