# American Institute of Mathematical Sciences

2012, 2(3): 631-654. doi: 10.3934/naco.2012.2.631

## Perturbation feedback control: A geometric interpretation

 1 Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Missouri, 63130-4899 2 Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026

Received  September 2011 Revised  March 2012 Published  August 2012

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.
Citation: Heinz Schättler, Urszula Ledzewicz. Perturbation feedback control: A geometric interpretation. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 631-654. doi: 10.3934/naco.2012.2.631
##### References:
 [1] A. A. Agrachev and Y. Sachkov, "Control Theory from the Geometric Viewpoint," Springer Verlag, Berlin, 2004. [2] 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. [3] 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. [4] B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory," Mathématiques & Applications, Vol. 40, Springer Verlag, Paris, 2003. [5] 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. [6] A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control," American Institute of Mathematical Sciences (AIMS), 2007. [7] A. E. Bryson, jr. and Y. C. Ho, "Applied Optimal Control," Revised Printing, Hemisphere Publishing Company, New York, 1975. [8] 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. [9] M. Golubitsky and V. Guillemin, "Stable Mappings and their Singularities," Springer-Verlag, New York, 1973. doi: 10.1007/978-1-4615-7904-5. [10] 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. [11] T. Kailath, "Linear Systems," Prentice Hall, Englewood Cliffs, NJ, 1980. [12] H. W. Knobloch and H. Kwakernaak, "Lineare Kontrolltheorie," Springer Verlag, Berlin, 1985. doi: 10.1007/978-3-642-69884-2. [13] 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. [14] 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. [15] 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. [16] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," MacMillan, New York, 1964. [17] 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. [18] H. Schättler and U. Ledzewicz, "Geometric Optimal Control," Springer Verlag, New York, 2012. [19] 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. [20] H. Schättler and U. Ledzewicz, Lyapunov-Schmidt reduction for optimal control problems, Discrete and Continuous Dynamical Systems, Series B, 17 (2012), 2201-2223. [21] 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.

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##### References:
 [1] A. A. Agrachev and Y. Sachkov, "Control Theory from the Geometric Viewpoint," Springer Verlag, Berlin, 2004. [2] 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. [3] 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. [4] B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory," Mathématiques & Applications, Vol. 40, Springer Verlag, Paris, 2003. [5] 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. [6] A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control," American Institute of Mathematical Sciences (AIMS), 2007. [7] A. E. Bryson, jr. and Y. C. Ho, "Applied Optimal Control," Revised Printing, Hemisphere Publishing Company, New York, 1975. [8] 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. [9] M. Golubitsky and V. Guillemin, "Stable Mappings and their Singularities," Springer-Verlag, New York, 1973. doi: 10.1007/978-1-4615-7904-5. [10] 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. [11] T. Kailath, "Linear Systems," Prentice Hall, Englewood Cliffs, NJ, 1980. [12] H. W. Knobloch and H. Kwakernaak, "Lineare Kontrolltheorie," Springer Verlag, Berlin, 1985. doi: 10.1007/978-3-642-69884-2. [13] 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. [14] 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. [15] 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. [16] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," MacMillan, New York, 1964. [17] 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. [18] H. Schättler and U. Ledzewicz, "Geometric Optimal Control," Springer Verlag, New York, 2012. [19] 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. [20] H. Schättler and U. Ledzewicz, Lyapunov-Schmidt reduction for optimal control problems, Discrete and Continuous Dynamical Systems, Series B, 17 (2012), 2201-2223. [21] 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.
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