Optimal control for discrete-time affine non-linear systems using general value iteration

In this study, the authors propose a novel adaptive dynamic programming scheme based on general value iteration (VI) to obtain near optimal control for discrete-time affine non-linear systems with continuous state and control spaces. First, the selection of initial value function is different from the traditional VI, and a new method is introduced to demonstrate the convergence property and convergence speed of value function. Then, the control law obtained at each iteration can stabilise the system under some conditions. At last, an error-bound-based condition is derived considering the approximation errors of neural networks, and then the error between the optimal and approximated value functions can also be estimated. To facilitate the implementation of the iterative scheme, three neural networks with Levenberg-Marquardt training algorithm are used to approximate the unknown system, the value function and the control law. Two simulation examples are presented to demonstrate the effectiveness of the proposed scheme.