Stability analysis of heuristic dynamic programming algorithm for nonlinear systems

In this paper, a value-iteration based heuristic dynamic programming (HDP) algorithm is developed to solve the optimal control for the continuous time affine nonlinear systems. First, a rigorous convergence proof of the HDP algorithm is given. Second, stability issues of the HDP algorithm for nonlinear systems are investigated. It is commonly believed that the main drawback of the HDP algorithm is that only the limit function of the iterative control sequence is proved to be stabilized, thus infinite iterations are executed. To confront this problem, we present a novel stability result for the HDP algorithm, which indicates that the resulting iterative control laws after finite iterations can guarantee the closed-loop stability. A similar stability result is also obtained for the discrete time nonlinear systems. Therefore, the practicality of the HDP algorithm is greatly improved. Single neural network (NN) structure is employed to implement the algorithm. It should be pointed that the algorithm can be implemented without knowing the internal dynamics of the systems. Finally, two numerical examples are given to demonstrate the effectiveness of the developed methods.