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Global Stability of a Class of Difference Equations on Solvable Lie Algebras

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Date

2020-06-17

Authors

McCarthy, Philip James
Nielsen, Christopher

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Publisher

Springer

Abstract

Motivated by the ubiquitous sampled-data setup in applied control, we examine the stability of a class of difference equations that arises by sampling a right- or left-invariant flow on a solvable matrix Lie group. The map defining such a difference equation has three key properties that facilitate our analysis: (1) its Lie series expansion enjoys a type of strong convergence; (2) the origin is an equilibrium; (3) the algebraic ideals enumerated in the lower central series of the Lie algebra are dynamically invariant. We show that certain global stability properties are implied by stability of the Jacobian linearization of the dynamics at the origin, in particular, global asymptotic stability. If the Lie algebra is nilpotent, then the origin enjoys semiglobal exponential stability.

Description

This is a post-peer-review, pre-copyedit version of an article published in Mathematics of Control, Signals, and Systems. The final authenticated version is available online at: http://dx.doi.org/https://doi.org/10.1007/s00498-020-00259-7

Keywords

stability, difference equations, discrete-time, Lie algebras, sampled-data systems

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URI

https://doi.org/10.1007/s00498-020-00259-7
 http://hdl.handle.net/10012/17485

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Waterloo Research
Electrical and Computer Engineering