Table of Contents
ISRN Applied Mathematics
Volume 2013 (2013), Article ID 856404, 9 pages
http://dx.doi.org/10.1155/2013/856404
Research Article

Adaptive Estimation of Biological Rhythm in Crassulacean Acid Metabolism with Critical Manifold

1Department of Architecture and Mechatronics, Oita University, 700 Dannoharu, Oita 870-1192, Japan
2Department of Electronics and Control, Kitakyushu National College of Technology 5-20-1 Shii, Kokuraminami-ku, Kitakyushu, Fukuoka 802-0985, Japan

Received 28 April 2013; Accepted 30 May 2013

Academic Editors: W. Huang, X. Meng, J. Shen, and L. You

Copyright © 2013 Takami Matsuo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The mechanism of endogenous circadian photosynthesis oscillations of plants performing crassulacean acid metabolism (CAM) is investigated in terms of a nonlinear theoretical model. Blasius et al. used throughout continuous time differential equations which adequately reflect the CAM dynamics. The model shows regular endogenous limit cycle oscillations that are stable for a wide range of temperatures in a manner that complies well with experimental data. In this paper, we pay attention to the approximation of the fast modes of the CAM dynamics. Using the zero-epsilon approximation of the slow manifold, we derive the critical manifold that is defined by two algebraic nonlinear equations. The critical manifold allows us to give the algebraic estimate of the order of the tonoplast membrane. The dynamic equation of the order of the tonoplast membrane includes the nonlinear function that gives the equilibrium value of the lipid order of tonoplast functions as a hysteresis switch. We identify the nonlinear function with the measurement signals. Using the basis function expansion of the nonlinear and the critical manifold, we propose an adaptive observer to estimate the tonoplast order and the nonlinear function.