Differential Transformation and Its Application to Nonlinear Optimal Control

2009 ◽  
Vol 131 (5) ◽  
Author(s):  
Inseok Hwang ◽  
Jinhua Li ◽  
Dzung Du
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Optimal Control Problems ◽  
Minimum Principle ◽  
Nonlinear Optimal Control ◽  
Control Problems ◽  
Algebraic Equations ◽  
Differential Transformation ◽  
Nonlinear Algebraic Equations ◽  
Transformation Algorithm

A novel numerical method based on the differential transformation is proposed for solving nonlinear optimal control problems in this paper. The differential transformation is a linear operator that transforms a function from the original time and/or space domain into another domain in order to simplify the differential calculations. The optimality conditions for the optimal control problems can be represented by algebraic and differential equations. Using the differential transformation, these algebraic and differential equations with their boundary conditions are first converted into a system of nonlinear algebraic equations. Then the numerical optimal solutions are obtained in the form of finite-term Taylor series by solving the system of nonlinear algebraic equations. The differential transformation algorithm is similar to the spectral element methods in that the computational region splits into several subregions but it uses polynomials of high degrees by keeping a small number of subregions. The differential transformation algorithm could solve the finite- (or infinite-) time horizon optimal control problems formulated as either the algebraic and ordinary differential equations using Pontryagin’s minimum principle or the Hamilton–Jacobi–Bellman partial differential equation using dynamic programming in one unified framework. In addition, the differential transformation algorithm can efficiently solve optimal control problems with the piecewise continuous dynamics and/or nonsmooth control. The performance is demonstrated through illustrative examples.

scholarly journals Shifted ultraspherical pseudo-Galerkin method for approximating the solutions of some types of ordinary fractional problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohamed Abdelhakem ◽  
Doha Mahmoud ◽  
Dumitru Baleanu ◽  
Mamdouh El-kady
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Optimal Control Problems ◽  
Approximate Solutions ◽  
Collocation Methods ◽  
Control Problems ◽  
Algebraic Equations ◽  
Fractional Optimal Control ◽  
Ultraspherical Polynomials ◽  
Fractional Optimal Control Problems

AbstractIn this work, a technique for finding approximate solutions for ordinary fraction differential equations (OFDEs) of any order has been proposed. The method is a hybrid between Galerkin and collocation methods. Also, this method can be extended to approximate fractional integro-differential equations (FIDEs) and fractional optimal control problems (FOCPs). The spatial approximations with their derivatives are based on shifted ultraspherical polynomials (SUPs). Modified Galerkin spectral method has been used to create direct approximate solutions of linear/nonlinear ordinary fractional differential equations, a system of ordinary fraction differential equations, fractional integro-differential equations, or fractional optimal control problems. The aim is to transform those problems into a system of algebraic equations. That system will be efficiently solved by any solver. Three spaces of collocation nodes have been used through that transformation. Finally, numerical examples show the accuracy and efficiency of the investigated method.

A modified pseudospectral method for indirect solving a class of switching optimal control problems

2020 ◽  
Vol 234 (9) ◽  
pp. 1531-1542
Author(s):  
Mohammad A Mehrpouya
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Minimum Principle ◽  
Pseudospectral Method ◽  
Piecewise Constant Function ◽  
Control Problems ◽  
Time Interval ◽  
Algebraic Equations ◽  
Control Functions ◽  
And Control

In the present paper, an efficient pseudospectral method for solving the Hamiltonian boundary value problems arising from a class of switching optimal control problems is presented. For this purpose, based on the Pontryagin's minimum principle, the first-order necessary conditions of optimality are derived. Then, by partitioning the time interval related to the problem under study into some subintervals, the states (and costates) and control functions are approximated on each subintervals with piecewise interpolating polynomials based on Legendre–Gauss–Radau points and a piecewise constant function, respectively. As a result, solution of the problem is turned into solution of a number of algebraic equations, in which the values of the states (and costates) and control functions at Legendre–Gauss–Radau points as well as switching and terminal points are allowed to be unknown. Numerical examples are presented at the end to show the efficiency of the proposed method.

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