# Flat input based canonical form observers for non-integrable nonlinear systems

## DOI:

https://doi.org/10.52846/stccj.2022.2.1.31## Keywords:

nonlinear systems, observer design, observer error linearization, differential flatness, flat inputs## Abstract

In this contribution, the design of canonical form observers for nonlinear non-integrable systems is investigated. These systems cannot be transformed into observer canonical form, therefore an exact observer error linearization cannot be achieved. However, using flat inputs and dynamic compensators, the original dynamics can be rendered into an integrable flat system. For this modified flat input system a canonical form observer can be designed. By utilizing a state transformation, it is then possible to obtain an estimate of the original state, where the observer error is approximately linearized. This procedure is exemplified by the Rössler system. Furthermore, we illustrate the relationship of this approach with high-gain observers.

## References

M. Zeitz, “Canonical forms for nonlinear systems,” in [47], ser. IFAC Symp. Ser., 1990, pp. 33–38.

D. Bestle and M. Zeitz, “Canonical form observer design for non-linear time-variable systems,” Int. J. Control, vol. 38, pp. 419–431, 1983.

A. Krener and A. Isidori, “Linearization by output injection and nonlinear observers,” Syst. Control Lett., vol. 3, pp. 47–52, 1983.

N. Jo and J. Seo, “Observer design for non-linear systems that are not uniformly observable,” Int. J. Control, vol. 75, no. 5, pp. 369–380, 2002.

K. R¨obenack, “Extended Luenberger observer for nonuniformly observable nonlinear systems,” in [48], 2005, pp. 19–34.

K. R¨obenack, “Observer design for a class of nonlinear systems with non-full relative degree,” Nonlinear Dyn. Syst. Theory, vol. 7, no. 4, pp. 399–408, 2007.

Y. Wang and A. Lynch, “Time scaling of multi-output observer form,” IEEE Trans. Automat. Contr., vol. 55, no. 4, pp. 966–971, 2010.

M. Zeitz, “The extended lueberger observer for nonlinear systems,” Syst. Control Lett., vol. 9, pp. 914–156, 1987.

F. Plestan and A. Glumineau, “Linearization by generlized input-output injection,” Syst. Control Lett., vol. 31, pp. 115–128, 1997.

R. Mishkov, “Nonlinear observer design by reduced generlized observer canonical form,” Int. J. Control, vol. 87, no. 3, pp. 172–185, 2005.

J. Back, H. Shim, and J. Seo, “An algorithm for system immersion into nonlinear observer form: forced systems,” IFAC Proc. Volumes, vol. 38, no. 1, 2005, 16th IFAC World Congress, Prague, Czech Republic.

S. Karahan, Higher degree linear approximation of nonlinear systems (dissertation). University of California, Davis, 1989.

S. Nicosia, P. Tomei, and A. Tornamb´e, “An approximate observer for a class of nonlinear systems,” Syst. Control Lett., vol. 12, pp. 43–51, 1989.

A. Krener, M. Hubbard, S. Karaham, A. Phelps, and B. Maag, “Poincar´e’s linearization method applied to the design of nonlinear compensators,” in [49], 1991, pp. 76–114.

S. Bortoff and B. Lynch, “Synthesis of optimal nonlinear observers,” Proc. IEEE Conf. Decis. Control (CDC), New Orleans, Louisiana, pp. 95–100, 1995.

A. Banaszuk and W. Sluis, “On nonlinear observers with approximately linear error dynamics,” Proc. Am. Control Conf. (ACC), Albuquerque, New Mexico, pp. 3460–3464, 1997.

J. Moreno, “Approximate observer error linearization by dissipativity,”, 2005, pp. 35–51.

J. Birk and M. Zeitz, “Extended Luenberger observer for non-linear multivariable systems,” Int. J. Control, vol. 47, no. 6, pp. 1823–1836, 1988.

M. Fliess, J. L´evine, P. Martin, and P. Rouchon, “Flatness and defect of non-linear systems: introductory theory and examples,” Int. J. Control, vol. 61, no. 6, pp. 1327–1361, 1995.

M. Sch¨oberl and K. Schlacher, “Construction of flat outputs by reduction and elimination,” IFAC Proc. Volumes, vol. 40, no. 12, pp. 693–698, 2007, 7th IFAC Symp. Nonlinear Control Systems (NOLCOS), Pretoria, South Africa.

F. Antritter and J. L´evine, “Towards a computer algebraic algorithm for flat output determination,” Proc. 21st Int. Symp. Symbolic and Algebraic Computation (ISSAC), Hagenberg, pp. 7–14, 2008.

J. L´evine, “On necessary and sufficient conditions for differential flatness,” Applicable Algebra in Engineering, Communication and Computing, vol. 22, no. 1, pp. 47–90, 2011.

M. Franke and K. R¨obenack, “On the computation of flat outputs for nonlinear control systems,” Proc. Eur. Control Conf. (ECC), Z¨urich, pp. 167–172, 2013.

K. Fritzsche, M. Franke, C. Knoll, and K. R¨obenack, “On the systematic computation of flat outputs for nonlinear systems with multiple inputs (in german),” at - Automatisierungstechnik, vol. 64, no. 12, pp. 948–960, 2016.

