Fast Approximate Dynamic Programming for Input-Affine Dynamics

TL;DR

This paper introduces two new numerical schemes for approximate dynamic programming in finite-horizon optimal control of discrete-time systems with input-affine dynamics, reducing computational complexity significantly.

Contribution

The paper presents novel algorithms that discretize state and input spaces and transform the DP minimization problem, achieving lower computational complexity with error bounds.

Findings

Reduced time complexity from O(XU) to O(X+U) for certain problems

Error bounds provided for the algorithms

Applicable to systems with separable data in state and input variables

Abstract

We propose two novel numerical schemes for approximate implementation of the dynamic programming~(DP) operation concerned with finite-horizon, optimal control of discrete-time systems with input-affine dynamics. The proposed algorithms involve discretization of the state and input spaces and are based on an alternative path that solves the dual problem corresponding to the DP operation. We provide error bounds for the proposed algorithms, along with a detailed analysis of their computational complexity. In particular, for a specific class of problems with separable data in the state and input variables, the proposed approach can reduce the typical time complexity of the DP operation from $O(XU)$ to $O (X+U)$, where $X$ and $U$ denote the size of the discrete state and input spaces, respectively. This reduction is achieved by an algorithmic transformation of the minimization in the DP operation to an addition via discrete conjugation.