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Feedback Linearization of Nonlinear Systems

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Encyclopedia of Systems and Control

Abstract

Effective methods exist for the control of linear systems but this is less true for nonlinear systems. Therefore, it is very useful if a nonlinear system can be transformed into or approximated by a nonlinear system. Linearity is not invariant under nonlinear changes of state coordinates and nonlinear state feedback. Therefore, it may be possible to convert a nonlinear system into a linear one via these transformations. This is called feedback linearization. This entry surveys feedback linearization and related topics.

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Bibliography

  • Arnol’d VI (1983) Geometrical methods in the theory of ordinary differential equations. Springer, Berlin

    Book  Google Scholar 

  • Banaszuk A, Hauser J (1995a) Feedback linearization of transverse dynamics for periodic orbits. Syst Control Lett 26:185–193

    Article  MATH  MathSciNet  Google Scholar 

  • Banaszuk A, Hauser J (1995b) Feedback linearization of transverse dynamics for periodic orbits in R 3 with points of transverse controllability loss. Syst Control Lett 26:95–105

    Article  MATH  MathSciNet  Google Scholar 

  • Barbot JP, Monaco S, Normand-Cyrot D (1995) Linearization about an equilibrium manifold in discrete time. In: Proceedings of IFAC NOLCOS 95, Tahoe City. Pergamon

    Google Scholar 

  • Barbot JP, Monaco S, Normand-Cyrot D (1997) Quadratic forms and approximate feedback linearization in discrete time. Int J Control 67:567–586

    Article  MATH  MathSciNet  Google Scholar 

  • Bestle D, Zeitz M (1983) Canonical form observer design for non-linear time-variable systems. Int J Control 38:419–431

    Article  MATH  Google Scholar 

  • Blankenship GL, Quadrat JP (1984) An expert system for stochastic control and signal processing. In: Proceedings of IEEE CDC, Las Vegas. IEEE, pp 716–723

    Google Scholar 

  • Brockett RW (1978) Feedback invariants for nonlinear systems. In: Proceedings of the international congress of mathematicians, Helsinki, pp 1357–1368

    Google Scholar 

  • Brockett RW (1981) Control theory and singular Riemannian geometry. In: Hilton P, Young G (eds) New directions in applied mathematics. Springer, New York, pp 11–27

    Google Scholar 

  • Brockett RW (1983) Nonlinear control theory and differential geometry. In: Proceedings of the international congress of mathematicians, Warsaw, pp 1357–1368

    Google Scholar 

  • Brockett RW (1996) Characteristic phenomena and model problems in nonlinear control. In: Proceedings of the IFAC congress, Sidney. IFAC

    Google Scholar 

  • Byrnes CI, Isidori A (1984) A frequency domain philosophy for nonlinear systems. In: Proceedings, IEEE conference on decision and control, Las Vegas. IEEE, pp 1569–1573

    Google Scholar 

  • Byrnes CI, Isidori A (1988) Local stabilization of minimum-phase nonlinear systems. Syst Control Lett 11:9–17

    Article  MATH  MathSciNet  Google Scholar 

  • Charlet R, Levine J, Marino R (1989) On dynamic feedback linearization. Syst Control Lett 13:143–151

    Article  MATH  MathSciNet  Google Scholar 

  • Charlet R, Levine J, Marino R (1991) Sufficient conditions for dynamic state feedback linearization. SIAM J Control Optim 29:38–57

    Article  MATH  MathSciNet  Google Scholar 

  • Gardner RB, Shadwick WF (1992) The GS-algorithm for exact linearization to Brunovsky normal form. IEEE Trans Autom Control 37:224–230

    Article  MATH  MathSciNet  Google Scholar 

  • Hunt LR, Su R (1981) Linear equivalents of nonlinear time varying systems. In: Proceedings of the symposium on the mathematical theory of networks and systems, Santa Monica, pp 119–123

    Google Scholar 

  • Hunt LR, Su R, Meyer G (1983) Design for multi-input nonlinear systems. In: Millman RS, Brockett RW, Sussmann HJ (eds) Differential geometric control theory. Birkhauser, Boston, pp 268–298

    Google Scholar 

  • Isidori A (1995) Nonlinear control systems. Springer, Berlin

    Book  MATH  Google Scholar 

  • Isidori A, Krener AJ (1982) On the feedback equivalence of nonlinear systems. Syst Control Lett 2:118–121

    Article  MATH  Google Scholar 

  • Isidori A, Ruberti A (1984) On the synthesis of linear input–output responses for nonlinear systems. Syst Control Lett 4:17–22

    Article  MATH  MathSciNet  Google Scholar 

  • Jakubczyk B (1987) Feedback linearization of discrete-time systems. Syst Control Lett 9:17–22

    Article  MathSciNet  Google Scholar 

  • Jakubczyk B, Respondek W (1980) On linearization of control systems. Bull Acad Polonaise Sci Ser Sci Math 28:517–522

