Optimal Control of Nonlinear Systems Using the Homotopy Perturbation Method

TL;DR

This paper introduces a novel approach combining the Homotopy Perturbation Method with optimal control theory to efficiently solve nonlinear control problems by transforming them into linear subproblems.

Contribution

The paper develops a new method that transforms nonlinear optimal control problems into linear TPBVPs using HPM, enabling recursive solutions for control laws and trajectories.

Findings

Method effectively solves nonlinear control problems.

Produces accurate sub-optimal control laws.

Demonstrates simplicity and efficiency in an example.

Abstract

This paper presents a new method for solving a class of nonlinear optimal control problems with a quadratic performance index. In this method, first the original optimal control problem is transformed into a nonlinear two-point boundary value problem (TPBVP) via the Pontryagin's maximum principle. Then, using the Homotopy Perturbation Method (HPM) and introducing a convex homotopy in topologic space, the nonlinear TPBVP is transformed into a sequence of linear time-invariant TPBVP's. By solving the presented linear TPBVP sequence in a recursive manner, the optimal control law and the optimal trajectory are determined in the form of infinite series. Finally, in order to obtain an accurate enough sub-optimal control law, an iterative algorithm with low computational complexity is introduced. An illustrative example demonstrates the simplicity and efficiency of proposed method.