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.
Highlights
IntroductionBiological rhythm is characterized by free-running, endogenous rhythms, ranging from periods of seconds (e.g., heart beat) to years (e.g., population dynamics)
Biological rhythm is characterized by free-running, endogenous rhythms, ranging from periods of seconds to years
The adaptive observer can estimate the dynamics of z, and its estimation performance is robust against the temperature changes
Summary
Biological rhythm is characterized by free-running, endogenous rhythms, ranging from periods of seconds (e.g., heart beat) to years (e.g., population dynamics). Blasius et al investigated the mechanism of endogenous circadian photosynthesis oscillations of plants performing CAM in terms of a nonlinear theoretical model [3,4,5]. From the viewpoint of real measurements, we presented an adaptive observer to estimate the states and the nonlinear function in the dynamics of the tonoplast order assuming that the available signal is only the internal CO2 concentration [10]. We derive an algebraic relationship among the internal CO2 concentration, w, the malate concentration in the vacuole, y, and the order of the tonoplast membrane, z This equation is available as an estimate of z, when w and y are measurable. We design a first-order adaptive observer for online estimation of the nonlinear function in the dynamics of the tonoplast order using the output signal as the algebraic estimate of z. The observer can attenuate the estimation error of the tonoplast order against the measurement noises
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