Fixed final time and free final state optimal control problem for fractional dynamic systems – linear quadratic discrete-time case

Abstract

Abstract. The optimization problem for fractional discrete-time systems with a quadratic performance index has been formulated and solved. The case of fixed final time and a free final state has been considered. A method for numerical computation of optimization problems has been presented. The presented method is a generalization of the well-known method for discrete-time systems of integer order. The efficiency of the method has been demonstrated on numerical examples and illustrated by graphs. Graphs also show the differences between the fractional and classical (standard) systems theory. Results for other cases of the fractional system order (coefficient α) and not illustrated with numerical examples have been obtained through a computer algorithm written for this purpose.

Discrete-Time Fractional Optimal Control

“…At 1   , these results recover the solutions obtained with the integer order problem. While comparing these results with the results presented in [69,72] in terms of the Riemann-Liouville fractional derivative, we observe that for …”

“…A general method for solving the state and co-state equations in terms of the Reimann-Liouville derivative is presented in [68,69,72]. Here, the same method is used for solving the present problem defined in terms of Caputo fractional derivatives.…”

“…Discrete-time fractional optimal control problems that exist in the literature is presented in terms of Riemann-Liouville derivative [68][69][70][71][72]. It has been observed that the Caputo fractional derivative is preferred as it requires natural initial conditions.…”

“…Dzielinski and Czyronis present a general formulation and solution scheme for discrete-time fractional optimal control problems with fixed final state [68] and free final state [69] in terms of the Reimann-Liouville fractional derivative. Dynamic programming methods for solving discrete-time fractional optimal control problems have been proposed by Dzielinski and Czyronis in [70,71].…”

“…The necessary and transversality conditions are obtained using a Hamiltonian approach. A numerical method [68,69,72] is used for the solution of the resulting equations obtained from the formulation.…”