Solving quadratic linear optimal control problems under interval uncertainty

Document Type : Original Article

Authors

1 Mathematics Faculty, University of Sistan and Baluchestan, Zahedan, Iran

2 Faculty of Industry and Mining (Khash), University of Sistan and Baluchestan, Zahedan, Iran

3 Mathematics Faculty, zahedan. Iran

Abstract

The main aim of this paper is to find an optimal interval control law for quadratic linear problems under interval uncertainty. For this purpose, using Bellman's optimality principle and interval Hamilton-Jacobi-Bellman inequalities, the interval optimal control problem is transformed into a system of interval differential inequalities. These inequalities are called Riccati's differential inequalities, which is a result of the dynamic programming method. To solve this system of inequalities, we use inclusion relations and interval arithmetic. By this method, we can obtain the upper and lower bounds of the solutions. We also use Hokuhara's generalized difference to reduce errors in the interval arithmetic. We apply the presented method for solving some interval quadratic linear optimal control problems by using MATLAB software. The obtained results show the efficiency of the proposed method.

Keywords

Main Subjects

References

[1] الله دادی،  م.  (1400) برنامه ریزی خطی بازه ­ای،  انتشارات دانشگاه سیستان و بلوچستان.
 
[2] Bertin, E., Brendel, E., H´eriss´e, B., Sandretto, J.A.D. and Chapoutot, A. (2021). Prospects on solving an optimal control problem with bounded uncertainties on parameters using interval arithmetics. Acta Cybernetica, 1-25.
 
[3] Bliss, G.A. (1914) The Weierstrass E-function for problems of the calculus of variations in space. Transactions of the American Mathematical Society. 15(4), 369-378.
 
[4] Bolza, O. Lectures on the Calculus of Variations. Courier Dover Publications. 2018.
 
[5] Brunt, B.V. The Calculus of Variations. Springer-Verlag, Heidelberg. 2004.
 
[6] Campos, J.R., Assunção, E., Silva, G.N., Lodwick, W.A., Teixeira, M.C. (2019) Discrete-time interval optimal control problem. International Journal of Control. 92(8), 1778-1784.
 
[7] Campos, J.R., Assuncao, E., Silva, G.N., Lodwick, W.A., Teixeira, M.C.M., Maqui-Huamn, G.G. (2020) Fuzzy interval optimal control problem. Fuzzy sets and systems. 385, 169-181.
 
[8] Farhadinia, B. (2014) Pontryagin’s minimum principle for fuzzy optimal control problems. Iranian Journal of Fuzzy Systems. 11(2), 27-43.
 
[9] Ghosh, D., Chauhan, R.S., Mesiar, R., Debnath, A.K. (2020) Generalized Hukuhara Gateaux and Frechet derivatives of interval-valued functions and their application in optimization with interval-valued functions. Information Sciences. 510, 317-340.
 
[10] Hansen E, Walster GW, eds Global optimization using interval analysis: revised and expanded (264). CRC Press. 2003.
 
[11] Huang, Y., Lu, W.M. (1996) Nonlinear optimal control: Alternatives to HamiltonJacobi equation. In Proceedings of 35th IEEE conference on decision and control IEEE. 4, 3942-3947.
 
[12] Kamien, M.I., Schwartz, N.L. Dynamic optimization: the calculus of variations and optimal control in economics and management. courier corporation. 2012.
 
[13] Kirk, D.E. (1967) An introduction to dynamic programming. IEEE Transactions on Education. 10(4), 212-219.
 
[14] Leal, U.A.S. Incerteza intervalar em otimização e controle [Interval uncertainty in optimization problems and control] (Doctoral dissertation). Universidade Estadual Paulista, São José do Rio Preto (in Portuguese). 2015.
 
[15] Lodwick, W.A., Jamison, K.D. (2018) A Constraint Fuzzy Interval Analysis approach to fuzzy optimization. Information Sciences. 426, 38-49.
 
[16] Lupulescu, V., Hoa, N.V. (2017) Interval Abel integral equation. Soft Computing. 21, 2777–2784.
 
[17] Najariyan M., Farahi M.H. (2013) Optimal control of fuzzy linear controlled system with fuzzy initial conditions. Iranian Journal of Fuzzy Systems. 10(3): 21-35.
 
[18] Razmjooy, N., and Ramezani, M. (2019) Interval structure of Runge-Kutta methods for solving optimal control problems with uncertainties. Computational Methods for Differential Equations. 7(2), 235-251.
 
[19] Stefanini, L. (2010) A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets and Systems. 161, 1564-1584.
 
[20] Stefanini, L., Bede, B. (2009) Generalized Hukuhara differentiability of intervalvalued functions and interval differential equations. Nonlinear Analysis: Theory, Methods Applications. 71(3-4), 1311-1328.
 
[21] Van Kampen, E. Global optimization using interval analysis: interval optimization for aerospace applications. 2010.
 
[22] Wang, H., Rodriguez-Lopez, R. (2023) On the first-order autonomous intervalvalued difference equations under gH-difference. Iranian Journal of Fuzzy Systems. 20(2), 21-32.
 
[23] Wang, H., Rodríguez-López, R., Khastan, A. (2021) On the stopping time problem of interval-valued differential equations under generalized Hukuhara differentiability. Information Sciences. 579, 776-795.
 
[24] Zarei, H., Khastan, A., Rodríguez-López, R. (2023) Suboptimal control of linear fuzzy systems. Fuzzy Sets and Systems. 453, 130-163.
Fuzzy Systems and its Applications
Volume 6, Issue 1 - Serial Number 12
September 2023
Pages 245-267

Files

History

  • Receive Date: 11 March 2023
  • Revise Date: 06 July 2023
  • Accept Date: 21 July 2023

Share

How to cite

Statistics