Towards a computer algebraic algorithm for flat output determination

ABSTRACT

This contribution deals with nonlinear control systems. More precisely, we are interested in the formal computation of a so-called flat output, a particular generalized output whose property is, roughly speaking, that all the integral curves of the system may be expressed as smooth functions of the components of this flat output and their successive time derivatives up to a finite order (to be determined). Recently, a characterization of such flat output has been obtained in the framework of manifolds of jets of infinite order that yields an abstract algorithm for its computation. In this paper it is discussed how these conditions can be checked using computer algebra. All steps of the algorithm are discussed for the simple (but rich enough) example of a non holonomic car.

REFERENCES

• 1: E. Aranda-Bricaire, C. Moog, and J.--B. Pomet. A linear algebraic framework for dynamic feedback linearization. IEEE Trans. Automat. Contr., 40(1):127--132, 1995.
• 2: B. Charlet , J. Lévine , R. Marino, Sufficient conditions for dynamic state feedback linearization, SIAM Journal on Control and Optimization, v.29 n.1, p.38-57, Jan. 1991 [doi>10.1137/0329002]
• 3: S. Chern, W. Chen, and K. Lam. Lectures on Differential Geometry, volume 1 of Series on University Mathematics. World Scientific, 2000.
• 4: V. Chetverikov. New flatness conditions for control systems. In Proceedings of NOLCOS'01, St. Petersburg, pages 168--173, 2001.
• 5: P. Cohn. Free Rings and Their Relations. Academic Press, London, 1985.
• 6: Michael Fliess, A remark on Willems' trajectory characterization of linear controllability, Systems & Control Letters, v.19 n.1, p.43-45, July 1992 [doi>10.1016/0167-6911(92)90038-T]
• 7: M. Fliess, J. Lévine, P. Martin, and P. Rouchon. Sur les systèmes non linéaires différentiellement plats. C.R. Acad. Sci. Paris, I--315:619--624, 1992.
• 8: M. Fliess, J. Lévine, P. Martin, and P. Rouchon. Flatness and defect of nonlinear systems: introductory theory and examples. Int. J. Control, 61(6):1327--1361, 1995.
• 9: M. Fliess, J. Lévine, P. Martin, and P. Rouchon. A Lie-Backlund approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control, 44(5):922--937, 1999.
• 10: J. Franch. Flatness, Tangent Systems and Flat Outputs. PhD thesis, Universitat Politècnica de Catalunya Jordi Girona, 1999.
• 11: I. Gohberg, P. Lancaster, and L. Rodman. Matrix Polynomials. Academic Press, Berlin, 1982.
• 12: B. Jakubczyk. Invariants of dynamic feedback and free systems. In Proc. ECC'93, Groningen, pages 1510--1513, 1993.
• 13: I. S. Krasil'shchik, V. V. Lychagin, and A. M. Vinogradov. Geometry of Jet Spaces and Nonlinear Partial Differential Equations. Gordon and Breach, New York, 1986.
• 14: J. Lévine. On flatness necessary and sufficient conditions. In Proc. of IFAC NOLCOS 2004 Conference, Stuttgart, 2004.
• 15: J. Lévine. On necessary and sufficient conditions for differential flatness. arXiv:math.OC/0605405v2, pages 1--28, 2006.
• 16: J. Lévine and D. Nguyen. Flat output characterization for linear systems using polynomial matrices. Systems & Control Letters, 48:69--75, 2003.
• 17: P. Martin. Contribution à l'étude des systèmes diffèrentiellement plats. PhD thesis, École des Mines de Paris, 1992.
• 18: P. Martin, R. Murray, and P. Rouchon. Flat systems. In G. Bastin and M. Gevers, editors, Plenary Lectures and Minicourses, Proc. ECC 97, Brussels, pages 211--264, 1997.
• 19: P. Martin and P. Rouchon. Systems without drift and flatness. In Proc. MTNS 93, Regensburg, Germany, August 1993.
• 20: Paulo SérgioPereira da, Relative Flatness and Flatness of Implicit Systems, SIAM Journal on Control and Optimization, v.39 n.6, p.1929-1951, 2000 [doi>10.1137/S0363012998349832]
• 21: P. S. Pereira da Silva. Flatness of nonlinear control systems : a Cartan-Kahler approach. In Proc. Mathematical Theory of Networks and Systems (MTNS'2000), Perpignan, pages 1--10, 2000.
• 22: J.-B. Pomet. A differential geometric setting for dynamic equivalence and dynamic linearization. In B. Jakubczyk, W. Respondek, and T. Rzezuchowski, editors, Geometry in Nonlinear Control and Differential Inclusions, pages 319--339. Banach Center Publications, Warsaw, 1993.
• 23: J.-B. Pomet. On dynamic feedback linearization of four-dimensional affine control systems with two inputs. ESAIM-COCV, 1997. http://www.emath.fr/Maths/Cocv/Articles/articleEng.html.
• 24: Muruhan Rathinam , Richard M. Murray, Configuration Flatness of Lagrangian Systems Underactuated by One Control, SIAM Journal on Control and Optimization, v.36 n.1, p.164-179, Jan. 1998 [doi>10.1137/S0363012996300987]
• 25: P. Rouchon. Necessary condition and genericity of dynamic feedback linearization. J. Math. Systems Estim. Control, 4(2):257--260, 1994.
• 26: P. Rouchon, M. Fliess, J. Lévine, and P. Martin. Flatness and motion planning: the car with n-trailers. In Proc. ECC'93, Groningen, pages 1518--1522, 1993.
• 27: P. Rouchon, M. Fliess, J. Lévine, and P. Martin. Flatness, motion planning and trailer systems. In Proc. IEEE Conf. Decision and Control, San Antonio, December 1993.
• 28: W. F. Shadwick, Absolute equivalence and dynamic feedback linearization, Systems & Control Letters, v.15 n.1, p.35-39, Jul. 1990 [doi>10.1016/0167-6911(90)90041-R]
• 29: Willem M. Sluis, A necessary condition for dynamic feedback linearization, Systems & Control Letters, v.21 n.4, p.277-283, Oct. 1993 [doi>10.1016/0167-6911(93)90069-I]
• 30: M. van Nieuwstadt , M. Rathinam , R. M. Murray, Differential Flatness and Absolute Equivalence of Nonlinear Control Systems, SIAM Journal on Control and Optimization, v.36 n.4, p.1225-1239, July 1, 1998 [doi>10.1137/S0363012995274027]
• 31: V. Zharinov. Geometrical Aspect of Partial Differential Equations. World Scientific, Singapore, 1992.

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.4 Applications

Additional Classification:
  G. Mathematics of Computing
  G.4 MATHEMATICAL SOFTWARE

General Terms:
Theory

Keywords:
control systems, differential flatness