Article An efficient numerical method for the optimal control of fractional-order dynamic systems Ehsan Mohammadzadeh 1 , Naser Pariz 1 , Seyed Kamal Hosseini Sani 1 and Amin Jajarmi 2 Abstract This paper aims to investigate an efficient numerical scheme for the optimal control of fractional-order dynamic systems. By using the Gru ¨nwald–Letnikov approximation for the fractional derivatives and introducing a new transformation in the calculus of variations, the fractional optimal control problem under consideration is converted into a linear programming problem. Then, the internal model principle is employed in order to extend the new scheme for the fractional dynamic systems affected by the external persistent disturbances. Numerical examples and comparative results verify the validity and applicability of the new technique. Keywords Fractional derivative, optimal control, Gru ¨nwald–Letnikov approximation, linear programming, persistent disturbance 1. Introduction Fractional calculus is developing fast and its various applications are extensively used in many fields of sci- ence and engineering (Baleanu et al., 2012; Hilfer, 2000; Kilbas et al., 2006; Podlubny, 1999; Yang et al., 2015). In addition, the application of fractional calculus in the optimal control problems (OCPs) has become a strong topic to be considered. In this field, Agrawal (2004) formulated and solved the fractional optimal control problems (FOCPs) in terms of the Riemann–Liouville fractional derivative. In (Agrawal, 2008), this fractional problem (in the Caputo sense) was converted into a system of algebraic equations by substituting the frac- tional differential equations with Volterra-type integral equations. In (Agrawal et al., 2010), the FOCPs were considered in the Riemann–Liouville sense, and the fractional derivatives were approximated by using the Gru¨nwald–Letnikov definitions. In (Baleanu et al., 2009), a central difference formula was derived in order to modify the Gru¨nwald–Letnikov definition for the FOCPs. In (O ¨ zdemir et al., 2009), the fractional optimal control of distributed systems was investigated by using an eigenfunctions expansion based approach. In (O ¨ zdemir and Avci, 2014), an OCP was formulated and solved numerically for a space–time fractional diffusion process. Recently, some more noticeable efforts have also been carried out by Akbarian and Keyanpour (2013), Doha et al. (2015), Alizadeh and Effati (2016), Keshavarz et al. (2016), and Sahu and Saha Ray (2016). In more recent works, Phang et al. (2017) proposed an efficient numerical scheme for solving FOCPs via a Genocchi operational matrix of integration. Behroozifar and Habibi (2017) provided a numerical approach based on operational matrix Bernoulli polynomials to solve this problem effectively. A new framework of solving FOCPs using fractional pseudospectral methods was investigated by Tang et al. (2017). The solution of fuzzy FOCPs with the Caputo derivative was examined by Alinezhad and Allahviranloo (2017). Formulation and numerical simulation of distributed-order FOCPs were provided by Zaky and Machado (2017). The numerical solution 1 Department of Electrical Engineering, Ferdowsi University of Mashhad, Iran 2 Department of Electrical Engineering, University of Bojnord, Iran Corresponding author: Naser Pariz, Department of Electrical Engineering, Ferdowsi University of Mashhad, P.O. Box, 91775-1111, Mashhad, Iran. Email: n-pariz@um.ac.ir Received: 20 July 2017; accepted: 11 December 2017 Journal of Vibration and Control 1–9 ! The Author(s) 2018 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1077546317751755 journals.sagepub.com/home/jvc