Dynamic Feedback Linearization of Control Systems with Symmetry

Jeanne N. Clelland ; Taylor J. Klotz ; Peter J. Vassiliou

Symmetry, integrability and geometry: methods and applications, Tome 20 (2024)

Résumé

Control systems of interest are often invariant under Lie groups of transformations. For such control systems, a geometric framework based on Lie symmetry is formulated, and from this a sufficient condition for dynamic feedback linearizability obtained. Additionally, a systematic procedure for obtaining all the smooth, generic system trajectories is shown to follow from the theory. Besides smoothness and the existence of symmetry, no further assumption is made on the local form of a control system, which is therefore permitted to be fully nonlinear and time varying. Likewise, no constraints are imposed on the local form of the dynamic compensator. Particular attention is given to the consideration of geometric (coordinate independent) structures associated to control systems with symmetry. To show how the theory is applied in practice we work through illustrative examples of control systems, including the vertical take-off and landing system, demonstrating the significant role that Lie symmetry plays in dynamic feedback linearization. Besides these, a number of more elementary pedagogical examples are discussed as an aid to reading the paper. The constructions have been automated in the Maple package DifferentialGeometry.

Détail

Keywords: Lie symmetry reduction, contact geometry, static feedback linearization, explicit integrability, flat outputs, principal bundle.

Bibliographie

[1] Anderson I., Torre C., The DifferentialGeometry package http://digitalcommons.usu.edu.au/dg

[2] Anderson I.M., Fels M.E., “Exterior differential systems with symmetry”, Acta Appl. Math., 87 (2005), 3–31 | DOI | MR | Zbl

[3] Aranda-Bricaire E., Moog C.H., Pomet J.-B., “A linear algebraic framework for dynamic feedback linearization”, IEEE Trans. Automat. Control, 40 (1995), 127–132 | DOI | MR | Zbl

[4] Aranda-Bricaire E., Pomet J.-B., “Some explicit conditions for a control system to be feedback equivalent to extended Goursat form”, IFAC Proc. Vol., 28 (1995), 489–494 | DOI

[5] Avanessoff D., Pomet J.-B., “Flatness and Monge parameterization of two-input systems, control-affine with 4 states or general with 3 states”, ESAIM Control Optim. Calc. Var., 13 (2007), 237–264, arXiv: math.OC/0505443 | DOI | MR | Zbl

[6] Battilotti S., Califano C., “A geometric approach to dynamic feedback linearization”, Analysis and Design of Nonlinear Control Systems, Springer, Berlin, 2008, 397–411 | DOI | MR | Zbl

[7] Brunovský P., “A classification of linear controllable systems”, Kybernetika (Prague), 6 (1970), 173–188 | MR | Zbl

[8] Bryant R.L., Some aspects of the local and global theory of Pfaffian systems, Ph.D. Thesis, The University of North Carolina at Chapel Hill, 1979 | MR

[9] Charlet B., Lévine J., Marino R., “Sufficient conditions for dynamic state feedback linearization”, SIAM J. Control Optim., 29 (1991), 38–57 | DOI | MR | Zbl

[10] Chetverikov V.N., “New flatness conditions for control systems”, IFAC Proc. Vol., 34 (2001), 191–196 | DOI

[11] Chetverikov V.N., “Flatness of dynamically linearized systems”, Differ. Equ., 40 (2004), 1747–1756 | DOI | MR

[12] Clelland J.N., Hu Y., “On absolute equivalence and linearization I”, Geom. Dedicata, 217 (2023), 78, 31 pp., arXiv: 2005.00643 | DOI | MR | Zbl

[13] De Doná J., Tehseen N., Vassiliou P.J., “Symmetry reduction, contact geometry, and partial feedback linearization”, SIAM J. Control Optim., 56 (2018), 201–230, arXiv: 1510.05761v1 | DOI | MR | Zbl

[14] D'Souza R.S., Nielsen C., “An algorithm for local transverse feedback linearization”, SIAM J. Control Optim., 61 (2023), 1248–1272, arXiv: 2109.00641 | DOI | MR | Zbl

