Multi-input control-affine systems static feedback equivalent to a triangular form and their flatness

References

• Bryant, R. (1979). Some aspects of the local and global theory of Pfaffian systems (Unpublished doctoral dissertation). University of North Carolina, Chapel Hill.
   Google Scholar
• Bryant, R., Chern, S., Gardner, R., Goldschmidt, H., & Griffiths, P. (1991). Exterior differential systems. New York, NY: Springer-Verlag.
   Google Scholar
• Cartan, M. (1914). Sur l’équivalence absolue de certains systèmes d’équations différentielles et sur certaines familles de courbes [On the absolute equivalence of certain systems of differential equations and on certain families of curves]. Bulletin de la Société mathématique de France, 2(42), 12–48.
   Google Scholar
• Fliess, M., Levine, J., Martin, P., & Rouchon, P. (1992). Sur les systemes non linéaires différentiellement plats [On nonlinear differential flat systems]. Comptes Rendus de l’Académie des Sciences Paris Séries I - Mathematics, 315(5), 619–624.
   Google Scholar
• Fliess, M., Lévine, J., Martin, P., & Rouchon, P. (1995). Flatness and defect of non-linear systems: introductory theory and examples. International Journal of Control, 61(6), 1327–1361.
   Web of Science ®Google Scholar
• Fliess, M., Lévine, J., Martin, P., & Rouchon, P. (1999). A Lie-Bäcklund approach equivalence and flatness of nonlinear systems. IEEE Transactions on Automatic Control, 44(5), 922–937.
   Web of Science ®Google Scholar
• Giaro, A., Kumpera, A., & Ruiz, C. (1978). Sur la lecture correcte d'un résultat d’Elie Cartan [On a correct lecture of a result of Elie Cartan]. Comptes Rendus de l’Académie des Sciences Paris, 287, 241–244.
   Google Scholar
• Isidori, A., Moog, C., & De Luca, A. (1986). A sufficient condition for full linearization via dynamic state feedback. In Proceedings of 25th IEEE CDC (pp. 203–208). Athens.
   Google Scholar
• Jakubczyk, B. (1993). Invariants of dynamic feedback and free systems. In Proceedings of 2nd ECC (pp. 1510–1513). Groningen.
   Google Scholar
• Kai, T. (2006). Extended chained forms and their application to nonholonomic kinematic systems with affine constraints: Control of a coin on a rotating table. In Proceedings of 45th IEEE CDC (pp. 6104–6109). San Diego, CA.
   Google Scholar
• Kumpera, A., & Ruiz, C. (1982). Sur l’équivalence locale des systemes de Pfaff en drapeau. In F. Gherardelli (Ed.), Monge–Ampere Equations and Related Topics, Ist. Naz. Alta Math. F. Severi (pp. 201–247). Rome.
   Google Scholar
• Lévine, J. (2009). Analysis and control of nonlinear systems: A flatness-based approach. Berlin: Springer.
   Google Scholar
• Li, S., & Respondek, W. (2011). The geometry, controllability, and flatness property of the n-bar system. International Journal of Control, 84(5), 834–850.
   Web of Science ®Google Scholar
• Li, S., & Respondek, W. (2012). Flat outputs of two-input driftless control systems. ESAIM: Control, Optimisation and Calculus of Variations, 18, 774–798.
   Web of Science ®Google Scholar
• Li, S., Xu, C., & Chu, J. (2013). Characterization and flatness of the extended chained system. In Proceedings of 32th Chinese Control Conference (pp. 1047–1051). Xi'an, China.
   Google Scholar
• Martin, P. (1992). Contribution à l’étude des systèmes différentiellement plats (Unpublished doctoral dissertation). Paris: l’Ecole Nationale Supérieure de Mines de Paris.
   Google Scholar
• Martin, P., & Rouchon, P. (1993). Feedback linearization and driftless systems (CAS Internal Report(446)). Fontainebleau: Ecole des Mines de Paris.
   Google Scholar
• Martin, P., & Rouchon, P. (1994). Feedback linearization and driftless systems. Mathematics of Control, Signals and Systems, 7(3), 235–254.
   Web of Science ®Google Scholar
• Martin, P., Rouchon, P., & Murray, R. (2003). Flat systems, equivalence and trajectory generation (CDS Technical Report). Fontainebleau: Ecole des Mines de Paris.
