Jacobian linearisation in a geometric setting

• Published in: 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475) Vol. 6; pp. 6084 - 6089 Vol.6
• Main Authors: Tyner, D.R., Lewis, A.D.
• Format: Conference Proceeding
• Language: English
• Published: IEEE 2003
• Subjects: Control systems; Control theory; Controllability; Design engineering; Design methodology; Jacobian matrices; Mathematics; Optimal control; Statistical analysis; System testing
• DOI: 10.1109/CDC.2003.1272230
• ISBN: 9780780379244; 0780379241
• ISSN: 0191-2216

Linearisation is a common technique in control applications, putting useful analysis and design methodologies at the disposal of the control engineer. In this paper, linearisation is studied from a differential geometric perspective. First it is pointed out that the "naive" Jacobian techniques do not make geometric sense along nontrivial reference trajectories, in that they are dependent on a choice of coordinates. A coordinate-invariant setting for linearisation is presented to address this matter. The setting here is somewhat more complicated than that seen in the naive setting. The controllability of the geometric linearisation is characterised by giving an alternate version of the usual controllability test for time-varying linear systems. The problems of stability, stabilisation, and quadratic optimal control are discussed as topics for future work.