Research Papers

The Use of the Differential Transform Method to Solve Time-Invariant State-Feedback Optimal Control Problems

[+] Author and Article Information
H. Saberi Nik

Department of Mathematics,
Neyshabur Branch,
Islamic Azad University,
Neyshabur, Iran
e-mail: saberi_hssn@yahoo.com

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received July 24, 2011; final manuscript received February 19, 2013; published online May 10, 2013. Assoc. Editor: Rama K. Yedavalli.

J. Dyn. Sys., Meas., Control 135(4), 041005 (May 10, 2013) (5 pages) Paper No: DS-11-1222; doi: 10.1115/1.4023974 History: Received July 24, 2011; Revised February 19, 2013

In this paper, the differential transform method (DTM) is applied for solving time-invariant state-feedback control problems. The optimal equations are obtained using the Pontryagin's maximum principle (PMP) and Bellman's Dynamic Programming. We present the closed-loop optimal control of linear plants with quadratic performance index. The results reveal that the proposed methods are very effective and simple. Comparisons are made between the results of two proposed methods and the exact solutions.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Pontryagin, L. S., 1962, The Mathematical Theory of Optimal Processes, Interscience Publishers, New York.
Bellman, R. E., 1957, Dynamic Programming, Princeton University Press, Princeton, NJ.
Yousefi, S. A., Dehghan, M., and Lotfi, A., 2010, “Finding the Optimal Control of Linear Systems via He's Variational Iteration Method,” Int. J. Comput. Math., 87, pp. 1042–1050. [CrossRef]
Fakharian, A., Hamidi Beheshti, M. T., and Davari, A., 2010, “Solving the Hamilton–Jacobi–Bellman Equation Using Adomian Decomposition Method,” Int. J. Comput. Math., 87, pp. 2769–2785. [CrossRef]
Saberi Nik, H., and Shateyi, S., 2013, “Application of Optimal HAM for Finding Feedback Control of Optimal Control Problems,” Math. Prob. Eng., 2013, p. 914741. [CrossRef]
Effati, S., Saberi Nik, H., and Shirazian, M., 2013, “An Improvement to the Homotopy Perturbation Method for Solving the Hamilton–Jacobi–Bellman Equation,” IMA J. Math. Control Info., (in press). [CrossRef]
Zhang, Z. Y., Li, Y. X., Liu, Z. H., Miao, X. J., 2011, “New Exact Solutions to the Perturbed Nonlinear Schrödinger's Equation With Kerr Law Nonlinearity via Modified Trigonometric Function Series Method,” Commun. Nonlinear Sci. Numer. Simul., 16(8), pp. 3097–3106. [CrossRef]
Zhou, J. K., 1986, Differential Transformation and Its Applications for Electrical Circuits, Huazhong University Press, Wuhan, PRC (in Chinese).
Ayaz, F., 2004, “Solutions of the System of Differential Equations by Differential Transform Method,” Appl. Math. Comput., 147, pp. 547–567. [CrossRef]
Ayaz, F., 2004, “Application of Differential Transform Method to Differential-Algebraic Equations,” Appl. Math. Comput., 152, pp. 649–657. [CrossRef]
Arikoglu, A., and Ozkol, I., 2005, “Solution of Boundary Value Problems for Integro-Differential Equations by Using Differential Transform Method,” Appl. Math. Comput., 168, pp. 1145–1158. [CrossRef]
Arikoglu, A., and Ozkol, I., 2006, “Solution of Differential-Difference Equations by Using Differential Transform Method,” Appl. Math. Comput., 181(1), pp. 153–162. [CrossRef]
Bildik, N., Konuralp, A., Bek, F., and Kucukarslan, S., 2006, “Solution of Different Type of the Partial Differential Equation by Differential Transform Method and Adomian's Decomposition Method,” Appl. Math. Comput., 127, pp. 551–567. [CrossRef]
Hassan, I. H., 2008, “Comparison Differential Transformation Technique With Adomian Decomposition Method for Linear and Nonlinear Initial Value Problems,” Chaos, Solitons Fract., 36(1), pp. 53–65. [CrossRef]
Thongmoon, M., and Pusjuso, S., 2010, “The Numerical Solutions of Differential Transform Method and the Laplace Transform Method for a System of Differential Equations,” Nonlinear Anal.: Hybrid Syst., 4, pp. 425–431. [CrossRef]
Pinch, E. R., 1993, Optimal Control and the Calculus of Variations, Oxford University Press, London.
Kirk, D. E., 1970, Optimal Control Theory: An Introduction, Prentice-Hall, Upper Saddle River, NJ.


Grahic Jump Location
Fig. 1

The exact and DTM solutions of Example 4.1 for n = 15

Grahic Jump Location
Fig. 2

The exact and DTM solutions of Example 4.2 for n = 10

Grahic Jump Location
Fig. 3

The exact and DTM solutions of Example 4.3 for n = 10



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In