[15] Elkin V.I., Reduction of nonlinear control systems, Math. Appl., 472, Kluwer Academic Publishers, Dordrecht, 1999 | DOI | MR | Zbl

[16] Fliess M., Lévine J., Martin P., Rouchon P., “Flatness and defect of non-linear systems: introductory theory and examples”, Internat. J. Control, 61 (1995), 1327–1361 | DOI | MR | Zbl

[17] Fliess M., Lévine J., Martin P., Rouchon P., “A Lie–Bäcklund approach to equivalence and flatness of nonlinear systems”, IEEE Trans. Automat. Control, 44 (1999), 922–937 | DOI | MR | Zbl

[18] Franch J., Fossas E., “Linearization by prolongations: new bounds on the number of integrators”, Eur. J. Control, 11 (2005), 171–179 | DOI | MR | Zbl

[19] Gardner R.B., Shadwick W.F., “An algorithm for feedback linearization”, Differential Geom. Appl., 1 (1991), 153–158 | DOI | MR | Zbl

[20] Gardner R.B., Shadwick W.F., “The GS algorithm for exact linearization to Brunovsky normal form”, IEEE Trans. Automat. Control, 37 (1992), 224–230 | DOI | MR | Zbl

[21] Grizzle J.W., Marcus S.I., “The structure of nonlinear control systems possessing symmetries”, IEEE Trans. Automat. Control, 30 (1985), 248–258 | DOI | MR | Zbl

[22] Gstöttner C., Kolar B., Schöberl M., “A structurally flat triangular form based on the extended chained form”, Internat. J. Control, 95 (2022), 1144–1163, arXiv: 2007.09935 | DOI | MR | Zbl

[23] Gstöttner C., Kolar B., Schöberl M., “Necessary and sufficient conditions for the linearisability of two-input systems by a two-dimensional endogenous dynamic feedback”, Internat. J. Control, 96 (2023), 800–821, arXiv: 2106.14722 | DOI | MR | Zbl

[24] Guay M., McLellan P.J., Bacon D.W., “A condition for dynamic feedback linearization of control-affine nonlinear systems”, Internat. J. Control, 68 (1997), 87–106 | DOI | MR | Zbl

[25] Hauser J., Sastry S., Meyer G., “Nonlinear control design for slightly nonminimum phase systems: application to V/STOL aircraft”, Automatica J. IFAC, 28 (1992), 665–679 | DOI | MR | Zbl

[26] Hunt L.R., Su R.J., Meyer G., “Design for multi-input nonlinear systems”, Differential Geometric Control Theory (Houghton, Mich., 1982), Progr. Math., 27, Birkhäuser, Boston, MA, 1983, 268–298 | MR

[27] Jakubczyk B., Respondek W., “On linearization of control systems”, Bull. Acad. Polon. Sci. Sér. Sci. Math., 28, 1980, 517–522 | MR | Zbl

[28] Klotz T.J., “Geometry of cascade feedback linearizable control systems”, Differential Geom. Appl., 90 (2023), 102044, 37 pp., arXiv: 2102.08521 | DOI | MR | Zbl

[29] Kobayashi S., Nomizu K., Foundations of differential geometry, v. I, Interscience Publishers, New York, 1963 | MR | Zbl

[30] Lévine J., “On the equivalence between differential flatness and dynamic feedback linearizability”, IFAC Proc. Vol., 40 (2007), 338–343 | DOI

[31] Lévine J., Analysis and control of nonlinear systems: A flatness-based approach, Math. Eng., Springer, Berlin, 2009 | DOI | MR | Zbl

[32] Li S., Nicolau F., “Respondek W., Multi-input control-affine systems static feedback equivalent to a triangular form and their flatness”, Internat. J. Control, 89 (2016), 1–24, arXiv: 1411.6282 | DOI | MR | Zbl

[33] Marino R., “On the largest feedback linearizable subsystem”, Systems Control Lett., 6 (1986), 345–351 | DOI | MR | Zbl

