Document Type : Research Article

**Author**

Payame Noor University (PNU), Tehran

**Abstract**

We apply the Adomian decomposition method (ADM) to obtain a subop timal control for linear time-varying systems with multiple state and control delays and with quadratic cost functional. In fact, the nonlinear two-point boundary value problem, derived from Pontryagin’s maximum principle, is solved by ADM. For the first time, we present here a convergence proof for ADM. In order to use the proposed method, a control design algorithm with low computational complexity is presented. Through the finite iterations of algorithm, a suboptimal control law is obtained for the linear time-varying multi-delay systems. Some illustrative examples are employed to demonstrate the accuracy and efficiency of the proposed methods.

**Keywords**

1. Adomian, G. Nonlinear stochastic systems theory and applications to physics, Kluwer Academic Publishers, 1989, Boston.

2. Adomian, G. A review of the decomposition method in applied mathematics, J. Math. Anal. Appl. 135(2) (1989), 501–544.

3. Alizadeh, A. and Effati, S. Numerical schemes for fractional optimal con trol problems, J. Dyn. Sys. Meas. Contr. 4 (2017), 1–14.

4. Balasubramaniam, P., Krishnasamy, R. and Rakkiyappan, R. Delay dependent stability of neutral systems with time-varying delays using delay decomposition approach, Appl. Math. Model. 36 (2012), 2253–2261.

5. Banks, H.T. and Burns, J.A. Hereditary control problem: Numerical methods based on averaging approximations, SIAM J. Contr. Optim. 16(2) (1978), 169–208.

6. Basin, M. and Rodriguez-Gonzalez, J. Optimal control of linear systems with multiple time delays in control input, IEEE Trans. Automat. Contr. 51(1) (2006), 91–97.

7. Dadkhah, M. and Farahi, M.H. Optimal control of time delay systems via hybrid of block-pulse functions and orthogonal Taylor series, Int. J. Appl. Comput. Math. 2(1) (2016), 137–152.

8. Edrisi-Tabriz, Y., Lakestani, M. and Heydari, A. Two numerical methods for nonlinear constrained quadratic optimal control problems using linear B-spline functions, Iranian J. Numer. Anal. Optim. 6(2) (2016), 17–37.

9. Gollmann, L., Kern, D. and Maurer, H. Optimal control problems with delays in state and control variables subject to mixed control state con straints, Optim. Contr. Appl. Meth. 30 (2009), 341–365.

10. Gollmann, L. and Maurer, H. Theory and applications of optimal control problems with mul-tiple time delays, J. Ind. Manag. Optim. 10(2) (2014), 413–441.

11. Haddadi, N., Ordokhani, Y. and Razzaghi, M.Optimal control of delay systems by using a hybrid functions approximation, J. Optim. Theor. Appl. 153 (2012), 338–356.

12. Hwang, G. and Chen, M.Y. Suboptimal control of linear time-varying multi-delay systems via shifted Legendre polynomials, Int. J. Syst. Sci. 16(12) (1985), 1517–1537.

13. Jajarmi, A., Dehghan-Nayyeri, M. and Saberi-Nik, H. A novel feedforward-feedback suboptimal control of linear time-delay systems via shifted Legendre polynomials, J. Complex. 35 (2016), 46–62.

14. Jamshidi, M. and Wang, C.M. A computational algorithm for large-scale nonlinear time-delays systems, IEEE Trans. Syst. Man. Cyber. SMC. 14 (1984), 2–9.

15. Kharatishvili, G.L. The maximum principle in the theory of optimal process with time-lags, Doklady Akademii Nauk SSSR. 136 (1961), 39–42.

16. Khellat, F. and Vasegh, N. Suboptimal control of linear systems with delays in state and input by orthogonal basis, Int. J. Comput. Math. 88(4) (2011), 781–794.

17. Koshkouei, A.J., Farahi, M.H. and Burnham, K.J. An almost optimal control design method for nonlinear time-delay systems, Int. J. Contr. 85(2) (2012), 147–158.

18. Malek-Zavarei, M. Near-optimum design of nonstationary linear systems with state and control delays, J. Optim. Theor. Appl. 30 (1980), 73–88.

19. Marzban, H.R. Optimal control of linear multi-delay systems based on a multi-interval decomposition scheme, Optim. Contr. Appl. Meth. 37(1) (2016), 190–211.

20. Marzban, H.R. and Hoseini, S.M. An efficient discretization scheme for solving nonlinear optimal control problems with multiple time delays, Optim. Contr. Appl. Meth. 37(4) (2016), 682–707.

21. Marzban, H.R. and Razzaghi, M. Optimal control of linear delay systems via hybrid of block-pulse and Legendre polynomials, J. Franklin Inst. 341 (2004), 279–293.

22. Marzban, H.R. and Pirmoradian, H. A direct approach for the solution of nonlinear optimal control problems with multiple delays subject to mixed state-control constraints , Appl. Math. Model. 53 (2018), 189–213.

23. Marzban, H.R. and Pirmoradian, H. A novel approach for the numerical investigation of optimal control problems containing multiple delays, Optim. Contr. Appl. Meth. 39(1) (2018), 302–325.

24. Mehne, H.H., and Farahi, M.H. Transformation to a fixed domian in LP modelling for a class of optimal shape design problems, Iranian J. Numer. Anal. Optim. 9(1) (2019), 1–16.

25. Mirhosseini-Alizamini, S.M. Numerical solution of the controlled harmonic oscillator by homotopy perturbation method, Contr. Optim. Appl Math. 2(1) (2017), 77–91.

26. Mirhosseini-Alizamini, S.M. and Effati, S. An iterative method for sub optimal control of a class of nonlinear time-delayed systems, Int. J. Contr. (2018), DOI: 10.1080/00207179.2018.1463456.

27. Mirhosseini-Alizamini, S.M., Effati, S. and Heydari, A. An iterative method for suboptimal control of linear time-delayed systems, Syst. Contr. Lett. 82 (2015), 40–50.

28. Mirhosseini-Alizamini, S.M., Effati, S. and Heydari, A. Solution of linear time-varying multi-delay systems via variational iteration method, J. Math. Comput. Sci. 16 (2016), 282–297.

29. Nazemi, A. and Mansoori, M. Solving optimal control problems of the time-delayed systems by Haar wavelet, J. Vib. Contr. 22(11) (2014), 2657–2670.

30. Nazemi, A. and Shabani, M.M. Numerical solution of the time-delayed optimal control problems with hybrid functions, IMA J. Math. Contr. Inform. 32(3) (2015), 623–638.

31. Richard, J.P. Time-delay systems: An overview of some recent advances and open problems, Automatica, 39 (2003), 1667–1694.

32. Saberi Nik, H., Rebelo, P. and Zahedi, S. Solution of infinite horizon nonlinear optimal control problems by piecewise Adomian decomposition method, Math. Model. Anal. 18(4) (2013), 543–560.

33. Shehata, M.M. A study of some nonlinear partial differential equations by using Adomian decomposition method and variational iteration method, Am. J. Comput. Math. 5 (2015), 195–203.

34. Vanderbei, R.J. and Shanno, D.F. An interior-point algorithm for non convex nonlinear programming, Comput. Optim. Appl. 13 (1999), 231–252.

35. Wachter, A. and Biegler, L.T. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program. 106 (2006), 25–57.

36. Wang, X.T. Numerical solutions of optimal control for linear time varying systems with delays via hybrid functions, J. Franklin Inst. 344 (2007), 941–953.

Summer and Autumn 2019

Pages 165-183