Research Papers

Differential Transformation and Its Application to Nonlinear Optimal Control

[+] Author and Article Information
Inseok Hwang1

School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907ihwang@purdue.edu

Jinhua Li

School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907li31@purdue.edu

Dzung Du

School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907ddu@purdue.edu


Corresponding author.

J. Dyn. Sys., Meas., Control 131(5), 051010 (Aug 19, 2009) (11 pages) doi:10.1115/1.3155013 History: Received August 05, 2008; Revised May 07, 2009; Published August 19, 2009

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.

Copyright © 2009 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 1

Comparison of solutions computed by Algorithm 1 with those by MATLAB command bvp4c

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Figure 5

Solution obtained by using the differential transformation algorithm

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Figure 2

Comparison of solutions computed by Algorithm 2 with the analytic solutions

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Figure 3

Comparison of solution obtained by using Algorithm 3 with the analytic solution

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Figure 4

Comparison of a solution computed by the differential transformation algorithm with that of Galerkin’s method



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