2010, 2(4): 397-440. doi: 10.3934/jgm.2010.2.397

Geometric Jacobian linearization and LQR theory

1. 

Department of Mathematics and Statistics, Queen's University, Kingston, ON K7L 3N6, Canada, Canada

Received  April 2010 Revised  December 2010 Published  January 2011

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.
Citation: Andrew D. Lewis, David R. Tyner. Geometric Jacobian linearization and LQR theory. Journal of Geometric Mechanics, 2010, 2 (4) : 397-440. doi: 10.3934/jgm.2010.2.397
References:
[1]

R. Abraham, J. E. Marsden and T. S. Ratiu, "Manifolds, Tensor Analysis, and Applications,'' 2nd edition,, Number 75 in Applied Mathematical Sciences, (1988).

[2]

C. D. Aliprantis and K. C. Border, "Infinite-dimensional Analysis,'' 2nd edition,, Springer-Verlag, (1999).

[3]

M. Athans and P. L. Falb, "Optimal Control. An Introduction to the Theory and its Applications,'', McGraw-Hill, (1966).

[4]

R. M. Bianchini and G. Stefani, Controllability along a trajectory: A variational approach,, SIAM Journal on Control and Optimization, 31 (1993), 900. doi: 10.1137/0331039.

[5]

R. W. Brockett, "Finite Dimensional Linear Systems,'', John Wiley and Sons, (1970).

[6]

R. M. Hirschorn and A. D. Lewis, Geometric local controllability: Second-order conditions,, Preprint, (2002).

[7]

M. Ikeda, H. Maeda and S. Kodama, Stabilization of linear systems,, Journal of the Society of Industrial and Applied Mathematics, 10 (1972), 716.

[8]

R. E. Kalman, Contributions to the theory of optimal control,, Boletín de la Sociedad Matemática Mexicana. Segunda Serie, 5 (1960), 102.

[9]

E. B. Lee and L. Markus, "Foundations of Optimal Control Theory,'', John Wiley and Sons, (1967).

[10]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "Matematicheskaya Teoriya Optimal' nykh Protsessov,'', Gosudarstvennoe izdatelstvo fiziko-matematicheskoi literatury, (1961).

[11]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,'', Classics of Soviet Mathematics. Gordon & Breach Science Publishers, (1986).

[12]

E. D. Sontag, "Mathematical Control Theory: Deterministic Finite Dimensional Systems,'' 2nd edition,, Number 6 in Texts in Applied Mathematics, (1998).

[13]

H. J. Sussmann, An introduction to the coordinate-free maximum principle,, in, (1997), 463.

[14]

D. R. Tyner, "Geometric Jacobian Linearisation,'', PhD thesis, (2007).

[15]

M. Vidyasagar, "Nonlinear Systems Analysis,'' 2nd edition,, Number 42 in Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, (2002).

[16]

K. Yano and S. Ishihara, "Tangent and Cotangent Bundles,'', Number 16 in Pure and Applied Mathematics. Dekker Marcel Dekker, (1973).

show all references

References:
[1]

R. Abraham, J. E. Marsden and T. S. Ratiu, "Manifolds, Tensor Analysis, and Applications,'' 2nd edition,, Number 75 in Applied Mathematical Sciences, (1988).

[2]

C. D. Aliprantis and K. C. Border, "Infinite-dimensional Analysis,'' 2nd edition,, Springer-Verlag, (1999).

[3]

M. Athans and P. L. Falb, "Optimal Control. An Introduction to the Theory and its Applications,'', McGraw-Hill, (1966).

[4]

R. M. Bianchini and G. Stefani, Controllability along a trajectory: A variational approach,, SIAM Journal on Control and Optimization, 31 (1993), 900. doi: 10.1137/0331039.

[5]

R. W. Brockett, "Finite Dimensional Linear Systems,'', John Wiley and Sons, (1970).

[6]

R. M. Hirschorn and A. D. Lewis, Geometric local controllability: Second-order conditions,, Preprint, (2002).

[7]

M. Ikeda, H. Maeda and S. Kodama, Stabilization of linear systems,, Journal of the Society of Industrial and Applied Mathematics, 10 (1972), 716.

[8]

R. E. Kalman, Contributions to the theory of optimal control,, Boletín de la Sociedad Matemática Mexicana. Segunda Serie, 5 (1960), 102.

[9]

E. B. Lee and L. Markus, "Foundations of Optimal Control Theory,'', John Wiley and Sons, (1967).

[10]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "Matematicheskaya Teoriya Optimal' nykh Protsessov,'', Gosudarstvennoe izdatelstvo fiziko-matematicheskoi literatury, (1961).

[11]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,'', Classics of Soviet Mathematics. Gordon & Breach Science Publishers, (1986).

[12]

E. D. Sontag, "Mathematical Control Theory: Deterministic Finite Dimensional Systems,'' 2nd edition,, Number 6 in Texts in Applied Mathematics, (1998).

[13]

H. J. Sussmann, An introduction to the coordinate-free maximum principle,, in, (1997), 463.

[14]

D. R. Tyner, "Geometric Jacobian Linearisation,'', PhD thesis, (2007).

[15]

M. Vidyasagar, "Nonlinear Systems Analysis,'' 2nd edition,, Number 42 in Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, (2002).

[16]

K. Yano and S. Ishihara, "Tangent and Cotangent Bundles,'', Number 16 in Pure and Applied Mathematics. Dekker Marcel Dekker, (1973).

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