[34] Marino R., Boothby W.M., Elliott D.L., “Geometric properties of linearizable control systems”, Math. Systems Theory, 18 (1985), 97–123 | DOI | MR | Zbl

[35] Martin P., Devasia S., Paden B., “A different look at output tracking: control of a VTOL aircraft”, Automatica J. IFAC, 32 (1996), 101–107 | DOI | Zbl

[36] Montgomery R., Zhitomirskii M., “Geometric approach to Goursat flags”, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 18 (2001), 459–493 | DOI | MR | Zbl

[37] Nicolau F., Respondek W., “Flatness of two-input control-affine systems linearizable via a two-fold prolongation”, 2016 IEEE 55th Conference on Decision and Control (CDC), IEEE, 2016, 3862–3867 | DOI | MR

[38] Nicolau F., Respondek W., “Flatness of multi-input control-affine systems linearizable via one-fold prolongation”, SIAM J. Control Optim., 55 (2017), 3171–3203 | DOI | MR | Zbl

[39] Olver P.J., Applications of Lie groups to differential equations, Grad. Texts Math., 107, 2nd ed., Springer, New York, 1993 | DOI | MR | Zbl

[40] Olver P.J., Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995 | DOI | MR | Zbl

[41] Palais R.S., A global formulation of the Lie theory of transformation groups, Mem. Amer. Math. Soc., 22, 1957, iii+123 pp. | DOI | MR | Zbl

[42] Pereira da Silva P.S., “Some remarks on static-feedback linearization for time-varying systems”, Automatica J. IFAC, 44 (2008), 3219–3221 | DOI | MR | Zbl

[43] Pomet J.-B., “A differential geometric setting for dynamic equivalence and dynamic linearization”, Geometry in Nonlinear Control and Differential Inclusions (Warsaw, 1993), Banach Center Publ., 32, Polish Acad. Sci. Inst. Math., Warsaw, 1995, 319–339 | DOI | MR | Zbl

[44] Pomet J.-B., “On dynamic feedback linearization of four-dimensional affine control systems with two inputs”, ESAIM Control Optim. Calc. Var., 2 (1997), 151–230 | DOI | MR | Zbl

[45] Respondek W., “Symmetries and minimal flat outputs of nonlinear control systems”, New Trends in Nonlinear Dynamics and Control, and their Applications, Lect. Notes Control Inf. Sci., 295, Springer, Berlin, 2003, 65–86 | DOI | MR | Zbl

[46] Shadwick W.F., “Absolute equivalence and dynamic feedback linearization”, Systems Control Lett., 15 (1990), 35–39 | DOI | MR | Zbl

[47] Silveira H.B., Pereira da Silva P.S., Rouchon P., “A flat triangular form for nonlinear systems with two inputs: necessary and sufficient conditions”, Eur. J. Control, 22 (2015), 17–22, arXiv: 1312.3527 | DOI | MR | Zbl

[48] Sluis W.M., Tilbury D.M., “A bound on the number of integrators needed to linearize a control system”, Proceedings of 1995 34th IEEE Conference on Decision and Control, IEEE, 1995, 602–607 | DOI | MR

[49] van der Schaft A.J., “Linearization and input-output decoupling for general nonlinear systems”, Systems Control Lett., 5 (1984), 27–33 | DOI | MR | Zbl

[50] van Nieuwstadt M., Rathinam M., Murray R.M., “Differential flatness and absolute equivalence of nonlinear control systems”, SIAM J. Control Optim., 36 (1998), 1225–1239 | DOI | MR | Zbl

[51] Vassiliou P.J., “A constructive generalised Goursat normal form”, Differential Geom. Appl., 24 (2006), 332–350, arXiv: math.DG/0404377 | DOI | MR | Zbl

[52] Vassiliou P.J., “Efficient construction of contact coordinates for partial prolongations”, Found. Comput. Math., 6 (2006), 269–308, arXiv: math.DG/0406234 | DOI | MR | Zbl

[53] Vassiliou P.J., “Cascade linearization of invariant control systems”, J. Dyn. Control Syst., 24 (2018), 593–623 | DOI | MR | Zbl