Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-11T10:41:52.917Z Has data issue: false hasContentIssue false
  • English
  • Français

Computing Switching Surfaces in Optimal Control Based on Triangular Decomposition

Published online by Cambridge University Press:  16 July 2018

Xiaoliang Li ,
Yanli Huang ,
Zewei Zheng  and
Wanyou Cheng
Xiaoliang Li
Affiliation:
School of Computer Science, Dongguan University of Technology, Dongguan, Guangdong 523808, China. Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Guangxi University for Nationalities, Nanning 530006, China.
Yanli Huang*
Affiliation:
School of Computer Science and Software Engineering, Tianjin Polytechnic University, Tianjin 300387, China. Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Guangxi University for Nationalities, Nanning 530006, China.
Zewei Zheng
Affiliation:
School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China.
Wanyou Cheng
Affiliation:
School of Computer Science, Dongguan University of Technology, Dongguan, Guangdong 523808, China.
*
*Corresponding author.Email address:huangyanlibuaa@gmail.com
Get access

Abstract

Various algorithms for optimal control require the explicit determination of switching surfaces. However, switching strategies may be very complicated, such that the computation of switching surfaces is quite challenging. General methods are proposed here to compute switching surfaces systematically, based on algebraic computational tools such as triangular decomposition. Our methods are highly complex compared to some widely-used numerical options, but they can be made feasible for realtime applications by moving the computational burden off-line. The tutorial-style presentation is intended to introduce potentially powerful symbolic computation methods to system scientists in particular, and an illustrative example of time-optimal control is given to show the effectiveness and generality of our approach.

