The Boundedness Conditions for Model-Free HDP(lambda)

This paper provides the stability analysis for a model-free action-dependent heuristic dynamic programing (HDP) approach with an eligibility trace long-term prediction parameter (lambda). HDP(lambda) learns from more than one future reward. Eligibility traces have long been popular in Q-learning. This paper proves and demonstrates that they are worthwhile to use with HDP. In this paper, we prove its uniformly ultimately bounded (UUB) property under certain conditions. Previous works present a UUB proof for traditional HDP [HDP(lambda = 0)], but we extend the proof with the lambda parameter. By using Lyapunov stability, we demonstrate the boundedness of the estimated error for the critic and actor neural networks as well as learning rate parameters. Three case studies demonstrate the effectiveness of HDP(lambda). The trajectories of the internal reinforcement signal nonlinear system are considered as the first case. We compare the results with the performance of HDP and traditional temporal difference [TD(lambda)] with different lambda values. The second case study is a single-link inverted pendulum. We investigate the performance of the inverted pendulum by comparing HDP(lambda) with regular HDP, with different levels of noise. The third case study is a 3-D maze navigation benchmark, which is compared with state action reward state action, Q(lambda), HDP, and HDP(lambda). All these simulation results illustrate that HDP(lambda) has a competitive performance; thus this contribution is not only UUB but also useful in comparison with traditional HDP.