Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T12:50:22.155Z Has data issue: false hasContentIssue false
  • English
  • Français

A WAVELET COLLOCATION SCHEME FOR SOLVING SOME OPTIMAL PATH PLANNING PROBLEMS

Part of: Existence theories Numerical methods in Fourier analysis

Published online by Cambridge University Press:  12 May 2016

MARZIYEH MORTEZAEE  and
ALIREZA NAZEMI
MARZIYEH MORTEZAEE*
Affiliation:
Department of Mathematics, School of Mathematical Sciences, Shahrood University of Technology, P.O. Box 3619995161-316, Shahrood, Iran email mortezaee91@gmail.com, nazemi20042003@yahoo.com
ALIREZA NAZEMI
Affiliation:
Department of Mathematics, School of Mathematical Sciences, Shahrood University of Technology, P.O. Box 3619995161-316, Shahrood, Iran email mortezaee91@gmail.com, nazemi20042003@yahoo.com
*
✉mortezaee91@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider an approximation scheme using Haar wavelets for solving optimal path planning problems. The problem is first expressed as an optimal control problem. A computational method based on Haar wavelets in the time domain is then proposed for solving the obtained optimal control problem. A Haar wavelets integral operational matrix and a direct collocation method are used to find an approximate optimal trajectory of the original problem. Numerical results are also presented for several examples to demonstrate the applicability and efficiency of the proposed method.

Keywords

optimal path planning problemsapproximationrationalized Haar functionsnonlinear programming

MSC classification

Primary: 49J15: Optimal control problems involving ordinary differential equations
Secondary: 65T60: Wavelets 90C30: Nonlinear programming
Type
Research Article
Information
The ANZIAM Journal , Volume 57 , Issue 4 , April 2016 , pp. 461 - 481
Copyright
© 2016 Australian Mathematical Society 

