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Hybrid functions approach for nonlinear constrained optimal control problems. (English) Zbl 1239.49043
Summary: In this paper, a new numerical method for solving the nonlinear constrained optimal control with quadratic performance index is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrix of integration is introduced. This matrix is then utilized to reduce the solution of the nonlinear constrained optimal control to a nonlinear programming one to which existing well-developed algorithms may be applied. Illustrative examples are included to demonstrate the validity and applicability of the technique.

49M37Methods of nonlinear programming type in calculus of variations
90C30Nonlinear programming
11B68Bernoulli and Euler numbers and polynomials
Full Text: DOI
[1] Kirk, D. E.: Optimal control theory, (1970)
[2] Kleiman, D. L.; Fortmann, T.; Athans, M.: On the design of linear systems with piecewise-constant feedback gains, IEEE trans automat contr 13, 354-361 (1968)
[3] Drefus, S. F.: Variational problems with state variable inequality constraints, J math anal appl 4, 291-301 (1962) · Zbl 0119.16005 · doi:10.1016/0022-247X(62)90056-2
[4] Mehra, R. K.; Davis, R. E.: A generalized gradient method for optimal control problems with inequality constraints and singular arcs, IEEE trans automat contr 17, 69-72 (1972) · Zbl 0268.49038 · doi:10.1109/TAC.1972.1099881
[5] Pananisamy, K. R.; Rao, G. P.: Minimum energy control of time-delay systems via Walsh functions, Optim contr appl methods 4, 213-226 (1983) · Zbl 0513.93033 · doi:10.1002/oca.4660040303
[6] Chang, R. Y.; Wang, M. L.: Shifted Legendre direct method for variational problems, J optim theory appl 39, 299-307 (1983) · Zbl 0481.49004 · doi:10.1007/BF00934535
[7] Hwang, C.; Shih, Y. P.: Optimal control of delay systems via block-pulse functions, J optim theory appl 45, 101-112 (1985) · Zbl 0541.93031 · doi:10.1007/BF00940816
[8] Horng, I. R.; Chou, J. H.: Shifted Chebyshev direct method for solving variational problems, Int J syst sci 16, 855-861 (1985) · Zbl 0568.49019 · doi:10.1080/00207728508926718
[9] Vlassenbroeck, J.: A Chebyshev polynomial method for optimal control with constraints, Automatica 24, 499-506 (1988) · Zbl 0647.49023 · doi:10.1016/0005-1098(88)90094-5
[10] Vlassenbroeck, J.; Van Dooren, R.: A Chebyshev technique for solving nonlinear optimal control problems, IEEE trans automat contr 33, 333-340 (1988) · Zbl 0643.49027 · doi:10.1109/9.192187
[11] Yen, V.; Nagurka, M.: Linear quadratic optimal control via Fourier-based state parameterization, J dyn syst measure contr 11, 206-215 (1991) · Zbl 0765.49022 · doi:10.1115/1.2896367
[12] Elnegar, G. N.; Kazemi, M. A.: Pseudospectral Chebyshev optimal control of constrained nonlinear dynamical systems, Comput optim applica 11, 195-217 (1998) · Zbl 0914.93024 · doi:10.1023/A:1018694111831
[13] Marzban, H. R.; Razzaghi, M.: Hybrid functions approach for linearly constrained quadratic optimal control problems, Appl math modell 27, 471-485 (2003) · Zbl 1020.49025 · doi:10.1016/S0307-904X(03)00050-7
[14] Ordokhani, Y.; Razzaghi, M.: Linear quadratic optimal control problems with inequality constraints via rationalized Haar functions, Dyn contin discrete impul syst ser B 12, 761-773 (2005) · Zbl 1081.49026
[15] Marzban, H. R.; Razzaghi, M.: Rationalized Haar approach for nonlinear constrined optimal control problems, Appl math modell 34, 174-183 (2010) · Zbl 1185.49032 · doi:10.1016/j.apm.2009.03.036
[16] Razzaghi, M.; Elnagar, G.: Linear quadratic optimal control problems via shifted Legendre state parametrization, Int J syst sci 25, 393-399 (1994) · Zbl 0810.49037 · doi:10.1080/00207729408928967
