Discrete-Time Fractional Optimal Control

Tirumalasetty Chiranjeevi,Raj Biswas

doi:10.3390/math5020025

Tirumalasetty Chiranjeevi, Raj Biswas

Open Access

https://doi.org/10.3390/math5020025

Copy DOI

Journal: Mathematics	Publication Date: Apr 19, 2017
Citations: 22	License type: CC BY 4.0

Affiliation: National Institute Of Technology Silchar

Abstract

A formulation and solution of the discrete-time fractional optimal control problem in terms of the Caputo fractional derivative is presented in this paper. The performance index (PI) is considered in a quadratic form. The necessary and transversality conditions are obtained using a Hamiltonian approach. Both the free and fixed final state cases have been considered. Numerical examples are taken up and their solution technique is presented. Results are produced for different values of α .

Highlights

Fractional calculus is a generalization of classical calculus
This paper presents a formulation and solution scheme of a discrete-time fractional optimal control problem defined in terms of the Caputo fractional derivative
A discrete-time fractional optimal control problem formulation and solution schemescheme isequation presented presented in this paper

Summary

Introduction

Fractional calculus is a generalization of classical calculus. It has been reported in the literature that systems described using fractional derivatives give more realistic behavior [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. This paper presents a formulation and solution scheme of a discrete-time fractional optimal control problem defined in terms of the Caputo fractional derivative. Dzielinski and Czyronis present a general formulation and solution scheme for discrete-time fractional optimal control problems with fixed final state [68] and free final state [69] in terms of the Reimann-Liouville fractional derivative. Discrete-time fractional optimal control problems that exist in the literature is presented in terms of Riemann-Liouville derivative [68,69,70,71,72]. In this paper the optimal control of the discrete-time fractional order system in terms of the Caputo fractional derivative has been considered Both the fixed and free final state cases are considered. A numerical method [68,69,72] is used for the solution of the resulting equations obtained from the formulation

Problem Formulation

Free Final State Problem

Fixed Final State Problem

Conclusions