This paper aims to investigate an efficient numerical scheme for the optimal control of fractional-order dynamic systems. By using the Grünwald–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.
AgrawalOP (1989) General formulation for the numerical solution of optimal control problems. International Journal of Control50(2): 627–638.
2.
AgrawalOP (2004) A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dynamics38(1): 323–337.
3.
AgrawalOP (2008) A quadratic numerical scheme for fractional optimal control problems. Journal of Dynamic Systems, Measurement and Control130(1): 011010(1–6).
4.
AgrawalOPDefterliOBaleanuD (2010) Fractional optimal control problems with several state and control variables. Journal of Vibration and Control16(13): 1967–1976.
5.
AkbarianTKeyanpourM (2013) A new approach to the numerical solution of fractional order optimal control problems. Applications and Applied Mathematics8(2): 523–534.
6.
AlinezhadMAllahviranlooT (2017) On the solution of fuzzy fractional optimal control problems with the Caputo derivative. Information Sciences421: 218–236.
7.
AlizadehAEffatiS (2016) An iterative approach for solving fractional optimal control problems. Journal of Vibration and Control. 1–19. DOI: https://doi.org/10.1177/1077546316633391.
8.
BaleanuDDefteliOAgrawalOP (2009) A central difference numerical scheme for fractional optimal control problems. Journal of Vibration and Control15(4): 583–597.
9.
BaleanuDDiethelmKScalasEet al.(2012) Fractional Calculus: Models and Numerical Methods, Berlin, Germany: World Scientific.
10.
BazaraaMSSheraliHDShettyCM (2006) Nonlinear Programming: Theory and Algorithms., 3rd edition. Hoboken, NJ: John Wiley & Sons.
11.
BazaraaMSJarvisJJSheraliHD (2010) Linear Programming and Network Flows., 4th edition. Hoboken, NJ: John Wiley & Sons.
12.
BehroozifarMHabibiN (2017) A numerical approach for solving a class of fractional optimal control problems via operational matrix Bernoulli polynomials. Journal of Vibration and Control. 1–18. DOI: https://doi.org/10.1177/1077546316688608.
13.
BhrawyAHEzz-EldienSSDohaEHet al.(2017) Solving fractional optimal control problems within a Chebyshev–Legendre operational technique. International Journal of Control90(6): 1230–1244.
14.
CamachoEFBordonsC (2007) Nonlinear Model Predictive Control: An Introductory Review. Assessment and Future Directions of Nonlinear Model Predictive Control, Verlag Berline Heidelberg: Springer.
15.
DohaEHBhrawyAHBaleanuDet al.(2015) An efficient numerical scheme based on the shifted orthonormal Jacobi polynomials for solving fractional optimal control problems. Advances in Difference Equations2015(1): 1–17.
16.
EjlaliNHosseiniSM (2017) A pseudospectral method for fractional optimal control problems. Journal of Optimization Theory and Applications174(1): 83–107.
17.
Ezz-EldienSSDohaEHBaleanuDet al.(2017) A numerical approach based on Legendre orthonormal polynomials for numerical solutions of fractional optimal control problems. Journal of Vibration and Control23(1): 16–30.
18.
HilferR (2000) Applications of Fractional Calculus in Physics, Singapore: World Scientific.
19.
KeshavarzEOrdokhaniYRazzaghiM (2016) A numerical solution for fractional optimal control problems via Bernoulli polynomials. Journal of Vibration and Control22(18): 3889–3903.
20.
KilbasAASrivastavaHHTrujilloJJ (2006) Theory and Applications of Fractional Differential Equations, New York, NY: Elsevier.
21.
ÖzdemirNAgrawalOPİskenderBBet al.(2009) Fractional optimal control of a 2-dimensional distributed system using eigenfunctions. Nonlinear Dynamics55: 251–260.
22.
ÖzdemirNAvciD (2014) Optimal control of a linear time-invariant space–time fractional diffusion process. Journal of Vibration and Control20(3): 370–380.
23.
PhangCIsmailNFIsahAet al.(2017) A new efficient numerical scheme for solving fractional optimal control problems via a Genocchi operational matrix of integration. Journal of Vibration and Control. 1–13. DOI: https://doi.org/10.1177/1077546317698909.
24.
PodlubnyI (1999) Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications, New York, NY: Academic Press.
25.
PoosehSAlmeidaRTorresDFM (2014) Fractional order optimal control problems with free terminal time. Journal of Industrial and Management Optimization10(2): 363–381.
26.
RebarberRWeissG (2003) Internal model based tracking and disturbance rejection for stable well-posed systems. Automatica39(9): 1555–1569.
27.
SahuPKSaha RayS (2016) Comparison on wavelets techniques for solving fractional optimal control problems. Journal of Vibration and Control.. 1–17. DOI: https://doi.org/10.1177/1077546316659611.
28.
TangXShiYWangLL (2017) A new framework for solving fractional optimal control problems using fractional pseudospectral methods. Automatica78: 333–340.
29.
TussetAMBalthazarJMChavaretteFRet al.(2012) On energy transfer phenomena, in a nonlinear ideal and nonideal essential vibrating systems, coupled to a (MR) magneto-rheological damper. Nonlinear Dynamics69(4): 1859–1880.
30.
YangXJBaleanuDSrivastavaHM (2015) Local Fractional Integral Transforms and Their Applications, London, UK: Academic Press.
31.
ZakyMAMachadoJAT (2017) On the formulation and numerical simulation of distributed-order fractional optimal control problems. Communications in Nonlinear Science and Numerical Simulation52: 177–189.
