Basic Control Theory for Linear Fractional Differential Equations With Constant Coefficients

Sebastián Buedo-Fernández,Juan J Nieto

doi:10.3389/fphy.2020.00377

Abstract

In this paper we present an analogous result of the famous Kalman controllability criterion for first order linear ordinary differential equations with constant coefficients that applies to the case of linear differential equations of fractional order with constant coefficients. We present some simple examples, including a linear fractional harmonic oscillator, to illustrate our results. Moreover, another simple linear system of incommensurate fractional orders is suggested as an open problem.

Highlights

Controllability is a mathematical problem consisting in determining the targets to which one can drive the state of a dynamical system by means of a control input appearing in the equation
In the context of the latter application of fractional calculus, we present the current work, which deals with the controllability of a linear fractional differential equation with constant coefficients
We have studied the controllability of the linear fractional differential equation

Summary

INTRODUCTION

Controllability is a mathematical problem consisting in determining the targets to which one can drive the state of a dynamical system by means of a control input appearing in the equation. Given a future time, an initial state and a target state, is it possible to find a control function such that the solution of the system starting from the initial state reaches the desirable state at the prescribed future time? This criterion is known since 1996 [19] we give another approach and use some elements of fractional calculus and a different proof to obtain the results. We reveal some interesting connections between linear differential equations of fractional order, control problems, linear algebra, Mittag-Leffler functions, geometry and physics. The calculation of the Gramian is useful to find a control to steer a given initial state to another prescribed final state

CLASSICAL LINEAR CONTROL

LINEAR CONTROL OF FRACTIONAL ORDER

PROOF OF THE FRACTIONAL CONTROL

CONCLUSIONS

DATA AVAILABILITY STATEMENT