Composite control of nonlinear singularly perturbed systems via approximate feedback linearization

  • Aleksey KabanovEmail author
  • Vasiliy Alchakov
Research Article


This article is devoted to the problem of composite control design for continuous nonlinear singularly perturbed (SP) system using approximate feedback linearization (AFL) method. The essence of AFL method lies in the feedback linearization only of a certain part of the original nonlinear system. According to AFL approach, we suggest to solve feedback linearization problems for continuous nonlinear SP system by reducing it to two feedback linearization problems for slow and fast subsystems separately. The resulting AFL control is constructed in the form of asymptotic composition (composite control). Standard procedure for the composite control design consists of the following steps: 1) system decomposition, 2) solution of control problem for fast subsystem, 3) solution of control problem for slow subsystem, 4) construction of the resulting control in the form of the composition of slow and fast controls. The main difficulty during system decomposition is associated with dynamics separation condition for nonlinear SP system. To overcome this, we propose to change the sequence of the design procedure: 1) solving the control problem for fast state variables part, 2) system decomposition, 3) solving the control problem for slow state variables part, 4) construction of the resulting composite control. By this way, fast feedback linearizing control is chosen so that the dynamics separation condition would be met and the fast subsystem would be stabilizable. The application of the proposed approach is illustrated through several examples.


