### Title: Optimal fractional order control for nonlinear systems represented by the Euler-Lagrange formulation

Addresses: College of Computer and Information Sciences (CCIS), Prince Sultan University, Riyadh, 11586, Saudi Arabia; Faculty of Computers and Artificial Intelligence, Benha University, Benha 13511, Egypt ' Universidad Tecnológica Centroamericana (UNITEC), Tegucigalpa, 3530, Honduras ' Faculty of Engineering, Cairo University, Giza 12613, Egypt

Abstract: In this paper a novel control strategy is shown for the control of fractional order systems established in the Euler-Lagrange formulation. The objective is to find an optimal fractional order law which drives the state variables in Lagrangian formulation to a desired final value in finite time, while minimising the control and energy effort. The methodology used in this study consists of all the inductive-deductive and axiomatic considerations. This strategy is based on the design of an optimal controller considering a fractional order system based in the Euler-Lagrangian formulation because this allows more degrees of freedom in the system establishment and the optimal controller design. The design procedure consists in establishing a performance index and then, by finding the gradient of this index, an optimal control law is obtained with the initial and final conditions of the system. It is important to note that there is a limited number of studies related to this topic found in literature. Finally, in order to test the theoretical results obtained in this work a numerical example which consists in the stabilisation of a two links robotic manipulator is shown. It is concluded that the proposed control strategy provides an accurate response with a lower control effort results, this time, for the control of a robot mechanism that is used as a benchmark but that can be extended to other kinds of physical system. A major finding is that a different strategy was used, different than a Lyapunov approach, not only obtaining a suitable theoretical framework but also ensuring the stability of the closed loop system, either in the equilibrium points or for reference tracking.

Keywords: fractional order systems; optimal control; optimal fractional order control; Lagrangian systems; robotic manipulator; fractional order sliding mode control; fractional order PID control.

International Journal of Modelling, Identification and Control, 2021 Vol.37 No.1, pp.1 - 9