Keywords

13P1513P25Optimal controlswitching surfacetriangular decompositionsemi-algebraic system
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 4 , Issue 4 , November 2014 , pp. 345 - 367
Copyright
Copyright © Global-Science Press 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Äthans, M. and Falb, P.L., Optimal Control: An Introduction to the Theory and Its Applications, McGraw-Hill, New York (1966).Google Scholar
[2] Chai, F., Gao, X.-S. and Yuan, C., A characteristic set method for solving Boolean equations and applications in cryptanalysis of stream ciphers, J. Systems Science and Complexity 21, 191–208 (2008).CrossRefGoogle Scholar
[3] Chen, C. and Moreno Maza, M., Algorithms for computing triangular decompositions of polynomial systems, Proc. 36th International Symposium on Symbolic and Algebraic Computation (Leykin, A., ed.), pp. 83–90, ACM, New York (2011).Google Scholar
[4] Chou, S.-C. and Gao, X.-S., Ritt-Wu's decomposition algorithm and geometry theorem proving, Proc. 10th International Conference on Automated Deduction (Stickel, M.E., ed.), pp. 207–220, Springer, Berlin (1990).Google Scholar
[5] Collins, G.E. and Hong, H., Partial cylindrical algebraic decomposition for quantifier elimination, J. Symbolic Comp. 12, 299–328 (1991).CrossRefGoogle Scholar
[6] Collins, G.E. and Loos, R., Real zeros of polynomial, in Computer Algebra: Symbolic and Algebraic Computation (Buchberger, B., Collins, G.E. and Loos, R., eds.), pp. 83–94, Springer, New York (1983).Google Scholar
[7] Deuflhard, P., Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Springer, Berlin (2011).CrossRefGoogle Scholar
[8] Fiene, J. and Niemeyer, G., Toward switching motor control, IEEE/ASME Trans. Mechatronics 11, 27–34 (2006).Google Scholar
[9] Gallo, G. and Mishra, B., Efficient algorithms and bounds for Wu-Ritt characteristic sets, in Effective Methods in Algebraic Geometry (Mora, T. and Traverso, C., eds.), pp. 119–142. Birkhauser, Berlin (1991).Google Scholar
[10] Gallo, G. and Mishra, B., Wu-Ritt characteristic sets and their complexity, in Discrete and Computational Geometry: Papers from the DIMACS Special Year (Goodman, J.E., Pollack, R.D. and Steiger, W., eds), pp. 111–136, American Mathematical Society, Providence (1991).CrossRefGoogle Scholar
[11] Gao, X.-S. and Chou, S.-C.. On the dimension of an arbitrary ascending chain. Chinese Science Bulletin, 38(5):396–399, 1993.Google Scholar
[12] Gao, X.-S. and Huang, Z., Characteristic set algorithms for equation solving in finite fields, J. Symbolic Comp. 47, 655–679 (2012).Google Scholar
[13] Ge, T. and Chang, J.S., Bang-Bang control class-d amplifiers: power-supply noise, IEEE Trans. Circuits and Systems II: Express Briefs 55, 723–727 (2008).Google Scholar
[14] Jin, M., Li, X. and Wang, D., A new algorithmic scheme for computing characteristic sets, J. Symbolic Comp. 51, 433–449 (2012).Google Scholar
[15] Kalkbrener, M., A generalized Euclidean algorithm for computing triangular representations of algebraic varieties, J. Symbolic Comp. 15, 143–167 (1993).CrossRefGoogle Scholar
[16] Kumar, B.S. and Ng, A., A Bang-Bang control approach to manoeuver spacecraft in a formation with differential drag, Proc. AIAA Guidance, Navigation and Control Conference and Exhibit, pp. 18–21, American Institute of Aeronautics and Astronautics, Reston (2008).Google Scholar
[17] Lazard, D., A new method for solving algebraic systems of positive dimension, Discrete Applied Math. 33 147–160 (1991).Google Scholar
[18] Lee, E.B. and Markus, L., Foundations of Optimal Control Theory, Krieger, Malabar (1967).Google Scholar
[19] Li, X., Mou, C. and Wang, D., Decomposing polynomial sets into simple sets over finite fields: the zero-dimensional case, Computers and Mathematics with Applications 60, 2983–2997 (2010).Google Scholar
[20] Li, X. and Wang, D., Computing equilibria of semi-algebraic economies using triangular decomposition and real solution classification, ArXiv e-prints arXiv:1308.5029 (2013).Google Scholar
[21] Li, Y., Some properties of triangular sets and improvement upon algorithm CharSer, in Artificial Intelligence and Symbolic Computation (Calmet, J., Ida, T., and Wang, D., eds), pp. 82–93, Springer, New York (2006).Google Scholar
[22] Mishra, B., Algorithmic Algebra, Springer, New York (1993).Google Scholar
[23] Moreno Maza, M., On triangular decompositions of algebraic varieties, Technical Report 4/99, NAG, UK. (Presented at the MEGA-2000 Conference, Bath.)Google Scholar
[24] Pao, L.Y. and Franklin, G.F., Proximate time-optimal control of third-order servomechanisms, IEEE Trans. Automatic Control 38, 560–580 (1993).CrossRefGoogle Scholar
[25] Ritt, J.F., Differential Algebra, American Mathematical Society, Providence (1950).CrossRefGoogle Scholar
[26] Rubagotti, M., Della Vedova, M.L, and Ferrara, A., Time-optimal sliding-mode control of a mobile robot in a dynamic environment, IET Control Theory and Applications 5, 1916–1924 (2011).CrossRefGoogle Scholar
[27] El Din, M.S., Testing sign conditions on a multiv ariate polynomial and applications, Mathematics in Comp. Sc. 1, 177–207 (2007).CrossRefGoogle Scholar
[28] El Din, M.S. and Schost, E., Polar varieties and computation of one point in each connected component of a smooth real algebraic set, Proc. 2003 International Symposium on Symbolic and Algebraic Computation, pp. 224–231. ACM, New York (2003).Google Scholar
[29] Sommese, A.J. and Wampler, C.W., The Numerical Solution of Systems of Polynomials Arising in Engineering and Science, World Scientific, Singapore (2005).CrossRefGoogle Scholar
[30] Walther, U., Georgiou, T.T. and Tannenbaum, A., On the computation of switching surfaces in optimal control: a Gröbner basis approach, IEEE Trans. Automatic Control 46, 534–540 (2001).CrossRefGoogle Scholar
[31] Wang, D., An elimination method for polynomial systems, J. Symbolic Comp. 16, 83–114 (1993).Google Scholar
[32] Wang, D., Decomposing polynomial systems into simple systems, J. Symbolic Comp. 25, 295–314 (1998).CrossRefGoogle Scholar
[33] Wang, D., Computing triangular systems and regular systems, J. Symbolic Comp. 30, 221–236 (2000).Google Scholar
[34] Wang, D., A generalized algorithm for computing characteristic sets, in Computer Mathematics: Proc. Fifth Asian Symposium (Shirayanagi, K. and Yokoyama, K., eds.), pp. 165–174, Springer, Berlin (2001).CrossRefGoogle Scholar
[35] Wang, D., A strategy for speeding-up the computation of characteristic sets., Proc. 17th International Symposium on Mathematical Foundations of Computer Science (Havel, I.M. and Koubek, V., eds.), pp. 504–510, Springer, Berlin (1992).Google Scholar
[36] Wang, D., Elimination Methods, Springer, New York (2001).CrossRefGoogle Scholar
[37] Wu, W.-T., Basic principles of mechanical theorem proving in elementary geometries, J. Automated Reasoning 2, 221–252 (1986).Google Scholar
[38] Wu, W.-T., On zeros of algebraic equations: an application of Ritt principle, Kexue Tongbao 31, 1–5 (1986).Google Scholar
[39] Yang, L. and Zhang, J.-Z., Searching dependency between algebraic equations: an algorithm applied to automated reasoning, in Artificial Intelligence in Mathematics (Johnson, J., McKee, S. and Vella, A., eds.), pp. 147–156, Oxford University Press, Oxford (1994).Google Scholar
[40] Yang, L., Hou, X. and Xia, B., A complete algorithm for automated discovering of a class of inequality-type theorems, Science in China Series F 44, 33–49 (2001).Google Scholar
[41] Zhao, Y. and Tsiotras, P., Time-optimal path following for fixed-wing aircraft, J. Guidance, Control and Dynamics 36, 83–95 (2012).CrossRefGoogle Scholar