Approximate feedback linearization (AFL) composite control nonlinear singularly perturbed system order reduction decomposition 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. L. Fradkov, I. V. Miroshnik, V. O. Nikiforov. Nonlinear and Adaptive Control of Complex Systems, Dordrecht, Netherlands: Kluwer, 1999.CrossRefzbMATHGoogle Scholar
  2. [2]
    A. Isidori. Nonlinear Control Systems, London, UK: Springer-Verlag, 1995.CrossRefzbMATHGoogle Scholar
  3. [3]
    S. S. Sastry. Nonlinear Systems: Analysis, Stability, and Control, New York, USA: Springer-Verlag, 2010.Google Scholar
  4. [4]
    G. O. Guardabassi, S. M. Savaresi. Approximate linearization via feedbackan overview. Automatica, vol. 37, no. 1, pp. 1–15, 2001.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    A. J. Krener. Approximate linearization by state feedback and coordinate change. System & Control Letters, vol.5, no. 3, 181–185, 1984.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    A. A. Kabanov. Full-state linearization of systems via feedback using similarity transformation. In Proceedings of International Siberian Conference on Control and Communications, IEEE, Moscow, Russia, pp. 1–5, 2016.Google Scholar
  7. [7]
    W. Kang. Approximate linearization of nonlinear control systems. Systems & Control Letters, vol. 23, no. 1, pp. 43–52, 1994.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    J. W. Son, J. T. Lim. Stabilization of approximately feedback linearizable systems using singular perturbation. IEEE Transactions on Automatic Control, vol. 53, no. 6, pp. 1499–1503, 2008.MathSciNetCrossRefGoogle Scholar
  9. [9]
    P. V. Kokotovic, H. K. Khalil, J. O’Reilly. Singular Perturbation Methods in Control: Analysis and Design, Orlando, USA: Academic Press, 1986.zbMATHGoogle Scholar
  10. [10]
    D. S. Naidu. Singular Perturbation Methodology in Control Systems, London, UK: Peregrinus on behalf of the Institution of Electrical Engineers, 1988.CrossRefzbMATHGoogle Scholar
  11. [11]
    Y. Zhang, D. S. Naidu, C. X. Cai, Y. Zou. Singular perturbations and time scales in control theories and applications: An overview 2002–2012. International Journal of Information and Systems Sciences, vol. 9, no. 1, pp. 1–36, 2014.CrossRefGoogle Scholar
  12. [12]
    M. G. Dmitriev, G. A. Kurina. Singular perturbations in control problems. Automation and Remote Control, vol. 67, no. 1, pp. 1–43, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    A. A. Kabanov. Optimal control of mobile robot’s trajectory movement. WSEAS Transactions on Systems & Control, vol. 9, pp. 398–404, 2014.Google Scholar
  14. [14]
    K. Khorasani. On linearization of nonlinear singularity perturbed systems. IEEE Transactions on Automatic Control, vol. 32, no. 3, pp. 256–260, 1987.CrossRefzbMATHGoogle Scholar
  15. [15]
    H. L. Choi, Y. S. Shin, J. T. Lim. Control of nonlinear singularly perturbed systems using feedback linearisation. IEE Proceedings-Control Theory and Applications, vol. 152, no. 1, pp. 91–94, 2005.CrossRefGoogle Scholar
  16. [16]
    P. D. Christofides, P. Daoutidis. Compensation of measurable disturbances for two-time-scale nonlinear systems. Automatica, vol. 32, no. 11, pp. 1553–1573, 1996.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    P. D. Christofides, P. Daoutidis. Feedback control of two-time-scale nonlinear systems. International Journal of Control, vol. 63, no. 5, pp. 965–994, 1996.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    P. D. Christofides, A. R. Teel, P. Daoutidis. Robust semiglobal output tracking for nonlinear singularly perturbed systems. International Journal of Control, vol. 65, no. 4, pp. 639–666, 1996.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    K. Khorasani. A slow manifold approach to linear equivalents of nonlinear singularly perturbed systems. Automatica, vol. 25, no. 2, pp. 301–306, 1989.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    A. A. Kabanov. Composite control for nonlinear singularly perturbed systems based on feedback linearization method. WSEAS Transactions on Systems, vol. 14, pp. 215–221, 2015.Google Scholar
  21. [21]
    A. A. Kabanov. Approximate feedback linearization based on the singular perturbations approach. Mekhatronika, Avtomatizatsiya, Upravlenie, vol. 16, no. 8, pp. 515–522, 2015. (in Russian)CrossRefGoogle Scholar
  22. [22]
    J. W. Son, J. T. Lim. Feedback linearisation of nonlinear singularly perturbed systems with non-separate slow-fast dynamics. IET Control Theory & Applications, vol. 2, no. 8, pp. 728–735, 2008.MathSciNetCrossRefGoogle Scholar
  23. [23]
    H. K. Khalil. Nonlinear Systems, 3rd ed., New Jersey, USA: Prentice Hall, 2002.zbMATHGoogle Scholar
  24. [24]
    V. A. Sobolev. Integral manifolds and decomposition of singularly perturbed systems. System & Control Letter, vol.5, no. 3, pp. 169–179, 1984.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    M. A. Henson, D. E. Seborg. Nonlinear Process Control, New Jersey, USA: Prentice Hall, 1997.zbMATHGoogle Scholar
  26. [26]
    A. I. Klimushchev, N. N. Krasovskii. Uniform asymptotic stability of systems of differential equations with a small parameter in the derivative terms. Journal of Applied Mathematics and Mechanics, vol. 25, no. 4, pp. 1011–1025, 1961. (in Russian)MathSciNetCrossRefGoogle Scholar
  27. [27]
    S. A. Dubovik. Robustness property and stability resource of linear systems. Journal of Automation and Information Sciences, vol. 39, no. 11, pp. 23–30, 2007.CrossRefGoogle Scholar
  28. [28]
    S. A. Dubovik, A. A. Kabanov. A measure of stability against singular perturbations and robust properties of linear systems. Journal of Automation and Information Sciences, vol. 42, no. 6, pp. 55–66, 2010.CrossRefGoogle Scholar
  29. [29]
    W. N. Feng. Characterization and computation for the bound ξ∗ in linear time-variantsingularly perturbed systems. Systems & Control Letters, vol. 11, no. 3, pp. 195–202, 1988.MathSciNetCrossRefGoogle Scholar
  30. [30]
    S. J. Chen, J. L. Lin. Maximal stability bounds of singularly perturbed systems. Journal of the Franklin Institute, vol. 336, no. 8, pp. 1209–1218, 1999.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    S. J. Chen, J. L. Lin. Maximal stability bounds of discretetime singularly perturbed systems. Control and Cybernetics, vol. 33, no. 1, pp. 95–108, 2004.MathSciNetGoogle Scholar
  32. [32]
    A. Ye¸sildirek, F. L. Lewis. Adaptive feedback linearization using efficient neural networks. Journal of Intelligent and Robotic Systems, vol. 31, no. 1–3, pp. 253–281, 2001.CrossRefzbMATHGoogle Scholar
  33. [33]
    H. Deng, H. X. Li, Y. H. Wu. Feedback-linearization-based neural adaptive control for unknown nonaffine nonlinear discrete-time systems. IEEE Transactions on Neural Networks, vol. 19, no. 9, pp. 1615–1625, 2008.CrossRefGoogle Scholar
  34. [34]
    M. Bahita, K. Belarbi. Neural feedback linearization adaptive control for affine nonlinear systems based on neural network estimator. Serbian Journal of Electrical Engineering, vol. 8, no. 3, pp. 307–323, 2011.CrossRefGoogle Scholar

Copyright information

© Institute of Automation, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Sevastopol State UniversitySevastopolRussian Federation

Personalised recommendations