References

Borzabadi, A. H., Kamyad, A. V., Farahi, M. H. and Mehne, H. H., “Solving some optimal path planning problems using an approach based on measure theory”, Appl. Math. Comput. 170 (2005) 1418–1435; doi:10.1016/j.amc.2005.01.035.Google Scholar
Burrus, C. S, Gopinath, R. A. and Guo, H., Introduction to wavelets and wavelet transforms: a primer (Prentice Hall, Upper Saddle River, NJ, 1998).Google Scholar
Chang, R. Y. and Wang, M. L., “Legendre polynomials approximation to dynamic linear state equations with initial or boundary value conditions”, Internat. J. Control 40 (1984) 215–232;doi:10.1080/00207178408933269.Google Scholar
Chen, C. F. and Hsiao, C. H., “A state-space approach to Walsh series solution of linear systems”, Internat. J. Systems Sci. 6 (1975) 833–858; doi:10.1080/00207727508941868.Google Scholar
Dai, R. and Cochran, J. E., “Wavelet collocation method for optimal control problems”, J. Optim. Theory Appl. 143 (2009) 265–278; doi:10.1007/s10957-009-9565-9.Google Scholar
Horng, I. R. and Chou, J. H., “Analysis, parameter estimation and optimal control of time-delay systems via Chebyshev series”, Internat. J. Control 41 (1985) 1221–1234;doi:10.1080/0020718508961193.Google Scholar
Hsiao, C. H. and Wang, W. J., “State analysis and parameter estimation of bilinear systems via Haar wavelets”, IEEE Trans. Circuits Syst. I. Fundam. Theor. Appl. 47 (2000) 246–250;doi:10.1109/81.828579.Google Scholar
Hwang, C. and Shih, Y. P., “Laguerre operational matrices for fractional calculus and applications”, Internat. J. Control 34 (1981) 577–584; doi:10.1080/00207178108922549.Google Scholar
Karimi, H. R., Lohmann, B., Jabehdar Maralani, P. and Moshiri, B., “A computational method for solving optimal control and parameter estimation of linear systems using Haar wavelets”, Int. J. Comput. Math. 81 (2004) 1121–1132; doi:10.1080/03057920412331272225.Google Scholar
Karimi, H. R., Moshiri, B., Lohmann, B. and Jabehdar Maralani, P., “Haar wavelet-based approach for optimal control of second-order linear systems in time domain”, J. Dyn. Control Syst. 11 (2005) 237–252; doi:10.1007/s10883-005-4172-z.Google Scholar
Latombe, J. C., Robot motion planning (Kluwer Academic, Norwell, MA, 1991).Google Scholar
Li, B., Teo, K. L., Zhao, G. H. and Duan, G. R., “An efficient computational approach to a class of minimax optimal control problems with applications”, ANZIAM J. 51 (2009) 162–177;doi:10.1017/S1446181110000040.Google Scholar
Li, B., Xu, C., Teo, K. L. and Chu, J., “Time optimal Zermelo’s navigation problem with moving and fixed obstacles”, Appl. Math. Comput. 224 (2013) 866–875; doi:10.1016/j.amc.2013.08.092.Google Scholar
Liu, C., Loxton, R. and Teo, K. L., “A computational method for solving time-delay optimal control problems with free terminal time”, Systems Control Lett. 72 (2014) 53–60;doi:10.1016/j.sysconle.2014.07.001.Google Scholar
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;doi:10.1016/j.jfranklin.2003.12.011.Google Scholar
Marzban, H. R. and Razzaghi, M., “Solution of time-varying delay systems by hybrid functions”, Math. Comput. Simulation 64 (2004) 597–607; doi:10.1016/j.matcom.2003.10.003.Google Scholar
Marzban, H. R. and Razzaghi, M., “Rationalized Haar approach for nonlinear constrained optimal control problems”, Appl. Math. Model. 34 (2010) 174–183; doi:10.1016/j.apm.2009.03.036.Google Scholar
Mashayekhi, S., Ordokhani, Y. and Razzaghi, M., “Hybrid functions approach for nonlinear constrained optimal control problems”, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 1831–1843; doi:10.1016/j.cnsns.2011.09.008.Google Scholar
Nagura, T., Yamazato, T., Katayama, M., Yendo, T., Fujii, T. and Okada, H., “Improved decoding methods of visible light communication system for ITS using LED array and high-speed camera”, in: Proc. 71st IEEE Vehicular Technology Conference (VTC 2010 Spring), Taipei, Taiwan, 16–19 May 2010, (IEEE, 2010) 1–5; doi:10.1109/VETECS.2010.5493958.Google Scholar
Ohkita, M. and Kobayashi, Y., “An application of rationalized Haar functions to solution of linear differential equations”, IEEE Trans. Circuits Syst. I. Regul. Pap. 33 (1986) 853–862;doi:10.1109/TCS.1986.1086019.Google Scholar
Ohkita, M. and Kobayashi, Y., “An application of rationalized Haar functions to solution of linear partial differential equations”, Math. Comput. Simulation 30 (1988) 419–428;doi:10.1016/0378-4754(88)90055-9.CrossRefGoogle Scholar
Rao, G. P., Piecewise constant orthogonal functions and their application to systems and control (Springer, Berlin–Heidelberg, 2003).Google Scholar
Razzaghi, M. and Nazarzadeh, J., “Walsh functions”, in: Wiley encyclopedia of electrical and electronics engineering (1999); doi:10.1002/047134608X.W2465.Google Scholar
Razzaghi, M. and Ordokhani, Y., “Solution of differential equations via rationalized Haar functions”, Math. Comput. Simulation 56 (2001) 235–246; doi:10.1016/S0378-4754(01)00278-6.Google Scholar
Razzaghi, M. and Ordokhani, Y., “A rationalized Haar functions method for nonlinear Fredholm– Hammerstein integral equations”, Int. J. Comput. Math. 79 (2002) 333–343;doi:10.1080/00207160211932.Google Scholar
Razzaghi, M. and Razzaghi, M., “Fourier series direct method for variational problems”, Internat. J. Control 48 (1988) 887–895; doi:10.1080/00207178808906224.Google Scholar
Rosenthal, R. E. and Brooke, A., GAMS: a user’s guide (GAMS Development Corporation, Washington, DC, 2004) www.GAMS.com.Google Scholar
Wang, Y., Lane, D. M. and Falconer, G. J., “Two novel approaches for unmanned under water vehicle path planning: constrained optimisation and semi-infinite constrained optimisation”, Robotica 18 (2000) 123–142; doi:10.1017/S0263574799002015.Google Scholar
Zamirian, M., Farahi, M. H. and Nazemi, A. R., “An applicable method for solving the shortest path problems”, Appl. Math. Comput. 190 (2007) 1479–1486; doi:10.1016/j.amc.2007.02.057.Google Scholar