Abstract
In this paper, an algorithm for solving optimal control of linear time-varying systems with quadratic performance indices is presented. By using important matrices which are derived from Chebyshev wavelets properties, the original problem is converted to a quadratic programming one. This parameter optimization method is applied on both constrained and unconstrained control systems having linear state equations of integer and fractional orders. The computing time saved by this approach is much better than with other methods in which there is no need to calculate the optimal costs of systems by substituting the approximations of the state and control vectors and their values are default outputs of the quadprog solvers.
References
[1] H. Adibi and P. Assari, Chebyshev wavelet method for numerical solution of Fredholm integral equations of the first kind, Math. Probl. Eng. 2010 (2010), Article ID 138408. 10.1155/2010/138408Search in Google Scholar
[2] B. Chachuat, Nonlinear and dynamic optimization: From theory to practice, LA-TEACHING-2007-001, École Polytechnique Fédeérale de Lausanne, 2007. Search in Google Scholar
[3] C. F. Chen and C. H. Hsiao, A Walsh series direct method for solving variational problems, J. Franklin Inst. 300 (1975), no. 4, 265–280. 10.1016/0016-0032(75)90199-4Search in Google Scholar
[4] J. H. Chou and I. R. Horng, Application of Chebyshev polynomials to the optimal control of time-varying linear systems, Internat. J. Control 41 (1985), no. 1, 135–144. 10.1080/0020718508961115Search in Google Scholar
[5] P. R. Clement, Laguerre functions in signal analysis and parameter identification, J. Franklin Inst. 313 (1982), 85–95. 10.1016/0016-0032(82)90070-9Search in Google Scholar
[6] G. N. Elnagar, State-control spectral Chebyshev parameterization for linearly constrained quadratic optimal control problems, J. Comput. Appl. Math. 79 (1997), no. 1, 19–40. 10.1016/S0377-0427(96)00134-3Search in Google Scholar
[7] C. H. Hsiao and W. J. Wang, Optimal control of linear time-varying systems via Haar wavelets, J. Optim. Theory Appl. 103 (1999), no. 3, 641–655. 10.1023/A:1021740209084Search in Google Scholar
[8] H. Jaddu, Numerical Methods for solving optimal control problems using Chebyshev polynomials, Ph.D. thesis, Japan Advanced Institute of Science and Technology, Japan, 1998. Search in Google Scholar
[9] H. Jaddu, Optimal control of time-varying linear systems using wavelets, Ph.D. thesis, Japan Advanced Institute of Science and Technology, Japan, 2006. Search in Google Scholar
[10] E. Keshavarz, Y. Ordokhani and M. Razzaghi, A numerical solution for fractional optimal control problems via Bernoulli polynomials, J. Vib. Control 22 (2016), no. 18, 3889–3903. 10.1177/1077546314567181Search in Google Scholar
[11] D. E. Kirk, Optimal Control Theory: An Introduction, Dover Books, Mineola, 2012. Search in Google Scholar
[12] Y. Liao, H. Li and W. Bao, Indirect Radau pseudospectral method for the receding horizon control problem, Chinese J. Aeronautics 1 (2016), no. 29, 215–227. 10.1016/j.cja.2015.12.023Search in Google Scholar
[13] C. Liu and Y. Shih, Analysis and optimal control of time varying systems via Chebyshev polynomials, Internat. J. Control 38 (1983), 1003–1012. 10.1080/00207178308933124Search in Google Scholar
[14] P. Lu, Closed-form control laws for linear time-varying systems, IEEE Trans. Automat. Control 45 (2000), no. 3, 537–542. 10.1109/9.847739Search in Google Scholar
[15] I. Malmir, Optimal control of linear time-varying systems with state and input delays by Chebyshev wavelets, Stat. Optim. Inf. Comput. 4 (2017), no. 5, 302–324. 10.19139/soic.v5i4.341Search in Google Scholar
[16] I. Malmir, A novel wavelet-based optimal linear quadratic tracker for time-varying systems with multiple delays, preprint (2018), https://arxiv.org/abs/1802.05618. 10.19139/soic-2310-5070-1228Search in Google Scholar
[17] I. Malmir, A new fractional integration operational matrix of Chebyshev wavelets in fractional delay systems, Fractal Fract. 3 (2019), 10.3390/fractalfract3030046. 10.3390/fractalfract3030046Search in Google Scholar
[18] I. Malmir, Novel Chebyshev wavelets algorithms for optimal control and analysis of general linear delay models, Appl. Math. Model. 69 (2019), 621–647. 10.1016/j.apm.2018.12.009Search in Google Scholar
[19] J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman & Hall/CRC, Boca Raton, 2003. 10.1201/9781420036114Search in Google Scholar
[20] Z. Rafiei, B. Kafash and S. M. Karbassi, A new approach based on using Chebyshev wavelets for solving various optimal control problems, Comput. Appl. Math. 37 (2018), S144–S157. 10.1007/s40314-017-0419-zSearch in Google Scholar
[21] A. V. Rao, A survey of numerical methods for optimal control, Adv. Astronautical Sci. 1 (2009), no. 135, 497–528. Search in Google Scholar
[22] M. Razzaghi, Optimal control of linear time-varying systems via Fourier series, J. Optim. Theory Appl. 65 (1990), no. 2, 375–384. 10.1007/BF01102353Search in Google Scholar
[23] M. Razzaghi and G. N. Elnagar, Linear quadratic optimal control problems via shifted Legendre state parametrization, Internat. J. Systems Sci. 25 (1994), no. 2, 393–399. 10.1080/00207729408928967Search in Google Scholar
[24] P. K. Sahu and S. Saha Ray, Chebyshev wavelet method for numerical solutions of integro-differential form of Lane–Emden type differential equations, Int. J. Wavelets Multiresolut. Inf. Process. 15 (2017), no. 2, Article ID 1750015. 10.1142/S0219691317500151Search in Google Scholar
[25] S. J. Tan and W. X. Zhong, Numerical solutions of linear quadratic control for time-varying systems via symplectic conservative perturbation, Appl. Math. Mech. 28 (2007), no. 3, 253–262. 10.1007/s10483-007-0301-1Search in Google Scholar
[26] H. L. Tidke, Some theorems on fractional semilinear evolution equations, J. Appl. Anal. 18 (2012), no. 2, 209–224. 10.1515/jaa-2012-0014Search in Google Scholar
[27] J. Vlassenbroeck, A Chebyshev polynomial method for optimal control with state constraints, Automatica J. IFAC 24 (1988), no. 4, 499–506. 10.1016/0005-1098(88)90094-5Search in Google Scholar
[28] J. Vlassenbroeck and R. Van Dooren, A Chebyshev technique for solving nonlinear optimal control problems, IEEE Trans. Automat. Control 33 (1988), no. 4, 333–340. 10.1109/9.192187Search in Google Scholar
[29] S.-K. Wang and M. L. Nagurka, A Chebyshev-based state representation for linear quadratic optimal control, J. Dynam. Sys. Measurement Control 1 (1993), no. 115, 1–6. 10.1115/1.2897400Search in Google Scholar
[30] W. Zhang and H. Ma, The Chebyshev–Legendre collocation method for a class of optimal control problems, Int. J. Comput. Math. 85 (2008), no. 2, 225–240. 10.1080/00207160701417381Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