    MATH  MathSciNet  Google Scholar 

  • Kang W (1994) Approximate linearization of nonlinear control systems. Syst Control Lett 23: 43–52

    Article  MATH  Google Scholar 

  • Korobov VI (1979) A general approach to the solution of the problem of synthesizing bounded controls in a control problem. Math USSR-Sb 37:535

    Article  Google Scholar 

  • Krener AJ (1973) On the equivalence of control systems and the linearization of nonlinear systems. SIAM J Control 11:670–676

    Article  MATH  MathSciNet  Google Scholar 

  • Krener AJ (1984) Approximate linearization by state feedback and coordinate change. Syst Control Lett 5:181–185

    Article  MATH  MathSciNet  Google Scholar 

  • Krener AJ (1986) The intrinsic geometry of dynamic observations. In: Fliess M, Hazewinkel M (eds) Algebraic and geometric methods in nonlinear control theory. Reidel, Amsterdam, pp 77–87

    Chapter  Google Scholar 

  • Krener AJ (1990) Nonlinear controller design via approximate normal forms. In: Helton JW, Grunbaum A, Khargonekar P (eds) Signal processing, Part II: control theory and its applications. Springer, New York, pp 139–154

    Chapter  Google Scholar 

  • Krener AJ, Isidori A (1983) Linearization by output injection and nonlinear observers. Syst Control Lett 3:47–52

    Article  MATH  MathSciNet  Google Scholar 

  • Krener AJ, Maag B (1991) Controller and observer design for cubic systems. In: Gombani A, DiMasi GB, Kurzhansky AB (eds) Modeling, estimation and control of systems with uncertainty. Birkhauser, Boston, pp 224–239

    Chapter  Google Scholar 

  • Krener AJ, Respondek W (1985) Nonlinear observers with linearizable error dynamics. SIAM J Control Optim 23:197–216

    Article  MATH  MathSciNet  Google Scholar 

  • Krener AJ, Karahan S, Hubbard M, Frezza R (1987) Higher order linear approximations to nonlinear control systems. In: Proceedings, IEEE conference on decision and control, Los Angeles. IEEE, pp 519–523

    Google Scholar 

  • Krener AJ, Karahan S, Hubbard M (1988) Approximate normal forms of nonlinear systems. In: Proceedings, IEEE conference on decision and control, San Antonio. IEEE, pp 1223–1229

    Google Scholar 

  • Krener AJ, Hubbard M, Karahan S, Phelps A, Maag B (1991) Poincare’s linearization method applied to the design of nonlinear compensators. In: Jacob G, Lamnahbi-Lagarrigue F (eds) Algebraic computing in control. Springer, Berlin, pp 76–114

    Chapter  Google Scholar 

  • Lee HG, Arapostathis A, Marcus SI (1986) Linearization of discrete-time systems. Int J Control 45:1803–1822

    Article  MathSciNet  Google Scholar 

  • Murray RM (1995) Nonlinear control of mechanical systems: a Lagrangian perspective. In: Krener AJ, Mayne DQ (eds) Nonlinear control systems design. Pergamon, Oxford

    Google Scholar 

  • Nonlinear Systems Toolbox available for down load at http://www.math.ucdavis.edu/~krener/1995

  • Sommer R (1980) Control design for multivariable non-linear time varying systems. Int J Control 31:883–891

    Article  MATH  MathSciNet  Google Scholar 

  • Su R (1982) On the linear equivalents of control systems. Syst Control Lett 2:48–52

    Article  MATH  Google Scholar 

  • Xia XH, Gao WB (1988a) Nonlinear observer design by canonical form. Int J Control 47: 1081–1100

    Article  MATH  MathSciNet  Google Scholar 

  • Xia XH, Gao WB (1988b) On exponential observers for nonlinear systems. Syst Control Lett 11:319–325

    Article  MATH  MathSciNet  Google Scholar 

  • Xia XH, Gao WB (1989) Nonlinear observer design by observer error linearization. SIAM J Control Optim 27:199–216

    Article  MATH  MathSciNet  Google Scholar 

  • Zeitz M (1987) The extended Luenberger observer for nonlinear systems. Syst Control Lett 9: 149–156

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Naval Postgrauate School, 93943, Monterey, CA, USA

    A. J. Krener

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  1. A. J. Krener
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Correspondence to A. J. Krener .

Editor information

Editors and Affiliations

  1. Electrical and Computer Engineering, Boston University, Boston, Massachusetts, USA

    John Baillieul

  2. Automation and Control Solutions, Honeywell, Golden Valley, Minnesota, USA

    Tariq Samad

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Krener, A.J. (2013). Feedback Linearization of Nonlinear Systems. In: Baillieul, J., Samad, T. (eds) Encyclopedia of Systems and Control. Springer, London. https://doi.org/10.1007/978-1-4471-5102-9_81-1

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  • DOI: https://doi.org/10.1007/978-1-4471-5102-9_81-1

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  • Publisher Name: Springer, London

  • Online ISBN: 978-1-4471-5102-9

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