K. Fritzsche, C. Knoll, M. Franke, and K. R¨obenack, “Unimodular completion and direct flat representation in the context of differential flatness,” Proc. Appl. Math. Mech. (PAMM), vol. 16, no. 1, pp. 807– 808, 2016.

S. Waldherr and M. Zeitz, “Conditions for the existence of a flat input,” Int. J. Control, vol. 81, no. 3, pp. 439–443, 2008.

——, “Flat inputs in the MIMO case,” IFAC Proc. Volumes, vol. 43, no. 13, pp. 695–700, 2010, 8th IFAC Symp. Nonlinear Control Systems (NOLCOS), Bologna, Italy.

F. Nicolau, W. Respondek, and J.-P. Barbot, “Constructing flat inputs for two-output systems,” 23rd Int. Symp. Mathematical Theory of Networks and Systems (MTNS), Hong Kong, pp. 414–421, 2018.

K. Fritzsche and K. R¨obenack, “On the computation of differentially flat inputs,” Proc. Int. Conf. Syst. Theory Control Comput. (ICSTCC), Sinaia, Romania, pp. 12–19, 2018.

F. Nicolau, W. Respondek, and J.-P. Barbot, “Flat inputs: theory and applications,” SIAM J. Control Optim., vol. 58, no. 6, pp. 3293–3321, 2020.

K. Fritzsche and K. R¨obenack, “On a generalized flat input definition and physical realizability,” IFAC-PapersOnLine, vol. 53, no. 2, pp. 5994– 5999, 2020.

J.-F. Stumper, F. Svaricek, and R. Kennel, “Trajectory tracking control with flat inputs and a dynamic compensator,” Proc. Eur. Control Conf. (ECC), Budapest, Hungary, pp. 248–253, 2009.

K. Fritzsche, Y. Guo, and K. R¨obenack, “Nonlinear control of nonobservable non-flat MIMO state space systems using flat inputs,” Proc. Int. Conf. Syst. Theory Control Comput. (ICSTCC), Sinaia, Romania, pp. 173–179, 2019.

J. O. A. L. Filho, E. L. F. Fortaleza, and M. C. M. M. de Campos, “A derivative-free nonlinear Kalman filtering approach using flat inputs,” Int. J. Control, 2021.

K. Fritzsche, Y. Guo, and K. R¨obenack, “Canonical form observers for non-integrable nonlinear single-output systems using flat inputs and dynamic compensators,” Proc. Int. Conf. Syst. Theory Control Comput. (ICSTCC), Ias, i, Romania, pp. 31–38, 2021.

M. Fliess, J. L´evine, P. Martin, and P. Rouchon, “Index and decomposition of nonlinear implicit differential equations,” IFAC Proc. Volumes, vol. 28, no. 8, pp. 37–42, 1995, IFAC Conf. Syst. Structure and Control 1995, Nantes, France.

R. Marino, “Adaptive observers for single output nonlinear systems,” IEEE Trans. Automat. Contr., vol. 35, no. 9, pp. 1054–1058, 1990.

A. van der Schaft, “Representing a nonlinear state-space system as set of higher-order differential equations in the inputs and outputs,” Syst. Control Lett., vol. 12, pp. 151–160, 1989.

O. E. R¨ossler, “An equation for continuous chaos,” Phys. Lett. 57A, p. 397, 1976.

——, “Continuous chaos – four prototype equations,”, 1979, pp. 376–392.

C. Letellier, L. A. Aguirre, and J. Maquet, “Relation between observability and differential embeddings for nonlinear dynamics,” Physical Review E, vol. 71, no. 6, p. 066213, 2005.

D. Gerbet and K. R¨obenack, “On global and local observability of nonlinear polynomial systems: a decidable criterion,” at– Automatisierungstechnik, vol. 68, no. 6, pp. 395–409, 2020.

E. N. Lorenz, “Deterministic nonperiodic flow,” J. Atmos. Sci., vol. 20, pp. 130–141, 1963.

K. R¨obenack, “High gain obervers using an approximate observer normal form,” Proc. Appl. Math. Mech. (PAMM), no. 6, pp. 837–838, 2006.

P. Denbig, System Analysis and Signal Processing. Boston: Addison- Wesley, 1998.

T. J. Cavicchi, Digital Signal Processing. Wiley, 2000.

A. Isidori, Ed., Nonlinear Control Systems Design 1989. Pergamon, Oxford, 1990.

T. Meurer, K. Graichen, and E. D. Gilles, Eds., Control and observer design for nonlinear finite and infinite dimensional systems. Springer, Berlin Heidelberg, 2005.

G. Jacob and F. Lagarrigue, Eds., Algebraic computing in control. Springer, 1991.

O. Gruel and R¨ossler, Bifurcation Theory and Applications in Scientific Disciplines. Ann. N. Y. Acad. Sci. Vol. 316, 1979.

## Downloads

## Published

## How to Cite

*Syst. Theor. Control Comput. J.*, vol. 2, no. 1, pp. 13–21, Jun. 2022.