# American Institute of Mathematical Sciences

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

## Linearisation of tautological control systems

 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.
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. [2] A. A. Agrachev and R. V. Gamkrelidze, The exponential representation of flows and the chronological calculus, Math. USSR-Sb., 107 (1978), 467-532. [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. [4] C. O. Aguilar, Local Controllability of Affine Distributions, PhD thesis, Queen's University, Kingston, Kingston, ON, Canada, 2010. [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. [6] R. Beckmann and A. Deitmar, Strong vector valued integrals, 2011,, , (). [7] N. Bourbaki, Algebra I, Elements of Mathematics, Springer-Verlag, New York-Heidelberg-Berlin, 1989. [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. [9] R. W. Brockett, Finite Dimensional Linear Systems, John Wiley and Sons, New York, 1970. [10] D. L. Cohn, Measure Theory, Birkhäuser, Boston-Basel-Stuttgart, 1980. [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. [12] H. Federer, Geometric Measure Theory, Reprint of 1969 edition, Classics in Mathematics, Springer-Verlag, New York-Heidelberg-Berlin, 1996. [13] R. Godement, Topologie Algébrique et Théorie Des Faisceaux, Publications de l'Institut de mathématique de l'Université de Strasbourg, 13, Hermann, Paris, 1958. [14] M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, 33, Springer-Verlag, New York-Heidelberg-Berlin, 1976. [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. [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. [17] S. Jafarpour and A. D. Lewis, Locally Convex Topologies and Control Theory, Submitted to Mathematics of Control, Signals and Systems, 2015. [18] H. Jarchow, Locally Convex Spaces, Mathematical Textbooks, Teubner, Leipzig, 1981. [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. [20] H. K. Khalil, Nonlinear Systems, 3rd edition, Prentice-Hall, Englewood Cliffs, NJ, 2001. [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. [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. [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. [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. [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. [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. [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. [28] S. Ramanan, Global Calculus, Graduate Studies in Mathematics, 65, American Mathematical Society, Providence, RI, 2005. [29] W. Rudin, Functional Analysis, 2nd edition, International Series in Pure and Applied Mathematics, McGraw-Hill, New York, 1991. [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. [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. [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. [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. [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. [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. [36] Stacks Project Authors, Stacks project, http://stacks.math.columbia.edu, 2014. [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. [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. [39] K. Yano and S. Ishihara, Tangent and Cotangent Bundles, Pure and Applied Mathematics, 16, Dekker Marcel Dekker, New York, 1973. [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.

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. [2] A. A. Agrachev and R. V. Gamkrelidze, The exponential representation of flows and the chronological calculus, Math. USSR-Sb., 107 (1978), 467-532. [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. [4] C. O. Aguilar, Local Controllability of Affine Distributions, PhD thesis, Queen's University, Kingston, Kingston, ON, Canada, 2010. [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. [6] R. Beckmann and A. Deitmar, Strong vector valued integrals, 2011,, , (). [7] N. Bourbaki, Algebra I, Elements of Mathematics, Springer-Verlag, New York-Heidelberg-Berlin, 1989. [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. [9] R. W. Brockett, Finite Dimensional Linear Systems, John Wiley and Sons, New York, 1970. [10] D. L. Cohn, Measure Theory, Birkhäuser, Boston-Basel-Stuttgart, 1980. [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. [12] H. Federer, Geometric Measure Theory, Reprint of 1969 edition, Classics in Mathematics, Springer-Verlag, New York-Heidelberg-Berlin, 1996. [13] R. Godement, Topologie Algébrique et Théorie Des Faisceaux, Publications de l'Institut de mathématique de l'Université de Strasbourg, 13, Hermann, Paris, 1958. [14] M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, 33, Springer-Verlag, New York-Heidelberg-Berlin, 1976. [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. [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. [17] S. Jafarpour and A. D. Lewis, Locally Convex Topologies and Control Theory, Submitted to Mathematics of Control, Signals and Systems, 2015. [18] H. Jarchow, Locally Convex Spaces, Mathematical Textbooks, Teubner, Leipzig, 1981. [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. [20] H. K. Khalil, Nonlinear Systems, 3rd edition, Prentice-Hall, Englewood Cliffs, NJ, 2001. [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. [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. [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. [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. [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. [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. [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. [28] S. Ramanan, Global Calculus, Graduate Studies in Mathematics, 65, American Mathematical Society, Providence, RI, 2005. [29] W. Rudin, Functional Analysis, 2nd edition, International Series in Pure and Applied Mathematics, McGraw-Hill, New York, 1991. [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. [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. [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. [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. [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. [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. [36] Stacks Project Authors, Stacks project, http://stacks.math.columbia.edu, 2014. [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. [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. [39] K. Yano and S. Ishihara, Tangent and Cotangent Bundles, Pure and Applied Mathematics, 16, Dekker Marcel Dekker, New York, 1973. [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.
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