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March  2016, 8(1): 99-138. doi: 10.3934/jgm.2016.8.99

Linearisation of tautological control systems

Andrew D. Lewis 1,

1. 

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

Received  May 2014 Revised  October 2015 Published  February 2016

The framework of tautological control systems is one where ``control'' in the usual sense has been eliminated, with the intention of overcoming the issue of feedback-invariance. Here, the linearisation of tautological control systems is described. This linearisation retains the feedback-invariant character of the tautological control systems framework and so permits, for example, a well-defined notion of linearisation of a system about an equilibrium point, something which has surprisingly been missing up to now. The linearisations described are of systems, first, and then about reference trajectories and reference flows.
Keywords: linearisation., Geometric control theory, tautological control systems.
Mathematics Subject Classification: Primary: 93A30, 93B18; Secondary: 46E10, 93B9.
Citation: Andrew D. Lewis. Linearisation of tautological control systems. Journal of Geometric Mechanics, 2016, 8 (1) : 99-138. doi: 10.3934/jgm.2016.8.99
References:
[1]

R. Abraham, J. E. Marsden and T. S. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd edition, Applied Mathematical Sciences, 75, Springer-Verlag, 1988. doi: 10.1007/978-1-4612-1029-0.  Google Scholar

[2]

A. A. Agrachev and R. V. Gamkrelidze, The exponential representation of flows and the chronological calculus, Math. USSR-Sb., 107 (1978), 467-532.  Google Scholar

[3]

A. A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopedia of Mathematical Sciences, 87, Springer-Verlag, New York-Heidelberg-Berlin, 2004. doi: 10.1007/978-3-662-06404-7.  Google Scholar

[4]

C. O. Aguilar, Local Controllability of Affine Distributions, PhD thesis, Queen's University, Kingston, Kingston, ON, Canada, 2010.  Google Scholar

[5]

M. Barbero-Liñán and A. D. Lewis, Geometric interpretations of the symmetric product in affine differential geometry, Int. J. Geom. Methods Mod. Phys., 9 (2012), 1250073, 33pp. doi: 10.1142/S0219887812500739.  Google Scholar

[6]

R. Beckmann and A. Deitmar, Strong vector valued integrals, 2011,, , ().   Google Scholar

[7]

N. Bourbaki, Algebra I, Elements of Mathematics, Springer-Verlag, New York-Heidelberg-Berlin, 1989. Google Scholar

[8]

G. E. Bredon, Sheaf Theory, 2nd edition, Graduate Texts in Mathematics, 170, Springer-Verlag, New York-Heidelberg-Berlin, 1997. doi: 10.1007/978-1-4612-0647-7.  Google Scholar

[9]

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

[10]

D. L. Cohn, Measure Theory, Birkhäuser, Boston-Basel-Stuttgart, 1980.  Google Scholar

[11]

C. T. J. Dodson and T. Poston, Tensor Geometry, Graduate Texts in Mathematics, 130, Springer-Verlag, New York-Heidelberg-Berlin, 1991. doi: 10.1007/978-3-642-10514-2.  Google Scholar

[12]

H. Federer, Geometric Measure Theory, Reprint of 1969 edition, Classics in Mathematics, Springer-Verlag, New York-Heidelberg-Berlin, 1996.  Google Scholar

[13]

R. Godement, Topologie Algébrique et Théorie Des Faisceaux, Publications de l'Institut de mathématique de l'Université de Strasbourg, 13, Hermann, Paris, 1958.  Google Scholar

[14]

M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, 33, Springer-Verlag, New York-Heidelberg-Berlin, 1976.  Google Scholar

[15]

A. Isidori, Nonlinear Control Systems, 3rd edition, Communications and Control Engineering Series, Springer-Verlag, New York-Heidelberg-Berlin, 1995. doi: 10.1007/978-1-84628-615-5.  Google Scholar

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S. Jafarpour and A. D. Lewis, Time-varying Vector Fields and Their Flows, To appear in Springer Briefs in Mathematics, 2014. doi: 10.1007/978-3-319-10139-2.  Google Scholar

[17]

