The Numerical Solution for Quadratic Optimal Control Problems by Using Chebyshev and Legendre Polynomials

Abstract

The purpose of this paper is to solve quadratic optimal control problems (QOCP) numerically with the assist of once Chebyshev and Legendre polynomials as basic functions to find the solution for optimal control (QOC) approximately. We will explain the algorithms of solution by examples and use the Mathcad’s Program to reach the exact result.

Introduction

          The optimal control problem is to find a control  which minimizes a given performance index while satisfying the system state equations and constraints.     [1]

We use the approximation methods to solve the optimal control problem depending on the Chebyshev polynomials in the first time and Legendre polynomials, after that we will approximate these solutions of continuous time linear. To reach the approximate solutions we use the linear multi- term differential equations of  and  for both Chebyshev and Legendre polynomials and make the terms of these equations as square matrix to find these values by matrices system.

When we use these polynomials in approximate solutions, the results were evaluated by using index with  

We will explain these algorithms by taking some examples for the quadratic control problems.

The linear quadratic problem is stated as follows;

Minimize the quadratic continuous time

Cost function          …(1)

Subject to the linear system state equations;

                                      …(2)

 where the initial condition  and the matrices  and  are