OPTIMAL CONTROL OF FUZZY LINEAR CONTROLLED SYSTEM WITH FUZZY INITIAL CONDITIONS

Document Type: Research Paper

Authors

1 Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran

2 Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran and The center of Excellence on Modelling and Control Systems (CEMCS)

Abstract

In this article we found the solution of fuzzy linear controlled system
with fuzzy initial conditions by using -cuts and presentation of numbers
in a more compact form by moving to the eld of complex numbers. Next, a
fuzzy optimal control problem for a fuzzy system is considered to optimize the
expected value of a fuzzy objective function. Based on Pontryagin Maximum
Principle, a constructive equation for the problem is presented. In the last
section, three examples are used to show that the method in e ective to solve
fuzzy and fuzzy optimal linear controlled systems.

Keywords


1. T. Allahviranloo and M. A. Kermani, Numerical methods for fuzzy linear partial di erential
equations under new de nition for derivative, Iranian Journal of Fuzzy Systems, 7 (2010),
33-50.
2. B. Bede, I. J. Rudas and A. L. Bencsik, First order linear fuzzy di erential equation sunder
generalized di erentiability, Information Science, 177 (2007), 1648-1662.
3. R. C. Dorf and R. H. Bishop, Modern control systems, Person Education, Inc. Upper Saddle
River, New Jersey, 07458 (2011).
4. P. Diamond and P. E. Kloeden, Metric space of Fuzzy sets, Theory And Applications, World
scienti c publishing, 1994.
5. O. S. Fard and A. V. Kamyad, Modi ed k-step method for solving fuzzy initial value problems,
Iranian Journal of Fuzzy Systems, 8 (2011), 49-63.
6. D. Filev and P. Angelove, Fuzzy optimal control, Fuzzy Sets and Systems, 47 (1992), 151-56.
7. D. N. Georgiou, J. J. Nieto and R. Rodriguez-Lopez, Initial value problems for higher-order
fuzzy di erential equations, Nonlinear Analysis, 63 (2005), 587-600.
8. A. Khastan, J. J. Nieto and R. Rodriguez-Lopez, Variation of constant formula for rst order
fuzzy di erential equations, Fuzzy Sets and Systems, 177 (2011), 20-33.
9. A. Khastan and J. J. Nieto, A boundary value problem for second order fuzzy di erential
equations, Nonlinear Analysis, 72 (2010), 3583-3593.
10. J. J. Nieto, R. Rodriguez-Lopez and M. Villanueva-Pesqueira, Exact solution to the periodic
boundary value problem for a rst-order linear fuzzy di erential equation with impulses, Fuzzy
Optimization Decision Making, 10 (2011), 323-339.
11. J. J. Nieto, A. Khastan and K. Ivaz, Numerical solution of fuzzy di erential equations under
generalized di erentiability, Nonlinear Analysis: Hybrid Systems, 3 (2009), 700-707.
12. J. H. Park, J. S. Park and Y. C. Kwun, Controllability for the semilinear fuzzy integrodi er-
ential equations with nonlocal conditions, Lecture Notes in Arti cial Intelligence, LNAI 4223
(2006), 221-230.
13. D. W. Pearson, A property of linear fuzzy di erential equations, Appl. Math. Lett., 10 (1997),
99-103.
14. E. R. Pinch, Optimal control and the calculuse of variations, Oxford University Press Inc.,
New Yourk, 1995.
15. Z. Qin, Time-homogeneous fuzzy optimal control problems, http://www.orsc.edu.cn/process/
080415.pdf.
16. S. Ramezanzadeh and A. Heydari, Optimal control with fuzzy chance constraint, Iranian
Journal of fuzzy systems, 8 (2011), 35-43.
17. S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems, 24 (1987), 319-330.
18. J. Xu, Z. Liao and J. J. Nieto, A class of di erential dynamical systems with fuzzy matrices,
Math. Anal. Appl., 368 (2010), 54-68.
19. J. Xu, Z. Liao and Z. Hu, A class of linear di erential dynamical systems with fuzzy initial
condition, Fuzzy Sets and Systems, 158 (2007), 2339-2358.
20. Y. Zhu, A fuzzy optimal control model, Journal of uncertain systems, 3 (2009), 270-279.
21. Y. Zhu, Fuzzy optimal control with application to portfolio selection, http://www.orsc.edu.cn/
process/080117.pdf.