Controllability of finite-dimensional linear fractional systems under uncertain parameters

TL;DR

This paper studies the controllability of finite-dimensional linear fractional systems with uncertain parameters, establishing new controllability criteria and control design methods that generalize classical results for integer-order systems.

Contribution

It introduces average controllability conditions based on the average Kalman rank and Gramian, extending classical controllability concepts to fractional systems with uncertainties.

Findings

Average controllability characterized by average Kalman rank and Gramian.

Open-loop control with minimal energy designed using the average Gramian.

Numerical validation on fractional R"ossler system confirms theoretical results.

Abstract

This paper investigates the controllability of finite-dimensional linear fractional systems involving an uncertain parameter. We establish new results on the simultaneous and average controllability. In particular, we show that average controllability can be characterized by the so-called average Kalman rank condition and the average Gramian matrix. Moreover, using the average Gramian matrix, we design an open-loop control with minimal energy. These results can be seen as a natural generalization of the classical results known for systems with integer-order derivatives. Finally, numerical simulations are provided to robustly validate the theoretical findings, with a focus on the fractional R\"ossler system.