Advanced Search
Article Contents
Article Contents

Perturbation feedback control: A geometric interpretation

Abstract Related Papers Cited by
  • Perturbation feedback control is a classical procedure in control engineering that is based on linearizing a nonlinear system around some locally optimal nominal trajectory. In the presence of terminal constraints defined by a $k$-dimensional embedded submanifold, the corresponding flow of extremals for the underlying system gives rise to a canonical foliation in the $(t,x)$-space consisting of $(n-k+1)$-dimensional leaves and $k$-dimensional cross sections. In this paper, the connections between the formal computations in the engineering literature and the geometric meaning underlying these constructions are described.
    Mathematics Subject Classification: Primary: 49K15, 49L99; Secondary: 93C10.


    \begin{equation} \\ \end{equation}
  • [1]

    A. A. Agrachev and Y. Sachkov, "Control Theory from the Geometric Viewpoint," Springer Verlag, Berlin, 2004.


    A. Agrachev, G. Stefani and P. L. Zezza, A Hamiltonian approach to strong minima in optimal control, in "Differential Geometry and Control," (Eds. G. Ferreyra, R. Gardner, H. Hermes and H. Sussmann), American Mathematical Society, (1999), 11-22.


    A. Agrachev, G. Stefani and P. L. Zezza, Strong optimality for a bang-bang trajectory, SIAM J. Control and Optimization, 41 (2002), 991-1014.doi: 10.1137/S036301290138866X.


    B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory," Mathématiques & Applications, Vol. 40, Springer Verlag, Paris, 2003.


    J. V. Breakwell, J. L. Speyer and A. E. Bryson, jr., Optimization and control of nonlinear systems using the second variation, SIAM J. Control, 1 (1963), 193-223.


    A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control," American Institute of Mathematical Sciences (AIMS), 2007.


    A. E. Bryson, jr. and Y. C. Ho, "Applied Optimal Control," Revised Printing, Hemisphere Publishing Company, New York, 1975.


    M. E. Fisher, W. J. Grantham and K. L. Teo, Neighbouring extremals for nonlinear systems with control constraints, Dynamics and Control, 5 (1995), 225-240.doi: 10.1007/BF01968675.


    M. Golubitsky and V. Guillemin, "Stable Mappings and their Singularities," Springer-Verlag, New York, 1973.doi: 10.1007/978-1-4615-7904-5.


    D. H. Jacobson, D. H. Martin, M. Pachter and T. Geveci, "Extensions of Linear-Quadratic Control Theory," Lecture Notes in Control and Information Sciences, Springer Verlag, Berlin, 1980.


    T. Kailath, "Linear Systems," Prentice Hall, Englewood Cliffs, NJ, 1980.


    H. W. Knobloch and H. Kwakernaak, "Lineare Kontrolltheorie," Springer Verlag, Berlin, 1985.doi: 10.1007/978-3-642-69884-2.


    H. Maurer and H. J. Oberle, Second order sufficient conditions for optimal control problems with free final time: the Riccati approach, SIAM J. on Control and Optimization, 41 (2002), 380-403.doi: 10.1137/S0363012900377419.


    J. Noble and H. Schättler, Sufficient conditions for relative minima of broken extremals, J. of Mathematical Analysis and Applications, 269 (2002), 98-128.doi: 10.1016/S0022-247X(02)00008-2.


    U. Ledzewicz, A. Nowakowski and H. Schättler, Stratifiable families of extremals and sufficient conditions for optimality in optimal control problems, J. of Optimization Theory and Applications (JOTA), 122 (2004), 105-130.doi: 10.1023/B:JOTA.0000042525.50701.9a.


    L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," MacMillan, New York, 1964.


    H. Schättler, On classical envelopes in optimal control theory, in "Proc. of the 49th IEEE Conference on Decision and Control," Atlanta, USA, December 2010, 1879-1884.


    H. Schättler and U. Ledzewicz, "Geometric Optimal Control," Springer Verlag, New York, 2012.


    H. Schättler and U. Ledzewicz, Synthesis of optimal controlled trajectories with chattering arcs, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms, 19 (2012), 161-186.


    H. Schättler and U. Ledzewicz, Lyapunov-Schmidt reduction for optimal control problems, Discrete and Continuous Dynamical Systems, Series B, 17 (2012), 2201-2223.


    H. J. Sussmann, Envelopes, high-order optimality conditions and Lie brackets, in "Proc. of the 28th IEEE Conference on Decision and Control," Tampa, Florida, December 1989, 1107-1112.doi: 10.1109/CDC.1989.70305.

  • 加载中

Article Metrics

HTML views() PDF downloads(72) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint