Linearization by State Immersion


In this chapter, symmetries, homogeneity and semi-invariants are used as tools to solve the problem of determining a state immersion that renders linear (and, in general, of higher order) a given nonlinear system. The problem is dealt with both for continuous-time and discrete-time systems. The proposed technique allows, under some assumptions, also the computation of a linearizing diffeomorphism (that does not alter the dimension of the state vector), when it exists. The last two sections particularize the previous results to Hamiltonian systems, for which, due to their structure, a very strong characterization of the problem can be given.


Hamiltonian System Arbitrary Function Poisson Bracket Hamiltonian Function Positive Eigenvalue 


  1. 12.
    Back, J., Seo, J.H.: Immersion of nonlinear systems into linear systems up to output injection: characteristic equation approach. Int. J. Control 77(8), 723–734 (2004) MathSciNetMATHCrossRefGoogle Scholar
  2. 34.
    Cicogna, G., Gaeta, G.: Symmetry and Perturbation Theory in Nonlinear Dynamics. Lecture Notes in Physics Monographs, vol. 57. Springer, Berlin (1999) MATHGoogle Scholar
  3. 72.
    Jouan, P.: Immersion of nonlinear systems into linear systems modulo output injection. SIAM J. Control Optim. 41(6), 1756–1778 (2003) MathSciNetMATHCrossRefGoogle Scholar
  4. 87.
    Menini, L., Tornambè, A.: Linearization of Hamiltonian systems through state immersion. In: Proceedings of the 47th IEEE Conference on Decision and Control, pp. 1261–1266 (2008) Google Scholar
  5. 89.
    Menini, L., Tornambè, A.: Linearization through state immersion of nonlinear systems admitting Lie symmetries. Automatica 45(8), 1873–1878 (2009) MATHCrossRefGoogle Scholar
  6. 95.
    Menini, L., Tornambè, A.: Linearization of discrete-time nonlinear systems through state immersion and Lie symmetries. In: Proceedings of the NOLCOS 2010, Bologna, pp. 197–202 (2010) Google Scholar
  7. 100.
    Nijmeijer, H., van der Schaft, A.J.: Nonlinear Dynamical Control Systems. Springer, New York (1990) MATHGoogle Scholar
  8. 101.
    Ohtsuka, T.: Model structure simplification of nonlinear systems via immersion. IEEE Trans. Autom. Control 50(5), 607–618 (2005) MathSciNetCrossRefGoogle Scholar
  9. 112.
    Svoronos, S., Stephanopoulos, G., Aris, R.: Bilinear approximation of general non-linear dynamic systems with linear inputs. Int. J. Control 31(1), 109–126 (1980) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Dipto. Informatica Sistemi e ProduzioneUniversità di Roma-Tor VergataRomeItaly

Personalised recommendations