S. Jafarpour and A. D. Lewis, Locally Convex Topologies and Control Theory, Submitted to Mathematics of Control, Signals and Systems, 2015. Google Scholar

[18]

H. Jarchow, Locally Convex Spaces, Mathematical Textbooks, Teubner, Leipzig, 1981.  Google Scholar

[19]

M. Kashiwara and P. Schapira, Sheaves on Manifolds, Grundlehren der Mathematischen Wissenschaften, 292, Springer-Verlag, New York-Heidelberg-Berlin, 1990. doi: 10.1007/978-3-662-02661-8.  Google Scholar

[20]

H. K. Khalil, Nonlinear Systems, 3rd edition, Prentice-Hall, Englewood Cliffs, NJ, 2001. Google Scholar

[21]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Volume I, Interscience Tracts in Pure and Applied Mathematics, 15, Interscience Publishers, New York, 1963.  Google Scholar

[22]

I. Kolář, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, New York-Heidelberg-Berlin, 1993. doi: 10.1007/978-3-662-02950-3.  Google Scholar

[23]

A. D. Lewis, Fundamental problems of geometric control theory, in Proceedings of the 51st IEEE Conference on Decision and Control, IEEE, Maui, HI, 2012, 7511-7516. doi: 10.1109/CDC.2012.6427046.  Google Scholar

[24]

A. D. Lewis, Tautological Control Systems, Springer Briefs in Electrical and Computer Engineering-Control, Automation and Robotics, Springer-Verlag, New York-Heidelberg-Berlin, 2014. doi: 10.1007/978-3-319-08638-5.  Google Scholar

[25]

A. D. Lewis and D. R. Tyner, Geometric Jacobian linearization and LQR theory, J. Geom. Mech., 2 (2010), 397-440. doi: 10.3934/jgm.2010.2.397.  Google Scholar

[26]

K. C. H. Mackenzie, The General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, New York-Port Chester-Melbourne-Sydney, 2005. doi: 10.1017/CBO9781107325883.  Google Scholar

[27]

H. Nijmeijer and A. J. van der Schaft, Nonlinear Dynamical Control Systems, Springer-Verlag, New York-Heidelberg-Berlin, 1990. doi: 10.1007/978-1-4757-2101-0.  Google Scholar

[28]

S. Ramanan, Global Calculus, Graduate Studies in Mathematics, 65, American Mathematical Society, Providence, RI, 2005.  Google Scholar

[29]

W. Rudin, Functional Analysis, 2nd edition, International Series in Pure and Applied Mathematics, McGraw-Hill, New York, 1991.  Google Scholar

[30]

S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tôhoku Math. J. (2), 10 (1958), 338-354. doi: 10.2748/tmj/1178244668.  Google Scholar

[31]

S. Sastry, Nonlinear Systems: Analysis, Stability, and Control, Interdisciplinary Applied Mathematics, 10, Springer-Verlag, New York-Heidelberg-Berlin, 1999. doi: 10.1007/978-1-4757-3108-8.  Google Scholar

[32]

D. J. Saunders, The Geometry of Jet Bundles, London Mathematical Society Lecture Note Series, 142, Cambridge University Press, New York-Port Chester-Melbourne-Sydney, 1989. doi: 10.1017/CBO9780511526411.  Google Scholar

[33]

H. H. Schaefer and M. P. Wolff, Topological Vector Spaces, 2nd edition, Graduate Texts in Mathematics, 3, Springer-Verlag, New York-Heidelberg-Berlin, 1999. doi: 10.1007/978-1-4612-1468-7.  Google Scholar

[34]

F. Schuricht and H. von der Mosel, Ordinary Differential Equations with Measurable Right-Hand Side and Parameter Dependence, Technical Report Preprint 676, Universität Bonn, SFB 256, 2000. Google Scholar

[35]

E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd edition, Texts in Applied Mathematics, 6, Springer-Verlag, New York-Heidelberg-Berlin, 1998. doi: 10.1007/978-1-4612-0577-7.  Google Scholar

[36]

Stacks Project Authors, Stacks project, http://stacks.math.columbia.edu, 2014. Google Scholar

[37]

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

[38]

