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

Geometric Jacobian linearization and LQR theory

Andrew D. Lewis 1, and  David R. Tyner 1,

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.
Keywords: Linearization, stability, linear controllability, stabilization., linear quadratic optimal control.
Mathematics Subject Classification: Primary: 49K05, 93B05, 93B18, 93B27, 93B52; Secondary: 37C1.
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, Springer-Verlag, 1988.  Google Scholar

[2]

C. D. Aliprantis and K. C. Border, "Infinite-dimensional Analysis,'' 2nd edition, Springer-Verlag, New York-Heidelberg-Berlin, 1999.  Google Scholar

[3]

M. Athans and P. L. Falb, "Optimal Control. An Introduction to the Theory and its Applications,'' McGraw-Hill, New York, 1966.  Google Scholar

[4]

R. M. Bianchini and G. Stefani, Controllability along a trajectory: A variational approach, SIAM Journal on Control and Optimization, 31 (1993), 900-927. doi: 10.1137/0331039.  Google Scholar

[5]

R. W. Brockett, "Finite Dimensional Linear Systems,'' John Wiley and Sons, New York, New York, 1970. Google Scholar

[6]

R. M. Hirschorn and A. D. Lewis, Geometric local controllability: Second-order conditions, Preprint, June 2002, available online at http://www.mast.queensu.ca/ andrew/. Google Scholar

[7]

M. Ikeda, H. Maeda and S. Kodama, Stabilization of linear systems, Journal of the Society of Industrial and Applied Mathematics, Series A Control, 10 (1972), 716-729.  Google Scholar

[8]

R. E. Kalman, Contributions to the theory of optimal control, Boletín de la Sociedad Matemática Mexicana. Segunda Serie, 5 (1960), 102-119.  Google Scholar

[9]

E. B. Lee and L. Markus, "Foundations of Optimal Control Theory,'' John Wiley and Sons, New York, New York, 1967.  Google Scholar

[10]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "Matematicheskaya Teoriya Optimal' nykh Protsessov,'' Gosudarstvennoe izdatelstvo fiziko-matematicheskoi literatury, Moscow, 1961. Reprint of translation: [11].  Google Scholar

[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, New York, 1986. Reprint of 1962 translation from the Russian by K. N. Trirogoff.  Google Scholar

[12]

E. D. Sontag, "Mathematical Control Theory: Deterministic Finite Dimensional Systems,'' 2nd edition, Number 6 in Texts in Applied Mathematics, Springer-Verlag, New York-Heidelberg-Berlin, 1998.  Google Scholar

[13]

H. J. Sussmann, An introduction to the coordinate-free maximum principle, in "Geometry of Feedback and Optimal Control'' (eds. B. Jakubczyk and W. Respondek), Dekker Marcel Dekker, New York, (1997), 463-557.  Google Scholar

[14]

D. R. Tyner, "Geometric Jacobian Linearisation,'' PhD thesis, Queen's University, Kingston, Kingston, ON, Canada, 2007. Google Scholar

[15]

M. Vidyasagar, "Nonlinear Systems Analysis,'' 2nd edition, Number 42 in Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 2002. Reprint of 1993 Prentice Hall second edition.  Google Scholar

[16]

K. Yano and S. Ishihara, "Tangent and Cotangent Bundles,'' Number 16 in Pure and Applied Mathematics. Dekker Marcel Dekker, New York, 1973.  Google Scholar

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, Springer-Verlag, 1988.  Google Scholar

[2]

C. D. Aliprantis and K. C. Border, "Infinite-dimensional Analysis,'' 2nd edition, Springer-Verlag, New York-Heidelberg-Berlin, 1999.  Google Scholar

[3]

M. Athans and P. L. Falb, "Optimal Control. An Introduction to the Theory and its Applications,'' McGraw-Hill, New York, 1966.  Google Scholar

[4]

R. M. Bianchini and G. Stefani, Controllability along a trajectory: A variational approach, SIAM Journal on Control and Optimization, 31 (1993), 900-927. doi: 10.1137/0331039.  Google Scholar

[5]

R. W. Brockett, "Finite Dimensional Linear Systems,'' John Wiley and Sons, New York, New York, 1970. Google Scholar

[6]

R. M. Hirschorn and A. D. Lewis, Geometric local controllability: Second-order conditions, Preprint, June 2002, available online at http://www.mast.queensu.ca/ andrew/. Google Scholar

[7]

M. Ikeda, H. Maeda and S. Kodama, Stabilization of linear systems, Journal of the Society of Industrial and Applied Mathematics, Series A Control, 10 (1972), 716-729.  Google Scholar

[8]

R. E. Kalman, Contributions to the theory of optimal control, Boletín de la Sociedad Matemática Mexicana. Segunda Serie, 5 (1960), 102-119.  Google Scholar

[9]

E. B. Lee and L. Markus, "Foundations of Optimal Control Theory,'' John Wiley and Sons, New York, New York, 1967.  Google Scholar

[10]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "Matematicheskaya Teoriya Optimal' nykh Protsessov,'' Gosudarstvennoe izdatelstvo fiziko-matematicheskoi literatury, Moscow, 1961. Reprint of translation: [11].  Google Scholar

[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, New York, 1986. Reprint of 1962 translation from the Russian by K. N. Trirogoff.  Google Scholar

[12]

E. D. Sontag, "Mathematical Control Theory: Deterministic Finite Dimensional Systems,'' 2nd edition, Number 6 in Texts in Applied Mathematics, Springer-Verlag, New York-Heidelberg-Berlin, 1998.  Google Scholar

[13]

H. J. Sussmann, An introduction to the coordinate-free maximum principle, in "Geometry of Feedback and Optimal Control'' (eds. B. Jakubczyk and W. Respondek), Dekker Marcel Dekker, New York, (1997), 463-557.  Google Scholar

[14]

D. R. Tyner, "Geometric Jacobian Linearisation,'' PhD thesis, Queen's University, Kingston, Kingston, ON, Canada, 2007. Google Scholar

[15]

M. Vidyasagar, "Nonlinear Systems Analysis,'' 2nd edition, Number 42 in Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 2002. Reprint of 1993 Prentice Hall second edition.  Google Scholar

[16]

K. Yano and S. Ishihara, "Tangent and Cotangent Bundles,'' Number 16 in Pure and Applied Mathematics. Dekker Marcel Dekker, New York, 1973.  Google Scholar

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