Generalized value iteration for discounted optimal control with stability analysis

Abstract

Abstract In this work, the generalized value iteration with a discount factor is developed for optimal control of discrete-time nonlinear systems, which is initialized with a positive definite value function rather than zero. The convergence analysis of the discounted value function sequence is provided. The condition for the discount factor is given to guarantee the stability of the controlled plants. With this operation, the iterative control policy that asymptotically stabilizes the closed-loop system can be determined. The introduction of a discount factor has eased some conditions that the system dynamics and the initialization of the generalized value iteration need to fulfill. It is not required that the initial control policy is stabilizing. In the iteration process, under some conditions, if the system is asymptotically stable at the current iteration, then it can be guaranteed that the iterative control policies after this current iteration step also are stabilizing. It is convenient and practical to evaluate the asymptotic stability of the closed-loop system using the iterative control policy. A numerical example with physical background is carried out to validate the present results.