W. M. Wonham, Linear Multivariable Control, A Geometric Approach, 3rd edition, Applications of Mathematics, 10, Springer-Verlag, New York-Heidelberg-Berlin, 1985. doi: 10.1007/978-1-4612-1082-5.  Google Scholar

[39]

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

[40]

K. Yano and S. Kobayashi, Prolongations of tensor fields and connections to tangent bundles I. General theory, J. Math. Soc. Japan, 18 (1966), 194-210. doi: 10.2969/jmsj/01820194.  Google Scholar

show all references

References:
[1]

R. Abraham, J. E. Marsden and T. S. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd edition, Applied Mathematical Sciences, 75, Springer-Verlag, 1988. doi: 10.1007/978-1-4612-1029-0.  Google Scholar

[2]

A. A. Agrachev and R. V. Gamkrelidze, The exponential representation of flows and the chronological calculus, Math. USSR-Sb., 107 (1978), 467-532.  Google Scholar

[3]

A. A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopedia of Mathematical Sciences, 87, Springer-Verlag, New York-Heidelberg-Berlin, 2004. doi: 10.1007/978-3-662-06404-7.  Google Scholar

[4]

C. O. Aguilar, Local Controllability of Affine Distributions, PhD thesis, Queen's University, Kingston, Kingston, ON, Canada, 2010.  Google Scholar

[5]

M. Barbero-Liñán and A. D. Lewis, Geometric interpretations of the symmetric product in affine differential geometry, Int. J. Geom. Methods Mod. Phys., 9 (2012), 1250073, 33pp. doi: 10.1142/S0219887812500739.  Google Scholar

[6]

R. Beckmann and A. Deitmar, Strong vector valued integrals, 2011,, , ().   Google Scholar

[7]

N. Bourbaki, Algebra I, Elements of Mathematics, Springer-Verlag, New York-Heidelberg-Berlin, 1989. Google Scholar

[8]

G. E. Bredon, Sheaf Theory, 2nd edition, Graduate Texts in Mathematics, 170, Springer-Verlag, New York-Heidelberg-Berlin, 1997. doi: 10.1007/978-1-4612-0647-7.  Google Scholar

[9]

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

[10]

D. L. Cohn, Measure Theory, Birkhäuser, Boston-Basel-Stuttgart, 1980.  Google Scholar

[11]

C. T. J. Dodson and T. Poston, Tensor Geometry, Graduate Texts in Mathematics, 130, Springer-Verlag, New York-Heidelberg-Berlin, 1991. doi: 10.1007/978-3-642-10514-2.  Google Scholar

[12]

H. Federer, Geometric Measure Theory, Reprint of 1969 edition, Classics in Mathematics, Springer-Verlag, New York-Heidelberg-Berlin, 1996.  Google Scholar

[13]

R. Godement, Topologie Algébrique et Théorie Des Faisceaux, Publications de l'Institut de mathématique de l'Université de Strasbourg, 13, Hermann, Paris, 1958.  Google Scholar

[14]

M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, 33, Springer-Verlag, New York-Heidelberg-Berlin, 1976.  Google Scholar

[15]

A. Isidori, Nonlinear Control Systems, 3rd edition, Communications and Control Engineering Series, Springer-Verlag, New York-Heidelberg-Berlin, 1995. doi: 10.1007/978-1-84628-615-5.  Google Scholar

[16]

S. Jafarpour and A. D. Lewis, Time-varying Vector Fields and Their Flows, To appear in Springer Briefs in Mathematics, 2014. doi: 10.1007/978-3-319-10139-2.  Google Scholar

[17]

S. Jafarpour and A. D. Lewis, Locally Convex Topologies and Control Theory, Submitted to Mathematics of Control, Signals and Systems, 2015. Google Scholar

[18]

H. Jarchow, Locally Convex Spaces, Mathematical Textbooks, Teubner, Leipzig, 1981.  Google Scholar

[19]

M. Kashiwara and P. Schapira, Sheaves on Manifolds, Grundlehren der Mathematischen Wissenschaften, 292, Springer-Verlag, New York-Heidelberg-Berlin, 1990. doi: 10.1007/978-3-662-02661-8.  Google Scholar

