Theoretical and Numerical Analysis of Approximate Dynamic Programming with Approximation Errors

TL;DR

This paper analyzes how approximation errors in Approximate Dynamic Programming influence the convergence, stability, and near-optimality of solutions in deterministic nonlinear control problems, providing theoretical bounds and practical verification.

Contribution

It offers a theoretical framework for understanding the impact of approximation errors in ADP, including convergence conditions, stability criteria, and application to orbital maneuver problems.

Findings

Convergence of Value Iteration with errors is established.

Conditions for stability and near-optimality are derived.

Implementation on orbital maneuver verifies theoretical assumptions.

Abstract

This study is aimed at answering the famous question of how the approximation errors at each iteration of Approximate Dynamic Programming (ADP) affect the quality of the final results considering the fact that errors at each iteration affect the next iteration. To this goal, convergence of Value Iteration scheme of ADP for deterministic nonlinear optimal control problems with undiscounted cost functions is investigated while considering the errors existing in approximating respective functions. The boundedness of the results around the optimal solution is obtained based on quantities which are known in a general optimal control problem and assumptions which are verifiable. Moreover, since the presence of the approximation errors leads to the deviation of the results from optimality, sufficient conditions for stability of the system operated by the result obtained after a finite number of value iterations, along with an estimation of its region of attraction, are derived in terms of a calculable upper bound of the control approximation error. Finally, the process of implementation of the method on an orbital maneuver problem is investigated through which the assumptions made in the theoretical developments are verified and the sufficient conditions are applied for guaranteeing stability and near optimality.