Asymptotic tracking by a reinforcement learning-based adaptive critic controller

Robust Integral of Neural Network and Error Sign Control of MIMO Nonlinear Systems

This paper presents a novel state-feedback control scheme for the tracking control of a class of multi-input multioutput continuous-time nonlinear systems with unknown dynamics and bounded disturbances. First, the control law consisting of the robust integral of a neural network (NN) output plus sign of the tracking error feedback multiplied with an adaptive gain is introduced. The NN in the control law learns the system dynamics in an online manner, while the NN residual reconstruction errors and the bounded disturbances are overcome by the error sign signal. Since both of the NN output and the error sign signal are included in the integral, the continuity of the control input is ensured. The controller structure and the NN weight update law are novel in contrast with the previous effort, and the semiglobal asymptotic tracking performance is still guaranteed by using the Lyapunov analysis. In addition, the NN weights and all other signals are proved to be bounded simultaneously. The proposed approach also relaxes the need for the upper bounds of certain terms, which are usually required in the previous designs. Finally, the theoretical results are substantiated with simulations.

Approximate N-Player Nonzero-Sum Game Solution for an Uncertain Continuous Nonlinear System

An approximate online equilibrium solution is developed for an N -player nonzero-sum game subject to continuous-time nonlinear unknown dynamics and an infinite horizon quadratic cost. A novel actor-critic-identifier structure is used, wherein a robust dynamic neural network is used to asymptotically identify the uncertain system with additive disturbances, and a set of critic and actor NNs are used to approximate the value functions and equilibrium policies, respectively. The weight update laws for the actor neural networks (NNs) are generated using a gradient-descent method, and the critic NNs are generated by least square regression, which are both based on the modified Bellman error that is independent of the system dynamics. A Lyapunov-based stability analysis shows that uniformly ultimately bounded tracking is achieved, and a convergence analysis demonstrates that the approximate control policies converge to a neighborhood of the optimal solutions. The actor, critic, and identifier structures are implemented in real time continuously and simultaneously. Simulations on two and three player games illustrate the performance of the developed method.

Online learning control using adaptive critic designs with sparse kernel machines

In the past decade, adaptive critic designs (ACDs), including heuristic dynamic programming (HDP), dual heuristic programming (DHP), and their action-dependent ones, have been widely studied to realize online learning control of dynamical systems. However, because neural networks with manually designed features are commonly used to deal with continuous state and action spaces, the generalization capability and learning efficiency of previous ACDs still need to be improved. In this paper, a novel framework of ACDs with sparse kernel machines is presented by integrating kernel methods into the critic of ACDs. To improve the generalization capability as well as the computational efficiency of kernel machines, a sparsification method based on the approximately linear dependence analysis is used. Using the sparse kernel machines, two kernel-based ACD algorithms, that is, kernel HDP (KHDP) and kernel DHP (KDHP), are proposed and their performance is analyzed both theoretically and empirically. Because of the representation learning and generalization capability of sparse kernel machines, KHDP and KDHP can obtain much better performance than previous HDP and DHP with manually designed neural networks. Simulation and experimental results of two nonlinear control problems, that is, a continuous-action inverted pendulum problem and a ball and plate control problem, demonstrate the effectiveness of the proposed kernel ACD methods.