[20]

H. K. Khalil, Nonlinear Systems, 3rd edition, Prentice-Hall, Englewood Cliffs, NJ, 2001. Google Scholar

[21]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Volume I, Interscience Tracts in Pure and Applied Mathematics, 15, Interscience Publishers, New York, 1963.  Google Scholar

[22]

I. Kolář, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, New York-Heidelberg-Berlin, 1993. doi: 10.1007/978-3-662-02950-3.  Google Scholar

[23]

A. D. Lewis, Fundamental problems of geometric control theory, in Proceedings of the 51st IEEE Conference on Decision and Control, IEEE, Maui, HI, 2012, 7511-7516. doi: 10.1109/CDC.2012.6427046.  Google Scholar

[24]

A. D. Lewis, Tautological Control Systems, Springer Briefs in Electrical and Computer Engineering-Control, Automation and Robotics, Springer-Verlag, New York-Heidelberg-Berlin, 2014. doi: 10.1007/978-3-319-08638-5.  Google Scholar

[25]

A. D. Lewis and D. R. Tyner, Geometric Jacobian linearization and LQR theory, J. Geom. Mech., 2 (2010), 397-440. doi: 10.3934/jgm.2010.2.397.  Google Scholar

[26]

K. C. H. Mackenzie, The General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, New York-Port Chester-Melbourne-Sydney, 2005. doi: 10.1017/CBO9781107325883.  Google Scholar

[27]

H. Nijmeijer and A. J. van der Schaft, Nonlinear Dynamical Control Systems, Springer-Verlag, New York-Heidelberg-Berlin, 1990. doi: 10.1007/978-1-4757-2101-0.  Google Scholar

[28]

S. Ramanan, Global Calculus, Graduate Studies in Mathematics, 65, American Mathematical Society, Providence, RI, 2005.  Google Scholar

[29]

W. Rudin, Functional Analysis, 2nd edition, International Series in Pure and Applied Mathematics, McGraw-Hill, New York, 1991.  Google Scholar

[30]

S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tôhoku Math. J. (2), 10 (1958), 338-354. doi: 10.2748/tmj/1178244668.  Google Scholar

[31]

S. Sastry, Nonlinear Systems: Analysis, Stability, and Control, Interdisciplinary Applied Mathematics, 10, Springer-Verlag, New York-Heidelberg-Berlin, 1999. doi: 10.1007/978-1-4757-3108-8.  Google Scholar

[32]

D. J. Saunders, The Geometry of Jet Bundles, London Mathematical Society Lecture Note Series, 142, Cambridge University Press, New York-Port Chester-Melbourne-Sydney, 1989. doi: 10.1017/CBO9780511526411.  Google Scholar

[33]

H. H. Schaefer and M. P. Wolff, Topological Vector Spaces, 2nd edition, Graduate Texts in Mathematics, 3, Springer-Verlag, New York-Heidelberg-Berlin, 1999. doi: 10.1007/978-1-4612-1468-7.  Google Scholar

[34]

F. Schuricht and H. von der Mosel, Ordinary Differential Equations with Measurable Right-Hand Side and Parameter Dependence, Technical Report Preprint 676, Universität Bonn, SFB 256, 2000. Google Scholar

[35]

E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd edition, Texts in Applied Mathematics, 6, Springer-Verlag, New York-Heidelberg-Berlin, 1998. doi: 10.1007/978-1-4612-0577-7.  Google Scholar

[36]

Stacks Project Authors, Stacks project, http://stacks.math.columbia.edu, 2014. Google Scholar

[37]

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

[38]

W. M. Wonham, Linear Multivariable Control, A Geometric Approach, 3rd edition, Applications of Mathematics, 10, Springer-Verlag, New York-Heidelberg-Berlin, 1985. doi: 10.1007/978-1-4612-1082-5.  Google Scholar

[39]

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

[40]

K. Yano and S. Kobayashi, Prolongations of tensor fields and connections to tangent bundles I. General theory, J. Math. Soc. Japan, 18 (1966), 194-210. doi: 10.2969/jmsj/01820194.  Google Scholar

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