   Google Scholar
• Montgomery, R., & Zhitomirskii, M. (2001). Geometric approach to Goursat flags. Annales de l’Institut Henri Poincaré. Analyse non linéaire, 18(4), 459–493.
   Web of Science ®Google Scholar
• Mormul, P. (2000). Goursat flags: Classification of codimension-one singularities. Journal of Dynamical and Control Systems, 6(3), 311–330.
   Google Scholar
• Mormul, P. (2004). Multi-dimensional Cartan prolongation and special k-flags. In H. Hironaka, S. Janeczko, & S. Lojasiewicz (Eds. ), Geometric singularity theory (pp. 157–178). Warsaw: Banach Center Publications
   Google Scholar
• Murray, R. (1994). Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems. Mathematics of Control, Signals and Systems, 7(1), 58–75.
   Web of Science ®Google Scholar
• Olver, P. (1995). Equivalence, invariants and symmetry. Cambridge: Cambridge University Press.
   Google Scholar
• Pasillas-Lépine, W., & Respondek, W. (2000). On geometry of control systems equivalent to canonical contact systems: regular points, singular points, and flatness. In Proceedings of 39th IEEE CDC (pp. 5151–5156). Sydney.
   Google Scholar
• Pasillas-Lépine, W., & Respondek, W. (2001a). Contact systems and corank one involutive subdistributions. Acta Applicandae Mathematica, 69(2), 105–128.
   Web of Science ®Google Scholar
• Pasillas-Lépine, W., & Respondek, W. (2001b). On the geometry of Goursat structures. ESAIM: Control, Optimisation and Calculus of Variations, 6, 119–181.
   Web of Science ®Google Scholar
• Pereira da Silva, P. (2001). Flatness of nonlinear control systems and exterior differential systems. In A. Isidori, F. Lamnabhi-Lagarrigue, & W. Respondek (Eds.). Nonlinear control in the year 2000 (Vol. 2, pp. 205–227). London: Springer.
   Google Scholar
• Pomet, J. (1995). A differential geometric setting for dynamic equivalence and dynamic linearization. Banach Center Publications, 32, 319–339.
   Google Scholar
• Pomet, J. (1997). On dynamic feedback linearization of four-dimensional affine control systems with two inputs. ESAIM: Control, Optimisation and Calculus of Variations, 2, 151–230.
   Google Scholar
• Respondek, W. (2001). Transforming nonholonomic control systems into the canonical contact form. In Proceedings of 40th IEEE CDC (pp. 1781–1786). Orlando, FL.
   Google Scholar
• Respondek, W. (2003). Symmetries and minimal flat outputs of nonlinear control systems. In W. Kang, M. Xiao & C. Borges (Eds.), New trends in nonlinear dynamics and control and their applications (pp. 65–86). Berlin: Springer.
   Google Scholar
• Respondek, W., & Pasillas-Lépine, W. (2001). Canonical contact systems for curves: A survey. In A. Anzaldo, B. Bonnard, J.P. Gauthier, & F. Monroy (Eds.), Contemporary trends in geometric control theory and applications (pp. 77–112). Singapore: World Scientific
   Google Scholar
• Schlacher, K., & Schoeberl, M. (2007). Construction of flat outputs by reduction and elimination. Proceedings of Nolcos 2007, 8(1), 666–671.
   Google Scholar
• Shibuya, K., & Yamaguchi, K. (2009). Drapeau theorem for differential systems. Differential Geometry and its Applications, 27(6), 793–808.
   Web of Science ®Google Scholar
• Silveira, H. (2010). Formas triangulares para sistemas não-lineares com duas entradas e controle de sistemas sem arrasto em SU(n) com aplicações em mecânica quântica (Unpublished doctoral dissertation). Sao Paulo: Escola Politécnica da Universidade de São Paulo.
   Google Scholar
• Silveira, H., Pereira da Silva, P., & Rouchon, P. (2013). A flat triangular form for nonlinear systemes with two inputs: Necessary and sufficient conditions. European Journal of Control, 22, 17–22.
   Web of Science ®Google Scholar
• Vinogradov, A., Krasilchchik, I., & Lychagin, V. (1986). Geometry of jet spaces and nonlinear partial differential equations. New York, NY: Gordon and Breach.
   Google Scholar
• Yamaguchi, K. (1982). Contact geometry of higher order. Japanese Journal of Mathematics, 8(1), 109–176.
   Google Scholar