Inverse optimal control with polynomial optimization

In the context of optimal control, we consider the inverse problem of Lagrangian identification given system dynamics and optimal trajectories. Many of its theoretical and practical aspects are still open. Potential applications are very broad as a reliable solution to the problem would provide a powerful modeling tool in many areas of experimental science. We propose to the Hamilton-Jacobi-Bellman sufficient optimality conditions as a tool for analyzing the inverse problem and propose a general method that attempts at numerically solving it, with techniques of polynomial optimization and linear matrix inequalities. The relevance of the method is illustrated based on academic examples.

[1]  Vladimir Vapnik,et al.  An overview of statistical learning theory , 1999, IEEE Trans. Neural Networks.

[2]  P. Kokotovic,et al.  Inverse Optimality in Robust Stabilization , 1996 .

[3]  Johannes P. Schlöder,et al.  Estimating Parameters in Optimal Control Problems , 2012, SIAM J. Sci. Comput..

[4]  B. Anderson,et al.  Nonlinear regulator theory and an inverse optimal control problem , 1973 .

[5]  J. Andrew Bagnell,et al.  Maximum margin planning , 2006, ICML.

[6]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[7]  Timothy Bretl,et al.  A convex approach to inverse optimal control and its application to modeling human locomotion , 2012, 2012 IEEE International Conference on Robotics and Automation.

[8]  Emmanuel Trélat,et al.  Robust optimal stabilization of the Brockett integrator via a hybrid feedback , 2005, Math. Control. Signals Syst..

[9]  S. Kahne,et al.  Optimal control: An introduction to the theory and ITs applications , 1967, IEEE Transactions on Automatic Control.

[10]  Jean-Paul Gauthier,et al.  How humans fly , 2013 .

[11]  R. E. Kalman,et al.  When Is a Linear Control System Optimal , 1964 .

[12]  Jean-Paul Laumond,et al.  An Optimality Principle Governing Human Walking , 2008, IEEE Transactions on Robotics.

[13]  Paolo Mason,et al.  On Inverse Optimal Control Problems of Human Locomotion: Stability and Robustness of the Minimizers , 2013 .

[14]  Ruggero Frezza,et al.  Linear Optimal Control Problems and Quadratic Cost Functions Estimation , 2004 .

[15]  G. Darboux,et al.  Leçons sur la Théorie Générale des Surfaces et les Applications Géométriques Du Calcul Infinitésimal , 2001 .

[16]  David Saunders.  Thirty years of the inverse problem in the calculus of variations , 2010 .

[17]  Michael Athans,et al.  Optimal Control , 1966 .

[18]  Michael A Babyak,et al.  What You See May Not Be What You Get: A Brief, Nontechnical Introduction to Overfitting in Regression-Type Models , 2004, Psychosomatic medicine.

[19]  A. Jameson,et al.  Inverse Problem of Linear Optimal Control , 1973 .

[20]  Pieter Abbeel,et al.  Apprenticeship learning via inverse reinforcement learning , 2004, ICML.

[21]  Stephen P. Boyd,et al.  Imputing a convex objective function , 2011, 2011 IEEE International Symposium on Intelligent Control.

[22]  Mary Ellen Soper.  What You See May Not Be What You Get:: Errors in Online Bibliographic Records for Serials , 1990 .

[23]  B. Anderson,et al.  Linear Optimal Control , 1971 .

[24]  Jean B. Lasserre.  Inverse polynomial optimization , 2011, IEEE Conference on Decision and Control and European Control Conference.

[25]  Jean-Paul Laumond,et al.  From human to humanoid locomotion—an inverse optimal control approach , 2010, Auton. Robots.

[26]  F. Thau.  On the inverse optimum control problem for a class of nonlinear autonomous systems , 1967, IEEE Transactions on Automatic Control.

[27]  Enzo Tonti,et al.  Variational formulation for every nonlinear problem , 1984 .

[28]  Takao Fujii,et al.  A complete optimally condition in the inverse problem of optimal control , 1984, The 23rd IEEE Conference on Decision and Control.

[29]  Anton van den Hengel,et al.  Semidefinite Programming , 2014, Computer Vision, A Reference Guide.

[30]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[31]  J. Casti.  On the general inverse problem of optimal control theory , 1980 .

[32]  J. Douglas.  Solution of the Inverse Problem of the Calculus of Variations. , 1939, Proceedings of the National Academy of Sciences of the United States of America.

[33]  Johan Löfberg,et al.  Pre- and Post-Processing Sum-of-Squares Programs in Practice , 2009, IEEE Transactions on Automatic Control.

[34]  Victor M. Becerra,et al.  Optimal control , 2008, Scholarpedia.

[35]  Emmanuel Trélat,et al.  Nonlinear Optimal Control via Occupation Measures and LMI-Relaxations , 2007, SIAM J. Control. Optim..