A model-based characterization of the long-term asymptotic behavior of nonlinear discrete-time processes using map invariance

• Published in: 2004 American Control Conference Proceedings; Volume 2 of 6 Vol. 2; pp. 1731 - 1736 vol.2
• Main Authors: Kazantzis, N., Good, T.A.
• Format: Conference Proceeding Journal Article
• Language: English
• Published: Piscataway NJ IEEE 01-01-2004; Evanston IL American Automatic Control Council
• Subjects: Applied sciences; Chemical engineering; Computer science; control theory; systems; Control theory. Systems; Difference equations; Distributed parameter systems; Exact sciences and technology; Jacobian matrices; Manifolds; Nonlinear dynamical systems; Nonlinear equations; Partial differential equations; Performance analysis; Time series analysis; Invariant; Asymptotic behavior; Functional equation; Non linear control; Discrete time processes; Invarying system; Modeling
• ISBN: 9780780383357; 0780383354
• ISSN: 0743-1619; 2378-5861
• DOI: 10.23919/ACC.2004.1386829

The present research work proposes a new approach to the problem of quantitatively characterizing the long-term dynamic behavior of nonlinear discrete-time processes. The formulation of the problem of interest can be naturally realized through a system of nonlinear functional equations (NFEs), for which a rather general set of conditions for the existence and uniqueness of a locally analytic solution is derived. The solution to the system of NFEs is then proven to represent a locally analytic invariant manifold for the nonlinear discrete-time process of interest. The local analyticity property of the invariant manifold map enables the development of a series solution method for the above system of NFEs, which can be easily implemented using MAPLE. Under a certain set of conditions, it is shown that the invariant manifold attracts all system trajectories, and therefore, the long-term dynamic behavior is determined through the restriction of the discrete-time process dynamics on the invariant manifold.