Optimal Control Problem Solution with Phase Constraints for Group of Robots by Pontryagin Maximum Principle and Evolutionary Algorithm

Askhat Diveev,Ivan Zelinka,Elena Sofronova

doi:10.3390/math8122105

Abstract

A numerical method based on the Pontryagin maximum principle for solving an optimal control problem with static and dynamic phase constraints for a group of objects is considered. Dynamic phase constraints are introduced to avoid collisions between objects. Phase constraints are included in the functional in the form of smooth penalty functions. Additional parameters for special control modes and the terminal time of the control process were introduced. The search for additional parameters and the initial conditions for the conjugate variables was performed by the modified self-organizing migrating algorithm. An example of using this approach to solve the optimal control problem for the oncoming movement of two mobile robots is given. Simulation and comparison with direct approach showed that the problem is multimodal, and it approves application of the evolutionary algorithm for its solution.

Highlights

The optimal control belongs to complex computational problems for which there are no universal solution algorithms
The goal of solving the boundary-value problem is to find the initial conditions for conjugate variables such that the vector of state variables falls into a given terminal condition
For this problem, there is no guarantee that the functional for the boundary-value problem is not unimodal and convex on the space of initial conditions of conjugate variables

Summary

Introduction

The optimal control belongs to complex computational problems for which there are no universal solution algorithms. The goal of solving the boundary-value problem is to find the initial conditions for conjugate variables such that the vector of state variables falls into a given terminal condition. For this problem, there is no guarantee that the functional for the boundary-value problem is not unimodal and convex on the space of initial conditions of conjugate variables. Phase constraints are included in the functional, so they are included in the system of equations for conjugate variables This greatly complicates the analysis of the problem on the convexity and unimodality of the target functional

Objectives

Methods

Conclusion