[17] Razzaghi, M.; Razzaghi, M.: Instabilities in the solution of a heat conduction problem using Taylor series and alternative approaches, J Frank instit 32, 683-690 (1989) · Zbl 0684.34012 · doi:10.1016/0016-0032(89)90026-4
[18] Razzaghi, M.; Marzban, H. R.: Direct method for variational problems via hybrid of block-pulse and Chebyshev functions, Math prob eng 6, 85-97 (2000) · Zbl 0987.65055 · doi:10.1155/S1024123X00001265
[19] Kajani, M. T.; Vencheh, A. H.: Solving second kind integral equations with hybrid Chebyshev and block-pulse functions, Appl math comput 163, 71-77 (2005) · Zbl 1067.65151 · doi:10.1016/j.amc.2003.11.044
[20] Wang, X. T.; Li, Y. M.: Numerical solutions of integro differential systems by hybrid of general block-pulse functions and the second Chebyshev polynomials, Appl math comput 209, 266-272 (2009) · Zbl 1161.65099 · doi:10.1016/j.amc.2008.12.044
[21] Marzban, H. R.; Razzaghi, M.: Optimal control of linear delay systems via hybrid of block-pulse and Legendre polynomials, J Frank inst 341, 279-293 (2004) · Zbl 1070.93028 · doi:10.1016/j.jfranklin.2003.12.011
[22] Hsiao, C. H.: Hybrid function method for solving Fredholm and Volterra integral equations of the second kind, J comput appl math 230, 59-68 (2009) · Zbl 1167.65473 · doi:10.1016/j.cam.2008.10.060
[23] Singh, V. K.; Pandey, R. K.; Singh, S.: A stable algorithm for Hankel transforms using hybrid of block-pulse and Legendre polynomials, Comput phys commun 181, 1-10 (2010) · Zbl 1205.65328 · doi:10.1016/j.cpc.2009.08.002
[24] Marzban, H. R.; Razzaghi, M.: Analysis of time-delay systems via hybrid of block-pulse functions and Taylor series, J vibr contr 11, 1455-1468 (2005) · Zbl 1182.93067 · doi:10.1177/1077546305058662
[25] Marzban, H. R.; Razzaghi, M.: Solution of multi-delay systems using hybrid of block-pulse functions and Taylor series, J sound vibr 292, 954-963 (2006) · Zbl 1243.93050
[26] Avrile, M.: Nonlinear programming, Analysis and methods (1976)
[27] Belegundu, A. D.; Arora, J. S.: A study of mathematical programming methods for structural optimization. Part II, Int J numer methods eng 21, 1601-1623 (1985) · Zbl 0585.73160 · doi:10.1002/nme.1620210905
[28] Gill, P. E.; Murray, W.: Linearly constrained problems including linear and quadratic programming, The state of the art in numerical analysis (1977)
[29] Nelder, J. A.; Mead, R. A.: A simplex method for function minimization, Comput J 7, 308-313 (1965) · Zbl 0229.65053
[30] Powell, M. J. D.: An efficient method for finding the minimum of a function of several variables without calculating the derivatives, Comput J 7, 155-162 (1964) · Zbl 0132.11702 · doi:10.1093/comjnl/7.2.155
[31] Schittkowskki, K.: NLPQL: a Fortran subroutine for solving constrained nonlinear programming problems, Ann operat res 5, 485-500 (1985)
[32] Costabile, F.; Dellaccio, F.; Gualtieri, M. I.: A new approach to Bernoulli polynomials, Rendiconti di matemat 26, 1-12 (2006) · Zbl 1105.11002
[33] Arfken, G.: Mathematical methods for physicists, (1985) · Zbl 0135.42304
[34] Kreyszig, E.: Introductory functional analysis with applications, (1978) · Zbl 0368.46014
[35] Lancaster, P.: Theory of matrixes, (1969) · Zbl 0186.05301
[36] Jaddu, H.: Direct solution of nonlinear optimal control problems using quasilinearization and Chebyshev polynomials, J Frank inst 339, 479-498 (2002) · Zbl 1010.93507 · doi:10.1016/S0016-0032(02)00028-5
[37] Teo, K. L.; Wong, K. H.: Nonlinearly constrained optimal control problems, J austral math soc ser B 33, 507-530 (1992) · Zbl 0764.49017 · doi:10.1017/S0334270000007207