A General Framework for Optimal Control of Fractional Nonlinear Delay Systems by Wavelets

Abstract

An iterative procedure to find the optimal solutions of general fractional nonlinear delay systems with quadraticperformance indices is introduced. The derivatives of state equations are understood in the Caputo sense. By presenting and applying a general framework, we use the Chebyshev wavelet method developed for fractional linear optimal control to convert fractional nonlinear optimal control problems as a sequence of quadratic programming ones. The concepts and computational procedure that are used for fractional linear optimal control are applied on fractional nonlinear optimal control. Different types of nonlinear optimal control problems with fractional or integer order can be solved. To see this, some numerical examples are solved. Another operational property of Chebyshev wavelets is presented.

Caputo fractional derivative operational matrices of Legendre and Chebyshev wavelets in fractional delay optimal control

“…Case 1 of this problem was studied in [31] by using the Riemann-Liouville integral operational matrix which the main problem was replaced with a sequence of linear problems. We see the proposed general framework by using the Caputo derivative operational matrices of the both wavelets, converges again to a solution and the results of J * are reported in Table 5.…”

“…We have R i = 0.5, k i = 0.5, , F = 0.5, V i = 0.5, δ 1 (t, x 1 (t − h 1 )) = θ 3 x 1 (t − h 1 ), h i = 0.25, t f = 5, δ 2 (x 2 (t − h 2 )) = 0.5θ 4 x 2 2 (t − h 2 )e 0.01x2(t−h2) , θ = [8, −8] , θ 1 = θ 2 = 2 and θ 3 = θ 4 = 1; the system is constrained by the following constraint x 2 1 (t) − x 2 2 (t) + u(t) ≤ 0, 0.25 ≤ t ≤ 5. For more details, see [31].…”

“…Considering these facts, we need a method which can handle all the mentioned types of optimal control systems. In [25][26][27][28][29][30][31], we transformed the optimization of different types of control systems into QP problems. But in all of them, we have used the integration operations of the state equations by using the integration operational matrices, that is, first we have integrated both sides of state equations and then we have replaced the approximated terms; especially in fractional systems, we have utilized the fractional integral operator.…”

“…In this work, we are going to present a general method for fractional/integer order linear/nonlinear delay/free-delay optimal control problems. Several works for optimal control of fractional delayed systems were used the fractional integral operator where some of them have been cited in [29,31]. In some of them, one can see there are certain issues of the accuracy of the final solutions, for example, the initial condition x(0) were not satisfied, or their results for α = 1 (in which we have an integer order optimal control problem) did not match with the previous results presented in many literature.…”

Necessary Conditions to A Fractional Variational Problem

“…In the last years numerous works have been developed tending to extend the theory of the variational calculus in order to be able to be applied to problems of fractional variational calculus. This is fundamentally due, on the one hand, to an important development of the fractional calculus both from the mathematical point of view and its applications in other areas (electricity, magnetism, mechanics, dynamics of fluids, medicine, etc [5,13,31]), which has led to great growth in its study in recent decades. On the other hand, the fractional differential equations establish models far superior to those that use differential equations with integer derivatives because they incorporate into the model issues of memory [11,21] or later effects that are neglected in the models with classical derivative.…”