New Methods for Shaping the Response of a Nonlinear System

  • Christopher I. Byrnes
  • Alberto Isidori
Part of the Progress in Systems and Control Theory book series (PSCT, volume 9)


Shaping the response of a control system has long been a central problem in the analysis and design of feedback systems. The widespread use of both frequency domain techniques and state-space methods is at least in part due to the relative ease and intuitive content of these methods in addressing problems such as asymptotic tracking and disturbance attenuation for linear systems. Recently, a combination of methods drawn from geometric nonlinear control theory and from nonlinear dynamics was developed to give an admittedly unanticipated local solution to the nonlinear regulator problem, yielding necessary and sufficient conditions for nonlinear regulation for the class of detectable and stabilizable nonlinear systems ([1], [2]). In section 2, we state the basic nonlinear regulator problem and give conditions for solvability of the problem in terms of the solvability of a system of nonlinear partial differential equations. In the linear case, these “regulator equations” coincide with the linear equations derived by Francis [3] in his rather complete treatment of the linear multivariable regulator problem. The derivation of the nonlinear regulator equations and the consequent design of a nonlinear controller repose on two essential problems: feedback stabilization for nonlinear systems, a research area currently enjoying intense activity and success, and an analysis of the “steady-state response” of a nonlinear system to a driving input. In section 3, we sketch our solution to the problem of existence of such a steady-state response using center manifold methods, from which the regulator equations can be derived mutatis mutandis.


Center Manifold Output Regulation Stable Limit Cycle Linear Quadratic Regulator Regulator Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    C.I. Byrnes and A. Isidori, Règulation asymptotique des systèmes nonlinedres, C.R. Acad. Sci. Paris 309 (1989), 527–530.Google Scholar
  2. 2.
    A. Isidori and C.I. Byrnes, Output Regulation of Nonlinear Systems, IEEE Trans. Aut. Contr. AC-35 (1990), 131–140.Google Scholar
  3. 3.
    B.A. Francis, The Linear Multivariable Regulator Problems, SIAM J. Contr. Optimiz. 115 (1977), 486–505.Google Scholar
  4. 4.
    C.I. Byrnes, “Some Partial Differential Equations Arising in Nonlinear Control, Computation and Control, II,” (K. Bowers, J. Lund, eds.),Birkhaiiser-Boston, to appear..Google Scholar
  5. 5.
    E.J. Davison, The Output Control of Linear Time-Invariant Multi-Variable System with Unmeasurable Arbitrary Disturbances, IEEE Trans. Aut Contr AC-17 (1972), 621–630.Google Scholar
  6. 6.
    B.A. Francis and W.M. Wonham, The Internal Model Principle for Linear Multivariable Regulators, J. Appl. Math. Optimiz. 2 (1975), 170–194.CrossRefGoogle Scholar
  7. 7.
    C.I. Byrnes and A. Isidori, Steady State Response, Separation Principle and the Output Regulation of Nonlinear Systems, Proceedings of the 28th IEEE Conference on Decision and Control, Tampa (1989), 2247–2251.Google Scholar
  8. 8.
    J.E. Marsden and M. McCracken, “The Hopf Bifurcation and Its Applications,” Springer-Verlag, New York, Heidelberg, Berlin, 1976.CrossRefGoogle Scholar
  9. 9.
    M. Hautus, Linear Matrix Equations with Application to the Regulator Problem,Outils and Modeles Mathematique pour l’Automatiquechrw(133)(I.D.Landau ed.),C.N.R.S. (1983), 399–412.Google Scholar
  10. 10.
    C.I. Byrnes and A. Isidori, A Frequency Domain Philosophy for Nonlinear Systems with Applications to Stabilization and Adaptive Control, Proc. of 23rd IEEE Conf. on Dec. and Control, Las Vegas, NV (1984).Google Scholar
  11. 11.
    A.J. Krener, A. Isidori, Nonlinear Zero Distributions, Proc. of the 19th IEEE Conf. on Dec. and Control, Albuquerque (1980).Google Scholar
  12. 12.
    C.I. Byrnes and A. Isidori, “Heuristics for Nonlinear Control,” in Modelling and Adaptive Control, (Proc. of the IIASA Conf. Sopron, July 1986, C.I. Byrnes, A. Kurzhansky, eds.), Springer-Verlag, Berlin, 1988, pp. 48–70.CrossRefGoogle Scholar
  13. 13.
    A. Isidori and C. Moog, On the Nonlinear Equivalent of the Notion of Transmission Zeros, in Modelling and Adaptive Control, (Proc. of the IIASA Conf. Sopron, July 1986, C.I. Byrnes, A. Kurzhansky, eds.), Springer-Verlag, Berlin.Google Scholar
  14. 14.
    C.I. Byrnes and A. Isidori, Local Stabilization of Minimum-phase Nonlinear Systems, Systems and Control Letters 11 (1988), 9–17.CrossRefGoogle Scholar
  15. 15.
    J.P. Aubin, C.I. Byrnes and A. Isidori, Viability Kernels, Controlled Invariance and Zero Dynamics for Nonlinear Systems in Analysis and Optimization of Systems, (Proc. of 9th Int’l Conf., Antibes, June 1990, A. Bensoussan and J.L. Lions, eds) (1990), 821–832. Springer-Verlag, Berlin.Google Scholar
  16. 16.
    J.P. Aubin and H. Frankowska, Viability Kernel of Control Systems, Nonlinear Synthesis (C. I. Byrnes and A. Kurzhansky, eds ). Birkhâuser-Boston, 1991.Google Scholar
  17. 17.
    X.M. Hu, Robust Stabilization of Nonlinear Control Systems, Ph.D. dissertation (1989). Arizona State University.Google Scholar
  18. 18.
    A. Ben-Artzi and J.W. Helton, A Riccati Partial Differential Equation for Factoring Nonlinear Systems,preprint.Google Scholar
  19. 19.
    A.P. Willemstein, Optimal Regulation of Nonlinear Systems on a Finite Interval, SIAM J. Control Opt. 15 (1977), 1050–1069.CrossRefGoogle Scholar
  20. 20.
    D.L. Lukes, Optimal Regulation of Nonlienar Dynamical Systems, SIAM J. Control and Opt. 7 (1969), 75–100.CrossRefGoogle Scholar
  21. 21.
    E.G. Al’brekht, On the optimal stabilization of nonlinear systems, J. Appl. Math. Mech. 25 (1962), 1254–1266.Google Scholar
  22. 22.
    E.G. Al’brekht, Optimal stabilization of nonlinear systems,Mathematical Notes, vol. 4, no. 2, The Ural Mathematical Society, The Ural State University of A. M. Gor’kii, Sverdiovsk (1963). In Russian.Google Scholar
  23. 23.
    P. Brunovsky, On optimal stabilization of nonlinear systems, Mathematical Theory of Control, A. V. Balakrishnan and Lucien W. Neustadt, eds., Academic Press, New York and London (1967).Google Scholar
  24. 24.
    M.K. Sain (Ed), Applications of tensors to modelling and control, Control Systems Technical Report #38, Dept. of Elec. Eng., Notre Dame University (1985).Google Scholar
  25. 25.
    T. Yoshida and K.A. Loparo, Quadratic Regulatory Theory for Analytic Non-linear Systems with Additive Control, Automatics 25 (1989), 531–544.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Christopher I. Byrnes
    • 1
  • Alberto Isidori
    • 1
  1. 1.Department of Systems Science and MathematicsWashington UniversitySt. LouisUSA

Personalised recommendations