Transforming linear time-varying optimal control problems with quadratic criteria into quadratic programming ones via wavelets
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.
Bibliogr. 30 poz., wykr.
-  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.
-  B. Chachuat, Nonlinear and dynamic optimization: From theory to practice, LA-TEACHING-2007-001, École Polytechnique Fédeérale de Lausanne, 2007.
-  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.
-  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.
-  P. R. Clement, Laguerre functions in signal analysis and parameter identification, J. Franklin Inst. 313 (1982), 85-95.
-  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.
-  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.
-  H. Jaddu, Numerical Methods for solving optimal control problems using Chebyshev polynomials, Ph.D. thesis, Japan Advanced Institute of Science and Technology, Japan, 1998.
-  H. Jaddu, Optimal control of time-varying linear systems using wavelets, Ph.D. thesis, Japan Advanced Institute of Science and Technology, Japan, 2006.
-  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.
-  D. E. Kirk, Optimal Control Theory: An Introduction, Dover Books, Mineola, 2012.
-  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.
-  C. Liu and Y. Shih, Analysis and optimal control of time varying systems via Chebyshev polynomials, Internat. J. Control 38 (1983), 1003-1012.
-  P. Lu, Closed-form control laws for linear time-varying systems, IEEE Trans. Automat. Control 45 (2000), no. 3, 537-542.
-  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.
-  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.
-  I. Malmir, A new fractional integration operational matrix of Chebyshev wavelets in fractional delay systems, Fractal Fract. 3 (2019), DOI 10.3390/fractalfract3030046.
-  I. Malmir, Novel Chebyshev wavelets algorithms for optimal control and analysis of general linear delay models, Appl. Math. Model. 69 (2019), 621-647.
-  J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman & Hall/CRC, Boca Raton, 2003.
-  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.
-  A. V. Rao, A survey of numerical methods for optimal control, Adv. Astronautical Sci. 1 (2009), no. 135, 497-528.
-  M. Razzaghi, Optimal control of linear time-varying systems via Fourier series, J. Optim. Theory Appl. 65 (1990), no. 2, 375-384.
-  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.
-  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.
-  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.
-  H. L. Tidke, Some theorems on fractional semilinear evolution equations, J. Appl. Anal. 18 (2012), no. 2, 209-224.
-  J. Vlassenbroeck, A Chebyshev polynomial method for optimal control with state constraints, Automatica J. IFAC 24 (1988), no. 4, 499-506.
-  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.
-  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.
-  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.
Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).