Computers and Mathematics with Applications an Approximate Method for Numerically Solving Fractional Order Optimal Control Problems of General Form Optimal Control Time-optimal Control Fractional Calculus Fractional Order Optimal Control Fractional Dynamic Systems Riots_95 Optimal Control Toolbox

In this article, we discuss fractional order optimal control problems (FOCPs) and their solutions by means of rational approximation. The methodology developed here allows us to solve a very large class of FOCPs (linear/nonlinear, time-invariant/time-variant, SISO/MIMO, state/input constrained, free terminal conditions etc.) by converting them into a general, rational form of optimal control problem (OCP). The fractional differentiation operator used in the FOCP is approximated using Oustaloup's approximation into a state-space realization form. The original problem is then reformulated to fit the definition used in general-purpose optimal control problem (OCP) solvers such as RIOTS_95, a solver created as a Matlab toolbox. Illustrative examples from the literature are reproduced to demonstrate the effectiveness of the proposed methodology and a free final time OCP is also solved.

[1]  W. Ames.  Mathematics in Science and Engineering , 1999 .

[2]  E. Polak,et al.  Amatlab Toolbox for Solving Optimal Control Problems , 1997 .

[3]  YangQuan Chen,et al.  Linear Feedback Control: Analysis and Design with MATLAB , 2008 .

[4]  YangQuan Chen,et al.  Solution of fractional order optimal control problems using SVD-based rational approximations , 2009, 2009 American Control Conference.

[5]  Arthur E. Bryson,et al.  Applied Optimal Control , 1969 .

[6]  Yangquan Chen,et al.  RIOTS―95: a MATLAB toolbox for solving general optimal control problems and its applications to chemical processes , 2002 .

[7]  O. Agrawal,et al.  A Hamiltonian Formulation and a Direct Numerical Scheme for Fractional Optimal Control Problems , 2007 .

[8]  Christophe Tricaud,et al.  Solving Fractional Order Optimal Control Problems in RIOTS 95-A General-Purpose Optimal Control Problems Solver ⋆ , 2008 .

[9]  O. Agrawal.  A Quadratic Numerical Scheme for Fractional Optimal Control Problems , 2008 .

[10]  O. P. Agrawal,et al.  General formulation for the numerical solution of optimal control problems , 1989 .

[11]  K. Moore,et al.  Discretization schemes for fractional-order differentiators and integrators , 2002 .

[12]  Delfim F. M. Torres,et al.  Fractional conservation laws in optimal control theory , 2007, 0711.0609.

[13]  O. Agrawal.  Fractional Optimal Control of a Distributed System Using Eigenfunctions , 2007 .

[14]  O. Agrawal.  A General Formulation and Solution Scheme for Fractional Optimal Control Problems , 2004 .

[15]  Y. Chen,et al.  Continued Fraction Expansion Approaches to Discretizing Fractional Order Derivatives—an Expository Review , 2004 .

[16]  Elijah Polak,et al.  On the use of consistent approximations in the solution of semi-infinite optimization and optimal control problems , 1993, Math. Program..

[17]  Alain Oustaloup,et al.  Frequency-band complex noninteger differentiator: characterization and synthesis , 2000 .

[18]  Delfim F. M. Torres,et al.  Noether's theorem for fractional optimal control problems , 2006, math/0603598.

[19]  Delfim F. M. Torres,et al.  Fractional Optimal Control in the Sense of Caputo and the Fractional Noether's Theorem , 2007, 0712.1844.

[20]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[21]  Nasser Sadati,et al.  Fopid Controller Design for Robust Performance Using Particle Swarm Optimization , 2007 .