Geometric Jacobian linearization and LQR theory

The procedure of linearizing a control-affine system along a non-trivial reference trajectory is studied from a differential geometric perspective. A coordinate-invariant setting for linearization is presented. With the linearization in hand, the controllability of the geometric linearization is characterized using an alternative version of the usual controllability test for time-varying linear systems. The various types of stability are defined using a metric on the fibers along the reference trajectory and Lyapunov's second method is recast for linear vector fields on tangent bundles. With the necessary background stated in a geometric framework, linear quadratic regulator theory is understood from the perspective of the Maximum Principle. Finally, the resulting feedback from solving the infinite time optimal control problem is shown to uniformly asymptotically stabilize the linearization using Lyapunov's second method.

[1]  R. Abraham,et al.  Manifolds, Tensor Analysis, and Applications , 1983 .

[2]  R. Abraham,et al.  Manifolds, tensor analysis, and applications: 2nd edition , 1988 .

[3]  Kim C. Border,et al.  Infinite Dimensional Analysis: A Hitchhiker’s Guide , 1994 .

[4]  C. Langenhop.  On the stabilization of linear systems , 1964 .

[5]  A. D. Lewis,et al.  Geometric local controllability: second-order conditions , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[6]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[7]  Gianna Stefani,et al.  Controllability along a trajectory: a variational approach , 1993 .

[8]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[9]  R. E. Kalman,et al.  Contributions to the Theory of Optimal Control , 1960 .

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

[11]  T. J. Willmore,et al.  TANGENT AND COTANGENT BUNDLES , 1975 .

[12]  M. Vidyasagar,et al.  Nonlinear systems analysis (2nd ed.) , 1993 .

[13]  Eduardo D. Sontag,et al.  Mathematical control theory: deterministic finite dimensional systems (2nd ed.) , 1998 .

[14]  Horst Albach,et al.  Production Theory and Its Applications , 1977 .

[15]  C. A. Desoer,et al.  Nonlinear Systems Analysis , 1978 .

[16]  T. Başar.  Contributions to the Theory of Optimal Control , 2001 .

[17]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .

[18]  Frank L. Lewis,et al.  Optimal Control , 1986 .

[19]  E B Lee,et al.  Foundations of optimal control theory , 1967 .

[20]  M. Sain.  Finite dimensional linear systems , 1972 .